Abstract

A three-mirror system consisting of a light, movable, semitransparent mirror located inside an optical resonator of a normal mirror and a phase-conjugate mirror is theoretically analyzed. The phase-conjugate mirror is composed provided by nearly degenerate four-wave mixing in a Kerr medium. A scheme of confining the movable mirror in a mechanical potential well through radiation pressure is described, with particular attention to the resonant mode structure of the three-mirror system.

© 1985 Optical Society of America

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  1. A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, H. Walther, “Optical bistability and mirror confinement induced by radiation pressure,” Phys. Rev. Lett. 51, 1550 (1983).
    [CrossRef]
  2. J. D. McCullen, P. Meystre, E. M. Wright, “Mirror confinement and control through radiation pressure,” Opt. Lett. 9, 193 (1984).
    [CrossRef] [PubMed]
  3. A. Dorsel, W. J. Firth, A. Guzman de Garcia, J. D. McCullen, F. Marquis, P. Meystre, D. P. J. O’Brien, G. Reiner, M. Sargent, K. Ujihara, E. Vignes, H. Walther, E. M. Wright, “Empty nonlinear resonators,” in Nonequilibrium Quantum Statistical Physics, G. Moore, M. O. Scully, eds. (Plenum, New York, to be published).
  4. P. Meystre, E. M. Wright, J. D. McCullen, E. Vignes, “Theory of radiation-pressure-driven interferometers,” J. Opt. Soc. Am. B 2, 1830–1840 (1985).
    [CrossRef]
  5. J. AuYeung, D. Fekete, D. M. Pepper, A. Yariv, “A theoretical and experimental investigation of the modes of optical resonators with phase conjugate mirror,” IEEE J. Quantum Electron. QE-15, 1180 (1979).
    [CrossRef]
  6. P. Belanger, A. Hardy, A. E. Siegman, “Resonant modes of optical cavities with phase-conjugate mirrors,” Appl. Opt. 19, 602 (1980).
    [CrossRef] [PubMed]
  7. 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, 1983).
    [CrossRef]
  8. D. Z. Anderson, “Coupled resonators employing phase-conjugate and ordinary mirrors,” Opt. Lett. 9, 417 (1984).
    [CrossRef] [PubMed]
  9. K. Ujihara, P. Meystre, “On the finesse of a phase-conjugate Fabry–Perot resonator using nearly-degenerate four-wave mixing,” Opt. Commun. 53, 48 (1985).
    [CrossRef]
  10. K. Ujihara, “Mean field theory of optical bistability in a phase-conjugate resonator using nearly degenerate four-wave mixing,” IEEE J. Quantum Electron. (to be published).
  11. A. Yariv, D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16 (1977).
    [CrossRef] [PubMed]

1985 (2)

K. Ujihara, P. Meystre, “On the finesse of a phase-conjugate Fabry–Perot resonator using nearly-degenerate four-wave mixing,” Opt. Commun. 53, 48 (1985).
[CrossRef]

P. Meystre, E. M. Wright, J. D. McCullen, E. Vignes, “Theory of radiation-pressure-driven interferometers,” J. Opt. Soc. Am. B 2, 1830–1840 (1985).
[CrossRef]

1984 (2)

1983 (1)

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, H. Walther, “Optical bistability and mirror confinement induced by radiation pressure,” Phys. Rev. Lett. 51, 1550 (1983).
[CrossRef]

1980 (1)

1979 (1)

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

1977 (1)

Anderson, D. Z.

AuYeung, J.

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

Belanger, P.

Belanger, P. A.

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, 1983).
[CrossRef]

Dorsel, A.

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, H. Walther, “Optical bistability and mirror confinement induced by radiation pressure,” Phys. Rev. Lett. 51, 1550 (1983).
[CrossRef]

A. Dorsel, W. J. Firth, A. Guzman de Garcia, J. D. McCullen, F. Marquis, P. Meystre, D. P. J. O’Brien, G. Reiner, M. Sargent, K. Ujihara, E. Vignes, H. Walther, E. M. Wright, “Empty nonlinear resonators,” in Nonequilibrium Quantum Statistical Physics, G. Moore, M. O. Scully, eds. (Plenum, New York, to be published).

