Open access peer-reviewed chapter

Fractional Operators Approach and Fractional Boundary Conditions

By Eldar Veliev, Turab Ahmedov, Maksym Ivakhnychenko

Submitted: October 8th 2010Reviewed: March 16th 2011Published: June 21st 2011

DOI: 10.5772/16300

Downloaded: 2052

1. Introduction

Tools of fractional calculus including fractional operators and transforms have been utilized in physics by many authors (Hilfer, 2000). Fractional operators defined as fractionalizations of some commonly used operators allow describing of intermediate states. For example, fractional derivatives and integrals (Oldham & Spanier, 1974; Samko et al., 1993) are generalizations of derivative and integral. Fractional curl operator defined in (Engheta, 1998) is a fractionalized analogue of conventional curl operator used in many equations of mathematical physics. A fractionalized operator generalizes the original operator. The idea to use fractional operators in electromagnetic problems was formulated by N. Engheta (Engheta, 2000) and named “fractional paradigm in electromagnetic theory”.

Our purpose is to find possible applications of the use of fractional operators in the problems of electromagnetic wave diffraction. In this paper two-dimensional problems of diffraction by infinitely thin surfaces are considered: a strip, a half-plane and a strip resonator (Fig.1). Assume that an incident field is an E-polarized plane wave, described by the function

Ei=zEzi(x,y)=zeik(xcosθ+ysinθ)E1

Figure 1.

Geometry of the diffraction problems: a) strip, b) half-plane, c) two parallel strips.

where θis the incidence angle, k=2πλis the wavenumber. Here, the time dependence is assumed to be eiωtand omitted throughout the paper. There are three structures considered in this paper:

  1. a strip located in the plane y=0(x[a,a]) infinite along the axis z (Fig. 1a);

  2. a half-plane (y=0,x0) (Fig. 1b);

  3. two parallel strips infinite along the axis z (a strip resonator). The first strip is located aty=l, x[a,a], and the second one is aty=l, x[a,a](Fig. 1c).

One may ask what new features are that the fractional operators can bring to the theory of diffraction. The concept of intermediate states, obtained with the aid of fractional

derivatives and integrals, yields to various generalizations of commonly used models in electrodynamics such as:

  1. Intermediate waves. For instance, intermediate waves between plane and cylindrical waves (Engheta, 1996, 1999) can be obtained using fractional integral of scalar Green’s function:

Gα(x,y;k)12(DyαG2(x,y;k)DyαG2(x,y;k)),0α1E2

where G2 is two-dimensional Green’s function of the free space. Gα describes an intermediate case between one- and two-dimensional Green’s functions and have the following behavior in the far-zone (Engheta, 1999):

Gα~i4πcos(πα2)(ksin|φ|)α2πkρeikρiπ/4+i2kαΓ(α)eik|x|k|y|1α, kρ=kx2+y2,
φ0E3

This function consists of two waves: a cylindrical wave and a non-uniform plane wave propagating in the xdirection and behaving with yas|y|α1.

  1. Fractional Green’s function Gαdefined as a fractional derivative (integral) of the ordinary Green’s function of the free space -GαDkyαG. αdenotes the fractional order and varies from 0 to 1 (0α1). In two-dimensional case Gαis expressed as

Gα(xx',yy')=i4DkyαH0(1)(k(xx')2+(yy')2)E4
  1. Fractional Green’s theorem which involves fractional derivatives of ordinary Green’s function and fractional derivatives of the considered function on a boundary of a domain (Veliev & Engheta, 2003). The corresponding equations will be presented later in this paper.

  2. Fractional boundary conditions (FBC) defined via fractional derivatives of the tangential electric field componentsU(x,y). For an infinitely thin boundary Slocated in the planey=d, FBC are defined asDyαU(x,y)|yS=0,y±d.

The order of the fractional derivative αis assumed to be between 0 and 1. Fractional derivative Dαis applied along the direction normal to the surfaceS. Fractional boundary conditions describe an intermediate boundary between the perfect electric conductor (PEC) and the perfect magnetic conductor (PMC), obtained from FBC if the fractional order equals to 0 and 1, respectively.

We will use the symbol Dyαfto denote operator of fractional derivative or integralDyαf, which is defined by the integral of Riemann-Liouville on semi-infinite interval (Samko et al., 1993):(Dxαf)(x)=1Γ(1α)ddxxf(t)dt(xt)α,0<α<1.

where Γ(1α)is Gamma function.

This paper is devoted to the problems of diffraction by a strip, a strip resonator and a half-plane characterized with fractional boundary conditions with 0α1expressed asDkyαEz(x,y)=0, y±0,xL.

where L=(a,a)for a strip and L=(0,)for a half-plane. For convenience, fractional derivative is applied with respect to dimensionless variableky. The function Ez(x,y)denotes z-component of the total electric field, Ez(x,y)=Ezi+Ezs, that is the sum of the incident plane wave Ezi(x,y)and the scattered waveEzs(x,y).

