CN113835221B - Integrated modeling method for initial structure of multi-reflection off-axis optical system - Google Patents
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Abstract
The invention discloses an integrated modeling method for an initial structure of a multi-reflection off-axis optical system, which comprises the steps of firstly establishing a space rectangular coordinate system, and establishing a uniform surface type expression of a coaxial structure for each reflector based on a paraxial optical theory; then, combining the rotation matrix and the displacement matrix to obtain the mirror surface uniform expression of the off-axis structure after rotation and displacement; then constructing an inter-mirror relation expression with any mirror as a reference; and modeling an imaging light path of the off-axis reflecting system, and establishing a target function by taking the height difference of the intersection point of the emergent ray and the image surface as constraint to obtain the surface shape coefficient of the k-surface reflector. The invention skillfully simplifies the complex free-form surface modeling into quadratic curve modeling which can be combined with a transformation matrix to integrally express the relationship between the pose and the relative position, realizes the establishment of an integral model of the reflector surface shape at any pose in space, and is suitable for the design of an off-axis system of any multi-surface reflector.
Description
Technical Field
The invention relates to the technical field of optical system design, in particular to an integrated modeling method for an initial structure of a multi-reflection off-axis optical system.
Background
At present, in the design aspect of a multi-reflection off-axis optical system, a patent library matching design is most commonly used. The design method is to find an off-axis reflecting system with the same number of reflectors and similar indexes such as focal length or entrance pupil diameter in a design target in an optical patent library, gradually adjust optical parameters of the reflectors in the system on optical simulation software by depending on the experience of an optical designer, and then integrally optimize the optical performance of the system to enable the structure of the multi-reflection off-axis optical system to be closer to the design target.
For the off-axis reflection system, the initial structure available in the optical patent library is very limited, so after finding the number of off-axis reflection systems equal to the number of designed target mirrors, it is very likely that the optical system is greatly different from the designed target requirement and has to be abandoned; even if a system with the same number of reflectors and similar focal length or entrance pupil as the design target is found in the patent library, in order to make the system completely meet the requirements of the design target, the surface type and the pose of the reflectors still need to be continuously adjusted, the optical performance of the system is continuously optimized, and the adjusting and optimizing process needs high time cost and is easy to fail.
Disclosure of Invention
In order to overcome the defects of the prior art, the invention provides an integrated modeling method for an initial structure of a multi-reflection off-axis optical system.
In order to solve the technical problems, the technical scheme adopted by the invention is as follows: an integrated modeling method for an initial structure of a multi-reflection off-axis optical system comprises the following steps:
step 1: establishing a rectangular space coordinate system to make the center of the primary mirror at the origin of the coordinate, i.e. the distance d between the center of the primary mirror and the origin of the coordinatec0Is set to 0;
step 2: according to design requirements, a coaxial structure is obtained based on a paraxial optical theory, and a uniform surface type expression is established for each reflector, wherein the surface type expression is as follows:
A＝[x y 1]
wherein d isc(n-1)Is the distance between the center of the n-1 st surface reflector and the center of the nth surface reflector; rnThe curvature radius of the center point of the nth surface reflector; e.g. of the typenThe eccentricity of the nth surface reflector; k is the total number of the reflecting mirrors; a is a coordinate point matrix; b isnDescribing a matrix for the surface shape;
representing a surface type expression of the reflector in a coaxial structure; x is the x coordinate of any point on the reflector; 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, the focal length of the system, and the like.
And step 3: obtaining a corresponding rotation matrix C according to the inclination angle of each reflectornObtaining a corresponding displacement matrix according to the displacement condition of the central point of each reflector, and obtaining the mirror surface unified expression of the off-axis structure after rotation and displacement by combining the rotation matrix and the displacement matrix;
obtaining a corresponding rotation matrix C according to the inclination angle of each reflectornThe method comprises the following steps:
wherein anIs the inclination angle of the nth surface reflector.
The method for obtaining the corresponding displacement matrix according to the displacement condition of the central point of each reflector comprises the following steps:
where DM is a displacement matrix and Δ x and Δ y are the displacement of the mirror along the x and y axes, respectively.
The mirror surface of the off-axis structure is uniformly expressed as follows:
wherein, CnIs a rotation matrix, [ -x [ ]no,-yno,0]Is a displacement matrix of the nth mirror, where xnoAnd ynoRespectively being the centre point of the n-th mirrorx and y coordinates, dcjThe distance between the center point of the jth surface reflector and the center point of the jth +1 surface reflector is obtained; alpha is alphaiThe inclination angle of the ith surface reflector;
representing the surface form expression of the reflector in an off-axis structure.
