Abstract

We present a model for the polarization states of a quasi-isotropic laser. The model includes the polarization competition among the gain medium, general cavity anisotropies, and the anisotropy arising from weak but arbitrarily polarized feedback. Three examples of linearly polarized feedback are given, one with the axes of the internal and external anisotropies parallel to each other and two other examples with the axes inclined at 45°. The new calculations are in agreement both with earlier calculations based on a more restricted model and with existing experimental results for a He–Ne laser operating at 3.39 μm. An important feature of the calculations is a method of finding all the stationary solutions, even in the general case. For the three examples considered we find many stationary polarization states. A linear stability analysis shows that only two are stable and permits us to relate our calculations to additive pulse mode locking, to Casperson instabilities, and to Hopf bifurcations.

© 1992 Optical Society of America

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  1. W. E. Lamb, “Theory of an optical maser,” Phys. Rev. A 134, 1429 (1964).
  2. W. Van Haeringen, “Polarization properties of a single-mode operating gas laser in a small axial magnetic field,” Phys. Rev. 158, 256 (1967).
    [Crossref]
  3. M. Sargent, W. E. Lamb, and R. L. Fork, “Theory of a Zeeman laser I,” Phys. Rev. 164, 436 (1967).
    [Crossref]
  4. D. Lenstra, “On the theory of polarization effects in a gas laser,” Phy. Rep. 59, 299 (1980).
    [Crossref]
  5. A. D. May and G. Stephan, “Stability of polarized modes in a quasi-isotropic laser,” J. Opt. Soc. Am. B 6, 2355 (1989).
    [Crossref]
  6. W. Xiong, P. Glanznig, P. Paddon, A. D. May, M. Bourouis, S. Laniepce, and G. M. Stéphan, “Stability of polarized modes in a quasi-isotropic laser: experimental confirmation,” J. Opt. Soc. Am. B 8, 1236 (1991).
    [Crossref]
  7. M. S. Borisova and I. P. Mazanko, “Stability of the state of polarization of the emission of a single mode gas laser with slightly anisotropic resonator,” Opt. Spectrosk. 47, 126 (1979) [Opt. Spectrosc. (USSR) 47, 69 (1979)].
  8. V. S. Smirnov and A. M. Tumaikin, “Polarization phenomena in a gas laser with an anisotropic resonator,” Opt. Spektrosk. 40, 1030 (1976) [Opt. Spectrosc. (USSR) 40, 593 (1976)].
  9. G. Stéphan, A. D. May, R. E. Mueller, and B. Aissaoui, “Competition effects in the polarization of light in a quasi-isotropic laser,” J. Opt. Soc. Am. B 4, 1276 (1987).
    [Crossref]
  10. G. Stéphan, A. D. May, R. E. Mueller, and B. Aissaoui, “Effets de competition dans la polarisation de la lumière d’un laser quasi-isotrope,” J. Phys. (Paris) 48, C7-549 (1987). See also G. Stéphan and D. Hugon, “Light polarization of a quasi-isotropic laser with optical feedback,” Phys. Rev. Lett. 55, 703 (1985); S. T. Hendow, R. W. Dunn, W. W. Chow, and J. G. Small, “Observation of bistable behavior in the polarization of a laser,” Opt. Lett. 7, 356 (1982).
    [Crossref] [PubMed]
  11. P. J. Brannon, “Laser feedback: its effect on laser frequency,” Appl. Opt. 15, 1119 (1976).
    [Crossref] [PubMed]
  12. J. Mark, L. Y. Liu, K. L. Hall, H. A. Haus, and E. P. Ippen, “Femtosecond pulse generation in a laser with a nonlinear external resonator,” Opt. Lett. 14, 48 (1989).
    [Crossref] [PubMed]
  13. L. Casperson and A. Yariv, “Longitudinal modes in a high-gain laser,” Appl. Phys. Lett. 17, 259 (1970).
    [Crossref]
  14. E. Hopf, “Abzweigung einer periodischen losung von einer stationaren losung eines differentialsystem,” Ber. Math.-Phys. Kl. Sachs. Akad. Wiss. Leipzig (1942), p. 94.
  15. We have adhered to the terminology of G. R. Fowles, Introduction to Modern Optics, 2nd ed. (Dover, New York, 1989), pp. 28–31, for describing the polarization states of optical fields. We differ only by a factor of 2. Note, however, yet another typographical error in Ref. 5; there is a sign error in the definition of E+below Eq. (1) on p. 2356.
  16. R. Seydel, From Equilibrium to Chaos (Elsevier, New York, 1988).
  17. Many authors use the Stokes parameters when discussing quasi-isotropic lasers. We have not yet found them to be particularly useful. The relationships between our variables and the Stokes parameters are S0= 4I, S1= 4I(1 − η2)1/2cos ϕ0, S2= 4I(1 − η2)1/2sin ϕ0, and S3= 4Iη. For a definition of the Stokes parameters see M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 24–30.
  18. This excludes the 100% circularly polarized and the I= 0 solutions. However, their behavior may be treated by an examination of the equations in their original form, i.e., Eqs. (1′)–(4′). The only novel aspect of the I= 0 solutions is that one polarization mode generally comes above threshold before the other. This is easy to understand, since the optical length of the cavity depends on the polarization. This is important when one considers experiments in which one sweeps the laser back and forward across the gain profile by modulating the length of the laser.
  19. A. Abromowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, Applied Mathematics Series 55 (National Bureau of Standards, Washington, D.C.).
  20. M. Kitano, T. Yabuzaki, and T. Ogawa, “Optical tristability,” Phys. Rev. Lett. 46, 926 (1981).
    [Crossref]
  21. P. Paddon, P. Glanznig, A. D. May, and G. Stéphan, “Comportement catastrophique du domain de bistabilité d’un laser vectoriel,” Ann. Phys. (N.Y.) 15, 177 (1990).
  22. M. Bourouis, S. Laniepce, G. Stéphan, P. Glanznig, P. Paddon, W. Xiong, and A. D. May, “Looking for polarization catastrophies in quasi-isotropic lasers,” in 1990 OSA Proceedings on Dynamics in Optical Systems, N. B. Abraham, E. Garmire, and P. Mandel, eds. (Optical Society of America, Washington, D.C., 1991), pp. 283–286.
  23. In terms of our variables the observed intensity ℐ equals I[1 − (1 − η2)1/2cos(ϕ0− 2ψ].
  24. M. V. Tratnik and J. E. Sipe, “Stokes vectors and polarization lasers,” J. Opt. Soc. Am. B 2, 1690 (1985).
    [Crossref]
  25. In the case of ϕf0= 0, one could debate whether it is the same or a different mode, since the jump is to another part of the same branch. Some authors consider a turning point as the junction of two branches.
  26. M. V. Tratnik and J. E. Sipe, “Polarization eigenstates of a Zeeman laser,” J. Opt. Soc. Am. B 3, 1127 (1986).
    [Crossref]
  27. G. Stéphan, R. Le Naour, and A. Le Floch, “Experimental and theoretical study of the anisotropy induced in a gas laser by a saturating field,” Phys. Rev. A 17, 733 (1978).
    [Crossref]
  28. R. S. Gioggia and N. B. Abraham, “Anomalous mode pulling, instabilities and chaos in a single-mode, standing-wave 3.39-μ m He–Ne laser,” Phys. Rev. A 29, 1304 (1984).
    [Crossref]
  29. G. P. Puccioni, G. L. Lippi, N. B. Abraham, and F. T. Arecchi, “Differences in polarization dynamics of the electromagnetic field in xenon and neon lasers,” Opt. Commun. 72, 361 (1989)
    [Crossref]
  30. L. W. Casperson, “Stability criteria for high-intensity lasers,” Phys. Rev. A 21, 911 (1980).
    [Crossref]
  31. K. Ikeda and O. Akimoto, “Instability leading to periodic and chaotic self-pulsations in a bistable optical cavity,” Phys. Rev. Lett. 48, 617 (1982).
    [Crossref]
  32. H. Kubo and R. Nagata, “Stokes parameters representation of the light propagation equations in inhomogeneous anisotropic, optically active media,” Opt. Commun. 34, 306 (1980).
    [Crossref]
  33. A. Messiah, Quantum Mechanics VII (Wiley, New York, 1962), p. 545.
  34. Note the typographical error in Ref. 5 on p. 2357 just below Eq. (5). We have repeatedly checked that none of the typographical errors made it through to the computer code.

