Abstract

A new type of optical-fiber surface-plasmon-resonance (SPR) sensor based on a thin metallic film and long-period fiber gratings for measuring small changes of refractive index of analyte is presented. This sensor simply employs a long-period fiber grating with a proper period to couple a core mode (HE11) to the copropagating cladding mode that can excite a surface-plasmon wave (SPW). The mainly theoretical base used to analyze this new structure is the unconjugated form of coupled-mode equations. In this new SPR sensor, the variation of the refractive index of analyte is determined by monitoring the change of the transmitted core mode power, which is calculated by unconjugated two-mode coupled-mode equations at a fixed wavelength. The numerical results have demonstrated that this new and simple configuration may be used as a highly sensitive amplitude sensor. As far as the excitation of SPW, the model of numerical simulation, and the complexity of measurement equipment are concerned, this new structure is superior to the proposed sensor, consisting of a bent polished single-mode SPR optical fiber. Furthermore, the structure can be easily adapted for a SPR fiber optical probe if a mirror is deposited on the fiber tip.

© 2006 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  11. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

2003

2002

1999

R. Slavík, J. Homola, and J. Ctyroký, "Single-mode optical fiber surface plasmon resonance sensor," Sens. Actuators B 54, 74-79 (1999).
[CrossRef]

1997

S.-M. Tseng, K.-Y. Hsu, H.-S. Wei, and K.-F. Chen, "Analysis and experiment of thin metal-clad fiber polarizer with index overlay," IEEE Photonics Technol. Lett. 9, 628-630 (1997).
[CrossRef]

A. J. C. Tubb, F. P. Payne, R. B. Millington, and C. R. Lowe, "Single-mode optical fibre surface plasma wave chemical sensor," Sens. Actuators B 41, 71-79 (1997).
[CrossRef]

S. Maruo, O. Nakamura, and S. Kawata, "Evanescent-wave holography by use of surface-plasmon resonance," Appl. Opt. 36, 2343-2346 (1997).
[CrossRef] [PubMed]

T. Erdogan, "Cladding-mode resonances in short and long period fiber grating filters," J. Opt. Soc. Am. A 14, 1760-1773 (1997).
[CrossRef]

1995

J. Homola, "Optical fiber sensor based on surface plasmon excitation," Sens. Actuators B 29, 401-405 (1995).
[CrossRef]

Alonso, R.

Chen, K.-F.

S.-M. Tseng, K.-Y. Hsu, H.-S. Wei, and K.-F. Chen, "Analysis and experiment of thin metal-clad fiber polarizer with index overlay," IEEE Photonics Technol. Lett. 9, 628-630 (1997).
[CrossRef]

Ctyroký, J.

R. Slavík, J. Homola, and J. Ctyroký, "Single-mode optical fiber surface plasmon resonance sensor," Sens. Actuators B 54, 74-79 (1999).
[CrossRef]

Erdogan, T.

Esteban, Ó.

González-Cano, A.

Homola, J.

R. Slavík, J. Homola, and J. Ctyroký, "Single-mode optical fiber surface plasmon resonance sensor," Sens. Actuators B 54, 74-79 (1999).
[CrossRef]

J. Homola, "Optical fiber sensor based on surface plasmon excitation," Sens. Actuators B 29, 401-405 (1995).
[CrossRef]

Hsu, K.-Y.

S.-M. Tseng, K.-Y. Hsu, H.-S. Wei, and K.-F. Chen, "Analysis and experiment of thin metal-clad fiber polarizer with index overlay," IEEE Photonics Technol. Lett. 9, 628-630 (1997).
[CrossRef]

Kabashin, A. V.

Kawata, S.

Love, J. D.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Lowe, C. R.

A. J. C. Tubb, F. P. Payne, R. B. Millington, and C. R. Lowe, "Single-mode optical fibre surface plasma wave chemical sensor," Sens. Actuators B 41, 71-79 (1997).
[CrossRef]

Luong, H. T.

