Abstract

We introduce a time-domain source-model technique for analysis of two-dimensional, transverse-magnetic, plane-wave scattering by a photonic crystal slab composed of a finite number of identical layers, each comprising a linear periodic array of dielectric cylinders. The proposed technique takes advantage of the periodicity of the slab by solving the problem within a unit cell of the periodic structure. A spectral analysis of the temporal behavior of the fields scattered by the slab shows a clear agreement between frequency bands where the spectral density of the transmitted energy is low and the bandgaps of the corresponding two-dimensionally infinite periodic structure. The effect of the bandwidth of the incident pulse and its center frequency on the manner it is transmitted through and reflected by the slab is studied via numerical examples.

© 2008 Optical Society of America

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References

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  1. E. Yablonovitch, T. J. Gmitter, and K. M. Leung, "Photonic band structure: The face-centered-cubic case," Phys. Rev. Lett. 63, 1950-1957 (1989).
    [CrossRef] [PubMed]
  2. J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).
  3. H. Y. D. Yang, "Finite-difference analysis of 2-D photonic crystals," IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1970).
    [CrossRef]
  4. B. Gralak, S. Enoch, and G. Tayeb, "Anomalous refractive properties of photonic crystals," J. Opt. Soc. Am. A 17, 1012-1020 (2000).
    [CrossRef]
  5. A. Ludwig and Y. Leviatan, "Analysis of arbitrary defects in photonic crystals by use of the source-model technique," J. Opt. Soc. Am. A 21, 1334-1343 (2004).
    [CrossRef]
  6. L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, "Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory," Phys. Rev. E 70, 056606.1-056606.13 (2004).
    [CrossRef]
  7. M. C. Lin and R. F. Jao, "Finite element analysis of photon density of states for two-dimensional photonic crystals with in-plane light propagation," Opt. Express 15, 207-218 (2007).
    [CrossRef] [PubMed]
  8. A. G. Tijhuis, "Toward a stable marching-on-in-time method for two-dimensional transient electromagnetic scattering problems," Radio Sci. 19, 1311-1317 (1984).
    [CrossRef]
  9. B. P. Rynne, "Instabilities in time marching methods for scattering problems," Electromagnetics 6, 129-144 (1986).
    [CrossRef]
  10. P. D. Smith, "Instabilities in time marching methods for scattering: Cause and rectification," Electromagnetics 10, 439-451 (1990).
    [CrossRef]
  11. B. P. Rynne and P. D. Smith, "Stability of time marching algorithms for the electric field integral equations," J. Electromagn. Waves Appl. 4, 1181-1205 (1990).
    [CrossRef]
  12. P. J. Davies, "Numerical stability and convergence of approximations of retarded potential integral equations," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 31, 856-875 (1994).
    [CrossRef]
  13. A. Mekis, S. Fan, and J. D. Joannopoulos, "Absorbing boundary conditions for FDTD simulations of photonic crystal waveguides," IEEE Microw. Guid. Wave Lett. 9, 502-504 (1999).
    [CrossRef]
  14. A. J. Ward and J. B. Pendry, "A program for calculating photonic band structures, Green's functions and transmission/reflection coefficients using a non-orthogonal FDTD method," Comput. Phys. Commun. 128, 590-621 (2000).
    [CrossRef]
  15. H. Mosallaei and Y. Rahmat-Samii, "Periodic bandgap and effective dielectric materials in electromagnetics: characterization and applications in nanocavities and waveguides," IEEE Trans. Antennas Propag. 51, 549-563 (2003).
    [CrossRef]
  16. A. Lavrinenko, P. Borel, L. Frandsen, M. Thorhauge, A. Harpøth, M. Kristensen, T. Niemi, and H. Chong, "Comprehensive FDTD modelling of photonic crystal waveguide components," Opt. Express 12, 234-248 (2004).
    [CrossRef] [PubMed]
  17. N.-W. Chen, B. Shanker, and E. Michielssen, "Integral-equation-based analysis of transient scattering from periodic perfectly conducting structures," IEE Proc. Microwaves, Antennas Propag. 150, 120-124 (2003).
    [CrossRef]
  18. A. Ludwig and Y. Leviatan, "A source-model technique for the analysis of transient electromagnetic scattering by a periodic array of cylinders," IEEE Trans. Antennas Propag. 55, 2578-2590 (2007).
    [CrossRef]
  19. B. Shanker, A. A. Ergin, M. Lu, and E. Michielssen, "Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm," IEEE Trans. Antennas Propag. 51, 628-641 (2003).
    [CrossRef]
  20. N.-W. Chen, M. Lu, F. Capolino, B. Shanker, and E. Michielssen, "Floquet wave-based analysis of transient scattering from doubly periodic, discretely planar, perfectly conducting structures," Radio Sci. 40, RS4007.1-RS4007.21 (2005).
    [CrossRef]
  21. A. Boag, Y. Leviatan, and A. Boag, "Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model," Radio Sci. 23, 612-624 (1988).
    [CrossRef]
  22. Y. Leviatan, A. Boag, and A. Boag, "Generalized fomulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies--theory and numerical solutions," IEEE Trans. Antennas Propag. 36, 1722-1734 (1988).
    [CrossRef]
  23. F. Zolla, R. Petit, and M. Cadilhac, "Electromagnetic theory of diffraction by a system of parallel rods: the method of fictitious sources," J. Opt. Soc. Am. A 11, 1087-1096 (1994).
    [CrossRef]
  24. D. Maystre, M. Saillard, and G. Tayeb, "Special methods of wave diffraction," in Scattering: Scattering and Inverse Scattering in Pure and Applied Science, E.R.Pike and P.C.Sabatier, eds. (Academic, 2001), Chap. 1.5.6.
  25. D. I. Kaklamani and H. T. Anastassiu, "Aspects of the method of auxiliary sources (MAS) in computational electromagnetics," IEEE Antennas Propag. Mag. 44, 48-64 (2002).
    [CrossRef]
  26. F. Shubitidze, H. T. Anastassiu, and D. I. Kaklamani, "An improved accuracy version of the method of auxiliary sources for computational electromagnetics," IEEE Trans. Antennas Propag. 52, 302-309 (2004).
    [CrossRef]
  27. A. Ludwig and Y. Leviatan, "Towards a stable two-dimensional time-domain source-model solution by use of a combined source formulation," IEEE Trans. Antennas Propag. 54, 3010-3021 (2006).
    [CrossRef]
  28. A. Ludwig and Y. Leviatan, "Analysis of bandgap characteristics of two-dimensional periodic structures by using the source-model technique," J. Opt. Soc. Am. A 20, 1553-1562 (2003).
    [CrossRef]
  29. S. Eisler and Y. Leviatan, "Analysis of electromagnetic scattering from metallic and penetrable cylinders with edges using a multifilament current model," IEE Proc., Part H: Microwaves, Antennas Propag. 136, 431-438 (1989).
    [CrossRef]

