Abstract

Guided longitudinal-acoustic modes, which give rise to Brillouin gain, are theoretically clarified for a single-mode fiber with a GeO2-doped core and pure-silica cladding. Longitudinal-acoustic L0m modes are found from the theoretical analysis to interact with the electromagnetic field of the HE11 mode. Brillouin-gain spectra are measured for clarifying the theory. A few gain peaks in the Brillouin-gain spectra are successfully explained by taking account of the phase-velocity characteristics of the L01, L02 and L03 modes guided in the GeO2-doped core region. Furthermore, Brillouin frequency shifts per unit dopant concentration for GeO2 and F are experimentally obtained to confirm the dispersion characteristics of the guided acoustic modes. The evaluated frequency shifts per unit dopant concentration are 107 and 356 MHz/mol% for GeO2 and F, respectively, at a wavelength of 1550 nm.

© 1989 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21, 539–541 (1972).
    [Crossref]
  2. N. Shibata, R. P. Braun, and R. G. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1205–1210 (1987).
    [Crossref]
  3. H. Toba, K. Inoue, and K. Nosu, “450 Mbit/s optical frequency-division-multiplexing transmission with an 11 GHz channel spacing,” Electron. Lett. 21, 656–657 (1985).
    [Crossref]
  4. E. J. Bachus, R. P. Braun, W. Eutin, E. Grossmann, H. Foisel, K. Heimes, and B. Strebel, “Coherent optical fiber subscriber line,” Electron. Lett. 21, 1203–1205 (1985).
    [Crossref]
  5. D. W. Smith, “Techniques for multigigabit coherent optical transmission,” IEEE J. Lightwave Technol. LT-5, 1466–1478 (1987).
    [Crossref]
  6. P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fiber,” Phys. Rev. B 19, 4986–4998 (1979).
    [Crossref]
  7. C. K. Jen, A. Safaai-Jazi, and G. W. Farnell, “Leaky modes in weakly guiding fiber acoustic waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control UFFC-33, 634–643 (1986).
  8. R. W. Tkach, A. R. Charaplyvy, and R. M. Derosier, “Spontaneous Brillouin scattering for single-mode optical fibre characterization,” Electron. Lett. 22, 1011–1013 (1986).
    [Crossref]
  9. N. Shibata, R. G. Waarts, and R. P. Braun, “Brillouin-gain spectra for single-mode fibers having pure-silica, GeO2-doped, and P2O5-doped cores,” Opt. Lett. 12, 269–271 (1987).
    [Crossref] [PubMed]
  10. N. Shibata, Y. Azuma, T. Horiguchi, and M. Tateda, “Identification of longitudinal acoustic modes guided in the core region of a single-mode optical fiber by Brillouin gain spectra measurements,” Opt. Lett. 13, 595–597 (1988).
    [Crossref] [PubMed]
  11. R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B 31, 5244–5252 (1985).
    [Crossref]
  12. R. N. Thurston, “Elastic waves in rods and clad rods,” J. Acoust. Soc. Am. 64, 1–37 (1978).
    [Crossref]
  13. A. E. Armenàkas, “Propagation of harmonic waves in composite circular-cylindrical rods,” J. Acoust. Soc. Am. 47, 822–837 (1970).
    [Crossref]
  14. T. Okoshi, Optical Fibers (Academic, New York, 1982), Chap. 4.
  15. D. Gloge, “Weakly guiding fibers,” Appl. Opt. 10, 2252–2258 (1971).
    [Crossref] [PubMed]
  16. D. Cotter, “Stimulated Brillouin scattering in monomode optical fibre,” J. Opt. Commun. 1, 10–19 (1983).
  17. N. A. Olsson and J. P. van der Ziel, “Characteristics of a semiconductor laser pumped Brillouin amplifier with electronically controlled bandwidth,” IEEE J. Lightwave Technol. LT-5, 147–153 (1987).
    [Crossref]
  18. A. R. Charaplyvy and R. W. Tkach, “Narrowband tunable optical filter for channel selection in density packed WDM systems,” Electron. Lett. 22, 1084–1085 (1986).
    [Crossref]

1988 (1)

1987 (4)

