Fractional Operators Approach and Fractional Boundary Conditions (Electromagnetic Waves) Part 2

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 coefficients f^ . The scattered field Esz (x, y) in the far-zone kr in the cylindrical coordinate system is expressed as

where the upper sign is chosen for, and the lower one when. Using the stationary phase method forwe presentas

where

The function describes RP and can be expressed via the coefficients f^ as

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

in IE (20) we get the following expressions for

In the special case of a = 0.5 and arbitrary value of ka we get an analytical expression for the RP

Bi-static radar cross section (BRCS) is expressed from RP

We have the following representations in PO approximation

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

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

In order to obtain physical surface currents fromwe have to apply additional integration. In case of E-polarized incidentplane wave we have the following induced currents on a strip: electric currentand magnetic current expressed fromas

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).

H-polarization

In the case of the H -polarized incident plane wave where fractional boundary conditions as the method proposed above can be applied as well. We define fractional boundary conditions as

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

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

This equation can be solved by repeating all steps of the E -polarization case after changing a to 1 -a .

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

 

be scattered by a half-plane (y = 0, x > 0). The total field Ez = E’z + Esz must satisfy fractional boundary conditions

and Meixner’s edge conditions must be satisfied for x ^ 0 .

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

where f 1-a(x) is the unknown function, Ga is the fractional Green’s function (2).

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

The Fourier transform of f1 a (x) is defined as

Then the scattered field will be expressed via the Fourier transform as

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

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

For the limit cases of the fractional order a = 0 and a = 1 these 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 of a e [0,1].

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

The scattered field can be found in the following form:

In the general case of 0 < a < 1 the equations (34) can be reduced to SLAE. To do this we represent the unknown functionas a series in terms of the Laguerre polynomials with coefficients f^ :

Laguerre polynomials are orthogonal polynomials on the intervalwith the appropriate weight functions used in (35) . It can be shown from (35) thatsatisfies the following edge condition:

For the special cases of a= 0 and a = 1, the edge conditions are reduced to the well-known equations (Hon! 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)

is known.

is known.

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

After some transformations IE (37) is reduced to

Then we integrate both sides of equation (38) with appropriate weight functions, as

 

Using orthogonality of Laguerre polynomials we get the following SLAE:

with matrix coefficients

It can be shown that the coefficients f^ can be found with any desired accuracy by using the truncation of SLAE. Then the function f1-a (x) is found from (35) that allows obtaining the scattered field (33).

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 = l and y = -l. Let the E -polarized plane wave Elz (x,y) = e ‘k(xcos0+ysm0 (1) be the incident field. The total field Ez = E’z + Esz satisfies fractional boundary conditions on each strip:

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

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

where

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

Using Fourier transforms, defined as

the scattered field is expressed as

Fractional boundary conditions (30) correspond to two equations

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

Multiplying both equations with e-ikxT and integrating them in £ on the interval [-a,a], the system (45), (46) leads to

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 fj1-a(x) as a series in terms of the orthogonal polynomials. We represent the unknown functions as series in terms of the Gegenbauer polynomials:

For the Fourier transforms we have the representations (22). Substituting the representations for into the (47), using the formula (25), then integrating representations for

we obtain the following SLAE:

where the matrix coefficients are defined as

Consider the case of the physical optics approximation, where ka »1. In this case we can obtain the solution of (47) in the explicit form. Indeed, using the formula (28) we get

Finally, we obtain the solution as

Having expressions for Fj1 a(q) we can obtain the physical characteristics. The radiation pattern of the scattered field in the far zone (27) is expressed as

where

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.