Abstract

In a Space quantum-cryptography experiment a satellite pointing system is needed to send single photons emitted by the source on the satellite to the polarization analysis apparatus on Earth. In this paper a simulation is presented regarding how the satellite pointing systems affect the polarization state of the single photons, to help designing a proper compensation system.

© 2006 Optical Society of America

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References

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  1. W. T. Buttler, R. J. Hughes, P. G. Kwiat, S. K. Lamoreaux, C. G. Peterson, and C. M. Simmons, Practical free-space quantum key distribution over 1 km," Phys. Rev. Lett. 81, 3283-3286 (1998).
    [CrossRef]
  2. R. J. Hughes, J. E. Nordholt, D. Derkacs, J. C. Peterson, "Practical free-space quantum key distribution over 10 km in daylight and at night," New J. Phys. 4, 43.1-43.14 (2002).
    [CrossRef]
  3. C. Kurtsiefer, P. Zarda, M. Holder, H. Weinfurter, P. Gormar, P. R. Tapster, and J. G. Rarity, "A step toward global quantum key distribution," Nature 419, 450 (2002).
    [CrossRef] [PubMed]
  4. M. Aspelmeyer,  et al., "Long distance free-space distribution of quantum entanglement," Science 301, 621 (2003).
    [CrossRef] [PubMed]
  5. K. J. Resch,  et al., "Distributing entanglement and single photons through an intra-city, free-space quantum channel," Opt. Express 13, 202-209 (2005).
    [CrossRef] [PubMed]
  6. C.-Z. Peng, T. Yang, X.-H. Bao, J.-Zhang, X.-M. Jin, F.-J. 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] [PubMed]
  7. R. Ursin,  et al., "Free-space distribution of entanglement and single photons over 144 Km," quant-ph/0607182.
  8. J. E. Nordholt, R. J. Hughes, J. R. Morgan, C. G. Peterson, and C. C. Wipf, "Present and future quantum key distribution," in Free-Space Laser Communication Technologies XIV, G. Stephen Mecherle, ed. Proc. SPIE 4635, 116-126 (2002).
    [CrossRef]
  9. M. Aspelmeyer, T. Jennewein, M. Pfennigbauer, W. R. Leeb, A. Zeilinger, "Long distance quantum communication with entangled photons using satellites," IEEE J. Sel. Top. Quantum Electron. 9, 1541 (2003).
    [CrossRef]
  10. P. Villoresi, F. Tamburini, M. Aspelmeyer, T. Jennewein, R. Ursin, C. Pernechele, G. Bianco, A. Zeilinger, and C. Barbieri, "Space-to-ground quantum-communication using an optical ground station: a feasibility study," Proc. SPIE: Quantum Communications and Quantum Imaging, II conference in Denver (2004).
  11. J. G. Rarity, P. R. Tapster, P. M. Gorman, and P. Knight, "Ground to satellite secure key exchange using quantum cryptography," New J. Phys. 4, 82.1-82.21 (2002).
    [CrossRef]
  12. M. Er-Iong, H. Zheng-fu, G. Shun-sheng, Z. Tao, D. Da-sheng, and G. Guang-can, "Background noise of satellite-to-ground quantum key distribution," New J. Phys. 7, 215 (2005).
    [CrossRef]
  13. M. Pfennigbauer, M. Aspelmeyer, W. R. Leeb, G. Baister, T. Dreischer, T. Jennewein, G. Neckamm, J. M. Perdigues, H. Weinfurter, and A. Zeilinger, "Satellite-based quantum communication terminal employing stateof-the-art technology," JON 4, 549-560, (2005).
  14. W. Tittel and G. Weihs, "Photonic entanglement for fundamental tests and quantum communications," Quantum Inf. Comput. 1, 3-56 (2001).
  15. A. Sehat,  et al., "Quantum polarization properties of two-mode energy eigenstates," PRA 71, 033818 (2004).
  16. E. D. Palik (ed.), Handbook of optical constants of solids, (San Diego, Academic Press, 1998).
  17. M. Born and E. Wolf, Principles of Optics, sixth ed., (Pergamon Press, Oxford, England, 1993).
  18. D. H. Hoehn, "Depolarization of a laser beam at 6328 A due to atmospheric transmission," Appl. Opt. 8, 367 (1968).
    [CrossRef]
  19. S. Jorna, "Atmospheric depolarization and stimulated Brillouin scattering," Appl. Opt. 10, 2661 (1971).
    [CrossRef] [PubMed]
  20. W. E. Forsythe, Smithsonian Physical Tables, (9th Revised Edition, Knovel).

