Abstract

We investigate the propagation properties of a Bragg fiber with high-accuracy analytical representation. In this study, electromagnetic waves in the cladding are treated as genuine cylindrical waves, that is, Hankel functions. We apply the Bloch theorem in the cylindrical coordinates to the electromagnetic fields in the periodically stratified cladding structure. Then, effective indices are actually calculated for TE, TM, and hybrid (HE, EH) modes through eigenvalue equations. We show that these results are distinctly close to those by the multilayer division method that gives more accurate solutions for cylindrically symmetric fiber structures than the results by the asymptotic expansion method, even for the lowest mode HE11.

© 2011 Optical Society of America

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References

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  1. P. St. J. Russell, “Photonic-crystal fibers,” J. Lightwave Technol. 24, 4729–4749 (2006).
    [CrossRef]
  2. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  3. 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]
  4. P. Yeh, A. Yariv, and E. Marom, “Theory of Bragg fiber,” J. Opt. Soc. Am. 68, 1196–1201 (1978).
    [CrossRef]
  5. V. N. Melekhin and A. B. Manenkov, “Dielectric tube as low-loss waveguide,” Zh. Tekh. Fiz. 38, 2113–2115 (1968).
  6. Y. Xu, R. K. Lee, and A. Yariv, “Asymptotic analysis of Bragg fibers,” Opt. Lett. 25, 1756–1758 (2000).
    [CrossRef]
  7. 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] [PubMed]
  8. A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express 10, 1411–1417 (2002).
    [PubMed]
  9. 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]
  10. 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] [PubMed]
  11. D. V. Prokopovich, A. V. Popov, and A. V. Vinogradov, “Analytical and numerical aspects of Bragg fiber design,” Prog. Electromag. Res. B 6, 361–379 (2008).
    [CrossRef]
  12. M. Ibanescu, S. G. Johnson, M. Soljačić, J. D. Joannopoulos, and Y. Fink, “Analysis of mode structure in hollow dielectric waveguide fibers,” Phys. Rev. E 67, 046608 (2003).
    [CrossRef]
  13. J. Sakai and P. Nouchi, “Propagation properties of Bragg fiber analyzed by a Hankel function formalism,” Opt. Commun. 249, 153–163 (2005).
    [CrossRef]
  14. 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]
  15. 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] [PubMed]
  16. 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] [PubMed]
  17. J. Sakai and H. Niiro, “Confinement loss evaluation based on a multilayer division method in Bragg fibers,” Opt. Express 16, 1885–1902 (2008).
    [CrossRef] [PubMed]
  18. F. Bloch, “Über die Quantenmechanik der Elektronen in Kristallgittern,” Z. Phys. 52, 555–600 (1929).
    [CrossRef]
  19. A. Kitagawa and J. Sakai, “Bloch theorem in cylindrical coordinates and its application to a Bragg fiber,” Phys. Rev. A 80, 033802 (2009).
    [CrossRef]
  20. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables (Dover, New York, 1965), Chap. 9.

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 (2)

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

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 (2)

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] [PubMed]

M. Ibanescu, S. G. Johnson, M. Soljačić, J. D. Joannopoulos, and Y. Fink, “Analysis of mode structure in hollow dielectric waveguide fibers,” Phys. Rev. E 67, 046608 (2003).
[CrossRef]

2002 (3)

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] [PubMed]

A. Argyros, “Guided modes and loss in Bragg fibres,” Opt. Express 10, 1411–1417 (2002).
[PubMed]

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]

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]

1987 (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

1978 (1)

1968 (1)

V. N. Melekhin and A. B. Manenkov, “Dielectric tube as low-loss waveguide,” Zh. Tekh. Fiz. 38, 2113–2115 (1968).

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, New York, 1965), Chap. 9.

Argyros, A.

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] [PubMed]

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.

Bloch, F.

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

Broeng, J.

Deyerl, H.-J.

Engeness, T. D.

Fink, Y.

