Scattering by a chiral sphere above a half-space

Hasan Zamani

doi:10.1364/OE.434643

1. Introduction

Interaction of electromagnetic waves with objects in vicinity of an interface has various applications in fields such as optics, bio-photonics and remote sensing [1–4]. Mostly, the scattering properties of objects are determined in free-space and since in practical contexts, substrates are almost always present, their effect is an adverse one [5] that should be somehow suppressed. On the other hand, the substrate presence, if suitably considered in the analysis of the scattering problem, could provide new possibilities [6]. A clear understanding of this composite scattering problem is needed and therefore of interest.

New relevant theoretical studies include numerical modelling of scattering from 3D targets or bodies-of-revolution [7], above or buried in a half-space [8–10], multilayered [11], or anisotropic [12,13] medium. A large number of analytical and numerical techniques are proposed for the problem of scattering from objects above or below an interface, or targets embedded in multilayered media (see [14] and the references therein). Numerical techniques, although capable of handling arbitrary shaped targets, are computationally expensive and time-consuming. Furthermore they do not provide the required physical insight to the scattering mechanisms. Based on these facts and also the need for benchmark results to study the accuracy of the numerical techniques, analytical solutions are desired whenever possible.

The first attempts to analytically solve the problem of scattering from an object near an interface were reported in mid 80s. Bobbert [15] and Takemori [16] solved the problem of scattering from a sphere in a half-space, using an extension of the Mie solution. Although they both made use of transformations between PWs and spherical wave functions, they followed completely different approaches.

Bobbert [15] used two operators to denote the scattering by a sphere and the reflection of spherical wave functions by the interface. This way, he obtained the formal solution of the problem in a concise form [15]. Videen [17] used image and addition theorems to write the field reflected from the interface. Then, he applied the boundary conditions on the sphere, which resulted in the same equation as that of Bobbert. Considering the special case of PEC interface and normal incidence, and using the same formulation as that of Videen, Johnson [18] presented perhaps the first numerical results of the problem.

Fucile [19], assumed an external field consisted of the incident, reflected, scattered and scattered-reflected fields, calculated the scattered-reflected field using image and addition theorems and finally applied boundary conditions on the sphere surface. However, by introducing a general reflection rule for spherical functions, he was able to solve the problem without the approximations assumed by Videen. Almost at the same time, Wriedt [20] proposed a simpler exact solution based on the operator approach of Bobbert [15]. He expanded all fields in terms of VSFs and related the coefficients of scattered-reflected and scattered fields by a reflection matrix, which was essentially the same as Bobbert reflection operator. To derive the reflection matrix, instead of the complicated image and addition theorems, Wriedt used the PW expansion of VSFs. More recently Frezza [21] used a combination of these tools to solve the problem of scattering from a sphere below an interface.

As compared to Bobbert [15], Takemori [16] solved the problem from a conceptually different viewpoint, in which the scattering mechanisms were central. He first calculated the scattered field by the sphere due to the incident and the reflected PWs separately, where he used an operator ‘$t$’ to represent scattering [16]. Then by expanding the scattered field in terms of PWs and using Fresnel reflection coefficients, he calculated the resulting reflected set of PWs; another operator ‘$S$’ was used for reflection [16]. Reflected PWs are once again scattered and reflected and this repeats indefinitely. For any further scattering and reflection he introduced factors t and S, respectively and the total scattered field was constructed using an alternate sequence of operators t and S, acted on the first-order fields [16]. There are a few analytical works [22–24] which follow an approach similar to Takemori [16]. Lawrence [22] proposed a similar method for solution of 2D scattering from a circular cylinder below a slightly rough interface. The Lawrence 2D solution was recently extended to the 3D case [23]. Another connected work is that of Wang [24] who used successive interactions of the object and interface to obtain the scattering from a sphere buried in a half-space.

Scattering properties of the chiral objects is investigated extensively, using both analytical [25–27] and numerical [28–30] techniques. Contrary to the free-space scattering problem, the interaction of electromagnetic waves with chiral objects in presence of a half-space is rarely investigated [31–33]. None of these limited studies are analytical; indeed, they are all numerical studies based on the method of moments.

It is the aim of this paper to analytically solve the problem of scattering from a chiral sphere above a half-space. In the proposed method, the first-order scattering is determined using an extension of Mie solution and taking the incident and reflected PWs as the excitation. Higher order scattered fields are then obtained in terms of lower order ones by combining the PW expansion of vector spherical functions and the reflection coefficient of the half-space. The obtained solution is a series one which is conveniently converted to a non-recursive solution, called the complete solution. The complete solution is used for numerical analysis and the series one is utilized to study scattering mechanisms.

The configuration considered in this paper could for instance model the optical detection of viruses. Indeed, it is known that many viruses are chiral [34] and in many optical setups used in detection systems, the virus is fixed on a substrate [35]; hence, the problem is essentially that of scattering by a chiral particle above a half-space. Another application relevant to the content of this paper is the directional scattering by chiral particles [36]. It is known that the directional properties are drastically modified by the presence of substrates [37]. Therefore, investigation of the substrate effect on the directional scattering properties of chiral objects is desired. In this paper, the focus is not on these or other special applications; instead, the general theoretical problem of scattering from a chiral sphere above a half-space is investigated.

