Abstract

Bistability of the transverse configuration of the optical field is found for a helium–neon laser oscillating in the TEM01* hybrid optical-resonator mode. Optically induced switching between the mirror-symmetric field patterns occurs when a fraction of the spatially rotated output field is reinjected into the laser resonator. These phenomena are described theoretically with laser equations modeling the mutual interaction of the two first-order transverse modes of the optical resonator. The experimentally observed bistability and switching of the field patterns is in full qualitative agreement with the theoretical results.

© 1990 Optical Society of America

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References

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  1. H. Haken, Advanced Synergetics (Springer-Verlag, Berlin, 1987), p. 1.
  2. L. A. Lugiato, R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
    [CrossRef] [PubMed]
  3. L. A. Lugiato, C. Oldano, L. M. Narducci, J. Opt. Soc. Am. B 5, 879 (1988); L. A. Lugiato, G. L. Oppo, M. A. Pernigo, J. R. Tredicce, L. M. Narducci, D. K. Bandy, Opt. Commun. 68, 63 (1988).
    [CrossRef]
  4. I. M. Belousova, G. N. Vinokurov, O. B. Danilov, N. N. Rozanov, Sov. Phys. JETP 25, 761 (1967).
  5. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966).
    [CrossRef] [PubMed]
  6. W. W. Rigrod, Appl. Phys. Lett. 2, 51 (1963).
    [CrossRef]
  7. C. Tamm, Phys. Rev. A 38, 5960 (1988).
    [CrossRef] [PubMed]
  8. J. M. Vaughan, D. V. Willets, Opt. Commun. 30, 263 (1979).
    [CrossRef]
  9. P. Coullet, L. Gil, F. Rocca, Opt. Commun. 73, 403 (1989).
    [CrossRef]
  10. H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, T. Mello, Phys. Rev. A 37, 470 (1988).
    [CrossRef] [PubMed]
  11. D. C. Hanna, IEEE J. Quantum Electron. QE-5, 483 (1969).
    [CrossRef]
  12. The Guoy phase effect and its relation to the propagation properties of laser beams are discussed in, e.g., A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), p. 645–685.
  13. C. Tamm, “The field patterns of a hybrid mode laser: detecting the ‘hidden’ bistability of the optical phase pattern,” in Proceedings of the Workshop on Quantitative Characterization of Dynamical Complexity in Nonlinear Systems, N. B. Abraham, A. Albano, eds. (Plenum, New York, 1989).
  14. See, e.g., Reference 11, p. 491–495.
  15. R. Hauck, F. Hollinger, H. Weber, Opt. Commun. 47, 141 (1983); D. J. Biswas, V. Dev, U. K. Chatterjee, Phys. Rev. A 38, 555 (1988); W. Klische, C. O. Weiss, B. Wellegehausen, Phys. Rev. A 39, 919 (1989); J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
    [CrossRef] [PubMed]

1989 (1)

P. Coullet, L. Gil, F. Rocca, Opt. Commun. 73, 403 (1989).
[CrossRef]

1988 (3)

1987 (1)

L. A. Lugiato, R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef] [PubMed]

1983 (1)

R. Hauck, F. Hollinger, H. Weber, Opt. Commun. 47, 141 (1983); D. J. Biswas, V. Dev, U. K. Chatterjee, Phys. Rev. A 38, 555 (1988); W. Klische, C. O. Weiss, B. Wellegehausen, Phys. Rev. A 39, 919 (1989); J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[CrossRef] [PubMed]

1979 (1)

J. M. Vaughan, D. V. Willets, Opt. Commun. 30, 263 (1979).
[CrossRef]

1969 (1)

D. C. Hanna, IEEE J. Quantum Electron. QE-5, 483 (1969).
[CrossRef]

1967 (1)

I. M. Belousova, G. N. Vinokurov, O. B. Danilov, N. N. Rozanov, Sov. Phys. JETP 25, 761 (1967).

1966 (1)

1963 (1)

W. W. Rigrod, Appl. Phys. Lett. 2, 51 (1963).
[CrossRef]

Abraham, N. B.

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, T. Mello, Phys. Rev. A 37, 470 (1988).
[CrossRef] [PubMed]

Belousova, I. M.

I. M. Belousova, G. N. Vinokurov, O. B. Danilov, N. N. Rozanov, Sov. Phys. JETP 25, 761 (1967).

Coullet, P.

P. Coullet, L. Gil, F. Rocca, Opt. Commun. 73, 403 (1989).
[CrossRef]

Danilov, O. B.

I. M. Belousova, G. N. Vinokurov, O. B. Danilov, N. N. Rozanov, Sov. Phys. JETP 25, 761 (1967).

Gil, L.

P. Coullet, L. Gil, F. Rocca, Opt. Commun. 73, 403 (1989).
[CrossRef]

Haken, H.

H. Haken, Advanced Synergetics (Springer-Verlag, Berlin, 1987), p. 1.

Hanna, D. C.

D. C. Hanna, IEEE J. Quantum Electron. QE-5, 483 (1969).
[CrossRef]

Hauck, R.