And 4, step 4: from the central light path, the coordinates (x) of the image point are obtainedIo,yIo) The calculation formula is as follows:
wherein x iskoIs the x coordinate, y coordinate of the central point of the last mirror in the systemkoIs the center point y coordinate, l 'of the last mirror in the system'ckThe distance between the last reflector and the image point;
and 5: constructing an inter-mirror relation expression model taking any mirror as a reference, and comprising the following processes:
step 5.1: obtaining a corresponding relative rotation matrix D based on a coordinate transformation concept according to the inclination angle of the reference reflectoriThe calculation formula is as follows:
αi(i-1)＝αi-αi-1
α10＝α1
wherein D isiIs a relative rotation matrix of any mirror relative to the ith mirror when the ith mirror is taken as a reference; alpha is alphai(i-1)The difference of the inclination angles of the ith surface reflector and the previous surface, namely the ith-1 surface reflector; alpha is alphaiThe inclination angle of the ith surface reflector; alpha is alphai-1Is the inclination angle of the i-1 st surface reflector; alpha is alpha10The difference between the inclination angle of the first surface reflector and the inclination angle of the previous surface reflector; alpha is alpha1The inclination angle of the first surface reflector;
and step 5.2: constructing an inter-mirror relation expression model taking the primary mirror as a reference, wherein the expression of any mirror relative to the primary mirror is as follows:
wherein D is1Is a relative rotation matrix with the primary mirror as a reference;
an expression representing when any mirror is referenced to the primary mirror;
step 5.3: obtaining the position of the coordinate of the center point and the coordinate of the image point of any mirror relative to the main mirror when the main mirror is taken as a reference, wherein the calculation formula is as follows:
wherein, CPnA column vector formed by the coordinates of the center point of the nth surface reflector; CP (CP)n R1The line vector is formed by the coordinates of the central point of the nth surface reflector when the main reflector is taken as the reference; i isoA column vector formed by the coordinates of the image points; i iso R1A column vector formed by coordinates of image points when the primary mirror is used as a reference;
a coordinate transformation matrix when the primary mirror is taken as a reference;
step 5.4: according to the sequence from the 2 nd mirror to the 3 rd mirror to the k-th mirror, the position of the coordinate of the center point and the coordinate of the image point of any mirror is obtained, i is 2,3, …, k, and the calculation formula is:
wherein,
a column vector consisting of central point coordinates of the nth surface reflector based on the ith surface reflector;
a column vector consisting of central point coordinates of the nth surface reflector based on the i-1 st surface reflector; CP (CP)i R(i-1) is a column vector formed by central point coordinates of the ith surface reflector with the i-1 th surface reflector as a reference;
the column vector is formed by the coordinates of the image point when the ith surface reflector is used as the reference;
is the coordinate composition of the image point when the i-1 th surface reflector is used as the referenceThe column vector of (a) is,
a coordinate transformation matrix when the ith mirror is taken as a reference;
step 5.5: obtaining an expression of an arbitrary mirror with the ith mirror as a reference in the order from the 2 nd mirror and the 3 rd mirror to the kth mirror, i being 2,3, …, k:
wherein D isiIs a relative rotation matrix with the ith mirror as a reference;
an x coordinate of a central point of the ith surface reflector based on the ith-1 surface reflector;
is the y coordinate of the center point of the ith surface reflector based on the ith-1 surface reflector, AiIs a relative coordinate point matrix with the i-th mirror as a reference, Ai-1Is a relative coordinate point matrix with the i-1 th mirror as a reference, A1＝A；
Representing the expression of an arbitrary mirror with reference to the ith mirror.
