Generation and manipulation of various hollow arrays by light wave scattering

Zhongyuan Xie; Qihang Dai; Hange Wang; Xiaoling Ji; Tao Wang

doi:10.1364/OE.445139

1. Introduction

Due to its potential applications in areas such as medical diagnosis, remote sensing and diffraction tomography [1–3], light wave scattering has become an important method to explore the structural information of scattering media. Therefore, researchers have made a series of theoretical studies, which focused on the influence of structural characteristics of scattering medium on the optical properties of scattered field [4–9]. In 1986, Wolf found that the spectrum of partially spatially coherent light will change in the process of propagating in free space and explained the reason for this phenomenon [10]. After that, he extended this variation phenomenon to the weak scattering process and found that the far-field spectrum will change when poly-chromatic light waves were scattered by a random medium [11]. In the past few decades, great progress has been made in the study of weak scattering. On the one hand, the scatterer has been extended to a wider range of medium models [12–15], including particle scattered model [12], semi-soft boundary medium model [13], anisotropic medium model [14], etc. On the other hand, the incident light wave has been extended from the plane monochromatic wave to some more general light waves [16–19], including polychromatic plane wave [16], partially coherent plane-wave pulse beam [17], electromagnetic light wave [18], etc. Furthermore, the inverse problem of light wave scattering, namely reconstructing the structural characteristics of the scattering medium through the optical characteristics of the scattered field, is also attracted much attention [20–22]. For example, the information of scattering potential of random medium was determined by measurement of the scattered field [23–25].

Due to the fact that the density distribution of the far field is closely related to the structural characteristics of the light field correlation function in the source plane, new light sources can be designed by selecting different realizable correlation functions to obtain various far-field distributions [26–30]. Recently, this method has been generalized to producing array beams, for example, Gaussian schell-model arrays [31] and optical coherence grids [32]. By analogizing with the design of light source, Korotkova firstly designed some media to obtain the desired far-field distribution pattern in the scattered field [33,34]. After that, Li et al. proposed a method to describe a three-dimensional, non-uniformly correlated medium [35]. Ding et al. introduced a scattering medium model to produce the array distribution [36]. Zhu et al. designed a new layered random medium to produce scattered far field with annular coherent array distribution [37]. Pan et al. put forward a method to design new scattering medium by the convolution of two weight functions [38]. In this manuscript, we will propose a kind of random medium model to generate periodic array profile, and discuss the relationship between the properties of far-zone scattered field and the characteristics of scattering medium. Moreover, the possibility of manipulating the far-zone scattered field, including the lobe shape, the hollow size, and the distance between lobes will also be discussed by changing the structural parameters of the scattering medium.

2. Theory

In the weak scattering theory, the properties for a random medium can be described by its correlation function of scattering potentials, which is defined by [39]

(1)$${C_F}({{{{\mathbf r^{\prime}}}_1},{{{\mathbf r^{\prime}}}_2},\omega } )= \left\langle {{F^\ast }({{{\mathbf r^{\prime}}}_1},\omega )F({{{\mathbf r^{\prime}}}_2},\omega )} \right\rangle ,$$

where ${{\mathbf r^{\prime}}_1}$ and ${{\mathbf r^{\prime}}_2}$ are the three-dimensional position vectors of any point within the area of the scatterer, $\omega$ is the angular frequency, “*” represents the complex conjugate, and the angular brackets denote the average over the statistical ensemble of monochromatic realizations of the incident field. It is important to note that any genuine correlation function which makes sense of physics must satisfy the hermiticity and non-negative definiteness, and the non-negative condition is the most difficult to achieve. This problem was solved by expressing the cross-spectral density function as an integral form, which was discussed in detail by Gori [40], and a necessary and sufficient non-negative definiteness condition for a genuine cross-spectral density function was provided by Martínez-Herrero [41]. Recently, this method was introduced to the area of light wave scattering by korotkova [33], with a form of

(2)$${C_F}({{{{\mathbf r^{\prime}}}_1},{{{\mathbf r^{\prime}}}_2},\omega } )= \int {p({\mathbf v} )} H_0^ \ast ({{{{\mathbf r^{\prime}}}_1},{\mathbf v},\omega } ){H_0}({{{{\mathbf r^{\prime}}}_2},{\mathbf v},\omega } ){d^3}v,$$

where ${\mathbf v}$ is the three-dimensional vector, ${H_0}({{{{\mathbf r^{\prime}}}_{}},{\mathbf v},\omega } )$ is an arbitrary complex-valued kernel function that defines the type of scattering medium, and $p({\mathbf v})$ is an arbitrary non-negative function, i.e., weight function, which defines the profile of correlation function of the medium. For a Schell-model media, ${H_0}$ usually have a Fourier-like structure [40]

