Abstract

We studied the polarization evolution of a reflective telescope designed for the quantum satellite Micius. The change in polarization extinction ratio (PER) of quantum light was derived and calculated. The PER deterioration caused by increase of incidence angle was calculated to determine the boundary conditions for the system design. The performance of the Micius prototype was evaluated both theoretically and experimentally to verify the viability of our optical design. Minimum and maximum PERs of 38 and 55 dB, respectively, were recorded, which were mostly in good agreement with the numerical calculations. Our investigations have contributed to the success of Micius, which is a significant milestone for building a global security network.

© 2017 Optical Society of America

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2014 (1)

2013 (2)

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

M. Zhang, Z. He, and F. Qian, “Real time basis-deviation measurement system based on BB84 module,” Proc. SPIE 8906, 890629 (2013).
[Crossref]

2012 (1)

J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, X.-D. Cai, P. Xu, G.-S. Pan, J.-J. Jia, Y.-M. Huang, H. Yin, J.-Y. Wang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185–188 (2012).
[Crossref]

2011 (1)

M. Toyoshima, H. Takenaka, Y. Shoji, Y. Takayama, M. Takeoka, M. Fujiwara, and M. Sasaki, “Polarization-basis tracking scheme in satellite quantum key distribution,” Int. J. Opt. 2011, 1–8 (2011).
[Crossref]

2009 (2)

D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, and S. Ten, “High rate, long-distance quantum key distribution over 250  km of ultralow loss fibres,” New J. Phys. 11, 075003 (2009).
[Crossref]

M. Toyoshima, H. Takenaka, Y. Shoji, Y. Takayama, Y. Koyama, and H. Kunimori, “Polarization measurements through space-to-ground atmospheric propagation paths by using a highly polarized laser source in space,” Opt. Express 17, 22333–22340 (2009).
[Crossref]

2007 (3)

D. Rosenberg, J. W. Harrington, P. R. Rice, P. A. Hiskett, C. G. Peterson, R. J. Hughes, A. E. Lita, S. W. Nam, and J. E. Nordholt, “Long-distance decoy-state quantum key distribution in optical fiber,” Phys. Rev. Lett. 98, 010503 (2007).
[Crossref]

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, Th. Scheidl, J. Perdigues, Z. Sodnik, Ch. Kurtsiefer, J. Rarity, A. Zeilinger, and H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144  km,” Phys. Rev. Lett. 98, 010504 (2007).
[Crossref]

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Omer, M. Furst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144  km,” Nat. Phys. 3, 481–486 (2007).
[Crossref]

2006 (1)

2005 (2)

X.-B. Wang, “Decoy-state protocol for quantum cryptography with four different intensities of coherent light,” Phys. Rev. A 72, 012322 (2005).
[Crossref]

C. Z. Peng, T. Yang, X. H. Bao, J. Zhang, X. M. Jin, F. Y. Feng, B. Yang, J. Yang, J. Yin, Q. Zhang, N. Li, B. L. Tian, and J. W. Pan, “Experimental free-space distribution of entangled photon pairs over 13  km: towards satellite-based global quantum communication,” Phys. Rev. Lett. 94, 150501 (2005).
[Crossref]

2003 (2)

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Classical and quantum communication without a shared reference frame,” Phys. Rev. Lett. 91, 027901 (2003).
[Crossref]

W. Y. Hwang, “Quantum key distribution with high loss: toward global secure communication,” Phys. Rev. Lett. 91, 057901 (2003).
[Crossref]

2002 (1)

C. Kurtsiefer, P. Zarda, M. Halder, H. Weinfurter, P. M. Gorman, P. R. Tapster, and J. G. Rarity, “Quantum cryptography: a step towards global key distribution,” Nature 419, 450 (2002).
[Crossref]

2000 (1)

P. W. Shor and J. Preskill, “Simple proof of security of the BB84 quantum key distribution protocol,” Phys. Rev. Lett. 85, 441–444 (2000).
[Crossref]

1999 (1)

A. J. W. Alexander, S. Lee, and C. C. Chen, “Pointing and tracking concepts for deep-space missions,” Proc. SPIE 3615, 230–249 (1999).
[Crossref]

Alexander, A. J. W.

