Efficient finite difference analysis of microstructured optical fibers

P. Kowalczyk; M. Wiktor; M. Mrozowski

doi:10.1364/OPEX.13.010349

Efficient finite difference analysis of microstructured optical fibers

P. Kowalczyk, M. Wiktor, and M. Mrozowski

Optics Express
Vol. 13,
Issue 25,
pp. 10349-10359
(2005)
•doi: 10.1364/OPEX.13.010349

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P. Kowalczyk, M. Wiktor, and M. Mrozowski, "Efficient finite difference analysis of microstructured optical fibers," Opt. Express 13, 10349-10359 (2005)

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Abstract

A new technique of numerical analysis of microstructured optical fibers is presented. The technique combines a standard 2D finite difference equations with the discrete function expansion. By doing this one gets a matrix eigenvalue problem of a smaller size and a simple formulation of radiation boundary condition. The new algorithm was tested for the microstructures of different types and excellent agreement of the obtained results with other methods was achieved.

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References

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|
Author
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Publication

P. Russell , “ Photonic Crystal Fibers ,” Science 299 , 358 – 362 ( 2003 ).
[Crossref] [PubMed]
C. Kerbage and B. J. Eggleton , ” Numerical analysis and experimental design of tunable birefringence in microstruc-tured optical fiber ,” Opt. Express 10 , 246 – 255 ( 2002 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-5-246 .
[PubMed]
T.P. White , R.C. McPhedran , C.M. de Sterke , L.C. Botten , and M.J. Steel , “ Confinement losses in microstructured optical fibers ,” Opt. Lett. 26 , 1660 – 1662 ( 2001 ).
[Crossref]
T.P. White , B.T. Kuhlmey , R.C. McPhedran , D. Maystre , R. Ranversez , C.M. de Sterke , L.C. Botten , and M.J. Steel , “ Multipole method for microstructured optical fibers. I. Formulation ,” J. Opt. Soc. Am. B 19 , 2322 – 2330 ( 2002 ).
[Crossref]
D. Ferrarini , L. Vincetti , M. Zoboli , A. Cucinotta , and S. Selleri , “ Leakage properties of photonic crystal fibers ,” Opt. Express 10 , 1314 – 1319 ( 2002 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1314 .
[PubMed]
H.P. Uranus and H.J.W.M. Hoekstra , “ Modeling of microstructured waveguides using a finite-element-based vecto-rial mode solver with transparent boundary conditions ,” Opt. Express 12 , 2795 – 2809 ( 2004 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2795 .
[Crossref] [PubMed]
Z. Zhu and T.G. Brown , “ Full-vectorial finite-difference analysis of microstructured optical fibers ,” Opt. Express 10 , 853 – 864 ( 2002 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853 .
[PubMed]
Shangping Guo , Feng Wu , Sacharia Albin , Hsiang Tai , and Robert S. Rogowski , “ Loss and dispersion analysis of microstructured fibers by finite-difference method ,” Opt. Express 12 , 3341 – 3352 ( 2004 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3341 .
[Crossref] [PubMed]
J.P. Berenger , “ A perfectly matched layer for the absorption of electromagnetic waves ,” J. Comput. Phys. 114 , 185 – 200 ( 1994 ).
[Crossref]
J.P. Berenger , “ Perfectly matched layer for the FDTD solution of wave-structure. Interaction problems ,” IEEE Trans. Microwave Theory Tech. 44 , 110 – 117 ( 1996 ).
H. Rogier and D. De Zutter , “ Berenger and Leaky Modes in Microstrip Substrates Terminated by a Perfectly Matched Layer ,” IEEE Trans. Microwave Theory Tech. 49 , 712 – 715 ( 2001 ).
[Crossref]
A. Bayliss , M. Gunzburger , and E. Turkel , “ Boundary conditions for the numerical solution of elliptic equations in exterior regions ,” SIAM J. Appl. Math. 42 , 430 – 451 ( 1982 ).
[Crossref]
T. Weiland , “ Verlustbehaftete Wellenleiter mit beliebiger. Randkontur und Materialverteilung ,” AEU 33 , 170 – 174 ( 1979 ).
N. Kaneda , B. Houshmand , and T. Itoh , “ FDTD analysis of dielectric resonators with curved surfaces ,” IEEE Trans. Microwave Theory Tech. 45 , 1645 – 1649 ( 1997 ).
[Crossref]
M. Mrozowski , “ A Hybrid PEE-FDTD Algorithm for Accelerated Time Domain Analysis of Electromagnetic Waves ,” IEEE Microwave and Guided Wave Letters 4 , 323 – 325 ( 1994 ).
[Crossref]
M. Mrozowski , M. Okoniewski , and M.A. Stuchly , “ A hybrid PEE-FDTD method for efficient field modeling in cyllindrical coordinates ,” Electronics Letters 32 , 194 – 195 ( 1996 ).
[Crossref]
M. Wiktor and M. Mrozowski ,“ Efficient Analysis of Waveguide Components Using a Hybrid PEE-FDFD Algorithm ,” IEEE Microwave and Wireless Components Letters 13 , 396 – 398 ( 2003 ).
[Crossref]
C.D. Meyer , “ Matrix analysis and applied linear algebra ”, SIAM , Philadelphia ( 2000 ).
M. Wiktor and M. Mrozowski ,“ Discrete Projection for Finite Difference Methods ,” 20th Annual Review or Progress in Applied Computational Electromagnetics, Syracuse NY 2004 , Conf. proceedings, S04P08.
N.A. Issa and L. Poladian , “ Vector Wave Expansion Method for Leaky Modes of Microstructured Optical Fibers ,” J. Lightwave Technol. 21 , 1005 – 1012 ( 2003 ).
[Crossref]

