Abstract

We review the theory of optical interferometers in which a light mirror is suspended to swing as a pendulum and can therefore respond to radiation-pressure forces. The basic principles are elaborated on by considering a two-mirror system, and optical bistability and mirror confinement are predicted. A three-mirror system is also described that is capable of producing a far higher level of mirror confinement. White- and ground-noise analyses are given.

© 1985 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. P. Lebedev, Ann. Phys. (Leipzig) 6, 433 (1901); Astrophys. J. 31, 385 (1910).
  2. E. F. Nichols, G. F. Hull, Phys. Rev. 13, 307 (1901); Phys. Rev. 17, 26 (1903).
  3. A. Ashkin, Phys. Rev. Lett. 24, 156 (1970).
    [CrossRef]
  4. A. Ashkin, J. M. Dziedzic, Appl. Phys. Lett. 19, 283 (1971); Appl. Phys. Lett. 24, 586 (1974); Appl. Phys. Lett. 28, 333 (1976); Appl. Phys. Lett. 30, 202 (1977).
    [CrossRef]
  5. A. Ashkin, Phys. Rev. Lett. 40, 729 (1978); J. Dalibard, S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 17, 4577 (1984).
    [CrossRef]
  6. W. D. Phillips, ed., Progress in Quantum Electronics (Pergamon, New York, 1984), p. 115.
  7. M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).
  8. J. H. Marburger, F. S. Felber, Phys. Rev. A 17, 335 (1978).
    [CrossRef]
  9. A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, H. Walther, Phys. Rev. Lett. 51, 1550 (1983).
    [CrossRef]
  10. W. Lukosz, Opt. Lett. 10, 143 (1985).
    [CrossRef] [PubMed]
  11. J. D. Mc Cullen, P. Meystre, E. M. Wright, Opt. Lett. 9, 193 (1984).
    [CrossRef]
  12. A. Labeyrie, Astron. Astrophys. 77, L1 (1979).
  13. R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1965), Vol. 1, Chap. 34–9.
  14. C. Hayashe, Nonlinear Oscillations in Physical Systems (McGraw-Hill, New York, 1964).
  15. H. Risken, The Fokker-Planck Equation, Methods of Solutions and Applications, Springer Series in Synergetics (Springer-Verlag, Berlin, 1984).
    [CrossRef]
  16. J. Bendat, A. G. Piersol, Measurement and Analysis of Random Data (Wiley, New York, 1966).
  17. H. Billing, W. Winkler, R. Schilling, A. Rüdiger, K. Maischberger, L. Schnupp, in Quantum Optics, Experimental Gravitation and Measurement Theory, P. Meystre, M. O. Scully, eds. (Plenum, New York, 1983), p. 525.
    [CrossRef]
  18. Useful input on this question was provided by J. H. Eberly (Department of Physics and Astronomy, University of Rochester, Rochester, N.Y. 14627), J. D. Garrison (Lawrence Livermore Laboratory, Livermore, Calif. 94550), P. L. Knight (Imperial College, London, England), and S. Stenholm (University of Helsinki, Helsinki, Finland) (personal communications, 1985).
  19. R. Dändliker, T. Tschudi, Appl. Opt. 8, 119 (1969).
    [CrossRef]

1985 (1)

1984 (1)

1983 (1)

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, H. Walther, Phys. Rev. Lett. 51, 1550 (1983).
[CrossRef]

1979 (1)

A. Labeyrie, Astron. Astrophys. 77, L1 (1979).

1978 (2)

A. Ashkin, Phys. Rev. Lett. 40, 729 (1978); J. Dalibard, S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 17, 4577 (1984).
[CrossRef]

J. H. Marburger, F. S. Felber, Phys. Rev. A 17, 335 (1978).
[CrossRef]

1971 (1)

A. Ashkin, J. M. Dziedzic, Appl. Phys. Lett. 19, 283 (1971); Appl. Phys. Lett. 24, 586 (1974); Appl. Phys. Lett. 28, 333 (1976); Appl. Phys. Lett. 30, 202 (1977).
[CrossRef]

1970 (1)

A. Ashkin, Phys. Rev. Lett. 24, 156 (1970).
[CrossRef]

1969 (1)

R. Dändliker, T. Tschudi, Appl. Opt. 8, 119 (1969).
[CrossRef]

1901 (2)

P. Lebedev, Ann. Phys. (Leipzig) 6, 433 (1901); Astrophys. J. 31, 385 (1910).

E. F. Nichols, G. F. Hull, Phys. Rev. 13, 307 (1901); Phys. Rev. 17, 26 (1903).

Ashkin, A.

