Abstract

Using a radio frequency coherent modulation and demodulation technique, we explicitly measure both the amplitude and the phase response of Fabry–Perot interferometers in reflection. This allows us to differentiate clearly between overcoupled and undercoupled cavities and allows a detailed measurement of the full width at half-maximum, the free spectral range, and the finesse of the cavities.

© 2000 Optical Society of America

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  1. A. E. Siegman, Lasers, (University Science, Mill Valley, Calif., 1986).
  2. T. Andreae, W. König, R. Wynands, D. Leibfried, F. Schmidt-Kaler, C. Zimmermann, D. Meschede, T. W. Hänsch, “Absolute frequency measurement of the hydrogen 1S–2S transition and a new value of the Rydberg constant,” Phys. Rev. Lett. 69, 1923–1926 (1992).
    [CrossRef] [PubMed]
  3. T. Day, E. K. Gustafson, R. L. Byer, “Sub-hertz relative frequency stabilization of two diode pumped Nd:YAG lasers locked to a Fabry-Perot interferometer,” IEEE J. Quantum Electron. 28, 1106–1117 (1992).
    [CrossRef]
  4. P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors (World Scientific, Singapore, 1994).
  5. D. Z. Anderson, J. C. Frisch, C. S. Masser, “Mirror reflectometer based on optical cavity decay time,” Appl. Opt. 23, 1238–1245 (1984).
    [CrossRef] [PubMed]
  6. G. Rempe, R. J. Thompson, H. J. Kimble, “Measurement of ultralow losses in an optical interferometer,” Opt. Lett. 17, 363–365 (1992).
    [CrossRef] [PubMed]
  7. 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]
  8. N. Uehara, A. Ueda, K. Ueda, H. Sekiguchi, T. Mitake, K. Nakamura, N. Kitajima, I. Kataoka, “Ultralow-loss mirror of the parts-in-106 level at 1064 nm,” Opt. Lett. 20, 530–532 (1995).
    [CrossRef] [PubMed]
  9. N. Uehara, K. Ueda, “Accurate measurement of ultralow loss in a high-finesse Fabry–Perot interferometer using the frequency response functions,” Appl. Phys. B 61, 9–15 (1995).
    [CrossRef]
  10. N. Uehara, “Ring mode cleaner for the initial LIGO 10 watt laser,” (Stanford University, Stanford, Calif., 1997).
  11. P. K. Fritschel, “Techniques for laser interferometer gravitational wave detectors,” Ph.D. dissertation (Massachusetts Institute of Technology, Cambridge, Mass., 1992).
  12. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1993).
  13. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, H. Ward, “Laser phase and frequency stabilisation using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
    [CrossRef]
  14. M. B. Gray, D. A. Shaddock, C. C. Harb, H. A. Bachor, “Photodetector designs for low-noise, broadband, and high-power applications,” Rev. Sci. Instrum. 69, 3755–3762 (1998).
    [CrossRef]

1998

M. B. Gray, D. A. Shaddock, C. C. Harb, H. A. Bachor, “Photodetector designs for low-noise, broadband, and high-power applications,” Rev. Sci. Instrum. 69, 3755–3762 (1998).
[CrossRef]

1995

1992

G. Rempe, R. J. Thompson, H. J. Kimble, “Measurement of ultralow losses in an optical interferometer,” Opt. Lett. 17, 363–365 (1992).
[CrossRef] [PubMed]

T. Andreae, W. König, R. Wynands, D. Leibfried, F. Schmidt-Kaler, C. Zimmermann, D. Meschede, T. W. Hänsch, “Absolute frequency measurement of the hydrogen 1S–2S transition and a new value of the Rydberg constant,” Phys. Rev. Lett. 69, 1923–1926 (1992).
[CrossRef] [PubMed]

T. Day, E. K. Gustafson, R. L. Byer, “Sub-hertz relative frequency stabilization of two diode pumped Nd:YAG lasers locked to a Fabry-Perot interferometer,” IEEE J. Quantum Electron. 28, 1106–1117 (1992).
[CrossRef]

1984

1983

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

An, K.