In case of a strip resonator we have two equations to impose fractional boundary conditions:DkyαEz(x,y)=0, yl±0, x(a,a),DkyαEz(x,y)=0, yl±0,x(a,a).

From the one hand, introduction of new boundary conditions should describe a new physical boundary world, and from the other hand they must allow to build an effective computational algorithm to solve the stated problems with a desired accuracy. Simple mathematical description of the scattering properties of surfaces is a common problem in modeling in diffraction theory.

One of the well-studied boundaries, which can be treated as an intermediate state between PEC and PMC, is an impedance boundary defined by the equationn×E(r)=ηn×(n×H(r)),rS.

where nis the normal to the surfaceS. The value of the impedance ηvaries from 0 for PEC to ifor PMC.

There are many papers devoted to diffraction by impedance boundaries. Impedance boundary conditions (IBC) have been used for the modeling of the scattering properties of good conductors, gratings, etc. In each case there are formulas to define the value of the impedance as a function of material parameters. IBC are approximate BC and therefore they have limitations in usage and cannot describe all diversity of boundaries.

Further approximation of IBC can be made with the aid of derivatives of higher but integer orders or generalized boundary conditions (Hope & Rahmat-Samii, 1995; Senior & Volakis, 1995). A general methodology to obtain exact IBC of higher order in spectral domain is presented in (Hope & Rahmat-Samii, 1995), where flat covers (and also surfaces with curvature) consisting of homogeneous materials with an arbitrary (linear, bi-anisotropic) constitutive equations. It is possible to obtain exact IBC in the spectral domain that can be often done in an analytical form very often. However, it is not always possible to get IBC in the spatial domain in an exact form. That is why it is necessary to approximate IBC in the spectral domain in order to apply inverse Fourier transform.

Another boundary condition that generalizes the perfect boundaries like PEC and PMC was introduced in (Lindell & Sihvola, 2005a). The corresponding surface was named perfect electromagnetic conductor (PEMC) and the mentioned condition is defined asH+ME=0.

ForM=0, PEMC defines a PEC boundary and for M=we get a PMC. The physical model of PEMC boundary was proposed in (Lindell & Sihvola, 2005b) where it was shown that the PEMC condition can simulate reflection from an anisotropic layer for the normal incidence of the plane wave. Diffraction by a PEMC boundary has not been considered yet. Further generalization of PEMC can be made using concept of the generalized soft-and-hard surface (GSHS) (Haninnen et al., 2006):aE=0,bH=0.

wherea, bare complex vectors that satisfy equations na=nb=0andab=1. GSHS can transform an incident plane wave with any given polarization into any other polarization of the reflected plane wave if the vectorsa, bare chosen appropriately (Haninnen et al., 2006).

Fractional boundary conditions (FBC) can be compared with impedance boundary conditions (IBC). First of all FBC are intermediate between PEC and PMC as well as IBC. The value of fractional order α=0(α=1) corresponds to the value of impedance η=0(η=i), respectively. For other values of 0<α<1the deeper analysis is needed.

Physical analysis of the strip with FBC shows that the induced surface currents behave similarly to the currents on an impedance strip. Due to specific properties the strip with FBC is compared with the well-known impedance strip. It can be shown that for a wide range of input parameters the “fractional strip” behaves similarly to the impedance strip if the fractional order is chosen appropriately (Veliev et al., 2008b). The proposed method used for a “fractional strip” has some advantages over the known methods applied to the analysis of the wave scattering by an impedance strip.

The purpose of this work is to build an effective analytic-numerical method to solve two-dimensional diffraction problems for the boundaries described by fractional boundary conditions with α ∈ [0,1]. The method will be applied to two canonical scattering objects: a strip and a half plane. The method is based on presenting the scattered field via fractional Green’s function,Ezs(x,y)Lf1α(x')Gα(xx',y)dx',

where f1α(x)is the unknown function and Gα(xx',y)=i4DkyαH0(1)(k(xx')2+y2)is the fractional derivative of the Green’s function defined by equation (2). This presentation leads to the following dual integral equations (DIE) with respect to the Fourier transform F1α(q)=Lf1α(ξ)eikqξdξof the functionf1α(x)

{F1α(q)eikdLξq(1q2)α1/2dq=4πeiπ/2(1α)sinαθeikdLξcosθ, ξL,F1α(q)eikdLξqdq=0,             ξL,E5

where dL=aforL=(a,a), dL=1forL=(0,).

In the case of a strip resonator, we obtain more complicated set of integral equations which will be presented later in this paper.

The method generalizes the known method used for the PEC and PMC strip and half plane. As will be shown later, this method allows obtaining a solution for the value α=0.5in the explicit analytical form. For other values of α[0,1]the scattering problems are reduced to solving of the infinite systems of linear algebraic equations (SLAE). In order to discretize the DIE the function f1α(x)is represented as a series in terms of orthogonal polynomials: Gegenbauer polynomials for the strip and Laguerre polynomials for the half-plane. These representations result in a special kind of the edge conditions for the fractional current density functionf1α(x). The physical characteristics of the considered scattering objects can be found with any desired accuracy by solving SLAE.

