Patent Publication Number: US-11386246-B2

Title: Method for designing hybrid surface optical system

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     This application claims all benefits accruing under 35 U.S.C. § 119 from China Patent Application No. 201710059418.3, field on Jan. 24, 2017 in the China Intellectual Property Office, disclosure of which is incorporated herein by reference. The application is also related to copending applications entitled, “OFF-AXIS HYBRID SURFACE THREE-MIRROR OPTICAL SYSTEM”, filed on Dec. 14, 2017 (application Ser. No. 15/842,834). 
     FIELD 
     The present disclosure relates to a method for designing hybrid surface optical system. 
     BACKGROUND 
     In a condition of meeting the optical performance requirements of an optical system, using a spherical surface or an aspheric surface to replace a freeform surface of the optical system can reduce a cost and a difficulty of processing and testing. A hybrid surface optical system comprises different surfaces. Each surface of the hybrid surface optical system can be selected from spherical surface, aspheric surface, and freeform surfaces. Thus, compared to freeform surface optical systems, the hybrid surface optical system has low cost and small processing and testing difficulty. 
     However, conventional method for designing hybrid surface optical system is to obtain a coaxial spherical surface system by a paraxial optical theory first, and then the coaxial spherical surface system is transformed into an off-axis hybrid surface optical system during following optimization process. A deviation between the off-axis hybrid surface optical system obtained by conventional method and an ideal optical system is large. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Implementations of the present technology will now be described, by way of example only, with reference to the attached figures. 
         FIG. 1  is a schematic view of start point and end point of one feature ray while calculating the feature data points. 
         FIG. 2  is a position diagram of one embodiment of a global coordinate system and a first local coordinate system. 
         FIG. 3  is a schematic view of one embodiment of calculating an intermediate point G m . 
         FIG. 4  is a schematic view of one embodiment of calculating a feature data point P m+1 . 
         FIG. 5  is a schematic view of one embodiment of calculating an off-axis spherical surface three-mirror optical system. 
         FIG. 6  is a schematic view of one embodiment of determining the parameters of the first local coordinate system. 
         FIG. 7  is a schematic view of one embodiment of a configuration of an off-axis hybrid surface three-mirror optical system. 
         FIG. 8  is a position diagram of one embodiment of a primary mirror, a secondary mirror and a tertiary mirror of an off-axis hybrid surface three-mirror optical system. 
         FIG. 9  shows average RMS spot diameters of one embodiment of an off-axis hybrid surface three-mirror optical system. 
         FIG. 10  shows a light path schematic view of one embodiment of an off-axis hybrid surface three-mirror optical system after optimization. 
         FIG. 11  shows a modulation transfer function (MTF) curve of one embodiment of an off-axis hybrid surface three-mirror optical system after optimization. 
         FIG. 12  shows a RMS wavefront error diagram of one embodiment of an off-axis hybrid surface three-mirror optical system after optimization. 
     
    
    
     DETAILED DESCRIPTION 
     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 open-ended inclusion or membership in the so-described combination, group, series and the like. 
     A method for designing a hybrid surface optical system of one embodiment is provided. The method comprises the following steps: 
     step (S 1 ), establishing a first initial system, wherein the first initial system comprises a plurality of initial surfaces, and each of the plurality of initial surfaces corresponds to a surface of an off-axis hybrid surface optical system to be calculated; and selecting a plurality of feature rays R i  (i=1, 2 . . . K) from different fields and different aperture positions; 
     step (S 2 ), a spherical surface of the off-axis hybrid surface optical system to be calculated is defined as a spherical surface “a”, keeping the plurality of initial surfaces of the first initial system unchanged and calculating a plurality of first feature data points (P 1 , P 2 , . . . P m ) point by point, wherein m is less than K, and the plurality of first feature data points (P 1 , P 2 , . . . P m ) are m intersection points of the spherical surface “a” and m of the plurality of feature rays R i  (i=1, 2 . . . K); and surface fitting the plurality of first feature data points (P 1 , P 2 , . . . P m ) to obtain an initial spherical surface A m ; calculating a (m+1)th first feature data point P m+1  based on the initial spherical surface A m , and surface fitting the (m+1) first feature data points (P 1 , P 2 , . . . P m , P m+1 ) to obtain a spherical surface A m+1 ; repeating such steps until a Kth first feature data point P K  is calculated, and surface fitting the first feature data points (P 1 , P 2 , . . . P K ) to obtain a spherical surface A K , wherein the spherical surface A K  is the spherical surface “a”; repeating such steps until all spherical surfaces of the off-axis hybrid surface optical system to be calculated are obtained, to obtain a spherical surface optical system; 
     step (S 3 ), an aspheric surface of the off-axis hybrid optical system to be calculated is defined as an aspheric surface “b”, the spherical surface optical system is as a second initial system, keeping all spherical surfaces of the aspheric optical system unchanged and calculating a plurality of second feature data points (P′ 1 , P′ 2 , . . . P′ K ), wherein the plurality of second feature data points (P′ 1 , P′ 2 , . . . P′ K ) are the intersection points of the spherical surface “a” and the plurality of feature rays R i  (i=1, 2 . . . K); and surface fitting the plurality of second feature data points (P′ 1 , P′ 2 , . . . P′ K ) to obtain the aspheric surface “b”; repeating such steps until all aspheric surfaces of the off-axis hybrid surface optical system to be calculated are obtained, to obtain a first hybrid surface optical system; and 
     step (S 4 ), a freeform surface of the off-axis hybrid optical system to be calculated is defined as a freeform surface “c”, the first hybrid surface optical system is as a third initial system, keeping all aspheric surfaces of the first hybrid surface optical system unchanged and calculating a plurality of third feature data points (P″ 1 , P″ 2 , . . . P″ K ), wherein the plurality of third feature data points (P″ 1 , P″ 2 , . . . P″ K ) are the intersection points of the aspheric surface “b” and the plurality of feature rays R i  (i=1, 2 . . . K); and surface fitting the plurality of third feature data points (P″ 1 , P″ 2 , . . . P″ K ) to obtain the freeform surface “c”; repeating such steps until all freeform surface s of the off-axis hybrid surface optical system to be calculated are obtained. 
     In step (S 1 ), 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, fourteen fields are fixed in the construction process, the fourteen fields are (0°, −16°), (0°, −15°), (0°, −14°), (0°, −13°), (0°, −12°), (0°, −11°), (0°, −10°), (1.5°, −16°), (1.5°, −15°), (1.5°, −14°), (1.5°, −13°), (1.5°, −12°), (1.5°, −11°), and (1.5°, −10°); one hundred and twelve feature rays are fixed from each of the fourteen fields. Therefore, 1568 different feature rays correspond to different aperture positions and different fields are fixed. 
     Referring to  FIG. 1 , a surface Ω is defined as an unknown aspheric surface, a surface Ω′ is defined as a surface located adjacent to and before the surface Ω, and a surface Ω″ is defined as a surface located adjacent to and behind the surface Ω. Defining the intersection points of the plurality of feature rays R i  (i=1, 2 . . . K) with the surface Ω as the first feature data points P i  (i=1, 2 . . . K). The first feature data points P i  (i=1, 2 . . . K) can be obtained by the intersection points of the feature rays R i  (i=1, 2 . . . K) with the surface Ω′ and the surface Ω″. The plurality of feature rays R i  (i=1, 2 . . . K) are intersected with the surface Ω′ at a plurality of start points S i  (i=1, 2 . . . K), and intersected with the surface Ω″ at a plurality of end points E i  (i=1, 2 . . . K). When the surface Ω and the plurality of feature rays R i  (i=1, 2 . . . K) are determined, the plurality of start points S i  (i=1, 2 . . . K) of the feature rays R i  (i=1, 2 . . . K) can also be determined. The plurality of end points E i  (i=1, 2 . . . K) can be obtained based on the object-image relationship. Under ideal conditions, the feature rays R i  (i=1, 2 . . . K) emitted from the plurality of start points S i  (i=1, 2 . . . K) on the surface Ω′; pass through the first feature data points P i  (i=1, 2 . . . K) on the surface Ω; intersect with the surface Ω″ at the plurality of end points E i  (i=1, 2 . . . K); and finally intersect with the image plane at the plurality of ideal image points I i  (i=1, 2 . . . K). When the surface Ω is a tertiary mirror, a plurality of intersection points of the plurality of feature rays R i  (i=1, 2 . . . K) and a secondary mirror are the plurality of start points S i  (i=1, 2 . . . K), a plurality of intersection points of the plurality of feature rays R i  (i=1, 2 . . . K) and the image plane are the plurality of end points E i  (i=1, 2 . . . K), and the plurality of end points E i  (i=1, 2 . . . K) are the plurality of ideal image points I i  (i=1, 2 . . . K). When the surface Ω is the secondary mirror, a plurality of intersection points of the plurality of feature rays R i  (i=1, 2 . . . K) and a primary mirror are the plurality of start points S i  (i=1, 2 . . . K), the plurality of end points E i  (i=1, 2 . . . K) are the points on the tertiary mirror which minimizes an optical path length between the plurality of first feature data points P i  (i=1, 2 . . . K) and the plurality of ideal image points I i  (i=1, 2 . . . K). When the surface Ω is the primary mirror, the plurality of end points E i  (i=1, 2 . . . K) are the points on the secondary mirror which minimizes an optical path length between the plurality of feature data points P i  (i=1, 2 . . . K) and the plurality of ideal image points I i  (i=1, 2 . . . K). 
     In step (S 2 ), 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 surface 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 surface error. In one embodiment, the value of m is about K/2=784. 
     A method for calculating the plurality of first feature data points (P 1 , P 2 , . . . P m ) includes the following sub-steps: 
     Step (a): defining a first intersection point of a first feature ray R i  and the spherical surface “a” as the first feature data point P i ; 
     Step (b): when an ith (1≤i≤m−1) first 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) first feature data point P i  (1≤i≤m−1) can be calculated based on the vector form of Snell&#39;s Law; 
     Step (c): making a first tangent plane through the ith (1≤i≤m−1) first 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) first feature data point P i  (1≤i≤m−1) is defined as d i ; 
     Step (d): making a second tangent plane at (i−1) first feature data points that are obtained before the ith first 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 first feature data point P i+1  (1≤i≤m−1); otherwise, Q′ i+1  is taken as the next first feature data point P i+1  (1≤i≤m−1); and 
     Step (f): repeating steps from b to e, until the plurality of first feature data points P i  (i=1, 2 . . . m) are all calculated. 
     In step (b), the unit normal vector {right arrow over (N)} i  (1≤i≤m−1) at each of the first feature data point P i  (1≤i≤m−1) can be calculated based on the vector form of Snell&#39;s Law. When the surface Ω is a refractive surface, 
                   N   →     i     =           n   ′     ⁢       r   →     i   ′       -     n   ⁢       r   →     i                    n   ′     ⁢       r   →     i   ′       -     n   ⁢       r   →     i                  ,         
wherein
 
