Abstract

By using the Debye potentials, the exact eigenvalue equations and the corresponding field distributions of the core and cladding modes for three-layered, radially stratified, and dielectric unixial optical fibers are derived completely; the modes include TE, TM, and hybrid HE/EH modes. The strain characteristics of long-period fiber gratings’ (LPFGs’) with applied axial strain are investigated theoretically by studying three-layered unaxial optical fibers. When uniform axial strain is applied to fiber, the core, and inner and outer cladding become uniaxial crystal optically; that is, the optical axes are parallel to the fiber’s axis. Analytic expressions of the strain sensitivities of LPFGs are derived. The analysis reveals that the strain sensitivities of LPFGs based on various cladding modes, including the shift value and direction of the resonance wavelength, differ greatly.

© 2005 Optical Society of America

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  1. A. W. Snyder, F. Ruhl, “Single-model, single-polarization fiber made of birefringent material,” J. Opt. Soc. Am. 73, 1165–1174 (1983).
    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  4. A. W. Snyder, F. Ruhl, “Anisotropic fibers studied by the Green’s function method,” J. Lightwave Technol. 3, 284–191 (1984).
  5. M. Geshiro, M. Hotta, T. Kameshima, “Coupled-mode analysis of leaky waves in channel waveguides consisting of anisotropic material,” IEEE Trans. Microwave Theory Tech. 41, 1159–1163 (1993).
    [CrossRef]
  6. N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, 1972), pp. 142–150.
  7. A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983), pp. 272–279.
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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2003 (1)

2002 (1)

2000 (2)

Y. Jeong, B. Yang, B. Lee, H. S. Seo, S. Choi, K. Oh., “Electrically controllable long-period liquid crystal fiber gratings,” IEEE Photonics Technol. Lett. 12, 519–521 (2000).
[CrossRef]

T. Erdogan, “Cladding-mode resonance in short- and long-period fiber grating filter: errata,” J. Opt. Soc. Am. A 17, 2113 (2000).
[CrossRef]

1997 (3)

K. O. Hill, G. Meltz, “Fiber bragg grating technology fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

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

1996 (3)

V. Bhatia, A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21, 692–694 (1996).
[CrossRef] [PubMed]

A. Sneh, K. Johnson, “High-speed continuously tunable liquid crystal filter for WDM networks,” J. Lightwave Technol. 14, 1067–1079 (1996).
[CrossRef]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. SIpe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

1993 (1)

M. Geshiro, M. Hotta, T. Kameshima, “Coupled-mode analysis of leaky waves in channel waveguides consisting of anisotropic material,” IEEE Trans. Microwave Theory Tech. 41, 1159–1163 (1993).
[CrossRef]

1991 (1)

J. D. Dai, C. K. Jen, “Analysis of cladded uniaxial single-crystal fibers,” J. Opt. Soc. Am. A 8, 2022–2025 (1991).
[CrossRef]

1989 (2)

C. Tsao, D. Payne, W. Gambling, “Modal characteristics of the three-layered optical fiber waveguide: a modified approach,” J. Opt. Soc. Am. A 17, 555–563 (1989).
[CrossRef]

C. Shi, “Leaky mods in anisotropic optical fibers with finite cladding,” J. Opt. Soc. Am. A 6, 550–554 (1989).
[CrossRef]

1986 (1)

1984 (1)

A. W. Snyder, F. Ruhl, “Anisotropic fibers studied by the Green’s function method,” J. Lightwave Technol. 3, 284–191 (1984).

1983 (1)

1982 (1)

A. Toning, “Circular symmetric optical waveguide with strong anisotropy,” IEEE Trans. Microwave Theory Tech. 30, 790–784 (1982).
[CrossRef]

1979 (1)

1978 (1)

Ankiewicz, A.

Askins, C. G.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Bennion, I.

Bhatia, V.

V. Bhatia, A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21, 692–694 (1996).
[CrossRef] [PubMed]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. SIpe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Burke, J. J.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, 1972), pp. 142–150.

Butter, C. D.

Choi, S.

Y. Jeong, B. Yang, B. Lee, H. S. Seo, S. Choi, K. Oh., “Electrically controllable long-period liquid crystal fiber gratings,” IEEE Photonics Technol. Lett. 12, 519–521 (2000).
[CrossRef]

Cielo, P. G.

Dai, J. D.

J. D. Dai, C. K. Jen, “Analysis of cladded uniaxial single-crystal fibers,” J. Opt. Soc. Am. A 8, 2022–2025 (1991).
[CrossRef]

Davis, M. A.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Douay, M.

O. Duhem, J. F. Henninot, M. Warenghem, M. Douay, L. Rivoallim, “Long period fiber gratings modulation by liquid crystal cladding,” in Telecommunications 1998, 6th IEE Conference, Conf. Publ. No. 451, pp 195–197 (IEE, 1998)

Duhem, O.

O. Duhem, J. F. Henninot, M. Warenghem, M. Douay, L. Rivoallim, “Long period fiber gratings modulation by liquid crystal cladding,” in Telecommunications 1998, 6th IEE Conference, Conf. Publ. No. 451, pp 195–197 (IEE, 1998)

Erdogan, T.

Friebele, E. J.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Gambling, W.

C. Tsao, D. Payne, W. Gambling, “Modal characteristics of the three-layered optical fiber waveguide: a modified approach,” J. Opt. Soc. Am. A 17, 555–563 (1989).
[CrossRef]

Geshiro, M.

