Abstract

We predict and measure the temporal response of a Fabry–Perot cavity field to changes in cavity length and frequency of the incident laser field. We outline the theoretical differences between changes in the cavity-length and laser-frequency modulation and present a theoretical derivation of the time response of the resulting cavity field and its effect on the reflected field, the transmitted field, and the Pound–Drever–Hall error signal. We show that oscillations in the resulting signals are due to oscillations in the amplitude and the phase of the cavity field itself. Finally, we demonstrate how induced cavity-field oscillations may be used to determine the mirror velocity or the frequency change of the injected laser field.

© 1999 Optical Society of America

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References

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  1. C. Fabry and A. Perot, “Théorie et applications d’une nouvelle méthode de Spectroscopie Interférentielle,” Ann. de Chim. et de Phys. 16, 115 (1899).
  2. J. M. Vaughan, The Fabry-Perot Interferometer: History, Theory, Practice, and Applications (Hilger, London, 1989).
  3. H. J. Schmitt and H. Zimmer, “Fast sweep measurements of relaxation times in superconducting cavities,” IEEE Trans. Microwave Theory Tech. MTT-14, 206–207 (1966).
    [CrossRef]
  4. K. An, C. Yang, R. R. Dasari, and 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. 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, and M. E. Zucker, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
    [CrossRef] [PubMed]
  6. B. A. Paldus, C. C. Harb, T. G. Spence, B. Willke, J. Xie, J. S. Harris, and R. N. Zare, “Cavity-locked ring-down spectroscopy,” J. Appl. Phys. 83, 3993 (1998).
    [CrossRef]
  7. J. Camp, L. Sievers, R. Bork, and J. Hefner, “Guided lock acquisition in a suspended Fabry–Perot cavity,” Opt. Lett. 20, 2463–2465 (1995).
    [CrossRef]
  8. K. Kawabe, N. Mio, and K. Tsubono, “Automatic alignment-control system for a suspended Fabry–Perot cavity,” Appl. Opt. 33, 5498–5505 (1994).
    [CrossRef] [PubMed]
  9. E. Morrison, B. J. Meers, D. I. Robertson, and H. Ward, “Automatic alignment of optical interferometers,” Appl. Opt. 33, 5041–5049 (1994).
    [CrossRef] [PubMed]
  10. E. Morrison, B. J. Meers, D. I. Robertson, and H. Ward, “Experimental demonstration of an automatic alignment system for optical interferometers,” Appl. Opt. 33, 5037–5040 (1994).
    [CrossRef] [PubMed]
  11. R. W. P. Drever, J. L. Hall, F. V. Kowalski, J. Hough, G. M. Ford, A. J. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. 31, 97–105 (1983).
    [CrossRef]
  12. E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987), p. 368. We extend the term “Fabry–Perot” to include cavities with curved as well as flat mirrors in this paper.
  13. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 413–426.
  14. M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 327–328.
  15. A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 437.
  16. A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, N.Y., 1985), pp. 294–296.
  17. N. Uehara and K. Ueda, “Frequency stabilization of two diode-pumped Nd:YAG lasers locked to two Fabry–Perot cavities,” Jpn. J. Appl. Phys. 33, 1628–1633 (1994).
    [CrossRef]
  18. T. Day, E. K. Gustafson, and R. L. Byer, “Active frequency stabilization of a 1.062-μm, Nd:GGG, diode-laser-pumped nonplanar ring oscillator to less than 3 Hz of relative linewidth,” Opt. Lett. 15, 221–223 (1990).
    [CrossRef] [PubMed]

1998 (1)

B. A. Paldus, C. C. Harb, T. G. Spence, B. Willke, J. Xie, J. S. Harris, and R. N. Zare, “Cavity-locked ring-down spectroscopy,” J. Appl. Phys. 83, 3993 (1998).
[CrossRef]

1995 (2)

1994 (4)

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, and M. E. Zucker, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

1990 (1)

1983 (1)

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

1966 (1)

H. J. Schmitt and H. Zimmer, “Fast sweep measurements of relaxation times in superconducting cavities,” IEEE Trans. Microwave Theory Tech. MTT-14, 206–207 (1966).
[CrossRef]

1899 (1)

C. Fabry and A. Perot, “Théorie et applications d’une nouvelle méthode de Spectroscopie Interférentielle,” Ann. de Chim. et de Phys. 16, 115 (1899).

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, and 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, and 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.

Day, T.

