Method for designing offaxis aspheric optical system

0Associated
Cases 
0Associated
Defendants 
0Accused
Products 
0Forward
Citations 
0
Petitions 
1
Assignment
First Claim
1. A method for making an offaxis aspheric optical system comprising steps S1˜
 S7, wherein the steps S1˜
S6 are implemented by a computer processor, the method comprises;
step (S1), establishing an initial system, the initial system comprising a plurality of initial surfaces, and each of the plurality of initial surfaces corresponding to an aspheric surface of the offaxis aspheric optical system; and
selecting a plurality of feature rays R_{i }(i=1, 2 . . . K) from different fields and different aperture positions;
step (S2), keeping the plurality of initial surfaces unchanged, and solving a plurality of feature data points (P_{1}, P_{2}, . . . P_{m}) point by point based on given objectimage relationship and Snell'"'"'s law to obtain an initial offaxis aspheric surface A_{m }by surface fitting the plurality of feature data points (P_{1}, P_{2}, . . . P_{m}), wherein m is less than K;
step (S3), introducing an intermediate point G_{m }based on the initial offaxis aspheric A_{m }to solve a (m+1)th feature data point P_{m+1}, and fitting a plurality of feature data points (P_{1}, P_{2}, . . . P_{m}, P_{m+1}) to obtain an offaxis aspheric surface A_{m+1};
introducing an intermediate point G_{m+1 }based on the offaxis aspheric surface A_{m+1 }to solve a (m+2)th feature data point P_{m+2}, and fitting a plurality of feature data points (P_{1}, P_{2}, . . . P_{m}, P_{m+1}, P_{m+2}) to obtain an offaxis aspheric surface A_{m+2};
repeating such steps until a Kth feature data point P_{K }is solved, and fitting a plurality of feature data points (P_{1}, P_{2}, . . . P_{K}) to obtain an offaxis aspheric surface A_{K}, wherein the offaxis aspheric surface A_{K }is a first aspheric surface of the offaxis aspheric optical system;
step (S4), keeping the first aspheric surface and other initial surfaces except the initial surface corresponds to a second aspheric surface of the offaxis aspheric optical system unchanged, and solving a plurality of feature data points (P′
_{1}, P′
_{2}, . . . P′
_{m}) based on given objectimage relationship and Snell'"'"'s law to obtain an initial offaxis aspheric surface A′
_{m }by surface fitting the plurality of feature data points (P′
_{1}, P′
_{2}, . . . P′
_{m}), wherein m is less than K;
step (S5), introducing an intermediate point G′
_{m }based on the initial offaxis aspheric surface A_{m }to solve a (m+1)th feature data point P′
_{m+1}, and fitting a plurality of feature data points (P′
_{1}, P′
_{2}, . . . P′
_{m+1}) to obtain an offaxis aspheric surface A′
_{m+1};
introducing an intermediate point G_{m+1 }based on the offaxis aspheric surface A′
_{m+1 }to solve a (m+2)th feature data point P′
_{m+2}, and fitting a plurality of feature data points (P′
_{1}, P′
_{2}, . . . P′
_{m+1}, P′
_{m+2}) to obtain an offaxis aspheric surface A′
_{m+2};
repeating such steps until a Kth feature data point P′
_{K }is solved, and fitting a plurality of feature data points (P′
_{1}, P′
_{2}, . . . P′
_{K}) to obtain an offaxis aspheric surface A′
_{K}, wherein the offaxis aspheric A′
_{K }is the second aspheric surface of the offaxis aspheric optical system;
step (S6), repeating the steps (S2)˜
(S5) until all the aspheric surfaces of the offaxis aspheric optical system are obtained andstep (S7), making an offaxis aspheric optical system comprising a primary mirror, a secondary mirror and a tertiary mirror based on all the aspheric surfaces obtained in step (S6).
1 Assignment
0 Petitions
Accused Products
Abstract
A pointbypoint design method for offaxis aspheric optical system, in which feature light rays from different field angles and aperture coordinates are considered. Some of the feature data points are calculated first and surface fitted into an initial aspheric surface. Then, intermediate point calculations, feature data point calculations, and aspheric surface fitting were repeated continuously to calculate remaining feature data points and the desired aspheric surface are repeated continuously to calculate remaining feature data points and a desired aspheric surface. A leastsquares method with a local search algorithm is used for aspheric surface fitting and deviations in both the coordinates and normals are used to reduce error.
5 Citations
No References
Methods and apparatus of aligning surfaces  
Patent #
US 20070195084A1
Filed 08/29/2006

