CN113835221A - Integrated modeling method for initial structure of multi-reflection off-axis optical system - Google Patents

Abstract

The invention discloses an integrated modeling method for the initial structure of a multi-reverse off-axis optical system. First, a space rectangular coordinate system is established, and a coaxial structure is established for each reflecting mirror based on the paraxial optics theory. and the displacement matrix to obtain the uniform expression of the mirror surface of the rotated and displaced off-axis structure; then construct the expression of the relationship between the mirrors based on any mirror; then model the imaging optical path of the off-axis reflection system, and finally use the outgoing light and the image surface. The height difference of the intersection is used as a constraint to establish the objective function, and the surface shape coefficient of the k-surface mirror is obtained. The invention subtly simplifies the complex free-form surface modeling into a quadratic curve modeling that can express the position and relative position relationship in an integrated manner in combination with the transformation matrix, and realizes the establishment of an integrated model of the mirror surface shape of any position and attitude in space. Design of off-axis systems for arbitrary polygon mirrors.

Description

An integrated modeling method for the initial structure of a multi-reverse off-axis optical system

technical field

The invention relates to the technical field of optical system design, in particular to an integrated modeling method for the initial structure of a multi-reverse off-axis optical system.

Background technique

At present, in the design of multi-reflection off-axis optical systems, the patent library matching design is the most commonly used. This design method is to find an off-axis reflection system in the optical patent library with the same number of mirrors as the design target and similar indicators such as focal length or entrance pupil diameter, and rely on the experience of optical designers to gradually adjust the system on the optical simulation software. The optical parameters of the intermediate reflector, and then the overall optical performance of the system is optimized, so that the structure of the multi-reflection off-axis optical system is getting closer and closer to the design goal.

For off-axis reflective systems, the initial structures available in the optical patent library are very limited, so after finding an off-axis reflective system with the same number of mirrors as the design target, it is very likely to find that the optical system is very different from the design target requirements. And have to give up; even if a system with the same number of mirrors as the design target and similar focal length or entrance pupil is found in the patent library, in order to make the system fully meet the requirements of the design target, it is still necessary to continuously adjust the surface shape and pose of the mirrors. Constantly optimizing the optical performance of the system, this tuning and optimization process is time-intensive and prone to failure.

SUMMARY OF THE INVENTION

In view of the above-mentioned deficiencies of the prior art, the present invention provides an integrated modeling method for the initial structure of a multi-counter off-axis optical system.

In order to solve the above-mentioned technical problems, the technical solution adopted by the present invention is: a method for integrating the initial structure of a multi-reverse off-axis optical system, comprising the following steps:

Step 1: Establish a space rectangular coordinate system, so that the center of the primary mirror is at the origin of coordinates, that is, the value of the distance d _c0 from the center of the primary mirror to the origin of coordinates is set to 0;

Step 2: According to the design requirements, the coaxial structure is obtained based on the paraxial optical theory, and a unified surface expression is established for each mirror. The surface expression is as follows:

A=[x y 1]

表示同轴结构时反射镜的面型表达式；x为反射镜上任意点的x坐标；y为反射镜上任意点的y坐标。Among them, d _c(n-1) is the distance between the center of the n-1th mirror and the center of the nth mirror; R _n is the radius of curvature of the center point of the nth mirror; e _n is the nth surface The eccentricity of the mirror; k is the total number of mirrors; A is the coordinate point matrix; B _n is the surface description matrix;

Represents the surface expression of the mirror when the coaxial structure is present; x is the x-coordinate of any point on the mirror; y is the y-coordinate of any point on the mirror.

In this embodiment, the design requirements include the number of mirrors, the size of the entrance pupil, and the focal length of the system.

Step 3: Obtain the corresponding rotation matrix C _n according to the inclination angle of each mirror, obtain the corresponding displacement matrix according to the displacement of the center point of each mirror, and combine the rotation matrix and the displacement matrix to obtain the rotated and displaced off-axis structure Mirror unified expression;

The method for obtaining the corresponding rotation matrix C _n according to the inclination angle of each mirror is as follows:

where α _n is the inclination angle of the nth mirror.

The method for obtaining the corresponding displacement matrix according to the displacement of the center point of each mirror is as follows:

Among them, DM is the displacement matrix, and Δx and Δy are the displacements of the mirror along the x-axis and y-axis, respectively.

The mirror surface of the off-axis structure is uniformly expressed as follows:

表示离轴结构时反射镜的面型表达式。Among them, C _n is the rotation matrix, [-x _no ,-y _no ,0] is the displacement matrix of the nth mirror, where x _no and y _no are the x and y coordinates of the center point of the nth mirror, respectively , d _cj is the distance from the center point of the j-th mirror to the center point of the j+1-th mirror; α _i is the inclination angle of the i-th mirror;

Represents the surface expression for the mirror when the off-axis structure is present.