Maruo, S.

Meunier, M.

Millington, R. B.

A. J. C. Tubb, F. P. Payne, R. B. Millington, and C. R. Lowe, "Single-mode optical fibre surface plasma wave chemical sensor," Sens. Actuators B 41, 71-79 (1997).
[CrossRef]

Nakamura, O.

Navarrete, M. C.

Patskovsky, S.

Payne, F. P.

A. J. C. Tubb, F. P. Payne, R. B. Millington, and C. R. Lowe, "Single-mode optical fibre surface plasma wave chemical sensor," Sens. Actuators B 41, 71-79 (1997).
[CrossRef]

Slavík, R.

R. Slavík, J. Homola, and J. Ctyroký, "Single-mode optical fiber surface plasmon resonance sensor," Sens. Actuators B 54, 74-79 (1999).
[CrossRef]

Snyder, A. W.

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

Tsao, C.

C. Tsao, Optical Fibre Waveguide Analysis (Oxford, 1992).

Tseng, S.-M.

S.-M. Tseng, K.-Y. Hsu, H.-S. Wei, and K.-F. Chen, "Analysis and experiment of thin metal-clad fiber polarizer with index overlay," IEEE Photonics Technol. Lett. 9, 628-630 (1997).
[CrossRef]

Tubb, A. J. C.

A. J. C. Tubb, F. P. Payne, R. B. Millington, and C. R. Lowe, "Single-mode optical fibre surface plasma wave chemical sensor," Sens. Actuators B 41, 71-79 (1997).
[CrossRef]

Wei, H.-S.

S.-M. Tseng, K.-Y. Hsu, H.-S. Wei, and K.-F. Chen, "Analysis and experiment of thin metal-clad fiber polarizer with index overlay," IEEE Photonics Technol. Lett. 9, 628-630 (1997).
[CrossRef]

Appl. Opt.

IEEE Photonics Technol. Lett.

S.-M. Tseng, K.-Y. Hsu, H.-S. Wei, and K.-F. Chen, "Analysis and experiment of thin metal-clad fiber polarizer with index overlay," IEEE Photonics Technol. Lett. 9, 628-630 (1997).
[CrossRef]

J. Lightwave Technol.

J. Opt. Soc. Am. A

Sens. Actuators B

A. J. C. Tubb, F. P. Payne, R. B. Millington, and C. R. Lowe, "Single-mode optical fibre surface plasma wave chemical sensor," Sens. Actuators B 41, 71-79 (1997).
[CrossRef]

J. Homola, "Optical fiber sensor based on surface plasmon excitation," Sens. Actuators B 29, 401-405 (1995).
[CrossRef]

R. Slavík, J. Homola, and J. Ctyroký, "Single-mode optical fiber surface plasmon resonance sensor," Sens. Actuators B 54, 74-79 (1999).
[CrossRef]

Other

C. Tsao, Optical Fibre Waveguide Analysis (Oxford, 1992).

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman and Hall, 1983).

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Figures (8)

Fig. 1
Fig. 1

Scheme of an optical-fiber SPR sensor based on two long-period fiber gratings and a deposited metallic film in multiplexing.

Fig. 2
Fig. 2

Plot of the azimuthal magnetic fields for the renumbered cladding modes: (a) odd modes, (b) even modes.

Fig. 3
Fig. 3

Coupling constants for all of the propagating modes at a wavelength of 1550 nm . The color of the colormap is used to represent the magnitudes of all of the coupling constants.

Fig. 4
Fig. 4

Cross-coupling constants for 75 renumbered cladding modes and SPW at a wavelength of 1550 nm : (a) k ν - co , and (b) k co - μ .

Fig. 5
Fig. 5

Plot of the effective refractive indices of the cladding modes versus the refractive index of the analyte: (a) odd modes and (b) even modes.

