Abstract

A method of real-time dynamical simulation for laser interferometric gravitational wave detectors is presented. The method is based on a digital filtering approach and a number of important physical points understood by a step-by-step investigation of two-mirror cavities, a three-mirror coupled cavity, and a full-length power-recycled interferometer with mirrors having longitudinal motion. The final analytical representation used for the fast simulation of a full-length power-recycled interferometer is analogous to a two-mirror dynamical cavity with time-dependent reflectivities, when intracavity fields of the interferometer are expressed together in a state-vector representation. A detailed discussion establishes the relationships among physical effects pertaining to field evolution in two-mirror cavities and coupled cavities or to the full interferometer.

© 1998 Optical Society of America

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  1. Papers in Proceedings of the International Conference on Gravitational Waves: Sources and Detectors, I. Ciufolini, F. Fidecaro, eds. (World Scientific, Singapore, 1997).
  2. R. W. P. Drever, “Interferometer detectors for gravitational radiation,” in Gravitational Radiation, N. Deruelle, T. Piran, eds. (North-Holland, Amsterdam, 1983).
  3. J. Camp, L. Sievers, R. Bork, J. Heefner, “Guided lock acquisition in a suspended Fabry–Perot cavity,” Opt. Lett. 20, 2463–2465 (1995).
    [CrossRef]
  4. VIRGO collaboration, VIRGO: Final Design (Istituto Nazionale di Fisica Nucleare, Pisa, Italy, 1995).
  5. B. Bhawal, “Global Control Document: dynamical simulation of cavity using a digital filtering approach,” (Laboratoire de l’Accélérateur Linéaire, Orsay, France, 1996).
  6. S. Song, “Recycling and squeezing in high-precision optical measurements,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1994).
  7. A. Bateman, W. Yates, Digital Signal Processing Design (Pitman, London, 1988); L. B. Jackson, Digital Filters and Signal Processing (Kluwer Academic, Boston, 1996).
  8. F. Cavalier, P. Hello, J.-Y. Vinet, “Global Control Document: dynamical simulation of cavities,” (Laboratoire de l’Accélérateur Linéaire, Orsay, France, 1995).
  9. B. Bhawal, “Global Control Document: dynamical simulation of 3-mirror coupled cavity using a digital filtering approach,” (Laboratoire de l’Accélérateur Linéaire, Orsay, France, 1996).
  10. A. E. Seigmann, Lasers (University Science, Mill Valley, Calif., 1986), pp. 524–530.
  11. P. Fritschel, D. Shoemaker, R. Weiss, “Demonstration of light recycling in a Michelson interferometer with Fabry–Perot cavities,” Appl. Opt. 31, 1412–1418 (1992).
    [CrossRef] [PubMed]
  12. J. Mizuno, “Comparison of optical configurations for laser-interferometric gravitational-wave detectors,” Ph.D. dissertation (Max-Planck Institut für Quantenoptik, Garching, Germany, 1995), Chap. 4.
  13. P. K. Fritschel, “Techniques for laser interferometric gravitational wave detectors,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1992), Secs. 5.2, 5.3.
  14. M. W. Regehr, “Signal extraction and control for an interferometric gravitational wave detector,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1995), Sec. 5.4.
  15. D. Redding, “Mathematical description of the LIGO single-mode acquisition code,” (LIGO project, California Institute of Technology, Pasadena, Calif., 1996).
  16. F. Marion, in “Minutes of the electronics and software meeting at Annecy, France,” B. Mours, ed., (Laboratoire d’Annecy de Physique des Particules, Annecy, France, 1996).
  17. B. Bhawal, “Global Control Document: freezing-the-finesse—a technique used for fast simulation of 3-mirror coupled cavity,” (Laboratoire de l’Accélérator Linéaire, Orsay, France, 1996).
  18. B. Bhawal, “What happens when all mirrors move in a 3-mirror coupled cavity,” (Laboratoire de l’Accélérator Linéaire, Orsay, France, 1996).
  19. R. Weiss, “Electromagnetically coupled broadband gravitational antenna,” Q. Prog. Rep. Res. Lab. Electron. MIT 105, 54–76 (1972); C. N. Man, D. Shoemaker, M. Pham Tu, D. Dewey, “External modulation technique for sensitive interferometric detection of displacements,” Phys. Lett. A 148, 8–16 (1990); M. W. Regehr, F. J. Raab, S. E. Whitcomb, “Demonstration of power-recycled Michelson interferometer with Fabry–Perot arms by frontal modulation,” Opt. Lett. 20, 1507–1509 (1995); R. Flaminio, H. Heitmann, “Longitudinal control of an interferometer for the detection of gravitational waves,” Phys. Lett. A 214, 112–122 (1996).
    [CrossRef] [PubMed]
  20. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
    [CrossRef]
  21. B. Bhawal, in “Minutes of the control workshop,” at Rome, Italy, B. Caron, ed., (Laboratoire d’Annecy de Physique des Particules, Annecy, France, 1996).
  22. B. Bhawal, “Evolution of intracavity fields at a nonsteady state in a dual-recycled interferometer,” Appl. Opt. 35, 1041–1045 (1996).
    [CrossRef] [PubMed]
  23. B. J. Meers, “Recycling in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 38, 2317–2326 (1988).
    [CrossRef]
  24. K.-X. Sun, M. M. Fejer, E. Gustafson, R. L. Byer, “Sagnac interferometer for gravitational-wave detection,” Phys. Rev. Lett. 76, 3053–3056 (1996).
    [CrossRef] [PubMed]

