Abstract

For analyzing plane wave scattering from a multilayer periodic structure where each layer consists of a two-dimensional periodic array of spheres, a spherical wave least squares method is developed which extends and improves the earlier work by Matsushima et al. [PIER 69, 305 (2007)] . A number of techniques are used to speed up the method and to reduce the memory requirement. Spherical wave expansions are used in one unit cell containing a sphere in each layer, and quasi-periodic conditions are imposed on lateral surfaces of the unit cell in the least squares sense. Unlike the layer-Korringa–Kohn–Rostoker method [Physica A 141, 575 (1987) ], the method does not need lattice sums and it is relatively simple to implement.

© 2010 Optical Society of America

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  1. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).
  2. K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
    [CrossRef] [PubMed]
  3. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
    [CrossRef] [PubMed]
  4. K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
    [CrossRef]
  5. H. S. Sözüer and J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231–239 (1994).
    [CrossRef]
  6. K. Ohtaka, “Scattering theory of low-energy photon diffraction,” J. Phys. C 13, 667–680 (1980).
    [CrossRef]
  7. A. Modinos, “Scattering of electromagnetic waves by a plane of spheres—formalism,” Physica A 141, 575–588 (1987).
    [CrossRef]
  8. P. A. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles (Cambridge Univ. Press, 2006).
  9. M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).
  10. N. Stafanou and A. Modinos, “Scattering of light from a two-dimensional array of spherical particles on a substrate,” J. Phys. Condens. Matter 3, 8135–8148 (1991).
    [CrossRef]
  11. N. Stefanou, V. Karathanos, and A. Modinos, “Scattering of electromagnetic waves by periodic structures,” J. Phys. Condens. Matter 4, 7389–7400 (1992).
    [CrossRef]
  12. K. Ohtaka and Y. Tanabe, “Photonic bands using vector spherical waves. II. Reflectivity, coherence and local field,” J. Phys. Soc. Jpn. 65, 2276–2284 (1996).
    [CrossRef]
  13. N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998).
    [CrossRef]
  14. N. Stefanou, V. Yannopapas, and A. Modinos, “MULTEM 2: a new version of the program for transmission and band-structure calculations of photonic crystals,” Comput. Phys. Commun. 132, 189–196 (2000).
    [CrossRef]
  15. L. Li, “New formulation of the Fourier modal method for crossed surface-relief gratings,” J. Opt. Soc. Am. A 14, 2758–2767 (1997).
    [CrossRef]
  16. G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. 70, 1–34 (2010).
  17. E. Popov and M. Nevière, “Maxwell equations in Fourier space: a fast-converging formulation for diffraction by arbitrary shaped, periodic, anisotropic media,” J. Opt. Soc. Am. A 18, 2886–2894 (2001).
    [CrossRef]
  18. A. Matsushima, Y. Momoka, M. Ohtsu, and Y. Okuno, “Efficient numerical approach to electromagnetic scattering from three-dimensional periodic array of dielectric spheres using sequential accumulation,” PIER 69, 305–322 (2007).
    [CrossRef]
  19. C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (SIAM, 1995).
    [CrossRef]

2010

G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. 70, 1–34 (2010).

2007

A. Matsushima, Y. Momoka, M. Ohtsu, and Y. Okuno, “Efficient numerical approach to electromagnetic scattering from three-dimensional periodic array of dielectric spheres using sequential accumulation,” PIER 69, 305–322 (2007).
[CrossRef]

2006

P. A. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles (Cambridge Univ. Press, 2006).

2001

2000

N. Stefanou, V. Yannopapas, and A. Modinos, “MULTEM 2: a new version of the program for transmission and band-structure calculations of photonic crystals,” Comput. Phys. Commun. 132, 189–196 (2000).
[CrossRef]

1999

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).

1998

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998).
[CrossRef]

1997

1996

K. Ohtaka and Y. Tanabe, “Photonic bands using vector spherical waves. II. Reflectivity, coherence and local field,” J. Phys. Soc. Jpn. 65, 2276–2284 (1996).
[CrossRef]

1995

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (SIAM, 1995).
[CrossRef]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).

