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# TIME-DOMAIN MODELING OF ELECTROMAGNETIC WAVE PROPAGATION IN COMPLEX MATERIALS_1999(2014)

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### TIME-DOMAIN MODELING OF ELECTROMAGNETIC WAVE PROPAGATION IN COMPLEX MATERIALS_1999(2014)

1. 1. TIME-DOMAIN MODELING OF ELECTROMAGNETIC WAVE PROPAGATION IN COMPLEX MATERIALS J. Paul, C. Christopoulos and D. W. P. Thomas School of Electrical and Electronic Engineering University of Nottingham Nottingham, NG7 2RD, United Kingdom ABSTRACT In this study, the Transmission-Line Modeling (TLM) method is ex- tended and applied to the time-domain simulation of electromagnetic wave propagation in materials displaying magnetoelectric coupling. The formulation is derived from Maxwell’s equations and the constitutive relations using bilinear Z-transform methods leading to a general Pad´e system. This approach is applicable to all frequency-dependent lin- ear materials including anisotropic and bianisotropic media. The close agreements between the results obtained from the time-domain simula- tions and analyses for examples involving isotropic and uniaxial chiral materials indicates that the numerical approach can be applied with conﬁdence to problems having no analytic solution. 1. INTRODUCTION Both the Transmission-Line Modeling (TLM) method [1] and the Finite- Diﬀerence Time-Domain (FDTD) method [2] are diﬀerential techniques useful for the time-domain solution of electromagnetic problems. Previ- ous researchers have used FDTD for the simulation of electromagnetic wave propagation in anisotropic materials. As an example, in [3], a model of an anisotropic material with constant parameters was devel- oped. In [4], FDTD was extended to include the frequency-dependent anisotropic material properties of a magnetized plasma and to a mag- 1
2. 2. netized ferrite material in [5]. However, because of the oﬀsets between the electric and magnetic ﬁelds of half a space-step and half a time-step in a FDTD grid, as noted in [3, 4], the update scheme for 3-D prob- lems requires spatial and temporal averaging. This leads to diﬃculties in the description of material discontinuities, boundary conditions and materials exhibiting magnetoelectric coupling. One of the main diﬀerences between FDTD and TLM is that in TLM the electric and magnetic ﬁelds are solved at the same point in space- time. This leads to the proposition that TLM may be easier to apply than FDTD for the simulation of electromagnetic wave propagation in anisotropic materials. Also the condensed nature of the TLM space- time grid oﬀers the possibility of describing bianisotropic materials. Previous investigators have developed TLM procedures for anisotropic materials, in [6] an anisotropic medium with constant parameters was studied and in [7, 8], magnetized plasmas and ferrites were examined. The model detailed in this article is an extension of the isotropic formulation of Debye (ﬁrst-order) frequency-dependent materials [9] to include frequency-dependent bianisotropic materials [10]. In this ap- proach, the model is developed throughout by the systematic applica- tion of bilinear Z-transform techniques [11]. This leads to a general formulation as a Pad´e system which is applicable to the time-domain modeling of all linear frequency-dependent complex materials. The approach is validated using two straightforward examples in which the plane wave reﬂection and transmission of an isotropic second- order chiral slab having a second-order frequency dependence [10] and the reﬂectivity of a uniaxial second-order chiral material [12] are stud- ied. The results show excellent agreement between the time-domain models and frequency-domain analyses. Also presented are the time- domain results of pulse propagation in these materials. 2. FORMULATION OF THE TIME-DOMAIN MODEL Equation (1) expresses Maxwell’s curl equations in compact form using the notation for the ﬁelds, current and ﬂux densities of [13]. ∇ × H −∇ × E = Je Jm + ∂ ∂t D B (1) Equation (2) expresses the constitutive relations for the electric and 2
