Analysis of Depolarized Electromagnetic WavesPropagated Through Random Medium

by Using Perturbation Method

Yukihisa NANBUSasebo National College Technology

1-1 Okishin-chou,Sasebo 857-1193, JapanEmail: nanbu@sasebo.ac.jp

Mitsuo TATEIBAKyushu University

744 motooka, nishi-ku,Fukuoka 819-0395, Japan

Email: tateiba@csce.kyushu-u.ac.jp

Hosam El−OclaLakehead University

Oliver Rd, Thunder Bay,Ontario P7B 5E1, CanadaE-mail: hosam@lakeheadu.ca

Abstract—We have derived an integral equation using thedyadic Green’s function on the assumption that there existsa random medium screen of which the dielectric constant isfluctuating randomly. On the integral equation, the depolarizedEM wave has been analyzed by using a perturbation method, andwe have shown the representation of the first order perturbationof the depolarized EM wave.

Keywords—random media; dyadic Green’s function; perturba-tion method

I. INTRODUCTION

Studies of the electromagnetic(EM) wave propagationthrough random medium have been continued into the im-portant development of geophysical measurements, satellitecommunications, and so on[1]. It is mainly assumed in thestudies that the fluctuating intensity of continuous randommedium is so weak and the fluctuating scale-size is much largerthan the wave length of EM wave. In this case, the scalarapproximation has been used. When the optical path-lengthbecomes large in long propagation through random medium,however, then we have to consider the effect of depolarizationof EM waves. In previous studies, quantitative analysis of thedepolarization has not been investigated sufficiently.

In this paper, to analyze the depolarized EM waves, wefirst have derived an integral equation using the dyadic Green’sfunction on the assumption that there exists a random mediumscreen of which the dielectric constant is fluctuating randomly.Next we have modified the integral equation on the assumptionthat the observation point is very far from the screen. Onthe basis of this modified integral equation, the depolarizedEM wave has been analyzed by using a perturbation method.Finally we have represented the first order perturbation of thedepolarized EM wave to discuss quantitatively the depolar-ization of EM wave propagated through the random mediumscreen.

II. FORMULATION

Let us consider the problem of EM wave scattering bya random medium with volume V . When we designate anincident EM wave by Fin = [Ein,Hin]t, the scattered EMwave by Fs = [Es,Hs]t, and the total EM wave by F =

[E,H]t, then F satisfies the Maxwell’s equations in overallregion (V + V̄ ). Therefore we obtain[2][3]

LF = −T T M F ; L =

[∇× −jωμ01

jωε01 ∇×

], (1)

M(r) = jω

[0 −δμ(r)μ01

δε(r)ε01 0

](2)

where [1 + δε(r)]ε0 and [1 + δμ(r)]μ0 are the dielectricconstant and magnetic permeability of the random medium,respectively, δε(r) and δμ(r) are random function, and

[∇× ]≡[

0 −∂/∂z ∂/∂y∂/∂z 0 −∂/∂x−∂/∂y ∂/∂x 0

], (3)

T =

[0 −11 0

], 1=

[1 0 00 1 00 0 1

], 0=

[0 0 00 0 00 0 0

](4)

Using the dyadic Green’s functionG, we can express solutionsof Eq.(2) with Fin as these of the following integral equation:

F = Fin +

∫V

G(r− r′)� T M(r

′)F(r

′) dr

′(5)

Let us define a vector A as A = [Ae, Am]t whereAe = [Aex Aey Aez]t and Am = [Amx Amy Amz]t. Then,using G = Ge+Gm where Ge, Gm, respectively, are due toelectric and magnetic point sources, we define the operator �as follows:

G�A =∑

α=x,y,z

(GeαAeα +GmαAmα) (6)

where Ge is expressed as

Ge=∑α

Geα=∑α

[Ge

eαGm

eα

];Ge

eα=∑α

[Gex

eαGey

eαGez

eα

],Gm

eα=∑α

[Gmx

eαGmy

eαGmz

eα

](7)

and Gm is expressed by changing the subscript of Ge frome to m. Here it should be noted that Geβ

eα, Gmβeα , respectively,

are the β-directional electric and magnetic wave fields radiatedfrom the α-directional electric point source.

978-1-4799-2225-3/14/$31.00 ©2014 IEEE

Figure 1 shows geometry of the problem. Assuming TE-wave incidence shown in Fig.1 and applying Eq.(6) on A withAmα = 0 because of δμ(r) = 0, we have[

Ex(ρρ, z)Ey(ρρ, z)Ez(ρρ, z)

]=

[Eix(ρρ, z)

00

]

−jω∫V

∑α

[Gex

eα(ρρ−ρρ′, z−z′)δε(ρρ′, z′)ε0Eα(ρρ′, z′)

Geyeα(ρρ−ρρ′, z−z′)δε(ρρ′, z′)ε0Eα(ρρ

′, z′)Gez

eα(ρρ−ρρ′, z−z′)δε(ρρ′, z′)ε0Eα(ρρ′, z′)

]dr′ (8)

where ρρ = (x, y) and∫Vdr′ =

∫ z00

dz′∫S∞

dρρ′.