Anderson, D. Z.

Andreae, T.

T. Andreae, W. König, R. Wynands, D. Leibfried, F. Schmidt-Kaler, C. Zimmermann, D. Meschede, T. W. Hänsch, “Absolute frequency measurement of the hydrogen 1S–2S transition and a new value of the Rydberg constant,” Phys. Rev. Lett. 69, 1923–1926 (1992).
[CrossRef] [PubMed]

Bachor, H. A.

M. B. Gray, D. A. Shaddock, C. C. Harb, H. A. Bachor, “Photodetector designs for low-noise, broadband, and high-power applications,” Rev. Sci. Instrum. 69, 3755–3762 (1998).
[CrossRef]

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1993).

Byer, R. L.

T. Day, E. K. Gustafson, R. L. Byer, “Sub-hertz relative frequency stabilization of two diode pumped Nd:YAG lasers locked to a Fabry-Perot interferometer,” IEEE J. Quantum Electron. 28, 1106–1117 (1992).
[CrossRef]

Dasari, R. R.

Day, T.

T. Day, E. K. Gustafson, R. L. Byer, “Sub-hertz relative frequency stabilization of two diode pumped Nd:YAG lasers locked to a Fabry-Perot interferometer,” IEEE J. Quantum Electron. 28, 1106–1117 (1992).
[CrossRef]

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 stabilisation using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[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 stabilisation using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Frisch, J. C.

Fritschel, P. K.

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

Gray, M. B.

M. B. Gray, D. A. Shaddock, C. C. Harb, H. A. Bachor, “Photodetector designs for low-noise, broadband, and high-power applications,” Rev. Sci. Instrum. 69, 3755–3762 (1998).
[CrossRef]

Gustafson, E. K.

T. Day, E. K. Gustafson, R. L. Byer, “Sub-hertz relative frequency stabilization of two diode pumped Nd:YAG lasers locked to a Fabry-Perot interferometer,” IEEE J. Quantum Electron. 28, 1106–1117 (1992).
[CrossRef]

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 stabilisation using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Hänsch, T. W.

T. Andreae, W. König, R. Wynands, D. Leibfried, F. Schmidt-Kaler, C. Zimmermann, D. Meschede, T. W. Hänsch, “Absolute frequency measurement of the hydrogen 1S–2S transition and a new value of the Rydberg constant,” Phys. Rev. Lett. 69, 1923–1926 (1992).
[CrossRef] [PubMed]

Harb, C. C.

M. B. Gray, D. A. Shaddock, C. C. Harb, H. A. Bachor, “Photodetector designs for low-noise, broadband, and high-power applications,” Rev. Sci. Instrum. 69, 3755–3762 (1998).
[CrossRef]

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 stabilisation using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Kataoka, I.

Kimble, H. J.

Kitajima, N.

König, W.

T. Andreae, W. König, R. Wynands, D. Leibfried, F. Schmidt-Kaler, C. Zimmermann, D. Meschede, T. W. Hänsch, “Absolute frequency measurement of the hydrogen 1S–2S transition and a new value of the Rydberg constant,” Phys. Rev. Lett. 69, 1923–1926 (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 stabilisation using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Leibfried, D.

T. Andreae, W. König, R. Wynands, D. Leibfried, F. Schmidt-Kaler, C. Zimmermann, D. Meschede, T. W. Hänsch, “Absolute frequency measurement of the hydrogen 1S–2S transition and a new value of the Rydberg constant,” Phys. Rev. Lett. 69, 1923–1926 (1992).
[CrossRef] [PubMed]

Masser, C. S.

Meschede, D.