A. Ashkin, Phys. Rev. Lett. 40, 729 (1978); J. Dalibard, S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 17, 4577 (1984).
[CrossRef]

A. Ashkin, J. M. Dziedzic, Appl. Phys. Lett. 19, 283 (1971); Appl. Phys. Lett. 24, 586 (1974); Appl. Phys. Lett. 28, 333 (1976); Appl. Phys. Lett. 30, 202 (1977).
[CrossRef]

A. Ashkin, Phys. Rev. Lett. 24, 156 (1970).
[CrossRef]

Bendat, J.

J. Bendat, A. G. Piersol, Measurement and Analysis of Random Data (Wiley, New York, 1966).

Billing, H.

H. Billing, W. Winkler, R. Schilling, A. Rüdiger, K. Maischberger, L. Schnupp, in Quantum Optics, Experimental Gravitation and Measurement Theory, P. Meystre, M. O. Scully, eds. (Plenum, New York, 1983), p. 525.
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Dändliker, R.

R. Dändliker, T. Tschudi, Appl. Opt. 8, 119 (1969).
[CrossRef]

Dorsel, A.

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, H. Walther, Phys. Rev. Lett. 51, 1550 (1983).
[CrossRef]

Dziedzic, J. M.

A. Ashkin, J. M. Dziedzic, Appl. Phys. Lett. 19, 283 (1971); Appl. Phys. Lett. 24, 586 (1974); Appl. Phys. Lett. 28, 333 (1976); Appl. Phys. Lett. 30, 202 (1977).
[CrossRef]

Eberly, J. H.

Useful input on this question was provided by J. H. Eberly (Department of Physics and Astronomy, University of Rochester, Rochester, N.Y. 14627), J. D. Garrison (Lawrence Livermore Laboratory, Livermore, Calif. 94550), P. L. Knight (Imperial College, London, England), and S. Stenholm (University of Helsinki, Helsinki, Finland) (personal communications, 1985).

Felber, F. S.

J. H. Marburger, F. S. Felber, Phys. Rev. A 17, 335 (1978).
[CrossRef]

Feynman, R. P.

R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1965), Vol. 1, Chap. 34–9.

Garrison, J. D.

Useful input on this question was provided by J. H. Eberly (Department of Physics and Astronomy, University of Rochester, Rochester, N.Y. 14627), J. D. Garrison (Lawrence Livermore Laboratory, Livermore, Calif. 94550), P. L. Knight (Imperial College, London, England), and S. Stenholm (University of Helsinki, Helsinki, Finland) (personal communications, 1985).

Hayashe, C.

C. Hayashe, Nonlinear Oscillations in Physical Systems (McGraw-Hill, New York, 1964).

Hull, G. F.

E. F. Nichols, G. F. Hull, Phys. Rev. 13, 307 (1901); Phys. Rev. 17, 26 (1903).

Knight, P. L.

Useful input on this question was provided by J. H. Eberly (Department of Physics and Astronomy, University of Rochester, Rochester, N.Y. 14627), J. D. Garrison (Lawrence Livermore Laboratory, Livermore, Calif. 94550), P. L. Knight (Imperial College, London, England), and S. Stenholm (University of Helsinki, Helsinki, Finland) (personal communications, 1985).

Labeyrie, A.

A. Labeyrie, Astron. Astrophys. 77, L1 (1979).

Lebedev, P.

P. Lebedev, Ann. Phys. (Leipzig) 6, 433 (1901); Astrophys. J. 31, 385 (1910).

Leighton, R. B.

R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1965), Vol. 1, Chap. 34–9.

Lukosz, W.

Maischberger, K.

H. Billing, W. Winkler, R. Schilling, A. Rüdiger, K. Maischberger, L. Schnupp, in Quantum Optics, Experimental Gravitation and Measurement Theory, P. Meystre, M. O. Scully, eds. (Plenum, New York, 1983), p. 525.
[CrossRef]

Marburger, J. H.

J. H. Marburger, F. S. Felber, Phys. Rev. A 17, 335 (1978).
[CrossRef]

Mc Cullen, J. D.

McCullen, J. D.

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, H. Walther, Phys. Rev. Lett. 51, 1550 (1983).
[CrossRef]

Meystre, P.

