Abstract

We propose a novel and simple concept of dynamic switching of guided-mode resonant grating filters with quadratic electro-optic effect within a waveguide layer modulated by external fields due to comb-shaped electrodes that also behave as a grating. As the device has subwavelength structure, the performance must be analyzed electromagnetically. We describe numerical simulation with the finite-difference time-domain method specially modified so that it can treat inhomogeneous anisotropic media such as lead lanthanum zirconate titanate.

© 2005 Optical Society of America

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  1. L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
    [CrossRef]
  2. L. Mashev, E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
    [CrossRef]
  3. G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
    [CrossRef]
  4. I. A. Avrutskii, G. A. Gobulenko, V. A. Sychugov, A. V. Tishchenko, “Spectral and laser characteristics of a mirror with a corrugated waveguide,” Sov. J. Quantum Electron. 16, 1063–1065 (1986).
    [CrossRef]
  5. G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, E. Popov, L. Mashev, “Diffraction characteristics of planar corrugated waveguide,” Opt. Quantum Electron. 18, 123–128 (1986).
    [CrossRef]
  6. R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
    [CrossRef]
  7. A. Mizutani, H. Kikuta, K. Nakajima, K. Iwata, “Nonpolarizing guided-mode resonant grating filter for oblique incidence,” J. Opt. Soc. Am. A 18, 1261–1266 (2001).
    [CrossRef]
  8. A. Sharon, D. Rosenblatt, A. A. Friesem, H. G. Weber, H. Engel, R. Steingrueber, “Light modulation with resonant grating-waveguide structures,” Opt. Lett. 21, 1564–1566 (1996).
    [CrossRef] [PubMed]
  9. W. Suh, S. Fan, “Mechanically switchable photonic crystal filter with either all-pass transmission or flat-top reflection characteristics,” Opt. Lett. 28, 1763–1765 (2003).
    [CrossRef] [PubMed]
  10. R. R. Boye, R. W. Ziolkowski, R. K. Kostuk, “Resonant waveguide grating switching device with nonlinear optical material,” Appl. Opt. 38, 5181–5185 (1999).
    [CrossRef]
  11. A. Mizutani, H. Kikuta, K. Iwata, “Numerical study on an asymmetric guided-mode resonant grating with a Kerr medium for optical switching,” J. Opt. Soc. Am. A 22, 355–360 (2005).
    [CrossRef]
  12. We use the term “reflectance” here as a ratio of the reflected power to the incident power.
  13. J. Turunen, “Diffraction theory of microrelief gratings,” in Micro-Optics, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 31–52.
  14. A. Mizutani, H. Kikuta, K. Iwata, “Wave localization of doubly periodic guided-mode resonance grating filters,” Opt. Rev. 10, 13–18 (2003).
    [CrossRef]
  15. X. Yang, M. Aspelmeyer, L. T. Wood, J. H. Miller, “Diffraction from tunable periodic structures. II. Experimental observation of electric field-induced diffraction peaks,” Appl. Opt. 41, 5845–5850 (2002).
    [CrossRef] [PubMed]
  16. Well-summarized detail of electro-optic effect can be found in, for example, B. E.A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), pp. 696–736.
    [CrossRef]
  17. K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).
  18. S. G. García, T. M. Hung-Bao, R. G. Martin, B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
    [CrossRef]
  19. H. Nishihara, M. Haruna, T. Suhara, “Optical integrated circuits,” (McGraw-Hill, New York, 1989) p. 166.
  20. For example, Eq. (1) in H. Ichikawa, “Analysis of femtosecond-order optical pulses diffracted by periodic structure,” J. Opt. Soc. Am. A 16, 299–304 (1999).
    [CrossRef]
  21. E. Popov, B. Bozhkov, “Corrugated waveguides as resonance optical filters: advantages and limitations,” J. Opt. Soc. Am. A 18, 1758–1764 (2001).
    [CrossRef]
  22. R. R. Boye, R. K. Kostuk, “Investigation of the effect of finite grating size on the performance of guided-mode resonance filters,” Appl. Opt. 39, 3649–3653 (2000).
    [CrossRef]
  23. F. Lemarchand, A. Sentenac, E. Cambril, H. Giovannini, “Study of the resonant behaviour of waveguide gratings: increasing the angular tolerance of guided-mode filters,” Pure Appl. Opt. 1, 545–551 (1999).
    [CrossRef]

2005 (1)

2003 (2)

A. Mizutani, H. Kikuta, K. Iwata, “Wave localization of doubly periodic guided-mode resonance grating filters,” Opt. Rev. 10, 13–18 (2003).
[CrossRef]

