Abstract

The spectral characteristics of a fiber Bragg grating (FBG) with a transversely inhomogeneous refractive index profile, differs considerably from that of a transversely uniform one. Transmission spectra of inhomogeneous and asymmetric FBGs that have been inscribed with focused ultrashort pulses with the so-called point-by-point technique are investigated. The cladding mode resonances of such FBGs can span a full octave in the spectrum and are very pronounced (deeper than 20dB). Using a coupled-mode approach, we compute the strength of resonant coupling and find that coupling into cladding modes of higher azimuthal order is very sensitive to the position of the modification in the core. Exploiting these properties allows precise control of such reflections and may lead to many new sensing applications.

© 2011 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  5. C.-F. Chan, C. Chen, A. Jafari, A. Laronche, D. J. Thomson, and J. Albert, "Optical fiber refractometer using narrowband cladding-mode resonance shifts," Appl. Opt. 46, 1142-1149 (2007).
    [CrossRef] [PubMed]
  6. K. Zhou, L. Zhang, X. Chen, and I. Bennion, "Optic sensors of high refractive-index responsivity and low thermal cross sensitivity that use fiber Bragg gratings of 80◦ tilted structures," Opt. Lett. 31, 1193-1195 (2006).
    [CrossRef] [PubMed]
  7. T. Mizunami, T. Djambova, T. Niiho, and S. Gupta, "Bragg gratings in multimode and few-mode optical fibers," J. Lightwave Technol. 18, 230-235 (2000).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]

2010

2009

2008

T. Guo, A. Ivanov, C. Chen, and J. Albert, "Temperature-independent tilted fiber grating vibration sensor based on cladding-core recoupling," Opt. Lett. 33, 1004-1006 (2008).
[CrossRef] [PubMed]

S. Ramachandran, J. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, "Ultra-large effective area, higher-order mode fibers: A new strategy for high-power lasers," Laser Photon. Rev. 2, 429-447 (2008).
[CrossRef]

2007

C.-F. Chan, C. Chen, A. Jafari, A. Laronche, D. J. Thomson, and J. Albert, "Optical fiber refractometer using narrowband cladding-mode resonance shifts," Appl. Opt. 46, 1142-1149 (2007).
[CrossRef] [PubMed]

J. Thomas, E. Wikszak, T. Clausnitzer, and U. Fuchs, "Inscription of fiber Bragg gratings with femtosecond pulses using a phase mask scanning technique," Appl. Phys., A Mater. Sci. Process. 86, 153-157 (2007).

2006

A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, "Photoinduced modifications in fiber gratings inscribed directly by infrared femtosecond irradiation," IEEE Photon. Technol. Lett. 18, 2266-2268 (2006).
[CrossRef]

K. Zhou, L. Zhang, X. Chen, and I. Bennion, "Optic sensors of high refractive-index responsivity and low thermal cross sensitivity that use fiber Bragg gratings of 80◦ tilted structures," Opt. Lett. 31, 1193-1195 (2006).
[CrossRef] [PubMed]

2005

2004

2000

1999

1997

1996

T. Erdogan, and J. E. Sipe, "Tilted fiber phase gratings," J. Opt. Soc. Am. A 13, 296-313 (1996).
[CrossRef]

S. Hewlett, J. Love, G. Meltz, T. Bailey, and W. Morey, "Coupling characteristics of photo-induced Bragg gratings in depressed-and matched-cladding fibre," Opt. Quantum Electron. 28, 1641-1654 (1996).
[CrossRef]

1995

S. J. Hewlett, J. D. Love, G. Meltz, T. J. Bailey, and W. W. Morey, "Cladding-mode coupling characteristics of Bragg gratings in depressed-cladding fibre," Electron. Lett. 31, 820-822 (1995).
[CrossRef]

1993

V. Mizrahi, and J. E. Sipe, "Optical properties of photosensitive fiber phase gratings," J. Lightwave Technol. 11, 1513-1517 (1993).
[CrossRef]

1989

1979

H. Kogelnik, "Theory of dielectric waveguides," Top. Appl. Phys. 7, 15-83 (1979).

1978

Albert, J.

Bailey, T.

S. Hewlett, J. Love, G. Meltz, T. Bailey, and W. Morey, "Coupling characteristics of photo-induced Bragg gratings in depressed-and matched-cladding fibre," Opt. Quantum Electron. 28, 1641-1654 (1996).
[CrossRef]

Bailey, T. J.