                 r   →     i     =           P   i     ⁢     S   i       ⇀                P   i     ⁢     S   i       ⇀                  
the surface Ω;
 
                 r   →     i   ′     =           E   i     ⁢     P   i       ⇀                E   i     ⁢     P   i       ⇀                  
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, 
     
       
         
           
             
               
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     The unit normal vector {right arrow over (N)} i  at the first 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 first feature data points P i  (i=1, 2 . . . m) can be obtained. 
     Referring to  FIG. 2 , a global coordinate system is defined by a primary mirror location. A beam propagation direction is defined as a Z-axis. A plane perpendicular to the Z-axis is defined as an xy plane. 
     Referring to  FIG. 3 , during calculating the plurality of first feature data points (P 1 , P 2 , . . . P m ), a intermediate point G m  is calculated. A method for calculating the intermediate point G m  comprises: a tangent plane T m  at the first feature data point P m  is calculated, the tangent plane T m  intersects with the initial spherical surface A m  at an intersection line L m ; and in the global coordinate system, a first feature data point located on the intersection line L m  whose x coordinate is the same as an x coordinate of the first feature data point P m  is defined as the intermediate point G m . 
     The first feature data point P m+1  is an intersection point between its corresponding feature ray and a tangent plane of the first feature data points (P 1 , P 2 , . . . P m ) that is closest to the feature data point P m+1 . 
     Referring to  FIG. 4 , a method for calculating the (m+1)th first feature data point P m+1  comprises the sub steps: first, finding the feature ray corresponding to the (m+1)th first feature data point P m+1 , wherein the feature ray corresponding to the (m+1)th first feature data point P m+1  is nearest to the intermediate point G m  in the remaining K−m characteristic rays. Since the intermediate point G m  is located on the initial spherical surface A m , a normal vector n m  and a tangent plane of the intermediate point G m  at the initial spherical surface A m  can be obtained according to an spherical surface expression. (K−m) intersection points can be obtained by the tangent plane of the intermediate point G m  intersecting with remaining (K−m) feature rays, an intersection point O m , which is nearest to the intermediate point G m  is fixed from the (K−m) intersection points; the feature ray where the intersection point O m  is located on is the feature ray R m+1  that corresponds to the feature data point P m+1 . Second, m intersection points O i  (1≤i≤m) can be obtained by the feature ray R m+1  intersects with each of the tangent planes of the first feature data points (P 1 , P 2  . . . P m ), and an intersection point G′ m  can be obtained by the feature ray R m+1  intersects with the intermediate point G m . The first feature data points (P 1 , P 2  . . . P m ) and the intermediate point G m  are defined as “F”; the m intersection points O i  (1≤i≤m) and the intersection point G′ m  are defined as “F”. A pair of (F−F′) which has shortest distance is found from (P 1 , P 2  . . . P m )−O i  and G′ m −G m . The point F is closest to the first feature data point P m+1 , and the point F′ is the next first feature data point P m+1 . 
     The methods for calculating the first feature data points P m+2  and P K  are the same as the method for calculating the first feature data point P m+1 . A method for calculating an intermediate point G m+1  in a process of calculating the first feature data point P m+2  and a method for calculating an intermediate point G K−1  in a process of calculating the first feature data point P K  are the same as the method for calculating the intermediate point G m . 
     A method for surface fitting the first feature data points (P 1 , P 2  . . . P m ) to obtain the initial spherical surface A m , a method for surface fitting the (m+1) first feature data points (P 1 , P 2 , . . . P m , P m+1 ) to obtain the spherical surface A m+1 , and a method for surface fitting the first feature data points (P 1 , P 2 , . . . P K ) to obtain the spherical surface A K  are all least squares method. 
     A coordinate of the first feature data point is (x i , y i , z i ), and its corresponding normal vector is (u i , v i , −1). When a sphere center is (A, B, C) and a radius is r, An equation of the spherical surface can be expressed by equation (1):
 
( x   i   −A ) 2 +( y   i   −B ) 2 +( z   i   C ) 2   =r   2   (1).
 
     Calculating a derivation of the equation (1) for x and y, to obtain an expression of a normal vector u i  in an x-axis direction and an expression of a normal vector v i  in a y-axis direction. 
     