Fekete, D.

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

Firth, W. J.

A. Dorsel, W. J. Firth, A. Guzman de Garcia, J. D. McCullen, F. Marquis, P. Meystre, D. P. J. O’Brien, G. Reiner, M. Sargent, K. Ujihara, E. Vignes, H. Walther, E. M. Wright, “Empty nonlinear resonators,” in Nonequilibrium Quantum Statistical Physics, G. Moore, M. O. Scully, eds. (Plenum, New York, to be published).

Guzman de Garcia, A.

A. Dorsel, W. J. Firth, A. Guzman de Garcia, J. D. McCullen, F. Marquis, P. Meystre, D. P. J. O’Brien, G. Reiner, M. Sargent, K. Ujihara, E. Vignes, H. Walther, E. M. Wright, “Empty nonlinear resonators,” in Nonequilibrium Quantum Statistical Physics, G. Moore, M. O. Scully, eds. (Plenum, New York, to be published).

Hardy, A.

P. Belanger, A. Hardy, A. E. Siegman, “Resonant modes of optical cavities with phase-conjugate mirrors,” 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, 1983).
[CrossRef]

Marquis, F.

A. Dorsel, W. J. Firth, A. Guzman de Garcia, J. D. McCullen, F. Marquis, P. Meystre, D. P. J. O’Brien, G. Reiner, M. Sargent, K. Ujihara, E. Vignes, H. Walther, E. M. Wright, “Empty nonlinear resonators,” in Nonequilibrium Quantum Statistical Physics, G. Moore, M. O. Scully, eds. (Plenum, New York, to be published).

McCullen, J. D.

P. Meystre, E. M. Wright, J. D. McCullen, E. Vignes, “Theory of radiation-pressure-driven interferometers,” J. Opt. Soc. Am. B 2, 1830–1840 (1985).
[CrossRef]

J. D. McCullen, P. Meystre, E. M. Wright, “Mirror confinement and control through radiation pressure,” Opt. Lett. 9, 193 (1984).
[CrossRef] [PubMed]

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, H. Walther, “Optical bistability and mirror confinement induced by radiation pressure,” Phys. Rev. Lett. 51, 1550 (1983).
[CrossRef]

A. Dorsel, W. J. Firth, A. Guzman de Garcia, J. D. McCullen, F. Marquis, P. Meystre, D. P. J. O’Brien, G. Reiner, M. Sargent, K. Ujihara, E. Vignes, H. Walther, E. M. Wright, “Empty nonlinear resonators,” in Nonequilibrium Quantum Statistical Physics, G. Moore, M. O. Scully, eds. (Plenum, New York, to be published).

Meystre, P.

P. Meystre, E. M. Wright, J. D. McCullen, E. Vignes, “Theory of radiation-pressure-driven interferometers,” J. Opt. Soc. Am. B 2, 1830–1840 (1985).
[CrossRef]

K. Ujihara, P. Meystre, “On the finesse of a phase-conjugate Fabry–Perot resonator using nearly-degenerate four-wave mixing,” Opt. Commun. 53, 48 (1985).
[CrossRef]

J. D. McCullen, P. Meystre, E. M. Wright, “Mirror confinement and control through radiation pressure,” Opt. Lett. 9, 193 (1984).
[CrossRef] [PubMed]

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, H. Walther, “Optical bistability and mirror confinement induced by radiation pressure,” Phys. Rev. Lett. 51, 1550 (1983).
[CrossRef]

A. Dorsel, W. J. Firth, A. Guzman de Garcia, J. D. McCullen, F. Marquis, P. Meystre, D. P. J. O’Brien, G. Reiner, M. Sargent, K. Ujihara, E. Vignes, H. Walther, E. M. Wright, “Empty nonlinear resonators,” in Nonequilibrium Quantum Statistical Physics, G. Moore, M. O. Scully, eds. (Plenum, New York, to be published).