M. Ibanescu, S. G. Johnson, M. Soljačić, J. D. Joannopoulos, and Y. Fink, “Analysis of mode structure in hollow dielectric waveguide fibers,” Phys. Rev. E 67, 046608 (2003).
[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] [PubMed]

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] [PubMed]

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] [PubMed]

Huang, Y.

Ibanescu, M.

Jacobs, S. A.

Jakobsen, C.

Jensen, J. B.

Joannopoulos, J. D.

M. Ibanescu, S. G. Johnson, M. Soljačić, J. D. Joannopoulos, and Y. Fink, “Analysis of mode structure in hollow dielectric waveguide fibers,” Phys. Rev. E 67, 046608 (2003).
[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] [PubMed]

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] [PubMed]

Johnson, S. G.

Kitagawa, A.

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

Lee, R. K.

Manenkov, A. B.

V. N. Melekhin and A. B. Manenkov, “Dielectric tube as low-loss waveguide,” Zh. Tekh. Fiz. 38, 2113–2115 (1968).

Marom, E.

Melekhin, V. N.

V. N. Melekhin and A. B. Manenkov, “Dielectric tube as low-loss waveguide,” Zh. Tekh. Fiz. 38, 2113–2115 (1968).

Mortensen, N. A.

Niiro, H.

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. Electromag. Res. B 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. Electromag. Res. B 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]

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, “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] [PubMed]

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]

J. Sakai and P. Nouchi, “Propagation properties of Bragg fiber analyzed by a Hankel function formalism,” Opt. Commun. 249, 153–163 (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, New York, 1965), Chap. 9.

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] [PubMed]

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. Electromag. Res. B 6, 361–379 (2008).
[CrossRef]

Weijun, L.

Weisberg, O.

Xu, Y.

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

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. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (1)

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] [PubMed]

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]

Phys. Rev. E (1)

M. Ibanescu, S. G. Johnson, M. Soljačić, J. D. Joannopoulos, and Y. Fink, “Analysis of mode structure in hollow dielectric waveguide fibers,” Phys. Rev. E 67, 046608 (2003).
[CrossRef]

Phys. Rev. Lett. (1)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Prog. Electromag. Res. B (1)

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

Z. Phys. (1)

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

Zh. Tekh. Fiz. (1)

V. N. Melekhin and A. B. Manenkov, “Dielectric tube as low-loss waveguide,” Zh. Tekh. Fiz. 38, 2113–2115 (1968).

Other (1)

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

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Figures (4)

Fig. 1
Fig. 1

Distribution of refractive index in Bragg fiber, which consists of core possessing n c and periodically stratified cladding made of bilayers a and b with n a and n b ( n a > n b > n c ). The core radius and thicknesses of layers a and b are set to r c , a, and b, and the period of cladding layer Λ = a + b is also introduced. r m A = r c + ( m 1 ) Λ + a and r m B = r c + m Λ are the interface position at mth layers a b and mth layer b ( m + 1 ) th layer a, respectively. A m D m ( A m D m ) are amplitude coefficients in cladding layer a (b).

Fig. 2
Fig. 2

Dependence of Re ( X ˜ m ( TE ) ) on layer indication m for HE 11 mode. In this evaluation, propagation constant under the QWS condition [14], which is obtained without solving the eigenvalue Eq. (44) of Bragg fiber, is employed. Refractive indices in the core and cladding layers a and b are set to n c = 1.0 , n a = 2.5 , and n b = 1.5 , respectively, for r c / λ 0 = 0.6 and 1.0. The cases of n a = 3.5 and 4.5 are also studied for r c / λ 0 = 0.3 , while n c and n b are fixed at the same value as above. Thicknesses of layers a and b are calculated in accordance with the QWS condition.

