- Email: [email protected]
Design and analysis of the four-mirror optical system
Optik 121 (2010) 1900–1903
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Design and analysis of the four-mirror optical system Meifang Zou, Jun Chang , M.M. Talha, Tingcheng Zhang, Yongtian Wang School of Optoelectronics Engineering, Beijing Institute of Technology, Beijing 100081, PR China
a r t i c l e in f o
a b s t r a c t
Article history: Received 22 January 2009 Accepted 9 May 2009
Based on the methods to design the three-mirror reflective system, the aberrations formulae for the four-mirror optical reflection system have been deduced and presented for the first time. The diversity of parameter ranges for the designed system for different structures is also studied to select the parameters to obtain the practicable initial structure. Example designs with proposed methods are presented here and their performance evaluations demonstrate that the image qualities of these examples attain the diffraction limits. & 2009 Elsevier GmbH. All rights reserved.
Keywords: Optical design All reflective Four-mirror optical system Large aperture
1. Introduction
2. Analysis of the system parameters
The manufacturing cost of a reflection optical system is decreasing day by day due to development of the machining and alignment techniques. A four-mirror reflection system has many advantages. It is free of chromatic aberrations, can allow wide spectrum range, has a large aperture, is light weight and small in volume, and has greater merits in heat tolerance. It is also helpful in solving obscuration problems. Due to the various system merit function variables, such system designs can improve image quality over a very large field of view (FOV). Such designs are also able to satisfy the requirements of remote sensing, infrared multispectral detection, and space photography. Using the four-mirror technique, it becomes easier as well as lighter to obtain fine image quality of an optical system. Very few research reports are available on this topic [4], which need to be re-examined and refined with regard to the scope of improvement. Some scholars like Rakich [5,6] have done some research work recently. The four-mirror optical system design presented in this paper is characterized by the use of a flat image plane, with good image qualities like low aberrations and low distortion. The methodology to obtain the parameters for the initial structure of a four-mirror system is given, and the physical meanings of these geometrical parameters are described. The parameter ranges of the systems with four different structures are also studied. Based on these studies, the initial structure has been worked out along with the aberration formulae. Finally, we will report the example designs with their proposed methods.
The layout of the four-mirror optical system is shown in Fig. 1. It comprises a primary mirror (M1), a secondary mirror (M2), a tertiary mirror (M3), and a quarternary mirror (M4). The conic coefficients employed in the structure of each mirror are e21, e22, e23, e24, the obscuration ratios of mirrors 1, 2, and 3 are a1 ¼ (l2/f0 1)E(h2/h1), a2 ¼ (l3/l0 2)E(h3/h2), and a3 ¼ (l4/l0 3)E(h4/h3), respectively, and the magnifications of mirrors 2, 3, and 4 are b2 ¼ (n2l0 2/n0 2l2) ¼ (l0 2/l2), b3 ¼ (n3l0 3/n0 3l3) ¼ (l0 3/l3), and b4 ¼ (n4l0 4/n0 4l4) ¼ (l0 4/l4), respectively. The image quality for the four-mirror system is decided by choosing the structure parameters a1, a2, a3, b2, b3, and b4. Assume that the total focal length of a four-mirror telescope system is f0 ; based on the definition of magnification and obstruction ratios, the expressions for the radii of curvature of different surfaces and their corresponding thickness values are given as follows: 2f 0 2a1 f 0 2a1 a2 f 0 2a3 a2 a1 f 0 R1 ¼ ; R3 ¼ ; R4 ¼ ; R2 ¼ b2 b3 b4 b3 b4 ðb2 1Þ b4 ð1 b3 Þ b4 1 f 0 ð1 a1 Þ a1 f 0 ða2 1Þ ð1 a3 Þa2 a1 f 0 ; d2 ¼ ; d3 ¼ d1 ¼
b2 b3 b4
b3 b4
b4
Assuming that the stop of the optical system is on the first mirror, various aberrations are given as follows: (i) Spherical aberration [2,3]: n 3 3 3 3 3 SI ¼ 14 b2 b3 b4 ðe21 1Þ þ a1 b3 b4 ðb2 1Þ ½ð1 þ b2 Þ2 e22 ðb2 1Þ2 3
Corresponding author.