R. Hauck, F. Hollinger, H. Weber, Opt. Commun. 47, 141 (1983); D. J. Biswas, V. Dev, U. K. Chatterjee, Phys. Rev. A 38, 555 (1988); W. Klische, C. O. Weiss, B. Wellegehausen, Phys. Rev. A 39, 919 (1989); J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[CrossRef] [PubMed]

Hoffer, L. M.

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, T. Mello, Phys. Rev. A 37, 470 (1988).
[CrossRef] [PubMed]

Hollinger, F.

R. Hauck, F. Hollinger, H. Weber, Opt. Commun. 47, 141 (1983); D. J. Biswas, V. Dev, U. K. Chatterjee, Phys. Rev. A 38, 555 (1988); W. Klische, C. O. Weiss, B. Wellegehausen, Phys. Rev. A 39, 919 (1989); J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[CrossRef] [PubMed]

Kogelnik, H.

Lefever, R.

L. A. Lugiato, R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef] [PubMed]

Li, T.

Lippi, G. L.

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, T. Mello, Phys. Rev. A 37, 470 (1988).
[CrossRef] [PubMed]

Lugiato, L. A.

Mandel, P.

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, T. Mello, Phys. Rev. A 37, 470 (1988).
[CrossRef] [PubMed]

Mello, T.

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, T. Mello, Phys. Rev. A 37, 470 (1988).
[CrossRef] [PubMed]

Narducci, L. M.

Oldano, C.

Rigrod, W. W.

W. W. Rigrod, Appl. Phys. Lett. 2, 51 (1963).
[CrossRef]

Rocca, F.

P. Coullet, L. Gil, F. Rocca, Opt. Commun. 73, 403 (1989).
[CrossRef]

Rozanov, N. N.

I. M. Belousova, G. N. Vinokurov, O. B. Danilov, N. N. Rozanov, Sov. Phys. JETP 25, 761 (1967).

Siegman, A. E.

The Guoy phase effect and its relation to the propagation properties of laser beams are discussed in, e.g., A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), p. 645–685.

Tamm, C.

C. Tamm, Phys. Rev. A 38, 5960 (1988).
[CrossRef] [PubMed]

C. Tamm, “The field patterns of a hybrid mode laser: detecting the ‘hidden’ bistability of the optical phase pattern,” in Proceedings of the Workshop on Quantitative Characterization of Dynamical Complexity in Nonlinear Systems, N. B. Abraham, A. Albano, eds. (Plenum, New York, 1989).

Vaughan, J. M.

J. M. Vaughan, D. V. Willets, Opt. Commun. 30, 263 (1979).
[CrossRef]

Vinokurov, G. N.

I. M. Belousova, G. N. Vinokurov, O. B. Danilov, N. N. Rozanov, Sov. Phys. JETP 25, 761 (1967).

Weber, H.

R. Hauck, F. Hollinger, H. Weber, Opt. Commun. 47, 141 (1983); D. J. Biswas, V. Dev, U. K. Chatterjee, Phys. Rev. A 38, 555 (1988); W. Klische, C. O. Weiss, B. Wellegehausen, Phys. Rev. A 39, 919 (1989); J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[CrossRef] [PubMed]

Willets, D. V.

J. M. Vaughan, D. V. Willets, Opt. Commun. 30, 263 (1979).
[CrossRef]

Zeghlache, H.

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, T. Mello, Phys. Rev. A 37, 470 (1988).
[CrossRef] [PubMed]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

W. W. Rigrod, Appl. Phys. Lett. 2, 51 (1963).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. C. Hanna, IEEE J. Quantum Electron. QE-5, 483 (1969).
[CrossRef]

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

Opt. Commun. (3)

J. M. Vaughan, D. V. Willets, Opt. Commun. 30, 263 (1979).
[CrossRef]

P. Coullet, L. Gil, F. Rocca, Opt. Commun. 73, 403 (1989).
[CrossRef]

R. Hauck, F. Hollinger, H. Weber, Opt. Commun. 47, 141 (1983); D. J. Biswas, V. Dev, U. K. Chatterjee, Phys. Rev. A 38, 555 (1988); W. Klische, C. O. Weiss, B. Wellegehausen, Phys. Rev. A 39, 919 (1989); J. R. Tredicce, E. J. Quel, A. M. Ghazzawi, C. Green, M. A. Pernigo, L. M. Narducci, L. A. Lugiato, Phys. Rev. Lett. 62, 1274 (1989).
[CrossRef] [PubMed]

Phys. Rev. A (2)

H. Zeghlache, P. Mandel, N. B. Abraham, L. M. Hoffer, G. L. Lippi, T. Mello, Phys. Rev. A 37, 470 (1988).
[CrossRef] [PubMed]

C. Tamm, Phys. Rev. A 38, 5960 (1988).
[CrossRef] [PubMed]

Phys. Rev. Lett. (1)

L. A. Lugiato, R. Lefever, Phys. Rev. Lett. 58, 2209 (1987).
[CrossRef] [PubMed]

Sov. Phys. JETP (1)

I. M. Belousova, G. N. Vinokurov, O. B. Danilov, N. N. Rozanov, Sov. Phys. JETP 25, 761 (1967).

Other (4)

H. Haken, Advanced Synergetics (Springer-Verlag, Berlin, 1987), p. 1.

The Guoy phase effect and its relation to the propagation properties of laser beams are discussed in, e.g., A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), p. 645–685.

