Abstract

Using the finite-difference time-domain (FDTD) method we have numerically investigated the transmission and reflection of both long and ultrashort elliptically polarized light pulses in periodic metamaterial made of polymer. In the first time we have analyzed the polarization evolution in the hodograph of the transmitted long pulses, and we demonstrated the behavior of the electric field in transmitted ultrashort pulses. The mechanisms of light-matter interaction in terms of the electromagnetic energy oscillation in polymeric metamaterial are shown. We studied the influence of all the parameters of metamaterial unit cell (a helix) on the transmission and reflection. Particularly, the increase of the amount of the helix cycles broadens the polarization-selective frequency range for the transmitted light.

© 2014 Optical Society of America

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References

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  1. J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
    [Crossref] [PubMed]
  2. T. Yoshioka, T. Ogata, T. Nonaka, M. Moritsugu, S.-N. Kim, and S. Kurihara, “Reversible-photon-mode full-color display by means of photochemical modulation of a helically cholesteric structure,” Adv. Mater. 17(10), 1226–1229 (2005).
    [Crossref]
  3. G. De Filpo, F. P. Nicoletta, and G. Chidichimo, “Cholesteric Emulsions for Colored Displays,” Adv. Mater. 17(9), 1150–1152 (2005).
    [Crossref]
  4. K. Claborn, E. Puklin-Faucher, M. Kurimoto, W. Kaminsky, and B. Kahr, “Circular dichroism imaging microscopy: application to enantiomorphous twinning in biaxial crystals of 1,8-dihydroxyanthraquinone,” J. Am. Chem. Soc. 125(48), 14825–14831 (2003).
    [Crossref] [PubMed]
  5. Z. Yang, M. Zhao, and P. Lu, “How to improve the signal-to-noise ratio for circular polarizers consisting of helical metamaterials?” Opt. Express 19(5), 4255–4260 (2011).
    [Crossref] [PubMed]
  6. Z. Y. Yang, M. Zhao, P. X. Lu, and Y. F. Lu, “Ultrabroadband optical circular polarizers consisting of double-helical nanowire structures,” Opt. Lett. 35(15), 2588–2590 (2010).
    [Crossref] [PubMed]
  7. M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19(2), 207–210 (2007).
    [Crossref]
  8. N. N. Potravkin, I. A. Perezhogin, and V. A. Makarov, “Numerical solution of Maxwell equations by a finite-difference time-domain method in a medium with frequency and spatial dispersion,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(5 Pt 2), 056706 (2012).
    [Crossref] [PubMed]
  9. G. A. Gryaznov, V. A. Makarov, I. A. Perezhogin, and N. N. Potravkin, “Modeling of nonlinear optical activity in propagation of ultrashort elliptically polarized laser pulses,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(1), 013306 (2014).
    [Crossref] [PubMed]
  10. M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1968).
  11. Y. Liu, “Fourier Analysis of Numerical Algorithms for the Maxwell Equations,” J. Comput. Phys. 124(2), 396–416 (1996).
    [Crossref]
  12. W. H. Press, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).
  13. R. Richtmyer and K. Morton, Difference Methods for Initial-Value Problems (Wiley, 1967).

2014 (1)

G. A. Gryaznov, V. A. Makarov, I. A. Perezhogin, and N. N. Potravkin, “Modeling of nonlinear optical activity in propagation of ultrashort elliptically polarized laser pulses,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(1), 013306 (2014).
[Crossref] [PubMed]

2012 (1)

N. N. Potravkin, I. A. Perezhogin, and V. A. Makarov, “Numerical solution of Maxwell equations by a finite-difference time-domain method in a medium with frequency and spatial dispersion,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(5 Pt 2), 056706 (2012).
[Crossref] [PubMed]

2011 (1)

2010 (1)

2009 (1)

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref] [PubMed]

2007 (1)

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19(2), 207–210 (2007).
[Crossref]

2005 (2)

T. Yoshioka, T. Ogata, T. Nonaka, M. Moritsugu, S.-N. Kim, and S. Kurihara, “Reversible-photon-mode full-color display by means of photochemical modulation of a helically cholesteric structure,” Adv. Mater. 17(10), 1226–1229 (2005).
[Crossref]