E-mail addresses: [email protected] (M. Zou), [email protected] (J. Chang). 0030-4026/$ - see front matter & 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.05.009
þa1 a2 b4 ½e23 ð1 b3 Þ3 þ ð1 þ b3 Þ2
ðb3 1Þ þ a1 a2 a3 ðb4 1Þ½ð1 þ b4 Þ2 þ e24 ðb4 1Þ2
M. Zou et al. / Optik 121 (2010) 1900–1903
1901
f 0 1 o0; a1 o0 ða1 a 1Þ; a2 41; b2 40; a3 41; b3 40; b4 o0
ð3Þ
f 0 1 o0; a1 o0 ða1 a 1Þ; 0oa2 o1; b2 o0; 0oa3 o1; b3 40; b4 40
ð4Þ
Fig. 1. The layout of the four-mirror optical system.
(ii) Coma:
a1 1
SII ¼
b23 b24 ðb2 1Þ½ð1 þ b2 Þ2 e22 ðb2 1Þ2
4b2
þ
a2 ða1 1Þ þ b2 ða2 1Þ
þ
a2 a3 ða1 1Þ þ a3 b2 ða2 1Þ þ b2 b3 ða3 1Þ
4b2 b3
1 2
b24 ð1 b3 Þ½e23 ð1 b3 Þ ð1 þ b3 Þ2
2.2. System with an intermediate image after the secondary mirror
4b2 b3 b4
ðb4 1Þ½ð1 þ b4 Þ2 e24 ðb4 1Þ2
To obtain an intermediate image immediately after the secondary mirror, the following conditions must be satisfied: a140, l0 2od2, then by solving these inequalities, the ranges of the various parameters for the optical system are obtained as follows:
(iii) Astigmatism: SIII ¼
ða1 1Þ2 b3 b4 2 1 b2
4a
There arise three limiting cases regarding the position of the intermediate image, and from the geometry shown below, we can obtain a relationship between a and b. Case I: When |f0 1| ¼ |d1||d2|, the intermediate image is formed at the center of the third mirror, and we obtain a relation b2(a21) ¼ 1. Case II: When |f0 1|o|d1||d2|, the intermediate image is formed between the primary mirror and the secondary mirror, and the relationship obtained is b2(a21)o1. Case III: When |f0 1|4|d1||d2|, the intermediate image is formed between the secondary mirror and the tertiary mirror, and the relationship between the two parameters is b2(a21)41.
f 0 1 o0; 0oa1 o1; a2 o0 ða2 a 1Þ; b2 40; 0oa3 o1; b3 o0; b4 40
ðb2 1Þ½ð1 þ b2 Þ2 e22 ð1 b2 Þ2
ð5Þ
2
þ
½a2 ða1 1Þ þ b2 ða2 1Þ 2
2
4a1 a2 b2 b3
b4 ð1 b3 Þ½e23 ð1 b3 Þ2 ð1 þ b3 Þ2 þ
½a2 a3 ða1 1Þ þ a3 b2 ða2 1Þ þ b2 b3 ða3 1Þ2 2
2
2
4a1 a2 a3 b2 b3 b4
ðb4 1Þ½ð1 þ b4 Þ2 e24 ðb4 1Þ2
a1 1 a ða 1Þ þ b2 ða2 1Þ b b ð1 b22 Þ 2 1 b4 ð1 b23 Þ a 1 b2 3 4 a1 a2 b2 b3
a2 a3 ða1 1Þ þ a3 b2 ða2 1Þ þ b2 b3 ða3 1Þ 2 ð1 b4 Þ a1 a2 a3 b2 b3 b4
b2 b3 b4 þ
b3 b4
a1
ðb2 1Þ þ
b4 ðb3 1Þ
a1 a2
þ
ðb4 1Þ
a1 a2 a3
(iv) Field curvature: SIV ¼ b2 b3 b4 þ
b3 b4
a1
ð1 b2 Þ þ
b4 ð1 b3 Þ
a1 a2
þ
ð1 b4 Þ
a1 a2 a3
f 0 1 o0; 0oa1 o1; a2 o0 ða2 a 1Þ; b2 40; a3 o0; b3 o0; b4 o0
ð6Þ
f 0 1 o0; 0oa1 o1; a2 o0 ða2 a 1Þ; b2 40; a3 41; b3 40; b4 o0
ð7Þ
f 0 1 o0; a1 41; a2 o0 ða2 a 1Þ; b2 40; 0oa3 o1; b3 o0; b4 40
ð8Þ
Again, there are three cases regarding the position of the intermediate image formed by the system, and the relationships between a and b could be obtained. Case I: When |l0 2| ¼ |d1|, the intermediate image is formed at the center of the first mirror, and the related relationship between two parameters is b2(1a1)/a1. Case II: When |l0 2|o|d1|, the position of intermediate image is between the first mirror and the third mirror, and the relationship b24(1a1)/a1 is obtained. Case III: When |l0 2|4|d1|, the position of the intermediate image is between the first mirror and the second mirror, and b2o(1a1)/a1 is the relationship between the two parameters. Specially, if the fourth mirror is placed at the position of the intermediate image, the relationship obtained is (1a3)b3 ¼ 1.