1996 (2)

B. Bhawal, “Evolution of intracavity fields at a nonsteady state in a dual-recycled interferometer,” Appl. Opt. 35, 1041–1045 (1996).
[CrossRef] [PubMed]

K.-X. Sun, M. M. Fejer, E. Gustafson, R. L. Byer, “Sagnac interferometer for gravitational-wave detection,” Phys. Rev. Lett. 76, 3053–3056 (1996).
[CrossRef] [PubMed]

1995 (1)

1992 (1)

1988 (1)

B. J. Meers, “Recycling in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 38, 2317–2326 (1988).
[CrossRef]

1983 (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

1972 (1)

R. Weiss, “Electromagnetically coupled broadband gravitational antenna,” Q. Prog. Rep. Res. Lab. Electron. MIT 105, 54–76 (1972); C. N. Man, D. Shoemaker, M. Pham Tu, D. Dewey, “External modulation technique for sensitive interferometric detection of displacements,” Phys. Lett. A 148, 8–16 (1990); M. W. Regehr, F. J. Raab, S. E. Whitcomb, “Demonstration of power-recycled Michelson interferometer with Fabry–Perot arms by frontal modulation,” Opt. Lett. 20, 1507–1509 (1995); R. Flaminio, H. Heitmann, “Longitudinal control of an interferometer for the detection of gravitational waves,” Phys. Lett. A 214, 112–122 (1996).
[CrossRef] [PubMed]

Bateman, A.

A. Bateman, W. Yates, Digital Signal Processing Design (Pitman, London, 1988); L. B. Jackson, Digital Filters and Signal Processing (Kluwer Academic, Boston, 1996).

Bhawal, B.

B. Bhawal, “Evolution of intracavity fields at a nonsteady state in a dual-recycled interferometer,” Appl. Opt. 35, 1041–1045 (1996).
[CrossRef] [PubMed]

B. Bhawal, in “Minutes of the control workshop,” at Rome, Italy, B. Caron, ed., (Laboratoire d’Annecy de Physique des Particules, Annecy, France, 1996).

B. Bhawal, “Global Control Document: freezing-the-finesse—a technique used for fast simulation of 3-mirror coupled cavity,” (Laboratoire de l’Accélérator Linéaire, Orsay, France, 1996).

B. Bhawal, “What happens when all mirrors move in a 3-mirror coupled cavity,” (Laboratoire de l’Accélérator Linéaire, Orsay, France, 1996).

B. Bhawal, “Global Control Document: dynamical simulation of cavity using a digital filtering approach,” (Laboratoire de l’Accélérateur Linéaire, Orsay, France, 1996).

B. Bhawal, “Global Control Document: dynamical simulation of 3-mirror coupled cavity using a digital filtering approach,” (Laboratoire de l’Accélérateur Linéaire, Orsay, France, 1996).

Bork, R.

Byer, R. L.