2004 (2)

H.P. Uranus and H.J.W.M. Hoekstra , “ Modeling of microstructured waveguides using a finite-element-based vecto-rial mode solver with transparent boundary conditions ,” Opt. Express 12 , 2795 – 2809 ( 2004 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-12-2795 .
[Crossref] [PubMed]

Shangping Guo , Feng Wu , Sacharia Albin , Hsiang Tai , and Robert S. Rogowski , “ Loss and dispersion analysis of microstructured fibers by finite-difference method ,” Opt. Express 12 , 3341 – 3352 ( 2004 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-12-15-3341 .
[Crossref] [PubMed]

2003 (3)

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

M. Wiktor and M. Mrozowski ,“ Efficient Analysis of Waveguide Components Using a Hybrid PEE-FDFD Algorithm ,” IEEE Microwave and Wireless Components Letters 13 , 396 – 398 ( 2003 ).
[Crossref]

N.A. Issa and L. Poladian , “ Vector Wave Expansion Method for Leaky Modes of Microstructured Optical Fibers ,” J. Lightwave Technol. 21 , 1005 – 1012 ( 2003 ).
[Crossref]

2002 (4)

C. Kerbage and B. J. Eggleton , ” Numerical analysis and experimental design of tunable birefringence in microstruc-tured optical fiber ,” Opt. Express 10 , 246 – 255 ( 2002 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-5-246 .
[PubMed]

T.P. White , B.T. Kuhlmey , R.C. McPhedran , D. Maystre , R. Ranversez , C.M. de Sterke , L.C. Botten , and M.J. Steel , “ Multipole method for microstructured optical fibers. I. Formulation ,” J. Opt. Soc. Am. B 19 , 2322 – 2330 ( 2002 ).
[Crossref]

D. Ferrarini , L. Vincetti , M. Zoboli , A. Cucinotta , and S. Selleri , “ Leakage properties of photonic crystal fibers ,” Opt. Express 10 , 1314 – 1319 ( 2002 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-23-1314 .
[PubMed]

Z. Zhu and T.G. Brown , “ Full-vectorial finite-difference analysis of microstructured optical fibers ,” Opt. Express 10 , 853 – 864 ( 2002 ), http://www.opticsexpress.org/abstract.cfm?URI=OPEX-10-17-853 .
[PubMed]

2001 (2)

T.P. White , R.C. McPhedran , C.M. de Sterke , L.C. Botten , and M.J. Steel , “ Confinement losses in microstructured optical fibers ,” Opt. Lett. 26 , 1660 – 1662 ( 2001 ).
[Crossref]

H. Rogier and D. De Zutter , “ Berenger and Leaky Modes in Microstrip Substrates Terminated by a Perfectly Matched Layer ,” IEEE Trans. Microwave Theory Tech. 49 , 712 – 715 ( 2001 ).
[Crossref]

1997 (1)

N. Kaneda , B. Houshmand , and T. Itoh , “ FDTD analysis of dielectric resonators with curved surfaces ,” IEEE Trans. Microwave Theory Tech. 45 , 1645 – 1649 ( 1997 ).
[Crossref]

1996 (2)

J.P. Berenger , “ Perfectly matched layer for the FDTD solution of wave-structure. Interaction problems ,” IEEE Trans. Microwave Theory Tech. 44 , 110 – 117 ( 1996 ).