1991 (1)

1990 (1)

P. Paddon, P. Glanznig, A. D. May, and G. Stéphan, “Comportement catastrophique du domain de bistabilité d’un laser vectoriel,” Ann. Phys. (N.Y.) 15, 177 (1990).

1989 (3)

1987 (2)

G. Stéphan, A. D. May, R. E. Mueller, and B. Aissaoui, “Competition effects in the polarization of light in a quasi-isotropic laser,” J. Opt. Soc. Am. B 4, 1276 (1987).
[Crossref]

G. Stéphan, A. D. May, R. E. Mueller, and B. Aissaoui, “Effets de competition dans la polarisation de la lumière d’un laser quasi-isotrope,” J. Phys. (Paris) 48, C7-549 (1987). See also G. Stéphan and D. Hugon, “Light polarization of a quasi-isotropic laser with optical feedback,” Phys. Rev. Lett. 55, 703 (1985); S. T. Hendow, R. W. Dunn, W. W. Chow, and J. G. Small, “Observation of bistable behavior in the polarization of a laser,” Opt. Lett. 7, 356 (1982).
[Crossref] [PubMed]

1986 (1)

1985 (1)

1984 (1)

R. S. Gioggia and N. B. Abraham, “Anomalous mode pulling, instabilities and chaos in a single-mode, standing-wave 3.39-μ m He–Ne laser,” Phys. Rev. A 29, 1304 (1984).
[Crossref]

1982 (1)

K. Ikeda and O. Akimoto, “Instability leading to periodic and chaotic self-pulsations in a bistable optical cavity,” Phys. Rev. Lett. 48, 617 (1982).
[Crossref]

1981 (1)

M. Kitano, T. Yabuzaki, and T. Ogawa, “Optical tristability,” Phys. Rev. Lett. 46, 926 (1981).
[Crossref]

1980 (3)

D. Lenstra, “On the theory of polarization effects in a gas laser,” Phy. Rep. 59, 299 (1980).
[Crossref]

H. Kubo and R. Nagata, “Stokes parameters representation of the light propagation equations in inhomogeneous anisotropic, optically active media,” Opt. Commun. 34, 306 (1980).
[Crossref]

L. W. Casperson, “Stability criteria for high-intensity lasers,” Phys. Rev. A 21, 911 (1980).
[Crossref]

1979 (1)

M. S. Borisova and I. P. Mazanko, “Stability of the state of polarization of the emission of a single mode gas laser with slightly anisotropic resonator,” Opt. Spectrosk. 47, 126 (1979) [Opt. Spectrosc. (USSR) 47, 69 (1979)].

1978 (1)

G. Stéphan, R. Le Naour, and A. Le Floch, “Experimental and theoretical study of the anisotropy induced in a gas laser by a saturating field,” Phys. Rev. A 17, 733 (1978).
[Crossref]

1976 (2)

V. S. Smirnov and A. M. Tumaikin, “Polarization phenomena in a gas laser with an anisotropic resonator,” Opt. Spektrosk. 40, 1030 (1976) [Opt. Spectrosc. (USSR) 40, 593 (1976)].

P. J. Brannon, “Laser feedback: its effect on laser frequency,” Appl. Opt. 15, 1119 (1976).
[Crossref] [PubMed]

1970 (1)

L. Casperson and A. Yariv, “Longitudinal modes in a high-gain laser,” Appl. Phys. Lett. 17, 259 (1970).
[Crossref]

1967 (2)

W. Van Haeringen, “Polarization properties of a single-mode operating gas laser in a small axial magnetic field,” Phys. Rev. 158, 256 (1967).
[Crossref]

M. Sargent, W. E. Lamb, and R. L. Fork, “Theory of a Zeeman laser I,” Phys. Rev. 164, 436 (1967).
[Crossref]

1964 (1)

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. A 134, 1429 (1964).

1942 (1)

E. Hopf, “Abzweigung einer periodischen losung von einer stationaren losung eines differentialsystem,” Ber. Math.-Phys. Kl. Sachs. Akad. Wiss. Leipzig (1942), p. 94.

Abraham, N. B.

G. P. Puccioni, G. L. Lippi, N. B. Abraham, and F. T. Arecchi, “Differences in polarization dynamics of the electromagnetic field in xenon and neon lasers,” Opt. Commun. 72, 361 (1989)
[Crossref]

R. S. Gioggia and N. B. Abraham, “Anomalous mode pulling, instabilities and chaos in a single-mode, standing-wave 3.39-μ m He–Ne laser,” Phys. Rev. A 29, 1304 (1984).
[Crossref]

Aissaoui, B.

G. Stéphan, A. D. May, R. E. Mueller, and B. Aissaoui, “Competition effects in the polarization of light in a quasi-isotropic laser,” J. Opt. Soc. Am. B 4, 1276 (1987).
[Crossref]

G. Stéphan, A. D. May, R. E. Mueller, and B. Aissaoui, “Effets de competition dans la polarisation de la lumière d’un laser quasi-isotrope,” J. Phys. (Paris) 48, C7-549 (1987). See also G. Stéphan and D. Hugon, “Light polarization of a quasi-isotropic laser with optical feedback,” Phys. Rev. Lett. 55, 703 (1985); S. T. Hendow, R. W. Dunn, W. W. Chow, and J. G. Small, “Observation of bistable behavior in the polarization of a laser,” Opt. Lett. 7, 356 (1982).
[Crossref] [PubMed]

Akimoto, O.

K. Ikeda and O. Akimoto, “Instability leading to periodic and chaotic self-pulsations in a bistable optical cavity,” Phys. Rev. Lett. 48, 617 (1982).
[Crossref]

Arecchi, F. T.

G. P. Puccioni, G. L. Lippi, N. B. Abraham, and F. T. Arecchi, “Differences in polarization dynamics of the electromagnetic field in xenon and neon lasers,” Opt. Commun. 72, 361 (1989)
[Crossref]

Borisova, M. S.

