Abstract

A system designed to apply Fabry–Perot interferometry to the measurement of displacements is described. Two adjacent modes of a Fabry–Perot cavity are probed, and both the absolute optical frequencies and their difference are used to determine displacements via changes in cavity length. Light is coupled to the cavity via an optical fiber, making the system ideal for remote sensing applications. Continuous interrogation is not necessary, as the cavity length is encoded in the free spectral range. The absolute uncertainty is determined to be below 10pm, which for the largest displacement measured corresponds to a relative uncertainty of 4×1010. To my knowledge this is the smallest relative uncertainty in a displacement measurement ever demonstrated.

© 2005 Optical Society of America

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References

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  1. N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
    [CrossRef]
  2. L. Howard, J. Stone, J. Fu, “Real-time displacement measurements with a Fabry-Perot cavity and a diode laser,” Precis. Eng. 25, 321–335 (2001).
    [CrossRef]
  3. H. V. Parks, J. E. Faller, D. S. Robertson, “A suspended laser interferometer for determining the Newtonian constant of gravitation,” IEEE Trans. Instrum. Meas. 50, 598–600 (2001).
    [CrossRef]
  4. W.-T. Ni, D.-K. Liu, T.-T. Liu, H.-H. Mei, S. Shi Pan, C.-P. Pang, H.-C. Yeh, “The application of laser metrology and resonant optical cavity techniques to the measurement of G,” Meas. Sci. Technol. 10, 495–498 (1999).
    [CrossRef]
  5. R. D. Deslattes, A. Henins, “X-ray to visible wavelength ratios,” Phys. Rev. Lett. 31, 972–975 (1973).
    [CrossRef]
  6. Z. Bay, “The use of microwave modulation of lasers for length measurements,” Natl. Bur. Stand. (U.S.) Spec. Publ. 343, 59–62 (1971).
  7. T. J. Dunn, T.-M. Lee, K. Jain, “Absolute distance measurement interferometry for alignment systems for advanced lithography tools,” J. Vac. Sci. Technol. B 14, 3960–3963 (1996).
    [CrossRef]
  8. R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. B31, 97–105 (1983).
    [CrossRef]
  9. E. D. Black, “An introduction to Pound-Drever-Hall laser frequency stabilization,” Am. J. Phys. 69, 79–87 (2001).
    [CrossRef]
  10. J. Lawall, J. M. Pedulla, Y. L. Coq, “Ultrastable laser array at 633 nm for real-time dimensional metrology,” Rev. Sci. Instrum. 72, 2879–2888 (2001).
    [CrossRef]
  11. JDS Uniphase Corporation, Model 1007P, http://www.jdsu.com. Certain commercial equipment, instru- ments, or materials are identified in this paper to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose.
  12. Gooch and Housego, Model FS040-2E, http://www. goochandhousego.com.
  13. Isomet, Model 1250C, http://www.isomet.com.
  14. Conoptics Inc., Model 380/4, http://www.conoptics.com.
  15. Perkin-Elmer Inc., Model 7280, http://www.signalrecovery.com.
  16. New Focus Inc., Model 8883, http://www.newfocus.com.
  17. A. Siegman, Lasers (University Science, 1986).
  18. Burleigh Corporation, Model TSE-150V, http://www.exfo.com/en/burleigh.asp.
  19. J. Lawall, E. Kessler, “Design and evaluation of a simple ultralow vibration vacuum environment,” Rev. Sci. Instrum. 73, 209–215 (2002).
    [CrossRef]
  20. H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
    [CrossRef]
  21. H. A. Macleod, Thin-Film Optical Filters (Institute of Physics, 2001).
    [CrossRef]
  22. The author is preparing a manuscript to be called “Accurate determination of radii of curvature using Fabry-Pérot interferometry.”
  23. J. Lawall, “Interferometry for accurate displacement metrology,” Opt. Photonics News 15, 40–46 (2004).
    [CrossRef]

2004

J. Lawall, “Interferometry for accurate displacement metrology,” Opt. Photonics News 15, 40–46 (2004).
[CrossRef]

2002

J. Lawall, E. Kessler, “Design and evaluation of a simple ultralow vibration vacuum environment,” Rev. Sci. Instrum. 73, 209–215 (2002).
[CrossRef]

2001

L. Howard, J. Stone, J. Fu, “Real-time displacement measurements with a Fabry-Perot cavity and a diode laser,” Precis. Eng. 25, 321–335 (2001).
[CrossRef]

