# Scattering from a PEC Slightly Rough Surface in Chiral Media

Authors: Haroon Akhtar Qureshi; Muhammad Arshad Fiaz; Muhammad Aqueel Ashraf
Journal: Advances in Materials Science and Engineering
 (2018)

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## Abstract

The 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. Zeroth-order 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.


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## Body

## 1. Introduction

Scattering from rough surfaces is an interdisciplinary research field which has many applications in optics, communication, and remote sensing. The simplest possible problem is an impenetrable rough interface, and the solution has widely been investigated [1]. The 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].The regions of validity of PM and KA are also well defined [7, 8]. In general, PM is applied to the surface of small height. There 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. They 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. Thermal 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. The effect 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]. Reflection from a chiral interface has been studied by many researchers [23–25]. 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. Specifically, reflection 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 flat. At some frequencies, the interface can behave rough. The roughness affects the wave propagation, radiation, and scattering properties. The applications of rough surface scattering can be found in antenna theory, communication, and remote sensing. In this paper, the problem of reflection from a PEC rough interface placed in chiral medium is presented. Rayleigh hypothesis is utilized to get the scattered field. Zeroth-order solution obtained by PMs is used to make a comparison with that of a PEC flat interface in chiral media. Section2 contains theoretical formulation, while numerical results are reported in Section 3. An e−iωt time dependence is assumed and suppressed throughout.


## 2. Theoretical Formulation

Consider a PEC rough surface placed in a chiral medium as shown in Figure1. The chiral medium is defined by the following constitutive relations [23]:(1)D=εE+β∇×E,B=μH+β∇×H,where ϵ, μ, and β are the permittivity, permeability, and chirality of the medium, respectively. The profile of the rough surface is defined as z=fx.Figure 1
Scattering from a PEC rough surface in chiral media.A left circularly polarized (LCP) plane wave is incident on the surface at an angleφi with the z-axis. Its electric field is given by(2)Ei=E0e^y+i−e^xαLiγ1+e^zκiγ1eiκix+αLiz,where E0 is the amplitude of incident plane wave and γ12=αLi2+κi2 is the wave number.The field scattered due to PEC rough surface can be written as(3)Es=∫−∞∞e^y+ie^xαLγ1+e^zκγ1RLκeiκx−αLzdκ+ar∫−∞∞e^y−ie^xαRγ2+e^zκγ2RRκeiκx−αRzdκ,where RLκ and RRκ are the unknown coefficients for LCP and RCP scattered fields, respectively, and ar=−iμ/ε.The perturbation series of unknown coefficients in spectral domain is(4)RLκ=∑m=0∞RLmκ,RRκ=∑m=0∞RRmκ.It is assumed that the following conditions are satisfied:(5)αLifx≪1,∂fx∂x≪1.Using the power series for exponentials, the scattered field can be written as(6)Es=∫−∞∞∑m=0∞RLmκe^y+ie^xαLγ1+e^zκγ1eiκx∑q=0∞−iαLzqq!dκ+ar∫−∞∞∑n=0∞RRmκe^y−ie^xαRγ2+e^zκγ2eiκx∑q=0∞−iαRzqq!dκ.The boundary condition is given by(7)e^z+∂f∂xe^x×Ei+Es=0.Applying the above boundary condition atz=fx,(8)E0eiκix∑q=0∞iαLifxqq!+∫−∞∞eiκx∑q=0∞∑m=0∞RLmκ−iαLfxqq!dκ+ar∫−∞∞eiκx∑q=0∞∑m=0∞RRmκ−iαRfxqq!dκ=0,−E0iαLiγ1+κiγ1∂f∂xeiκix∑q=0∞iαLizqq!+∫−∞∞iαLγ1−κγ1∂f∂xeiκx∑q=0∞∑m=0∞RLmκ−iαLfxqq!dκ−ar∫−∞∞iαRγ2−κγ2∂f∂xeiκx∑q=0∞∑m=0∞RRmκ−iαRfxqq!dκ=0.The above infinite series is solved for unknown coefficients. The zeroth-order terms of both the above equations are(9)E0eiκix+∫−∞∞eiκxRL0κdκ+ar∫−∞∞eiκxRR0κdκ=0,−E0iαLiγ1eiκix+∫−∞∞iαLγ1eiκxRL0κdκ−ar∫−∞∞iαRγ2eiκxRR0κdκ=0.Taking the Fourier transform of the above equations gives(10)E0δκ−κi+RL0κ+arRR0κ=0,−E0αLiγ1δκ−κi+αLγ1RL0κ−arαRγ2RR0κ=0.Solving the above equations, zero-order coefficients are obtained:(11)RL0κ=−1Qnγ2αLi−γ1αRE0δκ−κi,RR0κ=1Qnγ2αLi+αLalE0δκ−κi,where Qn=γ1αR+γ2αL and al=−1/ar.Putting in (3), zeroth-order scattered field is obtained:(12)Es=−e^y+ie^xαLγ1+ie^zκγ1γ2αLi−γ1αRiγ1αRi+γ2αLiE0eiκx−αLz+are^y−ie^xαRγ2−ie^zκγ2⋅2γ2αLiγ1αRi+γ2αLialE0eiκx−αRz.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(13)iE0αLifxeiκix+∫−∞∞RL1κeiκxdκ−∫−∞∞iαLfxRL0κeiκxdκ+ar∫−∞∞RR1κeiκxdκ−ar∫−∞∞iαRfxRR0κeiκxdκ=0,E0αLi2γ1fxeiκx−iE0κiγ1∂f∂xeiκix+∫−∞∞iαLγ1RL1κeiκxdκ+∫−∞∞αL2γ1fxRL0κeiκxdκ−∫−∞∞iκγ1∂f∂xRL0κeiκxdκ−ar∫−∞∞iαRγ2RR1κeiκxdκ−ar∫−∞∞αR2γ2fxRR0κeiκxdκ+ar∫−∞∞iκγ2∂f∂xRR0κeiκxdκ=0.Taking the Fourier transfer of the above equations,(14)RL1κ+arRR1κ=αLiC+arαRiD−E0αLiiFκ−κi,αLγ1RL1κ−arαRγ2RR1κ=E0γ1κκi−κi2−αLi2+1γ1αLi2+κκi−κi2C−arγ2αRi2+κκi−κi2DiFκ−κi,where(15)C=γ2αLi−γ1αRiγ1αRi+γ2αLiE0,D=2γ2αLiγ1αRi+γ2αLialE0.From the above equations, it can be noted that the first-order solution can be written in terms of zeroth-order solution. Solving the above equations, first-order coefficients may be written as(16)RL1κ=iQnE0γ2κκi−κi2−αLi2−αRαLiγ1+γ2αLi2+κκi−κi2+γ1αRαLiC−γ1αRi2+κκi−κi2−αRαRiarDFκ−κi,RR1κ=iarQnE0γ2κκi−αLi2−κi2−γ2αLαLi+γ2αLi2+κκi−κi2+αLαLiC−γ1αRi2+κκi−κi2−γ2αLαRiDFκ−κi.Finally, the solution up to any order can be found via recursive computation.


