Abstract

The application of the Jones calculus to optical circuits in which counterpropagating light beams are present is discussed, with particular attention to the conventions associated with the Jones calculus for the description of the polarization state of an optical beam and to the reference system adopted. The reference system adopted here differs from that used by Jones, but it exploits a simpler formalism when used to describe complex optical systems where multiple reflections take place. The differences between these two reference systems are pointed out with particular attention to the behavior of nonreciprocal optical devices and reflectors. An application of the results to a simple optical circuit is presented.

© 1995 Optical Society of America

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References

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  1. R. C. Jones, “New calculus for the treatment of optical systems I. Description and discussion,” J. Opt. Soc. Am. 31, 488–503 (1941); “New calculus for the treatment of optical systems IV,” J. Opt. Soc. Am. 32, 486–493 (1942); “New calculus for the treatment of optical systems V. A more general formulation and description of another calculus,” J. Opt. Soc. Am. 37, 107–112 (1947); “New calculus for the treatment of optical systems VII. Properties of the N-matrices,” J. Opt. Soc. Am. 38, 671–685 (1948).
    [Crossref]
  2. W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962), Chap. 1, p. 1.
  3. C. D. Graves, “Radar polarization power scattering matrix,” Proc. IRE 44, 248–252 (1956).
    [Crossref]
  4. R. Bhandari, “Classical light waves and spinors,” in Analogies in Optics and Micro Electronics, W. van Haeringen, D. Lenstra, eds. (Kluwer Academic, Boston, 1990), p. 69.
    [Crossref]
  5. R. Bhandari, “Evolution of light beams in polarization and direction,” Phys. Lett. B 175, 111–122 (1991).
  6. N. Vansteenkiste, P. Vignolo, A. Aspect, “Optical reversibility theorems for polarization: application to remote control of polarization,” J. Opt. Soc. Am. A 10, 2240–2245 (1993).
    [Crossref]
  7. A. L. Fymat, “Jones’s matrix representation of optical instruments. I: Beam splitters,” Appl. Opt. 10, 2499–2505 (1971).
    [Crossref] [PubMed]
  8. R. J. Vernon, B. D. Huggins, “Extension of the Jones matrix formalism to reflection problems and magnetic materials,” J. Opt. Soc. Am. 70, 1364–1370 (1980).
    [Crossref]
  9. M. P. Silverman, “Reflection and refraction at the surface of a chiral medium: comparison of gyrotropic constitutive relations invariant or noninvariant under a duality transformation,” J. Opt. Soc. Am. A 3, 830–837 (1986).
    [Crossref]
  10. V. M. Gelikonov, D. D. Gusovskii, V. I. Leonov, M. A. Novikov, “Birefringence compensation in single-mode optical fibers,” Sov. Tech. Phys. Lett. 13, 322–323 (1987).
  11. N. C. Pistoni, M. Martinelli, “Vibration-insensitive fiber-optic current sensor,” Opt. Lett. 18, 314–316 (1993).
    [Crossref] [PubMed]
  12. M. J. Marone, A. D. Kersey, I. N. Duling, R. D. Esman, “Polarization independent current sensor using an orthoconjugating fiber loop mirror,” in Proceedings of the Ninth International Conference on Optical Fiber Sensors (Instituto di Ricerca sulle Onde Ellettromagnetiche-Consiglio Nazionale delle Ricerche, 1993), p. 419.

1993 (2)

1991 (1)

R. Bhandari, “Evolution of light beams in polarization and direction,” Phys. Lett. B 175, 111–122 (1991).

1987 (1)

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

1986 (1)

1980 (1)

1971 (1)

1956 (1)

C. D. Graves, “Radar polarization power scattering matrix,” Proc. IRE 44, 248–252 (1956).
[Crossref]

1941 (1)

Aspect, A.

Bhandari, R.