W. Suh, S. Fan, “Mechanically switchable photonic crystal filter with either all-pass transmission or flat-top reflection characteristics,” Opt. Lett. 28, 1763–1765 (2003).
[CrossRef] [PubMed]

2002 (1)

2001 (2)

2000 (1)

1999 (3)

1996 (2)

S. G. García, T. M. Hung-Bao, R. G. Martin, B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
[CrossRef]

A. Sharon, D. Rosenblatt, A. A. Friesem, H. G. Weber, H. Engel, R. Steingrueber, “Light modulation with resonant grating-waveguide structures,” Opt. Lett. 21, 1564–1566 (1996).
[CrossRef] [PubMed]

1992 (1)

R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

1986 (2)

I. A. Avrutskii, G. A. Gobulenko, V. A. Sychugov, A. V. Tishchenko, “Spectral and laser characteristics of a mirror with a corrugated waveguide,” Sov. J. Quantum Electron. 16, 1063–1065 (1986).
[CrossRef]

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, E. Popov, L. Mashev, “Diffraction characteristics of planar corrugated waveguide,” Opt. Quantum Electron. 18, 123–128 (1986).
[CrossRef]

1985 (2)

L. Mashev, E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[CrossRef]

1984 (1)

L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
[CrossRef]

1966 (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Aspelmeyer, M.

Avrutskii, I. A.

I. A. Avrutskii, G. A. Gobulenko, V. A. Sychugov, A. V. Tishchenko, “Spectral and laser characteristics of a mirror with a corrugated waveguide,” Sov. J. Quantum Electron. 16, 1063–1065 (1986).
[CrossRef]

Boye, R. R.

Bozhkov, B.

Cambril, E.

F. Lemarchand, A. Sentenac, E. Cambril, H. Giovannini, “Study of the resonant behaviour of waveguide gratings: increasing the angular tolerance of guided-mode filters,” Pure Appl. Opt. 1, 545–551 (1999).
[CrossRef]

Engel, H.

Fan, S.

Friesem, A. A.

García, S. G.

S. G. García, T. M. Hung-Bao, R. G. Martin, B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
[CrossRef]

Giovannini, H.

F. Lemarchand, A. Sentenac, E. Cambril, H. Giovannini, “Study of the resonant behaviour of waveguide gratings: increasing the angular tolerance of guided-mode filters,” Pure Appl. Opt. 1, 545–551 (1999).
[CrossRef]

Gobulenko, G. A.

I. A. Avrutskii, G. A. Gobulenko, V. A. Sychugov, A. V. Tishchenko, “Spectral and laser characteristics of a mirror with a corrugated waveguide,” Sov. J. Quantum Electron. 16, 1063–1065 (1986).
[CrossRef]

Golubenko, G. A.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, E. Popov, L. Mashev, “Diffraction characteristics of planar corrugated waveguide,” Opt. Quantum Electron. 18, 123–128 (1986).
[CrossRef]

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[CrossRef]

Haruna, M.

H. Nishihara, M. Haruna, T. Suhara, “Optical integrated circuits,” (McGraw-Hill, New York, 1989) p. 166.

Hung-Bao, T. M.

S. G. García, T. M. Hung-Bao, R. G. Martin, B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
[CrossRef]

Ichikawa, H.

Iwata, K.

Kikuta, H.

Kostuk, R. K.

Lemarchand, F.

F. Lemarchand, A. Sentenac, E. Cambril, H. Giovannini, “Study of the resonant behaviour of waveguide gratings: increasing the angular tolerance of guided-mode filters,” Pure Appl. Opt. 1, 545–551 (1999).
[CrossRef]

Magnusson, R.

R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

Martin, R. G.

S. G. García, T. M. Hung-Bao, R. G. Martin, B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
[CrossRef]

Mashev, L.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, E. Popov, L. Mashev, “Diffraction characteristics of planar corrugated waveguide,” Opt. Quantum Electron. 18, 123–128 (1986).
[CrossRef]

L. Mashev, E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
[CrossRef]

Miller, J. H.

Mizutani, A.

Nakajima, K.

Nishihara, H.

H. Nishihara, M. Haruna, T. Suhara, “Optical integrated circuits,” (McGraw-Hill, New York, 1989) p. 166.

Olmedo, B. G.

S. G. García, T. M. Hung-Bao, R. G. Martin, B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
[CrossRef]

Popov, E.

E. Popov, B. Bozhkov, “Corrugated waveguides as resonance optical filters: advantages and limitations,” J. Opt. Soc. Am. A 18, 1758–1764 (2001).
[CrossRef]

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, E. Popov, L. Mashev, “Diffraction characteristics of planar corrugated waveguide,” Opt. Quantum Electron. 18, 123–128 (1986).
[CrossRef]

L. Mashev, E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
[CrossRef]

Rosenblatt, D.