2007 (2)

A. Ludwig and Y. Leviatan, "A source-model technique for the analysis of transient electromagnetic scattering by a periodic array of cylinders," IEEE Trans. Antennas Propag. 55, 2578-2590 (2007).
[CrossRef]

M. C. Lin and R. F. Jao, "Finite element analysis of photon density of states for two-dimensional photonic crystals with in-plane light propagation," Opt. Express 15, 207-218 (2007).
[CrossRef] [PubMed]

2006 (1)

A. Ludwig and Y. Leviatan, "Towards a stable two-dimensional time-domain source-model solution by use of a combined source formulation," IEEE Trans. Antennas Propag. 54, 3010-3021 (2006).
[CrossRef]

2005 (1)

N.-W. Chen, M. Lu, F. Capolino, B. Shanker, and E. Michielssen, "Floquet wave-based analysis of transient scattering from doubly periodic, discretely planar, perfectly conducting structures," Radio Sci. 40, RS4007.1-RS4007.21 (2005).
[CrossRef]

2004 (4)

F. Shubitidze, H. T. Anastassiu, and D. I. Kaklamani, "An improved accuracy version of the method of auxiliary sources for computational electromagnetics," IEEE Trans. Antennas Propag. 52, 302-309 (2004).
[CrossRef]

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, "Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory," Phys. Rev. E 70, 056606.1-056606.13 (2004).
[CrossRef]

A. Lavrinenko, P. Borel, L. Frandsen, M. Thorhauge, A. Harpøth, M. Kristensen, T. Niemi, and H. Chong, "Comprehensive FDTD modelling of photonic crystal waveguide components," Opt. Express 12, 234-248 (2004).
[CrossRef] [PubMed]

A. Ludwig and Y. Leviatan, "Analysis of arbitrary defects in photonic crystals by use of the source-model technique," J. Opt. Soc. Am. A 21, 1334-1343 (2004).
[CrossRef]

2003 (4)

A. Ludwig and Y. Leviatan, "Analysis of bandgap characteristics of two-dimensional periodic structures by using the source-model technique," J. Opt. Soc. Am. A 20, 1553-1562 (2003).
[CrossRef]

B. Shanker, A. A. Ergin, M. Lu, and E. Michielssen, "Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm," IEEE Trans. Antennas Propag. 51, 628-641 (2003).
[CrossRef]

N.-W. Chen, B. Shanker, and E. Michielssen, "Integral-equation-based analysis of transient scattering from periodic perfectly conducting structures," IEE Proc. Microwaves, Antennas Propag. 150, 120-124 (2003).
[CrossRef]

H. Mosallaei and Y. Rahmat-Samii, "Periodic bandgap and effective dielectric materials in electromagnetics: characterization and applications in nanocavities and waveguides," IEEE Trans. Antennas Propag. 51, 549-563 (2003).
[CrossRef]

2002 (1)

D. I. Kaklamani and H. T. Anastassiu, "Aspects of the method of auxiliary sources (MAS) in computational electromagnetics," IEEE Antennas Propag. Mag. 44, 48-64 (2002).
[CrossRef]

2000 (2)

A. J. Ward and J. B. Pendry, "A program for calculating photonic band structures, Green's functions and transmission/reflection coefficients using a non-orthogonal FDTD method," Comput. Phys. Commun. 128, 590-621 (2000).
[CrossRef]

B. Gralak, S. Enoch, and G. Tayeb, "Anomalous refractive properties of photonic crystals," J. Opt. Soc. Am. A 17, 1012-1020 (2000).
[CrossRef]

1999 (1)

A. Mekis, S. Fan, and J. D. Joannopoulos, "Absorbing boundary conditions for FDTD simulations of photonic crystal waveguides," IEEE Microw. Guid. Wave Lett. 9, 502-504 (1999).
[CrossRef]

1994 (2)

P. J. Davies, "Numerical stability and convergence of approximations of retarded potential integral equations," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 31, 856-875 (1994).
[CrossRef]

F. Zolla, R. Petit, and M. Cadilhac, "Electromagnetic theory of diffraction by a system of parallel rods: the method of fictitious sources," J. Opt. Soc. Am. A 11, 1087-1096 (1994).
[CrossRef]

1990 (2)

P. D. Smith, "Instabilities in time marching methods for scattering: Cause and rectification," Electromagnetics 10, 439-451 (1990).
[CrossRef]