The four-mirror optical system can be divided into four types depending on the formation and the position of the intermediate image formed by the designed optical system [1].
2.3. System with an intermediate image after the tertiary mirror
2.1. System formaning an intermediate image after the primary mirror
To obtain an intermediate image immediately after the tertiary mirror, the following conditions must be satisfied: d1f0 140; |f0 1|4|d1| a140, l0 24d2, l0 3d240, |l0 3|o|d3|, and then by solving these inequalities, the ranges of the various parameters for the optical system are obtained as follows:
To obtain an intermediate image immediately after the primary mirror, the following conditions must be satisfied: d1f0 140; |f0 1|o|d1|. Then after solving these inequalities, the ranges of various parameters for the optical system are as follows: f 0 1 o0; a1 o0 ða1 a 1Þ; a2 41; b2 40; 0oa3 o1; b3 o0; b4 40
ð1Þ
f 0 1 o0; a1 o0 ða1 a 1Þ; a2 41; b2 40; a3 o0; b3 o0; b4 o0
ð2Þ
f 0 1 o0; a1 40 ða1 a1Þ; 0oa2 o1; b2 40; a3 o0 ða3 a 1Þ; b3 40; b4 o0
ð9Þ Also, three limiting cases regarding the position of the intermediate image formed by the optical system and the relationship between a and b are as follows:
1902
M. Zou et al. / Optik 121 (2010) 1900–1903
Table 1 First sample for the four-mirror system optical parameters.
Table 2 Second sample for the four-mirror system optical parameters.
Mirror
Radius
Thickness
Mirror
Radius
Thickness
First mirror M1 Second mirror M2
204.3720 1012.5
152.2571
First mirror M1 Second mirror M2
1053.1 6602.8
226.4
Third mirror M3
9235.5
432.1649 Third mirror M3
436.1879
1099.8
361.1636 Fourth mirror M4
766.5 Fourth mirror M4
95.1181
1414.3
58.14 MM Lens has no title.
Scale:0.43
15-Jan-09
Fig. 2. The first kind of the four-mirror system’s layout map. Fig. 4. The second kind of the four-mirror system’s layout map. Lens has no title. GEOMETRICAL MTF 15-Jan-09
DIFFRACTION LIMIT Y 0.0 FIELD (0.00°) WAVELENGTH WEIGHT X 692.0 NM 1 Y 0.7 FIELD (0.50°) 586.0 NM 1 X Y 1.0 FIELD (0.70°) 487.0 NM 1 X DEFOCUSING 0.00000
1.0 0.9
MODULATION
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 5.0
10.0
15.0 20.0 25.0 30.0 35.0 40.0 SPATIAL FREQUENCY (CYCLES/MM)
45.0
50.0
Fig. 3. MTF curves of the designed optical system.
Case I: When |l0 3| ¼ |d2||d1|, the intermediate image is formed at the center of the first mirror, and b3b2a2 ¼ b2b2a2(1/a1)+1 is the relationship between the two parameters. Case II: When |l0 3|o|d2||d1, the position of intermediate image is between the first mirror and the third mirror, and b3b2a24b2b2a2(1/a1)+1 is the relationship between two parameters. Case III: When |l0 3|4|d2||d1, the position of intermediate image is between the first mirror and the fourth mirror, and b3b2a2ob2b2a2(1/a1)+1 is the required relation.
Fig. 5. MTF curves of the designed optical system.
Table 3 Third kind of the four-mirror system optical parameters. Mirror
Radius
Thickness
First mirror M1 Second mirror M2
136.5682 106.5036
54.6273 11.0203
Third mirror M3
55.3833
Fourth mirror M4
30.0000
30.0000
3. Design example The above four aberration equations contain ten unknowns, including e21, e22, e23, e24, a1, a2, a3, b2, b3, and b4. The last six variables are related to the structure and size of the system; the
structure can be arranged conveniently, if the purpose is just to get rid of the spherical aberration, coma, and astigmatism. If the image plane required is flat, the parameters can be obtained by field curvature equation.
