DE102020215540B4 - Method for determining an imaging quality of an imaging system, device and projection exposure system - Google Patents

Abstract

A method for determining an imaging quality of an imaging system (2) is described. The imaging quality of the imaging system (2) is characterized by its effect on a wave front. Directional derivatives of the wavefront are determined in partial measurements. However, the wavefront is incorrectly reconstructed from its directional derivatives if it changes unnoticed between partial acquisitions. The error is avoided if measurements are taken in non-orthogonal directions or if the vector field is broken down into a gradient field and a rotator field.

Description

field of invention

The invention relates to a method for determining an imaging quality of an imaging system.

The invention also relates to a device for determining an imaging quality of an imaging system.

State of the art

In many areas of technology and research, optical imaging systems are used, which are subject to increasing demands in terms of their imaging quality. One example is the microlithographic production of semiconductor components and other finely structured components, in which structures in the submicron range can be produced with the aid of optical imaging systems in the form of projection lenses. Such imaging systems have a complex optical structure with a large number of optical elements, which generally makes it impossible to derive the real optical properties from theoretical calculations. Therefore, the optical properties of imaging systems must be reliably determined or measured.

For this purpose, interferometric measurement methods are often used. The patent specification describes a device for wavefront detection that works in the manner of a shearing interferometer, which enables fast, highly precise measurement of high-resolution photolithographic projection lenses EP 1 257 882 B1 described. In this device, a mask is arranged in an object plane of an optical imaging system to be measured. The mask comprises a rigid, transparent structural support on which a two-dimensional object pattern is applied. The mask is illuminated with incoherent radiation for the measurement. A reference pattern designed as a diffraction grating is arranged in an image plane of the optical imaging system. A coherence of the radiation running through the imaging system is set by the object pattern. A superimposition of the waves generated by diffraction at the diffraction grating results in a superimposition pattern in the form of an interferogram, which is recorded with the aid of a suitable, in particular spatially resolving, detector and then evaluated. In order to be able to calculate a two-dimensional phase distribution from the interferograms, several interferograms with different phase positions are detected. According to a suggestion from EP 1 257 882 B1 the diffraction grating preferably has a diffractive periodic structure for different directions. As a result, phase gradients or wavefront gradients in more than one direction can be determined from a single interferogram recorded by the spatially resolving detector. If these directions are orthogonal to each other, such as in a checkerboard or cross-grating diffraction grating, then, in terms of lateral shear interferometer theory, the shear caused by the diffraction grating will be simultaneous in a first direction, e.g. the x-direction, and in a direction orthogonal thereto second direction, for example in the y-direction. The wavefront is then reconstructed from the wavefront gradients generated in the x and y directions.

In addition to the type of device described above for determining an imaging quality of an optical imaging system, which works on the principle of shearing interferometry, a second type is often used, which is based on the principle of pupil division according to Shack-Hartmann. In the patent US 5,978,085A describes such a method, based on the Shack-Hartmann principle, for analyzing an optical imaging system, in particular an imaging lens system, by measuring wavefront aberrations. In this method, a mask or a reticle with a structure made up of a number of small openings is introduced into an object plane of the optical imaging system. An aperture stop with at least one opening is positioned at a suitable distance from this. The reticle is imaged through the aperture stop onto an image plane of the imaging system, in which a plurality of light spots are created. In one embodiment there, a structure of the light spots is recorded by means of a wafer coated with photoresist. By comparing the structures on the wafer with a reference plate that is superimposed on it and exposed to reference structures, the shifts in the measured center of gravity positions of the light spots compared to the ideal, diffraction-limited ones are determined. A wave front gradient in the pupil of the lens system to be measured and from this the aberration of the wave front are determined from these shifts.

One challenge is to ensure a reliable determination of the imaging quality of the imaging system when the wavefront changes during a determination of the imaging quality or when the wavefront or, in particular, a focus of the imaging system drifts during the determination of the imaging quality. If, for example, in the shear interferometer described above, the wavefront gradient in the first and the second direction orthogonal thereto is determined in chronological succession, as a result of the determination of the imaging quality, an astigmatic wavefront error in a wavefront reconstructed as a function of the first and the second wavefront gradient is simulated if the focus drifted or ran away.

Other devices for wavefront measurement are for example in WO 2005/069 079 A1 described.

the U.S. 2005/0200940 A1 describes a method for wavefront measurements. In this case, the wavefront gradient is first determined in one direction with the aid of moiré grating structures. The wavefront gradient is then determined in a different direction. A two-dimensional reconstruction of the wave front can be carried out by fitting calculation or integration.

the DE 10 2005 041 373 A1 discloses a method for calibrating wavefront measurement of an optical system using multiple measurement processes for a respective field location, in particular with twisted grating structures.

The aforementioned procedures U.S. 2005/0200940 A1 and the DE 10 2005 041 373 A1 assume that neither the measuring system nor the optics to be tested change over time.

object of the invention

Against the above background, it is an object of the present invention to provide a method and a device with which the aforementioned problems are solved, in particular with which the imaging quality is or can be reliably determined even if the wavefront changes during the determination of the imaging quality or if the focus drifts during the determination of the imaging quality.

This object is solved according to the features of the independent patent claims.

Disclosure of Invention

According to the invention, the method for determining the imaging quality of the imaging system, in particular the optical imaging system, is carried out with the following steps: a) irradiating a wave front into the imaging system; b) detecting the wavefront after the wavefront has passed through the imaging system, the detection of the wavefront comprising a first partial detection of the wavefront to determine a first wavefront gradient in a first direction and at least a second partial detection of the wavefront to determine a second wavefront gradient in a second direction, wherein the wavefront of the first partial acquisition and the wavefront of the second partial acquisition differ from each other, and wherein the first direction and the second direction are not orthogonal to each other; c) reconstructing the wavefront as a function of the first wavefront gradient and the second wavefront gradient. In the present case, “different from one another” means that the wavefront in the first partial acquisition has changed or changed in particular unintentionally and/or unnoticed in comparison to the wavefront in the second partial acquisition. The method according to the invention with the features of claim 1 has the advantage that within the framework of the reconstruction of the wavefront, wavefront errors, in particular astigmatic wavefront errors, in the case of a changing wavefront, in particular during a measurement process, in particular as a function of a changing focus position or a focus drift, cannot be feigned, but can be determined and eliminated. This ensures a particularly reliable determination of the imaging quality of the imaging system, since an erroneous determination of the wavefront and thus of the imaging quality is minimized or ruled out by simulating wavefront errors. The ability to reliably determine the imaging quality is ensured in particular by the fact that the reconstruction is carried out as a function of the wavefront gradients determined in directions that are not aligned orthogonally to one another. In particular, this ensures that a system of equations, on the basis of which the reconstruction takes place, can be solved. The system of equations can be solved in particular by adding at least one variable to the system of equations, which can be solved or eliminated in particular by calculation, and which represents the drift animals. The wavefront is thus correctly reconstructed from the wavefront gradients or directional derivatives, even if it has changed or changed unnoticed between the partial acquisitions.

According to one development, it is provided that the second partial acquisition is carried out after the first partial acquisition. The advantage here is that, in comparison to a simultaneous first and second partial detection, particularly precise methods for detecting the respective wavefront gradients can be used. A time interval between the first and the second partial acquisition can preferably be specified as desired. Alternatively, the first partial detection and the second partial detection occur simultaneously.

According to the invention, the wavefront of the first partial acquisition and the wavefront of the second partial acquisition differ from one another in that a focus of the imaging system drifts between the first partial acquisition and the second partial acquisition. The wavefront thus changes between the first partial acquisition and the second partial acquisition depending on a third of the focus of the imaging system. "Drift" means a change in focus over time, specifically a changing systematic deviation of focus from an earlier value. Such drifting occurs in particular unintentionally and/or unnoticed during operation of the imaging system or during a measurement process, in particular as a function of a change or change in position of a wavefront generation unit operatively connected to the imaging system and/or a change in at least one optical element of the imaging system. Equivalent to this, an input-side focal position of the imaging system or measuring system can also have run away or drifted.

According to the invention, it is provided that measurement errors, which are caused by the in particular unintentional drifting of the focus, are mathematically eliminated by reconstructing the wavefront. The advantage here is that wavefront errors, in particular astigmatic wavefront errors, are eliminated as part of the reconstruction of the wavefront. The reconstruction of the wavefront is therefore particularly accurate. The computational elimination takes place in particular by solving a corresponding system of equations, on the basis of which the reconstruction takes place. "Being mathematically eliminated" thus means that the system of equations, on the basis of which the reconstruction is based, is solved in such a way that at least one variable of the system of equations, which represents astigmatism components or astigmatic wavefront errors, is eliminated or equals zero.

According to a development it is provided that the first partial detection and the second partial detection takes place after an interaction of the wavefront with a diffractive element, the diffractive element having a first grating with a first grating vector and at least one second grating with a second grating vector, the first lattice vector is aligned along the first direction and the second lattice vector is aligned along the second direction. The advantage here is that the first and the second wavefront gradient can be generated by a single element. This ensures that the method can be carried out in a simple and, in particular, time- and cost-efficient manner. The diffractive element is preferably designed as a diffraction grating or as a perforated diaphragm. The diffractive element is preferably arranged in an image plane of the imaging system.

According to a further development it is provided that the first partial detection takes place after an interaction of the wave front with a first diffractive element and the second partial detection takes place after an interaction of the wave front with at least one second diffractive element, the first diffractive element being a first grating with a first grating vector oriented along the first direction, and wherein the second diffractive element comprises a second grating having a second grating vector, the second grating vector being oriented along the second direction.

According to one development, it is provided that the first grid vector is aligned along the first direction and the second grid vector is aligned along the second direction in such a way that the first grid vector and the second grid vector are aligned at an angle of 45° or 135° to one another. In other words: the first lattice vector and the second lattice vector form an angle of 45° or 135°. The advantage here is that wavefront errors are eliminated or resolved particularly efficiently as part of the reconstruction of the wavefront.

According to a further development it is provided that the first and the second wavefront gradient are detected as a function of an interference pattern or a point pattern. A diffraction grating is preferably used to generate the interference pattern, and a pinhole diaphragm is used to generate the point pattern.

According to a further development, it is provided that the detection of the wavefront includes at least a third partial detection of the wavefront to determine a third wavefront gradient in a third direction, with the first direction, the second direction and the third direction each being oriented differently from one another. Optionally, the acquisition of the wavefront includes any number of partial acquisitions for determining any number of wavefront gradients in any number of directions.

According to a further development it is provided that the first partial acquisition, the second partial acquisition and the third partial acquisition takes place after an interaction of the wavefront with a diffractive element, the diffractive element having a first grating with a first grating vector, a second grating with a second grating vector and at least a third lattice having a third lattice vector, the first lattice vector oriented along the first direction, the second lattice vector oriented along the second direction, and the third lattice vector oriented along the third direction. The diffractive element optionally has any number of gratings with any number of grating vectors, with the respective grating vectors each being aligned along different directions.

According to a further development it is provided that the first, second and the third partial acquisition are carried out one after the other.

Furthermore, the invention relates to a device for determining an imaging quality of an imaging system, in particular an optical imaging system, having: a) a radiation source for generating radiation; b) a wavefront generation unit for generating a wavefront from the radiation, the wavefront passing through the imaging system; c) a detector unit for detecting the wavefront after the wavefront has passed through the imaging system, with the detection of the wavefront including a first partial detection of the wavefront to determine a first wavefront gradient in a first direction and at least a second partial detection of the wavefront to determine a second wavefront gradient in a second direction, wherein the wavefront of the first partial acquisition and the wavefront of the second partial acquisition differ from each other, and wherein the first direction and the second direction are not orthogonal to each other; d) a reconstruction unit for reconstructing the wavefront as a function of the first wavefront gradient and the second wavefront gradient, with measurement errors caused by the drifting of the focus being mathematically eliminated by the reconstruction of the wavefront. This results in the advantages already mentioned. Further advantages and preferred features emerge from what has been described above and from the claims.

According to a development of the measuring device, it is provided that the device is designed to carry out the second partial acquisition in time after the first partial acquisition.

According to the invention, the wavefront of the first partial acquisition and the wavefront of the second partial acquisition differ from one another in that a focus of the imaging system drifts between the first partial acquisition and the second partial acquisition.

According to one development, it is provided that the device has a diffractive element, the diffractive element having a first grating with a first grating vector and at least one second grating with a second grating vector, the first grating vector being aligned along the first direction and the second grating vector being aligned is aligned along the second direction.

According to a further development it is provided that the device has a first diffractive element and a second diffractive element, wherein the first diffractive element has a first grating with a first grating vector, which is aligned along the first direction, and by at least one second diffractive element, wherein the second diffractive element has a second grating with a second grating vector aligned along the second direction.

