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

Full Article  |  PDF Article
OSA Recommended Articles
A new conformal radiation boundary condition for high accuracy finite difference analysis of open waveguides

P. Kowalczyk and M. Mrozowski
Opt. Express 15(20) 12605-12618 (2007)

Analysis of microstructured optical fibers using compact macromodels

P. Kowalczyk, L. Kulas, and M. Mrozowski
Opt. Express 19(20) 19354-19364 (2011)

Full-vectorial finite-difference analysis of microstructured optical fibers

Zhaoming Zhu and Thomas G. Brown
Opt. Express 10(17) 853-864 (2002)

More Recommended Articles

References

  • View by:
  • Article Order
  • |
  • Year
  • |
  • Author
  • |
  • Publication
  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)

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]

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

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.

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1.
Fig. 1.

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

Download Full Size | PPT Slide | PDF

Fig. 2.
Fig. 2.

The media interface

Download Full Size | PPT Slide | PDF

Fig. 3.
Fig. 3.

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

Download Full Size | PPT Slide | PDF

Fig. 4.
Fig. 4.

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

Download Full Size | PPT Slide | PDF

Fig. 5.
Fig. 5.

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

Download Full Size | PPT Slide | PDF

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.

Download Full Size | PPT Slide | PDF

Tables (3)

Tables Icon

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

View Table | View all tables in this article
Tables Icon

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

View Table | View all tables in this article
Tables Icon

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

View Table | View all tables in this article

Equations (32)

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

[ 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

Metrics