Step 4: According to the central optical path, the coordinates of the image point (x _Io , y _Io ) are obtained, and the calculation formula is:

Among them, x _ko is the x-coordinate of the center point of the last mirror in the system, y _ko is the y-coordinate of the center point of the last mirror in the system, and l' _ck is the distance between the last mirror and the image point;

Step 5: Construct an expression model of the relationship between mirrors based on any mirror. The process is as follows:

Step 5.1: According to the tilt angle of the reference mirror, based on the concept of coordinate transformation, obtain the corresponding relative rotation matrix D _i , the calculation formula is:

α _i(i-1) =α _i -α _i-1

α ₁₀ =α ₁

Among them, D _i is the relative rotation matrix of any mirror relative to the i-th mirror when the _i -th mirror is used as a reference; Angle difference; α _i is the inclination angle of the i-th mirror; α _i-1 is the inclination angle of the i-1 mirror; α ₁₀ is the inclination angle of the first mirror and the inclination of the upper mirror Angle difference; α ₁ is the inclination angle of the first mirror;

Step 5.2: Construct an expression model of the relationship between mirrors based on the primary mirror. Taking the primary mirror as the benchmark, the expression of any mirror relative to the primary mirror is:

表示任意镜以主镜为基准时的表达式；Among them, D ₁ is the relative rotation matrix when taking the primary mirror as the reference;

represents the expression when any mirror is based on the primary mirror;

Step 5.3: When taking the primary mirror as the benchmark, the coordinates of the center point of any mirror and the coordinates of the image point relative to the primary mirror are obtained. The calculation formula is:

为以主镜为基准时的坐标变换矩阵；Among them, CP _n is the column vector composed of the coordinates of the center point of the n-th mirror; CP _n ^R1 is the column vector composed of the center point coordinates of the n-th mirror when the primary mirror is used as the reference; I _o is the image point coordinate composition The column vector of ; I _o ^R1 is the column vector composed of the coordinates of the image point when taking the primary mirror as the reference;

is the coordinate transformation matrix based on the primary mirror;

Step 5.4: According to the order of the second mirror and the third mirror to the k-th mirror, obtain the coordinates of the center point of any mirror and the position of the image point coordinates when the i-th mirror is used as the reference, i=2,3,...,k, The calculation formula is:

为第n面反射镜以第i面反射镜为基准时的中心点坐标组成的列向量；

为第n面反射镜以第i-1面反射镜为基准时的中心点坐标组成的列向量；CP_i ^R(i-1)为第i面反射镜以第i-1面反射镜为基准时的中心点坐标组成的列向量；

为像点在以第i面反射镜为基准时的像点坐标组成的列向量；

为像点在以第i-1面反射镜为基准时的像点坐标组成的列向量，

为以第i镜为基准时的坐标变换矩阵；in,

is a column vector composed of the coordinates of the center point of the n-th mirror with the i-th mirror as the reference;

is the column vector composed of the center point coordinates of the n-th mirror with the i-1th mirror as the reference; CP _i ^R( i-1) is the i-th mirror with the i-1th mirror as the reference A column vector consisting of the coordinates of the center point of time;

is a column vector composed of the coordinates of the image point when the image point is based on the i-th mirror;

is the column vector of the image point coordinates when the image point is based on the i-1th mirror,

is the coordinate transformation matrix based on the i-th mirror;

Step 5.5: According to the order of the second mirror and the third mirror to the k-th mirror, obtain the expression of any mirror when the i-th mirror is used as the reference, i=2,3,...,k:

为第i面反射镜以第i-1面反射镜为基准时中心点的x坐标；

为第i面反射镜以第i-1面反射镜为基准时中心点的y坐标，A_i为以第i镜为基准时的相对坐标点矩阵，A_i-1为以第i-1镜为基准时的相对坐标点矩阵，A₁＝A；

表示任意镜在以第i镜为基准时的表达式。Wherein, D _i is the relative rotation matrix when taking the i-th mirror as a reference;

is the x-coordinate of the center point of the i-th mirror with the i-1-th mirror as the reference;

is the y-coordinate of the center point of the i-th mirror with the i-1th mirror as the reference, A _i is the relative coordinate point matrix with the i-th mirror as the reference, and A _i-1 is the i-1th mirror. is the relative coordinate point matrix when it is the reference, A ₁ =A;

An expression representing an arbitrary mirror when the i-th mirror is used as the reference.

Step 6: Model the imaging optical path of the off-axis reflection system, the process is as follows:

Step 6.1: Select the ray parallel to the optical axis and the intersection of the primary mirror at (x _in1 , y _in1 ), and calculate the height and angle of the ray exiting on the primary mirror based on the optical matrix equation based on the primary mirror. Calculate The formula is:

Among them, P ₁ is the column vector composed of the coordinates of the intersection of the light and the primary mirror; P ₁ ^R1 is the column vector composed of the coordinates of the intersection of the light and the primary mirror when the primary mirror is used as the reference; hl ^iM1 is the incident height of the light on the primary mirror; θ ^iM1 is the incident angle of the light on the primary mirror; ρ ^M1 is the radius of curvature of the reflector at the intersection of the light and the primary mirror; hl ^oM1 is the exit height of the light on the primary mirror; θ ^oM1 is the exit angle of the light on the primary mirror;

计算公式为：Step 6.2: Calculate the optical path between the primary mirror and the second mirror, and get the intersection of the light and the second mirror

