A method of convolution of two weight functions to design new types of scattering medium is proposed, and two novel media models are illustrated in detail. The first one is that the two functions involved in convolution are the same type with different parameters. In this case, the far-zone scattered spectral intensity will maintain a stable shape, but the profile size can be controlled by changing the value of parameters. The second one is that the two functions involved in convolution are different types. In this case, the profile of far-zone scattered spectral intensity depends not only on the type of the weight functions but also on the value of the parameters in each weight function. By examining the scattered property of these media, we demonstrate that various far-zone scattered spectral density, such as rectangular distribution and circular distribution, can be achieved. The method proposed in this manuscript may have potential applications in the manipulation of the far-zone scattered field.
1. Introduction
The coherence effect in weak scattering has attracted much attention due to its potential applications in areas such as remote sensing and medical diagnosis. During the past few decades, the researchers have shown great interest in investigating the relations between the behaviors of the far-zone scattered field and the structural characteristics of the scattering medium. For example, the spectrum distribution of light waves on scattering has been discussed, and spectral shifts and spectral switches have been observed when light waves are scattered by various media [1–11]. The reciprocity relationship of light waves on scattering from a random medium has been derived, and it is shown that the normalized spectral density of the scattering field is proportional to the Fourier transform of the normalized correlation coefficient of the scattering potential [12–14]. The equivalence theorem of light waves on scattering from different media has been discussed, and it is shown that, under some special circumstances, the far-zone scattered field from different media may demonstrate identical density distributions or identical spectral degree of coherence [15–17]. For a detailed review of the above-mentioned work, please see [18] and references therein. These researches may provide an effective way to determine the structural information of an unknown object.
Furthermore, since the seminal work made by Gori [19], researches on light field control technology, which focused on the manipulating of the far field distribution by simulating genuine correlation function, has been growing rapidly [20–29]. Owing to the well-known similarity between the source radiation and the light scattering, Korotkova et al. has extended this light field control technique to the weak scattering theory, and has made a lot of work to discuss how to design medium to obtain some special patterns, including the circular flat distribution, ring-like distribution, squared distribution, rectangular distribution, etc. [30,31]. After that, many researchers have made progress on this topic. For instance, Guo has provided a convenient method to control far-zone scattered field with prescribed weak scattering media [32], and Ding has introduced a novel class of random media to produce an optical coherence array [33]. However, to the best of our knowledge, all studies on this topic are built on a linear combination of weight functions. In this manuscript, we will generalize this discussion from linear combination to nonlinear combination, i.e., convolution. As example, two typical models will be illustrated. One is that the two functions involved in convolution are the same type with different parameters, the other is that the two functions involved in convolution are different types. By these media models, far-zone scattered field with stable or with tunable distributions will be obtained.
