Abstract

A single-mode resonator is constructed by joining together the ends of the coupled fiber of a 2 × 2 directional single-mode optical fiber coupler. Analytical expressions are obtained for the resonance conditions. The effects of different kinds of losses are investigated. The variation of the phase at resonance and the problem of mismatch are discussed. A formula for the finesse of the resonator is also given. Unlike reentry fiber resonators, this kind of resonator needs a directional coupler with only a small coupling coefficient. It is possible to achieve a high finesse and symmetrical outputs. Applications of such a resonator are discussed.

© 1988 Optical Society of America

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References

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  1. S. J. Petuchowski, T. G. Giallorenzi, S. K. Sheem, “A sensitive fiber-optic Fabry-Perot interferometer,” IEEE J. Quantum Electron. QE-17, 2168–2170 (1981).
    [CrossRef]
  2. S. Ezekiel, S. R. Balsamo, “Passive ring resonator laser gyroscope,” Appl. Phys. Lett. 30, 478–480 (1977).
    [CrossRef]
  3. D. M. Shupe, “Fiber resonator gyroscope: sensitivity and thermal nonreciprocity,” Appl. Opt. 20, 286–289 (1981).
    [CrossRef] [PubMed]
  4. K. R. Reston, “Simple spectral control technique for external cavity laser transmitters,” Electron. Lett. 18, 1092–1094 (1982).
    [CrossRef]
  5. S. R. Mallinson, “Fiber-coupled Fabry-Perot wavelength demultiplexer,” Electron. Lett. 21, 121–122 (1985).
    [CrossRef]
  6. L. F. Stokes, M. Chodorow, H. J. Shaw, “All-fiber stimulated Brillouin ring laser with submilliwatt pump threshold,” Opt. Lett. 7, 509–511 (1982).
    [CrossRef] [PubMed]
  7. L. F. Stokes, M. Chodorow, H. J. Shaw, “Sensitive all-single-mode-fiber resonant ring interferometer,” IEEE J. Lightwave Technol. LT-1, 110–115 (1983).
    [CrossRef]
  8. R. E. Mayer, S. Ezekiel, D. W. Stowe, V. J. Tekippe, “Passive fiber-optic ring resonator for rotation sensing,” Opt. Lett. 8, 644–646 (1983).
    [CrossRef]
  9. R. A. Bergh, H. C. Lefevre, H. J. Shaw, “An overview of fiber-optic gyroscope,” IEEE J. Lightwave Technol. LT-2, 91–107 (1984).
    [CrossRef]
  10. L. F. Stokes, M. Chodorow, H. J. Shaw, “All-single-mode fiber resonator,” Opt. Lett. 7, 288–290 (1982).
    [CrossRef] [PubMed]
  11. D. B. Hall, M. H. Yu, “Low loss fiber optic directional coupler,” in Fiber Optic and Laser Sensors II, E. L. Moore, O. G. Ramer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.478, 75–79 (1984).
    [CrossRef]
  12. M. H. Yu, D. B. Hall, “Low loss fiber ring resonator,” in Fiber Optic and Laser Sensors II, E. L. Moore, O. G. Ramer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.478, 104–108 (1984).
    [CrossRef]
  13. R. C. Youngquist, L. F. Stokes, H. J. Shaw, “Effects of normal mode loss in dielectric waveguide directional couplers and interferometers,” IEEE J. Quantum Electron. QE-19, 1888–1896 (1983).
    [CrossRef]
  14. L. F. Stokes, “Single-mode optical-fiber resonator and applications to sensing,” doctoral dissertation (Stanford University, Stanford, Calif., 1984).
  15. H. Golde, “Theory and measurement of Q in resonant ring circuits,” IRE Trans. Microwave Theory Tech. MTT-8, 560–564 (1960).
    [CrossRef]
  16. T. Findakly, C. L. Chen, “Optical directional couplers with variable spacing,” Appl. Opt. 17, 769–773 (1978).
    [CrossRef] [PubMed]
  17. M. Digonnet, H. J. Shaw, “Analysis of a tunable single mode optical fiber coupler,” IEEE J. Quantum Electron. QE-18, 746–754 (1982).
    [CrossRef]
  18. A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. A 62, 1267–1277 (1972).
    [CrossRef]
  19. L. Jeunhomme, Single-Mode Fiber Optics (Dekker, New York, 1983).
  20. J. M. Senior, Optical Fiber Communications (Prentice-Hall, London, 1985).
  21. T. Uenoya, Y. Murakami, I. Yamanouchi, “A new field-use single-mode fiber splicer using a direct core observation technique and its application,” in Fiber Optic Couplers, Connectors, and Splice Technology, D. W. Stowe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.574, 152–156 (1985).
    [CrossRef]