According to a further development it is provided that the device has a control unit which is designed to carry out the method according to one of Claims 1 to 11. This results in the advantages already mentioned. Further advantages and preferred features emerge from what has been described above and from the claims.

Furthermore, the invention relates to a method for determining an imaging quality of an imaging system, in particular an optical imaging system, having the following steps: a) irradiating a wavefront into the imaging system; b) detecting the wavefront after the wavefront has passed through the imaging system, the detection of the wavefront including a first partial detection of the wavefront to determine a first wave lenfront gradients in a first direction and at least a second partial acquisition of the wavefront for determining a second wavefront gradient in a second direction, wherein the wavefront of the first partial acquisition and the wavefront of the second partial acquisition differ from one another; c) determining a vector field as a function of the first wavefront gradient and the second wavefront gradient; d) determining a gradient field and a rotator field as a function of the vector field; e) determining a drift of the wave front as a function of the rotator field; f) Reconstructing the wavefront as a function of the gradient field and the drift. The advantage of this method is that during the reconstruction of the wavefront, wavefront errors, in particular astigmatic wavefront errors, are not simulated in the case of a changing wavefront, in particular as a function of a changing focus position or a focus drift, but can be determined and, in particular, eliminated. In the present case, the first and the second direction are aligned orthogonally or non-orthogonally, that is to say in any way with respect to one another. The advantage is that in the case of a wavefront that changes or has changed unnoticed between the partial acquisitions, the wavefront can be correctly reconstructed both when the first and the second direction are aligned orthogonally to one another and when the first and the second direction are not - are aligned orthogonally to each other.

According to one development, it is provided that the second partial acquisition is carried out after the first partial acquisition.

According to a further development it is provided that the first partial detection and the second partial detection takes place after an interaction of the wavefront with a diffractive element which has a first grating with a first grating vector and at least one second grating with a second grating vector, the first grating vector being along the is aligned in the first direction and the second lattice vector is aligned along the second direction.

According to a further development it is provided that the first partial detection takes place after an interaction of the wave front with a first diffractive element and the second partial detection takes place after an interaction of the wave front with at least one second diffractive element, the first diffractive element being a first grating with a first grating vector oriented along the first direction, and wherein the second diffractive element has a second grating having a second grating vector oriented along the second direction.

According to a further development it is provided that the first and the second wavefront gradient are detected as a function of an interference pattern or a point pattern.

Furthermore, the invention relates to a device for determining an imaging quality of an imaging system, in particular an optical imaging system, having: a) a radiation source for generating radiation; b) a wavefront generation unit for generating a wavefront from the radiation, the wavefront passing through the imaging system; c) a detector unit for detecting the wavefront after the wavefront has passed through the imaging system, with the detection of the wavefront including a first partial detection of the wavefront to determine a first wavefront gradient in a first direction and at least a second partial detection of the wavefront to determine a second wavefront gradient in a second direction, wherein the wavefront of the first partial acquisition and the wavefront of the second partial acquisition differ from each other; d) an evaluation unit for determining a vector field as a function of the first wavefront gradient and the second wavefront gradient and for determining a gradient field and a rotator field as a function of the vector field and e) a reconstruction unit for reconstructing the wavefront as a function of the gradient field and a drift determined as a function of the rotator field the wavefront. This results in the advantages already mentioned. Further advantages and preferred features emerge from what has been described above and from the claims.

Furthermore, the invention relates to a projection exposure system for microlithography, having a projection objective, wherein the projection objective comprises a device according to one of Claims 10 to 14 and/or a device according to Claim 16.

In addition, the invention relates to a projection exposure system for microlithography, having a projection objective, the projection objective being prepared for carrying out a method according to one of Claims 1 to 9 and/or a method according to Claim 15.

• 1 eine schematische Darstellung einer Vorrichtung zum Bestimmen einer Abbildungsqualität eines optischen Abbildungssystems gemäß einem Ausführungsbeispiel,
• 2 ein Ablaufdiagramm zur Darstellung eines Verfahrens zum Bestimmen der Abbildungsqualität des optischen Abbildungssystems gemäß einem ersten Ausführungsbeispiel,
• 3 ein Ablaufdiagramm zur Darstellung eines Verfahrens zum Bestimmen der Abbildungsqualität des optischen Abbildungssystems gemäß einem zweiten Ausführungsbeispiel,
• 4A eine schematische Darstellung eines diffraktiven Elements gemäß einem ersten Ausführungsbeispiel,
• 4B eine schematische Darstellung eines diffraktiven Elements gemäß einem zweiten Ausführungsbeispiel,
• 5A eine schematische Darstellung eines diffraktiven Elements gemäß einem dritten Ausführungsbeispiel,
• 5B eine schematische Darstellung eines diffraktiven Elements gemäß einem vierten Ausführungsbeispiel,
• 6A eine schematische Darstellung eines diffraktiven Elements gemäß einem fünften Ausführungsbeispiel,
• 6B eine schematische Darstellung eines diffraktiven Elements gemäß einem sechsten Ausführungsbeispiel,
• 7 eine schematische Darstellung des Grundprinzips der Scherinterferometrie,
• 8 eine erste Tabelle, bei welcher die Vektorpotenzen in quellfreie und wirbelfreie Anteile nach aufsteigendem Grad d gruppiert sind,
• 9 eine zweite Tabelle, welche Vektor-Zernikes in quellfreie und wirbelfreie Felder untergliedert,
• 10 eine dritte Tabelle, welche niedrige Vektor-Zernikes in Form von Vektor-Polynomen auflistet, und
• 11 eine vierte Tabelle, welche (skalare) Zernikes in Form von Polynomen auflistet.

In the following, the invention will be explained in more detail with reference to the drawings. to show

• 1 a schematic representation of a device for determining an imaging quality of an optical imaging system according to an embodiment,
• 2 a flow chart for representing a method for determining the imaging quality of the optical imaging system according to a first embodiment,
• 3 a flow chart for representing a method for determining the imaging quality of the optical imaging system according to a second embodiment,
• 4A a schematic representation of a diffractive element according to a first embodiment,
• 4B a schematic representation of a diffractive element according to a second embodiment,
• 5A a schematic representation of a diffractive element according to a third embodiment,
• 5B a schematic representation of a diffractive element according to a fourth embodiment,
• 6A a schematic representation of a diffractive element according to a fifth embodiment,
• 6B a schematic representation of a diffractive element according to a sixth embodiment,
• 7 a schematic representation of the basic principle of shearing interferometry,
• 8th a first table in which the vector powers are grouped into source-free and vortex-free parts according to increasing degree d,
• 9 a second table, which subdivides vector Zernikes into source-free and vortex-free fields,
• 10 a third table listing low vector Zernikes in terms of vector polynomials, and
• 11 a fourth table listing (scalar) Zernikes in terms of polynomials.

1 shows a device 1 for determining an imaging quality of an optical imaging system 2, with a radiation source 3 for generating radiation, for example electromagnetic radiation or particle radiation, and a wavefront generating unit 4 for generating a wavefront from the radiation. The imaging system 2 is preferably a projection lens or part of a projection lens of a projection exposure system for microlithography, not shown in detail here. As an alternative to the optical imaging system 2, the imaging system 2 is an electron-optical imaging system or some other imaging system, which preferably has at least one lens and/or at least one mirror for optical imaging.

The wavefront generation unit 4 comprises an optical element 6 which is to be arranged or is arranged in an object plane 5 of the imaging system 2 and which has an object-side, in particular periodic, structure which is illuminated by the radiation. The optical element 6 is designed in particular as a coherence mask or reticle for shaping the coherence of the radiation. The wavefront generated by the wavefront generation unit 4 passes through the imaging system 2.

Furthermore, the device 1 has a detector unit 7 for detecting the wavefront after the wavefront has passed through the imaging system 2 . The detector unit 7 is arranged in or at least close to an image plane 8 or pupil plane or plane conjugate thereto of the imaging system 2 . In the present case, the detector unit 7 comprises a diffractive element 9, in particular a diffractive optical element, with an image-side, in particular periodic, structure and also a detector element 10 or a sensor for detecting an overlay pattern, for example an interference pattern or a point pattern, of the imaged object-side structure and image-side structure Structure. The detector element 10 is arranged in or at least close to the image plane 8 or pupil plane or a plane of the imaging system 2 conjugated thereto. The detector element 10 preferably comprises an image sensor with a detector surface for reading out the overlay pattern. The detector element 10 is preferably designed as a camera. Alternatively, the diffractive element 9 is an element that is independent of the detector unit 7 .

Furthermore, the device has a reconstruction unit 11 for reconstructing the wavefront and optionally also an evaluation unit 12 for evaluating the superimposition pattern.

2 FIG. 1 shows a flow chart for representing a method for determining the imaging quality of the optical imaging system 2 according to a first exemplary embodiment.

The method is carried out in particular by a control unit. The control unit is preferably designed to carry out each of the method steps or only some of the method steps. The control unit is an integral or separate part of the device 1. The control unit is preferably functionally connected to the reconstruction unit 11 and/or the evaluation unit 12 in a wireless or wire-based manner.

In a first step S1, the diffractive element 9 is arranged in the image plane 8 or pupil plane of the imaging system 2. The diffractive element 9 has a first grating with a first grating vector φ ₁ and at least one second grating with a second grating vector φ ₂ , the first grating vector being aligned along a first direction and the second grating vector being aligned along a second direction. The first direction and the second direction are not orthogonal to each other. "Non-orthogonal" means that the angle between the first and second directions is not equal to 90°. Preferably, the first lattice vector is oriented along the first direction and the second lattice vector is oriented along the second direction such that the first lattice vector and the second lattice vector are oriented at an angle of 45° or 135° to one another. In other words: the first lattice vector and the second lattice vector form an angle of 45° or 135°. Optionally, the diffractive element 9 has a microlens array.

In a second step S2, a wave front is radiated into the imaging system 2. After passing through the imaging system 2, in particular after interacting with the diffractive element 9, the wavefront is detected by the detector element 10. The detection includes a first and at least a second partial detection.

In a third step S3, the detector unit 7 performs the first partial detection of the wavefront to determine a first wavefront gradient in the first direction. The wave front gradient is determined as a function of an interference pattern or a point pattern.

In a fourth step S4, the detector unit 7, in particular the detector element 10, performs the at least second partial detection of the wavefront to determine a second wavefront gradient in the second direction. As described above, the first direction and the second direction are not orthogonal to each other. The first partial acquisition and the second partial acquisition take place in particular one after the other. This ensures that particularly accurate methods for detecting the respective wavefront gradients can be used. In addition, a change in the wavefront over time as it passes through the imaging system can be taken into account or detected. The wavefront of the first partial acquisition and the wavefront of the second partial acquisition thus differ from one another due to a change in the wavefront over time between the first and the second partial acquisition. In particular, the wavefront of the first partial acquisition and the wavefront of the second partial acquisition differ from one another in that a focus of the imaging system 2 drifts, in particular unintentionally and/or unnoticed, between the first partial acquisition and the second partial acquisition. "Unintentional and/or unnoticed drifting of the focus" means in particular that the drifting is not intentionally brought about or generated. The drifting of the focus is caused in particular as a function of a change or change in position in relation to the wavefront generation unit 4 and/or in relation to at least one optical element, for example a lens or a mirror, of the imaging system 2. Optionally, the first partial acquisition and the second partial acquisition occur simultaneously. In the case of simultaneous partial acquisitions, the wavefront of the first partial acquisition and the wavefront of the second partial acquisition differ from one another, in particular due to a different spatial extension of the wavefront gradients.

In a fifth step S5, the wavefront is reconstructed as a function of the first wavefront gradient and the second wavefront gradient. The reconstruction is preferably carried out in such a way that measurement errors, which are caused in particular by the drifting of the focus, are eliminated by computation. The reconstruction is carried out in particular by a calculation method in which a system of equations is solved. In particular, the system of equations is solved in such a way that at least one variable or several variables of the system of equations, which represents/represent measurement errors, in particular astigmatism components or astigmatic wavefront errors, is/are mathematically eliminated. measuring Errors are caused in particular by the fact that the wavefront gradient, in particular of the focus, is incorrectly determined or measured when the focus drifts, with the astigmatism being incorrectly reconstructed as a function of the incorrectly measured wavefront gradient. The system of equations can be solved in particular in that at least one variable, which can be solved or eliminated in particular by calculation, is or is added to the system of equations, which variable represents the drift.

Optionally, the first partial detection takes place after an interaction of the wavefront with a first diffractive element 13, not shown here, and the second partial detection after an interaction of the wavefront with at least one second diffractive element 14, not shown here, with the first diffractive element 13 having a first grating with a having a first grating vector aligned along the first direction, and wherein the second diffractive element 14 has a second grating having a second grating vector aligned along the second direction. In this optional case, it is provided that in step S1 the first diffractive element 13 is arranged in the image plane 8 of the imaging system 2 and that in an intermediate step between steps S3 and S4 the second diffractive element 14 is exchanged for the first diffractive element 13 in the Image plane 8 of the imaging system 2 is arranged.