M. Zou et al. / Optik 121 (2010) 1900–1903
1903
Lens has no title. GEOMETRICAL MTF 15 - Jan - 09
DIFFRACTION LIMIT Y 0.0 FIELD (0.00°) X Y 0.7 FIELD (0.50°) X Y 1.0 FIELD (0.70°) X
WAVELENGTH WEIGHT 692.0 NM 1 1 586.0 NM 487.0 NM 1
DEFOCUSING 0.00000
1.0 0.9
14.71 MM Lens has no title.
Scale:1.70
15-Jan-09
Fig. 6. The third kind of the four-mirror system’s layout map.
MODULATION
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
Now, let us design a coaxial system with focal length value 500 mm, aperture diameter (D) 60 mm, and value of the field of view (FOV) 1.41. Firstly, a particular structure is chosen. It is particularly advantageous if the four-mirror system has an intermediate image. This leads to the fact that the system has an excellent ability to restrain stray light when an eliminating stray light stop is on the position of the intermediate image. We choose a system with an intermediate image after primary mirror first, as structure (1) described above. We chose a1 ¼ 0.49, a3 ¼ 0.1, b2 ¼ 0.91, b3 ¼ 0.84, and the first and third mirror surfaces as aspherical surfaces. With the help of MATLAB software a2 and b4 can be calculated from the above aberration equations. The original structural parameters of the system are shown in Table 1. The system layout is shown in Fig. 2. After optimization, the value of its optical MTF can reach 0.6 at 50 Lp/mm. The optical MTF curve is shown in Fig. 3. Then, we design a coaxial system with focal length value 500 mm, aperture diameter value (D) 150 mm, and value of the field of view (FOV) 2.41. We also choose a system with an intermediate image that is formed at the center of the fourth mirror. As in structure (7), we chose a1 ¼ 0.57, a2 ¼ 1.83, b2 ¼ 1.1, b2 ¼ 1.2, and the first and third mirror surfaces as aspherical surfaces. The value of a3 and b4 can be calculated with the help of MATLAB. The original structural parameters of the system are shown in Table 2. The system layout is shown in Fig. 4. After optimization, the value of its optical MTF can reach 0.5 at 50 Lp/mm. The optical MTF curve is shown in Fig. 5. Lastly, we choose a system with an intermediate image after a tertiary mirror, as structure (9) described above. We chose a1 ¼ 0.2, a3 ¼ 0.4, a3 ¼ 2, b4 ¼ 4, and the first and third mirror surfaces as aspherical surfaces. With the help of MATLAB software b2 and b2 can be calculated from the above aberration equations. The original structural parameters of the system are shown in Table 3. The system layout is shown in Fig. 6.
10.0
20.0 30.0 SPATIAL FREQUENCY (CYCLES/MM)
40.0
50.0
Fig. 7. MTF curves of the designed optical system.
After optimization, the value of its optical MTF can reach 0.55 at 50 Lp/mm. The optical MTF curve is shown in Fig. 7.
4. Conclusion In this paper, the method to obtain the original structural parameters of a four-mirror optical system was studied. The validation of the design method and the parameter ranges of the system with different structures were tested by design examples. From the MTF curves, it is evident that the designed optical system has achieved good image qualities by using only two aspherical surfaces. It demonstrates that the four-mirror system with a simpler structure can satisfy the requirements of its use. Thus it is hoped that with the development of the remote-sensing technology, the improvements of CAM and the techniques of alignments, the four-mirror system, which has unique advantages, will be used widely in future. References [1] Chang Jun, Weng Zhi Cheng, Jiang Hui Ling, Zhang Xin, Cong Xiao Jie, Optical design of three-mirror system used in space, Acta Opt. Sin. 23 (2) (2003) 216– 219. [2] Junhua Pun, A study of the optical system with three mirrors of second order surface, Acta Opt. Sin. 8 (8) (1988) 717–721. [3] Junhua Pun, Design, Machining and Test of Optical Aspherical Surface, vol. 4, Science Publishing Company, 1994. [4] M. Lampton, M. Sholl, Comparison of on-axis three-mirror-anastigmat telescopes, Proc. SPIE 6687 (2007) 1–8. [5] Andrew Rakich, Four-mirror anastigmats, part 1: a complete solution set for all-spherical telescopic systems, Opt. Eng. 46 (10) (2007) 1–12. [6] Andrew Rakich, Four-mirror anastigmats, part 3: all-spherical systems with elements larger than the entrance pupil, Opt. Eng. 47 (3) (2008) 1–12.
