Abstract

The operating characteristics of optical resonators with phase-conjugate mirrors are discussed. Degenerate four-wave mixing is assumed to be the process giving rise to the phase-conjugated phase fronts. We have found that these resonators are unconditionally stable, have no longitudinal mode conditions that invoke the length of the cavity, and have smooth transverse mode structures. Computer simulation results pertaining to the properties of the lowest-order modes and the effect of mirror errors on these modes are also discussed.

© 1980 Optical Society of America

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  1. J. M. Bel’dyugin, M. G. Galushkin, E. M. Zemskov, Sov. J. Quantum Electron. 9, 20 (1979).
    [CrossRef]
  2. J. AuYeung, D. Fekete, D. M. Pepper, A. Yariv, IEEE J. Quantum Electron. QE-15, 1180 (1979).
    [CrossRef]
  3. P. A. Belanger, A. Hardy, A. Siegman (unpublished).
  4. J. F. Lam, IEEE/OSA Conference on Laser Engineering and Applications, Washington, D.C., June1979; IEEE J. Quantum Electron. QE-15, 931 (1979).
  5. R. W. Hellwarth, J. Opt. Soc. Am. 67, 1 (1977).
    [CrossRef]
  6. H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
    [CrossRef]
  7. A more general approach is to formulate the integral equation satisfied by the field. This yields essentially the same results as the plane-wave analysis. In each case, the requirement that the wave replicate itself after one complete round trip yields a condition on the initial phase ϕ0 but does not yield an explicit condition on the length of the resonator. The integral equation approach yieldsϕ0=-(ϕnm+ϕγ-ϕpcm)/2+qπ,         q=0, ±1, ±2,…,where ϕγ is the phase of the eigenvalue associated with the solution of the integral equation. The plane-wave analysis yields the same condition except that ϕγ = 0 for a plane wave.
  8. J. H. Marburger, J. F. Lam, Appl. Phys. Lett. 34, 389 (1979); Appl. Phys. Lett. 35, 249 (1979).
    [CrossRef]
  9. D. Slepian, H. O. Pollack, Bell Syst. Tech. J. 40, 43 (1961).
  10. G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).
  11. A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

1979 (3)

J. M. Bel’dyugin, M. G. Galushkin, E. M. Zemskov, Sov. J. Quantum Electron. 9, 20 (1979).
[CrossRef]

J. AuYeung, D. Fekete, D. M. Pepper, A. Yariv, IEEE J. Quantum Electron. QE-15, 1180 (1979).
[CrossRef]

J. H. Marburger, J. F. Lam, Appl. Phys. Lett. 34, 389 (1979); Appl. Phys. Lett. 35, 249 (1979).
[CrossRef]

1977 (1)

1966 (1)

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

1961 (3)

D. Slepian, H. O. Pollack, Bell Syst. Tech. J. 40, 43 (1961).

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

AuYeung, J.

J. AuYeung, D. Fekete, D. M. Pepper, A. Yariv, IEEE J. Quantum Electron. QE-15, 1180 (1979).
[CrossRef]

Bel’dyugin, J. M.

J. M. Bel’dyugin, M. G. Galushkin, E. M. Zemskov, Sov. J. Quantum Electron. 9, 20 (1979).
[CrossRef]

Belanger, P. A.

P. A. Belanger, A. Hardy, A. Siegman (unpublished).

Boyd, G. D.

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

Fekete, D.

J. AuYeung, D. Fekete, D. M. Pepper, A. Yariv, IEEE J. Quantum Electron. QE-15, 1180 (1979).
[CrossRef]

Fox, A. G.

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Galushkin, M. G.

J. M. Bel’dyugin, M. G. Galushkin, E. M. Zemskov, Sov. J. Quantum Electron. 9, 20 (1979).
[CrossRef]

Gordon, J. P.

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

Hardy, A.

P. A. Belanger, A. Hardy, A. Siegman (unpublished).

Hellwarth, R. W.

Kogelnik, H.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Lam, J. F.

J. H. Marburger, J. F. Lam, Appl. Phys. Lett. 34, 389 (1979); Appl. Phys. Lett. 35, 249 (1979).
[CrossRef]

J. F. Lam, IEEE/OSA Conference on Laser Engineering and Applications, Washington, D.C., June1979; IEEE J. Quantum Electron. QE-15, 931 (1979).

Li, T.

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

Marburger, J. H.

J. H. Marburger, J. F. Lam, Appl. Phys. Lett. 34, 389 (1979); Appl. Phys. Lett. 35, 249 (1979).
[CrossRef]

Pepper, D. M.

J. AuYeung, D. Fekete, D. M. Pepper, A. Yariv, IEEE J. Quantum Electron. QE-15, 1180 (1979).
[CrossRef]

Pollack, H. O.

