Abstract

Birefringence affects all the optical circuits both in free-space and guided-wave optics. It perturbs the state of polarization of the propagating light, causing unwanted detrimental effects in many practical situations. A retracing circuit offers the potentiality of compensating the birefringence, but it is not universal because the birefringence is of different kinds, that is, either linear or circular, as well as either reciprocal or nonreciprocal, and because the common mirror does not hold all the requested symmetries. This paper reviews the compensation techniques suitable for each kind of birefringence, taking into account the introduction of two further generalized mirrors, the mirrored Faraday rotator and the mirrored quarter-wave plate. The main retracing schemes are analyzed and presented with the help of the Poincaré sphere and the Jones matrices. Examples of full compensation of all the main cases of birefringence that occurs in practical optical circuits are given. The different compensation properties of the three mirrors can be interpreted by means of a unified vision in the abstract space related to the Poincaré sphere representation.

© 2017 Optical Society of America

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  1. In fact, according to Noether’s theorem [2], whenever the system shows a symmetry, that is, an invariance property for a transformation (like a translation in time, a translation in space or a rotation), there is a conservation law (respectively of energy, linear momentum, and angular momentum).
  2. D. E. Neuenschwander, Emmy Noether’s Wonderful Theorem (Johns Hopkins University, 2011).
  3. D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1982).
  4. The spin conservation law states that the difference between the total number of photons with spin +1   minus the total number of photons with spin −1 is conserved; see as an example [5].
  5. M. C. Calkin, “An invariance property of the free electromagnetic field,” Am. J. Phys. 33, 958–960 (1965).
  6. M. Martinelli, L. Marazzi, P. Parolari, M. Brunero, and G. Gavioli, “Polarization in retracing circuits for WDM-PON,” IEEE Photon. Technol. Lett. 24, 1191–1193 (2012).
  7. P. Parolari, L. Marazzi, M. Brunero, M. Martinelli, R. Brenot, A. Maho, S. Barbet, G. Gavioli, G. Simon, F. Saliou, and P. Chanclou, “10  Gb/s operation of a colorless self-seeded transmitter over more than 70  km of SSMF,” IEEE Photon. Technol. Lett. 26, 599–602 (2014).
    [Crossref]
  8. A. D. Kersey, M. J. Marrone, and M. A. Davis, “Polarization-insensitive fibre optic Michelson interferometer,” Electron. Lett. 27, 518–520 (1991).
    [Crossref]
  9. C. H. Bennet and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in International Conference on Computers, Systems and Signal Processing, Bangalore, India (1984), p. 175.
  10. H. Zbinden, J. D. Gautier, N. Gisin, B. Huttner, A. Muller, and W. Tittel, “Interferometry with Faraday mirrors for quantum cryptography,” Electron. Lett. 33, 586–588 (1997).
    [Crossref]
  11. M. Martinelli, P. Martelli, and A. Fasiello, “A universal compensator of the non-reciprocal circular birefringence in a retracing path by a mirrored quarter-wave plate,” Opt. Commun. 372, 123–125 (2016).
    [Crossref]
  12. S. L. Altmann, Icons and Symmetries (Clarendon, 1992).
  13. J. F. Nye, Physical Properties of Crystals (Clarendon, 1957).
  14. R. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
    [Crossref]
  15. U. Fano, “Remarks on the classical and quantum-mechanics treatment of partial polarization,” J. Opt. Soc. Am. 39, 859–863 (1949).
    [Crossref]
  16. W. H. McMaster, “Polarization and Stokes parameters,” Am. J. Phys. 22, 351–362 (1954).
    [Crossref]
  17. W. H. McMaster, “Matrix representation of polarization,” Rev. Mod. Phys. 33, 8–28 (1961).
    [Crossref]
  18. E. Cartan, The Theory of Spinors (Dover, 1981).
  19. According to quantum mechanics, a particle characterized by a spin quantum number σ presents (2σ+1) spin eigenvalues, in the range with extreme values ±σ. Nevertheless, it results in σ=1 for the photon, but its spin eigenvalues are only the two values ±1, showing an exception to the previous rule. In actuality, the photon is a massless particle always moving at the velocity of the light, and owing to this constraint (called the transversality constraint) the value 0 is not a permitted spin eigenvalue for the photon; see [20] as an example.
  20. J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Addison-Wesley, 1955).
  21. D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, 1971).
  22. E. Collect, Polarized Light, Fundamentals and Applications (Marcel Dekker, 1993).
  23. C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).
  24. R. C. Jones, “A new calculus for the treatment of optical systems,” J. Opt. Soc. Am. 31, 488–493 (1941).
  25. A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 2003).
  26. V. A. Markelov, M. A. Novikov, and A. A. Turkin, “Experimental observation of a new nonreciprocal magneto-optical effect,” J. Exp. Phys. Theory Lett. 25, 378–380 (1977).
  27. M. A. Novikov and A. A. Khyshov, “Anisotropy of the nonreciprocal birefringence in crystals,” Tech. Phys. Lett. 24, 130–131 (1998).
    [Crossref]
  28. T. Godde, M. M. Glazov, I. A. Akimov, D. R. Yakovlev, H. Mariette, and M. Bayer, “Magnetic field induced nutation of the exciton-polariton polarization in (Cd, Zn)Te crystals,” Phys. Rev. B 88, 155203 (2013).
    [Crossref]
  29. A. J. Rogers, “The electrogyration effect in crystalline quartz,” Proc. R. Soc. London A 353, 177–192 (1977).
    [Crossref]
  30. R. Bhandari, “A useful generalization of the Martinelli effect,” Opt. Commun. 88, 1–5 (1992).
    [Crossref]
  31. F. E. Wright, “A spherical projection chart for use in the study of elliptically polarized light,” J. Opt. Soc. Am. 20, 529–564 (1930).
    [Crossref]
  32. M. Martinelli, “A universal compensator for polarization changes induced by birefringence on a retracing beam,” Opt. Commun. 72, 341–344 (1989).
    [Crossref]
  33. V. M. Gelikonov, D. D. Gusovskii, V. I. Leonov, and M. A. Novikov, “Birefringence compensation in single-mode optical fibers,” Sov. Tech. Phys. Lett. 13, 322–323 (1987).
  34. V. M. Gelikonov, V. I. Leonov, and M. A. Novikov, “Optical anisotropy in single mode optical waveguides during the round trip and methods of its compensation,” Sov. J. Quantum Electron. 19, 1227–1230 (1989).
    [Crossref]
  35. G. Giuliani and P. Ristori, “Polarization flip cavities: a new approach to laser resonators,” Opt. Commun. 35, 109–112 (1980).
    [Crossref]
  36. M. Martinelli, P. Martelli, and A. Fasiello, “A universal compensator for polarization changes induced by non-reciprocal circular birefringence on a retracing beam,” Opt. Commun. 366, 119–121 (2016).
    [Crossref]
  37. In general, the inversion of the direction of the motion and the connection with the cancellation of the propagation effects relate with the time-reversal property of the phenomenon. In optics, this was proved with the discovery of the phase-conjugated mirror (PCM), a device (based on non-linear effects, like the Brillouin effect) that some authors (see, for instance [38]) consider a time-reversal operator for the propagation. In fact, if a retracing circuit is considered, the presence of a common mirror does not cancel for the effects of the change in the momentum of the photons, while has been experimentally demonstrated that a PCM did (see also [39] for a general review). Although a PCM does not realize a true time-reversal operation on the system, both mathematics and experiments confirm analogies with the quantum mechanical operator, which is the transition to the complex conjugated K. In general, the PCM does not compensate for birefringence.
  38. R. H. Hellwarth, “Generation of time-reversed wave fronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977).
  39. B. Y. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, 1985).
  40. The anti-unitary matrix [Eq. (61)] of the MFR suggested to one of the authors that a connection could exist between the general compensating properties of the reciprocal birefringence shown by the MFR and the time-reversal operator T, which in our notation is T=iσ3K. For details see [41].
  41. M. Martinelli, “Time reversal for the polarization state in optical systems,” J. Mod. Opt. 39, 451–455 (1992).
  42. N. Pistoni and M. Martinelli, “Polarization noise suppression in retracing optical fiber circuits,” Opt. Lett. 16, 711–713 (1991).
    [Crossref]
  43. S. Yamashita, K. Hotate, and M. Ito, “Polarization properties of a reflective fiber amplifier employing a circulator and a Faraday rotator mirror,” J. Lightwave Technol. 14, 385–390 (1996).
    [Crossref]
  44. Y. Takushima, S. Yamashita, K. Kikuchi, and K. Hotate, “Polarization-stable and single-frequency fiber lasers,” J. Lightwave Technol. 16, 661–669 (1998).
    [Crossref]
  45. V. Secondi, F. Sciarrino, and F. De Martini, “Quantum spin-flipping by the Faraday mirror,” Phys. Rev. A 70, 040301 (2004).
    [Crossref]
  46. D. S. Bethune and W. P. Risk, “An autocompensating fiber-optic quantum cryptography system based on polarization splitting of light,” IEEE J. Quantum Electron. 36, 340–347 (2000).
    [Crossref]
  47. N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
    [Crossref]
  48. T. A. C. Maiolo, L. Martina, G. Ruggeri, and G. Soliani, “Recent aspects of entanglement with applications in quantum optics and quantum information,” Adv. Stud. Theory Phys. 2, 333–383 (2008).