Optionally, the detection of the wavefront includes at least a third partial detection of the wavefront to determine a third wavefront gradient in a third direction, with the first direction, the second direction and the third direction each being oriented differently from one another. The first partial acquisition, the second partial acquisition and the third partial acquisition preferably take place after an interaction of the wavefront with a diffractive element 15 (not shown here), the diffractive element 15 having a first grating with a first grating vector, a second grating with a second grating vector and at least one third lattice having a third lattice vector, wherein the first lattice vector is oriented along the first direction, the second lattice vector is oriented along the second direction, and the third lattice vector is oriented along the third direction. A diffractive element 15 designed in this way has, for example, a trigonal or hexagonal lattice. Alternatively, the diffractive element has a non-square rectangular grating. The first, second and the at least third partial acquisition are preferably carried out one after the other.

3 FIG. 1 shows a flow chart for representing a method for determining the imaging quality of the optical imaging system 2 according to a second exemplary embodiment.

This method is also carried out in particular by a control unit. The control unit is preferably designed to carry out each of the method steps or only some of the method steps. The control unit is an integral or separate part of the device 1.

In a first step S1, a diffractive element 17 is arranged in the image plane 8 of the imaging system 2. The diffractive element 17 has a first grating with a first grating vector and at least one second grating with a second grating vector, the first grating vector being aligned along a first direction and the second grating vector being aligned along a second direction. The first direction and the second direction are aligned orthogonally or non-orthogonally, in particular arbitrarily, to one another.

In a second step S2, a wave front is radiated into the imaging system 2. After passing through the imaging system 2, in particular after interacting with the diffractive element 17, the wavefront is detected by the detector unit 7, in particular the detector element 10. The detection includes a first and at least a second partial detection.

In a third step S3, the detector unit 7, in particular the detector element 10, performs the first partial detection of the wavefront to determine a first wavefront gradient in the first direction. The wave front gradient is determined as a function of an interference pattern or a point pattern.

In a fourth step S4, the detector unit 7, in particular the detector element 10, performs the at least second partial detection of the wavefront to determine a second wavefront gradient in the second direction. As described above, the first direction and the second direction are orthogonal or non-orthogonal to each other. The first partial acquisition and the second partial acquisition take place in particular one after the other. This ensures that particularly accurate methods for detecting the respective wavefront gradients can be used. In addition, this can cause a change in the waves over time front are taken into account or detected when passing through the imaging system. The wavefront of the first partial acquisition and the wavefront of the second partial acquisition thus differ from one another due to a change in the wavefront over time between the first and the second partial acquisition. In particular, the wavefront of the first partial acquisition and the wavefront of the second partial acquisition differ from one another in that a focus of the imaging system 2 drifts, in particular unintentionally and/or unnoticed, between the first partial acquisition and the second partial acquisition. The drifting of the focus is caused in particular as a function of a change or change in position in relation to the wavefront generation unit 4 or a reticle or in relation to at least one optical element, for example a lens or a mirror, of the imaging system 2. Optionally, the first partial acquisition and the second partial acquisition occur simultaneously. In the case of simultaneous partial acquisitions, the wavefront of the first partial acquisition and the wavefront of the second partial acquisition differ from one another, in particular due to a different spatial extension of the wavefront gradients.

In a fifth step S5, a vector field is determined as a function of the first wavefront gradient and the second wavefront gradient. Subsequently, an in particular eddy-free gradient field and in particular a source-free rotator field are determined as a function of the vector field. Depending on the rotator field, a change over time or a drift of the wavefront is determined.

In a sixth step S6, the wavefront is reconstructed as a function of the gradient field and the drift. The reconstruction is preferably carried out in such a way that measurement errors, which are caused in particular by the drifting of the focus or the wave front, are eliminated by computation. In particular, a system of equations, on the basis of which the reconstruction takes place, is solved in such a way that at least one variable or several variables of the system of equations, which represents measurement errors, in particular astigmatism components or astigmatic wavefront errors, is/are eliminated. The system of equations can be solved in particular in that at least one variable, which can be solved or eliminated in particular by calculation, is or is added to the system of equations, which variable represents the drift.

The diffractive element 17 can be formed or is formed in accordance with each of the diffractive elements 9, 13, 14 or 15 described. Alternatively, the diffractive element 17 is designed in such a way that it has a first grating vector aligned along the first direction and a second grating vector aligned along the second direction, with the first grating vector and the second grating vector being aligned orthogonally to one another.

Optionally, as described above, the first partial detection can take place after an interaction of the wave front with the first diffractive element 13 and the second partial detection after an interaction of the wave front with at least the second diffractive element 14 .

Furthermore, it can optionally be provided that the detection of the wavefront, as described above, includes at least a third partial detection of the wavefront to determine a third wavefront gradient in a third direction. The first partial acquisition, the second partial acquisition and the third partial acquisition preferably take place after an interaction of the wavefront with the diffractive element 15.

4A shows a diffractive element, in particular the structure of the diffractive element, according to a first embodiment, in which the first direction and the second direction are aligned orthogonally to one another. In other words: the first grating vector φ ₃ is oriented along the first direction and the second grating vector φ ₄ is oriented along the second direction, which is orthogonal to the first direction. The diffractive element is in particular an orthogonal shearing grating, which is particularly useful as a diffractive element 17 for carrying out the in 3 described method is suitable.

4B shows a diffractive element, in particular the structure of the diffractive element, according to a second embodiment, in which the grating vectors φ ₁ and φ ₂ or the first direction and the second direction are aligned orthogonally to one another. In contrast to the diffractive element 4A the lattice vectors are rotated by 45° in this case.

5A 12 shows a diffractive element, in particular the structure of the diffractive element, according to a third embodiment, in which the first direction and the second direction are not aligned orthogonally to one another. In other words, the first lattice vector φ ₃ is oriented along the first direction and the second lattice vector φ ₄ is oriented along those non-orthogonal to the first direction second direction oriented. In particular, the diffractive element is a non-orthogonal shearing grating. In particular, the diffractive element 9 is formed in accordance with the presently described diffractive element.

5B shows a diffractive element according to a fourth embodiment, in which the grating vectors φ ₁ and φ ₂ or the first direction and the second direction are not aligned orthogonally to one another. In contrast to the diffractive element 5A the grid vectors are rotated by 90° in this case. In particular, the diffractive element 9 is formed in accordance with the presently described diffractive element.

6A shows a diffractive element according to a fifth embodiment, in which the first grating vector φ ₁ , the second grating vector φ ₂ and the at least third grating vector φ ₃ or the first direction, the second direction and the at least third direction are not aligned orthogonally to one another. In particular, the diffractive element is a non-orthogonal shearing grating with a hexagonal lattice. In particular, the diffractive element 15 is formed in accordance with the presently described diffractive element.

6B shows a diffractive element according to a sixth embodiment, in which the first grating vector φ ₁ , the second grating vector φ ₂ and the at least third grating vector φ ₃ or the first direction, the second direction and the at least third direction are not aligned orthogonally to one another. In particular, the diffractive element is a non-orthogonal shearing grating with a triangular or trigonal grating. In particular, the diffractive element 15 is formed in accordance with the presently described diffractive element.

In connection with 7 a measurement process or a determination of the imaging quality is described in which a first partial acquisition and a second partial acquisition take place one after the other. As already described, it can happen that such a measurement process or a wavefront measurement is irreproducibly afflicted with a measurement error, in particular an astigmatic wavefront error, if the wavefront changes between the first and the second partial acquisition, for example due to unwanted and/or unnoticed drifting of the focus of the imaging system 2. The following describes how it is possible to eliminate this drift and thus the measurement error by suitably combining different partial acquisitions and evaluating them.

In the following, this approach is described mathematically in detail as an example using a focus drift of unknown strength. In principle, however, the procedure can be extended to any type of drift.

Illustrated for explanation 7 the basic principle of shearing interferometry. A wave WE with a wavefront W experiences a wavefront error in the imaging system 2 to be tested, preferably a projection lens of a projection exposure system. In the pixel BP of the imaging system, the wave W(ξ), in this case the first wave, is divided and offset by ∂ξ to form a second wave W(ξ+∂ξ). The two waves are finally superimposed to form an interferogram, with the path difference W(ξ+∂ξ)-W(ξ) between the first wave and the second wave being proportional to the wavefront gradient M _ξ =∂W/∂ξ. A diffractive optical element 9, 13, 14, 15, 17 with a particularly periodic grating in the image plane 8 or pupil plane is used for the shearing. For the sake of simplicity, only the diffractive element 9 is shown here. In this context it should be noted that in 1 each of the diffractive optical elements 9, 13, 14, 15 or 17, in particular in combination with the detector unit 7, can be arranged in or at least near the image plane 8 or pupil plane.

The inventor recognized that previously unnoticed problems can arise with the usual measurements. Solutions to avoid such problems are presented.

In the case of the shear interferometer considered here, the gradient of the wave front W(ξ,η) is measured successively in terms of components. If the focus ƒ drifts by Δz between the two measurements or partial acquisitions M _ξ and M _η or if the wavefront changes, an astigmatic wavefront error ΔW between the two directions ξ and η is simulated. Focus drift does not affect diagonal astigmatism.

$$\begin{array}{l}ƒ=n-\sqrt{n^2-{\xi }^2-n^2}={\displaystyle \sum _{k=1}^{\infty }\frac{2n\left(2k-2\right)!}{\left(k-1\right)!k!}}{\left(\frac{{\xi }^2+{\eta }^2}{4n^2}\right)}^2\\ =\frac{{\xi }^2+{\eta }^2}{2n}+\frac{{\left({\xi }^2+{\eta }^2\right)}^2}{8n^3}+\frac{{\left({\xi }^2+{\eta }^2\right)}^3}{16n^5}+\frac{5{\left({\xi }^2+{\eta }^2\right)}^4}{128n^7}+\dots \end{array}$$

From a mathematical point of view, a system of differential equations for the unknown wavefront W from the measurements M must be integrated. The following formula (1) describes the basic shear equation for the Gradients of the wavefront W with, in particular, an unknown amplitude Δz for the drift ƒ of the focus shift: $$\begin{array}{c}{M}_{\xi }\left(\xi ,n\right)=\frac{\partial W}{\partial \xi }+\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\cdot \frac{\partial ƒ}{\partial \xi }\\ M_n\left(\xi ,n\right)=\frac{\partial W}{\partial n}-\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\cdot \frac{\partial ƒ}{\partial n}\end{array}$$

$$\begin{array}{l}ƒ=n-\sqrt{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-{\xi }^2-{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}={\displaystyle \sum _{k=1}^{\infty }\frac{2n\left(2k-2\right)!}{\left(k-1\right)!k!}}{\left(\frac{{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{4n^2}\right)}^2\\ =\frac{{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{2n}+\frac{{\left({\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+n^2\right)}^2}{\mathrm{<figure-callout\; id="3"\; label="8th\; n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">8th</figure-callout>}{n}^{}{}^3}+\frac{{\left({\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+n^2\right)}^3}{16{\mathrm{<figure-callout\; id="5"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">n</figure-callout>}}^5}+\frac{5{\left({\xi }^2+n^2\right)}^4}{128n^7}+...\end{array}$$

The above formula (2) describes the wavefront of a spherical wave in an immersion medium with a refractive index n and thus the wavefront error for a focus error. As mentioned, the drift is described by Δz. The function ƒ(ξ,η) describes the pattern in the wave front caused by the drift.