D. Slepian, H. O. Pollack, Bell Syst. Tech. J. 40, 43 (1961).

Siegman, A.

P. A. Belanger, A. Hardy, A. Siegman (unpublished).

Slepian, D.

D. Slepian, H. O. Pollack, Bell Syst. Tech. J. 40, 43 (1961).

Yariv, A.

J. AuYeung, D. Fekete, D. M. Pepper, A. Yariv, IEEE J. Quantum Electron. QE-15, 1180 (1979).
[CrossRef]

Zemskov, E. M.

J. M. Bel’dyugin, M. G. Galushkin, E. M. Zemskov, Sov. J. Quantum Electron. 9, 20 (1979).
[CrossRef]

Appl. Phys. Lett. (1)

J. H. Marburger, J. F. Lam, Appl. Phys. Lett. 34, 389 (1979); Appl. Phys. Lett. 35, 249 (1979).
[CrossRef]

Bell Syst. Tech. J. (3)

D. Slepian, H. O. Pollack, Bell Syst. Tech. J. 40, 43 (1961).

G. D. Boyd, J. P. Gordon, Bell Syst. Tech. J. 40, 489 (1961).

A. G. Fox, T. Li, Bell Syst. Tech. J. 40, 453 (1961).

IEEE J. Quantum Electron. (1)

J. AuYeung, D. Fekete, D. M. Pepper, A. Yariv, IEEE J. Quantum Electron. QE-15, 1180 (1979).
[CrossRef]

J. Opt. Soc. Am. (1)

Proc. IEEE (1)

H. Kogelnik, T. Li, Proc. IEEE 54, 1312 (1966).
[CrossRef]

Sov. J. Quantum Electron. (1)

J. M. Bel’dyugin, M. G. Galushkin, E. M. Zemskov, Sov. J. Quantum Electron. 9, 20 (1979).
[CrossRef]

Other (3)

A more general approach is to formulate the integral equation satisfied by the field. This yields essentially the same results as the plane-wave analysis. In each case, the requirement that the wave replicate itself after one complete round trip yields a condition on the initial phase ϕ0 but does not yield an explicit condition on the length of the resonator. The integral equation approach yieldsϕ0=-(ϕnm+ϕγ-ϕpcm)/2+qπ,         q=0, ±1, ±2,…,where ϕγ is the phase of the eigenvalue associated with the solution of the integral equation. The plane-wave analysis yields the same condition except that ϕγ = 0 for a plane wave.

P. A. Belanger, A. Hardy, A. Siegman (unpublished).

J. F. Lam, IEEE/OSA Conference on Laser Engineering and Applications, Washington, D.C., June1979; IEEE J. Quantum Electron. QE-15, 931 (1979).

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Figures (2)

Fig. 1
Fig. 1

(a) Amplitude distribution at the normal mirror for a symmetric confocal system (L = R) and effective Fresnel number equal to unity (N = 1). (b) Amplitude distribution for L = −R and N = 1. (c) Phase distribution for L = −R and N = 1.

Fig. 2
Fig. 2

(a) Phase distortion at the surface of the ordinary mirror. (b) Phase of the output wave at the ordinary mirror when N = 2/π. (c) Phase of the output wave at the ordinary mirror when N = 15/π.

Equations (10)

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M pcm - ( 1 0 0 - 1 ) .
M = M prop M pcm M prop M mir = ( 1 L 0 1 ) ( 1 0 0 - 1 ) ( 1 L 0 1 ) ( 1 0 2 1 R ¯ ) = ( 1 0 2 - 1 R ¯ ) ,
ϕ pcm ( 1 ) - ϕ pcm ( 2 ) = 2 πm ,
u ( x , z ) = exp [ i ( π 4 - k z - z 0 2 ) ] λ z - z 0 × - a a d x 0 u ( x 0 , z 0 ) exp [ - i k 2 ( x - x 0 ) 2 z - z 0 ] ,
γ u ( x ) = - a a d x 0 u ( x 0 ) exp [ - i k 2 L g ( x 2 - x 0 2 ) ] × sin k 2 L b ( x - x 0 ) π ( x - x 0 ) ,
A n + 1 = exp ( - i k L g x 2 ) - a a d x 0 A n ( x 0 ) × sin k 2 L b ( x - x 0 ) π ( x - x 0 ) ,
γ u ( x ) = - a a d x 0 u ( x 0 ) exp [ - i k 2 L ( x 2 - x 0 2 ) ] × sin k 2 L b ( x - x 0 ) π ( x - x 0 ) ,
u m ( x ) α S m ( x , a b / λ L ) exp ( - i k x 2 2 L ) ,
γ m α [ R m ( 1 , a b / λ L ) ] 2 ,
ϕ0=-(ϕnm+ϕγ-ϕpcm)/2+qπ,         q=0,±1,±2,,

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