M. Geshiro, M. Hotta, T. Kameshima, “Coupled-mode analysis of leaky waves in channel waveguides consisting of anisotropic material,” IEEE Trans. Microwave Theory Tech. 41, 1159–1163 (1993).
[CrossRef]

Henninot, J. F.

O. Duhem, J. F. Henninot, M. Warenghem, M. Douay, L. Rivoallim, “Long period fiber gratings modulation by liquid crystal cladding,” in Telecommunications 1998, 6th IEE Conference, Conf. Publ. No. 451, pp 195–197 (IEE, 1998)

Hill, K. O.

K. O. Hill, G. Meltz, “Fiber bragg grating technology fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

Hocker, G. B.

Hotta, M.

M. Geshiro, M. Hotta, T. Kameshima, “Coupled-mode analysis of leaky waves in channel waveguides consisting of anisotropic material,” IEEE Trans. Microwave Theory Tech. 41, 1159–1163 (1993).
[CrossRef]

Ivanov, I. O.V.

Jen, C. K.

J. D. Dai, C. K. Jen, “Analysis of cladded uniaxial single-crystal fibers,” J. Opt. Soc. Am. A 8, 2022–2025 (1991).
[CrossRef]

Jeong, Y.

Y. Jeong, B. Yang, B. Lee, H. S. Seo, S. Choi, K. Oh., “Electrically controllable long-period liquid crystal fiber gratings,” IEEE Photonics Technol. Lett. 12, 519–521 (2000).
[CrossRef]

Johnson, K.

A. Sneh, K. Johnson, “High-speed continuously tunable liquid crystal filter for WDM networks,” J. Lightwave Technol. 14, 1067–1079 (1996).
[CrossRef]

Judkins, J. B.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. SIpe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Kameshima, T.

M. Geshiro, M. Hotta, T. Kameshima, “Coupled-mode analysis of leaky waves in channel waveguides consisting of anisotropic material,” IEEE Trans. Microwave Theory Tech. 41, 1159–1163 (1993).
[CrossRef]

Kapany, N. S.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, 1972), pp. 142–150.

Kersey, A. D.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Koo, K. P.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

LeBlanc, M.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Lee, B.

Y. Jeong, B. Yang, B. Lee, H. S. Seo, S. Choi, K. Oh., “Electrically controllable long-period liquid crystal fiber gratings,” IEEE Photonics Technol. Lett. 12, 519–521 (2000).
[CrossRef]

Lemaire, P. J.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. SIpe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Love, J. D.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983), pp. 272–279.

Meltz, G.

K. O. Hill, G. Meltz, “Fiber bragg grating technology fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

Oh., K.

Y. Jeong, B. Yang, B. Lee, H. S. Seo, S. Choi, K. Oh., “Electrically controllable long-period liquid crystal fiber gratings,” IEEE Photonics Technol. Lett. 12, 519–521 (2000).
[CrossRef]

Patrick, H. J.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Payne, D.

C. Tsao, D. Payne, W. Gambling, “Modal characteristics of the three-layered optical fiber waveguide: a modified approach,” J. Opt. Soc. Am. A 17, 555–563 (1989).
[CrossRef]

Putnam, M. A.

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

Rivoallim, L.

O. Duhem, J. F. Henninot, M. Warenghem, M. Douay, L. Rivoallim, “Long period fiber gratings modulation by liquid crystal cladding,” in Telecommunications 1998, 6th IEE Conference, Conf. Publ. No. 451, pp 195–197 (IEE, 1998)

Ruhl, F.

A. W. Snyder, F. Ruhl, “Anisotropic fibers studied by the Green’s function method,” J. Lightwave Technol. 3, 284–191 (1984).

A. W. Snyder, F. Ruhl, “Single-model, single-polarization fiber made of birefringent material,” J. Opt. Soc. Am. 73, 1165–1174 (1983).
[CrossRef]

Seo, H. S.

Y. Jeong, B. Yang, B. Lee, H. S. Seo, S. Choi, K. Oh., “Electrically controllable long-period liquid crystal fiber gratings,” IEEE Photonics Technol. Lett. 12, 519–521 (2000).
[CrossRef]

Shi, C.

Shu, X. W.

SIpe, J. E.

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. SIpe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Sneh, A.

A. Sneh, K. Johnson, “High-speed continuously tunable liquid crystal filter for WDM networks,” J. Lightwave Technol. 14, 1067–1079 (1996).
[CrossRef]

Snyder, A. W.

A. W. Snyder, A. Ankiewicz, “Polarizing anisotropic fibers and non-aligned optical (stress) axes,” J. Opt. Soc. Am. A 3, 856–863 (1986).
[CrossRef]

A. W. Snyder, F. Ruhl, “Anisotropic fibers studied by the Green’s function method,” J. Lightwave Technol. 3, 284–191 (1984).

A. W. Snyder, F. Ruhl, “Single-model, single-polarization fiber made of birefringent material,” J. Opt. Soc. Am. 73, 1165–1174 (1983).
[CrossRef]

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983), pp. 272–279.

Toning, A.

A. Toning, “Circular symmetric optical waveguide with strong anisotropy,” IEEE Trans. Microwave Theory Tech. 30, 790–784 (1982).
[CrossRef]

Tsao, C.

C. Tsao, D. Payne, W. Gambling, “Modal characteristics of the three-layered optical fiber waveguide: a modified approach,” J. Opt. Soc. Am. A 17, 555–563 (1989).
[CrossRef]

C. Tsao, Optical Fiber Waveguide Analysis (Oxford U. Press, New York, 1992), pp. 298–369.

Vengsarkar, A. M.