In a first approximation ƒ ≃ (ξ ² + η ² )/2n, the drift integrates into an astigmatism Z ₅ in the wavefront error ΔW $$\begin{array}{c}\frac{\partial \mathrm{\Delta }W}{\partial \xi }=+\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\cdot \frac{\partial ƒ}{\partial \xi }\simeq +\frac{\Delta \mathrm{e.g}}{2n}\cdot \xi \\ \frac{\partial \mathrm{\Delta }W}{\partial n}=-\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\cdot \frac{\partial ƒ}{\partial n}\simeq -\frac{\Delta \mathrm{e.g}}{2n}\cdot n\end{array}\Rightarrow \mathrm{\Delta }W\simeq \frac{\mathrm{\Delta }\mathrm{e.g}}{4n}\cdot \left({\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2-n^2\right)$$

$$\begin{array}{c}\frac{\partial \mathrm{\Delta }W}{\partial \xi }+\frac{\partial V}{\partial \eta }=+\frac{\mathrm{\Delta }z}{2}\cdot \frac{\partial ƒ}{\partial \xi }\simeq +\frac{\mathrm{\Delta }z}{6n^3}\cdot {\xi }^3+\frac{\mathrm{\Delta }z}{12n^3}\cdot \left(3\xi {\eta }^2+{\xi }^3\right)\\ \frac{\partial \mathrm{\Delta }W}{\partial \eta }-\frac{\partial V}{\partial \xi }=-\frac{\mathrm{\Delta }z}{2}\cdot \frac{\partial ƒ}{\partial \eta }\simeq -\frac{\mathrm{\Delta }z}{6n^3}\cdot {\eta }^3-\frac{\mathrm{\Delta }z}{12n^3}\cdot \left(3{\xi }^2\eta +{\eta }^3\right)\end{array}$$

For the higher orders, the vector field (M _ξ , M _η ) would first have to be broken down according to the rules of vector analysis into an eddy-free field or gradient field and a source-free field or rotator field. The former part can be integrated to a higher astigmatism (for example as a mixture of Z ₁₂ and Z ₅ ), while the latter can be represented as a rotation of a vector potential V, but is not considered important as a residual within the scope of this application. For the next higher order with ƒ ≃ (ξ ² + η ² ) ² /8n ³ this would result $$\Rightarrow \begin{array}{c}\mathrm{\Delta }W\simeq \frac{\mathrm{\Delta }\mathrm{e.g}}{24{\mathrm{<figure-callout\; id="3"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">n</figure-callout>}}^3}\cdot \left({\mathrm{<figure-callout\; id="4"\; label="\xi "\; filenames="DE102020215540B4\_0062.png,DE102020215540B4\_0065.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^4-n^2\right)\\ V\simeq \frac{\mathrm{\Delta }\mathrm{e.g}}{12{\mathrm{<figure-callout\; id="3"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">n</figure-callout>}}^3}\cdot \left(\mathrm{<figure-callout\; id="3"\; label="\xi \; n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">\xi </figure-callout>}{n}^{}{}^3+n{\xi }^3\right)\end{array}$$

$$\begin{array}{c}\frac{\partial \mathrm{\Delta }W}{\partial \xi }+\frac{\partial V}{\partial n}=+\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\cdot \frac{\partial ƒ}{\partial \xi }\simeq +\frac{\mathrm{\Delta }\mathrm{e.g}}{6{\mathrm{<figure-callout\; id="3"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">n</figure-callout>}}^3}\cdot {\mathrm{<figure-callout\; id="3"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^3+\frac{\mathrm{\Delta }\mathrm{e.g}}{12{\mathrm{<figure-callout\; id="3"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">n</figure-callout>}}^3}\cdot \left(3\mathrm{<figure-callout\; id="2"\; label="\xi \; n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}{n}^{}{}^2+{\xi }^3\right)\\ \frac{\partial \mathrm{\Delta }W}{\partial n}-\frac{\partial V}{\partial \xi }=-\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\cdot \frac{\partial ƒ}{\partial n}\simeq -\frac{\mathrm{\Delta }\mathrm{e.g}}{6{\mathrm{<figure-callout\; id="3"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">n</figure-callout>}}^3}\cdot {\mathrm{<figure-callout\; id="3"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">n</figure-callout>}}^3-\frac{\mathrm{\Delta }\mathrm{e.g}}{12{\mathrm{<figure-callout\; id="3"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">n</figure-callout>}}^3}\cdot \left(3{\xi }^{2}n+n^3\right)\end{array}$$

The problem can be applied to any type of wavefront drift f or its gradient drift (+ ∂ƒ /∂ξ - ∂ƒ /∂η).

The drift problem can be addressed as follows, for example: Crosstalk of the Z ₄ drift on the Z ₅ astigmatism is avoided in a coordinate system rotated by 45°, where Z ₅ and Z ₆ swap roles. For example, it is possible to measure the wave front again rotated by 45°. By combining all four partial measurements (0° and 90° with 45° and 135° directions), the two astigmatism components Z ₅ and Z ₆ can be correctly determined despite a slight drift of the focus Z ₄ .

In a first variant, the lens to be tested can be rotated by 45° to the shear interferometer. Equivalent to this would be the rotation of the interferometer assembly.

In the case of a field-parallel measurement (simultaneous measurement at several field points), the field points of the two rotary positions do not match. The field profiles would therefore have to be interpolated onto a common grid.

In a second variant, the lens is not rotated, but a second shearing in a diagonal direction is carried out. For this purpose, instead of a first orthogonal shear grating, as in 2A shown, a second shear lattice twisted by 45°, as in 2 B shown, can be accommodated, which can be pushed into a measuring position with a translation table. This procedure is particularly useful for non-round field areas where only a (usually rectangular) section is used.

It was found that orthogonal shear lattices with mutually perpendicular lattice vectors, φ ₁ ⊥ φ ₂ and φ ₃ ⊥ φ ₄ , respectively, as in 2A and 2 B shown, alone cannot completely distinguish between focus drift and astigmatism. All wavefront errors, in particular the Z ₄ , Z ₅ and Z ₆ components, can only be determined using a combination of the two.

This approach has some disadvantages. The disadvantage of a double measurement time (four instead of two shears) is a more complex measurement setup. More space is needed for the two shear grids, which not only entails the need for a larger relay optics, but also a need for a larger camera. Large image sensors are more difficult to manufacture, which ultimately translates into price.

The approaches described below avoid the disadvantages of the known procedures and those outlined above.

When the pupil is sheared (ξ, η) at any angle φ, the directional derivative of the wavefront W is calculated from the phase of the shearing interferogram, in particular by scaling with the lattice constant. When the focus ƒ drifts by Δz, the directional derivative of the focus occurs added. The following formula (3a) describes a shear interferogram along a first direction, in particular along a first grating vector, with focus drift Δz ₁ : $$M_1\left(\xi ,n\right)=\left(\frac{\partial W}{\partial \xi }+\mathrm{\Delta }{\mathrm{e.g}}_1\frac{\partial ƒ}{\partial \xi }\right)\text{cos}{\phi }_1+\left(\frac{\partial W}{\partial n}+\mathrm{\Delta }{\mathrm{e.g}}_1\frac{\partial ƒ}{\partial n}\right)\text{<figure-callout id="1" label="sin φ" filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png" state="{{state}}">sin</figure-callout>}{\phi }_{}{}_1$$

To reconstruct the wavefront, a shear interferogram is needed in at least a second direction where the focus may have drifted in other ways. The following formula (3b) describes a shear interferogram along the second direction or along a second grating vector with focus drift Δz ₂ : $$M_2\left(\xi ,n\right)=\left(\frac{\partial W}{\partial \xi }+\mathrm{\Delta }{\mathrm{e.g}}_2\frac{\partial ƒ}{\partial \xi }\right)\text{cos}{\phi }_2+\left(\frac{\partial W}{\partial n}+\mathrm{\Delta }{\mathrm{e.g}}_2\frac{\partial ƒ}{\partial n}\right)\text{<figure-callout id="2" label="sin φ" filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png" state="{{state}}">sin</figure-callout>}{\phi }_{}{}_2$$

Without loss of generality, the wave front refers to the mean drift position of the focus, i.e. in the case of two measurements: $$0=\mathrm{\Delta }{\mathrm{e.g}}_1+\mathrm{\Delta }{\mathrm{e.g}}_2\mathrm{\Delta }{\mathrm{e.g}}_1=+\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\mathrm{\Delta }{\mathrm{e.g}}_2=-\frac{\mathrm{\Delta }\mathrm{e.g}}{2}$$

The focus drift or image shell Z ₄ and the astigmatism Z ₅ and Z ₆ lead to linear profiles in the interferogram. The following formula (4) describes linearized shear interferograms that are derived from the quadratic curves of the wavefront: $$\begin{array}{c}{\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">M</figure-callout>}}_1\dot{=}{m}_1^{\left(\xi \right)}\xi +m_1^{\left(n\right)}+\mathrm{<figure-callout\; id="2"\; label="n\; M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}& \\ M_{}{}_2\dot{=}{m}_2^{\left(\xi \right)}\xi +m_2^{\left(n\right)}+n\end{array}$$

sowie der niedrigsten Ordnung der Fokusdrift aus Formel (2): $$ƒ\dot{=}\frac{{\xi }^2+{\eta }^2}{2n}$$

$$\frac{\partial ƒ}{\partial \xi }\dot{=}\frac{\xi }{n}$$

$$\frac{\partial ƒ}{\partial \eta }\dot{=}\frac{\eta }{n}$$

The following formula 5 describes quadratic terms of the wavefront together with their (linear) gradients: $$\begin{array}{c}W\dot{=}{\mathrm{e.g}}_4\left(\frac{2{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+2n^2}{N{A}^2}-1\right)+{\mathrm{e.g}}_5\frac{{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2-n^2}{\mathrm{<figure-callout\; id="2"\; label="N\; A"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">N</figure-callout>}{A}^{}{}^2}+{\mathrm{e.g}}_6\frac{2\xi n}{N{A}^2}\\ \frac{\partial W}{\partial \xi }\dot{=}\left(\frac{4{\mathrm{e.g}}_4}{\mathrm{<figure-callout\; id="2"\; label="N\; A"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">N</figure-callout>}{A}^{}{}^2}+\frac{4{\mathrm{e.g}}_5}{N{A}^2}\right)\xi +\frac{2{\mathrm{e.g}}_6}{N{A}^2}n\\ \frac{\partial W}{\partial n}\dot{=}\left(\frac{4{\mathrm{e.g}}_4}{N{A}^2}-\frac{4{\mathrm{e.g}}_5}{N{A}^2}\right)n+\frac{2{\mathrm{e.g}}_6}{\mathrm{<figure-callout\; id="2"\; label="N\; A"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">N</figure-callout>}{A}^{}{}^2}\xi \end{array}$$

and the lowest order of the focus drift from formula (2): $$ƒ\dot{=}\frac{{\xi }^2+{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{2n}$$

$$\frac{\partial ƒ}{\partial \xi }\dot{=}\frac{\xi }{n}$$

$$\frac{\partial ƒ}{\partial n}\dot{=}\frac{n}{n}$$

$$m_2^{\left(\xi \right)}=\left(\frac{4z_4}{N{A}^2}+\frac{2z_5}{N{A}^2}-\frac{\mathrm{\Delta }z}{2n}\right)\text{cos}{\phi }_2+\frac{2z_6}{N{A}^2}\text{sin}{\phi }_2$$

$$m_1^{\left(\eta \right)}=\left(\frac{4z_4}{N{A}^2}-\frac{2z_5}{N{A}^2}+\frac{\mathrm{\Delta }z}{2n}\right)\text{sin}{\phi }_1+\frac{2z_6}{N{A}^2}\text{cos}{\phi }_1$$

$$m_2^{\left(\eta \right)}=\left(\frac{4z_4}{N{A}^2}-\frac{2z_5}{N{A}^2}-\frac{\mathrm{\Delta }z}{2n}\right)\text{sin}{\phi }_2+\frac{2z_6}{N{A}^2}\text{cos}{\phi }_2$$

A comparison of the coefficients of the linear curves in ξ and η yields four equations for the four unknowns z ₄ , z ₅ , z ₆ , and Δz : $$m_1^{\left(\xi \right)}=\left(\frac{4{\mathrm{e.g}}_4}{\mathrm{<figure-callout\; id="2"\; label="N\; A"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">N</figure-callout>}{A}^{}{}^2}+\frac{2{\mathrm{e.g}}_5}{\mathrm{<figure-callout\; id="2"\; label="N\; A"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">N</figure-callout>}{A}^{}{}^2}+\frac{\mathrm{\Delta }\mathrm{e.g}}{2n}\right)\text{cos}{\phi }_1+\frac{2{\mathrm{e.g}}_6}{\mathrm{<figure-callout\; id="2"\; label="N\; A"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">N</figure-callout>}{A}^{}{}^2}\text{<figure-callout id="1" label="sin φ" filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png" state="{{state}}">sin</figure-callout>}{\phi }_{}{}_1$$

$$m_2^{\left(\xi \right)}=\left(\frac{4{\mathrm{e.g}}_4}{\mathrm{<figure-callout\; id="2"\; label="N\; A"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">N</figure-callout>}{A}^{}{}^2}+\frac{2{\mathrm{e.g}}_5}{N{A}^2}-\frac{\mathrm{\Delta }\mathrm{e.g}}{2n}\right)\text{cos}{\phi }_2+\frac{2{\mathrm{e.g}}_6}{\mathrm{<figure-callout\; id="2"\; label="N\; A"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">N</figure-callout>}{A}^{}{}^2}\text{<figure-callout id="2" label="sin φ" filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png" state="{{state}}">sin</figure-callout>}{\phi }_{}{}_2$$

$$m_1^{\left(n\right)}=\left(\frac{4{\mathrm{e.g}}_4}{N{A}^2}-\frac{2{\mathrm{e.g}}_5}{\mathrm{<figure-callout\; id="2"\; label="N\; A"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">N</figure-callout>}{A}^{}{}^2}+\frac{\mathrm{\Delta }\mathrm{e.g}}{2n}\right)\text{<figure-callout id="1" label="sin φ" filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png" state="{{state}}">sin</figure-callout>}{\phi }_{}{}_1+\frac{2{\mathrm{e.g}}_6}{N{A}^2}\text{cos}{\phi }_1$$

$$m_2^{\left(n\right)}=\left(\frac{4{\mathrm{e.g}}_4}{N{A}^2}-\frac{2{\mathrm{e.g}}_5}{N{A}^2}-\frac{\mathrm{\Delta }\mathrm{e.g}}{2n}\right)\text{<figure-callout id="2" label="sin φ" filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png" state="{{state}}">sin</figure-callout>}{\phi }_{}{}_2+\frac{2{\mathrm{e.g}}_6}{N{A}^2}\text{cos}{\phi }_2$$

