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Analysis of athermalizing performance of thermal infrared optical system with Cassegrain antenna
Optik 121 (2010) 1904–1907
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Analysis of athermalizing performance of thermal infrared optical system with Cassegrain antenna Rui-Qing Wu a,, Kai Huang b, Huajun Yang b, Jiandong Wang b, Yidong Liu b a b
School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China Institute of Applied physics, University of Electronic Science and Technology of China, Chengdu 610054, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 21 January 2009 Accepted 8 May 2009
The thermal infrared imaging system is of greatest importance in modern optics and the image quality closely relies on the accuracy of the optical system. Heat-induced mirror deformation influences the transmission performance of the optical antenna system significantly, which further degrades the accuracy of the whole system. In this work, the simulation on lens deformation circumstances under the thermal environment is carried out using the finite-element analysis software ANSYS. The focus shift property caused by the thermal effect is studied in the cases of reflective and refractive Cassegrain optical antennas, and the athermalizing performance of Cassegrain optical antenna system is also concluded. This work presents a direction to accurate designing of the thermal infrared optical system. & 2009 Published by Elsevier GmbH.
Keywords: Cassegrain antenna Thermal infrared optical system (TIOS) Focus shift Athermalizing performance
1. Introduction After the sustained development in the past decades the fiber optics and the satellite communications have achieved great success in the modern society and improved the information science and technology. Nowadays the fiber optics and the satellite communications play great roles in the high-speed information transmission [1,2]. Unfortunately, there are many adverse factors degrading the information transmission performance. For example, the thermal effect is usually inevitable in the real environment. In this case the heat-induced lens deformation leads to the light spot deflection and expansion at the receiving antenna system [3,4]. For the space laser communication link the lens deformation will lead to lower performance [5]. To evaluate the influence of the thermal effect on the optical system and give proper direction, this work analyzes the athermalizing performance of TIOS with Cassegrain antenna through focus shift of the optical antenna system under the thermal environment. The focus shift property caused by the thermal effect is studied in the cases of reflective and refractive Cassegrain optical antennas and the athermalizing performance of Cassegrain optical antenna system is also concluded. This work presents a direction to accurate designing of the thermal infrared optical systems
Corresponding author.
E-mail address: [email protected] (R.-Q. Wu). 0030-4026/$ - see front matter & 2009 Published by Elsevier GmbH. doi:10.1016/j.ijleo.2009.05.012
and has wide systems [6].
applications
in
the
optical
transmission
2. Structure design of Cassegrain antenna As transmitting and receiving antennas, Cassegrain antenna has following obvious advantages: (1) its aperture can be made relatively large, with no color difference and even wider range of available band, (2) adopting aspherical mirror structure, it can provide aberration-correction function, and (3) it has the transceiver unification ability [7]. The comparatively common composite structures of Cassegrain antenna include combinations of paraboloid and paraboloid, paraboloid and hyperboloid, and hyperboloid and hyperboloid, which have advantages and disadvantages. The Cassegrain antenna designed in this paper adopts the optimum composition method that is reflective confocal paraboloid structure (the structure of primary mirror is paraboloid, and the structure of secondary mirror is hyperboloid), and the material object of Cassegrain antenna is shown in Fig. 1. According to the design requirement for optical system with aspherical lens, if the RMS value of system is less than or equal to 0.067l, the performance is comparatively well. The RMS value of antenna system is tested using Zygo interferometer, and as a result, it is 0.037l, as shown in Fig. 2. In this condition, the Cassegrain antenna system with reflective confocal paraboloid structure has a very good value of RMS. When the infrared beams
R.-Q. Wu et al. / Optik 121 (2010) 1904–1907
1905
Fig. 1. The object material of Cassegrain antenna system. Fig. 3. The finite element model of lens.
Fig. 2. The RMS value of Cassegrain antenna system.
propagate through the Cassegrain antenna, the transmitting beam divergence angle of transmitting antenna is about 1.045e10 rad, so it is beneficial for long-distance transmission of infrared beams.
Fig. 4. The displacement of lens in the x direction when the temperature increases 50 1C.
3. Athermalization of Cassegrain antenna system under thermal environment 3.1. Lens thermal distortion and the change of mirror focus shift under thermal environment Lens thermal distortion refers to the expansion (as temperature increases) or contraction (as temperature falls) of lens in the direction of x, y, z when environment temperature changes. According to the thermoelasticity theory, the object deformation Dl caused by temperature change is mainly made up by three parts [8], including free thermal expansion Dl1 caused by temperature rising of material object, material Poisson’s ratiorelated deformation Dl2 caused by the incapability to freely expand after fixed boundary, and deformation Dl3 caused by thermal stress.