Step 6: modeling an imaging light path of the off-axis reflection system, wherein the process comprises the following steps:
step 6.1: the intersection point of the main mirror parallel to the optical axis is selected as (x)in1,yin1) Based on the optical matrix equation, the height and angle of the ray emergent from the primary mirror are calculated, and the calculation formula is as follows:
wherein, P1A column vector consisting of intersection point coordinates of the light rays and the primary mirror; p1 R1The line vector is formed by the intersection point coordinates of the light and the primary mirror when the primary mirror is taken as a reference; hl (high efficiency liquid chromatography)iM1The incident height of the light on the primary mirror; thetaiM1The incident angle of the light on the primary mirror; rhoM1The curvature radius of the reflector at the intersection point of the light ray and the primary mirror; hl (high pressure chemical vapor deposition)oM1The emergent height of the light on the primary mirror; theta.theta.oM1The emergent angle of the light on the primary mirror is shown;
step 6.2: calculating the light path between the primary mirror and the 2 nd mirror, and obtaining the intersection point of the light and the 2 nd mirror
The calculation formula is as follows:
wherein,
the optical path between the primary mirror and the secondary mirror is based on the primary mirror;
an expression of the secondary mirror with reference to the primary mirror, C2Is a rotation matrix of secondary mirrors, B2Describing the matrix for the facet shape of the secondary mirror, y2oIs the y coordinate of the secondary mirror center point; x is the number of2oIs the x coordinate of the secondary mirror center point;
step 6.3: based on an optical matrix equation and a conversion matrix, a 2 nd mirror and a 3 rd mirror, a 3 rd mirror and a 4 th mirror, … …, and a light path mathematical model between a k-1 th mirror and a k-th mirror are sequentially established, wherein the calculation formula is as follows:
θiMn＝θoM(n-1)-αn(n-1)
wherein hl isiMnThe incident height of the light on the nth surface reflector is shown; thetaiMnThe incident angle of the light on the nth surface reflector is shown; rhoMnThe curvature radius of the reflector at the intersection point of the light ray and the nth reflector; hl (high efficiency liquid chromatography)oMnThe emergent height of the light on the nth surface reflector is shown; thetaoMnThe emergent angle of the light on the nth surface reflector is shown;
the optical path between the nth surface reflector and the (n + 1) th surface reflector is based on the nth surface reflector;
is a column vector formed by the intersection point coordinates of the light and the n-th reflector when the n-th reflector is used as a reference;
for light rays with the n-1 th surface reflector as a referenceA column vector formed by intersection point coordinates of the n-th reflector and the n-th reflector;
a column vector consisting of central point coordinates of the nth surface reflector based on the (n-1) th surface reflector; hl (high pressure chemical vapor deposition)iMnThe incident height of the light on the nth surface reflector; thetaiMnThe incident angle of the light on the nth surface reflector is shown;
the x coordinate of the intersection point of the light ray and the nth mirror is taken as the reference;
is the y coordinate of the intersection point of the light ray and the nth mirror when the nth mirror is taken as the reference; thetaoM(n-1)Is the angle of incidence of the light on the n-1 th mirror, alphan(n-1)Is the difference of the inclination angles of the nth surface reflector and the (n-1) th surface reflector;
step 6.4: based on an optical matrix equation and a conversion matrix, establishing a mathematical model of a light path between the kth surface reflector and an image point, wherein the calculation formula is as follows:
wherein lkIThe distance between the kth surface reflector and the image point is taken as the reference;
the x coordinate of the intersection point of the light ray and the kth surface reflector when the kth surface reflector is taken as a reference;
the x coordinate of an image point with the k-th surface reflector as a reference; rhoMkIs a light ray and the firstThe curvature radius of the reflector at the intersection point of the k-plane reflector; hl (high pressure chemical vapor deposition)iMkThe incident height of the light on the k-th surface reflector; thetaiMkIs the incident angle of the light on the k-th mirror.
And 7: and (3) selecting m incident rays to repeatedly execute the step 6, wherein m is more than or equal to k, and obtaining the intersection point of the corresponding emergent ray and the image surface:
and establishing a target function by taking the height difference of the intersection point of the emergent ray and the image surface as constraint to obtain the surface shape coefficient of the k-surface reflector, wherein the calculation formula is as follows:
wherein,
the height of the intersection point of the ith light ray and the image surface is taken as the height of the intersection point;
the surface shape coefficient is the negative number of the eccentricity square;
the height of the intersection point of the (i + n) th ray and the image plane is taken as the height of the intersection point;
is an objective function.
Adopt the produced beneficial effect of above-mentioned technical scheme to lie in:
1. the invention provides a modeling method with strong expandability and low parameter coupling for the initial structure of the off-axis reflecting system, skillfully simplifies the complex free-form surface modeling into quadratic curve modeling which can be combined with a transformation matrix to integrally express the relationship between the pose and the relative position, and realizes the establishment of an integral model of the reflecting mirror surface shape with any pose in space;
2. the relative rotation matrix is added, so that the problem that the traditional optical matrix limits the relation between the mirror surface and the optical axis is solved, and the model representation of the relative position between the mirrors of any reference mirror is realized;
3. the method establishes mirror-mirror and mirror-image light path modeling based on the optical matrix and the conversion matrix so as to achieve the aim of integrally constructing an initial structure model of the off-axis reflection system;
4. the invention is designed directly based on the design target, the inclination angle and the position of each reflector are controllable, the operation experience of optical designers is not required, the construction process is simple, and the invention is suitable for off-axis systems with any multi-surface reflector, and the application range is relatively wider.
