Abstract

The Doppler effect in Fabry–Perot cavities with suspended mirrors is analyzed. The Doppler shift, which is intrinsically small, accumulates in the cavity and becomes comparable with or greater than the linewidth of the cavity if the cavity’s finesse is high or its length is large. As a result, damped oscillations of the cavity field occur when one of the mirrors passes a resonance position. A formula for this transient is derived. It is shown that the frequency of the oscillations is equal to the accumulated Doppler shift and that the relaxation time of the oscillations is equal to the storage time of the cavity. Comparison of the predicted and the measured Doppler shifts is discussed, and application of the analytical solution for measurement of the mirror velocity is described.

© 2001 Optical Society of America

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  1. A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. 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).
    [CrossRef] [PubMed]
  2. 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]
  3. N. A. Robertson, K. A. Strain, J. Hough, “Measurements of losses in high reflectance mirrors coated for λ = 514.5 nm,” Opt. Commun. 69, 345–348 (1989).
    [CrossRef]
  4. K. An, C. Yang, R. R. Dasari, M. S. Feld, “Cavity ring-down technique and its application to the measurement of ultraslow velocities,” Opt. Lett. 20, 1068–1070 (1995).
    [CrossRef] [PubMed]
  5. J. Camp, L. Sievers, R. Bork, J. Heefner, “Guided lock acquisition in a suspended Fabry–Perot cavity,” Opt. Lett. 20, 2463–2465 (1995).
    [CrossRef]
  6. M. J. Lawrence, B. Willke, M. E. Husman, E. K. Gustafson, R. L. Byer, “Dynamic response of a Fabry–Perot interferometer,” J. Opt. Soc. Am. B 16, 523–532 (1999).
    [CrossRef]
  7. H. Yamamoto, “Fringe structure of LIGO Hanford 2km Fabry–Perot cavity,” (California Institute of Technology, Pasadena, Calif., 1999).
  8. M. Rakhmanov, “Dynamics of laser interferometric gravitational wave detectors,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 2000).
  9. M. Rakhmanov, M. Evans, H. Yamamoto, “An optical vernier technique for in situ measurement of the length of long Fabry–Perot cavities,” Meas. Sci. Technol. 10, 190–194 (1999).
    [CrossRef]

1999 (2)

M. J. Lawrence, B. Willke, M. E. Husman, E. K. Gustafson, R. L. Byer, “Dynamic response of a Fabry–Perot interferometer,” J. Opt. Soc. Am. B 16, 523–532 (1999).
[CrossRef]

M. Rakhmanov, M. Evans, H. Yamamoto, “An optical vernier technique for in situ measurement of the length of long Fabry–Perot cavities,” Meas. Sci. Technol. 10, 190–194 (1999).
[CrossRef]

1995 (2)

1992 (1)

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. 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).
[CrossRef] [PubMed]

1989 (1)

N. A. Robertson, K. A. Strain, J. Hough, “Measurements of losses in high reflectance mirrors coated for λ = 514.5 nm,” Opt. Commun. 69, 345–348 (1989).
[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]

Abramovici, A.

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. 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).
[CrossRef] [PubMed]

Althouse, W. E.

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. 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).
[CrossRef] [PubMed]

An, K.

Bork, R.

Byer, R. L.

Camp, J.

Dasari, R. R.

Drever, R. W.

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. 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).
[CrossRef] [PubMed]

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]

Evans, M.

M. Rakhmanov, M. Evans, H. Yamamoto, “An optical vernier technique for in situ measurement of the length of long Fabry–Perot cavities,” Meas. Sci. Technol. 10, 190–194 (1999).
[CrossRef]

Feld, M. S.

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]

Gürsel, Y.

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. 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).
[CrossRef] [PubMed]

Gustafson, E. K.

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.

Hough, J.

N. A. Robertson, K. A. Strain, J. Hough, “Measurements of losses in high reflectance mirrors coated for λ = 514.5 nm,” Opt. Commun. 69, 345–348 (1989).
[CrossRef]

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]

Husman, M. E.

Kawamura, S.

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. 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).
[CrossRef] [PubMed]

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]

Lawrence, M. J.

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]

Raab, F. J.

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. 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).
[CrossRef] [PubMed]

Rakhmanov, M.

M. Rakhmanov, M. Evans, H. Yamamoto, “An optical vernier technique for in situ measurement of the length of long Fabry–Perot cavities,” Meas. Sci. Technol. 10, 190–194 (1999).
[CrossRef]

M. Rakhmanov, “Dynamics of laser interferometric gravitational wave detectors,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 2000).

Robertson, N. A.

N. A. Robertson, K. A. Strain, J. Hough, “Measurements of losses in high reflectance mirrors coated for λ = 514.5 nm,” Opt. Commun. 69, 345–348 (1989).
[CrossRef]

Shoemaker, D.

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. 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).
[CrossRef] [PubMed]

Sievers, L.

J. Camp, L. Sievers, R. Bork, J. Heefner, “Guided lock acquisition in a suspended Fabry–Perot cavity,” Opt. Lett. 20, 2463–2465 (1995).
[CrossRef]

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. 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).
[CrossRef] [PubMed]

Spero, R. E.

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. 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).
[CrossRef] [PubMed]

Strain, K. A.

N. A. Robertson, K. A. Strain, J. Hough, “Measurements of losses in high reflectance mirrors coated for λ = 514.5 nm,” Opt. Commun. 69, 345–348 (1989).
[CrossRef]

Thorne, K. S.

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. 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).
[CrossRef] [PubMed]

Vogt, R. E.

