DE102020008145A1 - Method and device for accelerated wavefront computation through a complex optical system - Google Patents

Abstract

The invention relates to a computer-implemented method, a computer program product and a device for simulating an optical system by means of wavefront calculation. The computer-implemented method comprises the steps:- computer-implemented setting up at least one wavefront transfer function for the optical system, with the wavefront transfer function being intended for wavefronts incident on the optical system, taking into account aberrations with an order greater than the order of a defocus in each case assign an associated emergent wavefront; and- computer-implemented evaluation of the at least one wavefront transfer function for at least one wavefront incident on the optical system.

Description

The invention relates to a method, a computer program product and a device for simulating an optical system by means of wavefront calculation. In particular, the invention relates to a method, a computer program product and a device for calculating and/or optimizing and for producing a spectacle lens.

For the production or optimization of spectacle lenses, in particular of individual spectacle lenses, each spectacle lens is manufactured in such a way that the best possible correction of a refractive error of the respective eye of the spectacle wearer is achieved for each desired viewing direction or each desired object point. In general, a lens is considered fully corrective for a given direction of gaze when the sphere, cylinder and axis values of the wavefront as it passes through the apex sphere agree with the sphere, cylinder and axis values of the prescription for the ametropic eye. When determining the refraction for an eye of a spectacle wearer, dioptric values (in particular sphere, cylinder, axial position - i.e. in particular sphero-cylindrical deviations) for a large (usually infinite) distance and, if necessary (for multifocal lenses or varifocal lenses) an addition or a complete Near refraction determined for a close distance (e.g. according to DIN 58208). With modern spectacle lenses, object distances that deviate from the norm and were used to determine the refraction can also be specified. This defines the prescription (in particular sphere, cylinder, axis position and, if applicable, addition or near refraction) that is sent to a spectacle lens manufacturer. Knowledge of a special or individual anatomy of the respective eye or the refractive index of the defective eye that actually exists in the individual case is not required for this.

However, a complete correction for all viewing directions at the same time is not normally possible. The spectacle lenses are therefore manufactured in such a way that they provide good correction of ametropia in the eye and only minor aberrations, especially in the main areas of use, particularly in the central vision areas, while larger aberrations are permitted in peripheral areas.

In order to be able to manufacture a spectacle lens in this way, the spectacle lens surfaces or at least one of the spectacle lens surfaces are first calculated in such a way that the desired distribution of the unavoidable aberrations is brought about as a result. This calculation and optimization is usually carried out using an iterative variation method by minimizing a target function. A function F with the following functional relationship to the spherical power S, the amount of the cylindrical power Z and the axial position of the cylinder α (also referred to as the "SZA" combination) is considered and minimized as a target function: $$f={\displaystyle \sum _{i=1}^m\left[G_{i,S\Delta }{\left(S_{\Delta ,i}-S_{\Delta ,i,SOll}\right)}^2+G_{i,Z\Delta }{\left(Z_{\Delta ,i}-Z_{\Delta ,i,SOll}\right)}^2+...\right]}$$

At least the actual refraction deficits of the spherical power S _Δ,i and the cylindrical power Z _Δ,i as well as target specifications for the refraction deficits of the spherical power S _Δ,i,target and the cylindrical power Z _Δ,i,target taken into account.

Already in DE 103 13 275 it was recognized that it is advantageous not to specify the target specifications as absolute values of the properties to be optimized, but as their deviations from the regulation, i.e. as a required local mismatch. The "actual" values of the properties to be optimized are therefore not included in the target function as absolute values of these optical properties, but rather as deviations from the regulation. This has the advantage that the target specifications are independent of the prescription (in particular Sph _v , Zyl _v , Axis _v , Pr _v , B _v ) and the target specifications do not have to be changed for each individual prescription.

The respective refraction deficits at the respective evaluation points are preferably taken into account with weighting factors g _i,SΔ or g _i,ZΔ . The target specifications for the refraction deficits of the spherical power S _Δ,i,target and/or the cylindrical power Z _Δ,i,target form the so-called spectacle lens design, in particular together with the weighting factors g _i,SΔ or g _i,ZΔ . In addition, other residues, in particular other quantities to be optimized, such as coma and/or spherical aberration and/or prism and/or magnification and/or anamorphic distortion, etc., must be taken into account, which is indicated in particular by the expression "+..." in the above formula for the objective function F.

In some cases, it can contribute to a significant improvement, especially an individual adjustment of a spectacle lens, if, when optimizing the spectacle lens, not only aberrations up to the second order (sphere, amount of astigmatism and axis position), but also higher order (e.g. coma, three-leaf aberration, spherical aberration etc.) must be taken into account.

It is known from the prior art to determine the shape of a wave front for optical elements and in particular for spectacle lenses that are delimited by at least two refracting, refractive boundary surfaces. This can be done, for example, by numerical calculation of a sufficient number of neighboring rays, combined with a subsequent fit of the wavefront using Zernike polynomials. Another approach is based on a local wavefront calculation during refraction (see WO 2008/089999 A1 ). Only one ray is calculated for each viewing point (the main ray) and the derivations of the versine heights of the wave front according to the transversal (perpendicular to the main ray) coordinates. These derivations can be formed up to a certain order, with the second derivation describing the local curvature properties of the wave front (such as refractive index, astigmatism) and the higher derivation being related to the higher-order aberrations.

When light is calculated through a spectacle lens, the local derivatives of the wave fronts are calculated at a suitable position in the beam path in order to compare them there with desired values that result from the refraction of the spectacle lens wearer. The apex sphere or, for example, the main plane of the eye in the corresponding viewing direction can be used as such a position at which an evaluation of the wave fronts takes place. Alternatively or additionally, the entrance pupil EP, the exit pupil AP and/or preferably the plane after the refraction at the rear surface L2 of the eye lens can be used to evaluate the wave fronts. It is assumed here that a spherical wave front emanates from the object point and propagates to the first lens surface. The wavefront is refracted there and then propagates to the second lens surface, where it is refracted again. The final propagation then takes place from the second interface to the vertex sphere (or the principal plane of the eye), where the wavefront is compared to predetermined values for the correction of the refraction of the eye of the spectacle wearer.

In order to carry out this comparison on the basis of the determined refraction data of the respective eye, the evaluation of the wavefront at the apex sphere is based on an established model of the ametropia eye, in which an ametropia (refraction deficit) is superimposed on a right-sighted basic eye. This has proven to be particularly useful because it does not require further knowledge of the anatomy or optics of the respective eye (e.g. distribution of the refractive powers, eye length, ametropia of length and/or ametropia of refractive power). Detailed descriptions of this model of spectacle lenses and refractive deficit can be found, for example, in Dr. Roland Enders "The optics of the eye and visual aids", Optical Specialist Publication GmbH, Heidelberg, 1995, pages 25 ff. and in Diepes, Blendowske "Optics and Technology of Glasses", Optical Specialist Publication GmbH, Heidelberg, 2002, pages 47 ff . The REINER correction model described therein is used as a proven model.

The lack or excess of refractive power of the optical system of the defective eye compared to a right-sighted eye of the same length (remaining eye) is regarded as a refraction deficit. In particular, the refractive index of the refraction deficit is approximately equal to the far point refraction with a negative sign. For complete correction of the ametropia, the spectacle lens and the refraction deficit together form a telescope system (afocal system). The remaining eye (defective eye without added refraction deficit) is assumed to be right-sighted. A spectacle lens is therefore considered to be fully corrective for distance vision if its focal point on the image side coincides with the far point of the defective eye and thus also with the focal point of the refraction deficit on the object side.

In the pamphlet DE 10 2017 007 975 A1 or. WO 2018/138140 A2 , a method and a device are described that make it possible to improve the calculation or optimization of a spectacle lens, the spectacle lens being very effectively adapted to the individual requirements of the spectacle wearer with simple measurements of individual, optical and eye anatomical data.

Further, according to the prior art, lens optimization can be performed by minimizing an objective function that judges wavefront aberration within an eye model. The wavefront aberration arises by comparing a reference wavefront with a wavefront that is determined by means of a wavefront calculation through the refracting components of an eye model. In this case, each wavefront must be alternately refracted and propagated. This procedure is based on the publications by G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, D. Uttenweiler: "Derivation of the refractive equations for higher order aberrations of local wavefronts by oblique incidence", J .Opt.Soc. At the. A 27, 218-237 (2010) and G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, D. Uttenweiler: "Derivation of the propagation equations for higher order aberrations of local wavefronts", J .Opt.Soc. At the. A 28, 2442-2458 (2011), and is in particular in the patent specifications DE 10 2012 000 390 A1 , US 9,910,294 B2 , DE 10 2011 101 923 A1 and WO 2008/089999 A1 described. Express reference is made herein to the publications mentioned above and their content is fully included in the present description.

It is also known that, as an alternative to calculating the wavefront, the wavefront aberrations can be assessed using a beam of rays. The course of each individual light beam of the bundle through the spectacle lens and the eye model has to be calculated, which requires longer computing times compared to the wavefront calculation.

Even if powerful methods are already available according to the state of the art for the individual calculation steps of the refraction and propagation of wavefronts, unacceptably long calculation times arise due to the number of refractions and propagations, especially if new wavefronts are repeatedly calculated through one and the same eye model have to. A multiple calculation of wave fronts is necessary e.g. in the case of a spectacle lens optimization due to the iterative steps for the optimization. Furthermore, changing viewing directions and/or object distances can occur when optimizing the spectacle lens, which can also require multiple calculations of wave fronts.

In order to optimize an optical system via simulation, the passage of light through the system must be described physically and then evaluated according to suitable criteria. Optimization means a targeted modification of the system in such a way that the light passing through comes as close as possible to a set goal with regard to the criteria. For example, the light passing through can be described by a scalar or a vectorial electromagnetic field. There is also the option of defining the areas of the same phase as wave fronts in this field and using these as a basis for evaluation.

If one neglects interference effects due to the finite wavelength of light, then one can describe the passage of light with geometric optics instead of with wave optics. According to the prior art, there are methods for calculating rays (ray tracing) in geometric optics. A simple condition to do ray tracing under is paraxial. A particularly simple form here is Gaussian optics, in which all rays remain in one meridian. The description then only refers to the optics within one meridian (in the case of rotationally symmetrical systems in which each meridian is equivalent, the entire system can then be described representatively in one meridian). A more general form suitable for two independent meridians is linear optics. Both in Gaussian optics and in linear optics, each system is defined in that there is an entrance plane perpendicular to the propagation of the main ray and another exit plane parallel to it. The optical system is characterized by how the coordinates and directions of an outgoing ray depend on the corresponding magnitudes of the incoming ray, this dependency being linear in the paraxial domain.

As an alternative to ray tracing, there is also an equivalent calculation of wave fronts (wave tracing), which is defined in geometric optics as the spatial surfaces through which each ray of a bundle passes perpendicularly. There are equivalent reformulations of wave fronts in geometric optics. For example, Fourier optics do not use spatial surfaces, but rather planes perpendicular to the light propagation of the main ray and describe the OPD (Optical Path Difference) as a function of the lateral coordinates of this plane, which separates each point of the plane from a given reference point.

Wavefronts or the equivalent OPD functions can be described in different ways. For example, these functions can be described by free-form surfaces, such as with B-splines. If a section of the wavefront delimited by a pupil is relevant, wavefronts are described by combining a sufficient number of Zernike polynomials (see eg Born and Wolf: "Principles of Optics", Oxford, Pergamon, 1970), which are weighted with the corresponding Zernike coefficients. A common local description for the vicinity of a main ray consists in the Taylor series expansion, i.e. local derivatives of the wavefront according to the lateral coordinates, which represent weights with which powers of the coordinates are linearly combined to form the wavefront.

In particular, it is state of the art to optimize spectacle lenses by calculating wave fronts through the spectacle lenses using wave tracing and then comparing them with certain specifications in order to evaluate a target function (see, for example, WO 2008/089999 A1 ). This target function can be constructed in such a way that minimizing it leads to an improvement in the spectacle lens.

In one embodiment of the prior art, only second-order wavefront errors (LOA, Lower Order Aberration) are used for the wavefronts. In a further embodiment of the prior art, both second-order wavefront errors (LOA) and third-order or higher order wavefront errors (HOA, Higher Order Aberration) are used. In the state of the art, there are methods for refraction (see G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, D. Uttenweiler: "For the calculation of wavefronts that are associated with such HOA, there are methods for refraction. Derivation of the refractive equations for higher order aberrations of local wavefronts by oblique incidence", J. Opt. Soc. Am. A 27, 218-237 (2010), as well as WO 2008/089999 A1 ) and for propagation see (see G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, D. Uttenweiler: "Derivation of the propagation equations for higher order aberrations of local wavefronts", J. Opt Soc Am A 28, 2442-2458 (2011), and DE 10 2011 101 923 A1 ). The great technical advantage of these methods is that they are based on analytical formulas and are therefore not dependent on the computationally time-consuming numerics of ray tracing methods.

Furthermore, the state of the art discloses methods according to which wave fronts are only calculated through the spectacle lens itself in order to optimize the spectacle lens (see DE 10 2011 101 923 A1 ), as well as methods according to which wavefronts are calculated both through the lens to be optimized and through an eye model (see DE 10 2012 000 390 A1 , US 9,910,294 B2 , DE 10 2017 007 975 B4 ).

However, the state of the art does not yet offer a method for analytically calculating light in the form of a wavefront including the HOA through a complex optical system in general, so that for such a system one has to resort to ray tracing methods that require a lot of computing time or has to fall back on a repeatedly applied calculation of the individual components by means of wave tracing, which is also computationally time-consuming.

It is therefore an object of the present invention to provide a method for accelerated wavefront computation through a complex optical system. In particular, it is an object of the present invention to provide an efficient method for calculating and/or optimizing a spectacle lens and for producing a spectacle lens. In addition, it is an object of the present invention to provide a related computer program product and a related device. These objects are solved by the subjects of the independent claims. Advantageous embodiments are the subject matter of the dependent claims.

• - computerimplementiertes Aufstellen zumindest einer Wellenfront-Transferfunktion für das optische System, wobei die Wellenfront-Transferfunktion dafür bestimmt ist, in das optische System einfallenden Wellenfronten unter Berücksichtigung von Abbildungsfehlern mit einer Ordnung größer als die Ordnung eines Defocus jeweils eine zugehörige ausfallende Wellenfront zuzuordnen; und
• - computerimplementiertes Auswerten der zumindest einen Wellenfront-Transferfunktion für zumindest eine in das optische System einfallende Wellenfront.

A first independent aspect of solving the problem relates to a computer-implemented method for simulating an optical system by means of a wavefront calculation, the optical system being in particular a complex optical system whose effect goes beyond a single refraction, a single propagation or a single reflection, comprising the Steps:

• - Computer-implemented setting up of at least one wavefront transfer function for the optical system, the wavefront transfer function being intended to assign an associated emerging wavefront to wavefronts incident on the optical system, taking into account aberrations with an order greater than the order of a defocus; and
• - Computer-implemented evaluation of the at least one wavefront transfer function for at least one wavefront incident on the optical system.

A “complex optical system” in the context of this invention is in particular an optical system whose effect on any wavefront or alternatively on any light beam is not reflected either by refraction or reflection on a single surface or by a single propagation between two different levels can be described. Rather, a complex optical system comprises a plurality of components which, when light or a wave or wavefront passes through, lead to at least two refractions and/or to at least two propagations and/or to at least one refraction and at least one propagation.

In the context of the invention, the term “setting up a function” includes a determination and/or definition of the function. A function is set up automatically or in a computer-implemented manner with the aid of a processor or computer. In particular, establishing the at least one wavefront transfer function includes assigning coefficients to the at least one wavefront transfer function. The creation of at least one wavefront transfer function can include the creation of only a single wavefront transfer function or the creation of several, in particular two or three, wavefront transfer functions. For example, a variety (i.e., two, three, four, five, etc.) of wavefront transfer functions can be established for a variety of different optical system configurations. Different configurations of an optical system can generally include different parameters that characterize or describe the optical system. In particular, different configurations of an optical system can also include different positions and/or orientations of components of the optical system in relation to one another and/or different positions and/or orientations of the optical system (e.g. relative to another optical system).

In particular, all wavefront transfer functions that have been set up can therefore differ from one another.

The term “evaluating” the at least one wavefront transfer function for an incident wavefront means in particular that the at least one wavefront transfer function is applied to an incident wavefront. In other words, by “evaluating” the at least one wavefront transfer function, an associated emergent or outgoing wavefront is assigned to a predetermined incident wavefront (into the optical system).

The at least one wavefront transfer function is designed or defined in order to assign an associated emergent or outgoing wavefront to wavefronts incident in the optical system, taking into account aberrations with an order greater than the order of a defocus. In particular, the at least one wavefront transfer function is intended to generally describe a change in wavefronts incident on the optical system, taking into account aberrations, with an order greater than the order that corresponds to a defocus. In particular, the at least one wavefront transfer function is designed and/or defined in order to convert the at least one incident wavefront into an associated emergent or outgoing wavefront. In particular, the wavefront transfer function is intended to assign an associated emergent or outgoing wavefront to each wavefront incident on the optical system. The at least one wavefront transfer function also takes into account aberrations or a component that goes beyond a defocus and an astigmatism (corresponds to the "order" of a defocus), i.e. depending on the representation, in particular beyond a sphere and a cylinder (including the axis position). /goes out In a description of the aberrations according to Zernike (i.e. using Zernike polynomials) and/or in a description of the aberrations using a Taylor expansion, this means in particular that the at least one wavefront transfer function has aberrations with an order greater than two, i.e. higher order Aberrations (HOA) or higher-order aberrations are taken into account. In particular, the at least one wavefront transfer function takes into account not only aberrations up to the second order (sphere, amount of astigmatism and axial position), but also higher-order aberrations, such as spherical aberration, coma, trefoil errors, etc.

• - Aufstellen einer Wellenfront-Transferfunktion für das optische System;
• - Auswerten der Wellenfront-Transferfunktion für eine erste in das optische System einfallende Wellenfront;
• - Auswerten der Wellenfront-Transferfunktion für eine zweite in das optische System einfallende Wellenfront, die von der ersten Wellenfront verschieden ist;

wobei die Wellenfront-Transferfunktion

• - die einfallende Wellenfront in eine ausfallende Wellenfront überführt und
• - die Änderung einfallender Wellenfronten durch das optische System allgemein beschreibt und
• - dabei zumindest eine Komponente berücksichtigt, deren Ordnung über die Ordnung einer Defokus-Komponente hinausgeht und
• - wobei die Wirkung des optischen Systems über den einer einzelnen Brechung, einer einzelnen Propagation oder einer einzelnen Reflexion hinausgeht.

The invention thus relates in particular to a method for simulating an optical system, using a method for calculating wavefronts, which comprises the following steps:

• - Establishing a wavefront transfer function for the optical system;
• - evaluating the wavefront transfer function for a first incident wavefront in the optical system;
• - evaluating the wavefront transfer function for a second wavefront incident on the optical system, which is different from the first wavefront;

where is the wavefront transfer function

• - converts the incoming wavefront into an outgoing wavefront and
• - describes the change of incident wavefronts through the optical system in general and
• - taking into account at least one component whose order goes beyond the order of a defocus component and
• - where the action of the optical system goes beyond that of a single refraction, a single propagation, or a single reflection.

According to the invention, it has been recognized that for a conventional step-by-step wavefront calculation, the same intermediate steps are run through unnecessarily often. In all cases in which only the wavefront emerging at the last surface of an optical system (e.g. an eye model) is of interest, the explicit calculation of all wavefronts at the intermediate surfaces is superfluous and can be saved.

The present invention therefore breaks with the procedure known from the prior art and instead proposes a method with which an incident wavefront can be converted into an emerging wavefront with a single operation. In particular, corresponding transfer functions can be set up or defined for any optical system from any number of refracting surfaces and propagations if the parameters of such a complex system are given. The complex optical system, e.g. in the case of a "gradient index" object (GRIN object), does not have to be traceable to a finite number of pure refractions and/or propagations. Instead, it is sufficient to specify the optical system using the ray transfer function, which uniquely assigns an outgoing ray to every incoming ray. The steps to build this transfer function can be cumbersome because they only need to be performed once. The decisive factor is that the computational effort involved in evaluating the transfer function must be less than the computational effort involved in evaluating the individual sub-steps of propagation and refraction in a complex optical system.

The state of the art does not yet offer a method for analytically calculating light in the form of a wavefront including the HOA through a complex optical system in general, so that for such a system one has to resort to computationally time-intensive ray tracing methods or has to resort to a time-consuming repeated application of a wave tracing calculation on the individual components. However, the present invention now solves this problem by means of an analytical method that saves a great deal of computing time. The technical advantage of the invention consists in particular in the fact that a large number of different wavefronts can be calculated very efficiently by a given complex optical system by exploiting the fact that the optical system itself does not change and the calculation step for setting up a wavefront transfer function only needs to be run once beforehand.

In a preferred embodiment, the invention relates to a method, in particular a computer-implemented method, for optimizing an overall optical system, the optical system representing a second subsystem of the overall optical system and the overall optical system additionally comprising a first subsystem. The first subsystem and/or the second subsystem can be varied in particular in the course of the optimization. “Optimization” within the meaning of the present invention is understood in particular to mean the calculation and/or optimization of a spectacle lens (to be produced) for correcting a defective vision of a spectacle wearer.

In a further preferred embodiment, the first subsystem is a spectacle lens and the second subsystem is a model eye.

A “model eye” within the meaning of the present invention is in particular a data set with eye model parameters that describe a real eye. The model eye preferably also has provided (in particular measured) individual refraction data of a spectacle wearer. For example, the eye model parameters can be assumed or specified, at least in part, on the basis of standard or average values. The eye model parameters can also be measured, at least in part.

In particular, the method can include defining an individual eye model, which individually defines at least certain specifications regarding geometric and optical properties of a model eye. In an individual eye model, for example, at least one shape (topography) and/or effect of the cornea, in particular a front surface of the cornea, of the model eye, a corneal Lens distance d _CL (this distance between the cornea and a lens front surface of the model eye is also called anterior chamber depth), parameters of the lens of the model eye, which in particular at least partially define the optical effect of the lens of the model eye, and a lens-retina distance d _LR (this distance between the lens, in particular the rear surface of the lens, and the retina of the model eye is also referred to as the vitreous body length) in a specific way, in particular in such a way that the model eye has the individual refraction data provided, i.e. that a point in the model eye from a point on the retina the wavefront emerging from the model eye corresponds (up to a desired accuracy) to the wavefront determined (eg measured or otherwise determined) for the real eye of the spectacle wearer. As parameters of the lens of the model eye (lens parameters), for example, either geometric parameters (shape of the lens surfaces and their distance) and preferably material parameters (e.g. refractive indices of the individual components of the model eye) can be completely defined so that they at least partially define an optical effect of the lens. Alternatively or additionally, lens parameters can also be defined for the model eye, which directly describe the optical effect of the lens of the model eye. With regard to the cornea, the shape of the anterior surface of the cornea is usually measured, but alternatively or additionally the effect of the cornea as a whole (no differentiation between anterior and posterior surface) can also be specified. A posterior surface of the cornea and/or a corneal thickness can also be specified. Furthermore, if individual intraocular lens data are known or are provided, the parameters of the lens of the model eye can also be defined using the provided intraocular lens data.

In the simplest case of an eye model, the refraction of the eye is determined by the optical system consisting of the front surface of the cornea, the eye lens and the retina. In this simple model, the refraction of light at the anterior surface of the cornea and the power of the lens of the eye (preferably including spherical, astigmatic and higher-order aberrations) together with their positioning relative to the retina determine the refraction of the model eye. In this case, the individual variables (parameters) of the model eye are appropriately defined using individual measured values for the eye of the spectacle wearer and/or using standard values and/or using individual refraction data provided. In particular, some of the parameters (e.g. the topography of the anterior surface of the cornea and/or the depth of the anterior chamber and/or at least one curvature of a lens surface, etc.) can be provided directly as individual measured values. Other values can also be taken from values of standard models for a human eye, especially when it comes to parameters whose individual measurement is very complex. Overall, however, not all (geometric) parameters of the model eye have to be specified from individual measurements or from standard models. Rather, an individual adjustment can be made for one or more (free) parameters by calculation taking into account the specified parameters in such a way that the resulting model eye then has the individual refraction data provided. Depending on the number of parameters contained in the provided individual refraction data, a corresponding number of (free) parameters of the eye model can be individually adjusted (fitted).

For the calculation or optimization of the spectacle lens, a first surface and a second surface of the spectacle lens can be specified in particular as starting surfaces with a specified (individual) position relative to the model eye. In a preferred embodiment, only one of the two surfaces is optimized. This is preferably the back surface of the spectacle lens. A corresponding starting surface is preferably specified for both the front surface and the rear surface of the spectacle lens. In a preferred embodiment, however, only one surface is iteratively changed or optimized during the optimization process. The other surface of the spectacle lens can be, for example, a simple spherical or rotationally symmetrical aspherical surface. However, it is also possible to optimize both surfaces.

Based on the two specified surfaces, the method for calculating or optimizing can include determining the course of a principal ray through at least one visual point (i) of at least one surface of the spectacle lens to be calculated or optimized into the model eye. The principal ray describes the geometric course of the ray starting from an object point through the two spectacle lens surfaces, the front surface of the cornea and the lens of the model eye, preferably up to the retina of the model eye.

In addition, the method for calculating or optimizing may include evaluating an aberration of a wavefront resulting along the chief ray from a spherical wavefront incident on the first surface of the spectacle lens at an evaluation surface within the model eye in comparison with a wavefront converging at a point on the retina of the eye model ( Reference light) include. In particular, a spherical wavefront (wo) impinging on the first surface (front surface) of the spectacle lens along the chief ray can be specified for this purpose. This spherical wave front describes the light emanating from an object point (object light). The curvature of the spherical wavefront when hitting the first surface of the lens corresponds to the reciprocal of the object distance. The method thus preferably includes specifying an object distance model, which assigns an object distance to each viewing direction or each visual point of the at least one surface of the spectacle lens to be optimized. This preferably describes the individual usage situation in which the spectacle lens to be produced is to be used.

The wavefront impinging on the spectacle lens is now preferably refracted for the first time on the front surface of the spectacle lens. The wavefront then propagates along the principal ray within the lens from the front surface to the back surface, where it is refracted a second time. The wavefront transmitted through the spectacle lens now preferably propagates further along the principal ray to the front surface of the cornea of the eye, where it is preferably refracted again. After further propagation within the eye up to the lens of the eye, the wavefront is preferably refracted there again in order to finally propagate preferably up to the retina of the eye. Depending on the optical properties of the individual optical elements (spectacle lens surfaces, front surface of the cornea, lens of the eye), each refraction process and each propagation process also leads to a deformation of the wave front.

In order to achieve an exact mapping of the object point to an image point on the retina, the wavefront would have to leave the lens of the eye preferably as a converging spherical wavefront, the curvature of which corresponds exactly to the reciprocal of the distance to the retina. A comparison of the wave front emanating from the object point with a wave front (reference light) converging in a point on the retina (ideally perfect imaging) thus allows the evaluation of a mismatch. This comparison and thus the evaluation of the wavefront of the object light in the individual eye model can take place at different points along the course of the principal ray, in particular between the second surface of the optimizing spectacle lens and the retina. In particular, the evaluation surface can thus lie at different positions, in particular between the second surface of the spectacle lens and the retina. The refraction and propagation of the light emanating from the object point in the individual eye model is preferably calculated correspondingly far for each visual point. The evaluation surface can refer either to the actual beam path or to a virtual beam path, such as is used to construct the exit pupil AP. In the case of the virtual beam path, after refraction through the back surface of the eye lens, the light must be propagated back to a desired plane (preferably to the plane of the AP), whereby the refractive index used must correspond to the medium of the vitreous body and not to the eye lens. If the evaluation surface is provided behind the lens or after the refraction on the rear surface of the lens of the model eye, or if the evaluation surface is reached by back propagation along a virtual beam path (as in the case of the AP), then the resulting wavefront of the object light can preferably be simply measured with a spherical wavefront of the reference light are compared. To this end, the method preferably includes specifying a spherical wavefront impinging on the first surface of the spectacle lens, determining a wavefront resulting from the spherical wavefront due to the action of at least the first and second surfaces of the spectacle lens, the front surface of the cornea and the lens of the model eye in the at least an eye, and an aberration evaluation of the resulting wavefront compared to a spherical wavefront converging on the retina. If, on the other hand, an evaluation area is to be provided within the lens or between the lens of the model eye and the spectacle lens to be calculated or optimized, a reverse propagation from a point on the retina through the individual components of the model eye to the evaluation area is simply simulated as a reference light. to compare the object light with the reference light.

However, as already mentioned at the outset, a complete correction of the refraction of the eye simultaneously for all viewing directions of the eye, ie for all visual points of the at least one spectacle lens surface to be optimized, is generally not possible. Depending on the viewing direction, an intentional mismatch of the spectacle lens is thus preferably specified, which, depending on the application situation, is particularly low in the areas of the spectacle lens that are mainly used (e.g. central visual points) and somewhat higher in the areas that are rarely used (e.g. peripheral visual points). The principle of this procedure is already known from conventional optimization methods.

In a further preferred embodiment, the at least one wavefront incident on the optical system is determined on the basis of a predetermined test wavefront which passes through the first subsystem.

• - Bewerten des optischen Gesamtsystems auf Basis des Ergebnisses des Auswertens der zumindest einen Wellenfront-Transferfunktion für die zumindest eine in das optische System einfallende Wellenfront, wobei das optische Gesamtsystem unter Variation des ersten Teilsystems so lange bewertet wird, bis die Bewertung eine vorgegebene Bedingung erfüllt.

In a further preferred embodiment, the method further comprises the step:

• - Evaluation of the overall optical system based on the result of the evaluation of the at least one wavefront transfer function for the at least one wavefront incident on the optical system, with the overall optical system being evaluated by varying the first subsystem until the evaluation fulfills a predetermined condition.

verwendet werden. Da das Minimieren einer Zielfunktion sowie dafür verwendete iterative Variationsverfahren dem Fachmann wohlbekannt sind, wird im Rahmen dieser Erfindung nicht näher darauf eingegangen. Die vorgegebene Bedingung für die Bewertung kann insbesondere ein vorgegebener Schwellenwert der Zielfunktion sein. Unterschreitet die Zielfunktion bei einer bestimmten Variation bzw. Konfiguration des ersten Teilsystems diesen Schwellenwert, so ist das Ziel, nämlich das Gesamtsystem zu optimieren, erreicht. Eine weitere Variation und/oder ein weiteres Bewerten ist dann nicht mehr notwendig.In the context of this description, “evaluating” a system is understood in particular to mean evaluating the system with the aid of a functional or a target function. In particular, evaluating a system can include minimizing a functional or a target function, for example by means of an iterative variation method. In the case of spectacle lens optimization, the function mentioned at the outset can be used as a target function, for example $f={\displaystyle {\sum }_{i=1}^m\left[G_{i,S\Delta }{\left(S_{\Delta ,i}-S_{\Delta ,i,SOll}\right)}^2+G_{i,Z\Delta }{\left(Z_{\Delta ,i}-Z_{\Delta ,i,SOll}\right)}^2+...\right]}$

be used. Since the minimization of a target function and iterative variation methods used for this are well known to the person skilled in the art, this will not be discussed in any more detail within the scope of this invention. The predefined condition for the evaluation can in particular be a predefined threshold value of the target function. If the target function falls below this threshold value for a specific variation or configuration of the first subsystem, then the goal, namely to optimize the overall system, has been achieved. A further variation and/or a further evaluation is then no longer necessary.

In particular, the variation of the first subsystem includes a change in at least one refracting surface and/or at least a distance between refracting surfaces of the first subsystem, and/or a tilting and/or shifting of the first subsystem relative to the second subsystem. If the first subsystem is a spectacle lens, a variation of the first subsystem or spectacle lens can include, for example, a change in the shape of at least one spectacle lens surface (front and/or rear surface).

In particular, according to a preferred embodiment, the invention thus relates to a method for optimizing an overall optical system, the optical system representing a second subsystem of the overall optical system and the overall optical system additionally comprising a first subsystem, which can be varied in the course of the optimization, and
wherein the wavefront incident on the optical system is determined by passing a test wavefront through the first subsystem and the overall optical system is evaluated on the basis of the result of the evaluation of the wavefront transfer function for the wavefront incident on the optical system and in which the first subsystem is so varies for a long time and the overall optical system is evaluated until the evaluation satisfies a predetermined condition.

If the first subsystem is a spectacle lens, in order to optimize the spectacle lens, e.g. the at least one surface of the spectacle lens to be calculated or optimized can be iteratively varied until an aberration of the resulting wavefront corresponds to a specified target aberration, i.e. in particular by specified aberration values deviates from the wavefront of the reference light (e.g. a spherical wavefront whose center of curvature lies on the retina). The wavefront of the reference light is also referred to here as the reference wavefront. For this purpose, the method preferably includes minimizing a target function F, in particular analogously to the target function already described at the outset.

In a further preferred embodiment, the overall optical system is evaluated on the basis of the result of the evaluation of the at least one wavefront transfer function for a first incident wavefront in the optical system and also on the basis of the result of the evaluation of the at least one wavefront transfer function for a second incident wavefront Wavefront made, wherein the first subsystem is in a first position and alignment to the second subsystem when the first wavefront strikes, wherein the first subsystem is in a second position and alignment to the second subsystem when the second wavefront strikes, and wherein the first Position differs from the second position and / or the first orientation from the second orientation. With this procedure, e.g. different visual points can be considered or evaluated.

In particular, according to a preferred embodiment, the invention can thus relate to a method for optimizing an overall optical system, with the evaluation of the overall optical system in addition to the result of the evaluation of the wavefront transfer function for the wavefront incident in the optical system, the result of an additional evaluation of the wavefront transfer function for a further incident wavefront,
wherein the further wavefront incident on the optical system is determined by passing the test wavefront through the first subsystem, wherein the first subsystem is in a second position and an orientation to the second subsystem which differs from the position and/or the first orientation of the second position or orientation when evaluating the wavefront transfer function for the incident wavefront.

In particular, according to a further preferred embodiment, the invention thus relates to a method for optimizing an overall optical system, the optical system representing a second subsystem of the overall optical system and the overall optical system additionally comprising a first subsystem which can be varied in the course of the optimization,
wherein the first wavefront incident on the optical system is determined by passing a test wavefront through the first subsystem, the first subsystem being in a first configuration, and the overall optical system based on the result of the evaluation of the wavefront transfer function for the first in the optical system incident wavefront is assessed, and
wherein the second wavefront incident on the optical system is determined by passing the test wavefront through the first subsystem, wherein the first subsystem is in a second configuration that is determined on the basis of the evaluation, and the overall optical system on the basis of the result of the evaluation of the wavefront transfer function for the second wavefront incident on the optical system.

In particular, additional variations of the first subsystem and assessments of the overall optical system can be carried out in the method until the assessment satisfies a predetermined condition.

In particular, the invention can relate to a method for optimizing an overall optical system, the optical system representing a second subsystem of the overall optical system and the overall optical system additionally comprising a first subsystem, which can be varied in the course of the optimization, and
wherein the first wavefront incident on the optical system is determined by passing a test wavefront through the first subsystem, the first subsystem being in a first position and a first orientation to the second subsystem, and
wherein the second wavefront incident on the optical system is determined by passing the test wavefront through the first subsystem, wherein the first subsystem is in a second position and a second orientation to the second subsystem, wherein the first position and/or the first orientation differ from the second position or orientation, and where
the assessment of the overall optical system includes the result of evaluating the wavefront transfer function for the first wavefront incident on the optical system and the result of evaluating the wavefront transfer function for the second wavefront incident on the optical system.