Fig. 6
Fig. 6

Plot of the power transmission versus the length of a long-period fiber grating for (a) LPG1 and (b) LPG2.

Fig. 7
Fig. 7

Plot of the power transmission versus the refractive index of analyte for range (a) from 1.35619 to 1.35659 and (b) from 1.36041 to 1.36081.

Fig. 8
Fig. 8

Comparison of unconjugated two-mode coupled-mode equations with unconjugated four-mode coupled-mode equations for range (a) from 1.35619 to 1.35659 and (b) from 1.36041 to 1.36081.

Equations (88)

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ζ co = ζ co ,
ζ co = U 2 [ J 1 ( u 1 a 1 ) U J 1 ( u 1 a 1 ) + K 1 ( w 1 a 1 ) W K 1 ( w 1 a 1 ) ] β co w ϵ 1 ( V 2 W 2 ) ,
ζ co = β co w μ ( V 2 W 2 ) U 2 [ J 1 ( u 1 a 1 ) U J 1 ( u 1 a 1 ) + ϵ 2 ϵ 1 K 1 ( w 1 a 1 ) W K 1 ( w 1 a 1 ) ] .
U = u 1 a 1 ,
W = w 2 a 1 ,
u 1 = ( 2 π λ ) ( n 1 2 n eff co 2 ) 1 2 ,
w 1 = ( 2 π λ ) ( n eff co 2 n 2 2 ) 1 2 .
ζ cl = ζ cl ,
ζ cl = R 1 H 1 + T 1 H 2 + U 1 H 3 V 1 H 4 R 1 M 1 T 1 M 2 U 1 M 3 + V 1 M 4 ,
ζ cl = R 2 H 3 + T 2 H 4 + U 2 H 1 V 2 H 2 R 2 M 3 T 2 M 4 U 2 M 1 + V 2 M 2 .
H 1 = [ β cl w ϵ 4 u 3 a 3 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) ( w 4 2 u 3 2 1 ) ] [ u 3 u 2 I 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) u 3 2 u 2 2 I 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) ] [ β cl w ϵ 4 u 3 a 3 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) ( w 4 2 u 3 2 1 ) ] [ u 3 u 2 K 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) u 3 2 u 2 2 K 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) ] + [ β cl w ϵ 3 u 2 a 2 I 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) ( u 3 2 u 2 2 1 ) ] [ w 4 u 3 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) w 4 2 ϵ 3 u 3 2 ϵ 4 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) ] [ β cl w ϵ 3 u 2 a 2 K 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) ( u 3 2 u 2 2 1 ) ] [ w 4 u 3 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) w 4 2 ϵ 3 u 3 2 ϵ 4 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) ] ,
H 2 = [ β cl w ϵ 4 u 3 a 3 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) ( w 4 2 u 3 2 1 ) ] [ u 3 u 2 I 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) u 3 2 u 2 2 I 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) ] [ β cl w ϵ 4 u 3 a 3 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) ( w 4 2 u 3 2 1 ) ] [ u 3 u 2 K 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) u 3 2 u 2 2 K 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) ] + [ β cl w ϵ 3 u 2 a 2 I 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) ( u 