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Design and analysis of the four-mirror optical system Meifang Zou, Jun Chang , M.M. Talha, Tingcheng Zhang, Yongtian Wang School of Optoelectronics Engineering, Beijing Institute of Technology, Beijing 100081, PR China
a r t i c l e in f o
a b s t r a c t
Article history: Received 22 January 2009 Accepted 9 May 2009
Based on the methods to design the three-mirror reflective system, the aberrations formulae for the four-mirror optical reflection system have been deduced and presented for the first time. The diversity of parameter ranges for the designed system for different structures is also studied to select the parameters to obtain the practicable initial structure. Example designs with proposed methods are presented here and their performance evaluations demonstrate that the image qualities of these examples attain the diffraction limits. & 2009 Elsevier GmbH. All rights reserved.
Keywords: Optical design All reflective Four-mirror optical system Large aperture
1. Introduction
2. Analysis of the system parameters
The manufacturing cost of a reflection optical system is decreasing day by day due to development of the machining and alignment techniques. A four-mirror reflection system has many advantages. It is free of chromatic aberrations, can allow wide spectrum range, has a large aperture, is light weight and small in volume, and has greater merits in heat tolerance. It is also helpful in solving obscuration problems. Due to the various system merit function variables, such system designs can improve image quality over a very large field of view (FOV). Such designs are also able to satisfy the requirements of remote sensing, infrared multispectral detection, and space photography. Using the four-mirror technique, it becomes easier as well as lighter to obtain fine image quality of an optical system. Very few research reports are available on this topic [4], which need to be re-examined and refined with regard to the scope of improvement. Some scholars like Rakich [5,6] have done some research work recently. The four-mirror optical system design presented in this paper is characterized by the use of a flat image plane, with good image qualities like low aberrations and low distortion. The methodology to obtain the parameters for the initial structure of a four-mirror system is given, and the physical meanings of these geometrical parameters are described. The parameter ranges of the systems with four different structures are also studied. Based on these studies, the initial structure has been worked out along with the aberration formulae. Finally, we will report the example designs with their proposed methods.
The layout of the four-mirror optical system is shown in Fig. 1. It comprises a primary mirror (M1), a secondary mirror (M2), a tertiary mirror (M3), and a quarternary mirror (M4). The conic coefficients employed in the structure of each mirror are e21, e22, e23, e24, the obscuration ratios of mirrors 1, 2, and 3 are a1 ¼ (l2/f0 1)E(h2/h1), a2 ¼ (l3/l0 2)E(h3/h2), and a3 ¼ (l4/l0 3)E(h4/h3), respectively, and the magnifications of mirrors 2, 3, and 4 are b2 ¼ (n2l0 2/n0 2l2) ¼ (l0 2/l2), b3 ¼ (n3l0 3/n0 3l3) ¼ (l0 3/l3), and b4 ¼ (n4l0 4/n0 4l4) ¼ (l0 4/l4), respectively. The image quality for the four-mirror system is decided by choosing the structure parameters a1, a2, a3, b2, b3, and b4. Assume that the total focal length of a four-mirror telescope system is f0 ; based on the definition of magnification and obstruction ratios, the expressions for the radii of curvature of different surfaces and their corresponding thickness values are given as follows: 2f 0 2a1 f 0 2a1 a2 f 0 2a3 a2 a1 f 0 R1 ¼ ; R3 ¼ ; R4 ¼ ; R2 ¼ b2 b3 b4 b3 b4 ðb2 1Þ b4 ð1 b3 Þ b4 1 f 0 ð1 a1 Þ a1 f 0 ða2 1Þ ð1 a3 Þa2 a1 f 0 ; d2 ¼ ; d3 ¼ d1 ¼
b2 b3 b4
b3 b4
b4
Assuming that the stop of the optical system is on the first mirror, various aberrations are given as follows: (i) Spherical aberration [2,3]: n 3 3 3 3 3 SI ¼ 14 b2 b3 b4 ðe21 1Þ þ a1 b3 b4 ðb2 1Þ ½ð1 þ b2 Þ2 e22 ðb2 1Þ2 3
Corresponding author.
