Abstract

An unfolded equivalent model is constructed for the round-trip transfer operator of a phase-conjugate mirror (PCM) resonator with a cavity described by complex ABCD ray matrix elements, taking into account hard-edge mirror apertures and the possible frequency shift of the wave reflected by the PCM. The treatment makes use of recently developed canonical operator techniques and extends these methods further to incorporate wavelength variation and the effect of apertures.

© 1983 Optical Society of America

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References

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  1. M. Nazarathy and J. Sharmir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–158 (1980).
    [Crossref]
  2. M. Nazarathy and J. Shamir, “Holography described by operator algebra,” J. Opt. Soc. Am. 71, 529–541 (1981).
    [Crossref]
  3. M. Nazarathy and J. Shamir, “Wavelength variation in Fourier optics and holography described by operator algebra,” Isr. J. Technol. 18, 224–231 (1980).
  4. M. Nazarathy and J. Shamir, “First order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982).
    [Crossref]
  5. M. Nazarathy and J. Shamir, “First order optics—operator representation for systems with loss or gain,” J. Opt. Soc. Am. 72, 1398–1408 (1982).
    [Crossref]
  6. M. Nazarathy, A. Hardy, and J. Shamir, “Generalized mode propagation in first-order optical systems with loss or gain,” J. Opt. Soc. Am. 72, 1409–1420 (1982).
    [Crossref]
  7. J. AuYeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1980).
  8. J. F. Lam and W. P. Brown, “Optical resonators with phase-conjugate mirrors,” Opt. Lett. 5, 61–63 (1980).
    [Crossref] [PubMed]
  9. 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]
  10. P. A. Belanger, A. Hardy, and A. E. Siegman, “Resonant modes of optical cavities with phase conjugate mirrors: higher order modes,” Appl. Opt. 19, 479–481 (1980).
    [Crossref]
  11. A. E. Siegman, P. A. Belanger, and A. Hardy, “Optical resonators using phase conjugate mirrors,” in Optical Phase Conjugation, R. H. Fisher, ed. (Academic, New York, 1983).
    [Crossref]
  12. A. Hardy, P. A. Belanger, and A. E. Siegman, “Orthogonality properties of phase conjugate optical resonators,” Appl. Opt. 21, 1122–1124 (1982).
    [Crossref] [PubMed]
  13. W. Shao-Min and H. Weber, “Aspherical resonator equivalent to arbitrary phase-conjugate resonators,” Opt. Commun. 41, 360–362 (1982).
    [Crossref]
  14. M. Nazarathy, J. Shamir, and A. Hardy, “Phase-conjugate-mirror resonators—a canonical operator analysis,” J. Opt. Soc. Am. 72, 410(A) (1982).
  15. M. Nazarathy, A. Hardy, and J. Shamir, “Generalized mode theory of conventional and phase-conjugate resonators,” J. Opt. Soc. Am. 73, 576–586 (1983).
    [Crossref]
  16. A. Yariv, “Phase conjugate optics and real-time holography,” IEEE J. Quantum Electron. QE-14, 650–660 (1978).
    [Crossref]
  17. E. Wolf, “Phase conjugacy and symmetries in spatially band-limited wavefields containing no evanescent components,” J. Opt. Soc. Am. 70, 1311–1319 (1980).
    [Crossref]
  18. E. Wolf and W. H. Carter, “Comments on the theory of phase-conjugated waves,” Opt. Commun. 40, 397–400 (1982).
    [Crossref]
  19. A. Yariv, “Reply to the paper Comments on the theory of phase-conjugated waves by E. Wolf and W. H. Carter,” Opt. Commun. 40, 401 (1982).
    [Crossref]
  20. A. E. Siegman, “A canonical formulation for analyzing multielement unstable resonators,” IEEE J. Quantum Electron. QE-12, 35–40 (1976).
    [Crossref]
  21. Notice that, because of our reduced definition of the ray-transfer matrix and optical direction cosines [Eqs. (21) and (23) of Ref. 4], the velocity of light in vacuum, c, may be used in Eq. (7) without loss of generality, even in cases when the input and output planes are immersed in some dielectric medium.
  22. A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

1983 (1)

1982 (8)

E. Wolf and W. H. Carter, “Comments on the theory of phase-conjugated waves,” Opt. Commun. 40, 397–400 (1982).
[Crossref]

A. Yariv, “Reply to the paper Comments on the theory of phase-conjugated waves by E. Wolf and W. H. Carter,” Opt. Commun. 40, 401 (1982).
[Crossref]

M. Nazarathy and J. Shamir, “First order optics—a canonical operator representation: lossless systems,” J. Opt. Soc. Am. 72, 356–364 (1982).
[Crossref]

M. Nazarathy and J. Shamir, “First order optics—operator representation for systems with loss or gain,” J. Opt. Soc. Am. 72, 1398–1408 (1982).
[Crossref]