1994

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

H. S. Sözüer and J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231–239 (1994).
[CrossRef]

1992

N. Stefanou, V. Karathanos, and A. Modinos, “Scattering of electromagnetic waves by periodic structures,” J. Phys. Condens. Matter 4, 7389–7400 (1992).
[CrossRef]

1991

N. Stafanou and A. Modinos, “Scattering of light from a two-dimensional array of spherical particles on a substrate,” J. Phys. Condens. Matter 3, 8135–8148 (1991).
[CrossRef]

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[CrossRef] [PubMed]

1990

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

1987

A. Modinos, “Scattering of electromagnetic waves by a plane of spheres—formalism,” Physica A 141, 575–588 (1987).
[CrossRef]

1980

K. Ohtaka, “Scattering theory of low-energy photon diffraction,” J. Phys. C 13, 667–680 (1980).
[CrossRef]

Bao, G.

G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. 70, 1–34 (2010).

Biswas, R.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

Born, M.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).

Chan, C. T.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Dowling, J. P.

H. S. Sözüer and J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231–239 (1994).
[CrossRef]

Gmitter, T. J.

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[CrossRef] [PubMed]

Hanson, R. J.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (SIAM, 1995).
[CrossRef]

Ho, K. M.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Joannopoulos, J. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).

Karathanos, V.

N. Stefanou, V. Karathanos, and A. Modinos, “Scattering of electromagnetic waves by periodic structures,” J. Phys. Condens. Matter 4, 7389–7400 (1992).
[CrossRef]

Lawson, C. L.

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (SIAM, 1995).
[CrossRef]

Leung, K. M.

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[CrossRef] [PubMed]

Li, L.

Li, P.

G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. 70, 1–34 (2010).

Martin, P. A.

P. A. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles (Cambridge Univ. Press, 2006).

Matsushima, A.

A. Matsushima, Y. Momoka, M. Ohtsu, and Y. Okuno, “Efficient numerical approach to electromagnetic scattering from three-dimensional periodic array of dielectric spheres using sequential accumulation,” PIER 69, 305–322 (2007).
[CrossRef]

Meade, R. D.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).

Modinos, A.

N. Stefanou, V. Yannopapas, and A. Modinos, “MULTEM 2: a new version of the program for transmission and band-structure calculations of photonic crystals,” Comput. Phys. Commun. 132, 189–196 (2000).
[CrossRef]

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998).
[CrossRef]

N. Stefanou, V. Karathanos, and A. Modinos, “Scattering of electromagnetic waves by periodic structures,” J. Phys. Condens. Matter 4, 7389–7400 (1992).
[CrossRef]

N. Stafanou and A. Modinos, “Scattering of light from a two-dimensional array of spherical particles on a substrate,” J. Phys. Condens. Matter 3, 8135–8148 (1991).
[CrossRef]

A. Modinos, “Scattering of electromagnetic waves by a plane of spheres—formalism,” Physica A 141, 575–588 (1987).
[CrossRef]

Momoka, Y.

A. Matsushima, Y. Momoka, M. Ohtsu, and Y. Okuno, “Efficient numerical approach to electromagnetic scattering from three-dimensional periodic array of dielectric spheres using sequential accumulation,” PIER 69, 305–322 (2007).
[CrossRef]

Nevière, M.

Ohtaka, K.

K. Ohtaka and Y. Tanabe, “Photonic bands using vector spherical waves. II. Reflectivity, coherence and local field,” J. Phys. Soc. Jpn. 65, 2276–2284 (1996).
[CrossRef]

K. Ohtaka, “Scattering theory of low-energy photon diffraction,” J. Phys. C 13, 667–680 (1980).
[CrossRef]

Ohtsu, M.