In particular, the evaluation of the overall optical system comprises a first evaluation based on the result of the evaluation of the at least one wavefront transfer function for a first wavefront incident on the optical system, with the first partial system being in a first Configuration is where
the evaluation of the overall optical system includes a second evaluation based on the result of the evaluation of the at least one wavefront transfer function for a second wavefront incident on the optical system, wherein the first partial system is in a second configuration when the second wavefront is incident on the optical system, which is determined based on the first evaluation, and preferably wherein
one or more further variations of the configuration of the first subsystem are additionally made on the basis of the second assessment and the overall optical system is evaluated for each of these further configurations of the first subsystem until the assessment satisfies a predetermined condition, wherein
the different configurations of the first subsystem differ in particular in at least one refracting surface and/or at least one distance between refracting surfaces of the first subsystem, and/or in a position and/or orientation of the first subsystem relative to the second subsystem.

If the first subsystem is a spectacle lens, a configuration of the first subsystem can be characterized, for example, by the shape of at least one spectacle lens surface. A variation of the first subsystem or spectacle lens can, for example, include a change in the shape of at least one spectacle lens surface.

In a further preferred embodiment, the overall optical system is evaluated on the basis of the result of the evaluation of a first wavefront transfer function for a first wavefront incident on the optical system and also on the basis of the result of the evaluation of a second wavefront transfer function for a further, in particular second or third, incident wavefront, wherein the first subsystem is in a first position and orientation to the second subsystem when the first wavefront is incident, wherein the first subsystem is in a second position and orientation to the second subsystem when the other wavefront is incident, wherein the first position differs from the second position and/or the first orientation differs from the second orientation, and the second wavefront transfer function differs from the first wavefront transfer function.

In particular, the first wavefront transfer function is set up for a first configuration of the second subsystem and the second wavefront transfer function for a second configuration of the second subsystem that differs from the first configuration of the second subsystem, with the first configuration of the second subsystem being, for example, a first accommodation of the Model eye and the second configuration of the second subsystem describes a second accommodation of the eye or model eye.

In particular, in a preferred embodiment, the invention can thus relate to a method for optimizing a spectacle lens,
where the assessment of the overall optical system includes the result of the evaluation of a comprises a further wavefront transfer function for a third wavefront incident on the optical system, and
wherein the third wavefront incident on the optical system is determined by passing the test wavefront through the first subsystem, wherein the first subsystem is in a further position and a further orientation to the second subsystem, wherein the first position and/or the first orientation differ from the further position or orientation and wherein the further wavefront transfer function differs from the wavefront transfer function because the optical system has a different configuration than in the evaluation of the wavefront transfer function in the evaluation for the first incident in the optical system wavefront, and
wherein the change in configuration of the optical system may reflect accommodation of the eye.

This embodiment, in which two configurations of the eye are taken into account, can advantageously be used in particular for the optimization of progressive lenses.

In a further preferred embodiment, the first sub-system is a spectacle lens and the second sub-system is a model eye, with eye movements of the model eye causing a change in the position of the penetration point of the main ray through the spectacle lens surfaces and/or a change in the angle of incidence on a spectacle lens surface as a change the position and/or the orientation of the spectacle lens in the coordinate system of the eye. In this way, in particular, different visual points can be taken into account by an eye movement.

In a further preferred embodiment, the optical system is a GRIN system or comprises at least one GRIN element, where GRIN is the abbreviation for “Gradient Index”.

Both the incident wavefront and the respectively associated emerging wavefront are preferably represented by coefficients for the basis elements of a (common) basis system. The at least one wavefront transfer function assigns the incident wavefronts to the respective emerging wavefront, preferably in such a way that for one (in particular for each) basic element represented in the representation of an emerging wavefront, it calculates the coefficient of the emerging wavefront for this basic element at least as a function of the coefficient of the associated incident wavefront to the same base element is determined.

The basic elements are particularly preferably classified according to at least one order parameter and the at least one wavefront transfer function assigns the incident wavefronts to the respective emerging wavefront, in particular in such a way that they assign the coefficients for one (in particular each) basic element represented in the representation of an emerging wavefront this base element represented in the representation of the emerging wavefront is determined at least as a function of those coefficients of the associated incident wavefront for those base elements whose value of the order parameter corresponds to the value of the order parameter of the respective base element represented in the representation of the emerging wavefront.

In addition, it is particularly preferred if the at least one wavefront transfer function assigns the respective emerging wavefront to the incident wavefronts in such a way that, for one (in particular each) basic element represented in the representation of an emerging wavefront, it calculates the coefficient for this in the representation of the emerging wavefront represented basic element depending on coefficients of the associated incident wavefront determined for a plurality (in particular all) of those basic elements whose value of the order parameter is less than or equal to the value of the order parameter of the respective represented in the representation of the emerging wavefront basic element. In particular, coefficients of the associated incident wave front can be taken into account for a plurality of base elements with different values of their order parameters.

In particular, each incident wavefront as well as the respectively associated emerging wavefront can be represented with reference to a basic system, with basic elements of each incident wavefront being classified according to at least one order parameter, and with the at least one wavefront transfer function for a predetermined value of a first order parameter being given in that for each wave front incident on the optical system, it converts at least one of the basic elements whose order parameter is less than or equal to the specified value into an associated basic element of the associated emerging wave front whose order is less than the specified value. In particular, the base elements for each value of the first order parameter are additionally classified according to at least one second order parameter, the value range of which depends on the value of the first order parameter.

In particular, in a preferred embodiment, both each incident wavefront and the respective associated emerging wavefront are represented by coefficients that are associated with the basic elements of a basic system, the basic elements being classified according to at least one order parameter, and the at least one wavefront transfer function being given thereby is that, for each wave front incident on the optical system, it converts at least one coefficient, which is assigned to a base element with an order parameter less than or equal to a predetermined value, into a coefficient of the emerging wave front, which is associated with the same base element. In particular, at least the coefficients for base elements of the incident wavefront whose order parameters are less than or equal to the specified value can be taken into account for this purpose. In other words, at least the coefficients for basic elements of the incident wavefront whose order parameters are less than or equal to the specified value can be converted into a coefficient of the outgoing wavefront whose order parameter is less than or equal to the specified value.

In particular, the at least one wavefront transfer function can be given in that, for each wavefront incident on the optical system, it converts at least the coefficients for the base elements whose order parameters are less than or equal to a predetermined value into a coefficient for the base element of the associated emerging wavefront, whose order parameter is equal to the specified value. In this case, the coefficients of base elements of the incident wavefront whose order parameter is greater than the specified value can also be taken into account.

With a suitable choice of the base system, mixing terms of different orders can advantageously be neglected, so that a coefficient of the incident wavefront associated with a base element of a specific order can be converted into a coefficient of the emerging wavefront associated with the same base element.

In a further preferred embodiment, the order parameter is a first order parameter, with the basis elements of the basis system also being classified according to at least one second order parameter, the value range of which depends on the value of the first order parameter.

der ersten p Ordnungen jeder in das optische System einfallenden zweidimensionalen Wellenfront in zumindest eine der Aberrationen $E{\text{'}}_{n_x,n_y}$

der ersten p Ordnungen der ausfallenden zweidimensionalen Wellenfront überführt, wobei
$2\le n_x+n_y\le p\text{sowie}{q}_x,q_y\ge 0\text{und}{n}_x,n_y\ge 0$

gilt.The at least one wavefront transfer function can be given, for example, for a given order p by converting at least one of the aberrations E ₂ , E ₃ , ..., E _p of the first p orders of each wavefront incident on the optical system into one of the Aberrations E' ₂ , E' ₃ , ..., E' _p of the first p orders of the emerging wave front are transferred. Alternatively, aberrations can also be used in three dimensions, in which case the at least one wavefront transfer function for each given order p is given by the fact that for 2 ≤ q _x + q _y ≤ p the aberrations $E_{q_x,q_y}$

of the first p orders of each two-dimensional wavefront incident on the optical system into at least one of the aberrations $E{\text{'}}_{n_x,n_y}$

of the first p orders of the emerging two-dimensional wavefront, where
$2\le n_x+n_y\le p\text{such as}{q}_x,q_y\ge 0\text{and}{n}_x,n_y\ge 0$

is applicable.

In a further preferred embodiment (two-dimensional case), the basic system is therefore a decomposition according to aberrations (or the basic system is defined or specified by a decomposition according to aberrations), the order parameter being an order p of the aberrations, the to the basic elements of the basic system associated coefficients are given in that a p-th order coefficient of the incident wavefront is an incident wavefront aberration E _p and a p-th order coefficient of the associated outgoing wavefront is an aberration E' _p of the associated outgoing wavefront, and where p ≥ 2.

In a further preferred embodiment (three-dimensional case), the basic system is a decomposition according to aberrations (or the basic system is defined or specified by a decomposition according to aberrations), the first order parameter being the sum p of orders p _x and p _y of the aberrations, where the second order parameter is one of the orders p _x and p _y , where the coefficients associated with the basis elements of the basis system are given in that p-th order coefficients of the incident wavefront are aberrations E _px,py of the incident wavefront and that coefficients p- th order of the associated emergent wavefront are aberrations E' _px,py of the associated emergent wavefront, and where p ≥ 2 and p _x ≥ 0 and p _y ≥ 0.

auf, wobei die Indizes des Tupels k = (k₁,k₂,...,k_p-1) über den Bereich P(k*) ≤ p - 2 und 0 ≤ k₁ ≤ 2(p - P(k*) - 2) + δ_P(k*),0 laufen, wobei $P\left(k*\right)={\displaystyle {\sum }_{j=1}^{p-2}j{k}_{j+1}},$

P(k*) und wobei $-{\overline{r}}_1^0\left(p-1\right)=p-{\delta }_{\left(p-1\right)\mathrm{,1}}$

und $-\Delta {\overline{r}}_1\left(p-\mathrm{1,}k*\right)=\left(p-3\right)+{\delta }_{\left(p-1\right)\mathrm{,1}}-P\left(k**\right)$

gelten, und wobei β = (-BE₂ + A)^-1 als Funktion der zumindest einen einfallenden Wellenfront und des optischen Systems gegeben ist, und wobei A, B und die Wellenfront-TransferKoeffizienten ${\overline{b}}_{pk}$

als Funktion der Komponenten des optischen Systems gegeben sind.In a further preferred embodiment, the at least one wavefront transfer function has the form $$E{\text{'}}_p={\beta }^{-{\overline{\mathrm{right}}}_1^0\left(p-1\right)}{\displaystyle {\sum }_{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102020008145A1\_0383.png,DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_1,{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2,...,k_{p-1}}{\overline{b}}_{pk}}{\beta }^{-\Delta {\overline{\mathrm{right}}}_1\left(p-\mathrm{1,}k*\right)}{E}_2^{k_1}{E}_3^{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}\cdots E_p^{k_{p-1}},p=\mathrm{2,3,4,}...$$

where the indices of the tuple k = (k ₁ ,k ₂ ,...,k _p-1 ) over the range P(k*) ≤ p - 2 and 0 ≤ k ₁ ≤ 2(p - P(k *) - 2) + δ _P(k*),0 , where $P\left(k*\right)={\displaystyle {\sum }_{j=1}^{p-2}j{k}_{j+1}},$

P(k*) and where $-{\overline{\mathrm{right}}}_1^0\left(p-1\right)=p-{\delta }_{\left(p-1\right)\mathrm{,1}}$

and $-\Delta {\overline{\mathrm{right}}}_1\left(p-\mathrm{1,}k*\right)=\left(p-3\right)+{\delta }_{\left(p-1\right)\mathrm{,1}}-P\left(k**\right)$

and where β = (-BE ₂ + A) ^-1 is given as a function of the at least one incident wavefront and the optical system, and where A, B and the wavefront transfer coefficients ${\overline{b}}_{pk}$

are given as a function of the components of the optical system.

der ersten p Ordnungen jeder in das optische System einfallenden zweidimensionalen Wellenfront in zumindest eine der Taylor-Ableitungen $w{\text{'}}^{\left(n_x,n_y\right)}$

der ersten p Ordnungen der ausfallenden zweidimensionalen Wellenfront überführt, wobei $2\le n_x+n_y\le p\text{sowie}{q}_x,q_y\ge 0\text{und}{n}_x,n_y\ge 0$

gilt.The at least one wavefront transfer function can also be given for each given order p by introducing the Taylor derivatives w ⁽²⁾ , w ⁽³⁾ , . . . , w ^(p) of the first p orders of each into the optical system incident wavefront into at least one of the Taylor derivatives w' ⁽²⁾ , w' ⁽³⁾ , ..., w' ^(p) of the first p orders of the emerging wavefront. Alternatively, Taylor derivatives can also be used in three dimensions, in which case the at least one wavefront transfer function for each given order p is given by the fact that it is the Taylor derivatives for 2≦ _qx + _qy ≦p $w^{\left(q_x,q_y\right)}$

of the first p orders of each two-dimensional wavefront incident on the optical system into at least one of the Taylor derivatives $w{\text{'}}^{\left(n_x,n_y\right)}$

of the first p orders of the emerging two-dimensional wavefront, where $2\le n_x+n_y\le p\text{such as}{q}_x,q_y\ge 0\text{and}{n}_x,n_y\ge 0$

is applicable.

In a further preferred embodiment (two-dimensional case), the basic system is a decomposition according to Taylor derivatives (or the basic system is defined or specified by a decomposition according to Taylor derivatives), the order parameter being an order p of the Taylor derivatives, where the coefficients associated with the basis elements of the basis system are given in that a p-th order coefficient of the incident wavefront is a Taylor derivative w ^(p) of the incident wavefront and that the p-th order coefficient of the associated emerging wavefront is a Taylor derivative w' ^(p) of the associated outgoing wavefront, and where p ≥ 2. In the Taylor derivations These can be Taylor derivations of the wavefront arrow heights or Taylor derivations of the wavefront OPD (OPD = Optical Path Difference).

In a preferred embodiment (two-dimensional case), the basic system is therefore a decomposition according to Taylor derivatives of the wavefront arrow heights, the order parameter being an order p of the Taylor derivatives of the wavefront arrow heights, the coefficients associated with the basic elements of the basic system being given by this are that a p-th order coefficient of the incident wavefront is a Taylor derivative w ^(p) of the versine of the incident wavefront and that the p-th order coefficient of the associated outgoing wavefront is a Taylor derivative w' ^(p) of the versine of the associated outgoing wavefront, and where p ≥ 2. The designation w ^(p) means the Taylor derivation of order p of the function w of the incident wavefront, preferably at point 0. Accordingly, w ^(p) means the Taylor derivation of order p of the function w' of the emerging wavefront, preferably at the position 0.

In an alternative preferred embodiment (two-dimensional case), the basic system is a decomposition according to Taylor derivatives of a wavefront OPD (optical path difference), where the order parameter is an order p of the Taylor derivatives of the wavefront OPD, where the basic elements of the coefficients associated with the base system are given in that a p-th order coefficient of the incident wavefront is a Taylor derivative OPD ^(p) of the incident wavefront and that the p-th order coefficient of the associated outgoing wavefront is a Taylor derivative OPD' ^(p) of the associated outgoing wavefront, and where p ≥ 2. The designation OPD ^(p) means the Taylor derivative of order p of the function OPD of the incident wave front, preferably at point 0. Accordingly, OPD' ^(p) means the Taylor derivative of order p of the function OPD' of the emerging wave front, preferably at position 0.

In a further preferred embodiment (three-dimensional case), the basic system is a decomposition according to Taylor derivatives (or the basic system is defined or specified by a decomposition according to Taylor derivatives), the first order parameter being the sum p of orders p _x and p _y of the Taylor derivatives, where the second order parameter is one of the orders p _x and p _y , where the coefficients associated with the basis elements of the basis system are given by p-th order coefficients of the incident wavefront Taylor derivatives w ^{(px,py )} of the incident wavefront and that the p-th order coefficients of the associated outgoing wavefront are Taylor derivatives w' ^(px,py) of the associated outgoing wavefront, and where p ≥ 2 and p _x ≥ 0 and p _y ≥ 0. Here, too, the Taylor derivations can be Taylor derivations of the wavefront arrow heights or Taylor derivations of the wavefront OPD.

The basic system is thus in a preferred embodiment (three-dimensional case) a decomposition according to Taylor derivatives of the wavefront arrow heights, where the first order parameter is the sum p of orders p _x and p _y of the Taylor derivatives of the wavefront arrow heights, where the second order parameter is one of the orders p _x and p _y , where the coefficients associated with the basis elements of the basis system are given by the p-th order coefficients of the incident wavefront being Taylor derivatives w ^(px,py) of the versine of the incident wavefront and that the p-th order coefficients of the associated emergent wavefront are Taylor derivatives w' ^(px,py) of the versine of the associated emergent wavefront, and where p ≥ 2 and p _x ≥ 0 and p _y ≥ 0. The designation w ^(px,py) means the Taylor derivative of the order p=px+py of the function w( _x , _y ) of the incident wavefront, preferably at the point x=0, y=0. Accordingly, w means ' ^(px,py) the Taylor derivative of the order p = p _x + p _y of the function w'(x',y') of the emerging wave front, preferably at the point x' = 0, y' = 0. In a In an alternative preferred embodiment (three-dimensional case), the basic system is a decomposition according to Taylor derivatives of a wavefront OPD, the first order parameter being the sum p of orders p _x and p _y of the Taylor derivatives of the wavefront OPD, the second order parameter being is one of the orders p _x and p _y , where the coefficients associated with the basis elements of the basis system are given in that p-th order coefficients of the incident wavefront are Taylor derivatives OPD ^(px,py) of the incident wavefront and that the coefficients p -th order of the associated emerging wave nfront are Taylor derivatives OPD' ^(px,py) of the associated emergent wavefront, and where p ≥ 2 and p _x ≥ 0 and p _y ≥ 0. The designation OPD ^(px,py) means the Taylor derivative of the order p = p _x + p _y of the function OPD of the incident wave front, preferably at the point x = 0, y = 0. Accordingly, OPD' ^{(px,py )} the Taylor derivative of the order p = p _x + p _y of the function OPD' of the emerging wave front, preferably at the point x' = 0, y' = 0.

der ersten i = p - 1 Ordnungen der ausfallenden zweidimensionalen Wellenfront überführt, wobei 1 ≤ n_x + n_y ≤ i sowie j_x,j_y ≥ 0 und n_x,n_y ≥ 0 gilt.The at least one wavefront transfer function can also be given for each given order p by taking the derivatives t ^(1), t ^(2), . . . , t ⁽ⁱ⁾ of the first i=p-1 orders of a direction function t (x) converts each wavefront incident into the optical system into at least one of the derivatives t' ⁽¹⁾ , t' ⁽²⁾ , ..., t' ⁽ⁱ⁾ of the first i = p - 1 orders of a directional function of the emerging wavefront. Alternatively, local directions of the wavefront can also be used in three dimensions, in which case the at least one wavefront transfer function for each given order i = p - 1 is given by the fact that for 2 ≤ j _x + j _y ≤ i the Taylor -derivatives t _x ^(jx,jy) , t _y ^(jx,jy) of the first i = p - 1 orders of each two-dimensional wavefront incident in the optical system in at least one of the derivatives $t{\text{'}}_x{}^{\left(n_x,n_y\right)},t{\text{'}}_y{}^{\left(n_x,n_y\right)}$

of the first i = p - 1 orders of the emerging two-dimensional wavefront, where 1 ≤ n _x + n _y ≤ i as well as j _x ,j _y ≥ 0 and n _x ,n _y ≥ 0.

In a further preferred embodiment (two-dimensional case), the basic system is a decomposition according to derivatives of directional functions (or the basic system is defined or specified by a decomposition according to derivatives of directional functions), the order parameter being an order i of the derivatives of the directional functions, where the coefficients associated with the basis elements of the basis system are given in that an i-th order coefficient of the incident wavefront is a derivative t ⁽ⁱ⁾ of a directional function t(x) of the incident wavefront and that an i-th order coefficient of the associated emerging wavefront is a derivative t ⁽ⁱ⁾ of a directional function t'(x') of the associated outgoing wavefront, and where i ≥ 1.

In a further preferred embodiment (three-dimensional case), the basic system is a decomposition according to derivatives of directional functions (or the basic system is defined or specified by a decomposition according to derivatives of directional functions), the first order parameter being the sum i of orders i _x and i _y of the derivatives of the directional functions, with the second order parameter being one of the orders i _x and i _y , with the coefficients associated with the basis elements of the basis system being given by the fact that the i-th order coefficients of the incident wavefront are derivatives t _x ^(ix,iy) , t _y ^(ix,iy) of directional functions t _x (x,y), t _y (x,y) of the incident wavefront and that coefficients i-th order of the associated emerging wavefront are derivatives t' _x ^(ix,iy) , t' _y ^(ix,iy) of directional functions t' _x (x',y'), t' _y (x',y') of the associated emergent wavefront, and where i ≥ 1 and i _x ≥ 0 and i _y ≥ 0 applies.

der ersten p radialen Ordnungen jeder in das optische System einfallenden zweidimensionalen Wellenfront in zumindest einen der Zernike- Koeffizienten $Z_n^{im}$

der ersten p radialen Ordnungen der ausfallenden zweidimensionalen Wellenfront überführt, wobei 2 ≤ n ≤ p, und wobei n und q die radialen Ordnungen sowie m und r die in Schritten von 2 variierenden azimutalen Ordnungen sind, wobei -q ≤ r ≤ q und -n ≤ m ≤ n ist, wobei sich die Zernike Koeffizienten insbesondere auf eine festgelegte Pupille beziehen.Furthermore, Zernike polynomials can be used in two dimensions. The at least one wavefront transfer function for each given order p can be given by converting the Zernike coefficients Z ₂ , Z _{3 ,} ..., Z _p of the first p orders of each wavefront incident on the optical system into at least one of the _Zernike coefficients Z′ ₂ , Z′ ₃ , . Alternatively, Zernike polynomials can be used in three dimensions, in which case the at least one wavefront transfer function for any given order p is given by the Zernike coefficients for 2≦q≦p $Z_q^{\mathrm{right}}$

of the first p radial orders of any two-dimensional wavefront incident on the optical system into at least one of the Zernike coefficients $Z_n^{im}$

of the first p radial orders of the outgoing two-dimensional wavefront, where 2 ≤ n ≤ p, and where n and q are the radial orders and m and r are the azimuthal orders varying in steps of 2, where -q ≤ r ≤ q and -n ≤ m ≤ n, where the Zernike coefficients relate in particular to a fixed pupil.

In a further preferred embodiment (two-dimensional case), the basic system is a decomposition according to Zernike polynomials (or the basic system is defined or specified by a decomposition according to Zernike polynomials), the order parameter being a radial order n of the Zernike polynomials, where the coefficients associated with the basis elements of the basis system are given in that an nth-order coefficient of the incident wavefront is a Zernike coefficient Z _n of the incident wavefront and that an nth-order coefficient of the outgoing wavefront is a Zernike coefficient Z' _n of the associated emergent wavefront, where n ≥ 2, and in particular where the Zernike coefficients relate to a fixed pupil.

In a further preferred embodiment (three-dimensional case), the basic system is a decomposition according to Zernike polynomials (or the basic system is defined or specified by a decomposition according to Zernike polynomials), the first order parameter being a radial order n of the Zernike polynomials, where the second order parameter is an azimuthal order m of the Zernike polynomials, where the coefficients associated with the basis elements of the basis system are given by the fact that nth-order coefficients of the incident wavefront are Zernike coefficients Z _n ^m of the incident wavefront and that nth-order coefficients of the associated emerging wavefront are Zernike coefficients Z' _n ^m der where n ≥ 2 and -n ≤ m ≤ n, where m is even for even n, m is odd for n odd, and in particular where the Zernike coefficients relate to a fixed pupil.

A further independent aspect for solving the problem relates to a computer program product which comprises machine-readable program code which, when it is loaded on a computer, is suitable for executing the method according to the invention described above. In particular, a computer program product is a program stored on a data carrier. In particular, the program code is stored on a data carrier. In other words, the computer program product comprises computer-readable instructions which, when loaded into a memory of a computer and executed by the computer, cause the computer to perform an inventive method as described above. In particular, the computer program product may comprise a computer-readable storage medium having code stored thereon, which code, when executed by a processor, causes the processor to implement a method according to the invention. In particular, the computer program product can also include or be a storage medium with a computer program stored thereon, the computer program being designed to carry out a method according to the invention when loaded and executed on a computer. In particular, the invention offers a computer program product, in particular in the form of a storage medium or a data stream, which contains program code which, when loaded and executed on a computer, is designed to carry out a method according to the invention, in particular in a preferred embodiment.

• - ein Modellierungsmodul zum Bereitstellen zumindest einer Wellenfront-Transferfunktion für das optische System, wobei die Wellenfront-Transferfunktion dafür bestimmt ist, jeder in das optische System einfallenden Wellenfront unter Berücksichtigung von Abbildungsfehlern mit einer Ordnung größer als die Ordnung eines Defocus eine zugehörige ausfallende Wellenfront zuzuordnen;
• - ein Auswertemodul zum Auswerten der zumindest einen Wellenfront-Transferfunktion für zumindest eine in das optische System einfallende Wellenfront.

Another independent aspect for solving the problem relates to a device for simulating an optical system by means of a wavefront calculation, the optical system being in particular a complex optical system whose effect goes beyond a single refraction, a single propagation or a single reflection, comprising:

• - a modeling module for providing at least one wavefront transfer function for the optical system, wherein the wavefront transfer function is intended to assign an associated emerging wavefront to each wavefront incident in the optical system, taking into account aberrations with an order greater than the order of a defocus;
• - An evaluation module for evaluating the at least one wavefront transfer function for at least one wavefront incident on the optical system.

The modeling module can comprise an interface and a storage device with which the wavefront transfer function can be provided and stored. Providing the wavefront transfer function is understood to mean in particular setting up the wavefront transfer function. In this case, the wavefront transfer function is provided or set up automatically or in a computer-implemented manner. Accordingly, the modeling module can include a computing unit or a processor with which the wavefront transfer function can be provided or set up (automatically). In particular, the device also includes a data interface for acquiring data of the optical system or overall system, an evaluation module for evaluating the overall optical system, and/or an optimization module for optimizing the first subsystem.

• Berechnen oder Optimieren eines Brillenglases unter Verwendung eines oben beschriebenen erfindungsgemäßen Verfahrens; und
• Bereitstellen von Fertigungsdaten des so berechneten oder optimierten Brillenglases, und/oder Fertigen des so berechneten oder optimierten Brillenglases.

In a further aspect, the invention relates to a method for producing a spectacle lens, comprising the steps:

• Calculating or optimizing a spectacle lens using a method according to the invention described above; and
• Providing production data for the spectacle lens calculated or optimized in this way, and/or manufacturing the spectacle lens calculated or optimized in this way.

• Berechnungs- oder Optimierungsmittel, welche ausgelegt sind, das Brillenglas unter Verwendung eines erfindungsgemäßen Verfahrens zu berechnen oder zu optimieren; und
• Bearbeitungsmittel, welche ausgelegt sind, das Brillenglas fertig zu bearbeiten.

In a further aspect, the invention relates to a device for producing a spectacle lens comprising:

• Calculation or optimization means which are designed to calculate or optimize the spectacle lens using a method according to the invention; and
• Processing means designed to finish the spectacle lens.

In a further aspect, the invention relates to the use of a spectacle lens manufactured according to the manufacturing method according to the invention in a predetermined average or individual usage position of the spectacle lens in front of the eyes of a specific spectacle wearer to correct ametropia of the spectacle wearer.

The statements made above or below regarding the embodiments of the first aspect also apply to the above-mentioned further independent aspects and in particular to preferred embodiments in this regard. In particular, the statements made above and below regarding the embodiments of the respective other independent aspects also apply to an independent aspect of the present invention and to related preferred embodiments.

In the following, individual embodiments for solving the problem are described by way of example with reference to the figures. Some of the individual embodiments described have features that are not absolutely necessary to implement the claimed subject matter, but which provide desired properties in certain applications. Thus, embodiments that do not have all the features of the embodiments described below are also to be regarded as being disclosed as falling under the technical teaching described. Furthermore, in order to avoid unnecessary repetition, certain features are only mentioned in relation to individual embodiments described below. It is pointed out that the individual embodiments should therefore not only be considered individually, but should also be viewed as a whole. Based on this synopsis, the person skilled in the art will recognize that individual embodiments can also be modified by incorporating individual or multiple features of other embodiments. It is pointed out that a systematic combination of the individual embodiments with individual or multiple features that are described in relation to other embodiments can be desirable and useful and should therefore be considered and should also be regarded as covered by the description.

character list

• 1 shows a schematic sketch for an exemplary optical system;
• 2 shows a schematic sketch for the specification of an optical system using a beam transfer function;
• 3 shows a schematic sketch for the transformation between wavefronts w(x),w'(x') in the w-representation and the functions t(x),t'(x) in the t-representation according to an embodiment of the present invention;
• 4 Figure 12 shows a schematic flow diagram according to an embodiment of the present invention based on a Taylor series in a meridian;
• 5 shows a schematic sketch of the relationship between the OPD τ(x) and the direction of a t(x) ray which is perpendicular to a wavefront w(x);
• 6 shows a schematic sketch of the ray transfer function for propagation;
• 7a and 7b show schematic sketches of the ray transfer function for refraction;
• 8th shows a schematic sketch of a Young diagram for the tuple k = (1,0,2);
• 9 Figure 12 shows a schematic sketch of a modified Gullstrand-Emsley eye (mGE eye) to illustrate an embodiment of the present invention.

Detailed description of the drawings

• • Strahl
  • Infinitesimales Lichtbündel, beschrieben als Gerade, Halbgerade oder Strecke im Raum, die bevorzugt anhand von Durchstoßpunkten durch Ebenen und Richtungsparameter beschrieben werden. Im Falle eines Meridians wird ein Strahl beschrieben durch
    • - x Positionsparameter eines Strahls
    • - t Richtungsparameter eines Strahls
    • - (x, t) Parameter eines einfallenden Strahls
    • - (x',t') Parameter eines ausfallenden Strahls
  • Im Falle von zwei Meridianen wird ein Strahl beschrieben durch
    • - x, y Positionsparameter eines Strahls
    • - t_x,t_y Richtungsparameter eines Strahls
    • - (x,y, t_x,t_y) Parameter eines einfallenden Strahls
    • - (x',y', t'_x, t'_y) Parameter eines ausfallenden Strahls
• • Wellenfront
  • Im Fall eines Meridians ist eine Wellenfront eine Kurve in dem betrachteten Meridian, die senkrecht auf Strahlen steht. Der Einfachheit halber wird eine Kurve dabei auch als Fläche bezeichnet.
    • - w(x) Funktion zur Beschreibung einer Wellenfront als Fläche
    • - w^(p) Ableitung der Ordnung p der Funktion w(x), bevorzugt an der Stelle x = 0
    • - E_p = nw^(p) Aberration der Ordnung p der Wellenfront w(x), wobei n der Brechungsindex ist
    • - t(x) Funktion zur Beschreibung einer Wellenfront durch Abhängigkeit des Richtungsparameters vom Positionsparameter
    • - t⁽ⁱ⁾ Ableitung der Ordnung i der Funktion t(x), bevorzugt an der Stelle x = 0
    • - w(x), w^(p), E_p, t(x), t⁽ⁱ⁾ Größen zur Beschreibung der einfallenden Wellenfront
    • - w'(x'), w'^(p) , E'_p, t'(x'), t⁽ⁱ⁾ Größen zur Beschreibung der ausfallenden Wellenfront
  • Im Fall von zwei Meridianen ist eine Wellenfront eine Fläche im Raum, die senkrecht auf Strahlen steht
    • - w(x,y) Funktion zur Beschreibung einer Wellenfront als Fläche
    • - $w^{\left(p_x,p_y\right)}$
      
      Ableitung der Ordnung p = p_x + p_y der Funktion w(x,y), bevorzugt an der Stelle x = 0, y = 0
    • - $E_{p_x,p_y}=n{w}^{\left(p_x,p_y\right)}$
      
      Aberration der Ordnung p = p_x + p_y der Wellenfront w(x,y), wobei n der Brechungsindex ist
    • - t_x(x,y), t_y(x,y) Funktionen zur Beschreibung einer Wellenfront durch Abhängigkeit des Richtungsparameters vom Positionsparameter
    • - $t_x^{\left(i_x,i_y\right)},t_x^{\left(i_x,i_y\right)}$
      
      Ableitung der Ordnung i = i_x + i_y der Funktionen t_x(x,y), ty(x,y), bevorzugt an der Stelle x = 0, y = 0
    • - w(x,y), $w^{\left(p_x,p_y\right)},E_{p_x,p_y},$
      
      t_x(x,y), t_y(x,y) $t_x^{\left(i_x,i_y\right)},t_y^{\left(i_x,i_y\right)}$
      
      Größen zur Beschreibung der einfallenden Wellenfront
    • - w'(x',y'), $w{\text{'}}^{\left(p_x,p_y\right)},E{\text{'}}_{p_x,p_y},$
      
      t’_x(x'y'), t'_y(x',y'), $t{\text{'}}_x^{\left(i_x,i_y\right)},t{\text{'}}_y^{\left(i_x,i_y\right)}$
      
      Größen zur Beschreibung der ausfallenden Wellenfront
• • Strahl-Transferfunktion
  • Im Fall eines Meridians Funktion f(x,t), die einem einfallenden Strahl mit Parametern (x, t) die Parameter (x',t') eines ausfallenden Strahls zuordnet
    • - f^prop(x,t) Strahl-Transferfunktion für die reine Propagation
    • - f^ref(x,t) Strahl-Transferfunktion für die reine Refraktion
  • Im Fall von zwei Meridianen Funktion f(x,y,t_x,t_y), die einem einfallenden Strahl mit Parametern (x,y,t_x,t_y) die Parameter (x',y',t'_x,t'_y) eines ausfallenden Strahls zuordnet
    • - f^prop(x,y,t_x,t_y) Strahl-Transferfunktion für die reine Propagation
    • - f^ref(x,y, t_x,t_y) Strahl-Transferfunktion für die reine Refraktion
• • Wellenfront-Transferfunktion
  • Erfindungsgemäß ermittelte Funktion, die bei einer gegebenen Strahl-Transferfunktion f(x,t) bzw. f(x,y,t_x,t_y) einer einfallenden Wellenfront eine ausfallende Wellenfront zuordnet
• • Wellenfront-Transferkoeffizienten
  • Erfindungsgemäß ermittelte Koeffizienten, die bei einer gegebenen Strahl-Transferfunktion f(x, t) eines optischen Systems den Transfer einer Wellenfront durch das System beschreiben
    • - ${\overline{b}}_{pk}$
      
      Koeffizient zur Bestimmung der Ableitung t'⁽ⁱ⁾ der Ordnung i einer ausfallenden Wellenfront, der die Abhängigkeit von den Ableitungen t⁽¹⁾, ..., t⁽ⁱ⁾ der Funktion einer einfallenden Wellenfront beschreibt, wobei sich das Tupel k = (k₁,k₂,...,k_i) auf den Beitrag des Produkts ${\left(t^{\left(1\right)}\right)}^{k_1}{\left(t^{\left(2\right)}\right)}^{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}\cdots {\left(t^{\left(i\right)}\right)}^{k_i}$
      
      bezieht;
    • - ${\overline{b}}_{pk}$
      
      Koeffizient zur Bestimmung der Ableitung w'^(p) bzw. der Aberration E'_p der Ordnung p einer ausfallenden Wellenfront, der die Abhängigkeit von den Ableitungen w⁽²⁾, ..., w^(p) bzw. von den Aberrationen E₂, ..., E_p einer einfallenden Wellenfront beschreibt, wobei sich das Tupel k = (k₁,k₂,..., k_p-1) auf den Beitrag des Produkts ${\left(w^{\left(2\right)}\right)}^{k_1}{\left(w^{\left(3\right)}\right)}^{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}\cdots {\left(w^{\left(p\right)}\right)}^{k_{p-1}}$
      
      bzw. $E_2^{k_1}{E}_3^{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}\cdots E_p^{k_{p-1}}$
      
      bezieht.