The calculation formula is:

为以主镜为基准时主次镜间的光路；

为以主镜为基准时次镜的表达式，C₂为次镜的旋转矩阵，B₂为次镜的面型描述矩阵，y_2o为次镜中心点的y坐标；x_2o为次镜中心点的x坐标；in,

is the optical path between the primary and secondary mirrors when the primary mirror is used as the reference;

is the expression of the secondary mirror when the primary mirror is the reference, C ₂ is the rotation matrix of the secondary mirror, B ₂ is the surface description matrix of the secondary mirror, y _2o is the y coordinate of the center point of the secondary mirror; x _2o is the center of the secondary mirror the x-coordinate of the point;

Step 6.3: Based on the optical matrix equation and the transformation matrix, establish the mathematical model of the optical path between the second mirror and the third mirror, the third mirror and the fourth mirror, ..., the k-1th mirror and the kth mirror, and calculate The formula is:

θ ^iMn = θ ^oM(n-1) -α _n(n-1)

为以第n面反射镜为基准时第n面反射镜和第n+1面反射镜之间的光路；

为以第n面反射镜为基准时光线与该镜交点坐标组成的列向量；

为以第n-1面反射镜为基准时光线与第n面反射镜交点坐标组成的列向量；

为第n面反射镜以第n-1面反射镜为基准时的中心点坐标组成的列向量；hl^iMn为光线在第n面反射镜上的入射高度；θ^iMn为光线在第n面反射镜上的入射角度；

为以第n镜为基准时，光线与第n镜交点的x坐标；

为以第n镜为基准时，光线与第n镜交点的y坐标；θ^oM(n-1)为光线在第n-1镜上的入射角度，α_n(n-1)为第n面反射镜与第n-1面反射镜的倾斜角度之差；Wherein, hl ^iMn is the incident height of the light on the nth mirror; θ ^iMn is the incident angle of the light on the nth mirror; ρ ^Mn is the radius of curvature of the mirror at the intersection of the light and the nth mirror; hl ^oMn is the exit height of the light on the nth mirror; θ ^oMn is the exit angle of the light on the nth mirror;

is the optical path between the nth mirror and the n+1th mirror when the nth mirror is used as the reference;

is the column vector composed of the coordinates of the intersection of the ray and the mirror when the nth mirror is used as the reference;

is the column vector composed of the coordinates of the intersection of the ray and the nth mirror when the n-1th mirror is used as the reference;

is the column vector composed of the center point coordinates of the n-th mirror with the n-1th mirror as the reference; hl ^iMn is the incident height of the light on the n-th mirror; θ ^iMn is the reflection of the light on the n-th surface the angle of incidence on the mirror;

is the x coordinate of the intersection of the ray and the nth mirror when the nth mirror is used as the benchmark;

is the y-coordinate of the intersection of the light and the nth mirror when the nth mirror is used as the benchmark; θ ^oM(n-1) is the incident angle of the light on the n-1th mirror, and α _n(n-1) is the nth surface The difference between the inclination angle of the mirror and the n-1th mirror;

Step 6.4: Based on the optical matrix equation and the transformation matrix, establish a mathematical model of the optical path between the k-th mirror and the image point. The calculation formula is:

为以第k面反射镜为基准时，光线与第k面反射镜交点的x坐标；

为以第k面反射镜为基准时像点的x坐标；ρ^Mk为光线与第k面反射镜交点处反射镜的曲率半径；hl^iMk为光线在第k面反射镜上的入射高度；θ^iMk为光线在第k面反射镜上的入射角度。Among them, l _kI is the distance between the k-th mirror and the image point when the k-th mirror is used as the reference;

is the x-coordinate of the intersection of the ray and the k-th mirror when the k-th mirror is used as the benchmark;

is the x-coordinate of the image point with the k-th mirror as the reference; ρ ^Mk is the radius of curvature of the mirror at the intersection of the light and the k-th mirror; hl ^iMk is the incident height of the light on the k-th mirror; θ ^iMk is the incident angle of the light on the k-th mirror.

Step 7: Select m incident rays and repeat step 6, m≥k, to obtain the intersection of the corresponding outgoing rays and the image plane:

The objective function is established with the height difference between the outgoing ray and the intersection of the image plane as the constraint, and the surface shape coefficient of the k-surface mirror is obtained. The calculation formula is:

为第i条光线与像面交点的高度；

为第k面反射镜离心率的平方，面形系数为离心率平方的负数；

为第i+n条光线与像面交点的高度；

为目标函数。in,

is the height of the intersection of the i-th ray and the image plane;

is the square of the eccentricity of the k-th mirror, and the shape coefficient is the negative number of the square of the eccentricity;

is the height of the intersection of the i+nth ray and the image plane;

is the objective function.

