A design method for square optical elements based on free-form surfaces

Abstract

The present invention relates to the field of optical design technology, and specifically to a design method for a square optical element based on a free-form surface. The present invention uses the Graham-Schmidt orthogonalization method on a square domain to establish a polynomial that satisfies the standard orthogonal property for a traditional XY polynomial, so that the integral of the scalar product of any two polynomials in this series of polynomials within this domain is zero, and the integral of the self-product of any polynomial within the domain is equal to 1. A quadratic surface basis term is superimposed on the basis of this series of orthogonal polynomials to realize a free-form surface suitable for the design of a square optical element. The method of the present invention is simple and easy to implement.

Description

Technical Field

[0001] The present invention relates to the technical field of optical design, and in particular to a design method of a square optical element based on a free-form surface. Background Art

[0002] With the development of optical technology, people have put forward higher requirements on the performance of optical systems. In order to improve the performance of the system, free-form surfaces are increasingly used in optical design. At present, the most commonly used optical free-form surface is the XY polynomial surface. However, the XY polynomial does not have an orthogonal characteristic, and it is not easy to achieve stable convergence when using the surface for optical system optimization design. Zernike polynomial surface is a polynomial surface with orthogonal characteristics in a circular domain, which plays an important role in the design of optical systems. For optical elements with a circular shape and rotational symmetry, Zernike polynomial surfaces are very suitable for the design of systems composed of these elements. With the development of optical technology, there are more and more systems with square lenses / pupils, but in the square domain, Zernike polynomial surfaces do not have orthogonality. Therefore, a surface with orthogonality in a square domain is needed to be suitable for the design of optical elements with a square shape. Summary of the invention

[0003] In view of this, the present invention provides a design method for a square optical element based on a free-form surface. The present invention explains how to generate a polynomial function with standard orthogonal characteristics in a square domain and similar to the Zernike polynomial composition, and realizes an optical free-form surface with orthogonal XY polynomials in a square domain. The free-form surface is particularly suitable for the design of optical elements with a square shape. The surface can improve the robustness of the optical system design and the imaging quality of the system while maintaining the design freedom, and is convenient for processing.

[0004] The technical solution of the present invention is:

[0005] A design method for a square optical element based on a free-form surface, the method refers to designing the surface shape of the square optical element, the surface shape of the square optical element is a free-form surface, and the free-form surface includes a basis term and an XY polynomial with a standard orthogonal property;

[0006] The base term is a sphere, a quadratic surface, a biquadric surface or a second-order surface;

[0007] The quadratic surface basis term has the following expression:

[0008]

[0009] Where c is a real number, representing the vertex curvature; k is a real number, representing the quadratic surface constant; r = (x 2 +y2 ) 0.5 , represents the distance from a point on the surface to the optical axis; z is a real number, representing the surface sag;

[0010] The construction process of the XY polynomial with standard orthogonal characteristics is as follows:

[0011] In the first step, a generalized polynomial consisting of a linear combination of a set of linearly independent function systems is selected;

[0012] The second step is to determine the domain of the XY polynomial function based on the generalized polynomial in the first step;

[0013] The third step is to calculate the integral value of the scalar product of any two polynomials in the generalized polynomial in the first step within the definition domain determined in the second step, the integral value being defined as the first integral value, and the integral value of the self-product of any polynomial within the definition domain, the integral value being defined as the second integral value, to obtain a system of equations;

[0014] Step 4, solving the coefficient of each monomial in the generalized polynomial in the first step according to the system of equations obtained in the third step, and obtaining an XY polynomial with standard orthogonal characteristics;

[0015] In the first step, the independent variables of the linearly independent function system are set to x, y, and the dependent variable is set to p(x, y), wherein the domain of the independent variable is a square;

[0016] The dependent variable p(x,y) has the following expression:

[0017] p(x,y):x,y,x 2 ,xy,y 2 , x 3 , x 2 y,xy 2 ,y 3 , x 4 ,…

[0018] The generalized polynomial consisting of the linear combination of the dependent variables p(x,y) is set to The specific expression of this set of generalized polynomials is:

[0019]

[0020]

[0021]

[0022]

[0023]

[0024] …

[0025]

[0026] Among them, a j ,b j ,c j ,…are real numbers, representing the coefficients of each monomial in the polynomial; j is a positive integer, representing the sequence number of the polynomial and its coefficients; each polynomial is composed of a series of functions in a linearly independent function system through linear combination;

[0027] In the second step, the domain of the XY polynomial function is normalized, and the domain of the XY polynomial function is {x∈[-1,1], y∈[-1,1]};

