## Abstract

The effect of stratified substrates on scattering from a chiral sphere is analytically and numerically examined. A combination of vector Mie solution and the field transformations between vector spherical functions (VSFs) and plane waves (PWs) is used to determine the scattered fields of different orders. The generalized reflection coefficients of the stratified half-space and vector Mie solution for the chiral sphere are used to calculate successive interactions of the sphere and the interface and a series solution is obtained. Commercial software FEKO is used to numerically validate the derived expressions for the multilayered substrate. Various numerical results are presented and discussed. Specifically, a slab with exponential profile is considered as an example of an inhomogeneous substrate and its scattering coefficients are compared with the homogeneous case.

© 2021 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement

## 1. Introduction

The role of substrates in the problem of scattering of electromagnetic waves from objects of interest is an important research theme in optics, bio-photonics and remote sensing [1–4]. The scattering properties of objects in free-space are extensively studied; but in practical applications, substrates are almost always present and their generally unwanted effect should be investigated [5,6]. Many related theoretical and applied studies are reported recently [7]. Specifically scattering from 2D, 3D or bodies-of-revolution above a half-space with multilayered or anisotropic media is numerically examined in recent years [8,9]. Furthermore, scattering from objects buried in multilayered media is also extensively studied, often using numerical techniques [10–12]. Analytical techniques, though restricted to the canonical structures, are extremely fast and provide physical insight to the scattering mechanisms and hence are desired. As discussed in [13], the analytical tool capable of solving such a complicated problem is a combination of the Mie solution for the sphere and transformations between spherical harmonics and plane waves. This could be done either by considering the total fields (the mathematical approach of Bobbert [14]) or using a series of partial fields representing multiple interactions of the sphere and the interface (the physical approach of Takemori [15]). Both techniques are briefly explained in [13].

Videen [16] and Johnson [17] used essentially the same method and approximations as Bobbert [14]. However, they applied boundary conditions on the sphere, instead of using operators. The first exact solution for the sphere-interface problem was that of Fucile [18], where use was made of a general reflection rule for spherical functions. Wriedt [19] used the PW expansion of VSFs and as a result avoided the complicated image and addition theorems. The case with the sphere behind the interface was more recently solved by Frezza [20]. On the other hand, the 2D scattering problem of a circular cylinder below a slightly rough interface was solved by Lawrence [21]. He considered multiple interactions of the cylinder and the rough interface and similar to Takemori [15], found a series solution. Wang [22] used the same technique for a sphere behind an interface and the case of a sphere below a rough interface is recently reported [23]. Another set of relevant analytical works are those in which only the single scattering effects are considered [24–26]. Chiu [24] investigated the interaction of a finite cylinder and a rough half-space, using reciprocity theorem. This method was later applied to the case of a sphere [25]. The accuracy of these so-called four-path models was analyzed by Johnson [26] and it was shown that the neglected higher order interactions could significantly change the solutions.

Many analytical and numerical techniques are reported to study the problem of scattering from chiral objects in free space [27–29]. However, very few studies consider chiral objects near an interface [30–32]. These studies numerically analyze the scattering problem in vicinity of a lossy half-space. Multilayered or inhomogeneous substrates are not considered, and the analytical aspects of the problem (e.g. the relative importance of different scattering orders) is not reported, yet.

As an extension of [13], in this paper, scattering from a chiral sphere above a general stratified half-space (including an inhomogeneous dielectric profile) is analytically solved. In the proposed method, which is essentially based on the idea of Takemori [15], the scattered field of each order is obtained in terms of that of the previous order. This recursive formulation is obtained using the transformations between PWs and SVFs. On the other hand, the first order scattered field is the scattered field of the chiral sphere, when the excitation field is a combination of the incident and the reflected PWs. A non-recursive solution, containing all interactions between the chiral sphere and the interface, is also constructed using the fact that the appearing summation could be performed analytically. In particular, here, the effect of substrate on the scattering properties of the chiral sphere is studied. It should be noted that, although it is well-known that the substrate could substantially affect the interaction of electromagnetic waves with nearby objects [1], it seems that the issue is not comprehensibly addressed in the literature and more analytical and numerical studies are required [33].

