Abstract

Phase-conjugate lasers with saturable gain medium are analyzed, taking into account the saturation of the reflectivity of the phase-conjugate mirror that is due to pump depletion. It is shown that the behavior of the laser is determined by the extent to which the reflectivity is saturated as compared with the saturation of the gain medium. For weakly saturated phase-conjugate mirrors, the intensities and gain are not much different from those in conventional lasers. For strongly saturated mirrors, the laser behaves more like a single-pass amplifier with output intensity proportional to the pump intensity of the phase-conjugate mirror.

© 1983 Optical Society of America

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  1. J. AuYeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase conjugating mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
    [CrossRef]
  2. J. Feinberg and R. W. Hellwarth, “Phase-conjugate mirrors with continuous-wave gain,” Opt. Lett. 5, 519–521 (1980).
    [CrossRef]
  3. H. Vanherzeele, J. L. Van Eck, and A. E. Siegman, “Mode-locked laser oscillation using self-pumped phase conjugate reflection,” Opt. Lett. 6, 467–469 (1981).
    [CrossRef] [PubMed]
  4. R. C. Lind and D. G. Steel, “Demonstration of the longitudinal modes and aberration-correction properties of a continuous-wave dye laser with a phase-conjugate mirror,” Opt. Lett. 6, 554–556 (1981).
    [CrossRef] [PubMed]
  5. A. Hardy, “Sensitivity of phase-conjugate resonators to intracavity phase perturbations,” IEEE J. Quantum Electron. QE-17, 1581–1585 (1981).
    [CrossRef]
  6. J. F. Lam and W. P. Brown, “Optical resonators with phase-conjugate mirrors,” Opt. Lett. 5, 61–63 (1980).
    [CrossRef] [PubMed]
  7. A. Hardy and S. Hochhauser, “Higher-order modes of phase-conjugate resonators,” Appl. Opt. 21, 2330–2338 (1982).
    [CrossRef] [PubMed]
  8. P. A. Belanger, A. Hardy, and A. E. Siegman, “Resonant modes of optical cavities with phase-conjugate mirrors,” Appl. Opt,  19, 602–609 (1980).
    [CrossRef] [PubMed]
  9. A. E. Siegman, P. A. Belanger, and A. Hardy, “Optical resonators using phase-conjugate mirrors,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983).
    [CrossRef]
  10. R. W. Hellwarth, “Generation of time-reversed wave-fronts by nonlinear refraction,” J. Opt. Soc. Am. 67, 1–3 (1977).
    [CrossRef]
  11. A. Yariv, “Phase-conjugate optics and real time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
    [CrossRef]
  12. A. Khizhnyak, V. Kondilenko, V. Kremenitski, S. Odoulov, and M. Soskin, “Degenerate four-wave mixing in nonlinear media with local response: free carrier and space charge gratings, strong coupling,” Proc. Soc. Photo-Opt. Instrum. Eng. 213, 18–25 (1979).
  13. J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
    [CrossRef]
  14. M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, “Exact solution of a nonlinear model of four-wave mixing and phase-conjugation,” Opt. Lett. 7, 313–315 (1982).
    [CrossRef] [PubMed]
  15. W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
    [CrossRef]
  16. O. Svelto, “Self-focusing, self-trapping, and self-phase modulation of laser beams,” in Progress in Optics XII, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Chap. 1.
    [CrossRef]
  17. H. Vanherzeele and J. L. Van Eck, “Pulse compression by intracavity degenerate four wave mixing,” Appl. Opt. 20, 524–525 (1981).
    [CrossRef] [PubMed]
  18. T. J. Karr and H. J. Hoffman, “Intraresonator phase conjugation,” Proc. Soc. Photo-Opt. Instrum. Eng. 293, 162–182 (1981).

1982 (2)

1981 (5)

1980 (3)

1979 (3)

A. Khizhnyak, V. Kondilenko, V. Kremenitski, S. Odoulov, and M. Soskin, “Degenerate four-wave mixing in nonlinear media with local response: free carrier and space charge gratings, strong coupling,” Proc. Soc. Photo-Opt. Instrum. Eng. 213, 18–25 (1979).

J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
[CrossRef]

J. AuYeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase conjugating mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[CrossRef]

1978 (1)

A. Yariv, “Phase-conjugate optics and real time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
[CrossRef]

1977 (1)

1965 (1)

W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[CrossRef]

AuYeung, J.

J. AuYeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase conjugating mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[CrossRef]

Belanger, P. A.

P. A. Belanger, A. Hardy, and A. E. Siegman, “Resonant modes of optical cavities with phase-conjugate mirrors,” Appl. Opt,  19, 602–609 (1980).
[CrossRef] [PubMed]

A. E. Siegman, P. A. Belanger, and A. Hardy, “Optical resonators using phase-conjugate mirrors,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983).
[CrossRef]

Brown, W. P.

