Random medium model for producing optical coherence lattice

Yi Ding; Daomu Zhao

doi:10.1364/OE.25.025222

1. Introduction

Due to the potential applications in remote sensing, diffraction tomography, medical diagnosis and so on, the weak scattering theory is a subject of considerable importance [1–7]. During the past three decades, numerous papers have been published to discuss the statistical properties of the scattered field within the accuracy of the first-order Born approximation and the far field approximation for various types of incident waves and the scatterers (for a review of this research, please see [8]). Recently, some important progresses have been made on this theory. For example, the scatterers have been generalized from a continuous medium to a particulate medium [9,10]. The incident light waves have been generalized from plane wave to more common light, including stochastic electromagnetic wave [11–13], partially coherence light [14], and plane wave pulse [15]. These works have greatly enriched the weak scattering theory.

With the rapid development of the manufacturing technology, it is possible to manufacture production with the spatial resolution on the order of the several microns. Not long ago, designing a random medium with prescribed correlation functions to produce the expected scattered intensity patterns has been proposed by Korotkova [16]. Subsequently, this topic has attracted many researchers’ attention [17–19]. It is shown that many prescribed scattered intensity patterns, such as, the circular flat, ring-like, squared, rectangular and so son, can be produced by designing the correlation functions of scattering potential of a random medium. Meanwhile, it is also indicated that a large number of random light source models can be applied to design medium models. Furthermore, another effective method for designing the scattering medium generating specified optical beam patterns based on the deterministic mode representation has also been introduced by Li and Korotkova [20]. It is shown that novel correlation functions of scattering potential can be expressed as an infinite sum of weighted eigenmodes, and the major advantage of this approach is its remarkable tractability in dealing with scattering calculations on random media. On the other hand, because of the important applications in optical trapping, material processing, and atmospheric communications, the optical coherence lattice has recently stimulated substantial interests in the partially coherence beam domain, including the generation and propagation of the optical coherence lattice [21,22].

In this paper, we will design a novel class of random media to scatter light producing optical coherence lattice in the far field. Besides, we will also introduce an alternatively new method, i.e., convolution of degree of potential correlation, to design random media producing controllable scattered intensity patterns. Finally, we will demonstrate that how convolution can be used for generation of a scattered field being a modulated version of another one by designing a novel family of random media.

2. Theory

In the weak potential scattering theory, the correlation function of scattering potential of a random media can be defined as the second moment of scattering potential $F (r^{'}, ω)$ , which is defined as [23]

C_{F} ({r^{'}}_{1}, {r^{'}}_{2}; ω) = 〈 F^{*} ({r^{'}}_{1}, ω) F ({r^{'}}_{2}, ω) 〉 .

Just like any genuine correlation function,

C_{F} ({r^{'}}_{1}, {r^{'}}_{2}; ω)

must obey hermiticity and non-negative definiteness conditions. Namely, the following the integral representation for

C_{F} ({r^{'}}_{1}, {r^{'}}_{2}; ω)

must be met [24]

C_{F} ({r^{'}}_{1}, {r^{'}}_{2}; ω) = \int H_{0}^{*} ({r^{'}}_{1}, u, ω) H_{0} ({r^{'}}_{2}, u, ω) p (u, ω) d^{3} u,

where

u

is a 3D vector,

H_{0} (r^{'}, u, ω)

is an arbitrary kernel function whose choice defines the correlation class of a scattering medium, and

p (u, ω)

is an arbitrary non-negative function whose choice defines the profile of correlation function of the medium. For the Schell-model media,

H_{0}

has the Fourier-like structure [18]

H_{0} (r^{'}, u; ω) = τ (r^{'}) \exp (- i r^{'} \cdot u),

where

τ (r^{'})

is an amplitude function of random scattering potential. On substituting from Eq. (3) into Eq. (2), then the Schell-model correlation function of scattering potential takes on the form [18]

C_{F} ({r^{'}}_{1}, {r^{'}}_{2}; ω) = τ^{*} ({r^{'}}_{1}) τ ({r^{'}}_{2}) μ_{F} ({r^{'}}_{2} - {r^{'}}_{1}, ω),

where

μ_{F} ({r^{'}}_{2} - {r^{'}}_{1}, ω) = \int_{D} p (u, ω) \exp [- i u \cdot ({r^{'}}_{2} - {r^{'}}_{1})] d^{3} u .