In the context of this invention, the following terms are used as follows (unless otherwise specified, described in a meridian):

• • Beam
  • Infinitesimal light beam, described as a straight line, half-line or line in space, which are preferably described using points of intersection through planes and directional parameters. In the case of a meridian, a ray is described by
    • - x position parameters of a ray
    • - t directional parameters of a ray
    • - (x, t) parameters of an incident ray
    • - (x',t') parameters of an emergent ray
  • In the case of two meridians, a ray is described by
    • - x, y position parameters of a ray
    • - t _x ,t _y direction parameters of a ray
    • - (x,y,t _x ,t _y ) parameters of an incident ray
    • - (x',y', t' _x , t' _y ) parameters of an emergent ray
• • Wavefront
  • In the case of a meridian, a wavefront is a curve in the meridian under consideration that is perpendicular to rays. For the sake of simplicity, a curve is also referred to as a surface.
    • - w(x) Function for describing a wavefront as a surface
    • - w ^(p) Derivative of order p of the function w(x), preferably at the point x = 0
    • - E _p = nw ^{(p) p} -order aberration of the wavefront w(x), where n is the refractive index
    • - t(x) function for describing a wavefront by depending on the direction parameter from the position parameter
    • - t ⁽ⁱ⁾ Derivative of order i of the function t(x), preferably at the point x = 0
    • - w(x), w ^(p) , E _p , t(x), t ⁽ⁱ⁾ quantities describing the incident wave front
    • - w'(x'), w' ^(p) , E' _p , t'(x'), t ⁽ⁱ⁾ quantities for describing the emerging wavefront
  • In the case of two meridians, a wavefront is an area in space perpendicular to rays
    • - w(x,y) Function for describing a wave front as a surface
    • - $w^{\left(p_x,p_y\right)}$
      
      Derivation of the order p = p _x + p _y of the function w(x,y), preferably at the point x = 0, y = 0
    • - $E_{p_x,p_y}=n{w}^{\left(p_x,p_y\right)}$
      
      Aberration of order p = p _x + p _y of the wavefront w(x,y), where n is the refractive index
    • - t _x (x,y), t _y (x,y) Functions for describing a wavefront by dependence of the direction parameter on the position parameter
    • - $t_x^{\left(i_x,i_y\right)},t_x^{\left(i_x,i_y\right)}$
      
      Derivation of the order i = i _x + i _y of the functions t _x (x,y), ty(x,y), preferably at the point x = 0, y = 0
    • - w(x,y), $w^{\left(p_x,p_y\right)},E_{p_x,p_y},$
      
      t _x (x,y), t _y (x,y) $t_x^{\left(i_x,i_y\right)},t_y^{\left(i_x,i_y\right)}$
      
      Quantities describing the incident wave front
    • - w'(x',y'), $w{\text{'}}^{\left(p_x,p_y\right)},E{\text{'}}_{p_x,p_y},$
      
      t' _x (x'y'), t' _y (x',y'), $t{\text{'}}_x^{\left(i_x,i_y\right)},t{\text{'}}_y^{\left(i_x,i_y\right)}$
      
      Quantities to describe the emerging wavefront
• • Ray transfer function
  • In the case of a meridian, function f(x,t) which assigns the parameters (x',t') of an exiting ray to an incoming ray with parameters (x,t).
    • - f ^prop (x,t) ray transfer function for pure propagation
    • - f ^ref (x,t) ray transfer function for pure refraction
  • In the case of two meridians function f(x,y,t _x ,t _y ) corresponding to an incident ray with parameters (x,y,t _x ,t _y ) the parameters (x',y',t' _x ,t ' _y ) of an emerging beam
    • - f ^prop (x,y,t _x ,t _y ) ray transfer function for pure propagation
    • - f ^ref (x,y,t _x ,t _y ) ray transfer function for pure refraction
• • Wavefront transfer function
  • Function determined according to the invention which, given a beam transfer function f(x,t) or f(x,y,t _x ,t _y ), assigns an emerging wave front to an incident wave front
• • Wavefront transfer coefficients
  • Coefficients determined according to the invention, which describe the transfer of a wave front through the system for a given beam transfer function f(x, t) of an optical system
    • - ${\overline{b}}_{pk}$
      
      Coefficient for determining the derivative t' ⁽ⁱ⁾ of order i of an outgoing wavefront, which describes the dependence on the derivatives t ⁽¹⁾ , ..., t ⁽ⁱ⁾ of the function of an incident wavefront, where the tuple k = ( k ₁ ,k ₂ ,...,k _i ) on the contribution of the product ${\left(t^{\left(1\right)}\right)}^{k_1}{\left(t^{\left(2\right)}\right)}^{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}\cdots {\left(t^{\left(i\right)}\right)}^{k_i}$
      
      relates;
    • - ${\overline{b}}_{pk}$
      
      Coefficient for determining the derivative w' ^(p) or the aberration E' _p of order p of an emerging wave front, which is the dependence on the derivatives w ⁽²⁾ , ..., w ^(p) or on the aberrations E ₂ , ..., E _p describes an incident wave front, where the tuple k = (k ₁ ,k ₂ ,..., k _p-1 ) refers to the contribution of the product ${\left(w^{\left(2\right)}\right)}^{k_1}{\left(w^{\left(3\right)}\right)}^{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}\cdots {\left(w^{\left(p\right)}\right)}^{k_{p-1}}$
      
      or. $E_2^{k_1}{E}_3^{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}\cdots E_p^{k_{p-1}}$
      
      relates.

In a very general form, an optical system is already fully specified if there is a rule that clearly defines an outgoing ray for every incident ray (this rule is called the ray transfer function). All other details of the optical system are then irrelevant to the specification of the optical behavior, so that the system can also be understood as a "black box". If a wavefront is incident on such a system alongside a main ray, then the emerging wavefront, including its HOA, can only be determined according to the prior art by using ray tracing to calculate a sufficiently selected bundle of neighboring rays and on the emerging one Page again determines a wavefront from it numerically, for example by a fit method.

A step-by-step analytical calculation from area to area using the methods WO 2008 089 999A1 and DE 10 2011 101 923 A1 does not solve the task of the present invention, because not every optical system actually has to consist of individual components that can be described by refracting surfaces or propagations in a homogeneous medium. For example, a GRIN system (Gradient Index System) has no internal refracting surfaces at all apart from the input and output surfaces and achieves its effect through the inhomogeneity of the material. Such a GRIN system also has a well-defined behavior with regard to the imaging of rays, but the wavefront calculation cannot be done with one in the WO 2008 089 999 A1 and DE 10 2011 101 923 A1 described procedures are carried out.

Even if the complex optical system consists only of a finite number of refracting surfaces and propagations in the homogeneous medium, as is shown schematically in 1 shown, deliver the procedures WO 2008 089 999 A1 and DE 10 2011 101 923 A1 a result, but only at the price of a very high computing time. This applies in particular when there is a large number of interfaces or when an optical system is to be repeatedly calculated with different wave fronts.

Although it might seem obvious to combine a sequential execution of the existing methods from the prior art by mutually inserting the analytical formulas into one analytical method, the number of terms produced would be so high that it would seem completely hopeless that they would be so suitable again to summarize that there is a computing time advantage. However, this problem is also solved by the present invention by eliminating the needless intermediate steps and providing a wavefront transfer rule that is directly based on the ray transfer function a priori.

the 2 shows schematically an optical system for defining the beam transfer function. In linear optics, it is common to define a system by having an entrance plane and an exit plane limit level. More generally, an optical system can also be delimited by two non-parallel planes or by any two surfaces. What is essential about the specification, however, is that there is (initially in the case of a single meridian) a coordinate x on the incident surface and a coordinate x' on the emergent surface, which serve to describe the penetration points of the ray. Furthermore, there must be a clear way of defining the direction of an incident ray towards the entrance surface (e.g. through an angle α), and to define the direction of the emerging ray towards the exit surface (e.g. through an angle a'). In the case of two meridians, there must be two coordinates x, y and two angles α _x , α _y on the incoming side, and coordinates x', y' and two angles α' _x , α' _y on the outgoing side.

dann ist das optische System eindeutig spezifiziert, wenn eine Strahl-Transferfunktion f: R² ↦ R² mit den Komponenten f_x, f_t gegeben ist, die p nach p' überführt: $$\rho \text{'}=f\left(\rho \right)\iff \left(\begin{array}{c}x\text{'}\\ t\text{'}\end{array}\right)=\left(\begin{array}{c}{f}_x\left(x,t\right)\\ f_t\left(x,t\right)\end{array}\right)$$

Is then defined as a ray vector for a meridian $$\rho :=\left(\begin{array}{c}x\\ t\end{array}\right):=\left(\begin{array}{c}x\\ ntana\end{array}\right)\rho \text{'}:=\left(\begin{array}{c}x\text{'}\\ t\text{'}\end{array}\right):=\left(\begin{array}{c}x\text{'}\\ n\text{'}tana\text{'}\end{array}\right)$$

then the optical system is uniquely specified if a ray transfer function f: R ² ↦ R ² with the components f _x , f _t is given, which converts p to p': $$\rho \text{'}=f\left(\rho \right)\iff \left(\begin{array}{c}x\text{'}\\ t\text{'}\end{array}\right)=\left(\begin{array}{c}{f}_x\left(x,t\right)\\ f_t\left(x,t\right)\end{array}\right)$$

In two meridians, the optical system is uniquely specified if there is a corresponding ray transfer function f:R ⁴ ↦ R ⁴ .

wobei die 2x2 Systemmatrix (auch Transferenz genannt), definiert ist durch $$T=Jac\left(f\right)=\left(\begin{array}{cc}\partial x\text{'}/\partial x& \partial x\text{'}/\partial t\\ \partial t\text{'}/\partial x& \partial t\text{'}/\partial t\end{array}\right)=\left(\begin{array}{cc}{f}_x^{\left(\mathrm{1,0}\right)}& f_x^{\left(\mathrm{0,1}\right)}\\ f_t^{\left(\mathrm{1,0}\right)}& f_t^{\left(\mathrm{0,1}\right)}\end{array}\right)=:\left(\begin{array}{cc}A& B\\ C& D\end{array}\right),$$

wobei Jac die Jacobi Matrix bezeichnet, wobei die Einträge A, B, C, D Konstanten sind, die das optische System charakterisieren, und wobei die Notation $$f^m=f^{\left(m_x,m_t\right)}={\partial }^{m}f=\left({\partial }^{m_x}/\partial x^{m_x}{\partial }^{m_t}/\partial t^{m_t}\right)f$$

verwendet wird. Weiter ist im Stand der Technik bekannt, dass in einem Meridian eine Wellenfront mit Krümmung k und Vergenz S = nk, die in das System eintritt, beim Austritt zu einer Wellenfront mit Vergenz S' = n'k' führt, wobei die ausfallende Vergenz durch $$S\text{'}=-\frac{C-DS}{A-BS}$$

gegeben ist.It is state of the art both in Gaussian optics (f: R ² ↦R ² ) and in linear optics (f: R ⁴ ↦R ⁴ ) that the effect of a system described with f is described by a system matrix T becomes: $$\rho \text{'}=T\rho ,\iff \left(\begin{array}{c}x\text{'}\\ t\text{'}\end{array}\right)=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)\left(\begin{array}{c}x\\ y\end{array}\right)$$

where the 2x2 system matrix (also called transference) is defined by $$T=Jac\left(f\right)=\left(\begin{array}{cc}\partial x\text{'}/\partial x& \partial x\text{'}/\partial t\\ \partial t\text{'}/\partial x& \partial t\text{'}/\partial t\end{array}\right)=\left(\begin{array}{cc}{f}_x^{\left(1.0\right)}& f_x^{\left(0.1\right)}\\ f_t^{\left(1.0\right)}& f_t^{\left(0.1\right)}\end{array}\right)=:\left(\begin{array}{cc}A& B\\ C& D\end{array}\right),$$

where Jac denotes the Jacobi matrix, where the entries A, B, C, D are constants characterizing the optical system, and where the notation $$f^m=f^{\left(m_x,m_t\right)}={\partial }^{m}f=\left({\partial }^{m_x}/\partial x^{m_x}{\partial }^{m_t}/\partial t^{m_t}\right)f$$

is used. It is also known in the prior art that, in a meridian, a wavefront with curvature k and vergence S=nk entering the system results in a wavefront with vergence S'=n'k' when exiting, with the exiting vergence passing through $$S\text{'}=-\frac{C-DS}{A-BS}$$

given is.

In the case of two meridians, a corresponding connection can be given, see Qiang L, Shaomin W, Alda J, Bernabeu E: "Transformation of nonsymmetric Gaussian beam into symmetric one by means of tensor ABCD law", Optic - International Journal for Light and Electron Optics (OPTIK) 85(2): 67-72 (1990). Furthermore, the system can also have non-linear components, such as a prismatic effect.

The description in Eq.(3) for treating rays by means of a matrix T can be understood as a first-order description, the description in Eq.(5) for treating the vergences of wavefronts as a second-order description. On the other hand, there is no available disclosure in the prior art for dealing with higher order wavefront (HOA) features such as coma or spherical aberration given only the ray transfer function f. This is where the present invention comes in.

According to the invention, it has been recognized that a wavefront w(x) has a function t(x) which results from w(x) by a unique and reversible transformation H. The fixed input variable t, which was originally developed for linear optics, is reinterpreted according to the invention in such a way that it can also be used for the nonlinear description of wavefronts by allowing a dependency t(x). If you vary x, you then get two functions x'(x) and t'(x) on the failing side according to Eq.(2), which implicitly define a relationship t'(x'). This function t'(x') clearly corresponds to an emergent wavefront w'(x'), which is obtained by applying the inverse transformation H ^-1 to H. This connection is in the 3 shown schematically.

Wave fronts can, for example, be represented as symbolic functions or as free-form surfaces. Furthermore, wavefronts can be developed according to basis systems, where the order can function as an order parameter when counting according to the basis systems. Order parameters can preferably be used in such a way that contributions decrease in their numerical magnitude as the order parameter increases, so that contributions are taken into account up to a certain value of the order parameter and are neglected for even higher values of the order parameter. A similar procedure can be followed if several order parameters are present.

Wavefronts are preferably represented by Zernike polynomials. In a particularly preferred embodiment of the invention, wavefronts are represented by Taylor series, ie they are characterized by their local derivatives at a reference position, which is preferably at x=0. In this embodiment, the order parameter is the order p of the derivative w' ^(p) ( _x ') to be determined, in two meridians the sum p=px+ _py of the orders of the derivative $w{\text{'}}^{\left(p_x,p_y\right)}\left(x\text{'},y\text{'}\right).$

In a specific embodiment, optical systems are considered (in a meridian) which have parallel entry and exit planes and whose ray transfer function f(x,t) has the property f(0,0)=0. A ray preferably impinges on such a system at the point x=0 with direction t=0, which because of f(0,0)=0 also leaves the system again with x′=0 and t′=0. According to a further feature of the embodiment, the basic procedure takes place using a Taylor series for the wave fronts, preferably evaluated at the point x=0. The incoming and outgoing wave fronts, which must be perpendicular to the rays, then automatically have vanishing first derivatives w ^{( 1)} (0) = 0 and w' ⁽¹⁾ (0) = 0.

the 4 12 shows a schematic flow diagram according to an embodiment of the present invention based on a Taylor series in a meridian. In this embodiment, after defining an order p, the ray transfer function f(x,t) in the form of all its partial derivatives f _x ^(1,0) (0,0), f _x ^(0,1) (0,0 ), f _t ^(1,0) (0,0), f _t ^(0'1) (0,0), f _x ^(2,0) (0,0),... up to order (p - 1) are present. If these are available, then in the next step the method for determining the wavefront transfer function can be carried out directly, preferably for assigning wavefront transfer coefficients for the wavefront calculation (see 4 ). However, if the derivatives of the ray transfer function are not available, they must be determined beforehand. If the optical system consists of a succession of refracting surfaces interspersed with homogeneous media, then the derivatives of f(x,t) can be determined from the shape and location of the surfaces and the refractive indices. All surfaces are preferably also perpendicular to the ray at x = 0, then the derivatives of f(x,t) can be determined from the derivatives of the surfaces and from their distances. However, if there are no interfaces (eg in the case of a GRIN material), then the derivatives of f(x,t) must be determined in some other way, for example by measurement or by simulation.

zur Durchrechnung in der t-Darstellung aus den Ableitungen von f(x,t) bis zur Ordnung i = p - 1 bestimmt werden. Dies kann numerisch oder bevorzugt symbolisch erfolgen. In einer bevorzugten Ausführungsform jedoch werden aus den Koeffizienten ${\overline{c}}_{ik}$

mit Hilfe der Transformation H zuerst die Koeffizienten ${\overline{b}}_{pk}$

zur Wellenfront-Durchrechnung in der w-Darstellung bis zur Ordnung p direkt aus den Koeffizienten ${\overline{c}}_{ik}$

und aus den Ableitungen von f(x,t) bestimmt, wobei dieser Schritt wieder numerisch oder bevorzugt symbolisch durchgeführt werden kann.The method can be carried out, for example, by using the coefficients ${\overline{c}}_{ik}$

can be determined for calculation in the t representation from the derivatives of f(x,t) up to the order i = p - 1. This can be done numerically or preferably symbolically. In a preferred embodiment, however, the coefficients become ${\overline{c}}_{ik}$

with the help of the transformation H first the coefficients ${\overline{b}}_{pk}$

for wavefront calculation in the w representation up to order p directly from the coefficients ${\overline{c}}_{ik}$

and from the derivatives determined by f(x,t), this step again being able to be carried out numerically or, preferably, symbolically.

bis zur Ordnung p einmal vor, dann kann die Wellenfrontdurchrechnung von Wellenfronten bis zur Ordnung p mit beliebig vielen Wellenfronten wiederholt werden.lie the coefficients ${\overline{b}}_{pk}$

up to order p once, then the wavefront calculation of wavefronts up to order p can be repeated with any number of wavefronts.

oder von Koeffizienten ${\overline{b}}_{pk}$

verwendet werden, oder die bereits bis zu einer Ordnung p ermittelte Wellenfront-Transferfunktion kann bei gegebenen Koeffizienten direkt invertiert werden, um die Ableitungen der einfallenden Wellenfront durch die Ableitungen der ausfallenden Wellenfront zu bestimmen.For a wavefront calculation with reverse light direction, the inverse function can alternatively be formed from the beam transfer function f and then the method according to the invention for determining coefficients ${\overline{c}}_{ik}$

or by coefficients ${\overline{b}}_{pk}$

can be used, or the wavefront transfer function already determined up to an order p can be directly inverted for given coefficients in order to determine the derivatives of the incident wavefront through the derivatives of the outgoing wavefront.

Embodiment for the transformation H

Every connection between a wavefront w(x) and a function t(x) is a transformation. A transformation can be described symbolically or numerically. A preferred embodiment consists in expanding the function t(x) also in a Taylor series and expressing the derivatives t ⁽ⁱ⁾ (x) by derivatives w ^(p) (x), preferably for x = 0.

The task of converting the description of a spatial area by w(x) into a function t(x), which is related to a plane, is solved according to the invention as follows. Unlike in the prior art, in which the OPD also gives rise to a function τ(x) that has a defined relationship to the wavefront w(x), t(x) designates a direction and not an OPD. Therefore, a connection between τ(x) and τ(x) must first be established here.

In the 5 a wavefront is shown together with its reference plane, both of which are pierced by a neighboring ray with direction t. The tangent of the direction angle a, given by definition by tan α = t/n, on the other hand, is equal to the aspect ratio of in 5 shown right triangle. If one sets the length of the hypotenuse equal to 1, then the length of the opposite side of a is given by τ _x /n (proportional to the derivative of the OPD). This gives the equation $$tana=:\frac{t\left(x\right)}{n}=-\frac{{\tau }_x\left(x\right)/\mathrm{<figure-callout\; id="1"\; label="n"\; filenames="DE102020008145A1\_0383.png,DE102020008145A1\_0384.png"\; state="\{\{state\}\}">n</figure-callout>}}{\sqrt{1-{\left({\tau }_x\left(x\right)/n\right)}^2}}$$

und Substitution der Ableitungen τ⁽²⁾, τ⁽³⁾, τ⁽⁴⁾, ... durch die Ausdrücke aus GI.(B6) in „Appendix B“ der Veröffentlichung von G. Esser, W. Becken, W. Müller, P. Baumbach, J. Arasa, D. Uttenweiler: „Derivation of the refractive equations for higher order aberrations of local wavefronts by oblique incidence“, J. Opt. Soc. Am. A 27, 218-237 (2010), ergibt für die Transformation H: $$\begin{array}{c}{t}^{\left(1\right)}=-n{w}^{\left(2\right)}\\ t^{\left(2\right)}=-n{w}^{\left(3\right)}\\ t^{\left(3\right)}=-n\left(w^{\left(4\right)}-6w^{{\left(2\right)}^3}\right)-3{\left(n{w}^{\left(2\right)}\right)}^3/n^2\\ =-n\left(w^{\left(4\right)}-3w^{{\left(2\right)}^3}\right)\\ t^{\left(4\right)}=-n\left(w^{\left(5\right)}-40w^{{\left(2\right)}^2}{w}^{\left(3\right)}\right)-18{\left(n{w}^{\left(2\right)}\right)}^2\left(n{w}^{\left(3\right)}\right)/n^2\\ =-n\left(w^{\left(5\right)}-22w^{{\left(2\right)}^2}{w}^{\left(3\right)}\right)\\ ⋮\end{array}$$

Repeated derivation of eq.(6) with respect to x and evaluation for x = 0 performs $$\begin{array}{c}{t}^{\left(1\right)}=-{\tau }^{\left(2\right)}\\ t^{\left(2\right)}=-{\tau }^{\left(3\right)}\\ t^{\left(3\right)}=-{\tau }^{\left(4\right)}-3{\tau }^{{\left(2\right)}^3}/n^2\\ t^{\left(4\right)}=-{\tau }^{\left(5\right)}-18{\tau }^{{\left(2\right)}^2}{\tau }^{\left(3\right)}/{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">n</figure-callout>}}^2\\ ⋮\end{array}$$

and substitution of the derivatives τ ⁽²⁾ , τ ⁽³⁾ , τ ⁽⁴⁾ , ... by the expressions from Eq.(B6) in "Appendix B" of the publication by G. Esser, W. Becken, W. Müller , P. Baumbach, J. Arasa, D. Uttenweiler: "Derivation of the refractive equations for higher order aberrations of local wavefronts by oblique incidence", J. Opt. Soc. At the. A 27, 218-237 (2010), gives for the transformation H: $$\begin{array}{c}{t}^{\left(1\right)}=-n{w}^{\left(2\right)}\\ t^{\left(2\right)}=-n{w}^{\left(3\right)}\\ t^{\left(3\right)}=-n\left(w^{\left(4\right)}-6w^{{\left(2\right)}^3}\right)-3{\left(n{w}^{\left(2\right)}\right)}^3/{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">n</figure-callout>}}^2\\ =-n\left(w^{\left(4\right)}-3w^{{\left(2\right)}^3}\right)\\ t^{\left(4\right)}=-n\left(w^{\left(5\right)}-40w^{{\left(2\right)}^2}{w}^{\left(3\right)}\right)-18{\left(n{w}^{\left(2\right)}\right)}^2\left(n{w}^{\left(3\right)}\right)/{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">n</figure-callout>}}^2\\ =-n\left(w^{\left(5\right)}-22w^{{\left(2\right)}^2}{w}^{\left(3\right)}\right)\\ ⋮\end{array}$$

By reversing these equations, one obtains the transformation H ^-1 : $$\begin{array}{c}{w}^{\left(2\right)}=-t^{\left(1\right)}/n\\ w^{\left(3\right)}=-t^{\left(2\right)}/n\\ w^{\left(4\right)}=-\left(t^{\left(3\right)}+3t^{{\left(1\right)}^3}/n^2\right)/n\\ w^{\left(5\right)}=-\left(t^{\left(4\right)}+22t^{{\left(1\right)}^2}{t}^{\left(2\right)}/n^2\right)/n\\ ⋮\end{array}$$

Allocation of the ray transfer functions for an elementary propagation and refraction

propagation

the 6 shows a schematic sketch of the ray transfer function for propagation. A ray propagating from an entrance plane to an exit plane at a distance d = τ/n, where τ is the OPD that the light would travel at normal incidence, exits there offset. The ray transfer function is therefore quite simple because the direction does not change and therefore t' = t. The position component satisfies (x' - x)/(τ/n) = tan α = t/n, so that x' = x + tτ/n ² . The ray transfer function is then $$\left(\begin{array}{c}x\text{'}\\ t\text{'}\end{array}\right)=f^{p\mathrm{right}Op}\left(x,t\right)=\left(\begin{array}{c}x+t\tau /n^2\\ t\end{array}\right),$$

von f^prop(x,t) sehr einfach und verschwinden für alle höheren Ordnungen n_x + n_t ≥ 2, wie in der folgenden Tabelle 1 gezeigt. Tabelle 1: Ableitungen der Strahl-Transferfunktion f^prop(x,t) an der Stelle (x, t) = (0,0) Wellenfrontordnung Abl.-Ord. n _x + n _t x Ord. n _x t Ord. n _t $f_x^{pro{p}^{\left(n_x,n_t\right)}}$

$f_t^{pro{p}^{\left(n_x,n_t\right)}}$

1 0 0 -0 0 0 2 1 1 0 1 0 2 1 0 1 τ/n ² 1 Alle höheren Ordnungen verschwinden Remarkably, the ray transfer function f ^prop (x,t) for the propagation in Eq. (10) depends only linearly on both x and t. Hence the derivatives $f^{p\mathrm{right}O{p}^{\left(n_x,n_t\right)}}\left(x,t\right)$

of f ^prop (x,t) very simply and vanish for all higher orders n _x + _nt ≥ 2, as shown in Table 1 below. Table 1: Derivatives of the ray transfer function f ^prop (x,t) at the point (x,t) = (0,0) wavefront order Ord. n _x + n _t x order n _x t order n _t $f_x^{p\mathrm{right}O{p}^{\left(n_x,n_t\right)}}$

$f_t^{p\mathrm{right}O{p}^{\left(n_x,n_t\right)}}$

1 0 0 -0 0 0 2 1 1 0 1 0 2 1 0 1 τ/ n ² 1 All higher orders disappear

In the special case of a vanishing propagation distance τ = 0, the ray transfer function f ^prop (x,t) must be the identity with Jacobi matrix Jac f ^prop = 1. The only entry by which the derivatives differ from this trivial case is $f_x^{p\mathrm{right}O{p}^{\left(0.1\right)}}=\tau /{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">n</figure-callout>}}^2.$

refraction

the 7 shows a schematic sketch of the ray transfer function for propagation. Refraction is much more difficult to deal with, and it is generally not possible to give a ray transfer function f ^ref (x,t) for closed-form refraction as a function of the properties of the refracting surface. This is because the point at which a neighboring ray pierces an arbitrary surface can only be determined iteratively. Although there is a possibility for wavefronts to transform the spatial surface w(x) into a property τ(x) by means of a transformation H, there is also no corresponding transformation for the refracting surface itself without the existing parallax wrong to treat. The refraction must therefore continue to take place at a point in space that is generally outside the entry plane.

Since there is no propagation for pure refraction, the exit plane is identical to the entry plane. The problem to be solved according to the invention is therefore: given a neighboring ray (x, t) which starts at the entrance plane in the direction of a refracting surface, which parameters (x', t') as a function of (x, t) then correspond to refracted ray related to the same plane?

bezeichnet, wobei die Durchstoßkoordinate durch die eindeutige Funktion $\overline{x}\left(x,t\right)$

gegeben sei. Der Richtungstangens der Flächennormale sei mit $\overline{t}$

bezeichnet (siehe 7). Er ist durch das Negative der ersten Ableitung der Fläche gegeben, $\overline{t}=-{\overline{z}}^{(1)},$

die ebenfalls eine Funktion von $\overline{x}$

ist und somit auch eine Funktion $\overline{t}\left(x,t\right)=\overline{t}\left(\overline{x}\left(x,t\right)\right).$

Die Richtungen t des einfallenden Strahls und $\overline{t}$

der Fläche geben entsprechend dem Snellius'schen Brechungsgesetz Anlass zu einer Richtung t' des gebrochenen Strahls, und die rückwärtige Verlängerung des gebrochenen Strahls schneidet die Eintrittsebene an der eindeutigen Position x' (siehe 7).The preferred area of application of the method concerns situations in which the ray (x,t) has a unique point of intersection with the surface. This intersection becomes with $\left(\overline{x},\mathrm{e.g}\left(\overline{x}\right)\right)$

denoted, where the puncture coordinate is given by the unique function $\overline{x}\left(x,t\right)$

given. The direction tangent of the surface normal is with $\overline{t}$

denoted (see 7 ). It is given by the negative of the first derivative of the area, $\overline{t}=-{\overline{\mathrm{e.g}}}^{(1)},$

which is also a function of $\overline{x}$

is and therefore also a function $\overline{t}\left(x,t\right)=\overline{t}\left(\overline{x}\left(x,t\right)\right).$

The directions t of the incident ray and $\overline{t}$

of the surface give rise to a direction t' of the refracted ray according to Snell's law of refraction, and the backward extension of the refracted ray intersects the entrance plane at the unique position x' (see 7 ).

ist, was nach Auflösen nach t' ergibt $$t\text{'}\left(x,t\right)=n\text{'}\frac{bn-w{\overline{z}}^{\left(1\right)}}{w+bn{\overline{z}}^{\left(1\right)}}$$

wobei $$w=\sqrt{n{\text{'}}^2-n^2{b}^2},b=\frac{t+n{\overline{z}}^{\left(1\right)}}{\sqrt{n^2+t^2}\sqrt{1+{\overline{z}}^{{\left(1\right)}^2}}}$$

ist, und ${\overline{z}}^{\left(1\right)}$

an der Position $\overline{x}\left(x,t\right)$

ausgewertet wird.Snell's law of refraction states that $$\begin{array}{l}n\text{'}sin\left(a\mathrm{right}ctan\left(t\text{'}/n\text{'}\right)-a\mathrm{right}c\overline{t}ant\right)=nsin\left(a\mathrm{right}ctan\left(t/n\right)-a\mathrm{right}c\overline{t}ant\right)\\ \iff \frac{t\text{'}-n\text{'}\overline{t}}{\sqrt{1+{\left(t\text{'}/n\text{'}\right)}^2}}=\frac{t-n\overline{t}}{\sqrt{1+{\left(t/n\right)}^2}}\end{array}$$

is, which results after solving for t' $$t\text{'}\left(x,t\right)=n\text{'}\frac{bn-w{\overline{\mathrm{e.g}}}^{\left(1\right)}}{w+bn{\overline{\mathrm{e.g}}}^{\left(1\right)}}$$

whereby $$w=\sqrt{n{\text{'}}^2-n^2{\mathrm{<figure-callout\; id="2"\; label="b"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">b</figure-callout>}}^2},b=\frac{t+n{\overline{\mathrm{e.g}}}^{\left(1\right)}}{\sqrt{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">n</figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="t"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">t</figure-callout>}}^2}\sqrt{1+{\overline{\mathrm{e.g}}}^{{\left(1\right)}^2}}}$$

is and ${\overline{\mathrm{e.g}}}^{\left(1\right)}$

at the position $\overline{x}\left(x,t\right)$

is evaluated.

und t'(x, t) bekannt sind, kann aus 7 geometrisch direkt abgelesen werden, dass $$x\text{'}\left(x,t\right)=\overline{x}\left(x,t\right)-\frac{t\text{'}\left(x,t\right)}{n\text{'}}\overline{z}\left(\overline{x}\left(x,t\right)\right)$$

sein muss. Die Strahl-Transferfunktion für Brechung lautet damit $$\left(\begin{array}{c}x\text{'}\\ t\text{'}\end{array}\right)=f^{ref}\left(x,t\right)=\left(\begin{array}{c}\overline{x}\left(x,t\right)-t\text{'}\left(x,t\right)\overline{z}\left(\overline{x}\left(x,t\right)\right)/n\text{'}\\ t\text{'}\left(x,t\right)\end{array}\right)$$

As soon $\overline{x}\left(x,t\right)$

and t'(x, t) are known 7 can be read off geometrically directly that $$x\text{'}\left(x,t\right)=\overline{x}\left(x,t\right)-\frac{t\text{'}\left(x,t\right)}{n\text{'}}\overline{\mathrm{e.g}}\left(\overline{x}\left(x,t\right)\right)$$

have to be. The ray transfer function for refraction is then $$\left(\begin{array}{c}x\text{'}\\ t\text{'}\end{array}\right)=f^{\mathrm{right}ef}\left(x,t\right)=\left(\begin{array}{c}\overline{x}\left(x,t\right)-t\text{'}\left(x,t\right)\overline{\mathrm{e.g}}\left(\overline{x}\left(x,t\right)\right)/n\text{'}\\ t\text{'}\left(x,t\right)\end{array}\right)$$

zur Verfügung gestellt. Die Richtung des einfallenden Strahls erfüllt die Beziehung tan α = t/n und daher $$\frac{t}{n}=\frac{\overline{x}\left(x,t\right)-x}{\overline{z}\left(\overline{x}\left(x,t\right)\right)}$$

According to the invention, a method for calculating the function $\overline{x}\left(x,t\right)$

made available. The direction of the incident ray satisfies the relationship tan α = t/n and therefore $$\frac{t}{n}=\frac{\overline{x}\left(x,t\right)-x}{\overline{\mathrm{e.g}}\left(\overline{x}\left(x,t\right)\right)}$$

aufgelöst werden kann, kann man die partiellen Ableitungen bezüglich (x, t) von GI.(16) aufstellen und diese sukzessive für die Ableitungen ${\overline{x}}^{\left(\mathrm{1,0}\right)}{\overline{x}}^{\left(\mathrm{0,1}\right)},{\overline{x}}^{\left(\mathrm{2,0}\right)},{\overline{x}}^{\left(\mathrm{1,1}\right)},{\overline{x}}^{\left(\mathrm{0,2}\right)},{\overline{x}}^{\left(\mathrm{3,0}\right)},$

etc. an der Position (x, t) = (0,0) lösen.Although Eq.(16) is not closed according to $\overline{x}\left(x,t\right)$

can be solved, one can set up the partial derivatives with respect to (x, t) from Eq. (16) and these successively for the derivatives ${\overline{x}}^{\left(1.0\right)}{\overline{x}}^{\left(0.1\right)},{\overline{x}}^{\left(2.0\right)},{\overline{x}}^{\left(1.1\right)},{\overline{x}}^{\left(0.2\right)},{\overline{x}}^{\left(3.0\right)},$

etc. at the position (x,t) = (0,0).

und der Einführung der Funktion $$\overline{x}\left(x,t\right)={\overline{x}}_{Ansatz}\left(x,t\right)+\delta \overline{x}\left(x,t\right)$$

in GI.(16). Die Funktion ${\overline{x}}_{Ansatz}\left(x,t\right)$

wird als konsistente Lösung von GI.(16) in der Ordnung k bezeichnet, und k wird als Konsistenzordnung der Funktion ${\overline{x}}_{Ansatz}\left(x,t\right)$

bezeichnet, wenn (k + 1) die niedrigste Ordnung ist, für die eine der Ableitungen $\delta {\overline{x}}^{(\mathrm{1,0})}\delta {\overline{x}}^{(\mathrm{0,1})},\delta {\overline{x}}^{(\mathrm{2,0})},\delta {\overline{x}}^{(\mathrm{1,1})},\delta {\overline{x}}^{(\mathrm{0,2})},\delta {\overline{x}}^{(\mathrm{3,0})},$

etc. nicht verschwindet. Die folgende Tabelle 2 zeigt die verschiedenen Ausführungsformen der Funktion ${\overline{x}}_{Ansatz}\left(x,t\right)$

und ihre Ordnungskonsistenz. Tabelle 2: ${\overline{x}}_{Ansatz}\left(x,t\right)$

Ordnungskonsistenz x 2 $x+t/n\overline{z}\left(x\right)$

4 $x+t/n\overline{z}\left(x+t/n\overline{z}\left(x\right)\right)$

6 $x+\frac{t/n\overline{z}\left(x\right)}{1-t/n{\overline{z}}^{\left(1\right)}\left(x\right)}$

