Abstract

A theoretical model is developed to predict the behavior of the TE and TM eigenstates of a laser when the Goos–Hänchen effect associated with a total internal reflection inserted inside the laser cavity is taken into account. The generalized Jones matrix describing the internal reflection on a plane interface is determined, and the dynamics of the linearly polarized eigenstates around the critical angle is investigated both theoretically and experimentally. Moreover, the strong coupling between the TE and TM eigenstates is shown to provide a direct visualization of the Goos–Hänchen lateral shift.

© 1992 Optical Society of America

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References

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  1. F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. (Leipzig) 1, 333–346 (1947);F. Goos, H. Lindberg-Hänchen, “Neumessung des Strahlversetzungeffektes bei Totalreflexion,” Ann. Phys. (Leipzig) 2, 87–102 (1949).
  2. A. Mazet, C. Imbert, S. Huard, “Effet Goos–Hänchen en lumière non polarisée: la réflexion totale sépare les états de polarisation rectiligne,” C. R. Acad.Sci. 273, 592–594 (1971).
  3. O. Costa de Beauregard, C. Imbert, Y. Levy, “Observation of shifts in total reflection of a light beam by a multilayered structure,” Phys. Rev. D 15, 3553–3562 (1977).
    [CrossRef]
  4. J. J. Cowan, B. Anicin, “Longitudinal and transverse displacement of a bounded microwave beam at total internal reflection,” J. Opt. Soc. Am. 67, 1307–1314 (1977).
    [CrossRef]
  5. K. Artmann, “Berechnung der Seitenversetzung des totalreflektieren Strahles,” Ann. Phys. (Leipzig) 2, 87–102 (1948).
  6. B. R. Horowitz, T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am. 61, 586–594 (1971).
    [CrossRef]
  7. M. McGuirk, C. K. Carniglia, “An angular spectrum approach to the Goos–Hänchen shift,” J. Opt. Soc. Am. 67, 103–107 (1977).
    [CrossRef]
  8. T. Tamir, “Nonspecular phenomena in beam fields reflected by multilayered media,” J. Opt. Soc. Am. A 3, 558–565 (1986).
    [CrossRef]
  9. H. M. Lai, F. C. Cheng, W. K. Tang, “Goos–Hänchen effect around and off the critical angle,” J. Opt. Soc. Am. A 3, 550–557 (1986).
    [CrossRef]
  10. W. Nasalski, “Modified reflectance and geometrical deformations of Gaussian beams reflected at dielectric interface,” J. Opt. Soc. Am. A 6, 1447–1454 (1989).
    [CrossRef]
  11. A. Le Floch, R. Le Naour, “Polarization effects in Zeeman lasers with x-y-type loss anisotropics,” Phys. Rev. A 4, 290–295 (1971).
    [CrossRef]
  12. A. Le Floch, G. Ropars, J. M. Lenormand, R. Le Naour, “Dynamics of laser eigenstates,” Phys. Rev. Lett. 52, 918–921 (1984).
    [CrossRef]
  13. R. C. Jones, “A new calculus for the treatment of optical systems. I. Description and discussion of the calculus,” J. Opt. Soc. Am. 31, 488–493 (1941).
    [CrossRef]
  14. F. Bretenaker, A. Le Floch, “The dynamics of spatially resolved laser eigenstates,” IEEE J. Quantum Electron. 26, 1451–1454 (1990).
    [CrossRef]
  15. F. Bretenaker, A. Le Floch, “Laser eigenstates in the framework of a spatially generalized Jones matrix formalism,” J. Opt. Soc. Am. B 8, 230–238 (1991).
    [CrossRef]
  16. F. Bretenaker, A. Le Floch, J. Davit, J. M. Chiquier, “One- and two-eigenstate stability domains in laser systems,” IEEE J. Quantum Electron. 28, 348–354 (1992).
    [CrossRef]
  17. J. C. Cotteverte, F. Bretenaker, A. Le Floch, “Dynamics of circularly polarized eigenstates in lasers with non-weak atomic coupling,” Opt. Lett. 16, 572–574 (1991).
    [CrossRef] [PubMed]
  18. W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N.J., 1964).
  19. H. De Lang, “Polarization properties of optical resonators, passive and active,” Ph.D. dissertation (University of Utrecht, Utrecht, The Netherlands, 1966).
  20. H. Greenstein, “Some properties of a Zeeman laser with anisotropic mirrors,” Phys. Rev. 178, 585–589 (1969).
    [CrossRef]
  21. M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, U.K., 1965).
  22. M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).
  23. J. C. Cotteverte, F. Bretenaker, A. Le Floch, “Study of the dynamical behaviour of the polarization of a quasiisotropic laser in the earth magnetic field,” Opt. Commun. 79, 321–327 (1990).
    [CrossRef]
  24. J. C. Cotteverte, F. Bretenaker, A. Le Floch, “Jones matrices of a tilted plate for Gaussian beams,” Appl. Opt. 30, 305–311 (1991).
    [CrossRef] [PubMed]
  25. G. Ropars, A. Le Floch, R. Le Naour, “Polarization slaving mechanisms in vectorial bistable lasers,” Europhys. Lett. 3, 695–703 (1987).
    [CrossRef]
  26. F. Bretenaker, A. Le Floch, L. Dutriaux, “Direct measurement of the optical Goos–Hänchen effect in lasers,” Phys. Rev. Lett. 68, 931–933 (1992).
    [CrossRef] [PubMed]