III. REPRESENTATION OF EM WAVE USINGPERTURBATION METHOD

Here we use a perturbation method to estimate effects ofdepolarization due to the propagation in the random medium.Equation(8) can be expressed by the following formal equa-tion.

E = Ein + LE (9)

where E = [Ex, Ey, Ez]t, Ein = [Eix, Eiy, Eiz]t, and L =−jω ∫

Vdr′ · [∑α Ge

eα(r− r′)δε(r′)]. Here, L is divided intoL = L0 + ΔL, where L0 and ΔL are the unperturbed andperturbed operators, respectively. In this case, E can also bedefined as E =

∑∞n=0 Δ

nE(n). According as the power of Δ,the field equations may be expressed by

Δ0 : E(0)=Ein+L0E(0) (10)

: :

Δ(n) : E(n)=E(n)in +L0E

(n); E(n)in =ΔLE(n−1) (11)

: :

In above equations, E(0) is the unperturbed field which isnot depolarized through propagation in the random medium.By putting Δ(n) = 1, then E can be expressed as

E =∞∑

n=0

E(n) = E(0) +E(1)+E(2)+· · ·+E(n) + · · · . (12)

On the other hand, E can be expressed by

E=(Ein+L0E(0))+(E

(1)in +L0E

(1))+· · ·= Ein+(L0+ΔL)(E(0)+E(1)+· · ·)=Ein+LE (13)

Because Ein = [Eix, 0, 0]t as shown in Fig.1, the unper-turbed wave E(0) is given as the solution of the followingequation.

E(0)x (ρρ, z)=Eix(ρρ, z)−k2

∫ z0

0

dz′∫S∞dρρ′ · [G(ρρ−ρρ′, z−z′)

·Rexex(ρρ−ρρ′, z−z′)δε(ρρ′, z′)E(0)

x (ρρ′, z′)] (14)

where G(ρρ, z) is the scalar Green’s function in free space,Rex

ex(ρρ, z)={x2/(ρ2+z2)} − 1 and E(0)y (ρρ, z) = E

(0)z (ρρ, z) =

0. On the other hand, we can obtain the first order perturbedwave E(1) as the solution of the following equation.

E(1)α′ (ρρ, z)=E

(1)iα′(ρρ, z)−k2

∫ z0

0

dz′∫S∞dρρ′ · [G(ρρ− ρρ′, z − z′)

·Reα′eα′(ρρ− ρρ′, z − z′)δε(ρρ′, z′)E(1)

α′ (ρρ′, z′)] ; (15)

Fig.1 Geometry of the problem

E(1)iα′(ρρ, z) = −k2

∫ z0

0

dz′∫S∞

dρρ′ · [G(ρρ− ρρ′, z − z′)

·Reα′ex (ρρ− ρρ′, z − z′)δε(ρρ′, z′)E(0)

x (ρρ′, z′)] (16)

where Reα′eα′(ρρ, z) = {(α′)2/(ρ2 + z2)} − 1, Reα′

ex (ρρ, z) =xα′/(ρ2+z2), and α′ = y, z.

If z0 � z, then on the para-axis of z, Rexex ≈ −1 in

Eq.(14), Reyey ≈ −1 and Rez

ez ≈ 0 in Eq.(15). In this case,we have obtained E(0): the solution of Eq.(14) approximatelyin a compact form[4]. Therefore we can discuss quantitativelythe depolarization through the analysis of Eq.(12). From theapproximate solutions of Eqs.(14) and (15), we have theapproximate fields of E(ρρ, z):

E(ρρ, z)|z0�z ≈ E(0) +E(1) = [E(0)x , E(1)

y , E(1)z ]t . (17)

Here we have |E(0)x | � |E(1)

y | � |E(1)z | in case that the

fluctuating intensity of continuous random medium is so weakand the fluctuating scale-size is much larger than the wavelength of EM wave.

IV. CONCLUSION

We derived an integral equation using the dyadic Green’sfunction on the assumption that there exists a continuousrandom medium screen of which the dielectric constant isfluctuating randomly. To solve the depolarization problem, wemodified the integral equation under the condition that theobservation point is very far from the screen. On this modifiedintegral equation, the depolarized EM wave has been analyzedby using a perturbation method. Finally we have shown therepresentation of the first order perturbation of the depolarizedEM wave. Because the representation is written in a compactform using an ordered exponential function[4], this result isuseful for the analysis of the depolarization of EM wavepropagated through a random medium.

REFERENCES

[1] M. Tateiba: Theoretical study on wave propagation and scattering inrandom medium and its application (invited), IEICE Transactions onElectronics, Vol.E93-C, No.1, pp.3–8, 2010.

[2] Y. Nanbu, M. Tateiba, and Hosam El-Ocla: A Perturbative Representa-tion of Depolarized Electromagnetic Waves Propagated Through RandomMedium Screen, Proc. ISAP2011, (USB flash drive), 2011.

[3] Y. Nanbu, M. Tateiba, and Hosam El- Ocla: Analysis of Depolarized Elec-tromagnetic Waves Propagated Through Random Media, Proc. PIERS2011, pp.648 652, 2011.

[4] M. Tateiba: Moment equation of a wave propagating through randommedia, Memories of the Faculty of Engineering, Kyushu University,Vol.33, No.4, pp.129–137, 1974.