2016 (2)

M. Martinelli, P. Martelli, and A. Fasiello, “A universal compensator of the non-reciprocal circular birefringence in a retracing path by a mirrored quarter-wave plate,” Opt. Commun. 372, 123–125 (2016).
[Crossref]

M. Martinelli, P. Martelli, and A. Fasiello, “A universal compensator for polarization changes induced by non-reciprocal circular birefringence on a retracing beam,” Opt. Commun. 366, 119–121 (2016).
[Crossref]

2014 (1)

P. Parolari, L. Marazzi, M. Brunero, M. Martinelli, R. Brenot, A. Maho, S. Barbet, G. Gavioli, G. Simon, F. Saliou, and P. Chanclou, “10  Gb/s operation of a colorless self-seeded transmitter over more than 70  km of SSMF,” IEEE Photon. Technol. Lett. 26, 599–602 (2014).
[Crossref]

2013 (1)

T. Godde, M. M. Glazov, I. A. Akimov, D. R. Yakovlev, H. Mariette, and M. Bayer, “Magnetic field induced nutation of the exciton-polariton polarization in (Cd, Zn)Te crystals,” Phys. Rev. B 88, 155203 (2013).
[Crossref]

2012 (1)

M. Martinelli, L. Marazzi, P. Parolari, M. Brunero, and G. Gavioli, “Polarization in retracing circuits for WDM-PON,” IEEE Photon. Technol. Lett. 24, 1191–1193 (2012).

2008 (1)

T. A. C. Maiolo, L. Martina, G. Ruggeri, and G. Soliani, “Recent aspects of entanglement with applications in quantum optics and quantum information,” Adv. Stud. Theory Phys. 2, 333–383 (2008).

2004 (1)

V. Secondi, F. Sciarrino, and F. De Martini, “Quantum spin-flipping by the Faraday mirror,” Phys. Rev. A 70, 040301 (2004).
[Crossref]

2002 (1)

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

2000 (1)

D. S. Bethune and W. P. Risk, “An autocompensating fiber-optic quantum cryptography system based on polarization splitting of light,” IEEE J. Quantum Electron. 36, 340–347 (2000).
[Crossref]

1998 (2)

M. A. Novikov and A. A. Khyshov, “Anisotropy of the nonreciprocal birefringence in crystals,” Tech. Phys. Lett. 24, 130–131 (1998).
[Crossref]

Y. Takushima, S. Yamashita, K. Kikuchi, and K. Hotate, “Polarization-stable and single-frequency fiber lasers,” J. Lightwave Technol. 16, 661–669 (1998).
[Crossref]

1997 (1)

H. Zbinden, J. D. Gautier, N. Gisin, B. Huttner, A. Muller, and W. Tittel, “Interferometry with Faraday mirrors for quantum cryptography,” Electron. Lett. 33, 586–588 (1997).
[Crossref]

1996 (1)

S. Yamashita, K. Hotate, and M. Ito, “Polarization properties of a reflective fiber amplifier employing a circulator and a Faraday rotator mirror,” J. Lightwave Technol. 14, 385–390 (1996).
[Crossref]

1992 (2)

M. Martinelli, “Time reversal for the polarization state in optical systems,” J. Mod. Opt. 39, 451–455 (1992).

R. Bhandari, “A useful generalization of the Martinelli effect,” Opt. Commun. 88, 1–5 (1992).
[Crossref]

1991 (2)

A. D. Kersey, M. J. Marrone, and M. A. Davis, “Polarization-insensitive fibre optic Michelson interferometer,” Electron. Lett. 27, 518–520 (1991).
[Crossref]

N. Pistoni and M. Martinelli, “Polarization noise suppression in retracing optical fiber circuits,” Opt. Lett. 16, 711–713 (1991).
[Crossref]

1989 (2)

M. Martinelli, “A universal compensator for polarization changes induced by birefringence on a retracing beam,” Opt. Commun. 72, 341–344 (1989).
[Crossref]

V. M. Gelikonov, V. I. Leonov, and M. A. Novikov, “Optical anisotropy in single mode optical waveguides during the round trip and methods of its compensation,” Sov. J. Quantum Electron. 19, 1227–1230 (1989).
[Crossref]

1987 (1)

V. M. Gelikonov, D. D. Gusovskii, V. I. Leonov, and M. A. Novikov, “Birefringence compensation in single-mode optical fibers,” Sov. Tech. Phys. Lett. 13, 322–323 (1987).

1980 (1)

G. Giuliani and P. Ristori, “Polarization flip cavities: a new approach to laser resonators,” Opt. Commun. 35, 109–112 (1980).
[Crossref]

1977 (3)

A. J. Rogers, “The electrogyration effect in crystalline quartz,” Proc. R. Soc. London A 353, 177–192 (1977).
[Crossref]

V. A. Markelov, M. A. Novikov, and A. A. Turkin, “Experimental observation of a new nonreciprocal magneto-optical effect,” J. Exp. Phys. Theory Lett. 25, 378–380 (1977).