A. Matsushima, Y. Momoka, M. Ohtsu, and Y. Okuno, “Efficient numerical approach to electromagnetic scattering from three-dimensional periodic array of dielectric spheres using sequential accumulation,” PIER 69, 305–322 (2007).
[CrossRef]

Okuno, Y.

A. Matsushima, Y. Momoka, M. Ohtsu, and Y. Okuno, “Efficient numerical approach to electromagnetic scattering from three-dimensional periodic array of dielectric spheres using sequential accumulation,” PIER 69, 305–322 (2007).
[CrossRef]

Popov, E.

Sigalas, M.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

Soukoulis, C. M.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

Sözüer, H. S.

H. S. Sözüer and J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231–239 (1994).
[CrossRef]

Stafanou, N.

N. Stafanou and A. Modinos, “Scattering of light from a two-dimensional array of spherical particles on a substrate,” J. Phys. Condens. Matter 3, 8135–8148 (1991).
[CrossRef]

Stefanou, N.

N. Stefanou, V. Yannopapas, and A. Modinos, “MULTEM 2: a new version of the program for transmission and band-structure calculations of photonic crystals,” Comput. Phys. Commun. 132, 189–196 (2000).
[CrossRef]

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998).
[CrossRef]

N. Stefanou, V. Karathanos, and A. Modinos, “Scattering of electromagnetic waves by periodic structures,” J. Phys. Condens. Matter 4, 7389–7400 (1992).
[CrossRef]

Tanabe, Y.

K. Ohtaka and Y. Tanabe, “Photonic bands using vector spherical waves. II. Reflectivity, coherence and local field,” J. Phys. Soc. Jpn. 65, 2276–2284 (1996).
[CrossRef]

Winn, J. N.

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).

Wolf, E.

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).

Wu, H.

G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. 70, 1–34 (2010).

Yablonovitch, E.

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[CrossRef] [PubMed]

Yannopapas, V.

N. Stefanou, V. Yannopapas, and A. Modinos, “MULTEM 2: a new version of the program for transmission and band-structure calculations of photonic crystals,” Comput. Phys. Commun. 132, 189–196 (2000).
[CrossRef]

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998).
[CrossRef]

Comput. Phys. Commun.

N. Stefanou, V. Yannopapas, and A. Modinos, “Heterostructures of photonic crystals: frequency bands and transmission coefficients,” Comput. Phys. Commun. 113, 49–77 (1998).
[CrossRef]

N. Stefanou, V. Yannopapas, and A. Modinos, “MULTEM 2: a new version of the program for transmission and band-structure calculations of photonic crystals,” Comput. Phys. Commun. 132, 189–196 (2000).
[CrossRef]

J. Mod. Opt.

H. S. Sözüer and J. P. Dowling, “Photonic band calculations for woodpile structures,” J. Mod. Opt. 41, 231–239 (1994).
[CrossRef]

J. Opt. Soc. Am. A

J. Phys. C

K. Ohtaka, “Scattering theory of low-energy photon diffraction,” J. Phys. C 13, 667–680 (1980).
[CrossRef]

J. Phys. Condens. Matter

N. Stafanou and A. Modinos, “Scattering of light from a two-dimensional array of spherical particles on a substrate,” J. Phys. Condens. Matter 3, 8135–8148 (1991).
[CrossRef]

N. Stefanou, V. Karathanos, and A. Modinos, “Scattering of electromagnetic waves by periodic structures,” J. Phys. Condens. Matter 4, 7389–7400 (1992).
[CrossRef]

J. Phys. Soc. Jpn.

K. Ohtaka and Y. Tanabe, “Photonic bands using vector spherical waves. II. Reflectivity, coherence and local field,” J. Phys. Soc. Jpn. 65, 2276–2284 (1996).
[CrossRef]

Math. Comput.

G. Bao, P. Li, and H. Wu, “An adaptive edge element method with perfectly matched absorbing layers for wave scattering by biperiodic structures,” Math. Comput. 70, 1–34 (2010).

Phys. Rev. Lett.