[0028] In the third step, the method of obtaining the system of equations is:

[0029] In a generalized polynomial, the integral of the scalar product of any two polynomials in the domain is zero, and the integral of the self-product of any polynomial in the domain is 1; the integral value is set to I m,n , the integral value expression is:

[0030]

[0031] Wherein, m and n represent the mth term and nth term of the generalized polynomial respectively, and m and n are positive integers;

[0032] The system of equations obtained based on the integral value is:

[0033]

[0034] Among them, when m≠n, I m,n Represents the first integral value. The above formula represents the integral of the scalar product of two different polynomials in the domain of definition. When m=n, I m,n represents the second integral value, and the above formula represents the integral of a polynomial multiplied by itself;

[0035] In the fourth step, according to the equations The method for calculating the coefficients of the monomials in is as follows: when j = 1,

[0036]

[0037] Solved

[0038] When j = 2,

[0039]

[0040]

[0041] Solved b 2 =0;

[0042] The coefficient values ​​of the high-order polynomials are calculated in sequence to obtain a series of polynomials that satisfy the standard orthogonal characteristics; the XY polynomial with the standard orthogonal characteristics is combined with a quadratic surface basis term to form a free-form surface that can be used for optical system design. The expression of the free-form surface is:

[0043]

[0044] Where c is a real number, representing the vertex curvature; k is a real number, representing the quadratic surface constant; r = (x 2 +y 2 ) 0.5 , represents the distance from a point on the surface to the optical axis; z is a real number, representing the surface vector height; j is a positive integer, representing the serial number of each standard orthogonal polynomial and its coefficients; C j is a real number, representing the coefficient of each standard orthogonal polynomial;

[0045] The domain is {x∈[-1,1],y∈[-1,1]}. To construct an XY polynomial with standard orthogonal properties, the basis function is a generalized polynomial of a linear combination of any set of linearly independent function systems. The generalized polynomial is generated by the function system p(x,y), and p(x,y) has the following expression:

[0046] p(x,y):x,y,x 2 ,xy,y 2 , x 3 , x 2 y,xy 2 ,y 3 , x 4 ,…

[0047] There is no upper limit on the highest power of p(x,y);

[0048] The function system p(x,y) is a generalized polynomial in a given domain:

[0049]

[0050]

[0051]

[0052]

[0053]

[0054] …

[0055]

[0056] Among them, a j ,b j ,c j ,…are real numbers, representing the coefficients of each monomial in the polynomial; j is a positive integer, representing the sequence number of the polynomial and its coefficients;

[0057] In order to make the generalized polynomials have orthogonal characteristics, according to the properties of orthogonal polynomials: let the integral of the scalar product of any two polynomials in the domain be zero, and the integral of any one of the polynomials in the domain be not equal to 0. In order to obtain polynomials with standard orthogonal characteristics, the integral of any one of the polynomials is equal to 1.

[0058] The integral of the polynomial scalar product in the domain of definition is:

[0059]

[0060] According to the formula, all coefficients a in the generalized polynomial are solved in turn. j ,b j ,c j ,…, and then determine the specific expression of each generalized polynomial in turn.

[0061] Optical freeform surface with orthogonal XY polynomials in a square domain, the equation is:

[0062]

[0063] Where c is a real number, representing the vertex curvature; k is a real number, representing the quadratic surface constant; r = (x 2 +y 2 ) 0.5 , represents the distance from a point on the surface to the optical axis; z is a real number, representing the surface sag; C j are real numbers representing the coefficients of each standard orthogonal polynomial.

[0064] Beneficial Effects

[0065] (1) The present invention selects a set of linearly independent function systems and obtains a set of generalized polynomials through linear combination, so that the integral of the scalar product of any two different generalized polynomials in the square definition domain is zero, and the integral of the self-product of any generalized polynomial in the square definition domain is 1, thereby obtaining a set of XY polynomials with orthogonality in the square domain. A quadratic surface is superimposed on the above-mentioned set of XY polynomials to generate an optical free-form surface, which is particularly suitable for the design of optical elements with a square shape.

[0066] (2) In the method of the present invention, the quadratic surface basis term describes the main shape of the optical element, which is beneficial to the rough processing of the optical element blank. The orthogonal XY polynomial describes the parts other than the main shape, providing additional variables and degrees of freedom for the design of the optical system.

[0067] (3) In the method of the present invention, the basis function system for constructing the orthogonal polynomial is a general XY polynomial, which is very convenient for data interaction with the processing equipment and is easy to process. The XY polynomial with standard orthogonal characteristics constructed in this way also has this convenience in processing.