       
         
           
             
               
                 
                   
                     
                       
                         
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                                   ) 
                                 
                               
                             
                           
                         
                         ] 
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               ∑ 
                               
                                 
                                   U 
                                   i 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       U 
                                       i 
                                     
                                     - 
                                     
                                       U 
                                       _ 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               ∑ 
                               
                                 
                                   U 
                                   i 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       y 
                                       i 
                                     
                                     - 
                                     
                                       y 
                                       _ 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 
                                   U 
                                   i 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       y 
                                       i 
                                     
                                     - 
                                     
                                       y 
                                       _ 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               ∑ 
                               
                                 
                                   y 
                                   i 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       y 
                                       i 
                                     
                                     - 
                                     
                                       y 
                                       _ 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                       
                       ] 
                     
                     [ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             A 
                           
                         
                       
                       
                         
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             B 
                           
                         
                       
                       
                         
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             C 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                       
                     
                       
                         [ 
                         
                           
                             
                               
                                 ∑ 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           U 
                                           i 
                                         
                                         ⁢ 
                                         
                                           x 
                                           i 
                                         
                                       
                                       + 
                                       
                                         y 
                                         i 
                                         2 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         U 
                                         i 
                                       
                                       - 
                                       
                                         U 
                                         _ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 ∑ 
                                 
                                   
                                     ( 
                                     
                                       
                                         
                                           U 
                                           i 
                                         
                                         ⁢ 
                                         
                                           x 
                                           i 
                                         
                                       
                                       + 
                                       
                                         y 
                                         i 
                                         2 
                                       
                                     
                                     ) 
                                   
                                   ⁢ 
                                   
                                     ( 
                                     
                                       
                                         y 
                                         i 
                                       
                                       - 
                                       
                                         y 
                                         _ 
                                       
                                     
                                     ) 
                                   
                                 
                               
                             
                           
                           
                             
                               0 
                             
                           
                         
                         ] 
                       
                       , 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               ∑ 
                               
                                 
                                   x 
                                   i 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       x 
                                       i 
                                     
                                     - 
                                     
                                       x 
                                       _ 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               ∑ 
                               
                                 
                                   V 
                                   i 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       x 
                                       i 
                                     
                                     - 
                                     
                                       x 
                                       _ 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 
                                   V 
                                   i 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       x 
                                       i 
                                     
                                     - 
                                     
                                       x 
                                       _ 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               ∑ 
                               
                                 
                                   V 
                                   i 
                                 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     
                                       V 
                                       i 
                                     
                                     - 
                                     
                                       V 
                                       _ 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                       
                       ] 
                     
                     [ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             A 
                           
                         
                       
                       
                         
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             B 
                           
                         
                       
                       
                         
                           
                             2 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             C 
                           
                         
                       
                     
                     ] 
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               ∑ 
                               
                                 
                                   ( 
                                   
                                     
                                       x 
                                       i 
                                       2 
                                     
                                     + 
                                     
                                       
                                         V 
                                         i 
                                       
                                       ⁢ 
                                       
                                         y 
                                         i 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       x 
                                       i 
                                     
                                     - 
                                     
                                       x 
                                       _ 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               ∑ 
                               
                                 
                                   ( 
                                   
                                     
                                       x 
                                       i 
                                       2 
                                     
                                     + 
                                     
                                       
                                         V 
                                         i 
                                       
                                       ⁢ 
                                       
                                         y 
                                         i 
                                       
                                     
                                   
                                   ) 
                                 
                                 ⁢ 
                                 
                                   ( 
                                   
                                     
                                       V 
                                       i 
                                     
                                     - 
                                     
                                       V 
                                       _ 
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             0 
                           
                         
                       
                       ] 
                     
                     . 
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
           
         
       
     
     The normal vector (u i , v i , −1) decides a direction of light rays, thus, both a coordinate error and a normal error during the surface fitting should be considered to obtain an accurate spherical surface. The coordinate error and the normal error are linearly weighted to calculate the sphere center (A, B, C) and the radius r.
 
Equation (4)+ω×equation (5)+ω×equation (6)  (7),
 
Equation (1)+ω×equation (2)+ω×equation (3)  (8),
 
     wherein, ω is a weight of the normal error. The sphere center (A, B, C) can be obtained by equation (7), and the radius r can be obtained by equation (8). 
     After the spherical surface “a” is obtained, the radius of the spherical surface “a” can be further changed to obtain a spherical surface “a′”, an optical power of the spherical surface “a” is changed. In one embodiment, r a ′=ε a ×r a , ε a =0.5˜1.5, wherein, r a  is the radius of the spherical surface “a”, and r a ′ is the radius of the spherical surface “a′”. The radius of each of the spherical surfaces of the off-axis hybrid surface optical system can be further changed to change the optical power each of the spherical surfaces. Referring to  FIG. 5 , in one embodiment, a spherical surface “C” is obtained by the method like in step S 2 , a spherical surface “C′” is obtained by changing a radius of the spherical surface “C”; the primary mirror initial plane and the spherical surface “C′” are kept unchanged to solve a spherical surface “B”, and a spherical surface “B′” is obtained by changing a radius of the spherical surface “B”; the spherical surface “B′” and the spherical surface “C′” are kept unchanged to solve a spherical surface “A”, and a spherical surface “A′” is obtained by changing a radius of the spherical surface “A”, therefore, a spherical three-mirror optical system comprising the spherical surface “A′”, the spherical surface “B′” and the spherical surface “C′” is obtained. 
     In step (S 3 ), the normal vector N=(U,V,−1) determines a deflection direction of light rays. If a normal surface 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 surface fitting the plurality of second feature data points (P′ 1 , P′ 2 , P′ K ) should be considered. A unit normal vector of each of the plurality of second feature data points (P′ 1 , P′ 2 , P′ K ) is obtained based on the object-image relationship, and the coordinates and normal of the K second feature data points are surface fitted. 
     A method for calculating the unit normal vector at each of the second feature data points is the same as the method for calculating the unit normal vector at each of the first feature data points. 
     A first local coordinate system is defined with a center of the aspheric surface (0, a′, b′) as an origin and an aspherical symmetry axis as a Z-axis. The first local coordinate system xyz can be described by the parameters (a, b, θ), wherein θ is a rotation angle of the first local coordinate system relative to the global coordinate system. 
     The coordinate error and the normal error are linearly weighted, and an error evaluation function J is proposed: 
     
       
         
           
             
               
                 
                   
                     
                       
                         J 
                         = 
                           
                         ⁢ 
                         
                           
                             ∑ 
                             
                               
                                 ( 
                                 
                                   Z 
                                   - 
                                   z 
                                 
                                 ) 
                               
                               2 
                             
                           
                           + 
                           
                             ω 
                             × 
                             
                               ∑ 
                               
                                 
                                   ( 
                                   
                                     U 
                                     - 
                                     u 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                           + 
                           
                             ω 
                             × 
                             
                               ∑ 
                               
                                 
                                   ( 
                                   
                                     V 
                                     - 
                                     v 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                         ⁢ 
                         
                           
                             
                               
                                 ( 
                                 
                                   z 
                                   - 
                                   
                                     
                                       A 
                                       1 
                                     
                                     ⁢ 
                                     P 
                                   
                                 
                                 ) 
                               
                               T 
                             
                             ⁢ 
                             
                               ( 
                               
                                 z 
                                 - 
                                 
                                   
                                     A 
                                     1 
                                   
                                   ⁢ 
                                   P 
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               
                                 ω 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     u 
                                     - 
                                     
                                       
                                         A 
                                         2 
                                       
                                       ⁢ 
                                       P 
                                     
                                   
                                   ) 
                                 
                               
                               T 
                             
                             ⁢ 
                             
                               ( 
                               
                                 v 
                                 - 
                                 
                                   
                                     A 
                                     2 
                                   
                                   ⁢ 
                                   P 
                                 
                               
                               ) 
                             
                           
                           + 
                         
                       
                     
                   
                   
                     
                       
                           
                         ⁢ 
                         
                           
                             
                               
                                 ω 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     v 
                                     - 
                                     
                                       
                                         A 
                                         3 
                                       
                                       ⁢ 
                                       P 
                                     
                                   
                                   ) 
                                 
                               
                               T 
                             
                             ⁢ 
                             
                               ( 
                               
                                 v 
                                 - 
                                 
                                   
                                     A 
                                     3 
                                   
                                   ⁢ 
                                   P 
                                 
                               
                               ) 
                             
                           
                           , 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     wherein Z is a surface fitting coordinate, (U, V, −1) is the surface fitting normal vector, z is a real coordinate, (u, v, −1) is a real normal vector, ω is a weight of the normal error, P is an coefficient matrix (A, B, C, D . . . ) of the aspheric surface “b”, 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 calculating 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 )   (11).
 