(3)$${H_0}({{\mathbf r^{\prime}},{\mathbf v},\omega } )= \tau ({{\mathbf r^{\prime}}} )\exp ({ - 2\pi i{\mathbf v} \cdot {\mathbf r^{\prime}}} ),$$

where $\tau ({\mathbf r^{\prime}})$ is a complex amplitude profile of random scattering potential. Upon substituting from Eq. (3) into Eq. (2), and then one can immediately get the Schell-model correlation function of scattering potential, which is expressed as follows

(4)$${C_F}({{{{\mathbf r^{\prime}}}_1},{{{\mathbf r^{\prime}}}_2},\omega } )= {\tau ^ \ast }({{{{\mathbf r^{\prime}}}_1}} )\tau ({{{{\mathbf r^{\prime}}}_2}} ){\mu _F}({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1},\omega } ),$$

where

(5)$${\mu _F}({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1},\omega } )= \int {p({\mathbf v} )} \exp [{ - 2\pi i{\mathbf v} \cdot ({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1}} )} ]{d^3}v.$$

In the family of Schell-model media, a kind of basic random media is known as locally homogeneous medium, which was introduced by Silverman in the literature [42]. In addition, a more generalized model, namely, the quasi-homogeneous model medium was discussed by Carter and Wolf by analogizing with the concept of quasi-homogeneous source [43]. In this model, $\tau ({\mathbf r^{\prime}})$ changes more slowly with position ${\mathbf r^{\prime}}$ than the normalized correlation coefficient ${\mu _F}({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1},\omega } )$ changes with the position difference ${{\mathbf r^{\prime}}_2} - {{\mathbf r^{\prime}}_1}$. So that, its scattering potential strength ${I_F}({\mathbf r^{\prime}},\omega )$ almost remains constant, while the model of ${\mu _F}({{{{\mathbf r^{\prime}}}_{}},\omega } )$ varies greatly. In this case, the correlation function of scattering potential takes on the form [44]

(6)$${C_F}({{{{\mathbf r^{\prime}}}_1},{{{\mathbf r^{\prime}}}_2},\omega } )= {\tau ^2}\left( {\frac{{{{{\mathbf r^{\prime}}}_1} + {{{\mathbf r^{\prime}}}_2}}}{2},\omega } \right){\mu _F}({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1},\omega } ).$$

Within the accuracy of the first-order Born approximation and the far field approximation, the spectral density of the far-zone scattered field from a QH medium can be expressed as [39]

(7)$${S^{(s )}}({r{\mathbf s},\omega } )= \frac{{{S^{(i )}}(\omega )}}{{{r^2}}}{\tilde{I}_F}({0,\omega } ){\tilde{\mu }_F}[{k({{\mathbf s} - {{\mathbf s}_0}} ),\omega } ],$$

where ${{\mathbf s}_0}$ and ${\mathbf s}$ are unit vectors, which represent the direction of incident field and the direction of scattered field respectively, and

(8)$${\tilde{I}_F}({0,\omega } )= \int_D {{\tau ^2}} ({{\mathbf r^{\prime}}} ){d^3}r^{\prime}$$

is a measure of the average strength of the scattering potential at a single position ${\mathbf r^{\prime}}$, and

(9)$${\tilde{\mu }_F}[{k({{\mathbf s} - {{\mathbf s}_0}} ),\omega } ]= \int_D {{\mu _F}({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1},\omega } )\exp [{ - ik({{\mathbf s} - {{\mathbf s}_0}} )\cdot ({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1}} )} ]} {d^3}{r^{\prime}_1}{d^3}{r^{\prime}_2}$$

is the six-dimensional spatial Fourier transform of degree of potential correlation, and $k = \omega /c$, with $\omega$ being the frequency of the incident light wave and c being the speed of light.