Saleh, B. E.A.

Well-summarized detail of electro-optic effect can be found in, for example, B. E.A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), pp. 696–736.
[CrossRef]

Sentenac, A.

F. Lemarchand, A. Sentenac, E. Cambril, H. Giovannini, “Study of the resonant behaviour of waveguide gratings: increasing the angular tolerance of guided-mode filters,” Pure Appl. Opt. 1, 545–551 (1999).
[CrossRef]

Sharon, A.

Steingrueber, R.

Suh, W.

Suhara, T.

H. Nishihara, M. Haruna, T. Suhara, “Optical integrated circuits,” (McGraw-Hill, New York, 1989) p. 166.

Svakhin, A. S.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, E. Popov, L. Mashev, “Diffraction characteristics of planar corrugated waveguide,” Opt. Quantum Electron. 18, 123–128 (1986).
[CrossRef]

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[CrossRef]

Sychugov, V. A.

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, E. Popov, L. Mashev, “Diffraction characteristics of planar corrugated waveguide,” Opt. Quantum Electron. 18, 123–128 (1986).
[CrossRef]

I. A. Avrutskii, G. A. Gobulenko, V. A. Sychugov, A. V. Tishchenko, “Spectral and laser characteristics of a mirror with a corrugated waveguide,” Sov. J. Quantum Electron. 16, 1063–1065 (1986).
[CrossRef]

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[CrossRef]

Teich, M. C.

Well-summarized detail of electro-optic effect can be found in, for example, B. E.A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), pp. 696–736.
[CrossRef]

Tishchenko, A. V.

I. A. Avrutskii, G. A. Gobulenko, V. A. Sychugov, A. V. Tishchenko, “Spectral and laser characteristics of a mirror with a corrugated waveguide,” Sov. J. Quantum Electron. 16, 1063–1065 (1986).
[CrossRef]

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, E. Popov, L. Mashev, “Diffraction characteristics of planar corrugated waveguide,” Opt. Quantum Electron. 18, 123–128 (1986).
[CrossRef]

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[CrossRef]

Turunen, J.

J. Turunen, “Diffraction theory of microrelief gratings,” in Micro-Optics, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 31–52.

Wang, S. S.

R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

Weber, H. G.

Wood, L. T.

Yang, X.

Yee, K. S.

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

Ziolkowski, R. W.

Appl. Opt. (3)

Appl. Phys. Lett. (1)

R. Magnusson, S. S. Wang, “New principle for optical filters,” Appl. Phys. Lett. 61, 1022–1024 (1992).
[CrossRef]

IEEE Trans. Antennas Propag. (1)

K. S. Yee, “Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media,” IEEE Trans. Antennas Propag. AP-14, 302–307 (1966).

IEEE Trans. Microwave Theory Tech. (1)

S. G. García, T. M. Hung-Bao, R. G. Martin, B. G. Olmedo, “On the application of finite methods in time domain to anisotropic dielectric waveguides,” IEEE Trans. Microwave Theory Tech. 44, 2195–2206 (1996).
[CrossRef]

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

Opt. Commun. (2)

L. Mashev, E. Popov, “Diffraction efficiency anomalies of multicoated dielectric gratings,” Opt. Commun. 51, 131–136 (1984).
[CrossRef]

L. Mashev, E. Popov, “Zero order anomaly of dielectric coated gratings,” Opt. Commun. 55, 377–380 (1985).
[CrossRef]

Opt. Lett. (2)

Opt. Quantum Electron. (1)

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, E. Popov, L. Mashev, “Diffraction characteristics of planar corrugated waveguide,” Opt. Quantum Electron. 18, 123–128 (1986).
[CrossRef]

Opt. Rev. (1)

A. Mizutani, H. Kikuta, K. Iwata, “Wave localization of doubly periodic guided-mode resonance grating filters,” Opt. Rev. 10, 13–18 (2003).
[CrossRef]

Pure Appl. Opt. (1)

F. Lemarchand, A. Sentenac, E. Cambril, H. Giovannini, “Study of the resonant behaviour of waveguide gratings: increasing the angular tolerance of guided-mode filters,” Pure Appl. Opt. 1, 545–551 (1999).
[CrossRef]

Sov. J. Quantum Electron. (2)

G. A. Golubenko, A. S. Svakhin, V. A. Sychugov, A. V. Tishchenko, “Total reflection of light from a corrugated surface of a dielectric waveguide,” Sov. J. Quantum Electron. 15, 886–887 (1985).
[CrossRef]

I. A. Avrutskii, G. A. Gobulenko, V. A. Sychugov, A. V. Tishchenko, “Spectral and laser characteristics of a mirror with a corrugated waveguide,” Sov. J. Quantum Electron. 16, 1063–1065 (1986).
[CrossRef]

Other (4)

Well-summarized detail of electro-optic effect can be found in, for example, B. E.A. Saleh, M. C. Teich, Fundamentals of Photonics (Wiley, New York, 1991), pp. 696–736.
[CrossRef]

We use the term “reflectance” here as a ratio of the reflected power to the incident power.