6 Another embodiment, the advantage of which is a more compact notation, consists in choosing an appropriate approach ${\overline{x}}_{Ansat\mathrm{e.g}}\left(x,t\right)$

and the introduction of the function $$\overline{x}\left(x,t\right)={\overline{x}}_{Ansat\mathrm{e.g}}\left(x,t\right)+\delta \overline{x}\left(x,t\right)$$

in Eq.(16). The function ${\overline{x}}_{Ansat\mathrm{e.g}}\left(x,t\right)$

is called the consistent solution of eq.(16) in order k, and k is called the order of consistency of the function ${\overline{x}}_{Ansat\mathrm{e.g}}\left(x,t\right)$

denotes when (k+1) is the lowest order for one of the derivatives $\delta {\overline{x}}^{(1.0)}\delta {\overline{x}}^{(0.1)},\delta {\overline{x}}^{(2.0)},\delta {\overline{x}}^{(1.1)},\delta {\overline{x}}^{(0.2)},\delta {\overline{x}}^{(3.0)},$

etc. does not disappear. Table 2 below shows the different embodiments of the function ${\overline{x}}_{Ansat\mathrm{e.g}}\left(x,t\right)$

and their order consistency. Table 2: ${\overline{x}}_{Ansat\mathrm{e.g}}\left(x,t\right)$

order consistency x 2 $x+t/n\overline{\mathrm{e.g}}\left(x\right)$

4 $x+t/n\overline{\mathrm{e.g}}\left(x+t/n\overline{\mathrm{e.g}}\left(x\right)\right)$

6 $x+\frac{t/n\overline{\mathrm{e.g}}\left(x\right)}{1-t/n{\overline{\mathrm{e.g}}}^{\left(1\right)}\left(x\right)}$

6

und $f_t^{re{f}^{\left(\mathrm{0,1}\right)}}=1$

= 1 sind alle Einträge in Tabelle 3 proportional zu (n - n'). Das ist zu erwarten, weil sich f^ref(x,t) für n' = n auf die Identität reduzieren muss mit Jacobi-Matrix Jac f^ref = 1. Tabelle 3: Ableitungen der Strahl-Transferfunktion f^ref (x,t) an der Stelle (x, t) = (0,0) Ordnung der Wellen front Abl.-Ord. n _x + n _t x Ord. n _x t Ord. n _t $f_x^{re{f}^{\left(n_x,n_t\right)}}$

$f_t^{re{f}^{\left(n_x,n_t\right)}}$

1 0 0 0 0 0 2 1 1 0 1 $-\left(n^{\prime }-n\right){\overline{z}}^{(2)}$

2 1 0 1 0 1 3 2 2 0 0 $-\left(n^{\prime }-n\right){\overline{z}}^{(3)}$

3 2 1 1 0 0 3 2 0 2 0 0 4 3 3 0 $\left(n^{\prime }-n\right)\times 3/n^{\prime }{\overline{z}}^{{(2)}^2}$

$\begin{array}{l}-\left(n\text{'}-n\right)({\overline{z}}^{\left(4\right)}+3n/n{\text{'}}^2(n\\ -n\text{'}){\overline{z}}^{{\left(2\right)}^3})\end{array}$

4 3 2 1 $\left(n^{\prime }-n\right)/\left(n{n}^{\prime }\right){\overline{z}}^{(2)}$

$\begin{array}{l}-\left(n\text{'}-n\right)/\left(nn{\text{'}}^2\right)(3n^2-nn\text{'}\\ +n{\text{'}}^2){\overline{z}}^{{\left(2\right)}^2})\end{array}$

4 3 1 2 0 $-\left(n^{\prime }-n\right)/\left(n{n^{\prime }}^2\right)\left(3n+n^{\prime }\right){\overline{z}}^{\left(2\right)}$

4 3 0 3 0 $-\left(n^{\prime }-n\right)\times 3/\left(n^2{n^{\prime }}^2\right)\left(n+n^{\prime }\right)$

Analogously to Table 1, Table 3 below shows the derivations of the ray transfer function f ^ref (x,t) for refraction. In contrast to the case of f ^prop (x,t), the table for f ^ref (x,t) does not generally break off at any finite order. Except for $f_x^{\mathrm{right}e{f}^{\left(1.0\right)}}=1$

and $f_t^{\mathrm{right}e{f}^{\left(0.1\right)}}=1$

= 1, all entries in Table 3 are proportional to (n - n'). This is to be expected because f ^ref (x,t) for n' = n must reduce to the identity with Jacobian matrix Jac f ^ref = 1. Table 3: Derivatives of the ray transfer function f ^ref (x,t) an the place (x, t) = (0,0) order of the waves front Ord. n _x + n _t x order n _x t order n _t $f_x^{\mathrm{right}e{f}^{\left(n_x,n_t\right)}}$

$f_t^{\mathrm{right}e{f}^{\left(n_x,n_t\right)}}$

1 0 0 0 0 0 2 1 1 0 1 $-\left(n^{\prime }-n\right){\overline{\mathrm{e.g}}}^{(2)}$

2 1 0 1 0 1 3 2 2 0 0 $-\left(n^{\prime }-n\right){\overline{\mathrm{e.g}}}^{(3)}$

3 2 1 1 0 0 3 2 0 2 0 0 4 3 3 0 $\left(n^{\prime }-n\right)\times 3/n^{\prime }{\overline{\mathrm{e.g}}}^{{(2)}^2}$

$\begin{array}{l}-\left(n\text{'}-n\right)({\overline{\mathrm{e.g}}}^{\left(4\right)}+3n/n{\text{'}}^2(n\\ -n\text{'}){\overline{\mathrm{e.g}}}^{{\left(2\right)}^3})\end{array}$

4 3 2 1 $\left(n^{\prime }-n\right)/\left(n{n}^{\prime }\right){\overline{\mathrm{e.g}}}^{(2)}$

$\begin{array}{l}-\left(n\text{'}-n\right)/\left(nn{\text{'}}^2\right)(3{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-nn\text{'}\\ +n{\text{'}}^2){\overline{\mathrm{e.g}}}^{{\left(2\right)}^2})\end{array}$

4 3 1 2 0 $-\left(n^{\prime }-n\right)/\left(n{n^{\prime }}^2\right)\left(3n+n^{\prime }\right){\overline{\mathrm{e.g}}}^{\left(2\right)}$

4 3 0 3 0 $-\left(n^{\prime }-n\right)\times 3/\left(n^2{n^{\prime }}^2\right)\left(n+n^{\prime }\right)$

Actual calculation by the optical system

erfüllt.The real goal is to find a function t'(x') for the emitted light that, for a given function t(x), satisfies the equation $$f\left(x,t\left(x\right)\right)=\left(\begin{array}{c}{f}_x\left(x,t\left(x\right)\right)\\ f_t\left(x,t\left(x\right)\right)\end{array}\right)=\left(\begin{array}{c}x\text{'}\left(x\right)\\ t\text{'}\left(x\text{'}\left(x\right)\right)\end{array}\right)$$

Fulfills.

erreicht, mit denen GI.(18) die Form $$\left(\begin{array}{c}u\left(x\right)\\ v\left(x\right)\end{array}\right)=f\left(x,t\left(x\right)\right)$$

annimmt. Eine Kombination der beiden Gleichungen (19) und (20) ergibt $$v\left(x\right)=t\text{'}\left(u\left(x\right)\right)$$

was den Startpunkt des Verfahrens darstellt. Das Grundprinzip besteht im wiederholten Ableiten von GI.(21) $$\begin{array}{c}{v}^{\left(1\right)}=u^{\left(1\right)}t{\text{'}}^{\left(1\right)}\\ v^{\left(2\right)}={\left(u^{\left(1\right)}\right)}^{2}t{\text{'}}^{\left(2\right)}+u^{\left(2\right)}t{\text{'}}^{\left(1\right)}\\ v^{\left(3\right)}={\left(u^{\left(1\right)}\right)}^{3}t{\text{'}}^{\left(3\right)}+3u^{\left(1\right)}{u}^{\left(2\right)}t{\text{'}}^{\left(2\right)}+u^{\left(3\right)}t{\text{'}}^{\left(1\right)}\\ ⋮\end{array}$$

worin aus Gründen der besseren Lesbarkeit das Argument '(0)’ weggelassen ist. Der erste Schritt zum Erreichen des Ziels besteht darin, GI.(22) nacheinander nach den gesuchten Ableitungen t’⁽¹⁾, t’⁽²⁾, t⁽³⁾ .... aufzulösen und damit durch die Ableitungen von u und v auszudrücken: $$\begin{array}{c}t{\text{'}}^{\left(1\right)}=\frac{v^{\left(1\right)}}{u^{\left(1\right)}}\\ t{\text{'}}^{\left(2\right)}=\frac{u^{\left(1\right)}{v}^{\left(2\right)}-u^{\left(2\right)}{v}^{\left(1\right)}}{{\left(u^{\left(1\right)}\right)}^3}\\ t{\text{'}}^{\left(3\right)}=\frac{u^{\left(1\right)}{v}^{\left(3\right)}-u^{\left(3\right)}{v}^{\left(1\right)}}{{\left(u^{\left(1\right)}\right)}^4}-3u^{\left(2\right)}\frac{u^{\left(1\right)}{v}^{\left(2\right)}-u^{\left(2\right)}{v}^{\left(1\right)}}{{\left(u^{\left(1\right)}\right)}^5}\\ ⋮\end{array}$$

This goal is favored by introducing the intermediate variables $$\begin{array}{l}\mathrm{and}\left(x\right):=x\text{'}\left(x\right)\\ v\left(x\right):=t\text{'}\left(x\text{'}\left(x\right)\right)\end{array}$$

reached, with which eq. (18) the form $$\left(\begin{array}{c}\mathrm{and}\left(x\right)\\ v\left(x\right)\end{array}\right)=f\left(x,t\left(x\right)\right)$$

assumes A combination of the two equations (19) and (20) gives $$v\left(x\right)=t\text{'}\left(\mathrm{and}\left(x\right)\right)$$

which is the starting point of the process. The basic principle consists in the repeated derivation of eq.(21) $$\begin{array}{c}{v}^{\left(1\right)}={\mathrm{and}}^{\left(1\right)}t{\text{'}}^{\left(1\right)}\\ v^{\left(2\right)}={\left({\mathrm{and}}^{\left(1\right)}\right)}^{2}t{\text{'}}^{\left(2\right)}+{\mathrm{and}}^{\left(2\right)}t{\text{'}}^{\left(1\right)}\\ v^{\left(3\right)}={\left({\mathrm{and}}^{\left(1\right)}\right)}^{3}t{\text{'}}^{\left(3\right)}+3{\mathrm{and}}^{\left(1\right)}{\mathrm{and}}^{\left(2\right)}t{\text{'}}^{\left(2\right)}+{\mathrm{and}}^{\left(3\right)}t{\text{'}}^{\left(1\right)}\\ ⋮\end{array}$$

where the argument '(0)' is omitted for better readability. The first step to reach the goal is to solve eq.(22) for the required derivatives t' ⁽¹⁾ , t' ⁽²⁾ , t ⁽³⁾ .... and thus by the derivatives of u and v to express: $$\begin{array}{c}t{\text{'}}^{\left(1\right)}=\frac{v^{\left(1\right)}}{{\mathrm{and}}^{\left(1\right)}}\\ t{\text{'}}^{\left(2\right)}=\frac{{\mathrm{and}}^{\left(1\right)}{v}^{\left(2\right)}-{\mathrm{and}}^{\left(2\right)}{v}^{\left(1\right)}}{{\left({\mathrm{and}}^{\left(1\right)}\right)}^3}\\ t{\text{'}}^{\left(3\right)}=\frac{{\mathrm{and}}^{\left(1\right)}{v}^{\left(3\right)}-{\mathrm{and}}^{\left(3\right)}{v}^{\left(1\right)}}{{\left({\mathrm{and}}^{\left(1\right)}\right)}^4}-3{\mathrm{and}}^{\left(2\right)}\frac{{\mathrm{and}}^{\left(1\right)}{v}^{\left(2\right)}-{\mathrm{and}}^{\left(2\right)}{v}^{\left(1\right)}}{{\left({\mathrm{and}}^{\left(1\right)}\right)}^5}\\ ⋮\end{array}$$

wobei erneut die Argumente ‚(0)‘ und ‚(0,0)‘ fortgelassen sind. Ein vorläufiges Ergebnis für t'⁽ⁱ⁾, ausgedrückt durch t⁽ⁱ⁾, wird dann dadurch erhalten, dass man Gl.(24) in Gl.(23) einsetzt.The second step is to express the derivatives of u and v in terms of derivatives of the function t(x) (incident light) and derivatives of f(x, t) (properties of the optical system). Deriving from eq.(20) lists for u ⁽ⁱ⁾ . $$\begin{array}{c}{\mathrm{and}}^{\left(1\right)}=f_x^{\left(0.1\right)}{t}^{\left(1\right)}+f_x^{\left(1.0\right)}\\ {\mathrm{and}}^{\left(2\right)}=f_x^{\left(1.0\right)}{t}^{\left(2\right)}+f_x^{\left(2.0\right)}+2f_x^{\left(1.1\right)}{t}^{\left(1\right)}+f_x^{\left(0.2\right)}{t}^{{\left(1\right)}^2}\\ {\mathrm{and}}^{\left(3\right)}=f_x^{\left(0.1\right)}{t}^{\left(3\right)}+f_x^{\left(3.0\right)}+3\left(f_x^{\left(2.1\right)}+f_x^{\left(1.2\right)}{t}^{\left(1\right)}\right)t^{\left(1\right)}+3\left(f_x^{\left(1.1\right)}+f_x^{\left(0.2\right)}{t}^{\left(1\right)}\right)t^{\left(2\right)}+f_x^{\left(0.3\right)}{t}^{{\left(1\right)}^3}\\ ⋮\\ v^{\left(1\right)}={\mathrm{and}}^{\left(1\right)}\left(f_x\to f_t\right)\\ v^{\left(2\right)}={\mathrm{and}}^{\left(2\right)}\left(f_x\to f_t\right)\\ v^{\left(3\right)}={\mathrm{and}}^{\left(3\right)}\left(f_x\to f_t\right)\\ ⋮\end{array}$$

where again the arguments '(0)' and '(0,0)' are omitted. A preliminary result for t' ⁽ⁱ⁾ expressed by t ⁽ⁱ⁾ is then obtained by substituting Eq.(24) into Eq.(23).

Solutions in t-representation

If one inserts Eq. (24) into Eq. (23) for t' ⁽ⁱ⁾ , then with increasing order i, very many similar terms with mixed terms from powers of derivatives t ⁽ⁱ⁾ quickly arise, whose evaluation when inserting numerical values for t ⁽ⁱ⁾ requires a lot of computation time. Therefore, the task of setting up a computing-time-saving method that can be repeatedly evaluated with many different wavefronts is not yet solved by simply inserting eq.(24) into eq.(23). Rather, it has been recognized according to the invention that the symbolic expressions that describe the dependency of the solution t' ⁽ⁱ⁾ on the derivatives t ⁽ⁱ⁾ must be sorted and combined before inserting numerical values in such a way that only the minimum number of mixed terms are powers of derivatives t ⁽ⁱ⁾ must be evaluated numerically.

wobei man der Kürze halber die Abkürzung $$\frac{1}{\beta }:=u^{\left(1\right)}=f_x^{\left(\mathrm{0,1}\right)}{t}^{\left(1\right)}+f_x^{\left(\mathrm{1,0}\right)}=B{t}^{\left(1\right)}+A$$

benutzen kann.The order i = 1 directly results in a fraction $$t{\text{'}}^{\left(1\right)}=\frac{f_t^{\left(0.1\right)}{t}^{\left(1\right)}+f_t^{\left(1.0\right)}}{f_x^{\left(0.1\right)}{t}^{\left(1\right)}+f_x^{\left(1.0\right)}}=\frac{D{t}^{\left(1\right)}+C}{B{t}^{\left(1\right)}+A}=:\beta \left(D{t}^{\left(1\right)}+C\right)$$

using the abbreviation for brevity $$\frac{1}{\beta }:={\mathrm{and}}^{\left(1\right)}=f_x^{\left(0.1\right)}{t}^{\left(1\right)}+f_x^{\left(1.0\right)}=B{t}^{\left(1\right)}+A$$

can use.

d.h. also z.B. vier Beiträge zur Potenz $t^{{\left(1\right)}^2}$

die sich ausklammern lässt, und deren Vorfaktoren sich zum Vorfaktor $\left(A{f}_t^{\left(\mathrm{0,2}\right)}+2B{f}_t^{\left(\mathrm{1,1}\right)}-C{f}_x^{\left(\mathrm{0,2}\right)}-2D{f}_x^{\left(\mathrm{1,1}\right)}\right)$

zusammenfassen lassen.The next higher order t' ⁽²⁾ in eq. (23) results in: $$\begin{array}{c}t{\text{'}}^{\left(2\right)}={\beta }^3\left({\mathrm{and}}^{\left(1\right)}{v}^{\left(2\right)}-{\mathrm{and}}^{\left(2\right)}{v}^{\left(1\right)}\right)\\ ={\beta }^3\left[\begin{array}{c}\left(B{t}^{\left(1\right)}+A\right)\left(D{t}^{\left(2\right)}+f_t^{\left(2.0\right)}+2f_t^{\left(1.1\right)}{t}^{\left(1\right)}+f_t^{\left(0.2\right)}{t}^{{\left(1\right)}^2}\right)\\ -\left(D{t}^{\left(1\right)}+C\right)\left(B{t}^{\left(2\right)}+f_x^{\left(2.0\right)}+2f_x^{\left(1.1\right)}{t}^{\left(1\right)}+f_x^{\left(0.2\right)}{t}^{{\left(1\right)}^2}\right)\end{array}\right]\\ ={\beta }^3\left[\begin{array}{l}\left(AD-BC\right)t^{\left(2\right)}+\left(BD-DB\right)t^{\left(1\right)}{t}^{\left(2\right)}\hfill \\ +\left(A{f}_t^{\left(2.0\right)}-C{f}_x^{\left(2.0\right)}\right)+\left(B{f}_t^{\left(2.0\right)}+2A{f}_t^{\left(1.1\right)}-D{f}_x^{\left(2.0\right)}-2C{f}_x^{\left(1.1\right)}\right)t^{\left(1\right)}\hfill \\ +\left(A{f}_t^{\left(0.2\right)}+2B{f}_t^{\left(1.1\right)}-C{f}_x^{\left(0.2\right)}-2D{f}_x^{\left(1.1\right)}\right)t^{{\left(1\right)}^2}+\left(B{f}_t^{\left(0.2\right)}-D{f}_x^{\left(0.2\right)}\right)t^{{\left(1\right)}^3}\hfill \end{array}\right]\\ ={\beta }^3\left[\begin{array}{l}{t}^{\left(2\right)}+\left(A{f}_t^{\left(2.0\right)}-C{f}_x^{\left(2.0\right)}\right)+\left(B{f}_t^{\left(2.0\right)}+2A{f}_t^{\left(1.1\right)}-D{f}_x^{\left(2.0\right)}-2C{f}_x^{\left(1.1\right)}\right)t^{\left(1\right)}\hfill \\ +\left(A{f}_t^{\left(0.2\right)}+2B{f}_t^{\left(1.1\right)}-C{f}_x^{\left(0.2\right)}-2D{f}_x^{\left(1.1\right)}\right)t^{{\left(1\right)}^2}+\left(B{f}_t^{\left(0.2\right)}-D{f}_x^{\left(0.2\right)}\right)t^{{\left(1\right)}^3}\hfill \end{array}\right]\end{array}$$

ie, for example, four contributions to the power $t^{{\left(1\right)}^2}$

which can be excluded, and whose pre-factors become the pre-factor $\left(A{f}_t^{\left(0.2\right)}+2B{f}_t^{\left(1.1\right)}-C{f}_x^{\left(0.2\right)}-2D{f}_x^{\left(1.1\right)}\right)$

summarize.

According to the invention, the method can be used regardless of the value of the determinant det T=AD-BC. The invention preferably makes use of the fact that optical systems are symplectic and satisfy det T=AD−BC=1.

Continued insertion and summarization leads to solutions of the structure $$\begin{array}{l}t{\text{'}}^{\left(1\right)}=\beta \left[{\overline{c}}_{1.1}{t}^{\left(1\right)}+{\overline{c}}_{1.0}\right]\\ t{\text{'}}^{\left(2\right)}={\beta }^3\left[t^{\left(2\right)}+{\overline{c}}_{2.0}+{\overline{c}}_{2.1}{t}^{\left(1\right)}+{\overline{c}}_{2.2}{t}^{{\left(1\right)}^2}+{\overline{c}}_{2.3}{t}^{{\left(1\right)}^3}\right]\\ t{\text{'}}^{\left(3\right)}={\beta }^4\left[t^{\left(3\right)}+\beta \left(\begin{array}{l}\left({\overline{c}}_{3.0}+{\overline{c}}_{3.1}{t}^{\left(1\right)}+{\overline{c}}_{3.2}{t}^{{\left(1\right)}^2}+{\overline{c}}_{3.3}{t}^{{\left(1\right)}^3}+{\overline{c}}_{3.4}{t}^{{\left(1\right)}^4}+{\overline{c}}_{3.5}{t}^{{\left(1\right)}^5}\right)\\ +\left({\overline{c}}_{3.01}+{\overline{c}}_{\mathrm{3:11}}{t}^{\left(1\right)}+{\overline{c}}_{\mathrm{3:21}}{t}^{{\left(1\right)}^2}\right)t^{\left(2\right)}+{\overline{c}}_{3.02}{t}^{{\left(2\right)}^2}\end{array}\right)\right]\\ t{\text{'}}^{\left(4\right)}={\beta }^5\left[\begin{array}{l}{t}^{\left(4\right)}+{\beta }^2\left(\begin{array}{l}\left({\overline{c}}_{4.0}+{\overline{c}}_{4.1}{t}^{\left(1\right)}+...+{\overline{c}}_{4.7}{t}^{{\left(1\right)}^7}\right)\\ +\left({\overline{c}}_{4.01}+{\overline{c}}_{\mathrm{4:11}}{t}^{\left(1\right)}+...+{\overline{c}}_{4.41}{t}^{{\left(1\right)}^4}\right)t^{\left(2\right)}\\ +\left({\overline{c}}_{4.02}+{\overline{c}}_{\mathrm{4:12}}{t}^{\left(1\right)}+{\overline{c}}_{4.22}{t}^{{\left(1\right)}^2}\right)t^{{\left(2\right)}^2}+{\overline{c}}_{4.03}{t}^{{\left(2\right)}^3}\end{array}\right)\\ +\beta \left(\left({\overline{c}}_{4.001}+{\overline{c}}_{4.101}{t}^{\left(1\right)}+{\overline{c}}_{4.201}{t}^{{\left(1\right)}^2}\right)t^{\left(3\right)}+{\overline{c}}_{4.011}{t}^{\left(2\right)}{t}^{\left(3\right)}\right)\end{array}\right]\\ t{\text{'}}^{\left(5\right)}={\beta }^6\left[t^{\left(5\right)}+...\right]\end{array}$$

mit Koeffizienten c_ik gegeben, wobei die untere Zeile nur im symplektischen Fall gilt, und wobei k = (k₁, k₂,..., k_i) ein Tupel $k\in N_0^i$

aus Exponenten ist; k* = (k₂,..., k_i) ist ein Tupel, das aus k durch Fortlassen des ersten Elements entsteht, k** = (k₃,..., k_i) entsteht aus k durch Fortlassen der ersten beiden Elemente.In general, the solutions t' ⁽ⁱ⁾ are given by a sum approach of the form $$\begin{array}{l}t{\text{'}}^{\left(i\right)}={\beta }^{-{\overline{\mathrm{right}}}_1^0\left(i\right)}{\displaystyle \sum _{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102020008145A1\_0383.png,DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_1,{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2,...,k_i}{\overline{c}}_{i\mathbf{k}}{\beta }^{-\Delta {\overline{\mathrm{right}}}_1\left(i,\mathbf{k}*\right)}}{t}^{{\left(1\right)}^{k_1}}{t}^{{\left(2\right)}^{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}}\cdots t^{{\left(i\right)}^{k_i}}\\ ={\beta }^{-{\overline{\mathrm{right}}}_1^0\left(i\right)}\left[t^{\left(i\right)}+{\displaystyle \sum _{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102020008145A1\_0383.png,DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_1,{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2,...,k_{i-1}}{\overline{c}}_{i\mathbf{k}}{\beta }^{-\Delta {\overline{\mathrm{right}}}_1\left(i,\mathbf{k}*\right)}}{t}^{{\left(1\right)}^{k_1}}{t}^{{\left(2\right)}^{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}}\cdots t^{{\left(i-1\right)}^{k_{i-1}}}\right],i=\mathrm{1,2,3,}...\end{array}$$

given with coefficients c _ik , where the bottom row only applies in the symplectic case, and where k = (k ₁ , k ₂ ,..., k _i ) a tuple $k\in N_0^i$

is from exponents; k* = (k ₂ ,..., k _i ) is a tuple obtained from k by omitting the first element, k** = (k ₃ ,..., k _i ) arises from k by omitting the first both elements.

The exponents of β are given by $$\begin{array}{c}-{\overline{\mathrm{right}}}_1^0\left(i\right)=\left(i+1\right)-{\delta }_{i1}\\ -\mathrm{\Delta }{\overline{\mathrm{right}}}_1\left(i,k*\right)=\left(i-2\right)+{\delta }_{i1}-P\left(k**\right)\end{array}$$

sind in Tabelle 4 angegeben. Zur besseren Lesbarkeit kann eine Kurzschreibweise benutzt werden $$\begin{array}{l}{T}_{AC}^{\left(n_x,n_t\right)}:=A{f}_t^{\left(n_x,n_t\right)}-C{f}_x^{\left(n_x,n_t\right)}\\ T_{BD}^{\left(n_x,n_t\right)}:=B{f}_t^{\left(n_x,n_t\right)}-D{f}_x^{\left(n_x,n_t\right)}\end{array}$$

sowie eine Symmetrie-Transformation X, die jede Ableitung $f_x^{\left(n_x,n_t\right)},f_t^{\left(n_x,n_t\right)}$

durch $X\left(f_x^{\left(n_x,n_t\right)}\right)$

bzw. $X\left(f_t^{\left(n_x,n_t\right)}\right)$

ersetzt und in einer Vertauschung der Ordnungen n_x, n_t sowie einem Vorzeichen besteht: $$\begin{array}{l}X\left(f_x^{\left(n_x,n_t\right)}\right)={\left(-1\right)}^{n_x+n_t}{f}_x^{\left(n_t,n_x\right)}\\ X\left(f_t^{\left(n_x,n_t\right)}\right)={\left(-1\right)}^{n_x+n_t+1}{f}_t^{\left(n_t,n_x\right)}\end{array}$$

Gl. (32) impliziert direkt X(A) = -B, X(B) = -A, X(C) = D, X(D) = C, und für die Ausdrücke in GI. (31) $$\begin{array}{l}X\left(T_{AC}^{\left(n_x,n_t\right)}\right)={\left(-1\right)}^{n_x+n_t}{T}_{BD}^{\left(n_t,n_x\right)}\\ X\left(T_{BD}^{\left(n_x,n_t\right)}\right)={\left(-1\right)}^{n_x+n_t}{T}_{AC}^{\left(n_t,n_x\right)}\end{array}$$

The coefficients ${\overline{c}}_{ik}$

are given in Table 4. Abbreviated notation can be used for better legibility $$\begin{array}{l}{T}_{AC}^{\left(n_x,n_t\right)}:=A{f}_t^{\left(n_x,n_t\right)}-C{f}_x^{\left(n_x,n_t\right)}\\ T_{BD}^{\left(n_x,n_t\right)}:=B{f}_t^{\left(n_x,n_t\right)}-D{f}_x^{\left(n_x,n_t\right)}\end{array}$$

as well as a symmetry transformation X, which every derivative $f_x^{\left(n_x,n_t\right)},f_t^{\left(n_x,n_t\right)}$

through $X\left(f_x^{\left(n_x,n_t\right)}\right)$

or. $X\left(f_t^{\left(n_x,n_t\right)}\right)$

replaced and consists in an exchange of the orders n _x , n _t and a sign: $$\begin{array}{l}X\left(f_x^{\left(n_x,n_t\right)}\right)={\left(-1\right)}^{n_x+n_t}{f}_x^{\left(n_t,n_x\right)}\\ X\left(f_t^{\left(n_x,n_t\right)}\right)={\left(-1\right)}^{n_x+n_t+1}{f}_t^{\left(n_t,n_x\right)}\end{array}$$

Eq. (32) directly implies X(A) = -B, X(B) = -A, X(C) = D, X(D) = C, and for the expressions in eq. (31) $$\begin{array}{l}X\left(T_{AC}^{\left(n_x,n_t\right)}\right)={\left(-1\right)}^{n_x+n_t}{T}_{BD}^{\left(n_t,n_x\right)}\\ X\left(T_{BD}^{\left(n_x,n_t\right)}\right)={\left(-1\right)}^{n_x+n_t}{T}_{AC}^{\left(n_t,n_x\right)}\end{array}$$

The summation in Eq.(29) is constructed in such a way that the tuple k* runs over a region k* ∈ P _∪ (i - 1) which depends only on the order i, where the set P _∪ is defined by $$\begin{array}{r}\hfill P_{\cup }\left(p\right):=\left\{\mathbf{k}\in N_0^p|P\left(k\right)\le p\right\}={\displaystyle \underset{q=1}{\overset{p}{\cup }}P\left(q\right)}\\ \hfill wObeiP\left(p\right):=\left\{\mathbf{k}\in N_0^p|P\left(k\right)=p\right\}={\displaystyle \underset{m=1}{\overset{p}{\cup }}P\left(p,m\right)}\\ \hfill wObeiP\left(p,m\right):=\left\{\mathbf{k}\in N_0^p|P\left(k\right)=p\wedge M\left(k\right)=m\right\}\end{array},$$

The numbers used in Eq.(34) are the size M(k) and the partition order P(k) and are defined by $$\begin{array}{l}M\left(k\right):={\displaystyle \sum _{v=1}^p{k}_v}={\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102020008145A1\_0383.png,DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_1+{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2+...+k_p\hfill \\ P\left(k\right):={\displaystyle \sum _{v=1}^{p}v{k}_v}={\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102020008145A1\_0383.png,DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_1+2{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2+...+p{k}_p\hfill \end{array}$$

The index k ₁ runs over the range 0 ≤ k ₁ ≤ 2(i - P(k*) - 1) + δ _P(k*),0 , which depends on the order i and k*.

Spalten. Alle weiteren Spalten des Diagramms werden von rechts hinzugefügt, als nächstes ein Rechteck aus (i_max -1) Zeilen und $k_{i_{max}-1}$

Spalten, und das Diagramm endet an seiner rechten Seite mit einem Rechteck aus einer Zeile und k₁ Spalten.As an alternative to displaying numbers, a tuple can also be displayed graphically, preferably using Young diagrams (see Fig 7 ). A particularly preferred form of this representation consists in choosing the diagram for a given tuple k in such a way that the number of boxes chen equals P(k). If i _max is the index (i.e. the order) of the highest non-zero entry of k, then the Young diagram, as in the 8th shown, on its left side a rectangle of i _max rows and $k_{i_{max}}$

Columns. All further columns of the diagram are added from the right, next a rectangle of (i _max -1) rows and $k_{i_{max}-1}$

columns, and the diagram ends on its right side with a rectangle of 1 row and k ₁ columns.

für allgemeine optische Systeme angegeben sowie für eine einfache Propagation über eine Distanz d = τ/n sowie für eine Propagation durch eine Einzelfläche mit Flächenableitungen ${\overline{z}}^{(2)},{\overline{z}}^{(3)},{\overline{z}}^{(4)},\mathrm{...}$

zwischen zwei Medien mit Brechungsindizes n und n'. Tabelle 4: Ord Indizes Vorfaktor Term Koeffizient ${\overline{c}}_{ik}\equiv {\overline{c}}_{i,\left(k_1,k*\right)}$

i k _i k*. ${\beta }^{-\mathrm{\Delta }{\overline{r}}_1}$

Term Symbol Allgemeiner Fall einfache Propagation, Distanz τ/n Brechung an Einzelfläche (β=1) 1 0 0 1 1 ${\overline{c}}_{\mathrm{1,0}}$

C 0 $-\left(n\text{'}-n\right){\overline{z}}^{\left(2\right)}$

1 1 0 1 t ⁽¹⁾ ${\overline{c}}_{\mathrm{1,1}}$

$X\left({\overline{c}}_{\mathrm{1,1}}\right)=D$

1 1 2 0 0 1 1 ${\overline{c}}_{\mathrm{2,0}}$

$T_{AC}^{\left(\mathrm{2,0}\right)}$

0 $-\left(n\text{'}-n\right){\overline{z}}^{\left(3\right)}$

2 1 0 1 t ⁽¹⁾ ${\overline{c}}_{\mathrm{2,1}}$

$2T_{AC}^{\left(\mathrm{1,1}\right)}+T_{BD}^{\left(\mathrm{2,0}\right)}$

0 0 2 2 0 1 (t ⁽¹⁾)² ${\overline{c}}_{\mathrm{2,2}}$

$X\left({\overline{c}}_{\mathrm{2,1}}\right)=T_{AC}^{\left(\mathrm{2,0}\right)}+2T_{BD}^{\left(\mathrm{1,1}\right)}$

0 0 2 3 0 1 (t ⁽¹⁾)³ ${\overline{c}}_{\mathrm{2,3}}$

$X\left({\overline{c}}_{\mathrm{2,0}}\right)=T_{BD}^{\left(\mathrm{0,2}\right)}$

0 0 2 0 1 1 t ⁽²⁾ ${\overline{c}}_{\mathrm{2,01}}$

1 1 1 3 0 0 β 1 ${\overline{c}}_{\mathrm{3,0}}$

$-3T_{AC}^{\left(\mathrm{2,0}\right)}{f}_x^{\left(\mathrm{2,0}\right)}+A{T}_{AC}^{\left(\mathrm{3,0}\right)}$

0 $-\left(n\text{'}-n\right)\left(\begin{array}{l}{\overline{z}}^{\left(4\right)}+\\ 3\left(n^2/n{\text{'}}^2-1\right){\overline{z}}^{{\left(2\right)}^3}\end{array}\right)$

3 1 0 β t ⁽¹⁾ ${\overline{c}}_{\mathrm{3,1}}$

$\begin{array}{l}-6T_{AC}^{\left(\mathrm{2,0}\right)}{f}_x^{\left(\mathrm{1,1}\right)}-3\left(T_{BD}^{\left(\mathrm{2,0}\right)}+2T_{AC}^{\left(\mathrm{1,1}\right)}\right)f_x^{\left(\mathrm{2,0}\right)}\\ +3A{T}_{AC}^{\left(\mathrm{2,1}\right)}+B{T}_{AC}^{\left(\mathrm{3,0}\right)}+A{T}_{BD}^{\left(\mathrm{3,0}\right)}\end{array}$

0 $-3\left(n^{\prime }-n\right)\left(n^{\prime }+3n\right)/{n^{\prime }}^2{\overline{z}}^{{\left(2\right)}^3}$

3 2 0 β (t ⁽¹⁾)² ${\overline{c}}_{\mathrm{3,2}}$

$\begin{array}{l}3A\left(T_{AC}^{\left(\mathrm{1,2}\right)}+T_{BD}^{\left(\mathrm{2,1}\right)}\right)+B\left(3T_{AC}^{\left(\mathrm{2,1}\right)}+T_{BD}^{\left(\mathrm{3,0}\right)}\right)-3T_{AC}^{\left(\mathrm{2,0}\right)}{f}_x^{\left(\mathrm{0,2}\right)}\\ -6\left(2T_{AC}^{\left(\mathrm{1,1}\right)}+T_{BD}^{\left(\mathrm{2,0}\right)}\right)f_x^{\left(\mathrm{1,1}\right)}-3\left(T_{AC}^{\left(\mathrm{0,2}\right)}+2T_{BD}^{\left(\mathrm{1,1}\right)}\right)f_x^{\left(\mathrm{2,0}\right)}\end{array}$