The far-zone scattered spectral density is solely controlled by the shape of normalized correlation coefficient, while the density of the scattering potential only plays the role of proportionality factor, which is known as reciprocity relations [45]. Therefore, $\tau ({{\mathbf r^{\prime}},\omega } )$ can be chosen at will, and we assume it to be Gaussian [36]

(10)$$\tau ({{\mathbf r^{\prime}},\omega } )= A\exp \left( { - \frac{{{{{\mathbf r^{\prime}}}^2}}}{{4{\sigma^2}}}} \right),$$

where A is a constant, and $\sigma$ denotes effective width. On substituting from Eq. (10) into Eq. (8), one can find that

(11)$${\tilde{I}_F}({0,\omega } )= A{({2\pi } )^{\frac{3}{2}}}{\sigma ^3}.$$

Upon substituting from Eq. (11) into Eq. (7), we can obtain the following expression for scattered spectral density as

(12)$${S^{(s )}}({r{\mathbf s},\omega } )= \frac{{{S^{(i )}}(\omega )}}{{{r^2}}}A{({2\pi } )^{\frac{3}{2}}}{\sigma ^3}{\tilde{\mu }_F}[{k({{\mathbf s} - {{\mathbf s}_0}} ),\omega } ].$$

If the scattering medium takes the layer structure, the shape of normalized correlation coefficient along the scattering axis (z-axis) does not contribute to the scattered density distribution. In this case, the correlation function can be rewritten as [44]

(13)$${\mu _F}({{{{\mathbf r^{\prime}}}_2} - {{{\mathbf r^{\prime}}}_1},\omega } )= {\mu _F}({{{{\boldsymbol{\mathrm{\rho}}^{\prime}}}_2} - {{{\boldsymbol{\mathrm{\rho}}^{\prime}}}_1},\omega } )\delta ({{{z^{\prime}}_2} - {z_1}^\prime } ),$$

where $\delta$ denotes Dirac delta function, and ${\mu _F}({{{{\boldsymbol{\mathrm{\rho}}^{\prime}}}_2} - {{{\boldsymbol{\mathrm{\rho}}^{\prime}}}_1},\omega } )$ is the two-dimensional correlation function with ${\boldsymbol{\mathrm{\rho}} ^{\prime}}$ being any position vector in the x-y plane in the scattering region, which can be expressed as

(14)$${\mu _F}({{{{\boldsymbol{\mathrm{\rho}}^{\prime}}}_2} - {{{\boldsymbol{\mathrm{\rho}}^{\prime}}}_1},\omega } )\textrm{ = }\int {p({{{\mathbf v}_\rho }} )} \exp [{ - 2\pi i{{\mathbf v}_\rho } \cdot ({{{{\boldsymbol{\mathrm{\rho}}^{\prime}}}_2} - {{{\boldsymbol{\mathrm{\rho}}^{\prime}}}_1}} )} ]{d^2}{v_\rho }.$$

On substituting from Eq. (14) and Eq. (13) into Eq. (9), we will get the Fourier transform of the normalized correlation coefficients of the two-dimensional components of the transform vector ${\mathbf K}$, i.e.

(15)$${\tilde{\mu }_F}[{{{\mathbf {\rm K}}_\rho },\omega } ]= \int_D {{\mu _F}({{{{\boldsymbol{\mathrm{\rho}}^{\prime}}}_2} - {{{\boldsymbol{\mathrm{\rho}}^{\prime}}}_1},\omega } )\exp [{ - i{{\mathbf K}_\rho }({{{{\boldsymbol{\mathrm{\rho}}^{\prime}}}_2} - {{{\boldsymbol{\mathrm{\rho}}^{\prime}}}_1}} )} ]} {d^2}{\rho ^{\prime}_1}{d^2}{\rho ^{\prime}_2},$$

where ${{\mathbf K}_\rho } = [{k({{s_x} - {s_{0x}}} ),k({{s_y} - {s_{0y}}} )} ]$. On substituting from Eq. (15) into Eq. (12) and after some calculations, the spectral density of far field can be obtained, with the following form

(16)$${S^{(s )}}({r{{\mathbf s}_\rho },\omega } )= \frac{{{S^{(i )}}(\omega )}}{{{r^2}}}A{({2\pi } )^{\frac{3}{2}}}{\sigma ^3}{\tilde{\mu }_F}({{{\mathbf K}_\rho },\omega } ).$$

3. Linear combination of weight functions with the same types

In this section, two special far-field distributions, i.e., circular hollow array and rectangular hollow array, will be obtained through linear combination of weight functions with the same types. Some numerical results will also be presented to illustrate the dependence of far-zone distributions on the characteristics of the scattering medium.