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, and 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. Munley, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. 31, 97–105 (1983).
[CrossRef]

Fabry, C.

C. Fabry and A. Perot, “Théorie et applications d’une nouvelle méthode de Spectroscopie Interférentielle,” Ann. de Chim. et de Phys. 16, 115 (1899).

Feld, M. S.

Ford, G. M.

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

Harb, C. C.

B. A. Paldus, C. C. Harb, T. G. Spence, B. Willke, J. Xie, J. S. Harris, and R. N. Zare, “Cavity-locked ring-down spectroscopy,” J. Appl. Phys. 83, 3993 (1998).
[CrossRef]

Harris, J. S.

B. A. Paldus, C. C. Harb, T. G. Spence, B. Willke, J. Xie, J. S. Harris, and R. N. Zare, “Cavity-locked ring-down spectroscopy,” J. Appl. Phys. 83, 3993 (1998).
[CrossRef]

Hefner, J.

Hough, J.

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

Kawabe, K.

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, and 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, and H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. 31, 97–105 (1983).
[CrossRef]

Meers, B. J.

Mio, N.

Morrison, E.

Munley, A. J.

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

Paldus, B. A.

B. A. Paldus, C. C. Harb, T. G. Spence, B. Willke, J. Xie, J. S. Harris, and R. N. Zare, “Cavity-locked ring-down spectroscopy,” J. Appl. Phys. 83, 3993 (1998).
[CrossRef]

Perot, A.

C. Fabry and A. Perot, “Théorie et applications d’une nouvelle méthode de Spectroscopie Interférentielle,” Ann. de Chim. et de Phys. 16, 115 (1899).

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, and M. E. Zucker, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Robertson, D. I.

Schmitt, H. J.

H. J. Schmitt and H. Zimmer, “Fast sweep measurements of relaxation times in superconducting cavities,” IEEE Trans. Microwave Theory Tech. MTT-14, 206–207 (1966).
[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, and 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, and J. Hefner, “Guided lock acquisition in a suspended Fabry–Perot cavity,” Opt. Lett. 20, 2463–2465 (1995).
[CrossRef]

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, and M. E. Zucker, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Spence, T. G.

B. A. Paldus, C. C. Harb, T. G. Spence, B. Willke, J. Xie, J. S. Harris, and R. N. Zare, “Cavity-locked ring-down spectroscopy,” J. Appl. Phys. 83, 3993 (1998).
[CrossRef]

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, and 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, and M. E. Zucker, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Tsubono, K.

Ueda, K.

N. Uehara and K. Ueda, “Frequency stabilization of two diode-pumped Nd:YAG lasers locked to two Fabry–Perot cavities,” Jpn. J. Appl. Phys. 33, 1628–1633 (1994).
[CrossRef]

Uehara, N.

N. Uehara and K. Ueda, “Frequency stabilization of two diode-pumped Nd:YAG lasers locked to two Fabry–Perot cavities,” Jpn. J. Appl. Phys. 33, 1628–1633 (1994).
[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, and M. E. Zucker, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Ward, H.

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, and 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, and M. E. Zucker, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Willke, B.

B. A. Paldus, C. C. Harb, T. G. Spence, B. Willke, J. Xie, J. S. Harris, and R. N. Zare, “Cavity-locked ring-down spectroscopy,” J. Appl. Phys. 83, 3993 (1998).
[CrossRef]

Xie, J.

B. A. Paldus, C. C. Harb, T. G. Spence, B. Willke, J. Xie, J. S. Harris, and R. N. Zare, “Cavity-locked ring-down spectroscopy,” J. Appl. Phys. 83, 3993 (1998).
[CrossRef]

Yang, C.

Zare, R. N.

B. A. Paldus, C. C. Harb, T. G. Spence, B. Willke, J. Xie, J. S. Harris, and R. N. Zare, “Cavity-locked ring-down spectroscopy,” J. Appl. Phys. 83, 3993 (1998).
[CrossRef]

Zimmer, H.

H. J. Schmitt and H. Zimmer, “Fast sweep measurements of relaxation times in superconducting cavities,” IEEE Trans. Microwave Theory Tech. MTT-14, 206–207 (1966).
[CrossRef]

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, and M. E. Zucker, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Ann. de Chim. et de Phys. (1)

C. Fabry and A. Perot, “Théorie et applications d’une nouvelle méthode de Spectroscopie Interférentielle,” Ann. de Chim. et de Phys. 16, 115 (1899).