A. J. W. Alexander, S. Lee, and C. C. Chen, “Pointing and tracking concepts for deep-space missions,” Proc. SPIE 3615, 230–249 (1999).
[Crossref]

Aspelmeyer, M.

Bao, X. H.

C. Z. Peng, T. Yang, X. H. Bao, J. Zhang, X. M. Jin, F. Y. Feng, B. Yang, J. Yang, J. Yin, Q. Zhang, N. Li, B. L. Tian, and J. W. Pan, “Experimental free-space distribution of entangled photon pairs over 13  km: towards satellite-based global quantum communication,” Phys. Rev. Lett. 94, 150501 (2005).
[Crossref]

Barbieri, C.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Omer, M. Furst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144  km,” Nat. Phys. 3, 481–486 (2007).
[Crossref]

Bartlett, S. D.

S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Classical and quantum communication without a shared reference frame,” Phys. Rev. Lett. 91, 027901 (2003).
[Crossref]

Bennett, C. H.

C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing (1984), pp. 175–179.

Blauensteiner, B.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Omer, M. Furst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144  km,” Nat. Phys. 3, 481–486 (2007).
[Crossref]

Bonato, C.

Born, M.

M. Born and E. Wolf, Principles of Optics (Science, 1978), Chap. 1, Yang Jiasun Transl.

Brassard, G.

C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing (1984), pp. 175–179.

Cai, X.-D.

J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, X.-D. Cai, P. Xu, G.-S. Pan, J.-J. Jia, Y.-M. Huang, H. Yin, J.-Y. Wang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185–188 (2012).
[Crossref]

Cao, Y.

J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, X.-D. Cai, P. Xu, G.-S. Pan, J.-J. Jia, Y.-M. Huang, H. Yin, J.-Y. Wang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185–188 (2012).
[Crossref]

Chen, C. C.

A. J. W. Alexander, S. Lee, and C. C. Chen, “Pointing and tracking concepts for deep-space missions,” Proc. SPIE 3615, 230–249 (1999).
[Crossref]

Chen, K.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Chen, Y.-A.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, X.-D. Cai, P. Xu, G.-S. Pan, J.-J. Jia, Y.-M. Huang, H. Yin, J.-Y. Wang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185–188 (2012).
[Crossref]

Citrin, D. S.

Fan, A. Y.

R. Shintani, A. Y. Fan, and C. H. Kang, Polarized Light (Atomic Energy, 1994), Chap. 3.

Feng, F. Y.

C. Z. Peng, T. Yang, X. H. Bao, J. Zhang, X. M. Jin, F. Y. Feng, B. Yang, J. Yang, J. Yin, Q. Zhang, N. Li, B. L. Tian, and J. W. Pan, “Experimental free-space distribution of entangled photon pairs over 13  km: towards satellite-based global quantum communication,” Phys. Rev. Lett. 94, 150501 (2005).
[Crossref]

Fujiwara, M.

M. Toyoshima, H. Takenaka, Y. Shoji, Y. Takayama, M. Takeoka, M. Fujiwara, and M. Sasaki, “Polarization-basis tracking scheme in satellite quantum key distribution,” Int. J. Opt. 2011, 1–8 (2011).
[Crossref]

Furst, M.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Omer, M. Furst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144  km,” Nat. Phys. 3, 481–486 (2007).
[Crossref]

Fürst, M.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, Th. Scheidl, J. Perdigues, Z. Sodnik, Ch. Kurtsiefer, J. Rarity, A. Zeilinger, and H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144  km,” Phys. Rev. Lett. 98, 010504 (2007).
[Crossref]

Gisin, N.

D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, and S. Ten, “High rate, long-distance quantum key distribution over 250  km of ultralow loss fibres,” New J. Phys. 11, 075003 (2009).
[Crossref]

Gorman, P. M.

C. Kurtsiefer, P. Zarda, M. Halder, H. Weinfurter, P. M. Gorman, P. R. Tapster, and J. G. Rarity, “Quantum cryptography: a step towards global key distribution,” Nature 419, 450 (2002).
[Crossref]

Gray, S.

D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, and S. Ten, “High rate, long-distance quantum key distribution over 250  km of ultralow loss fibres,” New J. Phys. 11, 075003 (2009).
[Crossref]

Halder, M.

