Scattering from a PEC Slightly Rough Surface in Chiral Media

-e scattering of left circularly polarized wave from a perfectly electric conducting (PEC) rough surface in isotropic chiral media is investigated. Since a slightly rough interface is assumed, the solution is obtained using perturbation method. Zerothorder term corresponds to solution for a flat interface which helps in making a comparison with the results reported in the literature. First-order term gives the contribution from the surface perturbations, and it is used to define incoherent bistatic scattering coefficients for a Gaussian rough surface. Higher order solution is obtained in a recursive manner. Numerical results are reported for different values of chirality, correlation length, and rms height of the surface. Diffraction efficiency is defined for a sinusoidal grating.


Introduction
Scattering from rough surfaces is an interdisciplinary research field which has many applications in optics, communication, and remote sensing.e simplest possible problem is an impenetrable rough interface, and the solution has widely been investigated [1].e method of analysis can be analytical model such as perturbation methods (PMs) [2,3], Kirchhoff approximation (KA) [4,5], and extended boundary condition method [6].
e regions of validity of PM and KA are also well defined [7,8].In general, PM is applied to the surface of small height.ere can be an additional condition on small slopes.KA is applied to surfaces having larger radius of curvature compared with a wavelength.Numerical values of parameters such as rms height and correlation length of the surface vary with incident angle in defining the region of validity.
Numerical methods such as method of moments (MoM) [9], finite-difference time domain method (FDTDM) [10], and finite element method (FEM) [11] may also be selected to study rough surface scattering.ey offer many advantages in terms of modeling and accuracy, but analytical methods are mostly applied to get more understanding of the scattering phenomena.
Scattering from a penetrable surface has been discussed in [12].An application has been proposed to calculate the brightness temperature.ermal emission from a layered medium bounded by a slightly rough interface has been calculated in [13].Analytic height correlation function of rough surfaces derived from light scattering is reported in [14].Recently, the scattering of torsional guided waves from Gaussian rough surfaces in pipe work has been discussed [15].Full wave electromagnetic scattering from rough surfaces with buried inhomogeneities is presented by Duan and Moghaddam [16].
A chiral medium has widely been explored, and many applications have been proposed in chemistry, optics, elementary particle physics, and electromagnetics [17,18].Right-and left-hand circularly polarized waves have different refraction indices and different velocities in chiral medium.Consistency of hydrodynamic approximation for chiral media has been discussed by Avdoshkin et al. [19].Resonant absorption and amplification of circularly polarized waves in inhomogeneous chiral media is studied in [20].Bassiri et al. [21] studied the electromagnetic wave propagation and radiation in chiral media.e e ect of chirality on the polarization and intensity of the radiation from a dipole is discussed.Radiation from a layered chiral medium excited by an interior dipole is presented in [22].Re ection from a chiral interface has been studied by many researchers [23][24][25].In [23], the behavior of plane waves at the planar interfaces of mirror-conjugated chiral medium is explained.It is reported that the use of an imaging theory for chiral media is complicated for scattering problems in general since not only do the sources get imaged but the medium does also.Speci cally, reection from a PEC interface in chiral media has been evaluated in [24].It has been demonstrated that such a conducting plane could be used for focusing in the strong chiral medium.Electromagnetic scattering from a perfectly conducting obstacle in a homogeneous chiral environment is discussed in [25].
In all the above cited work, the interfaces are assumed to be at.At some frequencies, the interface can behave rough.
e roughness a ects the wave propagation, radiation, and scattering properties.
e applications of rough surface scattering can be found in antenna theory, communication, and remote sensing.In this paper, the problem of re ection from a PEC rough interface placed in chiral medium is presented.Rayleigh hypothesis is utilized to get the scattered eld.Zeroth-order solution obtained by PMs is used to make a comparison with that of a PEC at interface in chiral media.Section 2 contains theoretical formulation, while numerical results are reported in Section 3.An e −iωt time dependence is assumed and suppressed throughout.

