Abstract

A boundary integral method [1] for calculating leaky and guided modes of microstructured optical fibers is presented. The method is rapidly converging and can handle a large number of inclusions (hundreds) of arbitrary geometries. Both, solid and hollow core photonic crystal fibers can be treated efficiently. We demonstrate that for large systems featuring closely spaced inclusions the computational intensity of the boundary integral method is significantly smaller than that of the multipole method. This is of particular importance in the case of hollow core band gap guiding fibers. We demonstrate versatility of the method by applying it to several challenging problems.

© 2007 Optical Society of America

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  1. Matlab implementation of the code is available at http://www.photonics.phys.polymtl.ca/codes.html
  2. P. Russell“Photonic crystal fibers,” Science 299, 358–362 (2003).
    [Crossref] [PubMed]
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    [Crossref]
  9. K. Saitoh and M. Koshiba, “Full-Vectorial Imaginary-Distance Beam PropagationMethod Based on a Finite Element Scheme: Application to Photonic Crystal Fibers,” IEEE J. Quantum Electron. 38, 297 (2002).
    [Crossref]
  10. A. Cucinotta, S. Selleri, L. Vincent, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. 14, 1530–1532 (2002).
    [Crossref]
  11. X. Wang, J. Lou, C. Lu, C. L. Zhao, and W. T Ang, “Modeling of PCF with multiple reciprocity boundary element method,” Opt. Express 12, 961–966 (2004), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-5-961
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  13. T. Lu and D. Yevick, “A vectorial boundary element method analysis of integrated optical waveguides,” J. Lightwave Technol. 21, 1793–1807 (2003).
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  14. H. Cheng, W. Crutchfield, M. Doery, and L. Greengard, “Fast, accurate integral equation methods for the analysis of photonic crystal fibers I: Theory,” Opt. Express 12, 3791–3805 (2004),http://www.opticsinfobase.org/abstract.cfm?URI=oe-12-16-3791
    [Crossref] [PubMed]
  15. 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,” J. Opt. Soc. Am. B 19, 2322–2330 (2002).
    [Crossref]
  16. B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. Martijn 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).
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  17. S. Campbell, R. C. McPhedran, and C. Martijn de Sterke “Differential multipole method for microstructured optical fibers,” J. Opt. Soc. Am. B 21, 1919–1928 (2004).
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  18. M. Skorobogatiy, K. Saitoh, and M. Koshiba, “Coupling between two collinear air-core Bragg fibers,” J. Opt. Soc. Am. B 21, 2095–2101 (2004).
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  19. B. T. Kuhlmey, K. Pathmanandavel, and R. C. McPhedran, “Multipole analysis of photonic crystal fibers with coated inclusions,” Opt. Express 14, 10851–10864 (2006).
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  20. S. V. Boriskina, T.M. Benson., P. Sewell, and A. I. Nosich “Highly efficient full-vectorial integral equation solution for the bound, leaky and complex modes of dielectric waveguides,” IEEE J. Sel. Top. Quantum Electron. 8, 1225–1231 (2002).
    [Crossref]
  21. D. Colton and R. Kress “Integral equation methods in scattering theory,” John Wiley & Sons, New York, (1983).
  22. R. Kress “Linear integral equations,” Springer-Verlag, New York, (1989).
  23. S. V. Boriskina, P. Sewell, and T. M. Benson “Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization,” J. Opt. Soc. Am. A 21, 393–402 (2004).
    [Crossref]
  24. M. Abramowitz and I. A. Stegun “Handbook of mathematical functions,” Dover, New York, (1965).
  25. R. Rodriguez-Berral, F. Mesa, and F. Medina, “Systematic and efficient root finder for computing the modal spectrum of planar layered waveguides,” Int. J. RF Microw. Comp. Eng. 14, 73–83 (2004).
    [Crossref]
  26. M. S. Alam, K. Saitoh, and M. Koshiba “High group birefringence in air-core photonic bandgap fibers,” Opt. Lett. 30, 824–826 (2005).
    [Crossref] [PubMed]

2006 (1)

2005 (1)

2004 (6)

2003 (3)

2002 (5)

K. Saitoh and M. Koshiba, “Full-Vectorial Imaginary-Distance Beam PropagationMethod Based on a Finite Element Scheme: Application to Photonic Crystal Fibers,” IEEE J. Quantum Electron. 38, 297 (2002).
[Crossref]

A. Cucinotta, S. Selleri, L. Vincent, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. 14, 1530–1532 (2002).
[Crossref]