3.1 Scattered field with circular hollow lobe array distribution

To obtain a scattered field with $N \times M$ multi-Gaussian array profile, the spectral degree of coherence should be chosen as [46]

(17)$$\begin{array}{c} {\mu _{ct}}({\rho _2} - {\rho _1}) = \frac{1}{{{C_{0t}}NM}}\sum\limits_{l = 1}^{{L_t}} {\frac{{{{( - 1)}^{l - 1}}}}{l}} \left( \begin{array}{l} {L_t}\\ l \end{array} \right)\exp \left[ { - \frac{{{{({x_2} - {x_1})}^2}}}{{2l\delta_x^2}} - \frac{{{{({y_2} - {y_1})}^2}}}{{2l\delta_y^2}}} \right]\\ \times \sum\limits_{n ={-} P}^P {\cos \left[ {\frac{{2\pi n({x_2} - {x_1}){R_x}}}{{{\delta_x}}}} \right]} \sum\limits_{m ={-} Q}^Q {\cos \left[ {\frac{{2\pi m({y_2} - {y_1}){R_y}}}{{{\delta_y}}}} \right]} , \end{array}$$

where $t = 1,2$, ${C_{0t}}\textrm{ = }\sum\limits_{l = 1}^{{L_t}} {\frac{{{{( - 1)}^{l - 1}}}}{l}} \left( \begin{array}{l} {L_t}\\ l \end{array} \right)$ is the normalization factor, $\left( \begin{array}{l} {L_t}\\ l \end{array} \right)$ is binomial coefficients, ${\delta _x}$ and ${\delta _y}$ are the effective correlation widths, ${R_x}$ and ${R_y}$ are coherence parameters, $P = {{(N - 1)} / 2}$, and $Q = {{(M - 1)} / 2}$. By taking the Fourier-transform of Eq. (17), one can find the corresponding weight function, with a form of

(18)$$\begin{array}{c} {p_{ct}}({{\mathbf v}_\rho }) = \frac{{\delta _x^2\delta _y^2}}{{{C_{0t}}NM}}\sum\limits_{l = 1}^{{L_t}} {{{( - 1)}^{l - 1}}} \left( \begin{array}{l} {L_t}\\ l \end{array} \right)\exp \left( { - \frac{{l\delta_x^2v_x^2}}{2} - \frac{{l\delta_y^2v_y^2}}{2}} \right)\\ \times \sum\limits_{n ={-} P}^P {\cosh (2l{\delta _x}\pi n{R_x}{v_x})} \exp ({ - 2l{\pi^2}{n^2}R_x^2} )\\ \times \sum\limits_{m ={-} Q}^Q {\cosh (2l{\delta _y}\pi m{R_y}{v_y})} \exp ({ - 2l{\pi^2}{m^2}R_y^2} ), \end{array}$$

where $\cosh ({\cdot} )$ is the hyperbolic cosine function.

In order to generate a $N \times M$ hollow multi-Gaussian array in the far field, ${p_C}({{\mathbf v}_\rho })$ makes the following transformation

(19)$${p_C}({{\mathbf v}_\rho }) = {p_c}_1({{\mathbf v}_\rho }) - {p_{c2}}({{\mathbf v}_\rho }).$$

As shown in Eq. (19), Although ${p_{c1}}({{\mathbf v}_\rho }) \ge 0$ and ${p_{c2}}({{\mathbf v}_\rho }) \ge 0$, we have to make sure that the function $P({\mathbf v})$ is always non-negative for any values of the 2D vector ${{\mathbf v}_\rho }$, so we set the parameter ${L_1} > {L_2}$ in the following discussion. On substituting from Eq. (19) together with Eq. (18) into Eq. (14) then into Eq. (15), further substituting the corresponding result into Eq. (16), one can find the far-zone scattered spectral density as follows

(20)$$S_C^{(s)}(r{{\mathbf s}_\rho },\omega )\textrm{ = }S_{c1}^{(s)}(r{{\mathbf s}_\rho },\omega ) - S_{c2}^{(s)}(r{{\mathbf s}_\rho },\omega ),$$

where

(21)$$\begin{aligned}{S_{ct}}^{(s)}(r{{\mathbf s}_\rho },\omega ) &= \frac{{A{{(2\pi )}^{\frac{9}{2}}}{\sigma ^3}{S^{(i)}}(\omega )\delta _x^2\delta _y^2}}{{{r^2}{C_{0t}}NM}}\sum\limits_{l = 1}^{{L_t}} {{{( - 1)}^{l - 1}}} \left( \begin{array}{l} {L_t}\\ l \end{array} \right)\\ &\times \exp \left[ { - \frac{{{k^2}{{({s_x} - {s_{x0}})}^2}\delta_x^2l}}{2} - \frac{{ - {k^2}{{({s_y} - {s_{y0}})}^2}\delta_y^2l}}{2}} \right]\\ &\times \sum\limits_{n ={-} P}^P {\cosh [{2k({s_x} - {s_{x0}})n\pi {R_x}l{\delta_x}} ]} \exp ( - 2{n^2}{\pi ^2}R_x^2l)\\ &\times \sum\limits_{m ={-} Q}^Q {\cosh [{2k({s_y} - {s_{y0}})m\pi {R_y}l{\delta_y}} ]} \exp ( - 2{m^2}{\pi ^2}R_y^2l). \end{aligned}$$