M. S. Borisova and I. P. Mazanko, “Stability of the state of polarization of the emission of a single mode gas laser with slightly anisotropic resonator,” Opt. Spectrosk. 47, 126 (1979) [Opt. Spectrosc. (USSR) 47, 69 (1979)].

Born, M.

Many authors use the Stokes parameters when discussing quasi-isotropic lasers. We have not yet found them to be particularly useful. The relationships between our variables and the Stokes parameters are S0= 4I, S1= 4I(1 − η2)1/2cos ϕ0, S2= 4I(1 − η2)1/2sin ϕ0, and S3= 4Iη. For a definition of the Stokes parameters see M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 24–30.

Bourouis, M.

W. Xiong, P. Glanznig, P. Paddon, A. D. May, M. Bourouis, S. Laniepce, and G. M. Stéphan, “Stability of polarized modes in a quasi-isotropic laser: experimental confirmation,” J. Opt. Soc. Am. B 8, 1236 (1991).
[Crossref]

M. Bourouis, S. Laniepce, G. Stéphan, P. Glanznig, P. Paddon, W. Xiong, and A. D. May, “Looking for polarization catastrophies in quasi-isotropic lasers,” in 1990 OSA Proceedings on Dynamics in Optical Systems, N. B. Abraham, E. Garmire, and P. Mandel, eds. (Optical Society of America, Washington, D.C., 1991), pp. 283–286.

Brannon, P. J.

Casperson, L.

L. Casperson and A. Yariv, “Longitudinal modes in a high-gain laser,” Appl. Phys. Lett. 17, 259 (1970).
[Crossref]

Casperson, L. W.

L. W. Casperson, “Stability criteria for high-intensity lasers,” Phys. Rev. A 21, 911 (1980).
[Crossref]

Fork, R. L.

M. Sargent, W. E. Lamb, and R. L. Fork, “Theory of a Zeeman laser I,” Phys. Rev. 164, 436 (1967).
[Crossref]

Fowles, G. R.

We have adhered to the terminology of G. R. Fowles, Introduction to Modern Optics, 2nd ed. (Dover, New York, 1989), pp. 28–31, for describing the polarization states of optical fields. We differ only by a factor of 2. Note, however, yet another typographical error in Ref. 5; there is a sign error in the definition of E+below Eq. (1) on p. 2356.

Gioggia, R. S.

R. S. Gioggia and N. B. Abraham, “Anomalous mode pulling, instabilities and chaos in a single-mode, standing-wave 3.39-μ m He–Ne laser,” Phys. Rev. A 29, 1304 (1984).
[Crossref]

Glanznig, P.

W. Xiong, P. Glanznig, P. Paddon, A. D. May, M. Bourouis, S. Laniepce, and G. M. Stéphan, “Stability of polarized modes in a quasi-isotropic laser: experimental confirmation,” J. Opt. Soc. Am. B 8, 1236 (1991).
[Crossref]

P. Paddon, P. Glanznig, A. D. May, and G. Stéphan, “Comportement catastrophique du domain de bistabilité d’un laser vectoriel,” Ann. Phys. (N.Y.) 15, 177 (1990).

M. Bourouis, S. Laniepce, G. Stéphan, P. Glanznig, P. Paddon, W. Xiong, and A. D. May, “Looking for polarization catastrophies in quasi-isotropic lasers,” in 1990 OSA Proceedings on Dynamics in Optical Systems, N. B. Abraham, E. Garmire, and P. Mandel, eds. (Optical Society of America, Washington, D.C., 1991), pp. 283–286.

Hall, K. L.

Haus, H. A.

Hopf, E.

E. Hopf, “Abzweigung einer periodischen losung von einer stationaren losung eines differentialsystem,” Ber. Math.-Phys. Kl. Sachs. Akad. Wiss. Leipzig (1942), p. 94.

Ikeda, K.

K. Ikeda and O. Akimoto, “Instability leading to periodic and chaotic self-pulsations in a bistable optical cavity,” Phys. Rev. Lett. 48, 617 (1982).
[Crossref]

Ippen, E. P.

Kitano, M.

M. Kitano, T. Yabuzaki, and T. Ogawa, “Optical tristability,” Phys. Rev. Lett. 46, 926 (1981).
[Crossref]

Kubo, H.

H. Kubo and R. Nagata, “Stokes parameters representation of the light propagation equations in inhomogeneous anisotropic, optically active media,” Opt. Commun. 34, 306 (1980).
[Crossref]

Lamb, W. E.

M. Sargent, W. E. Lamb, and R. L. Fork, “Theory of a Zeeman laser I,” Phys. Rev. 164, 436 (1967).
[Crossref]

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. A 134, 1429 (1964).

Laniepce, S.

W. Xiong, P. Glanznig, P. Paddon, A. D. May, M. Bourouis, S. Laniepce, and G. M. Stéphan, “Stability of polarized modes in a quasi-isotropic laser: experimental confirmation,” J. Opt. Soc. Am. B 8, 1236 (1991).
[Crossref]

M. Bourouis, S. Laniepce, G. Stéphan, P. Glanznig, P. Paddon, W. Xiong, and A. D. May, “Looking for polarization catastrophies in quasi-isotropic lasers,” in 1990 OSA Proceedings on Dynamics in Optical Systems, N. B. Abraham, E. Garmire, and P. Mandel, eds. (Optical Society of America, Washington, D.C., 1991), pp. 283–286.

Le Floch, A.

G. Stéphan, R. Le Naour, and A. Le Floch, “Experimental and theoretical study of the anisotropy induced in a gas laser by a saturating field,” Phys. Rev. A 17, 733 (1978).
[Crossref]

Le Naour, R.

G. Stéphan, R. Le Naour, and A. Le Floch, “Experimental and theoretical study of the anisotropy induced in a gas laser by a saturating field,” Phys. Rev. A 17, 733 (1978).
[Crossref]

Lenstra, D.

D. Lenstra, “On the theory of polarization effects in a gas laser,” Phy. Rep. 59, 299 (1980).
[Crossref]

Lippi, G. L.

G. P. Puccioni, G. L. Lippi, N. B. Abraham, and F. T. Arecchi, “Differences in polarization dynamics of the electromagnetic field in xenon and neon lasers,” Opt. Commun. 72, 361 (1989)
[Crossref]

Liu, L. Y.

Mark, J.

May, A. D.

W. Xiong, P. Glanznig, P. Paddon, A. D. May, M. Bourouis, S. Laniepce, and G. M. Stéphan, “Stability of polarized modes in a quasi-isotropic laser: experimental confirmation,” J. Opt. Soc. Am. B 8, 1236 (1991).
[Crossref]

P. Paddon, P. Glanznig, A. D. May, and G. Stéphan, “Comportement catastrophique du domain de bistabilité d’un laser vectoriel,” Ann. Phys. (N.Y.) 15, 177 (1990).