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,” J. Opt. Soc. Am. B 19, 2322–2330 (2002).
[Crossref]

B. T. Kuhlmey, T. P. White, G. Renversez, D. Maystre, L. C. Botten, C. Martijn 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]

S. V. Boriskina, T.M. Benson., P. Sewell, and A. I. Nosich “Highly efficient full-vectorial integral equation solution for the bound, leaky and complex modes of dielectric waveguides,” IEEE J. Sel. Top. Quantum Electron. 8, 1225–1231 (2002).
[Crossref]

2000 (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. 6, 181–191 (2000).
[Crossref]

1999 (2)

1997 (1)

Abramowitz, M.

M. Abramowitz and I. A. Stegun “Handbook of mathematical functions,” Dover, New York, (1965).

Alam, M. S.

Andres, M.V.

Andres, P.

Ang, W. T

Barton, G.W.

M.C.J. Large, L. Poladian, G.W. Barton, and M.A. van Eijkelenborg, “Microstructured Polymer Optical Fibres,” Springer, Sydney, (2007)

Bennett, P.J.

Benson, T. M.

Benson., T.M.

S. V. Boriskina, T.M. Benson., P. Sewell, and A. I. Nosich “Highly efficient full-vectorial integral equation solution for the bound, leaky and complex modes of dielectric waveguides,” IEEE J. Sel. Top. Quantum Electron. 8, 1225–1231 (2002).
[Crossref]

Bjarklev, A.

A. Bjarklev, J. Broeng, and A.S. Bjarklev “Photonic crystal fibers,” Kluwer Academic Publishers, Boston, (2003).

Bjarklev, A.S.

A. Bjarklev, J. Broeng, and A.S. Bjarklev “Photonic crystal fibers,” Kluwer Academic Publishers, Boston, (2003).

Boriskina, S. V.

S. V. Boriskina, P. Sewell, and T. M. Benson “Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization,” J. Opt. Soc. Am. A 21, 393–402 (2004).
[Crossref]

S. V. Boriskina, T.M. Benson., P. Sewell, and A. I. Nosich “Highly efficient full-vectorial integral equation solution for the bound, leaky and complex modes of dielectric waveguides,” IEEE J. Sel. Top. Quantum Electron. 8, 1225–1231 (2002).
[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. 6, 181–191 (2000).
[Crossref]

Broderick, N.G.R.

Broeng, J.

A. Bjarklev, J. Broeng, and A.S. Bjarklev “Photonic crystal fibers,” Kluwer Academic Publishers, Boston, (2003).

Burks, T.A.

Campbell, S.

Cheng, H.

Colton, D.

D. Colton and R. Kress “Integral equation methods in scattering theory,” John Wiley & Sons, New York, (1983).

Crutchfield, W.

Cucinotta, A.

A. Cucinotta, S. Selleri, L. Vincent, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. 14, 1530–1532 (2002).
[Crossref]

Doery, M.

Ferrando, A.

Greengard, L.

Guan, N.

Habu, S.

Himeno, K.

Knight, J.C.

Koshiba, M.

Kress, R.

D. Colton and R. Kress “Integral equation methods in scattering theory,” John Wiley & Sons, New York, (1983).

R. Kress “Linear integral equations,” Springer-Verlag, New York, (1989).

Kuhlmey, B. T.

Large, M.C.J.

M.C.J. Large, L. Poladian, G.W. Barton, and M.A. van Eijkelenborg, “Microstructured Polymer Optical Fibres,” Springer, Sydney, (2007)

Lou, J.

Lu, C.

Lu, T.

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. 6, 181–191 (2000).
[Crossref]

Martijn de Sterke, C.

Maystre, D.

McPhedran, R. C.

Medina, F.

R. Rodriguez-Berral, F. Mesa, and F. Medina, “Systematic and efficient root finder for computing the modal spectrum of planar layered waveguides,” Int. J. RF Microw. Comp. Eng. 14, 73–83 (2004).
[Crossref]

Mesa, F.

R. Rodriguez-Berral, F. Mesa, and F. Medina, “Systematic and efficient root finder for computing the modal spectrum of planar layered waveguides,” Int. J. RF Microw. Comp. Eng. 14, 73–83 (2004).
[Crossref]

Miret, J.J.

Monro, T.M.

Nosich, A. I.