K.-X. Sun, M. M. Fejer, E. Gustafson, R. L. Byer, “Sagnac interferometer for gravitational-wave detection,” Phys. Rev. Lett. 76, 3053–3056 (1996).
[CrossRef] [PubMed]

Camp, J.

Cavalier, F.

F. Cavalier, P. Hello, J.-Y. Vinet, “Global Control Document: dynamical simulation of cavities,” (Laboratoire de l’Accélérateur Linéaire, Orsay, France, 1995).

Drever, R. W. P.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

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

Fejer, M. M.

K.-X. Sun, M. M. Fejer, E. Gustafson, R. L. Byer, “Sagnac interferometer for gravitational-wave detection,” Phys. Rev. Lett. 76, 3053–3056 (1996).
[CrossRef] [PubMed]

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Fritschel, P.

Fritschel, P. K.

P. K. Fritschel, “Techniques for laser interferometric gravitational wave detectors,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1992), Secs. 5.2, 5.3.

Gustafson, E.

K.-X. Sun, M. M. Fejer, E. Gustafson, R. L. Byer, “Sagnac interferometer for gravitational-wave detection,” Phys. Rev. Lett. 76, 3053–3056 (1996).
[CrossRef] [PubMed]

Hall, J. L.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Heefner, J.

Hello, P.

F. Cavalier, P. Hello, J.-Y. Vinet, “Global Control Document: dynamical simulation of cavities,” (Laboratoire de l’Accélérateur Linéaire, Orsay, France, 1995).

Hough, J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Kowalski, F. V.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Marion, F.

F. Marion, in “Minutes of the electronics and software meeting at Annecy, France,” B. Mours, ed., (Laboratoire d’Annecy de Physique des Particules, Annecy, France, 1996).

Meers, B. J.

B. J. Meers, “Recycling in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 38, 2317–2326 (1988).
[CrossRef]

Mizuno, J.

J. Mizuno, “Comparison of optical configurations for laser-interferometric gravitational-wave detectors,” Ph.D. dissertation (Max-Planck Institut für Quantenoptik, Garching, Germany, 1995), Chap. 4.

Munley, A. J.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Redding, D.

D. Redding, “Mathematical description of the LIGO single-mode acquisition code,” (LIGO project, California Institute of Technology, Pasadena, Calif., 1996).

Regehr, M. W.

M. W. Regehr, “Signal extraction and control for an interferometric gravitational wave detector,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1995), Sec. 5.4.

Seigmann, A. E.

A. E. Seigmann, Lasers (University Science, Mill Valley, Calif., 1986), pp. 524–530.

Shoemaker, D.

Sievers, L.

Song, S.

S. Song, “Recycling and squeezing in high-precision optical measurements,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1994).

Sun, K.-X.

K.-X. Sun, M. M. Fejer, E. Gustafson, R. L. Byer, “Sagnac interferometer for gravitational-wave detection,” Phys. Rev. Lett. 76, 3053–3056 (1996).
[CrossRef] [PubMed]

Vinet, J.-Y.

F. Cavalier, P. Hello, J.-Y. Vinet, “Global Control Document: dynamical simulation of cavities,” (Laboratoire de l’Accélérateur Linéaire, Orsay, France, 1995).

Ward, H.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Weiss, R.

P. Fritschel, D. Shoemaker, R. Weiss, “Demonstration of light recycling in a Michelson interferometer with Fabry–Perot cavities,” Appl. Opt. 31, 1412–1418 (1992).
[CrossRef] [PubMed]

R. Weiss, “Electromagnetically coupled broadband gravitational antenna,” Q. Prog. Rep. Res. Lab. Electron. MIT 105, 54–76 (1972); C. N. Man, D. Shoemaker, M. Pham Tu, D. Dewey, “External modulation technique for sensitive interferometric detection of displacements,” Phys. Lett. A 148, 8–16 (1990); M. W. Regehr, F. J. Raab, S. E. Whitcomb, “Demonstration of power-recycled Michelson interferometer with Fabry–Perot arms by frontal modulation,” Opt. Lett. 20, 1507–1509 (1995); R. Flaminio, H. Heitmann, “Longitudinal control of an interferometer for the detection of gravitational waves,” Phys. Lett. A 214, 112–122 (1996).
[CrossRef] [PubMed]

Yates, W.