2. Theory
Firstly, we recall the weak potential scattering theory. As shown in Fig. 1, suppose that a spatially coherent plane light wave, with a propagation direction specified by a unit vector ${{\mathbf s}_0}$, is incident on a scatterer. Within the accuracy of first-order Born approximation and the far field approximation, the far-zone spectral density of light waves on scattering from random medium can be expressed by [34]
(1)$${S^{(\textrm{s} )}}({r{\mathbf s},\omega } )= \frac{{{S^{(\textrm{i} )}}(\omega )}}{{{r^2}}}{\tilde{C}_F}[{ - k({{\mathbf s} - {{\mathbf s}_0}} ),k({{\mathbf s} - {{\mathbf s}_0}} ),\omega } ], $$
where ${S^{(\textrm{i} )}}(\omega )$ represents the spectrum of the incident wave, and (2)$${\tilde{C}_F}({{\mathbf K}_1},{{\mathbf K}_2},\omega ) = \int\!\!\!\int_D {{C_F}({{{\mathbf r^{\prime}}}_1},{{{\mathbf r^{\prime}}}_2},\omega )} \exp [{ - i({{\mathbf K}_1} \cdot {{{\mathbf r^{\prime}}}_1} + {{\mathbf K}_2} \cdot {{{\mathbf r^{\prime}}}_2})} ]{d^3}{r^{\prime}_\textrm{1}}{d^3}{r^{\prime}_\textrm{2}}$$
is the six-dimensional spatial Fourier transform of the correlation function of the scattering potential with ${{\mathbf K}_1} ={-} k({{\mathbf s} - {{\mathbf s}_0}} )$ and ${{\mathbf K}_2} = k({{\mathbf s} - {{\mathbf s}_0}} )$. For a random medium, the correlation function of scattering potential can be expressed as [34] (3)$${C_F}({{{{\mathbf r^{\prime}}}_1},{{{\mathbf r^{\prime}}}_2},\omega } )= \left\langle {{F^\ast }({{{\mathbf r^{\prime}}}_1},\omega )F({{{\mathbf r^{\prime}}}_2},\omega )} \right\rangle, $$
where ${{\mathbf r^{\prime}}_1}$ and ${{\mathbf r^{\prime}}_2}$ are the three-dimensional position vectors inside the scatterer, $\omega$ is angular frequency, and the asterisk denotes the complex conjugate. It should be noted that any genuine correlation function must obey realizable conditions, i.e., hermiticity and non-negative definiteness. The latter is the most difficult one for verification, which should be expressed as the integral form [19] (4)$${C_F}({{{{\mathbf r^{\prime}}}_1},{{{\mathbf r^{\prime}}}_2},\omega } )= \int {p({ {\boldsymbol {\mathrm {\nu}}} } )} H_0^ \ast ({{{{\mathbf r^{\prime}}}_1},{\boldsymbol {\mathrm {\nu}} },\omega } ){H_0}({{{{\mathbf r^{\prime}}}_2},{\boldsymbol {\mathrm {\nu}} },\omega } ){d^3}\nu, $$
where ${\boldsymbol {\mathrm {\nu}} }$ is a three-dimensional vector, ${H_0}({{\mathbf r^{\prime}},{\boldsymbol {\mathrm {\nu}} },\omega } )$ is an arbitrary “mode” function, i.e., kernel function, which determines the correlation type of scatterer, and $p({\mathbf v} )$ is a non-negative “shape” function, i.e., weight function, defining the profile of the correlation function. Assume that ${H_0}({{\mathbf r^{\prime}},{\boldsymbol {\mathrm {\nu}} },\omega } )$ is selected to be a Fourier-like structure, we generally refer it as the classic Schell-model medium [19] (5)$${H_0}({{\mathbf r^{\prime}},{\boldsymbol {\mathrm {\nu}} },\omega } )= \tau ({{\mathbf r^{\prime}}} )\exp ({ - 2\pi i{\boldsymbol {\mathrm {\nu}} } \cdot {\mathbf r^{\prime}}} ), $$
where $\tau ({{\mathbf r^{\prime}}} )$ is a profile function of random scattering potential. Upon substituting from Eq. (5) into Eq. (4), we can immediately obtain the following expression for the Schell-model correlation function of scattering potential, with a form of (6)$${C_F}({{{{\mathbf r^{\prime}}}_1},{{{\mathbf r^{\prime}}}_2},\omega } )= {\tau ^ \ast }({{{{\mathbf r^{\prime}}}_1}} )\tau ({{{{\mathbf r^{\prime}}}_2}} ){\mu _F}({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1},\omega } ), $$
