Abstract

A novel measurement principle for fiber-optic sensing is presented. Use of a cavity-ring-down scheme enables measurements of minute optical losses in high-finesse fiber-optic cavities. The loss may be induced by evanescent-field absorption, fiber bending, fiber degradation, Bragg gratings, or any other effect that might change the fiber transmission or cavity reflector properties. The principle is proved to be rather insensitive to ambient perturbations such as temperature changes. A high-sensitivity measurement of loss due to bending is presented as a proof-of-principle. With a cavity finesse of 627 a sensitivity for induced loss of 108 ppm (4.68 × 10-4 dB) is achieved. Preliminary measurements of evanescent-field absorption are also discussed.

© 2002 Optical Society of America

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2002 (1)

T. von Lerber, M. W. Sigrist, “Time constant extraction from noisy cavity-ring-down signals,” Chem. Phys. Lett. 353, 131–137 (2002).
[CrossRef]

2001 (1)

G. Stewart, K. Atherton, H. Yu, B. Culshaw, “An investigation of an optical fiber amplifier loop for intracavity and ring-down-cavity loss measurements,” Meas. Sci. Technol. 12, 843–849 (2001).
[CrossRef]

2000 (3)

M. D. Levenson, B. A. Paldus, T. G. Spence, C. C. Harb, R. N. Zare, M. J. Lawrence, R. L. Byer, “Frequency-switched heterodyne cavity-ring-down spectroscopy,” Opt. Lett. 25, 920–922 (2000).
[CrossRef]

G. Berden, R. Peeters, G. Meijer, “Cavity-ring-down spectroscopy: experimental schemes and applications,” Int. Rev. Phys. Chem. 19, 565–607 (2000).
[CrossRef]

Y. Zhu, E. Simova, P. Berini, C. Grover, “A comparison of wavelength-dependent polarization-dependent loss measurements in fiber gratings,” IEEE Trans. Instrum. Meas. 49, 1231–1239 (2000).
[CrossRef]

1999 (1)

1998 (4)

K. J. Schulz, W. R. Simpson, “Frequency-matched cavity-ring-down spectroscopy,” Chem. Phys. Lett. 297, 523–529 (1998).
[CrossRef]

E. J. Friebele, M. A. Putnam, H. J. Patrick, A. D. Kersey, A. S. Greenblatt, G. P. Ruthven, H. Krim, K. S. Gottschalck, “Ultrahigh-sensitivity fiber-optic strain and temperature sensor,” Opt. Lett. 23, 222–224 (1998).
[CrossRef]

R. Engeln, G. Berden, R. Peeters, G. Meijer, “Cavity enhanced absorption and cavity enhanced magnetic rotation spectroscopy,” Rev. Sci. Instrum. 69, 3763–3769 (1998).
[CrossRef]

M. D. Wheeler, S. M. Newman, A. J. Orr-Ewing, M. N. R. Ashfold, “Cavity-ring-down spectroscopy,” J. Chem. Soc. Faraday Trans. 94, 337–351 (1998).
[CrossRef]

1997 (4)

J. J. Scherer, J. B. Paul, A. O’Keefe, R. J. Saykally, “Cavity-ring-down laser absorption spectroscopy: history, development, and application to pulsed molecular beams,” Chem. Rev. 97, 25–51 (1997).
[CrossRef] [PubMed]

D. Romanini, A. A. Kachanov, N. Sadeghi, F. Stoeckel, “Continuous-wave cavity-ring-down spectroscopy,” Chem. Phys. Lett. 264, 316–322 (1997).
[CrossRef]

B. A. Paldus, J. S. Harris, J. Martin, J. Xie, R. N. Zare, “Laser diode cavity-ring-down spectroscopy using acousto-optic modulator stabilization,” J. Appl. Phys. 82, 3199–3204 (1997).
[CrossRef]

R. Engeln, G. Berden, E. van den Berg, G. Meijer, “Polarization-dependent cavity-ring-down spectroscopy,” J. Chem. Phys. 107, 4458–4467 (1997).
[CrossRef]

1996 (4)