2005 (4)

C.-Z. Peng, T. Yang, X.-H. Bao, J.-Zhang, X.-M. Jin, F.-J. 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] [PubMed]

M. Er-Iong, H. Zheng-fu, G. Shun-sheng, Z. Tao, D. Da-sheng, and G. Guang-can, "Background noise of satellite-to-ground quantum key distribution," New J. Phys. 7, 215 (2005).
[CrossRef]

M. Pfennigbauer, M. Aspelmeyer, W. R. Leeb, G. Baister, T. Dreischer, T. Jennewein, G. Neckamm, J. M. Perdigues, H. Weinfurter, and A. Zeilinger, "Satellite-based quantum communication terminal employing stateof-the-art technology," JON 4, 549-560, (2005).

K. J. Resch,  et al., "Distributing entanglement and single photons through an intra-city, free-space quantum channel," Opt. Express 13, 202-209 (2005).
[CrossRef] [PubMed]

2004 (1)

A. Sehat,  et al., "Quantum polarization properties of two-mode energy eigenstates," PRA 71, 033818 (2004).

2003 (2)

M. Aspelmeyer,  et al., "Long distance free-space distribution of quantum entanglement," Science 301, 621 (2003).
[CrossRef] [PubMed]

M. Aspelmeyer, T. Jennewein, M. Pfennigbauer, W. R. Leeb, A. Zeilinger, "Long distance quantum communication with entangled photons using satellites," IEEE J. Sel. Top. Quantum Electron. 9, 1541 (2003).
[CrossRef]

2002 (1)

C. Kurtsiefer, P. Zarda, M. Holder, H. Weinfurter, P. Gormar, P. R. Tapster, and J. G. Rarity, "A step toward global quantum key distribution," Nature 419, 450 (2002).
[CrossRef] [PubMed]

2001 (1)

W. Tittel and G. Weihs, "Photonic entanglement for fundamental tests and quantum communications," Quantum Inf. Comput. 1, 3-56 (2001).

1998 (1)

W. T. Buttler, R. J. Hughes, P. G. Kwiat, S. K. Lamoreaux, C. G. Peterson, and C. M. Simmons, Practical free-space quantum key distribution over 1 km," Phys. Rev. Lett. 81, 3283-3286 (1998).
[CrossRef]

1971 (1)

1968 (1)

Aspelmeyer, M.

M. Pfennigbauer, M. Aspelmeyer, W. R. Leeb, G. Baister, T. Dreischer, T. Jennewein, G. Neckamm, J. M. Perdigues, H. Weinfurter, and A. Zeilinger, "Satellite-based quantum communication terminal employing stateof-the-art technology," JON 4, 549-560, (2005).

M. Aspelmeyer, T. Jennewein, M. Pfennigbauer, W. R. Leeb, A. Zeilinger, "Long distance quantum communication with entangled photons using satellites," IEEE J. Sel. Top. Quantum Electron. 9, 1541 (2003).
[CrossRef]

M. Aspelmeyer,  et al., "Long distance free-space distribution of quantum entanglement," Science 301, 621 (2003).
[CrossRef] [PubMed]

Baister, G.

M. Pfennigbauer, M. Aspelmeyer, W. R. Leeb, G. Baister, T. Dreischer, T. Jennewein, G. Neckamm, J. M. Perdigues, H. Weinfurter, and A. Zeilinger, "Satellite-based quantum communication terminal employing stateof-the-art technology," JON 4, 549-560, (2005).

Bao, X.-H.

C.-Z. Peng, T. Yang, X.-H. Bao, J.-Zhang, X.-M. Jin, F.-J. 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] [PubMed]

Buttler, W. T.

W. T. Buttler, R. J. Hughes, P. G. Kwiat, S. K. Lamoreaux, C. G. Peterson, and C. M. Simmons, Practical free-space quantum key distribution over 1 km," Phys. Rev. Lett. 81, 3283-3286 (1998).
[CrossRef]

Da-sheng, D.

M. Er-Iong, H. Zheng-fu, G. Shun-sheng, Z. Tao, D. Da-sheng, and G. Guang-can, "Background noise of satellite-to-ground quantum key distribution," New J. Phys. 7, 215 (2005).
[CrossRef]

Dreischer, T.

