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
where is the incidence angle, is the wavenumber. Here, the time dependence is assumed to be and omitted throughout the paper. There are three structures considered in this paper:
a strip located in the plane () infinite along the axis z (Fig. 1a);
a half-plane (,) (Fig. 1b);
two parallel strips infinite along the axis z (a strip resonator). The first strip is located at, , and the second one is at, (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:
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):, ,
This function consists of two waves: a cylindrical wave and a non-uniform plane wave propagating in the direction and behaving with as.
Fractional Green’s function defined as a fractional derivative (integral) of the ordinary Green’s function of the free space -. denotes the fractional order and varies from 0 to 1 (). In two-dimensional case is expressed as
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.
Fractional boundary conditions (FBC) defined via fractional derivatives of the tangential electric field components. For an infinitely thin boundary located in the plane, FBC are defined as,.
The order of the fractional derivative is assumed to be between 0 and 1. Fractional derivative is applied along the direction normal to the surface. 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 to denote operator of fractional derivative or integral, which is defined by the integral of Riemann-Liouville on semi-infinite interval (Samko et al., 1993):,.
where 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 expressed as, ,.
where for a strip and for a half-plane. For convenience, fractional derivative is applied with respect to dimensionless variable. The function denotes z-component of the total electric field, , that is the sum of the incident plane wave and the scattered wave.
In case of a strip resonator we have two equations to impose fractional boundary conditions:, , ,, ,.
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 equation,.
where is the normal to the surface. The value of the impedance varies from 0 for PEC to for 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 as.
For, PEMC defines a PEC boundary and for 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):,.
where, are complex vectors that satisfy equations and. GSHS can transform an incident plane wave with any given polarization into any other polarization of the reflected plane wave if the vectors, are 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 () corresponds to the value of impedance (), respectively. For other values of the 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,,
where is the unknown function and 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 of the function
where for, for.
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 in the explicit analytical form. For other values of the scattering problems are reduced to solving of the infinite systems of linear algebraic equations (SLAE). In order to discretize the DIE the function 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 function. 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 function. The total field must satisfy fractional boundary conditions
where for a strip. For convenience, fractional derivative is applied with respect to a dimensionless variable. The function denotes the z-component of the total electric field that is the sum of the incident plane wavе and the scattered field. Solution to the diffraction by the screen is to be sought under the following conditions:
The total field must satisfy the Helmholtz equation everywhere outside the screen
The scattered field must satisfy Sommerfeld radiation condition at the infinity
The total field must satisfy the edge condition, i.e. the finiteness of energy in every local area near the edges of the screen (Honl et al., 1961).
The total field 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:
In (6), the function is the unknown function called the density of the fractional potential, and is 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 and representation (6) corresponds to the single-layer and double-layer potentials commonly used to present the scattered fields in diffraction problems:
2.1. Fractional Green’s theorem
Consider a function, which satisfies inhomogeneous scalar Helmholtz equation with the source density given by the function:
Besides, defineas the Green’s function of the Helmholtz equation:
Here, is the three-dimensional Dirac delta function, and are the position vectors for the observation and source points, respectively, is the Laplacian, and is a scalar constant. After applying fractional derivatives to equations (7) and (8) with respect to the variable, multiplying the first equation with, and the second with, subtracting one from another, integrating this over all source coordinates inside, and finally using the Green’s theorem, we obtain the following representation:
where. Operator denotes the operator of gradient in respect of variable. Here it was used the property of the fractional derivative of the Dirac delta function:
We use the uniform symbol (or) to denote both fractional derivatives and fractional integrals, and it defines a fractional derivative for and a fractional integral for.
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 is the whole space, the surface integrals in (9) vanish, and we have:
Originally function characterizes the field excited by the source with the volume density. From the other hand, for representation (11) means that the field is expressed through the distribution of fractional sources with density inside the volume and by using fractional integral of conventional Green’s function.
Assuming, we can obtain some other important representations:
From this representation we see that the fractional derivative of function 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.