C. Kurtsiefer, P. Zarda, M. Halder, H. Weinfurter, P. M. Gorman, P. R. Tapster, and J. G. Rarity, “Quantum cryptography: a step towards global key distribution,” Nature 419, 450 (2002).
[Crossref]

Harrington, J. W.

D. Rosenberg, J. W. Harrington, P. R. Rice, P. A. Hiskett, C. G. Peterson, R. J. Hughes, A. E. Lita, S. W. Nam, and J. E. Nordholt, “Long-distance decoy-state quantum key distribution in optical fiber,” Phys. Rev. Lett. 98, 010503 (2007).
[Crossref]

He, Z.

Hiskett, P. A.

D. Rosenberg, J. W. Harrington, P. R. Rice, P. A. Hiskett, C. G. Peterson, R. J. Hughes, A. E. Lita, S. W. Nam, and J. E. Nordholt, “Long-distance decoy-state quantum key distribution in optical fiber,” Phys. Rev. Lett. 98, 010503 (2007).
[Crossref]

Hu, X.-F.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Hu, Y.-H.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Huang, Y.-M.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, X.-D. Cai, P. Xu, G.-S. Pan, J.-J. Jia, Y.-M. Huang, H. Yin, J.-Y. Wang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185–188 (2012).
[Crossref]

Hughes, R. J.

D. Rosenberg, J. W. Harrington, P. R. Rice, P. A. Hiskett, C. G. Peterson, R. J. Hughes, A. E. Lita, S. W. Nam, and J. E. Nordholt, “Long-distance decoy-state quantum key distribution in optical fiber,” Phys. Rev. Lett. 98, 010503 (2007).
[Crossref]

Hwang, W. Y.

W. Y. Hwang, “Quantum key distribution with high loss: toward global secure communication,” Phys. Rev. Lett. 91, 057901 (2003).
[Crossref]

Jennewein, T.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Omer, M. Furst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144  km,” Nat. Phys. 3, 481–486 (2007).
[Crossref]

C. Bonato, M. Aspelmeyer, T. Jennewein, C. Pernechele, P. Villoresi, and A. Zeilinger, “Influence of satellite motion on polarization qubits in a Space–Earth quantum communication link,” Opt. Express 14, 10050–10059 (2006).
[Crossref]

Jia, J.

Jia, J.-J.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, X.-D. Cai, P. Xu, G.-S. Pan, J.-J. Jia, Y.-M. Huang, H. Yin, J.-Y. Wang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185–188 (2012).
[Crossref]

Jiang, H.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Jiang, Y.

J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, X.-D. Cai, P. Xu, G.-S. Pan, J.-J. Jia, Y.-M. Huang, H. Yin, J.-Y. Wang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185–188 (2012).
[Crossref]

Jin, X. M.

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P. W. Shor and J. Preskill, “Simple proof of security of the BB84 quantum key distribution protocol,” Phys. Rev. Lett. 85, 441–444 (2000).
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R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Omer, M. Furst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144  km,” Nat. Phys. 3, 481–486 (2007).
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D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, and S. Ten, “High rate, long-distance quantum key distribution over 250  km of ultralow loss fibres,” New J. Phys. 11, 075003 (2009).
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D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, and S. Ten, “High rate, long-distance quantum key distribution over 250  km of ultralow loss fibres,” New J. Phys. 11, 075003 (2009).
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T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, Th. Scheidl, J. Perdigues, Z. Sodnik, Ch. Kurtsiefer, J. Rarity, A. Zeilinger, and H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144  km,” Phys. Rev. Lett. 98, 010504 (2007).
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M. Toyoshima, H. Takenaka, Y. Shoji, Y. Takayama, M. Takeoka, M. Fujiwara, and M. Sasaki, “Polarization-basis tracking scheme in satellite quantum key distribution,” Int. J. Opt. 2011, 1–8 (2011).
[Crossref]

M. Toyoshima, H. Takenaka, Y. Shoji, Y. Takayama, Y. Koyama, and H. Kunimori, “Polarization measurements through space-to-ground atmospheric propagation paths by using a highly polarized laser source in space,” Opt. Express 17, 22333–22340 (2009).
[Crossref]

Trojek, P.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Omer, M. Furst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144  km,” Nat. Phys. 3, 481–486 (2007).
[Crossref]

Ursin, R.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Omer, M. Furst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144  km,” Nat. Phys. 3, 481–486 (2007).
[Crossref]

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, Th. Scheidl, J. Perdigues, Z. Sodnik, Ch. Kurtsiefer, J. Rarity, A. Zeilinger, and H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144  km,” Phys. Rev. Lett. 98, 010504 (2007).
[Crossref]

Vannel, F.