N. A. Olsson and J. P. van der Ziel, “Characteristics of a semiconductor laser pumped Brillouin amplifier with electronically controlled bandwidth,” IEEE J. Lightwave Technol. LT-5, 147–153 (1987).
[Crossref]

N. Shibata, R. P. Braun, and R. G. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1205–1210 (1987).
[Crossref]

D. W. Smith, “Techniques for multigigabit coherent optical transmission,” IEEE J. Lightwave Technol. LT-5, 1466–1478 (1987).
[Crossref]

N. Shibata, R. G. Waarts, and R. P. Braun, “Brillouin-gain spectra for single-mode fibers having pure-silica, GeO2-doped, and P2O5-doped cores,” Opt. Lett. 12, 269–271 (1987).
[Crossref] [PubMed]

1986 (3)

C. K. Jen, A. Safaai-Jazi, and G. W. Farnell, “Leaky modes in weakly guiding fiber acoustic waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control UFFC-33, 634–643 (1986).

R. W. Tkach, A. R. Charaplyvy, and R. M. Derosier, “Spontaneous Brillouin scattering for single-mode optical fibre characterization,” Electron. Lett. 22, 1011–1013 (1986).
[Crossref]

A. R. Charaplyvy and R. W. Tkach, “Narrowband tunable optical filter for channel selection in density packed WDM systems,” Electron. Lett. 22, 1084–1085 (1986).
[Crossref]

1985 (3)

R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B 31, 5244–5252 (1985).
[Crossref]

H. Toba, K. Inoue, and K. Nosu, “450 Mbit/s optical frequency-division-multiplexing transmission with an 11 GHz channel spacing,” Electron. Lett. 21, 656–657 (1985).
[Crossref]

E. J. Bachus, R. P. Braun, W. Eutin, E. Grossmann, H. Foisel, K. Heimes, and B. Strebel, “Coherent optical fiber subscriber line,” Electron. Lett. 21, 1203–1205 (1985).
[Crossref]

1983 (1)

D. Cotter, “Stimulated Brillouin scattering in monomode optical fibre,” J. Opt. Commun. 1, 10–19 (1983).

1979 (1)

P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fiber,” Phys. Rev. B 19, 4986–4998 (1979).
[Crossref]

1978 (1)

R. N. Thurston, “Elastic waves in rods and clad rods,” J. Acoust. Soc. Am. 64, 1–37 (1978).
[Crossref]

1972 (1)

E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21, 539–541 (1972).
[Crossref]

1971 (1)

1970 (1)

A. E. Armenàkas, “Propagation of harmonic waves in composite circular-cylindrical rods,” J. Acoust. Soc. Am. 47, 822–837 (1970).
[Crossref]

Armenàkas, A. E.

A. E. Armenàkas, “Propagation of harmonic waves in composite circular-cylindrical rods,” J. Acoust. Soc. Am. 47, 822–837 (1970).
[Crossref]

Azuma, Y.

Bachus, E. J.

E. J. Bachus, R. P. Braun, W. Eutin, E. Grossmann, H. Foisel, K. Heimes, and B. Strebel, “Coherent optical fiber subscriber line,” Electron. Lett. 21, 1203–1205 (1985).
[Crossref]

Bayer, P. W.

R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B 31, 5244–5252 (1985).
[Crossref]

Braun, R. P.

N. Shibata, R. P. Braun, and R. G. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1205–1210 (1987).
[Crossref]

N. Shibata, R. G. Waarts, and R. P. Braun, “Brillouin-gain spectra for single-mode fibers having pure-silica, GeO2-doped, and P2O5-doped cores,” Opt. Lett. 12, 269–271 (1987).
[Crossref] [PubMed]

E. J. Bachus, R. P. Braun, W. Eutin, E. Grossmann, H. Foisel, K. Heimes, and B. Strebel, “Coherent optical fiber subscriber line,” Electron. Lett. 21, 1203–1205 (1985).
[Crossref]

Charaplyvy, A. R.

R. W. Tkach, A. R. Charaplyvy, and R. M. Derosier, “Spontaneous Brillouin scattering for single-mode optical fibre characterization,” Electron. Lett. 22, 1011–1013 (1986).
[Crossref]

A. R. Charaplyvy and R. W. Tkach, “Narrowband tunable optical filter for channel selection in density packed WDM systems,” Electron. Lett. 22, 1084–1085 (1986).
[Crossref]

Cotter, D.