J. D. Mc Cullen, P. Meystre, E. M. Wright, Opt. Lett. 9, 193 (1984).
[CrossRef]

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, H. Walther, Phys. Rev. Lett. 51, 1550 (1983).
[CrossRef]

Nichols, E. F.

E. F. Nichols, G. F. Hull, Phys. Rev. 13, 307 (1901); Phys. Rev. 17, 26 (1903).

Piersol, A. G.

J. Bendat, A. G. Piersol, Measurement and Analysis of Random Data (Wiley, New York, 1966).

Risken, H.

H. Risken, The Fokker-Planck Equation, Methods of Solutions and Applications, Springer Series in Synergetics (Springer-Verlag, Berlin, 1984).
[CrossRef]

Rüdiger, A.

H. Billing, W. Winkler, R. Schilling, A. Rüdiger, K. Maischberger, L. Schnupp, in Quantum Optics, Experimental Gravitation and Measurement Theory, P. Meystre, M. O. Scully, eds. (Plenum, New York, 1983), p. 525.
[CrossRef]

Sands, M.

R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1965), Vol. 1, Chap. 34–9.

Schilling, R.

H. Billing, W. Winkler, R. Schilling, A. Rüdiger, K. Maischberger, L. Schnupp, in Quantum Optics, Experimental Gravitation and Measurement Theory, P. Meystre, M. O. Scully, eds. (Plenum, New York, 1983), p. 525.
[CrossRef]

Schnupp, L.

H. Billing, W. Winkler, R. Schilling, A. Rüdiger, K. Maischberger, L. Schnupp, in Quantum Optics, Experimental Gravitation and Measurement Theory, P. Meystre, M. O. Scully, eds. (Plenum, New York, 1983), p. 525.
[CrossRef]

Stenholm, S.

Useful input on this question was provided by J. H. Eberly (Department of Physics and Astronomy, University of Rochester, Rochester, N.Y. 14627), J. D. Garrison (Lawrence Livermore Laboratory, Livermore, Calif. 94550), P. L. Knight (Imperial College, London, England), and S. Stenholm (University of Helsinki, Helsinki, Finland) (personal communications, 1985).

Tschudi, T.

R. Dändliker, T. Tschudi, Appl. Opt. 8, 119 (1969).
[CrossRef]

Vignes, E.

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, H. Walther, Phys. Rev. Lett. 51, 1550 (1983).
[CrossRef]

Walther, H.

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, H. Walther, Phys. Rev. Lett. 51, 1550 (1983).
[CrossRef]

Winkler, W.

H. Billing, W. Winkler, R. Schilling, A. Rüdiger, K. Maischberger, L. Schnupp, in Quantum Optics, Experimental Gravitation and Measurement Theory, P. Meystre, M. O. Scully, eds. (Plenum, New York, 1983), p. 525.
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

Wright, E. M.

Ann. Phys. (Leipzig) (1)

P. Lebedev, Ann. Phys. (Leipzig) 6, 433 (1901); Astrophys. J. 31, 385 (1910).

Appl. Opt. (1)

R. Dändliker, T. Tschudi, Appl. Opt. 8, 119 (1969).
[CrossRef]

Appl. Phys. Lett. (1)

A. Ashkin, J. M. Dziedzic, Appl. Phys. Lett. 19, 283 (1971); Appl. Phys. Lett. 24, 586 (1974); Appl. Phys. Lett. 28, 333 (1976); Appl. Phys. Lett. 30, 202 (1977).
[CrossRef]

Astron. Astrophys. (1)

A. Labeyrie, Astron. Astrophys. 77, L1 (1979).

Opt. Lett. (2)

Phys. Rev. (1)

E. F. Nichols, G. F. Hull, Phys. Rev. 13, 307 (1901); Phys. Rev. 17, 26 (1903).

Phys. Rev. A (1)

J. H. Marburger, F. S. Felber, Phys. Rev. A 17, 335 (1978).
[CrossRef]

Phys. Rev. Lett. (3)

A. Dorsel, J. D. McCullen, P. Meystre, E. Vignes, H. Walther, Phys. Rev. Lett. 51, 1550 (1983).
[CrossRef]

A. Ashkin, Phys. Rev. Lett. 24, 156 (1970).
[CrossRef]

A. Ashkin, Phys. Rev. Lett. 40, 729 (1978); J. Dalibard, S. Reynaud, C. Cohen-Tannoudji, J. Phys. B 17, 4577 (1984).
[CrossRef]

Other (8)

W. D. Phillips, ed., Progress in Quantum Electronics (Pergamon, New York, 1984), p. 115.

M. Born, E. Wolf, Principles of Optics (Pergamon, Oxford, 1965).

R. P. Feynman, R. B. Leighton, M. Sands, The Feynman Lectures on Physics (Addison-Wesley, Reading, Mass., 1965), Vol. 1, Chap. 34–9.

