Abstract

The transmission of a mode guided by the core of an optical fiber through an ultraviolet-induced fiber grating when substantial coupling to cladding modes occurs is analyzed both experimentally and theoretically. A straightforward theory is presented that is based on the calculation of the modes of a three-layer step-index fiber geometry and on multimode coupled-mode theory that accurately models the measured transmission in gratings that support both counterpropagating (short-period) and co-propagating (long-period) interactions. These cladding-mode resonance filters promise unique applications for spectral filtering and sensing.

© 1997 Optical Society of America

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References

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  1. G. Meltz, W. W. Morey, W. H. Glenn, “Formation of Bragg gratings in optical fibers by a transverse holographic method,” Opt. Lett. 14, 823–825 (1989).
    [CrossRef] [PubMed]
  2. K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
    [CrossRef]
  3. V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
    [CrossRef]
  4. 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]
  5. R. Kashyap, R. Wyatt, R. J. Campbell, “Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating,” Electron. Lett. 29, 154–156 (1993).
    [CrossRef]
  6. V. Bhatia, A. M. Vengsarkar, “Optical fiber long-period grating sensors,” Opt. Lett. 21, 692–694 (1996).
    [CrossRef] [PubMed]
  7. T. Erdogan, J. E. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A 13, 296–313 (1996).
    [CrossRef]
  8. H. Kogelnik, “Theory of optical waveguides,” in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990).
  9. D. G. Hall, “Theory of waveguides and devices,” in Integrated Optical Circuits and Components, L. D. Hutcheson, ed. (Marcel Dekker, New York, 1987), Secs. 2.4.2 and 2.4.3.
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    [CrossRef] [PubMed]
  11. C. Tsao, Optical Fibre Waveguide Analysis (Oxford, New York, 1992).
  12. D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Boston, Mass., 1991), Chap. 2.
  13. A. Bjarklev, “Microdeformation losses of single-mode fibers with step-index profiles,” J. Lightwave Technol. LT-4, 341–346 (1986).
    [CrossRef]
  14. P. J. Lemaire, R. M. Atkins, V. Mizrahi, K. L. Walker, K. S. Kranz, W. A. Reed, “High pressure H2 loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2 doped optical fibers,” Electron. Lett. 29, 1191–1193 (1993).
    [CrossRef]

1996

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]

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

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

1993

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

R. Kashyap, R. Wyatt, R. J. Campbell, “Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating,” Electron. Lett. 29, 154–156 (1993).
[CrossRef]

P. J. Lemaire, R. M. Atkins, V. Mizrahi, K. L. Walker, K. S. Kranz, W. A. Reed, “High pressure H2 loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2 doped optical fibers,” Electron. Lett. 29, 1191–1193 (1993).
[CrossRef]

1990

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

1989

1986

A. Bjarklev, “Microdeformation losses of single-mode fibers with step-index profiles,” J. Lightwave Technol. LT-4, 341–346 (1986).
[CrossRef]

1971

Atkins, R. M.

P. J. Lemaire, R. M. Atkins, V. Mizrahi, K. L. Walker, K. S. Kranz, W. A. Reed, “High pressure H2 loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2 doped optical fibers,” Electron. Lett. 29, 1191–1193 (1993).
[CrossRef]

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]

Bilodeau, F.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Bjarklev, A.

A. Bjarklev, “Microdeformation losses of single-mode fibers with step-index profiles,” J. Lightwave Technol. LT-4, 341–346 (1986).
[CrossRef]

Campbell, R. J.

R. Kashyap, R. Wyatt, R. J. Campbell, “Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating,” Electron. Lett. 29, 154–156 (1993).
[CrossRef]

Erdogan, T.

T. Erdogan, J. E. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A 13, 296–313 (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]

Glenn, W. H.

Gloge, D.

Hall, D. G.

D. G. Hall, “Theory of waveguides and devices,” in Integrated Optical Circuits and Components, L. D. Hutcheson, ed. (Marcel Dekker, New York, 1987), Secs. 2.4.2 and 2.4.3.