0 $-3\left(n^{\prime }-n\right)\left(2n^{\prime }+3n\right)/\left(n{n^{\prime }}^2\right){\overline{z}}^{\left(2\right)}$

3 3 0 β (t ⁽¹⁾)³ ${\overline{c}}_{\mathrm{3,3}}$

$X\left({\overline{c}}_{\mathrm{3,2}}\right)$

0 $-3\left(n^{\prime }-n\right)\left(n^{\prime }+n\right)/\left(n^2{n^{\prime }}^2\right)$

3 4 0 β (t ⁽¹⁾)⁴ ${\overline{c}}_{\mathrm{3,4}}$

$X\left({\overline{c}}_{\mathrm{3,1}}\right)$

) 0 0 3 5 0 β (t ⁽¹⁾)⁵ ${\overline{c}}_{\mathrm{3,5}}$

$X\left({\overline{c}}_{\mathrm{3,0}}\right)$

0 0 3 0 1 β t ⁽²⁾ ${\overline{c}}_{\mathrm{3,01}}$

$-3\left(B{T}_{AC}^{\left(\mathrm{2,0}\right)}-A{T}_{AC}^{\left(\mathrm{1,1}\right)}+{ƒ}_x^{\left(\mathrm{2,0}\right)}\right)$

0 0 3 1 1 β t ⁽¹⁾t⁽²⁾ ${\overline{c}}_{\mathrm{3,11}}$

$-3\left(B{T}_{BD}^{\left(\mathrm{2,0}\right)}-A{T}_{AC}^{\left(\mathrm{0,2}\right)}+3{ƒ}_x^{\left(\mathrm{1,1}\right)}\right)$

0 0 3 2 1 β (t ⁽¹⁾)² t ⁽²⁾ ${\overline{c}}_{\mathrm{3,21}}$

$X\left({\overline{c}}_{\mathrm{3,01}}\right)$

0 0 3 0 2 β (t ⁽²⁾)² ${\overline{c}}_{\mathrm{3,02}}$

-3B -3τ/n ² 0 3 0 01 β t ⁽³⁾ ${\overline{c}}_{\mathrm{3,001}}$

1 1 1 4 0 0 β ² 1 ${\overline{c}}_{\mathrm{4,0}}$

$\begin{array}{l}15T_{AC}^{\left(\mathrm{2,0}\right)}{\left(f_x^{\left(\mathrm{2,0}\right)}\right)}^2\\ +A(A{T}_{AC}^{\left(\mathrm{4,0}\right)}-6T_{AC}^{\left(\mathrm{3,0}\right)}{f}_x^{\left(\mathrm{2,0}\right)}\\ -4T_{AC}^{\left(\mathrm{2,0}\right)}{f}_x^{\left(\mathrm{3,0}\right)})\end{array}$

0 $-\left(n\text{'}-n\right)\left(\begin{array}{l}{\overline{z}}^{\left(5\right)}+2\left(n-n\text{'}\right)/n{\text{'}}^2\times \\ \left(9n+11n\text{'}\right){\overline{z}}^{{\left(2\right)}^2}{\overline{z}}^{\left(3\right)}\end{array}\right)$

4 1 0 β ² t ⁽¹⁾ ${\overline{c}}_{\mathrm{4,1}}$

$\begin{array}{l}-2\left(9A{T}_{AC}^{\left(\mathrm{2,1}\right)}+5B{T}_{AC}^{\left(\mathrm{3,0}\right)}+A{T}_{BD}^{\left(\mathrm{3,0}\right)}\right)f_x^{\left(\mathrm{2,0}\right)}\\ +15\left(2T_{AC}^{\left(\mathrm{1,1}\right)}+T_{BD}^{\left(\mathrm{2,0}\right)}\right){\left(f_x^{\left(\mathrm{2,0}\right)}\right)}^2\\ -12\left(A{T}_{AC}^{\left(\mathrm{3,0}\right)}-5T_{AC}^{\left(\mathrm{2,0}\right)}{f}_x^{\left(\mathrm{2,0}\right)}\right)f_x^{\left(\mathrm{1,1}\right)}\\ +A\left(\begin{array}{l}4A{T}_{AC}^{\left(\mathrm{3,1}\right)}+2B{T}_{AC}^{\left(\mathrm{4,0}\right)}+A{T}_{BD}^{\left(\mathrm{4,0}\right)}-\\ 12T_{AC}^{\left(\mathrm{2,0}\right)}{f}_x^{\left(\mathrm{2,1}\right)}-8\left(T_{AC}^{\left(\mathrm{1,1}\right)}+T_{BD}^{\left(\mathrm{2,0}\right)}\right)f_x^{\left(\mathrm{3,0}\right)}\end{array}\right)f_x^{\left(\mathrm{1,1}\right)}\end{array}$

0 $\begin{array}{c}-2\left(n\text{'}-n\right)\cdot \\ \left(7n\text{'}+18n\right)/n{\text{'}}^2{\overline{z}}^{\left(2\right)}{\overline{z}}^{\left(3\right)}\end{array}$

4 2 0 β ² (t ⁽¹⁾)² ${\overline{c}}_{\mathrm{4,2}}$

$\begin{array}{l}{A}^2\left(6T_{AC}^{\left(\mathrm{2,2}\right)}+4T_{BD}^{\left(\mathrm{3,1}\right)}\right)+2AB\left(4T_{AC}^{\left(\mathrm{3,1}\right)}+T_{BD}^{\left(\mathrm{4,0}\right)}\right)\\ +60T_{AC}^{\left(\mathrm{2,0}\right)}{\left(f_x^{\left(\mathrm{1,1}\right)}\right)}^2-6\left(A{T}_{AC}^{\left(\mathrm{3,0}\right)}-5T_{AC}^{\left(\mathrm{2,0}\right)}{f}_x^{\left(\mathrm{2,0}\right)}\right)\\ +B^2{T}_{AC}^{\left(\mathrm{4,0}\right)}+4\left(\begin{array}{l}-9A{T}_{AC}^{\left(\mathrm{2,1}\right)}-B{T}_{AC}^{\left(\mathrm{3,0}\right)}-5A{T}_{BD}^{\left(\mathrm{3,0}\right)}\\ +15\left(2T_{AC}^{\left(\mathrm{2,1}\right)}+T_{BD}^{\left(\mathrm{2,0}\right)}{f}_x^{\left(\mathrm{2,0}\right)}\right)\end{array}\right)f_x^{\left(\mathrm{1,1}\right)}\\ -12A{T}_{AC}^{\left(\mathrm{2,0}\right)}{f}_x^{\left(\mathrm{1,2}\right)}+\\ 3\left(\begin{array}{l}\left(\begin{array}{l}2A\left(3T_{AC}^{\left(\mathrm{1,2}\right)}+5T_{BD}^{\left(\mathrm{2,1}\right)}\right)+2B\left(T_{AC}^{\left(\mathrm{2,1}\right)}+T_{BD}^{\left(\mathrm{3,0}\right)}\right)\\ -5\left(T_{AC}^{\left(\mathrm{0,2}\right)}+2T_{BD}^{\left(\mathrm{1,1}\right)}\right)f_x^{\left(\mathrm{2,0}\right)}\end{array}\right)f_x^{\left(\mathrm{2,0}\right)}\\ +8\left(A{T}_{AC}^{\left(\mathrm{1,1}\right)}+B{T}_{AC}^{\left(\mathrm{2,0}\right)}\right)f_x^{\left(\mathrm{2,1}\right)}\end{array}\right)\\ -4\left(A{T}_{AC}^{\left(\mathrm{0,2}\right)}+B\left(4T_{AC}^{\left(\mathrm{1,1}\right)}+T_{BD}^{\left(\mathrm{2,0}\right)}\right)\right)f_x^{\left(\mathrm{3,0}\right)}\end{array}$

0 $\begin{array}{l}-2\left(n^{\prime }-n\right)\cdot \\ \left(5n^{\prime }+9n\right)/\left(n{n^{\prime }}^2\right){\overline{z}}^{\left(3\right)}\end{array}$

4 3 0 β ² (t ⁽¹⁾)³ ${\overline{c}}_{\mathrm{4,3}}$

$\begin{array}{l}{A}^2\left(4T_{AC}^{\left(\mathrm{1,3}\right)}+6T_{BD}^{\left(\mathrm{2,2}\right)}\right)+4AB\left(3T_{AC}^{\left(\mathrm{2,2}\right)}+2T_{BD}^{\left(\mathrm{3,1}\right)}\right)\\ +B^2\left(4T_{AC}^{\left(\mathrm{3,1}\right)}+T_{BD}^{\left(\mathrm{4,0}\right)}\right)-A{T}_{AC}^{\left(\mathrm{2,0}\right)}{f}_x^{\left(\mathrm{0,3}\right)}+\\ 2\left(\begin{array}{l}-5B{T}_{AC}^{\left(\mathrm{3,0}\right)}-A\left(9T_{AC}^{\left(\mathrm{2,1}\right)}+T_{BD}^{\left(\mathrm{2,0}\right)}\right)+\\ 30T_{AC}^{\left(\mathrm{2,0}\right)}{f}_x^{\left(\mathrm{1,1}\right)}+15\left(2T_{AC}^{\left(\mathrm{1,1}\right)}+T_{BD}^{\left(\mathrm{2,0}\right)}\right)f_x^{\left(\mathrm{2,0}\right)}\end{array}\right)f_x^{\left(\mathrm{0,2}\right)}\\ +3\left(\begin{array}{l}20\left(2T_{AC}^{\left(\mathrm{1,1}\right)}+T_{BD}^{\left(\mathrm{2,0}\right)}\right){\left(f_x^{\left(\mathrm{1,1}\right)}\right)}^2\\ -8\left(A{T}_{AC}^{\left(\mathrm{1,1}\right)}+B{T}_{AC}^{\left(\mathrm{2,0}\right)}\right)f_x^{\left(\mathrm{1,2}\right)}+\\ \left(-2\left(\begin{array}{l}A{T}_{AC}^{\left(\mathrm{0,3}\right)}+B{T}_{AC}^{\left(\mathrm{1,2}\right)}\\ +5A{T}_{BD}^{\left(\mathrm{1,2}\right)}+3B{T}_{BD}^{\left(\mathrm{2,1}\right)}\end{array}\right)+5T_{BD}^{\left(\mathrm{0,2}\right)}{f}_x^{\left(\mathrm{2,0}\right)}\right)f_x^{\left(\mathrm{2,0}\right)}\\ -4\left(\begin{array}{l}3A{T}_{AC}^{\left(\mathrm{1,2}\right)}+B{T}_{AC}^{\left(\mathrm{2,1}\right)}+5A{T}_{BD}^{\left(\mathrm{2,1}\right)}\\ +B{T}_{BD}^{\left(\mathrm{3,0}\right)}-5\left(T_{AC}^{\left(\mathrm{0,2}\right)}+2T_{BD}^{\left(\mathrm{1,1}\right)}\right)f_x^{\left(\mathrm{2,0}\right)}\end{array}\right)f_x^{\left(\mathrm{1,1}\right)}\\ -4\left(A{T}_{AC}^{\left(\mathrm{0,2}\right)}+4B{T}_{AC}^{\left(\mathrm{1,1}\right)}+B{T}_{BD}^{\left(\mathrm{2,0}\right)}\right)f_x^{\left(\mathrm{2,1}\right)}\end{array}\right)\\ -8\left(A{T}_{AC}^{\left(\mathrm{0,2}\right)}+B{T}_{BD}^{\left(\mathrm{1,1}\right)}\right)f_x^{\left(\mathrm{3,0}\right)}\end{array}$

0 0 4 4 0 β ² (t ^(l))⁴ ${\overline{c}}_{\mathrm{4,4}}$

$X\left({\overline{c}}_{\mathrm{4,3}}\right)$

0 0 4 5 0 β ² t ⁽¹⁾⁵ ${\overline{c}}_{\mathrm{4,5}}$

$X\left({\overline{c}}_{\mathrm{4,2}}\right)$

0 0 4 6 0 β ² (t ⁽¹⁾)⁶ ${\overline{c}}_{\mathrm{4,6}}$

$X\left({\overline{c}}_{\mathrm{4,1}}\right)$

0 0 4 7 0 β ² (t ⁽¹⁾)⁷ ${\overline{c}}_{\mathrm{4,7}}$

$X\left({\overline{c}}_{\mathrm{4,0}}\right)$

0 0 4 0 1 β ² t ⁽²⁾ ${\overline{c}}_{\mathrm{4,01}}$

$\begin{array}{l}6A^2{T}_{AC}^{\left(\mathrm{2,1}\right)}-12A{T}_{AC}^{\left(\mathrm{2,0}\right)}{f}_x^{\left(\mathrm{1,1}\right)}+30B{T}_{AC}^{\left(\mathrm{2,0}\right)}{f}_x^{\left(\mathrm{2,0}\right)}-\\ 18A{T}_{AC}^{\left(\mathrm{1,1}\right)}{f}_X^{\left(\mathrm{2,0}\right)}+15{\left(f_X^{\left(\mathrm{2,0}\right)}\right)}^2-6AB{T}_{AC}^{\left(\mathrm{3,0}\right)}+4A{f}_x^{\left(\mathrm{3,0}\right)}\end{array}$

0 $-6\left(n\text{'}-n\right)\left(2n\text{'}+3n\right)\text{/}n{\text{'}}^2{\overline{z}}^{{\left(2\right)}^2}$

4 1 1 β ² t ⁽¹⁾ t ⁽²⁾ ${\overline{c}}_{\mathrm{4,11}}$

$\begin{array}{l}-60A{T}_{AC}^{\left(\mathrm{1,1}\right)}{f}_x^{\left(\mathrm{1,1}\right)}+12A^2{T}_{AC}^{\left(\mathrm{1,2}\right)}{f}_x^{\left(\mathrm{1,1}\right)}+30B{T}_{BD}^{\left(\mathrm{2,0}\right)}{f}_x^{\left(\mathrm{2,0}\right)}\\ -10B^2{T}_{AC}^{\left(\mathrm{3,0}\right)}-2AB{T}_{BD}^{\left(\mathrm{3,0}\right)}-18A{T}_{AC}^{\left(\mathrm{0,2}\right)}{f}_x^{\left(\mathrm{2,0}\right)}-\\ 12A{T}_{AC}^{\left(\mathrm{2,0}\right)}{f}_x^{\left(\mathrm{0,2}\right)}+24B{T}_{AC}^{\left(\mathrm{1,1}\right)}{f}_x^{\left(\mathrm{2,0}\right)}+\\ 36B{T}_{AC}^{\left(\mathrm{2,0}\right)}{f}_x^{\left(\mathrm{1,1}\right)}+90f_x^{\left(\mathrm{2,0}\right)}{f}_x^{\left(\mathrm{1,1}\right)}-18A{f}_x^{\left(\mathrm{21,1}\right)}\end{array}$

0 $\begin{array}{l}-6\left(n\text{'}-n\right)\cdot \\ \left(5n^{\prime }+6n\right)/\left(n{n^{\prime }}^2\right){\overline{z}}^{\left(2\right)}\end{array}$

4 2 1 β ² (t ⁽¹⁾)² t ⁽²⁾ ${\overline{c}}_{\mathrm{4,21}}$

$6\left(\begin{array}{l}{A}^2{T}_{AC}^{\left(\mathrm{0,3}\right)}-3B^2{T}_{AC}^{\left(\mathrm{2,1}\right)}-B^2{T}_{BD}^{\left(\mathrm{3,0}\right)}-7A{T}_{AC}^{\left(\mathrm{1,1}\right)}{ƒ}_x^{\left(\mathrm{0,2}\right)}\\ -8A{T}_{AC}^{\left(\mathrm{0,2}\right)}{ƒ}_x^{\left(\mathrm{1,1}\right)}+3A^2{T}_{BD}^{\left(\mathrm{1,2}\right)}+8B{T}_{BD}^{\left(\mathrm{2,0}\right)}{ƒ}_x^{\left(\mathrm{1,1}\right)}+\\ 7B{T}_{BD}^{\left(\mathrm{1,1}\right)}{ƒ}_x^{\left(\mathrm{2,0}\right)}+20{\left({ƒ}_x^{\left(\mathrm{1,1}\right)}\right)}^2-A{ƒ}_x^{\left(\mathrm{1,2}\right)}+\\ AB\left({ƒ}_x^{\left(\mathrm{0,2}\right)}{ƒ}_t^{\left(\mathrm{2,0}\right)}-{ƒ}_x^{\left(\mathrm{2,0}\right)}{ƒ}_t^{\left(\mathrm{0,2}\right)}\right)+10{ƒ}_x^{\left(\mathrm{2,0}\right)}{ƒ}_x^{\left(\mathrm{0,2}\right)}\\ -B{ƒ}_x^{\left(\mathrm{2,1}\right)}\end{array}\right)$

0 $\begin{array}{l}-18\left(n\text{'}-n\right)\cdot \\ \left(n^{\prime }+n\right)/\left(n^2{n^{\prime }}^2\right)\end{array}$

4 3 1 β ² (t ⁽¹⁾)³ t ⁽²⁾ ${\overline{c}}_{\mathrm{4,31}}$

$X\left({\overline{c}}_{\mathrm{4,11}}\right)$

0 0 4 4 1 β ² (t ⁽¹⁾)⁴ t ⁽²⁾ ${\overline{c}}_{\mathrm{4,41}}$

$X\left({\overline{c}}_{\mathrm{4,01}}\right)$

0 0 4 0 2 β ² t ⁽²⁾)² ${\overline{c}}_{\mathrm{4,02}}$

$3\left(\begin{array}{l}{A}^2{T}_{AC}^{\left(\mathrm{0,2}\right)}-6AB{T}_{AC}^{\left(\mathrm{1,1}\right)}+\\ 5B^2{T}_{AC}^{\left(\mathrm{2,0}\right)}-4A{f}_x^{\left(\mathrm{1,1}\right)}+10B{f}_x^{\left(\mathrm{2,0}\right)}\end{array}\right)$

0 0 4 1 2 β ² t ⁽¹⁾(t ⁽²⁾)² ${\overline{c}}_{\mathrm{4,12}}$

$3\left(\begin{array}{l}-AB\left(3T_{AC}^{\left(\mathrm{0,2}\right)}+2T_{BD}^{\left(\mathrm{1,1}\right)}\right)+\\ 5B^2{T}_{AC}^{\left(\mathrm{2,0}\right)}-5A{f}_x^{\left(\mathrm{0,2}\right)}+20B{f}_x^{\left(\mathrm{1,1}\right)}\end{array}\right)$

0 0 4 2 2 β ² (t ⁽¹⁾)²(t ⁽²⁾)² ${\overline{c}}_{\mathrm{4,22}}$

$6B\left(-2A{T}_{BD}^{\left(\mathrm{0,2}\right)}+2B{T}_{BD}^{\left(\mathrm{1,1}\right)}+3f_X^{\left(\mathrm{0,2}\right)}\right)$

0 0 4 0 01 β t ⁽³⁾ ${\overline{c}}_{\mathrm{4,001}}$

$2\left(2A{T}_{AC}^{\left(\mathrm{1,1}\right)}-2B{T}_{AC}^{\left(\mathrm{2,0}\right)}-3f_x^{\left(\mathrm{2,0}\right)}\right)$

0 0 4 1 01 β t ⁽¹⁾ t ⁽³⁾ ${\overline{c}}_{\mathrm{4,101}}$

$4\left(A{T}_{AC}^{\left(\mathrm{0,2}\right)}-B{T}_{BD}^{\left(\mathrm{0,2}\right)}-4f_x^{\left(\mathrm{1,1}\right)}\right)$

0 0 4 2 01 β (t ⁽¹⁾)² t ⁽³⁾ ${\overline{c}}_{\mathrm{4,201}}$

$X\left({\overline{c}}_{\mathrm{4,001}}\right)$

0 0 4 0 3 β ² (t ⁽²⁾)³ ${\overline{c}}_{\mathrm{4,03}}$

15B ² 15τ²/n ⁴ 0 4 0 11 β t ⁽²⁾ t ⁽³⁾ ${\overline{c}}_{\mathrm{4,011}}$

-10B -10τ/n ² 0 4 0 001 1 t ⁽⁴⁾ ${\overline{c}}_{\mathrm{4,0001}}$

1 1 1 5 0 0 β ³ 1 ${\overline{c}}_{\mathrm{5,0}}$

... 0 $\begin{array}{l}-\left(n\text{'}-n\right)\times \\ \left(\begin{array}{l}{\overline{z}}^{\left(6\right)}+(\dots ){\overline{z}}^{{\left(2\right)}^5}+\\ (\dots ){\overline{z}}^{{\left(2\right)}^2}{\overline{z}}^{\left(4\right)}+(\dots ){\overline{z}}^{\left(2\right)}{\overline{z}}^{{\left(3\right)}^2}\end{array}\right)\end{array}$

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 5 0 0 1 1 ${\overline{c}}_{\mathrm{5,00001}}$

1 1 1 6 0 0 β ⁴ 1 ${\overline{c}}_{\mathrm{6,0}}$

0 $\begin{array}{l}-\left(n\text{'}-n\right)\times \\ \left(\begin{array}{l}{\overline{z}}^{\left(7\right)}+(\dots ){\overline{z}}^{{\left(2\right)}^4}{\overline{z}}^{\left(3\right)}+\\ (\dots ){\overline{z}}^{{\left(2\right)}^2}{\overline{z}}^{\left(5\right)}+(\dots ){\overline{z}}^{\left(2\right)}{\overline{z}}^{\left(3\right)}{\overline{z}}^{\left(4\right)}\\ +(\dots ){\overline{z}}^{{\left(3\right)}^3}\end{array}\right)\end{array}$

In Table 4 below are the coefficients ${\overline{c}}_{ik}$

given for general optical systems and for a simple propagation over a distance d = τ/n as well as for a propagation through a single surface with surface derivatives ${\overline{\mathrm{e.g}}}^{(2)},{\overline{\mathrm{e.g}}}^{(3)},{\overline{\mathrm{e.g}}}^{(4)},\mathrm{...}$

between two media with refractive indices n and n'. Table 4: ord indices prefactor term coefficient ${\overline{c}}_{ik}\equiv {\overline{c}}_{i,\left({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102020008145A1\_0383.png,DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_1,k*\right)}$

i k _i k *. ${\beta }^{-\mathrm{\Delta }{\overline{\mathrm{right}}}_1}$

term symbol general case simple propagation, distance τ / n Refraction at single plane (β=1) 1 0 0 1 1 ${\overline{c}}_{1.0}$

C 0 $-\left(n\text{'}-n\right){\overline{\mathrm{e.g}}}^{\left(2\right)}$

1 1 0 1 t ⁽¹⁾ ${\overline{c}}_{1.1}$

$X\left({\overline{c}}_{1.1}\right)=D$

1 1 2 0 0 1 1 ${\overline{c}}_{2.0}$

$T_{AC}^{\left(2.0\right)}$

0 $-\left(n\text{'}-n\right){\overline{\mathrm{e.g}}}^{\left(3\right)}$

2 1 0 1 t ⁽¹⁾ ${\overline{c}}_{2.1}$

$2T_{AC}^{\left(1.1\right)}+T_{BD}^{\left(2.0\right)}$

0 0 2 2 0 1 ( t ⁽¹⁾ ) ² ${\overline{c}}_{2.2}$

$X\left({\overline{c}}_{2.1}\right)=T_{AC}^{\left(2.0\right)}+2T_{BD}^{\left(1.1\right)}$

0 0 2 3 0 1 ( t ⁽¹⁾ ) ³ ${\overline{c}}_{2.3}$

$X\left({\overline{c}}_{2.0}\right)=T_{BD}^{\left(0.2\right)}$

0 0 2 0 1 1 t ⁽²⁾ ${\overline{c}}_{2.01}$

1 1 1 3 0 0 β 1 ${\overline{c}}_{3.0}$

$-3T_{AC}^{\left(2.0\right)}{f}_x^{\left(2.0\right)}+A{T}_{AC}^{\left(3.0\right)}$

0 $-\left(n\text{'}-n\right)\left(\begin{array}{l}{\overline{\mathrm{e.g}}}^{\left(4\right)}+\\ 3\left({\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">n</figure-callout>}}^2/n{\text{'}}^2-1\right){\overline{\mathrm{e.g}}}^{{\left(2\right)}^3}\end{array}\right)$

3 1 0 β t ⁽¹⁾ ${\overline{c}}_{3.1}$

$\begin{array}{l}-6T_{AC}^{\left(2.0\right)}{f}_x^{\left(1.1\right)}-3\left(T_{BD}^{\left(2.0\right)}+2T_{AC}^{\left(1.1\right)}\right)f_x^{\left(2.0\right)}\\ +3A{T}_{AC}^{\left(2.1\right)}+B{T}_{AC}^{\left(3.0\right)}+A{T}_{BD}^{\left(3.0\right)}\end{array}$

0 $-3\left(n^{\prime }-n\right)\left(n^{\prime }+3n\right)/{n^{\prime }}^2{\overline{\mathrm{e.g}}}^{{\left(2\right)}^3}$

3 2 0 β ( t ⁽¹⁾ ) ² ${\overline{c}}_{3.2}$

$\begin{array}{l}3A\left(T_{AC}^{\left(1.2\right)}+T_{BD}^{\left(2.1\right)}\right)+B\left(3T_{AC}^{\left(2.1\right)}+T_{BD}^{\left(3.0\right)}\right)-3T_{AC}^{\left(2.0\right)}{f}_x^{\left(0.2\right)}\\ -6\left(2T_{AC}^{\left(1.1\right)}+T_{BD}^{\left(2.0\right)}\right)f_x^{\left(1.1\right)}-3\left(T_{AC}^{\left(0.2\right)}+2T_{BD}^{\left(1.1\right)}\right)f_x^{\left(2.0\right)}\end{array}$

0 $-3\left(n^{\prime }-n\right)\left(2n^{\prime }+3n\right)/\left(n{n^{\prime }}^2\right){\overline{\mathrm{e.g}}}^{\left(2\right)}$

3 3 0 β ( t ⁽¹⁾ ) ³ ${\overline{c}}_{3.3}$

$X\left({\overline{c}}_{3.2}\right)$

0 $-3\left(n^{\prime }-n\right)\left(n^{\prime }+n\right)/\left(n^2{n^{\prime }}^2\right)$

3 4 0 β ( t ⁽¹⁾ ) ⁴ ${\overline{c}}_{3.4}$

$X\left({\overline{c}}_{3.1}\right)$

) 0 0 3 5 0 β ( t ⁽¹⁾ ) ⁵ ${\overline{c}}_{3.5}$

$X\left({\overline{c}}_{3.0}\right)$

0 0 3 0 1 β t ⁽²⁾ ${\overline{c}}_{3.01}$

$-3\left(B{T}_{AC}^{\left(2.0\right)}-A{T}_{AC}^{\left(1.1\right)}+{ƒ}_x^{\left(2.0\right)}\right)$

0 0 3 1 1 β t ⁽¹⁾ t ⁽²⁾ ${\overline{c}}_{\mathrm{3:11}}$

$-3\left(B{T}_{BD}^{\left(2.0\right)}-A{T}_{AC}^{\left(0.2\right)}+3{ƒ}_x^{\left(1.1\right)}\right)$

0 0 3 2 1 β ( t ⁽¹⁾ ) 2t ( ^{2 )} ${\overline{c}}_{\mathrm{3:21}}$

$X\left({\overline{c}}_{3.01}\right)$

0 0 3 0 2 β ( t ⁽²⁾ ) ² ${\overline{c}}_{3.02}$

-3 B -3τ/ n ² 0 3 0 01 β t ⁽³⁾ ${\overline{c}}_{3.001}$

1 1 1 4 0 0 β ² 1 ${\overline{c}}_{4.0}$

$\begin{array}{l}15T_{AC}^{\left(2.0\right)}{\left(f_x^{\left(2.0\right)}\right)}^2\\ +A(A{T}_{AC}^{\left(4.0\right)}-6T_{AC}^{\left(3.0\right)}{f}_x^{\left(2.0\right)}\\ -4T_{AC}^{\left(2.0\right)}{f}_x^{\left(3.0\right)})\end{array}$

0 $-\left(n\text{'}-n\right)\left(\begin{array}{l}{\overline{\mathrm{e.g}}}^{\left(5\right)}+2\left(n-n\text{'}\right)/n{\text{'}}^2\times \\ \left(9n+11n\text{'}\right){\overline{\mathrm{e.g}}}^{{\left(2\right)}^2}{\overline{\mathrm{e.g}}}^{\left(3\right)}\end{array}\right)$

4 1 0 β ² t ⁽¹⁾ ${\overline{c}}_{4.1}$

$\begin{array}{l}-2\left(9A{T}_{AC}^{\left(2.1\right)}+5B{T}_{AC}^{\left(3.0\right)}+A{T}_{BD}^{\left(3.0\right)}\right)f_x^{\left(2.0\right)}\\ +15\left(2T_{AC}^{\left(1.1\right)}+T_{BD}^{\left(2.0\right)}\right){\left(f_x^{\left(2.0\right)}\right)}^2\\ -12\left(A{T}_{AC}^{\left(3.0\right)}-5T_{AC}^{\left(2.0\right)}{f}_x^{\left(2.0\right)}\right)f_x^{\left(1.1\right)}\\ +A\left(\begin{array}{l}4A{T}_{AC}^{\left(3.1\right)}+2B{T}_{AC}^{\left(4.0\right)}+A{T}_{BD}^{\left(4.0\right)}-\\ 12T_{AC}^{\left(2.0\right)}{f}_x^{\left(2.1\right)}-\mathrm{8th}\left(T_{AC}^{\left(1.1\right)}+T_{BD}^{\left(2.0\right)}\right)f_x^{\left(3.0\right)}\end{array}\right)f_x^{\left(1.1\right)}\end{array}$

0 $\begin{array}{c}-2\left(n\text{'}-n\right)\cdot \\ \left(7n\text{'}+18n\right)/n{\text{'}}^2{\overline{\mathrm{e.g}}}^{\left(2\right)}{\overline{\mathrm{e.g}}}^{\left(3\right)}\end{array}$

4 2 0 β ² ( t ⁽¹⁾ ) ² ${\overline{c}}_{4.2}$

$\begin{array}{l}{A}^2\left(6T_{AC}^{\left(2.2\right)}+4T_{BD}^{\left(3.1\right)}\right)+2AB\left(4T_{AC}^{\left(3.1\right)}+T_{BD}^{\left(4.0\right)}\right)\\ +60T_{AC}^{\left(2.0\right)}{\left(f_x^{\left(1.1\right)}\right)}^2-6\left(A{T}_{AC}^{\left(3.0\right)}-5T_{AC}^{\left(2.0\right)}{f}_x^{\left(2.0\right)}\right)\\ +B^2{T}_{AC}^{\left(4.0\right)}+4\left(\begin{array}{l}-9A{T}_{AC}^{\left(2.1\right)}-B{T}_{AC}^{\left(3.0\right)}-5A{T}_{BD}^{\left(3.0\right)}\\ +15\left(2T_{AC}^{\left(2.1\right)}+T_{BD}^{\left(2.0\right)}{f}_x^{\left(2.0\right)}\right)\end{array}\right)f_x^{\left(1.1\right)}\\ -12A{T}_{AC}^{\left(2.0\right)}{f}_x^{\left(1.2\right)}+\\ 3\left(\begin{array}{l}\left(\begin{array}{l}2A\left(3T_{AC}^{\left(1.2\right)}+5T_{BD}^{\left(2.1\right)}\right)+2B\left(T_{AC}^{\left(2.1\right)}+T_{BD}^{\left(3.0\right)}\right)\\ -5\left(T_{AC}^{\left(0.2\right)}+2T_{BD}^{\left(1.1\right)}\right)f_x^{\left(2.0\right)}\end{array}\right)f_x^{\left(2.0\right)}\\ +\mathrm{8th}\left(A{T}_{AC}^{\left(1.1\right)}+B{T}_{AC}^{\left(2.0\right)}\right)f_x^{\left(2.1\right)}\end{array}\right)\\ -4\left(A{T}_{AC}^{\left(0.2\right)}+B\left(4T_{AC}^{\left(1.1\right)}+T_{BD}^{\left(2.0\right)}\right)\right)f_x^{\left(3.0\right)}\end{array}$

0 $\begin{array}{l}-2\left(n^{\prime }-n\right)\cdot \\ \left(5n^{\prime }+9n\right)/\left(n{n^{\prime }}^2\right){\overline{\mathrm{e.g}}}^{\left(3\right)}\end{array}$

4 3 0 β ² ( t ⁽¹⁾ ) ³ ${\overline{c}}_{4.3}$

$\begin{array}{l}{A}^2\left(4T_{AC}^{\left(1.3\right)}+6T_{BD}^{\left(2.2\right)}\right)+4AB\left(3T_{AC}^{\left(2.2\right)}+2T_{BD}^{\left(3.1\right)}\right)\\ +B^2\left(4T_{AC}^{\left(3.1\right)}+T_{BD}^{\left(4.0\right)}\right)-A{T}_{AC}^{\left(2.0\right)}{f}_x^{\left(0.3\right)}+\\ 2\left(\begin{array}{l}-5B{T}_{AC}^{\left(3.0\right)}-A\left(9T_{AC}^{\left(2.1\right)}+T_{BD}^{\left(2.0\right)}\right)+\\ 30T_{AC}^{\left(2.0\right)}{f}_x^{\left(1.1\right)}+15\left(2T_{AC}^{\left(1.1\right)}+T_{BD}^{\left(2.0\right)}\right)f_x^{\left(2.0\right)}\end{array}\right)f_x^{\left(0.2\right)}\\ +3\left(\begin{array}{l}20\left(2T_{AC}^{\left(1.1\right)}+T_{BD}^{\left(2.0\right)}\right){\left(f_x^{\left(1.1\right)}\right)}^2\\ -\mathrm{8th}\left(A{T}_{AC}^{\left(1.1\right)}+B{T}_{AC}^{\left(2.0\right)}\right)f_x^{\left(1.2\right)}+\\ \left(-2\left(\begin{array}{l}A{T}_{AC}^{\left(0.3\right)}+B{T}_{AC}^{\left(1.2\right)}\\ +5A{T}_{BD}^{\left(1.2\right)}+3B{T}_{BD}^{\left(2.1\right)}\end{array}\right)+5T_{BD}^{\left(0.2\right)}{f}_x^{\left(2.0\right)}\right)f_x^{\left(2.0\right)}\\ -4\left(\begin{array}{l}3A{T}_{AC}^{\left(1.2\right)}+B{T}_{AC}^{\left(2.1\right)}+5A{T}_{BD}^{\left(2.1\right)}\\ +B{T}_{BD}^{\left(3.0\right)}-5\left(T_{AC}^{\left(0.2\right)}+2T_{BD}^{\left(1.1\right)}\right)f_x^{\left(2.0\right)}\end{array}\right)f_x^{\left(1.1\right)}\\ -4\left(A{T}_{AC}^{\left(0.2\right)}+4B{T}_{AC}^{\left(1.1\right)}+B{T}_{BD}^{\left(2.0\right)}\right)f_x^{\left(2.1\right)}\end{array}\right)\\ -\mathrm{8th}\left(A{T}_{AC}^{\left(0.2\right)}+B{T}_{BD}^{\left(1.1\right)}\right)f_x^{\left(3.0\right)}\end{array}$

0 0 4 4 0 β ² ( t ^(l) ) ⁴ ${\overline{c}}_{4.4}$

$X\left({\overline{c}}_{4.3}\right)$

0 0 4 5 0 β ² t ⁽¹⁾⁵ ${\overline{c}}_{4.5}$

$X\left({\overline{c}}_{4.2}\right)$

0 0 4 6 0 β ² ( t ⁽¹⁾ ) ⁶ ${\overline{c}}_{4.6}$

$X\left({\overline{c}}_{4.1}\right)$

0 0 4 7 0 β ² ( t ⁽¹⁾ ) ⁷ ${\overline{c}}_{4.7}$

$X\left({\overline{c}}_{4.0}\right)$

0 0 4 0 1 β ² t ⁽²⁾ ${\overline{c}}_{4.01}$

$\begin{array}{l}6A^2{T}_{AC}^{\left(2.1\right)}-12A{T}_{AC}^{\left(2.0\right)}{f}_x^{\left(1.1\right)}+30B{T}_{AC}^{\left(2.0\right)}{f}_x^{\left(2.0\right)}-\\ 18A{T}_{AC}^{\left(1.1\right)}{f}_X^{\left(2.0\right)}+15{\left(f_X^{\left(2.0\right)}\right)}^2-6AB{T}_{AC}^{\left(3.0\right)}+4A{f}_x^{\left(3.0\right)}\end{array}$