H. V. Parks, J. E. Faller, D. S. Robertson, “A suspended laser interferometer for determining the Newtonian constant of gravitation,” IEEE Trans. Instrum. Meas. 50, 598–600 (2001).
[CrossRef]

E. D. Black, “An introduction to Pound-Drever-Hall laser frequency stabilization,” Am. J. Phys. 69, 79–87 (2001).
[CrossRef]

J. Lawall, J. M. Pedulla, Y. L. Coq, “Ultrastable laser array at 633 nm for real-time dimensional metrology,” Rev. Sci. Instrum. 72, 2879–2888 (2001).
[CrossRef]

1999

W.-T. Ni, D.-K. Liu, T.-T. Liu, H.-H. Mei, S. Shi Pan, C.-P. Pang, H.-C. Yeh, “The application of laser metrology and resonant optical cavity techniques to the measurement of G,” Meas. Sci. Technol. 10, 495–498 (1999).
[CrossRef]

1996

T. J. Dunn, T.-M. Lee, K. Jain, “Absolute distance measurement interferometry for alignment systems for advanced lithography tools,” J. Vac. Sci. Technol. B 14, 3960–3963 (1996).
[CrossRef]

1993

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

1983

R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. B31, 97–105 (1983).
[CrossRef]

1973

R. D. Deslattes, A. Henins, “X-ray to visible wavelength ratios,” Phys. Rev. Lett. 31, 972–975 (1973).
[CrossRef]

1971

Z. Bay, “The use of microwave modulation of lasers for length measurements,” Natl. Bur. Stand. (U.S.) Spec. Publ. 343, 59–62 (1971).

1966

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Bay, Z.

Z. Bay, “The use of microwave modulation of lasers for length measurements,” Natl. Bur. Stand. (U.S.) Spec. Publ. 343, 59–62 (1971).

Black, E. D.

E. D. Black, “An introduction to Pound-Drever-Hall laser frequency stabilization,” Am. J. Phys. 69, 79–87 (2001).
[CrossRef]

Bobroff, N.

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

Coq, Y. L.

J. Lawall, J. M. Pedulla, Y. L. Coq, “Ultrastable laser array at 633 nm for real-time dimensional metrology,” Rev. Sci. Instrum. 72, 2879–2888 (2001).
[CrossRef]

Deslattes, R. D.

R. D. Deslattes, A. Henins, “X-ray to visible wavelength ratios,” Phys. Rev. Lett. 31, 972–975 (1973).
[CrossRef]

Drever, R.

R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. B31, 97–105 (1983).
[CrossRef]

Dunn, T. J.

T. J. Dunn, T.-M. Lee, K. Jain, “Absolute distance measurement interferometry for alignment systems for advanced lithography tools,” J. Vac. Sci. Technol. B 14, 3960–3963 (1996).
[CrossRef]

Faller, J. E.

H. V. Parks, J. E. Faller, D. S. Robertson, “A suspended laser interferometer for determining the Newtonian constant of gravitation,” IEEE Trans. Instrum. Meas. 50, 598–600 (2001).
[CrossRef]

Ford, G.

R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. B31, 97–105 (1983).
[CrossRef]

Fu, J.

L. Howard, J. Stone, J. Fu, “Real-time displacement measurements with a Fabry-Perot cavity and a diode laser,” Precis. Eng. 25, 321–335 (2001).
[CrossRef]

Hall, J.

R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. B31, 97–105 (1983).
[CrossRef]

Henins, A.

R. D. Deslattes, A. Henins, “X-ray to visible wavelength ratios,” Phys. Rev. Lett. 31, 972–975 (1973).
[CrossRef]

Hough, J.

R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. B31, 97–105 (1983).
[CrossRef]

Howard, L.

L. Howard, J. Stone, J. Fu, “Real-time displacement measurements with a Fabry-Perot cavity and a diode laser,” Precis. Eng. 25, 321–335 (2001).
[CrossRef]

Jain, K.

T. J. Dunn, T.-M. Lee, K. Jain, “Absolute distance measurement interferometry for alignment systems for advanced lithography tools,” J. Vac. Sci. Technol. B 14, 3960–3963 (1996).
[CrossRef]

Kessler, E.

J. Lawall, E. Kessler, “Design and evaluation of a simple ultralow vibration vacuum environment,” Rev. Sci. Instrum. 73, 209–215 (2002).
[CrossRef]

Kogelnik, H.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Kowalski, F.

