Computing the Electric and Magnetic Green’s Functions in General Electrically Gyrotropic Media

1 Introduction

The Faraday effect,observed in 1945 in a piece of glass placed between the poles of a magnet,was the first of the magneto-optical effects to be discovered by Michael Faraday[Faraday(1933)].The result of the existence of magnetic fields inside of magneto-optical materials opens gyrotropic materials.Gyrotropic materials have been an active research topic because,using these materials,new microwave devices such as circulators,isolators,resonators,and optical devices such as modulators,sw itches,phase shifters can be designed[Eroglu(2003);Eroglu(2006a);Eroglu(2006b)].Moreover,the study of electromagnetic wave propagation in gyrotropic materials can be used in development of gyrotropic devices for ionospheric applications[Eroglu(2006b)].The electromagnetic fields observed in magnetically biased plasma or ferrite can be modeled as electromagnetic waves in gyrotropic materials[Eroglu(2006b)].Many problems of remote sensing,monolithic integrated circuits and optics,geophysical probing,microstrip circuits and antennas,submarine communication,opto-electronics etc.are connected with electromagnetic fields in gyro-electric materials[Eroglu(2010);Eroglu(2003);Eroglu(2006a);Eroglu(2006b);Prati(2003)].

If the electrical and/or magnetic properties of a medium depend upon the directions of field vectors,then the medium is called anisotropic medium and the relationships between fields have the follow ing form

Anisotropic materials may be divided into two classes,depending on whether the natural modes of propagation are linearly polarized or circularly polarized waves[Eroglu(2010)].In the first class,the permittivity and permeability components are symmetric;that isεik=εkiandµik=µki.For the second class,called gyrotropic media,the permittivity or permeability matrices are Herm itian,that isand,where∗is denoted as complex conjugate of the given element.

The gyrotropic medium becomeselectrically gyrotropicorgyro-electricif the permittivity matrixis Hermitian and the permeability matrix.The gyrotropic medium becomesmagnetically gyrotropicorgyro-magneticif the permeability matrixis Hermitian and[Eroglu(2010);Kong(1984)].

In the present paper we consider the gyro-electric materials in the general form when the permittivity matrix can be written as the following matrix(see,for example,[Eroglu(2010)])

Let us mention some works related to the construction of the Green’s functions for equations of electro magnetics.Ortner and Wagner[Ortner(2004)],Wagner[Wagner(2011)]have applied the Fourier meta-approach and found the explicit formulae of the Green’s functions for the systems of elasticity and Maxwell’s equations for some particular cases of anisotropy only by the Fourier transform meta-approach.These formulae can be used for the computer implementation.The approximate computation of the time-dependent three-dimensional Green’s functions(fundamental solutions)in general anisotropic materials by the Fourier transform metaapproach was suggested in the papers[Yakhno(2008);Yakhno(2011b)].Using the Lorentz reciprocity relation and the multiple scattering approach the dyadic Green’s functions for gyro-electric media are found in[Barkleshli(1993)].The dyadic Green’s functions for an electrically gyrotropic medium with particular form ofhave been derived by matrix method with dyadic decomposition in[Eroglu(2003);Eroglu(2006a);Eroglu(2006b)].The time-harmonic Green’s dyadics have been constructed in closed form for a particular case of homogeneous gyrotropic mate-rials by Fourier transform approach in[Olyslager(1997)].

However,the numerical computation of the Green’s functions(fundamental solutions)of Maxwell’s equations in general gyro-electric materials is not known.Moreover the numerical methods for the space of tempered distributions are not developed till now.

In our paper we suggest a method of an approximate(regularized)computation of the in finite-body Green’s functions for the time-harmonic Maxwell’s equations in the general gyro-electric materials.This method is based on the Fourier transform meta-approach where the Fourier image of the Green’s function is computed by some matrix transformations and symbolic computations in MATLAB.After that the inverse Fourier transform is computed in an regularized(approximate)form.The parameters of the regularization have been chosen by the comparison of the regularized Green’s function with Green’s function obtained by the explicit formula for the isotropic case.The approximate computation of the inverse Fourier transform has been implemented by MATLAB tools.The computational experiments are presented in the paper.

The paper is organized as follows.The equations for the time-harmonic electric and magnetic Green’s functions are written in the beginning of Section 2.Methods of computing the electric and magnetic Green’s functions are described in Section 2.1 and 2.2 respectively.Computational experiments are described in Section 3.

2 Electric and Magnetic Green’s Functions in General Gyro-electric Media

The electric Green’s function is a matrix function

whose columnssatisfy the equation

The magnetic Green’s function is a matrix function

whose columnssatisfy the equation

Herex=(x1,x2,x3)∈R3is the 3Dspace variable;ωis a fixed parameter(frequency);ε0=8.854×10-12F/m,µ0=1.257×10-6N/A2are positive constants(dielectric permittivity and magnetic permeability)of vacuum,respectively;e1=(1,0,0)T,e2=(0,1,0)T,e3=(0,0,1)Tare basis vectors of R3;iis the imaginary uniti2=-1;δ(x)=δ(x1)δ(x2)δ(x3),δ(xj)is the Dirac delta function concentrated atxj=0 forj=1,2,3.

