Abstract

The lowest-order self-consistent Gaussian transverse modes are derived, also the resonant frequencies of an optical resonator formed by conventional paraxial optical components plus a phase-conjugate mirror (PCM) on one end. The conventional optical elements are described by an over-all ABCD matrix. Cavities with purely real elements (no aperturing) have a continuous set of self-reproducing Gaussian modes described by a semicircular locus in the 1/q plane for one round trip; all Gaussian beams are self-reproducing after two round trips. Complex ABCD matrices, such as are produced by Gaussian aperturing in the cavity, lead to unique self-consistent perturbation-stable Gaussian modes. The resonant frequency spectrum of a PCM cavity consists of a central resonance at the driving frequency ω0 of the PCM element, independent of the cavity length L, plus half-axial sidebands spaced by Δωax = 2π(c/4L), with phase and amplitude constraints on each pair of upper and lower sidebands.

© 1980 Optical Society of America

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References

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  1. R. W. Hellwarth, J. Opt. Soc. Am. 67, 1 (Jan1977).
    [CrossRef]
  2. D. M. Bloom, G. C. Bjorklund, Appl. Phys. Lett. 31, 592 (1November1977).
    [CrossRef]
  3. A. Yariv, IEEE J. Quantum Electron. QE-14, 650 (1978); see also comments in A. Yariv, IEEE J. Quantum Electron. QE-15, 523 (1979).
    [CrossRef]
  4. J. AuYeung, D. Fekete, D. M. Pepper, A. Yariv, IEEE J. Quantum Electron. QE-15, 1180 (1979).
    [CrossRef]
  5. J. F. Lam, “Optical Resonators With Phase Conjugate Mirrors,” at 1979 IEEE/OSA Conference on Laser Engineering and Applications, Washington D.C., May 30–June 1 1979, Paper 11,1 (abstract in IEEE J. Quantum Electron. QE-15, 69D (1979).
  6. I. M. Beldyugin, M. G. Galushkin, E. M. Zemskov, Sov. J. Quantum Electron. 9, 20 (1979).
    [CrossRef]
  7. L. W. Casperson, IEEE J. Quantum Electron. QE-10, 629 (1974).
    [CrossRef]
  8. R. Trebino, A. E. Siegman, “Phase Conjugate Reflection at Arbitrary Angles Using TEM00 Pump Beams,” Opt. Commun. (in press).
  9. D. M. Pepper, R. L. Abrams, Opt. Lett. 3, 212 (1978).
    [CrossRef] [PubMed]

1979 (2)

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

I. M. Beldyugin, M. G. Galushkin, E. M. Zemskov, Sov. J. Quantum Electron. 9, 20 (1979).
[CrossRef]

1978 (2)

A. Yariv, IEEE J. Quantum Electron. QE-14, 650 (1978); see also comments in A. Yariv, IEEE J. Quantum Electron. QE-15, 523 (1979).
[CrossRef]

D. M. Pepper, R. L. Abrams, Opt. Lett. 3, 212 (1978).
[CrossRef] [PubMed]

1977 (2)

R. W. Hellwarth, J. Opt. Soc. Am. 67, 1 (Jan1977).
[CrossRef]

D. M. Bloom, G. C. Bjorklund, Appl. Phys. Lett. 31, 592 (1November1977).
[CrossRef]

1974 (1)

L. W. Casperson, IEEE J. Quantum Electron. QE-10, 629 (1974).
[CrossRef]

Abrams, R. L.

AuYeung, J.

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

Beldyugin, I. M.

I. M. Beldyugin, M. G. Galushkin, E. M. Zemskov, Sov. J. Quantum Electron. 9, 20 (1979).
[CrossRef]

Bjorklund, G. C.

D. M. Bloom, G. C. Bjorklund, Appl. Phys. Lett. 31, 592 (1November1977).
[CrossRef]

Bloom, D. M.

D. M. Bloom, G. C. Bjorklund, Appl. Phys. Lett. 31, 592 (1November1977).
[CrossRef]

Casperson, L. W.

L. W. Casperson, IEEE J. Quantum Electron. QE-10, 629 (1974).
[CrossRef]

Fekete, D.

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

Galushkin, M. G.

I. M. Beldyugin, M. G. Galushkin, E. M. Zemskov, Sov. J. Quantum Electron. 9, 20 (1979).
[CrossRef]

Hellwarth, R. W.

Lam, J. F.

J. F. Lam, “Optical Resonators With Phase Conjugate Mirrors,” at 1979 IEEE/OSA Conference on Laser Engineering and Applications, Washington D.C., May 30–June 1 1979, Paper 11,1 (abstract in IEEE J. Quantum Electron. QE-15, 69D (1979).

Pepper, D. M.

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

D. M. Pepper, R. L. Abrams, Opt. Lett. 3, 212 (1978).
[CrossRef] [PubMed]

Siegman, A. E.

R. Trebino, A. E. Siegman, “Phase Conjugate Reflection at Arbitrary Angles Using TEM00 Pump Beams,” Opt. Commun. (in press).

Trebino, R.