0 $-6\left(n\text{'}-n\right)\left(2n\text{'}+3n\right)\text{/}n{\text{'}}^2{\overline{\mathrm{e.g}}}^{{\left(2\right)}^2}$

4 1 1 β ² t ⁽¹⁾ t ⁽²⁾ ${\overline{c}}_{\mathrm{4:11}}$

$\begin{array}{l}-60A{T}_{AC}^{\left(1.1\right)}{f}_x^{\left(1.1\right)}+12A^2{T}_{AC}^{\left(1.2\right)}{f}_x^{\left(1.1\right)}+30B{T}_{BD}^{\left(2.0\right)}{f}_x^{\left(2.0\right)}\\ -10B^2{T}_{AC}^{\left(3.0\right)}-2AB{T}_{BD}^{\left(3.0\right)}-18A{T}_{AC}^{\left(0.2\right)}{f}_x^{\left(2.0\right)}-\\ 12A{T}_{AC}^{\left(2.0\right)}{f}_x^{\left(0.2\right)}+24B{T}_{AC}^{\left(1.1\right)}{f}_x^{\left(2.0\right)}+\\ 36B{T}_{AC}^{\left(2.0\right)}{f}_x^{\left(1.1\right)}+90f_x^{\left(2.0\right)}{f}_x^{\left(1.1\right)}-18A{f}_x^{\left(\mathrm{21:1}\right)}\end{array}$

0 $\begin{array}{l}-6\left(n\text{'}-n\right)\cdot \\ \left(5n^{\prime }+6n\right)/\left(n{n^{\prime }}^2\right){\overline{\mathrm{e.g}}}^{\left(2\right)}\end{array}$

4 2 1 β ² ( t ⁽¹⁾ ) 2t ( ^{2 )} ${\overline{c}}_{\mathrm{4:21}}$

$6\left(\begin{array}{l}{A}^2{T}_{AC}^{\left(0.3\right)}-3B^2{T}_{AC}^{\left(2.1\right)}-B^2{T}_{BD}^{\left(3.0\right)}-7A{T}_{AC}^{\left(1.1\right)}{ƒ}_x^{\left(0.2\right)}\\ -\mathrm{8th}A{T}_{AC}^{\left(0.2\right)}{ƒ}_x^{\left(1.1\right)}+3A^2{T}_{BD}^{\left(1.2\right)}+\mathrm{8th}B{T}_{BD}^{\left(2.0\right)}{ƒ}_x^{\left(1.1\right)}+\\ 7B{T}_{BD}^{\left(1.1\right)}{ƒ}_x^{\left(2.0\right)}+20{\left({ƒ}_x^{\left(1.1\right)}\right)}^2-A{ƒ}_x^{\left(1.2\right)}+\\ AB\left({ƒ}_x^{\left(0.2\right)}{ƒ}_t^{\left(2.0\right)}-{ƒ}_x^{\left(2.0\right)}{ƒ}_t^{\left(0.2\right)}\right)+10{ƒ}_x^{\left(2.0\right)}{ƒ}_x^{\left(0.2\right)}\\ -B{ƒ}_x^{\left(2.1\right)}\end{array}\right)$

0 $\begin{array}{l}-18\left(n\text{'}-n\right)\cdot \\ \left(n^{\prime }+n\right)/\left(n^2{n^{\prime }}^2\right)\end{array}$

4 3 1 β ² ( t ⁽¹⁾ ) 3t ( ^{2 )} ${\overline{c}}_{\mathrm{4:31}}$

$X\left({\overline{c}}_{\mathrm{4:11}}\right)$

0 0 4 4 1 β ² ( t ⁽¹⁾ ) ^4t ( ²⁾ ${\overline{c}}_{4.41}$

$X\left({\overline{c}}_{4.01}\right)$

0 0 4 0 2 β ² t ⁽²⁾ ) ² ${\overline{c}}_{4.02}$

$3\left(\begin{array}{l}{A}^2{T}_{AC}^{\left(0.2\right)}-6AB{T}_{AC}^{\left(1.1\right)}+\\ 5B^2{T}_{AC}^{\left(2.0\right)}-4A{f}_x^{\left(1.1\right)}+10B{f}_x^{\left(2.0\right)}\end{array}\right)$

0 0 4 1 2 β ² t ⁽¹⁾ ( t ⁽²⁾ ) ² ${\overline{c}}_{\mathrm{4:12}}$

$3\left(\begin{array}{l}-AB\left(3T_{AC}^{\left(0.2\right)}+2T_{BD}^{\left(1.1\right)}\right)+\\ 5B^2{T}_{AC}^{\left(2.0\right)}-5A{f}_x^{\left(0.2\right)}+20B{f}_x^{\left(1.1\right)}\end{array}\right)$

0 0 4 2 2 β ² ( t ⁽¹⁾ ) ² ( t ⁽²⁾ ) ² ${\overline{c}}_{4.22}$

$6B\left(-2A{T}_{BD}^{\left(0.2\right)}+2B{T}_{BD}^{\left(1.1\right)}+3f_X^{\left(0.2\right)}\right)$

0 0 4 0 01 β t ⁽³⁾ ${\overline{c}}_{4.001}$

$2\left(2A{T}_{AC}^{\left(1.1\right)}-2B{T}_{AC}^{\left(2.0\right)}-3f_x^{\left(2.0\right)}\right)$

0 0 4 1 01 β t ⁽¹⁾ t ⁽³⁾ ${\overline{c}}_{4.101}$

$4\left(A{T}_{AC}^{\left(0.2\right)}-B{T}_{BD}^{\left(0.2\right)}-4f_x^{\left(1.1\right)}\right)$

0 0 4 2 01 β ( t ⁽¹⁾ ) ^2t ( 3 ⁾ ${\overline{c}}_{4.201}$

$X\left({\overline{c}}_{4.001}\right)$

0 0 4 0 3 β ² ( t ⁽²⁾ ) ³ ${\overline{c}}_{4.03}$

15 B ^{2 15τ2 /} n4 0 4 0 11 β t ⁽²⁾ t ⁽³⁾ ${\overline{c}}_{4.011}$

-10B -10τ/ n ² 0 4 0 001 1 t ⁽⁴⁾ ${\overline{c}}_{4.0001}$

1 1 1 5 0 0 β ³ 1 ${\overline{c}}_{5.0}$

... 0 $\begin{array}{l}-\left(n\text{'}-n\right)\times \\ \left(\begin{array}{l}{\overline{\mathrm{e.g}}}^{\left(6\right)}+(...){\overline{\mathrm{e.g}}}^{{\left(2\right)}^5}+\\ (...){\overline{\mathrm{e.g}}}^{{\left(2\right)}^2}{\overline{\mathrm{e.g}}}^{\left(4\right)}+(...){\overline{\mathrm{e.g}}}^{\left(2\right)}{\overline{\mathrm{e.g}}}^{{\left(3\right)}^2}\end{array}\right)\end{array}$

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 5 0 0 1 1 ${\overline{c}}_{5.00001}$

1 1 1 6 0 0 β ⁴ 1 ${\overline{c}}_{6.0}$

0 $\begin{array}{l}-\left(n\text{'}-n\right)\times \\ \left(\begin{array}{l}{\overline{\mathrm{e.g}}}^{\left(7\right)}+(...){\overline{\mathrm{e.g}}}^{{\left(2\right)}^4}{\overline{\mathrm{e.g}}}^{\left(3\right)}+\\ (...){\overline{\mathrm{e.g}}}^{{\left(2\right)}^2}{\overline{\mathrm{e.g}}}^{\left(5\right)}+(...){\overline{\mathrm{e.g}}}^{\left(2\right)}{\overline{\mathrm{e.g}}}^{\left(3\right)}{\overline{\mathrm{e.g}}}^{\left(4\right)}\\ +(...){\overline{\mathrm{e.g}}}^{{\left(3\right)}^3}\end{array}\right)\end{array}$

Solutions in w-representation

The solutions for the derivatives of the wave fronts w' ^(p) are obtained by applying the transformation H (see 3 ^{) . _ _ _ _} ..to describe the incident wave front. For this purpose one can apply Eq. (9) to w' ^(p) , t' ⁽ⁱ⁾ instead of to w ^(p) , t ⁽ⁱ⁾ and obtains
$w{\text{'}}^{\left(2\right)}=-t{\text{'}}^{\left(1\right)}\text{/}n\text{'},w{\text{'}}^{\left(3\right)}=-t{\text{'}}^{\left(2\right)}\text{/}n\text{'},w{\text{'}}^{\left(4\right)}=-\left(t{\text{'}}^{\left(3\right)}+3t{\text{'}}^{{\left(1\right)}^3}\text{/}n{\text{'}}^2\right)\text{/}n\text{'},....$

Then one can substitute all derivatives t' ⁽ⁱ⁾ on the right-hand side of these equations by the solutions of the t-representation from Eq.(28),(29) and thus express them by the derivatives t ^(k) . Finally, the inverse transformation H ^-1 is then applied by replacing all derivatives t ^(k) by corresponding functions of derivatives w ^(k) according to Eq. (8).

wobei $$\beta ={\left(-B{E}_2+A\right)}^{-1}$$

ist, und wobei zur einfacheren Interpretation mit ophthalmischen Größen die Notation E'_p = n'w’^(p), E_p = nw^(p) verwendet wurde (per Definition gehört die niedrigste nichtverschwindende Ordnung, die der Krümmung entspricht, zur Ordnung p = 2 in der w-Darstellung, aber zu i = 1 in der t-Darstellung). Ein Vergleich von GI.(36) mit GI.(28) zeigt, dass die Lösungen E'_p für p = i + 1 genau die gleiche Struktur aufweise wie die Lösungen t'⁽ⁱ⁾. Tatsächlich lassen sich die Gln.(36) durch einen Summenansatz zusammenfassen: $$\begin{array}{l}E{\text{'}}_p={\beta }^{-{\overline{r}}_1^0\left(p-1\right)}{\displaystyle \sum _{k_1,k_2,\dots ,k_{p-1}}{\overline{b}}_{p\text{k}}}{\beta }^{-\mathrm{\Delta }{\overline{r}}_1\left(p-\mathrm{1,}\text{k}*\right)}{E}_2^{k_1}{E}_3^{k_2}\dots E_p^{k_{p-1}}\\ ={\beta }^{-{\overline{r}}_1^0\left(p-1\right)}\left[E_p+{\displaystyle \sum _{k_1,k_2,\dots ,k_{p-2}}{\overline{b}}_{p\text{k}}}{\beta }^{-\mathrm{\Delta }{\overline{r}}_1\left(p-\mathrm{1,}\text{k}*\right)}{E}_2^{k_1}{E}_3^{k_2}\dots E_{p-1}^{k_{p-2}}\right],p=\mathrm{2,3,4,}\dots \end{array}$$

wobei die untere Zeile wieder nur im symplektischen Fall gilt. Die Koeffizienten ${\overline{b}}_{pk}$

entstehen durch die Koeffizienten ${\overline{c}}_{ik}$

als Ergebnis der Transformationen H, H^-1 und lassen sich auf diese zurückführen, wie in der folgenden Tabelle 5 gezeigt ist. Tabelle 5: Ord Indizes Vorfaktor Aberrationen Koeffizient ${\overline{b}}_{pk}\equiv {\overline{b}}_{p,\left(k_1,k*\right)}$

p k ₁ k * ${\beta }^{-\Delta {\overline{r}}_1}$

Term Symbol Allgemeiner Fall Einfache Propagation, Distanz τ/n Brechung an Einzelfläche (β = 1) 2 0 0 1 1 ${\overline{b}}_{\mathrm{2,0}}$

-C 0 $\left(n^{\prime }-n\right){\overline{z}}^{\left(2\right)}$

2 1 0 1 E ₂ ${\overline{b}}_{\mathrm{2,1}}$

D 1 1 3 0 0 1 1 ${\overline{b}}_{\mathrm{3,0}}$

$-{\overline{c}}_{\mathrm{2,0}}=-T_{AC}^{\left(\mathrm{2,0}\right)}$

0 $\left(n^{\prime }-n\right){\overline{z}}^{\left(3\right)}$

3 1 0 1 E ₂ ${\overline{b}}_{\mathrm{3,1}}$

$+{\overline{c}}_{\mathrm{2,1}}=2T_{AC}^{\left(\mathrm{1,1}\right)}+T_{BD}^{\left(\mathrm{2,0}\right)}$

0 0 3 2 0 1 $E_2^2$

${\overline{b}}_{\mathrm{3,2}}$

$-{\overline{c}}_{\mathrm{2,2}}=-X\left({\overline{c}}_{\mathrm{2,1}}\right)=-\left(T_{AC}^{\left(\mathrm{0,2}\right)}+2T_{BD}^{\left(\mathrm{1,1}\right)}\right)$

0 0 3 3 0 1 $E_2^3$

${\overline{b}}_{\mathrm{3,3}}$

$+{\overline{c}}_{\mathrm{2,3}}=X\left({\overline{c}}_{\mathrm{2,0}}\right)=T_{BD}^{\left(\mathrm{0,2}\right)}$

0 0 3 0 1 1 E ₃ ${\overline{b}}_{\mathrm{3,01}}$

1 1 1 4 0 0 β 1 ${\overline{b}}_{\mathrm{4,0}}$

$-{\overline{c}}_{\mathrm{3,0}}-3A^2{C}^3/{n^{\prime }}^2$

0 $\begin{array}{l}(n\text{'}\\ -n)\left(\begin{array}{l}{\overline{z}}^{\left(4\right)}\\ -6n\left(n\text{'}-n\right)/n{\text{'}}^2{\overline{z}}^{{\left(2\right)}^3}\end{array}\right)\end{array}$

4 1 0 β E ₂ ${\overline{b}}_{\mathrm{4,1}}$

$+{\overline{c}}_{\mathrm{3,1}}-3A{C}^2\left(2BC+\text{3}AD\right)/{N^{\prime }}^2$

0 $6\left(n^{\prime }-n\right)\left(n^{\prime }-\text{3}n\right)/{n^{\prime }}^2{\overline{z}}^{{(2)}^2}$

4 2 0 β $E_2^2$

${\overline{b}}_{\mathrm{4,2}}$

$-{\overline{c}}_{\mathrm{3,2}}-3C\left(B^2{C}^2+\text{6}ABCD+3A^2{D}^2\right)/{N^{\prime }}^2$

0 $6\left(n^{\prime }-n\right)\left(n^{\prime }+\text{3}n\right)/\left(n{n^{\prime }}^2\right){\overline{z}}^{(2)}$

4 3 0 β $E_2^3$

${\overline{b}}_{\mathrm{4,3}}$

$+{\overline{c}}_{\mathrm{3,3}}-3A/n^2+3D\left(3B^2{C}^2+\text{6}ABCD+A^2{D}^2\right)/{N^{\prime }}^2$

0 $-6\left(n^{\prime }-n\right)\left(n^{\prime }+n\right)/\left(n^2{n^{\prime }}^2\right)$

4 4 0 β $E_2^4$

${\overline{b}}_{\mathrm{4,4}}$

$-{\overline{c}}_{\mathrm{3,4}}+3B/n^2+3B{D}^2\left(3BC+\text{2}AD\right)/{N^{\prime }}^2$

-3τ /n ⁴ 0 4 5 0 β $E_2^5$

${\overline{b}}_{\mathrm{4,5}}$

$+{\overline{c}}_{\mathrm{3,5}}+3B^2{D}^3/{N^{\prime }}^2$

3τ²/n ⁶ 0 4 0 1 β E ₃ ${\overline{b}}_{\mathrm{4,01}}$

$+{\overline{c}}_{\mathrm{3,01}}$

0 0 4 1 1 β E ₂ E ₃ ${\overline{b}}_{\mathrm{4,11}}$

$-{\overline{c}}_{\mathrm{3,11}}$

0 0 4 2 1 β $E_2^2{E}_3$

${\overline{b}}_{\mathrm{4,21}}$

$+{\overline{c}}_{\mathrm{3,21}}$

0 0 4 0 2 β $E_3^2$

${\overline{b}}_{\mathrm{4,02}}$

$-{\overline{c}}_{\mathrm{3,02}}=3B$

3τ/n ² 0 4 0 01 β E ₄ ${\overline{b}}_{\mathrm{4,001}}$

1 1 1 5 0 0 β ² 1 ${\overline{b}}_{\mathrm{5,0}}$

$-{\overline{c}}_{\mathrm{4,0}}-22A^2{C}^2{T}_{AC}^{\left(\mathrm{2,0}\right)}\text{/}n{\text{'}}^2$

0 $\left(n\text{'}-n\right)\left(\begin{array}{l}{\overline{z}}^{\left(5\right)}-40n\times \\ \left(n\text{'}-n\right)\text{/}n{\text{'}}^2{\overline{z}}^{{\left(2\right)}^2}{\overline{z}}^{\left(3\right)}\end{array}\right)$

5 1 0 β ² E ₂ ${\overline{b}}_{\mathrm{5,1}}$

$\begin{array}{l}+{\overline{c}}_{\mathrm{4,1}}+22AC(2\left(AD+BC\right)T_{AC}^{\left(\mathrm{2,0}\right)}+AC\left(2T_{AC}^{\left(\mathrm{1,1}\right)}+T_{BD}^{\left(\mathrm{2,0}\right)}\right)\\ \text{/}n{\text{'}}^2\end{array}$

0 $\begin{array}{l}10\left(n\text{'}-n\right)\left(3n\text{'}-8n\right)\\ \text{/}n{\text{'}}^2{\overline{z}}^{\left(2\right)}{\overline{z}}^{\left(3\right)}\end{array}$

5 2 0 β ² $E_2^2$

${\overline{b}}_{\mathrm{5,2}}$

$\begin{array}{l}-{\overline{c}}_{\mathrm{4,2}}-\\ 22\left(\begin{array}{l}\left(A^2{D}^2+6ABCD+B^2{C}^2\right)T_{AC}^{\left(\mathrm{2,0}\right)}+AC\times \\ \left(\begin{array}{l}4\left(AD+BC\right)T_{AC}^{\left(\mathrm{1,1}\right)}+AC\left(T_{AC}^{\left(\mathrm{0,2}\right)}+2T_{BD}^{\left(\mathrm{1,1}\right)}\right)\\ -BC{T}_{BD}^{\left(\mathrm{2,0}\right)}-2D{f}_x^{\left(\mathrm{2,0}\right)}\end{array}\right)\end{array}\right)\text{/}n{\text{'}}^2\end{array}$

0 $10\left(n\text{'}-n\right)\left(n\text{'}+4n\right)\text{/}\left(n{n^{\prime }}^2\right){\overline{z}}^{\left(3\right)}$

5 3 0 β ² $E_2^3$

${\overline{b}}_{\mathrm{5,3}}$

$\begin{array}{l}-{\overline{c}}_{\mathrm{4,3}}-\\ 22\left(\begin{array}{l}2AB{D}^2{T}_{AC}^{\left(\mathrm{2,0}\right)}+AC\times \\ \left(2BC\left(T_{AC}^{\left(\mathrm{0,2}\right)}+2T_{BD}^{\left(\mathrm{1,1}\right)}\right)+A\left(2D{T}_{AC}^{\left(\mathrm{0,2}\right)}+C{T}_{BD}^{\left(\mathrm{0,2}\right)}\right)\right)\\ +\left(A^2{D}^2+6ABCD+B^2{C}^2\right)\left(2T_{AC}^{\left(\mathrm{1,1}\right)}+T_{BD}^{\left(\mathrm{2,0}\right)}\right)\end{array}\right)\\ \text{/}n{\text{'}}^2\\ +6A\left(2B{T}_{AC}^{\left(\mathrm{2,0}\right)}-2A{T}_{AC}^{\left(\mathrm{1,1}\right)}+3f_x^{\left(\mathrm{2,0}\right)}\right)\text{/}{n}^2\end{array}$

0 0 5 4 0 β ² $E_2^4$

${\overline{b}}_{\mathrm{5,4}}$

$-{\overline{c}}_{\mathrm{4,4}}-$

0 0 $+6\left(\begin{array}{l}4B^2{T}_{AC}^{\left(\mathrm{2,0}\right)}-2A^2\left(T_{AC}^{\left(\mathrm{0,2}\right)}+T_{BD}^{\left(\mathrm{1,1}\right)}\right)\\ +6A{f}_x^{\left(\mathrm{1,1}\right)}+B{f}_x^{\left(\mathrm{2,0}\right)}\end{array}\right)\text{/}{n}^2$

5 5 0 β ² $E_2^5$

${\overline{b}}_{\mathrm{5,5}}$

$\begin{array}{l}+{\overline{c}}_{\mathrm{4,5}}+\\ 22\left(\begin{array}{l}\left(A^2{D}^2+6ABCD+B^2{C}^2\right)T_{BD}^{\left(\mathrm{0,2}\right)}+BD\times \\ \left(\begin{array}{l}BD\left(6T_{AC}^{\left(\mathrm{1,1}\right)}+T_{BD}^{\left(\mathrm{2,0}\right)}\right)+2AD{T}_{AC}^{\left(\mathrm{0,2}\right)}+4BC{T}_{BD}^{\left(\mathrm{1,1}\right)}\\ +2\left(C{f}_x^{\left(\mathrm{0,2}\right)}-2DB{f}_x^{\left(\mathrm{1,1}\right)}\right)\end{array}\right)\end{array}\right)\\ +6\left(\begin{array}{l}4A^2{T}_{BD}^{\left(\mathrm{0,2}\right)}-2B^2\left(T_{AC}^{\left(\mathrm{1,1}\right)}+T_{BD}^{\left(\mathrm{2,0}\right)}\right)\\ -A{f}_x^{\left(\mathrm{0,2}\right)}-6B{f}_x^{\left(\mathrm{1,1}\right)}\end{array}\right)\text{/}{n}^2\end{array}$

0 0 5 6 0 β ² $E_2^6$

${\overline{b}}_{\mathrm{5,6}}$

$\begin{array}{l}-{\overline{c}}_{\mathrm{4,6}}-\\ 22BD\left(B\left(3D{T}_{AC}^{\left(\mathrm{0,2}\right)}+2D{T}_{BD}^{\left(\mathrm{1,1}\right)}+2C{T}_{BD}^{\left(\mathrm{0,2}\right)}\right)-2D{f}_x^{\left(\mathrm{0,2}\right)}\right)\\ +6B\left(2B{T}_{BD}^{\left(\mathrm{1,1}\right)}-2A{T}_{BD}^{\left(\mathrm{0,2}\right)}\right)+3f_x^{\left(\mathrm{0,2}\right)})\text{/}{n}^2\end{array}$

0 0 5 7 0 β ² $E_2^7$

${\overline{b}}_{\mathrm{5,7}}$

$+{\overline{c}}_{\mathrm{4,7}}+22B^2{D}^2{T}_{BD}^{\left(\mathrm{0,2}\right)}\text{/}n{\text{'}}^2$

0 0 5 0 1 β ² E ₃ ${\overline{b}}_{\mathrm{5,01}}$

$+{\overline{c}}_{\mathrm{4,11}}+22A^2{C}^2/{n^{\prime }}^2$

0 $\begin{array}{l}10\left(n\text{'}-n\right)\left(n\text{'}-4n\right)\\ \text{/}\left(nn{\text{'}}^2\right){\overline{z}}^{{\left(2\right)}^2}\end{array}$

5 1 1 β ² E ₂ E ₃ ${\overline{b}}_{\mathrm{5,11}}$

$-{\overline{c}}_{\mathrm{4,11}}-\text{44}AC\left(AD+BC\right)/{n^{\prime }}^2$

0 $\begin{array}{l}10\left(n\text{'}-n\right)\left(3n\text{'}+8n\right)\\ \text{/}\left(nn{\text{'}}^2\right){\overline{z}}^{(2)}\end{array}$

5 2 1 β ² $E_2^2{E}_3$

${\overline{b}}_{\mathrm{5,21}}$

$+{\overline{c}}_{\mathrm{4,21}}+22\left(\left(A^2{D}^2+\text{4}ABCD+B^2{C}^2\right)/{n^{\prime }}^2-A^2/n^2\right)$

0 $-40\left(n\text{'}-n\right)\left(n\text{'}+n\right)\text{/}\left(n^{2}n{\text{'}}^2\right)$

5 3 1 β ² $E_2^3{E}_3$

${\overline{b}}_{\mathrm{5,31}}$

$-{\overline{c}}_{\mathrm{4,31}}+2B\left(22D\left(AD+BC\right)/{n^{\prime }}^2+\text{7}A/n^2\right)$

0 0 5 4 1 β ² $E_2^4{E}_3$

${\overline{b}}_{\mathrm{5,41}}$

$+{\overline{c}}_{\mathrm{4,41}}+2B^2\left(11D^2/{n^{\prime }}^2+\text{4}/n^2\right)$

0 0 5 0 2 β ² $E_3^2$

${\overline{b}}_{\mathrm{5,02}}$

$-{\overline{c}}_{\mathrm{4,02}}$

0 0 5 1 2 β ² $E_2{E}_3^2$

${\overline{b}}_{\mathrm{5,12}}$

$+{\overline{c}}_{\mathrm{4,12}}$

0 0 5 2 2 β ² $E_2^2{E}_3^2$

2 3 ${\overline{b}}_{\mathrm{5,22}}$

$-{\overline{c}}_{\mathrm{4,22}}$

0 0 5 0 01 β E ₄ ${\overline{b}}_{\mathrm{5,001}}$

$+{\overline{c}}_{\mathrm{4,001}}$

0 0 5 1 01 β E ₂ E ₄ ${\overline{b}}_{\mathrm{5,101}}$

$-{\overline{c}}_{\mathrm{4,101}}$

0 0 5 2 01 β $E_2^2{E}_4$

${\overline{b}}_{\mathrm{5,201}}$

$-{\overline{c}}_{\mathrm{4,201}}$

0 0 5 0 3 β ² $E_3^3$

${\overline{b}}_{\mathrm{5,03}}$

15B ² 15τ² /n ⁴ 0 5 0 11 β E ₃ E ₄ ${\overline{b}}_{\mathrm{5,011}}$

-10B -10τ /n ² 0 5 0 001 1 E ₅ ${\overline{b}}_{\mathrm{5,0001}}$

1 1 1 6 0 0 β ³ 1 ${\overline{b}}_{\mathrm{6,0}}$

... 0 $\begin{array}{l}\left(n\text{'}-n\right)\times \\ \left(\begin{array}{c}{\overline{z}}^{\left(6\right)}+(\dots ){\overline{z}}^{{\left(2\right)}^5}+\\ (\dots ){\overline{z}}^{{\left(2\right)}^2}{\overline{z}}^{\left(4\right)}+(\dots ){\overline{z}}^{\left(2\right)}{\overline{z}}^{{\left(2\right)}^3}\end{array}\right)\end{array}$

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 6 0 0 1 1 ${\overline{b}}_{\mathrm{6,00001}}$

1 1 1 7 0 0 β ⁴ 1 ${\overline{b}}_{\mathrm{7,0}}$

0 ... The result does not yet solve the task of optimally short computing time for the evaluation, because due to the transformations H, H^-1 Mixed terms arise that occur multiple times. According to the invention, however, these can be combined again in the form $$\begin{array}{c}E{\text{'}}_2=\beta \left[{\overline{b}}_{2.1}{E}_2+{\overline{b}}_{2.0}\right]\\ E{\text{'}}_3={\beta }^3\left[E_3+{\overline{b}}_{3.0}+{\overline{b}}_{3.1}{E}_2+{\overline{b}}_{3.2}{E}_2^2+{\overline{b}}_{3.3}{E}_2^3\right]\\ E_4^{\text{'}}={\beta }^4\left[E_4+\beta \left(\begin{array}{l}\left({\overline{b}}_{4.0}+{\overline{b}}_{4.1}{E}_2+{\overline{b}}_{4.2}{E}_2^2+{\overline{b}}_{4.3}{E}_2^3+{\overline{b}}_{4.4}{E}_2^4+{\overline{b}}_{4.5}{E}_2^5\right)\\ +\left({\overline{b}}_{4.01}+{\overline{b}}_{\mathrm{4:11}}{E}_2+{\overline{b}}_{\mathrm{4:21}}{E}_2^2\right)E_3+{\overline{b}}_{4.02}{E}_3^2\end{array}\right)\right]\\ E{\text{'}}_5={\beta }^5\left[\begin{array}{l}{E}_5+{\beta }^2\left(\begin{array}{l}\left({\overline{b}}_{5.0}+{\overline{b}}_{5.1}{E}_2+...+{\overline{b}}_{5.7}{E}_2^7\right)\\ +\left({\overline{b}}_{5.01}+{\overline{b}}_{\mathrm{5:11}}{E}_2+...+{\overline{b}}_{5.41}{E}_2^4\right)E_3\\ +\left({\overline{b}}_{5.02}+{\overline{b}}_{5.12}{E}_2+{\overline{b}}_{5.22}{E}_2^2\right)E_3^2+{\overline{b}}_{5.03}{E}_3^3\end{array}\right)\\ +\beta \left(\left({\overline{b}}_{5.001}+{\overline{b}}_{5.101}{E}_2+{\overline{b}}_{5.201}{E}_2^2\right)E_3+{\overline{b}}_{5.011}{E}_3{E}_4\right)\end{array}\right]\\ E{\text{'}}_6={\beta }^6\left[E_6+...\right]\end{array}$$

whereby $$\beta ={\left(-\mathrm{<figure-callout\; id="2"\; label="B\; E"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">B</figure-callout>}{E}_{}{}_2+A\right)}^{-1}$$

and where, for easier interpretation with ophthalmic quantities, the notation E'_p = n'w'^(p), E_p = nw^(p) was used (by definition, the lowest nonzero order corresponding to curvature belongs to order p = 2 in w-representation, but to i = 1 in t-representation). A comparison of eq.(36) with eq.(28) shows that the solutions E'_p for p = i + 1 has exactly the same structure as the solutions t'⁽ⁱ⁾. In fact, equations (36) can be summed up using a summation approach: $$\begin{array}{l}E{\text{'}}_p={\beta }^{-{\overline{\mathrm{right}}}_1^0\left(p-1\right)}{\displaystyle \sum _{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102020008145A1\_0383.png,DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_1,{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2,...,k_{p-1}}{\overline{b}}_{p\text{k}}}{\beta }^{-\mathrm{\Delta }{\overline{\mathrm{right}}}_1\left(p-\mathrm{1,}\text{k}*\right)}{E}_2^{k_1}{E}_3^{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}...E_p^{k_{p-1}}\\ ={\beta }^{-{\overline{\mathrm{right}}}_1^0\left(p-1\right)}\left[E_p+{\displaystyle \sum _{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102020008145A1\_0383.png,DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_1,{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2,...,k_{p-2}}{\overline{b}}_{p\text{k}}}{\beta }^{-\mathrm{\Delta }{\overline{\mathrm{right}}}_1\left(p-\mathrm{1,}\text{k}*\right)}{E}_2^{k_1}{E}_3^{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}...E_{p-1}^{k_{p-2}}\right],p=\mathrm{2,3,4,}...\end{array}$$

where the bottom line is again only valid in the symplectic case. The coefficients ${\overline{b}}_{pk}$

arise from the coefficients ${\overline{c}}_{ik}$

as a result of the transformations H, H^-1 and can be traced back to them as shown in Table 5 below. Table 5: ord indices prefactor aberrations coefficient ${\overline{b}}_{pk}\equiv {\overline{b}}_{p,\left({\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102020008145A1\_0383.png,DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_1,k*\right)}$

p k ₁ k * ${\beta }^{-\Delta {\overline{\mathrm{right}}}_1}$

term symbol general case Simple propagation, distance τ/ n Refraction at a single surface (β = 1) 2 0 0 1 1 ${\overline{b}}_{2.0}$

-C 0 $\left(n^{\prime }-n\right){\overline{\mathrm{e.g}}}^{\left(2\right)}$

2 1 0 1 E ₂ ${\overline{b}}_{2.1}$

D 1 1 3 0 0 1 1 ${\overline{b}}_{3.0}$

$-{\overline{c}}_{2.0}=-T_{AC}^{\left(2.0\right)}$

0 $\left(n^{\prime }-n\right){\overline{\mathrm{e.g}}}^{\left(3\right)}$

3 1 0 1 E ₂ ${\overline{b}}_{3.1}$

$+{\overline{c}}_{2.1}=2T_{AC}^{\left(1.1\right)}+T_{BD}^{\left(2.0\right)}$

0 0 3 2 0 1 ${\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">E</figure-callout>}}_2^2$

${\overline{b}}_{3.2}$

$-{\overline{c}}_{2.2}=-X\left({\overline{c}}_{2.1}\right)=-\left(T_{AC}^{\left(0.2\right)}+2T_{BD}^{\left(1.1\right)}\right)$

0 0 3 3 0 1 ${\mathrm{<figure-callout\; id="2"\; label="E"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">E</figure-callout>}}_2^3$

${\overline{b}}_{3.3}$

$+{\overline{c}}_{2.3}=X\left({\overline{c}}_{2.0}\right)=T_{BD}^{\left(0.2\right)}$

0 0 3 0 1 1 E ₃ ${\overline{b}}_{3.01}$

1 1 1 4 0 0 β 1 ${\overline{b}}_{4.0}$

$-{\overline{c}}_{3.0}-3A^2{C}^3/{n^{\prime }}^2$

0 $\begin{array}{l}(n\text{'}\\ -n)\left(\begin{array}{l}{\overline{\mathrm{e.g}}}^{\left(4\right)}\\ -6n\left(n\text{'}-n\right)/n{\text{'}}^2{\overline{\mathrm{e.g}}}^{{\left(2\right)}^3}\end{array}\right)\end{array}$

4 1 0 β E ₂ ${\overline{b}}_{4.1}$

$+{\overline{c}}_{3.1}-3A{C}^2\left(2BC+\text{3}AD\right)/{N^{\prime }}^2$

0 $6\left(n^{\prime }-n\right)\left(n^{\prime }-\text{3}n\right)/{n^{\prime }}^2{\overline{\mathrm{e.g}}}^{{(2)}^2}$

4 2 0 β $E_{}^{}$${}_2^2$

${\overline{b}}_{4.2}$

$-{\overline{c}}_{3.2}-3C\left(B^2{\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">C</figure-callout>}}^2+\text{6}ABCD+3A^2{D}^2\right)/{N^{\prime }}^2$

0 $6\left(n^{\prime }-n\right)\left(n^{\prime }+\text{3}n\right)/\left(n{n^{\prime }}^2\right){\overline{\mathrm{e.g}}}^{(2)}$

4 3 0 β $E_{}^{}$${}_2^3$

${\overline{b}}_{4.3}$

$+{\overline{c}}_{3.3}-3A/{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">n</figure-callout>}}^2+3D\left(3B^2{\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">C</figure-callout>}}^2+\text{6}ABCD+A^2{D}^2\right)/{N^{\prime }}^2$

0 $-6\left(n^{\prime }-n\right)\left(n^{\prime }+n\right)/\left(n^2{n^{\prime }}^2\right)$

4 4 0 β $E_{}^{}$${}_2^4$

${\overline{b}}_{4.4}$

$-{\overline{c}}_{3.4}+3B/{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">n</figure-callout>}}^2+3B{D}^2\left(3BC+\text{2}AD\right)/{N^{\prime }}^2$