Although the basic theory is the same as [13], two point are worth mentioning. First, the present extended formulation applies to the more realistic case of multilayered or inhomogeneous substrates. Second, the presented theory requires that the reflection coefficients be calculated not only for incident propagating plane waves, but also for incident evanescent plane waves. This requirement is related to the fact that in the presented theory, all the multiple interactions between the sphere and the surface are precisely formulated. Technically, the reflection coefficient is required to be known for all values of ${\bar{k}_ \bot }$. As a result, numerical calculation of the reflection coefficients (based for instance on Runge-Kutta method, which is the common option) which gives the reflection coefficients only for the propagating plane waves are not applicable in this formulation. Instead, here, analytical expressions which are valid for both propagating and evanescent incident plane waves, are used for a multilayered [34,35] ground or an inhomogeneous [36] finite substrate.

It should be noted that while in this paper the focus is not on special applications, the problem investigated here is relevant to applications such as virus detection [37,38] and directional scattering [39,40]. As explained in [13], in many optical detection systems, the virus is fixed on a generally multilayered substrate [37] and on the other hand many viruses are chiral [38]; therefore the electromagnetic problem is essentially that of scattering by a chiral particle in presence of a half-space. Directional scattering by small particles [39] in presence of substrates [40] is also known to be significantly different from the free-space case. The chiral particles are no exception and thus their scattering properties in vicinity of substrates are of interest.

## 2. Formulation

#### 2.1 Geometry and definitions

A chiral sphere with permittivity ${\epsilon _s}$, chirality ${\xi _s}$, and radius *a* is assumed to be centered at the origin of the coordinate system. The upper and the lower half-spaces are respectively characterized with ${\epsilon _0}$ and $\epsilon (z )$.

The chiral sphere is located a distance *d* above the general stratified half-space (Fig. 1) and an arbitrary PW ${\bar{E}_i}{e^{j\bar{k}_{0i}^ -{\cdot} \bar{r}}}$ is incident on the structure where $\bar{k}_0^ \pm{=} {\bar{k}_ \bot } \pm \hat{z}{k_{0z}}$, ${k_\rho } = |{{{\bar{k}}_ \bot }} |$, ${k_{0z}} = {({k_0^2 - k_\rho^2} )^{1/2}}$, and ${\bar{k}_ \bot } = \hat{x}{k_x} + \hat{y}{k_y}$. It is easily seen that the direction of propagation could be represented with $\bar{k}_ \bot ^i$ and the signs ${\pm} $ to distinguish between the up-going and down-going incident waves. A second representation of the direction of propagation is possible by using the azimuthal and elevation angles $({{\theta_i},{\phi_i}} )$, where $k_x^i = {k_0}\sin {\theta _i}\cos {\phi _i}$, $k_{0z}^i = {k_0}\sin {\theta _i}\sin {\phi _i}$ and $k_{0z}^i = {k_0}\cos {\theta _i}$. Both representations are used in subsequent subsections.

#### 2.2 Successive interactions

The general procedure of calculation of the scattered fields, based on the Takemori [15] idea was thoroughly explained in [13]. Here, for the sake of completeness, the main aspects are briefly reviewed. The first order Mie field $\bar{E}_1^M$ is the scattered field by the chiral sphere due to the superposition of the incident field and the PW reflected from the stratified half-space:

where $b_{n,1}^s = b_{n,1}^{sh}{P_{ih}} + b_{n,1}^{sv}{P_{iv}}$, $c_{n,1}^s = c_{n,1}^{sh}{P_{ih}} + c_{n,1}^{sv}{P_{iv}}$, and^{st}order Mie field could be briefly denoted by ${{\boldsymbol a}_1} = {\left[ {\begin{array}{ll} {{\boldsymbol b}_1^s}&{{\boldsymbol c}_1^s} \end{array}} \right]^t}$