Fig. 3
Fig. 3

Effective index n eff = β / k 0 as a function of r c / λ 0 . Refractive indices are set to n c = 1.0 , n a = 2.5 , and n b = 1.5 . Thicknesses of cladding layers a and b are determined by Eq. (51) and tentative index n t = 0.8 . Results obtained by present method are shown with solid lines for two lower orders of TE, TM, and hybrid (HE, EH) modes. Results with MLD [17] method are also indicated with various symbols for HE 11 , EH 11 , TE 01 , and TM 01 modes. The dashed lines indicate results calculated by the asymptotic expansion method [14]. In this figure, except for the HE 11 mode, no appreciable difference exists between results of present and asymptotic expansion methods.

Fig. 4
Fig. 4

Dependence of effective index on refractive index in cladding layer a, where n a = 2.5 , 3.5, and 4.5 are supposed. Thicknesses of cladding layers a and b are calculated with Eq. (51) and n t = 0.8 for each case. Solid and dashed lines show results calculated by pres ent and asymptotic expansion method [14]. Results by the MLD method [17] are indicated by various symbols for comparison.

Equations (51)

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( H z i E θ E z i H θ ) = U t z Ω ( θ ) D i ( r ) ( A i B i C i D i ) ,
Ω ( θ ) = diag . ( sin ( ν θ + θ in ) , sin ( ν θ + θ in ) , cos ( ν θ + θ in ) , cos ( ν θ + θ in ) ) ,
D i ( r ) = ( d 11 ( i ) d 12 ( i ) 0 0 d 21 ( i ) d 22 ( i ) d 23 ( i ) d 24 ( i ) 0 0 d 33 ( i ) d 34 ( i ) d 41 ( i ) d 42 ( i ) d 43 ( i ) d 44 ( i ) ) ,
d 11 ( i ) = d 33 ( i ) = H ν ( 2 ) ( κ i r ) , d 12 ( i ) = d 34 ( i ) = H ν ( 1 ) ( κ i r ) , d 21 ( i ) = d 43 ( i ) Y i 2 = ω μ 0 κ i H ν ( 2 ) ( κ i r ) , d 22 ( i ) = d 44 ( i ) Y i 2 = ω μ 0 κ i H ν ( 1 ) ( κ i r ) , d 23 ( i ) = d 41 ( i ) = ν β κ i 2 r H ν ( 2 ) ( κ i r ) , d 24 ( i ) = d 42 ( i ) = ν β κ i 2 r H ν ( 1 ) ( κ i r ) ,
κ i = [ ( n i k 0 ) 2 β 2 ] 1 / 2 ( i = c , a , b )
D i | ν = 0 = ( d 11 ( i ) d 12 ( i ) d 21 ( i ) d 22 ( i ) ) ( d 33 ( i ) d 34 ( i ) d 43 ( i ) d 44 ( i ) ) = D i ( TE ) D i ( TM ) ,