Drawings
FIG. 1 is a flowchart illustrating an integrated modeling method for an initial structure of a multi-reflection off-axis optical system according to an embodiment of the present invention;
FIG. 2 is a schematic diagram of the relative positions of a primary mirror and a secondary mirror in a multi-reflection off-axis optical system according to an embodiment of the present disclosure;
FIG. 3 is a schematic diagram of an initial structure of an off-axis three-mirror optical system according to an embodiment of the present invention;
FIG. 4 is a field diagram of a spot radius RMS spot column for the entire field of view at the image plane of the initial structure in an embodiment of the invention.
Detailed Description
The following detailed description of embodiments of the present invention is provided in connection with the accompanying drawings and examples. The following examples are intended to illustrate the invention but are not intended to limit the scope of the invention.
In the embodiment, an off-axis three-mirror optical system is designed for verification, and the design requirements are as follows: the entrance pupil is 150mm, the focal length is 900mm, and the number of mirrors is 3.
As shown in fig. 1, an integrated modeling method for an initial structure of a multi-reflection off-axis optical system in this embodiment is as follows.
Step 1: establishing a rectangular space coordinate system to make the center of the primary mirror at the origin of the coordinate, i.e. the distance d between the center of the primary mirror and the origin of the coordinatec0Is set to 0;
step 2: according to design requirements, a coaxial structure is obtained based on a paraxial optical theory, and a uniform surface type expression is established for each reflector, wherein the surface type expression is as follows:
A＝[x y 1]
wherein d isc(n-1)Is the distance between the center of the n-1 st surface reflector and the center of the nth surface reflector; rnThe curvature radius of the center point of the nth surface reflector; e.g. of the typenThe eccentricity of the nth surface reflector; k is the total number of the reflecting mirrors; a is a coordinate point matrix; b isnDescribing a matrix for the surface shape;
representing a surface type expression of the reflector in a coaxial structure; x is the x coordinate of any point on the reflector; y is the y coordinate of any point on the mirror.
And step 3: obtaining a corresponding rotation matrix C according to the inclination angle of each reflectornObtaining a corresponding displacement matrix according to the displacement condition of the central point of each reflector, and obtaining the mirror surface unified expression of the off-axis structure after rotation and displacement by combining the rotation matrix and the displacement matrix;
obtaining a corresponding rotation matrix C according to the inclination angle of each reflectornThe method comprises the following steps:
wherein, αnIs the inclination angle of the nth surface reflector.
The relative positions of the primary mirror and the secondary mirror are shown in FIG. 2, where (a) denotes that the mirror center of the primary mirror is at the origin and the tilt angle is α1(ii) a The coordinate of the center point of the secondary mirror is (x)2o,y2o) Angle of inclination of α2. Wherein (b) denotes that the secondary mirror is considered relative to the primary mirrorIn the position, the expression of the secondary mirror is written when a coordinate system is established by taking the center of the primary mirror as an origin and taking a normal perpendicular to the center of the primary mirror as a y axis.
The method for obtaining the corresponding displacement matrix according to the displacement condition of the central point of each reflector is as follows:
where DM is a displacement matrix and Δ x and Δ y are the displacement of the mirror along the x-axis and y-axis, respectively.
The mirror surface of the off-axis structure is uniformly expressed as follows:
wherein, CnIs a rotation matrix, [ -x [ ]no,-yno,0]Is a displacement matrix of the nth mirror, where xnoAnd ynoX and y coordinates of the center point of the nth mirror, dcjThe distance between the center point of the jth surface reflector and the center point of the jth +1 surface reflector is obtained; alpha is alphaiThe inclination angle of the ith surface reflector;
representing the surface form expression of the reflector in an off-axis structure.