Current Assignee
Hong Kong Polytechnic University

Sponsoring Entity
Hong Kong Polytechnic University

Method for designing a refractive or reflective optical system and method for designing a diffraction optical element  
Patent #
US 6,567,226 B2
Filed 03/01/1999

Current Assignee
Sumitomo Electric Industries Limited

Sponsoring Entity
Sumitomo Electric Industries Limited

METHOD FOR DESIGNING A REFRACTIVE OR REFLECTIVE OPTICAL SYSTEM AND METHOD FOR DESIGNING A DIFFRACTION OPTICAL ELEMENT  
Patent #
US 20020183994A1
Filed 03/01/1999

Current Assignee
Sumitomo Electric Industries Limited

Sponsoring Entity
Sumitomo Electric Industries Limited

OFFAXIAL THREEMIRROR OPTICAL SYSTEM WITH FREEFORM SURFACES  
Patent #
US 20150253552A1
Filed 12/16/2014

Current Assignee
Tsinghua University, Hon Hai Precision Industry Co. Ltd.

Sponsoring Entity
Tsinghua University, Hon Hai Precision Industry Co. Ltd.

METHOD FOR DESIGNING OFFAXIS THREEMIRROR IMAGING SYSTEM WITH FREEFORM SURFACES  
Patent #
US 20160170191A1
Filed 07/30/2015

Current Assignee
Tsinghua University, Hon Hai Precision Industry Co. Ltd.

Sponsoring Entity
Tsinghua University, Hon Hai Precision Industry Co. Ltd.