-3τ / n ⁴ 0 4 5 0 β $E_{}^{}$${}_2^5$

${\overline{b}}_{4.5}$

$+{\overline{c}}_{3.5}+3B^2{D}^3/{N^{\prime }}^2$

^{3τ2 /} n6 0 4 0 1 β E ₃ ${\overline{b}}_{4.01}$

$+{\overline{c}}_{3.01}$

0 0 4 1 1 β E2 E3 _{_ _} ${\overline{b}}_{\mathrm{4:11}}$

$-{\overline{c}}_{\mathrm{3:11}}$

0 0 4 2 1 β $E_{}^{}$${}_2^2{E}_3$

${\overline{b}}_{\mathrm{4:21}}$

$+{\overline{c}}_{\mathrm{3:21}}$

0 0 4 0 2 β $E_3^2$

${\overline{b}}_{4.02}$

$-{\overline{c}}_{3.02}=3B$

3τ/ n ² 0 4 0 01 β E ₄ ${\overline{b}}_{4.001}$

1 1 1 5 0 0 β ² 1 ${\overline{b}}_{5.0}$

$-{\overline{c}}_{4.0}-22A^2{C}^2{T}_{AC}^{\left(2.0\right)}\text{/}n{\text{'}}^2$

0 $\left(n\text{'}-n\right)\left(\begin{array}{l}{\overline{\mathrm{e.g}}}^{\left(5\right)}-40n\times \\ \left(n\text{'}-n\right)\text{/}n{\text{'}}^2{\overline{\mathrm{e.g}}}^{{\left(2\right)}^2}{\overline{\mathrm{e.g}}}^{\left(3\right)}\end{array}\right)$

5 1 0 β ² E ₂ ${\overline{b}}_{5.1}$

$\begin{array}{l}+{\overline{c}}_{4.1}+22AC(2\left(AD+BC\right)T_{AC}^{\left(2.0\right)}+AC\left(2T_{AC}^{\left(1.1\right)}+T_{BD}^{\left(2.0\right)}\right)\\ \text{/}n{\text{'}}^2\end{array}$

0 $\begin{array}{l}10\left(n\text{'}-n\right)\left(3n\text{'}-\mathrm{8th}n\right)\\ \text{/}n{\text{'}}^2{\overline{\mathrm{e.g}}}^{\left(2\right)}{\overline{\mathrm{e.g}}}^{\left(3\right)}\end{array}$

5 2 0 β ² $E_2^2$

${\overline{b}}_{5.2}$

$\begin{array}{l}-{\overline{c}}_{4.2}-\\ 22\left(\begin{array}{l}\left(A^2{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">D</figure-callout>}}^2+6ABCD+B^2{C}^2\right)T_{AC}^{\left(2.0\right)}+AC\times \\ \left(\begin{array}{l}4\left(AD+BC\right)T_{AC}^{\left(1.1\right)}+AC\left(T_{AC}^{\left(0.2\right)}+2T_{BD}^{\left(1.1\right)}\right)\\ -BC{T}_{BD}^{\left(2.0\right)}-2D{f}_x^{\left(2.0\right)}\end{array}\right)\end{array}\right)\text{/}n{\text{'}}^2\end{array}$

0 $10\left(n\text{'}-n\right)\left(n\text{'}+4n\right)\text{/}\left(n{n^{\prime }}^2\right){\overline{\mathrm{e.g}}}^{\left(3\right)}$

5 3 0 β ² $E_2^3$

${\overline{b}}_{5.3}$

$\begin{array}{l}-{\overline{c}}_{4.3}-\\ 22\left(\begin{array}{l}2AB{D}^2{T}_{AC}^{\left(2.0\right)}+AC\times \\ \left(2BC\left(T_{AC}^{\left(0.2\right)}+2T_{BD}^{\left(1.1\right)}\right)+A\left(2D{T}_{AC}^{\left(0.2\right)}+C{T}_{BD}^{\left(0.2\right)}\right)\right)\\ +\left(A^2{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">D</figure-callout>}}^2+6ABCD+B^2{C}^2\right)\left(2T_{AC}^{\left(1.1\right)}+T_{BD}^{\left(2.0\right)}\right)\end{array}\right)\\ \text{/}n{\text{'}}^2\\ +6A\left(2B{T}_{AC}^{\left(2.0\right)}-2A{T}_{AC}^{\left(1.1\right)}+3f_x^{\left(2.0\right)}\right)\text{/}{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">n</figure-callout>}}^2\end{array}$

0 0 5 4 0 β ² $E_2^4$

${\overline{b}}_{5.4}$

$-{\overline{c}}_{4.4}-$

0 0 $+6\left(\begin{array}{l}4B^2{T}_{AC}^{\left(2.0\right)}-2A^2\left(T_{AC}^{\left(0.2\right)}+T_{BD}^{\left(1.1\right)}\right)\\ +6A{f}_x^{\left(1.1\right)}+B{f}_x^{\left(2.0\right)}\end{array}\right)\text{/}{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">n</figure-callout>}}^2$

5 5 0 β ² $E_2^5$

${\overline{b}}_{5.5}$

$\begin{array}{l}+{\overline{c}}_{4.5}+\\ 22\left(\begin{array}{l}\left(A^2{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">D</figure-callout>}}^2+6ABCD+B^2{C}^2\right)T_{BD}^{\left(0.2\right)}+BD\times \\ \left(\begin{array}{l}BD\left(6T_{AC}^{\left(1.1\right)}+T_{BD}^{\left(2.0\right)}\right)+2AD{T}_{AC}^{\left(0.2\right)}+4BC{T}_{BD}^{\left(1.1\right)}\\ +2\left(C{f}_x^{\left(0.2\right)}-2DB{f}_x^{\left(1.1\right)}\right)\end{array}\right)\end{array}\right)\\ +6\left(\begin{array}{l}4A^2{T}_{BD}^{\left(0.2\right)}-2B^2\left(T_{AC}^{\left(1.1\right)}+T_{BD}^{\left(2.0\right)}\right)\\ -A{f}_x^{\left(0.2\right)}-6B{f}_x^{\left(1.1\right)}\end{array}\right)\text{/}{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">n</figure-callout>}}^2\end{array}$

0 0 5 6 0 β ² $E_2^6$

${\overline{b}}_{5.6}$

$\begin{array}{l}-{\overline{c}}_{4.6}-\\ 22BD\left(B\left(3D{T}_{AC}^{\left(0.2\right)}+2D{T}_{BD}^{\left(1.1\right)}+2C{T}_{BD}^{\left(0.2\right)}\right)-2D{f}_x^{\left(0.2\right)}\right)\\ +6B\left(2B{T}_{BD}^{\left(1.1\right)}-2A{T}_{BD}^{\left(0.2\right)}\right)+3f_x^{\left(0.2\right)})\text{/}{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">n</figure-callout>}}^2\end{array}$

0 0 5 7 0 β ² $E_2^7$

${\overline{b}}_{5.7}$

$+{\overline{c}}_{4.7}+22B^2{D}^2{T}_{BD}^{\left(0.2\right)}\text{/}n{\text{'}}^2$

0 0 5 0 1 β ² E ₃ ${\overline{b}}_{5.01}$

$+{\overline{c}}_{\mathrm{4:11}}+22A^2{\mathrm{<figure-callout\; id="2"\; label="C"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">C</figure-callout>}}^2/{n^{\prime }}^2$

0 $\begin{array}{l}10\left(n\text{'}-n\right)\left(n\text{'}-4n\right)\\ \text{/}\left(nn{\text{'}}^2\right){\overline{\mathrm{e.g}}}^{{\left(2\right)}^2}\end{array}$

5 1 1 β ² E2 E3 _{_ _} ${\overline{b}}_{\mathrm{5:11}}$

$-{\overline{c}}_{\mathrm{4:11}}-\text{44}AC\left(AD+BC\right)/{n^{\prime }}^2$

0 $\begin{array}{l}10\left(n\text{'}-n\right)\left(3n\text{'}+\mathrm{8th}n\right)\\ \text{/}\left(nn{\text{'}}^2\right){\overline{\mathrm{e.g}}}^{(2)}\end{array}$

5 2 1 β ² $E_2^2{E}_3$

${\overline{b}}_{5.21}$

$+{\overline{c}}_{\mathrm{4:21}}+22\left(\left(A^2{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">D</figure-callout>}}^2+\text{4}ABCD+B^2{C}^2\right)/{n^{\prime }}^2-A^2/n^2\right)$

0 $-40\left(n\text{'}-n\right)\left(n\text{'}+n\right)\text{/}\left(n^{2}n{\text{'}}^2\right)$

5 3 1 β ² $E_2^3{E}_3$

${\overline{b}}_{5.31}$

$-{\overline{c}}_{\mathrm{4:31}}+2B\left(22D\left(AD+BC\right)/{n^{\prime }}^2+\text{7}A/n^2\right)$

0 0 5 4 1 β ² $E_2^4{E}_3$

${\overline{b}}_{5.41}$

$+{\overline{c}}_{4.41}+2B^2\left(11{\mathrm{<figure-callout\; id="2"\; label="D"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">D</figure-callout>}}^2/{n^{\prime }}^2+\text{4}/n^2\right)$

0 0 5 0 2 β ² $E_3^2$

${\overline{b}}_{5.02}$

$-{\overline{c}}_{4.02}$

0 0 5 1 2 β ² $E_2{E}_3^2$

${\overline{b}}_{5.12}$

$+{\overline{c}}_{\mathrm{4:12}}$

0 0 5 2 2 β ² $E_2^2{E}_3^2$

2 3 ${\overline{b}}_{5.22}$

$-{\overline{c}}_{4.22}$

0 0 5 0 01 β E ₄ ${\overline{b}}_{5.001}$

$+{\overline{c}}_{4.001}$

0 0 5 1 01 β E2 E4 _{_ _} ${\overline{b}}_{5.101}$

$-{\overline{c}}_{4.101}$

0 0 5 2 01 β $E_2^2{E}_4$

${\overline{b}}_{5.201}$

$-{\overline{c}}_{4.201}$

0 0 5 0 3 β ² $E_3^3$

${\overline{b}}_{5.03}$

15 B ^{2 15τ2 /} n4 0 5 0 11 β _E3 E4 _ _{_} ${\overline{b}}_{5.011}$

-10B -10τ / n ² 0 5 0 001 1 E ₅ ${\overline{b}}_{5.0001}$

1 1 1 6 0 0 β ³ 1 ${\overline{b}}_{6.0}$

... 0 $\begin{array}{l}\left(n\text{'}-n\right)\times \\ \left(\begin{array}{c}{\overline{\mathrm{e.g}}}^{\left(6\right)}+(...){\overline{\mathrm{e.g}}}^{{\left(2\right)}^5}+\\ (...){\overline{\mathrm{e.g}}}^{{\left(2\right)}^2}{\overline{\mathrm{e.g}}}^{\left(4\right)}+(...){\overline{\mathrm{e.g}}}^{\left(2\right)}{\overline{\mathrm{e.g}}}^{{\left(2\right)}^3}\end{array}\right)\end{array}$

⋮ ⋮ ⋮ ⋮ ⋮ ⋮ ⋮ 6 0 0 1 1 ${\overline{b}}_{6.00001}$

1 1 1 7 0 0 β ⁴ 1 ${\overline{b}}_{7.0}$

0 ...

example

A simple example is the passage of light through a meridian of a Bennett and Rabbetts eye model, ie a modified Gullstrand-Emsley eye (mGE eye) adapted for biometric studies (see R. Rabetts: “Bennett &Rabbetts' Clinical Visual Optics”, Butterworth Heinemann Elsevier Health Sciences, 2007, ISBN: 9780750688741). The eyeball contains a cornea with refractive power S _C and an eye lens with refractive powers L ₁ and L ₂ on the anterior and posterior surfaces, respectively. The radii of curvature of the cornea and the two lens surfaces are r _C , r ₁ and r ₂ , respectively. The distance between the cornea and the anterior surface of the lens is given by the anterior chamber depth d _CL , the lens thickness is given by d _L , and the vitreous body depth by d _LR . The refractive index of the aqueous humor is n _CL , that of the eye lens n _L , and that of the vitreous body n _LR . According to the state of the art, the following values are customary for the biometric parameters: $$\begin{array}{l}{\mathrm{i.e}}_{CL}=3.6mm,\\ {\mathrm{i.e}}_L=3.7mm,\\ {\mathrm{i.e}}_{LR}=16.7859082mm,\\ {\mathrm{right}}_C=7.8mm,\\ {\mathrm{right}}_1=11.0mm,\\ {\mathrm{right}}_2=-6.47515mm,\\ n_{CL}=\mathrm{1,336}\\ n_L=\mathrm{1,422}\\ n_{LR}=\mathrm{1,336}\\ {\overline{\mathrm{e.g}}}_{L1}^{\left(2\right)}=1/{\mathrm{right}}_C=128.2051282m^{-1},S_C=\left(n_{CL}-1\right){\overline{\mathrm{e.g}}}_C^{\left(2\right)}=43.076923\mathrm{i.e}pt\\ {\overline{\mathrm{e.g}}}_{L1}^{\left(2\right)}=1/{\mathrm{right}}_1=90.9090909m^{-1},{\mathrm{<figure-callout\; id="1"\; label="L"\; filenames="DE102020008145A1\_0383.png,DE102020008145A1\_0384.png"\; state="\{\{state\}\}">L</figure-callout>}}_1=\left(n_L-n_{CL}\right){\overline{\mathrm{e.g}}}_{L1}^{\left(2\right)}=7.81818181\mathrm{i.e}pt\\ {\overline{\mathrm{e.g}}}_{L2}^{\left(2\right)}=1/{\mathrm{right}}_2=-154.436576m^{-1},{\mathrm{<figure-callout\; id="2"\; label="L"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">L</figure-callout>}}_2=\left(n_{LR}-n_L\right){\overline{\mathrm{e.g}}}_{L2}^{\left(2\right)}=13.28154560\mathrm{i.e}pt\end{array}$$

In eq. (39) none of the values with more than three decimal places come from a direct measurement, but from consistency considerations, with which the model is exactly emmetropic, i.e. a plane wave front must be mapped onto a spherical wave that converges exactly on the retina .

Since both the Gullstrand-Emsley eye and the Bennett-Rabbetts eye are paraxial according to the prior art and are therefore only specified up to second-order wavefront aberrations, these models are not directly suitable as exemplary embodiments for calculating higher-order aberrations , as long as no suitable specification for the HOA is added here. For the purpose of the present embodiment, the characteristic of the emmetropic eye should also be retained for HOA. The two lens surfaces, including the HOA, are preferably exact spherical surfaces, whereas the HOA of the cornea must then be selected in such a way that the eye meets the above emmetropia requirement.

wohingegen die geraden Ableitungen durch die Krümmung ${\overline{z}}^{\left(2\right)}$

eindeutig bestimmt sind in Form von $${\overline{z}}^{\left(4\right)}=3{\left({\overline{z}}^{\left(2\right)}\right)}^3,{\overline{z}}^{\left(6\right)}=45{\left({\overline{z}}^{\left(2\right)}\right)}^5$$

For every spherical surface, the odd derivatives of the function vanish $\overline{\mathrm{e.g}}\left(x\right),$

whereas the straight derivatives through the curvature ${\overline{\mathrm{e.g}}}^{\left(2\right)}$

are uniquely determined in the form of $${\overline{\mathrm{e.g}}}^{\left(4\right)}=3{\left({\overline{\mathrm{e.g}}}^{\left(2\right)}\right)}^3,{\overline{\mathrm{e.g}}}^{\left(6\right)}=45{\left({\overline{\mathrm{e.g}}}^{\left(2\right)}\right)}^5$$

Numerically, this means for four additional biometric parameters $$\begin{array}{l}{\overline{\mathrm{e.g}}}_{L1}^{\left(4\right)}=2.253944\times {10}^6{m}^{-3},{}_{\overline{\text{e.g}}}^{L1}=2.794145\times {10}^{11}{m}^{-5}\\ {\overline{\mathrm{e.g}}}_{L2}^{\left(4\right)}=1.105024\times {10}^7{m}^{-3},{\overline{\mathrm{e.g}}}_{L2}^{\left(6\right)}=-3.953332\times {10}^{12}{m}^{-5}\end{array}$$

der Cornea so zu wählen, dass die Wellenfront, die nach der Brechung an der Linsen-Rückfläche entsteht, genau auf die Retina konvergiert, also genau eine Kugelwelle mit Radius d_LR ist. Das Kriterium ist, dass eine einfallende ebene Welle (w^(k) = 0) auf die ausfallenden Ableitungen $${w^{\prime }}^{\left(4\right)}=3{\left({w^{\prime }}^{\left(2\right)}\right)}^3,{w^{\prime }}^{\left(6\right)}=45{\left({w^{\prime }}^{\left(2\right)}\right)}^5$$

führt, analog zu GI.(40). Eine besonders bevorzugte Alternative zu Ableitungen w'^(p) als Beschreibung von Wellenfronten sind Aberrationen E'_p = n'w'^(p), also im Fall der Kugelwelle $$\begin{array}{l}E{\text{'}}_4/n\text{'}=3{\left(E{\text{'}}_2/n\text{'}\right)}^3,E{\text{'}}_6/n\text{'}=45{\left(E{\text{'}}_2/n\text{'}\right)}^5\\ E{\text{'}}_4=3n_{LR}^{-2}E{\text{'}}_2^3,E{\text{'}}_6=45n_{LR}^{-4}E{\text{'}}_2^5\end{array}$$

wobei n' = n_LR. Andererseits ergeben sich die Aberrationen E'₂, E'₄, E'₆ durch Gl.(38) und GI. (36) in Abhängigkeit des optischen Systems, und eine einfallende ebene Welle bedeutet, dass die Aberrationen E_p = 0 für alle Ordnungen p sind. Für jede Ordnung p trägt also nur der niedrigste Koeffizient ${\overline{b}}_{pk}$

für k = 0 bei: $$E{\text{'}}_2=\beta {\overline{b}}_{\mathrm{2,0}},E{\text{'}}_4={\beta }^4\beta {\overline{b}}_{\mathrm{4,0}},E{\text{'}}_6={\beta }^6{\beta }^3{\overline{b}}_{\mathrm{6,0}}$$

wobei ${\overline{b}}_{\mathrm{2,0}}=-C$

aus Tabelle 5 ist, und wobei weiter ${\overline{b}}_{\mathrm{4,0}}$

eine Funktion des zu ermittelnden Parameters ${\overline{z}}_C^{\left(4\right)}$

ist, und wobei ${\overline{b}}_{\mathrm{4,0}}$

eine Funktion der beiden zu ermittelnden Parameter ${\overline{z}}_C^{\left(4\right)},{\overline{z}}_C^{\left(6\right)}$

ist. Durch Einsetzen von GI.(44) in GI.(43) ergibt sich die Forderung $${\overline{b}}_{\mathrm{4,0}}=3{\left(\beta n_{LR}\right)}^{-2}{\overline{b}}_{\mathrm{2,0}}^3,{\overline{b}}_{\mathrm{6,0}}=45{\left(\beta n_{LR}\right)}^{-4}{\overline{b}}_{\mathrm{2,0}}^5$$

The method according to the invention can now be used to order the higher orders ${\overline{\mathrm{e.g}}}_C^{\left(4\right)},{\overline{\mathrm{e.g}}}_C^{\left(6\right)}$

of the cornea in such a way that the wave front, which occurs after the refraction on the rear surface of the lens, converges exactly on the retina, i.e. it is exactly a spherical wave with radius d _LR . The criterion is that an incident plane wave (w ^(k) = 0) impacts the outgoing derivatives $${w^{\prime }}^{\left(4\right)}=3{\left({w^{\prime }}^{\left(2\right)}\right)}^3,{w^{\prime }}^{\left(6\right)}=45{\left({w^{\prime }}^{\left(2\right)}\right)}^5$$

leads, analogous to eq.(40). A particularly preferred alternative to derivations w' ^(p) as a description of wavefronts are aberrations E'p=n'w' ^(p) _, ie in the case of the spherical wave $$\begin{array}{l}E{\text{'}}_4/n\text{'}=3{\left(E{\text{'}}_2/n\text{'}\right)}^3,E{\text{'}}_6/n\text{'}=45{\left(E{\text{'}}_2/n\text{'}\right)}^5\\ E{\text{'}}_4=3n_{LR}^{-2}E{\text{'}}_2^3,E{\text{'}}_6=45n_{LR}^{-4}E{\text{'}}_2^5\end{array}$$

where n' = n _LR . On the other hand, the aberrations E' ₂ , E' ₄ , E' ₆ result from Eq. (38) and Eq. (36) depending on the optical system, and an incident plane wave means that the aberrations are E _p = 0 for all orders p. For each order p, only the lowest coefficient carries ${\overline{b}}_{pk}$

for k = 0 at: $$E{\text{'}}_2=\beta {\overline{b}}_{2.0},E{\text{'}}_4={\mathrm{<figure-callout\; id="4"\; label="\beta "\; filenames="DE102020008145A1\_0383.png"\; state="\{\{state\}\}">\beta </figure-callout>}}^4\beta {\overline{b}}_{4.0},E{\text{'}}_6={\beta }^6{\beta }^3{\overline{b}}_{6.0}$$

whereby ${\overline{b}}_{2.0}=-C$

from Table 5, and where further ${\overline{b}}_{4.0}$

a function of the parameter to be determined ${\overline{\mathrm{e.g}}}_C^{\left(4\right)}$

is, and where ${\overline{b}}_{4.0}$

a function of the two parameters to be determined ${\overline{\mathrm{e.g}}}_C^{\left(4\right)},{\overline{\mathrm{e.g}}}_C^{\left(6\right)}$

is. Inserting eq.(44) into eq.(43) results in the requirement $${\overline{b}}_{4.0}=3{\left(\beta n_{LR}\right)}^{-2}{\overline{b}}_{2.0}^3,{\overline{b}}_{6.0}=45{\left(\beta n_{LR}\right)}^{-4}{\overline{b}}_{2.0}^5$$

wobei sich β für E₂ = 0 nach GI.(37) auf β = A^-1 reduziert, und die Werte für A und C mit Hilfe von GI.(49) ermittelt werden.Substituting all known numerical values in Table 5 leads to the coefficients $$\begin{array}{l}{\overline{b}}_{4.0}\left({\overline{\mathrm{e.g}}}_2^{\left(4\right)}\right)=2.11526\times {10}^5{m}^{-3}+0.253296{\overline{\mathrm{e.g}}}_C^{\left(4\right)}\\ {\overline{b}}_{6.0}\left({\overline{\mathrm{e.g}}}_C^{\left(4\right)},{\overline{\mathrm{e.g}}}_C^{\left(6\right)}\right)=2.204136\times {10}^{11}{m}^{-5}-3.36879\times {10}^4{m}^{-2}{\overline{\mathrm{e.g}}}_C^{\left(4\right)}+3.36306m{\left({\overline{\mathrm{e.g}}}_C^{\left(4\right)}\right)}^2+0.143949{\overline{\mathrm{e.g}}}_C^{\left(6\right)}\end{array}$$

where β for E ₂ = 0 is reduced to β = A ^-1 according to eq.(37), and the values for A and C are determined using eq.(49).

gelöst, so dass zusammen mit GI.(41) und Gl. (39) alle biometrischen Parameters 6ter Ordnung des mGE-Auges ermittelt sind. Die Art, wie GI.(47) zustande kommt, zeigt die technischen Vorteile der Erfindung auf. Auch wenn dasselbe Ergebnis auch durch wiederholte Anwendung des aus dem Stand der Technik bekannten Verfahrens erhalten werden kann, ist der Weg über die Koeffizienten ${\overline{b}}_{\mathrm{4,0}},{\overline{b}}_{\mathrm{6,0}}$

direkter und eignet sich, die Abhängigkeit von den Parametern ${\overline{z}}_C^{\left(4\right)},{\overline{z}}_C^{\left(6\right)}$

darzustellen, während der Einfluss aller anderen Brechungen und Propagationen zu diesem Moment schon zusammengefasst und vor-ausgewertet ist.Equations (46) are given by $${\overline{\mathrm{e.g}}}_C^{\left(4\right)}=-2.05536\times {10}^4{m}^{-3},{\overline{\mathrm{e.g}}}_C^{\left(6\right)}=-1.51137\times {10}^{12}{m}^{-5}$$

solved, so that together with Eq.(41) and Eq. (39) all biometric parameters of the 6th order of the mGE eye are determined. The way eq.(47) comes about shows the technical advantages of the invention. Even if the same result can also be obtained by repeated application of the method known from the prior art, the way is via the coefficients ${\overline{b}}_{4.0},{\overline{b}}_{6.0}$

more direct and suitable, the dependence on the parameters ${\overline{\mathrm{e.g}}}_C^{\left(4\right)},{\overline{\mathrm{e.g}}}_C^{\left(6\right)}$

to represent, while the influence of all other refractions and propagations is already summarized and pre-evaluated at this moment.

$f_t^{C,re{f}^{\left(n_x,n_t\right)}}$

$f_x^{L\mathrm{1,}re{f}^{\left(n_x,n_t\right)}}$

$f_t^{L\mathrm{1,}re{f}^{\left(n_x,n_t\right)}}$

$f_x^{L\mathrm{2,}re{f}^{\left(n_x,n_t\right)}}$

$f_t^{L\mathrm{2,}re{f}^{\left(n_x,n_t\right)}}$

2 1 0 1 -43.0769 dpt 1 -7.81818 dpt 1 -13.2815 dpt 2 0 1 0 1 0 1 0 1 4 3 0 12401.2 m^-2 4.067615*10⁵ m^-3 1499.46 m^-2 -1.82825*10⁵ m^-3 -4605.89 m^-2 -1.01543*10⁶ m^-3 4 2 1 32.2432 m^-1 -10671.29 m^-2 4.11528 m^-1 -1440.95 m^-2 6.99105 m^-1 4809.53 m^-2 4 1 2 0 -104.645 m^-1 0 -15.7145 m^-1 0 -29.3143 m^-1 4 0 3 0 -1.31923 0 -0.197152 0 0.197152 6 5 0 -3.93649*10⁸ m^-4 4.80357*10¹¹ m^-5 1.78842*10⁸ m^-4 -2.14040*10¹⁰ m^-5 -1.72307*10⁹ m^-4 -3.88671*10¹¹ m^-5 6 4 1 1.24986*10⁷ m^-3 6.19886*10⁸ m^-4 1.06288*10⁶ m^-3 -1.70212*10⁸ m^-4 5.83689*10⁶ m^-3 1.82661*10⁹ m^-4 6 3 2 5.49284*10⁴ m^-2 -8.19587*10⁶ m^-3 4694.07 m^-2 -1.25104*10⁶ m^-3 -14721.4 m^-2 -8.08185*10⁶ m^-3 6 2 3 126.595 m^-1 -5.87327*10⁴ m^-2 12.604 m^-1 -8231.94 m^-2 22.7901 m^-1 33476.3 m^-2 6 1 4 0 98.8023 m^-1 0 -36.5481 m^-1 0 -117.301 m^-1 6 0 5 0 8.70184 0 0.194345 0 0.194345 Tabelle 7: Ableitungen der Strahl-Transfer-Funktionen f^e,prop, f^dCL,prop, t^dL,prop der Propagationen der Komponenten des Auges Ordnung Hornhautscheitelabstand e Vorderkammertiefe d _CL Linsendicke d _L p n _x n _t $f_x^{e,pro{p}^{\left(n_x,n_t\right)}}$

$f_t^{e,pro{p}^{\left(n_x,n_t\right)}}$

$f_x^{dCL,pro{p}^{\left(n_x,n_t\right)}}$

$f_t^{dCL,pro{p}^{\left(n_x,n_t\right)}}$

$f_x^{dL,pro{p}^{\left(n_x,n_t\right)}}$

$f_t^{dL,pro{p}^{\left(n_x,n_t\right)}}$

2 1 0 1 0 1 0 1 0 2 0 1 0.013 m 1 2.69461* 10^-3 m 1 2.60197*10^-3 m 1 Alle höheren Ordnungen verschwinden Based on the parameter values, all derivatives of the involved beam transfer functions can be combined up to order p = 6. In the following Table 6 all derivatives of the ray transfer functions of the refractive components of the mGE eye are shown, in Table 7 those for the propagations, and in Table 8 those of the whole mGE eye. The derivatives of the ray transfer functions f ^C,ref , f ^L1,ref , f ^L2,ref of the refractions can be determined from Table 3 by substituting those of the ray transfer functions f ^e,prop , f ^dCL,prop , f ^{dL, prop} of the propagations given Table 1. Since the mGE eye is symmetric, these derivatives vanish for odd orders p (i.e. for even values of the sum n _x + n _t of the orders of the derivatives), so Tables 6, 7 and 8 only have entries for even orders. Table 6: Derivatives of the ray transfer functions f ^C,ref , f ^L1,ref , f ^L2,ref of the refractions of the components of the eye Order Cornea S _C Front surface lens L ₁ Back surface lens L ₂ p nx _{_} n.t _{_} $f_x^{C,\mathrm{right}e{f}^{\left(n_x,n_t\right)}}$

$f_t^{C,\mathrm{right}e{f}^{\left(n_x,n_t\right)}}$

${\mathrm{<figure-callout\; id="1"\; label="f\; x\; L"\; filenames="DE102020008145A1\_0383.png,DE102020008145A1\_0384.png"\; state="\{\{state\}\}">f</figure-callout>}}_{xL}^{\mathrm{1,}\mathrm{right}e{f}^{\left(n_x,n_t\right)}}$

${\mathrm{<figure-callout\; id="1"\; label="f\; t\; L"\; filenames="DE102020008145A1\_0383.png,DE102020008145A1\_0384.png"\; state="\{\{state\}\}">f</figure-callout>}}_{tL}^{\mathrm{1,}\mathrm{right}e{f}^{\left(n_x,n_t\right)}}$

${\mathrm{<figure-callout\; id="2"\; label="f\; x\; L"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">f</figure-callout>}}_{xL}^{\mathrm{2,}\mathrm{right}e{f}^{\left(n_x,n_t\right)}}$

${\mathrm{<figure-callout\; id="2"\; label="f\; t\; L"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">f</figure-callout>}}_{tL}^{\mathrm{2,}\mathrm{right}e{f}^{\left(n_x,n_t\right)}}$

2 1 0 1 -43.0769D 1 -7.81818 dpt 1 -13.2815 dpt 2 0 1 0 1 0 1 0 1 4 3 0 12401.2 m ^-2 4.067615* ^105m - ^{3 1499.46m} -2 -1.82825* ^105m - ³ -4605.89 m ^-2 -1.01543*10 ^6m - ³ 4 2 1 ^32.2432m -1 -10671.29 m ^{-2 4.11528m} -1 -1440.95 m ^{-2 6.99105m} -1 ^4809.53m -2 4 1 2 0 ^-104,645m -1 0 -15.7145m ^-1 0 -29.3143m ^-1 4 0 3 0 -1.31923 0 -0.197152 0 0.197152 6 5 0 -3.93649* ^108m - ^{4 4.80357} *10 ^11m -5 1.78842* ^108m - ^{4 -2.14040} *10 ^10m -5 ^-1.72307 *10 ^9m -4 ^-3.88671 *10 ^11m -5 6 4 1 1.24986* ^107m - ³ 6.19886* ^108m - ⁴ 1.06288*10 ^6m - ³ -1.70212* ^108m - ⁴ 5.83689*10 ^6m - ^{3 1.82661} *109m- ⁴ 6 3 2 5.49284* ^104m - ² -8.19587* ^106m - ^{3 4694.07m} -2 -1.25104*10 ^6m - ³ -14721.4 m ^-2 -8.08185* ^106m - ³ 6 2 3 ^126,595m -1 -5.87327* ^104m - ^{2 12,604m} -1 ^-8231.94m -2 ^22.7901m -1 33476.3 m ^-2 6 1 4 0 98.8023 m ^-1 0 ^-36.5481m -1 0 ^-117,301m -1 6 0 5 0 8.70184 0 0.194345 0 0.194345 Table 7: Derivatives of the ray transfer functions f ^e,prop , f ^dCL,prop , t ^dL,prop of the propagations of the components of the eye Order corneal vertex distance e Anterior chamber depth d _CL Lens thickness d _L p nx _{_} n.t _{_} $f_x^{e,p\mathrm{right}O{p}^{\left(n_x,n_t\right)}}$

$f_t^{e,p\mathrm{right}O{p}^{\left(n_x,n_t\right)}}$

$f_x^{\mathrm{i.e}CL,p\mathrm{right}O{p}^{\left(n_x,n_t\right)}}$

$f_t^{\mathrm{i.e}CL,p\mathrm{right}O{p}^{\left(n_x,n_t\right)}}$

$f_x^{\mathrm{i.e}L,p\mathrm{right}O{p}^{\left(n_x,n_t\right)}}$

$f_t^{\mathrm{i.e}L,p\mathrm{right}O{p}^{\left(n_x,n_t\right)}}$

2 1 0 1 0 1 0 1 0 2 0 1 0.013m 1 2.69461* 10 ^-3 m 1 2.60197*10 ^-3m 1 All higher orders disappear

definiert ist, können alle Ableitungen von f^mGE bis zur Ordnung p = 6 mit Hilfe der Kettenregel aus den Ableitungen der komponentenweisen Strahl-Transferfunktionen bestimmt werden. Tabelle 8: Ableitungen der Strahl-Transfer-Funktionen f^mGE des modifizierten Gullstrand-Emsley-Auges Ordnung mod. Gullstrand-Emsley-Auge mGE p n _x n _t $f_x^{mGE\left(n_x,n_t\right)}$

$f_t^{mGE\left(n_x,n_t\right)}$

2 1 0 0.753858 -60.00001 dpt 2 0 1 5.24176*10^-3 m 0.909314 4 3 0 12221.93 m^-2 -9.72751*10⁵ m^-3 4 2 1 -32.6185 m^-1 -10883.5 m^-2 4 1 2 -0.595797 -110.553 m^-1 4 0 3 -7.82984*10^-3 m -1.56287 6 5 0 7.24284*10⁸ m^-4 -5.76462*10¹⁰ m^-5 6 4 1 1.20722*10⁷ m^-3 -3.549999*10⁸ m^-4 6 3 2 10955.5*10⁴ m^-2 -8.559098*10⁶ m^-3 6 2 3 -88.57369 m^-1 -4.62591*10⁴ m^-2 6 1 4 1.407511 271.8149 m^-1 6 0 5 0.056790 m 11.548181 The derivatives of the component-wise beam transfer functions f ^C,ref , f ^L1,ref , f ^L2,ref , f ^e,prop f ^dCL,prop , f ^dL,prop inside the mGE eye are the starting point to calculate the total beam transfer function f to determine ^mGE of the entire mGE eye. Since f ^mGE as concatenation $$f^{mGE}\left(\rho \right)=f^{L\mathrm{2,}\mathrm{right}ef}\left(f^{\mathrm{i.e}L,p\mathrm{right}Op}\left(f^{L\mathrm{1,}\mathrm{right}ef}\left(f^{\mathrm{i.e}CL,p\mathrm{right}Op}\left(f^{C,\mathrm{right}ef}\left(\rho \right)\right)\right)\right)\right),$$

is defined, all derivatives of f ^mGE up to order p = 6 can be determined using the chain rule from the derivatives of the component-wise beam transfer functions. Table 8: Derivatives of the ray transfer functions f ^mGE of the modified Gullstrand-Emsley eye Order model Gullstrand-Emsley eye mGE p nx _{_} n.t _{_} $f_x^{mGE\left(n_x,n_t\right)}$

$f_t^{mGE\left(n_x,n_t\right)}$

2 1 0 0.753858 -60.00001 dpt 2 0 1 5.24176*10 ^-3m 0.909314 4 3 0 12221.93 m ^-2 -9.72751* ^105m - ³ 4 2 1 ^-32.6185m -1 -10883.5 m ^-2 4 1 2 -0.595797 ^-110,553m -1 4 0 3 -7.82984*10 ^-3 m -1.56287 6 5 0 7.24284* ^108m - ⁴ -5.76462*10 ^{10m -5} 6 4 1 ^1.20722 *10 ^7m -3 -3.549999* ^108m - ⁴ 6 3 2 10955.5* ^104m - ² -8.559098*10 ^6m - ³ 6 2 3 -88.57369 m ^{-1 -4.62591} *10 ^4m -2 6 1 4 1.407511 ^271.8149m -1 6 0 5 0.056790 m 11.548181

des mGE-Auges bestimmt werden, die zur Auswertung der Wellenfrontdurchrechnung in GI.(38),(36) erforderlich sind.The prefactor β and the wavefront transfer coefficients can now be derived from the derivatives of the beam transfer function $${\overline{b}}_{pk}$$

of the mGE eye, which are required for the evaluation of the wavefront calculation in Eq.(38),(36).