S. J. Hewlett, J. D. Love, G. Meltz, T. J. Bailey, and W. W. Morey, "Cladding-mode coupling characteristics of Bragg gratings in depressed-cladding fibre," Electron. Lett. 31, 820-822 (1995).
[CrossRef]

Bennion, I.

A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, "Photoinduced modifications in fiber gratings inscribed directly by infrared femtosecond irradiation," IEEE Photon. Technol. Lett. 18, 2266-2268 (2006).
[CrossRef]

K. Zhou, L. Zhang, X. Chen, and I. Bennion, "Optic sensors of high refractive-index responsivity and low thermal cross sensitivity that use fiber Bragg gratings of 80◦ tilted structures," Opt. Lett. 31, 1193-1195 (2006).
[CrossRef] [PubMed]

A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, "Direct writing of fibre Bragg gratings by femtosecond laser," Electron. Lett. 40, 1170-1172 (2004).
[CrossRef]

Borrelli, N. F.

Chan, C.-F.

Chen, C.

Chen, X.

Clausnitzer, T.

J. Thomas, E. Wikszak, T. Clausnitzer, and U. Fuchs, "Inscription of fiber Bragg gratings with femtosecond pulses using a phase mask scanning technique," Appl. Phys., A Mater. Sci. Process. 86, 153-157 (2007).

Djambova, T.

Dubov, M.

A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, "Photoinduced modifications in fiber gratings inscribed directly by infrared femtosecond irradiation," IEEE Photon. Technol. Lett. 18, 2266-2268 (2006).
[CrossRef]

A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, "Direct writing of fibre Bragg gratings by femtosecond laser," Electron. Lett. 40, 1170-1172 (2004).
[CrossRef]

Erdogan, T.

Fini, J.

S. Ramachandran, J. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, "Ultra-large effective area, higher-order mode fibers: A new strategy for high-power lasers," Laser Photon. Rev. 2, 429-447 (2008).
[CrossRef]

Fuchs, U.

J. Thomas, E. Wikszak, T. Clausnitzer, and U. Fuchs, "Inscription of fiber Bragg gratings with femtosecond pulses using a phase mask scanning technique," Appl. Phys., A Mater. Sci. Process. 86, 153-157 (2007).

Fuerbach, A.

Gaeta, A. L.

Ghalmi, S.

S. Ramachandran, J. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, "Ultra-large effective area, higher-order mode fibers: A new strategy for high-power lasers," Laser Photon. Rev. 2, 429-447 (2008).
[CrossRef]

Grobnic, D.

Guo, T.

Gupta, S.

Hewlett, S.

S. Hewlett, J. Love, G. Meltz, T. Bailey, and W. Morey, "Coupling characteristics of photo-induced Bragg gratings in depressed-and matched-cladding fibre," Opt. Quantum Electron. 28, 1641-1654 (1996).
[CrossRef]

Hewlett, S. J.

S. J. Hewlett, J. D. Love, G. Meltz, T. J. Bailey, and W. W. Morey, "Cladding-mode coupling characteristics of Bragg gratings in depressed-cladding fibre," Electron. Lett. 31, 820-822 (1995).
[CrossRef]

Homoelle, D.

Ivanov, A.

Jafari, A.

Jáuregui, C.

Jovanovic, N.

Khrushchev, I.

A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, "Photoinduced modifications in fiber gratings inscribed directly by infrared femtosecond irradiation," IEEE Photon. Technol. Lett. 18, 2266-2268 (2006).
[CrossRef]

A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, "Direct writing of fibre Bragg gratings by femtosecond laser," Electron. Lett. 40, 1170-1172 (2004).
[CrossRef]

Kogelnik, H.

H. Kogelnik, "Theory of dielectric waveguides," Top. Appl. Phys. 7, 15-83 (1979).

Krug, P. A.

Laronche, A.

López-Higuera, J. M.

Love, J.

S. Hewlett, J. Love, G. Meltz, T. Bailey, and W. Morey, "Coupling characteristics of photo-induced Bragg gratings in depressed-and matched-cladding fibre," Opt. Quantum Electron. 28, 1641-1654 (1996).
[CrossRef]

Love, J. D.

S. J. Hewlett, J. D. Love, G. Meltz, T. J. Bailey, and W. W. Morey, "Cladding-mode coupling characteristics of Bragg gratings in depressed-cladding fibre," Electron. Lett. 31, 820-822 (1995).
[CrossRef]

Marshall, G.

Marshall, G. D.

Martinez, A.