2.1 Computing the Electric Green’s function

LetF xbe the operator of the Fourier transform with respect tox=(x1,x2,x3),i.e.

for the scalar integrable functionE(x),whereν=(ν1,ν2,ν3)is a 3Dparameter of the Fourier transform;ν·x=ν1x1+ν2x2+ν3x3.The operator of the Fourier transform is de fined in[Vladimirov(1979)]for any generalized function(tempered distribution).

we find

where

Letεbe a symmetric real 3×3 matrix andgbe an antisymmetric real 3×3 matrix de fined by

In the paper we suppose thatBis positive definite.

Remark 2.1Positive definiteness of B is natural for a wide class of gyrotropic materials because the matrix ε is always positive definite and the elements of matrix g essentially smaller then elements of ε(see,for example,[Freiser(1968)];[Pershan(1967)]).

Moreover under assumption of positive de finiteness of B the matrix¯ε is invertible,i.e.the inverse matrixexists.

Let us denoteas the real part ofjcomponent ofandas the imaginary part ofjcomponent of,i.e.and1,2,3.Then equality(6)can be written as a vector equation

whereApplying technique and computational tools of[Yakhno(2011b)],it is possible to compute a non-singular matrixT(ν)and diagonal matrixD(ν)=diag(d1(ν),d2(ν),...,d6(ν)),such that

(MATLAB code of this computation is given in Appendix A:).

Below the equation(10)is written in terms of a new unknown vector functionYkwhich is defined by

Substituting(11)into(10)we find

Multiplying the equation(12)byTT(ν),we find

or in a component form

whereAs a result,the solution of(10)is determined by

Finally,applying the inverse Fourier transform to(15),we find thek-column of the electric Green’s function as a tempered distribution,i.e.

whereis the operator of the inverse Fourier transform in the space of distributionS′(R3)[Vladimirov(1979)].

Usually the classical functions are de fined by a point-wise manner and we can draw their graphs.Unfortunately,this point-wise definition and its graphical presentation is not adequate to singular tempered distributions[Vladimirov(1979)].They are very often replaced by regularized functions which are classical and have graphic presentations.This regularization has a parameter of the regularization and the singular generalized function is a limit in sense of the generalized functions space,when the parameter of the regularization tends to+∞.The right hand side of(16)can be regularized by

whereAis the parameter of regularization.

We takeA=N∆and approximate the integral(17)by the integral sum

for the numerical computation.The parametersNand∆are determined by the procedure described in Section 3.2.

2.2 Computing the magnetic Green’s function

Applying the Fourier transform with respect toxto the equation(5)and using equalities

we find

Letbe the real parts ofandbe the imaginary parts ofrespectively.

Denotingandthe equation(19)can be written in the form

whereP(ν)is a 6×6 symmetric matrix defined by

Using the symbolic matrix transformation in MATLAB and the technique from[Yakhno(2011b)]we can compute an invertible matrixQ(ν)andM(ν)such that

whereM(ν)=diag(mn(ν),n=1,2,3,4,5,6).(MATLAB code of computation ofQ(ν)andM(ν)is given in Appendix B:)

Letthen

Using(23)the equation(20)can be written in the form

Multiplying the equation(24)byQ-1(ν),we find

wheren=1,2,...,6.As a result the solution of(20)is determined by

and the solution of(19)is found by

Applying the inverse Fourier transform to(27),we find thek-column of the magnetic Green’s function as a tempered distribution,i.e.

The right hand side of(28)can be regularized by

whereAis the parameter of regularization.

We takeA=N∆and approximate the integral(29)by the integral sum

for the numerical computation.The parametersNand∆are determined by the procedure described in Section 3.2.

3 Computational Examples

3.1 Computational accuracy of the Fourier transform of the electric Green’s function

The Fourier transform of the electric and magnetic Green’s functions can be found by exact formulas for the case of isotropic homogeneous materials.We use these formulas to compute the exact values of the Fourier transform of Green’s functions and then to compare them with values computed by our method.

These computational experiments have shown that values offound by our method and by explicit formulas are almost the same(the accuracy is around 10-6).

3.2 Determining parameters for the approximate computation of the Fourier transform in the space of generalized functions

The fundamental solution of the Helmholtz equation as well as the image of the Fourier transform with respect to a space variable are given by explicit formulas.We apply these explicit formulas to determine the parameters which we use for the numerical computation of the inverse Fourier transform for finding the electric and magnetic Green’s functions in gyro-electric media.