1992 (2)

F. Bretenaker, A. Le Floch, J. Davit, J. M. Chiquier, “One- and two-eigenstate stability domains in laser systems,” IEEE J. Quantum Electron. 28, 348–354 (1992).
[CrossRef]

F. Bretenaker, A. Le Floch, L. Dutriaux, “Direct measurement of the optical Goos–Hänchen effect in lasers,” Phys. Rev. Lett. 68, 931–933 (1992).
[CrossRef] [PubMed]

1991 (3)

1990 (2)

F. Bretenaker, A. Le Floch, “The dynamics of spatially resolved laser eigenstates,” IEEE J. Quantum Electron. 26, 1451–1454 (1990).
[CrossRef]

J. C. Cotteverte, F. Bretenaker, A. Le Floch, “Study of the dynamical behaviour of the polarization of a quasiisotropic laser in the earth magnetic field,” Opt. Commun. 79, 321–327 (1990).
[CrossRef]

1989 (1)

1987 (1)

G. Ropars, A. Le Floch, R. Le Naour, “Polarization slaving mechanisms in vectorial bistable lasers,” Europhys. Lett. 3, 695–703 (1987).
[CrossRef]

1986 (2)

1984 (1)

A. Le Floch, G. Ropars, J. M. Lenormand, R. Le Naour, “Dynamics of laser eigenstates,” Phys. Rev. Lett. 52, 918–921 (1984).
[CrossRef]

1977 (3)

1971 (3)

A. Mazet, C. Imbert, S. Huard, “Effet Goos–Hänchen en lumière non polarisée: la réflexion totale sépare les états de polarisation rectiligne,” C. R. Acad.Sci. 273, 592–594 (1971).

B. R. Horowitz, T. Tamir, “Lateral displacement of a light beam at a dielectric interface,” J. Opt. Soc. Am. 61, 586–594 (1971).
[CrossRef]

A. Le Floch, R. Le Naour, “Polarization effects in Zeeman lasers with x-y-type loss anisotropics,” Phys. Rev. A 4, 290–295 (1971).
[CrossRef]

1969 (1)

H. Greenstein, “Some properties of a Zeeman laser with anisotropic mirrors,” Phys. Rev. 178, 585–589 (1969).
[CrossRef]

1948 (1)

K. Artmann, “Berechnung der Seitenversetzung des totalreflektieren Strahles,” Ann. Phys. (Leipzig) 2, 87–102 (1948).

1947 (1)

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. (Leipzig) 1, 333–346 (1947);F. Goos, H. Lindberg-Hänchen, “Neumessung des Strahlversetzungeffektes bei Totalreflexion,” Ann. Phys. (Leipzig) 2, 87–102 (1949).