D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, and S. Ten, “High rate, long-distance quantum key distribution over 250  km of ultralow loss fibres,” New J. Phys. 11, 075003 (2009).
[Crossref]

Villoresi, P.

Walenta, N.

D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, and S. Ten, “High rate, long-distance quantum key distribution over 250  km of ultralow loss fibres,” New J. Phys. 11, 075003 (2009).
[Crossref]

Wan, X.

Wang, J.

Wang, J.-Y.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, X.-D. Cai, P. Xu, G.-S. Pan, J.-J. Jia, Y.-M. Huang, H. Yin, J.-Y. Wang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185–188 (2012).
[Crossref]

Wang, X.-B.

X.-B. Wang, “Decoy-state protocol for quantum cryptography with four different intensities of coherent light,” Phys. Rev. A 72, 012322 (2005).
[Crossref]

Weier, H.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, Th. Scheidl, J. Perdigues, Z. Sodnik, Ch. Kurtsiefer, J. Rarity, A. Zeilinger, and H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144  km,” Phys. Rev. Lett. 98, 010504 (2007).
[Crossref]

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Omer, M. Furst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144  km,” Nat. Phys. 3, 481–486 (2007).
[Crossref]

Weinfurter, H.

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, Th. Scheidl, J. Perdigues, Z. Sodnik, Ch. Kurtsiefer, J. Rarity, A. Zeilinger, and H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144  km,” Phys. Rev. Lett. 98, 010504 (2007).
[Crossref]

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Omer, M. Furst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144  km,” Nat. Phys. 3, 481–486 (2007).
[Crossref]

C. Kurtsiefer, P. Zarda, M. Halder, H. Weinfurter, P. M. Gorman, P. R. Tapster, and J. G. Rarity, “Quantum cryptography: a step towards global key distribution,” Nature 419, 450 (2002).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Science, 1978), Chap. 1, Yang Jiasun Transl.

Wu, J.

Wu, J.-C.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Wu, Y.-P.

J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, X.-D. Cai, P. Xu, G.-S. Pan, J.-J. Jia, Y.-M. Huang, H. Yin, J.-Y. Wang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185–188 (2012).
[Crossref]

Xu, P.

J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, X.-D. Cai, P. Xu, G.-S. Pan, J.-J. Jia, Y.-M. Huang, H. Yin, J.-Y. Wang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185–188 (2012).
[Crossref]

Yang, B.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

C. Z. Peng, T. Yang, X. H. Bao, J. Zhang, X. M. Jin, F. Y. Feng, B. Yang, J. Yang, J. Yin, Q. Zhang, N. Li, B. L. Tian, and J. W. Pan, “Experimental free-space distribution of entangled photon pairs over 13  km: towards satellite-based global quantum communication,” Phys. Rev. Lett. 94, 150501 (2005).
[Crossref]

Yang, J.

C. Z. Peng, T. Yang, X. H. Bao, J. Zhang, X. M. Jin, F. Y. Feng, B. Yang, J. Yang, J. Yin, Q. Zhang, N. Li, B. L. Tian, and J. W. Pan, “Experimental free-space distribution of entangled photon pairs over 13  km: towards satellite-based global quantum communication,” Phys. Rev. Lett. 94, 150501 (2005).
[Crossref]

Yang, S.

Yang, S.-J.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Yang, T.

C. Z. Peng, T. Yang, X. H. Bao, J. Zhang, X. M. Jin, F. Y. Feng, B. Yang, J. Yang, J. Yin, Q. Zhang, N. Li, B. L. Tian, and J. W. Pan, “Experimental free-space distribution of entangled photon pairs over 13  km: towards satellite-based global quantum communication,” Phys. Rev. Lett. 94, 150501 (2005).
[Crossref]

Yin, H.

