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Complex phase shift per cell, comprising Re Ψ (curves 1, 2, and 3) and Im Ψ (curves 4 and 5), in the case of a planar periodic photonic crystal with ε 1 = µ 1 = 1, ε 2 = 3 − jε 2 , and a = b = 1 for ε 2 = 0 (curves 1 and 4), ε 2 = 0.1 (curve 2), and ε 2 = 1.0 (curves 3 and 5). TABLE 1. Dependence of ImΨ on k 0 , α, and β for ∆ω/ω r = 0.001 in the case of a planar structure with parameters corresponding to ImΨ for k 0r = ω r √ ε 0 µ 0 = 1.0 k 0 α = 0.0 α = 1.0 β = 0 β = 1 β = −1 β = 0 β = −1 β = −2 0.85 0.000 0.000 0.000 0.242 0.242 0.242 1.00 0.219 0.504 −0.070 0.504 0.219 −0.070 1.15 0.500 0.500 0.500 0.827 0.827 0.828 ImΨ for k 0r = ω r √ ε 0 µ 0 = 1.15 k 0 α = 0.0 α = 1.0 β = 0 β = 1 β = −1 β = 0 β = −1 β = −2 0.85 0.000 0.000 0.000 0.242 0.242 0.242 1.00 0.219 0.219 0.219 0.504 0.504 0.504 1.15 0.500 0.828 0.172 0.828 0.500 0.172
Source publication
We obtain representations of scalar and dyadic Green’s functions for one-, two-, and three-dimensional periodic photonic crystals.
On this basis, volumetric, surface, and mixed surface-volumetric integral and integro-differential equations are deduced.
Numerical results of solution of the obtained equations for some types of structures are presente...
Contexts in source publication
Context 1
... can easily be verified that this expression, obtained by the matching method, also results from an exact solution of the corresponding one-dimensional integral equation. Figure 4 shows the results of study of the influence of losses on the complex phase shift. Table 1 gives similar results in the presence of an active layer, in which losses are specified by the expression ε 2 (ω) = α + β exp{− ln(2) [(ω − ω r )/∆ω] 2 }. ...
Context 2
... outside the resonant-absorption line with frequency ω r , β is the absorption intensity, and ∆ω is the half-power width of the spectral line. Simulation of the 1D structure of a PC with a finite number of layers by using Green's function (10) as well as by the method of transmission matrices yields almost complete coincidence of the results with Fig. 4 for N ≥ 20. The results of studies of 2D structures with inclusion of active elements are also in qualitative agreement with those presented ...
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Citations
... All real known LHM are bianisotropic with periodic metallic inclusions of complicated form (usually these elements are pins and split ring resonators, or -elements, spirals, helixes and some similar configurations). The electrophysical (electromagnetic) parameters of metamaterials must be obtained by homogenization [6,[33][34][35][36][37][38][39][40][41][42][43][44][45][46][47][48]. It is fulfilled by inverse problem solutions and averaging based on full-wave analysis of periodic structure. ...
... It is fulfilled by inverse problem solutions and averaging based on full-wave analysis of periodic structure. It is necessary for this to solve many times the direct problems for dispersion and field determination using the rigorous methods (for example, integral equation method, or plane wave expansion method) [48]. The homogenization is also based on the models of media, for example, in the form ...
... are resulted from the homogenization are dependent on averaging method and determined, at least, for wave length D , where D is the characteristic dimension connected with the region of averaging (for example, the cell period). The homogenization procedure in addition to calculation of averaged cell dipole moments may be based on least-square analysis (minimization) of rigorous (full-wave) and model DE solutions, or least-square analysis (minimization) for corresponding plane wave diffraction results for structure vacuum-metamaterial with different angles of hade and polarizations to boundary [10,45,48]. It is so as the Ewald-Oseen extinction theorem [43] in this case may be proven. ...
It has been shown that for left-handed metamaterials and generally for
negative refraction media the refraction index cannot be entered unequivocally
and cannot be considered as real, and especially as negative. This index for
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... Waveguide properties and dispersion in a variety of perfectly conducting and impedance two-dimensional periodic pin (wire) structures (see Fig. 1) have recently been investigated [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18]. Such structures are referred usually as metallic photonic crystals (PC) or wire media (WM). ...
... where S is the cross-section area, λ ε π ε / 2 0 = = k k , and 2D Green's function reads as [3,4] ...
... Dispersion equation is obtained multiplying (3) by E z (x,y) and integrating over the cross-section. Using the following relation in polar coordinates (see [3,4]) ...
Metamaterials, made in form of periodically arranged metal and dielectric cylindrical inclusions, have been investigated
on the basis of the integral equation method, based on periodic Green's function. Metal rods are described in terms of
complex permittivity [see manuscript]. Along the rods terahertz and infrared waves propagate substantially with the speed of
light c and small losses weakly depending on the transverse wave number, whereas in the optical range, in particular in
the shortwave part of a spectrum, they turn into the slow -waves of a dielectric waveguide.