**.**In (1)– (3), it is assumed that $l = \left[ {\sqrt {n - 1} } \right]$ and $m = ({n - 1} )- l({l + 1} )$.

The VSFs $\bar{M}_n^{h0}$ and $\bar{N}_n^{h0}$ could be expanded in terms of plane waves

The expansion coefficients of Mie field of each order is related to those of the previous order by the following relations [13]:

The expansion coefficients of the total Mie field are

which upon using (7) is transformed to the two following equivalent formSince ${{\boldsymbol a}_1}$ is known, ${\boldsymbol a}$ is either using the series solution of (11) or the complete solution of (12). Finally the scattering cross-sections are

where $\alpha ,\; \beta = h,\; v$ and where $\sigma _f^\alpha = \sigma _t^{\alpha + } + {R_\alpha }\sigma _t^{\alpha - }$ and $\sigma _t^{\alpha \pm } = \mathop \sum \limits_{n = 1}^\infty [{b_n^sD_n^{M\alpha \pm } + c_n^sD_n^{N\alpha \pm }} ]$.#### 2.3 Reflection from a stratified half-space

When the substrate is multilayered or inhomogeneous, the reflection coefficients in the presented formulation are no longer the Fresnel ones. Consider a multilayered medium for which the dielectric constants are given by ${\epsilon _1}$, ${\epsilon _2}$, …, ${\epsilon _{N - 1}}$ for different layers. The permittivity of the upper and the lower half-spaces are respectively denoted by ${\epsilon _0}$ and ${\epsilon _N}$. For $i = 1,\; 2,\; \ldots ,\; N - 1$, the upper and the lower boundaries of the layer with ${\epsilon _i}$ are $z = {z_i}$ and $z = {z_{i + 1}}$, respectively and the thickness is represented with ${\Delta _i}$. Moreover, assume $R_{m,N}^\alpha $ denotes the generalized reflection coefficient of a similar multilayered structure, in which all the layers above ${z_{m + 1}}$ are filled with ${\epsilon _m}$. In this notation, $\alpha = h$, $v$ which represent horizontal and vertical polarizations, respectively. The generalized reflection coefficient of the whole structure is given by the following recursive formula [34,35]:

A limiting form of the multilayered medium just discussed is a general continuous dielectric profile. In such a case, discrete expressions like $R_{0,N}^\alpha $, $R_{1,N}^\alpha $, … are replaced with a continuous function ${\tilde{R}_\alpha }(z )$ and Eqn. (15) is replaced with a first order nonlinear differential equation [34]:

Since ${\tilde{R}_\alpha }({{z_i}} )$ is the limit of $R_{i,N}^\alpha $, it gives the reflection coefficient of the stratified medium below $z = {z_{i + 1}}$ under the assumption that the region above ${z_{i + 1}}$ is filled with $\epsilon ({{z_i}} )$. Hence, ${\tilde{R}_\alpha }({{z_1}} )$ as given by solving (16) is the reflection coefficient of the stratified medium when its upper region is filled with $\epsilon ({{z_1}} )$. The reflection coefficient for the case in which the upper medium is filled with ${\epsilon _0}$ is

It should be noted that the dependence of the reflection coefficient on the direction of the incident wave is not shown in (15)–(18), for brevity.