A suitable linear combination eliminates the Zernike coefficients z ₄ to z ₆ , as given in formula (6a) below: $$m_1^{\left(\xi \right)}\text{cos}{\phi }_2-m_2^{\left(\xi \right)}\text{cos}{\phi }_1+m_1^{\left(n\right)}\text{<figure-callout id="2" label="sin φ" filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png" state="{{state}}">sin</figure-callout>}{\phi }_{}{}_2-m_2^{\left(n\right)}\text{<figure-callout id="1" label="sin φ" filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png" state="{{state}}">sin</figure-callout>}{\phi }_{}{}_1=\frac{\mathrm{\Delta }\mathrm{e.g}}{n}\text{cos}\left({\phi }_2-{\phi }_1\right)$$

$$\begin{array}{l}{m}_1^{\left(\xi \right)}\text{cos}{\phi }_2-m_2^{\left(\xi \right)}\text{cos}{\phi }_1+m_1^{\left(\eta \right)}\text{sin}{\phi }_2-m_2^{\left(\eta \right)}\text{sin}{\phi }_1\\ =\frac{\mathrm{\Delta }z}{n}\text{cos}\left({\phi }_2+{\phi }_1\right)-\frac{4z_6}{N{A}^2}\text{sin}\left({\phi }_2-{\phi }_1\right)\end{array}$$

With the focus drift Δz now known, the astigmatic wavefront errors or astigmatisms z ₅ and z ₆ can be calculated, as specified in the following formulas (6b) and (6c): $$\begin{array}{l}{m}_1^{\left(\xi \right)}\text{<figure-callout id="2" label="sin φ" filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png" state="{{state}}">sin</figure-callout>}{\phi }_{}{}_2-m_2^{\left(\xi \right)}\text{<figure-callout id="1" label="sin φ" filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png" state="{{state}}">sin</figure-callout>}{\phi }_{}{}_1+m_1^{\left(n\right)}\text{cos}{\phi }_2-m_2^{\left(n\right)}\text{cos}{\phi }_1\\ =\frac{\mathrm{\Delta }\mathrm{e.g}}{n}\text{sin}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2+{\phi }_1\right)+\frac{4{\mathrm{e.g}}_5}{\mathrm{<figure-callout\; id="2"\; label="N\; A"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">N</figure-callout>}{A}^{}{}^2}\text{sin}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2-{\phi }_1\right)\end{array}$$

$$\begin{array}{l}{m}_1^{\left(\xi \right)}\text{cos}{\phi }_2-m_2^{\left(\xi \right)}\text{cos}{\phi }_1+m_1^{\left(n\right)}\text{<figure-callout id="2" label="sin φ" filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png" state="{{state}}">sin</figure-callout>}{\phi }_{}{}_2-m_2^{\left(n\right)}\text{<figure-callout id="1" label="sin φ" filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png" state="{{state}}">sin</figure-callout>}{\phi }_{}{}_1\\ =\frac{\mathrm{\Delta }\mathrm{e.g}}{n}\text{cos}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2+{\phi }_1\right)-\frac{4{\mathrm{e.g}}_6}{\mathrm{<figure-callout\; id="2"\; label="N\; A"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">N</figure-callout>}{A}^{}{}^2}\text{sin}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2-{\phi }_1\right)\end{array}$$

The (mean) focus z ₄ is then obtained from the relationship according to the following formula (6d): $$m_1^{\left(\xi \right)}\text{<figure-callout id="2" label="sin φ" filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png" state="{{state}}">sin</figure-callout>}{\phi }_{}{}_2-m_2^{\left(\xi \right)}\text{<figure-callout id="1" label="sin φ" filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png" state="{{state}}">sin</figure-callout>}{\phi }_{}{}_1+m_1^{\left(n\right)}\text{cos}{\phi }_2-m_2^{\left(n\right)}\text{cos}{\phi }_1=\frac{\mathrm{8th}{\mathrm{e.g}}_4}{\mathrm{<figure-callout\; id="2"\; label="N\; A"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">N</figure-callout>}{A}^{}{}^2}\text{sin}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2-{\phi }_1\right)$$

However, the standard case of orthogonal shearing, in which the first shearing direction is perpendicular to the second shearing direction, leads to a dilemma, because the system of equations then becomes singular because of cos(φ ₂ - φ ₁ ) = 0 in formula (6a). There is no longer a (unique) solution. In particular, z ₅ becomes insoluble in a measurement in which the first direction and the second direction coincide with the axes ξ and η of the pupil coordinate system.

For the axis-parallel shear lattice (φ ₁ || ξ and φ ₂ || η) at least z ₆ can be determined from formula (6c) because of cos(φ ₂ + φ ₁ ) = 0, or for the diagonal shear lattice, as in 2A shown, at least z ₅ can be determined from formula (6b) because of sin(φ ₂ + (φ ₁ ) = 0. For any orthogonal shearing, at least the following linear combination can be calculated: $${\mathrm{e.g}}_5\text{cos}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2+{\phi }_1\right)-{\mathrm{e.g}}_6\text{sin}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2+{\phi }_1\right)$$

However, the system of equations becomes unsolvable for collinear shears with sin(φ ₂ - φ ₁ ) = 0.

According to one aspect of the invention described in this application, it is proposed to shear the wavefront with non-orthogonal lattice vectors. In other words: the wavefront is detected after the wavefront has passed through the imaging system, with the detection of the wavefront involving a first partial detection of the wavefront to determine a first wavefront gradient in a first direction and at least a second partial detection of the wavefront to determine a second wavefront gradient in a second includes direction, wherein the wavefront of the first partial acquisition and the wavefront of the second partial acquisition differ from each other, and wherein the first direction and the second direction are not orthogonal to each other. The two astigmatisms can then be determined from the two directional derivatives, even if the focus position has changed between the two shear measurements.

Examples of diffractive elements that have corresponding grating structures, i.e. a first grating with a first grating vector and at least one second grating with a second grating vector, the first grating vector being aligned along the first direction and the second grating vector being aligned along the direction not orthogonal to the first direction second direction are, but not limited to, in the 5A and 5B shown. The use of at least one such diffractive element ensures that a complete reconstruction of the wave front is possible even despite a possible change in the wave front or a possible focus drift.

Also according to the use of a diffractive element 4A or 4B is possible, in which case not all four wavefront derivatives are measured, but only two wavefront derivatives of two non-orthogonal directions (φ ₁ and φ ₃ or φ ₂ and φ ₄ ), which are distributed over the two grating structures. The advantage here is in particular a saving in measurement time. Optionally can be made with the grids 5A and 5B all four directions can also be measured.

Optionally other diffractive elements or grating types can be used, for example trigonal shear gratings as in 6A and 6B shown. The wavefront can already be completely reconstructed with two directions alone, even if the wavefront changes between the first and the second partial acquisition or if the focus changes. The wavefront gradient can be measured isotropically with the third direction. While the diffractive element corresponds to the lattice with hexagonal structural elements 6A characterized by fewer (only odd) and weaker harmonics, the diffractive element with the grating with triangular structural elements according to 6B more light through.

Even if all three directional derivatives of the wave front are measured here, the advantages are a compact installation space (compared to the double grating) and a reduced measurement time (compared to four directions).

An embodiment is discussed below in which a calculation or reconstruction of the wavefront based on Zernike coefficients, as described above, is not necessary. In particular, the wavefront is detected here after the wavefront has passed through the imaging system 2, with the detection of the wavefront involving a first partial detection of the wavefront to determine a first wavefront gradient in a first direction and at least a second partial detection of the wavefront to determine a second wavefront gradient in a second direction comprises, wherein the wavefront of the first partial acquisition and the wavefront of the second partial acquisition differ from each other, and wherein a vector field depending on the first wavefront gradient and the second wavefront gradient ten is determined, and wherein a gradient field and a rotator field are determined as a function of the vector field, wherein a drift or change over time in the wavefront is determined as a function of the rotator field, and the wavefront is reconstructed as a function of the gradient field and the drift. In this case, the wavefront reconstruction can be performed even if the first and second directions are orthogonal to each other.

The system of equations according to the formulas (6a to 6d) is based on the linearized form of the interferograms according to the formulas (3a and/or 3b), which can generally be separated into the two original directions ξ and η. The following formula (7) describes a transformed system of equations for the wavefront gradients under oblique shears and a change in the wavefront, in particular a drift of the focus: $$\begin{array}{c}{\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">M</figure-callout>}}_1\text{<figure-callout id="2" label="sin φ" filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png" state="{{state}}">sin</figure-callout>}{\phi }_{}{}_2-{\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">M</figure-callout>}}_2\text{<figure-callout id="1" label="sin φ" filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png" state="{{state}}">sin</figure-callout>}{\phi }_{}{}_1=\frac{\partial W}{\partial \xi }\text{sin}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2-{\phi }_1\right)+\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\frac{\partial ƒ}{\partial \xi }\text{sin}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2+{\phi }_1\right)+\mathrm{\Delta }\mathrm{e.g}\frac{\partial ƒ}{\partial n}\text{<figure-callout id="1" label="sin φ" filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png" state="{{state}}">sin</figure-callout>}{\phi }_{}{}_1\text{sin}{\phi }_2\\ {\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">M</figure-callout>}}_2\text{cos}{\phi }_1-{\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">M</figure-callout>}}_1\text{cos}{\phi }_2=\frac{\partial W}{\partial n}\text{sin}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2-{\phi }_1\right)-\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\frac{\partial ƒ}{\partial n}\text{sin}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2+{\phi }_1\right)-\mathrm{\Delta }\mathrm{e.g}\frac{\partial ƒ}{\partial \xi }\text{cos}{\phi }_1\text{cos}{\phi }_2\end{array}$$

Instead of fitting approximately linear curves according to formula (4), the focus drift Δz is preferably determined as the amplitude of with the derivatives ∂M ₁ /∂ξ and ∂M ₁ /∂η as well as ∂M ₂ /∂ξ and ∂M ₂ /∂η pupil courses. The following formula (8) gives the conditional equation for focus drift from the interferogram differential quotient under oblique shear: $$\begin{array}{l}\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">M</figure-callout>}}_1}{\partial \xi }\text{cos}{\phi }_2-\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">M</figure-callout>}}_2}{\partial \xi }\text{cos}{\phi }_1+\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">M</figure-callout>}}_1}{\partial n}\text{sin}{\phi }_2-\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">M</figure-callout>}}_2}{\partial \mathrm{<figure-callout\; id="1"\; label="n\; sin\; \phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">n</figure-callout>}}\text{sin}{\phi }_{}{}_1=\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\left(\frac{{\partial }^2ƒ}{\partial {\xi }^2}+\frac{{\partial }^2ƒ}{\partial n^2}\right)\text{cos}\left({\phi }_2-{\phi }_1\right)+\\ +\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\left(\frac{{\partial }^2ƒ}{\partial {\xi }^2}-\frac{{\partial }^2ƒ}{\partial n^2}\right)\text{cos}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2+{\phi }_1\right)+\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\left(\frac{2{\partial }^2ƒ}{\partial \xi \partial n}\right)\text{sin}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2+{\phi }_1\right)\end{array}$$

sowie die gemischten zweiten Ableitungen $$\frac{{\partial }^2ƒ}{\partial {\xi }^2}-\frac{{\partial }^2ƒ}{\partial {\eta }^2}=\frac{{\xi }^2-{\eta }^2}{{\sqrt{n^2-{\xi }^2-{\eta }^2}}^3}$$

beziehungsweise $$\frac{2{\partial }^2ƒ}{\partial \xi \partial \eta }=\frac{2\xi \eta }{{\sqrt{n^2-{\xi }^2-{\eta }^2}}^3}$$

