Abstract

Material patterns with temporal modulation of permittivity provide an unprecedented platform for active light manipulation and multifunctional devices. The ongoing discovery of new classes of tunable optical materials combined with various multiphysics paradigms demands an appropriate computational tool to characterize and design the wave manipulation patterns with the additional degree of freedom (time). Here we present, for the first time, a fast and accurate numerical technique to compute light propagation through patterned layers with spatial and temporal modulation of permittivity by extending the well-established rigorous coupled-wave analysis framework. We expand the electric and magnetic field components of an arbitrarily shaped space-time unit cell into a basis of spatio-temporal harmonics and compute their modal field profiles as a set of eigenmodes of the unit cell. The relation between the wave amplitudes of the incident, reflected, and transmitted spatio-temporal harmonics are then computed by enforcing the boundary conditions at each interface of the pattern. The validity of the proposed method is rigorously verified against the results of a time-variant finite-difference time-domain technique. The applicability of the method to study dielectric and plasmonic time-modulated metasurfaces is exemplified by considering two metasurfaces with spatially-variant phase and frequency of temporal modulation, based on electro-optical modulation of materials such as silicon and graphene. Utilizing the developed technique, we demonstrate some recently reported phenomena associated with time-modulated metasurfaces, such as anomalous steering of higher-order frequency harmonics, and dynamic optical beam generation. Even though we restrict ourselves to one-dimensional patterns for simplicity, the developed method can be flexibly employed for the design, characterization and optimization of two-dimensional spatio-temporal modulation patterns of arbitrary composition and shape.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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2019 (1)

S. Inampudi, V. Toutam, and S. Tadigadapa, “Robust visibility of graphene monolayer on patterned plasmonic substrates,” Nanotechnology 30, 015202 (2019).
[Crossref]

2018 (11)

M. Minkov and S. Fan, “Localization and time-reversal of light through dynamic modulation,” Phys. Rev. B 97, 060301 (2018).
[Crossref]

S. Taravati, “Aperiodic space-time modulation for pure frequency mixing,” Phys. Rev. B 97, 115131 (2018).
[Crossref]

A. Mock, D. Sounas, and A. Alù, “Tunable orbital angular momentum radiation from angular-momentum-biased microcavities,” Phys. Rev. Lett. 121, 103901 (2018).
[Crossref] [PubMed]

S. Taravati, “Giant linear nonreciprocity, zero reflection, and zero band gap in equilibrated space-time-varying media,” Phys. Rev. Appl. 9, 064012 (2018).
[Crossref]

A. Forouzmand and H. Mosallaei, “Dynamic beam control via mie-resonance based phase-change metasurface: a theoretical investigation,” Opt. express 26, 17948–17963 (2018).
[Crossref] [PubMed]

A. Forouzmand, M. M. Salary, S. Inampudi, and H. Mosallaei, “A tunable multigate indium-tin-oxide-assisted all-dielectric metasurface,” Adv. Opt. Mater. 6, 1701275 (2018).
[Crossref]

M. M. Salary, S. Jafar-Zanjani, and H. Mosallaei, “Time-varying metamaterials based on graphene-wrapped microwires: Modeling and potential applications,” Phys. Rev. B 97, 115421 (2018).
[Crossref]

M. M. Salary, S. Jafar-Zanjani, and H. Mosallaei, “Electrically tunable harmonics in time-modulated metasurfaces for wavefront engineering,” New J. Phys.: 1801.10575 (2018).

M. Liu, D. A. Powell, Y. Zarate, and I. V. Shadrivov, “HuygensâĂŹ metadevices for parametric waves,” Phys. Rev. X 8, 031077 (2018).

N. Chamanara, Y. Vahabzadeh, and C. Caloz, “Simultaneous control of the spatial and temporal spectra of light with space-time varying metasurfaces,” arXiv preprint arXiv: 1808.03385 (2018).

S. Taravati and A. A. Kishk, “Dynamic modulation yields one-way beam splitting,” arXiv preprint arXiv: 1809.00347 (2018).