In this paper, an exponential dielectric profile is considered as an example of an inhomogeneous substrate. For this special case, an analytical solution could be found for Eq. (16), as follows [36]. A layer of thickness *D* is considered between $z = 0$ and $z ={-} D$ and it is assumed that the dielectric constants are ${\epsilon _0}$ and ${\epsilon _1}$ above and below the slab, respectively. Furthermore, an exponential profile ${\epsilon _r}(z )= B{e^{\beta z}}$ is considered for the layer so that the permittivity for the upper and lower points of the slab are respectively ${\epsilon _u}$ and ${\epsilon _d}$; i.e. $B = {\epsilon _{ru}}$ and $\beta = \frac{1}{D}\ln \frac{{{\epsilon _u}}}{{{\epsilon _d}}}$. For this case, it could be shown that, apart from the phase compensation, the reflection coefficients are given by [36]:

*z*, computed in $z = 0$. Finally $m = \frac{{2{k_\rho }}}{\beta }$ and $q = \sqrt {1 + {m^2}} $, where ${k_\rho }$ is the component of the wave vector in the $xy$ plane.

## 3. Validation

In order to validate the results of the present paper, scattering from a dielectric sphere above a multilayered medium is considered and the solutions of the present paper are compared with those obtained using FEKO.

A dielectric sphere above a three-layered structure with and without a ground plane is considered and the RCS is calculated. The dielectric sphere is characterized with unit radius and ${\epsilon _{sr}} = 4$ and centered at origin. The upper medium, where the sphere is located in, is free-space and the middle and lower layers have respectively the dielectric constants ${\epsilon _{1r}} = 2.56$ and ${\epsilon _{2r}} = 6.5$. The upper interface is located at $z ={-} 2m$, and the thickness of the upper layer is ${\mathrm{\Delta }_1} = 1m$. The ground plane, when present, is assumed to be located at $z ={-} 4\; m$. The incident PW is in normal direction and of frequency $f = 100MHz$.

The results of the analytical solution of the present paper, together with those of the commercial software FEKO are presented in Fig. 2. Case I and Case II respectively represent the cases with and without the ground plane. Very good agreement is observed for both cases despite the fact that the two methods are very different. It should be noted that the dielectric constants for the multilayered ground are taken from [30], where a similar problem is numerically studied using the method of moments.

## 4. Discussion

In this section, using both the series and the complete solutions, a few observations are performed. First, the first-order and the second-order scattered fields are compared in monostatic backscattering case. Next, a dielectric sphere above a multilayered ground is considered and the effect of variation of the permittivity of the lower layer on scattering cross-sections is examined. Finally, an inhomogeneous dielectric profile is assumed for the half-space and under normal incidence, the bistatic scattering for a chiral sphere is presented. In all simulations of this section, the parameters of Case I of Sec. 3 are used, except otherwise stated. Notice that other sets of parameters were also considered (not included here, for brevity) and similar patterns were observed.

#### 4.1 Monostatic backscattering

The parameters of Case I of Sec. 3, together with the chirality factors of ${\xi _r} = 0$ and ${\xi _r} = 0.6$ for the sphere are considered. The scattered fields of the first two orders are calculated. The objective is to observe the relative values of the first-order and the second-order scattered fields as the incident angle varies. Thus, the monostatic backscattering case is examined and the results are presented in Fig. 3.

As could be seen, for $hh$ polarizations, when the incident angle passes a certain limit, the first-order and the second-order solutions are practically identical. This limit is around ${\theta _i} = \pi /3$ and ${\theta _i} = \pi /6$, respectively for the dielectric and the chiral spheres. Moreover, the second order contribution is generally smaller for the chiral sphere. On the contrary, for the $vv$ polarization, it is observed that the importance of the second order contribution is almost the same for different incident angles and there is no significant difference between the dielectric and the chiral cases.

This observation could be explained by the following simple argument. The scattering pattern of the sphere in free-space consists of a main big forward lobe and a smaller backward lobe. When the substrate is present and normal illumination is considered, the first-order contribution is the superposition of the backward lobe (Backscattered field, which we denote by “B”) and the forward lobe (the “Reflected and then Forward scattered field”, which we denote by “RF”) [13]. As the incidence angle is increased, the RF maximum departs from the backscattering direction and the first order contribution is dominated by B. On the other hand, the second order contribution is the superposition of four terms: FRF + RBRBR + RBRF + FRBR. For normal illumination, the maximum of all these contributions is in the backscattering direction. But, as the incident angle increases, the maximum of the first two contributions depart from the backscattering direction. Therefore, it seems that for the parameters considered in this simulation, the first two terms of the backscattered second order field are the main contributions for the $hh$ polarization while for the $vv$ polarization, the last two terms are dominant.