R. Trebino, A. E. Siegman, “Phase Conjugate Reflection at Arbitrary Angles Using TEM00 Pump Beams,” Opt. Commun. (in press).

Yariv, A.

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

A. Yariv, IEEE J. Quantum Electron. QE-14, 650 (1978); see also comments in A. Yariv, IEEE J. Quantum Electron. QE-15, 523 (1979).
[CrossRef]

Zemskov, E. M.

I. M. Beldyugin, M. G. Galushkin, E. M. Zemskov, Sov. J. Quantum Electron. 9, 20 (1979).
[CrossRef]

Appl. Phys. Lett. (1)

D. M. Bloom, G. C. Bjorklund, Appl. Phys. Lett. 31, 592 (1November1977).
[CrossRef]

IEEE J. Quantum Electron. (3)

A. Yariv, IEEE J. Quantum Electron. QE-14, 650 (1978); see also comments in A. Yariv, IEEE J. Quantum Electron. QE-15, 523 (1979).
[CrossRef]

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

L. W. Casperson, IEEE J. Quantum Electron. QE-10, 629 (1974).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Lett. (1)

Sov. J. Quantum Electron. (1)

I. M. Beldyugin, M. G. Galushkin, E. M. Zemskov, Sov. J. Quantum Electron. 9, 20 (1979).
[CrossRef]

Other (2)

R. Trebino, A. E. Siegman, “Phase Conjugate Reflection at Arbitrary Angles Using TEM00 Pump Beams,” Opt. Commun. (in press).

J. F. Lam, “Optical Resonators With Phase Conjugate Mirrors,” at 1979 IEEE/OSA Conference on Laser Engineering and Applications, Washington D.C., May 30–June 1 1979, Paper 11,1 (abstract in IEEE J. Quantum Electron. QE-15, 69D (1979).

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Figures (9)

Fig. 1
Fig. 1

(a) A typical PCM resonator and (b) the basic analytical model for discussion of the lowest-order Gaussian transverse mode of an optical cavity with a phase-conjugate mirror. The Gaussian q parameters are measured at a reference plane immediately in front of the PCM.

Fig. 2
Fig. 2

(a) Locus of self-consistent Gaussian beam modes in the |B|/q plane for real ABCD elements. (b) A single round trip inside the PCM cavity transforms an arbitrary initial point B/q to the geometrically inverted point B/q′ with respect to the displaced unit circle. A second round trip returns B/q′ to B/q.

Fig. 3
Fig. 3

Unfolded model for two successive round trips in a simple PCM cavity. The PCM acts like a converging lens for a diverging wave (solid lines) but like a diverging lens for a converging wave (dashed lines).

Fig. 4
Fig. 4

A PCM cavity with weak Gaussian aperturing has a unique lowest-order self-consistent Gaussian mode with a Gaussian parameter qsc corresponding to a single point located on or close to the displaced unit circle. A beam starting from any other initial value of q converges toward the value qsc on repeated bounces.

Fig. 5
Fig. 5

Example of a complex PCM optical cavity containing a plane mirror (PM), Gaussian aperture (GA), and phase-conjugate mirror (PCM): (a) Cavity model. (b) Self-consistent points in the 1/q plane for X = La/L ranging from 0 to 1. (c) Self-consistent mode for X → 0; the spot size approaches zero, at one end and infinity at the other. (d) Self-consistent mode for X → 1; the mode is essentially a half-confocal resonator mode.

Fig. 6
Fig. 6

(a) Optical cavity containing a plane mirror (PM), Gaussian duct (GD), Gaussian aperture (GA), and phase-conjugate mirror (PCM). (b) Locus of self-consistent solutions in the 1/q plane for various values of the relative-strength parameter Y = FηL2. Solutions on the solid line are perturbation-stable, while those along the dashed regions are confined but perturbation-unstable (for the case F > 0).

Fig. 7
Fig. 7

Locus of stable self-consistent solutions for a cavity length L containing a curved mirror of radius R0, a weak Gaussian aperture, and a phase conjugate mirror, with different values of the parameter R0/L.

Fig. 8
Fig. 8

Model for analyzing the resonant frequencies of a PCM cavity of length L. The PCM cell is pumped at frequency ω0 and has a reference plane at z = 0.

Fig. 9
Fig. 9

(a) A PCM cavity is initially filled with radiation at ω0 + Δω. (b) Reflection from the PCM converts the incident radiation at ω0 + Δω to ω0 − Δω, as shown here after ~¼ of a cavity round trip. (c) After ~1 complete round trip, the cavity is now nearly filled with radiation at ω0 − Δω, and the reverse process begins.