Because the exact Laplace operation of the focus drift in the pupil (cf. formula (2)) is known $$\frac{{\partial }^2ƒ}{\partial {\xi }^2}+\frac{{\partial }^2ƒ}{\partial n^2}=\frac{2{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-{\xi }^2-n^2}{{\sqrt{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-{\xi }^2-{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}}^3}$$

and the mixed second derivatives $$\frac{{\partial }^2ƒ}{\partial {\xi }^2}-\frac{{\partial }^2ƒ}{\partial n^2}=\frac{{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2-n^2}{{\sqrt{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-{\xi }^2-{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}}^3}$$

respectively $$\frac{2{\partial }^2ƒ}{\partial \xi \partial n}=\frac{2\xi n}{{\sqrt{n^2-{\xi }^2-{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}}^3}$$

According to one aspect of the invention described in this application, it is proposed that the measured shear interferograms M ₁ (ξ,η) and M ₂ (ξ,η) for errors! Reference source could not be found.) numerically according to ∂/∂ξ and ∂/∂η, then using formula (8) to fit the unknown focus drift Δz and finally to reconstruct the wavefront using the usual methods from formula (7). In contrast to the linear case, the conditional equation (8) for orthogonal shear lattices does not become singular because the other terms besides cos(φ ₂ - φ ₁ ) do not vanish at the same time.

$$\begin{array}{l}\frac{\partial M_1}{\partial \xi }\text{cos}{\phi }_2-\frac{\partial M_2}{\partial \xi }\text{cos}{\phi }_1-\frac{\partial M_1}{\partial \eta }\text{sin}{\phi }_2+\frac{\partial M_2}{\partial \eta }\text{sin}{\phi }_1=-\left(\frac{2{\partial }^{2}W}{\partial \xi \partial \eta }\right)\text{sin}\left({\phi }_2-{\phi }_1\right)+\\ +\frac{\mathrm{\Delta }z}{2}\left(\frac{{\partial }^2ƒ}{\partial {\xi }^2}+\frac{{\partial }^2ƒ}{\partial {\eta }^2}\right)\text{cos}\left({\phi }_2+{\phi }_1\right)+\frac{\mathrm{\Delta }z}{2}\left(\frac{{\partial }^2ƒ}{\partial {\xi }^2}-\frac{{\partial }^2ƒ}{\partial {\eta }^2}\right)\text{cos}\left({\phi }_2-{\phi }_1\right)\end{array}$$

$$\begin{array}{l}\frac{\partial M_1}{\partial \xi }\text{sin}{\phi }_2-\frac{\partial M_2}{\partial \xi }\text{sin}{\phi }_1-\frac{\partial M_1}{\partial \eta }\text{cos}{\phi }_2+\frac{\partial M_2}{\partial \eta }\text{cos}{\phi }_1=\left(\frac{{\partial }^{2}W}{\partial {\xi }^2}+\frac{{\partial }^{2}W}{\partial {\eta }^2}\right)\text{sin}\left({\phi }_2-{\phi }_1\right)+\\ +\frac{\mathrm{\Delta }z}{2}\left(\frac{{\partial }^2ƒ}{\partial {\xi }^2}-\frac{{\partial }^2ƒ}{\partial {\eta }^2}\right)\text{sin}\left({\phi }_2+{\phi }_1\right)-\frac{\mathrm{\Delta }z}{2}\left(\frac{2{\partial }^2ƒ}{\partial \xi \partial \eta }\right)\text{cos}\left({\phi }_2+{\phi }_1\right)\end{array}$$

Analogous to the linearized approximation, once the focus drift Δz has been determined, the four derivatives can be transformed into equivalent equations which separate the second derivatives of the wavefront. $$\begin{array}{l}\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">M</figure-callout>}}_1}{\partial \xi }\text{sin}{\phi }_2-\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">M</figure-callout>}}_2}{\partial \mathrm{<figure-callout\; id="1"\; label="\xi \; sin\; \phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">\xi </figure-callout>}}\text{sin}{\phi }_{}{}_1+\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">M</figure-callout>}}_1}{\partial n}\text{cos}{\phi }_2-\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">M</figure-callout>}}_2}{\partial n}\text{cos}{\phi }_1=\left(\frac{{\partial }^{2}W}{\partial {\xi }^2}-\frac{{\partial }^{2}W}{\partial n^2}\right)\text{sin}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2-{\phi }_1\right)+\\ +\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\left(\frac{{\partial }^2ƒ}{\partial {\xi }^2}+\frac{{\partial }^2ƒ}{\partial n^2}\right)\text{sin}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2+{\phi }_1\right)+\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\left(\frac{2{\partial }^2ƒ}{\partial \xi \partial n}\right)\text{cos}\left({\phi }_2-{\phi }_1\right)\end{array}$$

$$\begin{array}{l}\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">M</figure-callout>}}_1}{\partial \xi }\text{cos}{\phi }_2-\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">M</figure-callout>}}_2}{\partial \xi }\text{cos}{\phi }_1-\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">M</figure-callout>}}_1}{\partial \mathrm{<figure-callout\; id="2"\; label="n\; sin\; \phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}\text{sin}{\phi }_{}{}_2+\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">M</figure-callout>}}_2}{\partial \mathrm{<figure-callout\; id="1"\; label="n\; sin\; \phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">n</figure-callout>}}\text{sin}{\phi }_{}{}_1=-\left(\frac{2{\partial }^{2}W}{\partial \xi \partial n}\right)\text{sin}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2-{\phi }_1\right)+\\ +\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\left(\frac{{\partial }^2ƒ}{\partial {\xi }^2}+\frac{{\partial }^2ƒ}{\partial n^2}\right)\text{cos}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2+{\phi }_1\right)+\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\left(\frac{{\partial }^2ƒ}{\partial {\xi }^2}-\frac{{\partial }^2ƒ}{\partial n^2}\right)\text{cos}\left({\phi }_2-{\phi }_1\right)\end{array}$$

$$\begin{array}{l}\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">M</figure-callout>}}_1}{\partial \xi }\text{sin}{\phi }_2-\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">M</figure-callout>}}_2}{\partial \xi }\text{sin}{\phi }_1-\frac{\partial {\mathrm{<figure-callout\; id="1"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">M</figure-callout>}}_1}{\partial n}\text{cos}{\phi }_2+\frac{\partial {\mathrm{<figure-callout\; id="2"\; label="M"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">M</figure-callout>}}_2}{\partial n}\text{cos}{\phi }_1=\left(\frac{{\partial }^{2}W}{\partial {\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2}+\frac{{\partial }^{2}W}{\partial n^2}\right)\text{sin}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2-{\phi }_1\right)+\\ +\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\left(\frac{{\partial }^2ƒ}{\partial {\xi }^2}-\frac{{\partial }^2ƒ}{\partial n^2}\right)\text{sin}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2+{\phi }_1\right)-\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\left(\frac{2{\partial }^2ƒ}{\partial \xi \partial n}\right)\text{cos}\left({\mathrm{<figure-callout\; id="2"\; label="\phi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\phi </figure-callout>}}_2+{\phi }_1\right)\end{array}$$

$$\frac{{\partial }^{2}W}{\partial {\xi }^2}-\frac{{\partial }^{2}W}{\partial {\eta }^2}\doteq \frac{4z_5}{N{A}^2}$$

$$\frac{2{\partial }^{2}W}{\partial \xi \partial \eta }\doteq \frac{4z_6}{N{A}^2}$$

However, the second derivatives of the wavefront would need to be integrated twice to get the desired wavefront. This numerical operation would result in significant numerical overhead, except in the linearized case according to formula (5), where those would correspond directly to the Zernike coefficients: $$\frac{{\partial }^{2}W}{\partial {\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2}+\frac{{\partial }^{2}W}{\partial {\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}\doteq \frac{\mathrm{8th}{\mathrm{e.g}}_4}{\mathrm{<figure-callout\; id="2"\; label="N\; A"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">N</figure-callout>}{A}^{}{}^2}$$

$$\frac{{\partial }^{2}W}{\partial {\xi }^2}-\frac{{\partial }^{2}W}{\partial {\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}\doteq \frac{4{\mathrm{e.g}}_5}{\mathrm{<figure-callout\; id="2"\; label="N\; A"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">N</figure-callout>}{A}^{}{}^2}$$

$$\frac{2{\partial }^{2}W}{\partial \xi \partial n}\doteq \frac{4{\mathrm{e.g}}_6}{\mathrm{<figure-callout\; id="2"\; label="N\; A"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">N</figure-callout>}{A}^{}{}^2}$$

$$\frac{2{\partial }^2ƒ}{\partial \xi \partial \eta }=2\xi \eta {\displaystyle \sum _{k=1}^{\infty }\frac{4k\left(2k\right)!}{{\left(2n\right)}^{2k+1}k{!}^2}{\left({\xi }^2+{\eta }^2\right)}^{k-1}}=2\xi \eta \left[\frac{1}{n^3}+\frac{3\left({\xi }^2+{\eta }^2\right)}{2n^5}+\frac{15{\left({\xi }^2+{\eta }^2\right)}^2}{8n^7}+\cdots \right]$$

On the other hand, the coefficients of power series can be compared: $$\begin{array}{l}\frac{{\partial }^2ƒ}{\partial {\xi }^2}+\frac{{\partial }^2ƒ}{\partial n^2}={\displaystyle \sum _{k=0}^{\infty }\frac{4\left(k+1\right)\left(2k\right)!}{{\left(2n\right)}^{2k+1}k{!}^2}{\left({\xi }^2+n^2\right)}^k}=\frac{2}{n}+\frac{2\left({\xi }^2+n^2\right)}{{\mathrm{<figure-callout\; id="3"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">n</figure-callout>}}^3}+\frac{9{\left({\xi }^2+n^2\right)}^2}{4{\mathrm{<figure-callout\; id="5"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">n</figure-callout>}}^5}+\frac{5{\left({\xi }^2+n^2\right)}^3}{2n^7}+\cdots \\ \\ \frac{{\partial }^2ƒ}{\partial {\xi }^2}-\frac{{\partial }^2ƒ}{\partial n^2}=\left({\xi }^2-n^2\right){\displaystyle \sum _{k=1}^{\infty }\frac{4k\left(2k\right)!}{{\left(2n\right)}^{2k+1}k{!}^2}{\left({\xi }^2+n^2\right)}^{k-1}}\\ =\left({\xi }^2-n^2\right)\left[\frac{1}{{\mathrm{<figure-callout\; id="3"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">n</figure-callout>}}^3}+\frac{3\left({\xi }^2+n^2\right)}{2{\mathrm{<figure-callout\; id="5"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">n</figure-callout>}}^5}+\frac{15{\left({\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+n^2\right)}^2}{\mathrm{8th}{n}^7}+\cdots \right]\end{array}$$

$$\frac{2{\partial }^2ƒ}{\partial \xi \partial n}=2\xi n{\displaystyle \sum _{k=1}^{\infty }\frac{4k\left(2k\right)!}{{\left(2n\right)}^{2k+1}k{!}^2}{\left({\xi }^2+n^2\right)}^{k-1}}=2\xi n\left[\frac{1}{{\mathrm{<figure-callout\; id="3"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">n</figure-callout>}}^3}+\frac{3\left({\xi }^2+n^2\right)}{2{\mathrm{<figure-callout\; id="5"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">n</figure-callout>}}^5}+\frac{15{\left({\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+n^2\right)}^2}{\mathrm{8th}{n}^7}+\cdots \right]$$

As mentioned at the beginning, noticeably higher terms appear in the shear interferograms in highly open systems, which cannot all be attributed to a wavefront error. From their signature in the non-integrable components (∂M _ξ /∂η ≠ ∂M _η /∂ξ) the focus drift Δz can be deduced if the refractive index n is known with sufficient accuracy. The pupil coordinates (ξ,η)—in particular their scaling, that is, the numerical aperture NA—must be determined in a different way anyway. As soon as the focus drift Δz has been determined, its error ΔW can be subtracted from the wavefront W to be measured by calculation.

desto günstiger die Fehlerfortpflanzung ~ 1/NA² von den höheren Termen, aus denen Δz zu bestimmen ist, in den niedrigsten, der Z₅ enthält. Nur bei geringer numerischer Apertur werden die nicht-orthogonalen Scherungen benötigt, um die Messgenauigkeit für z₅ beziehungsweise z₆ zu halten.As a result, the astigmatism can be measured without additional shearing, either diagonally or trigonally, despite a change in the wavefront or focus drift, even in the lowest order Z ₅ . The higher the numerical aperture $\sqrt{{\xi }^2+n^2}\to n,$

the more favorable the error propagation ~ 1/NA ² from the higher terms, from which Δz is to be determined, to the lowest, which contains Z ₅ . The non-orthogonal shears are only required for low numerical apertures in order to maintain the measurement accuracy for z ₅ or z ₆ .