Let us consider the fundamental solution

of the Helmholtz equation

Applying the Fourier transform(the Fourier transform of generalized functions[Vladimirov(1979)])to equation(34),we find

whereν=(ν1,ν2,ν3)∈R3is the parameter of the Fourier transform,

Forwe have

The fundamental solutionU(x),de fined by(33),can be found by application of the inverse Fourier transform(as the inverse Fourier transform of generalized functions[Vladimirov(1979)])todetermined by(35).

For the approximate computationUN,∆(x)of the functionU(x)we apply the approximation of the inverse Fourier transform by

The parametersNand∆have been chosen using the empirical observation and natural logic.Namely,using the formula(36)we compute valuesUN,∆(x)for∆ =0.1,0.5,0.8,1.0,N=20,30,40,50,60,80 and so on numerically in MATLAB.We compare the computed values ofUN,∆(x)with values of the functionU(x),defined by(33).We have observed that the difference between the values ofUN,∆(x)andU(x)corresponding to∆=1 andN=30,40,50,60,80,100 becomes small and increment of the approximation for the parameterNis not essential,according to the case∆=1,N=30.For this reason we choose∆=1,N=30 as the suitable parameters for the calculation of the inverse Fourier transform by(36).

We have presented 1Dgraphs of the functionsU(x)andUN,∆(x)for the different values ofNand∆in Fig.1 and Fig.2.

Figure 1:1D plot of U(x)and UN,∆(x)for N=30,∆ =0.5

Figure 2:1D plot of U(x)and UN,∆(x)for N=30,∆ =1.0

3.3 The approximate computation of the electric and magnetic Green’s functions in general gyro-electric media

In this section we consider the computation of electric and magnetic Green’s functions for general gyro-electric media characterizing by

whereκ=10-1and 10-2.

Applying the method of Section2.1 we have computedT(ν),TT(ν),D(ν)and then using the formula(18)we have derived solutionsE1(x),E2(x),E3(x)of(6)numerically and applying the method of Section 2.2 we have computedQ(ν),Q-1(ν),M(ν)and then using the formula(30)we have derived solutionsH1(x),H2(x),H3(x)of(19)numerically.

Results of the computation of real and imaginary parts ofandforκ=10-1,10-2are presented in Fig.3,Figs.5-6 and Fig.4,Figs.7-8,respectively.

Figure 3:(a)1D plot of plot of in rectangular region for κ=10-1,10-2

Figure 4:ω =2c,κ=10-1,10-2(a)2D plot of(b)2D plot of

Figure 5:ω=2c,κ=10-1(a)2D plot of(b)3D plot of

Figure 6:ω=2c,κ=10-1(a)2D plot of(b)2D plot of

Figure 7:ω=2c,κ=10-1(a)2D plot of (b)2D plot of

Figure 8:ω=2c,κ=10-1(a)2D plot of(b)2D plot of

In Fig.3(a),the 1Dplots offorx1=x2=x,x3=0,ω=2cforκ=10-1andκ=10-2is given.Fig.3(b)presents the zoomed part of the graph of Fig.3(a)in the indicated rectangle.Figs.4(a),(b)present the 1Dgraphs ofandrespectively forκ=10-1,10-2.These graphs show the influence ofκon the components of the electric and magnetic Green’s functions.

The behaviour of real and imaginary parts of the computed components of the electric and magnetic Green’s functions is presented in Fig.3,Figs.5-6.The result of the simulationforω=2c,κ=10-1is presented in Fig.5.The 3Dplot ofis shown in Fig.5(b).Here the horizontal axes arex1andx2,respectively.The vertical axis is the magnitude ofFig.5(a)is a screen shot of 2Dlevel plot of the same surfacei.e.a view of the surfacepresented in Fig.5(b)from the top ofz-axis.Figs.6(a),(b)illustrate the 2Dlevel plots ofandrespectively.

The result of simulationandare presented in Fig.7.Figs.7(a),(b)are the 2Dplots ofandrespectively.The illustration ofandis given in Fig.8.

4 Conclusion

The method for the approximate computation of the electric and magnetic Green’s functions for the time-harmonic Maxwell’s equations in general gyro-electric materials has been developed in the paper.The method is based on the Fourier transform meta-approach.The Fourier transform with respect to the 3Dspace variable has been applied to partial differential equations for electric and magnetic Green’s functions.The images of the Fourier transform of Green’s functions were found from the obtained equations by the matrix transformations in MATLAB.The inverse Fourier transform of these images has been done numerically in the regularized(approximate)form in MATLAB.The parameters of this regularization have been chosen using the explicit formulae of the Green’s function and its Fourier image for the Helmholtz equation.Two types of the computational experiments were presented in the paper.The first one demonstrates the high level of computational accuracy for the Fourier images and the inverse Fourier transform.The second one has been done for computing the electric and magnetic Green’s functions in a general gyro-electric material.These computational experiments con firm the robustness of the method.

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