A. Bateman, W. Yates, Digital Signal Processing Design (Pitman, London, 1988); L. B. Jackson, Digital Filters and Signal Processing (Kluwer Academic, Boston, 1996).

Appl. Opt. (2)

Appl. Phys. B (1)

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. D (1)

B. J. Meers, “Recycling in laser-interferometric gravitational-wave detectors,” Phys. Rev. D 38, 2317–2326 (1988).
[CrossRef]

Phys. Rev. Lett. (1)

K.-X. Sun, M. M. Fejer, E. Gustafson, R. L. Byer, “Sagnac interferometer for gravitational-wave detection,” Phys. Rev. Lett. 76, 3053–3056 (1996).
[CrossRef] [PubMed]

Q. Prog. Rep. Res. Lab. Electron. MIT (1)

R. Weiss, “Electromagnetically coupled broadband gravitational antenna,” Q. Prog. Rep. Res. Lab. Electron. MIT 105, 54–76 (1972); C. N. Man, D. Shoemaker, M. Pham Tu, D. Dewey, “External modulation technique for sensitive interferometric detection of displacements,” Phys. Lett. A 148, 8–16 (1990); M. W. Regehr, F. J. Raab, S. E. Whitcomb, “Demonstration of power-recycled Michelson interferometer with Fabry–Perot arms by frontal modulation,” Opt. Lett. 20, 1507–1509 (1995); R. Flaminio, H. Heitmann, “Longitudinal control of an interferometer for the detection of gravitational waves,” Phys. Lett. A 214, 112–122 (1996).
[CrossRef] [PubMed]

Other (17)

B. Bhawal, in “Minutes of the control workshop,” at Rome, Italy, B. Caron, ed., (Laboratoire d’Annecy de Physique des Particules, Annecy, France, 1996).

Papers in Proceedings of the International Conference on Gravitational Waves: Sources and Detectors, I. Ciufolini, F. Fidecaro, eds. (World Scientific, Singapore, 1997).

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

VIRGO collaboration, VIRGO: Final Design (Istituto Nazionale di Fisica Nucleare, Pisa, Italy, 1995).

B. Bhawal, “Global Control Document: dynamical simulation of cavity using a digital filtering approach,” (Laboratoire de l’Accélérateur Linéaire, Orsay, France, 1996).

S. Song, “Recycling and squeezing in high-precision optical measurements,” Ph.D. dissertation (University of Southern California, Los Angeles, Calif., 1994).

A. Bateman, W. Yates, Digital Signal Processing Design (Pitman, London, 1988); L. B. Jackson, Digital Filters and Signal Processing (Kluwer Academic, Boston, 1996).

F. Cavalier, P. Hello, J.-Y. Vinet, “Global Control Document: dynamical simulation of cavities,” (Laboratoire de l’Accélérateur Linéaire, Orsay, France, 1995).

B. Bhawal, “Global Control Document: dynamical simulation of 3-mirror coupled cavity using a digital filtering approach,” (Laboratoire de l’Accélérateur Linéaire, Orsay, France, 1996).

A. E. Seigmann, Lasers (University Science, Mill Valley, Calif., 1986), pp. 524–530.

J. Mizuno, “Comparison of optical configurations for laser-interferometric gravitational-wave detectors,” Ph.D. dissertation (Max-Planck Institut für Quantenoptik, Garching, Germany, 1995), Chap. 4.

P. K. Fritschel, “Techniques for laser interferometric gravitational wave detectors,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1992), Secs. 5.2, 5.3.

M. W. Regehr, “Signal extraction and control for an interferometric gravitational wave detector,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 1995), Sec. 5.4.

D. Redding, “Mathematical description of the LIGO single-mode acquisition code,” (LIGO project, California Institute of Technology, Pasadena, Calif., 1996).

F. Marion, in “Minutes of the electronics and software meeting at Annecy, France,” B. Mours, ed., (Laboratoire d’Annecy de Physique des Particules, Annecy, France, 1996).

B. Bhawal, “Global Control Document: freezing-the-finesse—a technique used for fast simulation of 3-mirror coupled cavity,” (Laboratoire de l’Accélérator Linéaire, Orsay, France, 1996).