Hill, K. O.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Johnson, D. C.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[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]

Kashyap, R.

R. Kashyap, R. Wyatt, R. J. Campbell, “Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating,” Electron. Lett. 29, 154–156 (1993).
[CrossRef]

Kogelnik, H.

H. Kogelnik, “Theory of optical waveguides,” in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990).

Kranz, K. S.

P. J. Lemaire, R. M. Atkins, V. Mizrahi, K. L. Walker, K. S. Kranz, W. A. Reed, “High pressure H2 loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2 doped optical fibers,” Electron. Lett. 29, 1191–1193 (1993).
[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]

P. J. Lemaire, R. M. Atkins, V. Mizrahi, K. L. Walker, K. S. Kranz, W. A. Reed, “High pressure H2 loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2 doped optical fibers,” Electron. Lett. 29, 1191–1193 (1993).
[CrossRef]

Malo, B.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Marcuse, D.

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Boston, Mass., 1991), Chap. 2.

Meltz, G.

Mizrahi, V.

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

P. J. Lemaire, R. M. Atkins, V. Mizrahi, K. L. Walker, K. S. Kranz, W. A. Reed, “High pressure H2 loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2 doped optical fibers,” Electron. Lett. 29, 1191–1193 (1993).
[CrossRef]

Morey, W. W.

Reed, W. A.

P. J. Lemaire, R. M. Atkins, V. Mizrahi, K. L. Walker, K. S. Kranz, W. A. Reed, “High pressure H2 loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2 doped optical fibers,” Electron. Lett. 29, 1191–1193 (1993).
[CrossRef]

Sipe, J. E.

T. Erdogan, J. E. Sipe, “Tilted fiber phase gratings,” J. Opt. Soc. Am. A 13, 296–313 (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]

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

Skinner, I.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Tsao, C.

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

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]

Vineberg, K. A.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

Walker, K. L.

P. J. Lemaire, R. M. Atkins, V. Mizrahi, K. L. Walker, K. S. Kranz, W. A. Reed, “High pressure H2 loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2 doped optical fibers,” Electron. Lett. 29, 1191–1193 (1993).
[CrossRef]

Wyatt, R.

R. Kashyap, R. Wyatt, R. J. Campbell, “Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating,” Electron. Lett. 29, 154–156 (1993).
[CrossRef]

Appl. Opt.

Electron. Lett.

K. O. Hill, B. Malo, K. A. Vineberg, F. Bilodeau, D. C. Johnson, I. Skinner, “Efficient mode conversion in telecommunication fibre using externally written gratings,” Electron. Lett. 26, 1270–1272 (1990).
[CrossRef]

R. Kashyap, R. Wyatt, R. J. Campbell, “Wideband gain flattened erbium fibre amplifier using a photosensitive fibre blazed grating,” Electron. Lett. 29, 154–156 (1993).
[CrossRef]

P. J. Lemaire, R. M. Atkins, V. Mizrahi, K. L. Walker, K. S. Kranz, W. A. Reed, “High pressure H2 loading as a technique for achieving ultrahigh UV photosensitivity and thermal sensitivity in GeO2 doped optical fibers,” Electron. Lett. 29, 1191–1193 (1993).
[CrossRef]

J. Lightwave Technol.

A. Bjarklev, “Microdeformation losses of single-mode fibers with step-index profiles,” J. Lightwave Technol. LT-4, 341–346 (1986).
[CrossRef]

V. Mizrahi, J. E. Sipe, “Optical properties of photosensitive fiber phase gratings,” J. Lightwave Technol. 11, 1513–1517 (1993).
[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]

J. Opt. Soc. Am. A

Opt. Lett.

Other

H. Kogelnik, “Theory of optical waveguides,” in Guided-Wave Optoelectronics, T. Tamir, ed. (Springer-Verlag, Berlin, 1990).