R. H. Hellwarth, “Generation of time-reversed wave fronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977).

1965 (1)

M. C. Calkin, “An invariance property of the free electromagnetic field,” Am. J. Phys. 33, 958–960 (1965).

1961 (1)

W. H. McMaster, “Matrix representation of polarization,” Rev. Mod. Phys. 33, 8–28 (1961).
[Crossref]

1954 (1)

W. H. McMaster, “Polarization and Stokes parameters,” Am. J. Phys. 22, 351–362 (1954).
[Crossref]

1949 (1)

1941 (1)

1936 (1)

R. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
[Crossref]

1930 (1)

Akimov, I. A.

T. Godde, M. M. Glazov, I. A. Akimov, D. R. Yakovlev, H. Mariette, and M. Bayer, “Magnetic field induced nutation of the exciton-polariton polarization in (Cd, Zn)Te crystals,” Phys. Rev. B 88, 155203 (2013).
[Crossref]

Altmann, S. L.

S. L. Altmann, Icons and Symmetries (Clarendon, 1992).

Barbet, S.

P. Parolari, L. Marazzi, M. Brunero, M. Martinelli, R. Brenot, A. Maho, S. Barbet, G. Gavioli, G. Simon, F. Saliou, and P. Chanclou, “10  Gb/s operation of a colorless self-seeded transmitter over more than 70  km of SSMF,” IEEE Photon. Technol. Lett. 26, 599–602 (2014).
[Crossref]

Bayer, M.

T. Godde, M. M. Glazov, I. A. Akimov, D. R. Yakovlev, H. Mariette, and M. Bayer, “Magnetic field induced nutation of the exciton-polariton polarization in (Cd, Zn)Te crystals,” Phys. Rev. B 88, 155203 (2013).
[Crossref]

Bennet, C. H.

C. H. Bennet and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in International Conference on Computers, Systems and Signal Processing, Bangalore, India (1984), p. 175.

Beth, R.

R. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
[Crossref]

Bethune, D. S.

D. S. Bethune and W. P. Risk, “An autocompensating fiber-optic quantum cryptography system based on polarization splitting of light,” IEEE J. Quantum Electron. 36, 340–347 (2000).
[Crossref]

Bhandari, R.

R. Bhandari, “A useful generalization of the Martinelli effect,” Opt. Commun. 88, 1–5 (1992).
[Crossref]

Brassard, G.

C. H. Bennet and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in International Conference on Computers, Systems and Signal Processing, Bangalore, India (1984), p. 175.

Brenot, R.

P. Parolari, L. Marazzi, M. Brunero, M. Martinelli, R. Brenot, A. Maho, S. Barbet, G. Gavioli, G. Simon, F. Saliou, and P. Chanclou, “10  Gb/s operation of a colorless self-seeded transmitter over more than 70  km of SSMF,” IEEE Photon. Technol. Lett. 26, 599–602 (2014).
[Crossref]

Brosseau, C.

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

Brunero, M.

P. Parolari, L. Marazzi, M. Brunero, M. Martinelli, R. Brenot, A. Maho, S. Barbet, G. Gavioli, G. Simon, F. Saliou, and P. Chanclou, “10  Gb/s operation of a colorless self-seeded transmitter over more than 70  km of SSMF,” IEEE Photon. Technol. Lett. 26, 599–602 (2014).
[Crossref]

M. Martinelli, L. Marazzi, P. Parolari, M. Brunero, and G. Gavioli, “Polarization in retracing circuits for WDM-PON,” IEEE Photon. Technol. Lett. 24, 1191–1193 (2012).

Calkin, M. C.

M. C. Calkin, “An invariance property of the free electromagnetic field,” Am. J. Phys. 33, 958–960 (1965).

Cartan, E.

E. Cartan, The Theory of Spinors (Dover, 1981).

Chanclou, P.

P. Parolari, L. Marazzi, M. Brunero, M. Martinelli, R. Brenot, A. Maho, S. Barbet, G. Gavioli, G. Simon, F. Saliou, and P. Chanclou, “10  Gb/s operation of a colorless self-seeded transmitter over more than 70  km of SSMF,” IEEE Photon. Technol. Lett. 26, 599–602 (2014).
[Crossref]

Clarke, D.

D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, 1971).

Collect, E.

E. Collect, Polarized Light, Fundamentals and Applications (Marcel Dekker, 1993).

Davis, M. A.

A. D. Kersey, M. J. Marrone, and M. A. Davis, “Polarization-insensitive fibre optic Michelson interferometer,” Electron. Lett. 27, 518–520 (1991).
[Crossref]

De Martini, F.

V. Secondi, F. Sciarrino, and F. De Martini, “Quantum spin-flipping by the Faraday mirror,” Phys. Rev. A 70, 040301 (2004).
[Crossref]

Fano, U.

Fasiello, A.

M. Martinelli, P. Martelli, and A. Fasiello, “A universal compensator for polarization changes induced by non-reciprocal circular birefringence on a retracing beam,” Opt. Commun. 366, 119–121 (2016).
[Crossref]

M. Martinelli, P. Martelli, and A. Fasiello, “A universal compensator of the non-reciprocal circular birefringence in a retracing path by a mirrored quarter-wave plate,” Opt. Commun. 372, 123–125 (2016).
[Crossref]

Gautier, J. D.

H. Zbinden, J. D. Gautier, N. Gisin, B. Huttner, A. Muller, and W. Tittel, “Interferometry with Faraday mirrors for quantum cryptography,” Electron. Lett. 33, 586–588 (1997).
[Crossref]

Gavioli, G.

P. Parolari, L. Marazzi, M. Brunero, M. Martinelli, R. Brenot, A. Maho, S. Barbet, G. Gavioli, G. Simon, F. Saliou, and P. Chanclou, “10  Gb/s operation of a colorless self-seeded transmitter over more than 70  km of SSMF,” IEEE Photon. Technol. Lett. 26, 599–602 (2014).
[Crossref]

M. Martinelli, L. Marazzi, P. Parolari, M. Brunero, and G. Gavioli, “Polarization in retracing circuits for WDM-PON,” IEEE Photon. Technol. Lett. 24, 1191–1193 (2012).

Gelikonov, V. M.

V. M. Gelikonov, V. I. Leonov, and M. A. Novikov, “Optical anisotropy in single mode optical waveguides during the round trip and methods of its compensation,” Sov. J. Quantum Electron. 19, 1227–1230 (1989).
[Crossref]

V. M. Gelikonov, D. D. Gusovskii, V. I. Leonov, and M. A. Novikov, “Birefringence compensation in single-mode optical fibers,” Sov. Tech. Phys. Lett. 13, 322–323 (1987).

Gisin, N.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

H. Zbinden, J. D. Gautier, N. Gisin, B. Huttner, A. Muller, and W. Tittel, “Interferometry with Faraday mirrors for quantum cryptography,” Electron. Lett. 33, 586–588 (1997).
[Crossref]

Giuliani, G.

G. Giuliani and P. Ristori, “Polarization flip cavities: a new approach to laser resonators,” Opt. Commun. 35, 109–112 (1980).
[Crossref]

Glazov, M. M.

