CN114488522B

CN114488522B - Optical description method for smoothly splicing free-form surfaces

Info

Publication number: CN114488522B
Application number: CN202210096176.6A
Authority: CN
Inventors: 程德文; 陈海龙; 杨通; 王涌天
Original assignee: Beijing Institute of Technology BIT
Current assignee: Beijing Institute of Technology BIT
Priority date: 2022-01-26
Filing date: 2022-01-26
Publication date: 2023-09-05
Anticipated expiration: 2042-01-26
Also published as: CN114488522A

Abstract

The invention discloses an optical description method for smoothly splicing free curved surfaces. The optical curved surface is formed by splicing the two sub-curved surfaces, and a special constraint condition for ensuring that the two sub-curved surfaces have the first-order continuous characteristic at the splicing position is obtained by establishing an equation set for enabling the equation set to meet the first-order continuous characteristic and solving the equation set, so that the smooth spliced optical free-form curved surface is finally generated. The curved surface description method is simple and easy to realize. The curved surface is particularly suitable for an imaging optical system with complex functions, on one hand, more optimized variables can be provided for system design, and the imaging quality of the system is improved; on the other hand, the curved surface has the characteristic of smoothness and continuity, so that the processing and testing difficulty can be reduced, and the cost can be saved.

Description

Optical description method for smoothly splicing free-form surfaces

Technical Field

The invention relates to the technical field of optical design, in particular to an optical description method for smoothly splicing free curved surfaces.

Background

Generally, in the case of performing an optimal design of an optical system, if design indexes are not changed, the main factors affecting the quality of the design result are the number of variables and the structural form of the system. The splice curves are a very good choice from the standpoint of adding design variables to the optical system. For systems with annular segmented aperture, sasian presents an annular surface, and the results of application to practical optical system designs indicate that the use of such a curved surface can improve the imaging quality of the system, however, the center of the curved surface has sharp corners, does not have continuity, and the processing problems are not discussed. The continuity of the spliced optical curved surface has important significance for the application, firstly, the continuous surface is beneficial to design, the generation of stray light can be reduced, the discontinuity of wave fronts is avoided, and secondly, the difficulty and cost of processing and detecting the spliced optical curved surface can be reduced. The continuity of the curved surfaces shows the smoothness of the curved surfaces, the zero-order continuity indicates that the two curved surfaces have the same function value at the joint, and the first-order continuity indicates that the two curved surfaces have the same function value at the joint and the same first derivative value. It is generally believed that optical surfaces need to have at least first order continuous characteristics to meet design and processing requirements. Cheng et al propose an ASAS (continuous smooth stitching) surface suitable for annular aperture which not only increases the number of variables for the optical system design, but also maintains the first order continuity between adjacent sub-surfaces, providing additional optimization variables for the system, while also providing excellent processability. But for non-annular aperture systems, there is no one spliced free-form surface with a first order continuity.

Disclosure of Invention

In view of this, the invention provides an optical description method for smoothly splicing free-form surfaces, which makes a spliced surface formed by splicing two sub-surfaces have a first-order continuous characteristic by constraining coefficients of an equation, so that more optimized variables can be provided for optical system design, and meanwhile, the processing and the testing are convenient.

In order to achieve the above purpose, the technical scheme of the invention comprises the following steps:

step 1, randomly generating a curved surface alpha by using a group of polynomials, and dividing the curved surface into two sub-curved surfaces alpha by using a plane gamma ₁ And alpha ₂ 。

And 2, maintaining the polynomial coefficient of one of the sub-surfaces unchanged, and adding a disturbance term to the polynomial coefficient of the other sub-surface to obtain two discontinuous and unsmooth sub-surfaces.

And step 3, splicing the two discontinuous and unsmooth self-curved surfaces, constructing a boundary condition equation set by taking the first-order continuous property of the two sub-curved surfaces at the splicing position as a constraint condition, and solving the boundary condition equation set.

And 4, superposing a base curved surface on the basis of two sub-curved surfaces with first-order continuous properties to obtain the description of the smooth spliced free curved surface.

Preferably, the set of polynomials in step 1 has the following specific equations:

z(x，y)＝∑C _m，n x ^m y ⁿ ，(m+n)≥1.

wherein m and n are both nonnegative integers, C _m,n For polynomial coefficients, (x, y) is the coordinates on the curved surface α, and z (x, y) represents the curved surface α.