2. Preliminaries

2.1 Geometry and definitions

A chiral sphere with radius a and centered at the origin of the coordinate system is assumed to be located a distance d above a lossy half-space, as shown in Fig. 1. The permittivity and chirality of the sphere are denoted by ${\epsilon _s}$ and ${\xi _s}$, respectively. The upper half-space is assumed to be free space and the half-space is characterized with the dielectric constant ${\epsilon _1}$. The structure is illuminated from above by an incident PW with arbitrary angle and polarization and the scattered fields by the chiral sphere are to be determined. The incident field ${\bar{E}_i}{e^{j\bar{k}_{0i}^ -{\cdot} \bar{r}}}$ is characterized by the polarization vector ${\bar{E}_i}$ and the propagation vector $\bar{k}_{0i}^ - $. In this notation following definitions are used: $\bar{k}_0^ \pm{=} {\bar{k}_ \bot } \pm \hat{z}{k_{0z}}$, ${k_\rho } = |{{{\bar{k}}_ \bot }} |$, ${k_{0z}} = {({k_0^2 - k_\rho^2} )^{1/2}}$, and ${\bar{k}_ \bot } = \hat{x}{k_x} + \hat{y}{k_y}$. Moreover, a subscript i in $\bar{k}_{0i}^ - $ means that $\bar{k}_0^ \pm $ is evaluated at ${\bar{k}_ \bot } = \bar{k}_ \bot ^i$, which characterizes the incident wave direction. In other words, $\bar{k}_0^ - $ is a function of direction ($\theta ,\; \phi $); the quantity $\bar{k}_{0i}^ - $ is the same function when evaluated at (${\theta _i},\; {\phi _i}$), i.e. the incident direction.

Fig. 1. The cross-section of the 3D geometry.

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2.2 Field transformations

In the following section, in order to formulate the multiple interactions of the half-space and the chiral sphere, transformations between PWs and vector spherical functions are required. The vector spherical functions $\bar{M}_{l,m}^{j0}$, $\bar{N}_{l,m}^{j0}$, $\bar{M}_{l,m}^{h0}$ and $\bar{N}_{l,m}^{h0}$ are defined in [23,38], where superscripts j and h denote respectively the spherical Bessel and Hankel functions of order l.

The VSFs could be expanded in terms of a continuous spectrum of PWs [39]:

(1)$$\bar{A}_{l,m}^{h0}({R,\theta ,\phi } )= \mathop \int \limits_{ - \infty }^{ + \infty } d{\bar{k}_ \bot }\bar{D}_{l,m}^{A \pm }({{{\bar{k}}_ \bot }} )\; {e^{j\bar{k}_0^ \pm{\cdot} \bar{r}}},$$

where $A = M,\; N$ and the coefficients are

(2)$$\bar{D}_{l,m}^{A \pm }({{{\bar{k}}_ \bot }} )= D_{l,m}^{Ah \pm }\hat{h} + D_{l,m}^{Av \pm }\hat{v}_0^ \pm ,$$

where $\hat{h} = ({\hat{k}_0^ \pm{\times} \hat{z}} )/|{\hat{k}_0^ \pm{\times} \hat{z}} |$ and $\hat{v}_0^ \pm{=} \hat{h} \times \hat{k}_0^ \pm $ and $\hat{k}_0^ \pm{=} \bar{k}_0^ \pm{/}{k_0}$. The components in (2) are given by

(3)$$D_{l,m}^{Mh \pm } ={-} \frac{{{{({ - j} )}^l}}}{{2\pi {k_0}{k_{0z}}}}\left[ {{{\left( {\frac{{{k_x} + j{k_y}}}{{{k_\rho }}}} \right)}^m}\tau_l^{ - m}\left( {\frac{{ \pm {k_{0z}}}}{{{k_0}}}} \right)} \right],$$

(4)$$D_{l,m}^{Nh \pm } = \; \; \; \; \frac{{{{({ - j} )}^l}}}{{2\pi {k_0}{k_{0z}}}}\left[ {{{\left( {\frac{{{k_x} + j{k_y}}}{{{k_\rho }}}} \right)}^m}\pi_l^{ - m}\left( {\frac{{ \pm {k_{0z}}}}{{{k_0}}}} \right)} \right],$$

and $D_{l,m}^{Mv \pm } ={-} jD_{l,m}^{Nh \pm }$, $D_{l,m}^{Nv \pm } ={-} jD_{l,m}^{Mh \pm }$. The functions $\tau _l^m({\cdot} )$ and $\pi _l^m({\cdot} )$ are related to the associated Legendre function as defined in [23,38]. These expressions are equivalent to those given e.g. in [40] in terms of the azimuthal and elevation angles. Conversely, an up-going PW could be expanded in terms of VSFs [38]:

(5)$${\bar{P}_i}{e^{j\bar{k}_{0i}^ +{\cdot} \bar{r}}} = \mathop \sum \limits_{l,m}^{} [{b_{l,m}^{i + }\bar{M}_{l,m}^{j0}({R,\theta ,\phi } )+ c_{l,m}^{i + }\bar{N}_{l,m}^{j0}({R,\theta ,\phi } )} ],$$

where the expansion coefficients are as follows.