## 3. Numerical Analysis

Numerical implementation of the theoretical formulation is done in this section. A rough surface profile 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 defined as(17)fx=Acos2πλsx,where A and λs are the amplitude and period of the surface, respectively. Using the Fourier transform of sinusoidal surface, the scattered field can be written as(18)Es=∑p=−11RLsinκpxeiκpxx−αLpzz,where(19)RLsinκpx=γ2αLi−γ1αRiγ1αRi+γ2αLiE0,p=0,RLsinκpx=iγ1αRpz+γ2αLpz×γ2κpxκi−κi2−αLi2−αRpzαLiγ1E0+γ2αLi2+κpxκi−κi2+γ1αRpzαLiC−γ1αRi2+κpxκi−κi2−αRpzαRiarD,p=±1,κpx=κi+2πpλs,αLpz=γ12−κpx2,αRpz=γ22−κpx2.In a similar way, the expressions for RCP scattered field can be written. Thepth order diffraction efficiency ηp is defined as the fraction of the incident energy which is reflected in the κpx direction, and it is given by(20)ηpL=ReαLpzαLiRLsinκpx2.Now, consider a rough surface with Gaussian roughness spectrum. The power spectral densityWkx is given by(21)Wkx=h2lc2πe−κ2lc2/4,where lc is the correlation length and h is the root mean square (rms) height of the rough surface. The rms slope is given by [1](22)s=2hlc.Since both LCP and RCP waves are scattered for LCP incidence, the LCP and RCP incoherent bistatic scattering coefficients can be expressed as(23)σLκ=αLαLiWκ−κiγL2,σRκ=αRαLiWκ−κiγR2,where(24)γL=iQnγ2κκi−κi2−αLi2−αRαLiγ1E0+γ2αLi2+κκi−κi2+γ1αRαLiC−γ1αRi2+κκi−κi2−αRαRiarD,γR=iarQnγ2κκi−αLi2−κi2−γ2αLαLiE0+γ2αLi2+κκi−κi2+αLαLiC−γ1αRi2+κκi−κi2−γ2αLαRiD.Figure2 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. The smaller h corresponds to a flat interface. As expected, the scattered field increases with the increase in rms height. The angular behavior of LCP and RCP scattered fields is same. The effect on scattering pattern for different values of correlation length is shown in Figure 4. The effect of slope (correlation length) can be observed on the scattering pattern away from the specular direction. The 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. The 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.Figure 2
LCP and RCP scattering coefficients for different values of chirality, where φi = 30°, εr = 4, h = 0.05, and lc = 0.35.
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(b)Figure 3
LCP and RCP scattering coefficients for different values of rms height, where φi = 30°, εr = 4, β = 0.03, and lc = 0.35.
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(b)Figure 4
LCP and RCP scattering coefficients for different values of correlation length, where φi = 30°, εr = 4, β = 0.03, and h = 0.05.
(a)
(b)


## 4. 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. This analysis can also be used for a triangular grating.



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