B. P. Rynne and P. D. Smith, "Stability of time marching algorithms for the electric field integral equations," J. Electromagn. Waves Appl. 4, 1181-1205 (1990).
[CrossRef]

1989 (2)

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, "Photonic band structure: The face-centered-cubic case," Phys. Rev. Lett. 63, 1950-1957 (1989).
[CrossRef] [PubMed]

S. Eisler and Y. Leviatan, "Analysis of electromagnetic scattering from metallic and penetrable cylinders with edges using a multifilament current model," IEE Proc., Part H: Microwaves, Antennas Propag. 136, 431-438 (1989).
[CrossRef]

1988 (2)

A. Boag, Y. Leviatan, and A. Boag, "Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model," Radio Sci. 23, 612-624 (1988).
[CrossRef]

Y. Leviatan, A. Boag, and A. Boag, "Generalized fomulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies--theory and numerical solutions," IEEE Trans. Antennas Propag. 36, 1722-1734 (1988).
[CrossRef]

1986 (1)

B. P. Rynne, "Instabilities in time marching methods for scattering problems," Electromagnetics 6, 129-144 (1986).
[CrossRef]

1984 (1)

A. G. Tijhuis, "Toward a stable marching-on-in-time method for two-dimensional transient electromagnetic scattering problems," Radio Sci. 19, 1311-1317 (1984).
[CrossRef]

1970 (1)

H. Y. D. Yang, "Finite-difference analysis of 2-D photonic crystals," IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1970).
[CrossRef]

Comput. Phys. Commun. (1)

A. J. Ward and J. B. Pendry, "A program for calculating photonic band structures, Green's functions and transmission/reflection coefficients using a non-orthogonal FDTD method," Comput. Phys. Commun. 128, 590-621 (2000).
[CrossRef]

Electromagnetics (2)

B. P. Rynne, "Instabilities in time marching methods for scattering problems," Electromagnetics 6, 129-144 (1986).
[CrossRef]

P. D. Smith, "Instabilities in time marching methods for scattering: Cause and rectification," Electromagnetics 10, 439-451 (1990).
[CrossRef]

IEE Proc. Microwaves, Antennas Propag. (1)

N.-W. Chen, B. Shanker, and E. Michielssen, "Integral-equation-based analysis of transient scattering from periodic perfectly conducting structures," IEE Proc. Microwaves, Antennas Propag. 150, 120-124 (2003).
[CrossRef]

IEE Proc., Part H: Microwaves, Antennas Propag. (1)

S. Eisler and Y. Leviatan, "Analysis of electromagnetic scattering from metallic and penetrable cylinders with edges using a multifilament current model," IEE Proc., Part H: Microwaves, Antennas Propag. 136, 431-438 (1989).
[CrossRef]

IEEE Antennas Propag. Mag. (1)

D. I. Kaklamani and H. T. Anastassiu, "Aspects of the method of auxiliary sources (MAS) in computational electromagnetics," IEEE Antennas Propag. Mag. 44, 48-64 (2002).
[CrossRef]

IEEE Microw. Guid. Wave Lett. (1)

A. Mekis, S. Fan, and J. D. Joannopoulos, "Absorbing boundary conditions for FDTD simulations of photonic crystal waveguides," IEEE Microw. Guid. Wave Lett. 9, 502-504 (1999).
[CrossRef]

IEEE Trans. Antennas Propag. (6)

Y. Leviatan, A. Boag, and A. Boag, "Generalized fomulations for electromagnetic scattering from perfectly conducting and homogeneous material bodies--theory and numerical solutions," IEEE Trans. Antennas Propag. 36, 1722-1734 (1988).
[CrossRef]

A. Ludwig and Y. Leviatan, "A source-model technique for the analysis of transient electromagnetic scattering by a periodic array of cylinders," IEEE Trans. Antennas Propag. 55, 2578-2590 (2007).
[CrossRef]

B. Shanker, A. A. Ergin, M. Lu, and E. Michielssen, "Fast analysis of transient electromagnetic scattering phenomena using the multilevel plane wave time domain algorithm," IEEE Trans. Antennas Propag. 51, 628-641 (2003).
[CrossRef]

H. Mosallaei and Y. Rahmat-Samii, "Periodic bandgap and effective dielectric materials in electromagnetics: characterization and applications in nanocavities and waveguides," IEEE Trans. Antennas Propag. 51, 549-563 (2003).
[CrossRef]

F. Shubitidze, H. T. Anastassiu, and D. I. Kaklamani, "An improved accuracy version of the method of auxiliary sources for computational electromagnetics," IEEE Trans. Antennas Propag. 52, 302-309 (2004).
[CrossRef]

A. Ludwig and Y. Leviatan, "Towards a stable two-dimensional time-domain source-model solution by use of a combined source formulation," IEEE Trans. Antennas Propag. 54, 3010-3021 (2006).
[CrossRef]

IEEE Trans. Microwave Theory Tech. (1)

H. Y. D. Yang, "Finite-difference analysis of 2-D photonic crystals," IEEE Trans. Microwave Theory Tech. 44, 2688-2695 (1970).
[CrossRef]

J. Electromagn. Waves Appl. (1)

B. P. Rynne and P. D. Smith, "Stability of time marching algorithms for the electric field integral equations," J. Electromagn. Waves Appl. 4, 1181-1205 (1990).
[CrossRef]

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

Opt. Express (2)

Phys. Rev. E (1)

L. C. Botten, T. P. White, A. A. Asatryan, T. N. Langtry, C. M. de Sterke, and R. C. McPhedran, "Bloch mode scattering matrix methods for modeling extended photonic crystal structures. I. Theory," Phys. Rev. E 70, 056606.1-056606.13 (2004).
[CrossRef]

Phys. Rev. Lett. (1)