2. Diffraction by a strip with fractional boundary conditions

Assume that an E-polarized plane wave is characterized with the functionEi=zEzi(x,y)=zeik(xcosθ+ysinθ). The total field E=zEz(x,y)must satisfy fractional boundary conditions

DkyαEz(x,y)=0,y±0,xLE6

where L=(a,a)for a strip. For convenience, fractional derivative Dkyαis applied with respect to a dimensionless variableky. The function Ez(x,y)denotes the z-component of the total electric field Ez(x,y)=Ezi+Ezsthat is the sum of the incident plane wavе Ezi(x,y)and the scattered fieldEzs(x,y). Solution to the diffraction by the screen S={(x,y):y=0,a<x<a}is to be sought under the following conditions:

  1. The total field Emust satisfy the Helmholtz equation everywhere outside the screen

(2x2+2y2+k2)Ez(x,y)=0E7
  1. The scattered field Ezs(x,y)must satisfy Sommerfeld radiation condition at the infinity

limrr(EzsriEzs)=0,r=x2+y2E8
  1. The total field Emust satisfy the edge condition, i.e. the finiteness of energy in every local area near the edges of the screen (Honl et al., 1961).

  2. The total field Ez(x,y)must satisfy the boundary conditions (3).

The method is based on representation of the scattered field with the aid of the fractional derivative of the Green’s function:

Ezs(x,y)Lf1α(x')Gα(xx',y)dx'E9

In (6), the function f1α(x)is the unknown function called the density of the fractional potential, and Gαis the fractional derivative of two-dimensional the Green’s function of the free space defined by equation (2).

For the limit cases of the fractional order with α=0and α=1representation (6) corresponds to the single-layer and double-layer potentials commonly used to present the scattered fields in diffraction problems:

Ezs(x,y)={i4aaf'(x')H0(1)(k(xx')2+y2)dx',          α=0i4aaf(x')yH0(1)(k(xx')2+y2)dx',        α=1

More general representations (6) can be derived from the fractional Green’s theorem (Veliev & Engheta, 2003) which generalizes the ordinary Green’s theorem.

2.1. Fractional Green’s theorem

Consider a functionψ(r), which satisfies inhomogeneous scalar Helmholtz equation with the source density given by the functionρ(r):

Δψ(r)+k2ψ(r)=4πρ(r)E10

Besides, defineG(r,r0)as the Green’s function of the Helmholtz equation:

ΔG(r,r0)+k2G(r,r0)=4πδ(rr0)E11

Here, δ(rr0)is the three-dimensional Dirac delta function, rand r0are the position vectors for the observation and source points, respectively, Δ=2x2+2y2+2z2is the Laplacian, and kis a scalar constant. After applying fractional derivatives to equations (7) and (8) with respect to the xvariable, multiplying the first equation withDxνG(r,r0), and the second withDxμψ(r), subtracting one from another, integrating this over all source coordinates x0,y0,z0insideS, and finally using the Green’s theorem, we obtain the following representation:

Dxβψ(r)={VDx0βνρ(r0)·Dx0νG(r,r0)dv0++14πS[Dx0νG(r,r0)·0Dx0βνψ(r0)Dx0βνψ(r0)·0Dx0νG(r,r0)]·ds0,  rV 0,          rVE12

whereμ+ν=β. Operator 0denotes the operator of gradient in respect of variabler0(x0,y0,z0). Here it was used the property of the fractional derivative of the Dirac delta function:

VF(r0)Dx0νδ(r0r)dv0=DxνF(r)E13

We use the uniform symbol Dxα(orDxα) to denote both fractional derivatives and fractional integrals, and it defines a fractional derivative for 0<α<1and a fractional integral forα<0.

Equation (9) is a generalization of well-known Green’s theorem for the case of fractional derivatives.

Consider some important particular cases, which can be obtained from (9).

In the case of excitation in a free space so that the volume Vis the whole space, the surface integrals in (9) vanish, and we have:

Dxβψ(r)=VDx0βνρ(r0)·Dx0νG(r,r0)dv0E14

Originally function ψ(r)characterizes the field excited by the source with the volume densityρ(r). From the other hand, for β=0representation (11) means that the field ψ(r)is expressed through the distribution of fractional sources with density Dνρ(r0)inside the volume Vand by using fractional integral of conventional Green’s functionDνG(r0,r).

Assumingρ(r)=0, we can obtain some other important representations:

Dxβψ(r)={14πS[Dx0βG(r,r0)·0ψ(r0)ψ(r0)·0Dx0βG(r,r0)]ds0,   if  ν=β,μ=014πS[G(r,r0)·0Dx0βψ(r0)Dx0βψ(r0)·0G(r,r0)]ds0,   if  ν=0 E15

From this representation we see that the fractional derivative of function ψ(r)is expressed either via the value of the function and its first derivative at the boundary and the fractional derivatives of Green’s function, or by the fractional derivatives of the function at the boundary and the usual Green’s function.