C. Hayashe, Nonlinear Oscillations in Physical Systems (McGraw-Hill, New York, 1964).

H. Risken, The Fokker-Planck Equation, Methods of Solutions and Applications, Springer Series in Synergetics (Springer-Verlag, Berlin, 1984).
[CrossRef]

J. Bendat, A. G. Piersol, Measurement and Analysis of Random Data (Wiley, New York, 1966).

H. Billing, W. Winkler, R. Schilling, A. Rüdiger, K. Maischberger, L. Schnupp, in Quantum Optics, Experimental Gravitation and Measurement Theory, P. Meystre, M. O. Scully, eds. (Plenum, New York, 1983), p. 525.
[CrossRef]

Useful input on this question was provided by J. H. Eberly (Department of Physics and Astronomy, University of Rochester, Rochester, N.Y. 14627), J. D. Garrison (Lawrence Livermore Laboratory, Livermore, Calif. 94550), P. L. Knight (Imperial College, London, England), and S. Stenholm (University of Helsinki, Helsinki, Finland) (personal communications, 1985).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (8)

Fig. 1
Fig. 1

Pendulum mirror with incident and transmitted fields.

Fig. 2
Fig. 2

Two-mirror-system. Note that E2 = 0 (see Fig. 1).

Fig. 3
Fig. 3

Characteristic curves. Parameter values are those given in Section 3.A, and ϕ0 = −2.0.

Fig. 4
Fig. 4

Potential curves for a) the two-mirror case; b) the three-mirror case with one-sided pumping; c) the three-mirror case with symmetric pumping. The arrow in curve a) indicates the position of the first potential minimum. Parameter values are R = 0.99, R′ = 0.95, and K2 = 109.

Fig. 5
Fig. 5

Phase-space separatrices of the moving mirror trajectories, constructed as described in text. The shaded areas delimit the region of initial conditions evolving toward (a) the first, nonresonant potential minimum [arrow in Fig. 4a], (b) the second potential minimum (first resonant minimum for increasing ξ). In both cases, the trajectory drawn illustrates the approach to the next potential minimum.

Fig. 6
Fig. 6

Three-mirror system. The system comprises two subcavities 1 and 2 as marked.

Fig. 7
Fig. 7

Numerical (solid curves) and linearized theory (dashed curves) results for the frequency response of the three-mirror system with symmetric pumping.

Fig. 8
Fig. 8

Probability density Q(ξ) as a function of ξ and uth/u0 together with the corresponding contour plot for (a), (b) the two-mirror and (c), (d) three-mirror systems. These results correspond to the deepest potential wells in Figs. 4a) and 4b).

Equations (45)