M. Nazarathy, A. Hardy, and J. Shamir, “Generalized mode propagation in first-order optical systems with loss or gain,” J. Opt. Soc. Am. 72, 1409–1420 (1982).
[Crossref]

A. Hardy, P. A. Belanger, and A. E. Siegman, “Orthogonality properties of phase conjugate optical resonators,” Appl. Opt. 21, 1122–1124 (1982).
[Crossref] [PubMed]

W. Shao-Min and H. Weber, “Aspherical resonator equivalent to arbitrary phase-conjugate resonators,” Opt. Commun. 41, 360–362 (1982).
[Crossref]

M. Nazarathy, J. Shamir, and A. Hardy, “Phase-conjugate-mirror resonators—a canonical operator analysis,” J. Opt. Soc. Am. 72, 410(A) (1982).

1981 (1)

1980 (7)

1978 (1)

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

1976 (1)

A. E. Siegman, “A canonical formulation for analyzing multielement unstable resonators,” IEEE J. Quantum Electron. QE-12, 35–40 (1976).
[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-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1980).

Belanger, P. A.

Brown, W. P.

Carter, W. H.

E. Wolf and W. H. Carter, “Comments on the theory of phase-conjugated waves,” Opt. Commun. 40, 397–400 (1982).
[Crossref]

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-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1980).

Hardy, A.

Lam, J. F.

Nazarathy, M.

Papoulis, A.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

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-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1980).

Shamir, J.

Shao-Min, W.

W. Shao-Min and H. Weber, “Aspherical resonator equivalent to arbitrary phase-conjugate resonators,” Opt. Commun. 41, 360–362 (1982).
[Crossref]

Sharmir, J.

Siegman, A. E.

Weber, H.

W. Shao-Min and H. Weber, “Aspherical resonator equivalent to arbitrary phase-conjugate resonators,” Opt. Commun. 41, 360–362 (1982).
[Crossref]

Wolf, E.

Yariv, A.

A. Yariv, “Reply to the paper Comments on the theory of phase-conjugated waves by E. Wolf and W. H. Carter,” Opt. Commun. 40, 401 (1982).
[Crossref]

J. AuYeung, D. Fekete, D. M. Pepper, and A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1980).

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

Appl. Opt. (3)

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-conjugate mirrors,” IEEE J. Quantum Electron. QE-15, 1180–1188 (1980).

A. E. Siegman, “A canonical formulation for analyzing multielement unstable resonators,” IEEE J. Quantum Electron. QE-12, 35–40 (1976).
[Crossref]

Isr. J. Technol. (1)

M. Nazarathy and J. Shamir, “Wavelength variation in Fourier optics and holography described by operator algebra,” Isr. J. Technol. 18, 224–231 (1980).

J. Opt. Soc. Am. (8)

Opt. Commun. (3)

E. Wolf and W. H. Carter, “Comments on the theory of phase-conjugated waves,” Opt. Commun. 40, 397–400 (1982).
[Crossref]

A. Yariv, “Reply to the paper Comments on the theory of phase-conjugated waves by E. Wolf and W. H. Carter,” Opt. Commun. 40, 401 (1982).
[Crossref]

W. Shao-Min and H. Weber, “Aspherical resonator equivalent to arbitrary phase-conjugate resonators,” Opt. Commun. 41, 360–362 (1982).
[Crossref]

Opt. Lett. (1)

Other (3)

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

Notice that, because of our reduced definition of the ray-transfer matrix and optical direction cosines [Eqs. (21) and (23) of Ref. 4], the velocity of light in vacuum, c, may be used in Eq. (7) without loss of generality, even in cases when the input and output planes are immersed in some dielectric medium.

A. Papoulis, Systems and Transforms with Applications in Optics (McGraw-Hill, New York, 1968).

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Figures (3)

Fig. 1
Fig. 1

Distorting filter to be cascaded at the output of an ideal first-order system to represent the effect of inserting an aperture at the input. The rectangle represents a linear shift-invariant system with its impulse response denoted inside; ⊗’s represent multipliers.

Fig. 2
Fig. 2

(a) General first-order PCM resonator. CM denotes the conventional mirror. (b) Equivalent resonator with plane mirrors and effective pupil functions A 1 and A 2. The mirror curvature and Gaussian reflectivity profiles are absorbed into the ABCD elements.

Fig. 3
Fig. 3

Equivalent system for the round-trip complex amplitude transformation through a nonideal resonator. The input signal is transmitted through the following blocks: a phase conjugator, a first-order system with ray matrix MII, a quadratic-phase multiplier, a linear shift invariant system, and, finally, another multiplier.

