Abstract

We study the material loss in a Bragg fiber possessing a hollow core and stratified periodic cladding layers through a perturbative method. In the present scheme, electromagnetic fields are treated via the approximate Bloch theorem in cylindrical coordinates for a loss-free Bragg fiber, and then dissipation is added as a perturbation in complex refractive indices. Analytical representation of material loss is described for TE, TM, and hybrid (HE, EH) modes, and some numerical examples are given. They are compared with results obtained by the multilayer division method, which gives very accurate solutions for cylindrically symmetric fiber structures. Results obtained by those two methods mostly agree with each other even for the lowest mode, that is, HE11 mode.

© 2013 Optical Society of America

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  1. P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729–4749 (2006).
    [CrossRef]
  2. T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
    [CrossRef]
  3. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196–1201 (1978).
    [CrossRef]
  4. Y. Xu, R. K. Lee, and A. Yariv, “Asymptotic analysis of Bragg fibers,” Opt. Lett. 25, 1756–1758 (2000).
    [CrossRef]
  5. Y. Xu, G. X. Ouyang, R. K. Lee, and A. Yariv, “Asymptotic matrix theory of Bragg fibers,” J. Lightwave Technol. 20, 428–440 (2002).
    [CrossRef]
  6. J. Sakai and P. Nouchi, “Propagation properties of Bragg fiber analyzed by a Hankel function formalism,” Opt. Commun. 249, 153–163 (2005).
    [CrossRef]
  7. J. Sakai, “Hybrid modes in a Bragg fiber: general properties and formulas under the quarter-wave stack condition,” J. Opt. Soc. Am. B 22, 2319–2330 (2005).
    [CrossRef]
  8. S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljačić, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001).
    [CrossRef]
  9. W. Zhi, R. Guobin, L. Shuqin, L. Weijun, and S. Guo, “Compact supercell method based on opposite parity for Bragg fibers,” Opt. Express 11, 3542–3549 (2003).
    [CrossRef]
  10. J. Sakai and H. Niiro, “Confinement loss evaluation based on a multilayer division method in Bragg fibers,” Opt. Express 16, 1885–1902 (2008).
    [CrossRef]
  11. D. V. Prokopovich, A. V. Popov, and A. V. Vinogradov, “Analytical and numerical aspects of Bragg fiber design,” Prog. Electromagn. Res. 6, 361–379 (2008).
    [CrossRef]
  12. B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
    [CrossRef]
  13. G. Vienne, Y. Xu, C. Jakobsen, H.-J. Deyerl, J. B. Jensen, T. Sørensen, T. P. Hansen, Y. Huang, M. Terrel, R. K. Lee, N. A. Mortensen, J. Broeng, H. Simonsen, A. Bjarklev, and A. Yariv, “Ultra-large bandwidth hollow-core guiding in all-silica Bragg fibers with nano-supports,” Opt. Express 12, 3500–3508 (2004).
    [CrossRef]
  14. F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555–600 (1929).
    [CrossRef]
  15. A. Kitagawa and J. Sakai, “Bloch theorem in cylindrical coordinates and its application to a Bragg fiber,” Phys. Rev. A 80, 033802 (2009).
    [CrossRef]
  16. A. Kitagawa and J. Sakai, “High-accuracy representation of propagation properties of hybrid modes in a Bragg fiber based on Bloch theorem in cylindrical coordinates,” J. Opt. Soc. Am. B 28, 613–621 (2011).
    [CrossRef]
  17. J. Sakai and N. Nishida, “Confinement loss, including cladding material loss effects, in Bragg fibers,” J. Opt. Soc. Am. B 28, 379–386 (2011).
    [CrossRef]
  18. J. Sakai, “Analytical expression of confinement loss in Bragg fibers and its relationship with generalized quarter-wave stack condition,” J. Opt. Soc. Am. B 28, 2740–2754 (2011).
    [CrossRef]
  19. J. Sakai, “Analytical expression of core and cladding material losses in Bragg fibers using the perturbation theory,” J. Opt. Soc. Am. B 28, 2755–2764 (2011).
    [CrossRef]
  20. J. Sakai and Y. Suzuki, “Equivalence between in-phase and antiresonant reflection conditions in Bragg fiber and its application to antiresonant reflecting optical waveguide-type fibers,” J. Opt. Soc. Am. B 28, 183–192 (2011).
    [CrossRef]
  21. J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993).
    [CrossRef]
  22. K. J. Rowland, S. V. Afshar, and T. M. Monro, “Bandgaps and antiresonances in integrated-ARROW and Bragg fibers; a simple model,” Opt. Express 16, 17935–17951 (2008).
    [CrossRef]
  23. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965), Chap. 9.