... The homogenization method based on determining the reflection coefficient for ε ˜ ε ˜ normal incidence of a planar wave was used; the base material was infinite and not semiiinfinite, which complicates the application of the results obtained in [10]. Our study is based on the method of Green's functions of periodically arranged sources, as well as on integral and integroodifferential equations11121314. We are using the homogenization methods described in567891011121314. ...
... Our study is based on the method of Green's functions of periodically arranged sources, as well as on integral and integroodifferential equations11121314. We are using the homogenization methods described in567891011121314. ...
... To obtain a numerr ical result, we will use the approximation of small elecc trical sizes of inclusions. Inclusions emit as secondary sources with polarization current (1) Further, we will use the scalar Green function (GF) with periodically arranged phased sources (periodic GF)11121314. The free (natural) waves emitted photoo nic crystals produced at inclusion polarization curr rents with a phase shift, which in turn sustain (excite) a wave. ...
The method of integral equations, which is based on the Green function
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and dielectric inclusions into a rectangular cubic lattice in a
dielectric medium (matrix). Dielectric inclusions in the form of
parallelepipeds and cubes are considered, as well as similar metallic
inclusions (perfectly conducting metal rods) and 1D nanosized structures
with metallic layers. Metal inclusions are investigated in the case of
ideal conduction and in the case of penetration of a field into the
metal, which is simulated as an electron plasma. The results are
applicable from microwave of optical frequencies.
... В ряде работ, например, [1][2][3] были исследованы электрофизические свойства металлических проволочных фотонных кристаллов (ФК) -периодических метаматериалов (рис. 1), построены их зонные структуры. ...
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... Most studies of MPCs are based on various approxii mate models or software packages based on the grid method or the finite element method. The approach applied in this work is based on the Green's function (GF) method and the integroodifferential (IDE) or integral equation (IE) method [29, 30] in the thin wire approximation developed in313233343536. In MPC investigations, it is important to calculate a band structure (Brillouin diagram) in order to deterr mine the presence and shape of bandgaps and to obtain the electrophysical parameters29303132333435363738. ...
... The approach applied in this work is based on the Green's function (GF) method and the integroodifferential (IDE) or integral equation (IE) method [29, 30] in the thin wire approximation developed in313233343536. In MPC investigations, it is important to calculate a band structure (Brillouin diagram) in order to deterr mine the presence and shape of bandgaps and to obtain the electrophysical parameters29303132333435363738. The calculation of such diagrams requires much time; therefore, it is important to choose optimal values for the parameters that substantially affect the calculation rate, namely, the number of current harmonics taken into account in calculations and the number of space harmonics taken into account in GF. ...
... Here, k s = (2s – 1)π/l, and the origin of coordinates is shifted toward the center of the wire as compared toFig. 1 so that the current at the ends is zero, and z 0 is the unit vector in the z direction. For MPC, we use the scalar GF of periodically arranged innphase sources (periodic GF) [29, 30]. The free waves (eigenwaves) in an MPC generate phaseeshifted currents on the metall lic rods (wires), which, in turn, maintain (excite) a wave. ...
The band structures of three-dimensional metallic photonic crystals in the form of thin wires are calculated using integral equations and periodic Green’s functions with allowance for several tens of current harmonics and several thousands of space harmonics. The convergence of the algorithm, the accuracy of the results, and the dependence of the bandgap width are studied.
... PSs describe different physical phenomena [1]. Especially it is concerning towards following areas: crystalline semi-conductors and metals in solid-state and semiconductor physics [2], slow-wave structures (SWS) and periodic waveguides in electromagnetics and electronics [3], crystals in crystal optics [4], photonic crystals (PC) [5,6], periodic metamaterials (artificial media) [7] in electrodynamics, optics, and acoustoelectronics, X-ray and particle diffraction by crystals. So, recently the considerable interest to PPS arose in different branch of knowledge (electronics, slow-wave, electrodynamic PC and metamaterials structures, photonics, optics, nanostructure physics). ...
... The PC and SWS loss leads to the wave damping along the propagation direction, i.e. to space aperiodicity. At the same time the eigenmode dispersion curves are distorted and the bandgaps disappear, i.e. there is the wave propagation possibility with high attenuation [6]. This propagation exist when the finite SWS joins with the semi-infinite waveguides or for two-dimensional PC plate excitation. ...
... The metallic structures are described by surface IEs, and the a metallic-dielectric ones -by surface-volume IEs. The corresponding equations for the PSs are presented in [6]. Often it is necessary to consider the structures with some objects embedded into the dielectric background 'i. ...
