Abstract

We describe a multipole method for calculating the modes of microstructured optical fibers. The method uses a multipole expansion centered on each hole to enforce boundary conditions accurately and matches expansions with different origins by use of addition theorems. We also validate the method and give representative results.

© 2002 Optical Society of America

Full Article  |  PDF Article

Corrections

T. P. White, B. T. Kuhlmey, R. C. McPhedran, D. Maystre, G. Renversez, C. Martijn de Sterke, and L. C. Botten, "Multipole method for microstructured optical fibers. I. Formulation: errata," J. Opt. Soc. Am. B 20, 1581-1581 (2003)
https://www.osapublishing.org/josab/abstract.cfm?uri=josab-20-7-1581

References

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  1. H. Kubota, K. Suzuki, S. Kawanishi, M. Nakazawa, M. Tanaka, and M. Fujita, “Low-loss, 2 km-long photonic crystal fiber with zero GVD in the near IR suitable for picosecond pulse propagation at the 800 nm band,” in Conference on Lasers and Electro-Optics (CLEO 2001), Vol. 56 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper CPD3.
  2. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Group-velocity dispersion in photonic crystal fibers,” Opt. Lett. 23, 1662–1664 (1998).
    [CrossRef]
  3. T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (2000).
    [CrossRef]
  4. J. C. Knight, J. Broeng, T. A. Birks, and P. St. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998).
    [CrossRef] [PubMed]
  5. T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
    [CrossRef] [PubMed]
  6. A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1996).
  7. D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, San Diego, Calif., 1991), Chap. 2.
  8. M. Midrio, M. P. Singh, and C. G. Someda, “The space filling mode of holey fibres: an analytic vectorial solution,” J. Lightwave Technol. 18, 1031–1048 (2000).
    [CrossRef]
  9. F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fiber Technol. Mater., Devices Syst. 6, 181–191 (2000).
    [CrossRef]
  10. A. Ferrando, E. Silvestre, J. J. Miret, and P. Andrés, “Vector description of higher-order modes in photonic crystal fibers,” J. Opt. Soc. Am. A 17, 1333–1340 (2000).
    [CrossRef]
  11. A. A. Maradudin and A. R. McGurn, “Out of plane propagation of electromagnetic waves in two-dimensional periodic dielectric medium,” J. Mod. Opt. 41, 275–284 (1994).
    [CrossRef]
  12. J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguide guidance by the photonic bandgap effect in optical fibre,” J. Opt. Pure Appl Opt. 1, 477–482 (1999).
    [CrossRef]
  13. D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Localized function method for modeling defect modes in 2-D photonic crystals,” J. Lightwave Technol. 17, 2078–2081 (2000).
    [CrossRef]
  14. B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, and G. L. Burdge, “Cladding-mode-resonances in air–silica microstructure optical fibers,” J. Lightwave Technol. 18, 1084–1100 (2000).
    [CrossRef]
  15. J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Optical properties of high-delta air–silica microstructure optical fibers,” Opt. Lett. 25, 796–797 (2000).
    [CrossRef]
  16. K. M. Lo, R. C. McPhedran, I. M. Bassett, and G. W. Milton, “An electromagnetic theory of dielectric waveguides with multiple embedded cylinders,” J. Lightwave Technol. 12, 396–410 (1994).
    [CrossRef]
  17. E. Centeno and D. Felbacq, “Rigorous vector diffraction of electromagnetic waves by bidimensional photonic crystals,” J. Opt. Soc. Am. A 17, 320–327 (2000).
    [CrossRef]
  18. P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides. I. Summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975).
    [CrossRef]
  19. M. J. Steel, T. P. White, C. M. De Sterke, R. C. McPhedran, and L. C. Botten, “Symmetry and degeneracy in microstructured optical fibers,” Opt. Lett. 26, 488–490 (2001).
    [CrossRef]
  20. D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
    [CrossRef]
  21. E. Yamashita, S. Ozeki, and K. Atsuki, “Modal analysis method for optical fibers with symmetrically distributed multiple cores,” J. Lightwave Technol. 3, 341–346 (1985).
    [CrossRef]
  22. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. M. de Sterke, and R. C. McPhedran, “Multipole method for microstructured optical fibers. II. Implementation and results,” J. Opt. Soc. Am. B 19, 2331–2340 (2002).
    [CrossRef]
  23. W. Wijngaard, “Guided normal modes of two parallel circular dielectric rods,” J. Opt. Soc. Am. 63, 944–949 (1973).
    [CrossRef]
  24. C.-S. Chang and H.-C. Chang, “Theory of the circular harmonics expansion method for multiple-optical-fiber system,” J. Lightwave Technol. 12, 415–417 (1994).
    [CrossRef]
  25. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).
  26. D. Maystre and P. Vincent, “Diffraction d’une onde electromagnetique plane par an object cylindrique non infinitement conducteur,” Opt. Commun. 5, 327–330 (1972).
    [CrossRef]
  27. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Sec. 2.9.
  28. L. Schwartz, Mathematics for the Physical Sciences (Addison-Wesley, Reading, Mass., 1966).