Appl. Opt. (3)

Appl. Phys. B: Photophys. Laser Chem. (1)

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

IEEE Trans. Microwave Theory Tech. (1)

H. J. Schmitt and H. Zimmer, “Fast sweep measurements of relaxation times in superconducting cavities,” IEEE Trans. Microwave Theory Tech. MTT-14, 206–207 (1966).
[CrossRef]

J. Appl. Phys. (1)

B. A. Paldus, C. C. Harb, T. G. Spence, B. Willke, J. Xie, J. S. Harris, and R. N. Zare, “Cavity-locked ring-down spectroscopy,” J. Appl. Phys. 83, 3993 (1998).
[CrossRef]

Jpn. J. Appl. Phys. (1)

N. Uehara and K. Ueda, “Frequency stabilization of two diode-pumped Nd:YAG lasers locked to two Fabry–Perot cavities,” Jpn. J. Appl. Phys. 33, 1628–1633 (1994).
[CrossRef]

Opt. Lett. (3)

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, and M. E. Zucker, “LIGO: the laser interferometer gravitational-wave observatory,” Science 256, 325–333 (1992).
[CrossRef] [PubMed]

Other (6)

J. M. Vaughan, The Fabry-Perot Interferometer: History, Theory, Practice, and Applications (Hilger, London, 1989).

E. Hecht, Optics (Addison-Wesley, Reading, Mass., 1987), p. 368. We extend the term “Fabry–Perot” to include cavities with curved as well as flat mirrors in this paper.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), pp. 413–426.

M. Born and E. Wolf, Principles of Optics (Pergamon, Oxford, 1980), pp. 327–328.

A. E. Siegman, Lasers (University Science, Mill Valley, Calif., 1986), p. 437.

A. Yariv, Optical Electronics (Holt, Rinehart & Winston, New York, N.Y., 1985), pp. 294–296.

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Figures (7)

Fig. 1
Fig. 1

Laser light entering a Fabry–Perot cavity. The input field, E˜in, results in a cavity field, E˜cav. Transmission of the cavity field yields the transmitted field, E˜trans, while interference between the directly reflected input field and the transmitted cavity field produces the reflected field, E˜refl.

Fig. 2
Fig. 2

Phasor representation of the cavity field near the resonant condition and at the off-resonant condition. The phasor components are shown in gray and the cavity field in black. The horizontal axis represents signals in phase with the input field, and the vertical axis represents signals π/2 out of phase with the input field. Near the resonant condition, all phasors acquire a phase shift close to an integer multiple of 2π after each round trip, and they constructively interfere to produce a large cavity field. At the off-resonant condition, all phasors acquire a phase shift significantly different from an integer multiple of 2π after each round trip, and they destructively interfere to produce a small cavity field.

Fig. 3
Fig. 3

Cavity field at times and (n+1)τ with a moving end mirror. Since all phasors acquire the same loss and phase shift within a round trip, they can be replaced by their sum phasor. Instead of keeping track of the individual phasors, this sum phasor can then be reduced (multiplied by ρ), rotated by the round-trip phase, and then added to the transmitted input beam to yield the cavity field a round trip later in time.

Fig. 4
Fig. 4

Theoretical transmitted, reflected, and Pound–Drever–Hall error photodetector signals for νL=νω equal to (a) 0.01, (b) 0.5, (c) 2, and (d) 10 from Eqs. (11) and (16). The horizontal axes represent the distance sweep in units of fwhm cavity linewidths: Δωfwhm for frequency sweeps, and ΔLfwhm for length sweeps. This distance is given by half the normalized time, t/2τs, multiplied by νω or νL. Oscillations are visible when νω or νL approach 1. The curves are offset in the vertical direction for clarity. Also plotted for all figures are the series expressions, Eqs. (9) and (14). The curves are indistinguishable from the differential equation solutions.

Fig. 5
Fig. 5

Experimental apparatus used to measure the time response of the Fabry–Perot cavity for both the length- and the frequency-modulation schemes. The cavity had a round-trip length of 42 cm and a finesse of 4000 for S-polarized light and a finesse of 220 for P-polarized light. The P-polarized laser beam was used to lock the laser frequency to the cavity with the traditional Pound–Drever–Hall method. An acousto-optic modulator was used to change the frequency of the S-polarized light. For length modulation the S-polarized beam was unused and a piezoelectric actuator was used to change the cavity length.