K. K. Lehmann, D. Romanini, “The superposition principle and cavity-ring-down spectroscopy,” J. Chem. Phys. 105, 10263–10277 (1996).
[CrossRef]

J. T. Hodges, J. P. Looney, R. D. van Zee, “Response of a ring-down cavity to an arbitrary excitation,” J. Chem. Phys. 105, 10278–10288 (1996).
[CrossRef]

R. Engeln, G. von Helden, G. Berden, G. Meijer, “Phase-shift cavity-ring-down absorption spectroscopy,” Chem. Phys. Lett. 262, 105–109 (1996).
[CrossRef]

A. D. Kersey, “A review of recent developments in fiber-optic sensor technology,” Opt. Fiber Technol.: Mater. Devices Syst. 2, 291–317 (1996).
[CrossRef]

1995 (2)

P. Zalicki, R. N. Zare, “Cavity-ring-down spectroscopy for quantitative absorption measurements,” J. Chem. Phys. 102, 2708–2717 (1995).
[CrossRef]

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]

1989 (1)

1988 (2)

P. R. Morkel, M. C. Farries, D. N. Payne, “Losses in fiber laser cavities,” Electron. Lett. 24, 92–93 (1988).
[CrossRef]

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

1984 (1)

1980 (1)

1979 (1)

W. Gamblin, H. Matsumura, C. Ragdale, “Curvature andmicrobending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
[CrossRef]

An, K.

Anderson, D. Z.

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]

D. Z. Anderson, “Reflectometer based on optical cavity decay time,” U.S. patent4,571,085 (18February1986).

Ashfold, M. N. R.

M. D. Wheeler, S. M. Newman, A. J. Orr-Ewing, M. N. R. Ashfold, “Cavity-ring-down spectroscopy,” J. Chem. Soc. Faraday Trans. 94, 337–351 (1998).
[CrossRef]

Atherton, K.

G. Stewart, K. Atherton, H. Yu, B. Culshaw, “An investigation of an optical fiber amplifier loop for intracavity and ring-down-cavity loss measurements,” Meas. Sci. Technol. 12, 843–849 (2001).
[CrossRef]

Benard, D. J.

Berden, G.

G. Berden, R. Peeters, G. Meijer, “Cavity-ring-down spectroscopy: experimental schemes and applications,” Int. Rev. Phys. Chem. 19, 565–607 (2000).
[CrossRef]

R. Engeln, G. Berden, R. Peeters, G. Meijer, “Cavity enhanced absorption and cavity enhanced magnetic rotation spectroscopy,” Rev. Sci. Instrum. 69, 3763–3769 (1998).
[CrossRef]

R. Engeln, G. Berden, E. van den Berg, G. Meijer, “Polarization-dependent cavity-ring-down spectroscopy,” J. Chem. Phys. 107, 4458–4467 (1997).
[CrossRef]

R. Engeln, G. von Helden, G. Berden, G. Meijer, “Phase-shift cavity-ring-down absorption spectroscopy,” Chem. Phys. Lett. 262, 105–109 (1996).
[CrossRef]

Berini, P.

Y. Zhu, E. Simova, P. Berini, C. Grover, “A comparison of wavelength-dependent polarization-dependent loss measurements in fiber gratings,” IEEE Trans. Instrum. Meas. 49, 1231–1239 (2000).
[CrossRef]

Byer, R. L.

Culshaw, B.

G. Stewart, K. Atherton, H. Yu, B. Culshaw, “An investigation of an optical fiber amplifier loop for intracavity and ring-down-cavity loss measurements,” Meas. Sci. Technol. 12, 843–849 (2001).
[CrossRef]

Dasari, R. R.

Deacon, D. A. G.

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

Engeln, R.

R. Engeln, G. Berden, R. Peeters, G. Meijer, “Cavity enhanced absorption and cavity enhanced magnetic rotation spectroscopy,” Rev. Sci. Instrum. 69, 3763–3769 (1998).
[CrossRef]

R. Engeln, G. Berden, E. van den Berg, G. Meijer, “Polarization-dependent cavity-ring-down spectroscopy,” J. Chem. Phys. 107, 4458–4467 (1997).
[CrossRef]

R. Engeln, G. von Helden, G. Berden, G. Meijer, “Phase-shift cavity-ring-down absorption spectroscopy,” Chem. Phys. Lett. 262, 105–109 (1996).
[CrossRef]

Falco, L.