R. Bhandari, “Evolution of light beams in polarization and direction,” Phys. Lett. B 175, 111–122 (1991).

R. Bhandari, “Classical light waves and spinors,” in Analogies in Optics and Micro Electronics, W. van Haeringen, D. Lenstra, eds. (Kluwer Academic, Boston, 1990), p. 69.
[Crossref]

Duling, I. N.

M. J. Marone, A. D. Kersey, I. N. Duling, R. D. Esman, “Polarization independent current sensor using an orthoconjugating fiber loop mirror,” in Proceedings of the Ninth International Conference on Optical Fiber Sensors (Instituto di Ricerca sulle Onde Ellettromagnetiche-Consiglio Nazionale delle Ricerche, 1993), p. 419.

Esman, R. D.

M. J. Marone, A. D. Kersey, I. N. Duling, R. D. Esman, “Polarization independent current sensor using an orthoconjugating fiber loop mirror,” in Proceedings of the Ninth International Conference on Optical Fiber Sensors (Instituto di Ricerca sulle Onde Ellettromagnetiche-Consiglio Nazionale delle Ricerche, 1993), p. 419.

Fymat, A. L.

Gelikonov, V. M.

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

Graves, C. D.

C. D. Graves, “Radar polarization power scattering matrix,” Proc. IRE 44, 248–252 (1956).
[Crossref]

Gusovskii, D. D.

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

Huggins, B. D.

Jones, R. C.

Kersey, A. D.

M. J. Marone, A. D. Kersey, I. N. Duling, R. D. Esman, “Polarization independent current sensor using an orthoconjugating fiber loop mirror,” in Proceedings of the Ninth International Conference on Optical Fiber Sensors (Instituto di Ricerca sulle Onde Ellettromagnetiche-Consiglio Nazionale delle Ricerche, 1993), p. 419.

Leonov, V. I.

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

Marone, M. J.

M. J. Marone, A. D. Kersey, I. N. Duling, R. D. Esman, “Polarization independent current sensor using an orthoconjugating fiber loop mirror,” in Proceedings of the Ninth International Conference on Optical Fiber Sensors (Instituto di Ricerca sulle Onde Ellettromagnetiche-Consiglio Nazionale delle Ricerche, 1993), p. 419.

Martinelli, M.

Novikov, M. A.

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

Pistoni, N. C.

Shurcliff, W. A.

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962), Chap. 1, p. 1.

Silverman, M. P.

Vansteenkiste, N.

Vernon, R. J.

Vignolo, P.

Appl. Opt. (1)

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (2)

Opt. Lett. (1)

Phys. Lett. B (1)

R. Bhandari, “Evolution of light beams in polarization and direction,” Phys. Lett. B 175, 111–122 (1991).

Proc. IRE (1)

C. D. Graves, “Radar polarization power scattering matrix,” Proc. IRE 44, 248–252 (1956).
[Crossref]

Sov. Tech. Phys. Lett. (1)

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

Other (3)

R. Bhandari, “Classical light waves and spinors,” in Analogies in Optics and Micro Electronics, W. van Haeringen, D. Lenstra, eds. (Kluwer Academic, Boston, 1990), p. 69.
[Crossref]

W. A. Shurcliff, Polarized Light (Harvard U. Press, Cambridge, Mass., 1962), Chap. 1, p. 1.

M. J. Marone, A. D. Kersey, I. N. Duling, R. D. Esman, “Polarization independent current sensor using an orthoconjugating fiber loop mirror,” in Proceedings of the Ninth International Conference on Optical Fiber Sensors (Instituto di Ricerca sulle Onde Ellettromagnetiche-Consiglio Nazionale delle Ricerche, 1993), p. 419.

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Figures (3)

Fig. 1
Fig. 1

Definition of the azimuth angle of a linear birefringent or linear dichroic wave plate: The observer always looks toward the light source through the wave plate. Reversing the light propagation causes one to observe the light source from the other side of the wave plate. As a result the azimuth angle also changes from ρ to −ρ.

Fig. 2
Fig. 2

E s component of the electric field of the optical beam incident on an ideal mirror reflected without change. The component lying in plane of incidence E p is reversed in sign. When partial reflections at interfaces between different optical media occur, various effects may change the relative phases and amplitude of the reflected E s and E p .