(6)$$b_{l,m}^{i + } = {v_{l,m}}\bar{m}_{l,m}^{j\ast }({{\theta_i},{\phi_i}} )\cdot {\bar{P}_i},$$

(7)$$c_{l,m}^{i + } ={-} j{v_{l,m}}\bar{n}_{l,m}^{j\ast }({{\theta_i},{\phi_i}} )\cdot {\bar{P}_i}.$$

In (6) and (7) ${v_{l,m}} = {({ - 1} )^m}{(j )^l}\frac{{({2l + 1} )}}{{l({l + 1} )}}$, and ${\theta _i}$ and ${\phi _i}$ are the elevation and azimuthal angles of the incident field. The functions $\bar{m}_{l,m}^{j\ast }$ and $\bar{n}_{l,m}^{j\ast }$ are respectively the complex conjugates of $\bar{m}_{l,m}^j$ and $\bar{n}_{l,m}^j$, which are defined in [23,38]. A down-going PW have a similar expansion with coefficients $b_{l,m}^{i - }$ and $c_{l,m}^{i - }$, for which (6) and (7) should be modified by replacing ${\theta _i}$ with $\pi - {\theta _i}$.

C. Scattering by a Chiral Sphere

Consider a chiral sphere of radius a characterized with permittivity ${\epsilon _s}$ and chirality ${\xi _s}$ in free-space and illuminated by a PW characterized with electric field ${\bar{E}_i} = {\bar{P}_i}{e^{j\bar{k}_{0i}^ \pm{\cdot} \bar{r}}}$. The incident and the scattered electric fields could be expanded in terms of VSFs:

(8)$${\bar{E}_i} = \mathop \sum \limits_{l,m}^{} [{b_{l,m}^{i \pm }\bar{M}_{l,m}^{j0} + c_{l,m}^{i \pm }\bar{N}_{l,m}^{j0}} ],$$

(9)$${\bar{E}_s} = \sum\limits_{l,m}^{} {[{b_{l,m}^{s \pm }\bar{M}_{l,m}^{h0} + c_{l,m}^{s \pm }\bar{N}_{l,m}^{h0}} ]} ,$$

where the coefficients $b_{l,m}^{i \pm }$ and $c_{l,m}^{i \pm }$ are given by (6) and (7) and $b_{l,m}^{s \pm }$ and $c_{l,m}^{s \pm }$ are to be determined. Notice that the dependence of VSFs on $({R,\theta ,\phi } )$ are not shown, for brevity. In order to consider the chirality, the internal field of the sphere should be expanded in a different form [25]:

(10)$${\bar{E}_t} = \mathop \sum \limits_{l,m}^{} [{b_{l,m}^{t \pm }({\bar{M}_{l,m}^{jR} + \bar{N}_{l,m}^{jR}} )+ c_{l,m}^{t \pm }({\bar{M}_{l,m}^{jL} - \bar{N}_{l,m}^{jL}} )} ],$$

where $\bar{M}_{l,m}^{jR}$ and $\bar{M}_{l,m}^{jL}$ are defined similar to $\bar{M}_{l,m}^{j0}$, with ${k_0}$ being respectively replaced with ${k_R}$ and ${k_L}$. These quantities are defined as ${k_R} = {k_s} + \omega {\xi _s}$ and ${k_L} = {k_s} - \omega {\xi _s}$, in which ${k_s} = \omega \sqrt {{\epsilon _s}{\mu _0}} $ . The same notation is used for $\bar{N}_{l,m}^{jR}$ and $\bar{N}_{l,m}^{jL}$.

By applying boundary conditions, the unknown coefficients could be determined:

(11)$$b_{l,m}^{s \pm } = \frac{1}{\Delta }({{\zeta_{bb}}b_{l,m}^{i \pm } + {\zeta_{bc}}c_{l,m}^{i \pm }} ),$$

(12)$$c_{l,m}^{s \pm } = \frac{1}{\Delta }({{\zeta_{cb}}b_{l,m}^{i \pm } + {\zeta_{cc}}c_{l,m}^{i \pm }} ),$$

where

(13)$${\zeta _{bb}} = \alpha - [{{A_R} + {A_L}} ]({\kappa {k_0}{h_l}({{k_0}a} ){{ {{{[{R{j_l}({{k_0}R} )} ]}^{\prime}}} |}_a} + {k_s}{j_l}({{k_0}a} ){{ {{{[{R{h_l}({{k_0}R} )} ]}^{\prime}}} |}_a}} ),$$

(14)$${\zeta _{bc}} = {\zeta _{cb}} = {k_0}[{{A_R} - {A_L}} ]({{h_l}({{k_0}a} ){{ {{{[{R{j_l}({{k_0}R} )} ]}^{\prime}}} |}_a} - {{ {{j_l}({{k_0}a} ){{[{R{h_l}({{k_0}R} )} ]}^{\prime}}} |}_a}} ),$$

(15)$${\zeta _{cc}} = \alpha - [{{A_R} + {A_L}} ]({{k_s}{h_l}({{k_0}a} ){{ {{{[{R{j_l}({{k_0}R} )} ]}^{\prime}}} |}_a} + \kappa {k_0}{j_l}({{k_0}a} ){{ {{{[{R{h_l}({{k_0}R} )} ]}^{\prime}}} |}_a}} ),$$

and

(16)$$\Delta = \beta + [{{A_R} + {A_L}} ]({{k_s} + \kappa {k_0}} ){h_l}({{k_0}a} ){ {{{[{R{h_l}({{k_0}R} )} ]}^{\prime}}} |_a}.$$

In (13)-(16), the following definitions are used:

(17)$$\alpha ={-} 4{A_R}{A_L}k_0^2{h_l}({{k_0}a} ){j_l}({{k_0}a} )- { {{{[{R{j_l}({{k_0}R} )} ]}^{\prime}}} |_a}{ {{{[{R{h_l}({{k_0}R} )} ]}^{\prime}}} |_a},$$