According to Eq.(4) the following applies: $$T=\left(\begin{array}{cc}A& B\\ C& D\end{array}\right)=\left(\begin{array}{cc}0.753858& 0.00524176m\\ -60.00001\mathrm{i.e}pt& 0.909314\end{array}\right)$$

In addition, the prefactor β according to eq.(26) is specific for the incident wavefront t ⁽¹⁾ = -w ⁽²⁾ = -nE ₂ = E ₂ , where eq.(8) and the definition E _p = nw ^(p) can be exploited and the fact that the refractive index in the space in front of the eye is given by n=1. The prefactor β can then be determined from E ₂ according to Eq.(37) with the help of Eq.(49).

kann schließlich durch Einsetzen der numerischen Werte der Ableitungen der Strahl-Transferfunktion aus Tabelle 8 in Tabelle 5 durchgeführt werden. Das Ergebnis ist in der folgenden Tabelle 9 dargestellt. Tabelle 9: Wellenfront-Transfer-Koeffizienten ${\overline{b}}_{pk}$

für das modifizierte Gullstrand-Emsley-Auge Indizes Wellenfront-Ordnung p k* k ₁ 2 3 4 5 6 (0) 0 60.0000 dpt 0 2.06320*10⁵ m^-3 0 3.54733*10⁹ m^-5 1 0.909314 0 -28692.2 m^-2 0 2.16457*10⁹ m^-4 2 0 597.737 m^-1 0 -7.52551*10⁶ m^-3 3 0 -6.69794 0 -1.03962*10⁶ m^-2 4 2.26513*10^-2m 0 23021.1 m^-1 5 2.91006*10^-5m² 0 -237.259 6 0 1 .70495 m 7 0 -0.009055 m² 8 2.21606*10^-5 m³ 9 2.38306*10^-9 m⁴ (1) 0 1 0 -46287.1 m^-2 0 1 0 655.057 m^-1 0 2 0 -10.7069 0 3 -0.0343949 m 0 4 6.44793*10^-4 m² 0 5 0 6 0 (2) 0 0.0157253 m 0 -2547.464 m^-1 1 0 16.42902 2 0 -0.607000 m 3 0.00119945 m² 4 -98305.4 m⁴ (0,1) 0 1 0 1216.102 m⁵ 1 0 -17.7499 m⁶ 2 0 -0.0983983 m⁷ 3 0.00136332 m⁸ 4 -98305.4 m⁴ (3) 0 4.12141*10^-4 m² 0 1 0 2 0 (1,1) 0 0.0524176 m 0 1 0 2 0 (0,0,1) 0 1 0 1 0 2 0 (4) 0 1.51224*10^-5 m³ (2,1) 0 0.00288499 m² (0,2) 0 0.0524176 m (1,0,1) 0 0.0786265 m (0,0,0,1) 0 1 The determination of the wavefront transfer coefficient ${\overline{b}}_{pk}$

can finally be performed by substituting in Table 5 the numerical values of the derivatives of the ray transfer function from Table 8. The result is shown in Table 9 below. Table 9: Wavefront Transfer Coefficients ${\overline{b}}_{pk}$

for the modified Gullstrand-Emsley eye indices Wavefront Order p k * k ₁ 2 3 4 5 6 (0) 0 60,0000 dpt 0 2.06320* ^105m - ³ 0 3.54733* ^109m - ⁵ 1 0.909314 0 -28692.2 m ^-2 0 ^2.16457 *109m- ⁴ 2 0 ^597,737m -1 0 -7.52551*10 ^6m - ³ 3 0 -6.69794 0 -1.03962* ^106m - ² 4 2.26513*10 ^-2m 0 ^23021.1m -1 5 ^2.91006 * ^10-5m2 0 -237,259 6 0 1 .70495 m 7 0 ^-0.009055m2 8th ^2.21606 * ^10-5m3 9 2.38306*10 ^{-9m 4} (1) 0 1 0 -46287.1 m ^-2 0 1 0 ^655,057m -1 0 2 0 -10.7069 0 3 -0.0343949 m 0 4 6.44793* ^{10 -4m2} 0 5 0 6 0 (2) 0 0.0157253 m 0 ^-2547.464m -1 1 0 16.42902 2 0 -0.607000m 3 ^0.00119945m2 4 -98305.4 m ⁴ (0.1) 0 1 0 1216.102m ⁵ 1 0 -17.7499m ⁶ 2 0 -0.0983983m ⁷ 3 0.00136332m ⁸ 4 -98305.4 m ⁴ (3) 0 ^4.12141 * ^10-4m2 0 1 0 2 0 (1.1) 0 0.0524176 m 0 1 0 2 0 (0,0,1) 0 1 0 1 0 2 0 (4) 0 ^1.51224 * ^10-5m3 (2.1) 0 ^0.00288499m2 (0.2) 0 0.0524176 m (1,0,1) 0 0.0786265 m (0,0,0,1) 0 1

The actual goal, namely the repeated calculation of different wavefronts by the mGE eye, can now be achieved on the basis of the values from Eq. (49) and from Table 9.

The results for E' _p are shown for orders p ≤ 6 in Table 10 below. The first column shows the results for a plane incident wavefront, which is mapped onto a spherical wave after the construction of the emmetropic mGE eye. The second and third columns refer to an incident wavefront at a distance of 40 cm, with the third column corresponding to an exact spherical wave and the second column to the second-order approximation to this spherical wave. Finally, the fourth column shows an arbitrarily chosen wavefront whose HOA is only chosen as an example. Table 10: Aberrations E' ₂ , E' ₃ , E' ₄ , E' ₅ , E' ₆ of the emerging wavefront after the wavefront calculation of different wavefronts with aberrations E ₂ , E ₃ , E ₄ , E ₅ , E ₆ through the modified Gullstrand-Emsley eye aberrations and prefactor plane wavefront Wavefront from a distance of 40 cm arbitrary wavefront with HOA 2nd order wavefront (parabola) ball wave input E ₂ 0 -2.5 dpt -2.5 dpt -2.5 dpt E ₃ 0 0 0 -1.0* ^103m - ² E ₄ 0 0 -46,875m ^-3 7.0* ^{105m- 3} E ₅ 0 0 0 -2.0* ^108m - ⁴ E ₆ 0 0 -4.39453*10 ^{3m -5} 5.0*10 ^{10m -5} output β 1.32651 1.26348 1.26348 1.26348 E'2 _{_} 79.5906 dpt 75.4005 dpt 75.4005 dpt 75.4005 dpt E'3 _{_} 0 0 0 -1.50974* ^103m - ² E'4 _{_} 8.47409* ^105m - ³ 1.05563*10 ^6m - ³ 1.05551*10 ^6m - ³ 2.98485* ^106m - ³ E'5 _{_} 0 0 0 -8.09790* ^108m - ⁴ E'6 _{_} ^4.51123 *10 ^10m -5 - ^1.88462 *10 ^10m -5 - ^1.88111 *10 ^10m -5 ^3.31012 *10 ^11m -5

The method according to the invention can preferably be used to optimize spectacle lenses, in which case the actual biometric parameters of the individual eye are to be used instead of the mGE eye, and the input wavefront for the calculation is preferably a second-order wavefront, which is from the to be optimized glasses come from. The spectacle lens is to be determined in such a way that the output wave front comes as close as possible in relation to a metric of the spherical wave, which converges on the retina.

In particular, the invention relates to the following points:

• - Aufstellen zumindest einer Wellenfront-Transferfunktion für das optische System, wobei die Wellenfront-Transferfunktion dafür bestimmt ist, in das optische System einfallenden Wellenfronten unter Berücksichtigung von Abbildungsfehlern mit einer Ordnung größer als die Ordnung eines Defocus jeweils eine zugehörige ausfallende Wellenfront zuzuordnen; und
• - Auswerten der zumindest einen Wellenfront-Transferfunktion für zumindest eine in das optische System einfallende Wellenfront.

Item 1: Method for simulating an optical system by means of a wavefront calculation, the optical system being a complex optical system whose effect goes beyond a single refraction, a single propagation or a single reflection, comprising the steps:

• - Setting up at least one wavefront transfer function for the optical system, the wavefront transfer function being intended to assign an associated emerging wavefront to wavefronts incident on the optical system, taking into account aberrations with an order greater than the order of a defocus; and
• - Evaluation of the at least one wavefront transfer function for at least one wavefront incident on the optical system.

Point 2: Method according to point 1, wherein the method is a method for optimizing an overall optical system, wherein the optical system represents a second subsystem of the overall optical system and the overall optical system additionally comprises a first subsystem, in particular the first subsystem and/or the second subsystem can be varied in the course of optimization.

Point 3: Method according to point 2, the first sub-system being a spectacle lens and the second sub-system being a model eye (10).

Point 4: Procedure according to point 2 or 3, where
the at least one wavefront incident on the optical system is determined on the basis of a predetermined test wavefront which passes through the first subsystem.

• - Bewerten des optischen Gesamtsystems auf Basis des Ergebnisses des Auswertens der zumindest einen Wellenfront-Transferfunktion für die zumindest eine in das optische System einfallende Wellenfront, wobei das optische Gesamtsystem unter Variation des ersten Teilsystems so lange bewertet wird, bis die Bewertung eine vorgegebene Bedingung erfüllt, wobei

die Variation des ersten Teilsystems insbesondere eine Änderung zumindest einer brechenden Fläche und/oder zumindest eines Abstandes zwischen brechenden Flächen des ersten Teilsystems, und/oder ein Verkippen und/oder Verschieben des ersten Teilsystems gegenüber dem zweiten Teilsystem umfasst.Point 5: Method according to one of points 2 to 4, further comprising the step:

• - Evaluation of the overall optical system based on the result of the evaluation of the at least one wavefront transfer function for the at least one wavefront incident on the optical system, the overall optical system being evaluated while varying the first subsystem until the evaluation fulfills a predetermined condition, whereby

the variation of the first sub-system comprises in particular a change in at least one refracting surface and/or at least one distance between refracting surfaces of the first sub-system, and/or a tilting and/or shifting of the first sub-system in relation to the second sub-system.

Point 6: Procedure according to point 5, where
the overall optical system is evaluated on the basis of the result of the evaluation of the at least one wavefront transfer function for a first wavefront incident on the optical system and also on the basis of the result of the evaluation of the at least one wavefront transfer function for a second incident wavefront, with the first subsystem is in a first position and orientation to the second subsystem when the first wavefront is incident, the first subsystem being in a second position and orientation to the second subsystem when the second wavefront is incident, and the first position is different from the second position and/or differs the first orientation from the second orientation.

Point 7: Method according to one of points 5 to 6, wherein
the overall optical system is evaluated on the basis of the result of the evaluation of a first wavefront transfer function for a first wavefront incident on the optical system and also on the basis of the result of the evaluation of a second wavefront transfer function for a further incident wavefront, the first subsystem is in a first position and orientation to the second subsystem when the first wavefront is incident, the first subsystem being in a second position and orientation to the second subsystem when the other wavefront is incident, the first position being different from the second position and/or the first orientation differs from the second orientation, and wherein the second wavefront transfer function differs from the first wavefront transfer function.

Point 8: Method according to one of points 2 to 7, wherein
the first sub-system is a spectacle lens and the second sub-system is a model eye, and where visual movements of the model eye that cause a change in the position of the point where the main ray penetrates the spectacle lens surfaces and/or a change in the angle of incidence on a spectacle lens surface are considered a change in position and/ or the alignment of the spectacle lens in the coordinate system of the eye.

Item 9: Method according to one of the preceding items, wherein the optical system is a GRIN system or comprises at least one GRIN element.

aufweist und wobei die Indizes des Tupels k = (k₁,k₂,...,k_p-1) über den Bereich P(k*) ≤ p - 2 und 0 ≤ k₁ ≤ 2(p - P(k*) - 2) + δ_P(k*),0 laufen, wobei $P\left(k*\right)={\displaystyle {\sum }_{j=1}^{p-2}j{k}_{j+1},}$

und wobei $-{\overline{r}}_1^0\left(p-1\right)=p-{\delta }_{\left(p-1\right)\mathrm{,1}}$

und $-\Delta {\overline{r}}_1\left(p-\mathrm{1,}k*\right)=\left(p-3\right)+{\delta }_{\left(p-1\right)\mathrm{,1}}-P\left(k**\right)$

gelten, und wobei β = (~BE₂ + A)^-1 als Funktion der zumindest einen einfallenden Wellenfront und des optischen Systems gegeben ist, und wobei A,B und die Wellenfront-Transfer-Koeffizienten ${\overline{b}}_{pk}$

als Funktion der Komponenten des optischen Systems gegeben sind.Item 10: Method according to one of the preceding items, wherein the at least one wavefront transfer function has the form $$E{\text{'}}_p={\beta }^{-{\overline{\mathrm{right}}}_1^0\left(p-1\right)}{\displaystyle \sum {}_{{\mathrm{<figure-callout\; id="1"\; label="k"\; filenames="DE102020008145A1\_0383.png,DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_1,{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2,...k_{p-1}}}{\overline{b}}_{pk}{\beta }^{-\Delta {\overline{\mathrm{right}}}_1\left(p-\mathrm{1,}k*\right)}{E}_2^{k_1}{E}_3^{{\mathrm{<figure-callout\; id="2"\; label="k"\; filenames="DE102020008145A1\_0384.png"\; state="\{\{state\}\}">k</figure-callout>}}_2}\cdots E_p^{k_{p-1}},p=\mathrm{2,3,4,}...$$

and where the indices of the tuple k = (k ₁ ,k ₂ ,...,k _p-1 ) over the range P(k*) ≤ p - 2 and 0 ≤ k ₁ ≤ 2(p - P(k *) - 2) + δ _P(k*),0 , where $P\left(k*\right)={\displaystyle {\sum }_{j=1}^{p-2}j{k}_{j+1},}$

and where $-{\overline{\mathrm{right}}}_1^0\left(p-1\right)=p-{\delta }_{\left(p-1\right)\mathrm{,1}}$

and $-\Delta {\overline{\mathrm{right}}}_1\left(p-\mathrm{1,}k*\right)=\left(p-3\right)+{\delta }_{\left(p-1\right)\mathrm{,1}}-P\left(k**\right)$

apply, and where β = (~BE ₂ + A) ^-1 is given as a function of the at least one incident wavefront and the optical system, and where A,B and the wavefront transfer coefficients ${\overline{b}}_{pk}$

are given as a function of the components of the optical system.

Point 11: Method according to one of the preceding points, in which both the incident and the emerging wavefronts are each represented by coefficients for the basis elements of a basis system, the basis elements of which are classified according to at least one order parameter, and the at least one wavefront transfer function is given by that it assigns the respective emerging wavefront to the wavefronts incident in the optical system in such a way that for a basic element represented in the representation of an emerging wavefront, it assigns the coefficients for this basic element represented in the representation of the emerging wavefront as a function of coefficients of the associated incident wavefront a plurality of those base elements are determined whose value of the order parameter is less than or equal to the value of the order parameter of the respective base element represented in the representation of the emerging wave front .

Point 12: Method according to point 11, wherein the basic system is a decomposition according to aberrations, the order parameter being an order p of the aberrations, the coefficients associated with the basic elements of the basic system being given by a p-th order coefficient of the incident wavefront is an aberration E _p of the incident wavefront and that a p-th order coefficient of the associated outgoing wavefront is an aberration E' _p of the associated outgoing wavefront, and where p ≥ 2.

Item 13: Method according to item 11 or 12, wherein the order parameter is a first order parameter, and wherein the basic elements of the basic system are additionally classified according to at least one second order parameter, the value range of which depends on the value of the first order parameter.

Item 14: Method according to any one of items 11 to 13, wherein the basis system is a decomposition according to Taylor derivatives of wavefront arrow heights, the order parameter being an order p of the Taylor derivatives of the wavefront arrow heights, where the to the basis elements of the basis system associated coefficients are given in that a p-th order coefficient of the incident wavefront is a Taylor derivative w ^(p) of the sagittal height of the incident wavefront and that the p-th order coefficient of the associated outgoing wavefront is a Taylor derivative w' ^{(p )} is the versine of the associated emerging wavefront, and where p ≥ 2.

Point 15: Method according to one of points 11 to 13, wherein the basic system is a decomposition according to Taylor derivatives of a wavefront OPD, wherein the order parameter is an order p of the Taylor derivatives of the wavefront OPD, where the to the basic elements of the basic system associated coefficients are given in that a p-th order coefficient of the incident wavefront is a Taylor derivative OPD ^(p) of the incident wavefront and that the p-th order coefficient of the associated outgoing wavefront is a Taylor derivative OPD' ^(p) of associated outgoing wavefront, and where p ≥ 2.

Point 16: Method according to one of points 11 to 13, wherein the basic system is a decomposition according to derivatives of directional functions, the order parameter being an order i of the derivatives of the directional functions, the coefficients associated with the basic elements of the basic system being given in that a ith order coefficient of the incident wavefront is a derivative t ⁽ⁱ⁾ of a directional function t(x) of the incident wavefront and that an ith order coefficient of the associated outgoing wavefront is a derivative t' ⁽ⁱ⁾ of a directional function t'(x' ) of the associated outgoing wavefront, and where i ≥ 1.

Point 17: Method according to one of points 11 to 13, the basic system being a decomposition according to Zernike polynomials, the order parameter being a radial order n of the Zernike polynomials, the coefficients associated with the basic elements of the basic system being given in that an nth order coefficient of the incident wavefront is a Zernike coefficient Z _n of the incident wavefront and that an nth order coefficient of the outgoing wavefront is a Zernike coefficient Z' _n of the associated outgoing wavefront, where n ≥ 2, and in particular, the Zernike coefficients relate to a fixed pupil.

Item 18: Method according to Item 13, wherein the basic system is a decomposition according to aberrations, the first order parameter being the sum p of orders p _x and p _y of the aberrations, the second order parameter being one of the orders p _x and p _y , where the coefficients associated with the basis elements of the basis system are given in that p-th order coefficients of the incident wavefront are aberrations E _px,py of the incident wavefront and that p-th order coefficients of the associated emerging wavefront are aberrations E' _px,py of the associated emerging are wavefront, and where p ≥ 2 and p _x ≥ 0 and p _y ≥ 0.

Item 19: Method according to item 13, wherein the basic system is a decomposition according to Taylor derivatives of wavefront arrow heights, the first order parameter being the sum p of orders p _x and p _y of the Taylor derivatives of the wavefront arrow heights, the second order parameter is one of the orders p _x and p _y , where the coefficients associated with the basis elements of the basis system are given by the p-th order coefficients of the incident wavefront being Taylor derivatives w ^(px,py) of the versine of the incident wavefront and that the p-th order coefficients of the associated emergent wavefront are Taylor derivatives w' ^(px,py) of the versine of the associated emergent wavefront, and where p ≥ 2 and p _x ≥ 0 and p _y ≥ 0.

Point 20: Method according to point 13, wherein the basic system is a decomposition according to Taylor derivatives of a wavefront OPD, the first order parameter being the sum p of orders p _x and p _y of the Taylor derivatives of the wavefront OPD, the second order parameter is one of the orders p _x and p _y , where the coefficients associated with the basis elements of the basis system are given in that p-th order coefficients of the incident wavefront are Taylor derivatives OPD ^(px,py) of the incident wavefront and that the coefficients p-th order of the associated emergent wavefront are Taylor derivatives OPD' ^(px,py) of the associated emergent wavefront, and where p ≥ 2 and p _x ≥ 0 and p _y ≥ 0.

Point 21: Method according to point 13, wherein the basic system is a decomposition according to derivatives of directional functions, the first order parameter being the sum i of orders i _x and i _y of the derivatives of the directional functions, the second order parameter being one of the orders i _x and i _y , where the coefficients associated with the basis elements of the basis system are given by the fact that coefficients of the ith order of the incident wave front are derivatives t _x ^(ix,iy) , t _y ^(ix,iy) of directional functions t _x (x,y) , t _y (x,y) of the incident wavefront and that i-th order coefficients of the associated emerging wavefront are derivatives t' _x ^(ix,iy) , t' _y ^(ix,iy) of the directional functions t' _x (x', y'), t' _y (x',y') of the associated emergent wavefront, and where i ≥ 1 and i _x ≥ 0 and i _y ≥ 0.

Point 22: Method according to point 13, wherein the basic system is a decomposition according to Zernike polynomials, the first order parameter being a radial order n of the Zernike polynomials, the second order parameter being an azimuthal order m of the Zernike polynomials, the to The coefficients associated with the basis elements of the basis system are given by the fact that nth-order coefficients of the incident wavefront are Zernike coefficients Z _n ^m of the incident wavefront and that nth-order coefficients of the associated emerging wavefront are Zernike coefficients Z` _n ^m of the associated emerging are wavefront, where n ≥ 2 and -n ≤ m ≤ n, where m is even for even n and odd for n odd, and in particular where the Zernike coefficients relate to a fixed pupil.

Item 23: Computer program product comprising computer-readable instructions which, when loaded into a memory of a computer and executed by the computer, cause the computer to perform a method according to any one of items 1 to 22.

• - ein Modellierungsmodul zum Bereitstellen zumindest einer Wellenfront-Transferfunktion für das optische System, wobei die Wellenfront-Transferfunktion dafür bestimmt ist, jeder in das optische System einfallenden Wellenfront unter Berücksichtigung von Abbildungsfehlern mit einer Ordnung größer als die Ordnung eines Defocus eine zugehörige ausfallende Wellenfront zuzuordnen;
• - ein Auswertemodul zum Auswerten der zumindest einen Wellenfront-Transferfunktion für zumindest eine in das optische System einfallende Wellenfront.

Item 24: Device for simulating an optical system by means of a wavefront calculation, the optical system being a complex optical system whose effect goes beyond a single refraction, a single propagation or a single reflection, comprising:

• - a modeling module for providing at least one wavefront transfer function for the optical system, wherein the wavefront transfer function is intended to assign an associated emerging wavefront to each wavefront incident in the optical system, taking into account aberrations with an order greater than the order of a defocus;
• - An evaluation module for evaluating the at least one wavefront transfer function for at least one wavefront incident on the optical system.

• - Berechnen oder Optimieren eines Brillenglases unter Verwendung eines Verfahrens nach einem der Punkte 1 bis 22; und
• - Bereitstellen von Fertigungsdaten des so berechneten oder optimierten Brillenglases, und/oder Fertigen des so berechneten oder optimierten Brillenglases.

Item 25: Method for manufacturing a spectacle lens comprising:

• - Calculating or optimizing a spectacle lens using a method according to one of points 1 to 22; and
• - Providing production data of the spectacle lens calculated or optimized in this way, and/or manufacturing the spectacle lens calculated or optimized in this way.

• Berechnungs- oder Optimierungsmittel, welche ausgelegt sind, das Brillenglas unter Verwendung eines Verfahrens nach einem der Punkte 1 bis 22 zu berechnen oder zu optimieren; und
• Bearbeitungsmittel, welche ausgelegt sind, das Brillenglas fertig zu bearbeiten.

Item 26: Device for manufacturing a spectacle lens comprising:

• Calculation or optimization means, which are designed to calculate or optimize the spectacle lens using a method according to one of points 1 to 22; and
• Processing means designed to finish the spectacle lens.

Point 27: Use of a spectacle lens manufactured according to the manufacturing method according to point 25 in a predetermined average or individual usage position of the spectacle lens in front of the eyes of a specific spectacle wearer to correct a defective vision of the spectacle wearer.

QUOTES INCLUDED IN DESCRIPTION

This list of documents cited by the applicant was generated automatically and is included solely for the better information of the reader. The list is not part of the German patent or utility model application. The DPMA assumes no liability for any errors or omissions.

Patent Literature Cited

• DE 10313275 [0006]
• WO 2008/089999 A1 [0009, 0014, 0021, 0022]
• DE 102017007975 A1 [0013]
• WO 2018/138140 A2 [0013]
• DE 102012000390 A1 [0014, 0023]
• US 9910294 B2 [0014, 0023]
• DE 102011101923 A1 [0014, 0022, 0023, 0100, 0101]
• DE 102017007975 B4 [0023]
• WO 2008089999 A1 [0100, 0101]

Claims (24)

Computer-implemented method for simulating an optical system by means of a wavefront calculation, the optical system being a complex optical system whose effect goes beyond a single refraction, a single propagation or a single reflection, comprising the steps: - Computer-implemented setting up of at least one wavefront transfer function for the optical system, the wavefront transfer function being intended to assign an associated emerging wavefront to wavefronts incident on the optical system, taking into account aberrations with an order greater than the order of a defocus; and - Computer-implemented evaluation of the at least one wavefront transfer function for at least one wavefront incident on the optical system. procedure after claim 1 , wherein the method is a method for optimizing an overall optical system, wherein the optical system represents a second sub-system of the overall optical system and the overall optical system additionally comprises a first sub-system, with the first sub-system and/or the second sub-system in particular varying in the course of the optimization can be. procedure after claim 2 , wherein the first subsystem is a spectacle lens and the second subsystem is a model eye (10). procedure after claim 2 or 3 , wherein the at least one wavefront incident on the optical system is determined on the basis of a predetermined test wavefront, which passes through the first subsystem. Procedure according to one of claims 2 until 4 , further comprising the step of: - evaluating the overall optical system based on the result of the evaluation of the at least one wavefront transfer function for the at least one wavefront incident on the optical system, wherein the overall optical system is evaluated while varying the first subsystem until the Assessment satisfies a predetermined condition, with the variation of the first subsystem including, in particular, a change in at least one refracting surface and/or at least one distance between refracting surfaces of the first subsystem, and/or a tilting and/or shifting of the first subsystem relative to the second subsystem. procedure after claim 5 , wherein the overall optical system is evaluated on the basis of the result of the evaluation of the at least one wavefront transfer function for a first incident wavefront in the optical system and also on the basis of the result of the evaluation of the at least one wavefront transfer function for a second incident wavefront, wherein the first subsystem is in a first position and orientation to the second subsystem when the first wavefront is incident, wherein the first subsystem is in a second position and orientation to the second subsystem when the second wavefront is incident, and wherein the first position is different from the second position and/or the first orientation differs from the second orientation. Procedure according to one of Claims 5 until 6 , wherein the overall optical system is evaluated on the basis of the result of the evaluation of a first wavefront transfer function for a first wavefront incident on the optical system and also on the basis of the result of the evaluation of a second wavefront transfer function for a further incident wavefront, with the first subsystem is in a first position and orientation to the second subsystem when the first wavefront is incident, the first subsystem being in a second position and orientation to the second subsystem when the other wavefront is incident, the first position being different from the second position and /or the first orientation differs from the second orientation, and wherein the second wavefront transfer function differs from the first wavefront transfer function. Procedure according to one of claims 2 until 7 , wherein the first sub-system is a spectacle lens and the second sub-system is a model eye, and wherein eye movements of the model eye, which cause a change in the position of the penetration point of the main ray through the spectacle lens surfaces and/or a change in the angle of incidence on a spectacle lens surface, as a change in position and/or the alignment of the spectacle lens in the coordinate system of the eye. Method according to one of the preceding claims, wherein the optical system is a GRIN system or comprises at least one GRIN element. A method according to any one of the preceding claims, wherein the at least one wavefront transfer function is the form $$E{\text{'}}_p={\beta }^{-{\overline{\mathrm{right}}}_1^0\left(p-1\right)}{\displaystyle \sum {}_{k_1,k_2,...k_{p-1}}}{\overline{b}}_{pk}{\beta }^{-\Delta {\overline{\mathrm{right}}}_1\left(p-\mathrm{1,}k*\right)}{E}_2^{k_1}{E}_3^{k_2}\cdots E_p^{k_{p-1}},p=\mathrm{2,3,4,}...$$

and where the indices of the tuple k = (k ₁ ,k ₂ ,...,k _p-1 ) over the range P(k*) ≤ p - 2 and 0 ≤ k ₁ ≤ 2(p - P(k *) - 2) + δ _P(k*),0 , where $P\left(k*\right)={\displaystyle {\sum }_{j=1}^{p-2}j{k}_{j+1},}$

and where $-{\overline{\mathrm{right}}}_1^0\left(p-1\right)=p-{\delta }_{\left(p-1\right)\mathrm{,1}}$

and $-\Delta {\overline{\mathrm{right}}}_1\left(p-\mathrm{1,}k*\right)=\left(p-3\right)+{\delta }_{\left(p-1\right)\mathrm{,1}}-P\left(k**\right)$

apply, and where β = (-BF ₂ + A) ^-1 is given as a function of the at least one incident wavefront and the optical system, and where A, B and the wavefront transfer coefficients $${\overline{b}}_{pk}$$

are given as a function of the components of the optical system. Method according to one of the preceding claims, wherein both the incident and the emerging wavefronts are each represented by coefficients for the basis elements of a basis system, the basis elements of which are classified according to at least one order parameter, and the at least one wavefront transfer function is given in that they assigns the respective emerging wavefront to the wavefronts incident in the optical system in such a way that, for a basic element represented in the representation of an emerging wavefront, it converts the coefficients for this basic element represented in the representation of the emerging wavefront as a function of coefficients of the associated incident wavefront to a plurality of those Basic elements are determined whose value of the order parameter is less than or equal to the value of the order parameter of the respective basic element represented in the representation of the emerging wavefront. procedure after claim 11 , where the basis system is a decomposition according to aberrations, where the order parameter is an order p of the aberrations, the coefficients associated with the basis elements of the basis system being given in that a p-th order coefficient of the incident wavefront has an aberration E _p of the incident wavefront and that a p-th order coefficient of the associated outgoing wavefront is an aberration E' _p of the associated outgoing wavefront, and where p ≥ 2. procedure after claim 11 or 12 , where the order parameter is a first order parameter, and where the basis elements of the basis system are additionally classified according to at least one second order parameter, the value range of which depends on the value of the first order parameter. Procedure according to one of Claims 11 until 13 , where the basis system is a decomposition according to Taylor derivatives of wavefront arrow heights, where the order parameter is an order p of the Taylor derivatives of the wavefront arrow heights, the coefficients associated with the basis elements of the basis system being given by a coefficient p- th order of the incident wavefront is a Taylor derivative w ^(p) of the versine of the incident wavefront and that the pth order coefficient of the associated outgoing wavefront is a Taylor derivative w' ^(p) of the versine of the associated outgoing wavefront, and where p ≥ 2. Procedure according to one of Claims 11 until 13 , where the basis system is a decomposition according to Taylor derivatives of a wavefront OPD, with the order parameter being an order p of the Taylor derivatives of the wavefront OPD, with the coefficients associated with the basis elements of the basis system being given in that a coefficient p- th order of the incident wavefront is a Taylor derivative OPD ^(p) of the incident wavefront and that the pth order coefficient of the associated outgoing wavefront is a Taylor derivative OPD' ^(p) of the associated outgoing wavefront, and where p ≥ 2 . Procedure according to one of Claims 11 until 13 , where the basis system is a decomposition into derivatives of directional functions, where the order parameter is an order i of the derivatives of the is directional functions, where the coefficients associated with the basis elements of the basis system are given in that an i-th order coefficient of the incident wavefront is a derivative t ⁽ⁱ⁾ of a directional function t(x) of the incident wavefront and that an i-th order coefficient of the associated emerging wavefront is a derivative t' ⁽ⁱ⁾ of a directional function t`(x`) of the associated emerging wavefront, and where i ≥ 1. Procedure according to one of Claims 11 until 13 , where the basis system is a decomposition according to Zernike polynomials, where the order parameter is a radial order n of the Zernike polynomials, the coefficients associated with the basis elements of the basis system being given by an nth-order coefficient of the incident wavefront being a Zernike - Coefficient Z _n of the incident wavefront and that an nth order coefficient of the outgoing wavefront is a Zernike coefficient Z' _n of the associated outgoing wavefront, where n ≥ 2, and the Zernike coefficients relate in particular to a fixed pupil relate. procedure after Claim 13 , where the basis system is a decomposition according to aberrations, where the first order parameter is the sum p of orders p _x and p _y of the aberrations, where the second order parameter is one of the orders p _x and p _y , where the associated to the basis elements of the basis system coefficients are given in that p-th order coefficients of the incident wavefront are aberrations E _px,py of the incident wavefront and that p-th order coefficients of the associated emergent wavefront are aberrations E' _px,py of the associated emergent wavefront, and where p ≥ 2 and p _x ≥ 0 and p _y ≥ 0. procedure after Claim 13 , where the basis system is a decomposition into Taylor derivatives of wavefront arrow heights, where the first order parameter is the sum p of orders p _x and p _y of the Taylor derivatives of the wavefront arrow heights, where the second order parameter is one of the orders p _x and p _y , where the coefficients associated with the basis elements of the basis system are given by the p-th order coefficients of the incident wavefront being Taylor derivatives w ^(px,py) of the versine of the incident wavefront and the p-th order coefficients of the associated emergent wavefront are Taylor derivatives w' ^(px,py) of the versine of the associated emergent wavefront, and where p ≥ 2 and p _x ≥ 0 and p _y ≥ 0. procedure after Claim 13 , where the basis system is a Taylor derivative decomposition of a wavefront OPD, where the first order parameter is the sum p of orders p _x and p _y of the Taylor derivatives of the wavefront OPD, where the second order parameter is one of the orders p _x and p _y , where the coefficients associated with the basis elements of the basis system are given in that p-th order coefficients of the incident wavefront are Taylor derivatives OPD ^(px,py) of the incident wavefront and that the p-th order coefficients of the associated exiting Wavefront are Taylor derivatives OPD' ^(px,py) of the associated emergent wavefront, and where p ≥ 2 and p _x ≥ 0 and p _y ≥ 0. procedure after Claim 13 , where the basis system is a decomposition by derivatives of directional functions, where the first order parameter is the sum i of orders i _x and i _y of the derivatives of the directional functions, where the second order parameter is one of the orders i _x and i _y , where the to the Coefficients associated with basis elements of the basis system are given by the fact that coefficients of the i-th order of the incident wavefront are derivatives t _x ^(ix,iy) , t _y ^(ix,iy) of directional functions t _x (x,y), t _y (x,y ) of the incident wavefront and that i-th order coefficients of the associated emerging wavefront are derivatives t' _x ^(ix,iy) , t' _y ^(ix,iy) of the directional functions t' _x (x',y'), t' _y (x',y') of the associated outgoing wavefront, and where i ≥ 1 and i _x ≥ 0 and i _y ≥ 0. procedure after Claim 13 , where the basis system is a decomposition according to Zernike polynomials, where the first order parameter is a radial order n of the Zernike polynomials, where the second order parameter is an azimuthal order m of the Zernike polynomials, the coefficients associated with the basis elements of the basis system thereby are given that nth-order coefficients of the incident wavefront are Zernike coefficients Z _n ^m of the incident wavefront and that nth-order coefficients of the associated emergent wavefront are Zernike coefficients Z' _n ^m of the associated emergent wavefront, where n ≥ 2 and -n ≤ m ≤ n, where m is even for even n and odd for odd n, and in particular where the Zernike coefficients relate to a fixed pupil. A computer program product comprising computer-readable instructions which, when loaded into a memory of a computer and executed by the computer, cause the computer to carry out a method according to any one of Claims 1 until 22 performs. Device for simulating an optical system by means of a wavefront calculation, the optical system being a complex optical system whose effect goes beyond a single refraction, a single propagation or a single reflection, comprising: - A modeling module for the computer-implemented provision of at least one wavefront transfer function for the optical system, the wavefront transfer function being intended to assign an associated emerging wavefront to each incident wavefront in the optical system, taking into account aberrations with an order greater than the order of a defocus ; - An evaluation module for the computer-implemented evaluation of the at least one wavefront transfer function for at least one wavefront incident on the optical system.

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