Abstract

A numerical analysis based on the Prony algorithm was carried out to find the higher-order modes of phase conjugate optical resonators with hard-edged apertures. The mode patterns are nearly Hermite-Gaussians even for unstable resonator configurations. This indicates that there is not a phase conjugate analog of conventional unstable resonators. The eigenvalues and the extent to which the phase fronts match the surface of the conventional mirror were also calculated for a variety of resonator parameters. When there is one limiting aperture in the resonator and all others (including the phase conjugating mirror) can be considered as unbound, the eigenvalues and phase matching parameter are scalable by the ratio g/N, where N is the Fresnel number of the aperture and g = 1 − L/R as in conventional resonator theory.

© 1982 Optical Society of America

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References

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  1. A. E. Siegman, H. Y. Miller, Appl. Opt. 9, 2729 (1970).
    [CrossRef] [PubMed]
  2. A. E. Siegman, “A Prony Algorithm for Fitting Exponential Factors or Extracting Matrix Eigenvalues from a Sequence of Complex Numbers,” Preprint Ginzton Laboratory Report 3111, Stanford U., Stanford, Calif. (Mar.1980).
  3. A. G. Fox, T. Li, IEEE J. Quantum Electron. QE-4, 460 (1968).
    [CrossRef]
  4. A. Consortini, F. Pasqualetti, Opt. Acta 20, 793 (1973).
    [CrossRef]
  5. P.-A. Belanger, A. Hardy, A. E. Siegman, Appl. Opt. 19, 602 (1980).
    [CrossRef] [PubMed]
  6. J. F. Lam, W. P. Brown, Opt. Lett. 5, 61 (1980).
    [CrossRef] [PubMed]
  7. P.-A. Belanger, A. Hardy, A. E. Siegman, Appl. Opt. 19, 479 (1980).
    [CrossRef] [PubMed]
  8. H. Kogelnik, T. Li, Appl. Opt. 5, 1550 (1966);J. A. Arnaud, Beam and Fiber Optics (Academic, New York, 1976), pp. 122–123, 175.
    [CrossRef] [PubMed]
  9. A. Hardy, P.-A. Belanger, A. E. Siegman, Appl. Opt. 21, 1121 (1982).
  10. A. Hardy, IEEE J. Quantum Electron. QE-17, 1581 (1981).
    [CrossRef]
  11. A. Hardy, S. Hochhauser, Appl. Opt. 21, 1118 (1982).
    [CrossRef] [PubMed]
  12. A. E. Siegman, P.-A. Belanger, A. Hardy, “Optical Resonators Using Phase-Conjugate Mirrors,” in Optical Phase Conjugation, R. A. Fisher, Ed. (Academic, New York, 1982).
  13. J. AuYeung, D. Fekete, D. M. Pepper, A. Yariv, IEEE J. Quantum Electron. QE-15, 1180 (1979).
    [CrossRef]

1982 (2)

A. Hardy, P.-A. Belanger, A. E. Siegman, Appl. Opt. 21, 1121 (1982).

A. Hardy, S. Hochhauser, Appl. Opt. 21, 1118 (1982).
[CrossRef] [PubMed]

1981 (1)

A. Hardy, IEEE J. Quantum Electron. QE-17, 1581 (1981).
[CrossRef]

1980 (3)

1979 (1)

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

1973 (1)

A. Consortini, F. Pasqualetti, Opt. Acta 20, 793 (1973).
[CrossRef]

1970 (1)

1968 (1)

A. G. Fox, T. Li, IEEE J. Quantum Electron. QE-4, 460 (1968).
[CrossRef]

1966 (1)

AuYeung, J.

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

Belanger, P.-A.

A. Hardy, P.-A. Belanger, A. E. Siegman, Appl. Opt. 21, 1121 (1982).

P.-A. Belanger, A. Hardy, A. E. Siegman, Appl. Opt. 19, 602 (1980).
[CrossRef] [PubMed]

P.-A. Belanger, A. Hardy, A. E. Siegman, Appl. Opt. 19, 479 (1980).
[CrossRef] [PubMed]

A. E. Siegman, P.-A. Belanger, A. Hardy, “Optical Resonators Using Phase-Conjugate Mirrors,” in Optical Phase Conjugation, R. A. Fisher, Ed. (Academic, New York, 1982).