1985 (1)

S. R. Mallinson, “Fiber-coupled Fabry-Perot wavelength demultiplexer,” Electron. Lett. 21, 121–122 (1985).
[CrossRef]

1984 (1)

R. A. Bergh, H. C. Lefevre, H. J. Shaw, “An overview of fiber-optic gyroscope,” IEEE J. Lightwave Technol. LT-2, 91–107 (1984).
[CrossRef]

1983 (3)

R. C. Youngquist, L. F. Stokes, H. J. Shaw, “Effects of normal mode loss in dielectric waveguide directional couplers and interferometers,” IEEE J. Quantum Electron. QE-19, 1888–1896 (1983).
[CrossRef]

L. F. Stokes, M. Chodorow, H. J. Shaw, “Sensitive all-single-mode-fiber resonant ring interferometer,” IEEE J. Lightwave Technol. LT-1, 110–115 (1983).
[CrossRef]

R. E. Mayer, S. Ezekiel, D. W. Stowe, V. J. Tekippe, “Passive fiber-optic ring resonator for rotation sensing,” Opt. Lett. 8, 644–646 (1983).
[CrossRef]

1982 (4)

L. F. Stokes, M. Chodorow, H. J. Shaw, “All-single-mode fiber resonator,” Opt. Lett. 7, 288–290 (1982).
[CrossRef] [PubMed]

L. F. Stokes, M. Chodorow, H. J. Shaw, “All-fiber stimulated Brillouin ring laser with submilliwatt pump threshold,” Opt. Lett. 7, 509–511 (1982).
[CrossRef] [PubMed]

K. R. Reston, “Simple spectral control technique for external cavity laser transmitters,” Electron. Lett. 18, 1092–1094 (1982).
[CrossRef]

M. Digonnet, H. J. Shaw, “Analysis of a tunable single mode optical fiber coupler,” IEEE J. Quantum Electron. QE-18, 746–754 (1982).
[CrossRef]

1981 (2)

S. J. Petuchowski, T. G. Giallorenzi, S. K. Sheem, “A sensitive fiber-optic Fabry-Perot interferometer,” IEEE J. Quantum Electron. QE-17, 2168–2170 (1981).
[CrossRef]

D. M. Shupe, “Fiber resonator gyroscope: sensitivity and thermal nonreciprocity,” Appl. Opt. 20, 286–289 (1981).
[CrossRef] [PubMed]

1978 (1)

1977 (1)

S. Ezekiel, S. R. Balsamo, “Passive ring resonator laser gyroscope,” Appl. Phys. Lett. 30, 478–480 (1977).
[CrossRef]

1972 (1)

A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. A 62, 1267–1277 (1972).
[CrossRef]

1960 (1)

H. Golde, “Theory and measurement of Q in resonant ring circuits,” IRE Trans. Microwave Theory Tech. MTT-8, 560–564 (1960).
[CrossRef]

Balsamo, S. R.

S. Ezekiel, S. R. Balsamo, “Passive ring resonator laser gyroscope,” Appl. Phys. Lett. 30, 478–480 (1977).
[CrossRef]

Bergh, R. A.

R. A. Bergh, H. C. Lefevre, H. J. Shaw, “An overview of fiber-optic gyroscope,” IEEE J. Lightwave Technol. LT-2, 91–107 (1984).
[CrossRef]

Chen, C. L.

Chodorow, M.

Digonnet, M.

M. Digonnet, H. J. Shaw, “Analysis of a tunable single mode optical fiber coupler,” IEEE J. Quantum Electron. QE-18, 746–754 (1982).
[CrossRef]

Ezekiel, S.

R. E. Mayer, S. Ezekiel, D. W. Stowe, V. J. Tekippe, “Passive fiber-optic ring resonator for rotation sensing,” Opt. Lett. 8, 644–646 (1983).
[CrossRef]

S. Ezekiel, S. R. Balsamo, “Passive ring resonator laser gyroscope,” Appl. Phys. Lett. 30, 478–480 (1977).
[CrossRef]

Findakly, T.