     A method for surface fitting the second feature data points (P′ 1 , P′ 2 , . . . P′ K ) to obtain the aspheric surface “b” comprising the sub-step: 
     step (S 31 ), transferring the coordinates (x 0 , y 0 , z 0 ) of the second feature data points (P′ 1 , P′ 2 , . . . P′ K ) and their corresponding normal vectors (α 0 , β 0 , γ 0 ) in the global coordinate system to the coordinates (x, y, z) and their corresponding normal vectors (α, β, γ) in the first local coordinate system; and 
     step (S 32 ), performing the least squares fitting in the first local coordinate system xyz. 
     In step (S 31 ), 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 first local coordinate system xyz can be written as: 
             {             x   =     x   0                 y   =         (       y   0     -   a     )     ⁢           ⁢   cos   ⁢           ⁢   θ     -       (       z   0     -   b     )     ⁢   sin   ⁢           ⁢   θ                   z   =         (       y   0     -   a     )     ⁢   sin   ⁢           ⁢   θ     +       (       z   0     -   b     )     ⁢   cos   ⁢           ⁢   θ               .           
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 first local coordinate system xyz can be written as:
 
     
       
         
           
             { 
             
               
                 
                   
                     
                       u 
                       = 
                       
                         u 
                         0 
                       
                     
                   
                 
                 
                   
                     
                       v 
                       = 
                       
                         
                           
                             v 
                             0 
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                         - 
                         
                           
                             w 
                             0 
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                     
                   
                 
                 
                   
                     
                       w 
                       = 
                       
                         
                           
                             v 
                             0 
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                         + 
                         
                           
                             w 
                             0 
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                     
                   
                 
               
               . 
             
           
         
       
     
     In one embodiment, a number of the first local coordinate system positions are obtained by using a local search algorithm near the second feature data points, from which a first local coordinate system with the smallest surface fitting error is chosen, and a z-axis of the first local coordinate system with the smallest surface fitting error is as a symmetry axis of the aspheric surface “b”. 
     Referring to  FIG. 6 , a value of the aspheric vertex coordinates (0, a′, b′) is obtained by using a local search algorithm in a rectangle with a diagonal of the connection between the second feature data points P′ 1  and P′ k . In one embodiment, a grid mesh is generated in the rectangle with the diagonal of the connection between the second feature data points P′ 1  and P′ k , a grid node is a value of the vertex coordinates (0, a′, b′) of the aspheric surface “b”, and a grid size is a step size of (0, a′, b′). In one embodiment, the step size of (0, a′, b′) is about 0.2 mm. A value of the rotation angle θ can be ranged from about −π to about π for each set of (a′, b′). Therefore, a plurality of parameters (a′, b′, θ) correspond to a plurality of first local coordinate system positions can be obtained. 
     For each of the plurality of parameters (a′, b′, θ), the coefficient matrix P (A, B, C, D . . . ) can be obtained by equation (11), and its corresponding J is known by equation (10). Selecting a parameter (a′, b′, θ) whose corresponding J is the smallest and defining it as an optimal local coordinate position parameter, and the first local coordinate system is established using the optimal local coordinate position parameter. 
     The coefficient matrix P (A, B, C, D . . . ) is substituted into the following equation (12) to obtain the aspheric surface “b”, 
     
       
         
           
             
               
                 
                   
                     Z 
                     = 
                     
                       
                         
                           Ar 
                           2 
                         
                         + 
                         
                           Br 
                           4 
                         
                         + 
                         
                           Cr 
                           6 
                         
                         + 
                         
                           Dr 
                           8 
                         
                         + 
                         
                           … 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             r 
                             2 
                           
                         
                       
                       = 
                       
                         
                           
                             x 
                             2 
                           
                           + 
                           
                             
                               y 
                               2 
                             
                             ⁢ 
                             
                               
 
                             
                             ⁢ 
                             A 
                           
                         
                         = 
                         
                           
                             1 
                             2 
                           
                           ⁢ 
                           c 
                         
                       
                     
                   
                   , 
                   
                     B 
                     = 
                     
                       
                         a 
                         1 
                       
                       + 
                       
                         
                           1 
                           8 
                         
                         ⁢ 
                         
                           ( 
                           
                             k 
                             + 
                             1 
                           
                           ) 
                         
                         ⁢ 
                         
                           c 
                           3 
                         
                       
                     
                   
                   , 
                   
                     C 
                     = 
                     
                       
                         a 
                         2 
                       
                       + 
                       
                         
                           1 
                           
                             1 
                             ⁢ 
                             6 
                           
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                           2 
                         
                         ⁢ 
                         
                           c 
                           5 
                         
                       
                     
                   
                   , 
                   
                     
 
                   
                   ⁢ 
                   
                     D 
                     = 
                     
                       
                         a 
                         3 
                       
                       + 
                       
                         
                           5 
                           
                             1 
                             ⁢ 
                             2 
                             ⁢ 
                             8 
                           
                         
                         ⁢ 
                         
                           
                             ( 
                             
                               k 
                               + 
                               1 
                             
                             ) 
                           
                           3 
                         
                         ⁢ 
                         
                           c 
                           7 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
           
         
       
     
     wherein, L is a more than eighth order term about r. Since an order of L term is too high, a processing of the aspheric surface “b” is more difficult, so the L item can be directly omitted in the actual calculation. 
     In step (S 4 ), both a coordinate error and a normal error during surface fitting the plurality of third feature data points (P″ 1 , P″ 2 , . . . P″ K ) should be considered. A unit normal vector of each of the plurality of third feature data points (P″ 1 , P″ 2 , . . . P″ K ) is obtained based on the object-image relationship, and the coordinates and normal of the K third feature data points are surface fitted. 
     A method for calculating the unit normal vector at each of the third feature data points is the same as the method for calculating the unit normal vector at each of the first feature data points. 
     A method for surface fitting the plurality of third feature data points (P″ 1 , P″ 2 , . . . P″ K ) to obtain the freeform surface “c” comprising the sub-step: 
     step (S 41 ), surface fitting the plurality of third feature data points to a sphere in the global coordinate system, and obtaining a curvature c of the sphere and a curvature center (x c , y c , z c ) corresponding to the curvature c of the sphere; 
     step (S 42 ), defining a feature data point (x o , y o , z o ) corresponding to a chief ray of the central field angle among the entire field-of-view (FOV) as the vertex of the sphere, defining a second local coordinate system by the vertex of the sphere as origin and a line passing through the curvature center and the vertex of the sphere as a Z-axis; 
     step (S 43 ), transforming the coordinates (x i , y i , z i ) and the normal vector (α i , β i , γ i ), of the plurality of third feature data points in the global coordinate system, into the coordinates (x′ i , y′ i , z′ i ) and the normal vector (α′ i , β′ i , γ′ i ), of the plurality of third feature data points in the second local coordinate system; 
     step (S 44 ), surface fitting the plurality of third feature data points into the conic surface in the second local coordinate system, based on the coordinates (x′ i , y′ i , z′ i ) and the curvature c of the sphere, and obtaining the conic constant k; and 
     step (S 45 ), removing the coordinates and the normal vector of the plurality of third feature data points P i  (i=1, 2 . . . K), on the conic surface in the second local coordinate system, from the coordinates (x′ i , y′ i , z′ i ) and the normal vector (α′ i , β′ i , γ′ i ), to obtain a residual coordinate and a residual normal vector; and surface fitting the residual coordinate and the residual normal vector to obtain a freeform surface; the equation of the freeform surface “c” can be obtained by adding an conic surface equation and an freeform surface equation. 
     Generally, the optical systems are symmetric about the yoz plane. Therefore, a tilt angle θ of the sphere, in the yoz plane of the second local coordinate system relative to in the yoz plane of the global coordinate system, is: 
     
       
         
           
             θ 
             = 
             
               
                 arctan 
                 ⁡ 
                 
                   ( 
                   
                     
                       
                         y 
                         o 
                       
                       - 
                       
                         y 
                         c 
                       
                     
                     
                       
                         z 
                         o 
                       
                       - 
                       
                         z 
                         c 
                       
                     
                   
                   ) 
                 
               
               . 
             