A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, "Photoinduced modifications in fiber gratings inscribed directly by infrared femtosecond irradiation," IEEE Photon. Technol. Lett. 18, 2266-2268 (2006).
[CrossRef]

A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, "Direct writing of fibre Bragg gratings by femtosecond laser," Electron. Lett. 40, 1170-1172 (2004).
[CrossRef]

Meltz, G.

S. Hewlett, J. Love, G. Meltz, T. Bailey, and W. Morey, "Coupling characteristics of photo-induced Bragg gratings in depressed-and matched-cladding fibre," Opt. Quantum Electron. 28, 1641-1654 (1996).
[CrossRef]

S. J. Hewlett, J. D. Love, G. Meltz, T. J. Bailey, and W. W. Morey, "Cladding-mode coupling characteristics of Bragg gratings in depressed-cladding fibre," Electron. Lett. 31, 820-822 (1995).
[CrossRef]

Mermelstein, M.

S. Ramachandran, J. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, "Ultra-large effective area, higher-order mode fibers: A new strategy for high-power lasers," Laser Photon. Rev. 2, 429-447 (2008).
[CrossRef]

Mihailov, S. J.

Mizrahi, V.

V. Mizrahi, and J. E. Sipe, "Optical properties of photosensitive fiber phase gratings," J. Lightwave Technol. 11, 1513-1517 (1993).
[CrossRef]

Mizunami, T.

Morey, W.

S. Hewlett, J. Love, G. Meltz, T. Bailey, and W. Morey, "Coupling characteristics of photo-induced Bragg gratings in depressed-and matched-cladding fibre," Opt. Quantum Electron. 28, 1641-1654 (1996).
[CrossRef]

Morey, W. W.

S. J. Hewlett, J. D. Love, G. Meltz, T. J. Bailey, and W. W. Morey, "Cladding-mode coupling characteristics of Bragg gratings in depressed-cladding fibre," Electron. Lett. 31, 820-822 (1995).
[CrossRef]

Nicholson, J. W.

S. Ramachandran, J. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, "Ultra-large effective area, higher-order mode fibers: A new strategy for high-power lasers," Laser Photon. Rev. 2, 429-447 (2008).
[CrossRef]

Niiho, T.

Nolte, S.

Payne, D.

Quintela, A.

Ramachandran, S.

S. Ramachandran, J. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, "Ultra-large effective area, higher-order mode fibers: A new strategy for high-power lasers," Laser Photon. Rev. 2, 429-447 (2008).
[CrossRef]

Shao, L.

Sipe, J. E.

T. Erdogan, and J. E. Sipe, "Tilted fiber phase gratings," J. Opt. Soc. Am. A 13, 296-313 (1996).
[CrossRef]

V. Mizrahi, and J. E. Sipe, "Optical properties of photosensitive fiber phase gratings," J. Lightwave Technol. 11, 1513-1517 (1993).
[CrossRef]

Smelser, C. W.

Smith, C.

Snyder, A.

Steel, M. J.

Tam, H.-Y.

Thomas, J.

N. Jovanovic, J. Thomas, R. J. Williams, M. J. Steel, G. D. Marshall, A. Fuerbach, S. Nolte, A. Tünnermann, and M. J. Withford, "Polarization-dependent effects in point-by-point fiber Bragg gratings enable simple, linearly polarized fiber lasers," Opt. Express 17, 6082-6095 (2009).
[CrossRef] [PubMed]

J. Thomas, E. Wikszak, T. Clausnitzer, and U. Fuchs, "Inscription of fiber Bragg gratings with femtosecond pulses using a phase mask scanning technique," Appl. Phys., A Mater. Sci. Process. 86, 153-157 (2007).

Thomson, D. J.

Tsao, C.

Tünnermann, A.

Wielandy, S.

Wikszak, E.

J. Thomas, E. Wikszak, T. Clausnitzer, and U. Fuchs, "Inscription of fiber Bragg gratings with femtosecond pulses using a phase mask scanning technique," Appl. Phys., A Mater. Sci. Process. 86, 153-157 (2007).

Williams, R.

Williams, R. J.

Withford, M. J.

Yan, M. F.

S. Ramachandran, J. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, "Ultra-large effective area, higher-order mode fibers: A new strategy for high-power lasers," Laser Photon. Rev. 2, 429-447 (2008).
[CrossRef]

Young, W.

Zhang, L.

Zhou, K.

Appl. Opt.

Appl. Phys., A Mater. Sci. Process.