T. Andreae, W. König, R. Wynands, D. Leibfried, F. Schmidt-Kaler, C. Zimmermann, D. Meschede, T. W. Hänsch, “Absolute frequency measurement of the hydrogen 1S–2S transition and a new value of the Rydberg constant,” Phys. Rev. Lett. 69, 1923–1926 (1992).
[CrossRef] [PubMed]

Mitake, T.

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 stabilisation using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Nakamura, K.

Rempe, G.

Saulson, P. R.

P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors (World Scientific, Singapore, 1994).

Schmidt-Kaler, F.

T. Andreae, W. König, R. Wynands, D. Leibfried, F. Schmidt-Kaler, C. Zimmermann, D. Meschede, T. W. Hänsch, “Absolute frequency measurement of the hydrogen 1S–2S transition and a new value of the Rydberg constant,” Phys. Rev. Lett. 69, 1923–1926 (1992).
[CrossRef] [PubMed]

Sekiguchi, H.

Shaddock, D. A.

M. B. Gray, D. A. Shaddock, C. C. Harb, H. A. Bachor, “Photodetector designs for low-noise, broadband, and high-power applications,” Rev. Sci. Instrum. 69, 3755–3762 (1998).
[CrossRef]

Siegman, A. E.

A. E. Siegman, Lasers, (University Science, Mill Valley, Calif., 1986).

Thompson, R. J.

Ueda, A.

Ueda, K.

N. Uehara, A. Ueda, K. Ueda, H. Sekiguchi, T. Mitake, K. Nakamura, N. Kitajima, I. Kataoka, “Ultralow-loss mirror of the parts-in-106 level at 1064 nm,” Opt. Lett. 20, 530–532 (1995).
[CrossRef] [PubMed]

N. Uehara, K. Ueda, “Accurate measurement of ultralow loss in a high-finesse Fabry–Perot interferometer using the frequency response functions,” Appl. Phys. B 61, 9–15 (1995).
[CrossRef]

Uehara, N.

N. Uehara, K. Ueda, “Accurate measurement of ultralow loss in a high-finesse Fabry–Perot interferometer using the frequency response functions,” Appl. Phys. B 61, 9–15 (1995).
[CrossRef]

N. Uehara, A. Ueda, K. Ueda, H. Sekiguchi, T. Mitake, K. Nakamura, N. Kitajima, I. Kataoka, “Ultralow-loss mirror of the parts-in-106 level at 1064 nm,” Opt. Lett. 20, 530–532 (1995).
[CrossRef] [PubMed]

N. Uehara, “Ring mode cleaner for the initial LIGO 10 watt laser,” (Stanford University, Stanford, Calif., 1997).

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 stabilisation using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1993).

Wynands, R.

T. Andreae, W. König, R. Wynands, D. Leibfried, F. Schmidt-Kaler, C. Zimmermann, D. Meschede, T. W. Hänsch, “Absolute frequency measurement of the hydrogen 1S–2S transition and a new value of the Rydberg constant,” Phys. Rev. Lett. 69, 1923–1926 (1992).
[CrossRef] [PubMed]

Yang, C.

Zimmermann, C.

T. Andreae, W. König, R. Wynands, D. Leibfried, F. Schmidt-Kaler, C. Zimmermann, D. Meschede, T. W. Hänsch, “Absolute frequency measurement of the hydrogen 1S–2S transition and a new value of the Rydberg constant,” Phys. Rev. Lett. 69, 1923–1926 (1992).
[CrossRef] [PubMed]

Appl. Opt.

Appl. Phys. B

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

N. Uehara, K. Ueda, “Accurate measurement of ultralow loss in a high-finesse Fabry–Perot interferometer using the frequency response functions,” Appl. Phys. B 61, 9–15 (1995).
[CrossRef]

IEEE J. Quantum Electron.

T. Day, E. K. Gustafson, R. L. Byer, “Sub-hertz relative frequency stabilization of two diode pumped Nd:YAG lasers locked to a Fabry-Perot interferometer,” IEEE J. Quantum Electron. 28, 1106–1117 (1992).
[CrossRef]

Opt. Lett.