M. Mrozowski , M. Okoniewski , and M.A. Stuchly , “ A hybrid PEE-FDTD method for efficient field modeling in cyllindrical coordinates ,” Electronics Letters 32 , 194 – 195 ( 1996 ).
[Crossref]

1994 (2)

M. Mrozowski , “ A Hybrid PEE-FDTD Algorithm for Accelerated Time Domain Analysis of Electromagnetic Waves ,” IEEE Microwave and Guided Wave Letters 4 , 323 – 325 ( 1994 ).
[Crossref]

J.P. Berenger , “ A perfectly matched layer for the absorption of electromagnetic waves ,” J. Comput. Phys. 114 , 185 – 200 ( 1994 ).
[Crossref]

1982 (1)

A. Bayliss , M. Gunzburger , and E. Turkel , “ Boundary conditions for the numerical solution of elliptic equations in exterior regions ,” SIAM J. Appl. Math. 42 , 430 – 451 ( 1982 ).
[Crossref]

1979 (1)

T. Weiland , “ Verlustbehaftete Wellenleiter mit beliebiger. Randkontur und Materialverteilung ,” AEU 33 , 170 – 174 ( 1979 ).

Albin, Sacharia

Bayliss, A.

Berenger, J.P.

J.P. Berenger , “ Perfectly matched layer for the FDTD solution of wave-structure. Interaction problems ,” IEEE Trans. Microwave Theory Tech. 44 , 110 – 117 ( 1996 ).

J.P. Berenger , “ A perfectly matched layer for the absorption of electromagnetic waves ,” J. Comput. Phys. 114 , 185 – 200 ( 1994 ).
[Crossref]

Botten, L.C.

T.P. White , R.C. McPhedran , C.M. de Sterke , L.C. Botten , and M.J. Steel , “ Confinement losses in microstructured optical fibers ,” Opt. Lett. 26 , 1660 – 1662 ( 2001 ).
[Crossref]

Brown, T.G.

Cucinotta, A.

de Sterke, C.M.

T.P. White , R.C. McPhedran , C.M. de Sterke , L.C. Botten , and M.J. Steel , “ Confinement losses in microstructured optical fibers ,” Opt. Lett. 26 , 1660 – 1662 ( 2001 ).
[Crossref]

De Zutter, D.

Eggleton, B. J.

Ferrarini, D.

Gunzburger, M.

Guo, Shangping

Hoekstra, H.J.W.M.

Houshmand, B.

N. Kaneda , B. Houshmand , and T. Itoh , “ FDTD analysis of dielectric resonators with curved surfaces ,” IEEE Trans. Microwave Theory Tech. 45 , 1645 – 1649 ( 1997 ).
[Crossref]

Issa, N.A.

N.A. Issa and L. Poladian , “ Vector Wave Expansion Method for Leaky Modes of Microstructured Optical Fibers ,” J. Lightwave Technol. 21 , 1005 – 1012 ( 2003 ).
[Crossref]

Itoh, T.

N. Kaneda , B. Houshmand , and T. Itoh , “ FDTD analysis of dielectric resonators with curved surfaces ,” IEEE Trans. Microwave Theory Tech. 45 , 1645 – 1649 ( 1997 ).
[Crossref]

Kaneda, N.

N. Kaneda , B. Houshmand , and T. Itoh , “ FDTD analysis of dielectric resonators with curved surfaces ,” IEEE Trans. Microwave Theory Tech. 45 , 1645 – 1649 ( 1997 ).
[Crossref]

Kerbage, C.

Kuhlmey, B.T.

Maystre, D.

McPhedran, R.C.