Fig. 6
Fig. 6

Time response of the Fabry–Perot cavity’s reflected, transmitted, and Pound–Drever–Hall error photodetector currents for mirror velocities νL equal to (a) 0.14, (b) 0.8, (c) 1.6, and (d) 5.4 as a function of sweep time. The curves have been offset in the vertical direction for clarity, and the dashed lines indicate zero current. The dotted lines indicate the theoretical fits to the experimental results. The storage time of the cavity was 1.8 µs. The theoretical fits are in excellent agreement with the measured data, and, as predicted, oscillations are observed when the normalized sweep speed is of the order of 1.

Fig. 7
Fig. 7

Time response of the Fabry–Perot cavity’s reflected, transmitted, and Pound–Drever–Hall error photodetector currents for frequency sweep speeds νω equal to (a) 4.4, (b) 5.0, (c) 8.6, and (d) 13 as a function of sweep time. The curves have been offset in the vertical direction for clarity, and the dashed lines indicate zero current. The dotted lines indicate the theoretical fits to the experimental results. The storage time of the cavity was 1.8 µs. The theoretical fits are in good agreement with the measured data, although they diverge at longer times owing to the nonlinear frequency sweep produced by our experiment.

Equations (30)

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Δωfwhm=πcLF,
F=π(R1R2)1/41-R1R2,
ΔLfwhm=λ/2F=πcωF.
E˜in=E0exp(-iωt)+δ2{exp[-i(ω+ωm)t]-exp[-i(ω-ωm)t]}.
E˜trans=iT2E˜cav.
Itrans=12c0E˜transE˜trans*=12c0T2EcavEcav*,
itrans=RItrans=12c0RT2EcavEcav*,
E˜refl=R1E˜in exp-i2ωLc+iT1R2E˜cav,
E˜refl=R1E0 exp(-iωt)×1+δ2[exp(-iωmt)-exp(iωmt)]+iT1R2Ecav exp(-iωt)
Irefl=12c0E˜reflE˜refl*.
irefl=RIrefl=12c0RE˜reflE˜refl*.
irefl=12c0R[R1E02-2T1R1R2E0 Im(Ecav)+T1R2EcavEcav*].
ipdh=-c0RT1R1R2δE0 Re(Ecav).
E˜cav=iT1E0 exp[i(kx-ωt)].
E˜cav=iT1E0exp(-iωτ)exp{i[kx-ω(t-τ)]}+ρ expi1+2L˙τ/2c[kx-ω(t-τ)],
τ=2Lc,
E˜cav=iT1E0exp(-iωnτ)exp{i[kx-ω(t-nτ)]}+ρ exp[-iω(n-1)τ]expi1+2L˙(2n-1)τ/2c×[kx-ω(t-nτ)]+ρ2 exp[-iω(n-2)τ]expi1+2L˙(2n-1)τ/2c×1+2L˙(2n-3)τ/2c[kx-ω(t-nτ)]++ρn expi1+2L˙(2n-1)τ/2c×1+2L˙(2n-3)τ/2c1+2L˙τ/2c×[kx-ω(t-nτ)].
Ecav(t+τ)=iT1E0+ρ expi2ωLt+τ/2cEcav(t).
dEcavdt=ρ-1τ+i2ρωL˙τctEcav+iT1τE0.
τsFτπ=2FLπc.
dEcavdt=-(1-iνLt)Ecav+iT1FπE0,
νL=2FωL˙τsπc.
νL=L˙(ΔLfwhm/2)/τs.
E˜cav=iT1E0 exp[i(k0x-ω0t)],
E˜cav=iT1E0(exp(-iωττ)exp{i[kτx-iωτ(t-τ)]}+ρ exp{i[kx-ω0(t-τ)]}).
E˜cav=iT1E0{exp(-iωnτnτ)exp[iknτx-ωnτ(t-nτ)]+ρ exp[-iω(n-1)τ(n-1)τ]exp[ik(n-1)τx-ω(n-1)τ(t-nτ)]+ρ2 exp[-iω(n-2)τ(n-2)τ]exp[ik(n-2)τx-ω(n-2)τ(t-nτ)]++ρn exp[ik0x-ω0(t-nτ)]}.
dEcavdt=ρ-1τ+i2ρω˙LτctEcav+iT1τE0.
dEcavdt=-(1-iνωt)Ecav+iT1FπE0,
νω=2FLω˙τsπc.
νω=ω˙(Δωfwhm/2)/τs.

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