15 Claims
 1. A method for making an offaxis aspheric optical system comprising steps S1˜
 S7, wherein the steps S1˜
S6 are implemented by a computer processor, the method comprises;step (S1), establishing an initial system, the initial system comprising a plurality of initial surfaces, and each of the plurality of initial surfaces corresponding to an aspheric surface of the offaxis aspheric optical system; and
selecting a plurality of feature rays R_{i }(i=1, 2 . . . K) from different fields and different aperture positions;step (S2), keeping the plurality of initial surfaces unchanged, and solving a plurality of feature data points (P_{1}, P_{2}, . . . P_{m}) point by point based on given objectimage relationship and Snell'"'"'s law to obtain an initial offaxis aspheric surface A_{m }by surface fitting the plurality of feature data points (P_{1}, P_{2}, . . . P_{m}), wherein m is less than K; step (S3), introducing an intermediate point G_{m }based on the initial offaxis aspheric A_{m }to solve a (m+1)th feature data point P_{m+1}, and fitting a plurality of feature data points (P_{1}, P_{2}, . . . P_{m}, P_{m+1}) to obtain an offaxis aspheric surface A_{m+1};
introducing an intermediate point G_{m+1 }based on the offaxis aspheric surface A_{m+1 }to solve a (m+2)th feature data point P_{m+2}, and fitting a plurality of feature data points (P_{1}, P_{2}, . . . P_{m}, P_{m+1}, P_{m+2}) to obtain an offaxis aspheric surface A_{m+2};
repeating such steps until a Kth feature data point P_{K }is solved, and fitting a plurality of feature data points (P_{1}, P_{2}, . . . P_{K}) to obtain an offaxis aspheric surface A_{K}, wherein the offaxis aspheric surface A_{K }is a first aspheric surface of the offaxis aspheric optical system;step (S4), keeping the first aspheric surface and other initial surfaces except the initial surface corresponds to a second aspheric surface of the offaxis aspheric optical system unchanged, and solving a plurality of feature data points (P′
_{1}, P′
_{2}, . . . P′
_{m}) based on given objectimage relationship and Snell'"'"'s law to obtain an initial offaxis aspheric surface A′
_{m }by surface fitting the plurality of feature data points (P′
_{1}, P′
_{2}, . . . P′
_{m}), wherein m is less than K;step (S5), introducing an intermediate point G′
_{m }based on the initial offaxis aspheric surface A_{m }to solve a (m+1)th feature data point P′
_{m+1}, and fitting a plurality of feature data points (P′
_{1}, P′
_{2}, . . . P′
_{m+1}) to obtain an offaxis aspheric surface A′
_{m+1};
introducing an intermediate point G_{m+1 }based on the offaxis aspheric surface A′
_{m+1 }to solve a (m+2)th feature data point P′
_{m+2}, and fitting a plurality of feature data points (P′
_{1}, P′
_{2}, . . . P′
_{m+1}, P′
_{m+2}) to obtain an offaxis aspheric surface A′
_{m+2};
repeating such steps until a Kth feature data point P′
_{K }is solved, and fitting a plurality of feature data points (P′
_{1}, P′
_{2}, . . . P′
_{K}) to obtain an offaxis aspheric surface A′
_{K}, wherein the offaxis aspheric A′
_{K }is the second aspheric surface of the offaxis aspheric optical system;step (S6), repeating the steps (S2)˜
(S5) until all the aspheric surfaces of the offaxis aspheric optical system are obtained andstep (S7), making an offaxis aspheric optical system comprising a primary mirror, a secondary mirror and a tertiary mirror based on all the aspheric surfaces obtained in step (S6).  View Dependent Claims (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15)
 S7, wherein the steps S1˜
1 Specification
This application claims all benefits accruing under 35 U.S.C. § 119 from China Patent Application No. 201611107808.5, field on Dec. 5, 2016 in the China Intellectual Property Office, disclosure of which is incorporated herein by reference. The application is also related to copending applications entitled, “OFFAXIS ASPHERIC THREEMIRROR OPTICAL SYSTEM”, filed Aug. 9, 2017 Ser. No. 15/672,306.
The present disclosure relates to a method for designing an offaxis aspheric optical system.
Compared with spherical and quadric surfaces, aspheric surfaces have more degrees of design freedom. If aspheric surfaces are used in offaxis threemirror optical systems, the aberrations of the offaxis threemirror optical systems can be significantly reduced, while the freedom of the offaxis threemirror optical systems can be greatly improved.
However, conventional methods for designing offaxis aspheric optical system have limitations in the number of fields of view and the number of apertures to be considered.
Implementations of the present technology will now be described, by way of example only, with reference to the attached figures.
It will be appreciated that for simplicity and clarity of illustration, where appropriate, reference numerals have been repeated among the different figures to indicate corresponding or analogous elements. In addition, numerous specific details are set forth in order to provide a thorough understanding of the embodiments described herein. However, it will be understood by those of ordinary skill in the art that the embodiments described herein can be practiced without these specific details. In other instances, methods, procedures, and components have not been described in detail so as not to obscure the related relevant feature being described. Also, the description is not to be considered as limiting the scope of the embodiments described herein. The drawings are not necessarily to scale and the proportions of certain parts have been exaggerated to better illustrate details and features of the present disclosure.