E. Yablonovitch, T. J. Gmitter, and K. M. Leung, "Photonic band structure: The face-centered-cubic case," Phys. Rev. Lett. 63, 1950-1957 (1989).
[CrossRef] [PubMed]

Radio Sci. (3)

N.-W. Chen, M. Lu, F. Capolino, B. Shanker, and E. Michielssen, "Floquet wave-based analysis of transient scattering from doubly periodic, discretely planar, perfectly conducting structures," Radio Sci. 40, RS4007.1-RS4007.21 (2005).
[CrossRef]

A. Boag, Y. Leviatan, and A. Boag, "Analysis of two-dimensional electromagnetic scattering from a periodic grating of cylinders using a hybrid current model," Radio Sci. 23, 612-624 (1988).
[CrossRef]

A. G. Tijhuis, "Toward a stable marching-on-in-time method for two-dimensional transient electromagnetic scattering problems," Radio Sci. 19, 1311-1317 (1984).
[CrossRef]

SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. (1)

P. J. Davies, "Numerical stability and convergence of approximations of retarded potential integral equations," SIAM (Soc. Ind. Appl. Math.) J. Numer. Anal. 31, 856-875 (1994).
[CrossRef]

Other (2)

J. D. Joannopoulos, R. D. Meade, and J. N. Winn, Photonic Crystals (Princeton U. Press, 1995).

D. Maystre, M. Saillard, and G. Tayeb, "Special methods of wave diffraction," in Scattering: Scattering and Inverse Scattering in Pure and Applied Science, E.R.Pike and P.C.Sabatier, eds. (Academic, 2001), Chap. 1.5.6.

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Figures (10)

Fig. 1
Fig. 1

General two-dimensional problem of TM plane-wave pulse scattering by a photonic crystal slab.

Fig. 2
Fig. 2

Simulated equivalence for the exterior region in the simplified case of a one-layer slab.

Fig. 3
Fig. 3

Simulated equivalence for the unit-cell cylinder in the simplified case of a one-layer slab.

Fig. 4
Fig. 4

Photonic-crystal slab configuration investigated.

Fig. 5
Fig. 5

Band-structure diagram for the photonic-crystal structure composing the slab.

Fig. 6
Fig. 6

Normalized transmitted energy spectrum density versus normalized frequency for different number of layers when a δ BW = 1 2 incident pulse is impinging on the slab at normal angle. The spectral shape and temporal behavior of the incident pulse are shown below the upper panel.

Fig. 7
Fig. 7

Normalized transmitted energy spectrum density versus normalized frequency for different number of layers when a δ BW = 0.9 incident pulse is impinging on the slab at normal angle. The spectral shape and temporal behavior of the incident pulse are shown below the upper panel.

Fig. 8
Fig. 8

Transmitted energy spectrum density versus normalized frequency for different incidence angles. The upper subplot on the right is a duplication of the band-structure diagram of the infinite photonic-crystal shown in Fig. 5. The subplot below the band-structure diagram shows its projection on the k y axis. A rotated version of this projection is also shown below the main panel to facilitate comparison between the regions of zero transmission and the band-gap properties of the photonic crystal.

Fig. 9
Fig. 9

Normalized transmitted energy versus number of slab layers for incident pulses of different bandwidths. The dots correspond to the case where the incident Gaussian pulse is infinitely narrow. This result is obtained by use of a frequency-domain solver for a single frequency excitation ω ¯ = ω d 2 π c I = 0.38 . The subplots on the right depict the temporal/spatial shape of the incident pulse scaled in a manner that allows comparison with the photonic crystal slab structure.

Fig. 10
Fig. 10

Transient scattered field versus the shifted time parameter t ̃ for three δ BW = 1 32 incident pulses of different center frequencies impinging on a seven-layer slab at normal incidence. The upper right insert is a magnification of the pulse peaks, and the insert below it shows the normalized energy spectrum of the incident pulses. The three inserts on the left side of the figure show the original scattered pulses including the modulation around the center frequency.

Tables (2)

Tables Icon

Table 1 Numerical Parameters Used in Fig. 8 for the Different Values of ϕ inc Considered

Tables Icon

Table 2 Numerical Parameters Used in Fig. 9 for the Different Values of δ BW Considered

Equations (82)

