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DEVELOPMENT OF AN FDTD TOOL FOR MODELING OF DISPERSIVE MEDIA Part I. Material parameters and updating equations for fields. Jianmin Zhang, Jing Wu, Marina Y. Koledintseva, James L. Drewniak, University of Missouri-Rolla, USA

Konstantin Rozanov ITAE, Russian Academy of Sciences The implementation of the Debye, narrowband Lorentzian, and wideband Lorentzian dielectric and magnetic media in the numerical tool EZ-FDTD developed at the University of Missouri-Rolla (UMR is described. This tool allows efficient and robust full-wave finite-difference time-domain modeling of different complex electromagnetic structures. Algorithms for dispersive media use the linear recursive convolution (LRC) procedure. Fitting the experimentally obtained frequency characteristics of the materials by the Debye and Lorentzian dispersive laws is discussed. Criterion of which model is to be applied is introduced. The Debye

model can be used, if the Q-factor of the linear circuit analogue of the corresponding Lorentzian model is less than 0.8. If 0.8 < Q < 1, then the wideband Lorentzian model is appropriate. For Q > 1, the narrowband Lorentzian model must be applied. The narrowband Lorentzian model is suitable for materials exhibiting resonance effects in the microwave frequency range. The wideband Lorentzian curve is typical for composites filled with conducting fibers, and also for many microwave magnetic materials.

1.Introduction The finite-difference time-domain (FDTD) technique, based on the direct time integration of Maxwell's partial differential equations using the central finite difference method, has been widely used to analyze electromagnetic phenomena [1-3]. The FDTD method typically deals with media having constant conductivity, permittivity, and permeability. Dispersive parameters of dielectric, magnetic, and magneto-dielectric materials, including composites, can be incorporated into the FDTD codes in two alternative ways [1-3]. The first approach uses a linear recursive convolution (LRC) of magnetic or dielectric susceptibility and the corresponding field components in the time-domain, and the second is based on the discretization of an auxiliary differential equation (ADE) that relates the electric field vector with the displacement or polarization vector in the

dielectric case, and magnetic field intensity and magnetic flux density or magnetization in the magnetic case. For linear media, the LRC approach is computationally effective, straightforward to implement and, therefore, attractive. However, the frequency dependence of dielectric or magnetic susceptibility must have a causal inverse Fourier (or Laplace) transform. 2. Debye and Lorentzian material models and their combinations. The simplest for linear isotropic dielectric or magnetic material is a Debye model A (1) , χ (ω ) = ε ,µ

1 + jωτ

where τ is the characteristic relaxation time called the Debye constant, and A is a resonance amplitude parameter depending on the type of the dielectric or magnetic material. In the case of a dielectric material, (2) A = εS − ε∞ . where ε S is a relative static permittivity, and ε ∞ is an “optical” (or highfrequency) relative permittivity. In the case of a magnetic material, this is a static magnetic susceptibility, (3) A = χ0 . At the phenomenological description of frequency properties of polar dielectrics, the mathematical model, which likens the molecular dipoles to the solidr particles moving in viscous liquidr can be used. The polarization vector P and the electric field vector E are related by a first-order differential equation r dP 1 r A r + P= E, dt τ τ

(4)

the solution of which yields the complex absolute permittivity of the dielectric as the Debye function, (5) ⎛ A ⎞ ⎟⎟ . ε = ε 0 ⎜⎜1 + ⎝

1 + jωτ ⎠

The Debye dispersion law frequently governs low-frequency behavior of material parameters appearing due to such phenomena as the dipolar

polarization of molecules in dielectrics, and the domain wall motion in magnetic media. At higher frequencies, from microwaves to UV waves, many effects follow the Lorentzian, rather than the Debye dispersion law. The Lorentzian law is valid for ionic and electronic polarizability in dielectrics, as well as for magnetic spectra related to ferromagnetic resonance. For a singlecomponent linear, isotropic, homogeneous dielectric or magnetic material, a simplest Lorentzian susceptibility is described by a single-peak curve, A ω 02 . ω 02 − ω 2 + 2 jω δ

χ ε ,µ ( ω ) =

(6)

where ω 0 is the resonance frequency, δ is the Lorentzian resonance line halfwidth (at the –3 dB level), and A is the resonance amplitude parameter as in (2) or (3). If the polarization dynamics is described by the second-order differential equation, which arises when the dipole model is an oscillator with friction, r r r Aω 02 r (7) d 2 P 2 dP 2 + + ω = P E, 0 2 dt

τ dt

τ

where τ = 1 / δ , then the resultant complex absolute permittivity is governed by a Lorentzian law, (8) ⎛ ⎞ Aω 02 ⎟. ε = ε 0 ⎜1 + 2 2 ⎜ ⎟ ⎝

ω 0 − ω + 2 jωδ ⎠

Analogously, the Debye and the Lorentzian dependences are obtained for permeability of magnetic media. Typical Debye and Lorentzian curves are represented in Figures 1 (a, b). The Debye dispersive behavior has been introduced into the FDTD algorithm in the form of linear recursive convolution (LRC) [1,3], or piecewise linear recursive convolution (PLRC) [2]. The detailed description of the Debye material model for both dielectric and magnetic media incorporated into the EZ-FDTD codes, developed at the EMC Laboratory of the University of Missouri-Rolla, as well as some practical examples of these codes verification and comparison with experiments are described in our papers [4-6].