Further, in step 1, the plane γ is parallel to the XOZ plane, a space coordinate system is randomly constructed, the center point is O, three axes XYZ axes, and the passing point (x, y ₀ ,z)，y ₀ Is the intersection point of the plane gamma and the y axis; wherein the plane gamma intersects the curved surface alpha at a space curve l.

Further, in step 1, two sub-curved surfaces α ₁ And alpha ₂ The equation of (2) is expressed as:

wherein z is ₁ (x, y) represents a sub-curved surface alpha ₁ ；z ₂ (x, y) represents a sub-curved surface alpha ₂ The method comprises the steps of carrying out a first treatment on the surface of the The two sub-surfaces have a k-1 order continuity at the space curve l, and k is the sum of the maximum values of m and n at the time, namely the highest order degree of the polynomial; y is ₀ Is the intersection of plane gamma and y-axis.

Further, in step 2, after adding the disturbance term to the polynomial coefficient of the other sub-surface, two sub-surfaces α ₁ And alpha ₂ The equation of (2) is expressed as:

at this time, two sub-curved surfaces alpha ₁ And alpha ₂ There is no continuity at curve l.

Further, in step 3, the two discontinuous and unsmooth self-curved surfaces are spliced, and the two sub-curved surfaces have first-order continuous properties at the splicing position as constraint conditions, so that a boundary condition equation set is constructed as follows:

wherein;

in order to make the equation set have and have unique solutions, let y ₀ ＝0；

When the highest order of the polynomial is the 10 th order, the unique solution of the system of equations is:

preferably, in step 4, the curved surface of the substrate is one of a sphere, a quadric, a hyperboloid or other second order curved surface.

Further, in step 4, when the highest order of the polynomial is 10 th order and the base surface selects a quadric, the equation with the first-order continuous smooth stitched free-form surface is expressed as:

preferably, in step 4, the maximum order of 10 th order of the stitched free-form surface provides 45 more variables for the optical system design than the conventional XY polynomial free-form surface.

Further, in step 1, the plane γ is parallel to the YOZ plane, and the plane γ passes through the point (x ₀ Y, z), and then repeating the steps 2-4 to obtain the optical description method of the free-form surface which is smoothly spliced.

The beneficial effects are that:

the method constrains equation coefficients of two free-form surfaces by constructing and solving the first-order continuity equation set, so that the free-form surfaces have first-order continuity characteristics, a smooth spliced free-form surface description method is formed, the surfaces can provide more optimized variables for optical system design, the imaging quality of the system is improved, and the smooth characteristics of the surfaces are convenient for processing and testing.

Drawings

FIG. 1 shows a polynomial random surface alpha divided into alpha by a plane gamma in step 1 of the present invention ₁ And alpha ₂ Schematic of (2);

FIG. 2 is a diagram of the boundary conditions that need to be considered in constructing a first order continuity equation in the present invention;

FIG. 3 is a schematic diagram of an off-axis reflector system with a primary three-mirror co-substrate;

FIG. 4 is a schematic diagram of an off-axis reflector system with a principal three-mirror co-substrate and described by the same surface equation;

FIG. 5 is a wave aberration profile of the optical system of FIG. 4 over the full field of view;

FIG. 6 is a flow chart of an optical description method for smoothly splicing free-form surfaces.

Detailed Description

The invention will now be described in detail by way of example with reference to the accompanying drawings.

The existing freeform surface description modes are various, one of the most common modes is an XY polynomial optical freeform surface, and the XY polynomial optical freeform surface has good flexibility and excellent programmability and is widely used in the design of imaging optical systems. Taking an XY polynomial optical freeform surface with the highest order of 10 as an example, its sagittal height is mainly composed of two parts, one part is the sagittal height provided by the base term quadric surface and the other part is the additional sagittal height provided by the higher order term. Quadrics have rotational symmetry, and many complex surfaces have quadrics as the base term. While the curved shape described by the polynomial is more free. An equation for an XY polynomial optical free-form surface can be expressed as:

wherein C is the curvature of the surface vertex, k is the quadric constant, C _m,n As higher-order term coefficients, r ² ＝x ² +y ² 。

The invention provides a smooth spliced free-form surface optical description method.