S. V. Boriskina, T.M. Benson., P. Sewell, and A. I. Nosich “Highly efficient full-vectorial integral equation solution for the bound, leaky and complex modes of dielectric waveguides,” IEEE J. Sel. Top. Quantum Electron. 8, 1225–1231 (2002).
[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. 6, 181–191 (2000).
[Crossref]

Pathmanandavel, K.

Poladian, L.

M.C.J. Large, L. Poladian, G.W. Barton, and M.A. van Eijkelenborg, “Microstructured Polymer Optical Fibres,” Springer, Sydney, (2007)

Renversez, G.

Richardson, D.J.

Rodriguez-Berral, R.

R. Rodriguez-Berral, F. Mesa, and F. Medina, “Systematic and efficient root finder for computing the modal spectrum of planar layered waveguides,” Int. J. RF Microw. Comp. Eng. 14, 73–83 (2004).
[Crossref]

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. 6, 181–191 (2000).
[Crossref]

Russell, P.

P. Russell“Photonic crystal fibers,” Science 299, 358–362 (2003).
[Crossref] [PubMed]

Russell, P.S.J.

Saitoh, K.

Selleri, S.

A. Cucinotta, S. Selleri, L. Vincent, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. 14, 1530–1532 (2002).
[Crossref]

Sewell, P.

S. V. Boriskina, P. Sewell, and T. M. Benson “Accurate simulation of two-dimensional optical microcavities with uniquely solvable boundary integral equations and trigonometric Galerkin discretization,” J. Opt. Soc. Am. A 21, 393–402 (2004).
[Crossref]

S. V. Boriskina, T.M. Benson., P. Sewell, and A. I. Nosich “Highly efficient full-vectorial integral equation solution for the bound, leaky and complex modes of dielectric waveguides,” IEEE J. Sel. Top. Quantum Electron. 8, 1225–1231 (2002).
[Crossref]

Silvestre, E.

Skorobogatiy, M.

Stegun, I. A.

M. Abramowitz and I. A. Stegun “Handbook of mathematical functions,” Dover, New York, (1965).

Takenaga, K.

van Eijkelenborg, M.A.

M.C.J. Large, L. Poladian, G.W. Barton, and M.A. van Eijkelenborg, “Microstructured Polymer Optical Fibres,” Springer, Sydney, (2007)

Vincent, L.

A. Cucinotta, S. Selleri, L. Vincent, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. 14, 1530–1532 (2002).
[Crossref]

Wada, A.

Wang, X.

White, T. P.

Yevick, D.

Zhao, C. L.

Zoboli, M.

A. Cucinotta, S. Selleri, L. Vincent, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. 14, 1530–1532 (2002).
[Crossref]

IEEE J. Quantum Electron. (1)

K. Saitoh and M. Koshiba, “Full-Vectorial Imaginary-Distance Beam PropagationMethod Based on a Finite Element Scheme: Application to Photonic Crystal Fibers,” IEEE J. Quantum Electron. 38, 297 (2002).
[Crossref]

IEEE J. Sel. Top. Quantum Electron. (1)

S. V. Boriskina, T.M. Benson., P. Sewell, and A. I. Nosich “Highly efficient full-vectorial integral equation solution for the bound, leaky and complex modes of dielectric waveguides,” IEEE J. Sel. Top. Quantum Electron. 8, 1225–1231 (2002).
[Crossref]

IEEE Photon. Technol. Lett. (1)

A. Cucinotta, S. Selleri, L. Vincent, and M. Zoboli, “Holey fiber analysis through the finite element method,” IEEE Photon. Technol. Lett. 14, 1530–1532 (2002).
[Crossref]

Int. J. RF Microw. Comp. Eng. (1)

R. Rodriguez-Berral, F. Mesa, and F. Medina, “Systematic and efficient root finder for computing the modal spectrum of planar layered waveguides,” Int. J. RF Microw. Comp. Eng. 14, 73–83 (2004).
[Crossref]

J. Lightwave Technol. (3)

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

J. Opt. Soc. Am. B (4)

Opt. Express (3)

Opt. Fiber Technol. (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. 6, 181–191 (2000).
[Crossref]

Opt. Lett. (3)

Science (1)

P. Russell“Photonic crystal fibers,” Science 299, 358–362 (2003).
[Crossref] [PubMed]

Other (6)

A. Bjarklev, J. Broeng, and A.S. Bjarklev “Photonic crystal fibers,” Kluwer Academic Publishers, Boston, (2003).

M.C.J. Large, L. Poladian, G.W. Barton, and M.A. van Eijkelenborg, “Microstructured Polymer Optical Fibres,” Springer, Sydney, (2007)

Matlab implementation of the code is available at http://www.photonics.phys.polymtl.ca/codes.html

M. Abramowitz and I. A. Stegun “Handbook of mathematical functions,” Dover, New York, (1965).

D. Colton and R. Kress “Integral equation methods in scattering theory,” John Wiley & Sons, New York, (1983).

R. Kress “Linear integral equations,” Springer-Verlag, New York, (1989).

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Figures (7)

Fig. 1.
Fig. 1.

