Abstract

The phenomenon of ringing in swept Fabry–Perot cavities, which leads to discrepancies with respect to usual Airy peaks, exhibits three different regimes, depending on the values of the finesse, the sweep frequency, and the free spectral range of the cavity. In particular, the intermediate case in which the Fabry–Perot cavity transmission essentially oscillates is shown theoretically and experimentally to provide a new simple method to measure the finesse of the cavity. For cavities with intermediate finesses (F10 000) this method is experimentally shown to have a precision of the order of 1%.

© 1997 Optical Society of America

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  1. J. Brossel, “Multiple-beam localized fringes: Part I. Intensity distribution and localization,” Proc. Phys. Soc. London 59, 224–234 (1947).
    [CrossRef]
  2. M. Born and E. Wolf, Principle of Optics, 3rd ed. (Pergamon, Oxford, 1965), Chap. 7, pp. 351–360.
  3. J. Holden, “Multiple-beam interferometry: intensity distribution in the reflected system,” Proc. Phys. Soc. London, Sect. B 62, 405–417 (1949).
    [CrossRef]
  4. K. Kinosita, “Numerical evaluation of the intensity curve of a multiple-beam Fizeau fringe,” J. Phys. Soc. Jpn. 8, 219–225 (1953).
    [CrossRef]
  5. H. Boersch and G. Herziger, “Theoretical and experimental investigation of regenerative laser amplifiers and their applications,” IEEE J. Quantum Electron. QE-2, 549–552 (1966).
    [CrossRef]
  6. J. R. Greig and J. Cooper, “Rapid scanning with the Fabry–Perot étalon,” Appl. Opt. 7, 2166–2170 (1968).
    [CrossRef] [PubMed]
  7. J. H. Williamson and S. S. Medley, “On the interpretation of laser interferometer fringe patterns,” Can. J. Phys. 47, 515–519 (1969).
    [CrossRef]
  8. A. E. Dangor and S. J. Fielding, “The response of the Fabry–Perot interferometer to rapid changes in optical length,” J. Phys. D 3, 413–421 (1970).
    [CrossRef]
  9. Z. K. Ioannidis, P. M. Radmore, and I. P. Giles, “Dynamic response of an all-fiber ring resonator,” Opt. Lett. 13, 422–424 (1988).
    [CrossRef] [PubMed]
  10. Z. Li, R. G. T. Bennett, and G. E. Stedman, “Swept-frequency induced optical cavity ringing,” Opt. Commun. 86, 51–57 (1991).
    [CrossRef]
  11. Z. Li, G. E. Stedman, and H. R. Bilger, “Asymmetric response profile of a scanning Fabry–Perot interferometer,” Opt. Commun. 100, 240–246 (1993).
    [CrossRef]
  12. S. Balle, I. C. M. Littler, K. Bergman, and F. V. Kowalski, “Frequency shifted feedback dye laser operating at a small shift frequency,” Opt. Commun. 102, 166–174 (1993).
    [CrossRef]
  13. K. Hsu and C. H. Miller, “Theory and measurements of speed-of-light effects in long cavity fiber Fabry–Perot scanning interferometers,” J. Lightwave Technol. 11, 1204–1208 (1993).
    [CrossRef]
  14. 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]
  15. G. Rempe, R. J. Thompson, H. J. Kimble, and R. Lalezari, “Measurement of ultralow losses in an optical interferometer,” Opt. Lett. 17, 363–365 (1992).
    [CrossRef] [PubMed]
  16. D. Z. Anderson, J. C. Frish, and C. S. Masser, “Mirror reflectometer based on optical cavity decay time,” Appl. Opt. 23, 1238–1245 (1984).
    [CrossRef] [PubMed]
  17. A. Kastler, “Transmission d’une impulsion lumineuse par un interféromètre Fabry–Perot,” Nouv. Rev. Opt. 5, 133–139 (1974).
    [CrossRef]
  18. A. O. Keefe and D. A. G. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544–2551 (1988).
    [CrossRef]
  19. N. Uehara and 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]
  20. I. S. Gradshtein and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1979), Chap. 3, p. 307.
  21. D. Jacob, M. Vallet, F. Bretenaker, A. Le Floch, and M. Oger, “Supermirror phase anisotropy measurement,” Opt. Lett. 20, 671–673 (1995); D. Jacob, M. Vallet, F. Bretenaker, R. Le Naour, and M. Oger, “Small Faraday rotation measurement with a Fabry–Perot cavity,” Appl. Phys. Lett. 66, 3546–3548 (1995).
    [CrossRef] [PubMed]