The inclination angle employed in this embodiment is: alpha is alpha1＝-12.500°；α2＝-8.648°；α3＝-3.950°。
And 4, step 4: from the central light path, the coordinates (x) of the image point are obtainedIo,yIo) The calculation formula is as follows:
wherein x iskoIs the x coordinate, y coordinate of the central point of the last mirror in the systemkoIs the center point y coordinate, l 'of the last mirror in the system'ckThe distance between the last reflector and the image point;
and 5: constructing an inter-mirror relation expression model taking any mirror as a reference, and comprising the following processes:
step 5.1: obtaining a corresponding relative rotation matrix D based on a coordinate transformation concept according to the inclination angle of the reference reflectoriThe calculation formula is as follows:
αi(i-1)＝αi-αi-1
α10＝α1
wherein D isiIs a relative rotation matrix of any mirror relative to the ith mirror when the ith mirror is taken as a reference; alpha is alphai(i-1)The difference of the inclination angles of the ith surface reflector and the previous surface, namely the ith-1 surface reflector; alpha (alpha) ("alpha")iThe inclination angle of the ith surface reflector; alpha is alphai-1Is the inclination angle of the i-1 st surface reflector; alpha (alpha) ("alpha")10The difference between the inclination angle of the first surface reflector and the inclination angle of the previous surface reflector; alpha is alpha1The inclination angle of the first surface reflector;
step 5.2: constructing an inter-mirror relation expression model taking the primary mirror as a reference, wherein the expression of any mirror relative to the primary mirror is as follows:
wherein D is1Is a relative rotation matrix with the primary mirror as a reference;
an expression representing when any mirror is referenced to the primary mirror;
step 5.3: obtaining the position of the coordinate of the center point and the coordinate of the image point of any mirror relative to the main mirror when the main mirror is taken as a reference, wherein the calculation formula is as follows:
wherein, CPnA column vector formed by the coordinates of the center point of the nth surface reflector; CP (CP)n R1The line vector is formed by the coordinates of the central point of the nth surface reflector when the main reflector is taken as the reference; i isoA column vector formed by the coordinates of the image points; i iso R1A column vector formed by coordinates of image points when the primary mirror is used as a reference;
a coordinate transformation matrix when the primary mirror is taken as a reference;
step 5.4: according to the sequence from the 2 nd mirror to the 3 rd mirror to the k-th mirror, the position of the coordinate of the center point and the coordinate of the image point of any mirror is obtained, i is 2,3, …, k, and the calculation formula is:
wherein,
a column vector consisting of central point coordinates of the nth surface reflector based on the ith surface reflector;
a column vector consisting of central point coordinates of the nth surface reflector based on the i-1 st surface reflector; CP (CP)i R(i-1)A column vector consisting of central point coordinates of the ith surface reflector based on the ith-1 surface reflector;
the column vector is formed by the coordinates of the image point when the ith surface reflector is used as the reference;
is a column vector formed by the coordinates of image points when the i-1 th surface reflector is taken as the reference,
a coordinate transformation matrix when the ith mirror is taken as a reference;
step 5.5: obtaining an expression of an arbitrary mirror with the ith mirror as a reference in the order from the 2 nd mirror and the 3 rd mirror to the kth mirror, i being 2,3, …, k:
wherein D isiIs a relative rotation matrix with the ith mirror as a reference;
an x coordinate of a central point of the ith surface reflector based on the ith-1 surface reflector;
is the y coordinate of the center point of the ith surface reflector based on the ith-1 surface reflector, AiIs a relative coordinate point matrix with the i-th mirror as a reference, Ai-1Is a relative coordinate point matrix with the i-1 th mirror as a reference, A1＝A；
Representing the expression of an arbitrary mirror with reference to the ith mirror.
Step 6: modeling an imaging light path of the off-axis reflection system, wherein the process comprises the following steps:
step 6.1: the intersection point of the main mirror parallel to the optical axis is selected as (x)in1,yin1) Based on the optical matrix equation, the height and angle of the ray emergent from the primary mirror are calculated, and the calculation formula is as follows:
wherein, P1A column vector consisting of intersection point coordinates of the light rays and the primary mirror; p1 R1The line vector is formed by the intersection point coordinates of the light and the primary mirror when the primary mirror is taken as a reference; hl (high pressure chemical vapor deposition)iM1The incident height of the light on the primary mirror; theta.theta.iM1The incident angle of the light on the primary mirror; rhoM1The curvature radius of the reflector at the intersection point of the light ray and the primary mirror; hl (high pressure chemical vapor deposition)oM1The emergent height of the light on the primary mirror; thetaoM1The emergent angle of the light on the primary mirror is shown;
in the present embodiment, k is 3 rays, and the incident heights are: 75mm, -75mm and-72.5 mm.