Dl ¼ Dl1 þ Dl2 þ Dl3
ð1Þ
According to the finite-element method of thermal structure analysis based on the finite-element analysis software ANSYS, this paper made a thermal deformation finite-element simulation analysis on the lens with optical antenna system (germanium as the material). As shown in Figs. 3–6, Fig. 3 shows the finiteelement model of lens and Figs. 4–6 show lens displacement in
Fig. 5. The displacement of lens in the y direction when the temperature increases 50 1C.
the x direction, the y direction, and the z direction, respectively, when the temperature increases 50 1C. It can be shown that when the temperature increases 50 1C, mirror deformation is the largest in the z direction, and the maximum deformation is 0.169e5 mm. Therefore, lens deformation seriously influences the long-distance transmission performance of the antenna system. Focus shift refers to the movement of optical system focus point caused by temperature change. Optical system focus displacement contains two parts, focus displacement caused by lens thermal deformation and that caused by change in lens refractive index. The focus shift caused by lens heat
1906
R.-Q. Wu et al. / Optik 121 (2010) 1904–1907
Fig. 6. The displacement of lens in the z direction when the temperature increases 50 1C.
deformation is [9].
dr ¼ Lx L0 ;
ð2Þ
where L0 and Lx, respectively, are the focus before and after lens thermal deformation occurs. The relation of the absolute refractive index of glass with temperature is [10] " # dn n2 1 E0 þ Et DT ¼ D0 þ D1 DT þ D3 DT 2 þ 2 ; ð3Þ dt 2n l ltk where n is the relative refractive index of glass in the standard temperature and pressure conditions. DT is the temperature variation value relative to 20 1C. The additional six constants are provided by glass factories to describe glass thermal performance. For simple lens, it can be proved that when the environment temperature changes, the focus shift of lens caused by refractive index change is df ¼
f0 ; ðn 1Þðdn=dtÞDt
ð4Þ
where Dt is the temperature change, f0 the focus length before change, and n the refractive index. For the thermal infrared imaging system, assumed an f/1.5 germanium lens with 75 mm diameter, 0.00039/1C dn/dt, 112.5 mm focus length and 4 refractive index, when Dt ¼ 40 1C, lens axial deformation is 0.136e5 m, then lens thermal immersion will produce change of 0.585 mm in focus length. To take Rayleigh criterion of a quarter wavelength focus displacement (70.046 mm), as reference, the above-mentioned focus shift equals the shift of 3.1793 wavelengths. 12.7172 Rayleigh criterion is so large a number that it will influence the imaging quality of thermal infrared system. Hence, the total focus shift of lens is
Df ¼ df þ dr ¼ af ;
ð5Þ
where a and f are the thermal sensitivity and the initial focus of the lens, respectively. 3.2. Athermalization analysis of thermal infrared optical system with Cassegrain antenna Athermalizing performance means that when environment temperature changes, there is just little focus shift of optical system. It is the best method to solve the imaging quality problem of system caused by focus shift of optical system under thermal environment. When environment temperature changes, the mirrors of Cassegrain antenna deform, but due to the advantages of structure design, the total focus shift caused by mirror heat
deformation in the primary mirror and the secondary mirror is nearly zero. Both adopting reflective mirror structure, there will be no focus shift caused by the change of lens refractive index in the primary mirror and the secondary mirror as a whole. Hence, the thermal influence of Cassegrain antenna system as a whole is zero, and the athermalizing performance of thermal infrared optical system with Cassegrain antenna can be realized. Through a concrete example, a comparative analysis is given as follows. As shown in Fig. 7, there are two quite similar systems: the total refractive TIOS above the optical axis and the reflective TIOS using the Cassegrain antenna as primary elements below the optical axis. The thermal sensitivity of each element is given in Fig. 6. Both systems work at the LEIR waveband, which is within 73 1C central wavelength, and their entrance pupil diameters are 75 mm. Under 50 1C environment, the front lens shade of the refractive system above the optical axis has little optical power (approximately negative) and is used to make the image move 7.6 mm towards external focus. The first big element makes image move 1.7 mm towards internal focus and the next negative element makes image move 0.27 mm towards external focus. In the end, the two elements just produce little focus shift. Under 50 1C thermal immersion, the focus shift of the whole system is approximately the same as that of the large optical power element moving 1.71 mm towards internal focus, same as that of the first element. For the optical system below the optical axis, the front lens shade makes image move 7.6 mm towards external focus, and
Fairing Lens +0.0004 −0.071 Beams
Lens +0.0107
0 Mirror
Lenslet −0.0003
Lenslet −0.0011
−0.0007 Lenslet
−0.0017 Lenslet
Beams +0.0004
Mirror 0
Fig. 7. Thermal sensitivity of refractive system (above the optical axis) and refractive system (below the optical axis).