Equations (53)

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T [ M ] = T [ M ] ,
M = ( A B C D ) ,             M = ( D B C A ) .
u ( x ) = u * ( x ) ,
T [ M ] = T [ M * - 1 ] .
T [ M ] = T [ M * - 1 ] ,
T [ M ] = T [ M * - 1 ] = T [ M * - 1 ] ,
T [ M ] = T [ M * - 1 ] = T [ M * - 1 ] ,
T [ M ] = T [ M * - 1 ] = T [ M * - 1 ] ,
T [ M ] = T [ M * - 1 ] = T [ M * - 1 ] .
T ( ω ) [ M ] u i ( x 0 ) exp ( - j ω t ) = u 0 ( x i ) exp ( - j ω t ) ,
T ( ω ) [ A B C D ] = - d x exp ( j ω 2 c B ) × ( D x 0 2 - 2 x 0 x i + A x i 2 ) .
k = ω c ,
T ( ω ) [ A B C D ] = - d x exp ( j ω 2 c B ) × ( D x 0 2 - 2 x 0 x i + A x i 2 ) .
T ( ω ) [ A B C D ] = T ( ω ) [ A B ω ω C ω ω D ] .
( ω ) [ d ] = T ( ω ) [ 1 d 0 1 ] = T ( ω ) [ 1 d ω ω 0 1 ] = ( ω ) [ d ω ω ] .
T = T ( ω ) [ M ( ω ) ] .
M eff ( ω , ω ) = [ A ( ω ) B ( ω ) ω ω C ( ω ) ω ω D ( ω ) ] = [ 1 0 0 ω ω ] [ A ( ω ) B ( ω ) C ( ω ) D ( ω ) ] [ 1 0 0 ω ω ] ,
W ( y ) = [ 1 0 0 y ] ,
M eff ( ω , ω ) = W ( ω ω ) M ( ω ) W - 1 ( ω ω ) .
T ( ω ) [ M ( ω ) ] = T [ M eff ( ω , ω ) ] .
T = T [ M N ] A N T [ M 2 ] A 2 T [ M 1 ] A 1 .
T [ M ] = Q [ D B ] F ¯ [ 1 B ] Q [ A B ]
F ¯ [ m ] A = { F ¯ [ m ] A } F ¯ [ m ]
T [ M ] A = Q [ D B ] F ¯ [ 1 B ] A Q [ A B ] = Q [ D B ] { F [ 1 B ] A } F [ 1 B ] Q [ A B ] .
Q [ - D B ] Q [ D B ] = 1
T [ M ] A = Q [ D B ] { F [ 1 B ] A } Q [ - D B ] T [ M ] .
T [ M ] A = C [ A , M ] T [ M ] ,
C [ A , M ] = Q [ D B ] { F ¯ [ 1 B ] A } Q [ - D B ]
A T [ M ] = T [ M ] C [ A , M - 1 ] .
T [ M ] A = C [ A , M ] T [ M ]
A T [ M ] = T [ M ] C [ A , M - 1 ] .
ω 2 = ω 1 - Δ ω ,
Δ ω = 2 ( ω 1 - ω 0 ) ,
K = A 1 2 T ( ω b ) [ M ] A 2 A 2 T ( ω f ) [ M ] .
A 2 A 2 = A 2 ( A 2 ) = A 2 2 .
K = A 1 2 T ( ω b ) [ M ] A 2 2 T ( ω f ) [ M ] .
K = T s h K T s h - 1 .
K = A 1 2 C [ A 2 2 , M ] T ( ω b ) [ M ] T ( ω f ) [ M ] .
K = A 1 2 C [ A 2 2 , M ] K i ,
K i = T ( ω b ) [ M ] T ( ω f ) [ M ]
K i = T ( ω b ) [ M * - 1 ] T ( ω f ) [ M ] = T ( ω f ) [ W ( ω b ω f ) M * - 1 W - 1 ( ω b ω f ) ] T ( ω f ) [ M ] .
K i = T ( ω f ) [ M I ] ,
M I = W ( ω b ω f ) M * - 1 W - 1 ( ω b ω f ) M = ( A D * - B * C ω f ω b B * D - B * D ω f ω b A * C - A C * ω b ω f A * D - B C * ω b ω f ) .
M I = M * - 1 M .
M I = l ,
K i = ,
K i = T ( ω b ) [ M I I ] = T ( ω b ) [ M I I ] ,
M I I = MW ( ω f ω b ) M * - 1 W - 1 ( ω f ω b ) ,
M I I = ( A D * - B C * ω f ω b A * B - A B * ω b ω f C D * - C * D ω f ω b A * D - B * C ω b ω f ) .
K i = K i ( ω b , ω f ) ,
K i i = K i ( ω 1 , ω 2 ) K i ( ω 2 , ω 1 ) .
K i i = T ( ω 1 ) [ M I I ] T ( ω 1 ) [ M I ] = T ( ω 1 ) [ M I I M I ] .
M eq = M I I M I ,