Step 6.2: calculating the light path between the primary mirror and the 2 nd mirror, and obtaining the intersection point of the light and the 2 nd mirror
The calculation formula is as follows:
wherein,
the optical path between the primary mirror and the secondary mirror is based on the primary mirror;
an expression of the secondary mirror with reference to the primary mirror, C2Is a rotation matrix of secondary mirrors, B2Describing the matrix for the facet shape of the secondary mirror, y2oIs the y coordinate of the secondary mirror center point; x is the number of2oIs the x coordinate of the secondary mirror center point;
step 6.3: based on an optical matrix equation and a conversion matrix, a 2 nd mirror and a 3 rd mirror, a 3 rd mirror and a 4 th mirror, … …, and a light path mathematical model between a k-1 th mirror and a k-th mirror are sequentially established, wherein the calculation formula is as follows:
θiMn＝θoM(n-1)-αn(n-1)
wherein hl isiMnThe incident height of the light on the nth surface reflector is shown; thetaiMnThe incident angle of the light on the nth surface reflector is shown; rhoMnThe curvature radius of the reflector at the intersection point of the light ray and the nth reflector; hl (high pressure chemical vapor deposition)oMnThe emergent height of the light on the nth surface reflector is shown; thetaoMnThe emergent angle of the light on the nth surface reflector is shown;
the optical path between the nth surface reflector and the (n + 1) th surface reflector is based on the nth surface reflector;
the line vector is formed by the intersection point coordinates of the light and the n-th reflector when the n-th reflector is used as a reference;
the column vector is formed by the intersection point coordinates of the light and the nth surface reflector when the nth-1 surface reflector is taken as a reference;
a column vector consisting of central point coordinates of the nth surface reflector based on the nth-1 surface reflector; hl (high pressure chemical vapor deposition)iMnIs a light rayAn incident height on the nth mirror; thetaiMnThe incident angle of the light on the nth surface reflector is shown;
the x coordinate of the intersection point of the light ray and the nth mirror is taken as the reference;
is the y coordinate of the intersection point of the light ray and the nth mirror when the nth mirror is taken as the reference; thetaoM(n-1)Is the angle of incidence of the light on the n-1 th mirror, alphan(n-1)Is the difference of the inclination angles of the nth surface reflector and the (n-1) th surface reflector;
step 6.4: based on an optical matrix equation and a conversion matrix, establishing a mathematical model of a light path between the kth surface reflector and an image point, wherein the calculation formula is as follows:
wherein lkIThe distance between the kth surface reflector and the image point is taken as the reference;
the x coordinate of the intersection point of the light ray and the kth surface reflector when the kth surface reflector is taken as a reference;
the x coordinate of an image point with the k-th surface reflector as a reference; rhoMkThe curvature radius of the reflector at the intersection point of the ray and the kth reflector; hl (high pressure chemical vapor deposition)iMkThe incident height of the light on the k-th surface reflector; thetaiMkIs the incident angle of the light on the k-th mirror.
And 7: and (3) selecting m incident rays to repeatedly execute the step 6, wherein m is more than or equal to k, and obtaining the intersection point of the corresponding emergent ray and the image surface:
and establishing a target function by taking the height difference of the intersection point of the emergent ray and the image surface as constraint to obtain the surface shape coefficient of the k-surface reflector, wherein the calculation formula is as follows:
wherein,
the height of the intersection point of the ith light ray and the image surface is taken as the height of the intersection point;
the surface shape coefficient is the negative number of the eccentricity square;
the height of the intersection point of the (i + n) th ray and the image plane is taken as the height of the intersection point;
is an objective function.
The initial structural system layout of the off-axis three-mirror optical system designed by the method is shown in fig. 3, and it can be seen that light rays with different field angles and different aperture coordinates are basically focused on an image point. The spot radius RMS point-column field diagram of the whole field of view on the image plane of the initial structure is shown in FIG. 4, the minimum RMS value is 1.5275mm, and the maximum RMS value is 1.6894mm, which proves that the initial system has better light convergence in the full field of view. Thus, the initial system can serve as a starting point for further optimization.
Claims (7)
1. An integrated modeling method for an initial structure of a multi-reflection off-axis optical system is characterized by comprising the following steps:
step 1: establishing a rectangular space coordinate system to make the center of the primary mirror at the origin of the coordinate, i.e. the distance d between the center of the primary mirror and the origin of the coordinatec0Is set to 0;
and 2, step: obtaining a coaxial structure based on a paraxial optical theory according to design requirements, and establishing a uniform surface expression for each reflector;
and step 3: obtaining a corresponding rotation matrix C according to the inclination angle of each reflectornObtaining a corresponding displacement matrix according to the displacement condition of the central point of each reflector, and obtaining the mirror surface unified expression of the off-axis structure after rotation and displacement by combining the rotation matrix and the displacement matrix;
and 4, step 4: from the central light path, the coordinates (x) of the image point are obtainedIo,yIo) The calculation formula is as follows:
wherein x iskoIs the x coordinate, y coordinate of the central point of the last mirror in the systemkoIs the center point y coordinate, l 'of the last mirror in the system'ckIs the distance from the last mirror to the image point, aiThe inclination angle of the ith surface reflector;
and 5: constructing an inter-mirror relation expression model taking any mirror as a reference;
step 6: modeling an imaging light path of the off-axis reflecting system;
and 7: and (3) selecting m incident rays to repeatedly execute the step 6, wherein m is more than or equal to k, and obtaining the intersection point of the corresponding emergent ray and the image surface:
and establishing a target function by taking the height difference of the intersection point of the emergent ray and the image surface as constraint to obtain the surface shape coefficient of the k-surface reflector, wherein the calculation formula is as follows:
wherein,
the height of the intersection point of the ith light ray and the image surface is taken as the height of the intersection point;
the surface shape coefficient is the negative number of the eccentricity square;
the height of the intersection point of the (i + n) th ray and the image plane is taken as the height of the intersection point;
is an objective function.