Several definitions that apply throughout this disclosure will now be presented.
The term “substantially” is defined to be essentially conforming to the particular dimension, shape, or other feature that the term modifies, such that the component need not be exact. For example, “substantially cylindrical” means that the object resembles a cylinder, but can have one or more deviations from a true cylinder. The term “comprising,” when utilized, means “including, but not necessarily limited to”; it specifically indicates openended inclusion or membership in the sodescribed combination, group, series and the like.
A method for designing an offaxis aspheric optical system of one embodiment is provided. The method comprises the following steps:
step (S1), establishing an initial system, wherein the initial system comprises a plurality of initial surfaces, and each of the plurality of initial surfaces corresponds to one aspheric surface of the offaxis aspheric optical system; one aspheric surface of the offaxial aspheric optical system is defined as an aspheric surface “a”, another aspheric surface of the offaxial aspheric optical system is defined as an aspheric surface “b”; and selecting a plurality of feature rays R_{i }(i=1, 2 . . . K) from different fields and different aperture positions;
step (S2), keeping the plurality of initial surfaces unchanged, and solving a plurality of feature data points (P_{1}, P_{2}, . . . P_{m}) point by point based on given objectimage relationship and Snell'"'"'s law to obtain an initial offaxis aspheric surface A_{m }by surface fitting the plurality of feature data points (P_{1}, P_{2}, . . . P_{m}), wherein m is less than K;
step (S3), introducing an intermediate point G_{m }based on the initial offaxis aspheric A_{m }to solve a (m+1)th feature data point P_{m+1}, and fitting the (m+1) feature data points (P_{1}, P_{2}, . . . P_{m}, P_{m+1}) to obtain an offaxis aspheric surface A_{m+1}; introducing an intermediate point G_{m+1 }based on the offaxis aspheric surface A_{m+1 }to solve a (m+2)th feature data point P_{m+2}, and fitting the (m+2) feature data points (P_{1}, P_{2}, . . . P_{m}, P_{m+1}, P_{m+2}) to obtain an offaxis aspheric surface A_{m+2}; repeating such steps until a Kth feature data point P_{K }is solved, and fitting the feature data points (P_{1}, P_{2}, . . . P_{K}) to obtain an offaxis aspheric surface A_{K}, wherein the offaxis aspheric surface A_{K }is the aspheric surface “a”;
step (S4), keeping the aspheric surface “a” and other initial surfaces except the initial surface corresponds to the aspheric surface “b” unchanged, and solving a plurality of feature data points (P′_{1}, P′_{2}, . . . P′_{m}) point by point based on given objectimage relationship and Snell'"'"'s law to obtain an initial offaxis aspheric surface A′_{m }by surface fitting the plurality of feature data points (P′_{1}, P′_{2}, . . . P′_{m}), wherein m is less than K;
step (S5), introducing an intermediate point G′_{m }based on the initial offaxis aspheric surface A_{m }to solve a (m+1)th feature data point P′_{m+1}, and fitting the (m+1) feature data points (P′_{1}, P′_{2}, . . . P′_{m+1}) to obtain an offaxis aspheric surface A′_{m+1}; introducing an intermediate point G_{m+1 }based on the offaxis aspheric surface A′_{m+1 }to solve a (m+2)th feature data point P′_{m+2}, and fitting the (m+2) feature data points (P′_{1}, P′_{2}, . . . P′_{m+1}, P′_{m+2}) to obtain an offaxis aspheric surface A′_{m+2}; repeating such steps until a Kth feature data point P′_{K }is solved, and fitting the feature data points (P′_{1}, P′_{2}, . . . P′_{K}) to obtain an offaxis aspheric surface A′_{K}, wherein the offaxis aspheric A′_{K }is the aspheric surface “b”; and
step (S6), repeating the steps (S2)˜(S5) until all the aspheric surfaces of the offaxis aspheric optical system are obtained.
In step (S1), a premise for establishing the initial system is to eliminate obscuration. The initial surface can be a planar surface or a spherical surface. In one embodiment, the initial system comprises three initial surfaces; the three initial surfaces are a primary mirror initial surface, a secondary mirror initial surface and a tertiary mirror initial surface. Each of the primary mirror initial surface, the secondary mirror initial surface and the tertiary mirror initial surface is a planar surface with eccentricity and inclination.