V. Bhatia, A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21, 692–694 (1996).
[CrossRef] [PubMed]

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. SIpe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

Wang, L. A.

Warenghem, M.

O. Duhem, J. F. Henninot, M. Warenghem, M. Douay, L. Rivoallim, “Long period fiber gratings modulation by liquid crystal cladding,” in Telecommunications 1998, 6th IEE Conference, Conf. Publ. No. 451, pp 195–197 (IEE, 1998)

Yang, B.

Y. Jeong, B. Yang, B. Lee, H. S. Seo, S. Choi, K. Oh., “Electrically controllable long-period liquid crystal fiber gratings,” IEEE Photonics Technol. Lett. 12, 519–521 (2000).
[CrossRef]

Zhang, L.

Appl. Opt. (3)

IEEE Photonics Technol. Lett. (1)

Y. Jeong, B. Yang, B. Lee, H. S. Seo, S. Choi, K. Oh., “Electrically controllable long-period liquid crystal fiber gratings,” IEEE Photonics Technol. Lett. 12, 519–521 (2000).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (2)

M. Geshiro, M. Hotta, T. Kameshima, “Coupled-mode analysis of leaky waves in channel waveguides consisting of anisotropic material,” IEEE Trans. Microwave Theory Tech. 41, 1159–1163 (1993).
[CrossRef]

A. Toning, “Circular symmetric optical waveguide with strong anisotropy,” IEEE Trans. Microwave Theory Tech. 30, 790–784 (1982).
[CrossRef]

J. Lightwave Technol. (6)

A. Sneh, K. Johnson, “High-speed continuously tunable liquid crystal filter for WDM networks,” J. Lightwave Technol. 14, 1067–1079 (1996).
[CrossRef]

A. W. Snyder, F. Ruhl, “Anisotropic fibers studied by the Green’s function method,” J. Lightwave Technol. 3, 284–191 (1984).

A. M. Vengsarkar, P. J. Lemaire, J. B. Judkins, V. Bhatia, T. Erdogan, J. E. SIpe, “Long-period fiber gratings as band-rejection filters,” J. Lightwave Technol. 14, 58–65 (1996).
[CrossRef]

K. O. Hill, G. Meltz, “Fiber bragg grating technology fundamentals and overview,” J. Lightwave Technol. 15, 1263–1276 (1997).
[CrossRef]

A. D. Kersey, M. A. Davis, H. J. Patrick, M. LeBlanc, K. P. Koo, C. G. Askins, M. A. Putnam, E. J. Friebele, “Fiber grating sensors,” J. Lightwave Technol. 15, 1442–1463 (1997).
[CrossRef]

X. W. Shu, L. Zhang, I. Bennion, “Sensitivity characteristics of long-period fiber gratings,” J. Lightwave Technol. 20, 255–266 (2002).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Opt. Lett. (1)

Other (4)

C. Tsao, Optical Fiber Waveguide Analysis (Oxford U. Press, New York, 1992), pp. 298–369.

N. S. Kapany, J. J. Burke, Optical Waveguides (Academic, 1972), pp. 142–150.

A. W. Snyder, J. D. Love, Optical Waveguide Theory (Chapman & Hall, 1983), pp. 272–279.

O. Duhem, J. F. Henninot, M. Warenghem, M. Douay, L. Rivoallim, “Long period fiber gratings modulation by liquid crystal cladding,” in Telecommunications 1998, 6th IEE Conference, Conf. Publ. No. 451, pp 195–197 (IEE, 1998)

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Figures (5)

Fig. 1
Fig. 1

Three-layered step-index uniaxial optical fibers and the transverse index profile.

Fig. 2
Fig. 2

Plots of the effective elasto-optic coefficient ζ eff cl = ( 1 n eff cl , l ) d n eff cl , l d s for different cladding modes; λ = 1550 nm .

Fig. 3
Fig. 3

Plots of Δ β ( s , λ ) of LPFGs with applied axial strain: (a) l = 7 , (b) l = 27 .

Fig. 4
Fig. 4

Calculated sensitivities K L of LPFGs, plotted against cladding mode order; λ = 1550 nm .

Fig. 5
Fig. 5

Shifts of the resonance wavelength of LPFGs with applied axial strain whose periods are (a) 450 and (b) 400 and mode orders are (a) 7 and (b) 9.

Equations (123)