D. Cotter, “Stimulated Brillouin scattering in monomode optical fibre,” J. Opt. Commun. 1, 10–19 (1983).

Derosier, R. M.

R. W. Tkach, A. R. Charaplyvy, and R. M. Derosier, “Spontaneous Brillouin scattering for single-mode optical fibre characterization,” Electron. Lett. 22, 1011–1013 (1986).
[Crossref]

Eutin, W.

E. J. Bachus, R. P. Braun, W. Eutin, E. Grossmann, H. Foisel, K. Heimes, and B. Strebel, “Coherent optical fiber subscriber line,” Electron. Lett. 21, 1203–1205 (1985).
[Crossref]

Farnell, G. W.

C. K. Jen, A. Safaai-Jazi, and G. W. Farnell, “Leaky modes in weakly guiding fiber acoustic waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control UFFC-33, 634–643 (1986).

Foisel, H.

E. J. Bachus, R. P. Braun, W. Eutin, E. Grossmann, H. Foisel, K. Heimes, and B. Strebel, “Coherent optical fiber subscriber line,” Electron. Lett. 21, 1203–1205 (1985).
[Crossref]

Gloge, D.

Grossmann, E.

E. J. Bachus, R. P. Braun, W. Eutin, E. Grossmann, H. Foisel, K. Heimes, and B. Strebel, “Coherent optical fiber subscriber line,” Electron. Lett. 21, 1203–1205 (1985).
[Crossref]

Heimes, K.

E. J. Bachus, R. P. Braun, W. Eutin, E. Grossmann, H. Foisel, K. Heimes, and B. Strebel, “Coherent optical fiber subscriber line,” Electron. Lett. 21, 1203–1205 (1985).
[Crossref]

Horiguchi, T.

Inoue, K.

H. Toba, K. Inoue, and K. Nosu, “450 Mbit/s optical frequency-division-multiplexing transmission with an 11 GHz channel spacing,” Electron. Lett. 21, 656–657 (1985).
[Crossref]

Ippen, E. P.

E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21, 539–541 (1972).
[Crossref]

Jen, C. K.

C. K. Jen, A. Safaai-Jazi, and G. W. Farnell, “Leaky modes in weakly guiding fiber acoustic waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control UFFC-33, 634–643 (1986).

Levenson, M. D.

R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B 31, 5244–5252 (1985).
[Crossref]

Nosu, K.

H. Toba, K. Inoue, and K. Nosu, “450 Mbit/s optical frequency-division-multiplexing transmission with an 11 GHz channel spacing,” Electron. Lett. 21, 656–657 (1985).
[Crossref]

Okoshi, T.

T. Okoshi, Optical Fibers (Academic, New York, 1982), Chap. 4.

Olsson, N. A.

N. A. Olsson and J. P. van der Ziel, “Characteristics of a semiconductor laser pumped Brillouin amplifier with electronically controlled bandwidth,” IEEE J. Lightwave Technol. LT-5, 147–153 (1987).
[Crossref]

Rowell, N. L.

P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fiber,” Phys. Rev. B 19, 4986–4998 (1979).
[Crossref]

Safaai-Jazi, A.

C. K. Jen, A. Safaai-Jazi, and G. W. Farnell, “Leaky modes in weakly guiding fiber acoustic waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control UFFC-33, 634–643 (1986).

Shelby, R. M.

R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B 31, 5244–5252 (1985).
[Crossref]

Shibata, N.

Smith, D. W.

D. W. Smith, “Techniques for multigigabit coherent optical transmission,” IEEE J. Lightwave Technol. LT-5, 1466–1478 (1987).
[Crossref]

Stegeman, G. I.

P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fiber,” Phys. Rev. B 19, 4986–4998 (1979).
[Crossref]

Stolen, R. H.

E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21, 539–541 (1972).
[Crossref]

Strebel, B.