T.P. White , R.C. McPhedran , C.M. de Sterke , L.C. Botten , and M.J. Steel , “ Confinement losses in microstructured optical fibers ,” Opt. Lett. 26 , 1660 – 1662 ( 2001 ).
[Crossref]

Meyer, C.D.

C.D. Meyer , “ Matrix analysis and applied linear algebra ”, SIAM , Philadelphia ( 2000 ).

Mrozowski, M.

M. Wiktor and M. Mrozowski ,“ Efficient Analysis of Waveguide Components Using a Hybrid PEE-FDFD Algorithm ,” IEEE Microwave and Wireless Components Letters 13 , 396 – 398 ( 2003 ).
[Crossref]

M. Mrozowski , M. Okoniewski , and M.A. Stuchly , “ A hybrid PEE-FDTD method for efficient field modeling in cyllindrical coordinates ,” Electronics Letters 32 , 194 – 195 ( 1996 ).
[Crossref]

M. Mrozowski , “ A Hybrid PEE-FDTD Algorithm for Accelerated Time Domain Analysis of Electromagnetic Waves ,” IEEE Microwave and Guided Wave Letters 4 , 323 – 325 ( 1994 ).
[Crossref]

M. Wiktor and M. Mrozowski ,“ Discrete Projection for Finite Difference Methods ,” 20th Annual Review or Progress in Applied Computational Electromagnetics, Syracuse NY 2004 , Conf. proceedings, S04P08.

Okoniewski, M.

M. Mrozowski , M. Okoniewski , and M.A. Stuchly , “ A hybrid PEE-FDTD method for efficient field modeling in cyllindrical coordinates ,” Electronics Letters 32 , 194 – 195 ( 1996 ).
[Crossref]

Poladian, L.

N.A. Issa and L. Poladian , “ Vector Wave Expansion Method for Leaky Modes of Microstructured Optical Fibers ,” J. Lightwave Technol. 21 , 1005 – 1012 ( 2003 ).
[Crossref]

Ranversez, R.

Rogier, H.

Rogowski, Robert S.

Russell, P.

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

Selleri, S.

Steel, M.J.

T.P. White , R.C. McPhedran , C.M. de Sterke , L.C. Botten , and M.J. Steel , “ Confinement losses in microstructured optical fibers ,” Opt. Lett. 26 , 1660 – 1662 ( 2001 ).
[Crossref]

Stuchly, M.A.

M. Mrozowski , M. Okoniewski , and M.A. Stuchly , “ A hybrid PEE-FDTD method for efficient field modeling in cyllindrical coordinates ,” Electronics Letters 32 , 194 – 195 ( 1996 ).
[Crossref]

Tai, Hsiang

Turkel, E.

Uranus, H.P.

Vincetti, L.

Weiland, T.

T. Weiland , “ Verlustbehaftete Wellenleiter mit beliebiger. Randkontur und Materialverteilung ,” AEU 33 , 170 – 174 ( 1979 ).

White, T.P.

T.P. White , R.C. McPhedran , C.M. de Sterke , L.C. Botten , and M.J. Steel , “ Confinement losses in microstructured optical fibers ,” Opt. Lett. 26 , 1660 – 1662 ( 2001 ).
[Crossref]

Wiktor, M.

M. Wiktor and M. Mrozowski ,“ Efficient Analysis of Waveguide Components Using a Hybrid PEE-FDFD Algorithm ,” IEEE Microwave and Wireless Components Letters 13 , 396 – 398 ( 2003 ).
[Crossref]

Wu, Feng

Zhu, Z.

Zoboli, M.

AEU (1)

T. Weiland , “ Verlustbehaftete Wellenleiter mit beliebiger. Randkontur und Materialverteilung ,” AEU 33 , 170 – 174 ( 1979 ).