1995

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]

N. Uehara and 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]

1993

Z. Li, G. E. Stedman, and H. R. Bilger, “Asymmetric response profile of a scanning Fabry–Perot interferometer,” Opt. Commun. 100, 240–246 (1993).
[CrossRef]

S. Balle, I. C. M. Littler, K. Bergman, and F. V. Kowalski, “Frequency shifted feedback dye laser operating at a small shift frequency,” Opt. Commun. 102, 166–174 (1993).
[CrossRef]

K. Hsu and C. H. Miller, “Theory and measurements of speed-of-light effects in long cavity fiber Fabry–Perot scanning interferometers,” J. Lightwave Technol. 11, 1204–1208 (1993).
[CrossRef]

1992

1991

Z. Li, R. G. T. Bennett, and G. E. Stedman, “Swept-frequency induced optical cavity ringing,” Opt. Commun. 86, 51–57 (1991).
[CrossRef]

1988

A. O. Keefe and D. A. G. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544–2551 (1988).
[CrossRef]

Z. K. Ioannidis, P. M. Radmore, and I. P. Giles, “Dynamic response of an all-fiber ring resonator,” Opt. Lett. 13, 422–424 (1988).
[CrossRef] [PubMed]

1984

1974

A. Kastler, “Transmission d’une impulsion lumineuse par un interféromètre Fabry–Perot,” Nouv. Rev. Opt. 5, 133–139 (1974).
[CrossRef]

1970

A. E. Dangor and S. J. Fielding, “The response of the Fabry–Perot interferometer to rapid changes in optical length,” J. Phys. D 3, 413–421 (1970).
[CrossRef]

1969

J. H. Williamson and S. S. Medley, “On the interpretation of laser interferometer fringe patterns,” Can. J. Phys. 47, 515–519 (1969).
[CrossRef]

1968

1966

H. Boersch and G. Herziger, “Theoretical and experimental investigation of regenerative laser amplifiers and their applications,” IEEE J. Quantum Electron. QE-2, 549–552 (1966).
[CrossRef]

1953

K. Kinosita, “Numerical evaluation of the intensity curve of a multiple-beam Fizeau fringe,” J. Phys. Soc. Jpn. 8, 219–225 (1953).
[CrossRef]

1949

J. Holden, “Multiple-beam interferometry: intensity distribution in the reflected system,” Proc. Phys. Soc. London, Sect. B 62, 405–417 (1949).
[CrossRef]

1947

J. Brossel, “Multiple-beam localized fringes: Part I. Intensity distribution and localization,” Proc. Phys. Soc. London 59, 224–234 (1947).
[CrossRef]

An, K.

Anderson, D. Z.

Balle, S.

S. Balle, I. C. M. Littler, K. Bergman, and F. V. Kowalski, “Frequency shifted feedback dye laser operating at a small shift frequency,” Opt. Commun. 102, 166–174 (1993).
[CrossRef]

Bennett, R. G. T.

Z. Li, R. G. T. Bennett, and G. E. Stedman, “Swept-frequency induced optical cavity ringing,” Opt. Commun. 86, 51–57 (1991).
[CrossRef]

Bergman, K.

S. Balle, I. C. M. Littler, K. Bergman, and F. V. Kowalski, “Frequency shifted feedback dye laser operating at a small shift frequency,” Opt. Commun. 102, 166–174 (1993).
[CrossRef]

Bilger, H. R.

Z. Li, G. E. Stedman, and H. R. Bilger, “Asymmetric response profile of a scanning Fabry–Perot interferometer,” Opt. Commun. 100, 240–246 (1993).
[CrossRef]

Boersch, H.