2002 (1)

2001 (1)

2000 (8)

M. Midrio, M. P. Singh, and C. G. Someda, “The space filling mode of holey fibres: an analytic vectorial solution,” J. Lightwave Technol. 18, 1031–1048 (2000).
[CrossRef]

B. J. Eggleton, P. S. Westbrook, C. A. White, C. Kerbage, R. S. Windeler, and G. L. Burdge, “Cladding-mode-resonances in air–silica microstructure optical fibers,” J. Lightwave Technol. 18, 1084–1100 (2000).
[CrossRef]

T. M. Monro, D. J. Richardson, N. G. R. Broderick, and P. J. Bennett, “Holey optical fibers: an efficient modal model,” J. Lightwave Technol. 17, 1093–1102 (2000).
[CrossRef]

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Localized function method for modeling defect modes in 2-D photonic crystals,” J. Lightwave Technol. 17, 2078–2081 (2000).
[CrossRef]

E. Centeno and D. Felbacq, “Rigorous vector diffraction of electromagnetic waves by bidimensional photonic crystals,” J. Opt. Soc. Am. A 17, 320–327 (2000).
[CrossRef]

A. Ferrando, E. Silvestre, J. J. Miret, and P. Andrés, “Vector description of higher-order modes in photonic crystal fibers,” J. Opt. Soc. Am. A 17, 1333–1340 (2000).
[CrossRef]

J. K. Ranka, R. S. Windeler, and A. J. Stentz, “Optical properties of high-delta air–silica microstructure optical fibers,” Opt. Lett. 25, 796–797 (2000).
[CrossRef]

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fiber Technol. Mater., Devices Syst. 6, 181–191 (2000).
[CrossRef]

1999 (1)

J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguide guidance by the photonic bandgap effect in optical fibre,” J. Opt. Pure Appl Opt. 1, 477–482 (1999).
[CrossRef]

1998 (2)

J. C. Knight, J. Broeng, T. A. Birks, and P. St. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998).
[CrossRef] [PubMed]

D. Mogilevtsev, T. A. Birks, and P. St. J. Russell, “Group-velocity dispersion in photonic crystal fibers,” Opt. Lett. 23, 1662–1664 (1998).
[CrossRef]

1997 (1)

1994 (4)

C.-S. Chang and H.-C. Chang, “Theory of the circular harmonics expansion method for multiple-optical-fiber system,” J. Lightwave Technol. 12, 415–417 (1994).
[CrossRef]

K. M. Lo, R. C. McPhedran, I. M. Bassett, and G. W. Milton, “An electromagnetic theory of dielectric waveguides with multiple embedded cylinders,” J. Lightwave Technol. 12, 396–410 (1994).
[CrossRef]

A. A. Maradudin and A. R. McGurn, “Out of plane propagation of electromagnetic waves in two-dimensional periodic dielectric medium,” J. Mod. Opt. 41, 275–284 (1994).
[CrossRef]

D. Felbacq, G. Tayeb, and D. Maystre, “Scattering by a random set of parallel cylinders,” J. Opt. Soc. Am. A 11, 2526–2538 (1994).
[CrossRef]