The beneficial effects produced by the above technical solutions are:

1. The present invention provides a modeling method with strong scalability and low parameter coupling for the initial structure of the off-axis reflection system, and cleverly simplifies the modeling of complex free-form surfaces into one that can combine the transformation matrix to express the pose and relative The quadratic curve modeling of the position relationship realizes the establishment of an integrated model of the mirror surface shape of any position and orientation in space;

2. The present invention adds a relative rotation matrix, which solves the limitation of the traditional optical matrix on the relationship between the mirror surface and the optical axis, and realizes the model representation of the relative position between the mirrors of any reference mirror;

3. The present invention establishes mirror-mirror and mirror-image optical path modeling based on the optical matrix and the conversion matrix, so as to achieve the purpose of integrated construction of the initial structural model of the off-axis reflection system;

4. The present invention is directly designed based on the design target, and the tilt angle and position of each mirror are controllable, without relying on the operation experience of optical designers, the construction process is simple, and it is suitable for off-axis systems with any multi-faceted mirrors. Applicable, the scope of application is relatively wider.

Description of drawings

1 is a flowchart of an integrated modeling method for an initial structure of a multi-reflection off-axis optical system in an embodiment of the present invention;

2 is a schematic diagram of the relative positions of the primary mirror and the secondary mirror in the multi-reflection off-axis optical system according to the embodiment of the present invention;

3 is a schematic diagram of an initial structure of an off-axis three-mirror optical system in an embodiment of the present invention;

FIG. 4 is a field of view view of the spot radius RMS of the entire field of view on the image plane of the initial structure in the embodiment of the present invention.

Detailed ways

The specific embodiments of the present invention will be described in further detail below with reference to the accompanying drawings and embodiments. The following examples are intended to illustrate the present invention, but not to limit the scope of the present invention.

This embodiment is verified by designing an off-axis three-mirror optical system. The design requirements are: the entrance pupil is 150 mm, the focal length is 900 mm, and the number of mirrors is 3.

As shown in FIG. 1 , an integrated modeling method for the initial structure of a multi-reflection off-axis optical system in this embodiment is as follows.

Step 1: Establish a space rectangular coordinate system, so that the center of the primary mirror is at the origin of coordinates, that is, the value of the distance d _c0 from the center of the primary mirror to the origin of coordinates is set to 0;

Step 2: According to the design requirements, the coaxial structure is obtained based on the paraxial optical theory, and a unified surface expression is established for each mirror. The surface expression is as follows:

A=[x y 1]

表示同轴结构时反射镜的面型表达式；x为反射镜上任意点的x坐标；y为反射镜上任意点的y坐标。Among them, d _c(n-1) is the distance between the center of the n-1th mirror and the center of the nth mirror; R _n is the radius of curvature of the center point of the nth mirror; e _n is the nth surface The eccentricity of the mirror; k is the total number of mirrors; A is the coordinate point matrix; B _n is the surface description matrix;

Represents the surface expression of the mirror when the coaxial structure is present; x is the x coordinate of any point on the mirror; y is the y coordinate of any point on the mirror.

Step 3: Obtain the corresponding rotation matrix C _n according to the inclination angle of each mirror, obtain the corresponding displacement matrix according to the displacement of the center point of each mirror, and combine the rotation matrix and the displacement matrix to obtain the rotated and displaced off-axis structure Mirror unified expression;

The method for obtaining the corresponding rotation matrix C _n according to the inclination angle of each mirror is as follows:

where α _n is the inclination angle of the nth mirror.

The relative positions of the primary mirror and the secondary mirror are shown in Figure 2, where (a) indicates that the mirror center of the primary mirror is at the origin, and the tilt angle is α ₁ ; the coordinates of the center point of the secondary mirror are (x _2o , y _2o ), the tilt angle is The angle is α ₂ . (b) means that when considering the position of the secondary mirror relative to the primary mirror, it is equivalent to writing the expression of the secondary mirror when the coordinate system is established with the center of the primary mirror as the origin and the normal line perpendicular to the center of the primary mirror as the y-axis .

The method for obtaining the corresponding displacement matrix according to the displacement of the center point of each mirror is as follows:

Among them, DM is the displacement matrix, and Δx and Δy are the displacements of the mirror along the x-axis and y-axis, respectively.

The mirror surface of the off-axis structure is uniformly expressed as follows:

表示离轴结构时反射镜的面型表达式。Among them, C _n is the rotation matrix, [-x _no ,-y _no ,0] is the displacement matrix of the nth mirror, where x _no and y _no are the x and y coordinates of the center point of the nth mirror, respectively , d _cj is the distance from the center point of the j-th mirror to the center point of the j+1-th mirror; α _i is the inclination angle of the i-th mirror;

Represents the surface expression for the mirror when the off-axis structure is present.

The inclination angles adopted in this embodiment are: α ₁ =-12.500°; α ₂ =-8.648°; α ₃ =-3.950°.