(18)$$\beta = 4{A_R}{A_L}k_0^2{[{{h_l}({{k_0}a} )} ]^2} + {({{{ {{{[{R{h_l}({{k_0}R} )} ]}^{\prime}}} |}_a}} )^2},$$

where $\kappa = {{{k_0}} / {{k_s}}}$ and

(19)$${A_R} = \frac{{ - {{ {{{[{R{j_l}({{k_R}R} )} ]}^{\prime}}} |}_a}}}{{2{k_R}{j_l}({{k_R}a} )}},$$

(20)$${A_L} = \frac{{ - {{ {{{[{R{j_l}({{k_L}R} )} ]}^{\prime}}} |}_a}}}{{2{k_L}{j_l}({{k_L}a} )}}.$$

3. Formulation

3.1 General procedure

When a PW is incident on the structure shown in Fig. 1, it is either reflected by the half-space or scattered by the chiral sphere. The reflected part is another PW incident on the sphere and hence results in another contribution to the scattered field of the sphere. These two scattering contribution, i.e. the scattered fields of the sphere due to the incident and the reflected PWs, make the first-order Mie field, $\bar{E}_1^M$. This field contains only the first interaction with sphere. The first-order Mie field $\bar{E}_1^M$ is a combination of spherical waves which could be equivalently written as a set of up-going and down-going PWs. The down-going PWs experience a reflection from the interface and transform to a set of up-going PWs incident on the chiral sphere. The scattered field of the sphere due to these PWs, being the result of two interactions with the sphere, is the second-order Mie field, $\bar{E}_2^M$. This field is again a combination of spherical waves and based on the same arguments results in a scattered field, the third-order Mie field $\bar{E}_3^M$ and this cycle repeats indefinitely. The summation of the Mie fields of all orders makes the total Mie field, ${\bar{E}^M}$.

A schematic representation of the first few interactions between the sphere and the interface is presented in Fig. 2. The incident and reflected plane waves are shown with blue lines on arrows. Spherical waves are represented with green curves on arrows and the reflected integral of PWs are denoted by red thick arrows. The right panel shows the first interaction for which the initial waves are the incident (${\bar{E}_i}$) and the reflected (${\bar{E}_r}$) waves and the resulting waves are the first order Mie field ($\bar{E}_1^M$) and its reflection from the interface ($\bar{E}_1^{MR}$). The left panel shows the second interaction where the incident wave is $\bar{E}_1^{MR}$ and the resulting waves are the second order Mie field ($\bar{E}_2^M$) and its reflection from the interface ($\bar{E}_2^{MR}$). For clarity, in both panels the Mie fields are decomposed into their up-going ($\bar{E}_k^{M + }$) and down-going ($\bar{E}_k^{M - }$) parts.

Fig. 2. The first few interactions of the sphere and the interface.

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The Mie field ${\bar{E}^M}$ is not the only scattered field in the structure; there is another contribution which is the result of the reflection of the Mie field from the half-space. In what follows, this reflected field, which is represented by ${\bar{E}^{MR}}$ is easily obtained using the PW expansion of ${\bar{E}^M}$ and the reflection coefficients of the half-space. The superposition of ${\bar{E}^M}$ and ${\bar{E}^{MR}}$ is the scattered field by the sphere in the region above the half-space.

In the following, in Sec. III.B, the first-order Mie field is obtained using the results of Sec. II. Next, a matrix relation is derived between the coefficients of the Mie fields of order k and $k + 1$. Since, the coefficients of the 1^st order Mie field are known, using the derived matrix relation, the second and higher order coefficients could easily be calculated. Furthermore, this series could be summed analytically; hence the series solution is transformed to the complete solution.

3.2 First-order Mie field

The first order Mie field is simply the superposition of two fields: the scattered field by the chiral sphere due to the incident PW and scattered field by the chiral sphere due to the reflected PW. Using the formulations of the previous section and defining $l = \left[ {\sqrt {n - 1} } \right]$ (where $[{\cdot} ]$ denotes the integer part) and $m = ({n - 1} )- l({l + 1} )$, this field could easily be written:

(21)$$\bar{E}_1^M = \mathop \sum \limits_n^{} [{b_{n,1}^s\bar{M}_n^{h0} + c_{n,1}^s\bar{N}_n^{h0}} ],$$

where

(22)$$b_{n,1}^s = b_{n,1}^{sh}{P_{ih}} + b_{n,1}^{sv}{P_{iv}},$$

(23)$$c_{n,1}^s = c_{n,1}^{sh}{P_{ih}} + c_{n,1}^{sv}{P_{iv}},$$

and

(24)$$b_{n,1}^{s\alpha } = b_n^{s\alpha - }({\bar{k}_ \bot^i} )+ {R_\alpha }({\bar{k}_ \bot^i} )b_n^{s\alpha + }({\bar{k}_ \bot^i} ),$$

(25)$$c_{n,1}^{s\alpha } = c_n^{s\alpha - }({\bar{k}_ \bot^i} )+ {R_\alpha }({\bar{k}_ \bot^i} )c_n^{s\alpha + }({\bar{k}_ \bot^i} ),$$

where ${R_h}$ and ${R_v}$ are the Fresnel reflection coefficients. The first and the second terms of the right-hand sides of (24) and (25) are respectively associated with the incident and the reflected PWs and $b_n^{s\alpha \pm }$ and $c_n^{s\alpha \pm }$ are given in (11) and (12). Defining ${\boldsymbol b}_1^s = {[{b_{1,1}^s,\; b_{2,1}^s,\; \ldots \; } ]^t}$ and ${\boldsymbol c}_1^s = {[{c_{1,1}^s,\; c_{2,1}^s,\; \ldots \; } ]^t}$, the coefficients of the 1^st order Mie field could be briefly denoted by