M. Pfennigbauer, M. Aspelmeyer, W. R. Leeb, G. Baister, T. Dreischer, T. Jennewein, G. Neckamm, J. M. Perdigues, H. Weinfurter, and A. Zeilinger, "Satellite-based quantum communication terminal employing stateof-the-art technology," JON 4, 549-560, (2005).

Er-Iong, M.

M. Er-Iong, H. Zheng-fu, G. Shun-sheng, Z. Tao, D. Da-sheng, and G. Guang-can, "Background noise of satellite-to-ground quantum key distribution," New J. Phys. 7, 215 (2005).
[CrossRef]

Gormar, P.

C. Kurtsiefer, P. Zarda, M. Holder, H. Weinfurter, P. Gormar, P. R. Tapster, and J. G. Rarity, "A step toward global quantum key distribution," Nature 419, 450 (2002).
[CrossRef] [PubMed]

Guang-can, G.

M. Er-Iong, H. Zheng-fu, G. Shun-sheng, Z. Tao, D. Da-sheng, and G. Guang-can, "Background noise of satellite-to-ground quantum key distribution," New J. Phys. 7, 215 (2005).
[CrossRef]

Hoehn, D. H.

Holder, M.

C. Kurtsiefer, P. Zarda, M. Holder, H. Weinfurter, P. Gormar, P. R. Tapster, and J. G. Rarity, "A step toward global quantum key distribution," Nature 419, 450 (2002).
[CrossRef] [PubMed]

Hughes, R. J.

W. T. Buttler, R. J. Hughes, P. G. Kwiat, S. K. Lamoreaux, C. G. Peterson, and C. M. Simmons, Practical free-space quantum key distribution over 1 km," Phys. Rev. Lett. 81, 3283-3286 (1998).
[CrossRef]

Jennewein, T.

M. Pfennigbauer, M. Aspelmeyer, W. R. Leeb, G. Baister, T. Dreischer, T. Jennewein, G. Neckamm, J. M. Perdigues, H. Weinfurter, and A. Zeilinger, "Satellite-based quantum communication terminal employing stateof-the-art technology," JON 4, 549-560, (2005).

M. Aspelmeyer, T. Jennewein, M. Pfennigbauer, W. R. Leeb, A. Zeilinger, "Long distance quantum communication with entangled photons using satellites," IEEE J. Sel. Top. Quantum Electron. 9, 1541 (2003).
[CrossRef]

Jorna, S.

Kurtsiefer, C.

C. Kurtsiefer, P. Zarda, M. Holder, H. Weinfurter, P. Gormar, P. R. Tapster, and J. G. Rarity, "A step toward global quantum key distribution," Nature 419, 450 (2002).
[CrossRef] [PubMed]

Kwiat, P. G.

W. T. Buttler, R. J. Hughes, P. G. Kwiat, S. K. Lamoreaux, C. G. Peterson, and C. M. Simmons, Practical free-space quantum key distribution over 1 km," Phys. Rev. Lett. 81, 3283-3286 (1998).
[CrossRef]

Lamoreaux, S. K.

W. T. Buttler, R. J. Hughes, P. G. Kwiat, S. K. Lamoreaux, C. G. Peterson, and C. M. Simmons, Practical free-space quantum key distribution over 1 km," Phys. Rev. Lett. 81, 3283-3286 (1998).
[CrossRef]

Leeb, W. R.

M. Pfennigbauer, M. Aspelmeyer, W. R. Leeb, G. Baister, T. Dreischer, T. Jennewein, G. Neckamm, J. M. Perdigues, H. Weinfurter, and A. Zeilinger, "Satellite-based quantum communication terminal employing stateof-the-art technology," JON 4, 549-560, (2005).

M. Aspelmeyer, T. Jennewein, M. Pfennigbauer, W. R. Leeb, A. Zeilinger, "Long distance quantum communication with entangled photons using satellites," IEEE J. Sel. Top. Quantum Electron. 9, 1541 (2003).
[CrossRef]

Neckamm, G.

M. Pfennigbauer, M. Aspelmeyer, W. R. Leeb, G. Baister, T. Dreischer, T. Jennewein, G. Neckamm, J. M. Perdigues, H. Weinfurter, and A. Zeilinger, "Satellite-based quantum communication terminal employing stateof-the-art technology," JON 4, 549-560, (2005).

Peng, C.-Z.

C.-Z. Peng, T. Yang, X.-H. Bao, J.-Zhang, X.-M. Jin, F.-J. 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] [PubMed]

Perdigues, J. M.