E. J. Bachus, R. P. Braun, W. Eutin, E. Grossmann, H. Foisel, K. Heimes, and B. Strebel, “Coherent optical fiber subscriber line,” Electron. Lett. 21, 1203–1205 (1985).
[Crossref]

Tateda, M.

Thomas, P. J.

P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fiber,” Phys. Rev. B 19, 4986–4998 (1979).
[Crossref]

Thurston, R. N.

R. N. Thurston, “Elastic waves in rods and clad rods,” J. Acoust. Soc. Am. 64, 1–37 (1978).
[Crossref]

Tkach, R. W.

A. R. Charaplyvy and R. W. Tkach, “Narrowband tunable optical filter for channel selection in density packed WDM systems,” Electron. Lett. 22, 1084–1085 (1986).
[Crossref]

R. W. Tkach, A. R. Charaplyvy, and R. M. Derosier, “Spontaneous Brillouin scattering for single-mode optical fibre characterization,” Electron. Lett. 22, 1011–1013 (1986).
[Crossref]

Toba, H.

H. Toba, K. Inoue, and K. Nosu, “450 Mbit/s optical frequency-division-multiplexing transmission with an 11 GHz channel spacing,” Electron. Lett. 21, 656–657 (1985).
[Crossref]

van der Ziel, J. P.

N. A. Olsson and J. P. van der Ziel, “Characteristics of a semiconductor laser pumped Brillouin amplifier with electronically controlled bandwidth,” IEEE J. Lightwave Technol. LT-5, 147–153 (1987).
[Crossref]

van Driel, H. M.

P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fiber,” Phys. Rev. B 19, 4986–4998 (1979).
[Crossref]

Waarts, R. G.

N. Shibata, R. G. Waarts, and R. P. Braun, “Brillouin-gain spectra for single-mode fibers having pure-silica, GeO2-doped, and P2O5-doped cores,” Opt. Lett. 12, 269–271 (1987).
[Crossref] [PubMed]

N. Shibata, R. P. Braun, and R. G. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1205–1210 (1987).
[Crossref]

Appl. Opt. (1)

Appl. Phys. Lett. (1)

E. P. Ippen and R. H. Stolen, “Stimulated Brillouin scattering in optical fibers,” Appl. Phys. Lett. 21, 539–541 (1972).
[Crossref]

Electron. Lett. (4)

H. Toba, K. Inoue, and K. Nosu, “450 Mbit/s optical frequency-division-multiplexing transmission with an 11 GHz channel spacing,” Electron. Lett. 21, 656–657 (1985).
[Crossref]

E. J. Bachus, R. P. Braun, W. Eutin, E. Grossmann, H. Foisel, K. Heimes, and B. Strebel, “Coherent optical fiber subscriber line,” Electron. Lett. 21, 1203–1205 (1985).
[Crossref]

R. W. Tkach, A. R. Charaplyvy, and R. M. Derosier, “Spontaneous Brillouin scattering for single-mode optical fibre characterization,” Electron. Lett. 22, 1011–1013 (1986).
[Crossref]

A. R. Charaplyvy and R. W. Tkach, “Narrowband tunable optical filter for channel selection in density packed WDM systems,” Electron. Lett. 22, 1084–1085 (1986).
[Crossref]

IEEE J. Lightwave Technol. (2)

D. W. Smith, “Techniques for multigigabit coherent optical transmission,” IEEE J. Lightwave Technol. LT-5, 1466–1478 (1987).
[Crossref]

N. A. Olsson and J. P. van der Ziel, “Characteristics of a semiconductor laser pumped Brillouin amplifier with electronically controlled bandwidth,” IEEE J. Lightwave Technol. LT-5, 147–153 (1987).
[Crossref]

IEEE J. Quantum Electron. (1)

N. Shibata, R. P. Braun, and R. G. Waarts, “Phase-mismatch dependence of efficiency of wave generation through four-wave mixing in a single-mode optical fiber,” IEEE J. Quantum Electron. QE-23, 1205–1210 (1987).
[Crossref]

IEEE Trans. Ultrason. Ferroelectr. Freq. Control (1)

C. K. Jen, A. Safaai-Jazi, and G. W. Farnell, “Leaky modes in weakly guiding fiber acoustic waveguides,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control UFFC-33, 634–643 (1986).