T. Godde, M. M. Glazov, I. A. Akimov, D. R. Yakovlev, H. Mariette, and M. Bayer, “Magnetic field induced nutation of the exciton-polariton polarization in (Cd, Zn)Te crystals,” Phys. Rev. B 88, 155203 (2013).
[Crossref]

Godde, T.

T. Godde, M. M. Glazov, I. A. Akimov, D. R. Yakovlev, H. Mariette, and M. Bayer, “Magnetic field induced nutation of the exciton-polariton polarization in (Cd, Zn)Te crystals,” Phys. Rev. B 88, 155203 (2013).
[Crossref]

Grainger, J. F.

D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, 1971).

Gusovskii, D. D.

V. M. Gelikonov, D. D. Gusovskii, V. I. Leonov, and M. A. Novikov, “Birefringence compensation in single-mode optical fibers,” Sov. Tech. Phys. Lett. 13, 322–323 (1987).

Hellwarth, R. H.

Hotate, K.

Y. Takushima, S. Yamashita, K. Kikuchi, and K. Hotate, “Polarization-stable and single-frequency fiber lasers,” J. Lightwave Technol. 16, 661–669 (1998).
[Crossref]

S. Yamashita, K. Hotate, and M. Ito, “Polarization properties of a reflective fiber amplifier employing a circulator and a Faraday rotator mirror,” J. Lightwave Technol. 14, 385–390 (1996).
[Crossref]

Huttner, B.

H. Zbinden, J. D. Gautier, N. Gisin, B. Huttner, A. Muller, and W. Tittel, “Interferometry with Faraday mirrors for quantum cryptography,” Electron. Lett. 33, 586–588 (1997).
[Crossref]

Ito, M.

S. Yamashita, K. Hotate, and M. Ito, “Polarization properties of a reflective fiber amplifier employing a circulator and a Faraday rotator mirror,” J. Lightwave Technol. 14, 385–390 (1996).
[Crossref]

Jauch, J. M.

J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Addison-Wesley, 1955).

Jones, R. C.

Kersey, A. D.

A. D. Kersey, M. J. Marrone, and M. A. Davis, “Polarization-insensitive fibre optic Michelson interferometer,” Electron. Lett. 27, 518–520 (1991).
[Crossref]

Khyshov, A. A.

M. A. Novikov and A. A. Khyshov, “Anisotropy of the nonreciprocal birefringence in crystals,” Tech. Phys. Lett. 24, 130–131 (1998).
[Crossref]

Kikuchi, K.

Leonov, V. I.

V. M. Gelikonov, V. I. Leonov, and M. A. Novikov, “Optical anisotropy in single mode optical waveguides during the round trip and methods of its compensation,” Sov. J. Quantum Electron. 19, 1227–1230 (1989).
[Crossref]

V. M. Gelikonov, D. D. Gusovskii, V. I. Leonov, and M. A. Novikov, “Birefringence compensation in single-mode optical fibers,” Sov. Tech. Phys. Lett. 13, 322–323 (1987).

Maho, A.

P. Parolari, L. Marazzi, M. Brunero, M. Martinelli, R. Brenot, A. Maho, S. Barbet, G. Gavioli, G. Simon, F. Saliou, and P. Chanclou, “10  Gb/s operation of a colorless self-seeded transmitter over more than 70  km of SSMF,” IEEE Photon. Technol. Lett. 26, 599–602 (2014).
[Crossref]

Maiolo, T. A. C.

T. A. C. Maiolo, L. Martina, G. Ruggeri, and G. Soliani, “Recent aspects of entanglement with applications in quantum optics and quantum information,” Adv. Stud. Theory Phys. 2, 333–383 (2008).

Marazzi, L.

P. Parolari, L. Marazzi, M. Brunero, M. Martinelli, R. Brenot, A. Maho, S. Barbet, G. Gavioli, G. Simon, F. Saliou, and P. Chanclou, “10  Gb/s operation of a colorless self-seeded transmitter over more than 70  km of SSMF,” IEEE Photon. Technol. Lett. 26, 599–602 (2014).
[Crossref]

M. Martinelli, L. Marazzi, P. Parolari, M. Brunero, and G. Gavioli, “Polarization in retracing circuits for WDM-PON,” IEEE Photon. Technol. Lett. 24, 1191–1193 (2012).

Marcuse, D.

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1982).

Mariette, H.

T. Godde, M. M. Glazov, I. A. Akimov, D. R. Yakovlev, H. Mariette, and M. Bayer, “Magnetic field induced nutation of the exciton-polariton polarization in (Cd, Zn)Te crystals,” Phys. Rev. B 88, 155203 (2013).
[Crossref]

Markelov, V. A.

V. A. Markelov, M. A. Novikov, and A. A. Turkin, “Experimental observation of a new nonreciprocal magneto-optical effect,” J. Exp. Phys. Theory Lett. 25, 378–380 (1977).

Marrone, M. J.

A. D. Kersey, M. J. Marrone, and M. A. Davis, “Polarization-insensitive fibre optic Michelson interferometer,” Electron. Lett. 27, 518–520 (1991).
[Crossref]

Martelli, P.

M. Martinelli, P. Martelli, and A. Fasiello, “A universal compensator of the non-reciprocal circular birefringence in a retracing path by a mirrored quarter-wave plate,” Opt. Commun. 372, 123–125 (2016).
[Crossref]

M. Martinelli, P. Martelli, and A. Fasiello, “A universal compensator for polarization changes induced by non-reciprocal circular birefringence on a retracing beam,” Opt. Commun. 366, 119–121 (2016).
[Crossref]

Martina, L.

T. A. C. Maiolo, L. Martina, G. Ruggeri, and G. Soliani, “Recent aspects of entanglement with applications in quantum optics and quantum information,” Adv. Stud. Theory Phys. 2, 333–383 (2008).

Martinelli, M.

M. Martinelli, P. Martelli, and A. Fasiello, “A universal compensator for polarization changes induced by non-reciprocal circular birefringence on a retracing beam,” Opt. Commun. 366, 119–121 (2016).
[Crossref]

M. Martinelli, P. Martelli, and A. Fasiello, “A universal compensator of the non-reciprocal circular birefringence in a retracing path by a mirrored quarter-wave plate,” Opt. Commun. 372, 123–125 (2016).
[Crossref]

P. Parolari, L. Marazzi, M. Brunero, M. Martinelli, R. Brenot, A. Maho, S. Barbet, G. Gavioli, G. Simon, F. Saliou, and P. Chanclou, “10  Gb/s operation of a colorless self-seeded transmitter over more than 70  km of SSMF,” IEEE Photon. Technol. Lett. 26, 599–602 (2014).
[Crossref]

M. Martinelli, L. Marazzi, P. Parolari, M. Brunero, and G. Gavioli, “Polarization in retracing circuits for WDM-PON,” IEEE Photon. Technol. Lett. 24, 1191–1193 (2012).

M. Martinelli, “Time reversal for the polarization state in optical systems,” J. Mod. Opt. 39, 451–455 (1992).

N. Pistoni and M. Martinelli, “Polarization noise suppression in retracing optical fiber circuits,” Opt. Lett. 16, 711–713 (1991).
[Crossref]

M. Martinelli, “A universal compensator for polarization changes induced by birefringence on a retracing beam,” Opt. Commun. 72, 341–344 (1989).
[Crossref]

McMaster, W. H.