Electronics Letters (1)

M. Mrozowski , M. Okoniewski , and M.A. Stuchly , “ A hybrid PEE-FDTD method for efficient field modeling in cyllindrical coordinates ,” Electronics Letters 32 , 194 – 195 ( 1996 ).
[Crossref]

IEEE Microwave and Guided Wave Letters (1)

M. Mrozowski , “ A Hybrid PEE-FDTD Algorithm for Accelerated Time Domain Analysis of Electromagnetic Waves ,” IEEE Microwave and Guided Wave Letters 4 , 323 – 325 ( 1994 ).
[Crossref]

IEEE Microwave and Wireless Components Letters (1)

M. Wiktor and M. Mrozowski ,“ Efficient Analysis of Waveguide Components Using a Hybrid PEE-FDFD Algorithm ,” IEEE Microwave and Wireless Components Letters 13 , 396 – 398 ( 2003 ).
[Crossref]

IEEE Trans. Microwave Theory Tech. (3)

N. Kaneda , B. Houshmand , and T. Itoh , “ FDTD analysis of dielectric resonators with curved surfaces ,” IEEE Trans. Microwave Theory Tech. 45 , 1645 – 1649 ( 1997 ).
[Crossref]

J.P. Berenger , “ Perfectly matched layer for the FDTD solution of wave-structure. Interaction problems ,” IEEE Trans. Microwave Theory Tech. 44 , 110 – 117 ( 1996 ).

J. Comput. Phys. (1)

J.P. Berenger , “ A perfectly matched layer for the absorption of electromagnetic waves ,” J. Comput. Phys. 114 , 185 – 200 ( 1994 ).
[Crossref]

J. Lightwave Technol. (1)

N.A. Issa and L. Poladian , “ Vector Wave Expansion Method for Leaky Modes of Microstructured Optical Fibers ,” J. Lightwave Technol. 21 , 1005 – 1012 ( 2003 ).
[Crossref]

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

Opt. Express (5)

Opt. Lett. (1)

T.P. White , R.C. McPhedran , C.M. de Sterke , L.C. Botten , and M.J. Steel , “ Confinement losses in microstructured optical fibers ,” Opt. Lett. 26 , 1660 – 1662 ( 2001 ).
[Crossref]

Science (1)

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

SIAM J. Appl. Math. (1)

Other (2)

C.D. Meyer , “ Matrix analysis and applied linear algebra ”, SIAM , Philadelphia ( 2000 ).

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Fig. 1.

The discrete field components in the n-th cell (n = 1..K).

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Fig. 2.

The media interface

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Fig. 3.

Example of the structure used to illustrate the scheme of the DFE method implementation (different colors represents different media).

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Fig. 4.

Three different types of the microstructures used to testing presented technique.

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Fig. 5.

Illustration of the computation domain - DFE method for some outer and inner rings is used.

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Fig. 6.

The relative error of the calculated effective index: a)real part b)imaginary part. Current work - solid line, FD with rectangular mesh and PML [8] - dashed line.

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Tables (3)

Table 1. The effective indices of the structure with 6 circular holes obtained by different methods

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Table 2. The effective indices of the structure with 3 angular-shaped holes obtained by different methods.

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Table 3. The effective indices of the cored structure from Fig. 4(c).

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Equations (32)

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[\begin{matrix} ρ^{- 1} & 0 & 0 \\ 0 & ρ & 0 \\ 0 & 0 & ρ^{- 1} \end{matrix}] [\begin{matrix} 0 & γ & \frac{\partial}{\partial ϕ} \\ - γ & 0 & - \frac{\partial}{\partial ρ} \\ - \frac{\partial}{\partial ϕ} & \frac{\partial}{\partial ρ} & 0 \end{matrix}] [\begin{matrix} E_{ρ} \\ ρ E_{ϕ} \\ E_{z} \end{matrix}] = - jω μ_{0} [\begin{matrix} H_{ρ} \\ ρ H_{ϕ} \\ H_{z} \end{matrix}]

[\begin{matrix} ρ^{- 1} & 0 & 0 \\ 0 & ρ & 0 \\ 0 & 0 & ρ^{- 1} \end{matrix}] [\begin{matrix} 0 & γ & \frac{\partial}{\partial ϕ} \\ - γ & 0 & - \frac{\partial}{\partial ρ} \\ - \frac{\partial}{\partial ϕ} & \frac{\partial}{\partial ρ} & 0 \end{matrix}] [\begin{matrix} H_{ρ} \\ ρ H_{ϕ} \\ H_{z} \end{matrix}] = jω ε_{0} [\begin{matrix} ε_{ρ} & 0 & 0 \\ 0 & ε_{ϕ} & 0 \\ 0 & 0 & ε_{z} \end{matrix}] [\begin{matrix} E_{ρ} \\ ρ E_{ϕ} \\ E_{z} \end{matrix}]