1941 (1)

Anicin, B.

Artmann, K.

K. Artmann, “Berechnung der Seitenversetzung des totalreflektieren Strahles,” Ann. Phys. (Leipzig) 2, 87–102 (1948).

Ballard, S. S.

W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N.J., 1964).

Born, M.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, U.K., 1965).

Bretenaker, F.

F. Bretenaker, A. Le Floch, L. Dutriaux, “Direct measurement of the optical Goos–Hänchen effect in lasers,” Phys. Rev. Lett. 68, 931–933 (1992).
[CrossRef] [PubMed]

F. Bretenaker, A. Le Floch, J. Davit, J. M. Chiquier, “One- and two-eigenstate stability domains in laser systems,” IEEE J. Quantum Electron. 28, 348–354 (1992).
[CrossRef]

F. Bretenaker, A. Le Floch, “Laser eigenstates in the framework of a spatially generalized Jones matrix formalism,” J. Opt. Soc. Am. B 8, 230–238 (1991).
[CrossRef]

J. C. Cotteverte, F. Bretenaker, A. Le Floch, “Dynamics of circularly polarized eigenstates in lasers with non-weak atomic coupling,” Opt. Lett. 16, 572–574 (1991).
[CrossRef] [PubMed]

J. C. Cotteverte, F. Bretenaker, A. Le Floch, “Jones matrices of a tilted plate for Gaussian beams,” Appl. Opt. 30, 305–311 (1991).
[CrossRef] [PubMed]

J. C. Cotteverte, F. Bretenaker, A. Le Floch, “Study of the dynamical behaviour of the polarization of a quasiisotropic laser in the earth magnetic field,” Opt. Commun. 79, 321–327 (1990).
[CrossRef]

F. Bretenaker, A. Le Floch, “The dynamics of spatially resolved laser eigenstates,” IEEE J. Quantum Electron. 26, 1451–1454 (1990).
[CrossRef]

Carniglia, C. K.

Cheng, F. C.

Chiquier, J. M.

F. Bretenaker, A. Le Floch, J. Davit, J. M. Chiquier, “One- and two-eigenstate stability domains in laser systems,” IEEE J. Quantum Electron. 28, 348–354 (1992).
[CrossRef]

Costa de Beauregard, O.

O. Costa de Beauregard, C. Imbert, Y. Levy, “Observation of shifts in total reflection of a light beam by a multilayered structure,” Phys. Rev. D 15, 3553–3562 (1977).
[CrossRef]

Cotteverte, J. C.

Cowan, J. J.

Davit, J.

F. Bretenaker, A. Le Floch, J. Davit, J. M. Chiquier, “One- and two-eigenstate stability domains in laser systems,” IEEE J. Quantum Electron. 28, 348–354 (1992).
[CrossRef]

De Lang, H.

H. De Lang, “Polarization properties of optical resonators, passive and active,” Ph.D. dissertation (University of Utrecht, Utrecht, The Netherlands, 1966).

Dutriaux, L.

F. Bretenaker, A. Le Floch, L. Dutriaux, “Direct measurement of the optical Goos–Hänchen effect in lasers,” Phys. Rev. Lett. 68, 931–933 (1992).
[CrossRef] [PubMed]

Goos, F.

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. (Leipzig) 1, 333–346 (1947);F. Goos, H. Lindberg-Hänchen, “Neumessung des Strahlversetzungeffektes bei Totalreflexion,” Ann. Phys. (Leipzig) 2, 87–102 (1949).

Greenstein, H.

H. Greenstein, “Some properties of a Zeeman laser with anisotropic mirrors,” Phys. Rev. 178, 585–589 (1969).
[CrossRef]

Hänchen, H.