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

e = [ Ψ i r ϕ ( β ω ϵ i ) ( Φ i r ) ] r 0 [ Ψ i r + ( β ω ϵ i ) ( Φ i r ϕ ) ] ϕ 0 [ ( 1 j ω ϵ i ) ( k 2 n i 2 β ) Φ i ] z 0 ,
h = [ Φ i r ϕ + ( β ω ϵ i ) ( Ψ i r ) ] r 0 [ Φ i r ( β ω μ ) ( Ψ i r ϕ ) ] ϕ 0 + [ ( 1 j ω μ ) ( k 2 n i 2 β ) Ψ i ] z 0 .
( t 2 + k 2 n i z 2 n i z 2 n i t 2 β 2 ) Φ i = 0 , ( t 2 + k 2 n i t 2 β 2 ) Ψ i = 0 ,
i = 1 , 2 , 3 ,
Φ 1 = A 1 J v ( k 1 u 1 r ) exp ( j v ϕ ) ,
Φ 2 = [ A 2 J v ( k 2 u 2 r ) + B 2 I v ( k 2 u 2 r ) ] exp ( j v ϕ ) ,
Φ 3 = B 3 K v ( k 3 w 3 r ) exp ( j v ϕ ) ,
Ψ 1 = C 1 J v ( u 1 r ) exp ( j v ϕ ) ,
Ψ 2 = [ C 2 J v ( u 2 r ) + D 2 I v ( u 2 r ) ] exp ( j v ϕ ) ,
Ψ 3 = D 3 K v ( w 3 r ) exp ( j v ϕ ) ,
P v P v c 2 ( 1 α 2 W 2 2 ) 2 ( n 2 t 2 n 1 t n 3 t ) X 1 X 2 + X 1 2 X 2 2 [ J b ( K b P v R v α 2 W 2 ) + ( 1 W 2 ) ( K b Q v S v α 2 W 2 ) ] [ k 1 J b c ( k 3 K b c P v c k 2 s 23 R v c α 2 W 2 ) + ( k 2 s 21 W 2 ) ( k 3 K b c Q v c k 2 s 23 S v c α 2 W 2 ) ] X 1 2 ( J b P v + Q v W 2 ) ( k 1 J b c P v c + k 2 s 21 Q v c W 2 ) X 2 2 ( K b P v R v α 2 W 2 ) ( k 3 K b c P v c k 2 s 23 R v c α 2 W 2 ) = 0 ,
P v = P v ( r 2 ) , Q v = Q v ( r 2 ) , R v = R v ( r 2 ) , S v = S v ( r 2 ) ,
P v c = P v c ( r 2 ) , Q v c = Q v c ( r 2 ) , R v c = R v c ( r 2 ) , S v c = S v c ( r 2 ) ,
P v ( r ) = I v ( w 2 r ) K v ( w 2 r 1 ) I v ( w 2 r 1 ) K v ( w 2 r ) , Q v ( r ) = I v ( w 2 r ) K v ( w 2 r 1 ) I v ( w 2 r 1 ) K v ( w 2 r ) ,
R v ( r ) = I v ( w 2 r ) K v ( w 2 r 1 ) I v ( w 2 r 1 ) K v ( w 2 r ) , S v ( r ) = I v ( w 2 r ) K v ( w 2 r 1 ) I v ( w 2 r 1 ) K v ( w 2 r ) ,
P v c ( r ) = I v ( k 2 w 2 r ) K v ( k 2 w 2 r 1 ) I v ( k 2 w 2 r 1 ) K v ( k 2 w 2 r ) ,
Q v c ( r ) = I v ( k 2 w 2 r ) K v ( k 2 w 2 r 1 ) I v ( k 2 w 2 r 1 ) K v ( k 2 w 2 r ) ,
R v c ( r ) = I v ( k 2 w 2 r ) K v ( k 2 w 2 r 1 ) I v ( k 2 w 2 r 1 ) K v ( k 2 w 2 r ) ,
S v c ( r ) = I v ( k 2 w 2 r ) K v ( k 2 w 2 r 1 ) I v ( k 2 w 2 r 1 ) K v ( k 2 w 2 r ) ,
u 1 2 = k 2 ( n 1 t 2 n eff 2 ) , w 2 2 = k 2 ( n eff 2 n 2 t 2 ) , w 3 2 = k 2 ( n eff 2 n 3 t 2 ) ,
U 1 = u 1 r 1 , W 2 = w 2 r 1 , W 3 = w 3 r 2 , α 2 = r 2 r 1 , s 21 = n 2 t 2 n 1 t 2 , s 23 = n 2 t 2 n 3 t 2 ,
J b = J v ( U 1 ) U 1 J v ( U 1 ) , J b c = J v ( k 2 U 1 ) U 1 J v ( k 2 U 1 ) ,
K b = K v ( W 3 ) W 3 K v ( W 3 ) , K b c = K v ( k 3 W 3 ) W 3 K v ( k 3 W 3 ) ,
V 12 = k r 1 ( n 1 t 2 n 2 t 2 ) 1 2 , V 23 = k r 2 ( n 2 t 2 n 3 t 2 ) 1 2 ,
X 1 = n 1 t U 1 2 W 2 2 σ 0 V 12 2 , X 2 = n 2 t α 2 2 W 2 2 W 3 2 σ 0 V 23 2 , σ 0 2 = ( v β k ) 2 ,
( J b 2 + K b 2 ) ( k 1 n 1 t 2 J b 2 c + k 2 n 2 t 2 K b 2 c ) = ( l β k ) 2 ( V U W ) 4 ,