A method for selecting the plurality of feature rays R_{i }(i=1, 2 . . . K) from different fields and different aperture positions comprises steps of: M fields are selected according to the optical systems actual needs; an aperture of each of the M fields is divided into N equal parts; and, P feature rays at different aperture positions in each of the N equal parts are selected. As such, K=M×N×P different feature rays correspond to different aperture positions and different fields are fixed. The aperture can be circle, rectangle, square, oval or other shapes. In one embodiment, the aperture of each of the M fields is a circle, and a circular aperture of each of the M fields is divided into N angles with equal interval φ, as such, N=2π/φ; then, P different aperture positions are fixed along a radial direction of each of the N angles. Therefore, K=M×N×P different feature rays correspond to different aperture positions and different fields are fixed. In one embodiment, six fields are fixed in the construction process, the six fields are (0°, −16°), (0°, −13°), (0°, −10°), (1°, −16°), (1°, −13°), and (1°, −10°); one hundred and twelve feature rays are fixed from each of the six fields. Therefore, six hundred and seventytwo different feature rays correspond to different aperture positions and different fields are fixed.
Referring to
In step (S2), a value of m is selected to improve an accuracy of the aspheric surface fitting. In one embodiment, the value of m can be ranged from about K/3 to about 2K/3. If the value of m is too small, such as less than K/3, a fitting error will be large; on the contrary, if the value of m is too large, such as large than 2K/3, the subsequent feature data points are few, which is not conducive to further reduce the fitting error. In one embodiment, the value of m is about K/2.
The plurality of feature data points (P_{1}, P_{2}, . . . P_{m}) can be obtained by the following two calculating methods.
A first calculating method includes the following substeps:
Step (a): defining a first intersection point of a first feature ray R_{1 }and the initial surface corresponding to the aspheric surface “a” as the feature data point P_{1};
Step (b): when i (1≤i≤m−1) feature data points P_{i}(1≤i≤m−1) have been obtained, a unit normal vector {right arrow over (N)}_{i }(1≤i≤m−1) at each of the i (1≤i≤m−1) first feature data points P_{i }(1≤i≤m−1) can be calculated based on a vector form of Snell'"'"'s Law;
Step (c): making a first tangent plane at each of the i (1≤i≤m−1) feature data points P_{i}(1≤i≤m−1); thus i first tangent planes can be obtained, and i×(m−i) second intersection points can be obtained by the i first tangent planes intersecting with remaining (m−i) feature rays; and a second intersection point, which is nearest to the i (1≤i≤m−1) first feature data points P_{1}, is fixed from the i×(m−i) second intersection points as a next feature data point P_{i}+(1≤i≤m−1); and
Step (d): repeating steps b and c, until all the plurality of feature data points P_{i }(i=1, 2 . . . m) are calculated.
In step (b), the unit normal vector {right arrow over (N)}_{i }(1≤i≤m−1) at each of the feature data point P_{i }(1≤i≤m−1) can be calculated based on the vector form of Snell'"'"'s Law. When the surface Ω is a refractive surface,
is a unit vector along a direction of an incident ray of the surface Ω;
is a unit vector along a direction of an exit ray of the surface Ω; and n, n′ is refractive index of a media at two opposite sides of the surface Ω respectively.
Similarly, when the surface Ω is a reflective surface,
The unit normal vector {right arrow over (N)}_{i }at the feature data points P_{i }(i=1, 2 . . . m) is perpendicular to the first tangent plane at the feature data points P_{i }(i=1, 2 . . . m). Thus, the first tangent planes at the feature data points P_{i }(i=1, 2 . . . m) can be obtained.
The first calculating method includes a computational complexity formula of
When multifeature rays are used in a design, the first calculating method requires a long computation time.
A second calculating method includes the following substeps:
Step (a′): defining a first intersection point of a first feature ray R_{1 }and the aspheric “a” as the feature data point P_{1};
Step (b′): when an ith (1≤i≤m−1) feature data point P_{i}(1≤i≤m−1) has been obtained, a unit normal vector {right arrow over (N)}_{i }at the ith (1≤i≤m−1) feature data point P_{i }(1≤i≤m−1) can be calculated based on the vector form of Snell'"'"'s Law;
Step (c′): making a first tangent plane through the ith (1≤i≤m−1) feature data point P_{i}(1≤i≤m−1); and (m−i) second intersection points can be obtained by the first tangent plane intersects with remaining (m−i) feature rays; a second intersection point Q_{i+1}, which is nearest to the ith (1≤i≤m−1) feature data point P_{i}(1≤i≤m−1), is fixed; and a feature ray corresponding to the second intersection point Q_{i+1 }is defined as R_{i+1}, a shortest distance between the second intersection point Q_{i+1 }and the ith (1≤i≤m−1) feature data point P_{i}(1≤i≤m−1) is defined as d_{i};
Step (d′): making a second tangent plane at (i−1) feature data points that are obtained before the ith feature data point P_{i}(1≤i≤m−1) respectively; thus, (i−1) second tangent planes can be obtained, and (i−1) third intersection points can be obtained by the (i−1) second tangent planes intersecting with a feature ray R_{i+1}; in each of the (i−1) second tangent planes, each of the third intersection points and its corresponding feature data point form an intersection pair; the intersection pair, which has the shortest distance between a third intersection point and its corresponding feature data point, is fixed; and the third intersection point and the shortest distance is defined as Q′_{i+1 }and d′_{i }respectively;
