Abstract

I describe how exactly the intracavity fields in a dual-recycling cavity build up their power before achieving a steady-state value. The analysis is restricted to interferometers with lossless mirrors and a beam splitter. The complete series representation of intracavity lights at any stage of evolution in a nonsteady state is presented.

© 1996 Optical Society of America

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  1. A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. W. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational Wave Observatory,” Science 256, 325–333 (1992).
  2. R. W. P. Drever, “Interferometer detectors for gravitational radiation,” in Gravitational Radiation, N. Deruelle, T. Piran, eds. (North-Holland, Amsterdam, 1983), pp. 321–338.
  3. J.-Y. Vinet, B. J. Meers, C. N. Man, A. Brillet, “Optimization of long-baseline optical interferometers for gravitationalwave detection,” Phys. Rev. D 38, 433–447 (1988).
  4. B. J. Meers, “Recycling in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 38, 2317–2326 (1988);“The frequency response of interferometric gravitationalwave detectors,” Phys. Lett. A 142, 465–470 (1989).
  5. S. Wolfram, Mathematica: a System for doing Mathematics by Computer, 2nd ed. (Addison-Wesley, Redwood City, Calif., 1991).
  6. C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1991).
  7. A. Brillet, J. Gea-Banacloche, G. Leuches, C. N. Man, J.-Y. Vinet, “Advanced techniques: recycling and squeezing,” in The Detection of Gravitational Waves, D. G. Blair, ed. (Cambridge U. Press, Cambridge, U.K., 1991), pp. 369–405;V. Chickarmane, B. Bhawal, “Squeezing and dual recycling in laser interferometric gravitational wave detectors,” Phys. Lett. A 190, 22–28 (1994).
  8. B. J. Meers, A. Krolak, J. A. Lobo, “Dynamically tuned interferometers for the observation of gravitational waves from coalescing compact binaries,” Phys. Rev. D 47, 2184–2197 (1993).
  9. D. Shoemaker, “LIGO project,” Massachusetts Institute of Technology, Cambridge, Mass. 02139 (personal communication, 1995).
  10. P. Abbott, “Automatic summation,” Mathematica J. 5, (3) 33–35 (1995).

1995 (1)

P. Abbott, “Automatic summation,” Mathematica J. 5, (3) 33–35 (1995).

1993 (1)

B. J. Meers, A. Krolak, J. A. Lobo, “Dynamically tuned interferometers for the observation of gravitational waves from coalescing compact binaries,” Phys. Rev. D 47, 2184–2197 (1993).

1992 (1)

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. W. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational Wave Observatory,” Science 256, 325–333 (1992).

1991 (1)

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1991).

1988 (2)

J.-Y. Vinet, B. J. Meers, C. N. Man, A. Brillet, “Optimization of long-baseline optical interferometers for gravitationalwave detection,” Phys. Rev. D 38, 433–447 (1988).

B. J. Meers, “Recycling in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 38, 2317–2326 (1988);“The frequency response of interferometric gravitationalwave detectors,” Phys. Lett. A 142, 465–470 (1989).

Abbott, P.

P. Abbott, “Automatic summation,” Mathematica J. 5, (3) 33–35 (1995).

Abramovici, A.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. W. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational Wave Observatory,” Science 256, 325–333 (1992).

Althouse, W. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. W. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational Wave Observatory,” Science 256, 325–333 (1992).

Brillet, A.

J.-Y. Vinet, B. J. Meers, C. N. Man, A. Brillet, “Optimization of long-baseline optical interferometers for gravitationalwave detection,” Phys. Rev. D 38, 433–447 (1988).

A. Brillet, J. Gea-Banacloche, G. Leuches, C. N. Man, J.-Y. Vinet, “Advanced techniques: recycling and squeezing,” in The Detection of Gravitational Waves, D. G. Blair, ed. (Cambridge U. Press, Cambridge, U.K., 1991), pp. 369–405;V. Chickarmane, B. Bhawal, “Squeezing and dual recycling in laser interferometric gravitational wave detectors,” Phys. Lett. A 190, 22–28 (1994).

Caves, C. M.