2017 (12)

Y. Shi, A. Cerjan, and S. Fan, “Invited article: Acousto-optic finite-difference frequency-domain algorithm for first-principles simulations of on-chip acousto-optic devices,” APL Photonics 2, 020801 (2017).
[Crossref]

S. Taravati and C. Caloz, “Mixer-duplexer-antenna leaky-wave system based on periodic space-time modulation,” IEEE Transactions on Antennas Propag. 65, 442–452 (2017).
[Crossref]

S. Taravati, N. Chamanara, and C. Caloz, “Nonreciprocal electromagnetic scattering from a periodically space-time modulated slab and application to a quasisonic isolator,” Phys. Rev. B 96, 165144 (2017).
[Crossref]

D. L. Sounas and A. Alù, “Non-reciprocal photonics based on time modulation,” Nat. Photonics 11, 774 (2017).
[Crossref]

Y. Shi, S. Han, and S. Fan, “Optical circulation and isolation based on indirect photonic transitions of guided resonance modes,” ACS Photonics 4, 1639–1645 (2017).
[Crossref]

Z. Zhu, P. G. Evans, R. F. Haglund, and J. G. Valentine, “Dynamically reconfigurable metadevice employing nanostructured phase-change materials,” Nano letters 17, 4881–4885 (2017).
[Crossref] [PubMed]

M. C. Sherrott, P. W. Hon, K. T. Fountaine, J. C. Garcia, S. M. Ponti, V. W. Brar, L. A. Sweatlock, and H. A. Atwater, “Experimental demonstration of> 230 phase modulation in gate-tunable graphene–gold reconfigurable mid-infrared metasurfaces,” Nano letters 17, 3027–3034 (2017).
[Crossref]

J. Cheng, S. Inampudi, and H. Mosallaei, “Optimization-based dielectric metasurfaces for angle-selective multifunctional beam deflection,” Sci. reports 7, 12228 (2017).
[Crossref]

S. Colburn, A. Zhan, and A. Majumdar, “Tunable metasurfaces via subwavelength phase shifters with uniform amplitude,” Sci. reports 7, 40174 (2017).
[Crossref]

A. Forouzmand and H. Mosallaei, “All-dielectric c-shaped nanoantennas for light manipulation: Tailoring both magnetic and electric resonances to the desire,” Adv. Opt. Mater. 5, 1700147 (2017).
[Crossref]

Y. Yang, A. E. Miroshnichenko, S. V. Kostinski, M. Odit, P. Kapitanova, M. Qiu, and Y. S. Kivshar, “Multimode directionality in all-dielectric metasurfaces,” Phys. Rev. B 95, 165426 (2017).
[Crossref]

D. Ramaccia, D. L. Sounas, A. Alù, A. Toscano, and F. Bilotti, “Doppler cloak restores invisibility to objects in relativistic motion,” Phys. Rev. B 95, 075113 (2017).
[Crossref]

2016 (8)

J. S. Martínez-Romero, O. Becerra-Fuentes, and P. Halevi, “Temporal photonic crystals with modulations of both permittivity and permeability,” Phys. Rev. A 93, 063813 (2016).
[Crossref]

J. R. Reyes-Ayona and P. Halevi, “Electromagnetic wave propagation in an externally modulated low-pass transmission line,” IEEE Transactions on Microw. Theory Tech. 64, 3449–3459 (2016).
[Crossref]

T. J. Smy and S. Gupta, “Exact finite-difference time-domain modelling of broadband huygens’ metasurfaces with lorentz dispersions,” arXiv preprint arXiv: 1609.05575 (2016).

Y.-W. Huang, H. W. H. Lee, R. Sokhoyan, R. A. Pala, K. Thyagarajan, S. Han, D. P. Tsai, and H. A. Atwater, “Gate-tunable conducting oxide metasurfaces,” Nano letters 16, 5319–5325 (2016).
[Crossref] [PubMed]

J. Park, J.-H. Kang, S. J. Kim, X. Liu, and M. L. Brongersma, “Dynamic reflection phase and polarization control in metasurfaces,” Nano letters 17, 407–413 (2016).
[Crossref] [PubMed]

Y. Hadad, J. C. Soric, and A. Alu, “Breaking temporal symmetries for emission and absorption,”Proc. Natl. Acad. Sci. 13201517363 (2016).