Table 1 The focus shift of refractive TIOS (mm). Parameters
Initial focal
Refraction system
Fairing Large lens The second lens The first lenslet The second lenslet Total system
25.4 25.4 25.2 25.3 0.272 120.5
+0.01016 1.8034 +0.27 0.0076 0.0003 1.53114
Table 2 The focus shift of reflective TIOS (mm). Parameters
Initial focal
Reflection system
Fairing The first mirror The second mirror The first lenslet The second lenslet Total system
25.4 75 200 25.7 16.5 106.5
+0.01016 0 0 0.018 0.028 0.03584
R.-Q. Wu et al. / Optik 121 (2010) 1904–1907
the total focus shift of Cassegrain antenna system is nearly zero. In the end, the two elements just produce little focus shift as well. The specific experimental data is given in Tables 1 and 2. The following can be concluded from the above analysis: two reflecting mirrors of Cassegrain antenna will not cause thermal focus shift at all; the focus shift of the TIOS with Cassegrain antenna as a whole is 0.03584 mm, which is less than Rayleigh criterion at a quarter wavelengths. Therefore, the Athermalizing performance of TIOS has been achieved. Hence, in practical work, the thermal infrared optical system with Cassegrain antenna should be chosen to improve the work efficiency of the optical system.
4. Conclusion Based on the finite-element analysis software, ANSYS, this paper makes a simulation on lens thermal deformation of optical antenna system in thermal environment, and analyzes focus shift caused by the lens heat deformation and the change of refractive index. At the same time, it comparatively analyzes the focus shift change of refractive and reflection thermal infrared optical antenna systems and comes to a conclusion of the athermalizing performance of thermal infrared optical antenna systems with Cassegrain antenna, which lays a theoretical foundation for selecting highly efficient antenna system for optical communication.
1907
References [1] Pei Li, Tigang Ning, Tangjun Li, et al., Studies on the dispersion compensation of fiber Bragg grating in high-speed optical communication system, Acta Phys. Sin. 54 (2005) 1630 (in Chinese). [2] Jiamin Gong, Juan Liu, Qiang Fang, et al., The analytical model of SRS in singlemode silica fiber in density wavelength division multiplexed optical communication system, Acta Phys. Sin. 49 (2000) 1287 (in Chinese). [3] Chuanhua Wen, Yuquan Li, The optical system in space laser communication, Wireless Opt. Commun. 27 (7) (2003) 24–27. [4] Fuan Liu, The summary of space optical communication system, Space Electron. Technol. (3) (2003) 22–28 (in Chinese). [5] Hua Zhang, Xiaofeng Li, Wenshu Yang, Research on thermal distortion of mirror used for satellite-borne laser communication and its impact on optical system, Infrared 29 (4) (2008) 29 (in Chinese). [6] Shanghong Zhao, Introduction to Satellite of the Optical Communication, Xi’an University of Electronic Science and Technology Publishing House, China, 2005, pp. 3–4. (in Chinese). [7] Young-Min Cho, Hong-Jin Kong, Sang-Soo Lee, Cassegrainian – inverse Cassegrainian four-aspherical mirror system (magnification ¼ +1) derived from the solution of all zero third-order aberrations and suitable for deepultraviolet optical lithography [J], Opt. Eng. 33 (7) (1994) 2480–2487. [8] Yufeng Peng, Zuhai Cheng, Yaoning Zhang, Theoretical analysis of the thermal deformation resonator laser model, High Power Laser Part. Beams 12 (2003) 69–72 (in Chinese). [9] Bailei Zhang, Design of laser optical communications antenna and finite element analysis of lens thermal deformation, Master’s Dissertation of University of Electronic Science and Technology, Chendu, 2006, p. 62 (in Chinese). [10] ZEMAX Development Corporation, the Manual of the ZEMAX Soft, 2003, p. 360–390 (in Chinese).