[0068] (4) In the method of the present invention, the definition domain range is normalized, and orthogonal characteristics can be achieved in a rectangular domain of any range by scaling.

[0069] (5) In the method of the present invention, the integral of the scalar product of any two polynomials in the generalized polynomial in the square domain is set to zero, and the integral of the self-product of any polynomial in the square domain is set to 1. In this case, the polynomial has a standard orthogonal property.

[0070] (6) In the method of the present invention, the equation group constructed according to the orthogonal characteristics can be solved to obtain the coefficients of the generalized polynomial, which is easy to calculate.

[0071] (7) In the method of the present invention, the orthogonal characteristics of the polynomials can improve the robustness of the optical system design and enhance the imaging quality of the system. BRIEF DESCRIPTION OF THE DRAWINGS

[0072] Figure 1 is a three-dimensional graph of the first 20 terms of the polynomial with standard orthogonal properties in the present invention;

[0073] Figure 2 It is a contour map of the first 20 terms of the polynomial with standard orthogonal characteristics in the present invention;

[0074] Figure 3 The optical path diagram of the off-axis reflection system with square-shaped optical elements;

[0075] Figure 4 The error function curve of the optical system described by the traditional XY polynomial for the reflector;

[0076] Figure 5 The error function variation curve of the optical system is described by using a free-form surface with orthogonal XY polynomials in a square domain for the reflector. DETAILED DESCRIPTION

[0077] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, rather than all the embodiments. The drawings and specific embodiments are intended to further illustrate the present invention, but are not intended to limit the present invention. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without making creative work are within the scope of protection of the present invention.

[0078] There are many ways to describe free-form surfaces. The most commonly used one is the traditional XY polynomial optical free-form surface, which has good flexibility and excellent programmability and is widely used in the design of imaging optical systems. Taking a traditional XY polynomial optical free-form surface as an example, its sagitta mainly consists of two parts, one is the sagitta provided by the basis term quadratic surface, and the other is the additional sagitta provided by the high-order term. The quadratic surface itself has rotational symmetry, and many complex surfaces use the quadratic surface as the basis term. The shape of the surface described by the polynomial is more free. The equation of a traditional XY polynomial optical free-form surface can be expressed as:

[0079]

[0080] Where c is a real number, representing the vertex curvature; k is a real number, representing the quadratic surface constant; r = (x 2 +y 2 ) 0.5 , represents the distance from a point on the surface to the optical axis; z is a real number, representing the surface vector height; j is a positive integer, representing the sequence number of each coefficient; C j is a real number, representing the coefficient of each polynomial; w is a non-negative integer, representing the order of x; t is a non-negative integer, representing the order of y;

[0081] The optical free-form surface with orthogonal XY polynomials in a square domain in the present invention has a structure similar to that of a traditional XY polynomial, and is also composed of a basis term plus an XY polynomial with a standard orthogonal property.

[0082] The basis term may be a sphere, a quadratic surface, a biquadratic surface, or other second-order surfaces. Preferably, the basis term is a quadratic surface.

[0083] The XY polynomial with standard orthogonal characteristics, its construction process includes the following steps:

[0084] First, determine the domain of the polynomial function; second, select a generalized polynomial based on a linear combination of a set of linearly independent function systems; then calculate the integral value of the scalar product of any two polynomials in the generalized polynomial within the domain, as well as the integral value of the product of any function itself within the domain; finally, solve the system of equations formed by the integral results to obtain the coefficient of each monomial in the generalized polynomial, and obtain an XY polynomial with standard orthogonal characteristics.

[0085] Among them, the independent variables of the linearly independent function system are set to x, y, and the dependent variable is set to p(x, y), wherein the domain of the independent variable is a square. Preferably, the domain is normalized and the range of the independent variable is set to {x∈[-1,1], y∈[-1,1]}.

[0086] Among them, a set of linearly independent function systems p(x,y) are selected with the following expressions:

[0087] p(x,y):x,y,x 2 ,xy,y 2 , x 3 , x 2 y,xy 2 ,y 3 , x 4 ,…

[0088] This function system has the same form as the function system contained in the XY polynomial surface commonly used in optical systems. Due to its concise form, it facilitates data interaction between optical design software and processing equipment, and is therefore widely used. The order of the function contained in p(x, y) can be extended upwards all the time, and the highest order is set to 5 in the example of the present invention. At this time, the p(x, y) function system contains a total of 20 elements.