,εS

,ε

,ε

, εs

ω ω Figure 1. Dispersive laws: (a) Debye; (b) Lorentzian.

In many cases, spectra are of more complex nature, and frequency dependences of material parameters consist of several distinct Lorentzian peaks or are a superposition of Debye and Lorentzian curves. The example is given in Figure 2 for a fiber-filled composite with conducting fibers of two different lengths [7]. Two separate Lorentzian lines are clearly seen. In magnetic materials, complex magnetic spectra with multiple Lorentzian lines are also typical [8, 9], as shown in Figure 3. In general form, an absolute permittivity function can be written as N ( ε sk − ε ∞ )ω 02k ε si − ε ∞ jσ e + ε − , 0∑ 2 2 ωε 0 k =1 ω 0 k − ω + jω ( 2δ k ) i =1 1 + jωτ i M

ε ( ω ) = ε 0ε ∞ + ε 0 ∑

(9)

where the second term on the right-hand-side is responsible for the Lorentzian frequency dispersion, the third term introduces the Debye behavior, and the fourth term is responsible for the macroscopic ohmic loss in the material. In (9), εsk is the static dielectric constant for the k-th Lorentzian or Debye resonance line; ω0k and (2δk) are the resonance frequency and the width of k-the Lorentzian peak, respectively; τk is the loss constant for k-th Debye component; and σe is the ohmic (d.c.) conductivity. The sum on the right-hand-side of (9) is a rational function that can fit any actual dispersion law obeying the Kramers−Kronig causality relations [10] with any desired accuracy. In many cases the terms on the right-hand-side of (9) have a clear physical meaning.

Figure 3. Dispersive curves with two Lorentzian peaks of permeability of ferrite films on the Dacron substrate, thickness 0.3 µm [8].

Figure 2. Dispersive curves with multiple resonances for permittivity of fiber-filled composites [7].

Sometimes, however, the actual shape of the dispersion curve is formed by the spread in the parameters of the dispersion of individual scatters. This is true, for example, for the Cole-Cole dispersion law, used mainly for polymeric materials [11], as well as some composites, ε = ε∞ +

ε0 − ε∞ , 1−α 1 − ( jωτ )

(10)

where 0 < α < 1 . The Cole-Cole frequency dependence can be understood as an assembly of Debye scatterers with some distribution of the loss constant, and describes smoother dispersive behavior than that of the Debye law. To fit such dispersion dependences within a finite frequency range with the least number of parameters, non-physical representation of the dispersion curve is conventional. For example, the Cole-Cole dispersion dependence is frequently fitted as a sum of the third and fourth terms in (9). The more general dispersion law for polymers is described by the Havriliak-Negami function [12, 13], ε = ε∞ +

ε0 − ε∞

(1 − ( jωτ

)1-α

)

β

,

(11)

that eventually can be also approximated by the function (9). The permeability function of an isotropic magnetic material in general form is similar to the dependence (9), however, the last term responsible for ohmic loss, is absent,

N υ k Ak ω 02k υ i Ai + µ . 0∑ 2 2 k =1 ω 0 k − ω + jω ( 2δ k ) i =1 1 + jωτ i M

µ( ω ) = µ 0 + µ 0 ∑

(12)

where notations of the parameters are the same as in (9), and Ak is the resonance amplitude (static magnetic susceptibility) of the k-th component in the magnetic mixture. If a mixture of magnetic particles that almost do not interact with each other is modeled, then the demagnetization at the boundaries of the mixture phases can be neglected, and the resultant frequency characteristic is a weighted summation of partial dispersive curves, where υ k is a volumetric fraction of every phase (component), and n

∑υ k =1

k

= 1 [14]. If there is a single-component (or single-phase) magnetic

material, then the volumetric fraction is υ1 = 1 . Also, for approximation of the dispersive curve of an arbitrary magnetic material, the weight coefficients can be all accepted as υ k = 1 , but the amplitudes of the partial Debye and/or Lorentzian terms can be different. Luebbers et al. introduced both single- and multi-pole Lorentzian models into FDTD recursive convolution procedures [15]. Unlike the Debye case, where the convolution function of the susceptibility and field is real and straightforward to implement for a recursive procedure, in the Lorentzian model it is a complex function in the general case. Depending on the ratio of a Lorentzian resonance line half-width δ (at the -3 dB level), and the resonance frequency ω 0 , different recursive convolution equations and coefficients for field updating are needed [16]. It should be mentioned that ω 0 is the actual resonance frequency, and for a wideband material it can be different from the frequency where maximum of loss occurs, since essential loss factor not only widens the resonance line curve, but also shifts the maximum loss to the lower frequencies. When δ / ω 0 > 1 , it is a wideband Lorentzian material; and when δ / ω 0 < 1 , is a narrowband Lorentzian material suitable for resonance effects and highly absorbing media modeling. The borderline case δ = ω 0 is not subjected to recursive convolution, and can be modeled as either a wideband or a narrowband Lorentzian material. The circuit-theory analogue for a Debye model is an RC- or an RL-circuit, and for a Lorentzian model it is an RLCcircuit with a Q-factor Q = ω 0 / δ < 1 for a wideband material, and Q >1 for a narrowband one.

The wideband Lorentzian behavior of permittivity is observed in fiber-filled composites at microwave frequencies [7]. Frequency characteristics of some magnetic materials can also be described by the wideband Lorentzian dispersion law [8,9,17]. The frequency dependence in the wideband Lorentzian model for Q