In Schell-model media, an important class of random media is known as quasi-homogeneous (QH) medium whose correlation function of scattering potential is approximately described by the following form [18]

C_{F} ({r^{'}}_{1}, {r^{'}}_{2}; ω) = τ^{2} (\frac{{r^{'}}_{1} + {r^{'}}_{2}}{2}, ω) μ_{F} ({r^{'}}_{2} - {r^{'}}_{1}, ω) .

Within the accuracy of the first-order Born approximation and the far field approximation, the scattered spectral density of light waves on scattering from a QH medium is expressed as [23]

S^{(s)} (r s, ω) = \frac{1}{r^{2}} S^{(i)} (ω) {\tilde{I}}_{F} (0, ω) {\tilde{μ}}_{F} [k (s - s_{0}), ω],

where

{\tilde{μ}}_{F} [k (s - s_{0}), ω] = \int_{D} μ_{F} ({r^{'}}_{2} - {r^{'}}_{1}, ω) \exp [- i k (s - s_{0}) \cdot ({r^{'}}_{2} - {r^{'}}_{1})] d^{3} {r^{'}}_{1} d^{3} {r^{'}}_{2}

is the 3D Fourier transform of degree of potential correlation, and

{\tilde{I}}_{F} (0, ω) = \int_{D} τ^{2} (r^{'}) d^{3} r^{'}

is a measure of the average strength of the scattering potential at a single position

r^{'}

.

Equation (7) implies that the intensity distribution of the scattered field is solely governed by normalized correlation coefficient of scattering potential of the medium, while the potential strength of scattering potential of the medium only plays a coefficient role, which is known as reciprocity relations [25]. Therefore, $τ (r^{'})$ can be chosen at will, and we set it to be Gaussian

τ_{F} (r^{'}, ω) = A \exp (- \frac{{r^{'}}^{2}}{4 σ^{2}}),

where

A

is a constant, and

σ

is effective width of potential strength. On substituting from Eq. (10) into Eq. (9), after some calculations, one can obtain

{\tilde{I}}_{F} (0, ω) = A {(2 π)}^{\frac{3}{2}} σ^{3} .

Markov approximation is firstly introduced into weak potential scattering theory by Korotkova [17]. The approximation indicates that the correlation function along the scattering axis (z-axis) does not contribute to the scattered intensity distribution, namely, the degree of potential correlation can be expressed as [18]

μ_{F} ({r^{'}}_{2} - {r^{'}}_{1}, ω) = μ_{F} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω) δ ({z^{'}}_{2} - {z^{'}}_{1}),

where

μ_{F} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω)

is 2D correlation function with

ρ^{'}

being position vector in x-y plane within the domain of the scatterer, and

δ (\cdot \cdot \cdot)

denotes 1D Dirac delta function. It should be mentioned that the Markov approximation is reasonable if the medium is synthesized layer by layer, i.e., the adjacent layers can be treated as statistically independent [17].

On substituting from Eq. (12) first into Eq. (8), then together with Eq. (11) into Eq. (7), one can find that

S^{(s)} (r s, ω) = \frac{1}{r^{2}} A {(2 π)}^{\frac{3}{2}} σ^{3} S^{(i)} (ω) {\tilde{μ}}_{F} [K_{ρ}, ω],

where

{\tilde{μ}}_{F} (K_{ρ}, ω) = \int_{D} μ_{F} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω) \exp [- i K_{ρ} \cdot ({ρ^{'}}_{2} - {ρ^{'}}_{1})] d^{2} {ρ^{'}}_{1} d^{2} {ρ^{'}}_{2}

with

K_{ρ} = (k (s_{x} - s_{0 x}), k (s_{y} - s_{0 y}))

being 2D transform vector.