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. (Leipzig) 1, 333–346 (1947);F. Goos, H. Lindberg-Hänchen, “Neumessung des Strahlversetzungeffektes bei Totalreflexion,” Ann. Phys. (Leipzig) 2, 87–102 (1949).

Horowitz, B. R.

Huard, S.

A. Mazet, C. Imbert, S. Huard, “Effet Goos–Hänchen en lumière non polarisée: la réflexion totale sépare les états de polarisation rectiligne,” C. R. Acad.Sci. 273, 592–594 (1971).

Imbert, C.

O. Costa de Beauregard, C. Imbert, Y. Levy, “Observation of shifts in total reflection of a light beam by a multilayered structure,” Phys. Rev. D 15, 3553–3562 (1977).
[CrossRef]

A. Mazet, C. Imbert, S. Huard, “Effet Goos–Hänchen en lumière non polarisée: la réflexion totale sépare les états de polarisation rectiligne,” C. R. Acad.Sci. 273, 592–594 (1971).

Jones, R. C.

Lai, H. M.

Lamb, W. E.

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).

Le Floch, A.

F. Bretenaker, A. Le Floch, L. Dutriaux, “Direct measurement of the optical Goos–Hänchen effect in lasers,” Phys. Rev. Lett. 68, 931–933 (1992).
[CrossRef] [PubMed]

F. Bretenaker, A. Le Floch, J. Davit, J. M. Chiquier, “One- and two-eigenstate stability domains in laser systems,” IEEE J. Quantum Electron. 28, 348–354 (1992).
[CrossRef]

F. Bretenaker, A. Le Floch, “Laser eigenstates in the framework of a spatially generalized Jones matrix formalism,” J. Opt. Soc. Am. B 8, 230–238 (1991).
[CrossRef]

J. C. Cotteverte, F. Bretenaker, A. Le Floch, “Jones matrices of a tilted plate for Gaussian beams,” Appl. Opt. 30, 305–311 (1991).
[CrossRef] [PubMed]

J. C. Cotteverte, F. Bretenaker, A. Le Floch, “Dynamics of circularly polarized eigenstates in lasers with non-weak atomic coupling,” Opt. Lett. 16, 572–574 (1991).
[CrossRef] [PubMed]

J. C. Cotteverte, F. Bretenaker, A. Le Floch, “Study of the dynamical behaviour of the polarization of a quasiisotropic laser in the earth magnetic field,” Opt. Commun. 79, 321–327 (1990).
[CrossRef]

F. Bretenaker, A. Le Floch, “The dynamics of spatially resolved laser eigenstates,” IEEE J. Quantum Electron. 26, 1451–1454 (1990).
[CrossRef]

G. Ropars, A. Le Floch, R. Le Naour, “Polarization slaving mechanisms in vectorial bistable lasers,” Europhys. Lett. 3, 695–703 (1987).
[CrossRef]

A. Le Floch, G. Ropars, J. M. Lenormand, R. Le Naour, “Dynamics of laser eigenstates,” Phys. Rev. Lett. 52, 918–921 (1984).
[CrossRef]

A. Le Floch, R. Le Naour, “Polarization effects in Zeeman lasers with x-y-type loss anisotropics,” Phys. Rev. A 4, 290–295 (1971).
[CrossRef]

Le Naour, R.

G. Ropars, A. Le Floch, R. Le Naour, “Polarization slaving mechanisms in vectorial bistable lasers,” Europhys. Lett. 3, 695–703 (1987).
[CrossRef]

A. Le Floch, G. Ropars, J. M. Lenormand, R. Le Naour, “Dynamics of laser eigenstates,” Phys. Rev. Lett. 52, 918–921 (1984).
[CrossRef]

A. Le Floch, R. Le Naour, “Polarization effects in Zeeman lasers with x-y-type loss anisotropics,” Phys. Rev. A 4, 290–295 (1971).
[CrossRef]

Lenormand, J. M.