M. Pfennigbauer, M. Aspelmeyer, W. R. Leeb, G. Baister, T. Dreischer, T. Jennewein, G. Neckamm, J. M. Perdigues, H. Weinfurter, and A. Zeilinger, "Satellite-based quantum communication terminal employing stateof-the-art technology," JON 4, 549-560, (2005).

Peterson, C. G.

W. T. Buttler, R. J. Hughes, P. G. Kwiat, S. K. Lamoreaux, C. G. Peterson, and C. M. Simmons, Practical free-space quantum key distribution over 1 km," Phys. Rev. Lett. 81, 3283-3286 (1998).
[CrossRef]

Pfennigbauer, M.

M. Pfennigbauer, M. Aspelmeyer, W. R. Leeb, G. Baister, T. Dreischer, T. Jennewein, G. Neckamm, J. M. Perdigues, H. Weinfurter, and A. Zeilinger, "Satellite-based quantum communication terminal employing stateof-the-art technology," JON 4, 549-560, (2005).

M. Aspelmeyer, T. Jennewein, M. Pfennigbauer, W. R. Leeb, A. Zeilinger, "Long distance quantum communication with entangled photons using satellites," IEEE J. Sel. Top. Quantum Electron. 9, 1541 (2003).
[CrossRef]

Rarity, J. G.

C. Kurtsiefer, P. Zarda, M. Holder, H. Weinfurter, P. Gormar, P. R. Tapster, and J. G. Rarity, "A step toward global quantum key distribution," Nature 419, 450 (2002).
[CrossRef] [PubMed]

Resch, K. J.

Sehat, A.

A. Sehat,  et al., "Quantum polarization properties of two-mode energy eigenstates," PRA 71, 033818 (2004).

Shun-sheng, G.

M. Er-Iong, H. Zheng-fu, G. Shun-sheng, Z. Tao, D. Da-sheng, and G. Guang-can, "Background noise of satellite-to-ground quantum key distribution," New J. Phys. 7, 215 (2005).
[CrossRef]

Simmons, C. M.

W. T. Buttler, R. J. Hughes, P. G. Kwiat, S. K. Lamoreaux, C. G. Peterson, and C. M. Simmons, Practical free-space quantum key distribution over 1 km," Phys. Rev. Lett. 81, 3283-3286 (1998).
[CrossRef]

Tao, Z.

M. Er-Iong, H. Zheng-fu, G. Shun-sheng, Z. Tao, D. Da-sheng, and G. Guang-can, "Background noise of satellite-to-ground quantum key distribution," New J. Phys. 7, 215 (2005).
[CrossRef]

Tapster, P. R.

C. Kurtsiefer, P. Zarda, M. Holder, H. Weinfurter, P. Gormar, P. R. Tapster, and J. G. Rarity, "A step toward global quantum key distribution," Nature 419, 450 (2002).
[CrossRef] [PubMed]

Tittel, W.

W. Tittel and G. Weihs, "Photonic entanglement for fundamental tests and quantum communications," Quantum Inf. Comput. 1, 3-56 (2001).

Weihs, G.

W. Tittel and G. Weihs, "Photonic entanglement for fundamental tests and quantum communications," Quantum Inf. Comput. 1, 3-56 (2001).

Weinfurter, H.

M. Pfennigbauer, M. Aspelmeyer, W. R. Leeb, G. Baister, T. Dreischer, T. Jennewein, G. Neckamm, J. M. Perdigues, H. Weinfurter, and A. Zeilinger, "Satellite-based quantum communication terminal employing stateof-the-art technology," JON 4, 549-560, (2005).

C. Kurtsiefer, P. Zarda, M. Holder, H. Weinfurter, P. Gormar, P. R. Tapster, and J. G. Rarity, "A step toward global quantum key distribution," Nature 419, 450 (2002).
[CrossRef] [PubMed]

Yang, T.

C.-Z. Peng, T. Yang, X.-H. Bao, J.-Zhang, X.-M. Jin, F.-J. 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] [PubMed]

Zarda, P.

C. Kurtsiefer, P. Zarda, M. Holder, H. Weinfurter, P. Gormar, P. R. Tapster, and J. G. Rarity, "A step toward global quantum key distribution," Nature 419, 450 (2002).
[CrossRef] [PubMed]

Zeilinger, A.

M. Pfennigbauer, M. Aspelmeyer, W. R. Leeb, G. Baister, T. Dreischer, T. Jennewein, G. Neckamm, J. M. Perdigues, H. Weinfurter, and A. Zeilinger, "Satellite-based quantum communication terminal employing stateof-the-art technology," JON 4, 549-560, (2005).