2011 (5)

2009 (1)

A. Kitagawa and J. Sakai, “Bloch theorem in cylindrical coordinates and its application to a Bragg fiber,” Phys. Rev. A 80, 033802 (2009).
[CrossRef]

2008 (3)

2006 (1)

2005 (2)

J. Sakai and P. Nouchi, “Propagation properties of Bragg fiber analyzed by a Hankel function formalism,” Opt. Commun. 249, 153–163 (2005).
[CrossRef]

J. Sakai, “Hybrid modes in a Bragg fiber: general properties and formulas under the quarter-wave stack condition,” J. Opt. Soc. Am. B 22, 2319–2330 (2005).
[CrossRef]

2004 (1)

2003 (1)

2002 (2)

Y. Xu, G. X. Ouyang, R. K. Lee, and A. Yariv, “Asymptotic matrix theory of Bragg fibers,” J. Lightwave Technol. 20, 428–440 (2002).
[CrossRef]

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
[CrossRef]

2001 (1)

2000 (1)

1995 (1)

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
[CrossRef]

1993 (1)

J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993).
[CrossRef]

1978 (1)

1929 (1)

F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555–600 (1929).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965), Chap. 9.

Afshar, S. V.

Archambault, J. L.

J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993).
[CrossRef]

Atkin, D. M.

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
[CrossRef]

Benoit, G.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
[CrossRef]

Birks, T. A.

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
[CrossRef]

Bjarklev, A.

Black, R. J.

J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993).
[CrossRef]

Bloch, F.

F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555–600 (1929).
[CrossRef]

Broeng, J.

Bures, J.

J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993).
[CrossRef]

Deyerl, H.-J.

Engeness, T. D.

Fink, Y.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
[CrossRef]

S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljačić, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001).
[CrossRef]

Guo, S.

Guobin, R.

Hansen, T. P.

Hart, S. D.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
[CrossRef]

Huang, Y.

Ibanescu, M.

Jacobs, S. A.

Jakobsen, C.

Jensen, J. B.

Joannopoulos, J. D.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
[CrossRef]

S. G. Johnson, M. Ibanescu, M. Skorobogatiy, O. Weisberg, T. D. Engeness, M. Soljačić, S. A. Jacobs, J. D. Joannopoulos, and Y. Fink, “Low-loss asymptotically single-mode propagation in large-core OmniGuide fibers,” Opt. Express 9, 748–779 (2001).
[CrossRef]

Johnson, S. G.

Kitagawa, A.

Lacroix, S.

J. L. Archambault, R. J. Black, S. Lacroix, and J. Bures, “Loss calculations for antiresonant waveguides,” J. Lightwave Technol. 11, 416–423 (1993).
[CrossRef]

Lee, R. K.

Marom, E.

Monro, T. M.

Mortensen, N. A.

Niiro, H.

Nishida, N.

Nouchi, P.

J. Sakai and P. Nouchi, “Propagation properties of Bragg fiber analyzed by a Hankel function formalism,” Opt. Commun. 249, 153–163 (2005).
[CrossRef]

Ouyang, G. X.

Popov, A. V.

D. V. Prokopovich, A. V. Popov, and A. V. Vinogradov, “Analytical and numerical aspects of Bragg fiber design,” Prog. Electromagn. Res. 6, 361–379 (2008).
[CrossRef]

Prokopovich, D. V.

D. V. Prokopovich, A. V. Popov, and A. V. Vinogradov, “Analytical and numerical aspects of Bragg fiber design,” Prog. Electromagn. Res. 6, 361–379 (2008).
[CrossRef]

Roberts, P. J.

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
[CrossRef]

Rowland, K. J.

Russell, P. St. J.

P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729–4749 (2006).
[CrossRef]

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
[CrossRef]

Sakai, J.