A. Le Floch, G. Ropars, J. M. Lenormand, R. Le Naour, “Dynamics of laser eigenstates,” Phys. Rev. Lett. 52, 918–921 (1984).
[CrossRef]

Levy, Y.

O. Costa de Beauregard, C. Imbert, Y. Levy, “Observation of shifts in total reflection of a light beam by a multilayered structure,” Phys. Rev. D 15, 3553–3562 (1977).
[CrossRef]

Mazet, A.

A. Mazet, C. Imbert, S. Huard, “Effet Goos–Hänchen en lumière non polarisée: la réflexion totale sépare les états de polarisation rectiligne,” C. R. Acad.Sci. 273, 592–594 (1971).

McGuirk, M.

Nasalski, W.

Ropars, G.

G. Ropars, A. Le Floch, R. Le Naour, “Polarization slaving mechanisms in vectorial bistable lasers,” Europhys. Lett. 3, 695–703 (1987).
[CrossRef]

A. Le Floch, G. Ropars, J. M. Lenormand, R. Le Naour, “Dynamics of laser eigenstates,” Phys. Rev. Lett. 52, 918–921 (1984).
[CrossRef]

Sargent, M.

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).

Scully, M. O.

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).

Shurcliff, W. A.

W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N.J., 1964).

Tamir, T.

Tang, W. K.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, U.K., 1965).

Ann. Phys. (Leipzig) (2)

K. Artmann, “Berechnung der Seitenversetzung des totalreflektieren Strahles,” Ann. Phys. (Leipzig) 2, 87–102 (1948).

F. Goos, H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. (Leipzig) 1, 333–346 (1947);F. Goos, H. Lindberg-Hänchen, “Neumessung des Strahlversetzungeffektes bei Totalreflexion,” Ann. Phys. (Leipzig) 2, 87–102 (1949).

Appl. Opt. (1)

C. R. Acad.Sci. (1)

A. Mazet, C. Imbert, S. Huard, “Effet Goos–Hänchen en lumière non polarisée: la réflexion totale sépare les états de polarisation rectiligne,” C. R. Acad.Sci. 273, 592–594 (1971).

Europhys. Lett. (1)

G. Ropars, A. Le Floch, R. Le Naour, “Polarization slaving mechanisms in vectorial bistable lasers,” Europhys. Lett. 3, 695–703 (1987).
[CrossRef]

IEEE J. Quantum Electron. (2)

F. Bretenaker, A. Le Floch, “The dynamics of spatially resolved laser eigenstates,” IEEE J. Quantum Electron. 26, 1451–1454 (1990).
[CrossRef]

F. Bretenaker, A. Le Floch, J. Davit, J. M. Chiquier, “One- and two-eigenstate stability domains in laser systems,” IEEE J. Quantum Electron. 28, 348–354 (1992).
[CrossRef]

J. Opt. Soc. Am. (4)

J. Opt. Soc. Am. A (3)

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

J. C. Cotteverte, F. Bretenaker, A. Le Floch, “Study of the dynamical behaviour of the polarization of a quasiisotropic laser in the earth magnetic field,” Opt. Commun. 79, 321–327 (1990).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

H. Greenstein, “Some properties of a Zeeman laser with anisotropic mirrors,” Phys. Rev. 178, 585–589 (1969).
[CrossRef]

Phys. Rev. A (1)

A. Le Floch, R. Le Naour, “Polarization effects in Zeeman lasers with x-y-type loss anisotropics,” Phys. Rev. A 4, 290–295 (1971).
[CrossRef]

Phys. Rev. D (1)

O. Costa de Beauregard, C. Imbert, Y. Levy, “Observation of shifts in total reflection of a light beam by a multilayered structure,” Phys. Rev. D 15, 3553–3562 (1977).
[CrossRef]

Phys. Rev. Lett. (2)

A. Le Floch, G. Ropars, J. M. Lenormand, R. Le Naour, “Dynamics of laser eigenstates,” Phys. Rev. Lett. 52, 918–921 (1984).
[CrossRef]