E-mail addresses: [email protected] (M. Zou), [email protected] (J. Chang). 0030-4026/$ - see front matter & 2009 Elsevier GmbH. All rights reserved. doi:10.1016/j.ijleo.2009.05.009
þa1 a2 b4 ½e23 ð1 b3 Þ3 þ ð1 þ b3 Þ2
ðb3 1Þ þ a1 a2 a3 ðb4 1Þ½ð1 þ b4 Þ2 þ e24 ðb4 1Þ2
M. Zou et al. / Optik 121 (2010) 1900–1903
1901
f 0 1 o0; a1 o0 ða1 a 1Þ; a2 41; b2 40; a3 41; b3 40; b4 o0
ð3Þ
f 0 1 o0; a1 o0 ða1 a 1Þ; 0oa2 o1; b2 o0; 0oa3 o1; b3 40; b4 40
ð4Þ
Fig. 1. The layout of the four-mirror optical system.
(ii) Coma:
a1 1
SII ¼
b23 b24 ðb2 1Þ½ð1 þ b2 Þ2 e22 ðb2 1Þ2
4b2
þ
a2 ða1 1Þ þ b2 ða2 1Þ
þ
a2 a3 ða1 1Þ þ a3 b2 ða2 1Þ þ b2 b3 ða3 1Þ
4b2 b3
1 2
b24 ð1 b3 Þ½e23 ð1 b3 Þ ð1 þ b3 Þ2
2.2. System with an intermediate image after the secondary mirror
4b2 b3 b4
ðb4 1Þ½ð1 þ b4 Þ2 e24 ðb4 1Þ2
To obtain an intermediate image immediately after the secondary mirror, the following conditions must be satisfied: a140, l0 2od2, then by solving these inequalities, the ranges of the various parameters for the optical system are obtained as follows:
(iii) Astigmatism: SIII ¼
ða1 1Þ2 b3 b4 2 1 b2
4a
There arise three limiting cases regarding the position of the intermediate image, and from the geometry shown below, we can obtain a relationship between a and b. Case I: When |f0 1| ¼ |d1||d2|, the intermediate image is formed at the center of the third mirror, and we obtain a relation b2(a21) ¼ 1. Case II: When |f0 1|o|d1||d2|, the intermediate image is formed between the primary mirror and the secondary mirror, and the relationship obtained is b2(a21)o1. Case III: When |f0 1|4|d1||d2|, the intermediate image is formed between the secondary mirror and the tertiary mirror, and the relationship between the two parameters is b2(a21)41.
f 0 1 o0; 0oa1 o1; a2 o0 ða2 a 1Þ; b2 40; 0oa3 o1; b3 o0; b4 40
ðb2 1Þ½ð1 þ b2 Þ2 e22 ð1 b2 Þ2
ð5Þ
2
þ
½a2 ða1 1Þ þ b2 ða2 1Þ 2
2
4a1 a2 b2 b3
b4 ð1 b3 Þ½e23 ð1 b3 Þ2 ð1 þ b3 Þ2 þ
½a2 a3 ða1 1Þ þ a3 b2 ða2 1Þ þ b2 b3 ða3 1Þ2 2
2
2
4a1 a2 a3 b2 b3 b4
ðb4 1Þ½ð1 þ b4 Þ2 e24 ðb4 1Þ2
a1 1 a ða 1Þ þ b2 ða2 1Þ b b ð1 b22 Þ 2 1 b4 ð1 b23 Þ a 1 b2 3 4 a1 a2 b2 b3
a2 a3 ða1 1Þ þ a3 b2 ða2 1Þ þ b2 b3 ða3 1Þ 2 ð1 b4 Þ a1 a2 a3 b2 b3 b4
b2 b3 b4 þ
b3 b4
a1
ðb2 1Þ þ
b4 ðb3 1Þ
a1 a2
þ
ðb4 1Þ
a1 a2 a3
(iv) Field curvature: SIV ¼ b2 b3 b4 þ
b3 b4
a1
ð1 b2 Þ þ
b4 ð1 b3 Þ
a1 a2
þ
ð1 b4 Þ
a1 a2 a3
f 0 1 o0; 0oa1 o1; a2 o0 ða2 a 1Þ; b2 40; a3 o0; b3 o0; b4 o0
ð6Þ
f 0 1 o0; 0oa1 o1; a2 o0 ða2 a 1Þ; b2 40; a3 41; b3 40; b4 o0
ð7Þ
f 0 1 o0; a1 41; a2 o0 ða2 a 1Þ; b2 40; 0oa3 o1; b3 o0; b4 40
ð8Þ
Again, there are three cases regarding the position of the intermediate image formed by the system, and the relationships between a and b could be obtained. Case I: When |l0 2| ¼ |d1|, the intermediate image is formed at the center of the first mirror, and the related relationship between two parameters is b2(1a1)/a1. Case II: When |l0 2|o|d1|, the position of intermediate image is between the first mirror and the third mirror, and the relationship b24(1a1)/a1 is obtained. Case III: When |l0 2|4|d1|, the position of the intermediate image is between the first mirror and the second mirror, and b2o(1a1)/a1 is the relationship between the two parameters. Specially, if the fourth mirror is placed at the position of the intermediate image, the relationship obtained is (1a3)b3 ¼ 1.