In Fig. 1, the influence of parameters M and N on the far-field spectral density distribution is discussed. It follows from Fig. 1(a) and Fig. 1(b) that the number of rows will decrease when the value of parameter M decreases. Moreover, one can find from Fig. 1(a) and Fig. 1(c) that the number of columns will decrease as the value of parameter N decreases. Therefore, the number of rows and columns of the array can be manipulated by changing the values of parameters M and N. In Fig. 2, the influence of parameter ${L_2}$ on the size of circular hollow profile is discussed. It is shown that the hollow area of single lobe will increase with ${L_2}$ increasing gradually when keeps the value of ${L_1}$ unchanged. Therefore, the size of hollow area can be manipulated by changing the value of the parameter ${L_2}$.

Fig. 1. Influence of parameters M, N on the circular hollow lobes arrays of scattered spectral density. The parameters for calculations are ${L_1}\textrm{ = }5$, ${L_2}\textrm{ = }1$, ${R_x} = {R_y} = 1.1$, ${\delta _x} = {\delta _y} = 0.1mm$, and (a) $M = N = 3$; (b) $M = 1$, $N = 3$; (c) $M = 3$, $N = 1$.

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Fig. 2. Influence of parameter L on the circular hollow lobes arrays of scattered spectral density. The parameters for calculations are $M = N = 3$, ${R_x} = {R_y} = 1.1$, ${\delta _x} = {\delta _y} = 0.1mm$, and (a) ${L_1}\textrm{ = }5$, ${L_2}\textrm{ = }1$; (b) ${L_1}\textrm{ = }5$, ${L_2}\textrm{ = }2$; (c) ${L_1}\textrm{ = }5$, ${L_2}\textrm{ = }3$.

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In Fig. 3, the influence of parameters ${\delta _x}$ and ${\delta _y}$ on the far-field spectral density distribution is discussed. The numerical result shows that the value of $\delta$ will affect the shape of each lobe. Specifically, when the value of parameter ${\delta _x}$ is equal to the value of parameter ${\delta _y}$, as shown in Fig. 3(a), each lobe in the periodic array is isotropic. However, as shown in Fig. 3(b), when the value of parameter ${\delta _x}$ is smaller than the value of parameter ${\delta _y}$, each lobe in the periodic array is anisotropic and its long axis is in the x direction. As shown in Fig. 3(c), when the value of parameter ${\delta _x}$ is larger than the value of parameter ${\delta _y}$, each lobe in the periodic array is anisotropic and its long axis is in the y direction. Therefore, one can conclude that the shape and direction of each lobe can be manipulated by adjusting the value of $\delta$, and the long axis of each lobe is located in the direction with smaller effective correlation length. In Fig. 4, the dependence of the distance between adjacent lobes on the parameter R is discussed. As shown in Fig. 4(a), when setting ${R_x} = {R_y}$, the distance of adjacent lobes is equal in the x direction and in the y direction. However, as shown in Fig. 4(b), when the value of parameter ${R_x}$ is larger than the value of parameter ${R_y}$, the distance of adjacent lobes is larger in the x direction than in the y direction. When the value of parameter ${R_x}$ is smaller than the value of parameter ${R_y}$, the distance of adjacent lobes is smaller in the x direction than in the y direction, and the corresponding result is displayed in Fig. 4(c).

Fig. 3. Influence of parameter $\delta$ on the circular hollow lobes arrays of scattered spectral density. The parameters for calculations are $M = N = 3$, ${L_1}\textrm{ = }5$, ${L_2}\textrm{ = }1$, ${R_x} = {R_y} = 1.1$, and (a) ${\delta _x} = {\delta _y} = 0.1mm$; (b) ${\delta _x} = 0.07mm$, ${\delta _y} = 0.1mm$; (c) ${\delta _x} = 0.1mm$, ${\delta _y} = 0.07mm$.

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Fig. 4. Influence of parameter R on the circular hollow lobes arrays of scattered spectral density. The parameters for calculations are $M = N = 3$, ${L_1}\textrm{ = }5$, ${L_2}\textrm{ = }1$, ${\delta _x} = {\delta _y} = 0.1mm$, and (a) ${R_x} = {R_y} = 1.1$; (b) ${R_x} = 1.4$, ${R_y} = 1.1$; (c) ${R_x} = 1.1$, ${R_y} = 1.4$.