F. Bretenaker, A. Le Floch, L. Dutriaux, “Direct measurement of the optical Goos–Hänchen effect in lasers,” Phys. Rev. Lett. 68, 931–933 (1992).
[CrossRef] [PubMed]

Other (4)

M. Born, E. Wolf, Principles of Optics, 3rd ed. (Pergamon, Oxford, U.K., 1965).

M. Sargent, M. O. Scully, W. E. Lamb, Laser Physics (Addison-Wesley, Reading, Mass., 1974).

W. A. Shurcliff, S. S. Ballard, Polarized Light (Van Nostrand, Princeton, N.J., 1964).

H. De Lang, “Polarization properties of optical resonators, passive and active,” Ph.D. dissertation (University of Utrecht, Utrecht, The Netherlands, 1966).

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Figures (8)

Fig. 1
Fig. 1

The prism depicted here leads to a Goos–Hänchen effect at the total reflection from the prism to air. The lateral shift depends on the polarization.

Fig. 2
Fig. 2

(a) Absolute values of the reflection coefficients for the TE and TM polarizations versus angle of incidence i deduced from Fresnel’s laws for n = 1.409. Note the total reflection for i > ic (ic = 45.212°). (b) Phase shifts at total reflection undergone by the TE and TM polarizations (solid curves) versus angle of incidence i (n = 1.409). The dashed curve represents the difference between the two preceding curves, i.e., the phase anisotropy associated with the total reflection. (c) Goos–Hänchen lateral shifts DTE and DTM for the TE and TM polarizations versus angle of incidence i obtained from Artmann’s formulas with n = 1.409 and λ0 = 3.39 μm. These expressions are approximately valid for iic + θ.

Fig. 3
Fig. 3

Schematic representation of the prism as a box with two inputs and two outputs. The incident and output beams are represented by 4-vectors that are related by the spatially generalized 4 × 4 Jones matrix of the prism.

Fig. 4
Fig. 4

Experimental setup. The cavity may contain auxiliary phase and loss anisotropies Δϕ and Δp, respectively. Note the paths of the TE and TM eigenstates.

Fig. 5
Fig. 5

Experimental output power versus frequency profiles observed through a polarizer in the case in which no auxiliary anisotropy is introduced inside the cavity. (a) iic + 1′. The TE eigenstate is the only stable one because of the loss anisotropy associated with the nontotal reflection. (b) iic + 2′. Vectorial bistability in the inhibition mechanism. Note that the hysteresis cycle is not symmetric, indicating that the loss anisotropy still exists. (c) iic + 7′. Vectorial bistability in the inhibition mechanism. The fact that no peaks or dips can be observed on the output power when the polarizer is turned at 45° of the eigenstates proves that the polarization flips according to the inhibition mechanism. The loss anisotropy is now almost 0. (d) Theoretical TM output power profile corresponding to the experimental result (c) calculated with a phase anisotropy of 7.2°. (e) iic + 23′. Vectorial simultaneity. The phase anisotropy is now sufficient to permit the two eigenstates to oscillate simultaneously in a small region.

Fig. 6
Fig. 6

Same as Fig. 5, but when an auxiliary loss anisotropy is in the cavity. (a) iic − 30″. The loss anisotropy associated with the nontotal reflection is greater than the auxiliary loss anisotropy. The TE eigenstate is the only stable one. (b) iic + 30″. Vectorial bistability in the rotation mechanism. The two loss anisotropies perfectly compensate each other. The dips on the profile obtained when the polarizer is turned at 45° of the two eigenstates show that the polarization rotates continuously during the flip. (c) Theoretical TM output power profile corresponding to the experimental result (b) calculated with a phase anisotropy of 0.8°. (d) iic + 2′. The loss anisotropy due to the prism is smaller than the auxiliary loss anisotropy, leading to the stability of the TM eigenstate only.