J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, X.-D. Cai, P. Xu, G.-S. Pan, J.-J. Jia, Y.-M. Huang, H. Yin, J.-Y. Wang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185–188 (2012).
[Crossref]

Yin, J.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, X.-D. Cai, P. Xu, G.-S. Pan, J.-J. Jia, Y.-M. Huang, H. Yin, J.-Y. Wang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185–188 (2012).
[Crossref]

C. Z. Peng, T. Yang, X. H. Bao, J. Zhang, X. M. Jin, F. Y. Feng, B. Yang, J. Yang, J. Yin, Q. Zhang, N. Li, B. L. Tian, and J. W. Pan, “Experimental free-space distribution of entangled photon pairs over 13  km: towards satellite-based global quantum communication,” Phys. Rev. Lett. 94, 150501 (2005).
[Crossref]

Yong, H.-L.

J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, X.-D. Cai, P. Xu, G.-S. Pan, J.-J. Jia, Y.-M. Huang, H. Yin, J.-Y. Wang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185–188 (2012).
[Crossref]

Zarda, P.

C. Kurtsiefer, P. Zarda, M. Halder, H. Weinfurter, P. M. Gorman, P. R. Tapster, and J. G. Rarity, “Quantum cryptography: a step towards global key distribution,” Nature 419, 450 (2002).
[Crossref]

Zbinden, H.

D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, and S. Ten, “High rate, long-distance quantum key distribution over 250  km of ultralow loss fibres,” New J. Phys. 11, 075003 (2009).
[Crossref]

Zeilinger, A.

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Omer, M. Furst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144  km,” Nat. Phys. 3, 481–486 (2007).
[Crossref]

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, Th. Scheidl, J. Perdigues, Z. Sodnik, Ch. Kurtsiefer, J. Rarity, A. Zeilinger, and H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144  km,” Phys. Rev. Lett. 98, 010504 (2007).
[Crossref]

C. Bonato, M. Aspelmeyer, T. Jennewein, C. Pernechele, P. Villoresi, and A. Zeilinger, “Influence of satellite motion on polarization qubits in a Space–Earth quantum communication link,” Opt. Express 14, 10050–10059 (2006).
[Crossref]

Zhang, J.

C. Z. Peng, T. Yang, X. H. Bao, J. Zhang, X. M. Jin, F. Y. Feng, B. Yang, J. Yang, J. Yin, Q. Zhang, N. Li, B. L. Tian, and J. W. Pan, “Experimental free-space distribution of entangled photon pairs over 13  km: towards satellite-based global quantum communication,” Phys. Rev. Lett. 94, 150501 (2005).
[Crossref]

Zhang, L.

M. Zhang, L. Zhang, J. Wu, S. Yang, X. Wan, Z. He, J. Jia, D. S. Citrin, and J. Wang, “Detection and compensation of basis deviation in satellite-to-ground quantum communications,” Opt. Express 22, 9871–9886 (2014).
[Crossref]

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Zhang, M.

Zhang, Q.

C. Z. Peng, T. Yang, X. H. Bao, J. Zhang, X. M. Jin, F. Y. Feng, B. Yang, J. Yang, J. Yin, Q. Zhang, N. Li, B. L. Tian, and J. W. Pan, “Experimental free-space distribution of entangled photon pairs over 13  km: towards satellite-based global quantum communication,” Phys. Rev. Lett. 94, 150501 (2005).
[Crossref]

Zhong, B.

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Zhou, F.