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

E inc ( ρ , t ) = z ̂ E 0 exp [ ( t k ̂ inc ρ c I ) 2 8 π 2 σ 2 ] cos [ ω 0 ( t k ̂ inc ρ c I ) ] ,
H inc ( ρ , t ) = 1 η I k ̂ inc × E inc ( ρ , t )
η II 2 π C ext t P c II 1 c II 2 ( t t ) 2 P 2 J s ext , z ( ρ , t ) t d t d l = η I 2 π C int t P c I 1 c I 2 ( t t ) 2 P 2 J s int , z ( ρ , t ) t d t d l + E z inc ( ρ , t ) ρ C ,
1 2 π C ext t P c II c II ( t t ) n ̂ ( ρ ) × z ̂ × P ̂ P c II 2 ( t t ) 2 P 2 J s ext , z ( ρ , t ) t d t d l = 1 2 π C int t P c I c I ( t t ) n ̂ ( ρ ) × z ̂ × P ̂ P c I 2 ( t t ) 2 P 2 J s int , z ( ρ , t ) t d t d l + n ̂ ( ρ ) × H inc ( ρ , t ) ρ C ,
q Z : η II 2 π C q , ext t P c II 1 c II 2 ( t t ) 2 P 2 J s ext , z ( ρ , t ) t d t d l = η I 2 π q = { C q , int t P c I 1 c I 2 ( t t ) 2 P 2 J s int , z ( ρ , t ) t d t d l } + E z inc ( ρ , t ) ρ C q ,
q Z : 1 2 π C q , ext t P c II c II ( t t ) n ̂ ( ρ ) × z ̂ × P ̂ P c II 2 ( t t ) 2 P 2 J s ext , z ( ρ , t ) t d t d l = 1 2 π q = { C q , int t P c I c I ( t t ) n ̂ ( ρ ) × z ̂ × P ̂ P c I 2 ( t t ) 2 P 2 J s int , z ( ρ , t ) t d t d l } + n ̂ ( ρ ) × H inc ( ρ , t ) ρ C q .
{ ρ = ρ 0 + q d y y ̂ ρ = ρ 0 + q d y y ̂ } ,
{ J s int , z ( ρ 0 + q d y y ̂ , t ) = J s int , z ( ρ 0 , t Δ T q d ) J s ext , z ( ρ 0 + q d y y ̂ , t ) = J s ext , z ( ρ 0 , t Δ T q d ) } ,
Δ T q d = q k ̂ inc d y y ̂ c I .
η II 2 π C 0 , ext t P 0 c II 1 c II 2 ( t t ) 2 P 0 2 J s ext , z ( ρ 0 , t ) t d t d l = η I 2 π q = { C 0 , int t P q c I 1 c I 2 ( t t ) 2 P q 2 J s int , z ( ρ 0 , t Δ T q d ) t d t d l } + E z inc ( ρ 0 , t ) ρ 0 C 0 ,
1 2 π C 0 , ext t P 0 c II c II ( t t ) n ̂ ( ρ 0 ) × z ̂ × P ̂ 0 P 0 c II 2 ( t t ) 2 P 0 2 J s ext , z ( ρ 0 , t ) t d t d l = 1 2 π q = { C 0 , int t P q c I c I ( t t ) n ̂ ( ρ 0 ) × z ̂ × P ̂ q P q c I 2 ( t t ) 2 P q 2 J s int , z ( ρ 0 , t Δ T q d ) t d t d l } + n ̂ ( ρ 0 ) × H inc ( ρ 0 , t ) ρ 0 C 0 ,
J s Γ , z ( ρ 0 , t ) t = n k I Γ , n ( k ) δ s ( ρ 0 ρ 0 , Γ , n ) T Γ , n ( k ) ( t ) Γ { int , ext } ,
T Γ , n ( k ) ( t ) = { 1 k Δ t < t + Δ T Γ , n s < ( k + 1 ) Δ t 0 otherwise } ,
Γ { int , ext } .
Δ T Γ , n s = { P 0 , m ̃ int , n , int , n c I , Γ = int P 0 , m ̃ ext , n , ext , n c II , Γ = ext }
k = i i + i F [ Z E , int ( i k ) Z E , ext ( i k ) Z H , int ( i k ) Z H , ext ( i k ) ] [ I int ( k ) I ext ( k ) ] = [ V E ( i ) V H ( i ) ] k = 0 i 1 [ Z E , int ( i k ) Z E , ext ( i k ) Z H , int ( i k ) Z H , ext ( i k ) ] [ I int ( k ) I ext ( k ) ] ,
[ Z { E H } , Γ ( j ) ] m , n = q = [ Z ̃ { E H } , Γ ( j ) ] q , m , n , Γ = int ,
{ [ Z E , Γ ( j ) ] m , n = c I η II c II η I [ Z ̃ E , Γ ( j ) ] 0 , m , n [ Z H , Γ ( j ) ] m , n = c I c II [ Z ̃ H , Γ ( j ) ] 0 , m , n } , Γ = ext ,
[ Z ̃ { E H } , Γ ( j ) ] q , m , n = { W q , m , Γ , n { E H } ( j ) W q , m , Γ , n { E H } ( j 1 ) , j > κ q , m , Γ , n W q , m , Γ , n { E H } ( j ) , j = κ q , m , Γ , n 0 , otherwise } Γ { int , ext } , j = 0 , 1 , 2 , ,
I Γ ( j ) = [ I Γ , 1 ( j ) I Γ , 2 ( j ) I Γ , N Γ ( j ) ] Γ { int , ext } , j = 0 , 1 , 2 , ,
V E ( j ) = 2 π c I η I [ E z inc ( ρ 0 , 1 , ( j + 1 ) Δ t ) E z inc ( ρ 0 , 2 , ( j + 1 ) Δ t ) E z inc ( ρ 0 , M , ( j + 1 ) Δ t ) ] ,