3. 3. magnetic ﬂux densities D and B. D B = ε0 E µ0 H + ε0χe ξr/c ζr/c µ0χm ∗ E H (2) In (2), χe and χm are the electric and magnetic susceptibility tensors and the dimensionless tensors describing the magnetoelectric coupling are ξr and ζr. Also in (2), ε0 and µ0 are the free-space permittivity and permeability, c is the speed of light in free-space and ∗ denotes a time-domain convolution. The constitutive relations for the electric and magnetic current densities Je and Jm are given in (3). Je Jm = Jef Jmf + σe σem σme σm ∗ E H (3) In (3), Jef and Jmf are the free electric and magnetic current densities, σe and σm are the electric and magnetic conductivity tensors and σem and σme are the magnetoelectric conductivity tensors. Although all conductive materials can be described in (2), for time-domain modeling it is useful to express conduction separately as (3). For example, a frequency-dependent σe is used for describing plasmas [9] and σe and σm are used in absorbing boundaries. In the general case, the tensors of the constitutive relations of (2) and (3) contain elements describing causal time functions. Substitution of (2) and (3) into (1) yields ∇×H − Jef −∇×E −Jmf − ∂ ∂t ε0 E µ0 H = σe σem σme σm ∗ E H + ∂ ∂t ε0χe ξr/c ζr/c µ0χm ∗ E H (4) The model detailed in this paper is a discrete time solution of (4) in a Cartesian grid, solving for the ﬁelds E and H at each time-step. The possibility of modeling magnetoelectric coupling by TLM is allowed for by the normalization of E and H so that the circuit representations of these quantities V and i both have the dimensions of volts using E = −V/∆ℓ , H = −i/(∆ℓ η0) (5) In (5), ∆ℓ is the space step and η0 is the intrinsic impedance of free- space. Similarly the free current densities are normalized to quantities if and Vf both with the dimensions of volts using Jef = −if /(∆ℓ2 η0) , Jmf = −Vf /(∆ℓ2 ) (6) 3
4. 4. The conductivity tensors are normalized so that their circuit represen- tations ge, gem, gme and rm are dimensionless, i.e. σe = ge/(∆ℓ η0), σem = gem/∆ℓ, σme = gme/∆ℓ, σm = rm η0/∆ℓ (7) Also, the time and spatial derivatives are normalized using ∂ ∂t = 1 ∆t ∂ ∂¯t , (∇ × . . .) = 1 ∆ℓ (¯∇ × . . .) (8) In (8), ∆t is the time-step of the time-domain simulation and ¯t is the normalized time. In 1-D models, ∆ℓ/∆t = c and in 3-D models ∆ℓ/∆t = 2c [1, 14]. Using (5), (6), (7) and (8) in (4) for a 3-D model leads to ¯∇ × i − if −¯∇ × V − Vf −2 ∂ ∂¯t V i = ge gem gme rm ∗ V i + 2 ∂ ∂¯t χe ξr ζr χm ∗ V i (9) The left-hand side of (9) involves the curl operations, the free sources and the time derivative of the ﬁelds. The solution of the left-hand side of (9) is detailed for the 1-D and 3-D cases in Appendix A, leading to the TLM spatial transform ¯∇ × i − if −¯∇ × V − Vf − 2 ∂ ∂¯t V i TLM −→ 2 Fr − 4 F where F = V i (10) On the right-hand side of (10), the excitation vector Fr is a function of the incident voltages and any free sources. The deﬁnition of Fr is given in Appendix A. For the modeling of free-space, the right-hand side of (9) is 0, i.e. the null vector. However for the description of materials, using the complex variable z to represent the time-shift operator, the right-hand side of (9) is transformed to the Z-domain using the bilinear transform [11], i.e. ∂/∂¯t → 2(1 − z−1 )/(1 + z−1 ) ge gem gme rm ∗ V i +2 ∂ ∂¯t χe ξr ζr χm ∗ V i Z −→ σ(z)·F+4 1−z−1 1+z−1 M(z)·F(11) The matrices on the right-hand side of (11) are σ(z) = ge gem gme rm and M(z) = χe ξr ζr χm (12) The conductivity matrix σ(z) and the material matrix M(z) may con- tain frequency-dependent elements and this is indicated by explicitly 4
