Abstract

A method of solving problems of diffraction and dispersion in electromagnetic theory is presented. A modal expansion technique is used with a recursive R-matrix propagation scheme. This method retains the inherent R-matrix numerical stability and yet, contrary to some recent studies, is quite easy to implement for periodic structures (both two and three dimensional), including gratings and photonic crystal media. Grating structures may be multilayered structure, linear or crossed. Photonic media may be latticelike structures of finite or infinite depth. The eigenvalues of the modes are obtained by diagonalizing a matrix rather than searching for zeros of characteristic equations. Diffraction from dielectric and metallic sinusoidal gratings is calculated, and the results are compared with other published results. Transmission is calculated through a seven-layer-deep square arrangement of dielectric cylinders. Also, with the Floquet theorem, the bulk dispersion of the same cylinder geometry is calculated, and the results are compared with other published results. Of particular interest as a computational tool is a description of how a complex structure can be recursively added, whole structures at a time, after the initial structure has been calculated. This is very significant in terms of time savings, since most of the numerical work is done with the initial structure.

© 1995 Optical Society of America

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References

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  1. L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
    [CrossRef]
  2. D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976); J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
    [CrossRef]
  3. L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981); L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
    [CrossRef]
  4. E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
    [CrossRef] [PubMed]
  5. S. John, “Strong localization of photons in certain disordered dielectric super lattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
    [CrossRef] [PubMed]
  6. S. John, J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
    [CrossRef] [PubMed]
  7. K. M. Leung, Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
    [CrossRef] [PubMed]
  8. Z. Zhang, S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solutions of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
    [CrossRef] [PubMed]
  9. K. M. Ho, C. T. Chan, C. M. Soukoulis, “Existence of a photonic gap in periodic dielectric structure,” Phys. Rev. Lett. 65, 3152–3155 (1990).
    [CrossRef] [PubMed]
  10. K. M. Leung, Y. Qiu, “Multiple scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48, 7767–7771 (1993).
    [CrossRef]
  11. X. Wang, X. G. Zhang, Q. Yu, B. N. Harmon, “Multiple scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1992).
    [CrossRef]
  12. J. B. Pendry, A. MacKinnon, “Calculations of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
    [CrossRef] [PubMed]
  13. N. F. Johnson, P. M. Hui, “Theory of propagation of scalar waves in periodic and disordered composite structures,” Phys. Rev. B 48, 10118–10123 (1993).
    [CrossRef]
  14. J. B. Pendry, “Photonic band structures,” J. Mod. Opt. 41, 209–230 (1994).
    [CrossRef]
  15. W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Measurement of photonic band structure in a two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
    [CrossRef] [PubMed]
  16. R. A. Depine, “Perfectly conducting diffraction grating formalisms extended to good conductors via the surface impedance boundary condition,” Appl. Opt. 26, 2348–2354 (1987).
    [CrossRef] [PubMed]

1994 (1)

J. B. Pendry, “Photonic band structures,” J. Mod. Opt. 41, 209–230 (1994).
[CrossRef]

1993 (3)

K. M. Leung, Y. Qiu, “Multiple scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48, 7767–7771 (1993).
[CrossRef]

N. F. Johnson, P. M. Hui, “Theory of propagation of scalar waves in periodic and disordered composite structures,” Phys. Rev. B 48, 10118–10123 (1993).
[CrossRef]

L. Li, “Multilayer modal method for diffraction gratings of arbitrary profile, depth, and permittivity,” J. Opt. Soc. Am. A 10, 2581–2591 (1993).
[CrossRef]

1992 (3)

X. Wang, X. G. Zhang, Q. Yu, B. N. Harmon, “Multiple scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1992).
[CrossRef]

J. B. Pendry, A. MacKinnon, “Calculations of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Measurement of photonic band structure in a two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

1990 (4)

S. John, J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
[CrossRef] [PubMed]

K. M. Leung, Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

Z. Zhang, S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solutions of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

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

1987 (3)

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric super lattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

R. A. Depine, “Perfectly conducting diffraction grating formalisms extended to good conductors via the surface impedance boundary condition,” Appl. Opt. 26, 2348–2354 (1987).
[CrossRef] [PubMed]

1981 (1)

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981); L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

1976 (1)

D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976); J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

Adams, J. L.

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981); L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Andrewartha, J. R.

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981); L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Arjavalingam, G.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Measurement of photonic band structure in a two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

Botten, L. C.

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981); L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Brommer, K. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Measurement of photonic band structure in a two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

Chan, C. T.