Periodic structures (PSs) are interesting for many categories and fields of physics and chemistry, so correspondingly there is tremendous number of works devoted to their analysis. PSs describe different physical phenomena [1]. Especially it is concerning towards following areas: crystalline semi-conductors and metals in solid-state and semiconductor physics [2], slow-wave structures (SWS) and periodic waveguides in electromagnetics and electronics [3], crystals in crystal optics [4], photonic crystals (PC) [5,6], periodic metamaterials (artificial media) [7] in electrodynamics, optics, and acoustoelectronics, X-ray and particle diffraction by crystals. So, recently the considerable interest to PPS arose in different branch of knowledge (electronics, slow-wave, electrodynamic PC and metamaterials structures, photonics, optics, nanostructure physics). It is caused by the development of nanotechnology and also by such circumstance, that PS is the mathematical abstraction, and all real structures are the PPSs. The violation of periodicity arises owing to number of circumstance. The main is the finiteness of all real structures. This principal factor is essential here when the number of periods is not large. Its influence is sharply decreasing when the number of periods increases over a certain value, and the PPS is inherently similar (in the internal regions) to PS one. The periodicity-breaking factors also are the nonstationarity (aperiodicity) in time, the loss and amplification (generation). Notice, that in finite passive (lossy) or active PC and SWS structures the eigenfrequencies are complex. It means the time aperiodicity i.e. the nonstationarity.
... The IEs for different inclusions and GF (1) have been formulated in the paper [12]. The electromagnetic field is creating by the sources in the form of surface electric current density J on the S 0 and by the polarization currents in the volume V, using the electric and magnetic vector-potentials. ...
... But such problem is very complicated. To solve it by IE method it is necessary to present the fields in the plate using 2D-periodic source GF [12] and match with the fields of two half-spaces. Another way [5][6][7][8][9][10] is based on the perturbation theory and founded on field decomposition in the Maxwell equations by small perturbation parameters and by the averaging over the cell with getting zero-order, first-order and higher-order approximations. ...
... The AM is splitted or zoning by the cells i a N , which are numbered by the vector n We consider that the translation vectors form right-hand systems. To formulate the considered problems the more convenient is the periodic sources Green's function (GF) approach[12] which leads to the IE in one (usually in zero) cell 0 . Such scalar GF has the view[12] ...
The methods of homogenization for periodic metamaterials or artificial media with periodic magnetodielectric, semiconductor, metallic, and cavity objects included in the background medium have been considered using the based on Greenpsilas functions integrodifferential equation methods.
... To formulate the considered problems the more convenient is the periodic sources Green's function (GF) approach [26][27][28][29][30][31][32] which leads to the IE in one (usually in zero) cell 0 Ω . ...
... One more approach may be established on minimal mean-square discrepancy between the AS and the effective medium dispersion lows [31,32]. Since there is here the arbitrary assigned vector complex parameter k , the method allows to define the frequency-dependent tensors with taking into account the physical restriction in their elements in wide frequency range including the case a < λ . ...
The integrodifferential equations which describe the pseudo-periodic (quasi-periodic) structures in quantum mechanics and electromagnetics have been proposed and the algorithms of its solutions have been suggested.
... To formulate the considered problems the more convenient is the periodic sources Green's function (GF) approach [26][27][28][29][30][31][32] which leads to the IE in one (usually in zero) cell 0 Ω . ...
... One more approach may be established on minimal mean-square discrepancy between the AS and the effective medium dispersion lows [31,32]. Since there is here the arbitrary assigned vector complex parameter k , the method allows to define the frequency-dependent tensors with taking into account the physical restriction in their elements in wide frequency range including the case a < λ . ...
In this paper the method of spatio-temporal series for integration of Maxwell equations or wave equation has been developed using the finite element approach. As a result of development of nonstationary electromagnetics the necessity of effective numerical methods for nonstationary waves (pulses) propagation and diffraction is arising. The problems of pulse excitations and propagations are interesting both for linear and nonlinear dis- persive media. Usually the only set pulse propagation is considering using the method of propa- gator Green's function (GF) and rare - by the spectral integral calculation. These methods are not suitable for the diffraction on structures and for nonlinear media. Furthermore, the GFs are ex- plicitly known only for several (three) ideal (lossless) dispersion lows (1), whereas in the real dispersive media always there are losses, and the variety of the dispersion lows is sufficiently large. There are some difficulties to realize the mod-matching method for piecewise nonuniform media (or structures) and for nonhomogeneous ones in nonstationary case. At the same time the media must be nondispersive. The usage of inverse Fourier transform to inverse the spectral solu- tion is practically unreal even in the case when there is the simple analytical solution, not to men- tion even about numerical spectral algorithms. The most universal approach here is the special- time integral equation (IE) or integrodifferential equation (IDE) method based on special-time GF of free space or structures. It, particularly, eliminates paradoxes of "super-light velocity" in pulse propagation and tunneling (2-5). In this paper the method of spatio-temporal series for integration of Maxwell equations or wave equation has been developed using the finite element approach.