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1991).

Drever, R. W. P.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. W. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational Wave Observatory,” Science 256, 325–333 (1992).

R. W. P. Drever, “Interferometer detectors for gravitational radiation,” in Gravitational Radiation, N. Deruelle, T. Piran, eds. (North-Holland, Amsterdam, 1983), pp. 321–338.

Gea-Banacloche, J.

A. Brillet, J. Gea-Banacloche, G. Leuches, C. N. Man, J.-Y. Vinet, “Advanced techniques: recycling and squeezing,” in The Detection of Gravitational Waves, D. G. Blair, ed. (Cambridge U. Press, Cambridge, U.K., 1991), pp. 369–405;V. Chickarmane, B. Bhawal, “Squeezing and dual recycling in laser interferometric gravitational wave detectors,” Phys. Lett. A 190, 22–28 (1994).

Gürsel, Y.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. W. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational Wave Observatory,” Science 256, 325–333 (1992).

Kawamura, S.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. W. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational Wave Observatory,” Science 256, 325–333 (1992).

Krolak, A.

B. J. Meers, A. Krolak, J. A. Lobo, “Dynamically tuned interferometers for the observation of gravitational waves from coalescing compact binaries,” Phys. Rev. D 47, 2184–2197 (1993).

Leuches, G.

A. Brillet, J. Gea-Banacloche, G. Leuches, C. N. Man, J.-Y. Vinet, “Advanced techniques: recycling and squeezing,” in The Detection of Gravitational Waves, D. G. Blair, ed. (Cambridge U. Press, Cambridge, U.K., 1991), pp. 369–405;V. Chickarmane, B. Bhawal, “Squeezing and dual recycling in laser interferometric gravitational wave detectors,” Phys. Lett. A 190, 22–28 (1994).

Lobo, J. A.

B. J. Meers, A. Krolak, J. A. Lobo, “Dynamically tuned interferometers for the observation of gravitational waves from coalescing compact binaries,” Phys. Rev. D 47, 2184–2197 (1993).

Man, C. N.

J.-Y. Vinet, B. J. Meers, C. N. Man, A. Brillet, “Optimization of long-baseline optical interferometers for gravitationalwave detection,” Phys. Rev. D 38, 433–447 (1988).

A. Brillet, J. Gea-Banacloche, G. Leuches, C. N. Man, J.-Y. Vinet, “Advanced techniques: recycling and squeezing,” in The Detection of Gravitational Waves, D. G. Blair, ed. (Cambridge U. Press, Cambridge, U.K., 1991), pp. 369–405;V. Chickarmane, B. Bhawal, “Squeezing and dual recycling in laser interferometric gravitational wave detectors,” Phys. Lett. A 190, 22–28 (1994).

Meers, B. J.

B. J. Meers, A. Krolak, J. A. Lobo, “Dynamically tuned interferometers for the observation of gravitational waves from coalescing compact binaries,” Phys. Rev. D 47, 2184–2197 (1993).

B. J. Meers, “Recycling in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 38, 2317–2326 (1988);“The frequency response of interferometric gravitationalwave detectors,” Phys. Lett. A 142, 465–470 (1989).

J.-Y. Vinet, B. J. Meers, C. N. Man, A. Brillet, “Optimization of long-baseline optical interferometers for gravitationalwave detection,” Phys. Rev. D 38, 433–447 (1988).

Raab, F. J.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. W. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational Wave Observatory,” Science 256, 325–333 (1992).

Shoemaker, D.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. W. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational Wave Observatory,” Science 256, 325–333 (1992).

D. Shoemaker, “LIGO project,” Massachusetts Institute of Technology, Cambridge, Mass. 02139 (personal communication, 1995).

Sievers, L.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. W. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational Wave Observatory,” Science 256, 325–333 (1992).

Spero, R. W.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. W. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational Wave Observatory,” Science 256, 325–333 (1992).

Thorne, K. S.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. W. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational Wave Observatory,” Science 256, 325–333 (1992).

Vinet, J.-Y.