G. Kotrotsios, L. Falco, J. P. Jeanneret, O. Parriaux, “Radio frequency phase detection for intensity modulated fiber sensors,” in Fiber Optic Sensors I, H. J. Arditti, L. B. Jeunhomme, eds., Proc. SPIE586, 99–103 (1985).
[CrossRef]

Falco, L. G.

L. G. Falco, O. M. Parriaux, “Optical fiber detection system using an intensity-modulating sensor,” U.S. patent4,887,901 (19December1989).

Farries, M. C.

P. R. Morkel, M. C. Farries, D. N. Payne, “Losses in fiber laser cavities,” Electron. Lett. 24, 92–93 (1988).
[CrossRef]

Feld, M. S.

Friebele, E. J.

Frisch, J. C.

Gamblin, W.

W. Gamblin, H. Matsumura, C. Ragdale, “Curvature andmicrobending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
[CrossRef]

Gottschalck, K. S.

Greenblatt, A. S.

Grover, C.

Y. Zhu, E. Simova, P. Berini, C. Grover, “A comparison of wavelength-dependent polarization-dependent loss measurements in fiber gratings,” IEEE Trans. Instrum. Meas. 49, 1231–1239 (2000).
[CrossRef]

Hahn, J. W.

Harb, C. C.

Harris, J. S.

B. A. Paldus, J. S. Harris, J. Martin, J. Xie, R. N. Zare, “Laser diode cavity-ring-down spectroscopy using acousto-optic modulator stabilization,” J. Appl. Phys. 82, 3199–3204 (1997).
[CrossRef]

Herbelin, J. M.

Hodges, J. T.

J. T. Hodges, J. P. Looney, R. D. van Zee, “Response of a ring-down cavity to an arbitrary excitation,” J. Chem. Phys. 105, 10278–10288 (1996).
[CrossRef]

Hodgson, N.

N. Hodgson, H. Weber, Optical Resonators: Fundamentals, Advanced Concepts and Applications (Springer-Verlag, London, 1997).

Jeanneret, J. P.

G. Kotrotsios, L. Falco, J. P. Jeanneret, O. Parriaux, “Radio frequency phase detection for intensity modulated fiber sensors,” in Fiber Optic Sensors I, H. J. Arditti, L. B. Jeunhomme, eds., Proc. SPIE586, 99–103 (1985).
[CrossRef]

Kachanov, A. A.

D. Romanini, A. A. Kachanov, N. Sadeghi, F. Stoeckel, “Continuous-wave cavity-ring-down spectroscopy,” Chem. Phys. Lett. 264, 316–322 (1997).
[CrossRef]

Kersey, A. D.

Kim, J. W.

Kotrotsios, G.

G. Kotrotsios, L. Falco, J. P. Jeanneret, O. Parriaux, “Radio frequency phase detection for intensity modulated fiber sensors,” in Fiber Optic Sensors I, H. J. Arditti, L. B. Jeunhomme, eds., Proc. SPIE586, 99–103 (1985).
[CrossRef]

Krim, H.

Kwok, M. A.

Lawrence, M. J.

Lee, H. W.

Lee, J. Y.

Lehmann, K. K.

K. K. Lehmann, D. Romanini, “The superposition principle and cavity-ring-down spectroscopy,” J. Chem. Phys. 105, 10263–10277 (1996).
[CrossRef]

Levenson, M. D.

Looney, J. P.

J. T. Hodges, J. P. Looney, R. D. van Zee, “Response of a ring-down cavity to an arbitrary excitation,” J. Chem. Phys. 105, 10278–10288 (1996).
[CrossRef]

Martin, J.

B. A. Paldus, J. S. Harris, J. Martin, J. Xie, R. N. Zare, “Laser diode cavity-ring-down spectroscopy using acousto-optic modulator stabilization,” J. Appl. Phys. 82, 3199–3204 (1997).
[CrossRef]

Masser, C. S.

Matsumura, H.