Fig. 3
Fig. 3

A, Optical scheme of the classical setup used to reduce reflection from a specular surface. The light rays coming back after one or odd numbers of reflections are blocked by the linear polarizer. If the optical beam is backreflected by two or any even number of reflections, as in B, it is not blocked by the linear analyzer and the scheme is not able to isolate the source from the reflected light.

Tables (1)

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Table 1 Meaning of the N-Matrix Coefficients, as given by Jonesa

Equations (18)

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M ( β , ρ ) = S ( ρ ) L ( β ) S ( - ρ ) ,
S ( ρ ) = [ cos ( ρ ) - sin ( ρ ) sin ( ρ ) cos ( ρ ) ] ,
L ( β ) = [ exp ( i β / 2 ) 0 0 exp ( - i β / 2 ) ] .
D ( p ) = [ exp ( p ) 0 0 exp ( - p ) ] .
N = [ - κ + p 0 - i η + i ξ 0 - ω + p 45 - i q + i ξ 45 ω + p 45 + i q + i ξ 45 - κ - p 0 - i η - i ξ 0 ] .
ξ = ( ξ 0 2 + ξ 45 2 ) 1 / 2 , ρ = 1 2 arctan ( ξ 45 ξ 0 ) .
M = exp ( - i η - κ ) [ cosh Q N z + ( p 0 + i ξ 0 ) sinh Q N z Q N ( - ω + p 45 - i q + i ξ 45 ) sinh Q N z Q N ( ω + p 45 + i q + i ξ 45 ) sinh Q N z Q N cosh Q N z - ( p 0 + i ξ 0 ) sinh Q N z Q N ] , Q N 2 = ( p 0 + i ξ 0 ) 2 + ( 45 + i ξ 45 ) 2 - ( ω + i q ) 2 .
N r = [ - κ + p 0 - i η + i ξ 0 - ω - p 45 - i q - i ξ 45 ω - p 45 + i q + i ξ 45 - κ - p 0 - i η - i ξ 0 ] ,
M r = exp ( - i η - κ ) [ cosh Q N r z + ( p 0 + i ξ 0 ) sinh Q N r z Q N r ( - ω - p 45 - i q - i ξ 45 ) sinh Q N r z Q N r ( ω - p 45 + i q - i ξ 45 ) sinh Q N r z Q N r cosh Q N r z - ( p 0 + i ξ 0 ) sinh Q N r z Q N r ] , Q N r 2 = ( p 0 + i ξ 0 ) 2 + ( - p 45 + i ξ 45 ) 2 - ( ω + i q ) 2 = Q N 2 .
[ m 1 m 4 m 3 m 2 ] [ m 1 - m 3 - m 4 m 2 ] .
[ m 1 m 4 m 3 m 2 ] [ m 1 - m 4 - m 3 m 2 ] .
M = [ 1 0 0 - 1 ] .
MM = [ 1 0 0 - 1 ] [ 1 0 0 - 1 ] = [ 1 0 0 1 ] .
1 2 [ 1 0 0 0 ] [ 1 i i 1 ] [ 1 0 0 - 1 ] [ 1 - i - i 1 ] [ 1 0 0 0 ] = [ 0 0 0 0 ] .
1 2 [ 1 0 0 0 ] [ 1 i i 1 ] [ 1 0 0 - 1 ] [ 1 0 0 - 1 ] [ 1 - i - i 1 ] × [ 1 0 0 0 ] = [ 1 0 0 0 ] .
[ R s 0 0 - R p ] .
R s = sin ( Θ i - Θ t ) sin ( Θ i + Θ t ) , R p = tan ( Θ i - Θ t ) tan ( Θ i + Θ t ) ,
MFR = 1 2 [ 1 1 - 1 1 ] [ 1 0 0 - 1 ] [ 1 - 1 1 1 ] = [ 0 - 1 - 1 0 ] .

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