J. Acoust. Soc. Am. (2)

R. N. Thurston, “Elastic waves in rods and clad rods,” J. Acoust. Soc. Am. 64, 1–37 (1978).
[Crossref]

A. E. Armenàkas, “Propagation of harmonic waves in composite circular-cylindrical rods,” J. Acoust. Soc. Am. 47, 822–837 (1970).
[Crossref]

J. Opt. Commun. (1)

D. Cotter, “Stimulated Brillouin scattering in monomode optical fibre,” J. Opt. Commun. 1, 10–19 (1983).

Opt. Lett. (2)

Phys. Rev. B (2)

R. M. Shelby, M. D. Levenson, and P. W. Bayer, “Guided acoustic-wave Brillouin scattering,” Phys. Rev. B 31, 5244–5252 (1985).
[Crossref]

P. J. Thomas, N. L. Rowell, H. M. van Driel, and G. I. Stegeman, “Normal acoustic modes and Brillouin scattering in single-mode optical fiber,” Phys. Rev. B 19, 4986–4998 (1979).
[Crossref]

Other (1)

T. Okoshi, Optical Fibers (Academic, New York, 1982), Chap. 4.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Refractive-index profiles for test single-mode fibers 1–6.

Fig. 2
Fig. 2

Experimental arrangement for measuring Brillouin-gain spectra. Probe and pump sources are distributed-feedback lasers operating at a wavelength of 1550 nm; PC1 and PC2 are polarization controllers.

Fig. 3
Fig. 3

Brillouin-gain spectra for the test fibers 1, 4, and 6.

Fig. 4
Fig. 4

Brillouin-gain spectra for the sequentially spliced transmission line composed of the fibers 1–3.

Fig. 5
Fig. 5

Brillouin frequency-shift deviation δvB as a function of F concentration CF.

Fig. 6
Fig. 6

Phase-velocity characteristics for the longitudinal L0m modes. Open and filled circles are measured points.

Equations (62)