O’Brien, D. P. J.

A. Dorsel, W. J. Firth, A. Guzman de Garcia, J. D. McCullen, F. Marquis, P. Meystre, D. P. J. O’Brien, G. Reiner, M. Sargent, K. Ujihara, E. Vignes, H. Walther, E. M. Wright, “Empty nonlinear resonators,” in Nonequilibrium Quantum Statistical Physics, G. Moore, M. O. Scully, eds. (Plenum, New York, to be published).

Pepper, D. M.

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

A. Yariv, D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16 (1977).
[CrossRef] [PubMed]

Reiner, G.

A. Dorsel, W. J. Firth, A. Guzman de Garcia, J. D. McCullen, F. Marquis, P. Meystre, D. P. J. O’Brien, G. Reiner, M. Sargent, K. Ujihara, E. Vignes, H. Walther, E. M. Wright, “Empty nonlinear resonators,” in Nonequilibrium Quantum Statistical Physics, G. Moore, M. O. Scully, eds. (Plenum, New York, to be published).

Sargent, M.

A. Dorsel, W. J. Firth, A. Guzman de Garcia, J. D. McCullen, F. Marquis, P. Meystre, D. P. J. O’Brien, G. Reiner, M. Sargent, K. Ujihara, E. Vignes, H. Walther, E. M. Wright, “Empty nonlinear resonators,” in Nonequilibrium Quantum Statistical Physics, G. Moore, M. O. Scully, eds. (Plenum, New York, to be published).

Siegman, A. E.

P. Belanger, A. Hardy, A. E. Siegman, “Resonant modes of optical cavities with phase-conjugate mirrors,” 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, 1983).
[CrossRef]

Ujihara, K.

K. Ujihara, P. Meystre, “On the finesse of a phase-conjugate Fabry–Perot resonator using nearly-degenerate four-wave mixing,” Opt. Commun. 53, 48 (1985).
[CrossRef]

K. Ujihara, “Mean field theory of optical bistability in a phase-conjugate resonator using nearly degenerate four-wave mixing,” IEEE J. Quantum Electron. (to be published).

A. Dorsel, W. J. Firth, A. Guzman de Garcia, J. D. McCullen, F. Marquis, P. Meystre, D. P. J. O’Brien, G. Reiner, M. Sargent, K. Ujihara, E. Vignes, H. Walther, E. M. Wright, “Empty nonlinear resonators,” in Nonequilibrium Quantum Statistical Physics, G. Moore, M. O. Scully, eds. (Plenum, New York, to be published).

Vignes, E.

P. Meystre, E. M. Wright, J. D. McCullen, E. Vignes, “Theory of radiation-pressure-driven interferometers,” J. Opt. Soc. Am. B 2, 1830–1840 (1985).
[CrossRef]

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, H. Walther, “Optical bistability and mirror confinement induced by radiation pressure,” Phys. Rev. Lett. 51, 1550 (1983).
[CrossRef]

A. Dorsel, W. J. Firth, A. Guzman de Garcia, J. D. McCullen, F. Marquis, P. Meystre, D. P. J. O’Brien, G. Reiner, M. Sargent, K. Ujihara, E. Vignes, H. Walther, E. M. Wright, “Empty nonlinear resonators,” in Nonequilibrium Quantum Statistical Physics, G. Moore, M. O. Scully, eds. (Plenum, New York, to be published).

Walther, H.

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, H. Walther, “Optical bistability and mirror confinement induced by radiation pressure,” Phys. Rev. Lett. 51, 1550 (1983).
[CrossRef]

A. Dorsel, W. J. Firth, A. Guzman de Garcia, J. D. McCullen, F. Marquis, P. Meystre, D. P. J. O’Brien, G. Reiner, M. Sargent, K. Ujihara, E. Vignes, H. Walther, E. M. Wright, “Empty nonlinear resonators,” in Nonequilibrium Quantum Statistical Physics, G. Moore, M. O. Scully, eds. (Plenum, New York, to be published).