U 2 = ( k 2 n 1 t 2 β 2 ) r 1 2 , W 2 = ( β 2 k 2 n 2 t 2 ) r 1 2 ,
V 2 = k 2 r 1 2 ( n 1 t 2 n 2 t 2 )
J b 2 = J v ( U ) U J v ( U ) , J b 2 c = J v ( k 1 U ) U J v ( k 1 U ) ,
K b 2 = K v ( W ) W K v ( W ) , K b 2 c = K v ( k 2 W ) W K v ( k 2 W ) .
( J b P v + Q v W 2 ) ( K b P v R v α 2 W 2 ) = ( 1 α 2 W 2 2 ) 2 ,
( k 1 J b c P v c + k 2 s 21 Q v c w 2 r 1 ) ( k 2 K b c P v c k 2 s 23 R v c w 2 r 2 ) = ( 2 n 2 t 2 π n 1 t n 3 t w 2 2 r 1 r 2 ) 2
p ν p ν c + 2 ( 2 π α 2 U 2 2 ) 2 ( n 2 t 2 n 1 t n 3 t ) x 1 x 2 + x 1 2 x 2 2 [ J b ( K b p ν + r ν α 2 U 2 ) ( 1 U 2 ) ( K b q ν + s ν α 2 U 2 ) ] [ k 1 J b c ( k 3 K b c p ν c + k 2 s 23 r ν c α 2 U 2 ) ( k 2 s 21 U 2 ) ( k 3 K b c q ν c + k 2 s 23 s ν c α 2 U 2 ) ] x 1 2 ( J b p ν q ν U 2 ) ( k 1 J b c p ν c k 2 s 21 q ν c U 2 ) x 2 2 ( K b p ν + r ν α 2 U 2 ) ( k 3 K b c p ν c + k 2 s 23 r ν c α 2 U 2 ) = 0 ,
p ν = p ν ( r 2 ) , q ν = q ν ( r 2 ) , r ν = r ν ( r 2 ) , s ν = s ν ( r 2 ) ,
p ν c = p ν c ( r 2 ) , q ν c = q ν c ( r 2 ) , r ν c = r ν c ( r 2 ) , s ν c = s ν c ( r 2 ) ,
p ν ( r ) = J ν ( u 2 r ) Y ν ( u 2 r 1 ) J ν ( u 2 r 1 ) Y ν ( u 2 r ) ,
q ν ( r ) = J ν ( u 2 r ) Y ν ( u 2 r 1 ) J ν ( u 2 r 1 ) Y ν ( u 2 r ) ,
r ν ( r ) = J ν ( u 2 r ) Y ν ( u 2 r 1 ) J ν ( u 2 r 1 ) Y ν ( u 2 r ) ,
s ν ( r ) = J ν ( u 2 r ) Y ν ( u 2 r 1 ) J ν ( u 2 r 1 ) Y ν ( u 2 r ) ,
p ν c ( r ) = J ν ( k 2 u 2 r ) Y ν ( k 2 u 2 r 1 ) J ν ( k 2 u 2 r 1 ) Y ν ( k 2 u 2 r ) ,
q ν c ( r ) = J ν ( k 2 u 2 r ) Y ν ( k 2 u 2 r 1 ) J ν ( k 2 u 2 r 1 ) Y ν ( k 2 u 2 r ) ,
r ν c ( r ) = J ν ( k 2 u 2 r ) Y ν ( k 2 u 2 r 1 ) J ν ( k 2 u 2 r 1 ) Y ν ( k 2 u 2 r ) ,
s ν c ( r ) = J ν ( k 2 u 2 r ) Y ν ( k 2 u 2 r 1 ) J ν ( k 2 u 2 r 1 ) Y ν ( k 2 u 2 r ) ,
u 2 2 = k 0 2 ( n 2 t 2 n eff 2 ) , U 2 = u 2 r 1 , k i = n i z n i t , i = 1 , 2 , 3 ,
x 1 = n 1 t U 1 2 U 2 2 σ 0 V 12 2 , x 2 = n 2 t α 2 2 U 2 2 W 3 2 σ 0 V 23 2 ,
α 2 , s 21 , s 23 , u 1 , U 1 , w 3 , W 3 , J b , J b c , K b , K b c , V 12 , V 23 , σ 0
( J b p ν q ν W 2 ) ( K b p ν + r ν α 2 W 2 ) = ( 1 α 2 U 2 2 ) 2 ,
( k 1 J b c p ν c k 2 s 21 q ν c u 2 r 1 ) ( k 3 K b c p ν c + k 2 s 23 r ν c u 2 r 2 ) = ( 2 n 2 t 2 π n 1 t n 3 t u 2 2 r 1 r 2 ) 2 ,
r < r 1 :
e r = j E 0 co u 1 2 { J ν 1 ( u 1 r ) + J ν + 1 ( u 1 r ) + j ξ 0 k 1 Z 0 β k n 1 t 2 J ν ( u 1 r 1 ) J ν ( k 1 u 1 r 1 ) [ J ν 1 ( k 1 u 1 r ) J ν + 1 ( k 1 u 1 r ) ] } ,
e φ = E 0 co u 1 2 { J ν 1 ( u 1 r ) J ν + 1 ( u 1 r ) + j ξ 0 Z 0 β k n 1 t 2 J ν ( u 1 r 1 ) J ν ( k 1 u 1 r 1 ) [ J ν 1 ( k 1 u 1 r ) + J ν + 1 ( k 1 u 1 r ) ] } ,
e z = j E 0 co u 1 2 Z 0 ξ 0 k n 1 t 2 J ν ( u 1 r 1 ) J ν ( k 1 u 1 r 1 ) J ν ( k 1 u 1 r ) ,