3 2 u 2 2 1 ) ] [ w 4 u 3 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) w 4 2 ϵ 3 u 3 2 ϵ 4 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) ] [ β cl w ϵ 3 u 2 a 2 K 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) ( u 3 2 u 2 2 1 ) ] [ w 4 u 3 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) w 4 2 ϵ 3 u 3 2 ϵ 4 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) ] ,
H 3 = [ β cl w μ u 2 a 2 I 1 ( w 3 a 2 ) K 1 ( u 2 a 2 ) ( u 3 2 u 2 2 1 ) ] [ β cl w ϵ 4 u 3 a 3 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) w 4 2 u 3 2 1 ] [ β cl w μ u 2 a 2 K 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) ( u 3 2 u 2 2 1 ) ] [ β cl w ϵ 4 u 3 a 3 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) ( w 4 2 u 3 2 1 ) ] + [ u 3 u 2 I 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) u 3 2 ϵ 2 u 2 2 ϵ 3 I 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) ] [ w 4 u 3 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) w 4 2 ϵ 3 u 3 2 ϵ 4 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) ] [ u 3 u 2 K 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) u 3 2 ϵ 2 u 2 2 ϵ 3 K 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) ] [ w 4 u 3 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) w 4 2 ϵ 3 u 3 2 ϵ 4 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) ] ,
H 4 = [ β cl w μ u 2 a 2 I 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) ( u 3 2 u 2 2 1 ) ] [ β cl w ϵ 4 u 3 a 3 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) ( w 4 2 u 3 2 1 ) ] [ β cl w μ u 2 a 2 K 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) ( u 3 2 u 2 2 1 ) ] [ β cl w ϵ 4 u 3 a 3 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) ( w 4 2 u 3 2 1 ) ] + [ u 3 u 2 I 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) u 3 2 ϵ 2 u 2 2 ϵ 3 I 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) ] [ w 4 u 3 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) w 4 2 ϵ 3 u 3 2 ϵ 4 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) ] [ u 3 u 2 K 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) u 3 2 ϵ 2 u 2 2 ϵ 3 K 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) ] [ w 4 u 3 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) w 4 2 ϵ 3 u 3 2 ϵ 4 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) ] ,