           
         
       
     
     A relationship between the coordinates (x′ i , y′ i , z′ i ) and the coordinates (x i , y i , z i ) of each of the plurality of third feature data points can be expressed as following: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         x 
                         i 
                         ′ 
                       
                       = 
                       
                         
                           x 
                           i 
                         
                         - 
                         
                           x 
                           o 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         y 
                         i 
                         ′ 
                       
                       = 
                       
                         
                           
                             ( 
                             
                               
                                 y 
                                 i 
                               
                               - 
                               
                                 y 
                                 o 
                               
                             
                             ) 
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                         - 
                         
                           
                             ( 
                             
                               
                                 z 
                                 i 
                               
                               - 
                               
                                 z 
                                 o 
                               
                             
                             ) 
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         z 
                         i 
                         ′ 
                       
                       = 
                       
                         
                           
                             ( 
                             
                               
                                 y 
                                 i 
                               
                               - 
                               
                                 y 
                                 o 
                               
                             
                             ) 
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                         + 
                         
                           
                             ( 
                             
                               
                                 z 
                                 i 
                               
                               - 
                               
                                 z 
                                 o 
                               
                             
                             ) 
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                     
                   
                 
               
               . 
             
           
         
       
     
     A relationship between the normal vector (α′ i , β′ i , γ′ i ) and the normal vector (α i , β i , γ i ) of each of the plurality of third feature data points can be expressed as following: 
     
       
         
           
             { 
             
               
                 
                   
                     
                       
                         α 
                         i 
                         ′ 
                       
                       = 
                       
                         α 
                         i 
                       
                     
                   
                 
                 
                   
                     
                       
                         β 
                         i 
                         ′ 
                       
                       = 
                       
                         
                           
                             β 
                             i 
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                         - 
                         
                           
                             γ 
                             i 
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                     
                   
                 
                 
                   
                     
                       
                         γ 
                         i 
                         ′ 
                       
                       = 
                       
                         
                           
                             β 
                             i 
                           
                           ⁢ 
                           sin 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                         + 
                         
                           
                             γ 
                             i 
                           
                           ⁢ 
                           cos 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           θ 
                         
                       
                     
                   
                 
               
               . 
             
           
         
       
     
     In the second local coordinate system, the coordinates and the normal vector of the plurality of third feature data points on the conic surface are defined as (x′ i , y′ i , z′ is ) and (α′ is , β′ is , γ′ is ) respectively. An Z′-axis component of the normal vector is normalized to −1. The residual coordinate (x″ i , y″ i , z″ i ) and the residual normal vector (α″ i , β″ i , −1) can be obtained, wherein, (x″ i , y″ i , z″ i )=(x′ i , y′ i , z is ′) and 
     
       
         
           
             
               ( 
               
                 
                   α 
                   i 
                   ″ 
                 
                 , 
                 
                   β 
                   i 
                   ″ 
                 
                 , 
                 
                   - 
                   1 
                 
               
               ) 
             
             = 
             
               
                 ( 
                 
                   
                     
                       - 
                       
                         
                           α 
                           i 
                           ′ 
                         
                         
                           γ 
                           i 
                           ′ 
                         
                       
                     
                     + 
                     
                       
                         α 
                         is 
                         ′ 
                       
                       
                         γ 
                         is 
                         ′ 
                       
                     
                   
                   , 
                   
                     
                       - 
                       
                         
                           β 
                           i 
                           ′ 
                         
                         
                           γ 
                           i 
                           ′ 
                         
                       
                     
                     + 
                     
                       
                         β 
                         is 
                         ′ 
                       
                       
                         γ 
                         is 
                         ′ 
                       
                     
                   
                   , 
                   
                     - 
                     1 
                   
                 
                 ) 
               
               . 
             
           
         
       
     
     In step (S 45 ), a method of surface fitting the residual coordinate and the residual normal vector comprises: 
     step (S 451 ): in the second local coordinate system, expressing a polynomial surface by the polynomial surface equation leaving out the conic surface term, the polynomial surface can be expressed in terms of the following equation: 
               z   =       f   ⁡     (     x   ,     y   ;   P       )       =       ∑     j   =   1     J     ⁢           ⁢       P   j     ⁢       g   j     ⁡     (     x   ,   y     )               ,         
wherein g j (x, y) is one item of the polynomial, and P=(p 1 , p 2 , . . . p J ) T  is the coefficient sets;
 
     step (S 452 ): acquiring a first sum of squares d i (P), of residual coordinate differences in z′ direction between the residual coordinate value (x″ i , y″ i , z″ i )(i=1, 2, . . . , K) and the freeform surface; and a second sum of squares d 2 (P), of modulus of vector differences between the residual normal vector N i =(α″ i , β″ i , −1) (i=1, 2, . . . , K) and a normal vector of the freeform surface, wherein the first sum of squares d 1 (P) is expressed in terms of a first equation: 
                   d   1     ⁡     (   P   )       =         ∑     i   =   1     I     ⁢           ⁢       [       z   i     -     f   ⁡     (       x   i   ″     ,       y   i   ″     ;   P       )         ]     2       =         (     Z   -       A   1     ⁢   P       )     T     ⁢     (     Z   -       A   1     ⁢   P       )           ,         
and
 
     the second sum of squares d 2 (P) is expressed in terms of a second equation: 
     
       
         
           
             
               
                 
                   d 
                   2 
                 
                 ⁡ 
                 
                   ( 
                   P 
                   ) 
                 
               
               = 
               
                 
                   
                     ∑ 
                     
                       i 
                       = 
                       1 
                     
                     I 
                   
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     { 
                     
                       
                         
                           [ 
                           
                             
                               u 
                               i 
                             
                             - 
                             
                               
                                 f 
                                 
                                   x 
                                   ″ 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     i 
                                     ″ 
                                   
                                   , 
                                   
                                     
                                       y 
                                       i 
                                       ″ 
                                     
                                     ; 
                                     P 
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                         2 
                       
                       + 
                       
                         
                           [ 
                           
                             
                               v 
                               i 
                             
                             - 
                             
                               
                                 f 
                                 
                                   y 
                                   ″ 
                                 
                               
                               ⁡ 
                               
                                 ( 
                                 
                                   
                                     x 
                                     i 
                                     ″ 
                                   
                                   , 
                                   
                                     
                                       y 
                                       i 
                                       ″ 
                                     
                                     ; 
                                     P 
                                   
                                 
                                 ) 
                               
                             
                           
                           ] 
                         
                         2 
                       
                     
                     } 
                   
                 
                 = 
                 
                   
                     
                       
                         ( 
                         
                           U 
                           - 
                           
                             
                               A 
                               2 
                             
                             ⁢ 
                             P 
                           
                         
                         ) 
                       
                       T 
                     
                     ⁢ 
                     
                       ( 
                       
                         U 
                         - 
                         
                           
                             A 
                             2 
                           
                           ⁢ 
                           P 
                         
                       
                       ) 
                     
                   
                   + 
                   
                     
                       
                         ( 
                         
                           V 
                           - 
                           
                             
                               A 
                               3 
                             
                             ⁢ 
                             P 
                           
                         
                         ) 
                       
                       T 
                     
                     ⁢ 
                     
                       ( 
                       
                         V 
                         - 
                         
                           
                             A 
                             3 
                           
                           ⁢ 
                           P 
                         
                       
                       ) 
                     
                   
                 
               
             
             ; 
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               wherein 
               , 
               
                 
 
               
               ⁢ 
               
                 Z 
                 = 
                 
                   
                     ( 
                     
                       
                         z 
                         1 
                       
                       , 
                       
                         z 
                         2 
                       
                       , 
                       … 
                       ⁢ 
                       
                           
                       
                       , 
                       
                         z 
                         I 
                       
                     
                     ) 
                   
                   T 
                 
               
               , 
               
                 U 
                 = 
                 
                   
                     ( 
                     
                       
                         u 
                         1 
                       
                       , 
                       
                         u 
                         2 
                       
                       , 
                       … 
                       ⁢ 
                       
                           
                       