3. Scattered field with Gaussian Schell-model arrays

Based on the expression (13) in Sec. 2, now let us explore the possibility of designing a random medium which can scatter light producing $N \times M$ optical coherence lattice. In this case, we introduce a novel class of the degree of potential correlation [21]

\begin{array}{l} μ_{F} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω) = \frac{1}{N \times M} \exp [- \frac{{({x^{'}}_{2} - {x^{'}}_{1})}^{2}}{2 μ_{x}^{2}}] \exp [- \frac{{({y^{'}}_{2} - {y^{'}}_{1})}^{2}}{2 μ_{y}^{2}}] \sum_{n = - P}^{P} \cos [\frac{2 π n ({x^{'}}_{2} - {x^{'}}_{1}) R_{x}}{μ_{x}}] \\ \times \sum_{m = - Q}^{Q} \cos [\frac{2 π m ({y^{'}}_{2} - {y^{'}}_{1}) R_{y}}{μ_{y}}], \end{array}

where

μ_{i}

and

R_{i}

(

i = x, y

) denote correlation length of scattering potential and positive real constant along

i

axis, respectively,

P = (N - 1) / 2

and

Q = (M - 1) / 2

.

Figure 1 shows the behavior of the degree of potential correlation given by Eq. (15) as a function of $k ρ_{d}$ , where $ρ_{d} = | {ρ^{'}}_{2} - {ρ^{'}}_{1} |$ for several values of $N \times M$ with $k μ_{x} = k μ_{y} = 10$ and $R_{x} = R_{y} = 1$ (the first row), for several values of $R_{i}$ with $k μ_{x} = k μ_{y} = 10$ and $M = 7, N = 6$ (the second row), and for several values of $k μ_{i}$ with $R_{x} = R_{y} = 1$ and $M = 7, N = 6$ (the third row).

Fig. 1 Degree of the scattering potential correlation varying with $k ρ_{d}$ along $x$ (left columns) and $y$ (right columns) directions, respectively. $k = 10^{7}$ , (a-b) $M = N = 1$ (blue curve), $M = 7$ , $N = 6$ (red curve), and $M = 11$ , $N = 10$ (green curve); (c-d) $R_{x} = R_{y} = 0.25$ (blue curve), $R_{x} = R_{y} = 0. 5$ (red curve), and $R_{x} = R_{y} = 1$ (green curve); (e-f) $k μ_{x} = k μ_{y} = 15$ (blue curve), $k μ_{x} = k μ_{y} = 10$ (red curve), and $k μ_{x} = k μ_{y} = 5$ (green curve).

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On substituting from Eq. (15) first into Eq. (14), after some long but straightforward calculations, and then into Eq. (13), we obtain the spectral density of the scattered field as

\begin{array}{l} S^{(s)} (r s, ω) = \frac{S^{(i)} (ω) A σ^{3} {(2 π)}^{3} μ_{x} μ_{y}}{2 M N r^{2}} \exp (- \frac{k^{2} μ_{x}^{2} s_{x}^{2}}{2}) \exp (- \frac{k^{2} μ_{y}^{2} s_{y}^{2}}{2}) \\ \times \sum_{n = - P}^{P} \cos h (2 π n k s_{x} μ_{x} R_{x}) \exp (- 2 π^{2} n^{2} R_{x}^{2}) \\ \times \sum_{n = - Q}^{Q} \cos h (2 π n k s_{y} μ_{y} R_{y}) \exp (- 2 π^{2} n^{2} R_{y}^{2}) . \end{array}

Figure 2 shows the far-field spectral density distribution of light waves on scattering from the medium given by Eq. (15) as the function of the two dimensional components of direction vector $s$ . It is shown that light waves on scattering from such a medium can produce optical coherence lattice pattern with the identical and equivalent interval Gaussian profile lobes, and the optical lattice is symmetric about $x$ axis and $y$ axis. It should be emphasized that the optical lattice in the far-zone scattered field is not in rectangular distribution, as which is evaluated with respect to $s_{x}$ and $s_{y}$ variables, implying in a spherical observation surface. Furthermore, as shown in Fig. 3, it is possible to obtain other types of optical array with elliptical Gaussian lobes and with different dimensions by assigning different correlation length along $x$ and $y$ directions. In some particular cases, the optical lattice will change into the flat-top intensity and circular distribution displayed in Fig. 4. The flat-top intensity can be formed on changing parameters $R_{i}$ , and Fig. 4(a) shows an example of forming flat-top spectral density with same parameters as shown in Fig. 2 expect for $R_{x} = R_{y} = 0.25$ . When setting $M = N = 1$ , the intensity distribution of the scattered field can also reduce to circular distribution, and the corresponding result is displayed in Fig. 4(b).