Giallorenzi, T. G.

S. J. Petuchowski, T. G. Giallorenzi, S. K. Sheem, “A sensitive fiber-optic Fabry-Perot interferometer,” IEEE J. Quantum Electron. QE-17, 2168–2170 (1981).
[CrossRef]

Golde, H.

H. Golde, “Theory and measurement of Q in resonant ring circuits,” IRE Trans. Microwave Theory Tech. MTT-8, 560–564 (1960).
[CrossRef]

Hall, D. B.

D. B. Hall, M. H. Yu, “Low loss fiber optic directional coupler,” in Fiber Optic and Laser Sensors II, E. L. Moore, O. G. Ramer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.478, 75–79 (1984).
[CrossRef]

M. H. Yu, D. B. Hall, “Low loss fiber ring resonator,” in Fiber Optic and Laser Sensors II, E. L. Moore, O. G. Ramer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.478, 104–108 (1984).
[CrossRef]

Jeunhomme, L.

L. Jeunhomme, Single-Mode Fiber Optics (Dekker, New York, 1983).

Lefevre, H. C.

R. A. Bergh, H. C. Lefevre, H. J. Shaw, “An overview of fiber-optic gyroscope,” IEEE J. Lightwave Technol. LT-2, 91–107 (1984).
[CrossRef]

Mallinson, S. R.

S. R. Mallinson, “Fiber-coupled Fabry-Perot wavelength demultiplexer,” Electron. Lett. 21, 121–122 (1985).
[CrossRef]

Mayer, R. E.

Murakami, Y.

T. Uenoya, Y. Murakami, I. Yamanouchi, “A new field-use single-mode fiber splicer using a direct core observation technique and its application,” in Fiber Optic Couplers, Connectors, and Splice Technology, D. W. Stowe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.574, 152–156 (1985).
[CrossRef]

Petuchowski, S. J.

S. J. Petuchowski, T. G. Giallorenzi, S. K. Sheem, “A sensitive fiber-optic Fabry-Perot interferometer,” IEEE J. Quantum Electron. QE-17, 2168–2170 (1981).
[CrossRef]

Reston, K. R.

K. R. Reston, “Simple spectral control technique for external cavity laser transmitters,” Electron. Lett. 18, 1092–1094 (1982).
[CrossRef]

Senior, J. M.

J. M. Senior, Optical Fiber Communications (Prentice-Hall, London, 1985).

Shaw, H. J.

R. A. Bergh, H. C. Lefevre, H. J. Shaw, “An overview of fiber-optic gyroscope,” IEEE J. Lightwave Technol. LT-2, 91–107 (1984).
[CrossRef]

L. F. Stokes, M. Chodorow, H. J. Shaw, “Sensitive all-single-mode-fiber resonant ring interferometer,” IEEE J. Lightwave Technol. LT-1, 110–115 (1983).
[CrossRef]

R. C. Youngquist, L. F. Stokes, H. J. Shaw, “Effects of normal mode loss in dielectric waveguide directional couplers and interferometers,” IEEE J. Quantum Electron. QE-19, 1888–1896 (1983).
[CrossRef]

M. Digonnet, H. J. Shaw, “Analysis of a tunable single mode optical fiber coupler,” IEEE J. Quantum Electron. QE-18, 746–754 (1982).
[CrossRef]

L. F. Stokes, M. Chodorow, H. J. Shaw, “All-single-mode fiber resonator,” Opt. Lett. 7, 288–290 (1982).
[CrossRef] [PubMed]

L. F. Stokes, M. Chodorow, H. J. Shaw, “All-fiber stimulated Brillouin ring laser with submilliwatt pump threshold,” Opt. Lett. 7, 509–511 (1982).
[CrossRef] [PubMed]

Sheem, S. K.

S. J. Petuchowski, T. G. Giallorenzi, S. K. Sheem, “A sensitive fiber-optic Fabry-Perot interferometer,” IEEE J. Quantum Electron. QE-17, 2168–2170 (1981).
[CrossRef]

Shupe, D. M.

Snyder, A. W.

A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. A 62, 1267–1277 (1972).
[CrossRef]

Stokes, L. F.