(26)$${{\boldsymbol a}_1} = {\left[ {\begin{array}{{cc}} {{\boldsymbol b}_1^s}&{{\boldsymbol c}_1^s} \end{array}} \right]^t}.$$

3.3 Relation between successive scattering orders

In this subsection, the Mie field of order $k + 1$ is calculated assuming that the Mie field of order k is known. Hence, it is supposed that $b_{n,k}^s$ and $c_{n,k}^s$ are determined:

(27)$$\bar{E}_k^M = \mathop \sum \limits_n^{} [{b_{n,k}^s\bar{M}_n^{h0} + c_{n,k}^s\bar{N}_n^{h0}} ].$$

Using the down-going PW expansion of VSFs as given in (1), for negative values of z, $\bar{E}_k^M$ could be written as:

(28)$$\bar{E}_k^M = \mathop \int \limits_{ - \infty }^{ + \infty } d{\bar{k}_ \bot }{e^{j\bar{k}_0^ -{\cdot} \bar{r}}}({\sigma_k^h\hat{h} + \sigma_k^v\hat{v}_0^ - } ),$$

where for $\alpha = h,\; v$ the following definition is used:

(29)$$\sigma _k^\alpha = \mathop \sum \limits_n^{} [{b_{n,k}^sD_n^{M\alpha - } + c_{n,k}^sD_n^{N\alpha - }} ].$$

The down-going PWs of (28) experience a reflection from the half-space and transform to up-going PWs:

(30)$$\bar{E}_k^{MR} = \mathop \int \limits_{ - \infty }^{ + \infty } d{\bar{k}_ \bot }{e^{j\bar{k}_0^ +{\cdot} \bar{r}}}({\pi_k^h\hat{h} + \pi_k^v\hat{v}_0^ + } ),$$

where $\; \pi _k^h = {R_h}\sigma _k^h$ and $\pi _k^v = {R_v}\sigma _k^v$. For each up-going PW in (30), the Mie field could be written in component form; hence, for the continuous spectrum of (30), one obtains the Mie field of order $k + 1$ as:

(31)$$\bar{E}_{k + 1}^M = \mathop \int \limits_{ - \infty }^{ + \infty } d{\bar{k}_ \bot }\mathop \sum \limits_n^{} [{\pi_k^h({b_n^{sh + }\bar{M}_n^{h0} + c_n^{sh + }\bar{N}_n^{h0}} )+ \pi_k^v({b_n^{sv + }\bar{M}_n^{h0} + c_n^{sv + }\bar{N}_n^{h0}} )} ].$$

By reordering the terms, this equation could be written in the following form:

(32)$$\bar{E}_{k + 1}^M = \mathop \sum \limits_n^{} [{b_{n,k + 1}^s\bar{M}_n^{h0} + c_{n,k + 1}^s\bar{N}_n^{h0}} ],$$

where

(33)$$b_{n,k + 1}^s = \mathop \int \limits_{ - \infty }^{ + \infty } d{\bar{k}_ \bot }({b_n^{sh + }\pi_k^h + b_n^{sv + }\pi_k^v} ),$$

(34)$$c_{n,k + 1}^s = \mathop \int \limits_{ - \infty }^{ + \infty } d{\bar{k}_ \bot }({c_n^{sh + }\pi_k^h + c_n^{sv + }\pi_k^v} ).$$

Next, in summation (29) the dummy variable is changed to $n^{\prime}$ and the relations immediately following Eq. (30) are used. Substituting the resulting expressions for $\pi _k^h$ and $\pi _k^v$ in (33) and (34), and rearranging the terms, the following equation is obtained.

(35)$$b_{n,k + 1}^s = \mathop \sum \limits_{n^{\prime}}^{} [{I_{n,n^{\prime}}^{bb}b_{n^{\prime},k}^s + I_{n,n^{\prime}}^{bc}c_{n^{\prime},k}^s} ],$$

(36)$$c_{n,k + 1}^s = \mathop \sum \limits_{n^{\prime}}^{} [{I_{n,n^{\prime}}^{cb}b_{n^{\prime},k}^s + I_{n,n^{\prime}}^{cc}c_{n^{\prime},k}^s} ],$$

where

(37)$$I_{n,n^{\prime}}^{bb} = \mathop \int \limits_{ - \infty }^{ + \infty } d{\bar{k}_ \bot }[{{R_h}b_n^{sh + }D_{n^{\prime}}^{Mh - } + {R_v}b_n^{sv + }D_{n^{\prime}}^{Mv - }} ],$$

(38)$$I_{n,n^{\prime}}^{bc} = \mathop \int \limits_{ - \infty }^{ + \infty } d{\bar{k}_ \bot }[{{R_h}b_n^{sh + }D_{n^{\prime}}^{Nh - } + {R_v}b_n^{sv + }D_{n^{\prime}}^{Nv - }} ],$$