1985 (1)

E. Yamashita, S. Ozeki, and K. Atsuki, “Modal analysis method for optical fibers with symmetrically distributed multiple cores,” J. Lightwave Technol. 3, 341–346 (1985).
[CrossRef]

1975 (1)

P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides. I. Summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975).
[CrossRef]

1973 (1)

1972 (1)

D. Maystre and P. Vincent, “Diffraction d’une onde electromagnetique plane par an object cylindrique non infinitement conducteur,” Opt. Commun. 5, 327–330 (1972).
[CrossRef]

Andrés, P.

Atsuki, K.

E. Yamashita, S. Ozeki, and K. Atsuki, “Modal analysis method for optical fibers with symmetrically distributed multiple cores,” J. Lightwave Technol. 3, 341–346 (1985).
[CrossRef]

Barbeito, P. M.

J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguide guidance by the photonic bandgap effect in optical fibre,” J. Opt. Pure Appl Opt. 1, 477–482 (1999).
[CrossRef]

Barkou, S. E.

J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguide guidance by the photonic bandgap effect in optical fibre,” J. Opt. Pure Appl Opt. 1, 477–482 (1999).
[CrossRef]

Bassett, I. M.

K. M. Lo, R. C. McPhedran, I. M. Bassett, and G. W. Milton, “An electromagnetic theory of dielectric waveguides with multiple embedded cylinders,” J. Lightwave Technol. 12, 396–410 (1994).
[CrossRef]

Bennett, P. J.

Birks, T. A.

Bjarklev, A.

J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguide guidance by the photonic bandgap effect in optical fibre,” J. Opt. Pure Appl Opt. 1, 477–482 (1999).
[CrossRef]

Botten, L. C.

Brechet, F.

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fiber Technol. Mater., Devices Syst. 6, 181–191 (2000).
[CrossRef]

Broderick, N. G. R.

Broeng, J.

J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguide guidance by the photonic bandgap effect in optical fibre,” J. Opt. Pure Appl Opt. 1, 477–482 (1999).
[CrossRef]

J. C. Knight, J. Broeng, T. A. Birks, and P. St. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998).
[CrossRef] [PubMed]

Burdge, G. L.

Centeno, E.

Chang, C.-S.

C.-S. Chang and H.-C. Chang, “Theory of the circular harmonics expansion method for multiple-optical-fiber system,” J. Lightwave Technol. 12, 415–417 (1994).
[CrossRef]

Chang, H.-C.

C.-S. Chang and H.-C. Chang, “Theory of the circular harmonics expansion method for multiple-optical-fiber system,” J. Lightwave Technol. 12, 415–417 (1994).
[CrossRef]

de Sterke, C. M.

Eggleton, B. J.

Felbacq, D.

Ferrando, A.

Kerbage, C.

Knight, J. C.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998).
[CrossRef] [PubMed]

T. A. Birks, J. C. Knight, and P. St. J. Russell, “Endlessly single-mode photonic crystal fiber,” Opt. Lett. 22, 961–963 (1997).
[CrossRef] [PubMed]

Kuhlmey, B. T.

Lo, K. M.

K. M. Lo, R. C. McPhedran, I. M. Bassett, and G. W. Milton, “An electromagnetic theory of dielectric waveguides with multiple embedded cylinders,” J. Lightwave Technol. 12, 396–410 (1994).
[CrossRef]

Maradudin, A. A.

A. A. Maradudin and A. R. McGurn, “Out of plane propagation of electromagnetic waves in two-dimensional periodic dielectric medium,” J. Mod. Opt. 41, 275–284 (1994).
[CrossRef]

Marcou, J.

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fiber Technol. Mater., Devices Syst. 6, 181–191 (2000).
[CrossRef]

Maystre, D.

McGurn, A. R.

A. A. Maradudin and A. R. McGurn, “Out of plane propagation of electromagnetic waves in two-dimensional periodic dielectric medium,” J. Mod. Opt. 41, 275–284 (1994).
[CrossRef]

McIsaac, P. R.