V H ( j ) = 2 π c I [ z ̂ ( n ̂ ( ρ 0 , 1 ) × H inc ( ρ 0 , 1 , ( j + 1 ) Δ t ) ) z ̂ ( n ̂ ( ρ 0 , 2 ) × H inc ( ρ 0 , 2 , ( j + 1 ) Δ t ) ) z ̂ ( n ̂ ( ρ 0 , M ) × H inc ( ρ 0 , M , ( j + 1 ) Δ t ) ) ]
j = 0 , 1 , 2 , ,
i F = min q , Γ , m , n ( κ q , m , Γ , n ) Γ { int , ext } .
W q , m , Γ , n E ( x ) = acosh ( Δ t Δ T q , m , Γ , n t [ x + ( Δ T Γ , n s Δ T q d ) Δ t + 1 ] )
Γ { int , ext } ,
W q , m , Γ , n H ( x ) = z ̂ [ n ̂ ( ρ 0 , m ) × ( z ̂ × P ̂ q , m , Γ , n ) ] ( Δ t Δ T q , m , Γ , n t [ x + ( Δ T Γ , n s Δ T q d ) Δ t + 1 ] ) 2 1 , Γ { int , ext } ,
κ q , m , Γ , n = [ Δ T q , m , Γ , n t ( Δ T Γ , n s Δ T q d ) ] Δ t
Γ { int , ext } ,
Δ T q , m , Γ , n t = { P q , m , int , n c I Γ = int P q , m , ext , n c II Γ = ext } .
T Γ , n ( k ) ( t ) = { 1 k Δ t < t + Δ T Γ , n s t Γ , n r < ( k + 1 ) Δ t 0 otherwise } ,
Γ { int , ext } .
{ t Γ , n r = ( k ̂ inc y ̂ ) ρ 0 , m ̃ Γ , n y ̂ c I , Γ { int , ext } t m r = ( k ̂ inc y ̂ ) ρ 0 , m y ̂ c I } .
W q , m , Γ , n E ( x ) = acosh ( Δ t Δ T q , m , Γ , n t [ x + ( Δ T Γ , n s Δ T q d Δ T m , Γ , n r ) Δ t + 1 ] ) Γ { int , ext } ,
W q , m , Γ , n H ( x ) = z ̂ [ n ̂ ( ρ 0 , m ) × ( z ̂ × P ̂ q , m , Γ , n ) ] ( Δ t Δ T q , m , Γ , n t [ x + ( Δ T Γ , n s Δ T q d Δ T m , Γ , n r ) Δ t + 1 ] ) 2 1
Γ { int , ext } ,
κ q , m , Γ , n = [ Δ T q , m , Γ , n t ( Δ T Γ , n s Δ T q d Δ T m , Γ , n r ) ] Δ t ,
[ Z ͌ E , int ( 0 ) Z ͌ E , ext ( 0 ) Z ͌ H , int ( 0 ) Z ͌ H , ext ( 0 ) ] [ I int ( i ) I ext ( i ) ] = [ V E ( i ) V H ( i ) ] k = 0 i 1 [ Z ͌ E , int ( i k ) Z ͌ E , ext ( i k ) Z ͌ H , int ( i k ) Z ͌ H , ext ( i k ) ] [ I int ( k ) I ext ( k ) ] .
[ Z ͌ { E H } , Γ ( j ) ] = { k = 1 i F ( u = 1 N samp h ( N samp + k , u ) ( [ h ] 1 ) u , j + 1 ) [ Z { E H } , Γ ( k ) ] + [ Z { E H } , Γ ( j ) ] j N samp [ Z { E H } , Γ ( j ) ] otherwise } , Γ { int , ext } .
h ( k , u ) = sin [ ω max ( k u ) Δ t ] sin [ ω min ( k u ) Δ t ] π ( k u ) ,
F ( E z inc ) ω = ω min F ( E z inc ) ω = ω max δ 0 F ( E z inc ) ω = ω 0 ,
J s Γ , z ( ρ 0 , t ) t = n k I Γ , n ( k ) δ s ( ρ 0 ρ 0 , Γ , n ) T Γ , n ( k ) ( t )
Γ { int ( L ) , ext ( L ) } , L = 1 , 2 , , N .
T Γ , n ( k ) ( t ) = { 1 k Δ t < t + Δ T Γ , n s t Γ , n r < ( k + 1 ) Δ t 0 otherwise }
Γ { int ( L ) , ext ( L ) } , L = 1 , 2 , , N .
Δ T Γ , n s = { P 0 , L , m ̃ int ( L ) , n , int ( L ) , n c I Γ = int ( L ) P 0 , L , m ̃ ext ( L ) , n , ext ( L ) , n c II Γ = ext ( L ) } ,
L = 1 , 2 , , N ,
t Γ , n r = ( k ̂ inc y ̂ ) ρ 0 , L , m ̃ Γ , n y ̂ c I
Γ { int ( L ) , ext ( L ) } , L = 1 , 2 , , N ,
[ Z ͌ E ( 1 ) , int ( 1 ) ( 0 ) Z ͌ E ( 1 ) , int ( 2 ) ( 0 ) Z ͌ E ( 1 ) , int ( N ) ( 0 ) Z ͌ E ( 1 ) , ext ( 1 ) ( 0 ) Z ͌ E ( 2 ) , int ( 1 ) ( 0 ) Z ͌ E ( 2 ) , int ( 2 ) ( 0 ) Z ͌ E ( 2 ) , int ( N ) ( 0 ) Z ͌ E ( 2 ) , ext ( 2 ) ( 0 ) 0 0 Z ͌ E ( N ) , int ( 1 ) ( 0 ) Z ͌ E ( N ) , int ( 2 ) ( 0 ) Z ͌ E ( N ) , int ( N ) ( 0 ) Z ͌ E ( N ) , ext ( N ) ( 0 ) Z ͌ H ( 1 ) , int ( 1 ) ( 0 ) Z ͌ H ( 1 ) , int ( 2 ) ( 0 ) Z ͌ H ( 1 ) , int ( N ) ( 0 ) Z ͌ H ( 1 ) , ext ( 1 ) ( 0 ) Z ͌ H ( 2 ) , int ( 1 ) ( 0 ) Z ͌ H ( 2 ) , int ( 2 ) ( 0 ) Z ͌ H ( 2 ) , int ( N ) ( 0 ) Z ͌ H ( 2 ) , ext ( 2 ) ( 0 ) 0 0 Z ͌ H ( N ) , int ( 1 ) ( 0 ) Z ͌ H ( N ) , int ( 2 ) ( 0 ) Z ͌ H ( N ) , int ( N ) ( 0 ) Z ͌ H ( N ) , ext ( N ) ( 0 ) ] [ I int ( 1 ) ( i ) I int ( 2 ) ( i ) I int ( N ) ( i ) I ext ( 1 ) ( i ) I ext ( 2 ) ( i ) I ext ( N ) ( i ) ] = [ V E ( 1 ) ( i ) V E ( 2 ) ( i ) V E ( N ) ( i ) V H ( 1 ) ( i ) V H ( 2 ) ( i ) V H ( N ) ( i ) ] k = 0 i 1 [ Z ͌ E ( 1 ) , int ( 1 ) ( i k ) Z ͌ E ( 1 ) , int ( 2 ) ( i k ) Z ͌ E ( 1 ) , int ( N ) ( i k ) Z ͌ E ( 1 ) , ext ( 1 ) ( i k ) Z ͌ E ( 2 ) , int ( 1 ) ( i k ) Z ͌ E ( 2 ) , int ( 2 ) ( i k ) Z ͌ E ( 2 ) , int ( N ) ( i k ) Z ͌ E ( 2 ) , ext ( 2 ) ( i k ) 0 0 Z ͌ E ( N ) , int ( 1 ) ( i k ) Z ͌ E ( N ) , int ( 2 ) ( i k ) Z ͌ E ( N ) , int ( N ) ( i k ) Z ͌ E ( N ) , ext ( N ) ( i k ) Z ͌ H ( 1 ) , int ( 1 ) ( i k ) Z ͌ H ( 1 ) , int ( 2 ) ( i k ) Z ͌ H ( 1 ) , int ( N ) ( i k ) Z ͌ H ( 1 ) , ext ( 1 ) ( i k ) Z ͌ H ( 2 ) , int ( 1 ) ( i k ) Z ͌ H ( 2 ) , int ( 2 ) ( i k ) Z ͌ H ( 2 ) , int ( N ) ( i k ) Z ͌ H ( 2 ) , ext ( 2 ) ( i k ) 0 0 Z ͌ H ( N ) , int ( 1 ) ( i k ) Z ͌ H ( N ) , int ( 2 ) ( i k ) Z ͌ H ( N ) , int ( N ) ( i k ) Z ͌ H ( N ) , ext ( N ) ( i k ) ] [ I int ( 1 ) ( k ) I int ( 2 ) ( k ) I int ( N ) ( k ) I ext ( 1 ) ( k ) I ext ( 2 ) ( k ) I ext ( N ) ( k ) ] .
[ Z ͌ { E ( L ) H ( L ) } , Γ ( j ) ] = { k = 1 i F ( u = 1 N samp h ( N samp + k , u ) ( [ h ] 1 ) u , j + 1 ) [ Z { E ( L ) H ( L ) } , Γ ( k ) ] + [ Z { E ( L ) H ( L ) } , Γ ( j ) ] j N samp [ Z { E ( L ) H ( L ) } , Γ ( j ) ] otherwise } ,
Γ { int ( L ) , ext ( L ) } , L , L { 1 , 2 , , N } ;
[ Z { E ( L ) H ( L ) } , Γ ( j ) ] m , n = q = [ Z ̃ { E ( L ) H ( L ) } , Γ ( j ) ] q , m , n Γ = int ( L ) , L , L { 1 , 2 , , N } ;
{ [ Z E ( L ) , Γ ( j ) ] m , n = c I η II c II η I [ Z ̃ E ( L ) , Γ ( j ) ] 0 , m , n [ Z H ( L ) , Γ ( j ) ] m , n = c I c II [ Z ̃ H ( L ) , Γ ( j ) ] 0 , m , n } Γ = ext ( L ) , L , L { 1 , 2 , , N } ;
[ Z ̃ { E ( L ) H ( L ) } , Γ ( j ) ] q , m , n = { W q , m , Γ , n { E ( L ) H ( L ) } ( j ) W q , m , Γ , n { E ( L ) H ( L ) } ( j 1 ) j > κ q , m , Γ , n W q , m , Γ , n { E ( L ) H ( L ) } ( j ) j = κ q , m , Γ , n 0 otherwise } , Γ { int ( L ) , ext ( L ) } , L , L { 1 , 2 , , N } , j = 0 , 1 , 2 , ;
I Γ ( j ) = [ I Γ , 1 ( j ) I Γ , 2 ( j ) I Γ , N Γ ( j ) ] Γ { int ( L ) , ext ( L ) } , L = 1 , 2 , , N , j = 0 , 1 , 2 , ;
V E ( L ) ( j ) = 2 π c I η I [ E z inc ( ρ 0 , L , 1 , ( j + 1 ) Δ t ) E z inc ( ρ 0 , L , 2 , ( j + 1 ) Δ t ) E z inc ( ρ 0 , L , M , ( j + 1 ) Δ t ) ] , V H ( L ) ( j ) = 2 π c I [ z ̂ ( n ̂ ( ρ 0 , L , 1 ) × H inc ( ρ 0 , L , 1 , ( j + 1 ) Δ t ) ) z ̂ ( n ̂ ( ρ 0 , L , 2 ) × H inc ( ρ 0 , L , 2 , ( j + 1 ) Δ t ) ) z ̂ ( n ̂ ( ρ 0 , L , M ) × H inc ( ρ 0 , L , M , ( j + 1 ) Δ t ) ) ] L = 1 , 2 , , N , j = 0 , 1 , 2 , ;
i F = min q , L , m , Γ , n ( κ q , L , m , Γ , n ) Γ { int ( L ) , ext ( L ) } , L , L { 1 , 2 , , N } .
W q , m , Γ , n E ( L ) ( x ) = acosh ( Δ t Δ T q , L , m , Γ , n t [ x + ( Δ T Γ , n s Δ T q d Δ T L , m , Γ , n r ) Δ t + 1 ] ) Γ { int ( L ) , ext ( L ) } L , L { 1 , 2 , , N } ;
W q , m , Γ , n H ( L ) ( x ) = z ̂ [ n ̂ ( ρ 0 , L , m ) × ( z ̂ × P ̂ q , L , m , Γ , n ) ] ( Δ t Δ T q , L , m , Γ , n t [ x + ( Δ T Γ , n s Δ T q d Δ T L , m , Γ , n r ) Δ t + 1 ] ) 2 1