J. Thomas, E. Wikszak, T. Clausnitzer, and U. Fuchs, "Inscription of fiber Bragg gratings with femtosecond pulses using a phase mask scanning technique," Appl. Phys., A Mater. Sci. Process. 86, 153-157 (2007).

Electron. Lett.

S. J. Hewlett, J. D. Love, G. Meltz, T. J. Bailey, and W. W. Morey, "Cladding-mode coupling characteristics of Bragg gratings in depressed-cladding fibre," Electron. Lett. 31, 820-822 (1995).
[CrossRef]

A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, "Direct writing of fibre Bragg gratings by femtosecond laser," Electron. Lett. 40, 1170-1172 (2004).
[CrossRef]

IEEE Photon. Technol. Lett.

A. Martinez, M. Dubov, I. Khrushchev, and I. Bennion, "Photoinduced modifications in fiber gratings inscribed directly by infrared femtosecond irradiation," IEEE Photon. Technol. Lett. 18, 2266-2268 (2006).
[CrossRef]

J. Lightwave Technol.

V. Mizrahi, and J. E. Sipe, "Optical properties of photosensitive fiber phase gratings," J. Lightwave Technol. 11, 1513-1517 (1993).
[CrossRef]

T. Mizunami, T. Djambova, T. Niiho, and S. Gupta, "Bragg gratings in multimode and few-mode optical fibers," J. Lightwave Technol. 18, 230-235 (2000).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Laser Photon. Rev.

S. Ramachandran, J. Fini, M. Mermelstein, J. W. Nicholson, S. Ghalmi, and M. F. Yan, "Ultra-large effective area, higher-order mode fibers: A new strategy for high-power lasers," Laser Photon. Rev. 2, 429-447 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Opt. Quantum Electron.

S. Hewlett, J. Love, G. Meltz, T. Bailey, and W. Morey, "Coupling characteristics of photo-induced Bragg gratings in depressed-and matched-cladding fibre," Opt. Quantum Electron. 28, 1641-1654 (1996).
[CrossRef]

Top. Appl. Phys.

H. Kogelnik, "Theory of dielectric waveguides," Top. Appl. Phys. 7, 15-83 (1979).

Other

B. E. A. Saleh, M. C. Teich, and J. W. Goodman, Fundamentals of photonics, (Wiley, 1991) pp. 272-309.
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Transmission differential interference contrast microscope images of the index modifications viewed from (a) above (parallel to the direction of the writing laser beam) and (b) the side (perpendicular to the direction of the writing laser beam). (c) Schematic diagram of the cross-section of the femtosecond induced modifications within the core of radius a1, establishing the coordinate system, where z points along the fiber and x and y are perpendicular to the fiber axis.

Fig. 2
Fig. 2

Transmission spectrum of a PbP grating operating in second order at 1540 nm and showing strong cladding mode resonances. According to the notation laid out in section 3, the colored dots indicate the envelopes of resonances with HE l = 1 (red), EH l = 1 (blue) and HE l = 2 (green).The lower graphs show expanded scales to illustrate the HE/EH splitting of modes with high radial order m (lower left), and the existence of first-order cladding mode peaks right up to the second-order Bragg peak (lower right).

Fig. 3
Fig. 3

Hybrid mode parameter Plm for l = 1 to l = 6 as a function of the projected normalized wavevector Ulm. Blue and green circles denote modes labeled HE and EH respectively. The insets display the intensity of the cladding modes in the region r < 3a1 with the black circle denoting the core boundary (EH modes are shown above the x-axis, HE modes below it). Vertical lines indicate the virtual cutoffs Ũlm and are labelled on the upper edge of each graph. The change in character of the modes at each virtual cutoffs is clear.

Fig. 4
Fig. 4

Measured cladding mode spectra in transmission (bottom) and computed HE and EH resonances (top) for odd (red) and even (blue) azimuthal order l. (a) Complete measured wavelength range; (b) low-and (c) high-wavelength range in more detail.

Fig. 6
Fig. 6

Measured (a) and computed (b) transmission spectra for the micro void at dx = 0.8 dy = 0.7. The insets show the EH/HE doublets.

Fig. 5
Fig. 5

(a) Core-cladding mode coupling coefficients for different displacements dx of the microvoid in x-direction from the center of the fiber core. Coupling constants for l = 1 to l = 6 are plotted, m is ascending from top to bottom. Dotted lines indicate the virtual cutoffs. (b) Computed transmission spectra for different displacement of the modification in the x-direction.