M 1 = [ β cl w ϵ 3 u 2 a 2 I 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) ( u 3 2 u 2 2 1 ) ] [ β cl w μ u 3 a 3 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) ( w 4 2 u 3 2 1 ) ] [ β cl w ϵ 3 u 2 a 2 K 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) ( u 3 2 u 2 2 1 ) ] [ β cl w μ u 3 a 3 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) ( w 4 2 u 3 2 1 ) ] + [ u 3 u 2 I 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) u 3 2 u 2 2 I 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) ] [ w 4 u 3 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) w 4 2 u 3 2 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) ] [ u 3 u 2 K 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) u 3 2 u 2 2 K 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) ] [ w 4 u 3 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) w 4 2 u 3 2 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) ] ,
M 2 = [ β cl w ϵ 3 u 2 a 2 I 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) ( u 3 2 u 2 2 1 ) ] [ β cl w μ u 3 a 3 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) ( w 4 2 u 3 2 1 ) ] [ β cl w ϵ 3 u 2 a 2 K 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) ( u 3 2 u 2 2 1 ) ] [ β cl w μ u 3 a 3 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) ( w 4 2 u 3 2 1 ) ] + [ u 3 u 2 I 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) u 3 2 u 2 2 I 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) ] [ w 4 u 3 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) w 4 2 u 3 2 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) ] [ u 3 u 2 K 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) u 3 2 u 2 2 K 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) ] [ w 4 u 3 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) w 4 2 u 3 2 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) ] ,
M 3 = [ β cl w μ u 3 a 3 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) ( w 4 2 u 3 2 1 ) ] [ u 3 u 2 I 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) u 3 2 ϵ 2 u 2 2 ϵ 3 I 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) ] [ β cl w μ u 3 a 3 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) ( w 4 2 u 3 2 1 ) ] [ u 3 u 2 K 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) u 3 2 ϵ 2 u 2 2 ϵ 3 K 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) ] + [ β cl w μ u 2 a 2 I 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) ( u 3 2 u 2 2 1 ) ] [ w 4 u 3 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) w 4 2 u 3 2 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) ] [ β cl w μ u 2 a 2 K 1 ( u 3 a 2 ) K 1 ( u 2 a 2 ) ( u 3 2 u 2 2 1 ) ] [ w 4 u 3 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) w 4 2 u 3 2 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) ] ,