P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides. I. Summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975).
[CrossRef]

McPhedran, R. C.

Midrio, M.

Milton, G. W.

K. M. Lo, R. C. McPhedran, I. M. Bassett, and G. W. Milton, “An electromagnetic theory of dielectric waveguides with multiple embedded cylinders,” J. Lightwave Technol. 12, 396–410 (1994).
[CrossRef]

Miret, J. J.

Mogilevtsev, D.

Monro, T. M.

Ozeki, S.

E. Yamashita, S. Ozeki, and K. Atsuki, “Modal analysis method for optical fibers with symmetrically distributed multiple cores,” J. Lightwave Technol. 3, 341–346 (1985).
[CrossRef]

Pagnoux, D.

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fiber Technol. Mater., Devices Syst. 6, 181–191 (2000).
[CrossRef]

Ranka, J. K.

Renversez, G.

Richardson, D. J.

Roy, P.

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fiber Technol. Mater., Devices Syst. 6, 181–191 (2000).
[CrossRef]

Russell, P. St.

J. C. Knight, J. Broeng, T. A. Birks, and P. St. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998).
[CrossRef] [PubMed]

Russell, P. St. J.

Silvestre, E.

Singh, M. P.

Someda, C. G.

Sondergaard, T.

J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguide guidance by the photonic bandgap effect in optical fibre,” J. Opt. Pure Appl Opt. 1, 477–482 (1999).
[CrossRef]

Steel, M. J.

Stentz, A. J.

Tayeb, G.

Vincent, P.

D. Maystre and P. Vincent, “Diffraction d’une onde electromagnetique plane par an object cylindrique non infinitement conducteur,” Opt. Commun. 5, 327–330 (1972).
[CrossRef]

Westbrook, P. S.

White, C. A.

White, T. P.

Wijngaard, W.

Windeler, R. S.

Yamashita, E.

E. Yamashita, S. Ozeki, and K. Atsuki, “Modal analysis method for optical fibers with symmetrically distributed multiple cores,” J. Lightwave Technol. 3, 341–346 (1985).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

P. R. McIsaac, “Symmetry-induced modal characteristics of uniform waveguides. I. Summary of results,” IEEE Trans. Microwave Theory Tech. MTT-23, 421–429 (1975).
[CrossRef]

J. Lightwave Technol. (7)

J. Mod. Opt. (1)

A. A. Maradudin and A. R. McGurn, “Out of plane propagation of electromagnetic waves in two-dimensional periodic dielectric medium,” J. Mod. Opt. 41, 275–284 (1994).
[CrossRef]

J. Opt. Pure Appl Opt. (1)

J. Broeng, T. Sondergaard, S. E. Barkou, P. M. Barbeito, and A. Bjarklev, “Waveguide guidance by the photonic bandgap effect in optical fibre,” J. Opt. Pure Appl Opt. 1, 477–482 (1999).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

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

Opt. Commun. (1)

D. Maystre and P. Vincent, “Diffraction d’une onde electromagnetique plane par an object cylindrique non infinitement conducteur,” Opt. Commun. 5, 327–330 (1972).
[CrossRef]

Opt. Fiber Technol. Mater., Devices Syst. (1)

F. Brechet, J. Marcou, D. Pagnoux, and P. Roy, “Complete analysis of the characteristics of propagation into photonic crystal fibers, by the finite element method,” Opt. Fiber Technol. Mater., Devices Syst. 6, 181–191 (2000).
[CrossRef]

Opt. Lett. (4)

Science (1)

J. C. Knight, J. Broeng, T. A. Birks, and P. St. Russell, “Photonic band gap guidance in optical fibers,” Science 282, 1476–1478 (1998).
[CrossRef] [PubMed]

Other (6)

A. W. Snyder and J. D. Love, Optical Waveguide Theory (Chapman & Hall, London, 1996).

D. Marcuse, Theory of Dielectric Optical Waveguides, 2nd ed. (Academic, San Diego, Calif., 1991), Chap. 2.

H. Kubota, K. Suzuki, S. Kawanishi, M. Nakazawa, M. Tanaka, and M. Fujita, “Low-loss, 2 km-long photonic crystal fiber with zero GVD in the near IR suitable for picosecond pulse propagation at the 800 nm band,” in Conference on Lasers and Electro-Optics (CLEO 2001), Vol. 56 of OSA Trends in Optics and Photonics (Optical Society of America, Washington, D.C., 2001), paper CPD3.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, New York, 1965).