The four-mirror optical system can be divided into four types depending on the formation and the position of the intermediate image formed by the designed optical system [1].
2.3. System with an intermediate image after the tertiary mirror
2.1. System formaning an intermediate image after the primary mirror
To obtain an intermediate image immediately after the tertiary mirror, the following conditions must be satisfied: d1f0 140; |f0 1|4|d1| a140, l0 24d2, l0 3d240, |l0 3|o|d3|, and then by solving these inequalities, the ranges of the various parameters for the optical system are obtained as follows:
To obtain an intermediate image immediately after the primary mirror, the following conditions must be satisfied: d1f0 140; |f0 1|o|d1|. Then after solving these inequalities, the ranges of various parameters for the optical system are as follows: f 0 1 o0; a1 o0 ða1 a 1Þ; a2 41; b2 40; 0oa3 o1; b3 o0; b4 40
ð1Þ
f 0 1 o0; a1 o0 ða1 a 1Þ; a2 41; b2 40; a3 o0; b3 o0; b4 o0
ð2Þ
f 0 1 o0; a1 40 ða1 a1Þ; 0oa2 o1; b2 40; a3 o0 ða3 a 1Þ; b3 40; b4 o0
ð9Þ Also, three limiting cases regarding the position of the intermediate image formed by the optical system and the relationship between a and b are as follows:
1902
M. Zou et al. / Optik 121 (2010) 1900–1903
Table 1 First sample for the four-mirror system optical parameters.
Table 2 Second sample for the four-mirror system optical parameters.
Mirror
Radius
Thickness
Mirror
Radius
Thickness
First mirror M1 Second mirror M2
204.3720 1012.5
152.2571
First mirror M1 Second mirror M2
1053.1 6602.8
226.4
Third mirror M3
9235.5
432.1649 Third mirror M3
436.1879
1099.8
361.1636 Fourth mirror M4
766.5 Fourth mirror M4
95.1181
1414.3
58.14 MM Lens has no title.
Scale:0.43
15-Jan-09
Fig. 2. The first kind of the four-mirror system’s layout map. Fig. 4. The second kind of the four-mirror system’s layout map. Lens has no title. GEOMETRICAL MTF 15-Jan-09
DIFFRACTION LIMIT Y 0.0 FIELD (0.00°) WAVELENGTH WEIGHT X 692.0 NM 1 Y 0.7 FIELD (0.50°) 586.0 NM 1 X Y 1.0 FIELD (0.70°) 487.0 NM 1 X DEFOCUSING 0.00000
1.0 0.9
MODULATION
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1 5.0
10.0
15.0 20.0 25.0 30.0 35.0 40.0 SPATIAL FREQUENCY (CYCLES/MM)
45.0
50.0
Fig. 3. MTF curves of the designed optical system.
Case I: When |l0 3| ¼ |d2||d1|, the intermediate image is formed at the center of the first mirror, and b3b2a2 ¼ b2b2a2(1/a1)+1 is the relationship between the two parameters. Case II: When |l0 3|o|d2||d1, the position of intermediate image is between the first mirror and the third mirror, and b3b2a24b2b2a2(1/a1)+1 is the relationship between two parameters. Case III: When |l0 3|4|d2||d1, the position of intermediate image is between the first mirror and the fourth mirror, and b3b2a2ob2b2a2(1/a1)+1 is the required relation.
Fig. 5. MTF curves of the designed optical system.
Table 3 Third kind of the four-mirror system optical parameters. Mirror
Radius
Thickness
First mirror M1 Second mirror M2
136.5682 106.5036
54.6273 11.0203
Third mirror M3
55.3833
Fourth mirror M4
30.0000
30.0000
3. Design example The above four aberration equations contain ten unknowns, including e21, e22, e23, e24, a1, a2, a3, b2, b3, and b4. The last six variables are related to the structure and size of the system; the
structure can be arranged conveniently, if the purpose is just to get rid of the spherical aberration, coma, and astigmatism. If the image plane required is flat, the parameters can be obtained by field curvature equation.