W. H. McMaster, “Matrix representation of polarization,” Rev. Mod. Phys. 33, 8–28 (1961).
[Crossref]

W. H. McMaster, “Polarization and Stokes parameters,” Am. J. Phys. 22, 351–362 (1954).
[Crossref]

Muller, A.

H. Zbinden, J. D. Gautier, N. Gisin, B. Huttner, A. Muller, and W. Tittel, “Interferometry with Faraday mirrors for quantum cryptography,” Electron. Lett. 33, 586–588 (1997).
[Crossref]

Neuenschwander, D. E.

D. E. Neuenschwander, Emmy Noether’s Wonderful Theorem (Johns Hopkins University, 2011).

Novikov, M. A.

M. A. Novikov and A. A. Khyshov, “Anisotropy of the nonreciprocal birefringence in crystals,” Tech. Phys. Lett. 24, 130–131 (1998).
[Crossref]

V. M. Gelikonov, V. I. Leonov, and M. A. Novikov, “Optical anisotropy in single mode optical waveguides during the round trip and methods of its compensation,” Sov. J. Quantum Electron. 19, 1227–1230 (1989).
[Crossref]

V. M. Gelikonov, D. D. Gusovskii, V. I. Leonov, and M. A. Novikov, “Birefringence compensation in single-mode optical fibers,” Sov. Tech. Phys. Lett. 13, 322–323 (1987).

V. A. Markelov, M. A. Novikov, and A. A. Turkin, “Experimental observation of a new nonreciprocal magneto-optical effect,” J. Exp. Phys. Theory Lett. 25, 378–380 (1977).

Nye, J. F.

J. F. Nye, Physical Properties of Crystals (Clarendon, 1957).

Parolari, P.

P. Parolari, L. Marazzi, M. Brunero, M. Martinelli, R. Brenot, A. Maho, S. Barbet, G. Gavioli, G. Simon, F. Saliou, and P. Chanclou, “10  Gb/s operation of a colorless self-seeded transmitter over more than 70  km of SSMF,” IEEE Photon. Technol. Lett. 26, 599–602 (2014).
[Crossref]

M. Martinelli, L. Marazzi, P. Parolari, M. Brunero, and G. Gavioli, “Polarization in retracing circuits for WDM-PON,” IEEE Photon. Technol. Lett. 24, 1191–1193 (2012).

Pilipetsky, N. F.

B. Y. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, 1985).

Pistoni, N.

Ribordy, G.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Risk, W. P.

D. S. Bethune and W. P. Risk, “An autocompensating fiber-optic quantum cryptography system based on polarization splitting of light,” IEEE J. Quantum Electron. 36, 340–347 (2000).
[Crossref]

Ristori, P.

G. Giuliani and P. Ristori, “Polarization flip cavities: a new approach to laser resonators,” Opt. Commun. 35, 109–112 (1980).
[Crossref]

Rogers, A. J.

A. J. Rogers, “The electrogyration effect in crystalline quartz,” Proc. R. Soc. London A 353, 177–192 (1977).
[Crossref]

Rohrlich, F.

J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Addison-Wesley, 1955).

Ruggeri, G.

T. A. C. Maiolo, L. Martina, G. Ruggeri, and G. Soliani, “Recent aspects of entanglement with applications in quantum optics and quantum information,” Adv. Stud. Theory Phys. 2, 333–383 (2008).

Saliou, F.

P. Parolari, L. Marazzi, M. Brunero, M. Martinelli, R. Brenot, A. Maho, S. Barbet, G. Gavioli, G. Simon, F. Saliou, and P. Chanclou, “10  Gb/s operation of a colorless self-seeded transmitter over more than 70  km of SSMF,” IEEE Photon. Technol. Lett. 26, 599–602 (2014).
[Crossref]

Sciarrino, F.

V. Secondi, F. Sciarrino, and F. De Martini, “Quantum spin-flipping by the Faraday mirror,” Phys. Rev. A 70, 040301 (2004).
[Crossref]

Secondi, V.

V. Secondi, F. Sciarrino, and F. De Martini, “Quantum spin-flipping by the Faraday mirror,” Phys. Rev. A 70, 040301 (2004).
[Crossref]

Shkunov, V. V.

B. Y. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, 1985).

Simon, G.

P. Parolari, L. Marazzi, M. Brunero, M. Martinelli, R. Brenot, A. Maho, S. Barbet, G. Gavioli, G. Simon, F. Saliou, and P. Chanclou, “10  Gb/s operation of a colorless self-seeded transmitter over more than 70  km of SSMF,” IEEE Photon. Technol. Lett. 26, 599–602 (2014).
[Crossref]

Soliani, G.

T. A. C. Maiolo, L. Martina, G. Ruggeri, and G. Soliani, “Recent aspects of entanglement with applications in quantum optics and quantum information,” Adv. Stud. Theory Phys. 2, 333–383 (2008).

Takushima, Y.

Tittel, W.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

H. Zbinden, J. D. Gautier, N. Gisin, B. Huttner, A. Muller, and W. Tittel, “Interferometry with Faraday mirrors for quantum cryptography,” Electron. Lett. 33, 586–588 (1997).
[Crossref]

Turkin, A. A.

V. A. Markelov, M. A. Novikov, and A. A. Turkin, “Experimental observation of a new nonreciprocal magneto-optical effect,” J. Exp. Phys. Theory Lett. 25, 378–380 (1977).

Wright, F. E.

Yakovlev, D. R.

T. Godde, M. M. Glazov, I. A. Akimov, D. R. Yakovlev, H. Mariette, and M. Bayer, “Magnetic field induced nutation of the exciton-polariton polarization in (Cd, Zn)Te crystals,” Phys. Rev. B 88, 155203 (2013).
[Crossref]

Yamashita, S.

Y. Takushima, S. Yamashita, K. Kikuchi, and K. Hotate, “Polarization-stable and single-frequency fiber lasers,” J. Lightwave Technol. 16, 661–669 (1998).
[Crossref]

S. Yamashita, K. Hotate, and M. Ito, “Polarization properties of a reflective fiber amplifier employing a circulator and a Faraday rotator mirror,” J. Lightwave Technol. 14, 385–390 (1996).
[Crossref]

Yariv, A.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 2003).

Yeh, P.

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 2003).

Zbinden, H.

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

H. Zbinden, J. D. Gautier, N. Gisin, B. Huttner, A. Muller, and W. Tittel, “Interferometry with Faraday mirrors for quantum cryptography,” Electron. Lett. 33, 586–588 (1997).
[Crossref]

Zel’dovich, B. Y.

B. Y. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, 1985).

Adv. Stud. Theory Phys. (1)

T. A. C. Maiolo, L. Martina, G. Ruggeri, and G. Soliani, “Recent aspects of entanglement with applications in quantum optics and quantum information,” Adv. Stud. Theory Phys. 2, 333–383 (2008).