5. 5. writing their arguments (z). Combining the right-hand sides of (10) and (11) using (9) yields 2(1+z−1 )Fr = (1+z−1 )4 · F + (1+z−1 )σ(z) · F + (1−z−1 )4M(z) · F(13) where using 1 to represent the identity matrix, 4 = 4 1. In the model- ing of matrices σ(z) or M(z) consisting of causal frequency-dependent elements, the overall strategy is shift the frequency-dependence back to the previous time-step by taking the partial fraction expansions of (1+z−1 )σ(z) = σ0 + z−1 [σ1 + ¯σ(z)] (14) (1−z−1 )M(z) = M0 − z−1 [M1 + ¯M(z)] (15) In (14) and (15), depending on the type of material, matrices σ0, σ1, M0 and M1 contain constant (possibly zero) elements and matrices ¯σ(z) and ¯M(z) contain zero or frequency-dependent elements. Substituting (14) and (15) into (13) gives F = T · [2Fr + z−1 S] (16) where the forward gain matrix T = [4 + σ0 + 4M0]−1 . The main accu- mulator vector S is calculated using S = 2Fr + κ · F − ¯σ(z) · F + 4 ¯M(z) · F (17) where the feedback matrix κ = −[4 + σ1 − 4M1]. In the present model, (16) and (17) are used at each time-step to obtain the vector of total ﬁelds F from the excitation Fr . The process is summarized in the signal ﬂow diagram of Fig. 1. To generate a compact notation to describe general material functions, (16) and (17) are combined as F = t(z) · Fr (18) In (18), t(z) is a 6×6 matrix of transfer functions. 3. MODELING SECOND-ORDER CHIRAL MEDIA In this section, the formulation of section 2 is used to develop time- domain models for the description of electromagnetic propagation in isotropic and uniaxial chiral materials displaying a second-order fre- quency response. 5
6. 6. z-1 2 + + S + z-1 S F r F + _ __κ σ(z)_ _ __ M(z)4____ T__ + Figure 1: Field update system for a general material 3.1. Isotropic Chiral Medium In an isotropic chiral material [10], the susceptibility and magnetoelec- tric coupling tensors of (2) have the form: χe = χe1 , χm = χm1 , ξr = ξr1 , ζr = −ξr1 (19) One example is constructed by a dispersal of wire helices with ran- dom orientations in a low-loss isotropic dielectric background having a constant electric susceptibility χeb. In references [10, 15], a single helix was modeled as an intercon- nected dipole and loop and the polarizabilities were quantiﬁed using antenna theory. From this investigation, to a ﬁrst approximation the eﬀective material properties are second-order in form. In the Laplace domain, the electric susceptibility due to the helices is χec(s) = χe0 ω2 0 (s + δ)2 + β2 (20) In (20), s is the complex frequency [11], χe0 is the dc electric suscepti- bility, δ is the damping frequency, β is the natural frequency and the resonant frequency ω0 = √ δ2 + β2. The magnetic susceptibility is χmc(s) = −χm∞ s2 (s + δ)2 + β2 (21) where χm∞ is the magnetic susceptibility at high frequencies. In the analysis of [10, 15], the higher-order modes of the equivalent loop were neglected and above ω0, (21) goes negative. To obtain physical solu- tions, it is necessary to introduce a background magnetic susceptibility χmb ≥ χm∞ to account for high frequency eﬀects not described in (21). 6