R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. B31, 97–105 (1983).
[CrossRef]

Lawall, J.

J. Lawall, “Interferometry for accurate displacement metrology,” Opt. Photonics News 15, 40–46 (2004).
[CrossRef]

J. Lawall, E. Kessler, “Design and evaluation of a simple ultralow vibration vacuum environment,” Rev. Sci. Instrum. 73, 209–215 (2002).
[CrossRef]

J. Lawall, J. M. Pedulla, Y. L. Coq, “Ultrastable laser array at 633 nm for real-time dimensional metrology,” Rev. Sci. Instrum. 72, 2879–2888 (2001).
[CrossRef]

Lee, T.-M.

T. J. Dunn, T.-M. Lee, K. Jain, “Absolute distance measurement interferometry for alignment systems for advanced lithography tools,” J. Vac. Sci. Technol. B 14, 3960–3963 (1996).
[CrossRef]

Li, T.

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Liu, D.-K.

W.-T. Ni, D.-K. Liu, T.-T. Liu, H.-H. Mei, S. Shi Pan, C.-P. Pang, H.-C. Yeh, “The application of laser metrology and resonant optical cavity techniques to the measurement of G,” Meas. Sci. Technol. 10, 495–498 (1999).
[CrossRef]

Liu, T.-T.

W.-T. Ni, D.-K. Liu, T.-T. Liu, H.-H. Mei, S. Shi Pan, C.-P. Pang, H.-C. Yeh, “The application of laser metrology and resonant optical cavity techniques to the measurement of G,” Meas. Sci. Technol. 10, 495–498 (1999).
[CrossRef]

Macleod, H. A.

H. A. Macleod, Thin-Film Optical Filters (Institute of Physics, 2001).
[CrossRef]

Mei, H.-H.

W.-T. Ni, D.-K. Liu, T.-T. Liu, H.-H. Mei, S. Shi Pan, C.-P. Pang, H.-C. Yeh, “The application of laser metrology and resonant optical cavity techniques to the measurement of G,” Meas. Sci. Technol. 10, 495–498 (1999).
[CrossRef]

Munley, A.

R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. B31, 97–105 (1983).
[CrossRef]

Ni, W.-T.

W.-T. Ni, D.-K. Liu, T.-T. Liu, H.-H. Mei, S. Shi Pan, C.-P. Pang, H.-C. Yeh, “The application of laser metrology and resonant optical cavity techniques to the measurement of G,” Meas. Sci. Technol. 10, 495–498 (1999).
[CrossRef]

Pang, C.-P.

W.-T. Ni, D.-K. Liu, T.-T. Liu, H.-H. Mei, S. Shi Pan, C.-P. Pang, H.-C. Yeh, “The application of laser metrology and resonant optical cavity techniques to the measurement of G,” Meas. Sci. Technol. 10, 495–498 (1999).
[CrossRef]

Parks, H. V.

H. V. Parks, J. E. Faller, D. S. Robertson, “A suspended laser interferometer for determining the Newtonian constant of gravitation,” IEEE Trans. Instrum. Meas. 50, 598–600 (2001).
[CrossRef]

Pedulla, J. M.

J. Lawall, J. M. Pedulla, Y. L. Coq, “Ultrastable laser array at 633 nm for real-time dimensional metrology,” Rev. Sci. Instrum. 72, 2879–2888 (2001).
[CrossRef]

Robertson, D. S.

H. V. Parks, J. E. Faller, D. S. Robertson, “A suspended laser interferometer for determining the Newtonian constant of gravitation,” IEEE Trans. Instrum. Meas. 50, 598–600 (2001).
[CrossRef]

Shi Pan, S.

W.-T. Ni, D.-K. Liu, T.-T. Liu, H.-H. Mei, S. Shi Pan, C.-P. Pang, H.-C. Yeh, “The application of laser metrology and resonant optical cavity techniques to the measurement of G,” Meas. Sci. Technol. 10, 495–498 (1999).
[CrossRef]

Siegman, A.

A. Siegman, Lasers (University Science, 1986).

Stone, J.

L. Howard, J. Stone, J. Fu, “Real-time displacement measurements with a Fabry-Perot cavity and a diode laser,” Precis. Eng. 25, 321–335 (2001).
[CrossRef]

Ward, H.

R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. B31, 97–105 (1983).
[CrossRef]

Yeh, H.-C.