Contents lists available at ScienceDirect
Optik journal homepage: www.elsevier.de/ijleo
Analysis of athermalizing performance of thermal infrared optical system with Cassegrain antenna Rui-Qing Wu a,, Kai Huang b, Huajun Yang b, Jiandong Wang b, Yidong Liu b a b
School of Electronic Engineering, University of Electronic Science and Technology of China, Chengdu 610054, China Institute of Applied physics, University of Electronic Science and Technology of China, Chengdu 610054, China
a r t i c l e in f o
a b s t r a c t
Article history: Received 21 January 2009 Accepted 8 May 2009
The thermal infrared imaging system is of greatest importance in modern optics and the image quality closely relies on the accuracy of the optical system. Heat-induced mirror deformation influences the transmission performance of the optical antenna system significantly, which further degrades the accuracy of the whole system. In this work, the simulation on lens deformation circumstances under the thermal environment is carried out using the finite-element analysis software ANSYS. The focus shift property caused by the thermal effect is studied in the cases of reflective and refractive Cassegrain optical antennas, and the athermalizing performance of Cassegrain optical antenna system is also concluded. This work presents a direction to accurate designing of the thermal infrared optical system. & 2009 Published by Elsevier GmbH.
Keywords: Cassegrain antenna Thermal infrared optical system (TIOS) Focus shift Athermalizing performance
1. Introduction After the sustained development in the past decades the fiber optics and the satellite communications have achieved great success in the modern society and improved the information science and technology. Nowadays the fiber optics and the satellite communications play great roles in the high-speed information transmission [1,2]. Unfortunately, there are many adverse factors degrading the information transmission performance. For example, the thermal effect is usually inevitable in the real environment. In this case the heat-induced lens deformation leads to the light spot deflection and expansion at the receiving antenna system [3,4]. For the space laser communication link the lens deformation will lead to lower performance [5]. To evaluate the influence of the thermal effect on the optical system and give proper direction, this work analyzes the athermalizing performance of TIOS with Cassegrain antenna through focus shift of the optical antenna system under the thermal environment. The focus shift property caused by the thermal effect is studied in the cases of reflective and refractive Cassegrain optical antennas and the athermalizing performance of Cassegrain optical antenna system is also concluded. This work presents a direction to accurate designing of the thermal infrared optical systems
Corresponding author.
E-mail address: [email protected] (R.-Q. Wu). 0030-4026/$ - see front matter & 2009 Published by Elsevier GmbH. doi:10.1016/j.ijleo.2009.05.012
and has wide systems [6].
applications
in
the
optical
transmission
2. Structure design of Cassegrain antenna As transmitting and receiving antennas, Cassegrain antenna has following obvious advantages: (1) its aperture can be made relatively large, with no color difference and even wider range of available band, (2) adopting aspherical mirror structure, it can provide aberration-correction function, and (3) it has the transceiver unification ability [7]. The comparatively common composite structures of Cassegrain antenna include combinations of paraboloid and paraboloid, paraboloid and hyperboloid, and hyperboloid and hyperboloid, which have advantages and disadvantages. The Cassegrain antenna designed in this paper adopts the optimum composition method that is reflective confocal paraboloid structure (the structure of primary mirror is paraboloid, and the structure of secondary mirror is hyperboloid), and the material object of Cassegrain antenna is shown in Fig. 1. According to the design requirement for optical system with aspherical lens, if the RMS value of system is less than or equal to 0.067l, the performance is comparatively well. The RMS value of antenna system is tested using Zygo interferometer, and as a result, it is 0.037l, as shown in Fig. 2. In this condition, the Cassegrain antenna system with reflective confocal paraboloid structure has a very good value of RMS. When the infrared beams
R.-Q. Wu et al. / Optik 121 (2010) 1904–1907
1905
Fig. 1. The object material of Cassegrain antenna system. Fig. 3. The finite element model of lens.
Fig. 2. The RMS value of Cassegrain antenna system.
propagate through the Cassegrain antenna, the transmitting beam divergence angle of transmitting antenna is about 1.045e10 rad, so it is beneficial for long-distance transmission of infrared beams.
Fig. 4. The displacement of lens in the x direction when the temperature increases 50 1C.
3. Athermalization of Cassegrain antenna system under thermal environment 3.1. Lens thermal distortion and the change of mirror focus shift under thermal environment Lens thermal distortion refers to the expansion (as temperature increases) or contraction (as temperature falls) of lens in the direction of x, y, z when environment temperature changes. According to the thermoelasticity theory, the object deformation Dl caused by temperature change is mainly made up by three parts [8], including free thermal expansion Dl1 caused by temperature rising of material object, material Poisson’s ratiorelated deformation Dl2 caused by the incapability to freely expand after fixed boundary, and deformation Dl3 caused by thermal stress.