H. Boersch and G. Herziger, “Theoretical and experimental investigation of regenerative laser amplifiers and their applications,” IEEE J. Quantum Electron. QE-2, 549–552 (1966).
[CrossRef]

Brossel, J.

J. Brossel, “Multiple-beam localized fringes: Part I. Intensity distribution and localization,” Proc. Phys. Soc. London 59, 224–234 (1947).
[CrossRef]

Cooper, J.

Dangor, A. E.

A. E. Dangor and S. J. Fielding, “The response of the Fabry–Perot interferometer to rapid changes in optical length,” J. Phys. D 3, 413–421 (1970).
[CrossRef]

Dasari, R. R.

Deacon, D. A. G.

A. O. Keefe and D. A. G. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544–2551 (1988).
[CrossRef]

Feld, M. S.

Fielding, S. J.

A. E. Dangor and S. J. Fielding, “The response of the Fabry–Perot interferometer to rapid changes in optical length,” J. Phys. D 3, 413–421 (1970).
[CrossRef]

Frish, J. C.

Giles, I. P.

Greig, J. R.

Herziger, G.

H. Boersch and G. Herziger, “Theoretical and experimental investigation of regenerative laser amplifiers and their applications,” IEEE J. Quantum Electron. QE-2, 549–552 (1966).
[CrossRef]

Holden, J.

J. Holden, “Multiple-beam interferometry: intensity distribution in the reflected system,” Proc. Phys. Soc. London, Sect. B 62, 405–417 (1949).
[CrossRef]

Hsu, K.

K. Hsu and C. H. Miller, “Theory and measurements of speed-of-light effects in long cavity fiber Fabry–Perot scanning interferometers,” J. Lightwave Technol. 11, 1204–1208 (1993).
[CrossRef]

Ioannidis, Z. K.

Kastler, A.

A. Kastler, “Transmission d’une impulsion lumineuse par un interféromètre Fabry–Perot,” Nouv. Rev. Opt. 5, 133–139 (1974).
[CrossRef]

Keefe, A. O.

A. O. Keefe and D. A. G. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544–2551 (1988).
[CrossRef]

Kimble, H. J.

Kinosita, K.

K. Kinosita, “Numerical evaluation of the intensity curve of a multiple-beam Fizeau fringe,” J. Phys. Soc. Jpn. 8, 219–225 (1953).
[CrossRef]

Kowalski, F. V.

S. Balle, I. C. M. Littler, K. Bergman, and F. V. Kowalski, “Frequency shifted feedback dye laser operating at a small shift frequency,” Opt. Commun. 102, 166–174 (1993).
[CrossRef]

Lalezari, R.

Li, Z.

Z. Li, G. E. Stedman, and H. R. Bilger, “Asymmetric response profile of a scanning Fabry–Perot interferometer,” Opt. Commun. 100, 240–246 (1993).
[CrossRef]

Z. Li, R. G. T. Bennett, and G. E. Stedman, “Swept-frequency induced optical cavity ringing,” Opt. Commun. 86, 51–57 (1991).
[CrossRef]

Littler, I. C. M.

S. Balle, I. C. M. Littler, K. Bergman, and F. V. Kowalski, “Frequency shifted feedback dye laser operating at a small shift frequency,” Opt. Commun. 102, 166–174 (1993).
[CrossRef]

Masser, C. S.

Medley, S. S.

J. H. Williamson and S. S. Medley, “On the interpretation of laser interferometer fringe patterns,” Can. J. Phys. 47, 515–519 (1969).
[CrossRef]

Miller, C. H.

K. Hsu and C. H. Miller, “Theory and measurements of speed-of-light effects in long cavity fiber Fabry–Perot scanning interferometers,” J. Lightwave Technol. 11, 1204–1208 (1993).
[CrossRef]

Radmore, P. M.

Rempe, G.

Stedman, G. E.

Z. Li, G. E. Stedman, and H. R. Bilger, “Asymmetric response profile of a scanning Fabry–Perot interferometer,” Opt. Commun. 100, 240–246 (1993).
[CrossRef]

Z. Li, R. G. T. Bennett, and G. E. Stedman, “Swept-frequency induced optical cavity ringing,” Opt. Commun. 86, 51–57 (1991).
[CrossRef]

Thompson, R. J.