D. G. Hall, “Theory of waveguides and devices,” in Integrated Optical Circuits and Components, L. D. Hutcheson, ed. (Marcel Dekker, New York, 1987), Secs. 2.4.2 and 2.4.3.

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

D. Marcuse, Theory of Dielectric Optical Waveguides (Academic, Boston, Mass., 1991), Chap. 2.

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Figures (11)

Fig. 1
Fig. 1

Measured transmission through a typical short-period fiber grating under investigation where (a) the uncoated fiber is immersed in index-matching liquid to simulate an infinite cladding, (b) the fiber is immersed in glycerin, and (c) the bare fiber is surrounded by air and thus supports cladding modes.

Fig. 2
Fig. 2

Diagram of a cross section of the fiber geometry considered here, showing the coordinate system, the refractive indexes, and the radii of the core (a1) and of the cladding (a2).

Fig. 3
Fig. 3

Plots of the vector components of the electric field for the lowest-order (ν=1) cladding mode of a fiber with the structural parameters listed in the text: (a) n2(r) (for the unperturbed fiber) times the radial component (solid curve) and the azimuthal component (dashed curve), (b) the longitudinal component.

Fig. 4
Fig. 4

Plots of the local intensity Iz(r) as a function of radius for the four lowest-order l=1 cladding modes in a typical fiber. All modes are circularly symmetric and have been normalized to carry a power of 1 W.

Fig. 5
Fig. 5

Diagram of the ultraviolet-induced refractive-index change in the core for a grating with a Gaussian profile along the fiber (z) axis. The size of the grating period (Λ) relative to the grating width (w) has been exaggerated for clarity.

Fig. 6
Fig. 6

Coupling constant κ1ν-01cl-co divided by σ(z) for the 168 l=1 cladding modes in a typical fiber, showing odd and even modes separately.

Fig. 7
Fig. 7

Diagrams that illustrate the phase-matching conditions necessary for resonant coupling between two modes by a grating of period Λ. (a) For a short-period grating, counterpropagating coupling can occur between (top to bottom, longest to shortest wavelength) oppositely traveling similar core modes, two different core modes, a core mode and a cladding mode, and a core mode and radiation modes. (b) For a long-period grating, copropagating coupling can occur between (top to bottom, longest to shortest wavelength) a core mode and radiation modes, a core mode and a cladding mode, and two different core modes.

Fig. 8
Fig. 8

(a) Experimentally measured and (b) theoretically calculated transmission spectra through a relatively weak, Gaussian short-period grating, demonstrating both core-mode–core-mode and core-mode–cladding-mode coupling.

Fig. 9
Fig. 9

(a) Experimentally measured and (b) theoretically calculated transmission spectra through a strong, Gaussian short-period grating.

Fig. 10
Fig. 10

Theoretically calculated transmission spectra through (a) a relatively weak and (b) a relatively strong long-period grating, each designed to couple the LP01 core mode to the ν=5 cladding mode at 1550 nm.

Fig. 11
Fig. 11

(a) Experimentally measured and (b) theoretically calculated transmission spectra through a relatively weak, uniform long-period grating. The resonance is associated with coupling to the lowest-order (l=1, ν=1) cladding mode.  

Equations (86)