A. D. May and G. Stephan, “Stability of polarized modes in a quasi-isotropic laser,” J. Opt. Soc. Am. B 6, 2355 (1989).
[Crossref]

G. Stéphan, A. D. May, R. E. Mueller, and B. Aissaoui, “Competition effects in the polarization of light in a quasi-isotropic laser,” J. Opt. Soc. Am. B 4, 1276 (1987).
[Crossref]

G. Stéphan, A. D. May, R. E. Mueller, and B. Aissaoui, “Effets de competition dans la polarisation de la lumière d’un laser quasi-isotrope,” J. Phys. (Paris) 48, C7-549 (1987). See also G. Stéphan and D. Hugon, “Light polarization of a quasi-isotropic laser with optical feedback,” Phys. Rev. Lett. 55, 703 (1985); S. T. Hendow, R. W. Dunn, W. W. Chow, and J. G. Small, “Observation of bistable behavior in the polarization of a laser,” Opt. Lett. 7, 356 (1982).
[Crossref] [PubMed]

M. Bourouis, S. Laniepce, G. Stéphan, P. Glanznig, P. Paddon, W. Xiong, and A. D. May, “Looking for polarization catastrophies in quasi-isotropic lasers,” in 1990 OSA Proceedings on Dynamics in Optical Systems, N. B. Abraham, E. Garmire, and P. Mandel, eds. (Optical Society of America, Washington, D.C., 1991), pp. 283–286.

Mazanko, I. P.

M. S. Borisova and I. P. Mazanko, “Stability of the state of polarization of the emission of a single mode gas laser with slightly anisotropic resonator,” Opt. Spectrosk. 47, 126 (1979) [Opt. Spectrosc. (USSR) 47, 69 (1979)].

Messiah, A.

A. Messiah, Quantum Mechanics VII (Wiley, New York, 1962), p. 545.

Mueller, R. E.

G. Stéphan, A. D. May, R. E. Mueller, and B. Aissaoui, “Effets de competition dans la polarisation de la lumière d’un laser quasi-isotrope,” J. Phys. (Paris) 48, C7-549 (1987). See also G. Stéphan and D. Hugon, “Light polarization of a quasi-isotropic laser with optical feedback,” Phys. Rev. Lett. 55, 703 (1985); S. T. Hendow, R. W. Dunn, W. W. Chow, and J. G. Small, “Observation of bistable behavior in the polarization of a laser,” Opt. Lett. 7, 356 (1982).
[Crossref] [PubMed]

G. Stéphan, A. D. May, R. E. Mueller, and B. Aissaoui, “Competition effects in the polarization of light in a quasi-isotropic laser,” J. Opt. Soc. Am. B 4, 1276 (1987).
[Crossref]

Nagata, R.

H. Kubo and R. Nagata, “Stokes parameters representation of the light propagation equations in inhomogeneous anisotropic, optically active media,” Opt. Commun. 34, 306 (1980).
[Crossref]

Ogawa, T.

M. Kitano, T. Yabuzaki, and T. Ogawa, “Optical tristability,” Phys. Rev. Lett. 46, 926 (1981).
[Crossref]

Paddon, P.

W. Xiong, P. Glanznig, P. Paddon, A. D. May, M. Bourouis, S. Laniepce, and G. M. Stéphan, “Stability of polarized modes in a quasi-isotropic laser: experimental confirmation,” J. Opt. Soc. Am. B 8, 1236 (1991).
[Crossref]

P. Paddon, P. Glanznig, A. D. May, and G. Stéphan, “Comportement catastrophique du domain de bistabilité d’un laser vectoriel,” Ann. Phys. (N.Y.) 15, 177 (1990).

M. Bourouis, S. Laniepce, G. Stéphan, P. Glanznig, P. Paddon, W. Xiong, and A. D. May, “Looking for polarization catastrophies in quasi-isotropic lasers,” in 1990 OSA Proceedings on Dynamics in Optical Systems, N. B. Abraham, E. Garmire, and P. Mandel, eds. (Optical Society of America, Washington, D.C., 1991), pp. 283–286.

Puccioni, G. P.

G. P. Puccioni, G. L. Lippi, N. B. Abraham, and F. T. Arecchi, “Differences in polarization dynamics of the electromagnetic field in xenon and neon lasers,” Opt. Commun. 72, 361 (1989)
[Crossref]

Sargent, M.

M. Sargent, W. E. Lamb, and R. L. Fork, “Theory of a Zeeman laser I,” Phys. Rev. 164, 436 (1967).
[Crossref]

Seydel, R.

R. Seydel, From Equilibrium to Chaos (Elsevier, New York, 1988).

Sipe, J. E.

Smirnov, V. S.

V. S. Smirnov and A. M. Tumaikin, “Polarization phenomena in a gas laser with an anisotropic resonator,” Opt. Spektrosk. 40, 1030 (1976) [Opt. Spectrosc. (USSR) 40, 593 (1976)].

Stephan, G.

Stéphan, G.

P. Paddon, P. Glanznig, A. D. May, and G. Stéphan, “Comportement catastrophique du domain de bistabilité d’un laser vectoriel,” Ann. Phys. (N.Y.) 15, 177 (1990).

G. Stéphan, A. D. May, R. E. Mueller, and B. Aissaoui, “Competition effects in the polarization of light in a quasi-isotropic laser,” J. Opt. Soc. Am. B 4, 1276 (1987).
[Crossref]

G. Stéphan, A. D. May, R. E. Mueller, and B. Aissaoui, “Effets de competition dans la polarisation de la lumière d’un laser quasi-isotrope,” J. Phys. (Paris) 48, C7-549 (1987). See also G. Stéphan and D. Hugon, “Light polarization of a quasi-isotropic laser with optical feedback,” Phys. Rev. Lett. 55, 703 (1985); S. T. Hendow, R. W. Dunn, W. W. Chow, and J. G. Small, “Observation of bistable behavior in the polarization of a laser,” Opt. Lett. 7, 356 (1982).
[Crossref] [PubMed]

G. Stéphan, R. Le Naour, and A. Le Floch, “Experimental and theoretical study of the anisotropy induced in a gas laser by a saturating field,” Phys. Rev. A 17, 733 (1978).
[Crossref]

M. Bourouis, S. Laniepce, G. Stéphan, P. Glanznig, P. Paddon, W. Xiong, and A. D. May, “Looking for polarization catastrophies in quasi-isotropic lasers,” in 1990 OSA Proceedings on Dynamics in Optical Systems, N. B. Abraham, E. Garmire, and P. Mandel, eds. (Optical Society of America, Washington, D.C., 1991), pp. 283–286.

Stéphan, G. M.

Tratnik, M. V.

Tumaikin, A. M.

V. S. Smirnov and A. M. Tumaikin, “Polarization phenomena in a gas laser with an anisotropic resonator,” Opt. Spektrosk. 40, 1030 (1976) [Opt. Spectrosc. (USSR) 40, 593 (1976)].

Van Haeringen, W.

W. Van Haeringen, “Polarization properties of a single-mode operating gas laser in a small axial magnetic field,” Phys. Rev. 158, 256 (1967).
[Crossref]

Wolf, E.

Many authors use the Stokes parameters when discussing quasi-isotropic lasers. We have not yet found them to be particularly useful. The relationships between our variables and the Stokes parameters are S0= 4I, S1= 4I(1 − η2)1/2cos ϕ0, S2= 4I(1 − η2)1/2sin ϕ0, and S3= 4Iη. For a definition of the Stokes parameters see M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 24–30.

Xiong, W.