[\begin{matrix} T_{t}^{(e)} & 0 \\ 0 & T_{z}^{(e)} \end{matrix}] [\begin{matrix} γ R_{tt}^{(e)} & R_{tz}^{(e)} \\ R_{zt}^{(e)} & 0 \end{matrix}] [\begin{matrix} E_{t} \\ E_{z} \end{matrix}] = - jω μ_{0} [\begin{matrix} H_{t} \\ H_{z} \end{matrix}]

[\begin{matrix} T_{t}^{(h)} & 0 \\ 0 & T_{z}^{(h)} \end{matrix}] [\begin{matrix} γ R_{tt}^{(h)} & R_{tz}^{(h)} \\ R_{zt}^{(h)} & 0 \end{matrix}] [\begin{matrix} H_{t} \\ H_{z} \end{matrix}] = jω ε_{0} [\begin{matrix} P_{t} & 0 \\ 0 & P_{z} \end{matrix}] [\begin{matrix} E_{t} \\ E_{z} \end{matrix}]

T_{t}^{(e)} = diag (ρ_{1}^{{(h)}^{- 1}}, \dots, ρ_{K}^{{(h)}^{- 1}}, ρ_{1}^{(e)}, \dots, ρ_{K}^{(e)}), T_{z}^{(e)} = diag (ρ_{1}^{{(e)}^{- 1}}, \dots, ρ_{K}^{{(e)}^{- 1}})

T_{t}^{(h)} = diag (ρ_{1}^{{(e)}^{- 1}}, \dots, ρ_{K}^{{(e)}^{- 1}}, ρ_{1}^{(h)}, \dots, ρ_{K}^{(h)}), T_{z}^{(h)} = diag (ρ_{1}^{{(h)}^{- 1}}, \dots, ρ_{K}^{{(h)}^{- 1}})

P_{t} = diag (ε_{ρ 1}, \dots, ε_{ρK}, ε_{ϕ 1}, \dots, ε_{ϕK}), P_{z} = diag (ε_{z 1}, \dots, ε_{zK})

R_{tt}^{(e)} = [\begin{matrix} 0 & I \\ - I & 0 \end{matrix}] = R_{tt}^{(h)}, R_{tz}^{(e)} = [\begin{matrix} F \\ - R \end{matrix}] = R_{zt}^{{(h)}^{T}}, R_{zt}^{(e)} = [- F R] = R_{tz}^{{(h)}^{T}}

R = Δ ρ^{- 1} [\begin{matrix} - 1 & 1 \\ ⋱ & ⋱ \\ ⋱ & 1 \\ ⋱ \\ - 1 \end{matrix}], F = Δ ϕ^{- 1} [\begin{matrix} - 1 & 1 \\ ⋱ & ⋱ \\ ⋱ & ⋱ \\ ⋱ & 1 \\ - 1 \end{matrix}]

E_{t} = [\begin{matrix} E_{ρ 1} \\ ⋮ \\ E_{ρK} \\ ρ_{1}^{(e)} E_{ϕ 1} \\ ⋮ \\ ρ_{K}^{(e)} E_{ϕK} \end{matrix}], E_{z} = [\begin{matrix} E_{z 1} \\ ⋮ \\ E_{zK} \end{matrix}], H_{t} = [\begin{matrix} H_{ρ 1} \\ ⋮ \\ H_{ρK} \\ ρ_{1}^{(h)} H_{ϕ 1} \\ ⋮ \\ ρ_{K}^{(h)} H_{ϕK} \end{matrix}], H_{z} = [\begin{matrix} H_{z 1} \\ ⋮ \\ H_{zK} \end{matrix}]

γ^{2} T_{t}^{(h)} R_{tt}^{(h)} T_{t}^{(e)} R_{tt}^{(e)} E_{t} + γ T_{t}^{(h)} R_{tt}^{(h)} T_{t}^{(e)} R_{tz}^{(e)} E_{z} + T_{t}^{(h)} R_{tz}^{(h)} T_{z}^{(e)} R_{zt}^{(e)} E_{t} = k_{0}^{2} P_{t} E_{t}