(39)$$I_{n,n^{\prime}}^{cb} = \mathop \int \limits_{ - \infty }^{ + \infty } d{\bar{k}_ \bot }[{{R_h}c_n^{sh + }D_{n^{\prime}}^{Mh - } + {R_v}c_n^{sv + }D_{n^{\prime}}^{Mv - }} ],$$

(40)$$I_{n,n^{\prime}}^{cc} = \mathop \int \limits_{ - \infty }^{ + \infty } d{\bar{k}_ \bot }[{{R_h}c_n^{sh + }D_{n^{\prime}}^{Nh - } + {R_v}c_n^{sv + }D_{n^{\prime}}^{Nv - }} ].$$

Equations (35) and (36) give the Mie coefficients of order $k + 1$ in terms of the Mie coefficients of order k. For later use, these equations are written in matrix form:

(41)$${\boldsymbol b}_{k + 1}^s = {{\boldsymbol I}^{{\boldsymbol bb}}}{\boldsymbol b}_k^s + {{\boldsymbol I}^{{\boldsymbol bc}}}{\boldsymbol c}_k^s,$$

(42)$${\boldsymbol c}_{k + 1}^s = {{\boldsymbol I}^{{\boldsymbol cb}}}{\boldsymbol b}_k^s + {{\boldsymbol I}^{{\boldsymbol cc}}}{\boldsymbol c}_k^s,$$

where for instance, ${\boldsymbol b}_k^s = {[{b_{k,1}^s,\; b_{k,2}^s,\; \ldots \; } ]^t}$, in which t denotes the transpose operation. An even more concise form would be:

(43)$${{\boldsymbol a}_{k + 1}} = {\boldsymbol M}{{\boldsymbol a}_k},$$

where:

(44)$${\boldsymbol M} = \left[ {\begin{array}{{cc}} {{{\boldsymbol I}^{{\boldsymbol bb}}}}&{{{\boldsymbol I}^{{\boldsymbol bc}}}}\\ {{{\boldsymbol I}^{{\boldsymbol cb}}}}&{{{\boldsymbol I}^{{\boldsymbol cc}}}} \end{array}} \right],$$

and

(45)$${{\boldsymbol a}_k} = \left[ {\begin{array}{{c}} {{\boldsymbol b}_k^s}\\ {{\boldsymbol c}_k^s} \end{array}} \right].$$

Hence, according to (43), multiplying the Mie coefficients of each order ${{\boldsymbol a}_k}$ by matrix ${\boldsymbol M}$ gives the Mie coefficients of the subsequent order ${{\boldsymbol a}_{k + 1}}$.

3.4 Series and complete solutions

Now, using the first-order Mie coefficients (Eqs. (24) and (25)) and the relation between Mie coefficients of successive orders (Eqs. (35) and (36)), a series solution for the Mie field is obtained, as follows. The Mie field is the summation of Mie fields of different orders which could be written as:

(46)$${\bar{E}^M} = \mathop \sum \limits_k^{} \bar{E}_k^M = \mathop \sum \limits_k^{} \mathop \sum \limits_n^{} [{b_{n,k}^s\bar{M}_n^{h0} + c_{n,k}^s\bar{N}_n^{h0}} ],$$

where by changing the order of summations, one obtains:

(47)$${\bar{E}^M} = \mathop \sum \limits_n^{} [{b_n^s\bar{M}_n^{h0} + c_n^s\bar{N}_n^{h0}} ],$$

in which $b_n^s = \mathop \sum \limits_k^{} b_{n,k}^s$ and $c_n^s = \mathop \sum \limits_k^{} c_{n,k}^s$. The first order terms $b_{n,1}^s$, $c_{n,1}^s$ are given in (24) and (25), which represent the Mie field due to a single incidence of the wave on the chiral sphere. Next, using (35) and (36), $b_{n,2}^s$, $c_{n,2}^s$ are obtained in terms of $b_{n,1}^s$, $c_{n,1}^s$, which represent the Mie field due to double incidence of the wave on the sphere. Repeated application of (35) and (36) gives the higher order terms associated with multiple interactions of the half-space and the chiral sphere.

As was stated in Sec. II, the series solution could be analytically calculated. Using the matrix notation, the relations immediately after (47) could be written in the following form:

(48)$${\boldsymbol a} = \mathop \sum \limits_{k = 1}^\infty {{\boldsymbol a}_k},$$

where ${{\boldsymbol a}_k}$ is defined in (45), and ${\boldsymbol a} = \left[ {\begin{array}{cc} {{{\boldsymbol b}^s}}&{{{\boldsymbol c}^s}} \end{array}} \right]$, in which ${{\boldsymbol b}^s} = {[{b_1^s,\; b_2^s,\; \ldots \; } ]^t}$ and ${{\boldsymbol c}^s} = {[{c_1^s,\; c_2^s,\; \ldots \; } ]^t}$. Expanding (48),

(49)$${\boldsymbol a} = {{\boldsymbol a}_1} + {{\boldsymbol a}_2} + {{\boldsymbol a}_3} + \ldots ,$$

and using (43), one obtains:

(50)$${\boldsymbol a} = {{\boldsymbol a}_1} + {\boldsymbol M}{{\boldsymbol a}_1} + {\boldsymbol MM}{{\boldsymbol a}_1} + \ldots ,$$

which could be written as

(51)$${\boldsymbol a} = \mathop \sum \limits_{k = 0}^\infty {{\boldsymbol M}^k}{{\boldsymbol a}_1}.$$

The summation could easily be identified as the binomial expansion of ${({{\boldsymbol I} - {\boldsymbol M}} )^{ - 1}}$; hence:

(52)$${\boldsymbol a} = {({{\boldsymbol I} - {\boldsymbol M}} )^{ - 1}}{{\boldsymbol a}_1}.$$

Therefore, a solution containing all interactions is obtained: ${{\boldsymbol a}_1}$ is known from (26) and using (52), ${\boldsymbol a}$ which contains the coefficients of the total Mie field could be calculated.