Y. Shi, W. Shin, and S. Fan, “Multi-frequency finite-difference frequency-domain algorithm for active nanophotonic device simulations,” Optica 3, 1256–1259 (2016).
[Crossref]

D. Correas-Serrano, J. Gomez-Diaz, D. Sounas, Y. Hadad, A. Alvarez-Melcon, and A. Alù, “Nonreciprocal graphene devices and antennas based on spatiotemporal modulation,” IEEE Antennas Wirel. Propag. Lett. 15, 1529–1532 (2016).
[Crossref]

2015 (6)

A. Shaltout, A. Kildishev, and V. Shalaev, “Time-varying metasurfaces and lorentz non-reciprocity,” Opt. Mater. Express 5, 2459–2467 (2015).
[Crossref]

Y. Hadad, D. Sounas, and A. Alu, “Space-time gradient metasurfaces,” Phys. Rev. B 92, 100304 (2015).
[Crossref]

Z. Li, K. Yao, F. Xia, S. Shen, J. Tian, and Y. Liu, “Graphene plasmonic metasurfaces to steer infrared light,” Sci. reports 5, 12423 (2015).
[Crossref]

A. Tittl, A.-K. U. Michel, M. Schäferling, X. Yin, B. Gholipour, L. Cui, M. Wuttig, T. Taubner, F. Neubrech, and H. Giessen, “A switchable mid-infrared plasmonic perfect absorber with multispectral thermal imaging capability,” Adv. Mater. 27, 4597–4603 (2015).
[Crossref] [PubMed]

C. T. Phare, Y.-H. D. Lee, J. Cardenas, and M. Lipson, “Graphene electro-optic modulator with 30 ghz bandwidth,” Nat. Photonics 9, 511 (2015).
[Crossref]

J. Reyes-Ayona and P. Halevi, “Observation of genuine wave vector (k or β) gap in a dynamic transmission line and temporal photonic crystals,” Appl. Phys. Lett. 107, 074101 (2015).
[Crossref]

2014 (3)

A. O. Korotkevich, X. Ni, and A. V. Kildishev, “Fast eigensolver for plasmonic metasurfaces,” Opt. Mater. Express 4, 288–299 (2014).
[Crossref]

R. Magnusson, “Wideband reflectors with zero-contrast gratings,” Opt. letters 39, 4337–4340 (2014).
[Crossref]

B. Dana, L. Lobachinsky, and A. Bahabad, “Spatiotemporal coupled-mode theory in dispersive media under a dynamic modulation,” Opt. Commun. 324, 165–167 (2014).
[Crossref]

2013 (1)

D. L. Sounas, C. Caloz, and A. Alu, “Giant non-reciprocity at the subwavelength scale using angular momentum-biased metamaterials,” Nat. communications 4, 2407 (2013).
[Crossref]

2012 (3)

M. Liu, X. Yin, and X. Zhang, “Double-layer graphene optical modulator,” Nano letters 12, 1482–1485 (2012).
[Crossref] [PubMed]

H. Lira, Z. Yu, S. Fan, and M. Lipson, “Electrically driven nonreciprocity induced by interband photonic transition on a silicon chip,” Phys. review letters 109, 033901 (2012).
[Crossref]

V. Liu and S. Fan, “S4 : A free electromagnetic solver for layered periodic structures,” Comput. Phys. Commun. 183, 2233 – 2244 (2012).
[Crossref]

2011 (2)

L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and et al., “Graphene plasmonics for tunable terahertz metamaterials,” Nat. nanotechnology 6, 630 (2011).
[Crossref]

Y. Sivan and J. B. Pendry, “Time reversal in dynamically tuned zero-gap periodic systems,” Phys. review letters 106, 193902 (2011).
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2010 (4)

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M. S. Kang, A. Brenn, and P. S. J. Russell, “All-optical control of gigahertz acoustic resonances by forward stimulated interpolarization scattering in a photonic crystal fiber,” Phys. review letters 105, 153901 (2010).
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X. Ni, Z. Liu, A. Boltasseva, and A. V. Kildishev, “The validation of the parallel three-dimensional solver for analysis of optical plasmonic bi-periodic multilayer nanostructures,” Appl. Phys. A 100, 365–374 (2010).
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2009 (3)

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. photonics 3, 91 (2009).
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J. A. Dionne, K. Diest, L. A. Sweatlock, and H. A. Atwater, “Plasmostor: a metal-oxide-si field effect plasmonic modulator,” Nano Lett. 9, 897–902 (2009).
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2007 (3)

S. Longhi, “Stopping and time reversal of light in dynamic photonic structures via bloch oscillations,” Phys. Rev. E 75, 026606 (2007).
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2005 (2)