Brown, W. P.

Consortini, A.

A. Consortini, F. Pasqualetti, Opt. Acta 20, 793 (1973).
[CrossRef]

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, IEEE J. Quantum Electron. QE-4, 460 (1968).
[CrossRef]

Hardy, A.

A. Hardy, P.-A. Belanger, A. E. Siegman, Appl. Opt. 21, 1121 (1982).

A. Hardy, S. Hochhauser, Appl. Opt. 21, 1118 (1982).
[CrossRef] [PubMed]

A. Hardy, IEEE J. Quantum Electron. QE-17, 1581 (1981).
[CrossRef]

P.-A. Belanger, A. Hardy, A. E. Siegman, Appl. Opt. 19, 479 (1980).
[CrossRef] [PubMed]

P.-A. Belanger, A. Hardy, A. E. Siegman, Appl. Opt. 19, 602 (1980).
[CrossRef] [PubMed]

A. E. Siegman, P.-A. Belanger, A. Hardy, “Optical Resonators Using Phase-Conjugate Mirrors,” in Optical Phase Conjugation, R. A. Fisher, Ed. (Academic, New York, 1982).

Hochhauser, S.

Kogelnik, H.

Lam, J. F.

Li, T.

Miller, H. Y.

Pasqualetti, F.

A. Consortini, F. Pasqualetti, Opt. Acta 20, 793 (1973).
[CrossRef]

Pepper, D. M.

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

Siegman, A. E.

A. Hardy, P.-A. Belanger, A. E. Siegman, Appl. Opt. 21, 1121 (1982).

P.-A. Belanger, A. Hardy, A. E. Siegman, Appl. Opt. 19, 602 (1980).
[CrossRef] [PubMed]

P.-A. Belanger, A. Hardy, A. E. Siegman, Appl. Opt. 19, 479 (1980).
[CrossRef] [PubMed]

A. E. Siegman, H. Y. Miller, Appl. Opt. 9, 2729 (1970).
[CrossRef] [PubMed]

A. E. Siegman, “A Prony Algorithm for Fitting Exponential Factors or Extracting Matrix Eigenvalues from a Sequence of Complex Numbers,” Preprint Ginzton Laboratory Report 3111, Stanford U., Stanford, Calif. (Mar.1980).

A. E. Siegman, P.-A. Belanger, A. Hardy, “Optical Resonators Using Phase-Conjugate Mirrors,” in Optical Phase Conjugation, R. A. Fisher, Ed. (Academic, New York, 1982).

Yariv, A.

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

Appl. Opt. (6)

IEEE J. Quantum Electron. (3)

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

A. Hardy, IEEE J. Quantum Electron. QE-17, 1581 (1981).
[CrossRef]

A. G. Fox, T. Li, IEEE J. Quantum Electron. QE-4, 460 (1968).
[CrossRef]

Opt. Acta (1)

A. Consortini, F. Pasqualetti, Opt. Acta 20, 793 (1973).
[CrossRef]

Opt. Lett. (1)

Other (2)

A. E. Siegman, P.-A. Belanger, A. Hardy, “Optical Resonators Using Phase-Conjugate Mirrors,” in Optical Phase Conjugation, R. A. Fisher, Ed. (Academic, New York, 1982).

A. E. Siegman, “A Prony Algorithm for Fitting Exponential Factors or Extracting Matrix Eigenvalues from a Sequence of Complex Numbers,” Preprint Ginzton Laboratory Report 3111, Stanford U., Stanford, Calif. (Mar.1980).

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Figures (19)

Fig. 1
Fig. 1

General resonator configuration.

Fig. 2
Fig. 2

Plot of the eigenvalue squared vs the aperture's Fresnel number for indicated parameters: solid lines, symmetric modes; broken lines, antisymmetric modes.

Fig. 3
Fig. 3

Plot of the intensity pattern of the lowest-order symmetric mode across the unbound CM for the indicated parameters. Dashed vertical lines are the boundaries of the aperture. The abscissa is normalized to the PCM diameter (b = 1).