C. Tamm, “The field patterns of a hybrid mode laser: detecting the ‘hidden’ bistability of the optical phase pattern,” in Proceedings of the Workshop on Quantitative Characterization of Dynamical Complexity in Nonlinear Systems, N. B. Abraham, A. Albano, eds. (Plenum, New York, 1989).

See, e.g., Reference 11, p. 491–495.

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Figures (4)

Fig. 1
Fig. 1

Experimental setup: BS1, BS2, beam splitters; C1, C2, cylindrical lenses, forming the transverse-mode converter MC; Det, detector monitoring the local intensity in the bistable transverse intensity pattern—the detector position coincides with the nodal line of one pattern; Pr, rectangular reflecting prism—the plane of reflection of Pr is rotated out of the drawing plane by 45°; FI, Faraday isolator (optical diode); λ/2, λ/2 retardation plate; NDF, neutral density filter; PZT, piezoceramic transducer; M1, M2, M3, mirrors. For simplicity, mode matching and beam expanding lenses are not shown.

Fig. 2
Fig. 2

Calculated normalized intensities |F+|2 and |F|2 as a function of slowly varying feedback phase angle δ. Parameters: β = 0.5, γ = 10κ, γ = 20κ, C = 2Cth, sweep rate dδ/dtκ × 10−5 rad; for definition of symbols see text. Small random fluctuations were introduced into the numerical calculation to stimulate the decay of unstable states. The arrows indicate the path followed in the hysteresis cycle.

Fig. 3
Fig. 3

Photographs of (a) transverse intensity distribution of the laser output and (b), (c) bistable intensity patterns at output of mode converter. The cylindrical lenses of the mode converter (see text) are oriented horizontally with respect to patterns (b) and (c).

Fig. 4
Fig. 4

(a) Experimentally determined local intensity at output of the transverse-mode converter (see Fig. 1) as a function of feedback phase δ. (b), (c) The same signal as a function of time t in the transient regimes. In all cases the sweep rate is dδ/dt ≈ 0.1 rad/msec. For other experimental parameters, see text.

Equations (17)

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E ( r , t ) = e ˆ p N exp [ ( x 2 + y 2 ) / w 2 ] ( A + A * ) A = [ F 10 ( x / w ) + F 01 ( y / w ) ] exp [ i ( ω t k z + ϕ 0 ) ] .
I ( x , y ) = K exp [ 2 ( x 2 + y 2 ) / w 2 ] × [ | F 10 | 2 ( x 2 / w 2 ) + | F 01 | 2 ( y 2 / w 2 ) + ( F 10 F 01 * + F 10 * F 01 ) ( x y / w 2 ) ] .
F 10 t = κ [ F 10 + β exp ( i δ ) F 01 2 C P 10 ] ,
F 01 t = κ [ F 01 β exp ( i δ ) F 10 2 C P 01 ] ,
P 10 t = γ ( P 10 1 2 F 10 D 0 1 4 F 10 D 1 1 4 F 01 D 2 ) ,
P 01 t = γ ( P 01 1 2 F 01 D 0 1 4 F 01 D 1 1 4 F 10 D 2 ) ,
D 0 t = γ [ D 0 2 π + 1 4 ( F 10 P 10 * + F 10 * P 10 + F 01 P 01 * + F 01 * P 01 ) ] ,
D 1 t = γ [ D 1 + 1 4 ( F 10 P 10 * + F 10 * P 10 F 01 P 01 * F 01 * P 01 ) ] ,
D 2 t = γ [ D 2 + 1 4 ( F 10 P 01 * + F 10 * P 01 + F 01 P 10 * F 01 * P 10 ) ] .
[ D 0 D 1 D 2 ] = 2 π w 2 d x d y × D ( x , y ) exp [ 2 ( x 2 + y 2 ) / w 2 ] [ π ( x 2 + y 2 ) 2 ( x 2 y 2 ) 2 2 x y ] .
F + t = κ [ F + + β exp ( i δ ) F + 2 C P + ] ,
F t = κ [ F β exp ( i δ ) F 2 C P ] ,
P + t = γ ( P + 1 2 F + D 0 1 4 F d ) ,
P t = γ ( P 1 2 F D 0 1 4 F + d * ) ,
D 0 t = γ [ D 0 2 π + 1 4 ( F + P + * + F + * P + + F P * + F * P ) ] ,
d t = γ [ d + ( 1 / 2 ) ( F + P * + F * P + ) ] .
| β cos δ | 1 2 ( C C th 1 ) , cos δ > 0 for F + 0 , F = 0 F + = 0 , F 0 , cos δ < 0 for F + = 0 , F 0 F + 0 , F = 0 ,

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