Fig. 2 Spectral density distribution in far-zone scattered field for $M = 4$ , $N = 6$ , $R_{x} = R_{y} = 1$ , and $k μ_{x} = k μ_{y} = 20$ . The other parameters are $A = 1$ , $k σ = 30$ .

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Fig. 3 Optical lattice patterns with different dimensions and elliptical lobes in far-zone scattered field. The parameters are $k μ_{x} = 20$ , $k μ_{y} = 10$ , $R_{x} = R_{y} = 1$ , (a) $M = 6$ , $N = 1$ , and (b) $M = 6$ , $N = 5$ .

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Fig. 4 Rectangular and circular intensity distribution patterns in far-zone scattered field. The parameters are $k μ_{x} = k μ_{y} = 20$ , $R_{x} = R_{y} = 0 .25$ , (a) $M = 4, N = 6$ , and (b) $M = N = 1$ .

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4. A new method for designing media

From Figs. 2 and 3, it is not difficult to find that the optical lattices given by the Eq. (16) whose lobes intensity values cannot be conveniently changed. In this section, we will introduce a new method to construct the correlation function of scattering potential of the medium, and based on this method, we can obtain the new array with variable lobes intensity values. Furthermore, we will also show that the convolution of degrees of potential correlation with polar and Cartesian symmetries can result in scattered fields with different shapes along the $x$ and $y$ directions.

A. Convolution of degree of potential correlation

Let us now assume that a new degree of potential correlation is a convolution of any two legitimate degrees of potential correlation $μ_{F 1}$ and $μ_{F 2}$ , i.e.,

μ_{c} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω) = μ_{F 1} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω) \otimes μ_{F 2} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω),

where

\otimes

denotes convolution operation. This expression can be regarded as the analog of the synthesis of random sources, which is recently proposed in [26]. Since the Fourier transform of the convolution of two functions is the product of their Fourier transforms, and then kernel

p_{c} (u_{ρ}, ω)

corresponding to the new degree of coherence

μ_{c}

can be expressed as the product of two functions,

p_{1} (u_{ρ}, ω)

and

p_{2} (u_{ρ}, ω)

, i.e.,

p_{c} (u_{ρ}, ω) = p_{1} (u_{ρ}, ω) p_{2} (u_{ρ}, ω),

where

p_{j} (u_{ρ}, ω) = \int \int_{- \infty}^{+ \infty} μ_{j} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω) \exp [- i u_{ρ} \cdot ({ρ^{'}}_{2} - {ρ^{'}}_{1})] d^{2} {ρ^{'}}_{1} d^{2} {ρ^{'}}_{2},

with

j = 1, 2

. For function

p

to be non-negative, we must require

p_{1} \geq 0

and

p_{2} \geq 0

for any values of vector

u_{ρ}

being 2D vector.

Then, on substituting from Eq. (17) first into Eq. (14), after manipulating Fourier transform, and then combining with Eqs. (18) and (19) into Eq. (13), we arrive at

S_{c}^{(s)} (r s, ω) = \frac{1}{r^{2}} S^{(i)} (ω) {\tilde{I}}_{F} (0, ω) p_{1} (- K_{ρ}) p_{2} (- K_{ρ}) .

Equation (20) is one of the most valuable results in this manuscript. It should be mentioned that Eq. (20) can be viewed as the expected scattered spectral density corresponding to

p_{1}

is modulated by

p_{2}

, and vice versa. Thus, we provide an alternatively new method to construct the correlation function of a random medium producing the desirable scattering intensity.