J. Turunen, “Diffraction theory of microrelief gratings,” in Micro-Optics, H. P. Herzig, ed. (Taylor & Francis, London, 1997), pp. 31–52.

H. Nishihara, M. Haruna, T. Suhara, “Optical integrated circuits,” (McGraw-Hill, New York, 1989) p. 166.

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Figures (9)

Fig. 1
Fig. 1

Concept of the device. G, ITO comb-shaped grating electrodes. P, PLZT waveguide layer. S, sapphire substrate.

Fig. 2
Fig. 2

Grid-point map for TM polarization.

Fig. 3
Fig. 3

Finite-differencing scheme for anisotropic media. Solid circles, E x ( i , k + 1 2 ) ; arrows, averaging; dashed line connecting small solid squares, differencing. (a) E z , (b) H y x .

Fig. 4
Fig. 4

Analyzed device structure and refractive indices: n 1 , air; n 2 , ITO; n 3 , PLZT; n 4 , sapphire. The asterisk indicates the value at V = 0 or E = 0

Fig. 5
Fig. 5

Cross-sectional image of various distributions in the PLZT layer. Electrodes are shown only in (a). The color scale for each graph is provided. (a) Electric potential ϕ ( x , z ) , (b) static electric field E x ( x , z ) , (c) static electric field E z ( x , z ) , (d) element of the index tensor n x x ( x , z ) , (e) element of the index tensor n z z ( x , z ) .

Fig. 6
Fig. 6

Simulated reflection spectra for the proposed device. From right to left, V = 0 , 2, 3, 4, 5 and 6 V.

Fig. 7
Fig. 7

Resonance peak shift versus external static voltage.

Fig. 8
Fig. 8

Simulated reflection spectra for the devices with homogeneous waveguide layers. From right to left, n 3 = 2.54 , 2.53, 2.52, 2.51, 2.50, and 2.49.

Fig. 9
Fig. 9

Grid points for solving Laplace’s equation with finite-differencing.

Equations (32)