M. Aspelmeyer, T. Jennewein, M. Pfennigbauer, W. R. Leeb, A. Zeilinger, "Long distance quantum communication with entangled photons using satellites," IEEE J. Sel. Top. Quantum Electron. 9, 1541 (2003).
[CrossRef]

Zheng-fu, H.

M. Er-Iong, H. Zheng-fu, G. Shun-sheng, Z. Tao, D. Da-sheng, and G. Guang-can, "Background noise of satellite-to-ground quantum key distribution," New J. Phys. 7, 215 (2005).
[CrossRef]

Appl. Opt. (2)

IEEE J. Sel. Top. Quantum Electron. (1)

M. Aspelmeyer, T. Jennewein, M. Pfennigbauer, W. R. Leeb, A. Zeilinger, "Long distance quantum communication with entangled photons using satellites," IEEE J. Sel. Top. Quantum Electron. 9, 1541 (2003).
[CrossRef]

JON (1)

M. Pfennigbauer, M. Aspelmeyer, W. R. Leeb, G. Baister, T. Dreischer, T. Jennewein, G. Neckamm, J. M. Perdigues, H. Weinfurter, and A. Zeilinger, "Satellite-based quantum communication terminal employing stateof-the-art technology," JON 4, 549-560, (2005).

Nature (1)

C. Kurtsiefer, P. Zarda, M. Holder, H. Weinfurter, P. Gormar, P. R. Tapster, and J. G. Rarity, "A step toward global quantum key distribution," Nature 419, 450 (2002).
[CrossRef] [PubMed]

New J. Phys. (1)

M. Er-Iong, H. Zheng-fu, G. Shun-sheng, Z. Tao, D. Da-sheng, and G. Guang-can, "Background noise of satellite-to-ground quantum key distribution," New J. Phys. 7, 215 (2005).
[CrossRef]

Opt. Express (1)

Phys. Rev. Lett. (2)

W. T. Buttler, R. J. Hughes, P. G. Kwiat, S. K. Lamoreaux, C. G. Peterson, and C. M. Simmons, Practical free-space quantum key distribution over 1 km," Phys. Rev. Lett. 81, 3283-3286 (1998).
[CrossRef]

C.-Z. Peng, T. Yang, X.-H. Bao, J.-Zhang, X.-M. Jin, F.-J. 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] [PubMed]

PRA (1)

A. Sehat,  et al., "Quantum polarization properties of two-mode energy eigenstates," PRA 71, 033818 (2004).

Quantum Inf. Comput. (1)

W. Tittel and G. Weihs, "Photonic entanglement for fundamental tests and quantum communications," Quantum Inf. Comput. 1, 3-56 (2001).

Science (1)

M. Aspelmeyer,  et al., "Long distance free-space distribution of quantum entanglement," Science 301, 621 (2003).
[CrossRef] [PubMed]

Other (8)

W. E. Forsythe, Smithsonian Physical Tables, (9th Revised Edition, Knovel).

R. J. Hughes, J. E. Nordholt, D. Derkacs, J. C. Peterson, "Practical free-space quantum key distribution over 10 km in daylight and at night," New J. Phys. 4, 43.1-43.14 (2002).
[CrossRef]

E. D. Palik (ed.), Handbook of optical constants of solids, (San Diego, Academic Press, 1998).

M. Born and E. Wolf, Principles of Optics, sixth ed., (Pergamon Press, Oxford, England, 1993).

R. Ursin,  et al., "Free-space distribution of entanglement and single photons over 144 Km," quant-ph/0607182.

J. E. Nordholt, R. J. Hughes, J. R. Morgan, C. G. Peterson, and C. C. Wipf, "Present and future quantum key distribution," in Free-Space Laser Communication Technologies XIV, G. Stephen Mecherle, ed. Proc. SPIE 4635, 116-126 (2002).
[CrossRef]

P. Villoresi, F. Tamburini, M. Aspelmeyer, T. Jennewein, R. Ursin, C. Pernechele, G. Bianco, A. Zeilinger, and C. Barbieri, "Space-to-ground quantum-communication using an optical ground station: a feasibility study," Proc. SPIE: Quantum Communications and Quantum Imaging, II conference in Denver (2004).

J. G. Rarity, P. R. Tapster, P. M. Gorman, and P. Knight, "Ground to satellite secure key exchange using quantum cryptography," New J. Phys. 4, 82.1-82.21 (2002).
[CrossRef]

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Figures (6)

Fig. 1.
Fig. 1.