B. Bhawal, “What happens when all mirrors move in a 3-mirror coupled cavity,” (Laboratoire de l’Accélérator Linéaire, Orsay, France, 1996).

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Figures (33)

Fig. 1
Fig. 1

Configuration of a power-recycled interferometer. BS, beam splitter; EM, end mirror; IM, input mirror; PRM, power-recycling mirror.

Fig. 2
Fig. 2

Notation for a two-mirror cavity; rc(re) and tc(te) are amplitude reflectivity and transmittivity, respectively, for the input (end) mirror. The sign (+) or (-) indicates the phase factor acquired on reflection from that side of the mirror.

Fig. 3
Fig. 3

Typical resonance curve for a low-finesse cavity that can be drawn by using the SM, QSM, DEM, or DFA (with reasonable values of N in the last two cases), while the cavity length changes at a rate w=1 µm/s. Input power |A|2 is one unit. Note that the resolution of the plot cannot make any difference among curves obtained from these methods.

Fig. 4
Fig. 4

Relative error (%) of (a) QSM and (b) DFA based on perturbative calculation in comparison with the exact SM in computing the resonance curve of Fig. 3.

Fig. 5
Fig. 5

Maximum values of the relative error (%) of the DFA based on perturbative calculation in comparison with the exact SM (in computing the resonance curve of Fig. 3) plotted against N, the number of steps for two rates of change in the cavity length.

Fig. 6
Fig. 6

(a) Dynamical resonance curve for a high-finesse (rc =0.998733) two-mirror cavity while only the end mirror moves and so the cavity length changes at a rate of w=1 µm/s. Input power is one unit. (b) Quasi-static resonance curve for the same cavity.

Fig. 7
Fig. 7

Difference in power between the case when the input mirror moves (at a speed of 1 µm/s) and when it does not but the rate of change in cavity length w is the same in both cases (=1 µm/s). The latter case corresponds to Fig. 6(a).

Fig. 8
Fig. 8

(a) Error (in unit of power) and (b) relative error made by the DFA based on JAS in comparison with the SM for N=50 in calculating the resonance curve of Fig. 6(a).

Fig. 9
Fig. 9

Relative error (%) made by the DFA based on JAS in comparison with the SM for N=50 in calculating the resonance curve of Fig. 6(a). Plot 8(b) is merely a zoomed-in part of this plot. The relative error increases for a short time while the field amplitude is very small; this increase is thus unimportant.

Fig. 10
Fig. 10

Notation used for three-mirror coupled cavity; the (+) or (-) sign on two sides of a mirror indicates the phase (i.e., either zero or π) that a light beam acquires on reflection from that side of the mirror.

Fig. 11
Fig. 11

Exact dynamical code written for a three-mirror cavity for which the ratio of lengths of the arm cavity and the recycling cavity is an integer, ρ. The figure shows the case for ρ=4; For the equivalent case of the VIRGO interferometer, ρ has been assumed to be 250 (=3000/12). Note that at any moment of time the code needs to memorize 2ρ number of data for the evolution of phases in two cavities and ρ number of past values for the field D. RM, recycling mirror; IM, input mirror; EM, end mirror.

Fig. 12
Fig. 12

Exact dynamical double-resonance curve for the field B in a three-mirror cavity while only the end mirror moves with a velocity 1 µm/s and the round-trip phase in the recycling cavity θr is set to an integral multiple of 2π. Input power is one unit.

Fig. 13
Fig. 13

Equivalent amplitude reflectivity of the recycling mirror, rrec, is plotted as a function of round-trip phase offset Δθr in the recycling cavity.

Fig. 14
Fig. 14

(a) Dynamical double-resonance curve for field D in a three-mirror cavity drawn by using equations based on only the small-rec-cav assumption for wr=wc=1 µm/s. Input power is one unit. (b) Relative error by the DFA based on JAS and FTF with respect to only small-rec-cav in calculating the same curve for N=30. The relative error increases for a short time while the field amplitude is very small; this increase is thus unimportant.