If, i.e., we obtain a representation for the function itself:
This expression means that the function 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 on the surface as (or) then one of the surface integrals in (13) vanishes and we get a simple presentation for. 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 function, which characterizes a wave process, is presented as a superposition of waves radiated by elementary "fractional" sources distributed on the given surface. “Fractional” potentials, , , can be treated as a generalization of well-known single and double layer potentials.
2.2. Solution to integral equations
It is convenient to use the Fourier transform of the fractional potential density
where a new function is introduced:,
Then the scattered field is expressed via the Fourier transform as
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:
It can be shown that the equation (14) can be reduced to dual integral equations (DIE)
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
In the case of arbitrary the solutions can be obtained numerically. First, we modify the equations (17). After multiplying by and integrating in from -1 to 1, the first equation in (17) can be rewritten in the following form:
In order to discretize this equation, we present the unknown function as a uniformly convergent series in terms of the orthogonal polynomials with corresponding weight functions which allow satisfying the edge conditions:
where are the Gegenbauer polynomials and are the unknown coefficients. Gegenbauer polynomials can be treated as intermediate polynomials between Chebyshev polynomials of the first and second kind:,
The Fourier transform is expressed as the series
where is the Bessel function.
It must be noted that the edge conditions are chosen in the following form,
For special cases of and the edge conditions have the form as
These are well-known Meixner edge conditions in diffraction problems (Honl et al., 1961).
one can show that the homogenous equation in the set (17) is satisfied identically.
where the matrix coefficients are expressed as
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 can be found with any desired accuracy (within the machine precision) using the truncation of SLAE. The fractional density is computed by using (21) and the scattered field (6) and other physical characteristics can be obtained as series in terms of the found coefficients.
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. 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 coefficients. The scattered field in the far-zone 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 for we present as,
The function describes RP and can be expressed via the coefficients as
In physical optics (PO) approximation () has a simpler form. Using the following formula
in IE (20) we get the following expressions for and:
In the special case of and arbitrary value of we get an analytical expression for the RP
Bi-static radar cross section (BRCS) is expressed from RP as. MRCS is defined as.
We have the following representations in PO approximation
It must be noted that the density function in the integral (6) does not describe the density of physical surface currents on the strip for. The function is defined as the discontinuity of fractional derivatives of E-field at the plane:
For the limit cases of and 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 from 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 and magnetic current expressed from as
In the case of the -polarized incident plane wave, where, the method proposed above can be applied as well. We define fractional boundary conditions as,
The case of corresponds to diffraction of the -polarized plane wave on a PEC strip, while the case of describes diffraction of the -polarized plane wave on a PMC strip. As before, we represent the scattered field via the fractional Green’s function
This equation can be solved by repeating all steps of the -polarization case after changing to.
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 -polarized plane wave (1) be scattered by a half-plane (). The total field must satisfy fractional boundary conditions
and Meixner’s edge conditions must be satisfied for.
Following the idea used for the analysis of diffraction by a strip we represent the scattered field using the fractional Green’s function
where is the unknown function, is the fractional Green’s function (2).
The Fourier transform of is defined as
where for and for.
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:
For the limit cases of the fractional order and 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.
DIE allows an analytical solution in the special case of in the same manner as for a strip with fractional boundary conditions. Indeed, for we obtain the solution for any value of as
The scattered field can be found in the following form:, for
In the general case of the equations (34) can be reduced to SLAE. To do this we represent the unknown function as a series in terms of the Laguerre polynomials with coefficients:
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.
where 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
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 and. Let the -polarized plane wave (1) be the incident field. The total field satisfies fractional boundary conditions on each strip:
and Meixner’s edge conditions must be satisfied on the edges of both strips (,).
The scattered field consists of two parts
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,,
Using Fourier transforms, defined as
the scattered field is expressed as
Fractional boundary conditions (30) correspond to two equations
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 as a series in terms of the orthogonal polynomials. We represent the unknown functions as series in terms of the Gegenbauer polynomials:,
where the matrix coefficients are defined as
Finally, we obtain the solution as
Having expressions for we can obtain the physical characteristics. The radiation pattern of the scattered field in the far zone (27) is expressed as
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.