Fixed reference frame: the origin is set in the Earth center, the z direction is orthogonal to the equatorial plane and the y direction is on the intersection between the equatorial and the orbital planes

Fig. 2.
Fig. 2.

Satellite visibility: the satellite is visible from the ground station only if the angle ϕ between v, vector pointing from the ground station to the satellite, and M, normal to the Earth surface in the ground station position, is such that 0<ϕ<π/2

Fig. 3.
Fig. 3.

Procedure to determine whether the rotation of the s-polarization direction is clockwise or counterclockwise

Fig. 4.
Fig. 4.

Different passages of the satellite above the ground station (represented by the GS in the centre of the XY plane). The satellite comes along different trajectories and so the pointing mirrors must be tilted in order to send the photons to the ground station, whatever the position of the satellite is. This makes the reference frame of the satellite rotate in respect to the reference frame on the ground station, and changes the angle of incidence on the mirrors, resulting on a modification of the polarization states of the emitted photons.

Fig. 5.
Fig. 5.

Poincaré spheres showing the received polarization states for two different satellite passages on the sky and four different photon wavelengths. The source on the satellite emits a horizontally-polarized photon, whose polarization state, due to rotation of the reference frames determined by the satellite motion and to reflection on mirrors, is in general different from the emitted one and changes in time. Moreover, the polarization states of photons of different wavelengths change in different ways, because of the different responses of mirrors. Elliptical polarization states can be due to complex refractive indices of the mirrors. In particular this result also indicates that it is difficult to use a reference laser at a different wavelength for polarization compensation.

Fig. 6.
Fig. 6.

Poincaré sphere and its projections on the (S 1, S 2), (S 2, S 3) and (S 1, S 3) planes for 3000 satellite passages on the sky, starting from a horizontally polarized photon emitted by the source on the satellite. Strinkingly, the detected polarization state can be anywhere on the Poicaré sphere, with a higher probability to be on a strip near the equatorial plane.

Equations (26)

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

N orb = [ cos ξ 0 sin ξ 0 1 0 sin ξ 0 cos ξ ] [ 0 0 1 ] = [ sin ξ 0 cos ξ ]
x = R o ( cos ξ cos ω t , sin ω t , sin ξ cos ω t )
M = R e ( cos β cos α , cos β sin α ,sin β )
α 1 = M α = ( sin α , cos α , 0 )
α 2 = M β = ( sin β cos α , sin β sin α , cos β )
α 3 = M M = ( cos β cos α , cos β sin α , sin β )
L 1 = ( cos ξ sin ω t , cos ω t , sin ξ sin ω t )
s 0 = ( sin ξ , 0 , cos ξ )
L 2 = M x r
θ 1 = 1 2 arccos ( ( L 1 ) · L 2 )
[ r p ( λ , θ 1 ) 0 0 r s ( λ , θ 1 ) ]
s 1 = L 1 × L 2 L 1 × L 2
σ 01 = s 0 × s 1 · L 1 s 0 × s 1 · L 1
β 01 = σ 01 arccos ( s 0 · s 1 )
[ cos β 01 sin β 01 sin β 01 cos β 01 ]
L 3 = ( cos χ ) α 1 + ( sin χ ) α 2
s 2 = L 2 × L 3 L 2 × L 3
[ cos β 12 sin β 12 sin β 12 cos β 12 ] β 12 = σ 12 arccos s 1 · s 2
s 3 = α 1 × α 2 α 1 × α 2
β 23 = σ 23 arccos s 2 · s 3
[ cos β 23 sin β 23 sin β 23 cos β 23 ]
[ E p E s ] = [ cos β 23 sin β 23 sin β 23 cos β 23 ] [ r p ( λ , θ 2 ) 0 0 r s ( λ , θ 2 ) ] [ cos β 12 sin β 12 sin β 12 cos β 12 ]
[ r p ( λ , θ 1 ) 0 0 r s ( λ , θ 1 ) ] [ cos β 01 sin β 01 sin β 01 cos β 01 ]
1 E p 2 + E s 2 [ E p E s ]
r s ( λ , θ i ) = n o ( λ ) cos θ i n ( λ ) cos θ t n o ( λ ) cos θ i + n ( λ ) cos θ t r p ( λ , θ i ) = n o ( λ ) cos θ t n ( λ ) cos θ i n o ( λ ) cos θ t + n ( λ ) cos θ i
χ = V B d = 0.001 r a d

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