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

Craig, M. S.

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981); L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Depine, R. A.

Harmon, B. N.

X. Wang, X. G. Zhang, Q. Yu, B. N. Harmon, “Multiple scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1992).
[CrossRef]

Ho, K. M.

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

Hui, P. M.

N. F. Johnson, P. M. Hui, “Theory of propagation of scalar waves in periodic and disordered composite structures,” Phys. Rev. B 48, 10118–10123 (1993).
[CrossRef]

Joannopoulos, J. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Measurement of photonic band structure in a two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

John, S.

S. John, J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric super lattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

Johnson, N. F.

N. F. Johnson, P. M. Hui, “Theory of propagation of scalar waves in periodic and disordered composite structures,” Phys. Rev. B 48, 10118–10123 (1993).
[CrossRef]

Leung, K. M.

K. M. Leung, Y. Qiu, “Multiple scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48, 7767–7771 (1993).
[CrossRef]

K. M. Leung, Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

Li, L.

Light, J. C.

D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976); J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

Liu, Y. F.

K. M. Leung, Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

MacKinnon, A.

J. B. Pendry, A. MacKinnon, “Calculations of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

Mcphedran, R. C.

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981); L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Meade, R. D.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Measurement of photonic band structure in a two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

Pendry, J. B.

J. B. Pendry, “Photonic band structures,” J. Mod. Opt. 41, 209–230 (1994).
[CrossRef]

J. B. Pendry, A. MacKinnon, “Calculations of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

Qiu, Y.

K. M. Leung, Y. Qiu, “Multiple scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48, 7767–7771 (1993).
[CrossRef]

Rappe, A. M.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Measurement of photonic band structure in a two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

Robertson, W. M.

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Measurement of photonic band structure in a two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

Satpathy, S.

Z. Zhang, S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solutions of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

Soukoulis, C. M.

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

Wang, J.

S. John, J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
[CrossRef] [PubMed]

Wang, X.

X. Wang, X. G. Zhang, Q. Yu, B. N. Harmon, “Multiple scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1992).
[CrossRef]

Yablonovitch, E.

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

Yu, Q.

X. Wang, X. G. Zhang, Q. Yu, B. N. Harmon, “Multiple scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1992).
[CrossRef]

Zhang, X. G.

X. Wang, X. G. Zhang, Q. Yu, B. N. Harmon, “Multiple scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1992).
[CrossRef]

Zhang, Z.

Z. Zhang, S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solutions of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

Zvijac, D. J.

D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976); J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

Appl. Opt. (1)

Chem. Phys. (1)

D. J. Zvijac, J. C. Light, “R-matrix theory for collinear chemical reactions,” Chem. Phys. 12, 237–251 (1976); J. C. Light, R. B. Walker, “An R-matrix approach to the solution of coupled equations for atom–molecule reactive scattering,” J. Chem. Phys. 65, 4272–4282 (1976).
[CrossRef]

J. Mod. Opt. (1)

J. B. Pendry, “Photonic band structures,” J. Mod. Opt. 41, 209–230 (1994).
[CrossRef]

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

Opt. Acta (1)

L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The dielectric lamellar diffraction grating,” Opt. Acta 28, 413–428 (1981); L. C. Botten, M. S. Craig, R. C. Mcphedran, J. L. Adams, J. R. Andrewartha, “The finitely conducting lamellar diffraction grating,” Opt. Acta 28, 1087–1102 (1981).
[CrossRef]

Phys. Rev. B (3)

N. F. Johnson, P. M. Hui, “Theory of propagation of scalar waves in periodic and disordered composite structures,” Phys. Rev. B 48, 10118–10123 (1993).
[CrossRef]

K. M. Leung, Y. Qiu, “Multiple scattering calculation of the two-dimensional photonic band structure,” Phys. Rev. B 48, 7767–7771 (1993).
[CrossRef]

X. Wang, X. G. Zhang, Q. Yu, B. N. Harmon, “Multiple scattering theory for electromagnetic waves,” Phys. Rev. B 47, 4161–4167 (1992).
[CrossRef]

Phys. Rev. Lett. (8)

J. B. Pendry, A. MacKinnon, “Calculations of photon dispersion relations,” Phys. Rev. Lett. 69, 2772–2775 (1992).
[CrossRef] [PubMed]

W. M. Robertson, G. Arjavalingam, R. D. Meade, K. D. Brommer, A. M. Rappe, J. D. Joannopoulos, “Measurement of photonic band structure in a two-dimensional periodic dielectric array,” Phys. Rev. Lett. 68, 2023–2026 (1992).
[CrossRef] [PubMed]