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986), Sec. 2.9.

L. Schwartz, Mathematics for the Physical Sciences (Addison-Wesley, Reading, Mass., 1966).

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Figures (6)

Fig. 1
Fig. 1

Geometry of the MOFs considered, together with the contributions to the fields just outside a generic hole i. Regions of convergence of multipole expansions are indicated by dashed curves. Note that QP is rj in Eq. (8), and SP is rl and OP is r. Solid curves indicate physical boundaries; dashed curves indicate regions of convergence.

Fig. 2
Fig. 2

Logarithm of the magnitude of the determinant of M versus the real and the imaginary parts of the complex refractive index for the MOF given in the text.

Fig. 3
Fig. 3

Minimum sectors for waveguides with C6v symmetry. Mode classes p=1, 2, 7, 8 are nondegenerate; p=3, 4 and p=5, 6 are twofold degenerate. Solid lines indicate Dirichlet boundary conditions for the electric field; dashed lines indicate Neumann boundary conditions.

Fig. 4
Fig. 4

Normalized fields |Ez| and |Kz| and energy flow Sz for the degenerate fundamental mode class p=3 for a six-hole MOF, with data of Fig. 2 and neff=1.445395345+3.15×10-8 i.

Fig. 5
Fig. 5

Similar to Fig. 4 but for degenerate fundamental mode class with p=4 and neff=1.445395345+3.15×10-8 i.

Fig. 6
Fig. 6

Similar to Fig. 4 but for nondegenerate mode with p=2 and neff=1.438585801+4.986×10-7 i.

Tables (1)

Tables Icon

Table 1 Effective Index, Loss, Mode Class, and Degeneracy of the First 10 Modes of the MOF Given in the Text, Calculated for M=5

Equations (80)