J.-Y. Vinet, B. J. Meers, C. N. Man, A. Brillet, “Optimization of long-baseline optical interferometers for gravitationalwave detection,” Phys. Rev. D 38, 433–447 (1988).

A. Brillet, J. Gea-Banacloche, G. Leuches, C. N. Man, J.-Y. Vinet, “Advanced techniques: recycling and squeezing,” in The Detection of Gravitational Waves, D. G. Blair, ed. (Cambridge U. Press, Cambridge, U.K., 1991), pp. 369–405;V. Chickarmane, B. Bhawal, “Squeezing and dual recycling in laser interferometric gravitational wave detectors,” Phys. Lett. A 190, 22–28 (1994).

Vogt, R. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. W. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational Wave Observatory,” Science 256, 325–333 (1992).

Weiss, R.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. W. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational Wave Observatory,” Science 256, 325–333 (1992).

Whitcomb, S. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. W. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational Wave Observatory,” Science 256, 325–333 (1992).

Wolfram, S.

S. Wolfram, Mathematica: a System for doing Mathematics by Computer, 2nd ed. (Addison-Wesley, Redwood City, Calif., 1991).

Zucker, M. E.

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. W. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational Wave Observatory,” Science 256, 325–333 (1992).

Mathematica J. (1)

P. Abbott, “Automatic summation,” Mathematica J. 5, (3) 33–35 (1995).

Phys. Rev. D (4)

J.-Y. Vinet, B. J. Meers, C. N. Man, A. Brillet, “Optimization of long-baseline optical interferometers for gravitationalwave detection,” Phys. Rev. D 38, 433–447 (1988).

B. J. Meers, “Recycling in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 38, 2317–2326 (1988);“The frequency response of interferometric gravitationalwave detectors,” Phys. Lett. A 142, 465–470 (1989).

C. M. Caves, “Quantum-mechanical noise in an interferometer,” Phys. Rev. D 23, 1693–1708 (1991).

B. J. Meers, A. Krolak, J. A. Lobo, “Dynamically tuned interferometers for the observation of gravitational waves from coalescing compact binaries,” Phys. Rev. D 47, 2184–2197 (1993).

Science (1)

A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. W. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, M. E. Zucker, “LIGO: the Laser Interferometer Gravitational Wave Observatory,” Science 256, 325–333 (1992).

Other (4)

R. W. P. Drever, “Interferometer detectors for gravitational radiation,” in Gravitational Radiation, N. Deruelle, T. Piran, eds. (North-Holland, Amsterdam, 1983), pp. 321–338.

S. Wolfram, Mathematica: a System for doing Mathematics by Computer, 2nd ed. (Addison-Wesley, Redwood City, Calif., 1991).

D. Shoemaker, “LIGO project,” Massachusetts Institute of Technology, Cambridge, Mass. 02139 (personal communication, 1995).

A. Brillet, J. Gea-Banacloche, G. Leuches, C. N. Man, J.-Y. Vinet, “Advanced techniques: recycling and squeezing,” in The Detection of Gravitational Waves, D. G. Blair, ed. (Cambridge U. Press, Cambridge, U.K., 1991), pp. 369–405;V. Chickarmane, B. Bhawal, “Squeezing and dual recycling in laser interferometric gravitational wave detectors,” Phys. Lett. A 190, 22–28 (1994).

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Figures (4)

Fig. 1
Fig. 1

Optical arrangement for dual recycling: BS, beam splitter; EM, end mirror; PRM, power-recycling mirror; SRM, signal-recycling mirror; PD, photodetector. The light beams: a 0, input laser beam; b 0, ordinary or squeezed vacuum state (see text); x 1, x 4, intracavity fields in the PRC; y 1, y 4, intracavity fields in the SRC; x 1, x 2, y 2, y 3, light fields inside the arms; a 1, b 1, output beams through the PRM and SRM, respectively.

Fig. 2
Fig. 2

Mirror and light beams: a 0, b 0, input beams; a 1, b 1, output beams.

Fig. 3
Fig. 3

Flow chart of a Fabry–Perot cavity at a nonsteady state: CM, corner mirror; EM, end mirror. The circle with L inside represents phase change that is due to traversal of cavity length L.