7. 7. Note that another form of the magnetic susceptibility exists in the liter- ature: In references [16, 17], the frequency-dependence of the magnetic susceptibility is proposed to have the same form as (20). The frequency dependence of the magnetoelectric parameters obey Condon’s model [10, 18], ξr(s) = s τ ω2 0 (s + δ)2 + β2 (22) where τ is the chirality time constant. Because this medium is based on the wire and loop model, the chirality is not independent of the electric and magnetic susceptibilities. The time-domain model is developed by identifying the matrices of (12). For this material, σ = 0 where 0 is the null matrix and M = χeb1 0 0 χmb1 + χec1 ξr1 −ξr1 χmc1 = Mb + Mc (23) In (23), Mb is the matrix of background susceptibilities and Mc is the frequency-dependent matrix describing the coupled part of isotropic chiral material response. To simplify the notation, in the development of the time-domain model of M, the 1’s are suppressed. Using (20), (21) and (22) in (23) the frequency-dependent material matrix in the Laplace domain is Mc(s) = χec ξr −ξr χmc = 1 (s + δ)2 + β2 χe0 ω2 0 s τ ω2 0 −s τ ω2 0 −χm∞ s2 (24) Equation (24) is converted to the Z-domain using the bilinear trans- form as in (11), i.e. s → (2/∆t)(1 − z−1 )/(1 + z−1 ), yielding Mc(z) = b0 + z−1 b1 + z−2 b2 1 − z−1a1 − z−2a2 (25) where using D = [(2 + ∆t δ)2 + β2 ∆t2 ], the feedback gains are a1 = 2 D−1 [(2 + ∆t δ)(2 − ∆t δ) − β2 ∆t2 ] a2 = −D−1 [(2 − ∆t δ)2 + β2 ∆t2 ] and the forward gain matrices are b0 = D−1 χe0 ω2 0 ∆t2 2 ∆t τ ω2 0 −2 ∆t τ ω2 0 −4 χm∞ , b1 =2 D−1 χe0 ω2 0 ∆t2 0 0 4 χm∞ b2 = D−1 χe0 ω2 0 ∆t2 −2 ∆t τ ω2 0 2 ∆t τ ω2 0 −4 χm∞ 7
8. 8. Equation (25) is in the standard Pad´e form of a transfer function. The model follows by substituting (25) and (23) into (15) leading to (1 − z−1 )M(z) = M0 − z−1 [ M1 + ¯M(z)] = Mb +b0 − z−1  Mb + b′ 0/4 +z−1 b′ 1/4+z−2 b′ 2/4 1 − z−1a1 − z−2a2   (26) The matrices in the numerator of (26) are b′ 0/4 = b0 − b1 − a1b0 , b′ 1/4 = b1 − b2 − a2b0 , b′ 2/4 = b2 (27) From (26), the matrices of (15) are M0 = Mb + b0 , M1 = Mb , 4 ¯M(z) = b′ 0 + z−1 b′ 1 + z−2 b′ 2 1 − z−1a1 − z−2a2 (28) The total ﬁelds are obtained as in (16) using F = T · [2Fr + z−1 S] (29) where T = [4 + 4 Mb + 4 b0]−1 . The main accumulator vector S is calculated using (17) in the form S = 2Fr + κ · F + S1 (30) where κ = −(4 − 4 Mb). In (30), the material accumulator vector S1 is evaluated using S1 = 4 ¯M(z) · F =   b′ 0 + z−1 b′ 1 + z−2 b′ 2 1 − z−1a1 − z−2a2   · F (31) An eﬃcient technique for the solution of (31) is the phase-variable state- space method [11]. Deﬁning state vectors X1 and X2, the discrete state-space and output equations are X1 X2 = z−1 a1 a2 1 0 · X1 X2 + 1 0 F (32) S1 = b′ 0 (b′ 1 + z−1 b′ 2) · X1 X2 (33) The algorithm requires 4 backstores per ﬁeld component. The system described by (32) and (33) is illustrated in Fig. 2. 8
9. 9. z-1 z-1 X1 X2 X2 z-1 b0 == + + S 1 + b1 == b2 == ++ +F a2 a1 Figure 2: Phase-variable system for a second-order chiral medium Although in this section only a second-order system was consid- ered, because (31) is in a standard Pad´e form, it is straightforward to extend this formulation for the description of higher-order frequency- dependent material functions. 3.2. Uniaxial Chiral Medium In a uniaxial chiral material [10, 12], the helices are randomly dis- persed in the background material, but are aligned in a particular direc- tion. For example, consider 1-D propagation in ˆx with E and H trans- verse to ˆx (Appendix A.1), with the helices aligned in the ˆz-direction. Assuming the helices are embedded in an isotropic background hav- ing a frequency-independent electric susceptibility of χeb and have the frequency-dependent properties of (20), (21) and (22), the reduced ma- terial matrix of (12) is M =      χyy e χyz e ξyy r ξyz r χzy e χzz e ξzy r ξzz r −ξyy r −ξyz r χyy m χyz m −ξzy r −ξzz r χzy m χzz m      =      χeb 0 0 0 0 χeb 0 0 0 0 0 0 0 0 0 χmb      +      0 0 0 0 0 χec 0 ξr 0 0 0 0 0 −ξr 0 χmc      (34) As in the isotropic case of (23), (34) is written as M = Mb + Mc (35) where Mb and Mc in (35) apply to the uniaxial case. The development of the discrete-time model follows a similar approach to that detailed in equations (24) through to (33). 9