where (7)$${\mu _F}({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1},\omega } )= \int {p({\boldsymbol {\mathrm {\nu}} } )} \exp [{ - 2\pi i{\boldsymbol {\mathrm {\nu}} } \cdot ({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1}} )} ]{d^3}\nu. $$
It is no doubt that the correlation function ${\mu _F}$ plays a role of utmost importance in spectral intensity distribution of scattered field. Accordingly, in order to further manipulate far-zone scattered field and acquire the expected distribution pattern, we will simulate a series of random media by a novel form of ${\mu _F}$. So far, there are a number of methods to design medium by adjusting parameter of ${\mu _F}$ or linear combination of ${\mu _F}$. In the following, we will mainly discuss the influence of the product of the correlation function ${\mu _F}$ on the far field intensity distribution. It is assumed that ${\mu _F}$ can be rewritten as a product of ${\mu _{F1}}$ and ${\mu _{F2}}$. Thus, correlation function is redefined by
(8)$${\mu _{FP}}({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1},\omega } )= {\mu _{F1}}({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1},\omega } ){\mu _{F2}}({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1},\omega } ). $$
Since the Fourier transform of convolution of two functions is the product of the Fourier transform of two functions, with the help of Eq. (7), one can find that $P({\boldsymbol {\mathrm {\nu}} } )$ takes the following form
(9)$${P_C}({\boldsymbol {\mathrm {\nu}} } )= {P_1}({\boldsymbol {\mathrm {\nu}} } )\otimes {P_2}({\boldsymbol {\mathrm {\nu}} } ). $$
Here ${\otimes}$ is a convolution symbol, ${P_1}({\boldsymbol {\mathrm {\nu}} } )$ and ${P_2}({\boldsymbol {\mathrm {\nu}} } )$ are the arbitrary non-negative weight functions and they are governed by the Fourier transform of ${\mu _{F1}}$ and ${\mu _{F2}}$, respectively. Upon substituting from Eqs. (7), (8), (9) into Eq. (6), we obtain the reconstructed correlation function as (10)$${C_{FC}}({{{{\mathbf r^{\prime}}}_1},{{{\mathbf r^{\prime}}}_2},\omega } )= {\tau ^ \ast }({{{{\mathbf r^{\prime}}}_1}} )\tau ({{{{\mathbf r^{\prime}}}_2}} ){\tilde{P}_C}({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1}} ). $$
Considering that the medium is quasi-homogeneous, whose correlation function of scattering potential takes the form of [32] (11)$${C_F}({{{{\mathbf r^{\prime}}}_1},{{{\mathbf r^{\prime}}}_2},\omega } )= {\tau ^2}\left( {\frac{{{{{\mathbf r^{\prime}}}_1} + {{{\mathbf r^{\prime}}}_2}}}{2},\omega } \right){\mu _F}({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1},\omega } ). $$
Upon substituting from Eq. (11) first into Eq. (2) and then into Eq. (1), one can obtain the far-zone scattered spectral density as (12)$${S^{(\textrm{s} )}}({r{\mathbf s},\omega } )= \frac{{{S^{(\textrm{i} )}}(\omega )}}{{{r^2}}}{\tilde{I}_F}({0,\omega } ){\tilde{\mu }_F}[{k({{\mathbf s} - {{\mathbf s}_0}} ),\omega } ], $$
where (13)$${\tilde{I}_F}({0,\omega } )= \int_D {{\tau ^2}} ({{\mathbf r^{\prime}}} ){d^3}r^{\prime}$$
is an average strength of the scattering potential, and (14)$${\tilde{\mu }_F}[{k({{\mathbf s} - {{\mathbf s}_0}} ),\omega } ]= \int_D {{\mu _F}({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1},\omega } )\exp [{ - ik({{\mathbf s} - {{\mathbf s}_0}} )\cdot ({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1}} )} ]} {d^3}{r^{\prime}_1}{d^3}{r^{\prime}_2}$$