Three structures studied in the paper. (a) Hollow coreMOF with five rings of circular holes; the pitch is Λ=2.74µm, the hole diameter is d=.95Λ and the core diameter is dc =2.5d. (b) Elliptic hollow core MOF with three layers of circular holes; the pitch is Λ=2µm, the hole diameter is d=0.9Λ and the core principal axis are a=2.3µm and b=4.6µm. (c) Solid core MOF with six silver coated elliptical holes; the outer hole principal axis are ao =0.84µm and bo =0.76mm, the inner hole principal axis are ai =0.74µm and bi =0.66µm, the pitch is Λ=1.5mm.

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Fig. 2.
Fig. 2.

a) Schematic of aMOF cross section. b) Schematic of Re(G(s, s′)). Green’s function has a cusp when s → s′ it also exhibits oscillations ( γ = n g 2 − n e 2 ) . c ) Arbitrary shaped inclusion and a corresponding regularization circle.

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Fig. 3.
Fig. 3.

Convergence analysis and comparison with the multipole method for the three simple test structures: (a) six circular holes; diameter d=5µm, pitch L=6.75µm (b) six elliptic holes; axis a=2.5µm b=1.5µm, pitch L=6.75µm (c) six metal coated cylinders; outer diameter do =0.8µm, inner one di =0.7µm, pitch Λ = 1.5µm.

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Fig. 4.
Fig. 4.

Hollow core MOF with 5 rings of holes in the reflector. (a) Dispersion curve of the fundamental mode. (b) Loss as a function of the number of reflector layers.

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Fig. 5.
Fig. 5.

Birefringence of the fundamental mode of a PCF with elliptic hollow core. (b) Outset: Sz fluxes for the x and y polarizations of the fundamental mode at λ=1.42µm.

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Fig. 6.
Fig. 6.

Loss dispersion curves for the two polarizations of the fundamental mode of a MOF with one ring of metallized elliptic holes. Outset: Sz fluxes for the x and y polarizations of the fundamental mode at the wavelengths of the two plasmon excitation peaks.

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Fig. 7.
Fig. 7.

Schematic of a coated inclusion.

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Tables (4)

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Table 1. Performance comparison of the multipole and boundary integral methods.

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Table 2. Effective refractive index of a mode (of a symmetry class p=1 as defined in [15]) of a solid core MOF featuring one ring of six holes (see Fig. 3(a)).

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Table 3. Effective refractive index of a mode of a solid core MOF featuring one ring of six elliptic inclusions (see Fig. 3(b)). The results are for the fundamental mode where the nodal line of the Ez field is horizontal. For the other polarization the value 1.446429072+ 2.9898E-6i is obtained by us and 1.446427235+2.9601E-6i by [17].

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Table 4. Effective refractive index of a mode of a solid core MOF with one ring of six coated holes (see Fig. 3(c)). Results are for the fundamental core guided mode.

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Equations (35)