M 4 = [ β cl w μ u 3 a 3 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) ( w 4 2 u 3 2 1 ) ] [ u 3 u 2 I 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) u 3 2 ϵ 2 u 2 2 ϵ 3 I 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) ] [ β cl w μ u 3 a 3 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) ( w 4 2 u 3 2 1 ) ] [ u 3 u 2 K 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) u 3 2 ϵ 2 u 2 2 ϵ 3 K 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) ] + [ β cl w μ u 2 a 2 I 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) ( u 3 2 u 2 2 1 ) ] [ w 4 u 3 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) w 4 2 u 3 2 K 1 ( w 4 a 3 ) K 1 ( u 3 a 3 ) ] [ β cl w μ u 2 a 2 K 1 ( u 3 a 2 ) I 1 ( u 2 a 2 ) ( u 3 2 u 2 2 1 ) ] [ w 4 u 3 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) w 4 2 u 3 2 K 1 ( w 4 a 3 ) I 1 ( u 3 a 3 ) ] ,
R 1 = u 2 2 u 1 2 I 1 ( u 2 a 1 ) I 1 ( u 1 a 1 ) u 2 u 1 I 1 ( u 2 a 1 ) I 1 ( u 1 a 1 ) ,
T 1 = u 2 2 u 1 2 K 1 ( u 2 a 1 ) I 1 ( u 1 a 1 ) u 2 u 1 K 1 ( u 2 a 1 ) I 1 ( u 1 a 1 ) ,
U 1 = β cl w ϵ 2 u 1 a 1 I 1 ( u 2 a 1 ) I 1 ( u 1 a 1 ) ( u 2 2 u 1 2 1 ) ,
V 1 = β cl w ϵ 2 u 1 a 1 K 1 ( u 2 a 1 ) I 1 ( u 1 a 1 ) ( u 2 2 u 1 2 1 ) ,
R 2 = u 2 2 ϵ 1 u 1 2 ϵ 2 I 1 ( u 2 a 1 ) I 1 ( u 1 a 1 ) u 2 u 1 I 1 ( u 2 a 1 ) I 1 ( u 1 a 1 ) ,
T 2 = u 2 2 ϵ 1 u 1 2 ϵ 2 K 1 ( u 2 a 1 ) I 1 ( u 1 a 1 ) u 2 u 1 K 1 ( u 2 a 1 ) I 1 ( u 1 a 1 ) ,
U 2 = β cl w μ u 1 a 1 I 1 ( u 2 a 1 ) I 1 ( u 1 a 1 ) ( u 2 2 u 1 2 1 ) ,
V 2 = β cl w μ u 1 a 1 K 1 ( u 2 a 1 ) I 1 ( u 1 a 1 ) ( u 2 2 u 1 2 1 ) ,
u 1 = ( 2 π λ ) ( n eff cl 2 n 1 2 ) 1 2 ,
u 2 = ( 2 π λ ) ( n eff cl 2 n 2 2 ) 1 2 ,
u 3 = ( 2 π λ ) ( n eff cl 2 n 3 2 ) 1 2 ,
w 4 = ( 2 π λ ) ( n eff cl 2 n 4 2 ) 1 2 .
n ( z ) = n 1 + n 1 σ [ 1 + cos ( 2 π z Λ ) ] ,
d A μ d z = i ν A ν k ν μ [ 1 + cos ( 2 π z Λ ) ] exp [ i ( β ν β μ ) z ] ,
k ν μ = w ϵ 0 n 1 2 σ 0 2 π 0 a 1 r ( E r ν E r μ + E φ ν E φ μ E z ν E z μ ) d r d φ 0 2 π 0 r ( E r μ H φ μ E φ μ H r μ ) d r d φ ,
K ν μ = k ν μ [ 1 + cos ( 2 π z Λ ) ] .
d A co ( z ) d z = i k co co A co ( z ) + i k spw co 2 A spw ( z ) exp ( i 2 δ ) ,
d A spw ( z ) d z = i k co spw 2 A co ( z ) exp ( i 2 δ ) + i k spw spw A spw ( z )
δ = 1 2 ( β co β spw 2 π Λ ) ,
T = ( z ) = A co ( z ) 2 A co ( 0 ) 2 ,
T × ( z ) = A spw ( z ) exp ( i β spw z ) 2 A co ( 0 ) 2 .
Λ = Re ( 2 π k co co k spw spw + β co β spw ) .
d A co ( z ) d z = i k co co A co ( z ) [ 1 + cos ( 2 π z Λ ) ] + i k 71 co A 71 ( z ) [ 1 + cos ( 2 π z Λ ) ] exp [ i ( β 71 β co ) z ] + i k spw co A spw ( z ) [ 1 + cos ( 2 π z Λ ) ] exp [ i ( β spw β co ) z ] + i k 72 co A 72 ( z ) [ 1 + cos ( 2 π z Λ ) ] exp [ i ( β 72 β co ) z ] ,