The free-form surface is formed by splicing two sub-surfaces described by XY polynomials, and the XY polynomials describing the two sub-surfaces should have the same order and different coefficients. However, the simple splicing cannot ensure that the two sub-surfaces have the first-order continuous characteristic at the joint, and reasonable constraint needs to be made for the higher-order term coefficients of the two sub-surfaces so as to construct the smooth spliced optical free-form surface. The construction process is shown in fig. 6, and comprises the following steps:

step 1, randomly generating a curved surface alpha by using a group of polynomials, and dividing the curved surface into two sub-curved surfaces alpha by using a plane gamma ₁ And alpha ₂ . The specific equation for this set of polynomials can be expressed as:

z(x，y)＝∑C _m，n x ^m y ⁿ ，(m+n)≥1.

wherein m and n are both nonnegative integers, C _m,n Is a polynomial coefficient. m and n do not set an upper limit. For convenience of the following description, let m+n.ltoreq.10 here, i.e. the highest order of the set of polynomials is 10 th order. In order to obtain two discontinuous sub-curved surfaces, as shown in fig. 1, a plane gamma is first taken to pass through the curved surface alpha, the plane gamma intersects with the curved surface alpha to form a space curve l, and the curved surface alpha is divided into two sub-curved surfaces alpha ₁ And alpha ₂ . The plane γ may be parallel to the XOZ plane or perpendicular to the XOZ plane, and the derivation process in the subsequent steps is similar in both cases, and the following description is continued with the example that the plane γ is parallel to the XOZ plane, and the smooth splicing method when the plane γ is perpendicular to the XOZ plane may be derived only by simple modification.

When the plane γ is parallel to the XOZ plane, it is assumed that its passing point (x, y ₀ Z). The plane gamma is parallel to the XOZ plane, a spatial coordinate system is randomly constructed with a center point of O, three axes XYZ axes, and a pass point (x, y ₀ ,z)，y ₀ Is the intersection point of the plane gamma and the y axis; wherein the plane gamma intersects the curved surface alpha at a space curve l.

At this time, two sub-curved surfaces alpha ₁ And alpha ₂ The equation for (2) can be expressed as:

at this time, the two sub-surfaces can be regarded as different regions of the same surface, and thus still have k-1 order continuity at the curve l, where k is the polynomial highest order degree, i.e., 9 order continuity.

Step 2, to break the existing k-1 order continuity, one of the sub-surfaces α is maintained ₁ The polynomial coefficient of (a) is unchanged for another sub-surface alpha ₂ The polynomial coefficient of (2) increases the perturbation to obtain two discontinuous and non-smooth sub-surfaces. After adding the disturbance, two sub-curved surfaces alpha ₁ And alpha ₂ Is no longer equal, and its equation can be expressed as:

If no constraint is added, the two sub-surfaces may not have any order of continuity at curve l.

Step 3, in order to ensure two sub-curved surfaces alpha ₁ And alpha ₂ At least one first-order continuity is provided at the curve l, and an equation set which enables the equation set to meet the first-order continuity is established by using equations of two sub-curved surfaces and solved. As shown in fig. 2, to ensure first order continuity at the curved surface splicing position, the set of boundary condition equations to be established at the curve l can be expressed as:

wherein, the formula (1) shows that both the sub-curved surfaces pass through the space curve l, that is, when the formula (1) is established, the two sub-curved surfaces satisfy the characteristic of 0 th order continuity. Equations (2) and (3) show that the first order partial derivatives of the two sub-surfaces at curve l are equal, i.e., when equations (2) and (3) are true, the two sub-surfaces satisfy the 1 st order continuous characteristic. Through the above 3 equations, two sub-surfaces can be constrained to satisfy at curve lA first order continuous nature. Wherein C is _m,n Is known to D _m,n Is unknown. And expanding the equation set and combining the same-order terms to obtain an indefinite equation set with more unknowns than equations, and resolving the indefinite equation set without analysis.

Further, assume y ₀ And (0) simplifying the equation set according to the condition, and obtaining a positive equation set containing 20 equations and 20 unknowns after finishing, wherein the positive equation set has a unique analytical solution. In the case where the highest order of the polynomial is the 10 th order, the unique solution of the system of equations can be expressed as:

that is, in order to satisfy the first-order continuous characteristic of the two sub-curved surfaces at the joining position, i.e., curve l, it is necessary to have D subscripts (1, 0), (2, 0), (3, 0), (4, 0), (5, 0), (6, 0), (7, 0), (8, 0), (9, 0), (10, 0), (0, 1), (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (7, 1), (8, 1), (9, 1) for the subscripts (1, 0), (2, 0), (3, 0), (4, 0), (5, 0), (6, 0), (7, 0), (8, 0) _m,n And C _m,n Equal.