Fig. 15
Fig. 15

(a) Resonance peak of field D in a three-mirror cavity when the beam is on exact antiresonance in the arm and on resonance in the recycling cavity, as drawn by equations based on only the small-rec-cav assumption for wr=wc=1 µm/s. Input power is one unit. (b) Relative error by the DFA based on JAS and FTF with respect to only small-rec-cav in calculating the same curve for N=30.

Fig. 16
Fig. 16

Evolution of fields (a) D and (b) B in case aa, which corresponds to the double-resonance condition for first peaks.

Fig. 17
Fig. 17

Evolution of fields (a) D and (b) B in case dd.

Fig. 18
Fig. 18

Evolution of fields (a) D and (b) B in case ff.

Fig. 19
Fig. 19

Evolution of fields (a) D and (b) B in case gg.

Fig. 20
Fig. 20

Evolution of fields (a) D and (b) B in case hh.

Fig. 21
Fig. 21

Evolution of fields (a) D and (b) B in case ii.

Fig. 22
Fig. 22

Evolution of fields (a) D and (b) B in case jj.

Fig. 23
Fig. 23

Evolution of fields (a) D and (b) B in case kk.

Fig. 24
Fig. 24

Quasi-static curves for field D for (a) case ff and (b) case ii.

Fig. 25
Fig. 25

Equivalent reflectivity of the recycling cavity as a function of the phase offset in the same cavity. The points (ζ=) 324 and 8147 for the phase offset correspond to θr=2mπ and θr =(2m+1)π, respectively, as in Figs. 1623.

Fig. 26
Fig. 26

Reflected light in (a) case aa and (b) case ii.

Fig. 27
Fig. 27

Transmitted light in (a) case aa and (b) case ii.

Fig. 28
Fig. 28

Phase acquired on reflection [in excess of (2m+1)π] from the arm cavity as a function of the phase offset (in excess of 2nπ) in the arm cavity. These are expressed in units of phases acquired by recycling and arm cavities in a round-trip-time of the arm cavity due to mirror motion corresponding to wr =1.7 µm/s and wc=1 µm/s, respectively.

Fig. 29
Fig. 29

(a) Resonance peak for either of the fields Dy or Dx when the three-mirror systems become doubly resonant simultaneously. (b) Peak in power of either Dy or Dx when only the corresponding three-mirror system becomes doubly resonant but the other three-mirror system remains far out of resonance. In both figures the input power is one unit.

Fig. 30
Fig. 30

Notation for the mirror velocities with respect to the initial position of the beam splitter. The arrows indicate positive signs for V’s in Eqs. (84)–(87).

Fig. 31
Fig. 31

(a) Resonance peak for the field Dy when three-mirror system Y becomes doubly resonant ahead of system X, with a time gap equivalent to a phase difference of π/100, while the mirrors move such that Wry=Wrx=Wcx=Wcy=1 µm/s and the beam splitter is assumed to be static. Input power is one unit. (b) Relative error by the DFA based on JAS and FTF with respect to only the small-rec-cav approximation in computing such a peak. The relative error increases for a short time while the field amplitude is very small; this increase is thus unimportant.

Fig. 32
Fig. 32

Difference between resonant power levels of fields Dy and Dx under the condition described for Fig. 31.

Fig. 33
Fig. 33

(a) Peak in power of field Dy when three-mirror system Y gets to the condition of “resonance in recycling cavity but exact antiresonance in arm cavity” ahead of system X, with a time gap equivalent to a phase difference of π/100, while mirrors move such that Wry=Wrx=Wcx=Wcy=1 µm/s and the beam splitter is assumed to be static. Input power is one unit. (b) Relative error by the DFA based on JAS and FTF with respect to only the small-rec-cav approximation in computing such a peak.

Tables (4)

Tables Icon

Table 1 Approximate Values of the Peaks of the Relative Error Curves for Cavities with Various Values of the Finesse, with N fixed to 50

Tables Icon

Table 2 Numerical Values of Some Important Quantities for the Dynamical Simulation of Three-Mirror Coupled Cavities under Different Conditions

Tables Icon

Table 3 Numerical Values of Some Important Quantities for the Quasi-Static Curves of Three-Mirror Coupled Cavities under Different Conditions

Tables Icon

Table 4 Phase Offsets in Three-Mirror Coupled Cavity at Positions of the Two Peaks for Various Dynamical and Corresponding Quasi-Static Cases