Ifν=μ, i.e.β=0, we obtain a representation for the function ψ(r)itself:

ψ(r)=14πS[Dx0μG(r,r0)·0Dx0μψ(r0)Dx0μψ(r0)·0Dx0μG(r,r0)]ds0E16

This expression means that the function ψ(r)is represented through its fractional derivatives at the boundary and the fractional derivatives of Green's function. The equation (13) can be useful in scattering problems. If we have boundary conditions for the function ψ(r)on the surface Sas 0Dx0μψ(r0)|r0S=0(orDx0μψ(r0)|r0S=0) then one of the surface integrals in (13) vanishes and we get a simple presentation forψ(r). This fact will be used to present the scattered field in all diffraction problems considered in this paper (6). Equations (12), (13) generalize the Huygens principle in such a sense that the fractional derivative of the functionψ(r), which characterizes a wave process, is presented as a superposition of waves radiated by elementary "fractional" sources distributed on the given surface. “Fractional” potentials, SDx0βνψ(r0)·0Dx0νG(r,r0)·ds0, SDx0νG(r,r0)·0Dx0βνψ(r0)·ds0, can be treated as a generalization of well-known single and double layer potentials.

2.2. Solution to integral equations

Substituting the expression (6) for Ez(x,y)into fractional boundary conditions (3) we get the equation

limy0 DkyαLf1α(x')Gα(xx',y)dx'=limy0DkyαEzi(x,y)E17

It is convenient to use the Fourier transform of the fractional potential density f1α(x)

F1α(q)f˜1α(ξ)eikqξdξ=a11f1α(aξ)eikqξdξE18

where a new function f˜1α(ξ)is introduced:

f˜1α(ξ)af1α(aξ),
|ξ|<1E19
,f˜1α(ξ)0,
|ξ|1E20
.

Then the scattered field is expressed via the Fourier transform F1α(q)as

Ezs(x,y)=ie±iπα/24πF1α(q)eik(xq+|y|1q2)(1q2)(α1)/2dqE21

where the upper (lower) sign is chosen for y>0 (y<0). Here, in (15), the following representation for the fractional Green’s function was used:

Gα(xx',y)=iDkyαH0(1)(k(xx')2+y2)=iesign(y)iπα/24πeik((xx')q+|y|1q2)(1q2)(α1)/2dqE22

It can be shown that the equation (14) can be reduced to dual integral equations (DIE)

{F1α(q)eikaξq(1q2)α1/2dq=4πeiπ/2(1α)sinαθeikdLξcosθ, |ξ|<1,F1α(q)eikaξqdq=0,             |ξ|>1,E23

For the limit cases of the fractional order α = 0 and α = 1 the equations (17) are reduced to the well known integral equations used for PEC and PMC strips (Honl et al., 1961; Veliev & Veremey, 1993; Veliev & Shestopalov, 1988; Uflyand, 1977), respectively. In this paper the method to solve DIE (17) is proposed for arbitrary value of α ∈[0,1].

DIE (17) can be solved analytically for one special case of α = 0.5. In this case we get the solutions for any value of k as

f0.5(x)=2iksin1/2θeikxcosθ+iπ/4E24
F0.5(q)=4ieiπ/4sin1/2θsinka(q+cosθ)q+cosθE25

In the case of arbitrary αthe solutions can be obtained numerically. First, we modify the equations (17). After multiplying by eikaτξand integrating in ξfrom -1 to 1, the first equation in (17) can be rewritten in the following form:

F1α(q)sinka(qτ)qτ(1q2)α1/2dq=4πeiπ/2(1α)sinαθsinka(τ+cosθ)τ+cosθE26

In order to discretize this equation, we present the unknown function f˜1α(ξ)as a uniformly convergent series in terms of the orthogonal polynomials with corresponding weight functions which allow satisfying the edge conditions:

f˜1α(ξ)=(1ξ2)α1/2n=0fnα1αCnα(ξ)E27

where Cnα(x)are the Gegenbauer polynomials and fnαare the unknown coefficients. Gegenbauer polynomials can be treated as intermediate polynomials between Chebyshev polynomials of the first and second kind:

limα0Cnα(ξ)α={2nTn(ξ),n01,n=0,
limα1Cnα(ξ)α=Cn1(ξ)=Un(ξ)E28

The Fourier transform F1α(q)is expressed as the series

F1α(q)=2πΓ(α+1)n=0(i)nΓ(n+2α)Γ(n+1)Jn+α(kaq)(2kaq)αfnαE29

where Jn+α(kaq)is the Bessel function.

It must be noted that the edge conditions are chosen in the following form

f˜1α(ξ)=O((1ξ2)α1/2),
ξ±1E30

For special cases of α=0and α=1the edge conditions have the form as

f˜1α(ξ)={O((1ξ2)1/2),     α=0O((1ξ2)1/2),      α=1,ξ±1E31

These are well-known Meixner edge conditions in diffraction problems (Honl et al., 1961).