W.-T. Ni, D.-K. Liu, T.-T. Liu, H.-H. Mei, S. Shi Pan, C.-P. Pang, H.-C. Yeh, “The application of laser metrology and resonant optical cavity techniques to the measurement of G,” Meas. Sci. Technol. 10, 495–498 (1999).
[CrossRef]

Am. J. Phys.

E. D. Black, “An introduction to Pound-Drever-Hall laser frequency stabilization,” Am. J. Phys. 69, 79–87 (2001).
[CrossRef]

Appl. Phys. B: Photophys. Laser Chem.

R. Drever, J. Hall, F. Kowalski, J. Hough, G. Ford, A. Munley, H. Ward, “Laser phase and frequency stabilization using an optical resonator,” Appl. Phys. B: Photophys. Laser Chem. B31, 97–105 (1983).
[CrossRef]

IEEE Trans. Instrum. Meas.

H. V. Parks, J. E. Faller, D. S. Robertson, “A suspended laser interferometer for determining the Newtonian constant of gravitation,” IEEE Trans. Instrum. Meas. 50, 598–600 (2001).
[CrossRef]

J. Vac. Sci. Technol. B

T. J. Dunn, T.-M. Lee, K. Jain, “Absolute distance measurement interferometry for alignment systems for advanced lithography tools,” J. Vac. Sci. Technol. B 14, 3960–3963 (1996).
[CrossRef]

Meas. Sci. Technol.

W.-T. Ni, D.-K. Liu, T.-T. Liu, H.-H. Mei, S. Shi Pan, C.-P. Pang, H.-C. Yeh, “The application of laser metrology and resonant optical cavity techniques to the measurement of G,” Meas. Sci. Technol. 10, 495–498 (1999).
[CrossRef]

N. Bobroff, “Recent advances in displacement measuring interferometry,” Meas. Sci. Technol. 4, 907–926 (1993).
[CrossRef]

Natl. Bur. Stand. (U.S.) Spec. Publ.

Z. Bay, “The use of microwave modulation of lasers for length measurements,” Natl. Bur. Stand. (U.S.) Spec. Publ. 343, 59–62 (1971).

Opt. Photonics News

J. Lawall, “Interferometry for accurate displacement metrology,” Opt. Photonics News 15, 40–46 (2004).
[CrossRef]

Phys. Rev. Lett.

R. D. Deslattes, A. Henins, “X-ray to visible wavelength ratios,” Phys. Rev. Lett. 31, 972–975 (1973).
[CrossRef]

Precis. Eng.

L. Howard, J. Stone, J. Fu, “Real-time displacement measurements with a Fabry-Perot cavity and a diode laser,” Precis. Eng. 25, 321–335 (2001).
[CrossRef]

Proc. IEEE

H. Kogelnik, T. Li, “Laser beams and resonators,” Proc. IEEE 54, 1312–1329 (1966).
[CrossRef]

Rev. Sci. Instrum.

J. Lawall, E. Kessler, “Design and evaluation of a simple ultralow vibration vacuum environment,” Rev. Sci. Instrum. 73, 209–215 (2002).
[CrossRef]

J. Lawall, J. M. Pedulla, Y. L. Coq, “Ultrastable laser array at 633 nm for real-time dimensional metrology,” Rev. Sci. Instrum. 72, 2879–2888 (2001).
[CrossRef]

Other

JDS Uniphase Corporation, Model 1007P, http://www.jdsu.com. Certain commercial equipment, instru- ments, or materials are identified in this paper to specify the experimental procedure adequately. Such identification is not intended to imply recommendation or endorsement by the National Institute of Standards and Technology, nor is it intended to imply that the materials or equipment identified are necessarily the best available for the purpose.

Gooch and Housego, Model FS040-2E, http://www. goochandhousego.com.

Isomet, Model 1250C, http://www.isomet.com.

Conoptics Inc., Model 380/4, http://www.conoptics.com.

Perkin-Elmer Inc., Model 7280, http://www.signalrecovery.com.

New Focus Inc., Model 8883, http://www.newfocus.com.

A. Siegman, Lasers (University Science, 1986).

Burleigh Corporation, Model TSE-150V, http://www.exfo.com/en/burleigh.asp.

H. A. Macleod, Thin-Film Optical Filters (Institute of Physics, 2001).
[CrossRef]

The author is preparing a manuscript to be called “Accurate determination of radii of curvature using Fabry-Pérot interferometry.”