is the six-dimensional spatial Fourier transform of degree of potential correlation. It can be seen from Eq. (12) that scattered spectral density depends only on the normalized co$\sigma$rrelation coefficient of scattering potential, which is known as reciprocity relations [13]. Thus, without loss of generality, we can assume that the form of $\tau ({{\mathbf r^{\prime}}} )$ to be Gaussian [33] (15)$$\tau ({{\mathbf r^{\prime}},\omega } )= A\exp \left( { - \frac{{{{{\mathbf r^{\prime}}}^2}}}{{4{\sigma^2}}}} \right), $$
where A is a constant, and $\sigma$ is effective width of potential strength. Upon substituting from Eq. (15) into Eq. (13), we obtain that (16)$${\tilde{I}_F}({0,\omega } )= A{({2\pi } )^{\frac{3}{2}}}{\sigma ^3}. $$
If we approximately ignore the influence of the scattering axis ($z$axis) on the scattering field, the degree of potential correlation can be expressed as the following form, which is known as the Markov approximation [31]
(17)$${\mu _F}({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1},\omega } )= {\mu _F}({{{{\boldsymbol {\mathrm{\rho}}^{\prime}}}_2} - {{{\boldsymbol {\mathrm{\rho}}^{\prime}}}_1},\omega } )\delta ({{{z^{\prime}}_2} - {z_1}^\prime } ), $$
where ${\mu _F}({{{{\boldsymbol {\mathrm{\rho}}^{\prime}}}_2} - {{{\boldsymbol {\mathrm{\rho}}^{\prime}}}_1},\omega } )$ is a two-dimensional correlation function with ${\boldsymbol {\mathrm{\rho}} ^{\prime}}$ being the position vector in $xoy$ plane within the area of the scatterer. Upon substituting from Eq. (17) first into Eq. (14), then together with Eq. (16) into Eq. (12), and after some calculations, the scattered spectral density of far field takes the following form (18)$${S^{(\textrm{s} )}}({r{{\mathbf s}_\rho },\omega } )= \frac{{{S^{(\textrm{i} )}}(\omega )}}{{{r^2}}}A{({2\pi } )^{\frac{3}{2}}}{\sigma ^3}{\tilde{\mu }_F}({{{\mathbf K}_\rho },\omega } ), $$
where (19)$${\tilde{\mu }_F}({{{\mathbf K}_\rho },\omega } )= \int_D {{\mu _F}({{{{\boldsymbol {\mathrm{\rho}}^{\prime}}}_2} - {{{\boldsymbol {\mathrm{\rho}}^{\prime}}}_1},\omega } )} \exp [{ - i{{\mathbf K}_\rho } \cdot ({{{{\boldsymbol {\mathrm{\rho}}^{\prime}}}_2} - {{{\boldsymbol {\mathrm{\rho}}^{\prime}}}_1}} )} ]{d^2}{\rho ^{\prime}_1}{d^2}{\rho ^{\prime}_2}$$
with ${{\mathbf K}_\rho } = [{k({{s_x} - {s_{ox}}} ),k({{s_y} - {s_{oy}}} )} ]$ being a two-dimensional transform vector. Upon substituting from Eq. (8) first into Eq. (19), and then into Eq. (18), with the help Eq. (7), the scattered spectral density produced by convolution of two weight functions can be expressed as follows (20)$${S_C}^{(\textrm{s} )}({r{{\mathbf s}_\rho },\omega } )= \frac{{{S^{(\textrm{i} )}}(\omega )}}{{{r^2}}}A{({2\pi } )^{\frac{3}{2}}}{\sigma ^3}{P_C}({{{\mathbf K}_\rho }} ). $$
This equation is the main result of this manuscript, and based on the operation of Eq. (9), a series of new random scattering media can be designed to manipulate the far field pattern.3. Two special cases
In this section, two special cases, i.e., convolution of weight functions with same type or with different types, will be discussed in detail to further illustrate the above result.
3.1. Case 1: Convolution of weight function with same type
3.1.1. Convolution of Gaussian functions
We will first examine the variation trend of the far field density distribution with respect to the parameters corresponding to the proposed correlation function. Consider that both the two functions involved in the convolution are Gaussian functions, whose “shape” function $p({\mathbf v} )$ then should be expressed as [35]