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V1-b J1(V1-b)J0(V1-b)=Vb K1(Vb)K0(Vb),
ErcoiE01coJ0(V1-br/a1)×exp(iϕ)exp[i(βz-ωt)](ra1),
Eϕco-E01coJ0(V1-br/a1)×exp(iϕ)exp[i(βz-ωt)](ra1),
E01coZ0bπn21+2bΔ1/2 1a1J1(V1-b)
ζ0=ζ0,
ζ0=1σ2 u2JK+σ1σ2u21u32n22a1a2pl(a2)-Kql(a2)+Jrl(a2)-1u2 sl(a2)-u2u32n22a2 J-u21n12a1 Kpl(a2)+u32n12a2 ql(a2)+u21n12a1 rl(a2),
ζ0=σ1 u2u32a2 J-n32u21n22a1 Kpl(a2)+u32a2 ql(a2)+u21a1 rl(a2)u2n32n22 JK+σ1σ2u21u32n12a1a2pl(a2)-n32n12 Kql(a2)+Jrl(a2)-n22n12u2 sl(a2).
σ1ilneff/Z0,
σ2ilneffZ0,
u211u22-1u12,
u321w32+1u22,
uj2(2π/λ)2(nj2-neff2)[j(1, 2)],
w32(2π/λ)2(neff2-n32),
JJl(u1a1)u1Jl(u1a1),
KKl(w3a2)w3Kl(w3a2),
pl(r)Jl(u2r)Nl(u2a1)-Jl(u2a1)Nl(u2r),
ql(r)Jl(u2r)Nl(u2a1)-Jl(u2a1)Nl(u2r),
rl(r)Jl(u2r)Nl(u2a1)-Jl(u2a1)Nl(u2r),
sl(r)Jl(u2r)Nl(u2a1)-Jl(u2a1)Nl(u2r).
Ercl=iE1νcl u12 J2(u1r)+J0(u1r)-σ2ζ0n12 [J2(u1r)-J0(u1r)]exp(iϕ)exp[i(βz-ωt)]
(ra1),
Eϕcl=E1νcl u12 J2(u1r)-J0(u1r)-σ2ζ0n12 ×[J2(u1r)+J0(u1r)]exp(iϕ)exp[i(βz-ωt)]
(ra1),
Ezcl=E1νcl u12σ2ζ0n12β J1(u1r)exp(iϕ)exp[i(βz-ωt)]
(ra1),
Hrcl=E1νcl u12 {iσ1[J2(u1r)-J0(u1r)]-iζ0[J2(u1r)+J0(u1r)]}exp(iϕ)exp[i(βz-ωt)](ra1),
Hϕcl=-iE1νcl u12 {iσ1[J2(u1r)+J0(u1r)]+iζ0[J2(u1r)-J0(u1r)]}exp(iϕ)exp[i(βz-ωt)](ra1),
Hzcl=-iE1νcl u12iσ1β J1(u1r)exp(iϕ)exp[i(βz-ωt)]
(ra1).
Iz(r)=12 Re(E×H*)zˆ=12 Re(ErclHϕcl*-Hrcl*Eϕcl).
P=1/2 Re 02πdϕ0rdr(ErclHϕcl*-Hrcl*Eϕcl)=1 W.
n(r, z)=n1(z)=n11+σ(z)1+m cos2πΛ zra1n2a1<ra2n3r>a2.
σ(z)σ exp(-4 ln 2z2/w2),
Kνμt(z)=ω4 02πdϕ0rdrΔ(r, z)Eνt(r, ϕ)Eμt*(r, ϕ),
Kνμt(z)=κνμ(z)1+m cos2πΛ z,
κ01-01co-co(z)=ω0n12σ(z)2 02πdϕ0a1rdr(|Erco|2+|Eϕco|2).
κ01-01co-co(z)=σ(z) 2πλ n12bn21+2bΔ 1+J02(V1-b)J12(V1-b).
κν-01cl-co(z)=ω0n12σ(z)2 02πdϕ×0a1rdr(ErclErco*+EϕclEϕco*).
02πdϕ exp[i(l-1)ϕ]=2πδl1,
κ1ν-01cl-co(z)=σ(z) 2πλ πbZ0n21+2bΔ1/2 ×n12u1u12-V2(1-b)/a12×1+σ2ζ0n12E1νclu1J1(u1a1)×J0(V1-b)J1(V1-b)-V1-ba1 J0(u1a1).