Dl ¼ Dl1 þ Dl2 þ Dl3
ð1Þ
According to the finite-element method of thermal structure analysis based on the finite-element analysis software ANSYS, this paper made a thermal deformation finite-element simulation analysis on the lens with optical antenna system (germanium as the material). As shown in Figs. 3–6, Fig. 3 shows the finiteelement model of lens and Figs. 4–6 show lens displacement in
Fig. 5. The displacement of lens in the y direction when the temperature increases 50 1C.
the x direction, the y direction, and the z direction, respectively, when the temperature increases 50 1C. It can be shown that when the temperature increases 50 1C, mirror deformation is the largest in the z direction, and the maximum deformation is 0.169e5 mm. Therefore, lens deformation seriously influences the long-distance transmission performance of the antenna system. Focus shift refers to the movement of optical system focus point caused by temperature change. Optical system focus displacement contains two parts, focus displacement caused by lens thermal deformation and that caused by change in lens refractive index. The focus shift caused by lens heat
1906
R.-Q. Wu et al. / Optik 121 (2010) 1904–1907
Fig. 6. The displacement of lens in the z direction when the temperature increases 50 1C.
deformation is [9].
dr ¼ Lx L0 ;
ð2Þ
where L0 and Lx, respectively, are the focus before and after lens thermal deformation occurs. The relation of the absolute refractive index of glass with temperature is [10] " # dn n2 1 E0 þ Et DT ¼ D0 þ D1 DT þ D3 DT 2 þ 2 ; ð3Þ dt 2n l ltk where n is the relative refractive index of glass in the standard temperature and pressure conditions. DT is the temperature variation value relative to 20 1C. The additional six constants are provided by glass factories to describe glass thermal performance. For simple lens, it can be proved that when the environment temperature changes, the focus shift of lens caused by refractive index change is df ¼
f0 ; ðn 1Þðdn=dtÞDt
ð4Þ
where Dt is the temperature change, f0 the focus length before change, and n the refractive index. For the thermal infrared imaging system, assumed an f/1.5 germanium lens with 75 mm diameter, 0.00039/1C dn/dt, 112.5 mm focus length and 4 refractive index, when Dt ¼ 40 1C, lens axial deformation is 0.136e5 m, then lens thermal immersion will produce change of 0.585 mm in focus length. To take Rayleigh criterion of a quarter wavelength focus displacement (70.046 mm), as reference, the above-mentioned focus shift equals the shift of 3.1793 wavelengths. 12.7172 Rayleigh criterion is so large a number that it will influence the imaging quality of thermal infrared system. Hence, the total focus shift of lens is
Df ¼ df þ dr ¼ af ;
ð5Þ
where a and f are the thermal sensitivity and the initial focus of the lens, respectively. 3.2. Athermalization analysis of thermal infrared optical system with Cassegrain antenna Athermalizing performance means that when environment temperature changes, there is just little focus shift of optical system. It is the best method to solve the imaging quality problem of system caused by focus shift of optical system under thermal environment. When environment temperature changes, the mirrors of Cassegrain antenna deform, but due to the advantages of structure design, the total focus shift caused by mirror heat
deformation in the primary mirror and the secondary mirror is nearly zero. Both adopting reflective mirror structure, there will be no focus shift caused by the change of lens refractive index in the primary mirror and the secondary mirror as a whole. Hence, the thermal influence of Cassegrain antenna system as a whole is zero, and the athermalizing performance of thermal infrared optical system with Cassegrain antenna can be realized. Through a concrete example, a comparative analysis is given as follows. As shown in Fig. 7, there are two quite similar systems: the total refractive TIOS above the optical axis and the reflective TIOS using the Cassegrain antenna as primary elements below the optical axis. The thermal sensitivity of each element is given in Fig. 6. Both systems work at the LEIR waveband, which is within 73 1C central wavelength, and their entrance pupil diameters are 75 mm. Under 50 1C environment, the front lens shade of the refractive system above the optical axis has little optical power (approximately negative) and is used to make the image move 7.6 mm towards external focus. The first big element makes image move 1.7 mm towards internal focus and the next negative element makes image move 0.27 mm towards external focus. In the end, the two elements just produce little focus shift. Under 50 1C thermal immersion, the focus shift of the whole system is approximately the same as that of the large optical power element moving 1.71 mm towards internal focus, same as that of the first element. For the optical system below the optical axis, the front lens shade makes image move 7.6 mm towards external focus, and
Fairing Lens +0.0004 −0.071 Beams
Lens +0.0107
0 Mirror
Lenslet −0.0003
Lenslet −0.0011
−0.0007 Lenslet
−0.0017 Lenslet
Beams +0.0004
Mirror 0
Fig. 7. Thermal sensitivity of refractive system (above the optical axis) and refractive system (below the optical axis).