Substituting (22) into (17) and taking into account the properties of discontinuous integrals of Weber-Shafheitlin (Bateman & Erdelyi, 1953) and the following formula (Prudnikov et al., 1986)

1πJn+ν(εq)qνsinε(qβ)qβdq=Jn+ν(εβ)βνE32

one can show that the homogenous equation in the set (17) is satisfied identically.

The first equation of (17) written in the form (20) can be reduced to an infinite system of linear algebraic equations (SLAE) with respect to the unknown coefficientsfnα:

n=0(i)nΓ(n+2α)Γ(n+1)Cmnαfnα=Bmα,m=0,1,2,..,E33

where the matrix coefficients are expressed as

Cmnα=Jn+α(kaq)Jm+α(kaq)(1q2)α1/2q2αdqE34
Bmα=2Γ(α+1)(2ka)αeiπ/2(1α)sinαθJm+α(kacosθ)(cosθ)αE35

It can be shown that the SLAE (26) can be reduced to SLAE of the Fredholm type of the second kind (Veliev et al., 2008a). Then the coefficients fnαcan be found with any desired accuracy (within the machine precision) using the truncation of SLAE. The fractional density f1α(x)is computed by using (21) and the scattered field (6) and other physical characteristics can be obtained as series in terms of the found coefficientsfnα.

In order to solve the diffraction problem on a plane screen with fractional boundary conditions and obtain a convenient SLAE we applied several techniques. First of all, the fractional Green’s theorem presented above allowed searching the unknown scattered field as a potential with the fractional Green’s function. The order of the fractional Green’s function is defined from the fractional order of the boundary conditions. In general, the fractional derivative of Green’s function may have a complicated form, but we used the Fourier transform where application of the fractional derivative maps to a simple multiplication by(iq)α. Finally, utilization of the orthogonal Gegenbauer polynomials along with the specific form of the edge conditions allowed to reduce integral equations to SLAE in a convenient form. One can compare the method presented for fractional boundary conditions with the known methods applied to solve diffraction by an impedance strip. The impedance strip requires to consider two unknown densities in presentation of the scattered field as a sum of single- and double-layer potentials. The usage of two unknown functions leads to more complicated SLAE in spite of the SLAE obtained for fractional boundary conditions.

2.3. Physical characteristics

We consider such electrodynamic characteristics of the scattered field as the radiation pattern (RP), monostatic radar cross-section (MRCS) and surface current densities depending on the coefficientsfnα. The scattered field Ezs(x,y)in the far-zone krin the cylindrical coordinate system(r,ϕ), x=rcosφ,y=rsinφ, is expressed as

Ezs(r,φ)=i4π(±i)α+F1α(cosβ)eikrcos(φ±β)sinαβdβE36

where the upper sign is chosen forφ[0,π], and the lower one whenφ[π,2π]. Using the stationary phase method for krwe present Ezs(x,y)as

Ezs(x,y)A(kr)Φα(φ),
krE37

where

A(kr)=2πkreikriπ/4,
Φα(φ)=i4(±i)αF1α(cosφ)sinαφE38
.

The function Φα(φ)describes RP and can be expressed via the coefficients fnαas

Φα(φ)=πi(±i)α2Γ(α+1)tanαφn=0(i)nfnαΓ(n+2α)Γ(n+1)Jn+α(kacosφ)(2ka)αE39

In physical optics (PO) approximation (ka1) Φα(φ)has a simpler form. Using the following formula

limkasinka(αβ)αβ=πδ(αβ)E40

in IE (20) we get the following expressions for Fα(q)andΦα(φ):

F1α(q)4iαsin1αθ(1q2)(12α)/2sinka(qcosθ)qcosθE41
Φα(φ)(1)αsinφ(sinθsinφ)αsinka(cosφ+cosθ)cosφ+cosθE42

In the special case of α=0.5and arbitrary value of kawe get an analytical expression for the RP

Φ0.5(φ)=(1)1/2sinφsinθsinka(cosφ+cosθ)cosφ+cosθE43

Bi-static radar cross section (BRCS) is expressed from RP Φ(φ)asσ2dλ(φ)=2π|Φ(φ)|2. MRCS σ2Dmonois defined asσ2Dmono=σ2dλ(θ)=2π|Φ(θ)|2.

We have the following representations in PO approximation

σ2dλ=2πsin2φ(sinθsinφ)2α{sinka(cosφ+cosθ)cosφ+cosθ}2,
ka1E44
,σ2Dmono=2πsin2θ{sinka(2cosθ)2cosθ}2,
ka1E45
.