Step 4: According to the central optical path, the coordinates of the image point (x _Io , y _Io ) are obtained, and the calculation formula is:

Among them, x _ko is the x-coordinate of the center point of the last mirror in the system, y _ko is the y-coordinate of the center point of the last mirror in the system, and l' _ck is the distance between the last mirror and the image point;

Step 5: Construct an expression model of the relationship between mirrors based on any mirror. The process is as follows:

Step 5.1: According to the tilt angle of the reference mirror, based on the concept of coordinate transformation, obtain the corresponding relative rotation matrix D _i , the calculation formula is:

α _i(i-1) =α _i -α _i-1

α ₁₀ =α ₁

Among them, D _i is the relative rotation matrix of any mirror relative to the i-th mirror when the _i -th mirror is used as a reference; Angle difference; α _i is the inclination angle of the i-th mirror; α _i-1 is the inclination angle of the i-1 mirror; α ₁₀ is the inclination angle of the first mirror and the inclination of the upper mirror Angle difference; α ₁ is the inclination angle of the first mirror;

Step 5.2: Construct an expression model of the relationship between mirrors based on the primary mirror. Taking the primary mirror as the benchmark, the expression of any mirror relative to the primary mirror is:

表示任意镜以主镜为基准时的表达式；Among them, D ₁ is the relative rotation matrix when taking the primary mirror as the reference;

represents the expression when any mirror is based on the primary mirror;

Step 5.3: When taking the primary mirror as the benchmark, the coordinates of the center point of any mirror and the coordinates of the image point relative to the primary mirror are obtained. The calculation formula is:

为以主镜为基准时的坐标变换矩阵；Among them, CP _n is the column vector composed of the coordinates of the center point of the n-th mirror; CP _n ^R1 is the column vector composed of the center point coordinates of the n-th mirror when the primary mirror is used as the reference; I _o is the image point coordinate composition The column vector of ; I _o ^R1 is the column vector composed of the coordinates of the image point when taking the primary mirror as the reference;

is the coordinate transformation matrix based on the primary mirror;

Step 5.4: According to the order of the second mirror and the third mirror to the k-th mirror, obtain the coordinates of the center point of any mirror and the position of the image point coordinates when the i-th mirror is used as the reference, i=2,3,...,k, The calculation formula is:

为第n面反射镜以第i面反射镜为基准时的中心点坐标组成的列向量；

为第n面反射镜以第i-1面反射镜为基准时的中心点坐标组成的列向量；CP_i ^R(i-1)为第i面反射镜以第i-1面反射镜为基准时的中心点坐标组成的列向量；

为像点在以第i面反射镜为基准时的像点坐标组成的列向量；

为像点在以第i-1面反射镜为基准时的像点坐标组成的列向量，

为以第i镜为基准时的坐标变换矩阵；in,

is a column vector composed of the coordinates of the center point of the n-th mirror with the i-th mirror as the reference;

is the column vector composed of the center point coordinates of the n-th mirror with the i-1th mirror as the reference; CP _i ^R(i-1) is the i-th mirror with the i-1th mirror as the reference A column vector consisting of the coordinates of the center point of time;

is a column vector composed of the coordinates of the image point when the image point is based on the i-th mirror;

is the column vector of the image point coordinates when the image point is based on the i-1th mirror,

is the coordinate transformation matrix based on the i-th mirror;

Step 5.5: According to the order of the second mirror and the third mirror to the k-th mirror, obtain the expression of any mirror when the i-th mirror is used as the reference, i=2,3,...,k:

为第i面反射镜以第i-1面反射镜为基准时中心点的x坐标；

为第i面反射镜以第i-1面反射镜为基准时中心点的y坐标，A_i为以第i镜为基准时的相对坐标点矩阵，A_i-1为以第i-1镜为基准时的相对坐标点矩阵，A₁＝A；

表示任意镜在以第i镜为基准时的表达式。Wherein, D _i is the relative rotation matrix when taking the i-th mirror as a reference;

is the x-coordinate of the center point of the i-th mirror with the i-1-th mirror as the reference;

is the y-coordinate of the center point of the i-th mirror with the i-1th mirror as the reference, A _i is the relative coordinate point matrix with the i-th mirror as the reference, and A _i-1 is the i-1th mirror. is the relative coordinate point matrix when it is the reference, A ₁ =A;

An expression representing an arbitrary mirror when the i-th mirror is used as the reference.

Step 6: Model the imaging optical path of the off-axis reflection system, the process is as follows:

Step 6.1: Select the ray parallel to the optical axis and the intersection point with the primary mirror at (x _in1 , y _in1 ), and calculate the height and angle of the ray exiting the primary mirror based on the optical matrix equation based on the primary mirror, calculate The formula is:

Among them, P ₁ is the column vector composed of the coordinates of the intersection of the light and the primary mirror; P ₁ ^R1 is the column vector composed of the coordinates of the intersection of the light and the primary mirror when the primary mirror is used as the reference; hl ^iM1 is the incident height of the light on the primary mirror; θ ^iM1 is the incident angle of the light on the primary mirror; ρ ^M1 is the radius of curvature of the reflector at the intersection of the light and the primary mirror; hl ^oM1 is the exit height of the light on the primary mirror; θ ^oM1 is the exit angle of the light on the primary mirror;

In this embodiment, k=3 rays are selected, and the incident heights are: 75mm, -75mm and -72.5mm respectively.