Γ { int ( L ) , ext ( L ) } , L , L { 1 , 2 , , N } ;
κ q , L , m , Γ , n = [ Δ T q , L , m , Γ , n t ( Δ T Γ , n s Δ T q d Δ T L , m , Γ , n r ) ] Δ t ;
P ̂ q , L , m , Γ , n = P q , L , m , Γ , n P q , L , m , Γ , n Γ { int ( L ) , ext ( L ) } , L = 1 , 2 , , N ;
P q , L , m , Γ , n = [ ρ 0 , L , m ( ρ 0 , Γ , n + q d y y ̂ ) ] , Γ { int ( L ) , ext ( L ) } , L = 1 , 2 , , N .
Δ T L , m , Γ , n r = t Γ , n r t L , m r Γ { int ( L ) , ext ( L ) } , L = 1 , 2 , , N ;
Δ T q , L , m , Γ , n t = { P q , L , m , int ( L ) , n c I Γ = int ( L ) P q , L , m , ext ( L ) , n c II Γ = ext ( L ) } L , L { 1 , 2 , , N } .
E z s , ext ( ρ , ( i + 1 ) Δ t ) = η I 2 π c I q = { L [ n ( k = 0 i κ q , int ( L ) , n I int ( L ) , n ( k ) [ W q , int ( L ) , n s , E ( i k ) W q , int ( L ) , n s , E ( i k 1 ) ] + I int ( L ) , n ( i κ q , int ( L ) , n + 1 ) W q , int ( L ) , n s , E ( κ q , int ( L ) , n ) ) ] } .
E z s , int ( L ) ( ρ , ( i + 1 ) Δ t ) = η II 2 π c II q = { n ( k = 0 i κ q , ext ( L ) , n I ext ( L ) , n ( k ) [ W q , ext ( L ) , n s , E ( i k ) W q , ext ( L ) , n s , E ( i k 1 ) ] + I ext ( L ) , n ( i κ q , ext ( L ) , n + 1 ) W q , ext ( L ) , n s , E ( κ q , ext ( L ) , n ) ) } .
κ q , Γ , n = [ Δ T q , Γ , n t ( Δ T Γ , n s Δ T q d t Γ , n r ) ] Δ t Γ { int ( L ) , ext ( L ) } , L = 1 , 2 , , N ;
Δ T q , Γ , n t = { P q , int ( L ) , n c I Γ = int ( L ) P q , ext ( L ) , n c II Γ = ext ( L ) } , L = 1 , 2 , , N ;
P q , Γ , n = ρ ρ q , Γ , n Γ { int ( L ) , ext ( L ) } , L = 1 , 2 , , N ;
W q , Γ , n s , E ( x ) = acosh ( Δ t Δ T q , Γ , n t [ x + ( Δ T Γ , n s Δ T q d t Γ , n r ) Δ t + 1 ] ) Γ { int ( L ) , ext ( L ) } L = 1 , 2 , , N .
H s , ext ( ρ , ( i + 1 ) Δ t ) = 1 2 π c I q = { L [ n ( k = 0 i κ q , int ( L ) , n I int ( L ) , n ( k ) [ W q , int ( L ) , n s , H ( i k ) W q , int ( L ) , n s , H ( i k 1 ) ] + I int ( L ) , n ( i κ q , int ( L ) , n + 1 ) W q , int ( L ) , n s , H ( κ q , int ( L ) , n ) ) ] } .
H s , int ( L ) ( ρ , ( i + 1 ) Δ t ) = 1 2 π c II q = { n ( k = 0 i κ q , ext ( L ) , n I ext ( L ) , n ( k ) [ W q , ext ( L ) , n s , H ( i k ) W q , ext ( L ) , n s , H ( i k 1 ) ] + I ext ( L ) , n ( i κ q , ext ( L ) , n + 1 ) W q , ext ( L ) , n s , H ( κ q , ext ( L ) , n ) ) } .
W q , Γ , n s , H ( x ) = ( z ̂ × P ̂ q , Γ , n ) ( Δ t Δ T q , Γ , n t [ x + ( Δ T Γ , n s Δ T q d t Γ , n r ) Δ t + 1 ] ) 2 1 Γ { int ( L ) , ext ( L ) } , L = 1 , 2 , , N ;
P ̂ q , Γ , n = ( ρ ρ q , Γ , n ) P q , Γ , n Γ { int ( L ) , ext ( L ) } , L = 1 , 2 , , N .
Δ MAX = max L ( Δ E MAX ( L ) , Δ H MAX ( L ) ) ,
Δ E MAX ( L ) = max t ( max ρ C 0 , L E z inc ( ρ , t ) + E z s , ext ( ρ , t ) E z s , int ( L ) ( ρ , t ) ) max t ( max ρ C 0 , L E z inc ( ρ , t ) ) ,
Δ H MAX ( L )
= max t ( max ρ C 0 , L n ̂ ( ρ ) × [ H inc ( ρ , t ) + H s , ext ( ρ , t ) H s , int ( L ) ( ρ , t ) ] ) ) max t ( max ρ C 0 , L ( E z inc ( ρ , t ) ) ) .
F tran ( ω ) = Unit - Cell [ F ( E ( ρ , t ) ) × F * ( H ( ρ , t ) ) ] x ̂ d y ,
F ( f ( ρ , t ) ) ω k = 2 π k N t Δ t = Δ t i = 0 N t 1 f ( ρ , i Δ t ) exp ( j 2 π k i N t ) .

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