( H z i E θ E z i H θ ) = U t z Ω ( θ ) ( 2 J ν ( κ c r ) 0 0 0 2 ω μ 0 κ c J ν ( κ c r ) 0 2 ν β κ c 2 r J ν ( κ c r ) 0 0 0 2 J ν ( κ c r ) 0 2 ν β κ c 2 r J ν ( κ c r ) 0 2 ω ε 0 n c 2 κ c J ν ( κ c r ) 0 ) ( A c B c C c D c ) .
D b ( r m A ) ( A m B m C m D m ) = D a ( r m A ) ( A m B m C m D m ) ,
( A m B m C m D m ) = H a b ( r m A ) ( A m B m C m D m ) ,
H a b ( r m A ) = D b 1 ( r m A ) D a ( r m A ) .
[ H i j ( r ) ] 11 = [ H i j ( r ) ] 22 * = f i j ( TE ) ( r ) , [ H i j ( r ) ] 12 = [ H i j ( r ) ] 21 * = g i j ( TE ) ( r ) , [ H i j ( r ) ] 33 = [ H i j ( r ) ] 44 * = f i j ( TM ) ( r ) , [ H i j ( r ) ] 34 = [ H i j ( r ) ] 43 * = g i j ( TM ) ( r ) , [ H i j ( r ) ] 13 = [ H i j ( r ) ] 24 * = Y j 2 [ H i j ( r ) ] 31 = Y j 2 [ H i j ( r ) ] 42 * = p i j ( r ) , [ H i j ( r ) ] 14 = [ H i j ( r ) ] 23 * = Y j 2 [ H i j ( r ) ] 32 = Y j 2 [ H i j ( r ) ] 41 * = q i j ( r ) ,
f i j ( S ) ( r ) = i π κ j r 4 [ κ j κ i ζ j ζ i H ν ( 1 ) ( κ j r ) H ν ( 2 ) ( κ i r ) H ν ( 1 ) ( κ j r ) H ν ( 2 ) ( κ i r ) ] , g i j ( S ) ( r ) = i π κ j r 4 [ κ j κ i ζ j ζ i H ν ( 1 ) ( κ j r ) H ν ( 1 ) ( κ i r ) H ν ( 1 ) ( κ j r ) H ν ( 1 ) ( κ i r ) ] , p i j ( r ) = i π ν β 4 ω μ 0 ( 1 κ j 2 κ i 2 ) H ν ( 1 ) ( κ j r ) H ν ( 2 ) ( κ i r ) , q i j ( r ) = i π ν β 4 ω μ 0 ( 1 κ j 2 κ i 2 ) H ν ( 1 ) ( κ j r ) H ν ( 1 ) ( κ i r ) ,
ζ i = { 1 ( S = TE ) 1 / n i 2 ( i = a , b ) ( S = TM )
f j i ( S ) ( r ) = ( κ i κ j ) 2 ζ i ζ j f i j ( S ) * ( r ) , g j i ( S ) ( r ) = ( κ i κ j ) 2 ζ i ζ j g i j ( S ) ( r ) , p j i ( r ) = ( κ i κ j ) 2 p i j * ( r ) , q j i ( r ) = ( κ i κ j ) 2 q i j ( r ) .
det [ H i j ( r ) ] = ( κ j κ i ) 2 ( κ j n i κ i n j ) 2 .
( A m + 1 B m + 1 C m + 1 D m + 1 ) = H b a ( r m B ) ( A m B m C m D m ) .
( A m + 1 B m + 1 C m + 1 D m + 1 ) = S ( r m B ; r m A ) ( A m B m C m D m ) ,
S ( r m B ; r m A ) = H b a ( r m B ) H a b ( r m A ) = ( X m ( TE ) Y m ( TE ) s 13 s 14 Y m ( TE ) * X m ( TE ) * s 23 s 24 s 31 s 32 X m ( TM ) Y m ( TM ) s 41 s 42 Y m ( TM ) * X m ( TM ) * ) ,