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

E(r, θ, z, t)=E(r, θ)exp[i(βz-ωt)],
K(r, θ, z, t)=K(r, θ)exp[i(βz-ωt)],
[2+(ke)2]V=0
[2+(ki)2]V=0
Ez=m[AmElJm(kerl)+BmElHm(1)(kerl)]exp(imθl),
Ez=l=1Ncm BmElHm(1)(ke|rl|)exp[im arg(r-cl)]+m AmE0Jm(ker)exp(imθ).
m AmElJm(kerl)exp(imθl)
=j=1jlNm BmEjHm(1)(kerj)exp(imθj)+m AmE0Jm(ker)exp(imθ),
n AnEljJn(kerl)exp[in arg(rl)]
=m BmEjHm(1)(kerj)exp[im arg(rj)],
AnElj=m HnmljBmEj,
Hnmlj=Hn-m(1)(keclj)exp[-i(n-m)arg(clj)],
AElj=HljBEj.
n AnEl0Jn(kerl)exp[in arg(rl)]
=m AmE0Jm(ker)exp(imθ),
AEl0=Jl0AE0,
Jl0=[Jnml0]={(-1)(n-m)Jn-m(kecl)×exp[i(m-n)arg(cl)]}.
AEl=j=1jlNc AElj+AEl0=j=1jlNc HljBEj+Jl0AE0,
n BnE0jHn(1)(ker)exp(inθ)
=m BmEjHm(1)(kerj)exp[im arg(rj)],
BE0j=J0jBEj,
J0j=[Jnm0j]={Jn-m(kecj)exp[-i(n-m)arg(cj)]}, 
l=1Nc OEl=n BnE0Hn(1)(ker)exp(inθ)=OE0,
BE0=l=1Nc BE0l=l=1Nc J0lBEl,
BnEl=RnEElAnEl+RnEKlAnKl,
BnKl=RnKElAnEl+RnKKlAnKl,
BElBKl=REElREKlRKElRKKlAElAKl,
B˜l=R˜lA˜l,
A˜0=R˜0B˜0,
A˜l=j=1jlNcH˜ljB˜j+J˜l0A˜0,
A=J˜B+J˜B0A˜0,
J˜B0=[(J˜l0]=[(J˜10)T, (J˜20)T ,, (J˜Nc0)T]T,
B˜0=j=1NcJ˜0lB˜l=J˜0BB˜
J˜0B=[J˜0l]=[J˜01, J˜02 ,  J˜0Nc].
[I-R(H˜+J˜B0R˜0J˜0B)]BMB=0,
R=diag[R˜1, R˜2 , , R˜Nc].
L=20ln(10) 2πλ J(neff)×106,
U(x, y)=Ezr<R00elsewhere.
2U+k2U=s,
s=j=1NcSjδCj-TδC-·(nQδC),
AδC, φ=CA(M)φ(M)dM,
2U+(ke)2U=[(ke)2-(k)2]U+s,
U=Ge{s+[(ke)2-(k)2]U},
U=j=1NcGeDj+GeD,
Dj=SjδCj+[(ke)2-(kj)2]Uj,
D=-TδC-·(nQδC),
Uj=Uinthejthinclusion0elsewhere.
Vj=mBmEjHm(1)(kerj)exp(imθj).
Vinc=mAmE0Jm(ker)exp(imθ).
Hm(1)(kerj)exp[im arg(rj)]
=n=-Jn(krl)exp[in arg(rl)]Hn-m(1)(keclj)
×exp[-i(n-m)arg(clj)],
m=-BmjHm(1)(kerj)exp[im arg(rj)]
=n=-AnljJn(kerl)exp[in arg(rl)],
Jm(ker)exp(imθ)=n=- Jn(kerl)exp[in arg(rl)]×(-1)n-mJn-m(kecl)×exp[-i(n-m)arg(cl)],
m=- Am0Jm(ker)exp(imθ)
=n=- Al0Jn(kerl)exp[in arg(rl)],
Hm(1)(kerl)exp[im arg(rl)]
=n=- Hn(1)(ker)exp(inθ)Jn-m(kecl)×exp[-i(n-m)arg(cl)].
m=- BmlHm(1)(kerl)exp[im arg(rl)]
=n=- Bn0lHn(1)(ker)exp(inθ),
Ez(r, θ)=m=-[AmEJm(kr)+BmEHm(1)×(kr)]exp(imθ)
A˜-=T˜-A˜++R˜-B˜-,
B˜+=R˜+A˜++T˜+B˜-.
Eθ(r, θ)=ik2 βr Ezθ-k Kzr,
Kθ(r, θ)=ik2 βr Kzθ+kn2 Ezr,
AmE-Jm-+BmE-Hm-=AmE+Jm++BmE+Hm+, 
AmE-=RmEE-BmE-+RmEK-BmK-,
AmK-=RmKE-BmE-+RmKK-BmK-,
RmEE-=1δm[(αJ-H++-αH+J--)(n-2αH-H++-n+2αH+H--)-m2Jm-Hm-Hm+2τ2],
RmEK-=1δm 2mτπka k+k-Hm+2,
RmKE-=-n-2RmEK-,
RmKK-=1δm[(αH-H++-αH+H--)(n-2αJ-H++-n+2αH+J--)+m2Jm-Hm-Hm+2τ2],
αJ±H±±=k±kJm±Hm±,
δm=(αH+J---αJ-H++)(n-2αJ-H++-n+2αH+J--)+(mJm-Hm+τ2
τ=βak-k+(n+2-n-2).
RmEE-=1δm[(αJ-H++-αH+J--)(n-2αJ-J++-n+2αJ+J--)-m2Jm+Hm+Hm-2τ2],
RmEK+=1δm 2mτπka k-k+Jm-2,
RmKE+=-n+2RmEK+,
RmKK-=-1δm[(αJ-J++-αJ+J- -)(n-2αJ-H++-n+2αH+J--)-m2Jm+Hm+Hm-2τ2].

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