Equations (49)

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E inc ( x , y , z , t ) = Re E ˜ ( x , y , t ) exp [ j ( ω 0 t - k z ) ] ,
E refl ( x , y , z , t ) = Re κ ˜ E ˜ * ( x , y , t ) exp [ j ( ω 0 t + k z ) ] ,
1 q 1 R - j λ π w 2 ,
q 2 = - q 1 * .
q 1 = ( A q 2 + B ) / ( C q 2 + D ) .
B q 1 = A 2 - 1 - A B / q 1 * A - B / q 1 * ,
B 2 q q * - A B ( 1 q + 1 q * ) + A 2 - 1 = 0.
( B λ π w 2 ) 2 + ( B R - A ) 2 = 1.
( B / q - A ) ( B / q - A ) * = 1.
A = A 0 exp ( j θ A )             B = B 0 exp ( j θ B ) ,
B 0 R s c = A 0 cos ( θ B - θ A ) - sin 2 θ B 2 A 0 sin ( θ B - θ A ) ,
( B 0 λ π w s c 2 ) 2 = [ A 0 2 sin 2 ( θ B - θ A ) - sin 2 θ B ] [ A 0 2 sin 2 ( θ B - θ A ) + cos 2 θ B ] A 0 2 sin 2 ( θ B - θ A ) = cos 2 θ B + A 0 2 sin 2 ( θ B - θ A ) - [ sin 2 θ B 2 A 0 sin ( θ B - θ A ) ] 2 ,
sin 2 ( θ B - θ A ) sin 2 θ B A 0 2 ,
B R s c A 0 - θ B A 0 ( θ B - θ A ) , ( B 0 λ π w s c 2 ) 2 1 - [ θ B A 0 ( θ B - θ A ) ] 2 .
cos φ - θ B A 0 ( θ B - θ A )
| Δ q Δ q | 2 = [ Re ( A / B - 1 / q s c ) ] 2 + [ Im ( A / B - 1 / q s c ) ] 2 [ Re ( A / B - 1 / q s c ) ] 2 + [ Im ( A / B + 1 / q s c ) ] 2 = [ A 0 cos ( θ B - θ A ) - B 0 / R s c ] 2 + [ A 0 sin ( θ B - θ A ) - B 0 λ / π w s c 2 ] 2 [ A 0 cos ( θ B - θ A ) - B 0 / R s c ] 2 + [ A 0 sin ( θ B - θ A ) + B 0 λ / π w s c 2 ] 2 .
Im ( A / B ) 0
sin ( θ B - θ A ) 0.
F ( π a 2 ) / ( L λ ) ,             X ( L a ) / L .
A = 1 - 2 X ( 1 - X ) / F 2 - j 2 / F B = 2 L [ 1 - X ( 1 - X ) 2 / F 2 - j ( 1 - X 2 ) / F ] .
A 0 1 ,             B 0 2 L , θ a - 2 / F ,             θ B - ( 1 - X 2 ) / F .
L R s c 1 1 + X 2             L λ π w s c 2 X 1 + X 2 .
k 2 ( r ) = k 0 2 ( 1 - j η r 2 ) .
A = 1 - 2 j ( 1 / F + η L 2 )             B = 2 L ( 1 - 2 j η L 2 / 3 ) .
2 L R s c = 1 + ( Y 3 + 2 Y ) ( 2 L λ π w s c 2 ) 2 = 1 - ( Y 3 + 2 Y ) 2 } ,
Y F η L 2 .
Y F η L 2 - 1
Y F η L 2 - 3.
( 1 F + 2 η L 2 3 ) 0.
A = A r - j λ B r π a 2             B = B r ,
B r R s c = A r             ( B r λ π w s c 2 ) 2 = 1 + ( B r λ π a 2 ) 2 .
A r = 1 - 2 L / R 0             B r = 2 L ( 1 - L / R 0 ) .
R ( ω ) = R ( ω 0 ) | sin Δ ω L c / c Δ ω L c / c | 2 .
e ( t ) = Re E ˜ ( t ) exp ( j ω 0 t ) .
E ( t ) = Re κ ˜ E ˜ * ( t ) exp ( j ω 0 t ) .
E ( t + T ) = G 1 / 2 E ( t ) ,
E ˜ ( t + T ) exp ( j ω 0 T ) = exp ( j ψ ) E ˜ ( t ) = exp ( j ψ ) E ˜ * ( t ) .
E ˜ = E 0 exp ( j θ 0 ) ,
E 0 exp [ j ( θ 0 + ω 0 T ) ] = E 0 exp [ j ( ψ - θ 0 ) ] .
2 θ 0 = ψ - ω 0 T ( mod 2 π ) .
E ˜ ( t ) exp ( j ω m t ) .
E ˜ ( t ) E ˜ * ( t ) exp ( - j ω m t ) .
E ˜ ( t ) = exp ( j θ 0 ) [ c ˜ + exp ( j ω m t ) + c ˜ - exp ( - j ω m t ) ] ,
c ˜ + = exp ( - j ω m T ) c ˜ - * , c ˜ - = exp ( j ω m T ) c ˜ * * .
exp ( j 2 ω m T ) = exp ( j n 2 π ) ,
ω m = n ( π / T ) = n · 2 π ( c / 4 L ) .
E ( t ) = E ˜ ( t ) exp [ j ( ω 0 t + θ 0 ) ] = n = - c ˜ n exp [ j n ( π / T ) t ] exp [ j ( ω 0 t + θ 0 ) ] ,
c ˜ - n = ( - 1 ) n c ˜ n * .
E ˜ ( t ) = E 0 + j n = 1 , 3 , E n sin ( n π t / T + θ n ) + n = 2 , 4 E n cos ( n π t / T + θ n ) .

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