3.5 Scattered field

The scattered field is the superposition of the Mie field and its reflection from the half-space. To calculate the scattered field, using (1), the Mie field of (47) is written in terms of PWs:

(53)$${\bar{E}^M} = \mathop \int \limits_{ - \infty }^{ + \infty } d{\bar{k}_ \bot }{e^{j\bar{k}_0^ \pm{\cdot} \bar{r}}}({\sigma_t^{h \pm }\hat{h} + \sigma_t^{v \pm }\hat{v}_0^ \pm } ),$$

where the $+ \textrm{ / } - $ signs are used for observation points respectively with $z > 0$ and $z < 0$, and:

(54)$$\sigma _t^{\alpha \pm } = \mathop \sum \limits_{n = 1}^\infty [{b_n^sD_n^{M\alpha \pm } + c_n^sD_n^{N\alpha \pm }} ],$$

and $\alpha = h,\; v$. The down-going PWs in (53) are reflected from the half-space; the reflected field for all observation points above the interface is:

(55)$${\bar{E}^{MR}} = \mathop \int \limits_{ - \infty }^{ + \infty } d{\bar{k}_ \bot }{e^{j\bar{k}_0^ +{\cdot} \bar{r}}}({{R_h}\sigma_t^{h - }\hat{h} + {R_v}\sigma_t^{v - }\hat{v}_0^ + } ).$$

The scattered field for $- d < z < 0$ and for $z > 0$, is the sum of Eqs. (55) and (53) with the $- $ sign. Concentrating on observation points with $z > 0$, the scattered field would be:

(56)$${\bar{E}^s} = \mathop \int \limits_{ - \infty }^{ + \infty } d{\bar{k}_ \bot }{e^{j\bar{k}_0^ +{\cdot} \bar{r}}}[{\sigma_f^h\hat{h} + \sigma_f^v\hat{v}_0^ + } ],$$

where $\sigma _f^\alpha = \sigma _t^{\alpha + } + {R_\alpha }\sigma _t^{\alpha - }$. Using the far-field approximation, the scattering cross-sections would be

(57)$${\sigma _{\alpha \beta }} = {|{{S_{\alpha \beta }}} |^2},$$

where $\alpha ,\; \beta = h,\; v$ and

(58)$${S_{\alpha \beta }} ={-} 2\pi j{ {\sigma_f^{\alpha s}} |_{{P_{i\beta }} = 1}}.$$

4. Validation

In order to validate the formulation of the present paper, the numerical results of [31] is taken as the reference solution. The problem of scattering from a chiral sphere above a lossy half-space is solved in [31], using the method of moments. In Fig. 6 of [31], for a chiral sphere (${\epsilon _{sr}} = 3$ and ${\xi _{sr}} = 0.3,\; 0.8$) of radius $0.3\; m$ located at $0.7\; m$ above a half-space (${\epsilon _{1r}} = 5 + j0.8$), the co-polarized bistatic cross-sections are reported. The structure is illuminated with a normal PW of $f = 300MHz$. The numerical results of the present paper for the same case, together with those of [31], are depicted in Fig. 3. As could be seen, excellent agreement is observed between the results of the two methods.

Fig. 3. RCS for a chiral sphere above a lossy half-space; comparison with Ref. [31].

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5. Discussion

In this section, using the series solution of Eq. (50), a few observations regarding the role of different scattering orders are performed. In addition, using the complete solution of Eq. (52), some numerical examples are presented. First, it is shown that the series solution rapidly converges; in fact in many situations, only the first two terms are sufficient. Second, the ratio of the second-order to first-order scattered fields are computed for different values of chirality parameter. Finally, for a chiral sphere above a lossy half-space, the bistatic scattering for different incident angles is obtained. In all simulations of this section, the parameters of Sec. IV are used, except otherwise stated. It should be noted that simulations are repeated for other parameter values and generally similar variation patterns are obtained. However, for brevity, those results are not included.

5.1 Convergence of the series solution

A chirality factor of ${\xi _r} = 0.2$ is considered for the sphere and other parameters are the same as those of Sec. IV. Using the series solution of Eq. (50), the scattered fields of the first few orders are computed and depicted in Fig. 4. The complete solution of Eq. (52), which includes all multiple interactions between the sphere and the half-space is also plotted in the same figure. As could be observed, in this example the first two interactions basically determine the scattered field and hence the series solution is converged by considering the first two terms. Moreover, this converged solution is consistent with the complete one of Eq. (52).

Fig. 4. Convergence of the series solution and its consistency with the complete solution.

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As expected, the contribution of interactions of each order is considerably smaller than that of the previous one. Note that, as previous studies [22,23] also suggest, the permittivity contrast (or equivalently the reflection coefficient magnitude) is the main factor in determining the number of required interactions for the series solution to converge. Furthermore, as the permittivity contrast is increased, more spherical harmonics should be included [23]. Finally, it is worth mentioning that for the chirality parameter considered, the second-order contribution for the $hh$-polarization is considerable while that of $vv$-polarization is very small. When the simulations were repeated for large chirality values (e.g. ${\xi _r} = 0.8$), the situation was reversed. This point is further investigated in the following.