Wright, E. M.

P. Meystre, E. M. Wright, J. D. McCullen, E. Vignes, “Theory of radiation-pressure-driven interferometers,” J. Opt. Soc. Am. B 2, 1830–1840 (1985).
[CrossRef]

J. D. McCullen, P. Meystre, E. M. Wright, “Mirror confinement and control through radiation pressure,” Opt. Lett. 9, 193 (1984).
[CrossRef] [PubMed]

A. Dorsel, W. J. Firth, A. Guzman de Garcia, J. D. McCullen, F. Marquis, P. Meystre, D. P. J. O’Brien, G. Reiner, M. Sargent, K. Ujihara, E. Vignes, H. Walther, E. M. Wright, “Empty nonlinear resonators,” in Nonequilibrium Quantum Statistical Physics, G. Moore, M. O. Scully, eds. (Plenum, New York, to be published).

Yariv, A.

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

A. Yariv, D. M. Pepper, “Amplified reflection, phase conjugation, and oscillation in degenerate four-wave mixing,” Opt. Lett. 1, 16 (1977).
[CrossRef] [PubMed]

Appl. Opt. (1)

IEEE J. Quantum Electron. (1)

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

J. Opt. Soc. Am. B (1)

Opt. Commun. (1)

K. Ujihara, P. Meystre, “On the finesse of a phase-conjugate Fabry–Perot resonator using nearly-degenerate four-wave mixing,” Opt. Commun. 53, 48 (1985).
[CrossRef]

Opt. Lett. (3)

Phys. Rev. Lett. (1)

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, H. Walther, “Optical bistability and mirror confinement induced by radiation pressure,” Phys. Rev. Lett. 51, 1550 (1983).
[CrossRef]

Other (3)

A. Dorsel, W. J. Firth, A. Guzman de Garcia, J. D. McCullen, F. Marquis, P. Meystre, D. P. J. O’Brien, G. Reiner, M. Sargent, K. Ujihara, E. Vignes, H. Walther, E. M. Wright, “Empty nonlinear resonators,” in Nonequilibrium Quantum Statistical Physics, G. Moore, M. O. Scully, eds. (Plenum, New York, to be published).

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, 1983).
[CrossRef]

K. Ujihara, “Mean field theory of optical bistability in a phase-conjugate resonator using nearly degenerate four-wave mixing,” IEEE J. Quantum Electron. (to be published).

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Figures (8)

Fig. 1
Fig. 1

Arrangement of the PCM–SM–NM system. The PCM is pumped by waves E1 and E2 of frequency ω. Wave E3 of frequency ω + δ excites the system.

Fig. 2
Fig. 2

(a) The eight components of the field hitting or departing from the SM. (b) The corresponding components of the radiation pressure.

Fig. 3
Fig. 3

Resonance spectrum with respect to the optical frequency, |D(ω)|−2. The abscissa is scaled to (/l3). The vertical scale is arbitrary. ωl and other parameter values are given in the text.

Fig. 4
Fig. 4

Resonance spectrum with respect to the SM displacement l3, |D(l3)|−2. The abscissa is scaled to (π/k). The vertical scale is arbitrary. l1 and other parameter values are given in the text.

Fig. 5
Fig. 5

Examples of (a) regular, (b) irregular, and (c) complicated resonance spectra with respect to the frequency offset δ. The abscissa is scaled to δ0 = /l2, and the vertical scale is arbitrary. (a) l2/l3 = 1/2, cos 2kl3 = 1; (b) l2/l3 = 1.1/1.9, cos 2kl3 = 1; (C) l2/l3 = 1/2, cos 2kl3 = 0.5. Other parameter values are given in the text.

Fig. 6
Fig. 6

Arrangement of the resonances (ω) and (ω ± δ) and schematic of P+, P, and Pt.

Fig. 7
Fig. 7

Examples of the potential U and the total pressure Pt versus displacement of SM for different l0. (a) l0 = 2462 λ, (b) l0 = (2459 + ½) λ, (c) l0 = (2464 + ½) λ. The abscissa is in units of 10−2 λ, and the vertical scale is arbitrary but in the same units for (a)–(c). Other parameter values are given in the text.