Phys. Rev. Lett.

T. Andreae, W. König, R. Wynands, D. Leibfried, F. Schmidt-Kaler, C. Zimmermann, D. Meschede, T. W. Hänsch, “Absolute frequency measurement of the hydrogen 1S–2S transition and a new value of the Rydberg constant,” Phys. Rev. Lett. 69, 1923–1926 (1992).
[CrossRef] [PubMed]

Rev. Sci. Instrum.

M. B. Gray, D. A. Shaddock, C. C. Harb, H. A. Bachor, “Photodetector designs for low-noise, broadband, and high-power applications,” Rev. Sci. Instrum. 69, 3755–3762 (1998).
[CrossRef]

Other

A. E. Siegman, Lasers, (University Science, Mill Valley, Calif., 1986).

P. R. Saulson, Fundamentals of Interferometric Gravitational Wave Detectors (World Scientific, Singapore, 1994).

N. Uehara, “Ring mode cleaner for the initial LIGO 10 watt laser,” (Stanford University, Stanford, Calif., 1997).

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

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, London, 1993).

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Figures (6)

Fig. 1
Fig. 1

Theoretical frequency response for (a) an overcoupled cavity and (b) an undercoupled cavity. The real and imaginary components of the reflected field were plotted by use of Eqs. (3) and (4).

Fig. 2
Fig. 2

Corresponding FWHM of the transmitted field, by use of Eq. (1), and the turning points of the imaginary component of the reflected field, by use of Eq. (4).

Fig. 3
Fig. 3

Finesse ℱ as a function of the mirror reflectivity product q, showing that, for a finesse of 9 and above, the three techniques are identical. The traces a, b, and c refer to the responses from Eqs. (7), (9), and (10), respectively.

Fig. 4
Fig. 4

Schematic of the experimental system. We used a Lightwave-120 Nd:YAG laser that operates at 1.064 µm. The symbols not described in the text are FI, Faraday isolator; LPF, low-pass filter; PZT, piezoelectric transducer; LO, local oscillator; PD, photodetector; PC, Pockels cell; PBS, polarizing beam splitter; PID, proportional-integral-derivative servo amplifier.

Fig. 5
Fig. 5

Experimental results from the electric field amplitude and phase responses of the light reflected from the cavities around the cavity resonance. The resolution bandwidth was 1.2 kHz for these measurements.

Fig. 6
Fig. 6

Imaginary and real components of the reflected field obtained with the data from Fig. 5. The reflected field is normalized to the input field.

Tables (1)

Tables Icon

Table 1 Linewidth, Finesse, and Coupling of the Three Cavitiesa

Equations (14)

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

Et=tintout exp-iθ/21-rinrout exp-iθ Ei,
Er=rin-routrin2+tin2exp-iθ1-rinrout exp-iθ Ei,
ReEr/Ei=rin-rinq cosθ-z cosθ+zq1-2q cosθ+q2ER,Re,
ImEr/Ei=sinθz-rinq1-2q cosθ+q2ER,Im,
dER,Imdθ=z-rinqcosθ1+q2-2q1-2q cosθ+q22.
θIm=cos-12q1+q2.
a=FSRFWHM=2π2θIm=πcos-12q1+q2.
θhalf=cos-1-q2+4q-12q,
b=πcos-1-q2+4q-12q.
c=πrinrout1-rinrout=πq1-q.
Epd=E0+E+1 expiωmt+iϕ+E-1 exp-iωmt-iϕexp-iω0t,
Ipd=2ReE0*E+1 expiωmt+iϕ+E-1*E0 exp-iωmt-iϕ,
Ipd=2ReE0*E+1expiωmt+iϕ+exp-iωmt-iϕ,
Ipd=4 ReE0*E+1cosωmt+ϕ.

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