Step (e′): comparing d_{i }and d′_{i}, if d_{i}≤d′_{i}, Q_{i+1 }is taken as the next feature data point P_{i+1 }(1≤i≤m−1); otherwise, Q′_{i+1 }is taken as the next feature data point P_{i+1 }(1≤i≤m−1); and
Step (f′): repeating steps from b′ to e′, until the plurality of feature data points P_{i }(i=1, 2 . . . m) are all calculated.
In Step (b′), a calculating method of the unit normal vector {right arrow over (N)}_{i }at the ith (1≤i≤m−1) feature data point P_{i}(1≤i≤m−1) is same as the first calculation method.
A second calculating method includes a computational complexity formula of
When multifeature rays are used in a design, the computational complexity of the second calculating method is smaller than the computational complexity of the first calculating method. In one embodiment, constructing the plurality of first feature data points P_{i }(i=1, 2 . . . m) point by point using the second calculating method.
Referring to
The plurality of feature data points (P_{1}, P_{2}, . . . P_{m}) are fitted under the local coordinate system xyz to obtain the initial offaxis aspheric surface A_{m}.
In step (S3), referring to
The feature data point P_{m+1 }is an intersection point between its corresponding feature ray and a tangent plane of the feature data points (P_{1}, P_{2}, . . . P_{m}) that is closest to the feature data point P_{m+1}.
Referring to
The methods for solving the feature data points P_{m+2 }and P_{K }are the same as the method for solving the feature data point P_{m+1}. A method for solving an intermediate point G_{m+1 }in a process of solving the feature data point P_{m+2 }and a method for solving an intermediate point G_{K1 }in a process of solving the feature data point P_{K }are the same as the method for solving the intermediate point G_{m}.
The feature data points (P_{1}, P_{2 }. . . P_{K}) are fitted in the global coordinate system x_{0}y_{0}z_{0 }to obtain the offaxis aspheric surface A_{K}. An equation of the offaxis aspheric surface A_{K }can be expressed as follows:
Wherein
is a quadratic term, c represents surface curvature at quadratic surface vertex, k represents conic constant,
is an aspheric term, while a_{i }represents the ith term coefficient.
The equation of the offaxis aspheric surface A_{K }can be transformed into linear XY polynomials by a Taylor series expansion to simplify the fitting process. Thus, the equation of the offaxis aspheric surface A_{K }also be expressed as follows:
Wherein, L is a more than eighth order term about r. Since an order of L term is too high, a processing of the offaxis aspheric surface A_{K }is more difficult, so the L item can be directly omitted in the actual calculation.
A fitting normal vector N=(U, V, −1) of the feature data points determines a direction of the feature ray. If a normal fitting error is large, a propagation path of the feature ray will deviate from an expected direction. Therefore, both a coordinate error and a normal error during the fitting should be considered to obtain an accurate aspheric surface. The coordinate error and the normal error are linearly weighted, and an error evaluation function J is proposed:
Wherein Z is a fitting coordinate, (U, V, −1) is the fitting normal vector, z is a real coordinate, (u, v, −1) is a real normal vector, co is a weight of the normal error, P is an coefficient matrix (A, B, C, D . . . ) of unknown aspheric surface, A_{1 }is a matrix (r^{2}, r^{4}, r^{6}, r^{8 }. . . ), A_{2 }is a matrix obtained by calculating the partial derivatives of A_{1 }with respect to x, and A_{3 }is a matrix obtained by calculating the partial derivatives of A_{1 }with respect toy.
When the error evaluation function J is smallest, a solving equation of the coefficient matrix P is:
P=(A_{1}^{T}A_{1}+ωA_{2}^{T}A_{2}+ωA_{3}^{T}A_{3})^{−1}(A_{1}^{T}z+ωA_{2}^{T}u+ωA_{3}^{T}v) (4).
A method for fitting the feature data points (P_{1}, P_{2 }. . . P_{k}) to obtain the offaxis aspheric surface A_{K }comprising the substep:
step (S31), transferring the coordinates (x_{0}, y_{0}, z_{0}) of the feature data points P_{i }(i=1, 2 . . . K) and their corresponding normal vectors (α_{0}, β_{0}, γ_{0}) in the global coordinate system x_{0}y_{0}z_{0 }to the coordinates (x, y, z) and their corresponding normal vectors (α, β, γ) in the local coordinate system xyz; and
step (S32), performing the least squares fitting in the local coordinate system xyz.
In step (S31), a relationship between the coordinates (x_{0}, y_{0}, z_{0}) in the global coordinate system x_{0}y_{0}z_{0 }and the coordinates (x, y, z) in the local coordinate system xyz can be written as:
A relationship between the normal vectors (α_{0}, β_{0}, γ_{0}) in the global coordinate system and normal vectors (α, β, γ) in the local coordinate system xyz can be written as:
In one embodiment, a number of local coordinate system positions are obtained by using a local search algorithm near the feature data points, from which a local coordinate system with the smallest fitting error is chosen, and a zaxis of the local coordinate system with the smallest fitting error is as an aspherical symmetry axis.