X m ( S ) = f b a ( S ) ( r m B ) f a b ( S ) ( r m A ) + g b a ( S ) ( r m B ) g a b ( S ) * ( r m A ) + 1 Y b 2 ζ a ζ b [ p b a ( r m B ) p a b ( r m A ) + q b a ( r m B ) q a b * ( r m A ) ] , Y m ( S ) = f b a ( S ) ( r m B ) g a b ( S ) ( r m A ) + g b a ( S ) ( r m B ) f a b ( S ) * ( r m A ) + 1 Y b 2 ζ a ζ b [ p b a ( r m B ) q a b ( r m A ) + q b a ( r m B ) p a b * ( r m A ) ] , s 13 = s 24 * = f b a ( TE ) ( r m B ) p a b ( r m A ) + g b a ( TE ) ( r m B ) q a b * ( r m A ) + p b a ( r m B ) f a b ( TM ) ( r m A ) + q b a ( r m B ) g a b ( TM ) * ( r m A ) , s 14 = s 23 * = f b a ( TE ) ( r m B ) q a b ( r m A ) + g b a ( TE ) ( r m B ) p a b * ( r m A ) + p b a ( r m B ) g a b ( TM ) ( r m A ) + q b a ( r m B ) f a b ( TM ) * ( r m A ) , s 31 = s 42 * = 1 Y a 2 [ p b a ( r m B ) f a b ( TE ) ( r m A ) + q b a ( r m B ) g a b ( TE ) * ( r m A ) ] + 1 Y b 2 [ f b a ( TM ) ( r m B ) p a b ( r m A ) + g b a ( TM ) ( r m B ) q a b * ( r m A ) ] , s 32 = s 41 * = 1 Y a 2 [ p b a ( r m B ) g a b ( TE ) ( r m A ) + q b a ( r m B ) f a b ( TE ) * ( r m A ) ] + 1 Y b 2 [ f b a ( TM ) ( r m B ) q a b ( r m A ) + g b a ( TM ) ( r m B ) p a b * ( r m A ) ]
{ H z = U t z F ( r ) sin ( ν θ + θ in ) E z = U t z G ( r ) cos ( ν θ + θ in ) ,
κ ( r ) = { κ a ( r ( m 1 ) B r < r m A ) κ b ( r m A r < r m B ) .
d 2 Φ ( r ) d r 2 + κ 2 ( r ) Φ ( r ) = 0 ,
r + Λ F ( r + Λ ) = exp ( i K ˜ Λ ) r F ( r ) ,
F ( r ) = exp ( i K ˜ r ) u ( r ) r ,
H z ( m + 1 ) ( r m B ) = exp ( i K ˜ Λ ) r ( m 1 ) B r m B H z ( m ) ( r ( m 1 ) B ) ,
i E θ = 1 κ 2 ( β 1 r E z θ ω μ 0 H z r ) ,
E θ ( m + 1 ) ( r m B ) = exp ( i K ˜ Λ ) r ( m 1 ) B r m B E θ ( m ) ( r ( m 1 ) B ) .
r m B D a ( r m B ) ( A m + 1 B m + 1 C m + 1 D m + 1 ) = exp ( i K ˜ Λ ) r ( m 1 ) B D a ( r ( m 1 ) B ) ( A m B m C m D m ) ,
( A m + 1 B m + 1 C m + 1 D m + 1 ) = exp ( i K ˜ Λ ) Θ ( m + 1 ) , m ( A m B m C m D m ) ,
Θ m , m = r ( m 1 ) B r ( m 1 ) B D a 1 ( r ( m 1 ) B ) D a ( r ( m 1 ) B ) .
Θ m , m Θ m , m = Θ m , m .
Θ ( m + 1 ) , m diag . ( exp ( i κ a Λ ) , exp ( i κ a Λ ) , exp ( i κ a Λ ) , exp ( i κ a Λ ) ) ,
S ( r m B ; r m A ) ( A m B m C m D m ) = exp ( i K ˜ Λ ) Θ ( m + 1 ) , m ( A m B m C m D m ) .