Equations (98)

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

F(t)=tcA exp j2πλxc(t)+rcre×exp j[θ(t-τ/2)]F(t-τ),
θ(t-τ/2)=ϕ+2πλ2x(t-τ/2),
2x(t-τ/2)=-xc(t-τ)+2xe(t-τ/2)-xc(t),
F(t)=tcA exp j2πλxc(t)+R1+j2πλ2xt-τ2F(t-τ),
R=rcre exp j(ϕ).
F0=tcA1-R,
δF(t)=RδF(t-τ)+j4πλF0xt-τ2.
Hc(s)=δF(s)x(s)=jPF0exp(-τ2s)1-R exp(-τs)x(s),
P=(4π/λ)R.
H(s)=Y(s)X(s)=k=0KCk exp(-ksΔ)1-m=1MDm exp(-msΔ),
H¯(n)=Y¯(n)X¯(n)=k=0KCkz-k1-m=1MDmz-m,
Y¯(n)=k=0KCkX¯(n-k)+m=1MDmY¯(n-m).
F(t+Nτ)=t1A exp j[xc(t)]1-R(1+jP sum),
sum=n=1NRN-nxt+n-12τ.
Ck=jPF0Rk-1,k=1, 2, 3 ,.
ϕi=ϕ0+4πλx(iΔ),
wi=xi+1-xiNτ.
sum=(N-12)-(N+12)rerc exp j(ϕi)[1-rcre exp j(ϕi)]2wiτ+O(RN).
relativeerror(%)=valuepredictedbyY-valuegivenbyXvaluegivenbyX×100.
F(t+τ)=tcA+R exp j4πλx(t+τ/2)F(t),
F(t+Nτ)=tcA1+n=2NRn-1 k=N-n+2N expk+F(t)RNk=1N expk,
expk=expj4πλxk,
xkx(t+(k-1/2)τ),
expk=exp[ξ(k-1/2)],
F(t+Nτ)=tcA(1-RN)1-R+tcAξRi=1N-1Ri-1×k=N-i+1Nk-12+F(t)RN expξN22.
tcAξR1-Rk=2Nk-12RN-k-tcAξRN1-Rk=2Nk-12.
F(t+Nτ)=tcA(1-RN)1-R+tcAξR1-RS-tcAξRN2(1-R)(N2-1)+F(t)RN expξN22,
S=N-0.5-1.5RN-11-R-R-RN-1(1-R)2.
B(t)=t1A exp j[p1(t)]-r1r2B(t-2τr)×exp j[ϕr-p1(t-2τr)+2p2(t-tr)-p1(t)]+t2r1r3D(t-2τc-τr)×exp j[ϕc+ϕr/2-p2(t-2τc-τr)+2p3(t-τc-τr)-p1(t)],
D(t)=t2B(t-τr)exp j12ϕr-p1(t-τr)+p2(t)+r2r3D(t-2τc)exp j[ϕc-p2(t-2τc)+2p3(t-τc)-p2(t)],
B0=At1[1-r2r3 exp(jϕc)]/χ0,
D0=At1t2 exp(jϕr/2)/χ0,
χ0=1+r1r2 exp(jϕr)-r2r3 exp(jϕc)-r1r3(r22+t22)exp(jϕr+jϕc).
D(t)=trec(t)A+rrec(t)r3 exp[jθc(t-τc)]D(t-2τc),
rrec(t)=r2+r1(r22+t22)exp[jθr(t)]1+r1r2 exp[jθr(t)],
trec(t)=t1t2 exp j[θr(t)/2+p1(t)]1+r1r2 exp[jθr(t)].
B(t+τ)=1χ(t+τ)t1A exp j[p1(t+τ)]+t2r1r3D(t)exp jθr(t+τ)2+θc(t+τ/2),
D(t+τ)=trec(t+τ)A+rrec(t+τ)r3D(t)×exp j[θc(t+τ/2)],
trec(t)=1χ(t)t1t2 exp j12θr(t)+p1(t),
rrec(t)=r2+t22r1 exp j[θr(t)]χ(t),
θr(t)=ϕr-2p1(t)+2p2(t),