Fig. 8
Fig. 8

The potential U and (a) the total pressure Pt, (b) P+ and P, and (c) 1/|D|2 versus displacement of the SM corresponding to Fig. 7(a). The horizontal scale is in units of 10−2 λ. The vertical scales are arbitrary but are in the same units for (a) and (b). The locations of ω and ω ± δ resonances are indicated by bars.

Equations (22)

Equations on this page are rendered with MathJax. Learn more.

D = A + B cos 2 k l 3 ,
A = 1 + R S R N exp ( 4 i Δ k l 3 ) - η [ R S + R N exp ( 4 i Δ k l 3 ) ] e i θ ,
B = 2 ( R S R N ) 1 / 2 exp ( 2 i Δ k l 3 ) ( 1 - η e i θ ) ,
θ = 2 Ψ ¯ + 4 Δ k l 2 .
η = g 2 sin 2 K l 1 K 2 cos 2 K l 1 + ( n Δ k ) 2 sin 2 K l 1 ,
Ψ = tan - 1 [ ( Δ k / K ) tan ( K l 1 ) ] ,
ω = - i c 2 l 3 ( ln { - ( A / B ) ± [ ( A / B ) 2 - 1 ] 1 / 2 } + 2 i m π ) ,
Δ l 3 = - i 2 k ( ln { - ( A / B ) ± [ ( A / B ) 2 - 1 ] 1 / 2 } ) ,
P t = P + + P - ,
P + = 2 i c | in D | 2 η 2 T S T N × [ R S ( R N + 1 ) + 2 ( R S R N ) 1 / 2 cos 2 ( k + Δ k ) l 3 ] × R S + R N + 2 ( R S R N ) 1 / 2 cos 2 ( k - Δ k ) l 3 1 + R S R N + 2 ( R S R N ) 1 / 2 cos 2 ( k + Δ k ) l 3 ,
P - = 2 0 c | in D | 2 η T S T N × [ R S ( R N + 1 ] + 2 ( R S R N ) 1 / 2 cos 2 ( k - Δ k ) l 3 ] .
f 1 2 = + 2 ,
f 2 = - 2 ,
f 3 2 = | in T S T N + + Q exp [ i ( n - 1 ) k + l 1 ] 1 + R S R N exp ( 2 i k + l 3 ) | 2 ,
Q = R S exp [ 2 i k + ( l 1 + l 2 ) ] + R N exp [ 2 i k + ( l 1 + l 2 + l 3 ) ] ,
f 4 2 = | - [ R S + R N exp ( 2 i k - l 3 ) ] 1 + R S R N exp ( 2 i k - l 3 ) | 2 ,
f 5 2 = | + T S exp [ i ( n - 1 ) k + l 1 ] - in R S T N exp [ - 2 i k + ( l 1 + l 2 ) ] 1 + R S R N exp ( 2 i k + l 3 ) | 2 ,
f 6 2 = | - T S exp [ i ( n - 1 ) k - l 1 ] 1 + R S R N exp ( 2 i k - l 3 ) | 2 ,
f 7 2 = | + R N T S exp { i k + [ ( n + 1 ) l 1 + 2 ( l 2 + l 3 ) ] } + in T N 1 + R S R N exp ( 2 i k + l 3 ) | 2 ,
f 8 2 = | - R N T S 1 + R S R N exp ( 2 i k - l 3 ) | 2 ,
+ = in D η exp ( 2 i Ψ - 4 i n Δ k l 1 ) T S T N [ R S + R N exp ( - 2 i k - l 3 ) ] × exp i { [ ( n - 1 ) k + - 2 k - ] l 1 - 2 k - l 2 }
- = in * D * η exp ( i Ψ + 2 i n Δ k l 1 ) T S T N × [ 1 + R S R N exp ( 2 i k - l 3 ) ] exp [ i ( 1 - n ) k + l 1 ]

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