5.2 Chirality effect

To further examine the relative magnitude of the second-order field as compared to the first-order scattered field, another scenario is considered, as follows. The scattered fields of the first two orders are computed using (50) and the ratio of their respective scattering cross-sections, defined as ${\sigma _{21}} = {\sigma _2}/{\sigma _1}$, is calculated. Obviously, when the logarithm of this ratio tends to zero, the second-order contribution is negligible and the first-order interaction is sufficient. Such a first-order solution is basically equivalent to the four-path model [41]. The chirality parameter is changed from $0$ to $0.9$ by steps equal to $0.1$ and the results for both polarizations are plotted and inspected. For clarity, only a selection of these results are presented in Fig. 5.

Fig. 5. The ratio of second-order to first-order scattered fields for different values of chirality.

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By examination of the obtained results, the following observations are in order. First, for both polarizations, a similar trend could be seen: by increasing the chirality parameter from zero, the ratio generally decreases and reaches a minimum and then increases again. The minima for both polarizations occurs at ${\xi _r} = 0.5$. It is interesting that for the above-mentioned minima, the ratio is less than $0.5\; dB$ for all observation angles; i.e. the second-order contribution could be practically ignored. It should be noted that we performed such simulations with other parameters, e.g. for the case of substrates with different dielectric constants. Not only the same general pattern was observed, but also the minima occurred exactly at the same values for chirality parameter. It seems that more analytical investigations are needed to address this effect. Second, it is seen that for $hh$-polarization the maximum values of the ratio generally happens for the lowest values of the chirality parameter while, for the $vv$-polarization, the maximum values mostly occur at the highest values of the chirality parameter. This is consistent with the observations of the previous subsection. Notice that for cases with large ratio ${\sigma _{21}}$, considering a four-path model [41] and hence ignoring higher order interactions results in erroneous results.

5.3 Incident angle

Here, a chiral sphere above a lossy half-space with the same configuration as the one investigated in Sec. IV is considered. The chirality factors is set to be ${\xi _r} = 0.2$ for the sphere. The incident angle is changed from ${\theta _i} = 0$ to ${\theta _i} = {\pi / 3}$ and the bistatic scattering coefficients are calculated. The numerical results based on the complete solution are depicted in Fig. 6. Simulations were also done for the cases with chirality parameters of ${\xi _r} = 0$ and ${\xi _r} = 0.7$. For the sake of clarity, only the results of $hh$-polarization for ${\xi _r} = 0.2$ are shown.

Fig. 6. The co-polarized RCS for the different incident angles; a chiral sphere above a lossy half-space.

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The main features of the obtained results for this subsection could conveniently be explained using only the first interaction of the sphere and interface. Indeed, for a sphere in free-space, there is a main big forward lobe and a smaller backward lobe. In the configuration considered in this paper, the scattered fields in the backward direction are of interest. When a substrate is added, to the first-order, the scattered field is the superposition of the backward lobe (due to the incident PW) and the forward lobe (due to the reflected PW). For a given set of parameters, the superposition could be either constructive or destructive in different observation angles. For the parameters specified here, when normal incidence is considered, the superposition is constructive in the backscattering direction, as well as two near grazing angles. The maximum directions for the backward and forward lobes are respectively in the backscattering and specular directions which happen to coincide only for normal incidence. As the incident angle increases, the direction of these maxima depart and since the forward lobe is bigger, the maximum in forward direction clearly dominates, as could be seen for ${\theta _i} = {\pi / 3}$ in Fig. 6. As the chirality increases (not shown in Fig. 6), the side-lobes near grazing angles become dominant. Thus, the constructive superposition of the backward lobe and the respective dominant side-lobe could result in a large maximum in the backward direction, comparable to the one in forward direction.

6. Conclusions

In this paper, by using the extended Mie solution for a chiral sphere, and the transformations between PWs and VSFs, the fields scattered by a sphere above a half-space are analytically derived. The inputs are the dielectric constant, chirality, radius and height of the sphere, the dielectric constant of the half-space, and the characteristics of the incident PW. The outputs are scattered fields by the sphere in arbitrary points of space in a series form, given in (26), (44) and (50), and in a non-recursive form, given in (26), (44) and (52).

The complete solution (52) involves a matrix inversion and thus is essentially a numerical solution. This semi-analytical solution 1) is much faster than the traditional numerical techniques, 2) could easily be extended to the case of multilayered or inhomogeneous substrates, 3) could be extended to the case of objects with arbitrary shapes, using T-matrix method, and 3) could provide benchmark solutions, since only in the last step involves possible numerical errors. On the other hand, the series solution (50) is basically an approximate analytical solution, which 1) could be used to investigate different scattering mechanisms, and 2) could be used to analytically determine the impact of different parameters, at least in special cases.

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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Scattering by a chiral sphere above a half-space

Abstract

1. Introduction

2. Preliminaries

2.1 Geometry and definitions

2.2 Field transformations

3. Formulation

3.1 General procedure

3.2 First-order Mie field

3.3 Relation between successive scattering orders

3.4 Series and complete solutions

3.5 Scattered field

4. Validation

5. Discussion

5.1 Convergence of the series solution

5.2 Chirality effect

5.3 Incident angle

6. Conclusions

Disclosures

Data availability

References

Data availability

Cited By

Figures (6)

Equations (58)

Optics Express