Am. J. Phys. (2)

M. C. Calkin, “An invariance property of the free electromagnetic field,” Am. J. Phys. 33, 958–960 (1965).

W. H. McMaster, “Polarization and Stokes parameters,” Am. J. Phys. 22, 351–362 (1954).
[Crossref]

Electron. Lett. (2)

A. D. Kersey, M. J. Marrone, and M. A. Davis, “Polarization-insensitive fibre optic Michelson interferometer,” Electron. Lett. 27, 518–520 (1991).
[Crossref]

H. Zbinden, J. D. Gautier, N. Gisin, B. Huttner, A. Muller, and W. Tittel, “Interferometry with Faraday mirrors for quantum cryptography,” Electron. Lett. 33, 586–588 (1997).
[Crossref]

IEEE J. Quantum Electron. (1)

D. S. Bethune and W. P. Risk, “An autocompensating fiber-optic quantum cryptography system based on polarization splitting of light,” IEEE J. Quantum Electron. 36, 340–347 (2000).
[Crossref]

IEEE Photon. Technol. Lett. (2)

M. Martinelli, L. Marazzi, P. Parolari, M. Brunero, and G. Gavioli, “Polarization in retracing circuits for WDM-PON,” IEEE Photon. Technol. Lett. 24, 1191–1193 (2012).

P. Parolari, L. Marazzi, M. Brunero, M. Martinelli, R. Brenot, A. Maho, S. Barbet, G. Gavioli, G. Simon, F. Saliou, and P. Chanclou, “10  Gb/s operation of a colorless self-seeded transmitter over more than 70  km of SSMF,” IEEE Photon. Technol. Lett. 26, 599–602 (2014).
[Crossref]

J. Exp. Phys. Theory Lett. (1)

V. A. Markelov, M. A. Novikov, and A. A. Turkin, “Experimental observation of a new nonreciprocal magneto-optical effect,” J. Exp. Phys. Theory Lett. 25, 378–380 (1977).

J. Lightwave Technol. (2)

S. Yamashita, K. Hotate, and M. Ito, “Polarization properties of a reflective fiber amplifier employing a circulator and a Faraday rotator mirror,” J. Lightwave Technol. 14, 385–390 (1996).
[Crossref]

Y. Takushima, S. Yamashita, K. Kikuchi, and K. Hotate, “Polarization-stable and single-frequency fiber lasers,” J. Lightwave Technol. 16, 661–669 (1998).
[Crossref]

J. Mod. Opt. (1)

M. Martinelli, “Time reversal for the polarization state in optical systems,” J. Mod. Opt. 39, 451–455 (1992).

J. Opt. Soc. Am. (4)

Opt. Commun. (5)

M. Martinelli, P. Martelli, and A. Fasiello, “A universal compensator of the non-reciprocal circular birefringence in a retracing path by a mirrored quarter-wave plate,” Opt. Commun. 372, 123–125 (2016).
[Crossref]

R. Bhandari, “A useful generalization of the Martinelli effect,” Opt. Commun. 88, 1–5 (1992).
[Crossref]

M. Martinelli, “A universal compensator for polarization changes induced by birefringence on a retracing beam,” Opt. Commun. 72, 341–344 (1989).
[Crossref]

G. Giuliani and P. Ristori, “Polarization flip cavities: a new approach to laser resonators,” Opt. Commun. 35, 109–112 (1980).
[Crossref]

M. Martinelli, P. Martelli, and A. Fasiello, “A universal compensator for polarization changes induced by non-reciprocal circular birefringence on a retracing beam,” Opt. Commun. 366, 119–121 (2016).
[Crossref]

Opt. Lett. (1)

Phys. Rev. (1)

R. Beth, “Mechanical detection and measurement of the angular momentum of light,” Phys. Rev. 50, 115–125 (1936).
[Crossref]

Phys. Rev. A (1)

V. Secondi, F. Sciarrino, and F. De Martini, “Quantum spin-flipping by the Faraday mirror,” Phys. Rev. A 70, 040301 (2004).
[Crossref]

Phys. Rev. B (1)

T. Godde, M. M. Glazov, I. A. Akimov, D. R. Yakovlev, H. Mariette, and M. Bayer, “Magnetic field induced nutation of the exciton-polariton polarization in (Cd, Zn)Te crystals,” Phys. Rev. B 88, 155203 (2013).
[Crossref]

Proc. R. Soc. London A (1)

A. J. Rogers, “The electrogyration effect in crystalline quartz,” Proc. R. Soc. London A 353, 177–192 (1977).
[Crossref]

Rev. Mod. Phys. (2)

W. H. McMaster, “Matrix representation of polarization,” Rev. Mod. Phys. 33, 8–28 (1961).
[Crossref]

N. Gisin, G. Ribordy, W. Tittel, and H. Zbinden, “Quantum cryptography,” Rev. Mod. Phys. 74, 145–195 (2002).
[Crossref]

Sov. J. Quantum Electron. (1)

V. M. Gelikonov, V. I. Leonov, and M. A. Novikov, “Optical anisotropy in single mode optical waveguides during the round trip and methods of its compensation,” Sov. J. Quantum Electron. 19, 1227–1230 (1989).
[Crossref]

Sov. Tech. Phys. Lett. (1)

V. M. Gelikonov, D. D. Gusovskii, V. I. Leonov, and M. A. Novikov, “Birefringence compensation in single-mode optical fibers,” Sov. Tech. Phys. Lett. 13, 322–323 (1987).

Tech. Phys. Lett. (1)

M. A. Novikov and A. A. Khyshov, “Anisotropy of the nonreciprocal birefringence in crystals,” Tech. Phys. Lett. 24, 130–131 (1998).
[Crossref]

Other (17)

S. L. Altmann, Icons and Symmetries (Clarendon, 1992).

J. F. Nye, Physical Properties of Crystals (Clarendon, 1957).

In general, the inversion of the direction of the motion and the connection with the cancellation of the propagation effects relate with the time-reversal property of the phenomenon. In optics, this was proved with the discovery of the phase-conjugated mirror (PCM), a device (based on non-linear effects, like the Brillouin effect) that some authors (see, for instance [38]) consider a time-reversal operator for the propagation. In fact, if a retracing circuit is considered, the presence of a common mirror does not cancel for the effects of the change in the momentum of the photons, while has been experimentally demonstrated that a PCM did (see also [39] for a general review). Although a PCM does not realize a true time-reversal operation on the system, both mathematics and experiments confirm analogies with the quantum mechanical operator, which is the transition to the complex conjugated K. In general, the PCM does not compensate for birefringence.

B. Y. Zel’dovich, N. F. Pilipetsky, and V. V. Shkunov, Principles of Phase Conjugation (Springer-Verlag, 1985).

The anti-unitary matrix [Eq. (61)] of the MFR suggested to one of the authors that a connection could exist between the general compensating properties of the reciprocal birefringence shown by the MFR and the time-reversal operator T, which in our notation is T=iσ3K. For details see [41].

E. Cartan, The Theory of Spinors (Dover, 1981).

According to quantum mechanics, a particle characterized by a spin quantum number σ presents (2σ+1) spin eigenvalues, in the range with extreme values ±σ. Nevertheless, it results in σ=1 for the photon, but its spin eigenvalues are only the two values ±1, showing an exception to the previous rule. In actuality, the photon is a massless particle always moving at the velocity of the light, and owing to this constraint (called the transversality constraint) the value 0 is not a permitted spin eigenvalue for the photon; see [20] as an example.