W. Gamblin, H. Matsumura, C. Ragdale, “Curvature andmicrobending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
[CrossRef]

McKay, J. A.

Meijer, G.

G. Berden, R. Peeters, G. Meijer, “Cavity-ring-down spectroscopy: experimental schemes and applications,” Int. Rev. Phys. Chem. 19, 565–607 (2000).
[CrossRef]

R. Engeln, G. Berden, R. Peeters, G. Meijer, “Cavity enhanced absorption and cavity enhanced magnetic rotation spectroscopy,” Rev. Sci. Instrum. 69, 3763–3769 (1998).
[CrossRef]

R. Engeln, G. Berden, E. van den Berg, G. Meijer, “Polarization-dependent cavity-ring-down spectroscopy,” J. Chem. Phys. 107, 4458–4467 (1997).
[CrossRef]

R. Engeln, G. von Helden, G. Berden, G. Meijer, “Phase-shift cavity-ring-down absorption spectroscopy,” Chem. Phys. Lett. 262, 105–109 (1996).
[CrossRef]

Morkel, P. R.

P. R. Morkel, M. C. Farries, D. N. Payne, “Losses in fiber laser cavities,” Electron. Lett. 24, 92–93 (1988).
[CrossRef]

Newman, S. M.

M. D. Wheeler, S. M. Newman, A. J. Orr-Ewing, M. N. R. Ashfold, “Cavity-ring-down spectroscopy,” J. Chem. Soc. Faraday Trans. 94, 337–351 (1998).
[CrossRef]

O’Keefe, A.

J. J. Scherer, J. B. Paul, A. O’Keefe, R. J. Saykally, “Cavity-ring-down laser absorption spectroscopy: history, development, and application to pulsed molecular beams,” Chem. Rev. 97, 25–51 (1997).
[CrossRef] [PubMed]

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

Orr-Ewing, A. J.

M. D. Wheeler, S. M. Newman, A. J. Orr-Ewing, M. N. R. Ashfold, “Cavity-ring-down spectroscopy,” J. Chem. Soc. Faraday Trans. 94, 337–351 (1998).
[CrossRef]

Paldus, B. A.

M. D. Levenson, B. A. Paldus, T. G. Spence, C. C. Harb, R. N. Zare, M. J. Lawrence, R. L. Byer, “Frequency-switched heterodyne cavity-ring-down spectroscopy,” Opt. Lett. 25, 920–922 (2000).
[CrossRef]

B. A. Paldus, J. S. Harris, J. Martin, J. Xie, R. N. Zare, “Laser diode cavity-ring-down spectroscopy using acousto-optic modulator stabilization,” J. Appl. Phys. 82, 3199–3204 (1997).
[CrossRef]

Parriaux, O.

G. Kotrotsios, L. Falco, J. P. Jeanneret, O. Parriaux, “Radio frequency phase detection for intensity modulated fiber sensors,” in Fiber Optic Sensors I, H. J. Arditti, L. B. Jeunhomme, eds., Proc. SPIE586, 99–103 (1985).
[CrossRef]

Parriaux, O. M.

L. G. Falco, O. M. Parriaux, “Optical fiber detection system using an intensity-modulating sensor,” U.S. patent4,887,901 (19December1989).

Patrick, H. J.

Paul, J. B.

J. J. Scherer, J. B. Paul, A. O’Keefe, R. J. Saykally, “Cavity-ring-down laser absorption spectroscopy: history, development, and application to pulsed molecular beams,” Chem. Rev. 97, 25–51 (1997).
[CrossRef] [PubMed]

Payne, D. N.

P. R. Morkel, M. C. Farries, D. N. Payne, “Losses in fiber laser cavities,” Electron. Lett. 24, 92–93 (1988).
[CrossRef]

Peeters, R.

G. Berden, R. Peeters, G. Meijer, “Cavity-ring-down spectroscopy: experimental schemes and applications,” Int. Rev. Phys. Chem. 19, 565–607 (2000).
[CrossRef]

R. Engeln, G. Berden, R. Peeters, G. Meijer, “Cavity enhanced absorption and cavity enhanced magnetic rotation spectroscopy,” Rev. Sci. Instrum. 69, 3763–3769 (1998).
[CrossRef]

Putnam, M. A.