θc(t-τc)=ϕc-p2(t-2τc)+2p3(t-τc)-p2(t),
χ(t)=1+r1r2 exp j[θr(t)].
F(t+τ)=tc(t+τ)A+F(t)R(t+τ)exp[jθ(t+τ/2)],
wr=v2-v1,wc=v3-v2,
θr(t)=ϕr+4πλwrt,θc(t)=ϕc+4πλwc(t-τc).
ϕi=θc(ti),
rci=12[rrec(ti)+rrec(ti+1)],
tci=trec(ti+1).
θr(ζ)=Krζ+ϕr,Kr=2πλ2τwr,
θc(ζ)=Kcζ+ϕc,Kc=2πλ2τwc.
θr=2πm+KrΔir,θc=2πn+KcΔic,
θrefl=(2m+1)π+KrΔirefl
Bsh(t)=t1A+12r1t2r3Dy(t-τ)exp j[θcy(t-τ/2)]+12r1t2r3Dx(t-τ)exp j[θcx(t-τ/2)]-r1r2B(t),
Dx(t)=12t2B(t)+r2r3Dx(t-τ)×exp j[θcx(t-τ/2)],
Dy(t)=12t2B(t)+r2r3Dy(t-τ)×exp j[θcy(t-τ/2)].
Bsh(t)=r1t2r32(1+r1r2)Dy(t-τ)exp j[θcy(t-τ/2)]
+r1t2r32(1+r1r2)Dx(t-τ)exp j[θcx(t-τ/2)]+t1A(1+r1r2),
Dy(t)=A+r3βDy(t-τ)exp j[θcy(t-τ/2)]+r3CDx(t-τ)exp j[θcx(t-τ/2)],
Dx(t)=A+r3βDx(t-τ)exp j[θcx(t-τ/2)]+r3CDy(t-τ)exp j[θcy(t-τ/2)],
A=t1t2A2(1+r1r2),
β=r2+C=r2+r1t222(1+r1r2),
C=r1t222(1+r1r2).
rrec=β+C0.998733.
D(t+τ)=A+R E(t+τ/2)D(t),
D(t)=Dy(t)Dx(t),
A=A11,
R=r3βCCβexp j(ϕcy)00exp j(ϕcx),
E(t)=exp j4πλv3yt00exp j4πλv3xt,
D(t+τ)=A(t+τ)+R(t+τ)E(t+τ/2)D(t).
A(t)=t1Aχ(t)rbt2y exp j[Өry(t)]tbt2x exp j[Өrx(t)],
χ(t)=1+r1rb2r2y exp j[Өry(t)+Өry(t)]+r1tb2r2x exp j[2Өrx(t)],
R(t)=βyCxCyβxexp j(ϕcy)00exp j(ϕcx),
E(t)=exp j4πλWcyt00exp j4πλWcxt,
βy=r2yr3y+1χ(t)r1rb2t2y2r3y×exp j[Өry(t)+Өry(t)],
βx=r2xr3x+1χ(t)r1tb2t2x2r3x exp j[2Өrx(t)],
Cx=1χ(t)r1rbtbt2yt2xr3x exp j[Өrx(t)+Өry(t)],
Cy=1χ(t)r1rbtbt2yt2xr3y exp j[Өry(t)+Өrx(t)].
Өry(t)=ϕry/2+2πλ(Wryt+Vbxt),
Өry(t)=ϕry/2+2πλ(Wryt+Vbyt),
Өrx(t)=ϕrx/2+2πλWrxt,
θcy(t)=ϕcy+4πλWcyt,
θcx(t)=ϕcx+4πλWcxt,
Wry=V2y-V1,
Wrx=V2x-V1,
Wcy=V3y-V2y,
Wcx=V3x-V2x.
ϕcx=θcx(ti),
ϕcy=θcy(ti).
R=12[R(ti)+R(ti+1)],
A=A(ti+1).
D(t+Nτ)=RNED(t)+(U-R)-1[(U-RN)+RSξ-12(N2-1)RNξ]A,
U=1001,
E=exp(ξyN2/2)00exp(ξxN2/2),
ξ=ξy00ξx,
S=(U-R)-1[(N-12)U-1.5RN-1]+(U-R)-2(RN-1-R),
ξy=j4πWcyτλ,
ξx=j4πWcxτλ.

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