In the following, as an example, we will design a random medium producing the array with variable lobes intensity values. Here let $μ_{F 1} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω)$ take the form in Eq. (15) and $μ_{F 2} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω)$ take the following form [17],

\begin{array}{l} μ_{F 2} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω) = \frac{1}{C_{L_{x}} C_{L_{y}}} \sum_{l = 1}^{L_{x}} \frac{{(- 1)}^{l - 1}}{\sqrt{l}} (\begin{matrix} L_{x} \\ l \end{matrix}) \exp (- \frac{{({x^{'}}_{2} - {x^{'}}_{1})}^{2}}{2 l δ_{x}^{2}}) \\ \times \sum_{l = 1}^{L_{y}} \frac{{(- 1)}^{l - 1}}{\sqrt{l}} (\begin{matrix} L_{y} \\ l \end{matrix}) \exp (- \frac{{({y^{'}}_{2} - {y^{'}}_{1})}^{2}}{2 l δ_{y}^{2}}) \end{array}

with the normalization factors

C_{L_{x}} = \sum_{l = 1}^{L_{x}} \frac{{(- 1)}^{l - 1}}{\sqrt{l}} (\begin{matrix} L_{x} \\ l \end{matrix})

and

C_{L_{y}} = \sum_{l = 1}^{L_{y}} \frac{{(- 1)}^{l - 1}}{\sqrt{l}} (\begin{matrix} L_{y} \\ l \end{matrix}) .

Then, the new degree of potential correlation can be written as

\begin{array}{l} μ_{c} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω) = \frac{1}{N \times M} \exp [- \frac{{({x^{'}}_{2} - {x^{'}}_{1})}^{2}}{2 μ_{x}^{2}}] \exp [- \frac{{({y^{'}}_{2} - {y^{'}}_{1})}^{2}}{2 μ_{y}^{2}}] \sum_{n = - P}^{P} \cos [\frac{2 π n ({x^{'}}_{2} - {x^{'}}_{1}) R_{x}}{μ_{x}}] \\ \times \sum_{m = - Q}^{Q} \cos [\frac{2 π m ({y^{'}}_{2} - {y^{'}}_{1}) R_{y}}{μ_{y}}] \otimes \frac{1}{C_{L_{x}} C_{L_{y}}} \sum_{l = 1}^{L_{x}} \frac{{(- 1)}^{l - 1}}{\sqrt{l}} (\begin{matrix} L_{x} \\ l \end{matrix}) \exp (- \frac{{({x^{'}}_{2} - {x^{'}}_{1})}^{2}}{2 l δ_{x}^{2}}) \\ \times \sum_{l = 1}^{L_{y}} \frac{{(- 1)}^{l - 1}}{\sqrt{l}} (\begin{matrix} L_{y} \\ l \end{matrix}) \exp (- \frac{{({y^{'}}_{2} - {y^{'}}_{1})}^{2}}{2 l δ_{y}^{2}}) . \end{array}

Figure 5 presents the degree of potential correlation $μ_{F 1} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω)$ , $μ_{F 2} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω)$ and the results of their convolution for $k δ_{x} = k δ_{y} = 5$ and several values of $L$ . It is shown that relying on the choices for the parameters of $μ_{F 1}$ and $μ_{F 2}$ , their convolution represents a newly legitimate degree of potential correlation, which will account for a new scattered intensity distribution.

Fig. 5 First column is the degree of potential correlation corresponding to Eq. (15); the second column is the degree of potential correlation corresponding to Eq. (21); the third column is their convolution.

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The modulation to the degree of potential correlation can fulfill the modulation to the far-zone scattered intensity distribution. On evaluating the corresponding function $p_{c} (u_{ρ}, ω)$ of the degree of potential correlation $μ_{c}$ we obtain

\begin{array}{l} p_{c} (- K_{ρ}, ω) = p_{1} (- K_{ρ}, ω) p_{2} (- K_{ρ}, ω) \\ = \frac{μ_{x} μ_{y} δ_{x} δ_{y}}{2 M N C_{L_{x}} C_{L_{y}}} \exp (- \frac{k^{2} μ_{x}^{2} s_{x}^{2}}{2}) \exp (- \frac{k^{2} μ_{y}^{2} s_{y}^{2}}{2}) \\ \times \sum_{n = - P}^{P} \cos h (2 π n k s_{x} μ_{x} R_{x}) \exp (- 2 π^{2} n^{2} R_{x}^{2}) \\ \times \sum_{m = - Q}^{Q} \cos h (2 π m k s_{y} μ_{y} R_{y}) \exp (- 2 π^{2} m^{2} R_{y}^{2}) \\ \times \sum_{l = 1}^{L_{x}} \frac{{(- 1)}^{l - 1}}{\sqrt{l}} (\begin{matrix} L_{x} \\ l \end{matrix}) \exp [\frac{- l s_{x}^{2} k^{2} δ_{x}^{2}}{2}] \\ \times \sum_{l = 1}^{L_{y}} \frac{{(- 1)}^{l - 1}}{\sqrt{l}} (\begin{matrix} L_{y} \\ l \end{matrix}) \exp [\frac{- l s_{y}^{2} k^{2} δ_{y}^{2}}{2}] . \end{array}