$$-\frac{\mathrm{\Delta }z}{2}\cdot \frac{\partial ƒ}{\partial \eta }=-\frac{\mathrm{\Delta }z}{2}\cdot \frac{\eta }{\sqrt{n^2-{\xi }^2-{\eta }^2}}=\frac{\partial \mathrm{\Delta }W}{\partial \eta }-\frac{\partial V}{\partial \xi }$$

In a further embodiment, the two shear interferograms (M _ξ ,M _η ) from formula (1) are converted into an eddy-free gradient field (∂M _ξ /∂η = ∂M _η /ξ) and into a source-free rotator field (∂M _ξ / ∂ξ + ∂M _η /∂η = 0). In other words: a vector field is determined as a function of the first wavefront gradient and the second wavefront gradient, a gradient field and a rotator field being determined as a function of the vector field. The same decomposition or determination is applied to the change in wavefront, e.g. focus drift. In other words: A drift or change in the wavefront is determined as a function of the rotator field: $$+\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\cdot \frac{\partial ƒ}{\partial \xi }=+\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\cdot \frac{\mathrm{<figure-callout\; id="2"\; label="\xi \; n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}{\sqrt{n^{}}}=\frac{\partial \mathrm{\Delta }W}{\partial \xi }+\frac{\partial V}{\partial n}$$

$$-\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\cdot \frac{\partial ƒ}{\partial n}=-\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\cdot \frac{n}{\sqrt{n^2-{\xi }^2-{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}}=\frac{\partial \mathrm{\Delta }W}{\partial n}-\frac{\partial V}{\partial \xi }$$

$$\begin{array}{c}-\frac{\partial V}{\partial \xi }=\frac{\mathrm{\Delta }z}{2}[\frac{-2n^2}{3{\left({\xi }^2+{\eta }^2\right)}^2}\left(2n-\frac{2n^2-{\xi }^2-{\eta }^2}{\sqrt{n^2-{\xi }^2-{\eta }^2}}\right)\left(1-\frac{4{\xi }^2}{{\xi }^2+{\eta }^2}\right)\\ -\frac{2{\eta }^2}{3\left({\xi }^2+{\eta }^2\right)\sqrt{n^2-{\xi }^2-{\eta }^2}}]\eta \end{array}$$

The amplitude Δz of the drift is determined by comparing the source-free components: $$\begin{array}{c}+\frac{\partial V}{\partial n}=\frac{\mathrm{\Delta }\mathrm{e.g}}{2}[\frac{2{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{3{\left({\xi }^2+n^2\right)}^2}\left(2n-\frac{2{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-{\xi }^2-n^2}{\sqrt{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-{\xi }^2-n^2}}\right)\left(1-\frac{4{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+n^2}\right)\\ +\frac{2{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2}{3\left({\xi }^2+n^2\right)\sqrt{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-{\xi }^2-n^2}}]\xi \end{array}$$

$$\begin{array}{c}-\frac{\partial V}{\partial \xi }=\frac{\mathrm{\Delta }\mathrm{e.g}}{2}[\frac{-2{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{3{\left({\xi }^2+n^2\right)}^2}\left(2n-\frac{2{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-{\xi }^2-n^2}{\sqrt{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-{\xi }^2-n^2}}\right)\left(1-\frac{4{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2}{{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+n^2}\right)\\ -\frac{2{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{3\left({\xi }^2+n^2\right)\sqrt{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-{\xi }^2-n^2}}]n\end{array}$$

$$\begin{array}{c}\frac{\partial \mathrm{\Delta }W}{\partial \eta }=\frac{\mathrm{\Delta }z}{2}[\frac{-2n^2}{3{\left({\xi }^2+{\eta }^2\right)}^2}\left(2n-\frac{2n^2-{\xi }^2-{\eta }^2}{\sqrt{n^2-{\xi }^2-{\eta }^2}}\right)\left(1+\frac{2{\xi }^2-2{\eta }^2}{{\xi }^2+{\eta }^2}\right)\\ -\frac{3{\xi }^2+{\eta }^2}{3\left({\xi }^2+{\eta }^2\right)\sqrt{n^2-{\xi }^2-{\eta }^2}}]\eta \end{array}$$

With the amplitude Δz known in this way, the associated turbulence-free components are then subtracted from the shears from formula (1): $$\begin{array}{c}\frac{\partial \mathrm{\Delta }W}{\partial \xi }=\frac{\mathrm{\Delta }\mathrm{e.g}}{2}[\frac{2{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{3{\left({\xi }^2+n^2\right)}^2}\left(2n-\frac{2{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-{\xi }^2-n^2}{\sqrt{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-{\xi }^2-n^2}}\right)\left(1+\frac{2{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-2{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2}{{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+n^2}\right)\\ +\frac{{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+3{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{3\left({\xi }^2+n^2\right)\sqrt{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-{\xi }^2-n^2}}]\xi \end{array}$$

$$\begin{array}{c}\frac{\partial \mathrm{\Delta }W}{\partial n}=\frac{\mathrm{\Delta }\mathrm{e.g}}{2}[\frac{-2{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{3{\left({\xi }^2+n^2\right)}^2}\left(2n-\frac{2{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-{\xi }^2-n^2}{\sqrt{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-{\xi }^2-n^2}}\right)\left(1+\frac{2{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2-2{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+n^2}\right)\\ -\frac{3{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{3\left({\xi }^2+n^2\right)\sqrt{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-{\xi }^2-n^2}}]n\end{array}$$

The wavefront is then reconstructed from the measurement data using conventional methods, in particular as a function of the gradient field and the drift.

Alternatively, it is possible to subtract the wavefront error ΔW from the determined focus drift Δz after the wavefront has been reconstructed directly from the measured shear interferometers, i.e $$\mathrm{\Delta }W=\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\cdot \frac{2{\mathrm{<figure-callout\; id="3"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">n</figure-callout>}}^3-\left(2{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+n^2\right)\sqrt{{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-{\xi }^2-{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}}{3{\left({\xi }^2+n^2\right)}^2}\left({\xi }^2-n^2\right)$$

On a finite domain, however, the above decomposition is not unique, because the gradient of each harmonic function Z lacks both vortices and sources due to ∂ ² Z/∂ξ ² + ∂ ² Z/∂η ² = 0. Of the scalar Zernike polynomials, which ultimately describe the wave front, only the highest ripple of a radial order is harmonic, i.e. Z _2.3 , Z _5.6 , Z _10.11 , Z _17.18 , Z _{26, 27} and so on (compare those in the following 11 table shown). The Laplace operations of all other Zernike polynomials are linearly independent, so they cannot be combined to zero.

In terms of symmetry, the focus drift shows up as a two-wave error. Of all the harmonic Zernike polynomials, only the quadratic astigmatisms Z ₅ and Z ₆ have two waves. Therefore, the above decomposition with regard to focus drift becomes unambiguous if the linear terms in the shear interferograms are factored out. As already mentioned, the linear components with the higher orders are firmly linked to one another if the refractive index is known. If the aperture is opened wide enough, at least the quadratic terms from the measurement uncertainty appear.

$$-\frac{\partial V}{\partial \xi }=-\mathrm{\Delta }z\cdot \eta {\displaystyle \sum _{k=0}^{\infty }\frac{\left(2k\right)!\left[\left(2k+1\right){\xi }^2+{\eta }^2\right]}{\left(k+2\right)k{!}^2{\left(2n\right)}^{2k+1}}}{\left({\xi }^2+{\eta }^2\right)}^{k-1}$$

beziehungsweise für die ersten Potenzen: $$+\frac{\partial V}{\partial \eta }=+\mathrm{\Delta }z\cdot \xi \left[\frac{1}{4n}+\frac{{\xi }^2+3{\eta }^2}{12n^3}+3\frac{{\xi }^2+5{\eta }^2}{64n^5}\left({\xi }^2+{\eta }^2\right)+\frac{{\xi }^2+7{\eta }^2}{32n^7}{\left({\xi }^2+{\eta }^2\right)}^2+\cdots \right]$$

$$-\frac{\partial V}{\partial \xi }=-\mathrm{\Delta }z\cdot \eta \left[\frac{1}{4n}+\frac{3{\xi }^2+{\eta }^2}{12n^3}+3\frac{5{\xi }^2+{\eta }^2}{64n^5}\left({\xi }^2+{\eta }^2\right)+\frac{7{\xi }^2+{\eta }^2}{32n^7}{\left({\xi }^2+{\eta }^2\right)}^2+\cdots \right]$$

In general it is possible to use the relevant polynomials from the Taylor expansion instead of the analytical formulas. Finally, the wavefront is also broken down into (Zernike)polynomials. Specifically, the source-free portion of the focus drift is: $$+\frac{\partial V}{\partial n}=+\mathrm{\Delta }\mathrm{e.g}\cdot \xi {\displaystyle \sum _{k=0}^{\infty }\frac{\left(2k\right)!\left[{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+\left(2k+1\right)n^2\right]}{\left(k+2\right)k{!}^2{\left(2n\right)}^{2k+1}}}{\left({\xi }^2+n^2\right)}^{k-1}$$

$$-\frac{\partial V}{\partial \xi }=-\mathrm{\Delta }\mathrm{e.g}\cdot n{\displaystyle \sum _{k=0}^{\infty }\frac{\left(2k\right)!\left[\left(2k+1\right){\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+n^2\right]}{\left(k+2\right)k{!}^2{\left(2n\right)}^{2k+1}}}{\left({\xi }^2+n^2\right)}^{k-1}$$

or for the first powers: $$+\frac{\partial V}{\partial n}=+\mathrm{\Delta }\mathrm{e.g}\cdot \xi \left[\frac{1}{4n}+\frac{{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+3{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{12{\mathrm{<figure-callout\; id="3"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">n</figure-callout>}}^3}+3\frac{{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+5{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{64n^5}\left({\xi }^2+n^2\right)+\frac{{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+7{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{32n^7}{\left({\xi }^2+n^2\right)}^2+\cdots \right]$$

$$-\frac{\partial V}{\partial \xi }=-\mathrm{\Delta }\mathrm{e.g}\cdot n\left[\frac{1}{4n}+\frac{3{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{12{\mathrm{<figure-callout\; id="3"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">n</figure-callout>}}^3}+3\frac{5{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{64n^5}\left({\xi }^2+n^2\right)+\frac{7{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{32n^7}{\left({\xi }^2+n^2\right)}^2+\cdots \right]$$

$$\frac{\partial \mathrm{\Delta }W}{\partial \eta }=\mathrm{\Delta }z\cdot \eta {\displaystyle \sum _{k=0}^{\infty }\frac{\left(2k\right)!\left[\left(k-1\right){\xi }^2-\left(k+1\right){\eta }^2\right]}{\left(k+2\right)k{!}^2{\left(2n\right)}^{2k+1}}}{\left({\xi }^2+{\eta }^2\right)}^{k-1}$$

After the amplitude Δz of the focus drift has been fitted from the source-free portion of the shear interferograms (M _ξ , M _η ) from formula (1), in particular without a linear term, the associated eddy-free portions of the focus drift, in particular with a linear term, must be subtracted : $$\frac{\partial \mathrm{\Delta }W}{\partial \xi }=\mathrm{\Delta }\mathrm{e.g}\cdot \xi {\displaystyle \sum _{k=0}^{\infty }\frac{\left(2k\right)!\left[\left(k+1\right){\xi }^2-\left(k-1\right)n^2\right]}{\left(k+2\right)k{!}^2{\left(2n\right)}^{2k+1}}}{\left({\xi }^2+n^2\right)}^{k-1}$$

$$\frac{\partial \mathrm{\Delta }W}{\partial n}=\mathrm{\Delta }\mathrm{e.g}\cdot n{\displaystyle \sum _{k=0}^{\infty }\frac{\left(2k\right)!\left[\left(k-1\right){\xi }^2-\left(k+1\right)n^2\right]}{\left(k+2\right)k{!}^2{\left(2n\right)}^{2k+1}}}{\left({\xi }^2+n^2\right)}^{k-1}$$

$$\frac{\partial \mathrm{\Delta }W}{\partial \eta }=-\mathrm{\Delta }z\cdot \eta \left[\frac{1}{4n}+\frac{2{\eta }^2}{12n^3}+3\frac{3{\eta }^2-{\xi }^2}{64n^5}\left({\xi }^2+{\eta }^2\right)+\frac{4{\eta }^2-2{\xi }^2}{32n^7}{\left({\xi }^2+{\eta }^2\right)}^2+\cdots \right]$$