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

F RP = 1 c ( E 1 2 + E 3 2 - E 2 2 - E 4 2 ) ,
x ¨ + γ x ˙ + Ω P 2 x = 1 m F RP ,
ξ = x / ( λ / 2 ) τ = Ω P t , Γ = γ / Ω P ,
ξ ¨ + Γ ξ ˙ + ξ = ( 2 m Ω p 2 λ ) F RP = P ,
E 1 ( ξ ) 2 = 1 [ 1 + F 2 sin 2 ( ϕ / 2 ) ] T P in ( 1 - R R ) 2 ,
F 2 = 4 R R ( 1 - R R ) 2
P 2 ( ξ ) = K 2 1 + F 2 sin 2 [ ( 2 π ξ + ϕ 0 ) / 2 ] ,
K 2 = ( 1 m c λ Ω p 2 ) 4 R ( 1 - R ) ( 1 - R R ) 2 P in ,
ξ s { 1 + F 2 sin 2 [ ( 2 π ξ s + ϕ 0 ) / 2 ] } = K 2 .
P out = 1 1 + F 2 sin 2 [ ( 2 π ξ s + ϕ 0 ) / 2 ] ( 1 - R ) ( 1 - R ) ( 1 - R R ) 2 P in ,
ξ ¨ = P 2 ( ξ ) - ξ = - V 2 ξ ,
V 2 ( ξ ) = ξ 2 2 - 0 ξ d s P 2 ( s ) .
P 3 ( ξ ) = K 3 [ f ( 1 + R ) - 2 cos ( ϕ - ϕ T ) ] / D ,
D ( ξ ) = 1 + F 3 [ f cos ( ϕ - ϕ T ) / 2 - cos ( ϕ T / 2 ) ] × [ f R cos ( ϕ - ϕ T / 2 ) - cos ( ϕ T / 2 ) ] .
ϕ = 2 π ξ + ϕ 1 , F 3 = 4 R / ( 1 - R ) 2 , K 3 = 1 f ( 1 - f R 1 - R ) 2 K 2 .
P 3 s ( ξ ) = - 4 K 3 sin ( ϕ T / 2 ) sin ( ϕ - ϕ T / 2 ) / D ,
cos ( ϕ T / 2 ) = f ( 1 + R ) / 2.
η ¨ + Γ η + α 2 η = d ( τ ) ,
α 2 = 1 - P ξ | ξ e .
Ω eff = α Ω P .
L ( Δ ) = 1 ( α 2 + Δ 2 ) 2 + ( Γ Δ ) 2 .
P 3 s ξ = - 4 K 3 sin ( ϕ T / 2 ) × [ 2 π cos ( ϕ - ϕ T / 2 ) D - sin ( ϕ - ϕ T / 2 ) D 2 D ξ ] .
Ω eff Ω P { 1 + 4 π [ 4 - f 2 ( 1 + R ) 2 ] 1 / 2 K 3 / T } 1 / 2 .
Ω eff 4 ( π { R P [ 4 - f 2 ( 1 + R ) 2 ] } 1 / 2 T T 1 m c λ ) 1 / 2 P in .
ξ = u , u = - Γ u - V ξ + ζ ( τ ) ,
V ( ξ ) = ξ 2 2 - 0 ξ d s P ( s ) ,
ζ ( τ ) = 0 , ζ ( τ ) ζ ( τ ) = 2 D δ ( τ - τ ) .
D = Γ ( u t h / u 0 ) 2 ,
τ W ( u , ξ , τ ) = [ - u ξ + u ( Γ u + V ξ ) + D 2 u 2 ] W .
W ( u , ξ ) = N - 1 exp { - ( u 0 u t h ) 2 [ u 2 2 + V ( ξ ) ] } ,
Q ( ξ ) = exp [ - ( u 0 u t h ) 2 V ( ξ ) ] / - d s exp [ - ( u 0 u t h ) 2 V ( s ) ] ,
x ¨ 1 + γ 1 x ˙ 1 + Ω 1 2 x 1 - 1 m 1 F RP = - z ¨ , x ¨ 2 + γ 2 x ˙ 2 + Ω 2 2 x 2 - 1 m 2 F RP = - z ¨ .
S η ( ω ) = H ( ω ) 2 S z ( ω ) .
η 2 = d w S η ( ω ) .
H ( ω ) 2 = ω 4 ( Ω 1 2 - Ω 2 2 ) [ ( Ω + 2 - ω 2 ) 2 + γ 2 ω 2 ] [ ( Ω - 2 - ω 2 ) 2 + γ 2 ω 2 ] ,
Ω ± 2 = Ω 1 2 + Ω 2 2 + α 2 2 ± 1 2 [ ( Ω 1 2 - Ω 2 2 + α 2 ) 2 - 4 α 2 ( Ω 1 2 - Ω 2 2 ) 1 + m 2 / m 1 ] 1 / 2 ,
Ω + 2 = Ω 1 2 , Ω - 2 = Ω 2 2 + α 2 .
S z ( w ) ~ C / w 4 ,
H ( ω ) 2 ~ ω 4 [ ( Ω 2 2 + α 2 - ω 2 ) 2 + γ 2 ω 2 ]             for Ω 1 ω .
E 1 2 + E 3 2 - E 2 2 - E 4 2 ,
E 1 = R exp ( i δ 1 ) E 3 + ( T P in ) 1 / 2 , E 2 = R exp ( i δ 2 ) E 4 ,
δ 1 = ϕ 1 + 2 π ξ , δ 2 = ϕ 2 - 2 π ξ ,
E 3 = R E 1 + i T E 2 , E 4 = i T E 1 + R E 2 .
[ E 1 E 2 ] = [ R R exp ( i δ 1 ) i R T exp ( i δ 1 ) i R T exp ( i δ 2 ) R R exp ( i δ 2 ) ] [ E 1 E 2 ] + [ T P in 0 ] ,
E 1 = [ 1 - R R exp ( i δ 2 ) ] T P in Det ,

Metrics