2. The method of claim 1, wherein the method comprises the following steps: the face type expression in step 2 is as follows:
A＝[x y 1]
wherein d isc(n-1)Is the distance between the center of the n-1 st surface reflector and the center of the nth surface reflector; rnThe curvature radius of the center point of the nth surface reflector; e.g. of the typenThe eccentricity of the nth surface reflector; k is the total number of the reflecting mirrors; a is a coordinate point matrix; b isnDescribing a matrix for the surface shape;
representing a surface type expression of the reflector in a coaxial structure; x is the x coordinate of any point on the reflector; y is the y coordinate of any point on the mirror.
3. The method of integrally modeling an initial structure of a multi-reflection off-axis optical system according to claim 2, wherein: obtaining a corresponding rotation matrix C according to the inclination angle of each reflectornThe method comprises the following steps:
wherein anIs the inclination angle of the nth surface reflector.
4. The method of claim 3, wherein the method comprises the following steps: the mirror surface of the off-axis structure in the step 3 is uniformly expressed as follows:
wherein, CnIs a rotation matrix, [ -x [ ]no,-yno,0]Is a displacement matrix of the nth mirror, where xnoAnd ynoX and y coordinates of the center point of the nth mirror, dcjThe distance between the center point of the jth surface reflector and the center point of the jth +1 surface reflector is obtained; alpha (alpha) ("alpha")iThe inclination angle of the ith surface reflector;
representing the surface form expression of the reflector in an off-axis structure.
5. The method for integrally modeling the initial structure of the multi-reflection off-axis optical system according to claim 4, wherein: the method for obtaining the corresponding displacement matrix according to the displacement condition of the central point of each reflector is as follows:
where DM is a displacement matrix and Δ x and Δ y are the displacement of the mirror along the x-axis and y-axis, respectively.
6. The method of claim 5, wherein the method comprises the following steps: the process of the step 5 is as follows:
step 5.1: obtaining a corresponding relative rotation matrix D based on a coordinate transformation concept according to the inclination angle of the reference reflectoriThe calculation formula is as follows:
αi(i-1)＝αi-αi-1
α10＝α1
wherein D isiIs a relative rotation matrix of any mirror relative to the ith mirror when the ith mirror is taken as a reference; alpha is alphai(i-1)The difference of the inclination angles of the ith surface reflector and the previous surface, namely the ith-1 surface reflector; alpha is alphaiThe inclination angle of the ith surface reflector; alpha is alphai-1Is the inclination angle of the i-1 st surface reflector; alpha is alpha10The difference between the inclination angle of the first reflector and the inclination angle of the upper reflector; alpha is alpha1The inclination angle of the first surface reflector;
step 5.2: constructing an inter-mirror relation expression model taking the primary mirror as a reference, wherein the expression of any mirror relative to the primary mirror is as follows:
wherein D is1Is a relative rotation matrix with the primary mirror as a reference;
an expression representing when any mirror is referenced to the primary mirror;
step 5.3: obtaining the position of the coordinate of the center point and the coordinate of the image point of any mirror relative to the main mirror when the main mirror is taken as a reference, wherein the calculation formula is as follows:
wherein, CPnA column vector formed by the coordinates of the center point of the nth surface reflector; CP (CP)n R1The line vector is formed by the coordinates of the central point of the nth surface reflector when the main reflector is taken as the reference; i isoA column vector formed by the coordinates of the image points; i iso R1A column vector formed by coordinates of image points when the primary mirror is taken as a reference;
a coordinate transformation matrix when the primary mirror is taken as a reference;
step 5.4: according to the sequence from the 2 nd mirror to the 3 rd mirror to the k-th mirror, the position of the coordinate of the center point and the coordinate of the image point of any mirror is obtained, i is 2,3, …, k, and the calculation formula is:
wherein,
a column vector consisting of central point coordinates of the nth surface reflector based on the ith surface reflector;
a column vector consisting of central point coordinates of the nth surface reflector based on the i-1 st surface reflector;
a column vector consisting of central point coordinates of the ith surface reflector based on the ith-1 surface reflector;
a column vector formed by the coordinates of the image point when the image point takes the ith surface reflector as the reference;
is a column vector formed by the coordinates of the image point when the i-1 st surface reflector is used as the reference,
a coordinate transformation matrix when the ith mirror is taken as a reference;
step 5.5: obtaining an expression of an arbitrary mirror with the ith mirror as a reference in the order from the 2 nd mirror and the 3 rd mirror to the kth mirror, i being 2,3, …, k:
wherein D isiIs a relative rotation matrix with the ith mirror as a reference;
an x coordinate of a central point of the ith surface reflector based on the ith-1 surface reflector;
is the y coordinate of the center point of the ith surface reflector based on the ith-1 surface reflector, AiIs a relative coordinate point matrix with the i-th mirror as a reference, Ai-1Is a relative coordinate point matrix with the i-1 th mirror as a reference, A1＝A；
Representing the expression of an arbitrary mirror with reference to the ith mirror.