So in lowest powers: $$\frac{\partial \mathrm{\Delta }W}{\partial \xi }=+\mathrm{\Delta }\mathrm{e.g}\cdot \xi \left[\frac{1}{4n}+\frac{2{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2}{12{\mathrm{<figure-callout\; id="3"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">n</figure-callout>}}^3}+3\frac{3{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2-{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{64n^5}\left({\xi }^2+n^2\right)+\frac{4{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2-2{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{32n^7}{\left({\xi }^2+n^2\right)}^2+\cdots \right]$$

$$\frac{\partial \mathrm{\Delta }W}{\partial n}=-\mathrm{\Delta }\mathrm{e.g}\cdot n\left[\frac{1}{4n}+\frac{2{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{12{\mathrm{<figure-callout\; id="3"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">n</figure-callout>}}^3}+3\frac{3{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2}{64n^5}\left({\xi }^2+n^2\right)+\frac{4{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2-2{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2}{32n^7}{\left({\xi }^2+n^2\right)}^2+\cdots \right]$$

mit den niedrigsten Potenzen: $$\mathrm{\Delta }W=\frac{\mathrm{\Delta }z}{2}\cdot \left({\xi }^2-{\eta }^2\right)\left[\frac{1}{4n}+\frac{{\xi }^2+{\eta }^2}{12n^3}+3\frac{{\left({\xi }^2+{\eta }^2\right)}^2}{64n^5}+\frac{{\left({\xi }^2+{\eta }^2\right)}^3}{32n^7}+\cdots \right]$$

Alternatively, one can subtract the integrated Taylor expansion from the reconstructed wavefront, namely: $$\mathrm{\Delta }W=\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\cdot \left({\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2-n^2\right){\displaystyle \sum _{k=0}^{\infty }\frac{\left(2k\right)!}{\left(k+2\right)k{!}^2{\left(2n\right)}^{2k+1}}}{\left({\xi }^2+n^2\right)}^k$$

with the lowest powers: $$\mathrm{\Delta }W=\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\cdot \left({\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2-n^2\right)\left[\frac{1}{4n}+\frac{{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{12{\mathrm{<figure-callout\; id="3"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">n</figure-callout>}}^3}+3\frac{{\left({\xi }^2+n^2\right)}^2}{64{\mathrm{<figure-callout\; id="5"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">n</figure-callout>}}^5}+\frac{{\left({\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+n^2\right)}^3}{32n^7}+\cdots \right]$$

nebst niedrigsten Potenzen: $$V=\frac{\mathrm{\Delta }z}{2}\cdot 2\xi \eta \left[\frac{1}{4n}+\frac{{\xi }^2+{\eta }^2}{12n^3}+3\frac{{\left({\xi }^2+{\eta }^2\right)}^2}{64n^5}+\frac{{\left({\xi }^2+{\eta }^2\right)}^3}{32n^7}+\cdots \right]$$

For the sake of completeness, the vector potential of the source-free Taylor expansion is given: $$V=\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\cdot 2\xi n{\displaystyle \sum _{k=0}^{\infty }\frac{\left(2k\right)!}{\left(k+2\right)k{!}^2{\left(2n\right)}^{2k+1}}}{\left({\xi }^2+n^2\right)}^k$$

along with the lowest potencies: $$V=\frac{\mathrm{\Delta }\mathrm{e.g}}{2}\cdot 2\xi n\left[\frac{1}{4n}+\frac{{\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+{\mathrm{<figure-callout\; id="2"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">n</figure-callout>}}^2}{12{\mathrm{<figure-callout\; id="3"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0062.png"\; state="\{\{state\}\}">n</figure-callout>}}^3}+3\frac{{\left({\xi }^2+n^2\right)}^2}{64{\mathrm{<figure-callout\; id="5"\; label="n"\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0057.png"\; state="\{\{state\}\}">n</figure-callout>}}^5}+\frac{{\left({\mathrm{<figure-callout\; id="2"\; label="\xi "\; filenames="DE102020215540B4\_0056.png,DE102020215540B4\_0061.png"\; state="\{\{state\}\}">\xi </figure-callout>}}^2+n^2\right)}^3}{32n^7}+\cdots \right]$$

According to one embodiment, according to the usual methods, the higher powers are orthogonalized to the lower ones. Details can be found in Tables 1 to 4 below. This is not necessary for a reconstruction of the focus drift induced wavefront error.

8th shows a first table in which the vector powers are grouped into source-free and vortex-free components according to increasing degree d, the harmonic components are in the middle column “d+1”; the ripples are sorted into columns: rotationally symmetrical (reference w0), single-wave (reference w1), two-wave (reference w2), three-wave (reference w3), four-wave (reference w4); by rotation, the fields merge into each other in pairs according to the ripple, the left pair is symmetric to the ξ-axis, the right pair symmetric to the η-axis; the rotationally symmetrical portions remain unpaired; the various undulations are per se orthogonal on the circle; Powers of different degrees have not been post-orthogonalized here, so that they can be more easily assigned to the above Taylor series with summation index k=(d-1)/2.

9 shows a second table. This shows that after orthogonalization of the vector powers, the vector polynomials can be further divided into source-free, harmonic and eddy-free parts; for clarity, they are presented in a third table, shown below 10 , listed Vector Zernikes illustrated. The highest power of the polynomials agrees with the in 8th match the first table shown.

10 shows the third table: The vector Zernikes are sorted by degree and number; the number (second column) refers to the single vector component corresponding to the scalar Zernikes from the following 11 correspond to the fourth table shown. The ripple is one less for vector fields (= first order tensors). With negative ripple, the pairs transform in the negative direction of rotation; the lengths of the vectors are normalized to one on the unit circle. The respective reference symbols relate to an entire line (ie both columns “vertically symmetrical” and “horizontally symmetrical”).

11 shows the fourth table: The scalar Zernike polynomials are sorted by degree and number - ie different from the usual order in optics; for scalar fields (i.e. tensors of the zeroth order) the ripple corresponds to the number. The pair on the left takes the value +1 at the boundary point (ξ,η) = (1,0).

By deriving the scalar vector Zernikes from the fourth table ( 11 ) one generates the vector polynomials in the second table ( 9 ). The ripple does not change. The gradient is per se vortex-free, for example (∂/∂ξ,∂)Z ₁₃ = 4(Z _3,-1,- - Z _3,3,- ) + 2Z _1,-1,- . Analogously, the rotation is source-free, for example (∂/∂η, - ∂/∂ξ)Z ₉ = -8Z _3,1,- - 12Z _{1,1,_} . From the harmonic Zernikes (numerality = degree) the simultaneously source-free and vortex-free fields are derived, for example (∂/∂ξ,∂/∂η)Z ₁₀ = (∂/∂η, - ∂/∂ξZ ₁₁ = 3Z _2,-2,+ .

Claims (18)

Method for determining an imaging quality of an imaging system (2) with the following steps: a) Radiating a wavefront into the imaging system (2), b) detecting the wavefront after the wavefront has passed through the imaging system (2), the detection of the wavefront including a first partial detection of the wavefront to determine a first wavefront gradient in a first direction and at least a second partial detection of the wavefront to determine a second wavefront gradient in a second direction, the wavefront of the first partial acquisition and the wavefront of the second partial acquisition differing from one another, the wavefront of the first partial acquisition and the wavefront of the second partial acquisition differing from one another in that between the first partial acquisition and the second partial acquisition there is drifting of a focus of the imaging system (2), and wherein the first direction and the second direction are not aligned orthogonally to one another, c) reconstructing the wavefront as a function of the first wavefront gradient and the second wavefront gradient, with measurement errors caused by the drifting of the focus being eliminated by calculation by reconstructing the wavefront. procedure after claim 1 , characterized in that the second partial detection is carried out chronologically after the first partial detection. procedure after claim 1 or 2 , characterized in that the first partial detection and the second partial detection takes place after an interaction of the wave front with a diffractive element (9), the diffractive element (9) having a first grating with a first grating vector and at least one second grating with a second grating vector , wherein the first lattice vector is aligned along the first direction and the second lattice vector is aligned along the second direction. procedure after claim 1 or 2 , characterized in that the first partial detection takes place after an interaction of the wave front with a first diffractive element (13) and the second partial detection takes place after an interaction of the wave front with at least one second diffractive element (14), the first diffractive element (13) a first grating having a first grating vector aligned along the first direction, and the second diffractive element (14) having a second grating having a second grating vector aligned along the second direction. Method according to any of the preceding claims 3 or 4 , characterized in that the first lattice vector is oriented along the first direction and the second lattice vector is oriented along the second direction such that the first lattice vector and the second lattice vector form an angle of 45° or 135°. Method according to one of the preceding claims, characterized in that the first and the second wavefront gradient are detected as a function of an interference pattern or a point pattern. Method according to one of the preceding claims, characterized in that the detection of the wavefront comprises at least a third partial detection of the wavefront to determine a third wavefront gradient in a third direction, the first direction, the second direction and the third direction being aligned differently from one another. procedure after claim 7 , characterized in that the first partial detection, the second partial detection and the third partial detection takes place after an interaction of the wave front with a diffractive element (15), the diffractive element (15) having a first grating with a first grating vector, a second grating with a second lattice vector and at least one third lattice vector having a third lattice vector, wherein the first lattice vector is oriented along the first direction, the second lattice vector is oriented along the second direction, and the third lattice vector is oriented along the third direction. Method according to any of the preceding Claims 7 or 8th , characterized in that the first, second and third partial detection are carried out sequentially. Device (1) for determining an imaging quality of an imaging system (2), having: a) a radiation source (3) for generating radiation, b) a wavefront generation unit (4) for generating a wavefront from the radiation, the wavefront passing through the imaging system (2), c) a detector unit (7) for detecting the wavefront after the wavefront has passed through the imaging system (2), the detection of the wavefront including a first partial detection of the wavefront to determine a first wavefront gradient in a first direction and comprises at least a second partial detection of the wavefront to determine a second wavefront gradient in a second direction, wherein the wavefront of the first partial acquisition and the wavefront of the second partial acquisition differ from one another, the wavefront of the first partial acquisition and the wavefront of the second partial acquisition differing from one another in that a focus of the imaging system (2) drifts between the first partial acquisition and the second partial acquisition takes place, and where the first direction and the second direction are not aligned orthogonally to one another, d) a reconstruction unit (11) for reconstructing the wavefront as a function of the first wavefront gradient and the second wavefront gradient, set up in such a way that measurement errors caused by drifting of the focus are mathematically eliminated by the reconstruction of the wavefront. device after claim 10 , characterized in that the device (1) is designed to carry out the second partial detection chronologically after the first partial detection. Device according to one of the preceding Claims 10 or 11 , characterized by a diffractive element (9), wherein the diffractive element (9) has a first grating with a first grating vector and at least one second grating with a second grating vector, wherein the first grating vector is aligned along the first direction and the second grating vector along the second direction is aligned. Device according to one of the preceding Claims 10 or 11 , characterized by a first diffractive element (13), the first diffractive element having a first grating with a first grating vector aligned along the first direction, and by at least one second diffractive element (14), the second diffractive element a second lattice having a second lattice vector aligned along the second direction. Device according to one of the preceding Claims 10 until 13 , Characterized by a control unit which is designed to carry out the method according to one of Claims 1 until 9 to perform. Method for determining an imaging quality of an imaging system (2) with the following steps: a) irradiating a wavefront into the imaging system (2), b) detecting the wavefront after the wavefront has passed through the imaging system (2), the detection of the wavefront including a first partial detection of the wavefront to determine a first wavefront gradient in a first direction and at least a second partial detection of the wavefront to determine a second wavefront gradient in a second Includes direction, wherein the wavefront of the first partial detection and the wavefront of the second partial detection differ from each other, c) determining a vector field depending on the first wavefront gradient and the second wavefront gradient, d) determining a gradient field and a rotator field depending on the vector field, e) determining a drift of the wave front as a function of the rotator field, f) reconstructing the wave front as a function of the gradient field and the drift. Device (1) for determining an imaging quality of an imaging system (2), having: a) a radiation source (3) for generating radiation, b) a wavefront generation unit (4) for generating a wavefront from the radiation, the wavefront passing through the imaging system (2), c) a detector unit (7) for detecting the wavefront after the wavefront has passed through the imaging system (2), the detection of the wavefront including a first partial detection of the wavefront to determine a first wavefront gradient in a first direction and comprises at least a second partial detection of the wavefront to determine a second wavefront gradient in a second direction, wherein the wavefront of the first partial acquisition and the wavefront of the second partial acquisition differ from each other, d) an evaluation unit (12) set up to determine a vector field as a function of the first wavefront gradient and the second wavefront gradient and to determine a gradient field and a rotator field as a function of the vector field and e) a reconstruction unit (11) set up for reconstructing the wavefront as a function of the gradient field and a drift of the wavefront that is determined as a function of the rotator field. Projection exposure system for microlithography, having a projection objective, characterized in that the projection objective is a device (1) according to one of Claims 10 until 14 and/or a device (1). Claim 16 includes. Projection exposure system for microlithography, having a projection objective, characterized in that the projection objective for carrying out a method according to one of Claims 1 until 9 and/or a procedure claim 15 is prepared.

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