Table 1 The focus shift of refractive TIOS (mm). Parameters
Initial focal
Refraction system
Fairing Large lens The second lens The first lenslet The second lenslet Total system
25.4 25.4 25.2 25.3 0.272 120.5
+0.01016 1.8034 +0.27 0.0076 0.0003 1.53114
Table 2 The focus shift of reflective TIOS (mm). Parameters
Initial focal
Reflection system
Fairing The first mirror The second mirror The first lenslet The second lenslet Total system
25.4 75 200 25.7 16.5 106.5
+0.01016 0 0 0.018 0.028 0.03584
R.-Q. Wu et al. / Optik 121 (2010) 1904–1907
the total focus shift of Cassegrain antenna system is nearly zero. In the end, the two elements just produce little focus shift as well. The specific experimental data is given in Tables 1 and 2. The following can be concluded from the above analysis: two reflecting mirrors of Cassegrain antenna will not cause thermal focus shift at all; the focus shift of the TIOS with Cassegrain antenna as a whole is 0.03584 mm, which is less than Rayleigh criterion at a quarter wavelengths. Therefore, the Athermalizing performance of TIOS has been achieved. Hence, in practical work, the thermal infrared optical system with Cassegrain antenna should be chosen to improve the work efficiency of the optical system.
4. Conclusion Based on the finite-element analysis software, ANSYS, this paper makes a simulation on lens thermal deformation of optical antenna system in thermal environment, and analyzes focus shift caused by the lens heat deformation and the change of refractive index. At the same time, it comparatively analyzes the focus shift change of refractive and reflection thermal infrared optical antenna systems and comes to a conclusion of the athermalizing performance of thermal infrared optical antenna systems with Cassegrain antenna, which lays a theoretical foundation for selecting highly efficient antenna system for optical communication.
1907
References [1] Pei Li, Tigang Ning, Tangjun Li, et al., Studies on the dispersion compensation of fiber Bragg grating in high-speed optical communication system, Acta Phys. Sin. 54 (2005) 1630 (in Chinese). [2] Jiamin Gong, Juan Liu, Qiang Fang, et al., The analytical model of SRS in singlemode silica fiber in density wavelength division multiplexed optical communication system, Acta Phys. Sin. 49 (2000) 1287 (in Chinese). [3] Chuanhua Wen, Yuquan Li, The optical system in space laser communication, Wireless Opt. Commun. 27 (7) (2003) 24–27. [4] Fuan Liu, The summary of space optical communication system, Space Electron. Technol. (3) (2003) 22–28 (in Chinese). [5] Hua Zhang, Xiaofeng Li, Wenshu Yang, Research on thermal distortion of mirror used for satellite-borne laser communication and its impact on optical system, Infrared 29 (4) (2008) 29 (in Chinese). [6] Shanghong Zhao, Introduction to Satellite of the Optical Communication, Xi’an University of Electronic Science and Technology Publishing House, China, 2005, pp. 3–4. (in Chinese). [7] Young-Min Cho, Hong-Jin Kong, Sang-Soo Lee, Cassegrainian – inverse Cassegrainian four-aspherical mirror system (magnification ¼ +1) derived from the solution of all zero third-order aberrations and suitable for deepultraviolet optical lithography [J], Opt. Eng. 33 (7) (1994) 2480–2487. [8] Yufeng Peng, Zuhai Cheng, Yaoning Zhang, Theoretical analysis of the thermal deformation resonator laser model, High Power Laser Part. Beams 12 (2003) 69–72 (in Chinese). [9] Bailei Zhang, Design of laser optical communications antenna and finite element analysis of lens thermal deformation, Master’s Dissertation of University of Electronic Science and Technology, Chendu, 2006, p. 62 (in Chinese). [10] ZEMAX Development Corporation, the Manual of the ZEMAX Soft, 2003, p. 360–390 (in Chinese).