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Figures (9)

Fig. 1
Fig. 1

Frequency of the cavity mode labeled by N changes by d ν = 2 ν 2 ( N c ) d L when the cavity length changes by d L . The frequency difference between modes N and N 1 changes by d ν N . N is the number of half-wavelengths contained in the cavity, and varies from approximately 655,000 to 731,000 in this work.

Fig. 2
Fig. 2

Layout of laser system. B.E., beam expander; OI, (Faraday) optical isolator; VCO, voltage controlled oscillator; AOM, acousto-optic modulator; EOM, electro-optic modulator; BS, beam splitter; PBS, polarizing beam splitter; SMPM, single-mode polarization maintaining.

Fig. 3
Fig. 3

Scanning Fabry–Perot cavity, optical elements for mode matching light into the cavity, and detectors. All equipment is built on a vibration-isolated platform in vacuum.

Fig. 4
Fig. 4

Normalized transmission (dashed curve) and reflection (solid curve) as the laser is swept with the AOM. In the reflected case the PM is on and sidebands are visible. The FWHM in transmission is 370 kHz , corresponding to a finesse of 1960.

Fig. 5
Fig. 5

PDH error signals.

Fig. 6
Fig. 6

Audio-frequency spectrum of blue channel error signal.

Fig. 7
Fig. 7

Spectrum of beat between red and blue transmitted light. Frequency span 1 kHz ; sidebands at 380 Hz reflect mechanical oscillations of the cavity. FWHM of the central peak is 65 Hz . Inset: Similar, frequency span 6.2 kHz .

Fig. 8
Fig. 8

(a) Free spectral range measured by counting VCO frequencies; σ = 16 Hz . Lower trace: Counters driven directly from synthesizers for 10 s ; σ = 9 Hz . (b) Drift of cavity resonance; residuals from a linear fit have a standard deviation of 12.2 kHz . (c) Effective order number N eff = ν N Δ ν ; here σ = 0.015 .

Fig. 9
Fig. 9

Top: Effective order number N eff for 16 consecutive data sets in which the cavity length is changed with a piezoelectric transducer in increments of approximately λ 2 . N eff is not an integer but differences between distinct values of N eff are very nearly integral. Bottom: Departure of measured order number change N ( n ) N ( 1 ) from an integer as the cavity length is swept. Four separate data sets are shown taken at various times, each separated by approximately one week. The rms value for all data is 0.067.

Equations (55)