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. 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).
[CrossRef] [PubMed]

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.

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. 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).
[CrossRef] [PubMed]

Whitcomb, S. E.

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. 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).
[CrossRef] [PubMed]

Willke, B.

Yamamoto, H.

M. Rakhmanov, M. Evans, H. Yamamoto, “An optical vernier technique for in situ measurement of the length of long Fabry–Perot cavities,” Meas. Sci. Technol. 10, 190–194 (1999).
[CrossRef]

H. Yamamoto, “Fringe structure of LIGO Hanford 2km Fabry–Perot cavity,” (California Institute of Technology, Pasadena, Calif., 1999).

Yang, C.

Zucker, M. E.

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. 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).
[CrossRef] [PubMed]

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]

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

Meas. Sci. Technol. (1)

M. Rakhmanov, M. Evans, H. Yamamoto, “An optical vernier technique for in situ measurement of the length of long Fabry–Perot cavities,” Meas. Sci. Technol. 10, 190–194 (1999).
[CrossRef]

Opt. Commun. (1)

N. A. Robertson, K. A. Strain, J. Hough, “Measurements of losses in high reflectance mirrors coated for λ = 514.5 nm,” Opt. Commun. 69, 345–348 (1989).
[CrossRef]

Opt. Lett. (2)

Science (1)

A. Abramovici, W. E. Althouse, R. W. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. 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).
[CrossRef] [PubMed]

Other (2)

H. Yamamoto, “Fringe structure of LIGO Hanford 2km Fabry–Perot cavity,” (California Institute of Technology, Pasadena, Calif., 1999).

M. Rakhmanov, “Dynamics of laser interferometric gravitational wave detectors,” Ph.D. dissertation (California Institute of Technology, Pasadena, Calif., 2000).

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Figures (5)

Fig. 1
Fig. 1

Reflection of light off a moving mirror.

Fig. 2
Fig. 2

Schematic diagram of a Fabry–Perot cavity with a moving mirror.

Fig. 3
Fig. 3

Modeled response of the LIGO 4-km Fabry–Perot cavity (finesse 205). The two curves correspond to the slow (top) and the fast (bottom) motions of the mirror (v cr = 1.48 × 10-6 m/s).

Fig. 4
Fig. 4

Transient response of the Fabry–Perot cavity of the Caltech 40-m prototype interferometer (v/ v cr = 1.93).

Fig. 5
Fig. 5

Top, theoretical prediction (solid curve) and measurement (dashed curve) of the adjusted Pound–Drever (P.-D.) signal. Bottom, measured Doppler shift ν̅ n and linear fit ν̅(t).

Equations (66)

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ct-t=Xt-x.
refx, t=inXt, t.
inx, t=expiωt-kx,
refx, t=expiωt-kXt.
refx, t=expiωt+kxexp-2ikXt.
ωt=ω-2k dXdttt.
tt=cc+vt,
ωt=c-vtc+vt ω.
Etx, texp-iωt.
E2t=E1t-L/cexp-ikL,
refx, t=inx, t,
t=2t-t.
Ereft=Eintexp-2ikXt-x.
ωt1-2 vtcω,
T=L/c.
δωω-ω=-2kv.
δω, 2δω, 3δω, .
τ=2T|lnrarb|,
|δω| τ2T=ω vτcT.
vcr=λ2τπcλ4L2,
=πrarb1-rarb.
Xt=L+xt.
Et=-rbEt-2Texp-2ikxt-T,
Et=-raEt+taEint,
Etrt=tbEt-T,
Ereft=raEint+taEt,
Et=taEint+rarbEt-2Texp-2ikxt-T.
Et=taA+rarbEt-2Texp-2ikvt-T,
Et=Ct+Dt.
Ct-rarbCt-2Texp-2ikvt-T=taA,
Dt-rarbDt-2Texp-2ikvt-T=0.
CttaA1-rarb exp-2ikvt,
Dt=D0rarbt/2T expiϕt,
ϕt=ϕt-2T-2kvt-T.
ϕt=-kv2T t2.
Dt=D0 exp-tτ-i kv2T t2,
D0=taAiπ2kvT1/2 expiT2kvτ2.
Ωtdϕdt=k|v|T t.
Ωt=|δω| t2T,
Et=taA1-rarb exp-2ikvt+D0exp-tτ-i kv2T t2.
|Et|2=|Ct|2+|Dt|2+2 ReCt*Dt,
Ereft=ra2+ta2Eint-taEt/ra.
Ereft=Eint-taEt.
|Ereft|2=|Eint|2+ta2|Et|2-2taReEint*Et.
Vt=-ImexpiγEint*Et,
Vt=-A ImexpiγEt,
=-A ImexpiγCt+Dt.
VDt=-A|D0|exp-t-t0τ×sinγ+δ-kv2Tt-t02.
|VDt|  exp-t/τ.
=π2 sinhT/τ.
=1066±58.
kv2Ttn-t02=πn+γ+δ.
kv2Ttn+1-t02-tn-t02=π
Δtn=tn+1-tn,
t¯n=tn+tn+1/2.
ν¯n=12Δtn.
ν¯n=vλTt¯n-t0.
ν¯n=vλTt¯n-t0+δν¯n,
δν¯n=-4λvTt¯n-t02π2τ16v2t¯n-t04-λ2T2.
ν¯t=at+b,
a=86.8±0.6×106 Hz/s,
b=-0.5±1.0×103 Hz.
v=λTa,
t0=-b/a.
v=5.7±0.4×10-6 m/s,
t0=0.6±1.2×10-5 s.

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