Referring to
For each of the plurality of parameters (a, b, θ), the coefficient matrix P (A, B, C, D . . . ) can be obtained by Eq. (4), and its corresponding J is known by Eq. (3). Selecting a parameter (a, b, θ) whose corresponding J is the smallest and defining it as an optimal local coordinate position parameter, and the local coordinate system xyz is established using the optimal local coordinate position parameter. The coefficient matrix P (A, B, C, D . . . ) is substituted into Eq. (2) to obtain an optimal fitting aspheric surface.
In step (S4), the methods for solving the feature data points (P′_{1}, P′_{2}, . . . P′_{m}) on the aspheric surface “b” are the same as the methods for solving the feature data points (P_{1}, P_{2}, . . . P_{m}) on the aspheric surface “a”. An equation of the aspheric surface “b” is the same as the equation of the aspheric surface “a”. A method for fitting the feature data points (P_{1}, P_{2}, . . . P_{m+1}) to obtain the offaxis aspheric surface A_{m+1}, a method for fitting the feature data points (P_{1}, P_{2}, . . . P_{m+2}) to obtain the offaxis aspheric surface A_{m+2}, a method for fitting the feature data points (P′_{1}, P′_{2}, . . . P′_{m}) to obtain the offaxis aspheric surface A′_{m}, a method for fitting the feature data points (P′_{1}, P′_{2}, . . . P′_{m+1}) to obtain the offaxis aspheric surface A′_{m+1}, a method for fitting the feature data points (P′_{1}, P′_{2}, . . . P′_{m+2}) to obtain the offaxis aspheric surface A′_{m+2 }and a method for fitting the feature data points (P′_{1}, P′_{2}, . . . P′_{K}) to obtain the offaxis aspheric surface A′_{K }are all the same as the method for fitting the feature data points (P_{1}, P_{2}, . . . P_{K}) to obtain the offaxis aspheric surface A_{K}.
Furthermore, a step of optimizing the offaxis aspheric optical system obtained in step (S6) can be performed, and the offaxis aspheric optical system obtained in step (S6) can be as the initial system.
An order of steps (S2) to (S5) can be changed according to the actual needs.
Referring to
Table 1 parameters of the offaxial threemirror optical system 100
In one embodiment, six offaxial fields are selected during a designing process of the offaxis aspheric threemirror optical system 100. The six offaxial fields are (0°, −16°), (0°, −13°), (0°, −10°), (1°, −16°), (1°, −13°), and (1°, −10°). 112 feature rays are selected in each offaxial field, and 672 feature rays that corresponds different pupil positions and different offaxial fields are selected. The value of m is about 380, and the step size of (0, a, b) is about 0.2 mm.
Referring to
In space relative to the first threedimensional rectangular coordinates system (x_{1}, y_{1}, z_{1}), a second threedimensional rectangular coordinates system (x_{2}, y_{2}, z_{2}) is defined by a primary mirror location, a third threedimensional rectangular coordinates system (x_{3}, y_{3}, z_{3}) is defined by a secondary mirror location, a fourth threedimensional rectangular coordinates system (x_{4}, y_{4}, z_{4}) is defined by a tertiary mirror location, and a fifth threedimensional rectangular coordinates system (x_{5}, y_{5}, z_{5}) is defined by an image sensor location.
A second origin of the second threedimensional rectangular coordinates system (x_{2}, y_{2}, z_{2}) is in (0, −37.6707, 212.9768) position of the first threedimensional rectangular coordinates system (x_{1}, y_{1}, z_{1}), whose unit is millimeter. A z_{2}axis positive direction rotates about 5.2210 degrees along a clockwise direction relative to a z_{1}axis positive direction.
A third origin of the third threedimensional rectangular coordinates system (x_{3}, y_{3}, z_{3}) is in (0, −2.9078, −0.9909) position of the first threedimensional rectangular coordinates system (x_{1}, y_{1}, z_{1}), whose unit is millimeter. A z_{3}axis positive direction rotates about 3.2412 degrees along a clockwise direction relative to a z_{1}axis positive direction.
A fourth origin of the fourth threedimensional rectangular coordinates system (x_{4}, y_{4}, z_{4}) is in (0, −11.5288, 154.8088) position of the first threedimensional rectangular coordinates system (x_{1}, y_{1}, z_{1}), whose unit is millimeter. A z_{4}axis positive direction rotates about 2.3759 degrees along a clockwise direction relative to a z_{1}axis positive direction.
A fifth origin of the fifth threedimensional rectangular coordinates system (x_{5}, y_{6}, z_{5}) is in (0, −39.4349, −1.9678) position of the first threedimensional rectangular coordinates system (x_{1}, y_{1}, z_{1}), whose unit is millimeter. A z_{5}axis positive direction rotates about 5.1301 degrees along a clockwise direction relative to a z_{1}axis positive direction.
A length of the offaxis aspheric threemirror optical system 100 along the y_{1}axis is about 280 millimeters. A length of the offaxis aspheric threemirror optical system 100 along the z_{1}axis is about 220 millimeters