Cronin-Golomb, M.

Feinberg, J.

Fekete, D.

J. AuYeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase conjugating mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[CrossRef]

Fischer, B.

Hardy, A.

A. Hardy and S. Hochhauser, “Higher-order modes of phase-conjugate resonators,” Appl. Opt. 21, 2330–2338 (1982).
[CrossRef] [PubMed]

A. Hardy, “Sensitivity of phase-conjugate resonators to intracavity phase perturbations,” IEEE J. Quantum Electron. QE-17, 1581–1585 (1981).
[CrossRef]

P. A. Belanger, A. Hardy, and A. E. Siegman, “Resonant modes of optical cavities with phase-conjugate mirrors,” Appl. Opt,  19, 602–609 (1980).
[CrossRef] [PubMed]

A. E. Siegman, P. A. Belanger, and A. Hardy, “Optical resonators using phase-conjugate mirrors,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983).
[CrossRef]

Hellwarth, R. W.

Hochhauser, S.

Hoffman, H. J.

T. J. Karr and H. J. Hoffman, “Intraresonator phase conjugation,” Proc. Soc. Photo-Opt. Instrum. Eng. 293, 162–182 (1981).

Karr, T. J.

T. J. Karr and H. J. Hoffman, “Intraresonator phase conjugation,” Proc. Soc. Photo-Opt. Instrum. Eng. 293, 162–182 (1981).

Khizhnyak, A.

A. Khizhnyak, V. Kondilenko, V. Kremenitski, S. Odoulov, and M. Soskin, “Degenerate four-wave mixing in nonlinear media with local response: free carrier and space charge gratings, strong coupling,” Proc. Soc. Photo-Opt. Instrum. Eng. 213, 18–25 (1979).

Kondilenko, V.

A. Khizhnyak, V. Kondilenko, V. Kremenitski, S. Odoulov, and M. Soskin, “Degenerate four-wave mixing in nonlinear media with local response: free carrier and space charge gratings, strong coupling,” Proc. Soc. Photo-Opt. Instrum. Eng. 213, 18–25 (1979).

Kremenitski, V.

A. Khizhnyak, V. Kondilenko, V. Kremenitski, S. Odoulov, and M. Soskin, “Degenerate four-wave mixing in nonlinear media with local response: free carrier and space charge gratings, strong coupling,” Proc. Soc. Photo-Opt. Instrum. Eng. 213, 18–25 (1979).

Lam, J. F.

J. F. Lam and W. P. Brown, “Optical resonators with phase-conjugate mirrors,” Opt. Lett. 5, 61–63 (1980).
[CrossRef] [PubMed]

J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
[CrossRef]

Lind, R. C.

Marburger, J. H.

J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
[CrossRef]

Odoulov, S.

A. Khizhnyak, V. Kondilenko, V. Kremenitski, S. Odoulov, and M. Soskin, “Degenerate four-wave mixing in nonlinear media with local response: free carrier and space charge gratings, strong coupling,” Proc. Soc. Photo-Opt. Instrum. Eng. 213, 18–25 (1979).

Pepper, D. M.

J. AuYeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase conjugating mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[CrossRef]

Rigrod, W. W.

W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[CrossRef]

Siegman, A. E.

H. Vanherzeele, J. L. Van Eck, and A. E. Siegman, “Mode-locked laser oscillation using self-pumped phase conjugate reflection,” Opt. Lett. 6, 467–469 (1981).
[CrossRef] [PubMed]

P. A. Belanger, A. Hardy, and A. E. Siegman, “Resonant modes of optical cavities with phase-conjugate mirrors,” Appl. Opt,  19, 602–609 (1980).
[CrossRef] [PubMed]

A. E. Siegman, P. A. Belanger, and A. Hardy, “Optical resonators using phase-conjugate mirrors,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983).
[CrossRef]

Soskin, M.

A. Khizhnyak, V. Kondilenko, V. Kremenitski, S. Odoulov, and M. Soskin, “Degenerate four-wave mixing in nonlinear media with local response: free carrier and space charge gratings, strong coupling,” Proc. Soc. Photo-Opt. Instrum. Eng. 213, 18–25 (1979).

Steel, D. G.

Svelto, O.

O. Svelto, “Self-focusing, self-trapping, and self-phase modulation of laser beams,” in Progress in Optics XII, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Chap. 1.
[CrossRef]

Van Eck, J. L.

Vanherzeele, H.

White, J. O.

Yariv, A.