It must be noted that the density function f1α(x)in the integral (6) does not describe the density of physical surface currents on the strip for0<α<1. The function f1α(x)is defined as the discontinuity of fractional derivatives of E-field at the planey=0:

f1α(x)= Dky1αEz(x,y)|y=+0Dky1αEz(x,y)|y=0,x(a,a)E46

For the limit cases of α=0and α=1the equation (29) is reduced to well-known presentations for electric and magnetic surface currents, respectively, i.e.

f1α(x)={Ez(x,y)y|y=+0Ez(x,y)y|y=0=Hx(x,+0)Hx(x,0),    α=0Ez(x,+0)Ez(x,0),                    α=1

In order to obtain physical surface currents from f1α(x)we have to apply additional integration. In case of E-polarized incident plane wave we have the following induced currents on a strip: electric current jα(e)=zjzα(e)and magnetic current jα(m)=xjxα(m)expressed from f1α(x)as

jzα(e)(x)=2icos(πα2)i4π+F1α(q)eikax(1q2)α/2dqE47
jxα(m)(x)=2sin(πα2)i4π+F1α(q)eikax(1q2)α/21/2dqE48

The detailed analysis of the scattering properties of the strip with fractional boundary conditions one can find in papers (Veliev et al., 2008a; Veliev et al., 2008b).

2.4. H-polarization

In the case of the H-polarized incident plane waveHi(0,0,Hzi), whereHzi(x,y)=eik(xcosθ+ysinθ), the method proposed above can be applied as well. We define fractional boundary conditions as

Dky1αHz(x,y)|y±0=Dky1α[Hzi(x,y)+Hzs(x,y)]|y±0=0,
x(a,a)E49

The case of α=0corresponds to diffraction of the H-polarized plane wave on a PEC strip, while the case of α=1describes diffraction of the H-polarized plane wave on a PMC strip. As before, we represent the scattered field via the fractional Green’s function

Hzs(x,y)aafα(x)G1α(xx,y)dxE50

After substituting (18) into fractional boundary conditions (19) we get the equation

limy0Dky1αaafα(x)G1α(xx,y)dx=limy0Dky1αHzi(x,y)E51

This equation can be solved by repeating all steps of the E-polarization case after changing αto1α.

3. Diffraction by a half-plane with fractional boundary conditions

Another problem studied in this paper is the diffraction by a half-plane with fractional boundary conditions. The method introduced to solve the dual integral equation (DIE) for a finite object (a strip) will be modified to solve DIE for semi-infinite scatterers such as half-plane. There are many papers devoted to the classical problem of diffraction by a half-plane. The method to solve the scattering problem for a perfectly conducting half-plane is presented in (Honl et al., 1961). Usually, it is solved using Wiener-Hopf method. The first application of the method to a PEC half-plane can be referred to the papers of Copson (Copson, 1946) and independently to papers of Carlson and Heins (Carlson & Heins, 1947). In 1952 Senior first applied Wiener-Hopf method to the diffraction by an impedance half-plane (Senior, 1952) and later oblique incidence was considered (Senior, 1959). Diffraction by a resistive and conductive half-plane and also by various types of junctions is analyzed in details in (Senior & Volakis, 1995). We propose a new approach for the rigorous analysis of the considered problem which generalizes the results of (Veliev, 1999) obtained for the PEC boundaries and includes them as special cases.

Let an E-polarized plane wave Ezi(x,y)=eik(xcosθ+ysinθ)(1) be scattered by a half-plane (y=0,x>0). The total field Ez=Ezi+Ezsmust satisfy fractional boundary conditions

DkyαEz(x,y)=0,y±0,x>0E52

and Meixner’s edge conditions must be satisfied forx0.

Following the idea used for the analysis of diffraction by a strip we represent the scattered field using the fractional Green’s function

Ezs(x,y)0f1α(x)Gα(xx,y)dxE53

where f1α(x)is the unknown function, Gαis the fractional Green’s function (2).

After substituting the representation (31) into fractional boundary conditions (30) we get the equation

i4limy0Dky2α0f1α(x)H0(1)(k(xx)2+y2)dx=limy0DkyαEzi(x,y),x>0E54

The Fourier transform of f1α(x)is defined as

F1α(q)=f˜1α(ξ)eikqξdξ=0f1α(x)eikqxdxE55

where f˜1α(ξ)f1α(ξ)for ξ>0and f˜1α(ξ)0forξ<0.

Then the scattered field will be expressed via the Fourier transform F1α(q)as

Ezs(x,y)=ie±iπα/24πF1α(q)eik(xq+|y|1q2)(1q2)(α1)/2dqE56

Using the Fourier transform the equation (32) is reduced to the DIE with respect toF1α(q):

{F1α(q)eikξq(1q2)α1/2dq=4πeiπ/2(1α)sinαθeikξcosθ,ξ>0,F1α(q)eikξqdq=0,ξ<0.E57

The kernels in integrals (34) are similar to the ones in DIE (17) obtained for a strip if the constant dLis equal to 1 (L=(0,)in the case of a half-plane).

For the limit cases of the fractional order α=0and α=1these equations are reduced to well known integral equations used for the PEC and PMC half-planes (Veliev, 1999), respectively. In this paper the method to solve DIE (5) is proposed for arbitrary values ofα[0,1].