计算公式为：Step 6.2: Calculate the optical path between the primary mirror and the second mirror, and get the intersection of the light and the second mirror

The calculation formula is:

为以主镜为基准时主次镜间的光路；

为以主镜为基准时次镜的表达式，C₂为次镜的旋转矩阵，B₂为次镜的面型描述矩阵，y_2o为次镜中心点的y坐标；x_2o为次镜中心点的x坐标；in,

is the optical path between the primary and secondary mirrors when the primary mirror is used as the reference;

is the expression of the secondary mirror when the primary mirror is the reference, C ₂ is the rotation matrix of the secondary mirror, B ₂ is the surface description matrix of the secondary mirror, y _2o is the y coordinate of the center point of the secondary mirror; x _2o is the center of the secondary mirror the x-coordinate of the point;

Step 6.3: Based on the optical matrix equation and the transformation matrix, establish the mathematical model of the optical path between the second mirror and the third mirror, the third mirror and the fourth mirror, ..., the k-1th mirror and the kth mirror, and calculate The formula is:

θ ^iMn = θ ^oM(n-1) -α _n(n-1)

为以第n面反射镜为基准时第n面反射镜和第n+1面反射镜之间的光路；

为以第n面反射镜为基准时光线与该镜交点坐标组成的列向量；

为以第n-1面反射镜为基准时光线与第n面反射镜交点坐标组成的列向量；

为第n面反射镜以第n-1面反射镜为基准时的中心点坐标组成的列向量；hl^iMn为光线在第n面反射镜上的入射高度；θ^iMn为光线在第n面反射镜上的入射角度；

为以第n镜为基准时，光线与第n镜交点的x坐标；

为以第n镜为基准时，光线与第n镜交点的y坐标；θ^oM(n-1)为光线在第n-1镜上的入射角度，α_n(n-1)为第n面反射镜与第n-1面反射镜的倾斜角度之差；Wherein, hl ^iMn is the incident height of the light on the nth mirror; θ ^iMn is the incident angle of the light on the nth mirror; ρ ^Mn is the radius of curvature of the mirror at the intersection of the light and the nth mirror; hl ^oMn is the exit height of the light on the nth mirror; θ ^oMn is the exit angle of the light on the nth mirror;

is the optical path between the nth mirror and the n+1th mirror when the nth mirror is used as the reference;

is the column vector composed of the coordinates of the intersection of the ray and the mirror when the nth mirror is used as the reference;

is the column vector composed of the coordinates of the intersection of the ray and the nth mirror when the n-1th mirror is used as the reference;

is the column vector composed of the center point coordinates of the n-th mirror with the n-1th mirror as the reference; hl ^iMn is the incident height of the light on the n-th mirror; θ ^iMn is the reflection of the light on the n-th surface the angle of incidence on the mirror;

is the x coordinate of the intersection of the ray and the nth mirror when the nth mirror is used as the benchmark;

is the y-coordinate of the intersection of the light and the nth mirror when the nth mirror is used as the benchmark; θ ^oM(n-1) is the incident angle of the light on the n-1th mirror, and α _n(n-1) is the nth surface The difference between the inclination angle of the mirror and the n-1th mirror;

Step 6.4: Based on the optical matrix equation and the transformation matrix, establish a mathematical model of the optical path between the k-th mirror and the image point. The calculation formula is:

为以第k面反射镜为基准时，光线与第k面反射镜交点的x坐标；

为以第k面反射镜为基准时像点的x坐标；ρ^Mk为光线与第k面反射镜交点处反射镜的曲率半径；hl^iMk为光线在第k面反射镜上的入射高度；θ^iMk为光线在第k面反射镜上的入射角度。Among them, l _kI is the distance between the k-th mirror and the image point when the k-th mirror is used as the reference;

is the x-coordinate of the intersection of the ray and the k-th mirror when the k-th mirror is used as the benchmark;

is the x-coordinate of the image point with the k-th mirror as the reference; ρ ^Mk is the radius of curvature of the mirror at the intersection of the light and the k-th mirror; hl ^iMk is the incident height of the light on the k-th mirror; θ ^iMk is the incident angle of the light on the k-th mirror.

Step 7: Select m incident rays and repeat step 6, m≥k, to obtain the intersection of the corresponding outgoing rays and the image plane:

The objective function is established with the height difference between the outgoing ray and the intersection of the image plane as the constraint, and the surface shape coefficient of the k-surface mirror is obtained. The calculation formula is:

为第i条光线与像面交点的高度；

为第k面反射镜离心率的平方，面形系数为离心率平方的负数；

为第i+n条光线与像面交点的高度；

为目标函数。in,

is the height of the intersection of the i-th ray and the image plane;

is the square of the eccentricity of the k-th mirror, and the shape coefficient is the negative number of the square of the eccentricity;

is the height of the intersection of the i+nth ray and the image plane;

is the objective function.

The initial structural system layout of the off-axis three-mirror optical system designed by this method is shown in Figure 3. It can be seen that the light rays of different field angles and different aperture coordinates are basically focused on the image point. The spot radius RMS of the entire field of view on the initial structural image plane is shown in Figure 4. The minimum RMS value is 1.5275mm, and the maximum RMS value is 1.6894mm, which proves that the initial system has light in the full field of view. The degree of aggregation is better. Therefore, this initial system can serve as a starting point for further optimization.