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3. Scattered field with rectangular hollow lobe array distribution

In the above section, the far-field spectral density distribution demonstrates a circular hollow array. Here, let us further consider another case where a hollow array of lobe with a Cartesian symmetrical distribution will be obtained. In this case, we let ${p_R}({{\mathbf v}_\rho })$ take the following form

(22)$${p_R}({{\mathbf v}_\rho }) = {p_r}_1({{\mathbf v}_\rho }) - {p_{r2}}({{\mathbf v}_\rho }),$$

where

(23)$$\begin{aligned} {p_{rt}}({{\mathbf v}_\rho }) &= \frac{{\delta _x^2\delta _y^2}}{{{C_{0t}}NM}}\sum\limits_{l = 1}^{{L_t}} {\frac{{{{( - 1)}^{l - 1}}}}{{\sqrt l }}\frac{1}{{\sqrt {\frac{1}{l}} }}} \left( \begin{array}{l} {L_t}\\ l \end{array} \right)\exp \left( { - \frac{{l\delta_x^2v_x^2}}{2}} \right)\\ &\times \sum\limits_{l = 1}^{{L_t}} {\frac{{{{( - 1)}^{l - 1}}}}{{\sqrt l }}} \frac{1}{{\sqrt {\frac{1}{l}} }}\left( \begin{array}{l} {L_t}\\l \end{array} \right)\exp \left( { - \frac{{l\delta_y^2v_y^2}}{2}} \right)\\ &\times \sum\limits_{n ={-} P}^P {\cosh (2l{\delta _x}\pi n{R_x}{v_x})} \exp ({ - 2l{\pi^2}{n^2}R_x^2} )\\ &\times \sum\limits_{m ={-} Q}^Q {\cosh (2l{\delta _y}\pi m{R_y}{v_y})} \exp ({ - 2l{\pi^2}{m^2}R_y^2} ). \end{aligned}$$

On substituting from Eq. (23) together with Eq. (22) first into Eq. (14) and into Eq. (15), further substituting the corresponding result into Eq. (16), and one can find the rectangular far-field spectral density distribution as follows

(24)$$S_R^{(s)}(r{{\mathbf s}_\rho },\omega )\textrm{ = }S_{r1}^{(s)}(r{{\mathbf s}_\rho },\omega ) - S_{r2}^{(s)}(r{{\mathbf s}_\rho },\omega ),$$

where

(25)$$\begin{aligned} {S_{rt}}^{(s)}(r{{\mathbf s}_\rho },\omega ) &= \frac{{A{{(2\pi )}^{\frac{9}{2}}}{\sigma ^3}{S^{(i)}}(\omega )\delta _x^2\delta _y^2}}{{{r^2}{C_{0t}}NM}}\\& \times \sum\limits_{l = 1}^{{L_t}} {\frac{{{{( - 1)}^{l - 1}}}}{{\sqrt l }}\frac{1}{{\sqrt {\frac{1}{l}} }}} \left( \begin{array}{l} {L_t}\\ l \end{array} \right)\exp \left[ { - \frac{{{k^2}{{({s_x} - {s_{x0}})}^2}\delta_x^2l}}{2}} \right]\\& \times \sum\limits_{l = 1}^{{L_t}} {\frac{{{{( - 1)}^{l - 1}}}}{{\sqrt l }}\frac{1}{{\sqrt {\frac{1}{l}} }}} \left( \begin{array}{l} {L_t}\\ l \end{array} \right)\exp \left[ { - \frac{{ - {k^2}{{({s_y} - {s_{y0}})}^2}\delta_y^2l}}{2}} \right]\\& \times \sum\limits_{n ={-} P}^P {\cosh [{2k({s_x} - {s_{x0}})n\pi {R_x}l{\delta_x}} ]} \exp ( - 2{n^2}{\pi ^2}R_x^2l)\\& \times \sum\limits_{m ={-} Q}^Q {\cosh [{2k({s_y} - {s_{y0}})m\pi {R_y}l{\delta_y}} ]} \exp ( - 2{m^2}{\pi ^2}R_y^2l). \end{aligned}$$

Figure 5 illustrates the influence of parameter ${L_2}$ on the far-field spectral density distribution. Result shows that when the value of ${L_1}$ is fixed, the size of the central hollow of lobes will increase with the value of parameter ${L_2}$ increases. In addition, as the value of ${L_2}$ changes, the shape of the center will gradually transition from circular to rectangular. Figure 6 illustrates the influence of parameter $\delta$ on the far-field spectral density distribution. As shown in Fig. 6(a), we can clearly see that each lobe profile is isotropic when the effective correlation lengths, i.e., ${\delta _x}$ and ${\delta _y}$ take the same value. However, as shown in Fig. 6(b) and Fig. 6(c), when the parameters ${\delta _x}$ and ${\delta _y}$ take different values, each lobe profile in the array will demonstrate anisotropy.