h r = E 0 co u 1 2 { β k Z 0 [ J ν 1 ( u 1 r ) J ν + 1 ( u 1 r ) ] + j ξ 0 J ν ( u 1 r 1 ) J ν ( k 1 u 1 r 1 ) [ J ν 1 ( k 1 u 1 r ) + J ν + 1 ( k 1 u 1 r ) ] } ,
h φ = j E 0 co u 1 2 { β k Z 0 [ J ν 1 ( u 1 r ) + J ν + 1 ( u 1 r ) ] + j ξ 0 k 1 J ν ( u 1 r 1 ) J ν ( k 1 u 1 r 1 ) [ J ν 1 ( k 1 u 1 r ) J ν + 1 ( k 1 u 1 r ) ] } ,
h z = j E 0 co u 1 2 k Z 0 J ν ( u 1 r ) .
r 1 < r < r 2 :
e r = E 0 co 1 2 π r 1 u 1 2 J ν ( u 1 r 1 ) { j ν r [ F 2 P ν ( r ) + 1 u 2 Q ν ( r ) ] + Z 0 β k n 2 t 2 [ k 2 w 2 G 2 R ν c ( r ) + n 2 t 2 n 1 t 2 k 2 2 ξ 0 S ν c ( r ) ] } ,
e φ = E 0 co 1 2 π r 1 u 1 2 J ν ( u 1 r 1 ) { j ν r Z 0 β k n 2 t 2 [ G 2 p ν c ( r ) + 1 w 2 n 2 t 2 n 1 t 2 k 2 ξ 0 Q ν c ( r ) ] w 2 F 2 R ν ( r ) S ν ( r ) } ,
e z = j E 0 co π r 1 u 1 2 Z 0 2 k n 2 t 2 J ν ( u 1 r 1 ) [ G 2 P ν c ( r ) + 1 w 2 n 2 t 2 n 1 t 2 k 2 ξ 0 Q ν c ( r ) ] ,
h r = E 0 co 1 2 π r 1 u 1 2 J ν ( u 1 r 1 ) { j ν r [ G 2 P ν c ( r ) + 1 w 2 n 2 t 2 n 1 t 2 k 2 ξ 0 Q ν c ( r ) ] + β k Z 0 [ w 2 F 2 R ν ( r ) + S ν ( r ) ] } ,
h φ = E 0 co 1 2 π r 1 u 1 2 J ν ( u 1 r 1 ) { j ν r β k Z 0 [ F 2 P ν ( r ) + 1 w 2 Q ν ( r ) ] [ k 2 2 n 2 t 2 ξ 0 n 1 t 2 S ν c ( r ) + k 2 w 2 G 2 R ν c ( r ) ] } ,
h z = j E 0 co π r 1 u 1 2 w 2 2 2 k Z 0 J ν ( u 1 r 1 ) [ F 2 P ν ( r ) 1 w 2 Q ν ( r ) ] .
r > r 2 :
e r = E 0 co π r 1 u 1 2 w 2 2 4 w 3 { j F 3 K ν ( w 3 r 2 ) [ K ν 1 ( w 3 r ) K ν + 1 ( w 3 r ) ] G 3 K ν ( k 3 w 3 r 2 ) k 3 Z 0 β k n 3 t 2 [ K ν 1 ( k 3 w 3 r ) + K ν + 1 ( k 3 w 3 r ) ] } ,
e φ = E 0 co π r 1 u 1 2 w 2 2 4 w 3 { F 3 K ν ( w 3 r 2 ) [ K ν 1 ( w 3 r ) + K ν + 1 ( w 3 r ) ] j G 3 K ν ( k 3 w 3 r 2 ) Z 0 β k n 3 t 2 [ K ν 1 ( k 3 w 3 r ) K ν + 1 ( k 3 w 3 r ) ] } ,
e z = j E 0 co π r 1 u 1 2 w 2 2 Z 0 2 k n 3 t 2 K ν ( k 3 w 3 r 2 ) G 3 K ν ( k 3 w 3 r ) ,
h r = E 0 co π r 1 u 1 2 w 2 2 4 w 3 { β k Z 0 F 3 K ν ( w 3 r 2 ) [ K ν 1 ( w 3 r ) + K ν + 1 ( w 3 r ) ] + j G 3 K ν ( k 3 w 3 r 2 ) [ K ν 1 ( k 3 w 3 r ) K ν + 1 ( k 3 w 3 r ) ] } ,
h φ = E 0 co π r 1 u 1 2 w 2 2 4 w 3 { β k Z 0 j F 3 K ν ( w 3 r 2 ) [ K ν 1 ( w 3 r ) K ν + 1 ( w 3 r ) ] k 3 G 3 K ν ( k 3 w 3 r 2 ) [ K ν 1 ( k 3 w 3 r ) + K ν + 1 ( k 3 w 3 r ) ] } ,
h z = j E 0 co π r 1 u 1 2 w 2 2 2 k Z 0 K ν ( w 3 r 2 ) F 3 K ν ( w 3 r ) ,
J = J ν ( U 1 ) u 1 J ν ( U 1 ) , J c = J ν ( k 2 U 1 ) u 1 J ν ( k 2 U 1 ) ,
K = K ν ( W 3 ) w 3 K ν ( W 3 ) , K c = K ν ( k 3 W 3 ) w 3 K ν ( k 3 W 3 ) ,
σ 1 = j ν β k Z 0 , σ 2 = j ν β Z 0 k , u 21 = 1 u 1 2 + 1 w 2 2 ,
u 32 = 1 w 3 2 1 w 2 2 ,
F 2 = J u 21 σ 2 ξ 0 n 1 t 2 r 1 , G 2 = k 1 ξ 0 J c + u 21 σ 1 r 2 2 ,
G 3 = n 3 t 2 n 2 t 2 [ G 2 P ν c + k 2 n 2 t 2 ξ 0 n 1 t 2 w 2 Q ν c ] , F 3 = F 2 P ν 1 u 2 Q ν ,