K. M. Ho, C. T. Chan, and C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structures,” Phys. Rev. Lett. 65, 3152–3155 (1990).
[CrossRef] [PubMed]

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, “Photonic band structure: the face-centered-cubic case employing nonspherical atoms,” Phys. Rev. Lett. 67, 2295–2298 (1991).
[CrossRef] [PubMed]

Physica A

A. Modinos, “Scattering of electromagnetic waves by a plane of spheres—formalism,” Physica A 141, 575–588 (1987).
[CrossRef]

PIER

A. Matsushima, Y. Momoka, M. Ohtsu, and Y. Okuno, “Efficient numerical approach to electromagnetic scattering from three-dimensional periodic array of dielectric spheres using sequential accumulation,” PIER 69, 305–322 (2007).
[CrossRef]

Solid State Commun.

K. M. Ho, C. T. Chan, C. M. Soukoulis, R. Biswas, and M. Sigalas, “Photonic band gaps in three dimensions: new layer-by-layer periodic structures,” Solid State Commun. 89, 413–416 (1994).
[CrossRef]

Other

P. A. Martin, Multiple Scattering: Interaction of Time-Harmonic Waves with N Obstacles (Cambridge Univ. Press, 2006).

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge Univ. Press, 1999).

C. L. Lawson and R. J. Hanson, Solving Least Squares Problems (SIAM, 1995).
[CrossRef]

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals: Molding the Flow of Light (Princeton Univ. Press, 1995).

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Figures (4)

Fig. 1
Fig. 1

Cross sections S 1 and S 2 of the first two unit cells Ω 1 and Ω 2 , for square and triangular lattices, when the centers of spheres are shifted according to Eq. (13). The subdomains of S 1 and S 2 are needed for imposing the continuity conditions at z = z 1 . Possible discretization points in S 1 are also shown.

Fig. 2
Fig. 2

Transmission spectra of a normal incident plane wave for four layers of dielectric spheres. The spheres in each layer are located on a square lattice. (a) The centers of the spheres are not shifted. (b) The centers of the spheres are shifted according to Eq. (13).

Fig. 3
Fig. 3

Transmission spectrum of a normal incident plane wave for a dielectric slab containing air spheres on a square lattice with lattice constant a. The thickness of the slab is also a.

Fig. 4
Fig. 4

Transmission spectra of a normal incident plane wave for four layers of dielectric spheres. The spheres in each layer are located on a triangular lattice with lattice constant a. (a) The centers of the spheres are not shifted and the thickness of the layers is a. (b) The centers of the spheres are shifted according to Eq. (13) and the thickness of the layers is 2 / 3 a .

Tables (1)

Tables Icon

Table 1 Reflected and Transmitted Powers of a Normal Incident Plane Wave for Four Layers of Dielectric Spheres and ω a / ( 2 π c ) = 0.85 , Calculated Using Different Truncation and Discretization Parameters a

Equations (71)