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ϵ 0 i j ξ i j x i x j = 1 ,
ϵ 0 ξ i j ( E ) = ϵ 0 ξ i j ( 0 ) + k , l s i j k l E k E l ,
n i j ( E ) n 0 1 2 k , l s i j k l n 0 3 E k E l
ϵ st ( x , z ) ϕ ( x , z ) = 0 ,
E ( x , z ) = ϕ ( x , z ) ,
ϵ 0 ( ξ x x ξ y y ξ z z ξ y z ξ z x ξ x y ) = ( n 0 2 n 0 2 n 0 2 0 0 0 ) + [ s 11 s 12 s 12 0 0 0 s 12 s 11 s 12 0 0 0 s 12 s 12 s 11 0 0 0 0 0 0 s 44 0 0 0 0 0 0 s 44 0 0 0 0 0 0 s 44 ] ( E x 2 E y 2 E z 2 2 E y E z 2 E z E x 2 E x E y ) ,
ξ x x = ( n 0 2 + s 11 E x 2 + s 12 E z 2 ) ϵ 0 ,
ξ z z = ( n 0 2 + s 12 E x 2 + s 11 E z 2 ) ϵ 0 ,
ξ z x = ( s 11 s 12 ) E z E x ϵ 0 .
E t = ξ ( × H σ E ) ,
H t = 1 μ 0 × E ,
E x t = ξ x x ( H y z σ x E x ) + ξ z x ( H y x σ z E z ) .
E x t [ E x n + 1 ( i , k + 1 2 ) E x n ( i , k + 1 2 ) ] Δ t ,
E x [ E x n + 1 ( i , k + 1 2 ) + E x n ( i , k + 1 2 ) ] 2 ,
H y z [ H y n + 1 2 ( i , k + 1 ) H y n + 1 2 ( i , k ) ] Δ z ,
E z [ E z n + 1 ( i + 1 2 , k + 1 ) + E z n ( i + 1 2 , k + 1 ) + E z n + 1 ( i + 1 2 , k ) + E z n ( i + 1 2 , k ) + E z n + 1 ( i 1 2 , k + 1 ) + E z n ( i 1 2 , k + 1 ) + E z n + 1 ( i 1 2 , k ) + E z n ( i 1 2 , k ) ] 8 ,
H y x [ H y n + 1 2 ( i + 1 , k + 1 ) + H y n + 1 2 ( i + 1 , k ) + H y n + 1 2 ( i , k + 1 ) + H y n + 1 2 ( i , k ) H y n + 1 2 ( i , k + 1 ) H y n + 1 2 ( i , k ) H y n + 1 2 ( i 1 , k + 1 ) H y n + 1 2 ( i 1 , k ) ] 4 Δ x ,
E x n + 1 ( i , k + 1 2 ) = 1 ξ x x σ x Δ t 2 1 + ξ x x σ x Δ t 2 E x n ( i , k + 1 2 ) ξ x x Δ t 1 + ξ x x σ x Δ t 2 H y n + 1 2 ( i , k + 1 ) H y n + 1 2 ( i , k ) Δ z ξ x z σ z Δ t 1 + ξ x x σ x Δ t 2 { E z } + ξ x z Δ t 1 + ξ x x σ x Δ t 2 { H y x } Δ x .
E x n + 1 ( i , k + 1 2 ) = E x n ( i , k + 1 2 ) ξ x x Δ t H y n + 1 2 ( i , k + 1 ) H y n + 1 2 ( i , k ) Δ z + ξ x z Δ t { H y x } Δ x .
E z n + 1 ( i + 1 2 , k ) = E x n ( i + 1 2 , k ) ξ z z Δ t H y n + 1 2 ( i + 1 , k ) H y n + 1 2 ( i , k ) Δ x + ξ z x Δ t { H y z } Δ z ,
{ H y z } = [ H y n + 1 2 ( i + 1 , k + 1 ) + H y n + 1 2 ( i , k + 1 ) H y n + 1 2 ( i + 1 , k 1 ) H y n + 1 2 ( i , k 1 ) ] 4 .
H y n + 1 2 ( i , k ) = H y n 1 2 ( i , k ) Δ t μ 0 [ E x n ( i , k + 1 2 ) E x n ( i , k 1 2 ) Δ z E z n ( i + 1 2 , k ) E z n ( i 1 2 , k ) Δ x ] .
A m ( t ) = F T x { H y ( x , z 0 , t ) } ,
A 0 ( ν ) = F T t { A 0 ( t ) } ,
R ( λ ) = A 0 ( λ ) 2 .
x ( ϵ ϕ x ) + z ( ϵ ϕ z ) = 0 ,
ϵ x ϕ x + ϵ 2 ϕ x 2 + ϵ z ϕ z + ϵ 2 ϕ z 2 = 0 ,
( ϵ 1 , 0 ϵ 1 , 0 ) ( ϕ 1 , 0 ϕ 1 , 0 ) ( 2 Δ x ) 2 ,
[ ( ϵ 1 , 0 ϵ 0 , 0 ) ( ϕ 1 , 0 ϕ 0 , 0 ) + ( ϵ 0 , 0 ϵ 1 , 0 ) ( ϕ 0 , 0 ϕ 1 , 0 ) ] ( 2 Δ x ) 2 .
( ϵ 1 , 0 + ϵ 0 , 0 ) ( ϕ 1 , 0 ϕ 0 , 0 ) ( Δ x ) 2 ( ϵ 0 , 0 + ϵ 1 , 0 ) ( ϕ 0 , 0 ϕ 1 , 0 ) ( Δ x ) 2 + ( ϵ 0 , 1 + ϵ 0 , 0 ) ( ϕ 0 , 1 ϕ 0 , 0 ) ( Δ z ) 2 ( ϵ 0 , 0 + ϵ 0 , 1 ) ( ϕ 0 , 0 ϕ 0 , 1 ) ( Δ z ) 2 = 0 ,
ϕ 0 , 0 = [ ϵ 1 2 , 0 + ϵ 1 2 , 0 ( Δ x ) 2 + ϵ 0 , 1 2 + ϵ 0 , 1 2 ( Δ z ) 2 ] 1 × [ ϵ 1 2 , 0 ϕ 1 , 0 + ϵ 1 2 , 0 ϕ 1 , 0 ( Δ x ) 2 + ϵ 0 , 1 2 ϕ 0 , 1 + ϵ 0 , 1 2 ϕ 0 , 1 ( Δ z ) 2 ] .
ξ y y = [ n 0 2 + s 12 ( E x 2 + E z 2 ) ] ϵ 0 .

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