E. Yablonovitch, “Inhibited spontaneous emission in solid-state physics and electronics,” Phys. Rev. Lett. 58, 2059–2062 (1987).
[CrossRef] [PubMed]

S. John, “Strong localization of photons in certain disordered dielectric super lattices,” Phys. Rev. Lett. 58, 2486–2489 (1987).
[CrossRef] [PubMed]

S. John, J. Wang, “Quantum electrodynamics near a photonic band gap: photon bound states and dressed atoms,” Phys. Rev. Lett. 64, 2418–2421 (1990).
[CrossRef] [PubMed]

K. M. Leung, Y. F. Liu, “Full vector wave calculation of photonic band structures in face-centered-cubic dielectric media,” Phys. Rev. Lett. 65, 2646–2649 (1990).
[CrossRef] [PubMed]

Z. Zhang, S. Satpathy, “Electromagnetic wave propagation in periodic structures: Bloch wave solutions of Maxwell’s equations,” Phys. Rev. Lett. 65, 2650–2653 (1990).
[CrossRef] [PubMed]

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

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Figures (3)

Fig. 1
Fig. 1

TM polarization diffraction efficiency versus angle of incidence from a lossy sinusoidal grating. We compare our +1 and −3 diffracted order results (dashed curves) with those of Depine16 (solid curve). The dielectric constant is = (−5.28, 1.48), h/λ = 0.4, and d/λ = 2.0.

Fig. 2
Fig. 2

Schematic of a square array of infinitely long cylinders seven rows deep. The cylinders are infinitely long in the y direction and infinitely periodic in the x direction.

Fig. 3
Fig. 3

Transmission versus frequency of a photonic crystal array consisting of seven rows of infinitely long cylinders (see Fig. 2). The dashed–dotted curves show our results for transmission at normal incidence. The dashed curves are the experimental reference curve, and the solid curves are the product of the reference and the calculated transmission. The dots are measured data, and the polarization is (a) parallel and (b) perpendicular to the cylinder axes.

Tables (2)

Tables Icon

Table 1 Diffraction Efficiencies of Lossless Sinusoidal Grating

Tables Icon

Table 2 Diffraction Efficiencies of Lossy Sinusoidal Grating

Equations (38)