γ P_{z} E_{z} = D_{t} E_{t}

D_{t} = T_{z}^{(e)} ∙ [- R^{T} - F^{T}] ∙ T_{t}^{{(e)}^{- 1}}

A E_{t} = γ^{2} E_{t}

A = T_{t}^{(h)} R_{tt}^{(h)} T_{t}^{(e)} R_{tz}^{(e)} P_{z}^{- 1} D_{t} P_{t} + T_{t}^{(h)} R_{tz}^{(h)} T_{z}^{(e)} R_{zt}^{(e)} - k_{0}^{2} P_{t}

ε_{ρ} = Δ ϕ^{- 1} \ln \frac{ρ_{1}}{ρ_{2}} {[\int_{ρ_{1}}^{ρ_{2}} \frac{dρ}{ρ [(ϕ (ρ) - ϕ_{1}) ε_{a} + (ϕ_{2} - ϕ (ρ)) ε_{b}]}]}^{- 1}

ε_{ϕ} = Δϕ {[\ln \frac{ρ_{2}}{ρ_{1}} \int_{ϕ_{1}}^{ϕ_{2}} \frac{dϕ}{ε_{a} \ln \frac{ρ (ϕ)}{ρ_{1}} + ε_{b} \ln \frac{ρ_{2}}{ρ (ϕ)}}]}^{- 1}

ε_{z} = p ε_{b} + (1 - p) ε_{a}

{\begin{matrix} f_{1} = g_{1} \sin (\frac{π}{N}) + g_{2} \sin (\frac{2 π}{N}) + \dots + g_{q} \sin (\frac{qπ}{N}) \\ f_{2} = g_{1} \sin (\frac{π 2}{N}) + g_{2} \sin (\frac{2 π 2}{N}) + \dots + g_{q} \sin (\frac{qπ 2}{N}) \\ ⋮ \\ f_{N - 1} = g_{1} \sin (\frac{π (N - 1)}{N}) + g_{2} \sin (\frac{2 π (N - 1)}{N}) + \dots + g_{q} \sin (\frac{qπ (N - 1)}{N}) \end{matrix}

{\tilde{E}}_{t} = [E_{ρ 1} \dots E_{ρ (l - 1) ∙ N} {\tilde{E}}_{ρ 1} \dots {\tilde{E}}_{ρq} E_{ρ (l + 1) ∙ N} \dots E_{ρK}

ρ_{1}^{(e)} E_{ϕ 1} \dots ρ_{(l - 1) ∙ N}^{(e)} E_{ϕ (l - 1) ∙ N} ρ_{(l - 1) ∙ N + 1}^{(e)} {\tilde{E}}_{ϕ 1} \dots ρ_{l ∙ N}^{(e)} {\tilde{E}}_{ϕq}

{ρ_{l ∙ N + 1}^{(e)} E_{ϕ (l + 1) ∙ N} \dots ρ_{K}^{(e)} E_{ϕK}]}^{T}

E_{t} = \tilde{Q} {\tilde{E}}_{t}

\tilde{Q} = [\begin{matrix} {\tilde{Q}}_{ρ} \\ {\tilde{Q}}_{ϕ} \end{matrix}]

{\tilde{Q}}_{ρ} = [\begin{matrix} 1 \\ ⋱ \\ 1 \\ 0 & 0 & \dots & 0 \\ \sin \frac{π}{2 N - 1} & \sin \frac{3 π}{2 N - 1} & \dots & \sin \frac{(2 q - 1) π}{2 N - 1} \\ ⋮ & ⋮ & ⋮ \\ \sin \frac{π (N - 2)}{2 N - 1} & \sin \frac{3 π (N - 2)}{2 N - 1} & \dots & \sin \frac{(2 q - 1) π (N - 2)}{2 N - 1} \\ 1 \\ ⋱ \\ 1 \end{matrix}]

{\tilde{Q}}_{ϕ} = [\begin{matrix} 1 \\ ⋱ \\ 1 \\ \cos \frac{π}{2 (2 N - 1)} & \cos \frac{3 π}{2 (2 N - 1)} & \dots & \cos \frac{(2 q - 1) π}{2 (2 N - 1)} \\ ⋮ & ⋮ & ⋮ \\ \cos \frac{π (2 N - 2)}{2 (2 N - 1)} & \cos \frac{3 π (2 N - 2)}{2 (2 N - 1)} & \dots & \cos \frac{(2 q - 1) π (2 N - 2)}{2 (2 N - 1)} \\ 0 & 0 & \dots & 0 \\ 1 \\ ⋱ \\ 1 \end{matrix}]