ξ 0 = 1 σ 2 w 2 u 21 u 32 σ 1 σ 2 n 1 t 2 r 1 r 2 P ν c + 1 w 2 ( w 2 J a 2 b 2 ) u 32 n 2 t 2 r 2 ( k 1 w 2 J c P ν c + n 2 t 2 n 1 t 2 Q ν c ) + u 21 n 1 t 2 r 1 a 2 = σ 1 u 21 r 1 a 1 + u 32 r 2 ( w 2 J P ν + Q ν ) u 21 u 32 σ 1 σ 2 n 1 t 2 r 1 r 2 P ν + k 1 J c a 1 n 2 t 2 n 1 t 2 w 2 b 1 ,
a 1 = k 2 R ν c + n 3 t 2 n 2 t 2 k 3 w 2 K c P ν c , b 1 = k 2 2 S ν c n 3 t 2 n 2 t 2 k 2 k 3 w 2 K c Q ν c ,
a 2 = R ν + w 2 K P ν , b 2 = S ν w 2 K Q ν ,
r < r 1 :
e r = j E 0 cl u 1 2 { J ν 1 ( u 1 r ) + J ν + 1 ( u 1 r ) + j ξ 0 k 1 Z 0 β k n 1 t 2 J ν ( u 1 r 1 ) J ν ( k 1 u 1 r 1 ) [ J ν 1 ( k 1 u 1 r ) J ν + 1 ( k 1 u 1 r ) ] } ,
e φ = E 0 cl u 1 2 { J ν 1 ( u 1 r ) J ν + 1 ( u 1 r ) + j ξ 0 k 1 Z 0 β k n 1 t 2 J ν ( u 1 r 1 ) J ν ( k 1 u 1 r 1 ) [ J ν 1 ( k 1 u 1 r ) + J ν + 1 ( k 1 u 1 r ) ] } ,
e z = j E 0 cl u 1 2 Z 0 ξ 0 k n 1 t 2 J ν ( u 1 r 1 ) J ν ( k 1 u 1 r 1 ) J ν ( k 1 u 1 r ) ,
h r = E 0 cl u 1 2 { β k Z 0 [ J ν 1 ( u 1 r ) J ν + 1 ( u 1 r ) ] + j ξ 0 J ν ( u 1 r 1 ) J ν ( k 1 u 1 r 1 ) [ J ν 1 ( k 1 u 1 r ) + J ν + 1 ( k 1 u 1 r ) ] } ,
h φ = j E 0 cl u 1 2 { β k Z 0 [ J ν 1 ( u 1 r ) + J ν + 1 ( u 1 r ) ] + j ξ 0 k 1 J ν ( u 1 r 1 ) J ν ( k 1 u 1 r 1 ) [ J ν 1 ( k 1 u 1 r ) J ν + 1 ( k 1 u 1 r ) ] } ,
h z = j E 0 cl u 1 2 k Z 0 J ν ( u 1 r ) .
r 1 < r < r 2 :
e r = E 0 cl 1 2 π r 1 u 1 2 J ν ( u 1 r 1 ) { j ν r [ F 2 p ν ( r ) 1 u 2 J ν ( u 1 r 1 ) q ν ( r ) ] + Z 0 β k n 2 t 2 [ k 2 u 2 G 2 r ν c ( r ) n 2 t 2 n 1 t 2 k 2 2 ξ 0 s ν c ( r ) ] }
e φ = E 0 cl 1 2 π r 1 u 1 2 J ν ( u 1 r 1 ) { j ν r Z 0 β k n 2 t 2 [ G 2 p ν c ( r ) 1 u 2 n 2 t 2 n 1 t 2 k 2 ξ 0 q ν c ( r ) ] + u 2 F 2 r ν ( r ) J ν ( u 1 r 1 ) s ν ( r ) } ,
e z = j E 0 cl π r 1 u 1 2 u 2 2 Z 0 2 k n 2 t 2 J ν ( u 1 r 1 ) [ G 2 p ν c ( r ) 1 u 2 n 2 t 2 n 1 t 2 k 2 ξ 0 q ν c ( r ) ] ,
h r = E 0 cl 1 2 π r 1 u 1 2 J ν ( u 1 r 1 ) { j ν r [ G 2 p ν c ( r ) 1 u 2 n 2 t 2 n 1 t 2 k 2 ξ 0 q ν c ( r ) ] β k Z 0 [ u 2 F 2 r ν ( r ) s ν ( r ) ] } ,
h φ = E 0 cl 1 2 π r 1 u 1 2 J ν ( u 1 r 1 ) { j ν r β k Z 0 [ F 2 p ν ( r ) 1 u 2 q ν ( r ) ] [ k 2 2 n 2 t 2 ξ 0 n 1 t 2 s ν c ( r ) k 2 u 2 G 2 r ν c ( r ) ] } ,
h z = j E 0 cl π r 1 u 1 2 u 2 2 2 k Z 0 J ν ( u 1 r 1 ) [ F 2 p ν ( r ) 1 u 2 q ν ( r ) ] .
r > r 2 :
e r = E 0 cl π r 1 u 1 2 u 2 2 J ν ( u 1 r 1 ) 4 w 3 { j F 3 K ν ( w 3 r 2 ) [ K ν 1 ( w 3 r ) K ν + 1 ( w 3 r ) ] G 3 K ν ( k 3 w 3 r 2 ) k 3 Z 0 β k n 3 t 2 [ K ν 1 ( k 3 w 3 r ) + K ν + 1 ( k 3 w 3 r ) ] }
e φ = E 0 cl π r 1 u 1 2 u 2 2 J ν ( u 1 r 1 ) 4 w 3 { F 3 K ν ( w 3 r 2 ) [ K ν 1 ( w 3 r ) + K ν + 1 ( w 3 r ) ] j G 3 K ν ( k 3 w 3 r 2 ) Z 0 β k n 3 t 2 [ K ν 1 ( k 3 w 3 r ) K ν + 1 ( k 3 w 3 r ) ] }