Ueda, K.

N. Uehara and 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 and 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]

Williamson, J. H.

J. H. Williamson and S. S. Medley, “On the interpretation of laser interferometer fringe patterns,” Can. J. Phys. 47, 515–519 (1969).
[CrossRef]

Yang, C.

Appl. Opt.

Appl. Phys. B

N. Uehara and 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]

Can. J. Phys.

J. H. Williamson and S. S. Medley, “On the interpretation of laser interferometer fringe patterns,” Can. J. Phys. 47, 515–519 (1969).
[CrossRef]

IEEE J. Quantum Electron.

H. Boersch and G. Herziger, “Theoretical and experimental investigation of regenerative laser amplifiers and their applications,” IEEE J. Quantum Electron. QE-2, 549–552 (1966).
[CrossRef]

J. Lightwave Technol.

K. Hsu and C. H. Miller, “Theory and measurements of speed-of-light effects in long cavity fiber Fabry–Perot scanning interferometers,” J. Lightwave Technol. 11, 1204–1208 (1993).
[CrossRef]

J. Phys. D

A. E. Dangor and S. J. Fielding, “The response of the Fabry–Perot interferometer to rapid changes in optical length,” J. Phys. D 3, 413–421 (1970).
[CrossRef]

J. Phys. Soc. Jpn.

K. Kinosita, “Numerical evaluation of the intensity curve of a multiple-beam Fizeau fringe,” J. Phys. Soc. Jpn. 8, 219–225 (1953).
[CrossRef]

Nouv. Rev. Opt.

A. Kastler, “Transmission d’une impulsion lumineuse par un interféromètre Fabry–Perot,” Nouv. Rev. Opt. 5, 133–139 (1974).
[CrossRef]

Opt. Commun.

Z. Li, R. G. T. Bennett, and G. E. Stedman, “Swept-frequency induced optical cavity ringing,” Opt. Commun. 86, 51–57 (1991).
[CrossRef]

Z. Li, G. E. Stedman, and H. R. Bilger, “Asymmetric response profile of a scanning Fabry–Perot interferometer,” Opt. Commun. 100, 240–246 (1993).
[CrossRef]

S. Balle, I. C. M. Littler, K. Bergman, and F. V. Kowalski, “Frequency shifted feedback dye laser operating at a small shift frequency,” Opt. Commun. 102, 166–174 (1993).
[CrossRef]

Opt. Lett.

Proc. Phys. Soc. London

J. Brossel, “Multiple-beam localized fringes: Part I. Intensity distribution and localization,” Proc. Phys. Soc. London 59, 224–234 (1947).
[CrossRef]

Proc. Phys. Soc. London, Sect. B

J. Holden, “Multiple-beam interferometry: intensity distribution in the reflected system,” Proc. Phys. Soc. London, Sect. B 62, 405–417 (1949).
[CrossRef]

Rev. Sci. Instrum.

A. O. Keefe and D. A. G. Deacon, “Cavity ring-down optical spectrometer for absorption measurements using pulsed laser sources,” Rev. Sci. Instrum. 59, 2544–2551 (1988).
[CrossRef]

Other

I. S. Gradshtein and I. M. Ryzhik, Table of Integrals, Series, and Products (Academic, London, 1979), Chap. 3, p. 307.

D. Jacob, M. Vallet, F. Bretenaker, A. Le Floch, and M. Oger, “Supermirror phase anisotropy measurement,” Opt. Lett. 20, 671–673 (1995); D. Jacob, M. Vallet, F. Bretenaker, R. Le Naour, and M. Oger, “Small Faraday rotation measurement with a Fabry–Perot cavity,” Appl. Phys. Lett. 66, 3546–3548 (1995).
[CrossRef] [PubMed]

M. Born and E. Wolf, Principle of Optics, 3rd ed. (Pergamon, Oxford, 1965), Chap. 7, pp. 351–360.

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Figures (6)

Fig. 1
Fig. 1

Fabry–Perot cavity of variable length d(t)=d0+vt excited by an incident beam at wavelength λ.