10. 10. 4. RESULTS In this section, the time-domain models developed in sections 2 and 3 are validated against frequency-domain analysis for isotropic and uniax- ial chiral materials having a second-order frequency dependence. Also, time-domain results are presented for pulse propagation in these ma- terials. Although time-domain results for propagation in chiral media were presented in [16, 17], direct comparison with our technique would not be straightforward as these authors used a Beltrami description of the ﬁelds. 4.1. Isotropic Chiral Medium As in previous developments of novel formulations in the time-domain of frequency-dependent materials [2, 4, 5, 6], to ensure the model is giving the correct results and thus oﬀer a degree of validation, two basic problems involving an isotropic chiral medium with analytic solu- tions are investigated: Using both TLM and frequency-domain analysis [10], the reﬂection and transmission coeﬃcients of an isotropic chiral slab having a second-order frequency dependence between two isotropic free-spaces and the reﬂection coeﬃcient of the slab with a metal back- ing were obtained and compared. 4.1.1. Isotropic Chiral Slab—Frequency-domain Results The thickness of the slab was selected as 200mm with properties χe0 = χm∞ = ω0τ = 0.5, ω0 = 2π ×1000×106 and δ = 2π ×100×106 . The background electric and magnetic susceptibilities were selected as χeb = χmb = 1. Thus in the frequency-domain model, the overall relative permittivity and permeability were εr =1+χeb+χec and µr =1+χmb+χmc. The characteristic impedance was η=η0 µr/εr and the refractive index for circularly polarized waves was n± = √ µrεr ±jξr. The time-domain model had a space-step of ∆ℓ = 1mm and was excited with a ˆz-polarized pulse of electric ﬁeld traveling in the +ˆx- direction having unit magnitude. During the simulation, the time- domain co-polarized and cross-polarized reﬂected and transmitted elec- tric ﬁelds were saved for transformation to the frequency-domain for direct comparison with the analytic results. The transmission and re- ﬂection coeﬃcients obtained for circularly polarized (CP) waves [4, 5] 10
11. 11. are compared in Fig. 3. In this diagram, for example: Tlcp is the left-hand CP transmission coeﬃcient and Rrcp is the right-hand CP reﬂection coeﬃcient. -50 -40 -30 -20 -10 0 0 0.5 1 1.5 2 2.5 Reflection/transmissioncoefficientmagnitudes(dB) Frequency (GHz) Trcp (analytic) Trcp (TLM) Tlcp (analytic) Tlcp (TLM) Rrcp/Rlcp (analytic) Rrcp/Rlcp (TLM) Figure 3: Transmission and reﬂection coeﬃcients of a chiral slab 4.1.2. Isotropic Chiral Slab—Time-domain Results The time-domain response of the slab for a pulse plane wave excitation is shown in Fig. 4. The initial pulse was ˆz-polarized, traveling in the +ˆx direction, with a Gaussian proﬁle, having a maximum amplitude of 1V/m and a spatial width of 128 cells between the 0.001 amplitude truncation points [5]. -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 z-polarizedelecricfield(V/m) Distance (m) Free-space Free-space Isotropic chiral Initial condition 834ps 1668ps 2502ps -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 y-polarizedelectricfield(V/m) Distance (m) Free-space Free-space Isotropic chiral 834ps 1668ps 2502ps Figure 4: Time-domain response of an isotropic chiral slab In Fig. 4 the distribution of the ˆz- and ˆy-polarized electric ﬁelds at various times are indicated. As the incident wave enters the chiral slab, the coupling from the ˆz-polarization to the ˆy-polarization is seen. Also, in agreement with the frequency-domain analysis, no ˆy-polarized reﬂected electric ﬁeld is observed. 11