W. Xiong, P. Glanznig, P. Paddon, A. D. May, M. Bourouis, S. Laniepce, and G. M. Stéphan, “Stability of polarized modes in a quasi-isotropic laser: experimental confirmation,” J. Opt. Soc. Am. B 8, 1236 (1991).
[Crossref]

M. Bourouis, S. Laniepce, G. Stéphan, P. Glanznig, P. Paddon, W. Xiong, and A. D. May, “Looking for polarization catastrophies in quasi-isotropic lasers,” in 1990 OSA Proceedings on Dynamics in Optical Systems, N. B. Abraham, E. Garmire, and P. Mandel, eds. (Optical Society of America, Washington, D.C., 1991), pp. 283–286.

Yabuzaki, T.

M. Kitano, T. Yabuzaki, and T. Ogawa, “Optical tristability,” Phys. Rev. Lett. 46, 926 (1981).
[Crossref]

Yariv, A.

L. Casperson and A. Yariv, “Longitudinal modes in a high-gain laser,” Appl. Phys. Lett. 17, 259 (1970).
[Crossref]

Ann. Phys. (N.Y.) (1)

P. Paddon, P. Glanznig, A. D. May, and G. Stéphan, “Comportement catastrophique du domain de bistabilité d’un laser vectoriel,” Ann. Phys. (N.Y.) 15, 177 (1990).

Appl. Opt. (1)

Appl. Phys. Lett. (1)

L. Casperson and A. Yariv, “Longitudinal modes in a high-gain laser,” Appl. Phys. Lett. 17, 259 (1970).
[Crossref]

Ber. Math.-Phys. Kl. Sachs. Akad. Wiss. Leipzig (1)

E. Hopf, “Abzweigung einer periodischen losung von einer stationaren losung eines differentialsystem,” Ber. Math.-Phys. Kl. Sachs. Akad. Wiss. Leipzig (1942), p. 94.

J. Opt. Soc. Am. B (5)

J. Phys. (Paris) (1)

G. Stéphan, A. D. May, R. E. Mueller, and B. Aissaoui, “Effets de competition dans la polarisation de la lumière d’un laser quasi-isotrope,” J. Phys. (Paris) 48, C7-549 (1987). See also G. Stéphan and D. Hugon, “Light polarization of a quasi-isotropic laser with optical feedback,” Phys. Rev. Lett. 55, 703 (1985); S. T. Hendow, R. W. Dunn, W. W. Chow, and J. G. Small, “Observation of bistable behavior in the polarization of a laser,” Opt. Lett. 7, 356 (1982).
[Crossref] [PubMed]

Opt. Commun. (2)

G. P. Puccioni, G. L. Lippi, N. B. Abraham, and F. T. Arecchi, “Differences in polarization dynamics of the electromagnetic field in xenon and neon lasers,” Opt. Commun. 72, 361 (1989)
[Crossref]

H. Kubo and R. Nagata, “Stokes parameters representation of the light propagation equations in inhomogeneous anisotropic, optically active media,” Opt. Commun. 34, 306 (1980).
[Crossref]

Opt. Lett. (1)

Opt. Spectrosk. (1)

M. S. Borisova and I. P. Mazanko, “Stability of the state of polarization of the emission of a single mode gas laser with slightly anisotropic resonator,” Opt. Spectrosk. 47, 126 (1979) [Opt. Spectrosc. (USSR) 47, 69 (1979)].

Opt. Spektrosk. (1)

V. S. Smirnov and A. M. Tumaikin, “Polarization phenomena in a gas laser with an anisotropic resonator,” Opt. Spektrosk. 40, 1030 (1976) [Opt. Spectrosc. (USSR) 40, 593 (1976)].

Phy. Rep. (1)

D. Lenstra, “On the theory of polarization effects in a gas laser,” Phy. Rep. 59, 299 (1980).
[Crossref]

Phys. Rev. (2)

W. Van Haeringen, “Polarization properties of a single-mode operating gas laser in a small axial magnetic field,” Phys. Rev. 158, 256 (1967).
[Crossref]

M. Sargent, W. E. Lamb, and R. L. Fork, “Theory of a Zeeman laser I,” Phys. Rev. 164, 436 (1967).
[Crossref]

Phys. Rev. A (4)

W. E. Lamb, “Theory of an optical maser,” Phys. Rev. A 134, 1429 (1964).

L. W. Casperson, “Stability criteria for high-intensity lasers,” Phys. Rev. A 21, 911 (1980).
[Crossref]

G. Stéphan, R. Le Naour, and A. Le Floch, “Experimental and theoretical study of the anisotropy induced in a gas laser by a saturating field,” Phys. Rev. A 17, 733 (1978).
[Crossref]

R. S. Gioggia and N. B. Abraham, “Anomalous mode pulling, instabilities and chaos in a single-mode, standing-wave 3.39-μ m He–Ne laser,” Phys. Rev. A 29, 1304 (1984).
[Crossref]

Phys. Rev. Lett. (2)

K. Ikeda and O. Akimoto, “Instability leading to periodic and chaotic self-pulsations in a bistable optical cavity,” Phys. Rev. Lett. 48, 617 (1982).
[Crossref]

M. Kitano, T. Yabuzaki, and T. Ogawa, “Optical tristability,” Phys. Rev. Lett. 46, 926 (1981).
[Crossref]

Other (10)

A. Messiah, Quantum Mechanics VII (Wiley, New York, 1962), p. 545.

Note the typographical error in Ref. 5 on p. 2357 just below Eq. (5). We have repeatedly checked that none of the typographical errors made it through to the computer code.

In the case of ϕf0= 0, one could debate whether it is the same or a different mode, since the jump is to another part of the same branch. Some authors consider a turning point as the junction of two branches.

M. Bourouis, S. Laniepce, G. Stéphan, P. Glanznig, P. Paddon, W. Xiong, and A. D. May, “Looking for polarization catastrophies in quasi-isotropic lasers,” in 1990 OSA Proceedings on Dynamics in Optical Systems, N. B. Abraham, E. Garmire, and P. Mandel, eds. (Optical Society of America, Washington, D.C., 1991), pp. 283–286.

In terms of our variables the observed intensity ℐ equals I[1 − (1 − η2)1/2cos(ϕ0− 2ψ].

We have adhered to the terminology of G. R. Fowles, Introduction to Modern Optics, 2nd ed. (Dover, New York, 1989), pp. 28–31, for describing the polarization states of optical fields. We differ only by a factor of 2. Note, however, yet another typographical error in Ref. 5; there is a sign error in the definition of E+below Eq. (1) on p. 2356.

R. Seydel, From Equilibrium to Chaos (Elsevier, New York, 1988).

Many authors use the Stokes parameters when discussing quasi-isotropic lasers. We have not yet found them to be particularly useful. The relationships between our variables and the Stokes parameters are S0= 4I, S1= 4I(1 − η2)1/2cos ϕ0, S2= 4I(1 − η2)1/2sin ϕ0, and S3= 4Iη. For a definition of the Stokes parameters see M. Born and E. Wolf, Principles of Optics (Pergamon, New York, 1959), pp. 24–30.

This excludes the 100% circularly polarized and the I= 0 solutions. However, their behavior may be treated by an examination of the equations in their original form, i.e., Eqs. (1′)–(4′). The only novel aspect of the I= 0 solutions is that one polarization mode generally comes above threshold before the other. This is easy to understand, since the optical length of the cavity depends on the polarization. This is important when one considers experiments in which one sweeps the laser back and forward across the gain profile by modulating the length of the laser.

A. Abromowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, Applied Mathematics Series 55 (National Bureau of Standards, Washington, D.C.).

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Figures (13)

Fig. 1
Fig. 1

Schematic of the experimental arrangement used in Refs. 6 and 9 showing the orientation of the optical components. In this arrangement the passing direction of the polarizer, yf, determines the polarization of light reflected back from the detector into the laser.

Fig. 2
Fig. 2

Reproduction from Ref. 9 of the intensity observed through a polarizer inclined at 45° to the axis of birefringence of the laser. Weak feedback arises from light reflected from the detector face. The detector was approximately the length of the laser from the output mirror. The curves, which we refer to as bat’s ears, were obtained for different values of the phase origin ϕf0: (a) 0, (b) π, (c) −π/2, and (d) π/2 rad.

Fig. 3
Fig. 3

Reproduction from Ref. 9 of the intensity observed through a polarizer oriented parallel to the axis of birefringence of the laser tube. No feedback came from the detector. Strong feedback, through a polarizer oriented at (a) 45° and (b) 35° to the birefringence axis, was provided by a mirror placed 22 m from the laser. Curve (c) is an enlargement of part of (a).

Fig. 4
Fig. 4

Calculated solutions: (a) relative phase ϕ0, (b) ellipticity η, (c) intensity I, and (d) the largest Liapunov exponent λ for a He–Ne laser operating at 3.39 μm. In (a)–(c) the stable solutions, i.e., those with Re(λ) less than zero, are shown as solid curves. The internal and external axes are parallel.

Fig. 5
Fig. 5

Same conditions as for Fig. 4 except at higher gain. Note that in (b) and (c) the unstable (dashed) part of the curve for polarization state B is hidden by the solid curve for the stable polarization state A.

Fig. 6
Fig. 6

Same conditions as for Fig. 4 except at still higher gain. Note that in (b) and (c) the unstable (dashed) part of the curve for polarization state B is hidden by the solid curve for the stable polarization state A.

Fig. 7
Fig. 7

Calculated bat’s ears. The control parameters were chosen to match what we estimate to have been the experimental values appropriate to Fig. 2.

Fig. 8
Fig. 8

Calculated ellipticity corresponding to Fig. 7.

Fig. 9
Fig. 9

Calculated relative phase of the right- and left-handed components corresponding to Fig. 7. The angle ϕ0/2 gives the orientation of the major axis of the polarization ellipse.

Fig. 10
Fig. 10

Calculated intensity for the “porc-épic” effect. The values of the control parameters were chosen to match what we estimate to have been the experimental values corresponding to Fig. 3.

Fig. 11
Fig. 11

(a)–(c) Calculated stable solutions and (d) the real part of the largest Liapunov exponent for the small region shown in Fig. 10.

Fig. 12
Fig. 12

Same as Fig. 11 except for the other region marked in Fig. 10.

Fig. 13
Fig. 13

Calculated length versus frequency showing the artificial Casperson multiplication of the modes arising from the rapid variation in the phase of the feedback. Note the expanded length scales; the overall scale of the main figure covers two peaks in Fig. 10 or 11.

Tables (1)

Tables Icon

Table 1 Control Parameters Used in Calculating the Figuresa

Equations (72)