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

∇ 2 E z ( c , g ) + k 0 2 γ c , g 2 E z ( c , g ) = 0
∇ 2 H z ( c , g ) + k 0 2 γ c , g 2 H z ( c , g ) = 0 ,
E z ( c ) = E z ( g )
H z ( c ) = H z ( g )
E t ( c ) = E t ( g ) ⇒ i k 0 γ c 2 ( n e ∂ E z ( c ) ∂ τ − ∂ H z ( c ) ∂ n ) = i k 0 γ g 2 ( n e ∂ E z ( g ) ∂ τ − ∂ H z ( g ) ∂ n ) ,
H t ( c ) = H t ( g ) ⇒ i k 0 γ c 2 ( n e ∂ H z ( c ) ∂ τ + ε c ∂ E z ( c ) ∂ n ) = i k 0 γ g 2 ( n e ∂ H z ( g ) ∂ τ + ε g ∂ E z ( g ) ∂ n )
E z ( r → ) = ∫ L e ( r → s ) G ( r → , r → s ) d l s
H z ( r → ) = ∫ L h ( r → s ) G ( r → , r → s ) d l s .
G ( r → , r → s ) = i 4 H 0 ( 1 ) ( k 0 γ ∣ r → − r → s ∣ ) ,
1 : ∫ 0 2 π e c ( j ) ( s ′ ) G c ( s , s ′ ) J ( j ) ( s ′ ) d s ′ = ∑ k = 1 N c ∫ 0 2 π e g ( k ) ( s ′ ) G g ( s , s ′ ) J ( k ) ( s ′ ) d s ′
2 : ∫ 0 2 π h c ( j ) ( s ′ ) G c ( s , s ′ ) J ( j ) ( s ′ ) d s ′ = ∑ k = 1 N c ∫ 0 2 π h g ( k ) ( s ′ ) G g ( s , s ′ ) J ( k ) ( s ′ ) d s ′
3 : 1 γ c 2 ( n e ∫ 0 2 π e c ( j ) ( s ′ ) ∂ G c ( s , s ′ ) ∂ τ J ( j ) ( s ′ ) d s ′ − ∫ 0 2 π h c ( j ) ( s ′ ) ∂ G c ( s , s ′ ) ∂ τ J ( j ) ( s ′ ) ds ′ − h c ( j ) ( s ) 2 ) =
1 γ c 2 ( n e ∑ k = 1 N c ∫ 0 2 π e g ( k ) ( s ′ ) ∂ G g ( s , s ′ ) ∂ τ J ( k ) ( s ′ ) d s ′ − ∑ k = 1 N c ∫ 0 2 π h g ( k ) ( s ′ ) ∂ G g ( s , s ′ ) ∂ τ J ( k ) ( s ′ ) d s ′ + h c ( j ) ( s ) 2 ) ,
4 : 1 γ c 2 ( n e ∫ 0 2 π h c ( j ) ( s ′ ) ∂ G c ( s , s ′ ) ∂ τ J ( j ) ( s ′ ) d s ′ + ε c ∫ 0 2 π e c ( j ) ( s ′ ) ∂ G c ( s , s ′ ) ∂ n J ( j ) ( s ′ ) ds ′ + ε c e c ( j ) ( s ) 2 ) =
1 γ g 2 ( n e ∑ k = 1 N c ∫ 0 2 π h g ( k ) ( s ′ ) ∂ G g ( s , s ′ ) ∂ τ J ( k ) ( s ′ ) d s ′ + ε g ∑ k = 1 N c ∫ 0 2 π e g ( k ) ( s ′ ) ∂ G c ( s , s ′ ) ∂ n J ( k ) ( s ′ ) d s ′ − ε g e c ( j ) ( s ) 2 )
ψ ( k ) ( s ′ ) = ∑ t = 0 2 n ( k ) − 1 ( 1 2 n ( k ) ∑ m = − n ( k ) n ( k ) − 1 e i m ( s ′ − s t ) ) ψ ( k ) ( s t ) .
I ( j ) = ∫ 0 2 π ψ ( j ) ( s ′ ) Φ ( s , s ′ ) J ( j ) ( s ′ ) d s ′ ≃ ∑ t = 0 2 n ( j ) − 1 ( 1 2 π ∑ m = − n ( j ) n ( j ) − 1 e − i m s t ∫ 0 2 π e i m s ′ Φ ( s , s ′ ) a j d s ′ ) ψ ( j ) ( s t ) ,
∫ 0 2 π e i m s ′ G ( s , s ′ ) d s ' = i π 2 J m ( k 0 γ a j ) H m ( 1 ) ( k 0 γ a j ) e i m s
∫ 0 2 π e i m s ′ ∂ G ( s , s ' ) ∂ n d s ' = [ − 1 2 a j + i k 0 γ π 2 J m ′ ( k 0 γ a j ) H m ( 1 ) ( k 0 γ a j ) e i m s ] e i m s ,