12. 12. 4.1.3. Metal-backed Isotropic Chiral Slab In this example, to examine the eﬀect of the spatial discretization, the slab of the previous section was terminated with a metal back- ing and the simulation repeated using space-steps of 1mm, 5mm and 20mm. The co-polarized return loss magnitudes obtained using TLM and analysis are shown on the left-hand side of Fig. 5. -50 -45 -40 -35 -30 -25 -20 -15 -10 -5 0 0 0.5 1 1.5 2 2.5 Co-polarizedreturnloss(dB) Frequency (GHz) Analytic 1mm 5mm 20mm -4 -2 0 2 4 0 0.5 1 1.5 2 2.5 Error(dB) Frequency (GHz) 1mm 5mm 20mm Figure 5: Co-polarized return loss and error of an isotropic chiral slab with a metal backing The error between the time-domain model and analysis is shown on the right-hand side of Fig. 5. For the models using 1mm and 5mm cells, the worst case errors are ∼0.1dB and ∼1.5dB respectively, both at the chosen material resonant frequency (1GHz). The error in all cases is mainly due to the frequency error in the bilinear transform: This error can be reduced by applying the standard technique of prewarping the critical frequencies prior to bilinear transformation [11]. 4.2. Uniaxial Chiral Half-Space The formulation for uniaxial chiral materials developed in section 3.2 was validated by the simulation of the reﬂectivity of an inﬁnite half- space of this medium. The results obtained using TLM are compared with frequency-domain analysis [12] in Fig. 6. The material properties of the uniaxial chiral material were identical to that given in section 4.1.1 for the isotopic chiral medium, except that the helices were aligned with the ˆz direction. In Fig. 6, using the notation of [12], Ryz is the ˆy-polarized reﬂection coeﬃcient magnitude for a ˆz-polarized excita- tion and Rzz is the ˆz-polarized reﬂection coeﬃcient magnitude for a ˆz-polarized excitation. 12
13. 13. -40 -35 -30 -25 -20 -15 -10 -5 0 0 0.5 1 1.5 2 2.5 Reflectioncoefficientmagnitude(dB) Frequency (GHz) Rzz (analytic) Rzz (TLM) Ryz/Rzy (analytic) Ryz/Rzy (TLM) Ryy (analytic) Ryy (TLM) Figure 6: Reﬂection coeﬃcients of a uniaxial chiral half-space -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 z-polarizedelectricfield(V/m) Distance (m) Free-space Uniaxial chiral Initial condition 834ps 1668ps 2502ps -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 z-polarizedelectricfield(V/m) Distance (m) Free-space Uniaxial chiral Initial condition 834ps 1668ps 2502ps -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 y-polarizedelectricfield(V/m) Distance (m) Free-space Uniaxial chiral 834ps 1668ps 2502ps Figure 7: Time-domain response of a uniaxial chiral half-space for an incident wave polarized in ˆz -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 y-polarizedelectricfield(V/m) Distance (m) Free-space Uniaxial chiral Initial condition 834ps 1668ps 2502ps -0.4 -0.2 0 0.2 0.4 0.6 0.8 1 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 z-polarizedelectricfield(V/m) Distance (m) Free-space Uniaxial chiral 834ps 1668ps 2502ps Figure 8: As Fig. 7, but using a ˆy-polarized incident wave 13