R. C. Youngquist, L. F. Stokes, H. J. Shaw, “Effects of normal mode loss in dielectric waveguide directional couplers and interferometers,” IEEE J. Quantum Electron. QE-19, 1888–1896 (1983).
[CrossRef]

L. F. Stokes, M. Chodorow, H. J. Shaw, “Sensitive all-single-mode-fiber resonant ring interferometer,” IEEE J. Lightwave Technol. LT-1, 110–115 (1983).
[CrossRef]

L. F. Stokes, M. Chodorow, H. J. Shaw, “All-single-mode fiber resonator,” Opt. Lett. 7, 288–290 (1982).
[CrossRef] [PubMed]

L. F. Stokes, M. Chodorow, H. J. Shaw, “All-fiber stimulated Brillouin ring laser with submilliwatt pump threshold,” Opt. Lett. 7, 509–511 (1982).
[CrossRef] [PubMed]

L. F. Stokes, “Single-mode optical-fiber resonator and applications to sensing,” doctoral dissertation (Stanford University, Stanford, Calif., 1984).

Stowe, D. W.

Tekippe, V. J.

Uenoya, T.

T. Uenoya, Y. Murakami, I. Yamanouchi, “A new field-use single-mode fiber splicer using a direct core observation technique and its application,” in Fiber Optic Couplers, Connectors, and Splice Technology, D. W. Stowe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.574, 152–156 (1985).
[CrossRef]

Yamanouchi, I.

T. Uenoya, Y. Murakami, I. Yamanouchi, “A new field-use single-mode fiber splicer using a direct core observation technique and its application,” in Fiber Optic Couplers, Connectors, and Splice Technology, D. W. Stowe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.574, 152–156 (1985).
[CrossRef]

Youngquist, R. C.

R. C. Youngquist, L. F. Stokes, H. J. Shaw, “Effects of normal mode loss in dielectric waveguide directional couplers and interferometers,” IEEE J. Quantum Electron. QE-19, 1888–1896 (1983).
[CrossRef]

Yu, M. H.

D. B. Hall, M. H. Yu, “Low loss fiber optic directional coupler,” in Fiber Optic and Laser Sensors II, E. L. Moore, O. G. Ramer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.478, 75–79 (1984).
[CrossRef]

M. H. Yu, D. B. Hall, “Low loss fiber ring resonator,” in Fiber Optic and Laser Sensors II, E. L. Moore, O. G. Ramer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.478, 104–108 (1984).
[CrossRef]

Appl. Opt. (2)

Appl. Phys. Lett. (1)

S. Ezekiel, S. R. Balsamo, “Passive ring resonator laser gyroscope,” Appl. Phys. Lett. 30, 478–480 (1977).
[CrossRef]

Electron. Lett. (2)

K. R. Reston, “Simple spectral control technique for external cavity laser transmitters,” Electron. Lett. 18, 1092–1094 (1982).
[CrossRef]

S. R. Mallinson, “Fiber-coupled Fabry-Perot wavelength demultiplexer,” Electron. Lett. 21, 121–122 (1985).
[CrossRef]

IEEE J. Lightwave Technol. (2)

L. F. Stokes, M. Chodorow, H. J. Shaw, “Sensitive all-single-mode-fiber resonant ring interferometer,” IEEE J. Lightwave Technol. LT-1, 110–115 (1983).
[CrossRef]

R. A. Bergh, H. C. Lefevre, H. J. Shaw, “An overview of fiber-optic gyroscope,” IEEE J. Lightwave Technol. LT-2, 91–107 (1984).
[CrossRef]

IEEE J. Quantum Electron. (3)

M. Digonnet, H. J. Shaw, “Analysis of a tunable single mode optical fiber coupler,” IEEE J. Quantum Electron. QE-18, 746–754 (1982).
[CrossRef]

S. J. Petuchowski, T. G. Giallorenzi, S. K. Sheem, “A sensitive fiber-optic Fabry-Perot interferometer,” IEEE J. Quantum Electron. QE-17, 2168–2170 (1981).
[CrossRef]

R. C. Youngquist, L. F. Stokes, H. J. Shaw, “Effects of normal mode loss in dielectric waveguide directional couplers and interferometers,” IEEE J. Quantum Electron. QE-19, 1888–1896 (1983).
[CrossRef]

IRE Trans. Microwave Theory Tech. (1)

H. Golde, “Theory and measurement of Q in resonant ring circuits,” IRE Trans. Microwave Theory Tech. MTT-8, 560–564 (1960).
[CrossRef]