DIE allows an analytical solution in the special case of α=0.5in the same manner as for a strip with fractional boundary conditions. Indeed, for α=0.5we obtain the solution for any value of kas

F0.5(q)=2sin1/2θeiπ/4πkδ(q+cosθ)E58
f0.5(x)=2sin1/2θeiπ/4eikxcosθE59

The scattered field can be found in the following form:

Ezs(x,y)=i2ke±iπα/2eiπ/4sinα1/2θeik(cosθx+|y|sinθ),α=0.5, for
y>0   (y<0)E60

In the general case of 0<α<1the equations (34) can be reduced to SLAE. To do this we represent the unknown function f˜1α(ξ)as a series in terms of the Laguerre polynomials with coefficientsfnα:

f˜1α(x)=exxα1/2n=0fnαLnα1/2(2x)E61

Laguerre polynomials are orthogonal polynomials on the interval L=(0,)with the appropriate weight functions used in (35). It can be shown from (35) that f˜1α(ξ)satisfies the following edge condition:

f˜1α(ξ)=O(ξα1/2),ξ0E62

For the special cases of α = 0 and α = 1, the edge conditions are reduced to the well-known equations (Honl et al., 1961) used for a perfectly conducting half-plane.

After substituting (35) into the first equation of (34) we get an integral equation (IE)

n=0fnα[0ettα1/2Lnα1/2(2t)eikqtdt]×eikξq(1q2)α1/2dq=R(ξ)E63

where R(ξ)=4πeiπ/2(1α)sinαθeikξcosθis known.

Using the representation for Fourier transform of Laguerre polynomials (Prudnikov et al., 1986) we can evaluate the integral over dt as

0ettα1/2Lnα1/2(2t)eikqtdt=0et(1+ikq)tα1/2Lnα1/2(2t)dt=Γ(n+α+1/2)Γ(n+1)(ikq1)n(ikq+1)n+α+1/2E64

After some transformations IE (37) is reduced to

n=0fnαΓ(n+α+1/2)Γ(n+1)(ikq1)n(ikq+1)n+α+1/2(1q2)α1/2eikξqdq=R(ξ),ξ>0E65

Then we integrate both sides of equation (38) with appropriate weight functions, as0()eξξα1/2Lmα1/2(2ξ)dξ. Using orthogonality of Laguerre polynomials we get the following SLAE:

n=0fnαCmnα=Bmα,
m=0,1,2,...,E66

with matrix coefficients

Cmnα=Γ(n+α+1/2)Γ(n+1)(ikq+1)mnα1/2(ikq1)nmα1/2(1q2)α1/2dqE67
Bmα=4πeiπ/2α|sinθ|α(1ikcosθ)m(1+ikcosθ)m+α+1/2E68

It can be shown that the coefficients fnαcan be found with any desired accuracy by using the truncation of SLAE. Then the function f˜1α(x)is found from (35) that allows obtaining the scattered field (33).

4. Diffraction by two parallel strips with fractional boundary conditions

The proposed method to solve diffraction problems on surfaces described by fractional boundary conditions can be applied to more complicated structures. The interest to such structures is related to the resonance properties of scattering if the distance between the strips varies. Two strips of the width 2a infinite along the axis z are located in the planes y=landy=l. Let the E-polarized plane wave Ezi(x,y)=eik(xcosθ+ysinθ)(1) be the incident field. The total field Ez=Ezi+Ezssatisfies fractional boundary conditions on each strip:

DkyαEz(x,y)=0,y±l±0,x(a,a)E69

and Meixner’s edge conditions must be satisfied on the edges of both strips (y=±l,x±a).

The scattered field Ezs(x,y)consists of two parts

Ezs(x,y)Ez1s(x,y)+Ez2s(x,y)E70

where

Ezjs(x,yj)aafj1α(x')Gα(xx',yj)dx',j=1,2E71

Here, Gα is the fractional Green’s function defined in (2). y1,2 are the coordinates in the corresponding coordinate systems related to each strip,

y1=yl,
x1=xE72
y2=y+l,
x2=xE73

Using Fourier transforms, defined as

Fj1α(q)f˜j1α(ξ)eikqξdξ=a11fj1α(aξ)eikqξdξE74
f˜j1α(ξ)afj1α(aξ),
j=1,2E75

the scattered field is expressed as

Ez1s(x,y)=ie±iπα/24πF11α(q)eik[xq+|yl|1q2](1q2)(α1)/2dq,y>l,(y<l)E76
Ez2s(x,y)=ie±iπα/24πF21α(q)eik[xq+|y+l|1q2](1q2)(α1)/2dq,y>l,(y<l)E77

Fractional boundary conditions (30) correspond to two equations

DkyαEz(x,y)=0,yl±0,x(a,a)E78
DkyαEz(x,y)=0,yl±0,x(a,a)E79

After substituting expressions (41) and (42) into the equations (43) and (44) we obtain

F11α(q)eikxq(1q2)α1/2dq=4πieiπα/2sinαθeik(xcosθ+lsinθ)F21α(q)eik[xq+2l1q2](1q2)α1/2dqE80
F21α(q)eikxq(1q2)α1/2dq=4πieiπ/2αsinαθeik(xcosθlsinθ)F11α(q)eik[xq+2l1q2](1q2)α1/2dqE81

Multiplying both equations with e–ikxτ and integrating them in ζ on the interval [–a,a], the system (45), (46) leads to