T. Yasui, E. Saneyoshi, and T. Araki, “Asynchronous optical sampling terahertz time-domain spectroscopy for ultrahigh spectral resolution and rapid data acquisition,” Appl. Phys. Lett. 87, 061101 (2005).
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M. F. Yanik, W. Suh, Z. Wang, and S. Fan, “Stopping light in a waveguide with an all-optical analog of electromagnetically induced transparency,” Phys. review letters 93, 233903 (2004).
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J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-Gonzalez, “Reflection and transmission of a wave incident on aslab with a time-periodic dielectric function Ït (t),” Phys. Rev. A 79, 053821 (2009).
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J. Cheng, S. Inampudi, and H. Mosallaei, “Optimization-based dielectric metasurfaces for angle-selective multifunctional beam deflection,” Sci. reports 7, 12228 (2017).
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J. A. Dionne, K. Diest, L. A. Sweatlock, and H. A. Atwater, “Plasmostor: a metal-oxide-si field effect plasmonic modulator,” Nano Lett. 9, 897–902 (2009).
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J. A. Dionne, K. Diest, L. A. Sweatlock, and H. A. Atwater, “Plasmostor: a metal-oxide-si field effect plasmonic modulator,” Nano Lett. 9, 897–902 (2009).
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Z. Zhu, P. G. Evans, R. F. Haglund, and J. G. Valentine, “Dynamically reconfigurable metadevice employing nanostructured phase-change materials,” Nano letters 17, 4881–4885 (2017).
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A. Forouzmand, M. M. Salary, S. Inampudi, and H. Mosallaei, “A tunable multigate indium-tin-oxide-assisted all-dielectric metasurface,” Adv. Opt. Mater. 6, 1701275 (2018).
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Y. Shi, S. Han, and S. Fan, “Optical circulation and isolation based on indirect photonic transitions of guided resonance modes,” ACS Photonics 4, 1639–1645 (2017).
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L. Ju, B. Geng, J. Horng, C. Girit, M. Martin, Z. Hao, H. A. Bechtel, X. Liang, A. Zettl, Y. R. Shen, and et al., “Graphene plasmonics for tunable terahertz metamaterials,” Nat. nanotechnology 6, 630 (2011).
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M. S. Kang, A. Brenn, and P. S. J. Russell, “All-optical control of gigahertz acoustic resonances by forward stimulated interpolarization scattering in a photonic crystal fiber,” Phys. review letters 105, 153901 (2010).
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Y.-W. Huang, H. W. H. Lee, R. Sokhoyan, R. A. Pala, K. Thyagarajan, S. Han, D. P. Tsai, and H. A. Atwater, “Gate-tunable conducting oxide metasurfaces,” Nano letters 16, 5319–5325 (2016).
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Z. Li, K. Yao, F. Xia, S. Shen, J. Tian, and Y. Liu, “Graphene plasmonic metasurfaces to steer infrared light,” Sci. reports 5, 12423 (2015).
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Figures (8)