14. 14. The time-domain response of the uniaxial chiral medium is shown in Figs. 7 and 8 for both ˆz- and ˆy-polarized traveling pulse excitations. These ﬁgures give an insight into the dispersive behavior of the medium in the time-domain: The ˆz-polarized transmitted wave for a ˆz-polarized excitation is dispersed more heavily than the ˆy-polarized transmitted wave for a ˆy-polarized excitation. Also, the cross-polarized transmitted waves in both cases are of a similar magnitude. 5. CONCLUSIONS In this paper, discrete time-domain models of electromagnetic wave propagation in materials exhibiting magnetoelectric coupling have been developed and validated. The approach adopted here was based on bilinear Z-transform methods and the ﬁnal system was expressed in a standard Pad´e form. The essential details of the solution of the spatial network in 1-D and 3-D problems are detailed in an Appendix. Examples involving frequency-dependent chiral media have been studied and the close agreement between the analytic and modeled results demonstrate that the time-domain method may be applied to problems having no analytic solution. Also, time-domain pulse prop- agation in isotropic and uniaxial chiral materials were shown, giving a further insight into the properties of electromagnetic waves in these complex materials. The method described has all the advantages of a full 3-D numeri- cal model and can thus be used to study practical conﬁgurations with complex geometrical shapes and material properties. As the technique is implemented in the time-domain, it can also incorporate nonlineari- ties and hence is applicable to the most general electromagnetic ﬁeld- material interactions. APPENDIX A: BACKGROUND To complete the formulation of section 2, it is necessary to solve for the curl operations on the left-hand side of (10). This involves the derivation of the TLM algorithm for the time-domain solution of the spatial network. In this appendix, the computational algorithm for the 1-D case is derived and the 3-D method is shown to follow as an extension of the 1-D case. 14
15. 15. A.1. 1-D TLM To illustrate the general principles involved, initially, a 1-D model is studied: For propagation in the ˆx-direction, with E and H transverse to ˆx, where ˆx is the unit vector pointing in x, E · ˆx = 0 , H · ˆx = 0 (36) The left-hand side of Fig. 9 shows the 1-D spatial network applicable to the modeling of this situation. The node has four ports (V4, V5, V10 and V11) and the four transverse ﬁeld quantities (Ey, Ez, Hy and Hz) are indicated at the center of the cell. The curl operations are solved using Stokes’ theorem along the integration contours Cy and Cz. For consistency with a 3-D development in Appendix A.2, the port numbers used in the 1-D case are taken from the 3-D node shown on the right-hand side of Fig. 9. V5 V11 V4 V10 Cz Cy Ez Ey Hz Hy x y z Cz Cy V6 V0 V5 V11 V2 V8 V4 V10 V9 V3 V7 V1 Cx Figure 9: 1-D and 3-D TLM spatial networks Reduction of (4) to the 1-D case described above, gives      (∇×H)y (∇×H)z −(∇×E)y −(∇×E)z      −      Jefy Jefz Jmfy Jmfz      − ∂ ∂t      ε0Ey ε0Ez µ0Hy µ0Hz      = σe σem σme σm ∗      Ey Ez Hy Hz      + ∂ ∂t ε0χe ξr/c ζr/c µ0χm ∗      Ey Ez Hy Hz      (37) For modeling this particular 1-D case, the 3-D tensors have the following form: σe =    σxx e 0 0 0 σyy e σyz e 0 σzy e σzz e    (38) 15
16. 16. Thus in (37), we may write the tensors as 2×2, for example σe = σyy e σyz e σzy e σzz e (39) and (∇ × H)u = (∇ × H) · ˆu, where ˆu∈{ˆy,ˆz}. Using the ﬁeld-circuit equivalences of section 2 to transform (37) to a normalized form yields      V4 + V5 V10 + V11 V11 − V10 V4 − V5      −      ify ifz Vfy Vfz      − ∂ ∂¯t      Vy Vz iy iz      = ge gem gme rm ∗      Vy Vz iy iz      + ∂ ∂¯t χe ξr ζr χm ∗      Vy Vz iy iz      (40) Converting (40) to the traveling wave format [9], using superscript i to denote incident wave quantities and the notation of (12) gives 2      V4 + V5 V10 + V11 V11 − V10 V4 − V5      i −      ify ifz Vfy Vfz      −2      Vy Vz iy iz      = σ(t)∗      Vy Vz iy iz      + ∂ ∂¯t M(t)∗      Vy Vz iy iz      (41) The ﬁrst two terms on the left-hand side of (41) are deﬁned as the external excitation, 2      V4 + V5 V10 + V11 V11 − V10 V4 − V5      i −      ify ifz Vfy Vfz      = 2      Vy Vz −iy −iz      r = 2 Fr (42) where the superscript r denotes reﬂected wave quantities. By deﬁning a matrix RT 1 , the vector of free-sources Ff and the vector of incident voltages, Vi T = V4 V5 V10 V11 i (43) where the superscript T denotes a transposed vector, (42) can be writ- ten compactly as Fr = RT 1 · Vi − 0.5 Ff (44) Deﬁning the vector of total ﬁelds F = Vy Vz iy iz T (45) 16
