Abstract

We describe the accurate measurement of the radius of curvature of a concave mirror in a Fabry–Perot interferometer with a finesse of 78,100. The radius of curvature of the concave mirror is determined by measuring the free spectral range and the transverse-mode range with the frequency response functions. The radii of curvature at two orthogonal (x and y) axes on the mirror surface resulting from the polishing nonisotropy were accurately measured to be r x = 1008.46 mm and r y = 1006.94 mm, respectively, with an accuracy of 8 × 10−5. This accuracy is the best to our knowledge. The power dependence of the radii of curvature to the cavity internal intensity at a steady state was measured to be dr x/dI c = +60 μm/(MW/cm2) at the x axis and dr y/dI c = +47 μm/(MW/cm/2) at the y axis to an intensity of 2.1 MW/cm2.

© 1995 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. K. S. Thorne, “Gravitational radiation,” in 300 Years of Gravitation, S. W. Hawking, W. Israel, eds. (Cambridge U. Press, Cambridge, UK, 1987), Chap. 9, pp. 330–458.
  2. D. Shoemaker, R. Schilling, L. Schnupp, W. Winkler, K. Maischberger, A. Rüdiger, “Noise behavior of the Garching 30-meter prototype gravitational wave detector,” Phys. Rev. D 38, 423–432 (1988).
    [Crossref]
  3. A. Abramovici, W. E. Althouse, R. W. P. 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]
  4. W. Winkler, K. Danzmann, A. Rüdiger, R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1988).
    [Crossref]
  5. P. Hello, J. Y. Vinet, “Analytical models of transient thermoelastic deformations of mirrors heated by high power cw laser beams,” J. Phys. France 51, 2243–2261 (1990).
    [Crossref]
  6. O. Prakash, R. S. Ram, “Modification of the conventional method for determination of the focal length of convex mirrors,” Appl. Opt. 33, 2091–2094 (1994).
    [Crossref] [PubMed]
  7. H. Kogelnik, T. Li, “Laser beams and resonators,” Appl. Opt. 5, 1550–1567 (1966).
    [Crossref] [PubMed]
  8. A. Yariv, Optical Electronics, (CBS College Publishing, New York, 1985), Chap. 4, pp. 88–92.
  9. A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 19, pp. 761–766.
  10. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munleyand, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B 31, 97–105 (1983).
    [Crossref]
  11. N. Uehara, K. Ueda, “Ultrahigh-frequency stabilization of a diode-pumped Nd:YAG laser with a high-power-acceptance photodetector,” Opt. Lett. 19, 728–730 (1994).
    [Crossref] [PubMed]
  12. N. Uehara, K. Ueda, “Accurate measurement of ultralow loss in a high-finesse Fabry–Perot interferometer using the frequency response functions,” to be published in Appl. Phys. B.

1994 (2)

1992 (1)

A. Abramovici, W. E. Althouse, R. W. P. 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]

1990 (1)

P. Hello, J. Y. Vinet, “Analytical models of transient thermoelastic deformations of mirrors heated by high power cw laser beams,” J. Phys. France 51, 2243–2261 (1990).
[Crossref]

1988 (2)

D. Shoemaker, R. Schilling, L. Schnupp, W. Winkler, K. Maischberger, A. Rüdiger, “Noise behavior of the Garching 30-meter prototype gravitational wave detector,” Phys. Rev. D 38, 423–432 (1988).
[Crossref]

W. Winkler, K. Danzmann, A. Rüdiger, R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1988).
[Crossref]

1983 (1)

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

1966 (1)

Abramovici, A.

A. Abramovici, W. E. Althouse, R. W. P. 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. P. 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]

Danzmann, K.

W. Winkler, K. Danzmann, A. Rüdiger, R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1988).
[Crossref]

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. 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]

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

Ford, G. M.

R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munleyand, 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. P. 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]

Hall, J. L.

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

Hello, P.

P. Hello, J. Y. Vinet, “Analytical models of transient thermoelastic deformations of mirrors heated by high power cw laser beams,” J. Phys. France 51, 2243–2261 (1990).
[Crossref]

Hough, J.

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

Kawamura, S.

A. Abramovici, W. E. Althouse, R. W. P. 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]

Kogelnik, H.

Kowalski, F. V.

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

Li, T.

Maischberger, K.

D. Shoemaker, R. Schilling, L. Schnupp, W. Winkler, K. Maischberger, A. Rüdiger, “Noise behavior of the Garching 30-meter prototype gravitational wave detector,” Phys. Rev. D 38, 423–432 (1988).
[Crossref]

Munleyand, A. J.