On substituting from Eq. (25) together with Eq. (11) into Eq. (20), we arrive at

\begin{array}{l} S_{c}^{(s)} (r s, ω) = \frac{S^{(i)} (ω) A σ^{3} {(2 π)}^{3} μ_{x} μ_{y} δ_{x} δ_{y}}{2 M N C_{L_{x}} C_{L_{y}} r^{2}} \exp (- \frac{k^{2} μ_{x}^{2} s_{x}^{2}}{2}) \exp (- \frac{k^{2} μ_{y}^{2} s_{y}^{2}}{2}) \\ \times \sum_{n = - P}^{P} \cos h (2 π n k s_{x} μ_{x} R_{x}) \exp (- 2 π^{2} n^{2} R_{x}^{2}) \\ \times \sum_{m = - Q}^{Q} \cos h (2 π m k s_{y} μ_{y} R_{y}) \exp (- 2 π^{2} m^{2} R_{y}^{2}) \\ \times \sum_{l = 1}^{L_{x}} \frac{{(- 1)}^{l - 1}}{\sqrt{l}} (\begin{matrix} L_{x} \\ l \end{matrix}) \exp [\frac{- l s_{x}^{2} k^{2} δ_{x}^{2}}{2}] \\ \times \sum_{l = 1}^{L_{y}} \frac{{(- 1)}^{l - 1}}{\sqrt{l}} (\begin{matrix} L_{y} \\ l \end{matrix}) \exp [\frac{- l s_{y}^{2} k^{2} δ_{y}^{2}}{2}] . \end{array}

Figure 6 displays the optical lattice distribution generated by the novel medium with the degree of potential correlation calculated by Eq. (24). It is clearly seen from the Figs. 6(a)-(c) that the lobes intensity can be sequentially reducing from outside to inside by reducing boundary index $L$ . Furthermore, combining the modulation of the correlation length $δ_{i}$ , then the scattered array only consists of the central lobe corresponding to Fig. 6(d). The above same effects can be achieved by only increasing the value of correlation length $δ_{i}$ .

Fig. 6 Optical lattice patterns with adjustable lobes intensity calculated from Eq. (26). The medium with $M = N = 5$ , $k μ_{x} = k μ_{y} = 20$ , $R_{x} = R_{y} = 1$ , $k δ_{x} = k δ_{y} = 4$ , $k σ = 30$ . (a) $L = 40$ , (b) $L = 10$ , (c) $L = 1$ , (d) $L = 1$ with $k δ_{x} = k δ_{y} = 15$ .

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In what follows, we will demonstrate that the convolution of the degrees of potential correlation with polar and Cartesian symmetries can result in scattered fields with different shapes of distribution along the $x$ and $y$ directions. Figure 7 shows the degree of potential correlation $μ_{F 1} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω)$ with polar symmetry distribution, $μ_{F 2} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω)$ with Cartesian symmetry distribution, and the results of their convolution. The corresponding scattered intensity distributions are displayed in Fig. 8. It is shown that one can manipulate the circularly symmetric distribution only in one direction $x$ or $y$ on choosing $δ_{x} \geq μ$ or $δ_{y} \geq μ$ . Hence, the obtained scattered field has the shape of a circular distribution along one direction and rectangular distribution along the other. It should be noted that we can obtain more perfect rectangular distribution along one direction if we choose $μ_{F 1}$ as circular multi-Gaussian degree of potential correlation in [16].