M. Cronin-Golomb, J. O. White, B. Fischer, and A. Yariv, “Exact solution of a nonlinear model of four-wave mixing and phase-conjugation,” Opt. Lett. 7, 313–315 (1982).
[CrossRef] [PubMed]

J. AuYeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase conjugating mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[CrossRef]

A. Yariv, “Phase-conjugate optics and real time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
[CrossRef]

Appl. Opt (1)

P. A. Belanger, A. Hardy, and A. E. Siegman, “Resonant modes of optical cavities with phase-conjugate mirrors,” Appl. Opt,  19, 602–609 (1980).
[CrossRef] [PubMed]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

J. H. Marburger and J. F. Lam, “Nonlinear theory of degenerate four-wave mixing,” Appl. Phys. Lett. 34, 389–391 (1979).
[CrossRef]

IEEE J. Quantum Electron. (3)

A. Yariv, “Phase-conjugate optics and real time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
[CrossRef]

J. AuYeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase conjugating mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1979).
[CrossRef]

A. Hardy, “Sensitivity of phase-conjugate resonators to intracavity phase perturbations,” IEEE J. Quantum Electron. QE-17, 1581–1585 (1981).
[CrossRef]

J. Appl. Phys. (1)

W. W. Rigrod, “Saturation effects in high-gain lasers,” J. Appl. Phys. 36, 2487–2490 (1965).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (5)

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

T. J. Karr and H. J. Hoffman, “Intraresonator phase conjugation,” Proc. Soc. Photo-Opt. Instrum. Eng. 293, 162–182 (1981).

A. Khizhnyak, V. Kondilenko, V. Kremenitski, S. Odoulov, and M. Soskin, “Degenerate four-wave mixing in nonlinear media with local response: free carrier and space charge gratings, strong coupling,” Proc. Soc. Photo-Opt. Instrum. Eng. 213, 18–25 (1979).

Other (2)

O. Svelto, “Self-focusing, self-trapping, and self-phase modulation of laser beams,” in Progress in Optics XII, E. Wolf, ed. (North-Holland, Amsterdam, 1974), Chap. 1.
[CrossRef]

A. E. Siegman, P. A. Belanger, and A. Hardy, “Optical resonators using phase-conjugate mirrors,” in Optical Phase Conjugation, R. A. Fisher, ed. (Academic, New York, 1983).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Geometry and notation used for the analysis of the PCM.

Fig. 2
Fig. 2

Geometry and notation used for the analysis of the PCL.

Fig. 3
Fig. 3

Reflectivity curves of the PCM versus the normalized incident signal. Solid lines are the exact solution; dashed lines are approximate solutions as described in the text.

Fig. 4
Fig. 4

Intensity reflected from the PCM versus the normalized interaction length. The normalized incident signal is 0.01. The dashed line is the small-signal approximation of ~tan2(κL).

Fig. 5
Fig. 5

Reflectivity of the PCM in the laser cavity of Fig. 2 versus the unsaturated gain g0L. Here R1 = 0.9 and R2(0) = 1. The parameter α is proportional to the ratio of the saturation intensity of the PCM to that of the gain medium.

Fig. 6
Fig. 6

Laser saturated gain versus the unsaturated gain when R1 = 0.9 and R2 = 1.

Fig. 7
Fig. 7

Normalized output intensity through the conventional mirror versus the unsaturated gain when R1 = 0.9 and R2(0) = 1.

Fig. 8
Fig. 8

Reflectivity of the PCM versus the unsaturated gain when R1 = 0.9 and R2(0) = 10.

Fig. 9
Fig. 9

Saturated gain versus the unsaturated gain when R1 = 0.9 and R2(0) = 10.

Fig. 10
Fig. 10

Normalized output intensity through the conventional mirror versus the unsaturated gain when R1 = 0.9 and R2(0) = 10.

Equations (29)