Equations (38)

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

E z = E lm u 1 2 β P J l ( u 1 r ) sin ( l ϕ + φ ) e i ( β z ω t )
E r = i E lm u 1 2 [ ( 1 P ) J l 1 ( u 1 r ) + ( 1 + P ) J l + 1 ( u 1 r ) ] sin ( l ϕ + φ ) e i ( β z ω t )
E ϕ = i E lm u 1 2 [ ( 1 P ) J l 1 ( u 1 r ) ( 1 + P ) J l + 1 ( u 1 r ) ] cos ( l ϕ + φ ) e i ( β z ω t )
H z = E lm n ¯ Z 0 u 1 2 β J l ( u 1 r ) cos ( l ϕ + φ ) e i ( β z ω t )
H r = i E lm n ¯ Z 0 u 1 z [ ( 1 P n 1 2 n ¯ 2 ) J l 1 ( u 1 r ) + ( 1 + P n 1 2 n ¯ 2 ) J l + 1 ( u 1 r ) ] cos ( l ϕ + φ ) e i ( β z ω t )
H ϕ = i E lm n ¯ Z 0 u 1 2 [ ( 1 P n 1 2 n ¯ 2 ) J l 1 ( u 1 r ) ( 1 + P n 1 2 n ¯ 2 ) J l + 1 ( u 1 r ) ] sin ( l ϕ + φ ) e i ( β z ω t ) ,
P = P lm = n ¯ lm i ζ 0 n 1 2
U lm = 2 π a 1 λ n 1 n 2 n 2 2 n ¯ lm 2 ,
β 11 ± β lm 2 π / Λ = 0 .
β lm ( λ ) ( n 2 2 ( 2 π λ ) 2 A lm ) 1 / 2 with A lm = ( l + 2 m + 1 / 2 ) 2 π 2 4 a 2 2 ,
A ˜ lm = ( n 2 2 n ¯ lm 2 ) ( 2 π λ 0 ) 2 ,
λ lm = n ¯ 11 Λ ± [ ( n ¯ 11 Λ ) 2 + 1 4 π 2 ( A ˜ lm + ( 2 π Λ ) 2 ) ( n ¯ lm 2 ( n ¯ 11 ) 2 ) ] 1 / 2 1 4 π 2 ( A ˜ lm + ( 2 π Λ ) 2 ) .
κ lm = 4 π ν ω 4 0 2 π d ϕ 0 d r r Δ ɛ ( r , ϕ ) E 11 T E lm T * ,
K ( φ ) = Δ ɛ ( r , ϕ ) E 11 E lm u 11 u lm 2 J 0 ( u 11 r ) { ( 1 P lm ) J 1 1 ( u lm r ) cos [ ( l 1 ) ϕ + φ ] ( 1 + P lm ) J l + 1 ( u lm r ) cos [ ( l + 1 ) ϕ + φ ] }
Δ ɛ ( x , y ) = ɛ 0 ( n mod 2 n 1 2 ) θ ( x , y )
ζ 0 = ζ 0 ,
ζ 0 = 1 σ u 2 ( J K + σ 2 v 21 v 32 n 2 2 a 1 a 2 ) p l ( a 2 ) K q l ( a 2 ) + J r l ( a 2 ) 1 u 2 s l ( a 2 ) u 2 ( v 32 n 2 2 a 2 J v 21 n 1 2 a 1 K ) p l ( a 2 ) + v 32 n 1 2 a 1 q l ( a 2 ) + v 21 n 1 2 a 1 r l ( a 2 )
ζ 0 = σ u 2 ( v 32 a 2 J n 3 2 v 21 n 2 2 a 1 K ) p l ( a 2 ) + v 32 a 1 q l ( a 2 ) v 21 a 1 r l ( a 2 ) u 2 ( n 3 2 n 2 2 J K + σ 2 v 21 v 32 n 1 2 a 1 ) p l ( a 2 ) n 3 2 n 1 2 K q l ( a 2 ) + J r l ( a 2 ) n 2 2 n 1 2 u 2 s l ( a 2 ) .
u lm 2 = ( 2 π / λ ) 2 ( n 1 2 n ¯ 2 ) , u 2 2 = ( 2 π / λ ) 2 ( n 2 2 n ¯ 2 ) , w 3 2 ( 2 π / λ ) 2 ( n ¯ 2 n 3 2 ) ,
σ = il n ¯ , v 21 = 1 u 2 2 1 u lm 2 , v 32 = 1 w 3 2 + 1 u 2 2 , J = J l ( u lm a 1 ) u lm J l ( u lm a 1 ) , K = K l ( w 3 a 2 ) w 3 K l ( w 3 a 2 ) .