d A 71 ( z ) d z = i k co 71 A co ( z ) [ 1 + cos ( 2 π z Λ ) ] exp [ i ( β co β 71 ) z ] + i k 71 71 A 71 ( z ) [ 1 + cos ( 2 π z Λ ) ] + i k spw 71 A spw ( z ) [ 1 + cos ( 2 π z Λ ) ] exp [ i ( β spw β 71 ) z ] + i k 72 71 A 72 ( z ) [ 1 + cos ( 2 π z Λ ) ] exp [ i ( β 72 β 71 ) z ] ,
d A spw ( z ) d z = i K co spw A co ( z ) [ 1 + cos ( 2 π z Λ ) ] exp [ i ( β co β spw ) z ] + i K 71 spw A 71 ( z ) [ 1 + cos ( 2 π z Λ ) ] exp [ i ( β 71 β spw ) z ] + i K spw spw A spw ( z ) [ 1 + cos ( 2 π z Λ ) ] + i K 72 spw A 72 ( z ) [ 1 + cos ( 2 π z Λ ) ] exp [ i ( β 72 β spw ) z ] ,
d A 72 ( z ) d z = i K co 72 A co ( z ) [ 1 + cos ( 2 π z Λ ) ] exp [ i ( β co β 72 ) z ] + i K 71 72 A 71 ( z ) [ 1 + cos ( 2 π z Λ ) ] exp [ i ( β 71 β 72 ) z ] + i K spw 72 A spw ( z ) [ 1 + cos ( 2 π z Λ ) ] exp [ i ( β spw β 72 ) z ] + i K 72 72 A 72 ( z ) [ 1 + cos ( 2 π z Λ ) ] .
E r = [ 1 r A 1 co J 1 ( u 1 co r ) β co w ϵ 1 C 1 co u 1 co J 1 ( u 1 co r ) ] cos ( ϕ ) ,
E ϕ = [ A 1 co u 1 co J 1 ( u 1 co r ) β co w ϵ 1 1 r C 1 co J 1 ( u 1 co r ) ] sin ( ϕ ) ,
E z = u 1 co 2 j w ϵ 1 C 1 co J 1 ( u 1 co r ) cos ( ϕ ) ,
H r = [ 1 r C 1 co J 1 ( u 1 co r ) + β co w μ A 1 co u 1 co J 1 ( u 1 co r ) ] sin ( ϕ ) ,
H ϕ = [ C 1 co u 1 co J 1 ( u 1 co r ) β co w μ 1 r A 1 co J 1 ( u 1 co r ) ] cos ( ϕ ) ,
H z = u 1 co 2 j w μ A 1 co J 1 ( u 1 co r ) sin ( ϕ )
E r = [ 1 r B 2 co K 1 ( w 2 co r ) β co w ϵ 2 D 2 co w 2 co K 1 ( w 2 co r ) ] cos ( ϕ ) ,
E ϕ = [ B 2 co w 2 co K 1 ( w 2 co r ) β co w ϵ 2 1 r D 2 co K 1 ( w 2 co r ) ] sin ( ϕ ) ,
E z = w 2 co 2 j w ϵ 2 D 2 co K 1 ( w 2 co r ) cos ( ϕ ) ,
H r = [ 1 r D 2 co K 1 ( w 2 co r ) + β co w μ B 2 co w 2 co K 1 ( w 2 co r ) ] sin ( ϕ ) ,
H ϕ = [ D 2 co w 2 co K 1 ( w 2 co r ) β co w μ 1 r B 2 co K 1 ( w 2 co r ) ] cos ( ϕ ) ,
H z = w 2 co 2 j w μ B 2 co K 1 ( w 2 co r ) sin ( ϕ )
E r = [ 1 r C 1 cl I 1 ( u 1 cl r ) β cl w ϵ 1 A 1 cl u 1 cl I 1 ( u 1 cl r ) ] cos ( ϕ ) ,
E ϕ = [ C 1 cl u 1 cl I 1 ( u 1 cl r ) β cl w ϵ 1 1 r A 1 cl I 1 ( u 1 cl r ) ] sin ( ϕ ) ,
E z = u 1 cl 2 j w ϵ 1 A 1 cl I 1 ( u 1 cl r ) cos ( ϕ ) ,
H r = [ 1 r A 1 cl I 1 ( u 1 cl r ) + β cl w μ C 1 cl u 1 cl I 1 ( u 1 cl r ) ] sin ( ϕ ) ,
H ϕ = { A 1 cl u 1 cl I 1 ( u 1 r ) β cl w μ 1 r C 1 cl I 1 ( u 1 r ) } cos ( ϕ ) ,
H z = u 1 cl 2 j w μ C 1 cl I 1 ( u 1 cl r ) sin ( ϕ )
E r = { 1 r [ C 2 cl I 1 ( u 2 cl r ) + D 2 cl K 1 ( u 2 cl r ) ] β cl w ϵ 2 u 2 cl [ A 2 cl I 1 ( u 2 cl r ) + B 2 cl K 1 ( u 2 cl r ) ] } cos ( ϕ ) ,
E ϕ = { u 2 cl [ C 2 cl I 1 ( u 2 cl r ) + D 2 cl K 1 ( u 2 cl r ) ] β cl w ϵ 2 1 r [ A 2 cl I 1 ( u 2 cl r ) + B 2 cl K 1 ( u 2 cl r ) ] } sin ( ϕ ) ,
