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.

© 2005 Optical Society of America

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References

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  1. P. Russell , “ Photonic Crystal Fibers ,” Science   299 , 358 – 362 ( 2003 ).
    [Crossref] [PubMed]
  2. 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]
  3. 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]
  4. 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]
  5. 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]
  6. 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]
  7. 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]
  8. 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]
  9. J.P. Berenger , “ A perfectly matched layer for the absorption of electromagnetic waves ,” J. Comput. Phys.   114 , 185 – 200 ( 1994 ).
    [Crossref]
  10. J.P. Berenger , “ Perfectly matched layer for the FDTD solution of wave-structure. Interaction problems ,” IEEE Trans. Microwave Theory Tech.   44 , 110 – 117 ( 1996 ).
  11. 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]
  12. 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]
  13. T. Weiland , “ Verlustbehaftete Wellenleiter mit beliebiger. Randkontur und Materialverteilung ,” AEU   33 , 170 – 174 ( 1979 ).
  14. 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]
  15. 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]
  16. 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]
  17. 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]
  18. C.D. Meyer , “ Matrix analysis and applied linear algebra ”, SIAM , Philadelphia ( 2000 ).
  19. 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.
  20. 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)

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)

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.

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]

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.

Brown, T.G.

Cucinotta, A.

de Sterke, C.M.

De Zutter, D.

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]

Eggleton, B. J.

Ferrarini, D.

Gunzburger, M.

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]

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.

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.

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.

Ranversez, R.

Rogier, H.

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]

Rogowski, Robert S.

Russell, P.

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

Selleri, S.

Steel, M.J.

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.

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]

Uranus, H.P.

Vincetti, L.

Weiland, T.

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

White, T.P.

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]

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.

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 ).

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]

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)

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

Opt. Express (5)

Opt. Lett. (1)

Science (1)

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

SIAM J. Appl. Math. (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]

Other (2)

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.

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Figures (6)

Fig. 1.
Fig. 1.

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

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Fig. 2.
Fig. 2.

The media interface

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Fig. 3.
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.
Fig. 4.

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

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Fig. 5.
Fig. 5.

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

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Fig. 6.
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)