M. Zou et al. / Optik 121 (2010) 1900–1903
1903
Lens has no title. GEOMETRICAL MTF 15 - Jan - 09
DIFFRACTION LIMIT Y 0.0 FIELD (0.00°) X Y 0.7 FIELD (0.50°) X Y 1.0 FIELD (0.70°) X
WAVELENGTH WEIGHT 692.0 NM 1 1 586.0 NM 487.0 NM 1
DEFOCUSING 0.00000
1.0 0.9
14.71 MM Lens has no title.
Scale:1.70
15-Jan-09
Fig. 6. The third kind of the four-mirror system’s layout map.
MODULATION
0.8 0.7 0.6 0.5 0.4 0.3 0.2 0.1
Now, let us design a coaxial system with focal length value 500 mm, aperture diameter (D) 60 mm, and value of the field of view (FOV) 1.41. Firstly, a particular structure is chosen. It is particularly advantageous if the four-mirror system has an intermediate image. This leads to the fact that the system has an excellent ability to restrain stray light when an eliminating stray light stop is on the position of the intermediate image. We choose a system with an intermediate image after primary mirror first, as structure (1) described above. We chose a1 ¼ 0.49, a3 ¼ 0.1, b2 ¼ 0.91, b3 ¼ 0.84, and the first and third mirror surfaces as aspherical surfaces. With the help of MATLAB software a2 and b4 can be calculated from the above aberration equations. The original structural parameters of the system are shown in Table 1. The system layout is shown in Fig. 2. After optimization, the value of its optical MTF can reach 0.6 at 50 Lp/mm. The optical MTF curve is shown in Fig. 3. Then, we design a coaxial system with focal length value 500 mm, aperture diameter value (D) 150 mm, and value of the field of view (FOV) 2.41. We also choose a system with an intermediate image that is formed at the center of the fourth mirror. As in structure (7), we chose a1 ¼ 0.57, a2 ¼ 1.83, b2 ¼ 1.1, b2 ¼ 1.2, and the first and third mirror surfaces as aspherical surfaces. The value of a3 and b4 can be calculated with the help of MATLAB. The original structural parameters of the system are shown in Table 2. The system layout is shown in Fig. 4. After optimization, the value of its optical MTF can reach 0.5 at 50 Lp/mm. The optical MTF curve is shown in Fig. 5. Lastly, we choose a system with an intermediate image after a tertiary mirror, as structure (9) described above. We chose a1 ¼ 0.2, a3 ¼ 0.4, a3 ¼ 2, b4 ¼ 4, and the first and third mirror surfaces as aspherical surfaces. With the help of MATLAB software b2 and b2 can be calculated from the above aberration equations. The original structural parameters of the system are shown in Table 3. The system layout is shown in Fig. 6.
10.0
20.0 30.0 SPATIAL FREQUENCY (CYCLES/MM)
40.0
50.0
Fig. 7. MTF curves of the designed optical system.
After optimization, the value of its optical MTF can reach 0.55 at 50 Lp/mm. The optical MTF curve is shown in Fig. 7.
4. Conclusion In this paper, the method to obtain the original structural parameters of a four-mirror optical system was studied. The validation of the design method and the parameter ranges of the system with different structures were tested by design examples. From the MTF curves, it is evident that the designed optical system has achieved good image qualities by using only two aspherical surfaces. It demonstrates that the four-mirror system with a simpler structure can satisfy the requirements of its use. Thus it is hoped that with the development of the remote-sensing technology, the improvements of CAM and the techniques of alignments, the four-mirror system, which has unique advantages, will be used widely in future. References [1] Chang Jun, Weng Zhi Cheng, Jiang Hui Ling, Zhang Xin, Cong Xiao Jie, Optical design of three-mirror system used in space, Acta Opt. Sin. 23 (2) (2003) 216– 219. [2] Junhua Pun, A study of the optical system with three mirrors of second order surface, Acta Opt. Sin. 8 (8) (1988) 717–721. [3] Junhua Pun, Design, Machining and Test of Optical Aspherical Surface, vol. 4, Science Publishing Company, 1994. [4] M. Lampton, M. Sholl, Comparison of on-axis three-mirror-anastigmat telescopes, Proc. SPIE 6687 (2007) 1–8. [5] Andrew Rakich, Four-mirror anastigmats, part 1: a complete solution set for all-spherical telescopic systems, Opt. Eng. 46 (10) (2007) 1–12. [6] Andrew Rakich, Four-mirror anastigmats, part 3: all-spherical systems with elements larger than the entrance pupil, Opt. Eng. 47 (3) (2008) 1–12.