Querry, M. R.

Ragdale, C.

W. Gamblin, H. Matsumura, C. Ragdale, “Curvature andmicrobending losses in single-mode optical fibers,” Opt. Quantum Electron. 11, 43–59 (1979).
[CrossRef]

Romanini, D.

D. Romanini, A. A. Kachanov, N. Sadeghi, F. Stoeckel, “Continuous-wave cavity-ring-down spectroscopy,” Chem. Phys. Lett. 264, 316–322 (1997).
[CrossRef]

K. K. Lehmann, D. Romanini, “The superposition principle and cavity-ring-down spectroscopy,” J. Chem. Phys. 105, 10263–10277 (1996).
[CrossRef]

Romann, A.

T. von Lerber, A. Romann, “A method for measuring at least one physical parameter using an optical resonator,” European patent application EP00121314.9 (9October2000).

Ruthven, G. P.

Sadeghi, N.

D. Romanini, A. A. Kachanov, N. Sadeghi, F. Stoeckel, “Continuous-wave cavity-ring-down spectroscopy,” Chem. Phys. Lett. 264, 316–322 (1997).
[CrossRef]

Saykally, R. J.

J. J. Scherer, J. B. Paul, A. O’Keefe, R. J. Saykally, “Cavity-ring-down laser absorption spectroscopy: history, development, and application to pulsed molecular beams,” Chem. Rev. 97, 25–51 (1997).
[CrossRef] [PubMed]

Scherer, J. J.

J. J. Scherer, J. B. Paul, A. O’Keefe, R. J. Saykally, “Cavity-ring-down laser absorption spectroscopy: history, development, and application to pulsed molecular beams,” Chem. Rev. 97, 25–51 (1997).
[CrossRef] [PubMed]

Schulz, K. J.

K. J. Schulz, W. R. Simpson, “Frequency-matched cavity-ring-down spectroscopy,” Chem. Phys. Lett. 297, 523–529 (1998).
[CrossRef]

Sigrist, M. W.

T. von Lerber, M. W. Sigrist, “Time constant extraction from noisy cavity-ring-down signals,” Chem. Phys. Lett. 353, 131–137 (2002).
[CrossRef]

Simova, E.

Y. Zhu, E. Simova, P. Berini, C. Grover, “A comparison of wavelength-dependent polarization-dependent loss measurements in fiber gratings,” IEEE Trans. Instrum. Meas. 49, 1231–1239 (2000).
[CrossRef]

Simpson, W. R.

K. J. Schulz, W. R. Simpson, “Frequency-matched cavity-ring-down spectroscopy,” Chem. Phys. Lett. 297, 523–529 (1998).
[CrossRef]

Spence, T. G.

Spencer, D. J.

Stewart, G.

G. Stewart, K. Atherton, H. Yu, B. Culshaw, “An investigation of an optical fiber amplifier loop for intracavity and ring-down-cavity loss measurements,” Meas. Sci. Technol. 12, 843–849 (2001).
[CrossRef]

Stoeckel, F.

D. Romanini, A. A. Kachanov, N. Sadeghi, F. Stoeckel, “Continuous-wave cavity-ring-down spectroscopy,” Chem. Phys. Lett. 264, 316–322 (1997).
[CrossRef]

Uenten, R. H.

Urevig, D. S.

van den Berg, E.

R. Engeln, G. Berden, E. van den Berg, G. Meijer, “Polarization-dependent cavity-ring-down spectroscopy,” J. Chem. Phys. 107, 4458–4467 (1997).
[CrossRef]

van Zee, R. D.

J. T. Hodges, J. P. Looney, R. D. van Zee, “Response of a ring-down cavity to an arbitrary excitation,” J. Chem. Phys. 105, 10278–10288 (1996).
[CrossRef]

von Helden, G.

R. Engeln, G. von Helden, G. Berden, G. Meijer, “Phase-shift cavity-ring-down absorption spectroscopy,” Chem. Phys. Lett. 262, 105–109 (1996).
[CrossRef]

von Lerber, T.