G. De Filpo, F. P. Nicoletta, and G. Chidichimo, “Cholesteric Emulsions for Colored Displays,” Adv. Mater. 17(9), 1150–1152 (2005).
[Crossref]

2003 (1)

K. Claborn, E. Puklin-Faucher, M. Kurimoto, W. Kaminsky, and B. Kahr, “Circular dichroism imaging microscopy: application to enantiomorphous twinning in biaxial crystals of 1,8-dihydroxyanthraquinone,” J. Am. Chem. Soc. 125(48), 14825–14831 (2003).
[Crossref] [PubMed]

1996 (1)

Y. Liu, “Fourier Analysis of Numerical Algorithms for the Maxwell Equations,” J. Comput. Phys. 124(2), 396–416 (1996).
[Crossref]

Bade, K.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref] [PubMed]

Chidichimo, G.

G. De Filpo, F. P. Nicoletta, and G. Chidichimo, “Cholesteric Emulsions for Colored Displays,” Adv. Mater. 17(9), 1150–1152 (2005).
[Crossref]

Claborn, K.

K. Claborn, E. Puklin-Faucher, M. Kurimoto, W. Kaminsky, and B. Kahr, “Circular dichroism imaging microscopy: application to enantiomorphous twinning in biaxial crystals of 1,8-dihydroxyanthraquinone,” J. Am. Chem. Soc. 125(48), 14825–14831 (2003).
[Crossref] [PubMed]

De Filpo, G.

G. De Filpo, F. P. Nicoletta, and G. Chidichimo, “Cholesteric Emulsions for Colored Displays,” Adv. Mater. 17(9), 1150–1152 (2005).
[Crossref]

Decker, M.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref] [PubMed]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19(2), 207–210 (2007).
[Crossref]

Deubel, M.

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19(2), 207–210 (2007).
[Crossref]

Gansel, J. K.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref] [PubMed]

Gryaznov, G. A.

G. A. Gryaznov, V. A. Makarov, I. A. Perezhogin, and N. N. Potravkin, “Modeling of nonlinear optical activity in propagation of ultrashort elliptically polarized laser pulses,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(1), 013306 (2014).
[Crossref] [PubMed]

Kahr, B.

K. Claborn, E. Puklin-Faucher, M. Kurimoto, W. Kaminsky, and B. Kahr, “Circular dichroism imaging microscopy: application to enantiomorphous twinning in biaxial crystals of 1,8-dihydroxyanthraquinone,” J. Am. Chem. Soc. 125(48), 14825–14831 (2003).
[Crossref] [PubMed]

Kaminsky, W.

K. Claborn, E. Puklin-Faucher, M. Kurimoto, W. Kaminsky, and B. Kahr, “Circular dichroism imaging microscopy: application to enantiomorphous twinning in biaxial crystals of 1,8-dihydroxyanthraquinone,” J. Am. Chem. Soc. 125(48), 14825–14831 (2003).
[Crossref] [PubMed]

Kim, S.-N.

T. Yoshioka, T. Ogata, T. Nonaka, M. Moritsugu, S.-N. Kim, and S. Kurihara, “Reversible-photon-mode full-color display by means of photochemical modulation of a helically cholesteric structure,” Adv. Mater. 17(10), 1226–1229 (2005).
[Crossref]

Kurihara, S.

T. Yoshioka, T. Ogata, T. Nonaka, M. Moritsugu, S.-N. Kim, and S. Kurihara, “Reversible-photon-mode full-color display by means of photochemical modulation of a helically cholesteric structure,” Adv. Mater. 17(10), 1226–1229 (2005).
[Crossref]

Kurimoto, M.

K. Claborn, E. Puklin-Faucher, M. Kurimoto, W. Kaminsky, and B. Kahr, “Circular dichroism imaging microscopy: application to enantiomorphous twinning in biaxial crystals of 1,8-dihydroxyanthraquinone,” J. Am. Chem. Soc. 125(48), 14825–14831 (2003).
[Crossref] [PubMed]

Linden, S.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref] [PubMed]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19(2), 207–210 (2007).
[Crossref]

Liu, Y.

Y. Liu, “Fourier Analysis of Numerical Algorithms for the Maxwell Equations,” J. Comput. Phys. 124(2), 396–416 (1996).
[Crossref]

Lu, P.