Fig. 5. Influence of parameter L on the rectangular hollow lobes arrays of scattered spectral density. The parameters for calculations are $M = N = 3$, ${R_x} = {R_y} = 0.93$, ${\delta _x} = {\delta _y} = 0.1mm$, and (a) ${L_1}\textrm{ = }5$, ${L_2}\textrm{ = }1$; (b) ${L_1}\textrm{ = }5$, ${L_2}\textrm{ = }2$; (c) ${L_1}\textrm{ = }5$, ${L_2}\textrm{ = }4$.

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Fig. 6. Influence of parameter $\delta$ on the rectangular hollow lobes arrays of scattered spectral density. The parameters for calculations are $M = N = 3$, ${L_1}\textrm{ = }5$, ${L_2}\textrm{ = }4$, ${R_x}\textrm{ = }{R_y}\textrm{ = }0.93$, and (a) ${\delta _x}\textrm{ = }{\delta _y}\textrm{ = }0.1mm$; (b) ${\delta _x}\textrm{ = }0.1mm$, ${\delta _y}\textrm{ = }0.07mm$; (c) ${\delta _x}\textrm{ = }0.07mm$, ${\delta _y}\textrm{ = }0.1mm$.

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Figure 7 presents the influence of parameter R on the far-field spectral density distribution. We can clearly see that the distance between adjacent lobes in the array can be manipulated by the value of parameter R. As shown in Fig. 7(a), when the parameters ${R_x}$ and ${R_y}$ take same value, the distance between adjacent lobes is equal in any direction. Moreover, it can be clearly seen from Fig. 7(b) and Fig. 7(c) that when the value of parameter ${R_x}$ is larger than the value of parameter ${R_y}$, the distance of adjacent lobes along the x direction is larger than that along the y direction. On the contrary, when the value of parameter ${R_x}$ is smaller than the value of parameter ${R_y}$, the distance of adjacent lobes along the x direction is smaller than that along the y direction.

Fig. 7. Influence of parameter R on the rectangular hollow lobes arrays of scattered spectral density. The parameters for calculations are $M = N = 3$, ${L_1}\textrm{ = }5$, ${L_2}\textrm{ = }4$, ${\delta _x}\textrm{ = }{\delta _y}\textrm{ = }0.1mm$, and (a) ${R_x}\textrm{ = }{R_y}\textrm{ = }0.93$; (b) ${R_x}\textrm{ = 1}\textrm{.13}$, ${R_y}\textrm{ = }0.93$; (c) ${R_x}\textrm{ = 0}\textrm{.93}$, ${R_y}\textrm{ = }1.13$.

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4. Linear combination of weight functions with different types

In the following, a scattered far field with array lobe distribution may be obtained by introducing the linear superposition of different type of weight function. The superposition of different weight functions makes scattering medium to produce different scattered far fields. In this case, $p({{\mathbf v}_\rho })$ has the following form

(26)$$p({{\mathbf v}_\rho }) = a{p_c}_1({{\mathbf v}_\rho }) + b{p_{r2}}({{\mathbf v}_\rho }),$$

where a and b are the weight coefficients. After some calculations, we get the spectral density distribution of the scattered far field as

(27)$$S_{CR}^{(s)}(r{{\mathbf s}_\rho },\omega )\textrm{ = a}S_{c1}^{(s)}(r{{\mathbf s}_\rho },\omega ) + bS_{r2}^{(s)}(r{{\mathbf s}_\rho },\omega ),$$

where

(28)$$\begin{aligned} {S_{c1}}^{(s)}(r{{\mathbf s}_\rho },\omega ) &= \frac{{A{{(2\pi )}^{\frac{9}{2}}}{\sigma ^3}{S^{(i)}}(\omega )\delta _x^2\delta _y^2}}{{{r^2}{C_{01}}NM}}\sum\limits_{l = 1}^{{L_1}} {{{( - 1)}^{l - 1}}} \left( \begin{array}{l} {L_1}\\ l \end{array} \right)\\& \times \exp \left[ { - \frac{{{k^2}{{({s_x} - {s_{x0}})}^2}\delta_x^2l}}{2} - \frac{{ - {k^2}{{({s_y} - {s_{y0}})}^2}\delta_y^2l}}{2}} \right]\\& \times \sum\limits_{n ={-} P}^P {\cosh [{2k({s_x} - {s_{x0}})n\pi {R_x}l{\delta_x}} ]} \exp ( - 2{n^2}{\pi ^2}R_x^2l)\\& \times \sum\limits_{m ={-} Q}^Q {\cosh [{2k({s_y} - {s_{y0}})m\pi {R_y}l{\delta_y}} ]} \exp ( - 2{m^2}{\pi ^2}R_y^2l) \end{aligned}$$