17. 17. and by substituting (42) into (41) and transforming to the Laplace domain using ∂/∂¯t → ¯s = s ∆t yields 2 Fr = (2 + σ + ¯sM) · F (46) As in (18), by deﬁning a matrix of frequency-dependent transfer func- tions, t = 2(2 + σ + ¯sM)−1 and transforming to the Z-domain, (46) may be written as F = t(z) · Fr (47) In order to time-step the process, we require the reﬂected voltages on the transmission-lines. These are obtained using      V4 V5 V10 V11      r =      Vy − iz Vy + iz Vz + iy Vz − iy      −      V5 V4 V11 V10      i (48) Deﬁning the vector of reﬂected voltages Vr which is of the same form as Vi and the matrices R and P, (48) in concise form is Vr = R · F − P · Vi = R · F − ˜Vi (49) In (49), ˜Vi is the vector of voltages incident on the lines opposite those used to obtain Vi . In the ﬁnal step of the algorithm, the connection process, the reﬂected voltages are swapped between neighboring nodes and become the incident voltages of the next time-step. In summary, the 1-D method consists of the 3 steps of (44), (47) and (49) as illustrated in Fig. 10. As discussed in section 2, for the mod- eling of general material responses, only the block t(z) of this diagram requires further development. t(z)__ R__ Ff i V ~ r r VR T __V i __P 0.5 _ + + _ 1 F F Figure 10: Signal ﬂow graph of the general TLM process 17
18. 18. A.2. 3-D TLM The spatial network used for 3-D problems is illustrated on the right- hand side of Fig. 9 [14]. This node has 12 ports, (V0 . . . V11) and six total ﬁeld quantities (Ex, Ey, Ez, Hx, Hy and Hz) at the center of the cell. Extending the development of (42) from (37) to the 3-D case, the external excitation is:      Vx Vy Vz −ix −iy −iz      r =      ( V0 + V1 + V2 + V3 ) ( V4 + V5 + V6 + V7 ) ( V8 + V9 + V10 + V11 ) − ( V6 − V7 − V8 + V9 ) − ( V10 − V11 − V0 + V1 ) − ( V2 − V3 − V4 + V5 )      i − 1 2      ifx ify ifz Vfx Vfy Vfz      → Fr = RT 1 · Vi − 0.5 Ff (50) As in the 1-D development in (44), the right-hand side of (50) is ob- tained by deﬁning a matrix RT 1 , the free-source vector Ff and the vector of incident voltages, Vi T = V0 V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11 i (51) Deﬁning the vector of total ﬁelds F = Vx Vy Vz ix iy iz T (52) and using (44) and (52) in (41) written for the 3-D case and transform- ing to the Laplace domain using ∂/∂¯t → ¯s yields 2 Fr = (4 + σ + 2¯sM) · F → F = t(z) · Fr (53) The right-hand side of (53) follows by deﬁning a matrix of transfer functions, t = 2(4 + σ + 2¯sM)−1 and transforming to the Z-domain. The reﬂected voltages are obtained by extension of (48),                V0 V1 V2 V3 V4 V5 V6 V7 V8 V9 V10 V11                r =                Vx − iy Vx + iy Vx + iz Vx − iz Vy − iz Vy + iz Vy + ix Vy − ix Vz − ix Vz + ix Vz + iy Vz − iy                −                V1 V0 V3 V2 V5 V4 V7 V6 V9 V8 V11 V10                i → Vr = R · F − ˜Vi (54) The right-hand side of (54) is developed by deﬁning the vector of re- ﬂected voltages Vr of the same form as Vi , the matrix R and the reordered incident vector ˜Vi . Equations (50), (53) and (54) show that solution of the 3-D spatial network is simply an extended form of the 1-D case shown in Fig. 10. 18
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