A. Kitagawa and J. Sakai, “High-accuracy representation of propagation properties of hybrid modes in a Bragg fiber based on Bloch theorem in cylindrical coordinates,” J. Opt. Soc. Am. B 28, 613–621 (2011).
[CrossRef]

J. Sakai and N. Nishida, “Confinement loss, including cladding material loss effects, in Bragg fibers,” J. Opt. Soc. Am. B 28, 379–386 (2011).
[CrossRef]

J. Sakai and Y. Suzuki, “Equivalence between in-phase and antiresonant reflection conditions in Bragg fiber and its application to antiresonant reflecting optical waveguide-type fibers,” J. Opt. Soc. Am. B 28, 183–192 (2011).
[CrossRef]

J. Sakai, “Analytical expression of confinement loss in Bragg fibers and its relationship with generalized quarter-wave stack condition,” J. Opt. Soc. Am. B 28, 2740–2754 (2011).
[CrossRef]

J. Sakai, “Analytical expression of core and cladding material losses in Bragg fibers using the perturbation theory,” J. Opt. Soc. Am. B 28, 2755–2764 (2011).
[CrossRef]

A. Kitagawa and J. Sakai, “Bloch theorem in cylindrical coordinates and its application to a Bragg fiber,” Phys. Rev. A 80, 033802 (2009).
[CrossRef]

J. Sakai and H. Niiro, “Confinement loss evaluation based on a multilayer division method in Bragg fibers,” Opt. Express 16, 1885–1902 (2008).
[CrossRef]

J. Sakai and P. Nouchi, “Propagation properties of Bragg fiber analyzed by a Hankel function formalism,” Opt. Commun. 249, 153–163 (2005).
[CrossRef]

J. Sakai, “Hybrid modes in a Bragg fiber: general properties and formulas under the quarter-wave stack condition,” J. Opt. Soc. Am. B 22, 2319–2330 (2005).
[CrossRef]

Shepherd, T. J.

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
[CrossRef]

Shuqin, L.

Simonsen, H.

Skorobogatiy, M.

Soljacic, M.

Sørensen, T.

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965), Chap. 9.

Suzuki, Y.

Temelkuran, B.

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
[CrossRef]

Terrel, M.

Vienne, G.

Vinogradov, A. V.

D. V. Prokopovich, A. V. Popov, and A. V. Vinogradov, “Analytical and numerical aspects of Bragg fiber design,” Prog. Electromagn. Res. 6, 361–379 (2008).
[CrossRef]

Weijun, L.

Weisberg, O.

Xu, Y.

Yariv, A.

Yeh, P.

Zhi, W.

Electron. Lett. (1)

T. A. Birks, P. J. Roberts, P. St. J. Russell, D. M. Atkin, and T. J. Shepherd, “Full 2-D photonic bandgaps in silica/air structures,” Electron. Lett. 31, 1941–1943 (1995).
[CrossRef]

J. Lightwave Technol. (3)

J. Opt. Soc. Am. (1)

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

Nature (1)

B. Temelkuran, S. D. Hart, G. Benoit, J. D. Joannopoulos, and Y. Fink, “Wavelength-scalable hollow optical fibres with large photonic bandgaps for CO2 laser transmission,” Nature 420, 650–653 (2002).
[CrossRef]

Opt. Commun. (1)

J. Sakai and P. Nouchi, “Propagation properties of Bragg fiber analyzed by a Hankel function formalism,” Opt. Commun. 249, 153–163 (2005).
[CrossRef]

Opt. Express (5)

Opt. Lett. (1)

Phys. Rev. A (1)

A. Kitagawa and J. Sakai, “Bloch theorem in cylindrical coordinates and its application to a Bragg fiber,” Phys. Rev. A 80, 033802 (2009).
[CrossRef]

Prog. Electromagn. Res. (1)

D. V. Prokopovich, A. V. Popov, and A. V. Vinogradov, “Analytical and numerical aspects of Bragg fiber design,” Prog. Electromagn. Res. 6, 361–379 (2008).
[CrossRef]

Z. Phys. (1)

F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555–600 (1929).
[CrossRef]

Other (1)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, 1965), Chap. 9.

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Figures (10)

Fig. 1.
Fig. 1.

Distribution of the refractive index in a Bragg fiber composed of a hollow core (nc) and stratified cladding layers a and b (na, nb). Thicknesses of cladding layers a and b are set to a and b, respectively, and the period is Λ=a+b. rc, rmA, and rmB are interfaces between core and the first cladding layer a, the mth cladding layers a and b, and the mth cladding layer b and (m+1)th cladding layer a, respectively.

Fig. 2.
Fig. 2.

Dependence of material loss on core radius, where the wavelength and refractive index of cladding layer a are fixed to λ0=1.0μm and na=2.5, respectively. Dotted, dashed–dotted, solid, and dashed lines represent results of TE, TM, HE, and EH modes, respectively, by the present method. Symbols represent results obtained by the MLD method.