                       , 
                       
                         u 
                         I 
                       
                     
                     ) 
                   
                   T 
                 
               
               , 
               
                 V 
                 = 
                 
                   
                     ( 
                     
                       
                         v 
                         1 
                       
                       , 
                       
                         v 
                         2 
                       
                       , 
                       … 
                       ⁢ 
                       
                           
                       
                       , 
                       
                         v 
                         I 
                       
                     
                     ) 
                   
                   T 
                 
               
               , 
               
                 
 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   A 
                   1 
                 
                 = 
                 
                   ( 
                   
                     
                       
                         
                           
                             g 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 1 
                                 ″ 
                               
                               , 
                               
                                 y 
                                 1 
                                 ″ 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             g 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 1 
                                 ″ 
                               
                               , 
                               
                                 y 
                                 1 
                                 ″ 
                               
                             
                             ) 
                           
                         
                       
                       
                         … 
                       
                       
                         
                           
                             g 
                             J 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 1 
                                 ″ 
                               
                               , 
                               
                                 y 
                                 1 
                                 ″ 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             g 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 2 
                                 ″ 
                               
                               , 
                               
                                 y 
                                 2 
                                 ″ 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             g 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 2 
                                 ″ 
                               
                               , 
                               
                                 y 
                                 2 
                                 ″ 
                               
                             
                             ) 
                           
                         
                       
                       
                         … 
                       
                       
                         
                           
                             g 
                             J 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 2 
                                 ″ 
                               
                               , 
                               
                                 y 
                                 2 
                                 ″ 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         ⋮ 
                       
                       
                         ⋮ 
                       
                       
                         
                             
                         
                       
                       
                         ⋮ 
                       
                     
                     
                       
                         
                           
                             g 
                             1 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 I 
                                 ″ 
                               
                               , 
                               
                                 y 
                                 I 
                                 ″ 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             g 
                             2 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 I 
                                 ″ 
                               
                               , 
                               
                                 y 
                                 I 
                                 ″ 
                               
                             
                             ) 
                           
                         
                       
                       
                         … 
                       
                       
                         
                           
                             g 
                             J 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 I 
                                 ″ 
                               
                               , 
                               
                                 y 
                                 I 
                                 ″ 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   ) 
                 
               
               , 
               
                 
 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   A 
                   2 
                 
                 = 
                 
                   ( 
                   
                     
                       
                         
                           
                             g 
                             1 
                             x 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 1 
                                 ″ 
                               
                               , 
                               
                                 y 
                                 1 
                                 ″ 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             g 
                             2 
                             x 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 1 
                                 ″ 
                               
                               , 
                               
                                 y 
                                 1 
                                 ″ 
                               
                             
                             ) 
                           
                         
                       
                       
                         … 
                       
                       
                         
                           
                             g 
                             J 
                             x 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 1 
                                 ″ 
                               
                               , 
                               
                                 y 
                                 1 
                                 ″ 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         
                           
                             g 
                             1 
                             x 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 2 
                                 ″ 
                               
                               , 
                               
                                 y 
                                 2 
                                 ″ 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             g 
                             2 
                             x 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 2 
                                 ″ 
                               
                               , 
                               
                                 y 
                                 2 
                                 ″ 
                               
                             
                             ) 
                           
                         
                       
                       
                         … 
                       
                       
                         
                           
                             g 
                             J 
                             x 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 2 
                                 ″ 
                               
                               , 
                               
                                 y 
                                 2 
                                 ″ 
                               
                             
                             ) 
                           
                         
                       
                     
                     
                       
                         ⋮ 
                       
                       
                         ⋮ 
                       
                       
                         
                             
                         
                       
                       
                         ⋮ 
                       
                     
                     
                       
                         
                           
                             g 
                             1 
                             x 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 I 
                                 ″ 
                               
                               , 
                               
                                 y 
                                 I 
                                 ″ 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           
                             g 
                             2 
                             x 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 I 
                                 ″ 
                               
                               , 
                               
                                 y 
                                 I 
                                 ″ 
                               
                             
                             ) 
                           
                         
                       
                       
                         … 
                       
                       
                         
                           
                             g 
                             J 
                             x 
                           
                           ⁡ 
                           
                             ( 
                             
                               
                                 x 
                                 I 
                                 ″ 
                               
                               , 
                               
                                 y 
                                 I 
                                 ″ 
                               
                             
                             ) 
                           
                         
                       
                     
                   
                   ) 
                 
               
               , 
               
                 
 
               
               ⁢ 
               
                   
               
               ⁢ 
               
                 
                   
                     A 
                     3 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             
                               g 
                               1 
                               y 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   1 
                                   ″ 
                                 
                                 , 
                                 
                                   y 
                                   1 
                                   ″ 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               g 
                               2 
                               y 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   1 
                                   ″ 
                                 
                                 , 
                                 
                                   y 
                                   1 
                                   ″ 
                                 
                               
                               ) 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             
                               g 
                               J 
                               y 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   1 
                                   ″ 
                                 
                                 , 
                                 
                                   y 
                                   1 
                                   ″ 
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           
                             
                               g 
                               1 
                               y 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   2 
                                   ″ 
                                 
                                 , 
                                 
                                   y 
                                   2 
                                   ″ 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               g 
                               2 
                               y 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   2 
                                   ″ 
                                 
                                 , 
                                 
                                   y 
                                   2 
                                   ″ 
                                 
                               
                               ) 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             
                               g 
                               J 
                               y 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   2 
                                   ″ 
                                 
                                 , 
                                 
                                   y 
                                   2 
                                   ″ 
                                 
                               
                               ) 
                             
                           
                         
                       
                       
                         
                           ⋮ 
                         
                         
                           ⋮ 
                         
                         
                           
                               
                           
                         
                         
                           ⋮ 
                         
                       
                       
                         
                           
                             
                               g 
                               1 
                               y 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   I 
                                   ″ 
                                 
                                 , 
                                 
                                   y 
                                   I 
                                   ″ 
                                 
                               
                               ) 
                             
                           
                         
                         
                           
                             
                               g 
                               2 
                               y 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   I 
                                   ″ 
                                 
                                 , 
                                 
                                   y 
                                   I 
                                   ″ 
                                 
                               
                               ) 
                             
                           
                         
                         
                           … 
                         
                         
                           
                             
                               g 
                               J 
                               y 
                             
                             ⁡ 
                             
                               ( 
                               
                                 
                                   x 
                                   I 
                                   ″ 
                                 
                                 , 
                                 
                                   y 
                                   I 
                                   ″ 
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                     ) 
                   
                 
                 ; 
               
             
           
         
       
     
     step (S 453 ): obtaining an evaluation function,
 
 P =( A   1   T   A   1   +wA   2   T   A   2   +wA   3   T   A   3 ) −1 ·( A   1   T   Z+wA   2   T   U+wA   3   T   V ),
 
     wherein w is a weighting greater than 0; 
     step (S 454 ): selecting different weightings w and setting a gradient ∇f (P) of the evaluation function equal to 0, to obtain a plurality of different values of P and a plurality of freeform surface shapes z=f (x, y; P) corresponding to each of the plurality of different values of P; and 
     step (S 455 ): choosing a final freeform surface shape Ω opt  which has a best imaging quality from the plurality of freeform surface shapes z=f (x, y; P). 
     Furthermore, a step of optimizing the hybrid surface optical system obtained after step (S 4 ) can be performed, and the hybrid surface optical system obtained after step (S 4 ) can be as the initial system. 
     Referring to  FIG. 7 , an off-axis hybrid surface three-mirror optical system  100  is designed with the above method to prove the above method. The off-axis hybrid surface three-mirror optical system  100  comprises a primary mirror  120 , a secondary mirror  140 , a tertiary mirror  160  and an image sensor  180  located in that order and spaced from each other. The secondary mirror  140  is used as a stop surface. A surface shape of the primary mirror  120  is a freeform surface. A surface shape of the secondary mirror  140  is an aspheric surface. A surface shape of the tertiary mirror  160  is a spherical surface. The feature rays exiting from the light source would be successively reflected by the primary mirror  120 , the secondary mirror  140  and the tertiary mirror  160  to form an image on the image sensor  180 . The parameters of the off-axis hybrid surface three-mirror optical system  100  are shown in table 1 below. 
     