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

ε ( x + l 1 a 1 + l 2 a 2 , z ) = ε ( x , z ) ,
μ ( x + l 1 a 1 + l 2 a 2 , z ) = μ ( x , z ) ,
0 = z 0 < z 1 < < z Q = D .
( s q + l 1 a 1 + l 2 a 2 , z q 1 / 2 ) ,
× E = i k 0 μ H ̃ ,     × H ̃ = i k 0 ε E ,
γ 00 ( 1 ) = k 0 2 ε ( 1 ) μ ( 1 ) | α 00 | 2 > 0.
[ E ( i ) H ( i ) ] = [ n ( i ) t ( i ) ρ ( 1 ) t ( i ) ρ ( 1 ) n ( i ) ] [ cos   δ sin   δ ] exp [ i ( α 00 x γ 00 ( 1 ) z ) ] ,
n ( i ) = k ̂ ( i ) × z ̂ | k ̂ ( i ) × z ̂ | ,     t ( i ) = n ( i ) × k ̂ ( i ) .
a j b l = { 1 if   j = l 0 if   j l , }
α j k = α 00 + 2 π j b 1 + 2 π k b 2 ,
γ j k ( 1 ) = k 0 2 ε ( 1 ) μ ( 1 ) | α j k | 2 ,
γ j k ( 2 ) = k 0 2 ε ( 2 ) μ ( 2 ) | α j k | 2 .
[ E ( r ) H ( r ) ] = j k [ n j k ( r ) t j k ( r ) ρ ( 1 ) t j k ( r ) ρ ( 1 ) n j k ( r ) ] R j k   exp [ i ( α j k x γ j k ( 1 ) z ) ] ,
[ E ( t ) H ( t ) ] = j k [ n j k ( t ) t j k ( t ) ρ ( 2 ) t j k ( t ) ρ ( 2 ) n j k ( t ) ] T j k   exp [ i ( α j k x γ j k ( 2 ) z ) ] ,
k ̂ j k ( r ) = ( α j k , γ j k ( 1 ) ) k 0 ε ( 1 ) μ ( 1 ) ,     k ̂ j k ( t ) = ( α j k , γ j k ( 2 ) ) k 0 ε ( 2 ) μ ( 2 ) ,
n j k ( r ) = k ̂ j k ( r ) × z ̂ | k ̂ j k ( r ) × z ̂ | ,     n j k ( t ) = k ̂ j k ( t ) × z ̂ | k ̂ j k ( t ) × z ̂ | ,
t j k ( r ) = n j k ( r ) × k ̂ j k ( r ) ,     t j k ( t ) = n j k ( t ) × k ̂ j k ( t ) .
Ω q = { ( x , z ) : x S q , z q 1 < z < z q } .
S q = { x : | x s q | | x + l 1 a 1 + l 2 a 2 s q | } ,
| a 1 | | a 2 ± a 1 | ,     | a 2 | | a 2 ± a 1 | .
a j ( x s q ) = ± | a j | 2 2 ,     j = 1 , 2 , 3.
S q = { x : | a j ( x s q ) | < | a j | 2 2 ,     j = 1 , 2 , 3 } .
a 1 = ( a , 0 ) ,     a 2 = ( 0 , a ) ,
a 1 = ( a , 0 ) ,     a 2 = a 2 ( 1 , 3 ) ,     a 3 = a 2 ( 1 , 3 ) ,
Σ q , j ± = { ( x , z ) : a j ( x s q ) = ± | a j | 2 2 , z q 1 < z < z q } ,
E ( x + a j , z ) = ρ j E ( x , z ) ,     H ( x + a j , z ) = ρ j H ( x , z ) ,
a j × E ( x + a j , z ) = ρ j a j × E ( x , z ) ,
a j × H ( x + a j , z ) = ρ j a j × H ( x , z ) ,
z ̂ × E ( x , z q ) = z ̂ × E ( x , z q + ) ,     x S q ,
z ̂ × H ( x , z q ) = z ̂ × H ( x , z q + ) ,     x S q ,
S q + 1 ( l 1 , l 2 ) = S q + 1 { x + l 1 a 1 + l 2 a 2 x S q } = { x + l 1 a 1 + l 2 a 2 x S q ( l 1 , l 2 ) } .