The continuity of the spliced curved surface is not limited to the first order, and in order to obtain the spliced curved surface with the second-order continuity, the third-order continuity … and the like, an equation set of the continuity boundary condition can be expanded and solved, and the continuous curved surface is realized by further restraining coefficients of two sub-curved surfaces.

And 4, superposing a base curved surface on the basis of the two sub-curved surfaces with the first-order continuous property to obtain a description equation of the spliced free-form curved surface. The substrate curved surface can be a spherical surface, a quadric surface, a hyperboloid, other second-order curved surfaces and the like. When the highest order of the polynomial is 10 th order and the base surface selects one quadric, the equation of the smoothly spliced freeform surface can be expressed as:

a first order continuous stitched free-form surface with a 10 th order maximum may provide 45 more variables to the optical system than an XY polynomial free-form surface of the same order. As the highest order increases, so does the number of variables that can be provided additionally.

To use the surface description method in an actual optical system design, the surface description method may be instantiated by a computer programming language and compiled into a ". Dll" file that may be called by the optical design software.

The following describes the application process and beneficial effects of the smooth stitching freeform surface optical description method in the design of an actual optical system by an example.

By using a smooth spliced free-form surface optical description equation to design an off-axis three-reflecting system as an example, the feasibility of the spliced curved surface is verified, the imaging quality of the optical system can be improved by verifying the extra variable provided by the smooth spliced curved surface for the optical system, and the difficulty in processing, assembling and detecting the optical system can be saved by verifying the smooth spliced curved surface.

As shown in fig. 3, an off-axis three-mirror imaging optical system. The design wavelength is 0.4-0.8 microns, and the angle of view is 4 degrees by 4 degrees. The focal length of the system is 95mm, and the F number is equal to 2.4. The system includes three independent mirrors M1, M2, and M3, each described using an XY polynomial free-form surface. In order to reduce the degree of freedom of adjustment, the main mirror M1 and the three mirrors M3 may be processed on the same substrate. And when in adjustment, M1 and M3 can be used as a whole for adjustment, so that the adjustment difficulty is reduced. However, the equations and coefficients of the two curved surfaces M1 and M3 are still independent, and the two surfaces can be processed only by clamping twice during processing. Secondly, during detection, the two curved surfaces also need to be detected separately.

Considering the difficulty of processing and detection, on the basis of the optical path structure and design parameters of fig. 3, M1 and M3 are not only located on the same substrate, but also described by a curved surface equation, as shown in fig. 4. In general, one XY polynomial free-form surface can be used to describe M1/M3 simultaneously. Therefore, during processing, the processing of M1/M3 can be completed only by one clamping, and the detection of M1/M3 can also be completed at one time. The system reduces the difficulty of assembly and adjustment and simultaneously reduces the complexity of processing and detecting the main three mirrors.

Although describing the curved surface shape of the main three mirrors with only one expression does promote the practicality of the system, in the case of keeping the order of the XY polynomial free-form surface used the same, the XY polynomial free-form surface that can be used in the design of fig. 4 is changed from three to two compared with the design of fig. 3, thus reducing the number of optimization variables that can be used in the system, resulting in an increase in the difficulty of the design of the system, and difficulty in meeting the design requirement.

Compared with the traditional XY polynomial free-form surface with the same order, the smooth spliced free-form surface has more higher-order term coefficients, and the number of optimized variables of the system can be increased. Simultaneously, the smooth spliced free curved surface has a first-order continuous characteristic, and the two sub-curved surfaces can be processed through one-time clamping. The detection of the two sub-surfaces may also be performed simultaneously.

To further illustrate the beneficial effects of a smooth stitched free-form surface in an actual optical system design, two systems were designed according to the design parameters of the system of fig. 3. Firstly, M1/M3 is described by an XY polynomial free-form surface of order 10, and a first system is obtained after full optimization, as shown in fig. 4 (a); and describing M1/M3 by using a 10-order smoothly spliced free-form surface, and obtaining a second system after sufficient optimization, as shown in fig. 4 (b), wherein O represents the boundary position of two sub-surfaces included in the smoothly spliced free-form surface. In both systems, the diaphragm positions are identical, and M2 is described using a 5 th order XY polynomial free-form surface. In the optimization process, the same physical constraint condition is set for the two systems. In the two systems, the highest order of the XY polynomial free-form surface and the smooth spliced free-form surface is used to 10 orders, so that the comparability of design results is ensured. Since the optical system shown in fig. 4 has a symmetrical characteristic about the YOZ plane, all the odd term coefficients about x in the XY polynomial free-form surface and the smooth spliced free-form surface are set to 0, and do not participate in optimization.