7. The method of claim 6, wherein the method comprises the following steps: the process of the step 6 is as follows:
step 6.1: the intersection point of the main mirror parallel to the optical axis is selected as (x)in1,yin1) Based on the optical matrix equation, calculating the light ray that exits on the primary mirror based on the primary mirrorHeight and angle, the formula for calculation is:
wherein, P1A column vector is formed by the coordinates of the intersection point of the light ray and the primary mirror;
the line vector is formed by the intersection point coordinates of the light and the primary mirror when the primary mirror is taken as a reference; hl (high pressure chemical vapor deposition)iM1The incident height of the light on the primary mirror; thetaiM1The incident angle of the light on the primary mirror; rhoM1The curvature radius of the reflector at the intersection point of the light ray and the primary mirror; hl (high pressure chemical vapor deposition)oM1The emergent height of the light on the primary mirror; thetaoM1The emergent angle of the light on the primary mirror is shown;
step 6.2: calculating the light path between the primary mirror and the 2 nd mirror, and obtaining the intersection point of the light and the 2 nd mirror
The calculation formula is as follows:
wherein,
the optical path between the primary mirror and the secondary mirror is based on the primary mirror;
an expression of the secondary mirror with reference to the primary mirror, C2Is a rotation matrix of secondary mirrors, B2Describing the matrix for the facet shape of the secondary mirror, y2oIs the y coordinate of the secondary mirror center point; x is the number of2oIs the x coordinate of the secondary mirror center point;
step 6.3: based on an optical matrix equation and a conversion matrix, a 2 nd mirror and a 3 rd mirror, a 3 rd mirror and a 4 th mirror, … …, and a light path mathematical model between a k-1 th mirror and a k-th mirror are sequentially established, wherein the calculation formula is as follows:
θiMn＝θoM(n-1)-αn(n-1)
wherein hl isiMnThe incident height of the light on the nth surface reflector is shown; thetaiMnThe incident angle of the light on the nth surface reflector is shown; rhoMnThe curvature radius of the reflector at the intersection point of the light ray and the nth reflector; hl (high pressure chemical vapor deposition)oMnThe emergent height of the light on the nth surface reflector is shown; thetaoMnThe emergent angle of the light on the nth surface reflector is shown;
the nth surface reflector and the (n + 1) th surface reflector are used as referenceAn optical path therebetween;
the line vector is formed by the intersection point coordinates of the light and the n-th reflector when the n-th reflector is used as a reference;
the column vector is formed by the intersection point coordinates of the light and the nth surface reflector when the nth-1 surface reflector is taken as a reference;
a column vector consisting of central point coordinates of the nth surface reflector based on the nth-1 surface reflector; hl (high pressure chemical vapor deposition)iMnThe incident height of the light on the nth surface reflector is shown;
the x coordinate of the intersection point of the light ray and the nth mirror is taken as the reference;
the y coordinate of the intersection point of the light ray and the nth mirror is taken as the reference; thetaoM(n-1)Angle of incidence of light on the n-1 th mirror, αn(n-1)Is the difference of the inclination angles of the nth surface reflector and the (n-1) th surface reflector;
step 6.4: based on an optical matrix equation and a conversion matrix, establishing a mathematical model of a light path between the kth surface reflector and an image point, wherein the calculation formula is as follows:
wherein lkIWhen the k-th surface reflector is used as a reference, the k-th surface reflector is arranged between the image point and the k-th surface reflectorThe distance of (d);
the x coordinate of the intersection point of the light ray and the kth surface reflector when the kth surface reflector is taken as a reference;
the x coordinate of an image point with the k-th surface reflector as a reference; rhoMkThe curvature radius of the reflector at the intersection point of the ray and the kth reflector; hl (high pressure chemical vapor deposition)iMkThe incident height of the light on the k-th surface reflector; thetaiMkIs the incident angle of the light on the k-th mirror.
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