∫ 0 2 π e i m s ′ ∂ G ( s , s ' ) ∂ n d s ' = − m π 2 a j J m ( k 0 γ a j ) H m ( 1 ) ( k 0 γ a j ) e i m s
I ( k ) = ∫ 0 2 π ψ ( k ) ( s ′ ) Φ ( s , s ′ ) J ( k ) ( s ′ ) d s ′ ≃ ∑ t = 0 2 n ( k ) − 1 ( a k 2 π 2 n ( k ) Φ ( s , s t ) ) ψ ( k ) ( s t ) .
A ( n e ) · X = 0 .
Ψ ( k ) ( s ′ ) = ∑ t = 0 2 n ( k ) − 1 ( 1 2 n ( k ) ∑ m = − n ( k ) n ( k ) − 1 e i m ( s ′ − s t ) ) Ψ ( k ) ( s t ) .
I ( j ) = ∫ 0 2 π Ψ ( j ) ( s ′ ) Φ ( s , s ′ ) d s ′ ≃ ∑ t = 0 2 n ( j ) − 1 ( 1 2 n ( j ) ∑ m = − n ( j ) n ( j ) − 1 e − i m s t ∫ 0 2 π e i m s ′ Φ a ( s , s ′ ) d s ′ ) Ψ ( j ) ( s t ) ,
∫ 0 2 π e i m s ′ G a ( s , s ′ ) d s ′ = ∫ 0 2 π e i m s ′ [ G a ( s , s ' ) − G c ( s , s ′ ) ] d s ′ + ∫ 0 2 π e i m s ′ G c ( s , s ′ ) d s ′ ,
∫ 0 2 π e i m s ′ Φ a ( s , s ′ ) d s ′ = ∫ 0 2 π e i m s ′ [ Φ a ( s , s ' ) − a j J ( j ) ( s ) Φ c ( s , s ′ ) ] d s ′ + a j J ( j ) ( s ) ∫ 0 2 π e i m s ′ Φ c ( s , s ′ ) d s ′ ,
∂ H 0 ( 1 ) ( k 0 γ R ) ∂ n = k 0 γ x s ′ ( y s − y s ′ ) − y s ′ ( x s − x s ′ ) J ( s ) R H 1 ( 1 ) ( k 0 γ R ) .
∂ H 0 ( 1 ) ( k 0 γ R ) ∂ τ = − k 0 γ x s ′ ( x s − x s ′ ) + y s ′ ( y s − y s ′ ) J ( s ) R H 1 ( 1 ) ( k 0 γ R ) .
∂ H 0 ( 1 ) ( 2 a k 0 γ ∣ sin ⁡ s − s ′ 2 ∣ ) ∂ n = − k 0 γ ∣ sin ⁡ s − s ′ 2 ∣ H 1 ( 1 ) ( 2 a k 0 γ ∣ sin ⁡ s − s ′ 2 ∣ ) ,
∂ H 0 ( 1 ) ( 2 a k 0 γ ∣ sin ⁡ s − s ′ 2 ∣ ) ∂ τ = − k 0 γ sin ⁡ ( s − s ′ ) 2 ∣ sin ⁡ s − s ′ 2 ∣ H 1 ( 1 ) ( 2 a k 0 γ ∣ sin ⁡ s − s ′ 2 ∣ ) .
lim s ′ → s [ H 0 ( 1 ) ( k 0 γ R ) − H 0 ( 1 ) ( 2 a k 0 γ ∣ sin ⁡ s − s ′ 2 ∣ ) ] = 2 i π ln J ( s ) a
lim s ′ → s [ ∂ H 0 ( 1 ) ( k 0 γ R ) ∂ n − a J ( s ) ∂ H 0 ( 1 ) ( 2 a k 0 γ ∣ sin ⁡ s − s ′ 2 ∣ ) ∂ n ] = 1 − κ ( s ) J ( s ) i π J ( s ) ,
lim s ′ → s [ ∂ H 0 ( 1 ) ( k 0 γ R ) ∂ τ − a J ( s ) ∂ H 0 ( 1 ) ( 2 a k 0 γ ∣ sin ⁡ s − s ′ 2 ∣ ) ∂ τ ] = 0
∫ L o ( j ) e o ( j ) ( s ′ ) G m ( s , s ′ ) J ( s ′ ) d s ′ + ∫ L i ( j ) e i ( j ) ( s ′ ) G m ( s , s ′ ) J ( s ′ ) d s ′ = ∑ k = 1 N c ∫ L 0 ( k ) e g ( k ) ( s ′ ) G g ( s , s ′ ) J ( s ′ ) d s ′ ,
∫ L i ( j ) e c ( i ) ( s ′ ) G c ( s , s ′ ) J ( s ′ ) d s ′ = ∫ L i ( j ) e i ( j ) ( s ′ ) G m ( s , s ′ ) J ( s ′ ) d s ′ + ∫ L o ( j ) e o ( j ) ( s ′ ) G m ( s , s ′ ) J ( s ′ ) d s ′ .

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