Fig. 2
Fig. 2

Time evolution of the intensity transmitted by the FP cavity for Ii=1, d0=0.32 m, F=5000 (cavity decay time τ=1.70 µs), T=1-R, and d00[λ/2]: (a) Ω/ΩF=0.1/F2 (v=0.593 µm/s),(b) Ω/ΩF=2/F2 (v=11.86 µm/s), (c) Ω/ΩF=30/F2 (v=177.98 µm/s), (d) Ω/ΩF=100/F2 (v=593.25 µm/s), and (e) Ω/ΩF=1000/F2 (v=5932.5 µm/s). Notice that the horizontal and the vertical scales are different for the different profiles. The instant 2τ used in Fig. 3 is indicated on the horizontal axes of Figs. 2(b)2(e).

Fig. 3
Fig. 3

Contour plot of the function |erfc(Λ)|2 with Λ varying in the complex plane. The different contours correspond to |erfc(Λ)|2=1, 3, 4, 10, 102, 104, and 108, as indicated. The four oblique segments labeled a, b, c, and d correspond to the variation of Λ(t) for 0t2τ in the conditions of Figs. 2(a), 2(b), 2(c), and 2(d), respectively.

Fig. 4
Fig. 4

Solid line indicates theoretical evolution of πcΔt/Fd0 versus I1/I2 obtained from Eq. (22) (see text). Δt is the time interval between the two first maxima of the transmission versus time profile, and I1/I2 is the ratio of these maxima.

Fig. 5
Fig. 5

Experimental [(a)–(c)] and theoretical [(d)–(f)] evolution of the FP-cavity transmission versus time. The experimental profiles are recorded with (a) Ω/ΩF=2.1×10-7, (b) Ω/ΩF=3.25×10-7, and (c) Ω/ΩF=10.0×10-7. The corresponding theoretical profiles [(d)–(f)] are computed from Eq. (6) with F=8900 and with (d) Ω/ΩF=2.25×10-7, (e) Ω/ΩF=4.0×10-7, and (f) Ω/ΩF=8.5×10-7. For each experimental profile, one measures the ratio I1/I2 of the intensities of the two first maxima and the time delay Δt between these maxima. The introduction of these values in Eq. ( 22) leads to the determination of F.

Fig. 6
Fig. 6

Filled circles indicate experimental evolution of πcΔt/d0 versus I1/I2 obtained from different evolutions of the transmission versus time. The solid line indicates the theoretical fit obtained with Eq. (22), leading to F=8900±90.

Equations (23)

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

d(t)=d0+vt,
At(t)=n=1An(t)=n=1AiTRn-1 exp[iϕn(t)],
ϕn(t)=(2n-1)ϕ1(t)-πn(n-1)ΩΩF,
It(t)=At(t)At(t)*,
F=πR1-R.
It(t)=Ii4T2R2ΩFΩexp-πF-2ΩFΩϕ1(t)F|erfc[Λ(t)]|2,
Λ(t)=1-i22ΩFπΩ1/2πF-1+i22ΩFπΩ1/2×2ϕ1(t)-πΩΩF1/2.
2ΩFΩϕ1(t)F=πFcv+tτ,
τ=Fd0πc=F2πΩF,
It(t)exp-tτ|erfc[Λ(t)]|2.
ϕnr(t)nr=0,
nr=12+ΩFπΩϕ1,
ϕrϕnr=ΩFπΩϕ12+πΩ4ΩF.
at(t)=AiT exp(iϕ1)1-R exp(2iϕ1).
arg[at(t)]=ϕ1+arctanR sin 2ϕ11-R cos 2ϕ1.
arg[at(t)]=ϕ1+π2.
ΩFπΩ[ϕ1(p)]2-ϕ1(p)+πΩ4ΩF-2p-12=0,
ϕ1(p)=πΩ2ΩF1+2ΩFΩ(4p+1)1/2.
Δt=1πΩ[ϕ1(1)-ϕ1(0)]=5-1(2ΩΩF)1/2.
ΩΩF=4π2(3-5)F230.16F2.
I1I2=expΔtτeΔtτ,
πcd0Δt=2πΩFΔtFeI1I2.
πcd0Δt=F2I1I2+2-e.

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