#### 4.2 Substrate effect

In this subsection, a dielectric sphere above a multilayered ground is examined. The same parameters as Case I of Sec. 3 are considered. However, the dielectric constant of the lower layer is varied (${\epsilon _{r2}} = 2.5,\; 4.5,\; 6.5,\; 8.5$) and the bistatic scattering coefficients are determined. The numerical outcomes for these cases, together with that of a PEC lower layer, are shown in Fig. 4. When ${\epsilon _{r2}} = 2.5$, since the value of dielectric constants of the two layers are almost the same, the solution is basically determined by the reflection coefficient between free-space and the upper layer. As the permittivity of the lower layer increases, the interaction at the interface between the upper and the lower layers becomes stronger. For the parameters considered here, the superposition of this contribution with the main one is constructive and hence the scattering coefficients generally increase. The $vv$ polarization is more complicated (not shown here), but the general variations are the same.

#### 4.3 Inhomogeneous profile

Finally, a sphere above an inhomogeneous half-space is considered. The sphere is characterized with unit radius and ${\epsilon _{sr}} = 4$ with center at origin. The normally incident PW has the frequency of $100MHz$. An exponential dielectric profile ${\epsilon _r}(z )= B\exp \beta z$ is assumed for a layer with thickness $D = 1m$, where the values of the permittivity for upper and lower heights of the layer are respectively ${\epsilon _{ru}} = 2.56$ and ${\epsilon _{rd}} = 6.5$.

The region below the exponential slab is assumed to be filled with a homogenous material with ${\epsilon _{r1}} = 6.5$. The reflection coefficients for the exponential profile are obtained using the analytical expressions presented in Sec. 2. The bistatic scattering coefficients for a chiral sphere with ${\xi _r} = 0.7$ are presented in Fig. 5. Moreover, the results for the case with a homogeneous layer (${\epsilon _{r1}} = 2.56$) are also shown.

As could be clearly seen, the scattering coefficients for the case with inhomogeneous layer are much smaller than the case of a homogeneous layer. This could be attributed to the tapering effect in the former case. Indeed, in both cases the permittivity value for the upper and lower points of the layer are ${\epsilon _{ru}} = 2.56$ and ${\epsilon _{rd}} = 6.5$, respectively. However, for the homogeneous case, the transitions occurs abruptly, which result in large reflection coefficients. On the contrary, for the inhomogeneous case, the transition happens gradually and such a slow change prevents reflection. It should be noted that, as the observation angle increases, the difference becomes smaller between the two types of layers, especially for the $vv$ polarization. The simulations are repeated with other set of parameters (not shown for clarity) and similar variations were observed.

## 5. Conclusions

Scattering by a sphere above a general stratified half-space (including an inhomogeneous profile) is analytically investigated by combining the transformations between plane waves and vector spherical functions and the Mie solution for a chiral sphere. The inputs are the dielectric constants and thickness of different layers of the multilayered half-space or the permittivity profile of the inhomogeneous medium, together with the characteristics of the incident plane wave and the sphere. The outputs are fields scattered by the sphere in a series form and in a non-recursive form, respectively given in (11) and (12).

The recursive solution (11) is suitable for investigation of different scattering mechanisms and the non-recursive solution (12), being much faster than the traditional numerical techniques, could provide benchmark solutions. Moreover both solutions could straightforwardly be extended to the case of object with arbitrary shapes, using the T-matrix method.

## Disclosures

The author declares no conflicts of interest.

## Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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