p l ( r ) = J l ( u 2 r ) N l ( u 2 a 1 ) J l ( u 2 a 1 ) N l ( u 2 r ) ,
q l ( r ) = J l ( u 2 r ) N l ( u 2 a 1 ) J l ( u 2 a 1 ) N l ( u 2 r ) ,
r l ( r ) = J l ( u 2 r ) N l ( u 2 a 1 ) J l ( u 2 a 1 ) N l ( u 2 r ) ,
s l ( r ) = J l ( u 2 r ) N l ( u 2 a 1 ) J l ( u 2 a 1 ) N l ( u 2 r ) ,
E z cl = E lm C lm u 2 2 σ n 2 2 β l [ G 2 p l ( r ) n 2 2 ζ 0 n 1 2 u 2 q l ( r ) ] sin ( l ϕ + φ ) e i ( β z ω t )
E r cl = i E lm C lm ( l F 2 r p l ( r ) + 1 u 2 r q l ( r ) σ l n 2 2 [ u 2 G 2 r l ( r ) n 2 2 ζ 0 n 1 2 s l ( r ) ] ) sin ( l ϕ + φ ) e i ( β z ω t )
E ϕ cl = i E lm C lm ( σ n 2 2 [ G 2 r p l ( r ) n 2 2 ζ 0 n 1 2 u 2 r q l ( r ) ] + u 2 F 2 r l ( r ) s l ( r ) ) cos ( l ϕ + φ ) e i ( β z ω t )
H z cl = i E lm C lm 1 Z 0 u 2 2 σ β l [ F 2 p l ( r ) 1 u 2 q l ( r ) ] cos ( l ϕ + φ ) e i ( β z ω t )
H r cl = E lm C lm 1 Z 0 ( l G 2 r p l ( r ) l n 2 2 ζ 0 n 1 2 u 2 r q l ( r ) σ l [ u 2 F 2 r l ( r ) s l ( r ) ] ) cos ( l ϕ + φ ) e i ( β z ω t )
H ϕ cl = E lm C lm 1 Z 0 ( σ [ F 2 r p l ( r ) 1 u 2 r q l ( r ) ] u 2 G 2 r l ( r ) + n 2 2 ζ 0 n 1 2 s l ( r ) ) sin ( l ϕ + φ ) e i ( β z ω t )
C lm = π a 1 u lm 2 J 1 ( u lm a 1 ) 2 , F 2 = J u 21 σ ζ 0 n 1 2 a 1 , G 2 = ζ 0 J + v 21 σ a 1 .
E z cl = E lm C lm u 2 2 K l ( w 3 a 2 ) w 3 σ G 3 w 3 β l n 3 2 K l ( w 3 r ) sin ( l ϕ + φ ) e i ( β z ω t )
E r cl = i E lm C lm u 2 2 K l ( w 3 a 2 ) w 3 [ l F 3 w 3 r K l ( w 3 r ) + G 3 σ l n 3 2 K l ( w 3 r ) ] sin ( l ϕ + φ ) e i ( β z ω t )
E ϕ cl = i E lm C lm u 2 2 K l ( w 3 a 2 ) w 3 [ F 3 K l ( w 3 r ) + G 3 σ w 3 r n 3 2 K l ( w 3 r ) ] cos ( l ϕ + φ ) e i ( β z ω t )
H z cl = i E lm C lm 1 Z 0 u 2 2 K l ( w 3 a 2 ) w 3 σ F 3 w 3 β l cos ( l ϕ + φ ) e i ( β z ω t )
H r cl = E lm C lm 1 Z 0 u 2 2 K l ( w 3 a 2 ) w 3 [ l G 3 r w 3 K l ( w 3 r ) F 3 σ l K l ( w 3 r ) ] cos ( l ϕ + φ ) e i ( β z ω t )
H φ cl = E lm C lm 1 Z 0 u 2 2 K l ( w 3 a 2 ) w 3 [ G 3 K l ( w 3 r ) + F 3 σ w 3 r K l ( w 3 r ) ] sin ( l ϕ + φ ) e i ( β z ω t )
F 3 = F 2 p l ( a 2 ) + 1 u 2 q l ( a 2 ) , G 3 = n 3 2 n 2 2 [ G 2 p l ( a 2 ) n 2 2 ζ 0 n 1 2 u 2 q l ( a 2 ) ] .

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