J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, X.-D. Cai, P. Xu, G.-S. Pan, J.-J. Jia, Y.-M. Huang, H. Yin, J.-Y. Wang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185–188 (2012).
[Crossref]

Int. J. Opt. (1)

M. Toyoshima, H. Takenaka, Y. Shoji, Y. Takayama, M. Takeoka, M. Fujiwara, and M. Sasaki, “Polarization-basis tracking scheme in satellite quantum key distribution,” Int. J. Opt. 2011, 1–8 (2011).
[Crossref]

Nat. Photonics (1)

J.-Y. Wang, B. Yang, S.-K. Liao, L. Zhang, Q. Shen, X.-F. Hu, J.-C. Wu, S.-J. Yang, H. Jiang, Y.-L. Tang, B. Zhong, H. Liang, W.-Y. Liu, Y.-H. Hu, Y.-M. Huang, B. Qi, J.-G. Ren, G.-S. Pan, J. Yin, J.-J. Jia, Y.-A. Chen, K. Chen, C.-Z. Peng, and J.-W. Pan, “Direct and full-scale experimental verifications towards ground-satellite quantum key distribution,” Nat. Photonics 7, 387–393 (2013).
[Crossref]

Nat. Phys. (1)

R. Ursin, F. Tiefenbacher, T. Schmitt-Manderbach, H. Weier, T. Scheidl, M. Lindenthal, B. Blauensteiner, T. Jennewein, J. Perdigues, P. Trojek, B. Omer, M. Furst, M. Meyenburg, J. Rarity, Z. Sodnik, C. Barbieri, H. Weinfurter, and A. Zeilinger, “Entanglement-based quantum communication over 144  km,” Nat. Phys. 3, 481–486 (2007).
[Crossref]

Nature (2)

C. Kurtsiefer, P. Zarda, M. Halder, H. Weinfurter, P. M. Gorman, P. R. Tapster, and J. G. Rarity, “Quantum cryptography: a step towards global key distribution,” Nature 419, 450 (2002).
[Crossref]

J. Yin, J.-G. Ren, H. Lu, Y. Cao, H.-L. Yong, Y.-P. Wu, C. Liu, S.-K. Liao, F. Zhou, Y. Jiang, X.-D. Cai, P. Xu, G.-S. Pan, J.-J. Jia, Y.-M. Huang, H. Yin, J.-Y. Wang, Y.-A. Chen, C.-Z. Peng, and J.-W. Pan, “Quantum teleportation and entanglement distribution over 100-kilometre free-space channels,” Nature 488, 185–188 (2012).
[Crossref]

New J. Phys. (1)

D. Stucki, N. Walenta, F. Vannel, R. T. Thew, N. Gisin, H. Zbinden, S. Gray, C. R. Towery, and S. Ten, “High rate, long-distance quantum key distribution over 250  km of ultralow loss fibres,” New J. Phys. 11, 075003 (2009).
[Crossref]

Opt. Express (3)

Phys. Rev. A (1)

X.-B. Wang, “Decoy-state protocol for quantum cryptography with four different intensities of coherent light,” Phys. Rev. A 72, 012322 (2005).
[Crossref]

Phys. Rev. Lett. (6)

P. W. Shor and J. Preskill, “Simple proof of security of the BB84 quantum key distribution protocol,” Phys. Rev. Lett. 85, 441–444 (2000).
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S. D. Bartlett, T. Rudolph, and R. W. Spekkens, “Classical and quantum communication without a shared reference frame,” Phys. Rev. Lett. 91, 027901 (2003).
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D. Rosenberg, J. W. Harrington, P. R. Rice, P. A. Hiskett, C. G. Peterson, R. J. Hughes, A. E. Lita, S. W. Nam, and J. E. Nordholt, “Long-distance decoy-state quantum key distribution in optical fiber,” Phys. Rev. Lett. 98, 010503 (2007).
[Crossref]

C. Z. Peng, T. Yang, X. H. Bao, J. Zhang, X. M. Jin, F. Y. Feng, B. Yang, J. Yang, J. Yin, Q. Zhang, N. Li, B. L. Tian, and J. W. Pan, “Experimental free-space distribution of entangled photon pairs over 13  km: towards satellite-based global quantum communication,” Phys. Rev. Lett. 94, 150501 (2005).
[Crossref]

T. Schmitt-Manderbach, H. Weier, M. Fürst, R. Ursin, F. Tiefenbacher, Th. Scheidl, J. Perdigues, Z. Sodnik, Ch. Kurtsiefer, J. Rarity, A. Zeilinger, and H. Weinfurter, “Experimental demonstration of free-space decoy-state quantum key distribution over 144  km,” Phys. Rev. Lett. 98, 010504 (2007).
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W. Y. Hwang, “Quantum key distribution with high loss: toward global secure communication,” Phys. Rev. Lett. 91, 057901 (2003).
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Proc. SPIE (2)