J. Opt. Soc. Am. A (1)

A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. A 62, 1267–1277 (1972).
[CrossRef]

Opt. Lett. (3)

Other (6)

D. B. Hall, M. H. Yu, “Low loss fiber optic directional coupler,” in Fiber Optic and Laser Sensors II, E. L. Moore, O. G. Ramer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.478, 75–79 (1984).
[CrossRef]

M. H. Yu, D. B. Hall, “Low loss fiber ring resonator,” in Fiber Optic and Laser Sensors II, E. L. Moore, O. G. Ramer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.478, 104–108 (1984).
[CrossRef]

L. Jeunhomme, Single-Mode Fiber Optics (Dekker, New York, 1983).

J. M. Senior, Optical Fiber Communications (Prentice-Hall, London, 1985).

T. Uenoya, Y. Murakami, I. Yamanouchi, “A new field-use single-mode fiber splicer using a direct core observation technique and its application,” in Fiber Optic Couplers, Connectors, and Splice Technology, D. W. Stowe, ed., Proc. Soc. Photo-Opt. Instrum. Eng.574, 152–156 (1985).
[CrossRef]

L. F. Stokes, “Single-mode optical-fiber resonator and applications to sensing,” doctoral dissertation (Stanford University, Stanford, Calif., 1984).

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Figures (13)

Fig. 1
Fig. 1

Simple physical model of a DFR resonator: (a) DFR resonator; (b) equivalent model at optimum resonance (E3 = 0).

Fig. 2
Fig. 2

(a) Resonator output intensity ratio and (b) circulating intensity ratio as functions of the optical length βL of the loop. Coupler insertion loss, 0.1%; fiber loop length, 3 m; fiber loss, 1 dB/km; splice loss, 0.04 dB.

Fig. 3
Fig. 3

Resonance output and circulating intensities as funtions of the optical length of the loop for γ0 = 0.1, 1, 3.2%: (a) output intensity; (b) circulating intensity. L = 3 m, α = 1 dB/km, α0 = 0.04 dB.

Fig. 4
Fig. 4

Resonance output and circulating intensities as functions of the optical length of the loop for a0 = 0.005, 0.01, 0.04 dB: (a) output intensity; (b) circulating intensity. L = 3 m, α = 1 dB/km, γ0 = 0.1%.

Fig. 5
Fig. 5

Resonance output and circulating intensities as functions of the optical length of the loop for α = 0.5, 2, 5 dB/km: (a) output intensity, (b) circulating intensity. L = 3 m, a0 = 0.04 dB, γ0 = 0.1%.

Fig. 6
Fig. 6

Resonance output and circulating intensities as functions of the optical length of the loop for k = 0.005, 0.0154, 0.03: (a) output intensity; (b) circulating intensity. L = 3 m, α = 1 dB/km, a0 = 0.06 dB, γ0 = 0.1%.

Fig. 7
Fig. 7

Relative to the wave at port 1, the phase (a) at port 2, (b) at port 3, and (c) at port 4, as functions of the optical length of the loop.

Fig. 8
Fig. 8

Finesse as a function of the coupling coefficient.

Fig. 9
Fig. 9

Finesse as a function of (a) insertion loss γ0 for a0 = 0.01, 0.03, 0.06 dB and α = 1 dB/km; (b) splice loss a0 for γ0 = 0.05, 0.2, 1% and α = 1 dB/km; and (c) fiber loss α for γ0 = 0.08, 0.2, 1% and a0 = 0.01 dB.

Fig. 10
Fig. 10

Schematic of a fiber directional coupler.

Fig. 11
Fig. 11

Intensity coupling coefficient of a directional coupler as a function of the coupler geometry. Fiber: a = 3 μm, n1 = 1.458, n2 = 1.4551. (a) R = 25 cm; (b) R = 200 cm.

Fig. 12
Fig. 12

Microwave traveling-wave ring resonator: (a) schematic of resonator; (b) resonance ring circuit.

Fig. 13
Fig. 13

Schematic of CFR resonator.