Lu, P. X.

Lu, Y. F.

Makarov, V. A.

G. A. Gryaznov, V. A. Makarov, I. A. Perezhogin, and N. N. Potravkin, “Modeling of nonlinear optical activity in propagation of ultrashort elliptically polarized laser pulses,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(1), 013306 (2014).
[Crossref] [PubMed]

N. N. Potravkin, I. A. Perezhogin, and V. A. Makarov, “Numerical solution of Maxwell equations by a finite-difference time-domain method in a medium with frequency and spatial dispersion,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(5 Pt 2), 056706 (2012).
[Crossref] [PubMed]

Moritsugu, M.

T. Yoshioka, T. Ogata, T. Nonaka, M. Moritsugu, S.-N. Kim, and S. Kurihara, “Reversible-photon-mode full-color display by means of photochemical modulation of a helically cholesteric structure,” Adv. Mater. 17(10), 1226–1229 (2005).
[Crossref]

Nicoletta, F. P.

G. De Filpo, F. P. Nicoletta, and G. Chidichimo, “Cholesteric Emulsions for Colored Displays,” Adv. Mater. 17(9), 1150–1152 (2005).
[Crossref]

Nonaka, T.

T. Yoshioka, T. Ogata, T. Nonaka, M. Moritsugu, S.-N. Kim, and S. Kurihara, “Reversible-photon-mode full-color display by means of photochemical modulation of a helically cholesteric structure,” Adv. Mater. 17(10), 1226–1229 (2005).
[Crossref]

Ogata, T.

T. Yoshioka, T. Ogata, T. Nonaka, M. Moritsugu, S.-N. Kim, and S. Kurihara, “Reversible-photon-mode full-color display by means of photochemical modulation of a helically cholesteric structure,” Adv. Mater. 17(10), 1226–1229 (2005).
[Crossref]

Perezhogin, I. A.

G. A. Gryaznov, V. A. Makarov, I. A. Perezhogin, and N. N. Potravkin, “Modeling of nonlinear optical activity in propagation of ultrashort elliptically polarized laser pulses,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(1), 013306 (2014).
[Crossref] [PubMed]

N. N. Potravkin, I. A. Perezhogin, and V. A. Makarov, “Numerical solution of Maxwell equations by a finite-difference time-domain method in a medium with frequency and spatial dispersion,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(5 Pt 2), 056706 (2012).
[Crossref] [PubMed]

Potravkin, N. N.

G. A. Gryaznov, V. A. Makarov, I. A. Perezhogin, and N. N. Potravkin, “Modeling of nonlinear optical activity in propagation of ultrashort elliptically polarized laser pulses,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(1), 013306 (2014).
[Crossref] [PubMed]

N. N. Potravkin, I. A. Perezhogin, and V. A. Makarov, “Numerical solution of Maxwell equations by a finite-difference time-domain method in a medium with frequency and spatial dispersion,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(5 Pt 2), 056706 (2012).
[Crossref] [PubMed]

Puklin-Faucher, E.

K. Claborn, E. Puklin-Faucher, M. Kurimoto, W. Kaminsky, and B. Kahr, “Circular dichroism imaging microscopy: application to enantiomorphous twinning in biaxial crystals of 1,8-dihydroxyanthraquinone,” J. Am. Chem. Soc. 125(48), 14825–14831 (2003).
[Crossref] [PubMed]

Rill, M. S.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref] [PubMed]

Saile, V.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref] [PubMed]

Thiel, M.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref] [PubMed]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19(2), 207–210 (2007).
[Crossref]

von Freymann, G.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref] [PubMed]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19(2), 207–210 (2007).
[Crossref]

Wegener, M.

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref] [PubMed]

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19(2), 207–210 (2007).
[Crossref]

Yang, Z.

Yang, Z. Y.

Yoshioka, T.

T. Yoshioka, T. Ogata, T. Nonaka, M. Moritsugu, S.-N. Kim, and S. Kurihara, “Reversible-photon-mode full-color display by means of photochemical modulation of a helically cholesteric structure,” Adv. Mater. 17(10), 1226–1229 (2005).
[Crossref]

Zhao, M.