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

Prakash, O.

Raab, F. J.

A. Abramovici, W. E. Althouse, R. W. P. 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]

Ram, R. S.

Rüdiger, A.

D. Shoemaker, R. Schilling, L. Schnupp, W. Winkler, K. Maischberger, A. Rüdiger, “Noise behavior of the Garching 30-meter prototype gravitational wave detector,” Phys. Rev. D 38, 423–432 (1988).
[Crossref]

W. Winkler, K. Danzmann, A. Rüdiger, R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1988).
[Crossref]

Schilling, R.

W. Winkler, K. Danzmann, A. Rüdiger, R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1988).
[Crossref]

D. Shoemaker, R. Schilling, L. Schnupp, W. Winkler, K. Maischberger, A. Rüdiger, “Noise behavior of the Garching 30-meter prototype gravitational wave detector,” Phys. Rev. D 38, 423–432 (1988).
[Crossref]

Schnupp, L.

D. Shoemaker, R. Schilling, L. Schnupp, W. Winkler, K. Maischberger, A. Rüdiger, “Noise behavior of the Garching 30-meter prototype gravitational wave detector,” Phys. Rev. D 38, 423–432 (1988).
[Crossref]

Shoemaker, D.

A. Abramovici, W. E. Althouse, R. W. P. 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]

D. Shoemaker, R. Schilling, L. Schnupp, W. Winkler, K. Maischberger, A. Rüdiger, “Noise behavior of the Garching 30-meter prototype gravitational wave detector,” Phys. Rev. D 38, 423–432 (1988).
[Crossref]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 19, pp. 761–766.

Sievers, L.

A. Abramovici, W. E. Althouse, R. W. P. 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. P. 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]

Thorne, K. S.

A. Abramovici, W. E. Althouse, R. W. P. 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]

K. S. Thorne, “Gravitational radiation,” in 300 Years of Gravitation, S. W. Hawking, W. Israel, eds. (Cambridge U. Press, Cambridge, UK, 1987), Chap. 9, pp. 330–458.

Ueda, K.

N. Uehara, K. Ueda, “Ultrahigh-frequency stabilization of a diode-pumped Nd:YAG laser with a high-power-acceptance photodetector,” Opt. Lett. 19, 728–730 (1994).
[Crossref] [PubMed]

N. Uehara, K. Ueda, “Accurate measurement of ultralow loss in a high-finesse Fabry–Perot interferometer using the frequency response functions,” to be published in Appl. Phys. B.

Uehara, N.

N. Uehara, K. Ueda, “Ultrahigh-frequency stabilization of a diode-pumped Nd:YAG laser with a high-power-acceptance photodetector,” Opt. Lett. 19, 728–730 (1994).
[Crossref] [PubMed]

N. Uehara, K. Ueda, “Accurate measurement of ultralow loss in a high-finesse Fabry–Perot interferometer using the frequency response functions,” to be published in Appl. Phys. B.

Vinet, J. Y.

P. Hello, J. Y. Vinet, “Analytical models of transient thermoelastic deformations of mirrors heated by high power cw laser beams,” J. Phys. France 51, 2243–2261 (1990).
[Crossref]

Vogt, R. E.

A. Abramovici, W. E. Althouse, R. W. P. 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. Munleyand, 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. P. 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. P. 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]

Winkler, W.

D. Shoemaker, R. Schilling, L. Schnupp, W. Winkler, K. Maischberger, A. Rüdiger, “Noise behavior of the Garching 30-meter prototype gravitational wave detector,” Phys. Rev. D 38, 423–432 (1988).
[Crossref]

W. Winkler, K. Danzmann, A. Rüdiger, R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1988).
[Crossref]

Yariv, A.

A. Yariv, Optical Electronics, (CBS College Publishing, New York, 1985), Chap. 4, pp. 88–92.

Zucker, M. E.