\tilde{A} {\tilde{E}}_{t} = γ^{2} {\tilde{E}}_{t}

\tilde{A} = {\tilde{Q}}^{T} A \tilde{Q}

E_{ρ} = \sum_{m = 1}^{\infty} a_{m} H_{m}^{(2)} (κ ρ) \sin (m ϕ) e^{- γ z}

E_{ϕ} = \sum_{m = 0}^{\infty} b_{m} H_{m}^{(2)} (κ ρ) \cos (m ϕ) e^{- γ z}

{\tilde{E}}_{ρ m} = \frac{H_{m}^{(2)} (κ ρ_{n}^{(e)})}{H_{m}^{(2)} (κ ρ_{n + N}^{(e)})} {\tilde{E}}_{ρ m + q}

{\tilde{E}}_{ϕ m} = \frac{ρ_{n}^{(2)} H_{m}^{(2)} (κ ρ_{n}^{(h)})}{ρ_{n + N}^{(h)} H_{m}^{(2)} (κ ρ_{n + N}^{(h)})} {\tilde{E}}_{ϕ m + q}

MM		FEM		FDM-ABC		FD-DFE
ℜ(n_eff)	-ℑ(n_eff)	ℜ(n_eff)	-ℑ(n_eff)	ℜ(n_eff)	-ℑ(n_eff)	ℜ(n_eff)	-ℑ(n_eff)
1.445395	3.19E-08	1.445394	4.11E-08	1.445395	3.07E-08	1.445391	3.21E-08
1.438584	5.31E-07	1.438576	3.97E-07	1.438589	5.43E-07	1.438585	5.09E-07
1.438445	9.73E-07	1.438442	7.13E-07	1.438444	9.62E-07	1.438437	9.04E-07
1.429957	1.59E-05	1.429952	1.67E-05			1.429954	1.57E-05
1.429248	8.73E-06	1.429261	9.17E-06			1.429264	8.83E-06

FEM		FDM-ABC		FD-DFE
ℜ(n_eff)	-ℑ(n_eff)	ℜ(n_eff)	-ℑ(n_eff)	ℜ(n_eff)	-ℑ(n_eff)
1.35580	4.95E-05	1.35584	5.00E-05	1.35588	5.11E-05
1.21479	1.11E-03	1.21476	1.25E-03	1.21488	1.25E-03

FD-DFE
ℜ(n_eff)	-ℑ(n_eff)
1.435355	0
1.389851	1.29E-06
1.387892	5.44E-07
1.384163	4.13E-06

MM		FEM		FDM-ABC		FD-DFE
ℜ(n_eff)	-ℑ(n_eff)	ℜ(n_eff)	-ℑ(n_eff)	ℜ(n_eff)	-ℑ(n_eff)	ℜ(n_eff)	-ℑ(n_eff)
1.445395	3.19E-08	1.445394	4.11E-08	1.445395	3.07E-08	1.445391	3.21E-08
1.438584	5.31E-07	1.438576	3.97E-07	1.438589	5.43E-07	1.438585	5.09E-07
1.438445	9.73E-07	1.438442	7.13E-07	1.438444	9.62E-07	1.438437	9.04E-07
1.429957	1.59E-05	1.429952	1.67E-05			1.429954	1.57E-05
1.429248	8.73E-06	1.429261	9.17E-06			1.429264	8.83E-06

FEM		FDM-ABC		FD-DFE
ℜ(n_eff)	-ℑ(n_eff)	ℜ(n_eff)	-ℑ(n_eff)	ℜ(n_eff)	-ℑ(n_eff)
1.35580	4.95E-05	1.35584	5.00E-05	1.35588	5.11E-05
1.21479	1.11E-03	1.21476	1.25E-03	1.21488	1.25E-03

Abstract

References

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Albin, Sacharia

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Uranus, H.P.

Vincetti, L.

Weiland, T.

White, T.P.

Wiktor, M.

Wu, Feng

Zhu, Z.

Zoboli, M.

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