[0089] The generalized polynomial formed by the linear combination of this set of function systems p(x,y) is set to The specific expression of this set of generalized polynomials can be described as:

[0090]

[0091]

[0092]

[0093]

[0094]

[0095] …

[0096]

[0097] where a j ,b j ,c j ,…represents the coefficient of each monomial in the polynomial, and each polynomial is composed of a series of functions in the function system through simple linear combinations.

[0098] Among them, a set of generalized polynomials with standard orthogonal characteristics should have the following properties: in this set of polynomials, the integral of the scalar product of any two polynomials in the domain of definition is zero, and the integral of the self-product of any polynomial in the domain of definition is 1.

[0099] The integral value is set to I m,n , this property can be described by the expression:

[0100]

[0101] Wherein, m and n represent the mth term and nth term of the generalized polynomial respectively, and m and n are positive integers;

[0102] Among them, when m=n, the above formula represents the integral of a polynomial by itself, and when m≠n, the above formula represents the integral of the scalar product of two different polynomials within the domain of definition.

[0103] If the polynomial is not required to have standard orthogonal properties, just let m = n, I m,n ≠0. Construct the equation system based on the above integral and solve The coefficients of each monomial in .

[0104] Specifically, for example, when j=1,

[0105]

[0106] Solved therefore For example, when j = 2,

[0107]

[0108]

[0109] Solved b 2 = 0. Therefore

[0110] For example, when j=3,

[0111]

[0112]

[0113]

[0114] Solved b 3 =c 3 = 0. Therefore

[0115] The coefficient values ​​of the higher-order polynomials are calculated in sequence to obtain a series of polynomials that satisfy the standard orthogonal property.

[0116] The sum of the first 20 terms of the standard orthogonal polynomial can be expressed as:

[0117]

[0118] Where j is a positive integer, representing the serial number of each standard orthogonal polynomial and its coefficients; C j are real numbers representing the coefficients of each standard orthogonal polynomial.

[0119] Combining the XY polynomial with standard orthogonal properties with a quadratic surface basis term, a free-form surface that can be used in optical system design is constructed. Its expression can be described as:

[0120]

[0121] Where c is a real number, representing the vertex curvature; k is a real number, representing the quadratic surface constant; r = (x 2 +y 2 ) 0.5 , represents the distance from a point on the surface to the optical axis; z is a real number, representing the surface vector height; j is a positive integer, representing the serial number of each standard orthogonal polynomial and its coefficients; C j is a real number, representing the coefficient of each standard orthogonal polynomial;

[0122] The following example illustrates the application process and beneficial effects of optical free-form surfaces with orthogonal XY polynomials in a square domain in the design of actual optical systems.

[0123] By using an optical free-form surface with orthogonal XY polynomials in a square domain, an off-axis reflection imaging system with square optical elements is designed to verify the feasibility of the optical free-form surface and verify that the surface with orthogonal XY polynomials in a square domain can improve the robustness of the optical system design process, improve the imaging quality, and facilitate processing.

[0124] like Figure 3 As shown in FIG. 1 , it is an off-axis reflection imaging system, whose design wavelength is 550nm, the field of view angle only observes the central field of view, the system focal length is 5mm, the F number is 2.5, and the system only includes the aperture surface 11, the reflector 21, and the image surface 31. And the system is not symmetrical, and the lens data of the system are shown in Table 1.

[0125] Table 1

[0126] Surface serial number Surface type thickness Eccentric type Alpha Tilt Beta Tilt 11 Aperture 2 none 21 Polynomial Surface -5 Eccentricity and tilt 30 30 31 Image plane 0 none

[0127] The system was optimized using traditional XY polynomial surfaces and optical free-form surfaces with orthogonal XY polynomials. The vertex curvature c, quadratic surface constant k, and C3-C20 coefficients of the two surfaces were used as optimization variables, and no other variables were used. The same optimization starting point, the same optimization constraints, and the same image quality evaluation method were used.

[0128] As shown in Table 2, the parameters of the optimized orthogonal XY polynomial surface and the traditional XY polynomial surface are given.

[0129] Table 2

[0130]

[0131]

[0132] like Figure 4 and Figure 5 The figure shows the change of the system error function with the number of optimization iterations. Under the same design starting point, after three rounds of optimization, the error function of the system using the traditional XY polynomial surface converges to about 100, and the error function of the system using the orthogonal XY polynomial surface converges to within 5. The error function value is reduced by at least one order of magnitude, indicating that the surface is suitable for the design of optical systems with square optical elements and can improve the imaging quality of the optical system. The stable convergence of the error function shows that the surface has good robustness during the optimization process.