and

(29)$$\begin{aligned} {S_{r2}}^{(s)}(r{{\mathbf s}_\rho },\omega ) &= \frac{{A{{(2\pi )}^{\frac{9}{2}}}{\sigma ^3}{S^{(i)}}(\omega )\delta _x^2\delta _y^2}}{{{r^2}{C_{02}}NM}}\\& \times \sum\limits_{l = 1}^{{L_2}} {\frac{{{{( - 1)}^{l - 1}}}}{{\sqrt l }}\frac{1}{{\sqrt {\frac{1}{l}} }}} \left( \begin{array}{l} {L_2}\\ l \end{array} \right)\exp \left[ { - \frac{{{k^2}{{({s_x} - {s_{x0}})}^2}\delta_x^2l}}{2}} \right]\\& \times \sum\limits_{l = 1}^{{L_2}} {\frac{{{{( - 1)}^{l - 1}}}}{{\sqrt l }}\frac{1}{{\sqrt {\frac{1}{l}} }}} \left( \begin{array}{l} {L_2}\\ l \end{array} \right)\exp \left[ { - \frac{{ - {k^2}{{({s_y} - {s_{y0}})}^2}\delta_y^2l}}{2}} \right]\\& \times \sum\limits_{n ={-} P}^P {\cosh [{2k({s_x} - {s_{x0}})n\pi {R_x}l{\delta_x}} ]} \exp ( - 2{n^2}{\pi ^2}R_x^2l)\\& \times \sum\limits_{m ={-} Q}^Q {\cosh [{2k({s_y} - {s_{y0}})m\pi {R_y}l{\delta_y}} ]} \exp ( - 2{m^2}{\pi ^2}R_y^2l). \end{aligned}$$

As shown in Fig. 8, through the linear combination of two different types of weight functions, one can get more complex far-field distribution, i.e., the array scattered field which is composed of rectangular lobes with circular hollow and the array scattered field which is composed of circular lobes with rectangular hollow. Specifically, when the parameters are chosen as $a = 1$, $b ={-} 1$ and ${L_1} > {L_2}$, as shown in Fig. 8(a), the external shape of lobes depends on ${p_c}_1$ and the inner shape of lobes depends on ${p_r}_2$. In this case, a lobe shape of a rectangular hole inside a circle which is like Chinese ancient copper coins. When the parameters are chosen as $a ={-} 1$, $b = 1$ and ${L_2} > {L_1}$, as shown in Fig. 8(b), the external shape of lobes depends on ${p_r}_2$ and the inner shape of lobes depends on ${p_c}_1$. In this case, the lobe shape will demonstrate as a rectangular profile with a circular hollow distribution.

Fig. 8. The spectral density produced by superposition of weight functions with different types. The parameters for calculations are $M = N = 3$, ${R_x} = {R_y} = 1.2$, ${\delta _x} = {\delta _y} = 0.05mm$, and (a) $a = 1$, $b ={-} 1$, ${L_1} = 6$, ${L_2} = 2$; (b) $a ={-} 1$, $b = 1$, ${L_1} = 2$, ${L_2} = 6$.

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5. Conclusion

In summary, we have discussed a strategy of designing a series of novel random mediums, which can produce scattered field with hollow array distribution, including circular hollow lobes, rectangular hollow lobes, and their combinations. The possibility of manipulating the far field distributions, such as the shape of the lobes, the hollow size of the lobes, and the distance between the lobes, was discussed by adjusting the parameters of scattering medium. It is shown that by properly adjusting the structural parameters of scattering medium, various desired patterns with flexibility manipulating far-zone field can be obtained in the scattered field. This phenomenon may provide a method to produce some special patterns behind the scattering medium, which may have potential applications in areas such as biomedical imaging, medical diagnosis, and light wave manipulation, where the scattering process always exists.

Funding

National Natural Science Foundation of China (11404231, 61775152).

Disclosures

The authors declare that there are no conflicts of interest related to this article.

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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Generation and manipulation of various hollow arrays by light wave scattering

Abstract

1. Introduction

2. Theory

3. Linear combination of weight functions with the same types

3.1 Scattered field with circular hollow lobe array distribution

3. Scattered field with rectangular hollow lobe array distribution

4. Linear combination of weight functions with different types

5. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Equations (29)

Optics Express