Fig. 3.
Fig. 3.

Dependence of the relative difference between the material losses of the present and the MLD methods on core radius on percentage for TE01, TM01, HE11, and EH11 modes. Fiber parameters are the same as those in Fig. 2.

Fig. 4.
Fig. 4.

Dependence of material loss on refractive index of cladding layer a. The core radius is fixed at rc=2.0μm and the wavelength is supposed to be λ0=1.0μm. Dotted, dashed–dotted, solid, and dashed lines represent results of TE, TM, HE, and EH modes, respectively, obtained by the present method. Various kinds of symbols represent results obtained by the MLD method.

Fig. 5.
Fig. 5.

Wavelength dependence of the material loss of the fundamental PBG for TE01 mode. Thicknesses of cladding layers a and b are determined with the QWS condition at λ0=1.0μm for all cases.

Fig. 6.
Fig. 6.

Wavelength dependence of the material loss of the fundamental PBG for TM01 mode. Thicknesses of cladding layers a and b are determined similarly to the case of TE01 mode.

Fig. 7.
Fig. 7.

Wavelength dependence of the material loss of the fundamental PBG for HE11 mode. Thicknesses of cladding layers a and b are determined similarly to the case of TE01 mode.

Fig. 8.
Fig. 8.

Wavelength dependence of the material loss of the fundamental PBG for EH11 mode. Thicknesses of cladding layers a and b are determined similarly to the case of TE01 mode.

Fig. 9.
Fig. 9.

Wavelength dependence of Lmat of the lower five PBGs for TE01, TM01, HE11, and EH11 modes, respectively. Parameters are rc=2.0μm, na=2.5, and the thicknesses of cladding layer a and b are determined with the QWS condition (q1=q2=1) at λ0=1.0μm. The dip of the second PBG for TM01 is not here since it is out of range.

Fig. 10.
Fig. 10.

Dependence of the material loss on the thickness of the cladding layer a for TE01, TM01, HE11, and EH11 modes. Fiber parameters are rc=2.0μm, λ0=1.0μm, and na=2.5. Dips appear around a that is obtained from the generalized QWS condition, and they correspond to PBGs at the generalized QWS condition of q1=q2=1,2,3 from the left. Symbols indicate the points that satisfy the generalized QWS condition.

Equations (55)