Adv. Mater. (3)

M. Thiel, M. Decker, M. Deubel, M. Wegener, S. Linden, and G. von Freymann, “Polarization stop bands in chiral polymeric three-dimensional photonic crystals,” Adv. Mater. 19(2), 207–210 (2007).
[Crossref]

T. Yoshioka, T. Ogata, T. Nonaka, M. Moritsugu, S.-N. Kim, and S. Kurihara, “Reversible-photon-mode full-color display by means of photochemical modulation of a helically cholesteric structure,” Adv. Mater. 17(10), 1226–1229 (2005).
[Crossref]

G. De Filpo, F. P. Nicoletta, and G. Chidichimo, “Cholesteric Emulsions for Colored Displays,” Adv. Mater. 17(9), 1150–1152 (2005).
[Crossref]

J. Am. Chem. Soc. (1)

K. Claborn, E. Puklin-Faucher, M. Kurimoto, W. Kaminsky, and B. Kahr, “Circular dichroism imaging microscopy: application to enantiomorphous twinning in biaxial crystals of 1,8-dihydroxyanthraquinone,” J. Am. Chem. Soc. 125(48), 14825–14831 (2003).
[Crossref] [PubMed]

J. Comput. Phys. (1)

Y. Liu, “Fourier Analysis of Numerical Algorithms for the Maxwell Equations,” J. Comput. Phys. 124(2), 396–416 (1996).
[Crossref]

Opt. Express (1)

Opt. Lett. (1)

Phys. Rev. E Stat. Nonlin. Soft Matter Phys. (2)

N. N. Potravkin, I. A. Perezhogin, and V. A. Makarov, “Numerical solution of Maxwell equations by a finite-difference time-domain method in a medium with frequency and spatial dispersion,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 86(5 Pt 2), 056706 (2012).
[Crossref] [PubMed]

G. A. Gryaznov, V. A. Makarov, I. A. Perezhogin, and N. N. Potravkin, “Modeling of nonlinear optical activity in propagation of ultrashort elliptically polarized laser pulses,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 89(1), 013306 (2014).
[Crossref] [PubMed]

Science (1)

J. K. Gansel, M. Thiel, M. S. Rill, M. Decker, K. Bade, V. Saile, G. von Freymann, S. Linden, and M. Wegener, “Gold helix photonic metamaterial as broadband circular polarizer,” Science 325(5947), 1513–1515 (2009).
[Crossref] [PubMed]

Other (3)

M. Born and E. Wolf, Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light (Pergamon, 1968).

W. H. Press, Numerical Recipes in Fortran: The Art of Scientific Computing, 2nd ed. (Cambridge University, 1992).

R. Richtmyer and K. Morton, Difference Methods for Initial-Value Problems (Wiley, 1967).

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Figures (6)

Fig. 1
Fig. 1 Schematic image of a helix, representing the metamaterial unit cell. The definition of characteristic sizes is shown in the scheme.
Fig. 2
Fig. 2 Hodographs of the electric field vector in light pulses reflected from (a, c and e) and transmitted through (b, d, and f) the metamaterial, t=1075 fs, n=8 , M 0 =0 (а, b), M 0 =1 (c, d), M 0 =1 (e, f).
Fig. 3
Fig. 3 The dependencies of the ellipticity degree M (a, c, and e) and the angle of orientation of the polarization ellipse Ψ (b, d and f) on the longitudinal coordinate z along the temporal envelope of a pulse traversed through the metamaterial consisting of right-handed helices. The dependencies are shown at time moment t=1075 fs (in a far-field region after the metamaterial) in case of the incidence of (a and b) – linearly polarized pulse; (c and d) – RHCP pulse; (e and f) – LHCP pulse. Black curves correspond to n=2 ; red curves: n=4 ; blue curves: n=8 .
Fig. 4
Fig. 4 Spatial distributions of w e (x,y=0,z) and w h (x,y=0,z) in case of propagation of the long pulses at time values t p (а), (c) and after half of the oscillation period, at t p +π/ω (b), (d). RHCP incidence – (a) and (b); LHCP incidence – (c) and (d). The metamaterial consists of right-hand helices, n=8 .
Fig. 5
Fig. 5 The hodographs of the electric field vectors in (a) reflected and (b) transmitted pulses at t = 1075 fs after the incidence of ultrashort ( w 0 = 2 λ ) linearly polarized pulse on the metamaterial; z 0 = 12 λ .
Fig. 6
Fig. 6 (a) The frequency spectrum of the transmission coefficient and (b) the frequency spectrum of the ellipticity degree of the polarization ellipse in case of the incidence of RHCP (red curves) or LHCP (blue curves) pulse, when a metamaterial consists of right-handed helices with n=8 (solid lines) or n=4 (dashed lines). ω 0 =1.16 10 15 rad/s.