Fig. 1
Fig. 1 Space-time diffraction from a spatio-temporally modulated grating.
Fig. 2
Fig. 2 Comparison between the results obtained by the RSTWA, and a time-variant FDTD method (please see Appendix I), corresponding to a periodic arrangement of time-modulated dielectric bars with sinusoidal permittivity variation: ϵ(t) = ϵs[1+δ sin(2π fmt)]. Parameters: f0 = 3 THz (excitation frequency), D = 50 µm (width and height of the bars), Λ = 95 µm (periodicity), ϵs = 3.9, δ = 0.3, fm = 0.125 f0. (a) Magnitude of the transmission coefficient versus temporal harmonic index corresponding to the TM(p) polarization. The left pane represents the fundamental, and the right pane represents the ±1 spatial diffraction orders, respectively. (b), (c) Near-field plots of the magnitude of the x-component of the electric field, corresponding to the TM polarizations obtained by RSTWA and FDTD, respectively. Near-fields are plotted for −1, 0, +1 temporal harmonics (left, center, and right panes, respectively). (d)–(f) Same plot as (a)–(c) obtained for the TE(s) polarization.
Fig. 3
Fig. 3 Schematic depiction of a spatially phase-variant time-modulated dielectric metasurface. (a) An array of silicon nanobars with p-i-n configuration located on top of a layer of SiO2 backed by a silver mirror are modulated using an RF biasing network incorporating phase shifters. (b) The spatio-temporal profile of the refractive index with spatially variant phase shift, resembling a space-time diffraction grating. The dashed lines in (a) and (b) denote the line along which spatio-temporal refractive index profile is plotted. (c) Cross-sectional view of the unit cell.
Fig. 4
Fig. 4 (a) Frequency conversion efficiency of the dielectric time-modulated metasurface of Fig.3 in the logarithmic scale as a function of height and width of silicon nanobar. The dashed lines denote the branches of resonant modes. (b), (c-1), (d-1) and (e) depict the electric field distribution corresponding to the widths and heights of silicon nanobar at the resonant points denoted in the colormap of (a). The distributions exhibit characteristics of MD, ED-Fabry-Pérot, guided and guided-Fabry-Pérot resonant modes, respectively. (c-2) and (d-2) compare the spectrum of generated frequency harmonics at the resonant points of (c) and (d) corresponding to ED-Fabry-Pérot and guided mode resonances.
Fig. 5
Fig. 5 The wavefronts of the electric field corresponding to (a) down-modulated, (b) fundamental and (c) up-modulated frequency harmonics generated by the spatially phase-variant dielectric time-modulated metasurface. The fundamental frequency harmonic is specularly reflected while the first-order frequency harmonics are anomalously bent toward angles of θ±1 ≈±20°.
Fig. 6
Fig. 6 Schematic depiction of a spatially frequency-variant time-modulated plasmonic metasurface. (a) An array of graphene micro-ribbons located on top of a layer of SiO2 backed by a silver mirror are modulated with RF sources of different frequencies. (b) A spatially frequency-variant profile for surface conductivity. The dashed lines in (a) and (b) denote the line along which the surface conductivity profile is plotted. (c) Cross-sectional view of the unit cell.
Fig. 7
Fig. 7 (a) Frequency conversion efficiency of the time-modulated graphene metasurface as a function of width of the graphene micro-ribbons and height of the dielectric spacer layer. The dashed lines denote the plasmonic resonant regime. (b) Nearfield distribution of the electric field at the plasmonic resonance of the micro-ribbon, when w = 1 µm and h = 4.34 µm. (c) Generated frequency harmonic spectrum by the time-modulated metasurface at the plasmonic resonant regime.
Fig. 8
Fig. 8 Wavefront of the time-domain electric field at different time instants within a time frame of T = 1 ns corresponding to the spatially frequency-variant time-modulated graphene metasurface. Three dominant dynamic beams can be seen as a result of the coherent interference of fundamental, up-and down-modulated frequency harmonics. The beam resulting from the interference of fundamental harmonic is stationary while the other two beams are dynamic and scan the space in time with a 180° angle-of-view in opposite directions.

Equations (42)