Fig. 4
Fig. 4

Plot of the intensity pattern of the lowest-order antisymmetric mode across the CM. Dashed vertical lines are the boundaries of the aperture. The abscissa is normalized to the PCM diameter (b = 1).

Fig. 5
Fig. 5

Plot of the intensity distribution across the CM after 50 iterations of a symmetric starting wave with Na = 3.675. Dashed vertical lines are the boundaries of the aperture. The abscissa is normalized to the PCM diameter (b = 1).

Fig. 6
Fig. 6

Plot of the intensity distribution across the CM after 50 iterations of a symmetric starting wave with Na = 7.5. Dashed vertical lines are the boundaries of the aperture. The abscissa is normalized to the PCM diameter (b = 1).

Fig. 7
Fig. 7

Plot of the phase matching parameter of the iterated field vs the aperture's Fresnel number: solid line, symmetric starting wave; broken lines, antisymmetric starting wave.

Fig. 8
Fig. 8

Plot of the eigenvalue squared vs the aperture's Fresnel number: solid lines, symmetric modes; broken lines, antisymmetric modes.

Fig. 9
Fig. 9

Plot of the phase matching parameter of the iterated field vs the aperture's Fresnel number: solid line, symmetric starting wave; broken line, antisymmetric starting wave.

Fig. 10
Fig. 10

Plot of the eigenvalue squared vs the aperture's Fresnel number: solid lines, symmetric modes; broken lines, antisymmetric modes.

Fig. 11
Fig. 11

Plot of the phase matching parameter of the iterated field vs the aperture's Fresnel number: solid line, symmetric starting wave; broken line, antisymmetric starting wave.

Fig. 12
Fig. 12

Plot of the eigenvalue squared vs the aperture's Fresnel number: solid lines, symmetric modes; broken lines, antisymmetric modes.

Fig. 13
Fig. 13

Plot of the phase matching parameter of the iterated field vs the aperture's Fresnel number: solid line, symmetric starting wave; broken line, antisymmetric starting wave.

Fig. 14
Fig. 14

Plot of the eigenvalue squared vs the aperture's Fresnel number: solid lines, symmetric modes; broken lines, antisymmetric modes.

Fig. 15
Fig. 15

Plot of the phase matching parameter of the iterated field vs the aperture's Fresnel number: solid line, symmetric starting wave; broken line, antisymmetric starting wave.

Fig. 16
Fig. 16

Plot of the eigenvalue squared vs the aperture's Fresnel number: solid line (top), superposition of symmetric and antisymmetric modes; other solid lines, symmetric modes; broken lines, antisymmetric modes.

Fig. 17
Fig. 17

Plot of the intensity profile across the CM of an iterated symmetric wave after convergence for the indicated parameters. The abscissa is normalized to the PCM diameter (b = 1).

Fig. 18
Fig. 18

Plot of the intensity profile across the CM of an iterated symmetric wave after convergence for the indicated parameters. The abscissa is normalized to the PCM diameter (b = 1).

Fig. 19
Fig. 19

Schematic configuration of a large volume high mode discrimination phase conjugate resonator.

Equations (34)