Equations (24)

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

1 c B x t = E y z E z y 1 c D x t = H z y H y z 1 c B y t = E z x E x z ,
1 c D y t = H x z H z x 1 c B z t = E x y E y x 1 c D z t = H y x H x y ,
D i =ε(x,y,z) E i , B i = H i
E x (t=0)= ( I 0 /2) 1/2 [1 (1 M 0 2 ) 1/2 ] 1/2 exp[ (z z 0 ) 2 / w 0 2 ]× ×sign{ M 0 }sin(2π(z z 0 )/λ),
E y (t=0)= ( I 0 /2) 1/2 [1+ (1 M 0 2 ) 1/2 ] 1/2 exp[ (z z 0 ) 2 / w 0 2 ]× ×cos(2π(z z 0 )/λ).
|M( z ˜ m )|= 2 1/2 I 1/2 ( z ˜ m ) [I( z ¯ m )+I( z ¯ m+1 )] 1/2 I( z ˜ m )+[I( z ˜ m )+I( z ˜ m+1 )]/2 ,
Ψ( z ˜ m )=arctg[ E x ( z ˜ m )/ E y ( z ˜ m )],
β H α (μΔx,νΔy,ρΔz,(σ+1/2)Δt)= β H α,μ,ν,ρ (σ+1/2) =(1/Δβ) m=l r a m H α,μ+m δ μβ ,ν+m δ ν,ρ +m δ ρβ (σ+1/2) .
Δ β E α (μΔx,νΔy,ρΔz,σΔt)= Δ β E α,μ,ν,ρ (σ) =(1/Δβ) m=r l a m E α,μ+m δ μβ ,ν+m δ νβ ,ρ+m δ ρβ (σ) .
H x,μ,ν,ρ (σ+1/2) H x,μ,ν,ρ (σ1/2) cΔt = a 2 E y,μ,ν,ρ+2 (σ) + a 1 E y,μ,ν,ρ+1 (σ) + a 0 E y,μ,ν,ρ (σ) + a 1 E y,μ,ν,ρ1 (σ) Δz + + a 2 E z,μ,ν+2,ρ (σ) + a 1 E z,μ,ν+1,ρ (σ) + a 0 E z,μ,ν,ρ (σ) + a 1 E y,μ,ν1,ρ (σ) Δy ,
H y,μ,ν,ρ (σ+1/2) H y,μ,ν,ρ (σ1/2) cΔt = a 2 E z,μ+2,ν,ρ (σ) + a 1 E z,μ+1,ν,ρ (σ) + a 0 E z,μ,ν,ρ (σ) + a 1 E z,μ1,ν,ρ (σ) Δx + + a 2 E x,μ,ν,ρ+2 (σ) + a 1 E x,μ,ν,ρ+1 (σ) + a 0 E x,μ,ν,ρ (σ) + a 1 E x,μ,ν,ρ1 (σ) Δz ,
H z,μ,ν,ρ (σ+1/2) H z,μ,ν,ρ (σ1/2) cΔt = a 2 E x,μ,ν+2,ρ (σ) + a 1 E x,μ,ν+1,ρ (σ) + a 0 E x,μ,ν,ρ (σ) + a 1 E x,μ,ν1,ρ (σ) Δy + + a 2 E y,μ+2,ν,ρ (σ) + a 1 E y,μ+1,ν,ρ (σ) + a 0 E y,μ,ν,ρ (σ) + a 1 E y,μ1,ν,ρ (σ) Δx ,
D x,μ,ν,ρ (σ+1) D x,μ,ν,ρ (σ) cΔt = a 2 H z,μ,ν2,ρ (σ+1/2) + a 1 H z z,μ,ν1,ρ (σ+1/2) + a 0 H z,μ,ν,ρ (σ+1/2) + a 1 H z,μ,ν+1,ρ (σ+1/2) Δy a 2 H y,μ,ν,ρ2 (σ+1/2) + a 1 H y,μ,ν,ρ1 (σ+1/2) + a 0 H y,μ,ν,ρ (σ+1/2) + a 1 H y,μ,ν,ρ+1 (σ+1/2) Δz ,