Fig. 7
Fig. 7

Same as Fig. 5, but when an auxiliary phase anisotropy is in the cavity. (a) iic + 2′. The loss anisotropy of the prism leads to the stability of the TE eigenstate only. (b) iic + 3′. Vectorial bistability in the inhibition mechanism. Note that, contrary to the previous cases, νTE > νTM because the auxiliary phase anisotropy is larger than the one due to the prism, (c) iic + 5′. Same as (b), but without any remaining loss anisotropy. Note that the hysteresis cycle is larger than in (b), indicating that the total phase anisotropy has decreased. (d) iic + 6′. Vectorial bistability in the rotation mechanism. The overall phase anisotropy is now extremely low. (e) iic + 7′. Polarization instabilities due to the residual magnetic field. The two phase anisotropies compensate. (f) iic + 8′. Same as (d), but the total phase anisotropy has changed sign because the one due to the prism is now larger than the auxiliary phase anisotropy. (g) iic + 20′. Same as (c), but the total phase anisotrophy has changed sign.

Fig. 8
Fig. 8

Direct visualization of the Goos–Hänchen lateral shift. (a) Same conditions as in Fig. 6(b). (b) A knife edge is introduced between the prism and the plane mirror on the side of the TM eigenstate. The differential diffraction losses lead to the stability of the TE eigenstate only. (c) Same as (b), but when the knife edge is introduced on the side of the TE eigenstate.

Equations (24)

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R TE = sin ( i r ) sin ( i + r ) ,
R TM = tan ( i r ) tan ( i + r ) ,
sin r = n sin i .
tan ( ϕ TE 2 ) = ( sin 2 i 1 / n 2 ) 1 / 2 cos i ,
tan ( ϕ TM 2 ) = ( sin 2 i 1 / n 2 ) 1 / 2 ( 1 / n 2 ) cos i .
D TE = λ 0 π sin i ( n 2 sin 2 i 1 ) 1 / 2 ,
D TM = D TE ( n 2 + 1 ) sin 2 i 1 ,
= ( TE TM TE TM )
= ( TE TM TE TM )
E = P E ,
P = [ 0 0 | R TE | exp ( i ϕ TE ) 0 0 0 0 0 | R TE | exp ( i ϕ TE ) 0 0 0 0 0 0 | R TM | exp ( i ϕ TM ) ] .
M = P N Q A 2 Q N P ,
N = [ exp ( + i Δ ϕ / 2 ) 0 0 0 0 exp ( i Δ ϕ / 2 ) 0 0 0 0 exp ( + i Δ ϕ / 2 ) 0 0 0 0 exp ( i Δ ϕ / 2 ) ]
Q = [ 1 Δ p / 2 0 0 0 0 1 + Δ p / 2 0 0 0 0 1 Δ p / 2 0 0 0 0 1 + Δ p / 2 ]
A = a [ 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 ]
M = a 2 [ | R TE | 2 ( 1 Δ p / 2 ) 2 exp [ i ( 2 ϕ TE + Δ ϕ ) ] 0 0 0 0 0 0 0 0 0 0 0 0 0 0 | R TM | 2 ( 1 + Δ p / 2 ) 2 exp [ i ( 2 ϕ TM Δ ϕ ) ] ]
E TE = ( 1 0 0 0 ) ,
E TM = ( 0 0 0 1 ) ,
Λ TE = a 2 | R TE | 2 ( 1 Δ p / 2 ) 2 exp [ i ( 2 ϕ TE + Δ ϕ ) ] ,
Λ TM = a 2 | R TM | 2 ( 1 + Δ p / 2 ) 2 exp [ i ( 2 ϕ TM Δ ϕ ) ]
ν TE = c 2 L ( q + 2 ϕ TE + Δ ϕ 2 π ) ,
ν TM = c 2 L ( q + 2 ϕ TM Δ ϕ 2 π ) ,
Ė TE = E TE ( α TE β TE E TE 2 θ TE TM E TM 2 ) ,
Ė TM = E TM ( α TM β TM E TM 2 θ TM TE E TM 2 ) ,

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