Equations (44)

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E 2 = ( 1 a 0 ) 1 / 2 e α L E 4 ,
E 4 = ( 1 γ 0 ) 1 / 2 E 2 = ( 1 γ 0 ) 1 / 2 ( 1 a 0 ) 1 / 2 e α L E 4 ,
| E 4 | 2 | E 4 | 2 = | Δ E | 2 ,
k = | E 4 E 1 | E 2 = 0 2 = | E 1 E 4 | E 3 = 0 2 ,
| Δ E | 2 = k r | E 4 | 2 ,
k r = 1 ( 1 γ 0 ) ( 1 a 0 ) e 2 α L .
β L = p 2 π ( p = 1,2 , ) ,
1 γ 0 = ( | E 3 | 2 + | E 4 | 2 ) / ( | E 1 | 2 + | E 2 | 2 ) ,
E 3 = ( 1 γ 0 ) 1 / 2 [ ( 1 k ) 1 / 2 E 1 + j k 1 / 2 E 2 ] ,
E 4 = ( 1 γ 0 ) 1 / 2 [ j k 1 / 2 E 1 + ( 1 k ) 1 / 2 E 2 ] ,
E 2 = ( 1 a 0 ) 1 / 2 e α L e j β L E 4 ,
β n 1 ω c ,
E 3 / E 1 = ( 1 γ 0 ) 1 / 2 ( 1 k ) 1 / 2 ( 1 k 1 A 1 / 2 e j β L ) ,
A = ( 1 k ) ( 1 a 0 ) ( 1 γ 0 ) e 2 α L .
| E 3 / E 1 | 2 = ( 1 γ 0 ) [ 1 k ( 1 k A ) / ( 1 k ) 1 + A 2 A 1 / 2 cos β L ] .
E 4 / E 1 = j ( 1 γ 0 ) 1 / 2 k 1 / 2 1 A 1 / 2 e j β L ,
| E 4 / E 1 | 2 = ( 1 γ 0 ) k 1 + A 2 A 1 / 2 cos β L .
1 k A 1 / 2 cos β L j A 1 / 2 sin β L = 0.
β L = p 2 π ( p = 1,2,3 , ) .
k r = 1 ( 1 γ 0 ) ( 1 a 0 ) e 2 α L .
| E 3 / E 1 | 2 = 2 ( 1 γ 0 ) ( 1 k r ) ( 1 cos β L ) 1 + ( 1 k r ) 2 2 ( 1 k r ) cos β L ,
| E 4 / E 1 | 2 = ( 1 γ 0 ) k r 1 + ( 1 k r ) 2 2 ( 1 k r ) cos β L .
| E 3 / E 1 | 2 = 0 ,
| E 4 / E 1 | max 2 = 1 γ 0 k r .
ϕ 21 = π 2 + β L + tan 1 [ ( 1 k r ) sin β L 1 ( 1 k r ) cos β L ] ,
ϕ 31 = tan 1 [ k r sin β L ( 2 k r ) ( 1 cos β L ) ] ,
ϕ 41 = π 2 + tan 1 [ ( 1 k r ) sin β L 1 ( 1 k r ) cos β L ] ,
F = f p + 1 f p δ f 1 / 2 .
δ f 1 / 2 = 2 π c n 1 L sin 1 [ k r 2 ( 1 k r ) 1 / 2 ] ,
F = π 2 sin 1 [ k r 2 ( 1 k r ) 1 / 2 ] .
F π k r .
F π 1 ( 1 a 0 ) ( 1 γ 0 ) e 2 α L .
k = sin 2 ( c 0 L e ) ,
c 0 = λ 2 π n 1 U 2 a 2 V 2 K 0 ( W h 0 / a ) K 1 2 ( W ) ,
V = 2 π a λ ( n 1 2 n 2 2 ) 1 / 2 ,
W = a ( β 2 4 π 2 λ 2 n 2 2 ) 1 / 2 ,
L e ( π R a / W ) 1 / 2 .
c 0 L e = λ π a n 1 ( R h 0 ) 1 / 2 [ 1 ( W V ) 2 ] exp [ W ( 2 h 0 a ) ] ,
β l = 2 π m ( m = 1,2 , ) ,
k 2 2 = 1 e 2 α l ,
| b 2 / a 1 | 2 = 2 ( 1 k 2 2 ) ( 1 cos β l ) 1 + ( 1 k 2 2 ) 2 2 ( 1 k 2 2 ) cos β l ,
| b 3 / a 1 | 2 = k 2 2 1 + ( 1 k 2 2 ) 2 2 ( 1 k 2 2 ) cos β l ,
| b 3 / a 1 | max = 1 / k 2 2 .
Q L = π m k 2 2 ( λ g λ 0 ) 2 ,

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