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

ω 0 4 = λ 2 π 2 ( R L ) ( L ) ,
ω 1 4 = λ 2 π 2 R 2 L R L .
ϕ round trip = 2 [ 2 π L λ Φ ( L ) ] + ϕ 1 ( ν ) + ϕ 2 ( ν ) .
Φ ( L ) = tan 1 L z 0 ,
z 0 = π ω 0 2 λ ,
Φ ( L ) = 1 2 cos 1 [ 1 2 L R ] ,
= sin 1 L R .
ϕ R ( ν ) = ϕ 1 ( ν ) + ϕ 2 ( ν ) ,
ν N = c 2 L [ N + 1 π Φ ( L ) 1 2 π ϕ R ( ν N ) ] .
ϕ QWS = ϕ 0 + α ( ν ν 0 ) ,
ν N = c 2 L [ N + 1 2 π ( 2 Φ ( L ) ϕ 0 ) 1 2 π α ( ν N ν 0 ) ] ,
ν N = c 2 L 1 1 + α 2 π c 2 L [ N + 1 2 π ( 2 Φ ( L ) ϕ 0 + α ν 0 ) ] .
Δ ν = c 2 L ( 1 + α 2 π c 2 L ) 1 .
N eff = ν N Δ ν
= N + 1 2 π [ 2 Φ ( L ) ϕ 0 + α ν 0 ] .
L = c 2 [ 1 Δ ν α 2 π ] .
L ( 2 ) L ( 1 ) = c 2 [ 1 Δ ν ( 2 ) 1 Δ ν ( 1 ) ] .
L rf c 2 Δ ν ,
L = c 2 ν N [ N + 1 2 π ( 2 Φ ( L rf ) ϕ 0 + α ν 0 ) ] α c 4 π .
L = c 2 ν N N eff α c 4 π .
L ( 2 ) L ( 1 ) = c 2 [ N eff ( 2 ) ν N ( 2 ) N eff ( 1 ) ν N ( 1 ) ] .
L ( 2 ) L ( 1 ) = c 2 [ N eff ( 2 ) ν N ( 2 ) N eff ( 2 ) ν N ( 1 ) + N eff ( 2 ) ν N ( 1 ) N eff ( 1 ) ν N ( 1 ) ] .
L ( 2 ) L ( 1 ) = Δ L OC + Δ L IO ,
Δ L OC = c 2 ν N ( 1 ) [ N eff ( 2 ) N eff ( 1 ) ] ,
Δ L IO = c 2 [ 1 ν N ( 2 ) 1 ν N ( 1 ) ] N eff ( 2 ) = c 2 [ 1 ν N ( 2 ) 1 ν N ( 1 ) ] ν N ( 2 ) Δ ν ( 2 ) = c 2 ν N ( 1 ) ν N ( 1 ) ν N ( 2 ) Δ ν ( 2 ) .
N eff ( 2 ) N eff ( 1 ) = N ( 2 ) N ( 1 ) + 1 π [ Φ ( L rf ( 2 ) ) Φ ( L rf ( 1 ) ) ] .
N ( 2 ) N ( 1 ) = ν N ( 2 ) Δ ν ( 2 ) ν N ( 1 ) Δ ν ( 1 ) 1 π [ Φ ( L rf ( 2 ) ) Φ ( L rf ( 1 ) ) ] ,
Δ L OC = c 2 ν N ( 1 ) { N ( 21 ) + 1 π [ Φ ( L rf ( 2 ) ) Φ ( L rf ( 1 ) ) ] } .
Δ L = c 2 ν N ( 1 ) [ N ( 21 ) + 1 π { Φ ( L rf ( 2 ) ) Φ ( L rf ( 1 ) ) } + ν N ( 1 ) ν N ( 2 ) Δ ν ( 2 ) ] ,
δ N eff ν N Δ ν 2 δ Δ ν = N eff Δ ν δ Δ ν .
δ ( N ( 2 ) N ( 1 ) ) ( [ N eff ( 1 ) Δ ν ( 1 ) ] 2 + [ N eff ( 2 ) Δ ν ( 2 ) ] 2 ) 1 2 δ Δ ν .
665 , 000 < N eff < 736 , 000 ,
644 MHz < Δ ν < 723 MHz .
δ ( N ( 2 ) N ( 1 ) ) 2 N ¯ eff Δ ν ¯ δ Δ ν ,
δ Δ ν = 1 2 Δ ν ¯ N ¯ eff δ ( N ( 2 ) N ( 1 ) ) .
δ ( Δ L ) = c 2 δ Δ ν ( Δ ν ¯ ) 2 .
δ Δ L IO 2 λ 2 1 Δ ν ( 2 ) δ ν N ( 1 , 2 ) .
δ Δ L IO = 8.3 × 10 12 m .
Δ L R λ 8 π Δ L L ¯ 1 2 ( R L ¯ ) 3 2 ,
L R < 2 × 10 9 .
Δ L OC ν N ( 1 ) Δ L OC ν N ( 1 ) .
δ Δ L OC Δ L max OC δ ν N ( 1 ) ν N ( 1 ) = 6.4 × 10 13 m .
Δ L IO Δ ν ( 2 ) = Δ L IO Δ ν ( 2 ) .
δ Δ L IO λ 2 δ Δ ν ( 2 ) Δ ν ( 2 ) = 2.3 × 10 14 m .
t cav ( ν ) = t 1 t 2 e Γ 4 e i π ν Δ ν 1 r 1 r 2 e Γ 2 e 2 π i ν Δ ν ,
T cav ( ν ) = 4 T 1 T 2 ( T 1 + T 2 + Γ ) 2 + 16 sin 2 ( π ν Δ ν ) ,
F = 2 π T 1 + T 2 + Γ .
r cav ( ν ) = r 1 r 2 e Γ 2 e 2 π i ν Δ ν 1 r 1 r 2 e Γ 2 e 2 π i ν Δ ν .
r cav ( N Δ ν ) 2 = ( T 1 + T 2 + Γ ) 2 ( T 1 + T 2 + Γ ) 2
= F 2 ( T 1 + T 2 + Γ ) 2 4 π 2 .
T 1 = T 2 + Γ ,
ϵ ( ν ) = Im { r cav ( ν ) r cav * ( ν + ν mod ) r cav * ( ν ) r cav ( ν ν mod ) } .
ϵ ( ν ) = 2 Im { r cav ( ν ) } .
ϵ ( 0 ) = 16 π T 1 Δ ν ( T 1 + T 2 + Γ ) 2
= 4 F 2 T 1 π Δ ν .

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