T. von Lerber, M. W. Sigrist, “Time constant extraction from noisy cavity-ring-down signals,” Chem. Phys. Lett. 353, 131–137 (2002).
[CrossRef]

T. von Lerber, A. Romann, “A method for measuring at least one physical parameter using an optical resonator,” European patent application EP00121314.9 (9October2000).

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Figures (6)

Fig. 1
Fig. 1

Experimental arrangement for a FCRD scheme comprising a light source (ECDL), polarization controller, fiber cavity with a loop of varying diameter, patch fibers, InGaAs photodiode, transimpedance amplifier, oscilloscope, pulse generator, and function generator: ×, splice of two fibers. The wavelength of the ECDL is modulated at 10 Hz across 1 GHz to excite cavity modes. Simultaneously the amplitude of the ECDL is modulated by square waves with a pulse period of 3.0 µs. The oscilloscope triggers when the cavity is excited, and the pulse generator switches off the intensity of the light source.

Fig. 2
Fig. 2

Recorded transmission response of the 0.287-m-long fiber cavity with high-reflectivity dielectric coatings on both fiber ends. The wavelength of light was modulated (Δν = 1 GHz) by a 10-Hz triangular modulation. Two orthogonal linearly polarized cavity modes are present. By changing the state of polarization of the incoming beam, the mutual amplitude of the peaks can be changed.

Fig. 3
Fig. 3

Typical unaveraged cavity-ring-down signals for straight, i.e., infinite bending diameter (∅∞), and coiled (∅29-mm) fiber with appropriate fits.

Fig. 4
Fig. 4

Possible source of loss in a SMF-based high-finesse cavity. Part of the electromagnetic field is reflected into the cladding instead of the core, causing intrinsic cavity losses (∼0.42%/pass). The divergence of the output field is high owing to the small-mode field aperture of the SMF core (10.4 µm at 1550 nm).

Fig. 5
Fig. 5

Time constants and losses for fiber cavities (with length, ℓ = 1.11 m) coiled around mandrels of different diameters as well as for varying output powers. A diameter of infinity ∞ indicates straight-fiber measurements. For each bending diameter 100 averaged signals were acquired. On the right-hand side, axes are given for loss [L dB = 10 log(e)ℓn eff/(cτ)]. The error bars indicate the standard deviation of the measurement. •, bending-induced losses according to the fiber manufacturer’s data sheet; – – – –, measured loss versus output power (top horizontal axis) for a straight fiber.

Fig. 6
Fig. 6

Observed time constants of the evanescent-field sensing experiment. Time constants were measured while the 0.576-m-long stripped fiber cavity was etched in BHF. The evanescent field interacted with the surrounding BHF solution, thus decreasing the cavity finesse. The evanescent-field interaction is modeled for water (extinction coefficient, κ = 1.716 × 10-4) and for a hypothetical low-loss medium (κ = 4 × 10-8) still detectable with the current fiber probe at an outer fiber diameter, d = 63 µm. The extinction coefficient of water is marked to equal infinity because time-constant curves for κ = 4 × 10-8 and κ = ∞ are almost identical for the current measurement setup.

Equations (17)

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I=I0 exp-t/τ.
τ=neffc|lnTR|neffcL+1-R,
Δττ=1|lnTR|ΔRR-ΔL1-L+Δ+Δneffneff,
ΔLmin=|ln Reff|1-LΔττmin1-ReffΔττmin.
ΔLdB,min=10ln10 |ln Reff|Δττmin10ln10 ΔLmin.
Δν=-cλneff Δneff=-466 MHz.
F=νfsrδν=c/2neff1/2πτ=π|ln Reff|=627,
Reff=R1-L=0.995, Ttot=1-R21-L1-1-LR2=0.025.
Ez=E0 expiωt-zc,
1c=nc-iκc,
Ez=E0 expiωt-zncexp-ωzκc,
ur=AJ0kTr,racoreBK0γr,r>acladding.
kT=n12k2-β21/2,
γ=β2-n22k21/2,
κr=0,rρκ,r>ρ,
Ez=2πE00ρ rurdr+exp-ωzκcρ rurdr.
T=PP0=|E|2|E0|2.

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