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- 2 E z 2 + μ 0 0 2 E t 2 = - μ 0 2 P t 2 .
( E + E - ) { exp [ - i ( ω t - k z ) ] - exp [ - i ( ω t + k z ) ] } + c . c . ,
E ˙ + = [ α + - L + a - β + I + - θ + I - ] E + + b E - , E ˙ - = [ α - - L + d - β - I - - θ - I + ] E - + b E + ,
E ˙ + = E + [ α + r + a r - β + r I + - θ + r I - ] + E - [ b exp ( - i ϕ 0 ) ] r ,
E ˙ - = E - [ α - r + d r - β - r I - - θ - r I + ] + E + [ c exp ( i ϕ 0 ) ] r ,
E + ( ϕ ˙ + ϕ ˙ 0 / 2 ) = E + [ α + i + a i - β + i I + - θ + i I - ] + E - [ b exp ( - i ϕ 0 ) ] i ,
E - ( ϕ ˙ - ϕ ˙ 0 / 2 ) = E - [ α - i + d i - β - i I - - θ - i I + ] + E + [ c exp ( i ϕ 0 ) ] i ,
E ˙ + = E + [ C 1 - β + r I + - θ + r I - ] + E - Φ 1 ,
E ˙ - = E - [ C 2 - β - r I - - θ - r I + ] + E + Φ 2 ,
E + ( ϕ ˙ + ϕ ˙ 0 / 2 ) = E + [ C 3 - β + i I + - θ + i I - ] + E - Φ 3 ,
E - ( ϕ ˙ - ϕ ˙ 0 / 2 ) = E - [ C 4 - β - i I - - θ - i I + ] + E + Φ 4 ,
I ˙ / I = ( 1 + η ) [ C 1 - β + r ( 1 + η ) - θ + r I ( 1 - η ) ] + ( 1 - η ) [ C 2 - β - r I ( 1 - η ) - θ - r I ( 1 + η ) ] + ( Φ 1 + Φ 2 ) ( 1 - η 2 ) 1 / 2 ,
η ˙ = ( 1 - η 2 ) [ C 1 - β + r ( 1 + η ) - θ + r I ( 1 - η ) - C 2 + β - r I ( 1 - η ) + θ - r I ( 1 + η ) ] + [ ( Φ 1 + Φ 2 ) - η ( Φ 1 + Φ 2 ) ] ( 1 - η 2 ) 1 / 2 ,
ϕ ˙ 0 = [ C 3 - β + i I ( 1 + η ) - θ + i I ( 1 - η ) - C 4 + β - i I ( 1 - η ) + θ - i I ( 1 + η ) ] + Φ 3 [ ( 1 - η ) / ( 1 + η ) ] 1 / 2 - Φ 4 [ ( 1 + η ) / ( 1 - η ) ] 1 / 2 ,
2 ϕ ˙ = [ C 3 - β + i I ( 1 + η ) - θ + i I ( 1 - η ) - C 4 - β - i I ( 1 - η ) - θ - i I ( 1 + η ) ] + Φ 3 [ ( 1 - η ) / ( 1 + η ) ] 1 / 2 + Φ 4 [ ( 1 + η ) / ( 1 - η ) ] 1 / 2 .
0 = ( C 1 + C 2 ) + η ( C 1 - C 2 ) - I { [ ( β + r + β - r ) + ( θ + r + θ - r ) ] + 2 η ( β + r + β - r ) + η 2 [ ( β + r + β - r ) - ( θ + r + θ - r ) ] } + ( Φ 1 + Φ 2 ) ( 1 - η 2 ) 1 / 2 ,
0 = ( C 1 - C 2 ) - I { [ ( β + r - β - r ) + ( θ + r - θ - r ) ] + η [ ( β + r + β - r ) - ( θ + r + θ - r ) ] } + [ ( Φ 1 + Φ 2 ) - η ( Φ 1 + Φ 2 ) ] ( 1 - η 2 ) - 1 / 2 ,
0 = ( C 3 - C 4 ) - I { [ ( β + i - β - i ) + ( θ + i - θ - i ) ] + η [ ( β + i + β - i ) - ( θ + i + θ - i ) ] } + Φ 3 [ ( 1 - η ) / ( 1 + η ) ] 1 / 2 - Φ 4 [ ( 1 + η ) / ( 1 - η ) ] 1 / 2 ,
0 = ( C 3 + C 4 ) - I { [ ( β + i + β - i ) + ( θ + i + θ - i ) ] + η [ ( β + i - β - i ) - ( θ + i - θ - i ) ] } + Φ 3 [ ( 1 - η ) / ( 1 + η ) ] 1 / 2 + Φ 4 [ ( 1 + η ) / ( 1 - η ) ] 1 / 2 ,
0 = C 5 + C 6 η - I ( G 1 + η G 2 + η 2 G 3 ) + Φ 5 ( 1 - η 2 ) 1 / 2 ,
0 = C 6 - I ( G 4 + η G 3 ) + ( Φ 6 - η Φ 5 ) ( 1 - η 2 ) 1 / 2 ,
0 = C 7 - I ( G 5 + η G 6 ) + Φ 3 [ ( 1 - η ) / ( 1 + η ) ] 1 / 2 - Φ 4 [ ( 1 + η ) / ( 1 - η ) ] 1 / 2 ,
0 = C 8 - I ( G 7 + η G 8 ) + Φ 3 [ ( 1 - η ) / ( 1 + η ) ] 1 / 2 + Φ 4 [ ( 1 + η ) / ( 1 - η ) ] 1 / 2 ,
( G 4 + η G 3 ) / ( G 5 + η G 6 ) = [ C 6 ( 1 - η 2 ) 1 / 2 + Φ 6 - η Φ 5 ] / [ C 7 ( 1 - η 2 ) 1 / 2 + Φ 8 - η Φ 7 ] ,
E ˙ + = E + [ α r + w cos ( ϕ w ) + f cos ( ϕ f ) - β r I + - θ r I - ] - E + [ w cos ( ϕ w - ϕ 0 ) + f cos ( ϕ f - ϕ 0 - 2 ψ ) ] ,
E ˙ - = E - [ α r + w cos ( ϕ w ) + f cos ( ϕ f ) - β r I - - θ r I + ] - E + [ w cos ( ϕ w + ϕ 0 ) + f cos ( ϕ f + ϕ 0 + 2 ψ ) ] ,
E + ( ϕ ˙ + ϕ ˙ 0 / 2 ) = E + [ α i + w sin ( ϕ w ) + f sin ( ϕ f ) - β i I + - θ i I - ] - E - [ w sin ( ϕ w - ϕ 0 ) + f sin ( ϕ f - ϕ 0 - 2 ψ ) ] ,
E - ( ϕ ˙ + ϕ ˙ 0 / 2 ) = E - [ α i + w sin ( ϕ w ) + f sin ( ϕ f ) - β i I - - θ i I + ] - E + [ w sin ( ϕ w + ϕ 0 ) + f sin ( ϕ f + ϕ 0 + 2 ψ ) ] .
δ I ˙ / I = - { G 1 + η 2 G 3 } δ I - { I 2 η G 3 + Φ 5 η ( 1 - η 2 ) - 1 / 2 } δ η + { Φ 8 ( 1 - η 2 ) 1 / 2 } δ ϕ 0 ,
δ η ˙ / ( 1 - η 2 ) = - { η G 3 } δ I - { I G 3 + Φ 5 ( 1 - η 2 ) - 1 / 2 - η ( Φ 6 - η Φ 5 ) ( 1 - η 2 ) - 3 / 2 } δ η + { ( Φ 7 - η Φ 8 ) ( 1 - η 2 ) - 1 / 2 } δ ϕ 0 ,
δ ϕ ˙ 0 = - { η G 6 } δ I + { - I G 6 - Φ 3 [ ( 1 - η ) - 1 / 2 × ( 1 + η ) - 3 / 2 ] - Φ 4 [ ( 1 + η ) - 1 / 2 × ( 1 - η 2 ) - 3 / 2 } δ η - { Φ 1 [ ( 1 - η ) / ( 1 + η ) ] 1 / 2 + Φ 2 [ ( 1 + η ) / ( 1 - η ) ] 1 / 2 } δ ϕ 0 ,
2 δ ϕ ˙ = { - G 7 } δ I + { - Φ 3 [ ( 1 - η ) - 1 / 2 ( 1 + η ) - 3 / 2 ] + Φ 4 [ ( 1 + η ) - 1 / 2 ( 1 - η ) - 3 / 2 ] } δ η - { Φ 1 [ ( 1 - η ) / ( 1 + η ) ] 1 / 2 - Φ 2 [ ( 1 + η ) / ( 1 - η ) ] 1 / 2 } δ ϕ 0 ,
δ I ˙ / I = - { I G 1 } δ I / I - { 2 η I G 3 } δ η + { Φ 8 } δ ϕ 0 , δ η ˙ = - { η G 3 } δ I - { I G 3 + Φ 5 } δ η + { Φ 7 } δ ϕ 0 , δ ϕ ˙ 0 = - { η G 6 } δ I - { I G 6 + Φ 7 } δ η - { Φ 5 } δ ϕ 0 , 2 δ ϕ ˙ = - { G 7 } δ I - { Φ 8 } δ η - { Φ 6 } δ ϕ 0 .