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

E ˜ ( r , θ , z , t ) = v = 1 2 [ a v ( z ) E v ( i ) + b v ( z ) E v ( s ) ] ,
H ˜ ( r , θ , z , t ) = v = 1 2 [ a v ( z ) H v ( 1 ) + b v ( z ) H v ( s ) ] ,
E v ( i ) = e v ( r , θ ) exp { j [ ω i t - β v ( i ) z ] }             ( v = 1 , 2 ) ,
H v ( i ) = h v ( r , θ ) exp { j [ ω i t - β v ( i ) z ] }             ( v = 1 , 2 )
E v ( s ) = e v ( r , θ ) exp { j [ ω s t - β v ( s ) z ] }             ( v = 1 , 2 ) ,
H v ( s ) = h v ( r , θ ) exp { j [ ω s t - β v ( s ) z ] }             ( v = 1 , 2 ) ,
× E ˜ = - μ 0 H ˜ t ,
× H ˜ = ˜ E ˜ t .
d a μ d z + j ω s v = 1 2 b v 0 2 π 0 E μ ( i ) * · ( ˜ - ) E v ( s ) r d r d θ = 0             ( μ = 1 , 2 ) ,
β μ ( i ) β μ ( i ) d b μ d z + j ω i ( s ) v = 1 2 a v 0 2 π 0 E μ ( s ) * · ( ˜ - ) E v ( i ) r d r d θ = 0             ( μ = 1 , 2 ) ,
0 2 π 0 u ^ z · [ E v ( p ) × H μ ( q ) * - H v ( p ) × E μ ( q ) * ] r d r d θ = β v ( s ) β v ( s ) δ v μ δ ( ω p - ω q ) ,
δ = ( ˜ - ) = ½ X ( r , θ ) exp j ( Ω t - K z ) + c . c .
X r r = - 0 n 4 ( p 11 S r r + p 12 S θ θ + p 12 S z z ) ,
X θ θ = - 0 n 4 ( p 11 S r r + p 12 S θ θ + p 12 S z z ) ,
X r θ = - 0 n 4 p 44 S r θ ,
S r r = u r r ,
S θ θ = u r r + 1 r u θ θ ,
S z z = u z z ,
S r θ = 1 r u r θ + u θ r - u θ r ,
d b μ ( + ) d z = - j ω i 2 v = 1 2 a v exp { j ( ω i + Ω - ω s ) t - j × [ β v ( i ) + K - β μ ( i ) ] z } 0 2 π 0 e * μ · X e v r d r d θ - j ω i 2 v = 1 2 a v exp { j ( ω i - Ω - ω s ) t - j × [ β v ( i ) - K - β μ ( i ) ] z } 0 2 π 0 e * μ · X * e v r d r d θ             ( μ = 1 , 2 ) ,
d b μ ( - ) d z = j ω i 2 v = 1 2 a v exp { j ( ω i + Ω - ω s ) t - j × [ β v ( i ) + K + β μ ( i ) ] z } 0 2 π 0 e * μ · X e v r d r d θ + j ω i 2 v = 1 2 a v exp { j ( ω i - Ω - ω s ) t - j × [ β v ( i ) - K - β μ ( i ) ] z } 0 2 π 0 e * μ · X * e v r d r d θ ( μ = 1 , 2 ) ,
ω s = ω i + Ω .
b μ ( + ) ( 0 ) = 0 ,
b μ ( - ) ( L ) = 0.
b μ ( + ) ( z ) = - j ω i 2 v = 1 2 a v 0 z exp [ - j ( β v - K - β μ ) z ] d z × 0 2 π 0 e μ * · X * e v r d r d θ             ( μ = 1 , 2 )
b μ ( - ) ( z ) = - j ω i 2 v = 1 2 a v z z + L exp [ - j ( β v - K + β μ ) z ] d z × 0 2 π 0 e μ * · X * e v r d r d θ             ( μ = 1 , 2 ) ;
b μ ( + ) ( L ) = - j ω i 2 L v = 1 2 a v 0 2 π 0 e μ * · X * ( K = β v - β μ ) e v r d r d θ             ( μ = 1 , 2 ) ,
b μ ( - ) ( 0 ) = - j ω i 2 L v = 1 2 a v 0 2 π 0 e μ * · X * ( K = β v + β μ ) e v r d r d θ             ( μ = 1 , 2 ) .
e μ * · X * e v = X r r * e μ r * e v r + X θ r * ( e μ r * e v θ + e μ θ * e v r ) + X θ θ * e μ θ * e v θ .
b 1 ( + ) ( L ) = - j ω i 2 L a 1 0 2 π 0 e 2 * · X * ( K = 0 ) e 1 r d r d θ ,
b 2 ( + ) ( L ) = - j ω i 2 L a 1 0 2 π 0 e 2 * · X * ( K = β 1 - β 2 ) e 1 r d r d θ ,