[0133] The first 20 terms of the orthogonal polynomials are shown in Table 3. The three-dimensional graph of each polynomial in Table 3 is shown in Figure 1 As shown, the contour lines of each polynomial in Table 3 are as follows Figure 2 As shown;

[0134] Table 3

[0135]

[0136]

[0137] The above description is only a preferred embodiment of the present invention and is not intended to limit the implementation mode and protection scope of the present invention. Those skilled in the art should be aware that all solutions obtained by equivalent substitutions and obvious changes made using the description and illustrations of the present invention should be included in the protection scope of the present invention.

Claims

1. A design method for a square optical element based on a free-form surface, characterized in that: The method refers to designing the surface shape of a square optical element, wherein the surface shape of the square optical element is a free-form surface, and the free-form surface includes a basis term and an XY polynomial with a standard orthogonal property; The construction process of the XY polynomial with standard orthogonal characteristics is as follows: In the first step, a generalized polynomial consisting of a linear combination of a set of linearly independent function systems is selected; The second step is to determine the domain of the XY polynomial function based on the generalized polynomial in the first step; The third step is to calculate the integral value of the scalar product of any two polynomials in the generalized polynomial in the first step within the definition domain determined in the second step, the integral value being defined as the first integral value, and the integral value of the self-product of any polynomial within the definition domain, the integral value being defined as the second integral value, to obtain a system of equations; Step 4, solving the coefficient of each monomial in the generalized polynomial in the first step according to the system of equations obtained in the third step, and obtaining an XY polynomial with standard orthogonal characteristics; In the first step, the independent variables of the linearly independent function system are set to x, y, and the dependent variable is set to p(x, y), wherein the domain of the independent variable is a square; The dependent variable p(x,y) has the following expression: p(x,y):x,y,x 2 ,xy,y 2 ,x 3 ,x 2 y,xy 2 ,y 3 ,x 4 ,... The generalized polynomial consisting of the linear combination of the dependent variables p(x,y) is set to The specific expression of this set of generalized polynomials is: Among them, a j ,b j ,c j ,…are real numbers, representing the coefficients of each monomial in the polynomial; j is a positive integer, representing the sequence number of the polynomial and its coefficients; each polynomial is composed of a series of functions in a linearly independent function system through linear combination; In the third step, the method of obtaining the system of equations is: In a generalized polynomial, the integral of the scalar product of any two polynomials in the domain is zero, and the integral of the self-product of any polynomial in the domain is 1; the integral value is set to I m,n , the integral value expression is: Wherein, m and n represent the mth term and nth term of the generalized polynomial respectively, and m and n are positive integers; The system of equations obtained based on the integral value is: Among them, when m≠n, I m,n Represents the first integral value. The above formula represents the integral of the scalar product of two different polynomials in the domain of definition. When m=n, I m,n represents the second integral value, and the above formula represents the integral of a polynomial multiplied by itself; In the fourth step, according to the equations The method for calculating the coefficients of the monomials in is as follows: when j = 1, Solved When j = 2, Solved b2=0.

2. The design method of a square optical element based on a free-form surface according to claim 1, characterized in that: The basis term is a sphere, a quadratic surface, a biquadric surface or a second-order surface.

3. The design method of a square optical element based on a free-form surface according to claim 2, characterized in that: The quadratic surface basis term has the following expression: Where c is a real number, representing the vertex curvature; k is a real number, representing the quadratic surface constant; r = (x 2 +y 2 ) 0.5 , which represents the distance from a point on the surface to the optical axis; z is a real number, which represents the surface sag.

4. The design method of a square optical element based on a free-form surface according to claim 1, characterized in that: In the second step, the domain of the XY polynomial function is normalized, and the domain of the XY polynomial function is {x∈[-1,1], y∈[-1,1]}.

5. The design method of a square optical element based on a free-form surface according to claim 1, characterized in that: The coefficient values ​​of the high-order polynomials are calculated in sequence to obtain a series of polynomials that satisfy the standard orthogonal characteristics; the XY polynomial with the standard orthogonal characteristics is combined with a quadratic surface basis term to form a free-form surface that can be used for optical system design. The expression of the free-form surface is: Where c is a real number, representing the vertex curvature; k is a real number, representing the quadratic surface constant; r = (x 2 +y 2 ) 0.5 , represents the distance from a point on the surface to the optical axis; z is a real number, representing the surface vector height; j is a positive integer, representing the serial number of each standard orthogonal polynomial and its coefficients; C j are real numbers representing the coefficients of each standard orthogonal polynomial.

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