       
         
           
               
               
             
               
                 TABLE 1 
               
               
                   
               
               
                 Parameters 
                 Values 
               
               
                   
               
             
            
               
                 Field of View 
                 3° × 6° (an angle in a meridian direction is in a range 
               
               
                   
                 from −16° to −10°) 
               
               
                 F-number 
                 2.2 
               
               
                 entrance pupil 
                 100 mm 
               
               
                 diameter 
               
               
                 Wave Range 
                 Long-wave infrared (8-12 μm) 
               
               
                   
               
            
           
         
       
     
     In one embodiment, fourteen off-axial fields are selected during a designing process of the off-axis hybrid surface three-mirror optical system  100 . The fourteen off-axial fields are (0°, −16°), (0°, −15°), (0°, −14°), (0°, −13°), (0°, −12°), (0°, −11°), (0°, −10°), (1.5°, −16°), (1.5°, −15°), (1.5°, −14°), (1.5°, −13°), (1.5°, −12°), (1.5°, −11°, and (1.5°, −10°. 112 feature rays are selected in each off-axial field, and 1568 feature rays that corresponds different pupil positions and different off-axial fields are selected. 
     Referring to  FIG. 8 , a first three-dimensional rectangular coordinates system (x 1 , y 1 , z 1 ) is defined in space. A z 1 -axis is along an horizontal line, in the z 1 -axis, to the left is negative, and to the right is positive. A y 1 -axis is in a direction substantially perpendicular to the z 1 -axis, in the y 1 -axis, to the upward is positive, and to the downward is negative. An x 1 -axis is perpendicular to a y 1 z 1  plane, in the x 1 -axis, in a direction substantially perpendicular to the y 1 z 1  plane, to the inside is positive, and to the outside is negative. 
     In space relative to the first three-dimensional rectangular coordinates system (x 1 , y 1 , z 1 ), a second three-dimensional rectangular coordinates system (x 2 , y 2 , z 2 ) is defined by a primary mirror location, a third three-dimensional rectangular coordinates system (x 3 , y 3 , z 3 ) is defined by a secondary mirror location, a fourth three-dimensional rectangular coordinates system (x 4 , y 4 , z 4 ) is defined by a tertiary mirror location, and a fifth three-dimensional rectangular coordinates system (x 5 , y 5  z 5 ) is defined by an image sensor location. 
     A second origin of the second three-dimensional rectangular coordinates system (x 2 , y 2 , z 2 ) is in (0, 244.51, 193.62) position of the first three-dimensional rectangular coordinates system (x 1 , y 1 , z 1 ), whose unit is millimeter. A z 2 -axis positive direction rotates about 13.39 degrees along a counterclockwise direction relative to a z 1 -axis positive direction. 
     A third origin of the third three-dimensional rectangular coordinates system (x 3 , y 3 , z 3 ) is in (0, θ, −100) position of the first three-dimensional rectangular coordinates system (x 1 , y 1 , z 1 ), whose unit is millimeter. A z 3 -axis positive direction rotates about 15 degrees along a counterclockwise direction relative to a z 1 -axis positive direction. 
     A fourth origin of the fourth three-dimensional rectangular coordinates system (x 4 , y 4 , z 4 ) is in (0, −160.83, 209.49) position of the first three-dimensional rectangular coordinates system (x 1 , y 1 , z 1 ), whose unit is millimeter. A z 4 -axis positive direction rotates about 0 degrees along a counterclockwise direction relative to a z 1 -axis positive direction. 
     A fifth origin of the fifth three-dimensional rectangular coordinates system (x 5 , y 5 , z 5 ) is in (0, −145.13, −174.41) position of the first three-dimensional rectangular coordinates system (x 1 , y 1 , z 1 ), whose unit is millimeter. A z 5 -axis positive direction rotates about 10 degrees along a counterclockwise direction relative to a z 1 -axis positive direction. 
     A length of the off-axis hybrid surface three-mirror optical system  100  along the y 1 -axis is about 430 millimeters. A length of the off-axis hybrid surface three-mirror optical system  100  along the z 1 -axis is about 400 millimeters. 
     In the second three-dimensional rectangular coordinates system (x 2 , y 2 , z 2 ); a reflective surface of the primary mirror  120  is a sixth-order polynomial freeform surface of x 2 y 2 , and the sixth-order polynomial freeform surface of x 2 y 2  can be expressed as follows: 
     
       
         
           
             Z 
             = 
             
               
                 
                   cr 
                   2 
                 
                 
                   1 
                   + 
                   
                     
                       1 
                       - 
                       
                         
                           ( 
                           
                             1 
                             + 
                             k 
                           
                           ) 
                         
                         ⁢ 
                         
                           c 
                           2 
                         
                         ⁢ 
                         
                           r 
                           2 
                         
                       
                     
                   
                 
               
               + 
               
                 
                   b 
                   1 
                 
                 ⁢ 
                 y 
               
               + 
               
                 
                   b 
                   2 
                 
                 ⁢ 
                 
                   x 
                   2 
                 
               
               + 
               
                 
                   b 
                   3 
                 
                 ⁢ 
                 
                   y 
                   2 
                 
               
               + 
               
                 
                   b 
                   4 
                 
                 ⁢ 
                 
                   x 
                   2 
                 
                 ⁢ 
                 y 
               
               + 
               
                 
                   b 
                   5 
                 
                 ⁢ 
                 
                   y 
                   3 
                 
               
               + 
               
                 
                   b 
                   6 
                 
                 ⁢ 
                 
                   x 
                   4 
                 
               
               + 
               
                 
                   b 
                   7 
                 
                 ⁢ 
                 
                   x 
                   2 
                 
                 ⁢ 
                 
                   y 
                   2 
                 
               
               + 
               
                 
                   b 
                   8 
                 
                 ⁢ 
                 
                   y 
                   4 
                 
               
               + 
               
                 
                   b 
                   9 
                 
                 ⁢ 
                 
                   x 
                   6 
                 
               
               + 
               
                 
                   b 
                   10 
                 
                 ⁢ 
                 
                   x 
                   4 
                 
                 ⁢ 
                 
                   y 
                   2 
                 
               
               + 
               
                 
                   b 
                   11 
                 
                 ⁢ 
                 
                   x 
                   2 
                 
                 ⁢ 
                 
                   y 
                   4 
                 
               
               + 
               
                 
                   b 
                   12 
                 
                 ⁢ 
                 
                   y 
                   6 
                 
               
             
           
         
       
       
         
           
             
                 
             
             ⁢ 
             
               
                 
                   r 
                   2 
                 
                 = 
                 
                   
                     x 
                     2 
                   
                   + 
                   
                     y 
                     2 
                   
                 
               
               , 
             
           
         
       
     
     wherein, z represents surface sag, c represents surface curvature, k represents conic constant, while b i  (i=1, 2, . . . , 11, 12) represents the ith term coefficient. In one embodiment, c=4.57e-04, k=0.048, 1) 1 =0.0951, b 2 =5.26e-06, b 3 =7.90e-05, b 4 =3.41e-07, b 5 =3.97e-07, b 6 =5.54e-10, b 7 =1.29e-09, b 8 =6.84e-10, b 9 =0, b 10 =0, b 12 =0. 
     In the third three-dimensional rectangular coordinates system (x 3 , y 3 , z 3 ), the reflective surface of the secondary mirror  104  is a sixth-order polynomial aspheric surface of x 3 y 3 . The sixth-order polynomial aspheric surface of x 3 y 3  can be expressed as follows: 
     
       
         
           
             
               Z 
               = 
               
                 
                   