b 1 ( - ) ( 0 ) = - j ω i 2 L a 1 0 2 π 0 e 1 * · X * ( K = 2 β 1 ) e 1 r d r d θ ,
b 2 ( - ) ( 0 ) = - j ω i 2 L a 1 0 2 π 0 e 2 * · X * ( K = β 1 + β 2 ) e 1 r d r d θ .
e 1 * · X * e 1 = ½ R 2 ( r ) [ ( x r r * + X θ θ * ) + ( X r r * - X θ θ * ) × cos 2 θ - 2 X r θ * sin 2 θ ] ,
e 2 * · X * e 1 = ½ R 2 ( r ) [ ( X r r * - X θ θ * ) sin 2 θ + 2 X r θ * cos 2 θ ] ,
( X r r * + X θ θ * ) = - 0 n 4 [ ( p 11 + p 12 ) ( S r r * + S θ θ * ) + 2 p 12 S z z * ] U n m ( r ) cos ( n θ - Ψ 0 ) ,
( X r r * - X θ θ * ) = - 0 n 4 ( p 11 - p 12 ) ( S r r * + S θ θ * ) V n m ( r ) cos ( n θ - Ψ 0 ) ,
X r θ * = - 0 n 4 p 44 S r θ * W n m ( r ) sin ( n θ - Ψ 0 ) .
b 1 ( - ) ( 0 ) = - j ω i 2 L a 1 0 2 π 0 e 1 * · X * ( K = 2 β ) e 1 r d r d θ = ( - j ω i 2 L a 1 ) { [ 0 1 2 R 2 ( r ) U n m ( r ) r d r ] × [ 0 2 π cos ( n θ - Ψ 0 ) d θ ] + [ 0 1 2 R 2 ( r ) V n m ( r ) r d r ] × [ 0 2 π cos ( n θ - Ψ 0 ) cos 2 θ d θ ] - [ 0 R 2 ( r ) W n m ( r ) r d r ] × [ 0 2 π sin ( n θ - Ψ 0 ) sin 2 θ d θ ] }
b 2 ( - ) ( 0 ) = - j ω i 2 L a 1 0 2 π 0 e 2 * · X * ( K = 2 β ) e 1 r d r d θ = ( - j ω i 2 L a 1 ) { [ 0 1 2 R 2 ( r ) V n m ( r ) r d r ] × [ 0 2 π cos ( n θ - Ψ 0 ) sin 2 θ d θ ] - [ 0 R 2 ( r ) W n m ( r ) r d r ] × [ 0 2 π sin ( n θ - Ψ 0 ) cos 2 θ d θ ] } ,
R ( r ) = A J 0 ( u r / a )             for r < a , = A J 0 ( u ) K 0 ( w r / a ) / K 0 ( w )             for r > a .
b 1 ( - ) ( 0 ) = - j π ω i 2 L a 1 [ 0 R 2 ( r ) U 0 m ( r ) r d r ] ,
b 2 ( - ) ( 0 ) = 0 ,
b 1 ( - ) ( 0 ) = - j π ω i 4 L a 1 cos Ψ 0 × { 0 R 2 ( r ) [ V 2 m ( r ) - 2 W 2 m ( r ) ] r d r } ,
b 2 ( - ) ( 0 ) = - j π ω i 4 L a 1 sin Ψ 0 × { 0 R 2 ( r ) [ V 2 m ( r ) - 2 W 2 m ( r ) ] r d r } .
u r = [ d f d r + K g r ( r ) ] exp [ j ( Ω t - K z ) ] ,
u θ = 0 ,
u z = - j [ K f ( r ) + d g r d r + g r r ] exp [ j ( Ω t - K z ) ]
u r = [ d f d r + K g r ( r ) + 2 g z r ] cos ( 2 θ - Ψ 0 ) exp [ j ( Ω t - K z ) ] ,
u θ = [ - 2 r f ( r ) + K g r ( r ) - d g z d r ] sin ( 2 θ - Ψ 0 ) exp [ j ( Ω t - K z ) ] ,
u z = - j [ K f ( r ) + d g r d r + 3 g r r ] cos ( 2 θ - Ψ 0 ) exp [ j ( Ω t - K z ) ] ,
U 0 m ( r ) = - 0 n 4 { ( p 11 + p 12 ) [ d 2 f d r 2 + 1 r d f d r + K ( d g r d r + g r r ) ] - 2 p 12 K ( K f + d g r d r + g r r ) }
V 2 m ( r ) - 2 W 2 m ( r ) = - 0 n 4 ( p 11 - p 12 ) [ 2 r d f d r + ( 4 r 2 - ξ 2 ) f + 2 r d g z d r + ( 4 r 2 - η 2 ) g z ] ,
ξ 2 = Ω 2 V L 2 - K 2
η 2 = Ω 2 V S 2 - K 2 .
f ( r ) = A 1 J n ( ξ 1 r ) ,
g r ( r ) = B 1 J n + 1 ( η 1 r ) ,
g z ( r ) = C 1 J n ( η 1 r )
U 0 m ( r ) 2 0 n 4 p 12 K 2 A 1 J 0 ( ξ 1 r )
V 2 m ( r ) - 2 W 2 m ( r ) 0 n 4 ( p 11 - p 12 ) A 1 ξ 1 2 J 0 ( ξ 1 r ) .
{ [ b 1 ( - ) ( 0 ) / a 1 ] n = 2 / [ b 1 ( - ) ( 0 ) / a 1 ] n = 0 } 2 [ ( p 11 - p 12 4 p 12 ) ( ξ 2 K 2 ) ] 2 ,
V a = ( λ / 2 n ) v B .

Metrics