Claims (7)

1. a multi-reverse off-axis optical system initial structure integrated modeling method, is characterized in that, comprises the steps: Step 1: Establish a space rectangular coordinate system, so that the center of the primary mirror is at the origin of coordinates, that is, the value of the distance d _c0 from the center of the primary mirror to the origin of coordinates is set to 0; Step 2: According to the design requirements, the coaxial structure is obtained based on the paraxial optical theory, and a unified surface expression is established for each mirror; Step 3: Obtain the corresponding rotation matrix C _n according to the inclination angle of each mirror, obtain the corresponding displacement matrix according to the displacement of the center point of each mirror, and combine the rotation matrix and the displacement matrix to obtain the rotated and displaced off-axis structure Mirror unified expression; Step 4: According to the central optical path, the coordinates of the image point (x _Io , y _Io ) are obtained, and the calculation formula is:

Among them, x _ko is the x-coordinate of the center point of the last mirror in the system, y _ko is the y-coordinate of the center point of the last mirror in the system, and l' _ck is the distance between the last mirror and the image point; Step 5: Build an expression model of the relationship between mirrors based on any mirror; Step 6: Model the imaging optical path of the off-axis reflection system; Step 7: Select m incident rays and repeat step 6, m≥k, to obtain the intersection of the corresponding outgoing rays and the image plane: The objective function is established with the height difference between the outgoing ray and the intersection of the image plane as the constraint, and the surface shape coefficient of the k-surface mirror is obtained. The calculation formula is:

in,

is the height of the intersection of the i-th ray and the image plane;

is the square of the eccentricity of the k-th mirror, and the shape coefficient is the negative number of the square of the eccentricity;

is the height of the intersection of the i+nth ray and the image plane;

is the objective function. 2. The integrated modeling method for the initial structure of a multi-reflection off-axis optical system according to claim 1, wherein the surface shape in step 2 is expressed as follows:

A=[x y 1]

Among them, d _c(n-1) is the distance between the center of the n-1th mirror and the center of the nth mirror; R _n is the radius of curvature of the center point of the nth mirror; e _n is the nth surface The eccentricity of the mirror; k is the total number of mirrors; A is the coordinate point matrix; B _n is the surface description matrix; f _n ^C is the surface expression of the mirror in the coaxial structure; x is the arbitrary point on the mirror. x coordinate; y is the y coordinate of any point on the mirror. 3. The integrated modeling method for the initial structure of a multi-reflection off-axis optical system according to claim 1, wherein the method for obtaining the corresponding rotation matrix C _n according to the inclination angle of each mirror is as follows:

where α _n is the inclination angle of the nth mirror. 4. The integrated modeling method for the initial structure of a multi-reflection off-axis optical system according to claim 1, wherein the mirror surface of the off-axis structure described in step 3 is uniformly expressed as follows:

Among them, C _n is the rotation matrix, [-x _no ,-y _no ,0] is the displacement matrix of the nth mirror, where x _no and y _no are the x and y coordinates of the center point of the nth mirror, respectively , d _cj is the distance from the center point of the j-th mirror to the center point of the j+1-th mirror; α _i is the inclination angle of the i-th mirror;

Represents the surface expression for the mirror when the off-axis structure is present. 5. The integrated modeling method for the initial structure of a multi-reflection off-axis optical system according to claim 1, wherein the method for obtaining a corresponding displacement matrix according to the displacement of the center point of each mirror is as follows:

Among them, DM is the displacement matrix, and Δx and Δy are the displacements of the mirror along the x-axis and y-axis, respectively. 6. The integrated modeling method for the initial structure of a multi-reflection off-axis optical system according to claim 1, wherein the process of the step 5 is as follows: Step 5.1: According to the tilt angle of the reference mirror, based on the concept of coordinate transformation, obtain the corresponding relative rotation matrix D _i , the calculation formula is:

α _i(i-1) =α _i -α _i-1 α ₁₀ =α ₁ Among them, D _i is the relative rotation matrix of any mirror relative to the i-th mirror when the _i -th mirror is used as a reference; Angle difference; α _i is the inclination angle of the i-th mirror; α _i-1 is the inclination angle of the i-1 mirror; α ₁₀ is the inclination angle of the first mirror and the inclination of the upper mirror Angle difference; α ₁ is the inclination angle of the first mirror; Step 5.2: Construct an expression model of the relationship between mirrors based on the primary mirror. Taking the primary mirror as the benchmark, the expression of any mirror relative to the primary mirror is:

Among them, D ₁ is the relative rotation matrix when taking the primary mirror as the reference;

represents the expression when any mirror is based on the primary mirror; Step 5.3: When taking the primary mirror as the benchmark, the coordinates of the center point of any mirror and the coordinates of the image point relative to the primary mirror are obtained. The calculation formula is:

Among them, CP _n is the column vector composed of the coordinates of the center point of the n-th mirror; CP _n ^R1 is the column vector composed of the center point coordinates of the n-th mirror when the primary mirror is used as the reference; I _o is the image point coordinate composition The column vector of ; I _o ^R1 is the column vector composed of the coordinates of the image point when taking the primary mirror as the reference;

is the coordinate transformation matrix based on the primary mirror; Step 5.4: According to the order of the second mirror and the third mirror to the k-th mirror, obtain the coordinates of the center point of any mirror and the position of the image point coordinates when the i-th mirror is used as the reference, i=2,3,...,k, The calculation formula is:

in,

is a column vector composed of the coordinates of the center point of the n-th mirror with the i-th mirror as the reference;

is the column vector composed of the center point coordinates of the n-th mirror with the i-1th mirror as the reference; CP _i ^R(i-1) is the i-th mirror with the i-1th mirror as the reference A column vector consisting of the coordinates of the center point of time;

is a column vector composed of the coordinates of the image point when the image point is based on the i-th mirror;

is the column vector of the image point coordinates when the image point is based on the i-1th mirror,

is the coordinate transformation matrix based on the i-th mirror; Step 5.5: According to the order of the second mirror and the third mirror to the k-th mirror, obtain the expression of any mirror when the i-th mirror is used as the reference, i=2,3,...,k:

Wherein, D _i is the relative rotation matrix when taking the i-th mirror as a reference;

is the x-coordinate of the center point of the i-th mirror with the i-1-th mirror as the reference;

is the y-coordinate of the center point of the i-th mirror with the i-1th mirror as the reference, A _i is the relative coordinate point matrix with the i-th mirror as the reference, and A _i-1 is the i-1th mirror. is the relative coordinate point matrix when it is the reference, A ₁ =A;

An expression representing an arbitrary mirror when the i-th mirror is used as the reference. 7. The integrated modeling method for the initial structure of a multi-reflection off-axis optical system according to claim 1, wherein the process of the step 6 is as follows: Step 6.1: Select the ray parallel to the optical axis and the intersection of the primary mirror at (x _in1 , y _in1 ), and calculate the height and angle of the ray exiting on the primary mirror based on the optical matrix equation based on the primary mirror. Calculate The formula is:

Among them, P ₁ is the column vector composed of the coordinates of the intersection of the light and the primary mirror; P ₁ ^R1 is the column vector composed of the coordinates of the intersection of the light and the primary mirror when the primary mirror is used as the reference; hl ^iM1 is the incident height of the light on the primary mirror; θ ^iM1 is the incident angle of the light on the primary mirror; ρ ^M1 is the radius of curvature of the reflector at the intersection of the light and the primary mirror; hl ^oM1 is the exit height of the light on the primary mirror; θ ^oM1 is the exit angle of the light on the primary mirror; Step 6.2: Calculate the optical path between the primary mirror and the second mirror, and get the intersection of the light and the second mirror

The calculation formula is:

in,

is the optical path between the primary and secondary mirrors when the primary mirror is used as the reference;

is the expression of the secondary mirror when the primary mirror is the reference, C ₂ is the rotation matrix of the secondary mirror, B ₂ is the surface description matrix of the secondary mirror, y _2o is the y coordinate of the center point of the secondary mirror; x _2o is the center of the secondary mirror the x-coordinate of the point; Step 6.3: Based on the optical matrix equation and the transformation matrix, establish the mathematical model of the optical path between the second mirror and the third mirror, the third mirror and the fourth mirror, ..., the k-1th mirror and the kth mirror, and calculate The formula is:

θ ^iMn = θ ^oM(n-1) -α _n(n-1) Wherein, hl ^iMn is the incident height of the light on the nth mirror; θ ^iMn is the incident angle of the light on the nth mirror; ρ ^Mn is the radius of curvature of the mirror at the intersection of the light and the nth mirror; hl ^oMn is the exit height of the light on the nth mirror; θ ^oMn is the exit angle of the light on the nth mirror;

is the optical path between the nth mirror and the n+1th mirror when the nth mirror is used as the reference;

is the column vector composed of the coordinates of the intersection of the ray and the mirror when the nth mirror is used as the reference;

is the column vector composed of the coordinates of the intersection of the ray and the nth mirror when the n-1th mirror is used as the reference;

is the column vector composed of the center point coordinates of the n-th mirror with the n-1th mirror as the reference; hl ^iMn is the incident height of the light on the n-th mirror; θ ^iMn is the reflection of the light on the n-th surface the angle of incidence on the mirror;

is the x coordinate of the intersection of the ray and the nth mirror when the nth mirror is used as the benchmark;

is the y-coordinate of the intersection of the light and the nth mirror when the nth mirror is used as the benchmark; θ ^oM(n-1) is the incident angle of the light on the n-1th mirror, and α _n(n-1) is the nth surface The difference between the inclination angle of the mirror and the n-1th mirror; Step 6.4: Based on the optical matrix equation and the transformation matrix, establish a mathematical model of the optical path between the k-th mirror and the image point. The calculation formula is:

Among them, l _kI is the distance between the k-th mirror and the image point when the k-th mirror is used as the reference;

is the x-coordinate of the intersection of the ray and the k-th mirror when the k-th mirror is used as the benchmark;

is the x-coordinate of the image point with the k-th mirror as the reference; ρ ^Mk is the radius of curvature of the mirror at the intersection of the light and the k-th mirror; hl ^iMk is the incident height of the light on the k-th mirror; θ ^iMk is the incident angle of the light on the k-th mirror.

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