J. M. Jauch and F. Rohrlich, The Theory of Photons and Electrons (Addison-Wesley, 1955).

D. Clarke and J. F. Grainger, Polarized Light and Optical Measurement (Pergamon, 1971).

E. Collect, Polarized Light, Fundamentals and Applications (Marcel Dekker, 1993).

C. Brosseau, Fundamentals of Polarized Light: A Statistical Optics Approach (Wiley, 1998).

A. Yariv and P. Yeh, Optical Waves in Crystals (Wiley, 2003).

C. H. Bennet and G. Brassard, “Quantum cryptography: public key distribution and coin tossing,” in International Conference on Computers, Systems and Signal Processing, Bangalore, India (1984), p. 175.

In fact, according to Noether’s theorem [2], whenever the system shows a symmetry, that is, an invariance property for a transformation (like a translation in time, a translation in space or a rotation), there is a conservation law (respectively of energy, linear momentum, and angular momentum).

D. E. Neuenschwander, Emmy Noether’s Wonderful Theorem (Johns Hopkins University, 2011).

D. Marcuse, Light Transmission Optics (Van Nostrand Reinhold, 1982).

The spin conservation law states that the difference between the total number of photons with spin +1   minus the total number of photons with spin −1 is conserved; see as an example [5].

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Figures (21)

Figure 1
Figure 1

Although the polar vector and the axial vector are presented by the same icon, they behave very differently when reflected by a mirror.

Figure 2
Figure 2

By means of an analyzer and a quarter-wave plate it is possible to recover all three Stokes parameters.

Figure 3
Figure 3

Poincaré sphere.

Figure 4
Figure 4

Equivalent representation of the Stokes parameters, light polarization, photon spin, and Poincaré sphere.

Figure 5
Figure 5

Retracing optical circuit with a mirror (above) and a generalized mirror (below).

Figure 6
Figure 6

(a) Linear birefringence is represented by a vector lying on the equatorial plane of the Poincaré sphere that rotates clockwise around the sphere itself. Circular birefringence is represented by a vector whose direction coincides with the polar axis and rotates anti-clockwise around the sphere itself. (b) In the presence of negative birefringence, the vector is reversed.

Figure 7
Figure 7

Action of the linear birefringence vector on a generic state of polarization Pin: (a) all the sphere, including Pin, is rotated around the axis rr by the angle γ, and Pin migrates in Pout; (b) the sphere returns to the original position but leaving Pout in its position.

Figure 8
Figure 8

Action of the circular birefringence vector on a generic state of polarization Pin: (a) all the sphere, including Pin, is rotated around the polar axis by the angle 2δ, and Pin migrates in Pout; (b) the sphere returns to the original position, but leaving Pout in its position.

Figure 9
Figure 9

Effect of the mirror reflection on the Poincaré sphere. To maintain the right-hand set even in the returning beam, the y axis is inverted. In consequence, the Poincaré sphere turns around axis HV (axis S1) by an angle π. This operation is also called binary rotation.

Figure 10
Figure 10

(a) Original state of polarization P; (b) reflected state of polarization P by a MFR (the generalized mirror represented by the binary rotation around axis S2). The evolution on the Poincaré sphere of the state of polarization is highlighted by the dashed line.

Figure 11
Figure 11

(a) Original state of polarization P; (b) reflected state of polarization P by a MQW (the generalized mirror represented by the binary rotation around axis S3). The evolution on the Poincaré sphere of the state of polarization is evidenced by the dashed line.

Figure 12
Figure 12

Representation on the Poincaré sphere of the compensation of the nonreciprocal circular birefringence by the generalized mirror corresponding to a binary rotation around axis S3. The forward birefringence is described by the continuous purple rotation vector, which is opposite to the dashed purple vector obtained by equivalently moving the backward circular birefringence (continuous black vector) before the mirror.

Figure 13
Figure 13

Representation on the Poincaré sphere of the compensation of the reciprocal circular birefringence by the generalized mirror corresponding to a binary rotation around either axis S1 or axis S2. The forward birefringence is described by the continuous purple rotation vector, which is opposite to the dashed purple vector obtained by equivalently moving the backward circular birefringence (continuous black vector) before the mirror.

Figure 14
Figure 14

Compensation for the reciprocal circular birefringence by a common mirror. The rail vector represents the mirror action.

Figure 15
Figure 15

Compensation for the nonreciprocal linear birefringence by a common mirror. The rail vector represents the mirror action.

Figure 16
Figure 16

Compensation for the reciprocal circular birefringence by the MFR generalized mirror. The rail vector represents the MFR action. If point P coincides with the horizontal polarization, the light returns vertically polarized, with no trace of whatever birefringence.

Figure 17
Figure 17

Compensation for the reciprocal linear birefringence by the MFR generalized mirror. The rail vector represents the MFR action. If point P coincides with the horizontal polarization, the light returns vertically polarized, with no trace of whatever birefringence.

Figure 18
Figure 18

Compensation for the nonreciprocal circular birefringence by the MQW generalized mirror. The rail vector represents the MQW action. If point P coincides with the horizontal polarization, the light returns vertically polarized, with no trace of whatever birefringence.

Figure 19
Figure 19

Compensation for the reciprocal linear birefringence R[0,φ] by the MQW generalized mirror. The rail vector represents the MQW action. If point P coincides with the horizontal polarization, the light returns vertically polarized, with no trace of whatever birefringence.

Figure 20
Figure 20

Compensation for the reciprocal linear birefringence R[π/2,φ] by the MQW generalized mirror. The rail vector represents the MQW action.

Figure 21
Figure 21

In the abstract space of the Poincaré sphere, the action of the three mirrors (the common mirror and the two generalized mirrors MFR and MQW) appears equivalent. The figure shows also the effects of the mirror triad on the three couples of the polarization eigenstates (in the real space).

Tables (2)

Tables Icon

Table 1. Compensation Strategies for Linear and Circular Birefringenceab

Tables Icon

Table 2. Birefringence in the Propagation Spaceab

Equations (73)