Under the premise of considering that the curved surface has plane symmetry, compared with a 10-order XY polynomial free-form surface, the 10-order smooth spliced free-form surface can provide an additional 25 variables for the system. Fig. 5 shows the root mean square wave aberration over the full field of view of the first system and the second system, respectively. As shown in FIG. 5 (a), the average wavefront error at 546.1nm for a system using a conventional XY polynomial freeform surface is λ/6. And as shown in fig. 5 (b), the average wavefront error at the same wavelength for a system using a smoothly stitched free-form surface is λ/12. The design result shows that: first, the second system has better imaging quality than the first system because the smooth stitching of the free-form surfaces can provide more design variables to the system, increasing design flexibility. Secondly, the second system has similar difficulty in adjustment, processing and testing as the first system, and good mechanical properties are maintained because the smooth spliced free-form surface has first-order continuous characteristics.

The smooth spliced free-form surface optical description method can well complete the design of a complex optical system, improves the imaging quality of an actual imaging system, reduces the processing, adjustment and test difficulties, is convenient and simple, has strong applicability, and can be used for free-form surface imaging systems with various application and various system structures to obtain better system design results.

In summary, the above embodiments are only preferred embodiments of the present invention, and are not intended to limit the scope of the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention should be included in the protection scope of the present invention.

Claims

1. An optical description method of a smooth spliced freeform surface, comprising: the construction process comprises the following steps:

step 1, randomly generating a set of polynomialsA curved surface alpha, and dividing the curved surface into two sub-curved surfaces alpha by a plane gamma ₁ And alpha ₂ The method comprises the steps of carrying out a first treatment on the surface of the The plane gamma intersects with the curved surface alpha to form a space curve l; randomly constructing a spatial coordinate system such that plane gamma is parallel to the XOZ plane and passes through points (x, y ₀ Z); the set of polynomials, whose specific equations are:

z(x，y)＝∑C _m，n x ^m y ⁿ ，(m+n)≥l

wherein m and n are non-negative integers, C _m,n For polynomial coefficients, (x, y) is the coordinates on the curved surface α, z (x, y) represents the curved surface α;

two sub-curved surfaces alpha ₁ And alpha ₂ The equation of (2) is expressed as:

wherein z is ₁ (x, y) represents a sub-curved surface alpha ₁ ；z ₂ (x, y) represents a sub-curved surface alpha ₂ The method comprises the steps of carrying out a first treatment on the surface of the The two sub-surfaces have a k-1 order continuity at the space curve l, and k is the sum of the maximum values of m and n at the time, namely the highest order degree of the polynomial; y is ₀ Is the intersection point of the plane gamma and the y axis;

step 2, maintaining the polynomial coefficient of one of the sub-surfaces unchanged, adding a disturbance term to the polynomial coefficient of the other sub-surface to obtain two discontinuous and unsmooth sub-surfaces, wherein the two sub-surfaces are alpha ₁ And alpha ₂ The equation of (2) is expressed as:

at this time, two sub-curved surfaces alpha ₁ And alpha ₂ There is no continuity at curve l;

step 3, splicing the two discontinuous and unsmooth self-curved surfaces, and constructing a boundary condition equation set by taking the first-order continuous property of the two sub-curved surfaces at the splicing position as a constraint condition, wherein the boundary condition equation set is as follows:

and solving; in order to make the equation set have and have unique solutions, let y ₀ =0; when the highest order of the polynomial is 10 th order, the unique solution of the boundary condition equation set is:

step 4, superposing a base curved surface on the basis of two sub-curved surfaces with first-order continuous properties to obtain a description of a smooth spliced free curved surface, wherein the base curved surface is one of a spherical surface, a quadric surface, a hyperboloid or other second-order curved surfaces; when the highest order of the polynomial is 10 th order and the base surface selects a quadric surface, the equation with a first order continuous smooth stitched free-form surface is expressed as:

。

2. the method according to claim 1, wherein in the step 4, the maximum order of 10 is 45 more variables for the optical system design than the conventional XY polynomial free-form surface.

3. The method according to claim 1, wherein in the step 1, the plane γ is parallel to the YOZ plane, and the plane γ passes through the point (x ₀ Y, z), and then repeating the steps 2-4 to obtain the optical description method of the free-form surface which is smoothly spliced.