E z = u 2 cl 2 j w ϵ 2 [ A 2 cl I 1 ( u 2 cl r ) + B 2 cl K 1 ( u 2 cl r ) ] cos ( ϕ ) ,
H r = { 1 r [ A 2 cl I 1 ( u 2 cl r ) + B 2 cl K 1 ( u 2 cl r ) ] + β cl w μ u 2 cl [ C 2 cl I 1 ( u 2 cl r ) + D 2 cl K 1 ( u 2 cl r ) ] } sin ( ϕ ) ,
H ϕ = { u 2 cl [ A 2 cl I 1 ( u 2 cl r ) + B 2 cl K 1 ( u 2 cl r ) ] β cl w μ 1 r [ C 2 cl I 1 ( u 2 cl r ) + D 2 cl K 1 ( u 2 cl r ) ] } cos ( ϕ ) ,
H z = u 2 cl 2 j w μ [ C 2 cl I 1 ( u 2 cl r ) + D 2 cl K 1 ( u 2 cl r ) ] sin ( ϕ )
E r = { 1 r [ C 3 cl I 1 ( u 3 cl r ) + D 3 cl K 1 ( u 3 cl r ) ] β cl w ϵ 3 u 3 cl [ A 3 cl I 1 ( u 3 cl r ) + B 3 cl K 1 ( u 3 cl r ) ] } cos ( ϕ ) ,
E ϕ = { u 3 cl [ C 3 cl I 1 ( u 3 cl r ) + D 3 cl K 1 ( u 3 cl r ) ] β cl w ϵ 3 1 r [ A 3 cl I 1 ( u 3 cl r ) + B 3 cl K 1 ( u 3 cl r ) ] } sin ( ϕ ) ,
E z = u 3 cl 2 j w ϵ 3 [ A 3 cl I 1 ( u 3 cl r ) + B 3 cl K 1 ( u 3 cl r ) ] cos ( ϕ ) ,
H r = { 1 r [ A 3 cl I 1 ( u 3 cl r ) + B 3 cl K 1 ( u 3 cl r ) ] + β cl w μ u 3 cl [ C 3 cl I 1 ( u 3 cl r ) + D 3 cl K 1 ( u 3 cl r ) ] } sin ( ϕ ) ,
H ϕ = { u 3 cl [ A 3 cl I 1 ( u 3 cl r ) + B 3 cl K 1 ( u 3 cl r ) ] β cl w μ 1 r [ C 3 cl I 1 ( u 3 cl r ) + D 3 cl K 1 ( u 3 cl r ) ] } cos ( ϕ ) ,
H z = u 3 cl 2 j w μ [ C 3 cl I 1 ( u 3 cl r ) + D 3 cl K 1 ( u 3 cl r ) ] sin ( ϕ )
E r = [ 1 r D 4 cl K 1 ( w 4 cl r ) β cl w ϵ 4 B 4 cl w 4 cl K 1 ( w 4 cl r ) ] cos ( ϕ ) ,
E ϕ = [ D 4 cl w 4 cl K 1 ( w 4 cl r ) β cl w ϵ 4 1 r B 4 cl K 1 ( w 4 cl r ) ] sin ( ϕ ) ,
E z = w 4 cl 2 j w ϵ 4 B 4 cl K 1 ( w 4 cl r ) cos ( ϕ ) ,
H r = [ 1 r B 4 cl K 1 ( w 4 cl r ) + β cl w μ D 4 cl w 4 cl K 1 ( w 4 cl r ) ] sin ( ϕ ) ,
H ϕ = [ B 4 cl w 4 cl K 1 ( w 4 cl r ) β cl w μ 1 r D 4 cl K 1 ( w 4 cl r ) ] cos ( ϕ ) ,
H z = w 4 cl 2 j w μ D 4 cl K 1 ( w 4 cl r ) sin ( ϕ )
z A ( E 1 × H 2 E 2 × H 1 ) z ̂ d A = A ( E 1 × H 2 E 2 × H 1 ) d A ,
A E t ν × H t μ z ̂ d A = A E t μ × H t ν z ̂ d A = 0 for β ν ± β μ .
E 1 = ν A ν ( z ) E t ν ( r , ϕ ) exp ( i β v z ) + Δ ϵ ϵ Δ ϵ + ϵ ν A ν ( r , ϕ ) E z ν ( r , ϕ ) exp ( i β v z ) ,
H 1 = ν A ν ( z ) H t ν ( r , ϕ ) exp ( i β v z ) + ν A ν ( z ) H z ν ( r , ϕ ) exp ( i β ν z ) .
E 2 = [ E t μ ( r , ϕ ) E z μ ( r , ϕ ) ] exp ( i β μ z )
H 2 = [ H t μ ( r , ϕ ) + H z μ ( r , ϕ ) ] exp ( i β μ z ) .
d A μ d z = i ν A ν K ν μ exp [ i ( β ν β μ ) z ] ( ν and μ are any guiding modes in the fiber ) ,
K ν μ = w 2 Δ ϵ A r ( E r ν E r μ + E ϕ ν E ϕ μ ) d A Δ ϵ ( Δ ϵ + ϵ ) A r ( E z ν E z μ ) d A 0 2 π 0 r ( E r μ H ϕ μ E ϕ μ H r μ ) d r d ϕ .

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