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ε ( m , n ) = 1 Λ x 1 Λ t 0 Λ x 0 Λ t d x d t ϵ ( x , t ) exp ( i q x ; m x i q t ; n t )
ω n = ω 0 + n Ω , a n d k x , m = k x , 0 + m 2 π / Λ x ,
{ E , H } { x , y } ; l ( r ) = m = n = e i k x , m x e i ω n t { , } { x , y } ; l ( m , n ) ( z )
[ x ; l ( α ) ( z ) y ; l ( α ) ( z ) x ; l ( α ) ( z ) y ; l ( α ) ( z ) ] = [ W l W l V l V l ] [ a l ; s ( α ) + e i k z , l ( α ) ( z z l 1 ) a l ; p ( α ) + e i k z , l ( α ) ( z z l 1 ) a l ; s ( α ) + e i k z , l ( α ) ( z l 1 ) a l ; p ( α ) + e i k z , l ( α ) ( z l 1 ) ]
W l = [ O 1 ϵ 0 ϵ l W 1 K z ; l I O ] ; V l = [ 1 μ 0 μ l W 1 K z ; l O O I ]
W ( α , α ) = ω α ; K z ; l ( α , α ) = ( ω α / c ) 2 ϵ l ( k x , α ) 2 .
[ ˙ x ; l ( α ) ( z ) ˙ y ; l ( α ) ( z ) ˙ x ; l ( α ) ( z ) ˙ y ; l ( α ) ( z ) ] = [ O A l ( 1 ) A l ( 2 ) O ] [ x ; l ( α ) ( z ) y ; l ( α ) ( z ) x ; l ( α ) ( z ) y ; l ( α ) ( z ) ]
A l ( 1 ) = [ O i ( μ 0 μ r W 1 ϵ 0 K x Ξ l 1 W 1 K x ) i μ 0 μ r W O ] ; A l ( 2 ) = [ O i ( 1 μ 0 μ r K x W 1 K x ϵ 0 W Ξ l ) i ϵ 0 W Ξ l O ]
Ξ ( α , β ) = 1 Λ x Λ t 0 Λ x 0 Λ t d x d t ϵ l ( x , t , ω α β ) exp { i 2 π [ ( m α m β ) x / Λ x ( n α n β ) t / Λ t ] } ,
[ ¨ x ; l ( α ) ( z ) ¨ y ; l ( α ) ( z ) ] = A l ( 1 ) A l ( 2 ) [ x ; l ( α ) ( z ) y ; l ( α ) ( z ) ] .
( A l ( 1 ) A l ( 2 ) + K z ; l 2 I ) [ x ; l ( α ) ( z ) y ; l ( α ) ( z ) ] = O .
V l = A l ( 2 ) W l K z ; l 1
[ x ; l + 1 ( α ) ( z l ) y ; l + 1 ( α ) ( z l ) x ; l + 1 ( α ) ( z l ) y ; l + 1 ( α ) ( z l ) ] = [ x ; l ( α ) ( z l ) y ; l ( α ) ( z l ) x ; l ( α ) ( z l ) y ; l ( α ) ( z l ) ] .
[ a l ( α ) a l + 1 ( α ) + ] = [ r l + r l t l + t l ] [ a l ( α ) + e i k z , l ( α ) ( z l z l 1 ) a l + 1 ( α ) e i k z , l ( α ) ( z l + 1 z l ) ]
r l + = ( a + b ) 1 ( a b ) , t l = 2 ( a + b ) 1 t l + = 0.5 ( a + b ) + 0.5 ( a b ) r l + , r l = ( a b ) ( a + b ) 1
Y l = ( I r l e i K z , l + 1 ( z l + 1 z l ) R l + 1 ) 1 ( t l + e i K z , l + 1 ( z l z l 1 ) ) R l = r l + e i K z , l ( z l z l 1 ) + t l e i K z , l + 1 ( z l + 1 z l ) R l + 1 T l
sin ( θ ± 1 ( t ) ) = ± ( 2 π Δ f m k 0 Λ ± 1 ) t
× E = μ 0 μ r H t , × H = ϵ 0 t ( ϵ ( x , t ) E )
E y z = μ 0 μ r H x t
H y z = ϵ 0 t ( ϵ ( x , t ) E x )
E y x = μ 0 μ r H z t
H y x = ϵ 0 t ( ϵ ( x , t ) E z )
H x z H z x = ϵ 0 t ( ϵ ( x , t ) E y )
E z x E x z = μ 0 μ r H y t
x { E , H } { x , y , z } ( r ) = i α k x , α e i k x , α x e i ω α t { , } { x , y , z } ( α ) ( z )
t { E , H } { x , y , z } ( r ) = i α ω α e i k x , α x e i ω α t { , } { x , y , z } ( α ) ( z )
ϵ ( x , t ) E { x , y , z } ( r ) = α ϵ α e ( i q x , α x + i q t , α t ) α e i k x , α x e i ω α t { x , y , z } ( α ) ( z ) = α , β ϵ α , β e i k x , β x e i ω β t { x , y , z } ( β ) ( z )
t ϵ ( x , t ) E { x , y , z } ( r ) = i α , β ω β ϵ α , β e i k x , β x e i ω β t { x , y , z } ( β ) ( z )
z y α ( z ) = i ϵ 0 α , β ω β ϵ α , β x ( β ) ( z )
˙ y = i ϵ 0 W Ξ x .
z x α ( z ) = i μ 0 μ r ω α y ( α ) ( z ) + i k x , α z α
˙ x = i { μ 0 μ r W 1 ϵ 0 K x Ξ 1 W 1 K x } y
z y α ( z ) = i μ 0 μ r ω α x ( α ) ( z )
˙ x = i μ 0 μ r W x
z x α ( z ) = i α , β { 1 μ 0 μ r k x , α ω α 1 k x , α ϵ 0 ω β ϵ α , β } y ( β ) ( z )
˙ x = i { 1 μ 0 μ r K x W 1 K x ϵ 0 W Ξ } y
D t = × H J
D = ϵ ( r , t ) E = ϵ 0 ϵ r ( r , t ) E
J = σ e ( r , t ) E + J imp
t ( ϵ r ( r , t ) E ) = 1 ϵ 0 ( × H σ e ( r , t ) E J imp ) .
E n + 1 = C 1 E n + C 2 ( × H n + 1 / 2 J imp n + 1 / 2 )
C 1 = 2 ϵ 0 ϵ r n σ e n + 1 / 2 Δ t 2 ϵ 0 ϵ r n + 1 + σ e n + 1 / 2 Δ t ; C 2 = 2 Δ t 2 ϵ 0 ϵ r n + 1 + σ e n + 1 / 2 Δ t .

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