A. Abramovici, W. E. Althouse, R. W. P. 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. Opt. (2)

Appl. Phys. B (1)

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

J. Phys. France (1)

P. Hello, J. Y. Vinet, “Analytical models of transient thermoelastic deformations of mirrors heated by high power cw laser beams,” J. Phys. France 51, 2243–2261 (1990).
[Crossref]

Opt. Lett. (1)

Phys. Rev. A (1)

W. Winkler, K. Danzmann, A. Rüdiger, R. Schilling, “Heating by optical absorption and the performance of interferometric gravitational-wave detectors,” Phys. Rev. A 44, 7022–7036 (1988).
[Crossref]

Phys. Rev. D (1)

D. Shoemaker, R. Schilling, L. Schnupp, W. Winkler, K. Maischberger, A. Rüdiger, “Noise behavior of the Garching 30-meter prototype gravitational wave detector,” Phys. Rev. D 38, 423–432 (1988).
[Crossref]

Science (1)

A. Abramovici, W. E. Althouse, R. W. P. 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 (4)

K. S. Thorne, “Gravitational radiation,” in 300 Years of Gravitation, S. W. Hawking, W. Israel, eds. (Cambridge U. Press, Cambridge, UK, 1987), Chap. 9, pp. 330–458.

A. Yariv, Optical Electronics, (CBS College Publishing, New York, 1985), Chap. 4, pp. 88–92.

A. E. Siegman, Lasers (University Science Books, Mill Valley, Calif., 1986), Chap. 19, pp. 761–766.

N. Uehara, K. Ueda, “Accurate measurement of ultralow loss in a high-finesse Fabry–Perot interferometer using the frequency response functions,” to be published in Appl. Phys. B.

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 (9)

Fig. 1
Fig. 1

(a) Schematic diagram of a Fabry–Perot interferometer consisting of two mirrors. These mirrors, M1 and M2, have an intensity reflectance R, a transmittance T, and radii of curvature, r 1 and r 2. L is the interferometer length. A flat mirror (r 1 → ∞) and a concave mirror (r 2 = r) are assumed in this paper. (b) The upper part shows a resonant line shape of the FP interferometer in transmission. High-order transverse modes to the second-order TEM20 q modes are shown in this figure where q is the longitudinal-mode number. The lower part shows a phase-modulated incident electric field (E i ). ν L = ω L /2π and ν F = ω F /2π are the laser frequency and the modulation frequency, respectively.

Fig. 2
Fig. 2

Calculated result of a FSR and a TMR as a function of dr/r or dL/L in the case of a concave mirror of r = 1000 mm and an interferometer length of L = 200 mm. This cavity structure is used in the experiment. The continuous line shows dFSR as a function of dL/L, the dashed line shows ∂TMR r = (∂TMR/∂r)dr as a function of dr/r, and the dot–dashed line shows ∂TMR L = (∂TMR/∂L)dL as a function of dL/L.

Fig. 3
Fig. 3

Experimental setup for the accurate measurement of the radius of curvature in an FP interferometer. A laser frequency is locked to the qth TEM00 longitudinal mode of the FP interferometer by the Pound–Drever–Hall technique. EOM 1 is used to measure the frequency response functions of the interferometer; OSC is the oscillator.

Fig. 4
Fig. 4

Transmitted transverse-mode patterns (TEM00, TEM10, TEM01, TEM01, TEM20, TEM11, and TEM02) from the FP interferometer. The splitting of the degeneration of high-order transverse modes is observed. This is due to the nonisotropy of surface polishing (r x r y ).

Fig. 5
Fig. 5

Measured frequency response functions of high-order transverse modes: (a) first-order transverse modes (TEM10 and TEM01), (b) second-order transverse modes (TEM20, TEM11, and TEM02), (c) comparison of the measured response function and a calculated function.

Fig. 6
Fig. 6

(a) Measured frequency response function of the next TEM00 longitudinal modes. (b) Normalized response function of (a). The horizontal scale is normalized by x = 2(f − FSR)/Δν c , where the FSR and the Δν c are 749.3598 MHz and 9.60 kHz, respectively.

Fig. 7
Fig. 7

Power dependence of the resonant frequencies of the next TEM00 longitudinal mode and of the high-order TEM mn transverse modes as a function of the incident power (P i ). The vertical scale shows the relative frequency of f mn (0).

Fig. 8
Fig. 8

Power dependence of the radii of curvature as a function of an internal intensity (I c ) at the center (r = 0) corresponds to the TEM10, the TEM01, the TEM20, and the TEM02 modes. The vertical scale shows the change in dr mn (I c ) = (dr mn /dI c )I c .

Fig. 9
Fig. 9

Transmitted power (P T ) as a function of an incident power (P i ). The slope (η T ) is measured to be 36.5% to the maximum incident power of 93 mW.