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

E=ω,
p=k,
s=σ,
|ψ=a1|ψ1+a2|ψ2,
S0=ψ|σ0|ψ=[a1*a2*][1001][a1a2]=a1a1*+a2a2*,S1=ψ|σ1|ψ=[a1*a2*][1001][a1a2]=a1a1*a2a2*,S2=ψ|σ2|ψ=[a1*a2*][0110][a1a2]=a1a2*+a2a1*,S3=ψ|σ3|ψ=[a1*a2*][0ii0][a1a2]=i(a1a2*a2a1*),
S=[S0S1S2S3]=[ABCD]=[IMCS]=[IQUV].
S12+S22+S32=S02.
S1=σ1=+1horizontal linear state|H,S1=σ1=1vertical linear state|V,S2=σ2=+1diagonal linear state|Q,S2=σ2=1anti-diagonal linear state|Q,S3=σ3=+1right circular state|R,S3=σ3=1left circular state|L.
S1=S0cos(2χ)cos(2θ),S2=S0cos(2χ)sin(2θ),S3=S0sin(2χ),
tan(2θ)=S2S1,sin(2χ)=S3S0.
H|[1001]|H=σ1=+1with  |H=[10],V|[1001]|V=σ1=1with  |V=[01],Q|[0110]|Q=σ2=+1with  |Q=12[11],Q|[0110]|Q=σ2=1with  |Q=12[11],R|[0ii0]|R=σ3=+1with  |R=12[1i],L|[0ii0]|L=σ3=1with  |L=12[1i].
J=[ExEy]=[Ex0exp(iδx0)Ey0exp(iδy0)],
Ex0=amplitude of the field component along the axisx,Ey0=amplitude of the field component along the axisy,δx0=initial phase of the field component along the axisx,δy0=initial phase of the field component along the axisy,
|H[Ex0exp(iδx0)0]=Ex0exp(iδx0)[10],|V[0Ey0exp(iδy0)]=Ey0exp(iδy0)[01].
|Q[Ex0exp(iδx0)Ex0exp(iδx0)]=E0exp(iδx0)2[11],|Q[Ex0exp(iδx0)Ex0exp[i(δx0+π)]]=E0exp(iδx0)2[11].
|R[Ex0exp(iδx0)Ex0exp[i(δx0+π2)]]=E0exp(iδx0)2[1i],|L[Ex0exp(iδx0)Ex0exp[i(δx0π2)]]=E0exp(iδx0)2[1i].
I=[Ex0exp(iδx)Ey0exp(iδy)]*[Ex0exp(iδx)Ey0exp(iδy)]=(Ex0)2+(Ey0)2,
S0=[(Ex0)2+(Ey0)2],S1=[(Ex0)2(Ey0)2],S2=[2Ex0Ey0cos(δyδx)],S3=[2Ex0Ey0sin(δyδx)].
Jout=MJin.
Jout=MnM2M1Jin.
[DxDyDz]=ϵ0[ϵ11ϵ12ϵ13ϵ21ϵ22ϵ23ϵ31ϵ32ϵ33][ExEyEz],
[DxDyDz]=ϵ0[n112n122n132n122n222n232n132n232n332][ExEyEz].
[DxDyDz]=ϵ0[nx2000ny2000nz2][ExEyEz].
[DxDyDz]=ϵ0[n02000n02000n02][ExEyEz],
[DxDyDz]=ϵ0[no2000ne2000no2][ExEyEz],
Δn=nynx=neno,
φ=k0ΔnL=2πλ0ΔnL,
R[0,φ]=[exp(+iφ/2)00exp(iφ/2)],
[DxDyDz]=ϵ0[ϵxEx+i(α×E)xϵyEy+i(α×E)yϵzEz+i(α×E)z],
[DxDyDz]=ϵ0[ϵxiαziαyiαzϵyiαxiαyiαxϵz][ExEyEz]={ϵ0[nx2000ny2000nz2]+iϵ0[0αzαyαz0αxαyαx0]}[ExEyEz].
[DxDyDz]=ϵ0[n02iαz0iαzn02000nz2][ExEyEz]={ϵ0[n02000n02000nz2]+iϵ0[0αz0αz00000]}[ExEyEz].
C[δ]=[cosδsinδsinδcosδ],
n±=(n02±αz)1/2,
n±n0±αz2n0.
Δncirc=n+nαzn0.
2δ=k0ΔncircL=2πλ0ΔncircL
δ=k02n0αzL=πλ0n0αzL.
R[γ,φ]=C[γ]R[0,φ]C[γ],
R[γ,φ]=[cos(φ/2)+isin(φ/2)cos(2γ)isin(φ/2)sin(2γ)isin(φ/2)sin(2γ)cos(φ/2)isin(φ/2)cos(2γ)].
|H|H,|V|V,|Q|Q,|Q|Q,|R|L,|L|R,
Mmirror=[1001].
Rrec[γ,φ]inversion of propagationRrec[γ,φ],
Rnon-rec[γ,φ]inversion of propagationRnon-rec[γ,φ],
Crec[δ]inversion of propagationCrec[δ],
Cnonrec[δ]inversion of propagationCnonrec[δ].
Jout=Mbirefringence·Mmirror·Mbirefringence·Jin=Mbirefringence·[1001]·Mbirefringence·Jin,
Jout(t)=Mbirefringence(t)·[1001]·Mbirefringence(t)·Jin.
Jout=Mbirefringence(t)·Mgeneralizedmirror·Mbirefringence(t)·Jin=Mgeneralizedmirror·Mbirefringence1(t)·Mbirefringence(t)·Jin=Mgeneralizedmirror·Jin.
Mbirefringence(t)·Mgeneralizedmirror=Mgeneralizedmirror·Mbirefringence(t).
C[δ]·Π3·C[δ]=Π3·C[δ]·C[δ]=Π3,
C[δ]·Πn·C[δ]=Πn·C[δ]·C[δ]=Πn(for  n=1,2),
Mmirror=[1001]=σ1.
Mretracing=Crec[δ]·σ1·Crec[δ]=C[δ]·σ1·C[δ]=σ1,
C[δ]·σ1·C[δ]=[cosδsinδsinδcosδ]·[1001]·[cosδsinδsinδcosδ]=[1001]=σ1.
δ=gτ,
g=n22(p11p12)=n2p44,
Mretracing=Rnon-rec[γ,φ]·σ1·Rnon-rec[γ,φ]=C[γ]·R[0,φ]·C[γ]·σ1·C[γ]·R[0,φ]·C[γ].
R[0,φ]·σ1·R[0,φ]=σ1,
Mretracing=C[γ]·R[0,φ]·C[γ]·σ1·C[γ]·R[0,φ]·C[γ]=σ1,
ΔnH=3.57·1015  m/Afor  LiIO3,ΔnH=2.32·1016  m/Afor  KDP.
MMFR=[0110]=σ2,
Mretracing=Crec[δ]·σ2·C[δ]rec=C[δ]·σ2·C[δ]=σ2,
C[δ]·σ2·C[δ]=[cosδsinδsinδcosδ][0110][cosδsinδsinδcosδ]=[0110]=σ2.
Mretracing=Rrec[γ,φ]·σ2·Rrec[γ,φ]=C[γ]·R[0,φ]·C[γ]·σ2·C[γ]·R[0,φ]·C[γ]=σ2,
R[0,φ]·σ2·R[0,φ]=σ2.
MMQW=[0ii0]=σ3.
Mretracing=Cnon-rec[δ]·σ3·C[δ]non-rec=C[δ]·σ3·C[δ]=σ3,
σ3=i[0110]=iC[π/2].
C[δ]·σ3·C[δ]=C[δ]·C[δ]·σ3=σ3.
Mretracing=Rrec[0,φ]·σ3·Rrec[0,φ]=σ3.
Mretracing=Rnon-rec[π/4,φ]·σ3·Rnon-rec[π/4,φ]=C[π/4]·R[0,φ]·C[π/4]·σ3·C[π/4]·R[0,φ]·C[π/4].
C[π/4]·σ3·C[π/4]=C[π/4]·iC[π/2]·C[π/4]=iC[π]=iI,
Mretracing=iC[π/4]·R[0,φ]·R[0,φ]·C[π/4]=iC[π/2]=σ3.

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