E ( x + l 1 a 1 + l 2 a 2 , z ) = ρ 1 l 1 ρ 2 l 2 E ( x , z ) ,
ρ 1 l 1 ρ 2 l 2 z ̂ × E ( x , z q ) = z ̂ × E ( x + l 1 a 1 + l 2 a 2 , z q + ) ,
ρ 1 l 1 ρ 2 l 2 z ̂ × H ( x , z q ) = z ̂ × H ( x + l 1 a 1 + l 2 a 2 , z q + ) ,
s q = { ( 0 , 0 ) , if   q   is   odd ( a 1 + a 2 ) / 2 , if   q   is   even . }
A d = f ,
A = [ A 00 A 01 A ̂ 11 A 11 A 12 A ̂ 22 A ̂ Q Q A Q Q A Q , Q + 1 ] ,
d = [ d 0 d 1 d 2 d Q d Q + 1 ] ,     f = [ 0 0 0 f 2 Q ] .
min A d f .
[ A q q A q , q + 1 0 A ̂ q + 1 , q + 1 ] = U q [ B q q B q , q + 1 0 B ̂ q + 1 , q + 1 0 0 ] ,
min B d g ,
B = [ B 00 B 01 B ̂ 11 B 11 B 12 B ̂ 22 B ̂ Q Q B Q Q B Q , Q + 1 B ̂ Q + 1 , Q + 1 ] ,
g = [ 0 0 0 g 2 Q g 2 Q + 1 ] .
U Q f 2 Q = [ g 2 Q g 2 Q + 1 ] ,
[ B ̂ 11 0 B 11 B 12 ] = V 1 [ C 11 C 12 0 C ̂ 22 0 0 ] .
[ C ̂ q q 0 B ̂ q q 0 B q q B q , q + 1 ] = V q [ C q q C q , q + 1 0 C ̂ q + 1 , q + 1 0 0 ] .
V Q [ 0 0 g 2 Q ] = [ h Q h ̂ Q + 1 ] .
[ C ̂ Q + 1 , Q + 1 B ̂ Q + 1 , Q + 1 ] = V Q + 1 [ C Q + 1 , Q + 1 0 ] .
V Q + 1 [ h ̂ Q + 1 g 2 Q + 1 ] = [ h Q + 1 ] .
C [ d 0 d 1 d Q d Q + 1 ] = [ 0 0 h Q h Q + 1 ] ,
C = [ B 00 B 01 C 11 C 12 C Q Q C Q , Q + 1 C Q + 1 , Q + 1 ] .
C Q + 1 , Q + 1 d Q + 1 = h Q + 1 .
d 0 = Y q d q ,
Y 1 = B 00 1 B 01 ,
Y q + 1 = Y q C q q 1 C q , q + 1 ,     q = 0 , 1 , , Q 1.
ψ ( p ) ( r ) = { c ( p ) j l ( k 1 r ) , if   r < a a ( p ) j l ( k 2 r ) + b ( p ) h l ( 1 ) ( k 2 r ) , if   r > a , }
j l ( τ ) = π 2 τ J l + 1 / 2 ( τ ) ,     h l ( 1 ) ( τ ) = π 2 τ H l + 1 / 2 ( 1 ) ( τ ) .
Ψ ( p ) ( r , θ , ϕ ) = ψ ( p ) ( r ) Y l m ( θ , ϕ ) ,
Y l m ( θ , ϕ ) = [ ( 2 l + 1 ) ( l m ) ! 4 π ( l + m ) ! ] 1 / 2 P l m ( cos   θ ) e i m ϕ ,
E r ( 1 ) = l ( l + 1 ) i k 0 ε r Ψ ( 1 ) ,
E θ ( 1 ) = 1 i k 0 ε r 2 [ r Ψ ( 1 ) ] r θ ,
E ϕ ( 1 ) = 1 i k 0 ε r   sin   θ 2 [ r Ψ ( 1 ) ] r ϕ ,
H r ( 1 ) = 0 ,
H θ ( 1 ) = 1 sin   θ Ψ ( 1 ) ϕ ,
H ϕ ( 1 ) = Ψ ( 1 ) θ ,
E r ( 2 ) = 0 ,
E θ ( 2 ) = 1 sin   θ Ψ ( 2 ) ϕ ,
E ϕ ( 2 ) = Ψ ( 2 ) θ ,
H r ( 2 ) = l ( l + 1 ) i k 0 μ r Ψ ( 2 ) ,
H θ ( 2 ) = 1 i k 0 μ r 2 [ r Ψ ( 2 ) ] r θ ,
H ϕ ( 2 ) = 1 i k 0 μ r   sin   θ 2 [ r Ψ ( 2 ) ] r ϕ .

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