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E r ( r ) = 1 ( 2 π ) 2 d 2 K E r ( K ) exp ( i K · R ) exp ( i p z ) ,
H r ( r ) = 1 ( 2 π ) 2 d 2 K H r ( K ) exp ( i K · R ) exp ( i p z ) ,
E t ( r ) = 1 ( 2 π ) 2 d 2 K E t ( K ) exp ( i K · R ) exp ( - i p z ) ,
H t ( r ) = 1 ( 2 π ) 2 d 2 K H t ( K ) exp ( i K · R ) exp ( - i p z ) ,
E ( r ) = 1 ( 2 π ) 2 d 2 K E ( K , z ) exp ( i K · R ) ,
H ( r ) = 1 ( 2 π ) 2 d 2 K H ( K , z ) exp ( i K · R ) .
× E = i ω c H ,
× H = - i ω c D ,
D ( r ) = ( r ) E ( r ) ,
E x ( K , z ) z = - i { - ω c H y ( K , z ) + c K x ω 1 ( 2 π ) 2 d 2 K × - 1 ( K - K , z c ) [ K x H y ( K , z ) - K y H x ( K , z ) ] } ,
E y ( K , z ) z = - i { ω c H x ( K , z ) + c K y ω 1 ( 2 π ) 2 d 2 K × - 1 ( K - K , z c ) [ K x H y ( K , z ) - K y H x ( K , z ) ] } ,
H x ( K , z ) z = - i [ c K x K y ω E x ( K , z ) - c K x 2 ω E y ( K , z ) + ω c 1 ( 2 π ) 2 d 2 K ( K - K , z c ) E y ( K , z ) ] ,
H y ( K , z ) z = - i [ c K y 2 ω E x ( K , z ) - c K x K y ω E y ( K , z ) - ω c 1 ( 2 π ) 2 d 2 K ( K - K , z c ) E x ( K , z ) ] ,
( K - K , z c ) = d 2 R ( R , z c ) exp [ i ( K - K ) · R ] ,
- 1 ( K - K , z c ) = d 2 R [ ( R , z c ) ] - 1 exp [ i ( K - K ) · R ] .
A z = - i MA ,
A = ( E x ( K , z ) E y ( K , z ) H x ( K , z ) H y ( K , z ) )
A ( z ) = S e - i λ z C ,
S - 1 MS = Λ .
( E ˜ ( K , z ) E ˜ ( K , z + Δ z ) ) = r ( Δ z ) ( H ˜ ( K , z ) H ˜ ( K , z + Δ z ) ) ,
E ˜ = ( E x E y ) ,             H ˜ = ( H x H y )
( E ˜ ( K , z ) H ˜ ( K , z ) ) = [ S 11 S 12 S 21 S 22 ] [ e - i λ 1 z 0 0 e - i λ 2 z ] ( C 1 C 2 ) .
r ( Δ z ) = [ r 11 ( Δ z ) r 12 ( Δ z ) r 21 ( Δ z ) r 22 ( Δ z ) ] = [ S 11 S 12 S 11 e - i λ 1 Δ z S 12 e - i λ 2 Δ z ] × [ S 21 S 22 S 21 e - i λ 1 Δ z S 22 e - i λ 2 Δ z ] - 1 ,
( E ˜ ( K , z 1 ) E ˜ ( K , z 2 ) ) = R ( z 2 - z 1 ) ( H ˜ ( K , z 1 ) H ˜ ( K , z 2 ) ) ,
R ( z 2 - z 1 ) = [ R 11 ( z 2 - z 1 ) R 12 ( z 2 - z 1 ) R 21 ( z 2 - z 1 ) R 22 ( z 2 - z 1 ) ] .
R 11 ( z 2 + Δ z - z 1 ) = R 11 ( z 2 - z 1 ) + R 12 ( z 2 - z 1 ) [ r 11 ( Δ z ) - R 22 ( z 2 - z 1 ) ] - 1 R 21 ( z 2 - z 1 ) ,
R 12 ( z 2 + Δ z - z 1 ) = - R 12 ( z 2 - z 1 ) [ r 11 ( Δ z ) - R 22 ( z 2 - z 1 ) ] - 1 r 12 ( Δ z ) ,
R 21 ( z 2 + Δ z - z 1 ) = r 21 ( Δ z ) [ r 11 ( Δ z ) - R 22 ( z 2 - z 1 ) ] - 1 × R 21 ( z 2 - z 1 ) ,
R 22 ( z 2 + Δ z - z 1 ) = r 22 ( Δ z ) - r 21 ( Δ z ) [ r 11 ( Δ z ) - R 22 ( z 2 - z 1 ) ] - 1 r 12 ( Δ z ) .
( E ˜ t ( K , z 1 ) E ˜ r ( K , z 2 ) + E ˜ inc ( K inc , z 2 ) ) = R ( z 2 - z 1 ) × ( H ˜ t ( K , z 1 ) H ˜ r ( K , z 2 ) + H ˜ inc ( K inc , z 2 ) ) ,
H ˜ ( K , z ) = Z ( K , p ) E ˜ ( K , z ) ,
Z ( K , p ) = [ - K x K y ( ω / c ) p - p 2 + K y 2 ( ω / c ) p p 2 + K x 2 ( ω / c ) p K x K y ( ω / c ) p ] .
H ˜ t ( K , z ) = Z ( K , - p ) E ˜ t ( K , z ) ,
H ˜ r ( K , z ) = Z ( K , p ) E ˜ r ( K , z ) ,
[ I - R 11 Z ( K , - p ) - R 12 Z ( K , p ) - R 21 Z ( K , - p ) I - R 22 Z ( K , p ) ] ( E t ( K , z 1 ) E r ( K , z 2 ) ) = [ 0 R 12 - I R 22 ] ( E inc ( K inc , z 2 ) H inc ( K inc , z 2 ) ) ,
E ˜ ( K , z ) = S 11 C 1 ( z ) + S 12 C 2 ( z ) ,
H ˜ ( K , z ) = S 21 C 1 ( z ) + S 22 C 2 ( z ) .
( C 1 ( z 1 + L ) C 2 ( z 1 + L ) ) = exp ( i K z L ) ( C 1 ( z 1 ) C 2 ( z 1 ) ) = [ R 12 S 21 R 12 S 22 S 11 - R 22 S 21 S 12 - R 22 S 22 ] - 1 × [ S 11 - R 11 S 21 S 12 - R 11 S 22 R 21 S 21 R 21 S 22 ] × ( C 1 ( z 1 ) C 2 ( z 1 ) ) ,

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