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(HziEθEziHθ)=UtzΩ(θ)Di(r)(AiBiCiDi),
Di(r)=(d11(i)d12(i)00d21(i)d22(i)d23(i)d24(i)00d33(i)d34(i)d41(i)d42(i)d43(i)d44(i)),
d11(i)=d33(i)=Hν(2)(κir),d12(i)=d34(i)=Hν(1)(κir),d21(i)=d43(i)Yi2=ωμ0κiHν(2)(κir),d22(i)=d44(i)Yi2=ωμ0κiHν(1)(κir),d23(i)=d41(i)=νβκi2rHν(2)(κir),d24(i)=d42(i)=νβκi2rHν(1)(κir),
κi=[(nik0)2β2]1/2(i=c,a,b)
(AmBmCmDm)=S=TE,TMξ˜a,1(S)[η˜j(S)]m1Θm,1(a)(P˜1,j(S)Q˜1,j(S)R˜1,j(S)S˜1,j(S)),
(AmBmCmDm)=S=TE,TMξ˜b,1(S)[η˜j(S)]m1Θm,1(b)(P˜1,j(S)Q˜1,j(S)R˜1,j(S)S˜1,j(S)),
X˜a,m(S)=[cosκbbi2(ζbζaκbκa+ζaζbκaκb)sinκbb]exp(iκaa),
ζi={1(S=TE)1/ni2(i=a,b)(S=TM).
Re(X˜a,m(S))=Re(X˜b,m(S))=cosκaacosκbb12(ζbζaκbκa+ζaζbκaκb)sinκaasinκbb,
n˜i=niiki,kiαIi2k0(i=c,a,b),
εi=n˜i2ni22iniki=εi+δεi,
δβ=k02δε(r)|Ψ|2dS2β|Ψ|2dS,
δε(r)={0(0rrc)2inaka(r(m1)BrrmA)2inbkb(rmArrmB),
δβ=ik02βnakaIa+nbkbIbIc+Ia+Ib,
Ic=02πdθ0rcr(|Ez|2+|iEθ|2+|iEr|2)dr=Iz,c+Iθ,c+Ir,c,
Ia=m=102πdθr(m1)BrmAr(|Ez|2+|iEθ|2+|iEr|2)dr=Iz,a+Iθ,a+Ir,a,
Ib=m=102πdθrmArmBr(|Ez|2+|iEθ|2+|iEr|2)dr=Iz,b+Iθ,b+Ir,b.
Lmat[dB/km]=20ln10×109×βI[μm1],
κaa=πq1π2,κbb=π(q2q1)+π2,
κcrc=UQWS,
aλ0=2q114[na2nc2+(UQWSλ02πrc)2]1/2,
bλ0=2(q2q1)+14[nb2nc2+(UQWSλ02πrc)2]1/2.
Lmat(α[dB/km])=Lmat(α=10)×(α[dB/km]10)
S˜(rmB;rmA)(AmBmCmDm)=η˜j(S)(AmBmCmDm),
S˜(rmB;rmA)=[Θ(m+1),m(a)]1ba(rmB)ab(rmA),
ij(r)=Dj1(r)Di(r),
Θm,m(i)diag.[qimm,(qi*)mm,qimm,(qi*)mm],
qi=exp(iκiΛ).
η˜j(S)=Re(X˜a,m(S))±{[Re(X˜a,m(S))]21}1/2,
X˜a,m(S)={[S˜(rmB;rmA)]11(S=TE)[S˜(rmB;rmA)]33(S=TM),
S˜(r(m+1)A;rmB)(AmBmCmDm)=η˜j(S)(AmBmCmDm),
S˜(r(m+1)A;rmB)=[Θ(m+1),m(b)]1ab(r(m+1)A)ba(rmB),
η˜j(S)=Re(X˜b,m(S))±{[Re(X˜b,m(S))]21}1/2,
X˜b,m(S)={[S˜(r(m+1)A;rmB)]11(S=TE)[S˜(r(m+1)A;rmB)]33(S=TM).
(σ11σ120σ14σ21σ22σ23σ240σ32σ33σ34σ41σ42σ43σ44)(Acξ˜a,1(TE)Ccξ˜a,1(TM))=0.
CcAc=ωμ0κcrcνβJν(κcrc)Jν(κcrc)+κciκaΣj(TE)1+(κc/iκa)2=1Yc2νβωμ0κcrc1+(κc/iκa)2Jν(κcrc)Jν(κcrc)+κciκana2nc2Σj(TM)
(ξ˜a,1(TE)ξ˜a,1(TM))=(σ22σ24σ32σ34)1(σ21σ230σ33)(AcCc).
J1[Zν,Z˜ν](z)=zZν(z)Z˜ν(z)dz=z22[(1ν2z2)Zν(z)Z˜ν(z)+Zν(z)Z˜ν(z)],
J2[Zν,Z˜ν](z)=Zν(z)Z˜ν(z)zdz=12ν[Z0(z)Z˜0(z)+2ρ=1ν1Zρ(z)Z˜ρ(z)+Zν(z)Z˜ν(z)],
J3[Zν,Z˜ν](z)=zZν(z)Z˜ν(z)dz=12{J1[Zν1,Z˜ν1](z)+J1[Zν+1,Z˜ν+1](z)}ν2J2[Zν,Z˜ν](z),