S ˜ ( r m B ; r m A ) ( A m B m C m D m ) = exp ( i K ˜ Λ ) ( A m B m C m D m ) ,
S ˜ ( r m B ; r m A ) Θ ( m + 1 ) , m 1 S ( r m B ; r m A ) = ( X ˜ m ( TE ) Y ˜ m ( TE ) s ˜ t ˜ Y ˜ m ( TE ) * X ˜ m ( TE ) * t ˜ * s ˜ * u ˜ v ˜ X ˜ m ( TM ) Y ˜ m ( TM ) v ˜ * u ˜ * Y ˜ m ( TM ) * X ˜ m ( TM ) * )
[ η ˜ 2 2 Re ( X ˜ m ( TE ) ) η ˜ + ( | X ˜ m ( TE ) | 2 | Y ˜ m ( TE ) | 2 ) ] [ η ˜ 2 2 Re ( X ˜ m ( TM ) ) η ˜ + ( | X ˜ m ( TM ) | 2 | Y ˜ m ( TM ) | 2 ) ] = 0 ,
η ˜ j ( S ) = exp ( i K ˜ j ( S ) Λ ) = Re ( X ˜ m ( S ) ) ± { [ Re ( X ˜ m ( S ) ) ] 2 ( | X ˜ m ( S ) | 2 | Y ˜ m ( S ) | 2 ) } 1 / 2 ,
( A m B m C m D m ) = ξ ˜ m ( TE ) ( P ˜ m , j ( TE ) Q ˜ m , j ( TE ) R ˜ m , j ( TE ) S ˜ m , j ( TE ) ) + ξ ˜ m ( TM ) ( P ˜ m , j ( TM ) Q ˜ m , j ( TM ) R ˜ m , j ( TM ) S ˜ m , j ( TM ) ) ,
P ˜ m , j ( TE ) = Y ˜ m ( TE ) [ ( X ˜ m ( TM ) η ˜ j ( TE ) ) ( X ˜ m ( TM ) * η ˜ j ( TE ) ) | Y ˜ m ( TM ) | 2 ] , Q ˜ m , j ( TE ) = ( η ˜ j ( TE ) X ˜ m ( TE ) ) [ ( X ˜ m ( TM ) η ˜ j ( TE ) ) ( X ˜ m ( TM ) * η ˜ j ( TE ) ) | Y ˜ m ( TM ) | 2 ] , R ˜ m , j ( TE ) = Y ˜ m ( TE ) [ u ˜ ( X ˜ m ( TM ) * η ˜ j ( TE ) ) + v ˜ * Y ˜ m ( TM ) ] + ( X ˜ m ( TE ) η ˜ j ( TE ) ) [ u ˜ * Y ˜ m ( TM ) + v ˜ ( X ˜ m ( TM ) * η ˜ j ( TE ) ) ] , S ˜ m , j ( TE ) = Y ˜ m ( TE ) [ u ˜ Y ˜ m ( TM ) * v ˜ * ( X ˜ m ( TM ) η ˜ j ( TE ) ) ] + ( X ˜ m ( TE ) η ˜ j ( TE ) ) [ u ˜ * ( X ˜ m ( TM ) η ˜ j ( TE ) ) v ˜ Y ˜ m ( TM ) * ] , P ˜ m , j ( TM ) = [ s ˜ * Y ˜ m ( TE ) + t ˜ ( X ˜ m ( TE ) * η ˜ j ( TM ) ) ] ( X ˜ m ( TM ) η ˜ j ( TM ) ) + [ s ˜ ( X ˜ m ( TE ) * η ˜ j ( TM ) ) + t ˜ * Y ˜ m ( TE ) ] Y ˜ m ( TM ) , Q ˜ m , j ( TM ) = [ s ˜ * ( X ˜ m ( TE ) η ˜ j ( TM ) ) t ˜ Y ˜ m ( TE ) * ] ( X ˜ m ( TM ) η ˜ j ( TM ) ) + [ s ˜ Y ˜ m ( TE ) * t ˜ * ( X ˜ m ( TE ) η ˜ j ( TM ) ) ] Y ˜ m ( TM ) , R ˜ m , j ( TM ) = [ ( X ˜ m ( TE ) η ˜ j ( TM ) ) ( X ˜ m ( TE ) * η ˜ j ( TM ) ) | Y ˜ m ( TE ) | 2 ] Y ˜ m ( TM ) , S ˜ m , j ( TM ) = [ ( X ˜ m ( TE ) η ˜ j ( TM ) ) ( X ˜ m ( TE ) * η ˜ j ( TM ) ) | Y ˜ m ( TE ) | 2 ] ( η ˜ j ( TM ) X ˜ m ( TM ) ) .
( A m B m C m D m ) = S = TE , TM ξ ˜ 1 ( S ) exp [ i ( m 1 ) K ˜ j ( S ) Λ ] Θ ˜ m , 1 ( P ˜ 1 , j ( S ) Q ˜ 1 , j ( S ) R ˜ 1 , j ( S ) S ˜ 1 , j ( S ) ) ,