δ I ˙ = - { I G 1 } δ I , δ η ˙ = - { I G 3 + Φ 5 } δ η + { Φ 7 } δ ϕ 0 , δ ϕ ˙ 0 = - { I G 6 + Φ 7 } δ η - { Φ 5 } δ ϕ 0 , 2 δ ϕ ˙ = - { G 7 } δ I - { Φ 8 } δ η - { Φ 6 } δ ϕ 0 .
λ = - ( 1 / 2 ) { I G 3 + 2 Φ 5 } ± ( 1 / 2 ) { [ I G 3 ] 2 - 4 [ Φ 7 ] [ I G 6 + Φ 7 ] } 1 / 2 ,
λ = - Φ 5 + { [ ( S r ) 2 - Φ 7 2 - 2 S i Φ 7 ] 1 / 2 - S r } ,
Φ 5 = - 2 [ w cos ϕ w cos ϕ 0 + f cos ϕ f cos ( ϕ 0 + 2 ψ ) ] , Φ 7 = - 2 [ w sin ϕ w cos ϕ 0 + f sin ϕ f cos ( ϕ 0 + 2 ψ ) ] ,
Φ 5 = - 2 [ w cos ϕ w cos ϕ 0 + f cos ϕ f cos ϕ 0 ] , Φ 7 = - 2 [ w sin ϕ w cos ϕ 0 + f sin ϕ f cos ϕ 0 ] ,
1 2 [ 1 1 - i i ] ,
1 2 [ 1 i 1 - i ] ,
[ 1 0 0 1 ] ,             [ 1 0 0 - 1 ] ,             - [ 0 1 1 0 ] ,             i [ 0 1 - 1 0 ] .
[ 1 0 0 1 ] ,             [ 0 1 1 0 ] ,             [ 0 - i i 0 ] ,             [ 1 0 0 - 1 ] ,
M = [ exp ( i k x l ) 0 0 exp ( i k y l ) ] = exp ( i k l ) [ exp ( + i Δ b ) 0 0 exp ( - i Δ b ) ] = exp ( i k l ) [ 1 + i Δ b 0 0 1 - i Δ b ] = exp ( i k l ) { [ 1 0 0 1 ] + i Δ b [ 1 0 0 - 1 ] } ,
M b = exp ( i k l ) { [ 1 0 0 1 ] + i Δ b [ 0 1 1 0 ] } .
[ cos θ - sin θ sin θ cos θ ] ,
b [ cos 2 θ - sin 2 θ - sin 2 θ - cos 2 θ ] .
( b cos 2 θ ) [ 0 1 1 0 ] + ( b sin 2 θ ) [ 0 - i i 0 ]
b [ 0 exp ( - i 2 θ ) exp ( i 2 θ ) 0 ] ,
M b = exp ( i k l b ) { U + b [ 0 exp ( - i 2 θ ) exp ( i 2 θ ) 0 ] } .
M = [ exp ( i k x l ) 0 0 exp ( i k y l ) ] = exp ( i k l ) [ exp ( + i Δ ) 0 0 exp ( - i Δ ) ] = exp ( i k l ) [ 1 + i Δ 0 0 1 - i Δ ]             or M = exp ( i k l ) { [ 1 0 0 1 ] + i Δ [ 1 0 0 - 1 ] } .
M = exp ( i k l ) { [ 1 0 0 1 ] + [ 1 0 0 - 1 ] } .
M w = [ T 0 0 T + t ] = T { [ 1 0 0 1 ] + t T [ 0 0 0 1 ] } .
M w = T { [ 1 0 0 1 ] + t 2 T [ 1 0 0 1 ] - t 2 T [ 0 1 1 0 ] } .
M w = { [ w exp ( i ϕ w ) ] [ 1 0 0 1 ] - [ w exp ( i ϕ w ) ] [ 0 1 1 0 ] } .
[ R 0 0 R + r ] = R { [ 1 0 0 1 ] + r R [ 0 0 0 1 ] } ,
M f = R { [ 1 0 0 1 ] + r 2 R [ 1 0 0 1 ] - r 2 R [ 0 1 1 0 ] } .
M f = R { [ 1 0 0 1 ] + r 2 R [ 1 0 0 1 ] - r 2 R [ 0 exp ( - i 2 ψ ) exp ( i 2 ψ ) 0 ] } .
{ E + E - } { exp [ - i ( ω t - k z ) ] - exp [ - i ( ω t + k z ) ] } + c . c .
- 2 E z 2 + μ 0 σ · E t + μ 0 0 2 E t 2 = - μ 0 2 P t 2 ,
k 2 E + - i ω μ 0 σ + + E + - i ω μ 0 σ + - E - - μ 0 0 ω 2 E + - 2 i ω μ 0 0 ( E + / t ) = + μ 0 0 ω 2 ( χ + 1 + χ + 3 d I + + χ - 3 c I - ) E + ,
P + = 0 ( χ + 1 + χ + 3 d I + + χ - 3 c I - ) × E + { exp [ - i ( ω t - k z ) ] - exp [ - i ( ω t + k z ) ] } + c . c .
E + t = + i ω 2 { ( χ + 1 + χ + 3 d I + + χ - 3 c I - ) - k 2 c 2 ω 2 + 1 } × E + - σ + + 2 0 E + - σ + - 2 0 E -
E + t = { ( α + - β + I + - θ + I - ) } E + - σ + + 2 0 E + - σ + - 2 0 E - ,
{ E + E - } rt = exp ( i k i l i ) { U + i } { E + E - } 0 ,
{ E + E - } rt = exp ( - L ) { U + i } { E + E - } 0 .
( t ) c { E + E - } = { [ exp ( - L ) - 1 ] U + [ exp ( - L ) ] [ a i b i c i d i ] } × { E + E - } 0 ( c 2 L ) ,
( t ) c { E + E - } = { - L U + M c } { E + E - } 0 ( c 2 L ) ,
M c = [ a b c d ] = [ exp ( - L ) ] [ a i b i c i d i ] .
( t ) [ E + E - ] = { [ α + - β + I + - θ + I - 0 0 α - - β - I - - θ - I + ] + M c } [ E + E - ] ,
C 1 = α + r + a r α r + w cos ϕ w + f cos ϕ f , C 2 = α - r + d r α r + w cos ϕ w + f cos ϕ f , C 3 = α + i + a i α i + w sin ϕ w + f sin ϕ f , C 4 = α - i + d i α i + w sin ϕ w + f sin ϕ f , C 5 = C 1 + C 2 2 ( α r + w cos ϕ w + f cos ϕ f ) , C 6 = C 1 - C 2 0 , C 7 = C 3 - C 4 0 , C 8 = C 3 - C 4 2 ( α i + w sin ϕ w + f sin ϕ f ) ,
G 1 = ( β + r + β - r ) + ( θ + r + θ - r ) 2 ( β r + θ r ) , G 2 = 2 ( β + r - β - r ) 0 , G 3 = ( β + r + β - r ) - ( θ + r + θ - r ) 2 ( β r - θ r ) , G 4 = ( β + r - β - r ) + ( θ + r - θ - r ) 0 , G 5 = ( β + i - β - i ) + ( θ + i - θ - i ) 0 , G 6 = ( β + i + β - i ) - ( θ + i + θ - i ) 2 ( β i - θ i ) , G 7 = ( β + i + β - i ) + ( θ + i + θ - i ) 2 ( β i + θ i ) , G 8 = ( β + i - β - i ) - ( θ + i - θ - i ) 0.
Φ 1 = [ b exp ( - i ϕ 0 ) ] r - [ w cos ( ϕ w - ϕ 0 ) + f × cos ( ϕ f - ϕ 0 - 2 ψ ) ] , Φ 2 = [ c exp ( + i ϕ 0 ) ] r - [ w cos ( ϕ w + ϕ 0 ) + f × cos ( ϕ f + ϕ 0 + 2 ψ ) ] , Φ 3 = [ b exp ( - i ϕ 0 ) ] i - [ w sin ( ϕ w - ϕ 0 ) + f × sin ( ϕ f - ϕ 0 - 2 ψ ) ] , Φ 4 = [ c exp ( + i ϕ 0 ) ] i - [ w sin ( ϕ w + ϕ 0 ) + f × sin ( ϕ f + ϕ 0 + 2 ψ ) ] , Φ 5 = Φ 1 + Φ 2 - 2 [ w cos ϕ w cos ϕ 0 + f cos ϕ f × cos ( ϕ 0 + 2 ψ ) ] , Φ 6 = Φ 1 - Φ 2 - 2 [ w sin ϕ w sin ϕ 0 + f sin ϕ f × sin ( ϕ 0 + 2 ψ ) ] , Φ 7 = Φ 3 + Φ 4 - 2 [ w sin ϕ w cos ϕ 0 + f sin ϕ f × cos ( ϕ 0 + 2 ψ ) ] , Φ 8 = Φ 3 - Φ 4 + 2 [ w cos ϕ w sin ϕ 0 + f cos ϕ f × sin ( ϕ 0 + 2 ψ ) ] .

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