e z = j E 0 cl π r 1 u 1 2 u 2 2 Z 0 J ν ( u 1 r 1 ) 2 k n 3 t 2 K ν ( k 3 w 3 r 2 ) G 3 K ν ( k 3 w 3 r ) ,
h r = E 0 cl π r 1 u 1 2 u 2 2 J ν ( u 1 r 1 ) 4 w 3 { β k Z 0 F 3 K ν ( w 3 r 2 ) [ K ν 1 ( w 3 r ) + K ν + 1 ( w 3 r ) ] + j G 3 K ν ( k 3 w 3 r 2 ) [ K ν 1 ( k 3 w 3 r ) K ν + 1 ( k 3 w 3 r ) ] }
h φ = E 0 cl π r 1 u 1 2 u 2 2 J ν ( u 1 r 1 ) 4 w 3 { β k Z 0 j F 3 K ν ( w 3 r 2 ) [ K ν 1 ( w 3 r ) K ν + 1 ( w 3 r ) ] k 3 G 3 K ν ( k 3 w 3 r 2 ) [ K ν 1 ( k 3 w 3 r ) + K ν + 1 ( k 3 w 3 r ) ] }
h z = j E 0 cl π r 1 u 1 2 u 2 2 J ν ( u 1 r 1 ) 2 k Z 0 K ν ( w 3 r 2 ) F 3 K ν ( w 3 r ) ,
u 21 = 1 u 2 2 1 u 1 2 , u 32 = 1 w 3 2 + 1 u 2 2 ,
J = J ν ( U 1 ) u 1 J ν ( U 1 ) ,
J c = J ν ( k 2 U 1 ) u 1 J ν ( k 2 U 1 ) ,
K = K ν ( W 3 ) w 3 K ν ( W 3 ) ,
K c = K ν ( k 3 W 3 ) w 3 K ν ( k 3 W 3 ) ,
F 2 = J u 21 σ 2 ξ 0 n 1 t 2 r 1 , G 2 = k 1 ξ 0 r 1 J c + u 21 σ 1 r 2 2 ,
G 3 = n 3 t 2 n 2 t 2 [ G 2 p ν c k 2 n 2 t 2 ξ 0 n 1 t 2 u 2 q ν c ] , [ F 3 = F 2 p ν + 1 u 2 q ν ,
ξ 0 = 1 σ 2 u 2 u 21 u 32 σ 1 σ 2 n 1 t 2 r 1 r 2 p ν c + 1 u 2 ( u 2 J a 2 b 2 ) u 32 n 2 t 2 r 2 ( k 1 u 2 J c p ν c n 2 t 2 n 1 t 2 q ν c ) + u 21 n 1 t 2 r 1 a 2
= σ 1 u 21 r 1 a 1 + u 32 r 2 ( u 2 J p ν q ν ) u 21 u 32 σ 1 σ 2 n 1 t 2 r 1 r 2 p ν + k 1 J c a 1 n 2 t 2 n 1 t 2 u 2 b 1 ,
a 1 = k 2 r ν c + n 3 t 2 n 2 t 2 k 3 u 2 K c p ν c , b 1 = k 2 2 s ν c + n 3 t 2 n 2 t 2 k 2 k 3 u 2 K c q ν c ,
a 2 = r ν + u 2 K p ν , b 2 = s ν + u 2 K q ν ,
u 2 w 2 , J v ( u 2 r ) I ν ( w 2 r ) , Y ν ( u 2 r ) K ν ( w 2 r ) , J ν ( u 2 r ) I ν ( w 2 r ) , Y ν ( u 2 r ) K ν ( u 2 r ) ,
J ν ( k 2 u 2 r ) I ν ( k 2 w 2 r ) , Y ν ( k 2 u 2 r ) K ν ( k 2 w 2 r ) , J ν ( k 2 u 2 r ) I ν ( k 2 w 2 r ) , Y ν ( k 2 u 2 r ) K ν ( k 2 u 2 r ) ,
p ν P ν , q ν Q ν , r ν R ν , s ν S ν , p ν c P ν c ,
q ν c Q ν c , r ν c R ν c , s ν c S ν c , x 1 X 1 , x 2 X 2 , U 2 W 2 , V 23 V 23 , u 21 u 21 , u 32 u 32 .
n 1 t = n 1 1 2 n 1 3 [ p 1 , 12 μ ( p 1 , 11 + p 1 , 12 ) ] s ,
n 1 z = n 1 1 2 n 1 3 ( p 1 , 11 2 μ p 1 , 12 ) s ,
n 2 t = n 2 1 2 n 2 3 [ p 2 , 12 μ ( p 2 , 11 + p 2 , 12 ) ] s ,
n 2 z = n 2 1 2 n 2 3 ( p 2 , 11 2 μ p 2 , 12 ) s ,
n 3 t = n 3 1 2 n 3 3 [ p 3 , 12 μ ( p 3 , 11 + p 3 , 12 ) ] s ,
Δ β ( s , λ ) = β co ( s , λ ) β cl , l ( s , λ ) 2 π Λ ,
K L = d λ res ds = ( β co β cl , l ) + s ( β co β cl , l ) λ ( β co β cl , l ) ,
K L = λ res 1 + 1 n eff co n eff cl , l s ( n eff co n eff cl , l ) 1 λ res n eff co n eff cl , l λ ( n eff co n eff cl , l ) ,
K B = d λ res d s = β co + s β co λ β co = λ res 1 + 1 n eff co s n eff co 1 λ res n eff co λ n eff co .

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