Tables (2)

Tables Icon

Table 1 Resonant Frequencies of the Transverse Modes and the Radii of Curvaturea

Tables Icon

Table 2 Power Dependence of the Radii of Curvature to the Laser Incident Power P i

Equations (37)

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

E i ( t ) = E 0 exp ( i ω t ) ,
E t ( t ) = T 1 - R exp ( - i ω τ ) E i ( t ) ,
a ( ω ) = T 1 - R exp ( - i ω τ ) .
E t ( t ) = - a ( ω ) E ˜ i ( ω ) exp ( i ω t ) d ω
TMR = γ ( r 1 , r 2 , L ) FSR ,
γ ( r 1 , r 2 , L ) = 1 π cos - 1 [ ( 1 - L r 1 ) ( 1 - L r 2 ) ] 1 / 2 ,
FSR = c 2 n L ,
γ ( r , L ) = 1 π cos - 1 ( 1 - L r ) 1 / 2 .
ν m n q = ν 00 q + f m n ,
f m n = ( m + n ) TMR ,
E i ( t ) = E 0 i , j = 0 , 0 k = - ψ i j J k ( β ) exp [ i ( ω L + k ω F ) t ] ,
E i ( t ) = E 0 i , j ψ i j [ J 0 + J 1 exp ( i ω F t ) + J - 1 exp ( - i ω F t ) ] exp ( i ω L t ) .
E ˜ i ( ω ) = - E i ( t ) exp ( - i ω t ) d t = E 0 i , j ψ i j { J 0 δ ( ω - ω L ) + J 1 δ [ ω - ( ω L + ω F ) ] - J 1 δ [ ω - ( ω L - ω F ) ] } ,
E t ( t ) = - a ( ω ) E ˜ i ( ω ) exp ( i ω t ) d ω = E 0 i , j ψ i j [ J 0 a ( ω L ) + J 1 a ( ω L + ω F ) exp ( i ω F t ) - J 1 a ( ω L - ω F ) exp ( - i ω F t ) ] .
E t ( t ) = E 0 i , j ψ i j { J 0 a ( 0 ) + J 1 [ a ( ω F ) exp ( i ω F t ) - a ( - ω F ) exp ( - i ω F t ) ] } .
E t ( t ) = E 0 [ ψ 00 J 0 a ( 0 ) + ψ m n J 1 a ( ω F ) exp ( i ω F t ) ] .
I t ( ω F ) = E 0 2 ψ 00 * ψ m n J 0 J 1 a ( 0 ) a ( ω F ) exp ( i ω F t ) + ( c . c . ) .
a ( ω F ) = ( T 1 - R ) 1 1 + i [ 2 ( ω F - ω m n ) Δ ω c ] ,
I i ω F = E 0 2 ψ 00 ψ m n J 0 J 1 exp ( i ω F t ) + ( c . c . ) .
H m n ( ω F ) = Y ( ω F ) X ( ω F ) = I t ( ω F ) I i ω F = a ( 0 ) a ( ω F ) = a ( 0 ) 2 1 1 + i [ 2 ( ω F - ω m n ) Δ ω c ] ,
H m n ( ω F ) = a ( 0 ) 2 1 { 1 + [ 2 ( ω F - ω m n ) Δ ω c ] 2 } 1 / 2 ,
arg [ H m n ( ω F ) ] = tan - 1 [ 2 ( ω F - ω m n ) Δ ω c ] .
H 00 ( ω F ) = a ( 0 ) [ a ( ω F ) - a ( - ω F ) ] = a ( 0 ) 2 { 1 1 + i [ 2 ( ω F - ω FSR ) Δ ω c ] - 1 1 - i [ 2 ( ω F - ω FSR ) Δ ω c ] } ,
d FSR d L = - FSR L .
d TMR = ( TMR r ) d r + ( TMR L ) d L ,
( TMR r ) = - 1 2 FSR L π r 3 / 2 ( 1 - L / r ) 1 / 2 ,
( TMR L ) = 1 2 FSR π ( L r ) 1 / 2 ( 1 - L / r ) 1 / 2 + γ ( FSR L ) .
f 00 = ν 00 , q + 1 - ν 00 q = FSR ,
d f 00 = d FSR .
d f m n = ( m + n ) d TMR .
d FSR d L = - 3.75 MHz / mm ,
( TMR r ) = - 59.7 kHz / mm ,
( TMR L ) = - 255 kHz / mm .
H 00 , q ± 1 ( x ) = | 1 1 + i x - 1 1 - i x | .
f m n ( P i ) = f m n ( 0 ) + d f m n d P i P i .
g = P c P i = T ( 1 + R ) 2 ( 1 - R ) 2 ,
I c ( r ) = 2 P c π W 2 exp ( - 2 r 2 / W 2 ) ,

Metrics