Fig. 4
Fig. 4

Flow chart of a single delay-line dual-recycled cavity at a nonsteady state: PRM, power-recycling cavity; SRM, signal-recycling cavity; BS, beam splitter. Circles with l 1 or l 2 inside represent phase changes that are due to traversal of length l 1 or l 2 between the BS and the PRM or SRM, respectively.

Equations (13)

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( a 1 b 1 ) = ( t r r t ) ( b 0 a 0 ) ,
( x 3 [ j ] y 3 [ j ] ) = exp ( i 2 ω 0 L a c ) [ exp ( i θ ) 0 0 exp ( + i θ ) ] ( x 2 [ j ] y 2 [ j ] ) ,
x 4 [ j ] = b 0 [ i t 2 Q 0   sin   θ k = 0 K + 1 ( r 1 r 2 Q 0 2   sin 2   θ ) k × n = 1 N { n + k 1 C k } ( r 1 Q 1   cos   θ ) n 1 × m = 1 M { m + k 1 C k } ( r 2 Q 2   cos   θ ) m 1 ] + a 0 [ t 1 Q 1   cos   θ p = 1 j ( r 1 Q 1   cos   θ ) p 1 t 1 r 2 Q 0 2   sin 2 θ k = 0 K ( r 1 r 2 Q 0 2   sin 2 θ ) k × n = 1 N { n + k C k + 1 } ( r 1 Q 1   cos   θ ) n 1 × m = 1 M { m + k 1 C k } ( r 2 Q 2   cos ) m 1 ] ,
y 4 [ j ] = b 0 [ t 2 Q 2   cos   θ p = 1 j ( r 2 Q 2   cos   θ ) p 1 t 2 r 1 Q 0 2   sin 2 θ k = 0 K ( r 1 r 2 Q 0 2   sin 2 θ ) k × n = 1 N { n + k 1 C k } ( r 1 Q 1   cos   θ ) n 1 × m = 1 M { m + k C k + 1 } ( r 2 Q 2   cos   θ ) m 1 ] + a 0 [ i t 1 Q 1   sin θ k = 0 K + 1 ( r 1 r 2 Q 0 2   sin 2 θ ) k × n = 1 N { n + k 1 C k } ( r 1 Q 1   cos   θ ) n 1 × m = 1 M { m + k 1 C k } ( r 2 Q 2   cos   θ ) m 1 ] ,
Q 0 = exp [ i k 0 ( 2 L a + l 1 + l 2 ) ] , Q 1 = exp [ i k 0 ( 2 L a + 2 l 1 ) ] , Q 2 = exp [ i k 0 ( 2 L a + 2 l 2 ) ]
j = 2 K + M + N .
( r 1 cos θ ) ( i   sin θ ) r 2 ( i   sin θ ) t 1 + ( i   sin θ ) r 2 ( i   sin θ ) ( r 1 cos θ ) t 1 .
n = 1 { n + k 1 C k } ( r 1 Q 1   cos   θ ) n 1 = 1 ( 1 r 1 Q 1   cos   θ ) k + 1
x 4 ( steady ) = x 4 [ j ] = 1 χ [ a 0 t 1 Q 1 (   cos   θ r 2 Q 2 ) + i b 0 t 2 Q 0   sin θ ] ,
y 4 ( steady ) = y 4 [ j ] = 1 χ [ i a 0 t 1 Q 0   sin θ + b 0 t 2 Q 2 ( cos θ r 1 Q 1 ) ] ,
χ = 1 ( r 1 Q 1 + r 2 Q 2 ) cos θ + r 1 r 2 Q 1 Q 2 .
x 4 ( steady ) = a 0 t 1 Q 1 cos θ + x 4 ( steady ) r 1 Q 1 cos θ + i b 0 t 2 Q 0   sin θ + i y 4 ( steady ) r 2 Q 0   sin θ ,
y 4 ( steady ) = b 0 t 2 Q 2 cos θ + y 4 ( steady ) r 2 Q 2 cos θ + i a 0 t 1 Q 0   sin θ + i x 4 ( steady ) r 1 Q 0   sin θ .

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