{F11α(q)sinka(qτ)qτ(1q2)α1/2dq=4πieiπα/2sinαθsinka(τ+cosθ)τ+cosθeiklsinθF21α(q)sinka(qτ)qτei2kl1q2(1q2)α1/2dqF21α(q)sinka(qτ)qτ(1q2)α1/2dq=4πieiπα/2sinαθsinka(τ+cosθ)τ+cosθeiklsinθF11α(q)sinka(qτ)qτei2kl1q2(1q2)α1/2dqE82

Similarly to the method described for the diffraction by one strip, the set (47) can be reduced to a SLAE by presenting the unknown functions fj1α(x)as a series in terms of the orthogonal polynomials. We represent the unknown functions f˜j1α(ξ)as series in terms of the Gegenbauer polynomials:

f˜j1α(ξ)=(1ξ2)α1/2n=0fnj,α1αCnα(ξ),
j=1,2E83

For the Fourier transforms Fj1α(q)we have the representations (22). Substituting the representations for Fj1α(q)into the (47), using the formula (25), then integrating (.)Jm+α(kaτ)mαdτform=0,1,2,.., we obtain the following SLAE:

{n=0(i)nΓ(n+2α)Γ(n+1)Cmn11,αfn1,α+n=0(i)nΓ(n+2α)Γ(n+1)Cmn12,αfn2,α=Bm1,αn=0(i)nΓ(n+2α)Γ(n+1)Cmn21,αfn1,α+n=0(i)nΓ(n+2α)Γ(n+1)Cmn22,αfn2,α=Bm2,α,
m=0,1,2,..E84

where the matrix coefficients are defined as

Cmn11,α=Cmn22,α=Jm+α(kaτ)ταJn+α(kaτ)τα(1τ2)α1/2dτE85
Cmn12,α=Cmn21,α=Jm+α(kaτ)ταJn+α(kaτ)ταei2kl1τ2(1τ2)α1/2dτE86
Bm1,α=e2iklsinθBm2,α=2ieiπα/2Γ(α+1)sinαθeiklsinθ(2ka)αJm+α(kacosθ)(cosθ)αE87

Consider the case of the physical optics approximation, whereka1. In this case we can obtain the solution of (47) in the explicit form. Indeed, using the formula (28) we get

{πF11α(τ)(1τ2)α1/2==4πieiπα/2sinαθsinka(τ+cosθ)τ+cosθeiklsinθπF21α(τ)ei2kl1τ2(1τ2)α1/2πF21α(τ)(1τ2)α1/2==4πieiπα/2sinαθsinka(τ+cosθ)τ+cosθeiklsinθπF11α(τ)ei2kl1τ2(1τ2)α1/2E88

Finally, we obtain the solution as

{F11α(τ)=4ieiπα/2sinαθsinka(τ+cosθ)τ+cosθ1(1τ2)α1/2(eiklsinθei2kl1τ2eiklsinθ)(1ei4kl1τ2)F21α(τ)=4ieiπα/2sinαθsinka(τ+cosθ)τ+cosθ1(1τ2)α1/2(eiklsinθei2kl1τ2eiklsinθ)(1ei4kl1τ2)E89

Having expressions for Fj1α(q)we can obtain the physical characteristics. The radiation pattern of the scattered field in the far zone (27) is expressed as

Φα(φ)=Φ1α(φ)+Φ2α(φ)E90

where

Φ1α(φ)=i4e±iπ/2αF11α(cosφ)sinαφeiklcosφE91
Φ2α(φ)=i4e±iπ/2αF21α(cosφ)sinαφeiklcosφE92

5. Conclusion

The problems of diffraction by flat screens characterized by the fractional boundary conditions have been considered. Fractional boundary conditions involve fractional derivative of tangential field components. The order of fractional derivative is chosen between 0 and 1. Fractional boundary conditions can be treated as intermediate case between well known boundary conditions for the perfect electric conductor (PEC) and perfect magnetic conductor (PMC). A method to solve two-dimensional problems of scattering of the E-polarized plane wave by a strip and a half-plane with fractional boundary conditions has been proposed. The considered problems have been reduced to dual integral equations discretized using orthogonal polynomials. The method allowed obtaining the physical characteristics with a desired accuracy. One important feature of the considered integral equations has been noted: these equations can be solved analytically for one special value of the fractional order equal to 0.5 for any value of frequency. In that case the solution to diffraction problem has an analytical form. The developed method has been also applied to the analysis of a more complicated structure: two parallel strips. Introducing of fractional derivative in boundary conditions and the developed method of solving such diffraction problems can be a promising technique in modeling of scattering properties of complicated surfaces when the order of fractional derivative is defined from physical parameters of a surface.

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Eldar Veliev, Turab Ahmedov, Maksym Ivakhnychenko (June 21st 2011). Fractional Operators Approach and Fractional Boundary Conditions, Electromagnetic Waves, Vitaliy Zhurbenko, IntechOpen, DOI: 10.5772/16300. Available from:

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