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E j = A j exp ( i ω t - k j r ) + c . c . ,             j = 1 , 2 , 3 , 4.
n = n 0 + n 2 E 2 ,
A 1 ζ = - i ( 2 I 0 - I 1 ) A 1 - 2 i A 2 * A 3 A 4 , A 2 ζ = + i ( 2 I 0 - I 2 ) A 2 + 2 i A 1 * A 3 A 4 , A 3 ζ = - i ( 2 I 0 - I 3 ) A 3 - 2 i A 4 * A 1 A 2 , A 4 ζ = + i ( 2 I 0 - I 4 ) A 4 + 2 i A 3 * A 1 A 2 ,
ζ = 2 n 2 k 0 z / cos θ ,
J 1 = I 1 - I 4 , J 2 = I 2 + I 4 , J 3 = I 3 + I 4 , I 0 = J 1 + J 2 + J 3
a j = I j exp [ i ϕ j ( ζ ) ] ,             j = 1 , 2 , 3 , 4 ,
a j = A j exp ( i 2 I 0 ζ ) , j = 1 , 3 , a j = A j exp ( - i 2 I 0 ζ ) , j = 2 , 4.
ϕ ( ζ ) = ϕ 3 + ϕ 4 - ϕ 1 - ϕ 2 ,
d I j d ζ = 4 ( I 1 I 2 I 3 I 4 ) 1 / 2 sin ϕ ,             j = 1 , 4 d I j d ζ = - 4 ( I 1 I 2 I 3 I 4 ) 1 / 2 sin ϕ ,             j = 2 , 3 d ϕ d ζ = I 2 + I 3 - I 1 - I 4 + 2 cos ϕ ( I 1 I 2 I 3 I 4 ) 1 / 2 × ( I 1 I 2 I 3 + I 2 I 3 I 4 - I 1 I 2 I 4 - I 1 I 3 I 4 ) .
G 0 = cos ϕ [ I 4 ( J 1 + I 4 ) ( J 2 - I 4 ) ( J 3 - I 4 ) ] 1 / 2 + 1 4 J I 4 - 1 2 I 4 2 ,
d I 4 d ζ = ± 4 [ I 4 ( J 1 + I 4 ) ( J 2 - I 4 ) ( J 3 - I 4 ) - ( 1 4 J I 4 - 1 2 I 4 2 - G 0 ) 2 ] 1 / 2 .
d I 4 d ζ = ± [ I 4 ( J 1 + I 4 ) ( J 2 - I 4 ) ( J 3 - I 4 ) - ( 1 4 J I 4 - 1 2 I 4 2 ) 2 ] 1 / 2
I 4 ( ζ 0 ) = 0 , J 1 = I 1 ( 0 ) - I 4 ( 0 ) , J 2 = I 2 ( ζ 0 ) , J 3 = I 3 ( 0 ) + I 4 ( 0 ) , J = J 3 + J 2 - J 1 .
ζ 0 = 0 ζ 0 d ζ = ± I 4 ( 0 ) 0 [ I 4 ( J 1 + I 4 ) ( J 2 - I 4 ) ( J 3 - I 4 ) - ( 1 4 J I 4 - 1 2 I 4 2 ) 2 ] - 1 / 2 d I 4 .
± 1 β ± d β ± d z = g 0 1 + β + + β - ,
β 1 = R 1 β 4 , β 3 = R 2 ( β 2 ) β 2 ,
β 4 = β 2 [ R 2 ( β 2 ) / R 1 ] 1 / 2 , β 1 = β 2 [ R 2 ( β 2 ) R 1 ] 1 / 2 ,
β 2 = ( R 1 ) 1 / 2 { g 0 L + ln [ R 1 R 2 ( β 2 ) ] 1 / 2 } { ( R 1 ) 1 / 2 + [ R 2 ( β 2 ) ] 1 / 2 } { 1 - [ R 1 R 2 ( β 2 ) ] 1 / 2 } ,
g L = ln ( β 2 / β 1 ) .
g L = 0 L g ( z ) d z = 0 L g 0 d z 1 + β + + β - = ln ( β 4 / β 3 ) .
exp ( 2 g L ) R 1 R 2 ( β 2 ) = 1 ,
β out = β 4 ( 1 - R 1 ) = β 2 ( 1 - R 1 ) [ R 2 ( β 2 ) / R 1 ] 1 / 2 = [ R 2 ( β 2 ) ] 1 / 2 ( 1 - R 1 ) { g 0 L + ln [ R 1 R 2 ( β 2 ) ] 1 / 2 } { ( R 1 ) 1 / 2 + [ R 2 ( β 2 ) ] 1 / 2 } { 1 - [ R 1 R 2 ( β 2 ) ] 1 / 2 } .
R = R 0 1 + R 0 I 3 ( 0 ) / I p ,
I S M = I p / R 0 .
I 4 ( 0 ) = R I 3 ( 0 ) R 0 I 3 ( 0 ) 1 + R 0 I 3 ( 0 ) / I p = [ 1 + I p R 0 I 3 ( 0 ) ] - 1 I p < I p .
I out = ( 1 - R 1 ) exp ( g L ) I p [ 1 - 1 exp ( 2 g L ) R 1 R 2 ( 0 ) ] .
I out = ( 1 - R 1 ) [ 1 - 1 R 1 R 2 ( 0 ) ] I p
β out = ( 1 - R 1 ) [ 1 - 1 R 1 R 2 ( 0 ) ] α .
β out = { 1 - [ R 2 ( 0 ) ] - 1 / 2 } 2 α