Theoretical Formulation
Consider a PEC rough surface placed in a chiral medium as shown in Figure 1. e chiral medium is de ned by the following constitutive relations [23]: where ϵ, μ, and β are the permittivity, permeability, and chirality of the medium, respectively.e pro le of the rough surface is de ned as z f(x).
A left circularly polarized (LCP) plane wave is incident on the surface at an angle φ i with the z-axis.Its electric eld is given by where E 0 is the amplitude of incident plane wave and c 2 1 α 2 Li + κ 2 i is the wave number.e eld scattered due to PEC rough surface can be written as where R L (κ) and R R (κ) are the unknown coe cients for LCP and RCP scattered elds, respectively, and a r −i μ/ε .e perturbation series of unknown coe cients in spectral domain is It is assumed that the following conditions are satis ed: Using the power series for exponentials, the scattered eld can be written as e boundary condition is given by Figure 1: Scattering from a PEC rough surface in chiral media.
2 Advances in Materials Science and Engineering Applying the above boundary condition at z � f(x), e above infinite series is solved for unknown coefficients.e zeroth-order terms of both the above equations are Taking the Fourier transform of the above equations gives Solving the above equations, zero-order coefficients are obtained: where Putting in (3), zeroth-order scattered field is obtained: It corresponds to solution for a flat interface in chiral media, and the expressions are in agreement with those reported in [23].Balancing to the first order gives Taking the Fourier transfer of the above equations, where From the above equations, it can be noted that the firstorder solution can be written in terms of zeroth-order solution.Solving the above equations, first-order coefficients may be written as

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Finally, the solution up to any order can be found via recursive computation.

Numerical Analysis
Numerical implementation of the theoretical formulation is done in this section.A rough surface pro le has to be selected, and its Fourier transform is calculated.Two cases have been considered here: one is a sinusoidal surface and the other is a Gaussian rough surface.First, consider a sinusoidal surface de ned as where A and λ s are the amplitude and period of the surface, respectively.Using the Fourier transform of sinusoidal surface, the scattered eld can be written as 4 Advances in Materials Science and Engineering where In a similar way, the expressions for RCP scattered eld can be written.e pth order di raction e ciency η p is de ned as the fraction of the incident energy which is re ected in the κ px direction, and it is given by Now, consider a rough surface with Gaussian roughness spectrum.e power spectral density W(k x ) is given by where l c is the correlation length and h is the root mean square (rms) height of the rough surface.e rms slope is given by [1] Since both LCP and RCP waves are scattered for LCP incidence, the LCP and RCP incoherent bistatic scattering coe cients can be expressed as where

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Figure 2 shows the incoherent bistatic coefficients for different values of chirality parameter.It can be observed that RCP scattered field is smaller than LCP scattered field.Moreover, scattered field increases as the chirality increases.Chirality may change the angular behavior significantly at some scattered angles as described by the minima in Figure 2(a).Figure 3 shows the scattering pattern for the rms height.
e smaller h corresponds to a flat interface.As expected, the scattered field increases with the increase in rms height.
e angular behavior of LCP and RCP scattered fields is same.e effect on scattering pattern for different values of correlation length is shown in Figure 4. e effect of slope (correlation length) can be observed on the scattering pattern away from the specular direction.
e scattered field decreases away from specular direction φ � 30 °.It can also be observed that the slope has more effect on scattering behavior as compared to chirality.e numerical results show that slight roughness of the interface cannot be ignored in order to be close to the experimental results and for the design purposes.

Conclusion
Scattered field from a PEC rough surface in chiral media has been studied.PM is applied to obtain scattered field components.In general, the higher order scattered field can be obtained using lower order coefficients by utilizing the recursive nature of the problem.Two cases, sinusoidal and Gaussian rough surfaces, are considered, and the expressions of the LCP and RCP incoherent bistatic scattering coefficients have been reported.Scattering pattern is observed for chirality parameter, height, and correlation length of the Gaussian rough surface.Diffraction efficiency has been defined for a rough surface with sinusoidal profile. is analysis can also be used for a triangular grating.