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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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[ ρ − 1 0 0 0 ρ 0 0 0 ρ − 1 ] [ 0 γ ∂ ∂ ϕ − γ 0 − ∂ ∂ ρ − ∂ ∂ ϕ ∂ ∂ ρ 0 ] [ E ρ ρ E ϕ E z ] = − jω μ 0 [ H ρ ρ H ϕ H z ]
[ ρ − 1 0 0 0 ρ 0 0 0 ρ − 1 ] [ 0 γ ∂ ∂ ϕ − γ 0 − ∂ ∂ ρ − ∂ ∂ ϕ ∂ ∂ ρ 0 ] [ H ρ ρ H ϕ H z ] = jω ε 0 [ ε ρ 0 0 0 ε ϕ 0 0 0 ε z ] [ E ρ ρ E ϕ E z ]
[ T t ( e ) 0 0 T z ( e ) ] [ γ R tt ( e ) R tz ( e ) R zt ( e ) 0 ] [ E t E z ] = − jω μ 0 [ H t H z ]
[ T t ( h ) 0 0 T z ( h ) ] [ γ R tt ( h ) R tz ( h ) R zt ( h ) 0 ] [ H t H z ] = jω ε 0 [ P t 0 0 P z ] [ E t E z ]
T t ( e ) = diag ( ρ 1 ( h ) − 1 , ⋯ , ρ K ( h ) − 1 , ρ 1 ( e ) , ⋯ , ρ K ( e ) ) , T z ( e ) = diag ( ρ 1 ( e ) − 1 , ⋯ , ρ K ( e ) − 1 )
T t ( h ) = diag ( ρ 1 ( e ) − 1 , ⋯ , ρ K ( e ) − 1 , ρ 1 ( h ) , ⋯ , ρ K ( h ) ) , T z ( h ) = diag ( ρ 1 ( h ) − 1 , ⋯ , ρ K ( h ) − 1 )
P t = diag ( ε ρ 1 , … , ε ρK , ε ϕ 1 , … , ε ϕK ) , P z = diag ( ε z 1 , … , ε zK )
R tt ( e ) = [ 0 I − I 0 ] = R tt ( h ) , R tz ( e ) = [ F − R ] = R zt ( h ) T , R zt ( e ) = [ − F R ] = R tz ( h ) T
R = Δ ρ − 1 [ − 1 1 ⋱ ⋱ ⋱ 1 ⋱ − 1 ] , F = Δ ϕ − 1 [ − 1 1 ⋱ ⋱ ⋱ ⋱ ⋱ 1 − 1 ]
E t = [ E ρ 1 ⋮ E ρK ρ 1 ( e ) E ϕ 1 ⋮ ρ K ( e ) E ϕK ] , E z = [ E z 1 ⋮ E zK ] , H t = [ H ρ 1 ⋮ H ρK ρ 1 ( h ) H ϕ 1 ⋮ ρ K ( h ) H ϕK ] , H z = [ H z 1 ⋮ H zK ]
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 ⁡ ρ 1 ρ 2 [ ∫ ρ 1 ρ 2 dρ ρ [ ( ϕ ( ρ ) − ϕ 1 ) ε a + ( ϕ 2 − ϕ ( ρ ) ) ε b ] ] − 1
ε ϕ = Δϕ [ ln ρ 2 ρ 1 ∫ ϕ 1 ϕ 2 dϕ ε a ln ρ ( ϕ ) ρ 1 + ε b ln ρ 2 ρ ( ϕ ) ] − 1
ε z = p ε b + ( 1 − p ) ε a
{ f 1 = g 1 sin ( π N ) + g 2 sin ( 2 π N ) + ⋯ + g q sin ( qπ N ) f 2 = g 1 sin ( π 2 N ) + g 2 sin ( 2 π 2 N ) + ⋯ + g q sin ( qπ 2 N ) ⋮ f N − 1 = g 1 sin ( π ( N − 1 ) N ) + g 2 sin ( 2 π ( N − 1 ) N ) + ⋯ + g q sin ( qπ ( N − 1 ) N )
E ˜ t = [ E ρ 1 ⋯ E ρ ( l − 1 ) ∙ N E ˜ ρ 1 ⋯ E ˜ ρq E ρ ( l + 1 ) ∙ N ⋯ E ρK
ρ 1 ( e ) E ϕ 1 ⋯ ρ ( l − 1 ) ∙ N ( e ) E ϕ ( l − 1 ) ∙ N ρ ( l − 1 ) ∙ N + 1 ( e ) E ˜ ϕ 1 ⋯ ρ l ∙ N ( e ) E ˜ ϕq
ρ l ∙ N + 1 ( e ) E ϕ ( l + 1 ) ∙ N ⋯ ρ K ( e ) E ϕK ] T
E t = Q ˜ E ˜ t
Q ˜ = [ Q ˜ ρ Q ˜ ϕ ]
Q ˜ ρ = [ 1 ⋱ 1 0 0 … 0 sin π 2 N − 1 sin 3 π 2 N − 1 … sin ( 2 q − 1 ) π 2 N − 1 ⋮ ⋮ ⋮ sin π ( N − 2 ) 2 N − 1 sin 3 π ( N − 2 ) 2 N − 1 … sin ( 2 q − 1 ) π ( N − 2 ) 2 N − 1 1 ⋱ 1 ]
Q ˜ ϕ = [ 1 ⋱ 1 cos π 2 ( 2 N − 1 ) cos 3 π 2 ( 2 N − 1 ) … cos ( 2 q − 1 ) π 2 ( 2 N − 1 ) ⋮ ⋮ ⋮ cos π ( 2 N − 2 ) 2 ( 2 N − 1 ) cos 3 π ( 2 N − 2 ) 2 ( 2 N − 1 ) … cos ( 2 q − 1 ) π ( 2 N − 2 ) 2 ( 2 N − 1 ) 0 0 … 0 1 ⋱ 1 ]
A ˜ E ˜ t = γ 2 E ˜ t
A ˜ = Q ˜ T A Q ˜
E ρ = ∑ m = 1 ∞ a m H m ( 2 ) ( κ ρ ) sin ( m ϕ ) e − γ z
E ϕ = ∑ m = 0 ∞ b m H m ( 2 ) ( κ ρ ) cos ( m ϕ ) e − γ z
E ˜ ρ m = H m ( 2 ) ( κ ρ n ( e ) ) H m ( 2 ) ( κ ρ n + N ( e ) ) E ˜ ρ m + q
E ˜ ϕ m = ρ n ( 2 ) H m ( 2 ) ( κ ρ n ( h ) ) ρ n + N ( h ) H m ( 2 ) ( κ ρ n + N ( h ) ) E ˜ ϕ m + q

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