(21a)$${P_{G1}}({{{\boldsymbol {\mathrm {\nu}} }_\rho }} )= 2\pi {\delta _1}^2\exp ({ - 2{\pi^2}{\delta_1}^2{\boldsymbol {\mathrm {\nu}} }_\rho^2} ), $$
(21b)$${P_{G2}}({{{\boldsymbol {\mathrm {\nu}} }_\rho }} )= 2\pi {\delta _2}^2\exp ({ - 2{\pi^2}{\delta_2}^2{\boldsymbol {\mathrm {\nu}} }_\rho^2} ), $$
where ${P_{G1}}$ and ${P_{G2}}$ are the same type which both obey Gaussian distribution, with ${\delta _1}$ and ${\delta _2}$ being their effective width, respectively. Then, upon substituting from Eqs. (21) into Eq. (9), one can find the new “shape” function that we reconstructed with the following form (22)$${P_{C1}}({{{\boldsymbol {\mathrm {\nu}} }_\rho }} )= {P_G}_1({{{\boldsymbol {\mathrm {\nu}} }_\rho }} )\otimes {P_{G2}}({{{\boldsymbol {\mathrm {\nu}} }_\rho }} )= \frac{{4{\pi ^2}\delta _1^2\delta _2^2}}{{\delta _1^2 + \delta _2^2}}\exp \left( { - \frac{{2{\pi^2}\delta_1^2\delta_2^2}}{{\delta_1^2 + \delta_2^2}}{\boldsymbol {\mathrm {\nu}} }_\rho^2} \right). $$
Upon substituting from Eq. (22) into Eq. (20), expression of scattered spectral density can be derived as (23)$${S_{C1}}^{(\textrm{s} )}({r{{\mathbf s}_\rho },\omega } )= \frac{{{S^{(\textrm{i} )}}(\omega )}}{{{r^2}}}A{({2\pi } )^{\frac{5}{2}}}{\sigma ^3}\frac{{\delta _1^2\delta _2^2}}{{\delta _1^2 + \delta _2^2}}\exp \left( { - \frac{{2{\pi^2}\delta_1^2\delta_2^2}}{{\delta_1^2 + \delta_2^2}}{\mathbf K}_\rho^2} \right). $$
Figure 2 shows the degree of potential correlation varying with $k{x_d}$ and $k{y_d}$ (where ${\rho _\textrm{d}} = |{{{{\boldsymbol {\mathrm{\rho}}^{\prime}}}_2} - {{{\boldsymbol {\mathrm{\rho}}^{\prime}}}_1}\textrm{ }} |$), which is obtained by the Fourier transform of Eq. (22). While Fig. 3 shows the profile of normalized spectral intensity of light waves on scattering from the medium described by Fig. 2. It is shown that two-dimensional normalized spectral intensity patterns of far field demonstrate as a set of concentric circles, i.e., a Gaussian type. Since parameters $k{\delta _1}$ and $k{\delta _2}$ have the equal effects on the far field spectral density, we fix one of the parameters while changing the other one. It can be seen that, the normalized spectral density is still a set of concentric circle with increasing of $k{\delta _1}$. Yet, the profile becomes smaller, which means that the distribution of far field spectral intensity is more concentrated. That is to say, far field spectral density tends to be a beam-like structure.3.1.2. Convolution of rectangular functions
For the second example, we assume that the distribution of “shape” function $p({\mathbf v} )$ can be represented by two-dimensional rectangular function [35]
(24a)$${p_{R1}}({{{\boldsymbol {\mathrm {\nu}} }_\rho }} )= {\delta _1}^2\textrm{rect}({{\delta_1}{\nu_x}} )\textrm{rect}({{\delta_1}{\nu_y}} ), $$
(24b)$${p_{R2}}({{{\boldsymbol {\mathrm {\nu}} }_\rho }} )= {\delta _2}^2\textrm{rect}({{\delta_2}{\nu_x}} )\textrm{rect}({{\delta_2}{\nu_y}} ), $$
where ${P_{R1}}$ and ${P_{R2}}$ are the same type which both obey two-dimensional rectangular distribution with $\textrm{rect}({{{\boldsymbol {\mathrm {\nu}} }_\rho }} )$ being the rectangular function. Then, upon substituting from Eqs. (24) into Eq. (9), one can find that (25)$${p_{C2}}({{{\boldsymbol {\mathrm {\nu}} }_\rho }} )= {p_{R1}}({{{\boldsymbol {\mathrm {\nu}} }_\rho }} )\otimes {p_{R2}}({{{\boldsymbol {\mathrm {\nu}} }_\rho }} )= \frac{{{{({{\delta_1} + {\delta_2}} )}^2}}}{4}\textrm{Tri}\left( {\frac{{2{\delta_1}{\delta_2}}}{{{\delta_1} + {\delta_2}}}{\nu_x}} \right)\textrm{Tri}\left( {\frac{{2{\delta_1}{\delta_2}}}{{{\delta_1} + {\delta_2}}}{\nu_y}} \right), $$