dAμdz=i ν Aν(Kνμt+Kνμz)exp[i(βν-βμ)z]+i ν Bν(Kνμt-Kνμz)exp[-i(βν+βμ)z],
dBμdz=-i ν Aν(Kνμt-Kνμz)exp[i(βν-βμ)z]-i ν Bν(Kνμt+Kνμz)exp[-i(βν+βμ)z],
dAcodz=iκ01-01co-coAco+i m2 κ01-01co-coBco exp(-i2δ01-01co-coz)+i ν m2 κ1ν-01cl-coBνcl exp(-i2δ1ν-01cl-coz),
dBcodz=-iκ01-01co-coBco-i m2 κ01-01co-coAco exp(+i2δ01-01co-coz),
νdBνcldz=-i m2 κ1ν-01cl-coAco exp(+i2δ1ν-01cl-coz),
δ01-01co-co12 2β01co-2πΛ,
δ1ν-01cl-co12 β01co+β1νcl-2πΛ.
dAcodz=iκ01-01co-coAco+i ν m2 κ1ν-01cl-coAνcl exp(-i2δ1ν-01cl-coz),
νdAνcldz=+i m2 κ1ν-01cl-coAco exp(+i2δ1ν-01cl-coz),
δ1ν-01cl-co12 β01co-β1νcl-2πΛ.
δ01-01co-co+κ01-01co-co=0,
δ1ν-01cl-co+κ01-01co-co/2=0,
Δλλλκπnavg 1+πκL21/2.
ΔλλλΔnL 1+4κLπ1/2,
Ercl=iE1νcl πa1u12J1(u1a1)2 -F2r p1(r)+1u2r q1(r)-σ2n22 u2G2r1(r)-n22ζ0n12 s1(r)×exp[iϕ+i(βz-ωt)],
Eϕcl=E1νcl πa1u12J1(u1a1)2 σ2n22 G2r p1(r)-n22ζ0n12u2r q1(r)+u2F2r1(r)-s1(r)×exp[iϕ+i(βz-ωt)],
Ezcl=-E1νcl πa1u12u22σ2J1(u1a1)2n22β G2p1(r)+n22ζ0n12u2 q1(r)exp[iϕ+i(βz-ωt)]
Hrcl=E1νcl πa1u12J1(u1a1)2 -i G2r p1(r)+i n22ζ0n12u2r q1(r)+iσ1[u2F2r1(r)-s1(r)]×exp[iϕ+i(βz-ωt)],
Hϕcl=iE1νcl πa1u12J1(u1a1)2 iσ1F2r p1(r)-1u2r q1(r)-iu2G2r1(r)+i n22ζ0n12 s1(r)×exp[iϕ+i(βz-ωt)],
Hzcl=-iE1νcl πa1u12u22iσ1J1(u1a1)2β F2p1(r)-1u2 q1(r)exp[iϕ+i(βz-ωt)],
F2J-u21σ2ζ0n12a1,
G2ζ0J+u21σ1a1.
Ercl=iE1νcl πa1u12u22J1(u1a1)4w3K1(w3a2) -F3[K2(w3r)-K0(w3r)]+σ2G3n32 [K2(w3r)+K0(w3r)]×exp[iϕ+i(βz-ωt)],
Eϕcl=E1νcl πa1u12u22J1(u1a1)4w3K1(w3a2) -F3[K2(w3r)+K0(w3r)]+σ2G3n32 [K2(w3r)-K0(w3r)]×exp[iϕ+i(βz-ωt)],
Ezcl=E1νcl πa1u12u22σ2J1(u1a1)2n32βK1(w3a2) G3K1(w3r)×exp[iϕ+i(βz-ωt)].
Hrcl=E1νcl πa1u12u22J1(u1a1)4w3K1(w3a2) {-iσ1[K2(w3r)+K0(w3r)]-iG3[K2(w3r)-K0(w3r)]}×exp[iϕ+i(βz-ωt)],
Hϕcl=iE1νcl πa1u12u22J1(u1a1)4w3K1(w3a2) {iσ1[K2(w3r)-K0(w3r)]-G3[K2(w3r)+K0(w3r)]}×exp[iϕ+i(βz-ωt)],