D y,μ,ν,ρ (σ+1) D y,μ,ν,ρ (σ) cΔt = a 2 H x,μ,ν,ρ2 (σ+1/2) + a 1 H z x,μ,ν,ρ1 (σ+1/2) + a 0 H x,μ,ν,ρ (σ+1/2) + a 1 H x,μ,ν,ρ+1 (σ+1/2) Δz a 2 H z,μ2,ν,ρ (σ+1/2) + a 1 H z,μ1,ν,ρ (σ+1/2) + a 0 H z,μ,ν,ρ (σ+1/2) + a 1 H z,μ+1,ν,ρ (σ+1/2) Δx ,
D z,,μ,ν,ρ (σ+1) D z,,μ,ν,ρ (σ) cΔt = a 2 H y,μ2,ν,ρ (σ+1/2) + a 1 H y,μ1,ν,ρ (σ+1/2) + a 0 H y,,μ,ν,ρ (σ+1/2) + a 1 H y,μ+1,ν,ρ (σ+1/2) Δx a 2 H x,μ,ν2,ρ (σ+1/2) + a 1 H x,μ,ν1,ρ (σ+1/2) + a 0 H x,,μ,ν,ρ (σ+1/2) + a 1 H x,μ,ν+1,ρ (σ+1/2) Δy ,
E x,y,z (μΔx,νΔy,ρΔz,σΔt)= E x,y,z (0) exp[i( k x μΔx+ k y νΔy+ k z ρΔzωσΔt)],
H x,y,z (μΔx,νΔy,ρΔz,σΔt)= H x,y,z (0) exp[i( k x μΔx+ k y νΔy+ k z ρΔzωσΔt)].
( R 0 0 0 Q z Q y 0 R 0 Q z 0 Q x 0 0 R Q y Q x 0 0 G z G y R 0 0 G z 0 G x 0 R 0 G y G x 0 0 0 R )( E x (0) E y (0) E z (0) H x (0) H y (0) H z (0) )=( 0 0 0 0 0 0 ),
sin 2 (ωΔt/2)/ (cΔt/2) 2 =F( k x ,Δx)+F( k y ,Δy)+F( k z ,Δz).
F( k β ,Δβ)=[25+2cos(3 k β Δβ)18cos( k β Δβ)9cos(2 k β Δβ)]/[18 (Δβ) 2 ].
ω=ck[1+ (ckΔt) 2 /24(Δ x 4 k x 6 +Δ y 4 k y 6 +Δ z 4 k z 6 )/30 k 2 +o(Δ t 2 +Δ x 4 +Δ y 4 +Δ z 4 )].
( θ 1 θ 2 θ 3 0 θ 4 θ 5 θ 6 θ 7 θ 8 θ 4 0 θ 9 θ 10 θ 11 θ 12 θ 5 θ 9 0 0 θ 13 θ 14 1 0 0 θ 13 0 θ 15 0 1 0 θ 14 θ 15 0 0 0 1 )( E x,μ,ν,ρ (n) E y,μ,ν,ρ (n) E z,μ,ν,ρ (n) H x,μ,ν,ρ (n1/2) H y,μ,ν,ρ (n1/2) H z,μ,ν,ρ (n1/2) )=( E x,μ,ν,ρ (n+1) E y,μ,ν,ρ (n+1) E z,μ,ν,ρ (n+1) H x,μ,ν,ρ (n+1/2) H y,μ,ν,ρ (n+1/2) H z,μ,ν,ρ (n+1/2) ).
(Λ1) 2 ( Λ 2 Λ{( c 2 Δ t 2 /ε)[ G x Q x + G y Q y + G z Q z ]+2}+1 ) 2 =0,
( c 2 Δ t 2 /ε)[F( k x ,Δx)+F( k y ,Δy)+F( k z ,Δz)]4.

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