                     
                       cr 
                       2 
                     
                     
                       1 
                       + 
                       
                         
                           1 
                           - 
                           
                             
                               ( 
                               
                                 1 
                                 + 
                                 k 
                               
                               ) 
                             
                             ⁢ 
                             
                               c 
                               2 
                             
                             ⁢ 
                             
                               r 
                               2 
                             
                           
                         
                       
                     
                   
                   + 
                   
                     
                       a 
                       1 
                     
                     ⁢ 
                     
                       r 
                       4 
                     
                   
                   + 
                   
                     
                       a 
                       2 
                     
                     ⁢ 
                     
                       r 
                       6 
                     
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       r 
                       2 
                     
                   
                 
                 = 
                 
                   
                     x 
                     2 
                   
                   + 
                   
                     y 
                     2 
                   
                 
               
             
             , 
           
         
       
     
     wherein z represents surface sag, c represents surface curvature, k represents conic constant, a1 and a2 are polynomial coefficients. In one embodiment, c=5.73e-04, k=0, a 1 =1.21e-12, a 2 =−8.85e-16. 
     In the fourth three-dimensional rectangular coordinates system (x 4 , y 4 , z 4 ), the reflective surface of the tertiary mirror  106  is a spherical surface of x 4 y 4 , and the spherical surface of x 4 y 4  can be expressed as (x−A) 2 +(y−B) 2 +(z−C) 2 =r 2 . In one embodiment, A=0, B=−160.83, C=209.49, r=−988.48. 
     In other embodiments, the values of c, k, and A i  in the sixth-order polynomial freeform surface of x 2 y 2 , the values of z, c, k, a 1  and a 2  of sixth-order polynomial aspheric surface of x 3 y 3 , and the values of A, B, C and r of the spherical surface of x 4 y 4  can be selected according to actual needs. 
     Referring to  FIG. 9 , the RMS spot diameters of the off-axis hybrid surface three-mirror optical system  100  is shown. It can be seen that an average value of the RMS spot diameters is about 0.296 mm. It shows that the RMS spot diameters of the off-axis hybrid surface three-mirror optical system  100  is small, therefore the off-axis hybrid surface three-mirror optical system  100  can be used as a good initial structure for subsequent optimization. 
     The off-axis hybrid surface three-mirror optical system  100  can be optimized to improve the imaging quality and reduce the relative distortion. The off-axis hybrid surface three-mirror optical system is used as the initial structure for subsequent optimization. In one embodiment, the off-axis hybrid surface three-mirror optical system  100  is optimized via a CODEV software. In one embodiment, after the off-axis hybrid surface three-mirror optical system  100  is optimized, in the sixth-order polynomial freeform surface of x 2 y 2 , c=5.54e-04, k=90.31, b 1 =0, b 2 =0.0003, b 3 =0.0005, b 4 =9.07e-07, b 5 =1.03e-06, b 6 =−2.37e-10, b 7 =9.88e-10, b 8 =1.31e-09, b 9 =−4.64e-14, b 10 =−1.91e-13, b 11 =−2.05e-13 b 12 =−7.56e-14; in the sixth-order polynomial aspheric surface of x 3 y 3 , c=9.19e-04, k=0, α i =−4.49e-11, α 2 =−1.35e-15; in the spherical surface of x 4 y 4 , A=0, B=−266.75, C=313.84, r=−996.65. 
     In one embodiment, after the off-axis hybrid surface three-mirror optical system  100  is optimized, the second origin of the second three-dimensional rectangular coordinates system (x 2 , y 2 , z 2 ) is in (0, 18.57, 274.09) position of the first three-dimensional rectangular coordinates system (x 1 , y 1 , z 1 ), whose unit is millimeter. A z 2 -axis positive direction rotates about 7.63 degrees along a counterclockwise direction relative to a z 1 -axis positive direction. A third origin of the third three-dimensional rectangular coordinates system (x 3 , y 3 , z 3 ) is in (0, −227.00, −48.81) position of the first three-dimensional rectangular coordinates system (x 1 , y 1 , z 1 ), whose unit is millimeter. A z 3 -axis positive direction rotates about 18.03 degrees along a counterclockwise direction relative to a z 1 -axis positive direction. A fourth origin of the fourth three-dimensional rectangular coordinates system (x 4 , y 4 , z 4 ) is in (0, −266.75, 313.84) position of the first three-dimensional rectangular coordinates system (x 1 , y 1 , z 1 ), whose unit is millimeter. A z 4 -axis positive direction rotates about 10.77 degrees along a counterclockwise direction relative to a z 1 -axis positive direction. A fifth origin of the fifth three-dimensional rectangular coordinates system (x 5 , y 5 , z 5 ) is in (0, −365.47, −27.10) position of the first three-dimensional rectangular coordinates system (x 1 , y 1 , z 1 ), whose unit is millimeter. A z 5 -axis positive direction rotates about 9.74 degrees along a counterclockwise direction relative to a z 1 -axis positive direction. 
     Referring to  FIG. 10 , a light path schematic view of the off-axis hybrid surface three-mirror optical system  100  after optimization is shown. It is shown that the plurality of feature rays R i  (i=1, 2 . . . K) from different fields and different aperture positions substantially focus on the ideal target point, the image quality is improved. 
       FIG. 11  illustrates that a modulation transfer functions (MTF) in visible light band of partial field angles of the hybrid surface three-mirror optical system  100  after optimization are closes to the diffraction limitation, and the image quality is further improved. 
     Referring to  FIG. 12 , it can be seen that the average root mean square (RMS) of the pixel spot of the hybrid surface three-mirror optical system  100  after optimization is about 0.042λ, wherein λ=10000 nm, which meets the design requirements. 
     An effective entrance pupil diameter of the hybrid surface three-mirror optical system  100  is about 100 millimeter. 
     The hybrid surface three-mirror optical system  100  adopts an off-axis field of view in a vertical direction. A field angle of the hybrid surface three-mirror optical system  100  is about 3°×6°, wherein an angle in an horizontal direction is in a range from about −1.5° to about 1.5°, and an angle in the vertical direction is in a range from about −16° to about −10°. 
     A wavelength of the hybrid surface three-mirror optical system  100  is not limited, in one embodiment, the wavelength is in a range from about 8 microns to about 12 microns. 
     An effective focal length (EFL) of the hybrid surface three-mirror optical system  100  is about 220 millimeters. 
     F-number of the hybrid surface three-mirror optical system  100  is a D/f reciprocal. In one embodiment, the relative aperture (D/f) of the hybrid surface three-mirror optical system  100  is about 0.45, and the F-number is about 2.2. 
     The method for designing hybrid surface three-mirror optical system can have many advantages. First, the method is calculated directly in an off-axis state, an error produced during a process from the coaxial system transformed into the off-axis system can be avoided, thus, a deviation between the off-axis hybrid surface optical system obtained by the above method and an ideal optical system is small. Second, during calculating the spherical surface optical system, the surface fitting process is combined with the data point calculating process, the unknown surface is calculated by repeating the process of data point calculating-spherical surface fitting, which improve the accuracy of the method. Third, the optical power of the spherical surface is changed by changing the radius of the spherical surface, then the remaining spherical surface is calculated to compensate for changed optical power. Therefore, an optical power distribution of the spherical surface optical system is more uniform, and the spherical surface optical system can be a better initial system for the following steps. Fourth, the global coordinate system and the local coordinate system are defined, the coordinates and the normal vector of the plurality of feature data points in the global coordinate system are transformed into the local coordinate system, then the aspheric surfaces are obtained by surface fitting in the local coordinate system, thus, the method is more precisely compared to conventional methods. Fifth, when surface fitting the second plurality of feature data points and the third plurality of feature data points, both the coordinates and the normal vectors of the second plurality of feature data points and the third plurality of feature data points are considered; therefore, the shape of the freeform surface and shape of the aspheric surfaces are accurate. Finally, the point-by-point design method can be used to design hybrid surface systems comprise different surfaces, increasing the diversity of optical designs. 
     It is to be understood that the above-described 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 above-described embodiments illustrate the scope of the present disclosure but do not restrict the scope of the present disclosure.