Fig. 7 First row is circular Gaussian degree of potential correlation, the second row is rectangular multi-Gaussian degree of potential correlation, and the third row is their convolution.

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Fig. 8 Far-zone scattered fields calculated from Eq. (26). The medium with $M = N = 1$ , $k μ_{x} = k μ_{y} = 8$ , $R_{x} = R_{y} = 1$ , $L = 40$ . (a) $k δ_{x} = k δ_{y} = 5$ , (b) $k δ_{x} = 25$ , $k δ_{y} = 8$ , (c) $k δ_{x} = 8$ , $k δ_{y} = 25$ , and (d) $k δ_{x} = k δ_{y} = 25$ .

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From the above results, one can see that how convolution can be used for generation of a scattered field being a modulated version of another one, and the newly desirable scattered patterns can be obtained via the convolution method. Moreover, based on this method, one can steer scattered field more flexibly.

B. Generalization and discussion

We may readily generalize the previous derivation to a convolution of $N$ degrees of potential correlation. Indeed, assume $μ_{c} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω)$ is a legitimate convolution of $N$ degrees of potential correlation having the following form

μ_{c} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω) = μ_{F 1} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω) \otimes μ_{F 2} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω) \cdot \cdot \cdot \otimes μ_{F n} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω) .

The Fourier transform of a convolution of

N

functions is still a product of their Fourier transforms,

p_{c} (u_{ρ}, ω) = p_{1} (u_{ρ}, ω) p_{2} (u_{ρ}, ω) \cdot \cdot \cdot p_{n} (u_{ρ}, ω) .

All

p

can be same or different type, and for

p

to be non-negative, we must require

p_{n} \geq 0

for any values of vector

u_{ρ}

. At the same time, since

μ

and

p

are Fourier transform pairs, we may exchange the operation in Eqs. (27) and (28), i.e.,

p_{c} (u_{ρ}, ω) = p_{1} (u_{ρ}, ω) \otimes p_{2} (u_{ρ}, ω) \cdot \cdot \cdot \otimes p_{n} (u_{ρ}, ω)

. The corresponding result of the Fourier transform is related as

μ_{c} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω) = μ_{F 1} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω) μ_{F 2} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω) \cdot \cdot \cdot μ_{F n} ({ρ^{'}}_{2} - {ρ^{'}}_{1}, ω)

, which expresses a new degree of potential correlation of the medium as the product of

N

degrees of potential correlation. Hence, this is also a new way to design desirable media, which will be discussed in future. Thus, we have extended the method for designing weak media producing tunable scattered intensity to a convolution operation. In the end, we would like to point out that the new proposed methods are valid under the assumption that Markov approximation is made for scattering media. As a result, they only work well for sliced turbulence medium, such as SLM-produced turbulence, thin biological tissue samples, etc.

These results may have an important application in proposing new media. For example, by the aid of the convolution operation method, one can make use of the structure of the random resources to design a variety of novel media [27–31].

5. Conclusion

In summary, with the help of the Markov approximation, we have designed a novel class of random media which can produce a scattered field with optical lattice consisting of the same Gaussian-like lobes. It is shown that the array dimension, lobes intensity profile, and the periodicity of the optical lattice can be flexibly controlled by altering the correlation parameters of scattering potential of the random media. In addition, we have introduced an alternatively new method to design media. It is shown that the degree of potential correlation of a new medium can be expressed as the convolution of any two legitimate degrees of potential correlation, which can lead to a novel scattered intensity distribution. Finally, we have generalized the convolution of from two degrees of potential correlation to $N$ . It should be emphasized that the new method can be realized by programming 3D printing or liquid crystal light modulators. Hence, the methods for designing media we proposed are meaningful.

Funding

National Natural Science Foundation of China (NSFC) (11474253 and 11274273); Fundamental Research Funds for the Central Universities (2017FZA3005).

References and links

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Random medium model for producing optical coherence lattice

Abstract

1. Introduction

2. Theory

3. Scattered field with Gaussian Schell-model arrays

4. A new method for designing media

A. Convolution of degree of potential correlation

B. Generalization and discussion

5. Conclusion

Funding

References and links

Cited By

Figures (8)

Equations (28)

Optics Express