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E 3 ( x 1 ) = ( ik 2 π B ) 1 / 2 exp ( ik L 2 ) a a E 2 ( x 0 ) × exp [ ik 2 B ( A x 0 2 2 x 1 x 0 + D x 1 2 ) ] d x 0 ,
A = D = 1 , B = L 2 .
E 4 ( x 1 ) = { E 3 * ( x 1 ) b x 1 b , 0 elsewhere .
E 5 ( x ) = ( ik 2 π L 2 ) 1 / 2 exp ( ik L 2 ) b b E 4 ( x 1 ) × exp [ ik 2 L 2 ( x 1 2 2 x x 1 + x 2 ) ] d x 1 ,
E 5 ( x ) = a a E 2 * ( x 0 ) exp [ ik 2 L 2 ( x 2 x 0 2 ) ] × sin [ ( kb / L 2 ) ( x x 0 ) ] π ( x x 0 ) d x 0 .
E 5 ( η ) = 1 1 E 2 * ( ξ ) exp [ i π N 2 ( η 2 ξ 2 ) ] × sin [ 2 π N 2 ( b / a ) ( η ξ ) ] π ( η ξ ) d ξ ,
ξ = x 0 / a ,
η = x / a ,
N 2 = k a 2 2 π L 2 = a 2 λ L 2 .
E 2 ( ξ ) = ( i N 1 2 g ) 1 / 2 exp ( i 2 k L 1 ) 1 1 E 1 ( η ) × exp { i π N 1 2 g [ ( 2 g 1 ) η 2 2 η ξ + ( 2 g 1 ) ξ 2 ] } d η ,
A = D = 2 g 1 ,
B = 2 g L 1 .
E 5 ( η ) = γ E 1 ( η ) 1 η 1 ,
σ = ( 0 [ ϕ ( x ) ϕ ( 0 ) ] 2 | E ( x ) | 2 dx / 0 | E ( x ) | 2 dx ) 1 / 2 .
γ E 2 ( ξ ) = ( iN 2 g ) 1 / 2 exp ( i 2 kL ) 1 1 E 2 * ( η ) × exp { i π N 2 g [ ( 2 g 1 ) η 2 2 η ξ + ( 2 g 1 ) ξ 2 ] } d η
γ u ( ξ ) = ( iN 2 g ) 1 / 2 exp ( 2 ikL ) 1 1 u * ( η ) exp ( i π N g η ξ ) d η ,
u ( η ) = E 2 ( η ) exp [ i π N 2 g ( 2 g 1 ) η 2 ] .
[ A B C D ] = [ g L 1 / R 1 ] .
γ E ( x ) = E * ( x 0 ) exp [ ikg 2 L ( x 2 x 0 2 ) ] sin [ ka L ( x x 0 ) ] π ( x x 0 ) d x 0 ,
ξ = g x 0 / a ,
η = gx / a ,
u ( η ) = E ( x ) ,
γ u ( η ) = u * ( ξ ) exp [ i π N g ( η 2 ξ 2 ) ] sin [ 2 π N g ( η ξ ) ] π ( η ξ ) d ξ .
σ = { 3 4 π g / N lowest - order symmetric , 15 4 π g / N lowest - order antisymmetric ,
| γ m | 2 = { [ 1 + ( g / π N ) 2 ] 1 / 2 + | g / π N | } 2 m 1 ,
F m = ( v n v m n * ) ,
( a b ) = PCM a ( x ) b ( x ) dx .
v 0 = K 1 u 1 + K 2 u 2 + K 3 u 3 + , v 1 = K 1 * γ 1 u 1 + K 2 * γ 2 u 2 + K 3 * γ 3 u 3 + , v 2 = K 1 γ 1 2 u 1 + K 2 γ 2 2 u 2 + K 3 γ 3 2 u 3 + , v 3 = K 1 * γ 1 3 u 1 + K 2 * γ 2 3 u 2 + K 3 * γ 3 3 u 3 + ,
F 0 = ( v 0 v 0 * ) = | K 1 | 2 + | K 2 | 2 + | K 3 | 2 + , F 1 = ( v 0 v 1 * ) = K 1 2 γ 1 + K 2 2 γ 2 + K 3 2 γ 3 + , F ̅ 1 = ( v 0 * v 1 ) = ( K 1 * ) 2 γ 1 + ( K 2 * ) 2 γ 1 + ( K 3 * ) 2 γ 3 + = F 1 * , F 2 = ( v 0 v 2 * ) = | K 1 | 2 γ 1 2 + | K 2 | 2 γ 2 2 + | K 3 | 2 γ 3 2 + , F 3 = ( v 0 v 3 * ) = K 1 2 γ 1 3 + K 2 2 γ 2 3 + K 3 2 γ 3 3 + , F ̅ 3 = ( v 0 * v 3 ) = ( K 1 * ) 2 γ 1 3 + ( K 2 * ) 2 γ 2 3 + ( K 3 * ) 2 γ 3 3 + = F 3 * ,
ψ ( x ) = j α j u j ( x ) ,
P u j = γ u j ,
P ψ ( x ) = j α j * P u j ( x ) = γ j α j * u j ( x ) .
P 2 ψ ( x ) = γ j P ( α j * u j ) = γ 2 j α j u j = γ 2 ψ ( x ) ,
P ψ ( x ) = γ j ( α j u j ) * = γ ψ ( x ) * ,

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