( σ 11 σ 12 0 σ 14 σ 21 σ 22 σ 23 σ 24 0 σ 32 σ 33 σ 34 σ 41 σ 42 σ 43 σ 44 ) ( A c ξ ˜ 1 ( TE ) C c ξ ˜ 1 ( TM ) ) = 0 ,
σ 11 = σ 33 = 2 J ν ( κ c r c ) , σ 21 = 1 Y c 2 σ 43 = 2 ω μ 0 κ c J ν ( κ c r c ) , σ 23 = σ 41 = 2 ν β κ c 2 r c J ν ( κ c r c ) , σ 1 l = E ˜ j ( S ) , σ 2 l = ω μ 0 κ a G ˜ j ( S ) + ν β κ a 2 r c F ˜ j ( S ) , σ 3 l = F ˜ j ( S ) , σ 4 l = ν β κ a 2 r c E ˜ j ( S ) Y a 2 ω μ 0 κ a H ˜ j ( S ) , l = { 2 ( S = TE ) 4 ( S = TM ) .
E ˜ j ( S ) = P ˜ 1 , j ( S ) H ν ( 2 ) ( κ a r c ) + Q ˜ 1 , j ( S ) H ν ( 1 ) ( κ a r c ) , F ˜ j ( S ) = R ˜ 1 , j ( S ) H ν ( 2 ) ( κ a r c ) + S ˜ 1 , j ( S ) H ν ( 1 ) ( κ a r c ) , G ˜ j ( S ) = P ˜ 1 , j ( S ) H ν ( 2 ) ( κ a r c ) + Q ˜ 1 , j ( S ) H ν ( 1 ) ( κ a r c ) , H ˜ j ( S ) = R ˜ 1 , j ( S ) H ν ( 2 ) ( κ a r c ) + S ˜ 1 , j ( S ) H ν ( 1 ) ( κ a r c ) .
[ J ν ( κ c r c ) J ν ( κ c r c ) + κ c i κ a Σ j ( TE ) ] [ J ν ( κ c r c ) J ν ( κ c r c ) + κ c i κ a n a 2 n c 2 Σ j ( TM ) ] = ( ν β n c κ c r c k 0 ) 2 [ 1 + ( κ c i κ a ) 2 ] 2 ,
Σ j ( TE ) = i G ˜ j ( TE ) E ˜ j ( TE ) , Σ j ( TM ) = i H ˜ j ( TM ) F ˜ j ( TM ) ,
C c A c = ω 0 μ 0 κ c r c ν β J ν ( κ c r c ) J ν ( κ c r c ) + κ c i κ a Σ j ( TE ) 1 + ( κ c / i κ a ) 2 = 1 Y c 2 ν β ω μ 0 κ c r c 1 + ( κ c / i κ a ) 2 J ν ( κ c r c ) J ν ( κ c r c ) + κ c i κ a n a 2 n c 2 Σ j ( TM ) ,
ξ ˜ 1 ( TE ) 2 A c = J ν ( κ c r c ) E ˜ j ( TE ) + ω μ 0 κ c r c ν β [ 1 + ( κ c / i κ a ) 2 ] E ˜ j ( TM ) J ν ( κ c r c ) E ˜ j ( TE ) F ˜ j ( TM ) × [ J ν ( κ c r c ) J ν ( κ c r c ) + κ c i κ a Σ j ( TE ) ] ,
ξ ˜ 1 ( TM ) 2 C c = J ν ( κ c r c ) F ˜ j ( TM ) + Y c 2 ω μ 0 κ c r c ν β [ 1 + ( κ c / i κ a ) 2 ] F ˜ j ( TE ) J ν ( κ c r c ) E ˜ j ( TE ) F ˜ j ( TM ) × [ J ν ( κ c r c ) J ν ( κ c r c ) + κ c i κ a n a 2 n c 2 Σ j ( TM ) ] ,
i G ˜ j ( TM ) E ˜ j ( TM ) i G ˜ j ( TE ) E ˜ j ( TE ) = Σ j ( TE ) ,
λ 0 a = 2 [ 4 ( n a 2 n c 2 ) + ( U QWS π ) 2 ( λ 0 r c ) 2 ] 1 / 2 ,
a = π 2 k 0 n a 2 n t 2 , b = π 2 k 0 n b 2 n t 2 .

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