A. J. W. Alexander, S. Lee, and C. C. Chen, “Pointing and tracking concepts for deep-space missions,” Proc. SPIE 3615, 230–249 (1999).
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M. Zhang, Z. He, and F. Qian, “Real time basis-deviation measurement system based on BB84 module,” Proc. SPIE 8906, 890629 (2013).
[Crossref]

Other (3)

R. Shintani, A. Y. Fan, and C. H. Kang, Polarized Light (Atomic Energy, 1994), Chap. 3.

M. Born and E. Wolf, Principles of Optics (Science, 1978), Chap. 1, Yang Jiasun Transl.

C. H. Bennett and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in Proceedings of IEEE International Conference on Computers, Systems and Signal Processing (1984), pp. 175–179.

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Figures (9)

Fig. 1.
Fig. 1.

Elliptically polarized light with a basis deviation of θ. x/y and x/y are the coordinates of the test system and elliptically polarized light, respectively; θ is the basis deviation angle; α is the amplitude ratio angle in the coordinates of the test system; β is the elliptic angle; bx and by are the amplitudes of the x and y components, respectively; and αx and αy are amplitudes of the x and y components, respectively.

Fig. 2.
Fig. 2.

Model of reflection from the coated surface of a telescope.

Fig. 3.
Fig. 3.

Reflection characteristics of 45°-polarized light with different coating materials. Relationship between angle of incidence and (a) reflectivity and (b) phase delay.

Fig. 4.
Fig. 4.

Optical system of a telescope.

Fig. 5.
Fig. 5.

Comparison between the Micius telescope and a traditional on-axis telescope. (a) Clear apertures of the Micius telescope and the traditional on-axis telescope. Dotted line, Micius telescope; solid line, traditional on-axis telescope. (b) Relationship of angles of incidence on the primary and secondary mirrors. Blue curve: boundary condition for the two angles of incidence to maintain a PER of over 3000:1; red line: designed result for Micius.

Fig. 6.
Fig. 6.

Calculated PER maps within the clear aperture of a traditional on-axis telescope for (a) horizontally polarized light and (b) 45°-polarized light.

Fig. 7.
Fig. 7.

Laboratory experimental system. (a) A laterally movable light source, which is composed of a collimated laser and a polarizer. (b) Tested telescope for Micius. (c) PER analysis system, which is composed of a polarizer and a power meter. (d) Optical monitoring system, which is composed of a parallel light pipe and a beam analyzer.

Fig. 8.
Fig. 8.

Test position array on the primary mirror.

Fig. 9.
Fig. 9.

PER maps of the telescope designed for Micius. Measured results with (a) 45° and (b) 0° polarized light. Simulated results with (c) 45° and (d) 0° polarized light.

Tables (2)

Tables Icon

Table 1. Comparisons of Reflectivity and Maximum Viable Angle of Incidence with Different Coating Materials

Tables Icon

Table 2. ZEMAX Design Results of Telescope

Equations (12)

Equations on this page are rendered with MathJax. Learn more.

E=(axayeiδ),
{tan2θ=tan2αcosδsin2β=sin2αsinδ.
PER=(bxby)2=1tan2β.
PER=(axay)2=1tan2α=PERcos2θ+sin2θcos2θ+PERsin2θ.
QBER=1PER+1=1PER+1+PER1PER+1sin2θ.
{rs=ts1rs2ts3=sin(θθ˜)sin(θ+θ˜)sin2θsin2θisin2(θ+θi)rp=tp1rp2tp3=sin2θsin2θisin2(θ+θi)cos2(θθi)tan(θθ˜)tan(θ+θ˜),
(ExrsEyrp)=(|rs|Ex|rp|Eyeδ),
M1:z=x2+y2R1+R12(1+k1)(x2+y2),M2:zd1=x2+y2R2+R22(1+k2)(x2+y2).
θ1=cos1(kd1kn1|kd1||kn1|).
θ2=cos1(kd2kn2|kd2||kn2|).
J(Px,Py)=Tθ(rs200rp2)(rs100rp1)Tθ.
E45°(Px,Py)=J(Px,Py)(11).

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