In the second threedimensional rectangular coordinates system (x_{2}, y_{2}, z_{2}); a reflective surface of the primary mirror 120 is a sixthorder polynomial aspheric of x_{2}y_{2}. In the third threedimensional rectangular coordinates system (x_{3}, y_{3}, z_{3}); a reflective surface of the secondary mirror 140 is a sixthorder polynomial aspheric of x_{3}y_{3}. In the fourth threedimensional rectangular coordinates system (x_{4}, y_{4}, z_{4}); a reflective surface of the a tertiary mirror 160 is a sixthorder polynomial aspheric of x_{4}y_{4}. The sixthorder polynomial aspheric of x_{2}y_{2}, sixthorder polynomial aspheric of x_{3}y_{3}, and the sixthorder polynomial aspheric of x_{4}y_{4 }can be expressed as follows:
The term
is transformed into linear XY polynomials by a Taylor series expansion to simplify the fitting process. Thus, the equation (5) can be expressed as follows:
Z=Ar^{2}+Br^{4}+Cr^{6} (6).
The aspheric coefficients (A, B, C) and the positional parameters (a, b, θ) of the primary mirror 120, the secondary mirror 140 and the tertiary mirror 160 can be obtained by above method. The aspheric coefficients (A, B, C) can be transformed into the aspheric coefficients (c, a_{1}, a_{2}) by Eq (2) to use Eq. (5) as the aspheric expression in the subsequent optimization, in one embodiment, k=0, which can simplify the calculation. The aspheric coefficients (A, B, C) and the positional parameters (a, b, θ) of the primary mirror 120, the secondary mirror 140 and the tertiary mirror 160 are shown in table 2 below. However, the values of the aspheric coefficients (A, B, C) and the positional parameters (a, b, θ) of the primary mirror 120, the secondary mirror 140 and the tertiary mirror 160 are not limited to TABLE 2.
Referring to
Referring to
Referring to
The offaxis aspheric threemirror optical system 100 can be optimized to improve the imaging quality and reduce the relative distortion. The offaxis aspheric threemirror optical system 100 is used as the initial structure for subsequent optimization. The values of the aspheric coefficients (A, B, C) of the primary mirror 120, the secondary mirror 140 and the tertiary mirror 160 after optimization are shown in table 3 below. However, the values of the aspheric coefficients (A, B, C) of the primary mirror 120, the secondary mirror 140 and the tertiary mirror 160 after optimization are not limited to TABLE 3.
Referring to
Referring to
Referring to
Referring to
An effective entrance pupil diameter of the offaxis aspheric threemirror optical system 100 is about 100 millimeter.
The offaxis aspheric threemirror optical system 100 adopts an offaxis field of view in a vertical direction. A field angle of the offaxis aspheric threemirror optical system 100 is about 2°×6°, wherein an angle in an horizontal direction is in a range from about −1° to about 1°, and an angle in the vertical direction is in a range from about −16° to about −10°.
A wavelength of the offaxis aspheric threemirror optical system 100 is not limited, in one embodiment, the wavelength is in a range from about 4 microns to about 7 microns.
An effective focal length (EFL) of the offaxis aspheric threemirror optical system 100 is about 200 millimeters.
Fnumber of the offaxis aspheric threemirror optical system 100 is a D/f reciprocal, the Fnumber can up to a range from about 1.95 to about 2.05. In one embodiment, the relative aperture (D/f) of the offaxis aspheric threemirror optical system 100 is about 0.5, and the Fnumber is about 2.0.
Since the offaxis aspheric threemirror optical system has no center block, the aspheric threemirror optical system can achieve a larger field of view, and obtain the images close to an ideal imaging effect. The applications of the offaxis aspheric threemirror optical system 100 comprises earth observation, space target detection, astronomical observations, Multispectral thermal imaging, and dimensional mapping. The offaxis aspheric threemirror optical system can be used in the visible band or the infrared band.
The method for designing offaxis aspheric threemirror optical system can have many advantages. First, the plurality of feature rays R_{i }(i=1, 2 . . . K) from different fields and different aperture positions are considered, which overcome the limitations of the conventional methods in the number of fields and the number of apertures. Second, the plurality of feature data points were calculated first and fitted into an initial aspheric surface. Then, intermediate point calculations, feature data point calculations, and aspheric surface fitting were repeated continuously to calculate remaining feature data points and the desired aspheric surface, which improve the accuracy of the method. Third, the method considers both the coordinates and the normal vectors of the plurality of feature data points; therefore, the aspheric shape of the offaxis aspheric threemirror optical system designed by the method is accurate.
It is to be understood that the abovedescribed embodiments are intended to illustrate rather than limit the present disclosure. Variations may be made to the embodiments without departing from the spirit of the present disclosure as claimed. Elements associated with any of the above embodiments are envisioned to be associated with any other embodiments. The abovedescribed embodiments illustrate the scope of the present disclosure but do not restrict the scope of the present disclosure.