J4[Zν](z)=Zν(z)Zν(z)dz=12Zν2(z).
J5[Hν(1),Hν(2)](z)=Hν(1)(z)Hν(2)(z)dz=12Hν(1)(z)Hν(2)(z)2iπlnz.
Ki[Zν,Z˜ν](z2;z1)Ji[Zν,Z˜ν](z2)Ji[Zν,Z˜ν](z1)
Iz,c=4|Cc|2K1[Jν,Jν](κcrc;0)κc2,
Iθ,c=4{ω2μ02κc4K3[Jν,Jν](κcrc;0)|Ac|2+ωμ0νβκc4K4[Jν,Jν](κcrc;0)(AcCc*+Ac*Cc)+ν2β2κc4K2[Jν,Jν](κcrc;0)|Cc|2},
Ir,c=4{ω2μ02ν2κc4K2[Jν,Jν](κcrc;0)|Ac|2+ωμ0νβκc4K4[Jν,Jν](κcrc;0)(AcCc*+Ac*Cc)+β2κc4K3[Jν,Jν](κcrc;0)|Cc|2},
Iz,a=S,S=TE,TMξ˜a,1(S)*ξ˜a,1(S){2R˜1,j(S)*R˜1,j(S)L1(a)[Hν(1),Hν(2)](η˜j(S)η˜j(S))+R˜1,j(S)*S˜1,j(S)L1(a)[Hν(1),Hν(1)](η˜j(S)η˜j(S))+S˜1,j(S)*R˜1,j(S)L1(a)[Hν(2),Hν(2)](η˜j(S)η˜j(S))},
Iθ,a=ω2μ02κa2S,S=TE,TMξ˜a,1(S)*ξ˜a,1(S){2P˜1,j(S)*P˜1,j(S)L3(a)[Hν(1),Hν(2)](η˜j(S)η˜j(S))+P˜1,j(S)*Q˜1,j(S)L3(a)[Hν(1),Hν(1)](η˜j(S)η˜j(S))+Q˜1,j(S)*P˜1,j(S)L3(a)[Hν(2),Hν(2)](η˜j(S)η˜j(S))}4iπωμ0νβκa4m=1(AmCm*Am*Cm)lnrmAr(m1)B,
Ir,a=β2κa2S,S=TE,TMξ˜a,1(S)*ξ˜a,1(S){2R˜1,j(S)*R˜1,j(S)L3(a)[Hν(1),Hν(2)](η˜j(S)η˜j(S))+R˜1,j(S)*S˜1,j(S)L3(a)[Hν(1),Hν(1)](η˜j(S)η˜j(S))+S˜1,j(S)*R˜1,j(S)L3(a)[Hν(2),Hν(2)](η˜j(S)η˜j(S))}+4iπωμ0νβκa4m=1(AmCm*Am*Cm)lnrmAr(m1)B,
Iz,b=S,S=TE,TMξ˜b,1(S)*ξ˜b,1(S){2R˜1,j(S)*R˜1,j(S)L1(b)[Hν(1),Hν(2)](η˜j(S)η˜j(S))+R˜1,j(S)*S˜1,j(S)L1(b)[Hν(1),Hν(1)](η˜j(S)η˜j(S))+S˜1,j(S)*R˜1,j(S)L1(b)[Hν(2),Hν(2)](η˜j(S)η˜j(S))},
Iθ,b=ω2μ02κb2S,S=TE,TMξ˜b,1(S)*ξ˜b,1(S){2P˜1,j(S)*P˜1,j(S)L3(b)[Hν(1),Hν(2)](η˜j(S)η˜j(S))+P˜1,j(S)*Q˜1,j(S)L3(b)[Hν(1),Hν(1)](η˜j(S)η˜j(S))+Q˜1,j(S)*P˜1,j(S)L3(b)[Hν(2),Hν(2)](η˜j(S)η˜j(S))}4iπωμ0νβκb4m=1(AmCm*Am*Cm)lnrmBrmA,
Ir,b=β2κb2S,S=TE,TMξ˜b,1(S)*ξ˜b,1(S){2R˜1,j(S)*R˜1,j(S)L3(b)[Hν(1),Hν(2)](η˜j(S)η˜j(S))+R˜1,j(S)*S˜1,j(S)L3(b)[Hν(1),Hν(1)](η˜j(S)η˜j(S))+S˜1,j(S)*R˜1,j(S)L3(b)[Hν(2),Hν(2)](η˜j(S)η˜j(S))}+4iπωμ0νβκb4m=1(AmCm*Am*Cm)lnrmBrmA,
[Θm,1(i)]11r1ArmAHν(2)(κir1A)Hν(2)(κirmA)r1ArmAHν(2)(κir1A)Hν(2)(κirmA)
Li(a)[Zν,Z˜ν](x)=m=1{Ji[Zν,Z˜ν](κar1A)κa2r1ArmAxm1Ji[Zν,Z˜ν](κarc)κa2rcr(m1)Bxm1}=Ji[Zν,Z˜ν](κar1A)κa2r1Ar1A(1x)+Λx(1x)2Ji[Zν,Z˜ν](κarc)κa2rcrc(1x)+Λx(1x)2,
Li(b)[Zν,Z˜ν](x)=m=1{Ji[Zν,Z˜ν](κbr1B)κb2r1BrmBxm1Ji[Zν,Z˜ν](κbr1A)κb2r1ArmAxm1}=Ji[Zν,Z˜ν](κbr1B)κb2r1Br1B(1x)+Λx(1x)2Ji[Zν,Z˜ν](κbr1A)κb2r1Ar1A(1x)+Λx(1x)2,

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