where $\textrm{Tri}({{{\boldsymbol {\mathrm {\nu}} }_\rho }} )$ denotes the triangular function. On substituting from Eq. (25) into Eq. (20), we can obtain the scattered spectral density as (26)$${S_{C2}}^{(\textrm{s} )}({r{{\mathbf s}_\rho },\omega } )= \frac{{{S^{(\textrm{i} )}}(\omega )}}{{{r^2}}}A{({2\pi } )^{\frac{3}{2}}}{\sigma ^3}\frac{{{{({{\delta_1} + {\delta_2}} )}^2}}}{4}\textrm{Tri}\left( {\frac{{2{\delta_1}{\delta_2}}}{{{\delta_1} + {\delta_2}}}{\textrm{K}_x}} \right)\textrm{Tri}\left( {\frac{{2{\delta_1}{\delta_2}}}{{{\delta_1} + {\delta_2}}}{\textrm{K}_y}} \right). $$
Figure 4 presents the degree of potential correlation produced by the Fourier transform of Eq. (25), while Fig. 5 presents the normalized spectral intensity in the far-zone scattered field whose distribution depends on the medium described by Fig. 4. Similar to the above case, because $k{\delta _1}$ and $k{\delta _2}$ have the same status, we still fix one parameter while changing the other one. It is not difficult to find that, the profile of the far field normalized spectral intensity is shown to be a pyramid-shape. And with the increment of $k{\delta _1}$, the shape of the image remains the same but becomes smaller.3.2. Case 2: Convolution of weight function with different types
Case 1 provide a feasible method to produce scattered field, which is based on the convolution of two functions of the same type. In this section, the convolution of two different types of functions will be discussed to achieve far field modulation. Assume that ${p_1}({\boldsymbol {\mathrm {\nu}} } )= {p_G}({{{\boldsymbol {\mathrm {\nu}} }_\rho }} )$ and ${p_2}({\boldsymbol {\mathrm {\nu}} } )= {p_R}({{{\boldsymbol {\mathrm {\nu}} }_\rho }} )$, i.e.,
(27a)$${P_{G1}}({{{\boldsymbol {\mathrm {\nu}} }_\rho }} )= 2\pi {\delta _1}^2\exp ({ - 2{\pi^2}{\delta_1}^2{\boldsymbol {\mathrm {\nu}} }_\rho^2} ), $$
(27b)$${p_{R2}}({{{\boldsymbol {\mathrm {\nu}} }_\rho }} )= {\delta _2}^2\textrm{rect}({{\delta_2}{\nu_x}} )\textrm{rect}({{\delta_2}{\nu_y}} ), $$
where ${\delta _1}$ and ${\delta _2}$ are defined in Eqs. (21) and (24), respectively. Upon substituting from Eqs. (27) into Eq. (9), one can obtain (28)$$\begin{array}{l} {p_{C3}}({{{\boldsymbol {\mathrm {\nu}} }_\rho }} )= {p_{G1}}({{{\boldsymbol {\mathrm {\nu}} }_\rho }} )\otimes {p_{R2}}({{{\boldsymbol {\mathrm {\nu}} }_\rho }} )\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} = \frac{{\delta _2^2}}{4}\left\{ {\textrm{Erf}\left[ {\sqrt 2 \pi {\delta_1}\left( {{\nu_x} + \frac{1}{{2{\delta_2}}}} \right)} \right] - \textrm{Erf}\left[ {\sqrt 2 \pi {\delta_1}\left( {{\nu_x} - \frac{1}{{2{\delta_2}}}} \right)} \right]} \right\}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times {\kern 1pt} \left\{ {\textrm{Erf}\left[ {\sqrt 2 \pi {\delta_1}\left( {{\nu_y} + \frac{1}{{2{\delta_2}}}} \right)} \right] - \textrm{Erf}\left[ {\sqrt 2 \pi {\delta_1}\left( {{\nu_y} - \frac{1}{{2{\delta_2}}}} \right)} \right]} \right\} \end{array}, $$
where $\textrm{Erf}(x )$ is the Gaussian error function, with the form of (29)$$\textrm{Erf}(x) = \left( {{2 \mathord{\left/ {\vphantom {2 {\sqrt \pi }}} \right. } {\sqrt \pi }}} \right)\int_0^x {{e^{ - {t^2}}}} \textrm{d}t. $$