Hzcl=iE1νcl πa1u12u22iσ1J1(u1a1)2βK1(w3a2) F3K1(w3r)×exp[iϕ+i(βz-ωt)].
F3-F2p1(a2)+1u2 q1(a2),
G3-n32n22 G2p1(a2)+n22ζ0n12u2 q1(a2).
P=P1+P2+P3,
P1=(E1νcl)2 πa12u124 neffZ0-neffZ0ζ02n12+1+neff2n12Im(ζ0)[J22(u1a1)-J1(u1a1)×J3(u1a1)]+neffZ0-neffZ0ζ02n12-1+neff2n12×Im(ζ0)[J02(u1a1)+J12(u1a1)].
P2=(E1νcl)2 π3a12u14u22J12(u1a1)16×neffZ0 F22-neffZ0n22 G22(Q+Q˜)+1u22×neffZ0-neffZ0n22ζ02n14(R+R˜)+1+neff2n22F2 Im(G2)(Q-Q˜)+1+neff2n22 n22n12u22 Im(ζ0)(R-R˜)-1+neff2n22 n22 Im(ζ0)n12u2 F2+1u2 Im(G2)×(S-S˜)+2neffu2 Z0ζ0n12 G2-1Z0 F2(S+S˜).
QθJN12(u2a1)+θNJ12(u2a1)-2θJNJ1(u2a1)n1(u2a1),
Q˜θ˜JN12(u2a1)+θ˜NJ12(u2a1)-2θ˜JNJ1(u2a1)N1(u2a1),
R14 θJ[N2(u2a1)-N0(u2a1)]2+14 θN[J2(u2a1)-J0(u2a1)]2-12 θJN[N2(u2a1)-N0(u2a1)]×[J2(u2a1)-J0(u2a1)],
R˜14 θ˜J[N2(u2a1)-N0(u2a1)]2+14 θ˜N[J2(u2a1)-J0(u2a1)]2-12 θ˜JN[N2(u2a1)-N0(u2a1)]×[J2(u2a1)-J0(u2a1)],
S12 θJN1(u2a1)[N0(u2a1)-N2(u2a1)]+12 θNJ1(u2a1)[J0(u2a1)-J2(u2a1)]-12 θJN{N1(u2a1)[J0(u2a1)-J2(u2a1)]+J1(u2a1)[N0(u2a1)-N2(u2a1)]},
S˜12 θ˜JN1(u2a1)[N0(u2a1)-N2(u2a1)]+12 θ˜NJ1(u2a1)[J0(u2a1)-J2(u2a1)]-12 θ˜JN{N1(u2a1)[J0(u2a1)-J2(u2a1)]+J1(u2a1)[N0(u2a1)-N2(u2a1)]},
θJa22[J22(u2a2)-J1(u2a2)J3(u2a2)]-a12[J22(u2a1)-J1(u2a1)J3(u2a1)],
θNa22[N22(u2a2)-N1(u2a2)N3(u2a2)]-a12[N22(u2a1)-N1(u2a1)N3(u2a1)],
θJNa22{J2(u2a2)N2(u2a2)-12 [J1(u2a2)N3(u2a2)+J3(u2a2)N1(u2a2)]}-a12{J2(u2a1)N2(u2a1)-12 [J1(u2a1)N3(u2a1)+J3(u2a1)N1(u2a1)]},
θ˜Ja22[J22(u2a2)+J12(u2a2)]-a12[J22(u2a1)+J12(u2a1)],
θ˜Na22[N22(u2a2)+N12(u2a2)]-a12[N22(u2a1)+N12(u2a1)],
θ˜JNa22[J0(u2a2)N0(u2a2)+J1(u2a2)N1(u2a2)]-a12[J0(u2a1)N0(u2a1)+J1(u2a1)N1(u2a1)].
P3=(E1νcl)2 π3a12a22u14u24J12(u1a1)16w32K12(w3a2)×neffZ0n32 G32-neffZ0 F32-1+neff2n32F3 Im(G3)×[K22(w3a2)-K1(w3a2)K3(w3a2)]+neffZ0n32 G32-neffZ0 F32+1+neff2n32F3 Im(G3)×[K02(w3a2)-K12(w3a2)].

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