After tedious but straightforward calculations, the corresponding ${S_C}^{(\textrm{s} )}({r{{\mathbf s}_\rho },\omega } )$ takes the form (30)$$\begin{array}{l} {S_{C3}}^{(\textrm{s} )}({r{{\mathbf s}_\rho },\omega } )= \frac{{{S^{(\textrm{i} )}}(\omega )}}{{{r^2}}}A{({2\pi } )^{\frac{3}{2}}}\sigma \frac{{\delta _2^2}}{4}\left\{ {\textrm{Erf}\left[ {\sqrt 2 \pi {\delta_1}\left( {{\textrm{K}_x} + \frac{1}{{2{\delta_2}}}} \right)} \right] - \textrm{Erf}\left[ {\sqrt 2 \pi {\delta_1}\left( {{\textrm{K}_x} - \frac{1}{{2{\delta_2}}}} \right)} \right]} \right\}\\ {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} {\kern 1pt} \times {\kern 1pt} \left\{ {\textrm{Erf}\left[ {\sqrt 2 \pi {\delta_1}\left( {{\textrm{K}_y} + \frac{1}{{2{\delta_2}}}} \right)} \right] - \textrm{Erf}\left[ {\sqrt 2 \pi {\delta_1}\left( {{\textrm{K}_y} - \frac{1}{{2{\delta_2}}}} \right)} \right]} \right\}. \end{array}$$
Figure 6 illustrates the degree of potential correlation produced by the Fourier transform of Eq. (28), while Fig. 7 illustrates the normalized spectral intensity in the far-zone scattered field whose distribution obeys Eq. (30). In contrast to case 1, when we adjust the corresponding medium parameters, the profile of the normalized spectral intensity of the scattered field no longer maintains its original shape. The specific discussion is as follows: when the value of $k{\delta _1}$ is smaller than $k{\delta _2}$, the spectral distribution pattern displays a set of concentric circles that obeys the Gaussian distribution (see as Fig. 7(a)). When the value of $k{\delta _1}$ is larger than $k{\delta _2}$, the spectral distribution shows as obviously rectangular (see as Fig. 7(c)). In the case of Fig. 7(b) with $k{\delta _1} = k{\delta _2}$, the shape can be seen as an intermediary transformation from the Gaussian-shape to the rectangular-shape. This phenomenon can be explained as follows. It is well known that the spectral density of the scattered field is governed by the weight function $p({\boldsymbol {\mathrm {\nu}} } )$. In this case, the weight function $p({\boldsymbol {\mathrm {\nu}} } )$ is constructed by a convolution of Gaussian function and rectangular function, and the results of the convolution depend on the value of parameters which included in the two weight functions. Therefore, by properly choosing the values of parameters $k{\delta _1}$ and $k{\delta _2}$, one can obtain scattered field with various distributions, including the Gaussian distribution, the rectangular distribution, and the distribution between them.4. Conclusion
In summary, we have systematically modeled a series of random media that is capable of producing scattered field with stable or with tunable distributions. Specifically, we have discussed the relationship between the far-zone scattered field and the scattering media, whose weight function can take the form of convolution of the same type or of two different types, respectively. It is shown that, in the former case, the corresponding far-zone scattered field will exhibit a stable shape, but the profile size can be controlled by the correlation parameters. However, in the latter case, we obtained a group of far-zone scattered field patterns with a tunable shape whose profile depend not only on the type of weight functions, but also on the value of parameters in each weight function. By the method proposed in this manuscript, the far-zone scattered field with various distributions, such as rectangular, circular and some other patterns, are obtained. These results may have potential applications in the area of manipulation of the far-zone scattered field.
Funding
National Natural Science Foundation of China (11404231, 61475105, 61775152).
Acknowledgments
The authors are obliged to Dr. Ziyang Chen at Huaqiao University and Dr. Zhenfei Jiang at University of Missouri for theirs help in improving the English of this manuscript.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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