Tunable scattering intensity with prescribed weak media

Guo Zheng; Dong Ye; Xinyu Peng; Minmin Song; Qi Zhao

doi:10.1364/OE.24.024169

1. Introduction

A class of random media that play an important role in weak potential scattering theory are the so-called quasi-homogeneous media. It is shown in [1] that for quasi-homogeneous media, the far-zone intensity distribution of the scattered plane wave is proportional to the 3D spatial Fourier transform of the correlation coefficient of the scattering potential. For most of the past few decades, this reciprocity relation was used to study certain inverse problems [2] and was less well-known. However, in recent years, this reciprocity relation has become of considerable interest because of the fast development of modern medium manufacturing techniques, which makes it possible to produce a random medium with correlation functions as we proposed. With help of the reciprocity relation, random weak media scattering light with prescribed intensity distributions can be efficiently modeled. To the best of my knowledge, this work was researched by Korotkova and has led to circular flat, ring-like and rectangular scattered intensity patterns [3,4]. Thanks to her pioneer work, in this paper, we derive a more general and concise mathematical model for one to design the weak scattering media producing controllable intensity patterns.

Our paper is organized as follows. In Sec.2, we start this work by reviewing the general theory of scattering of scalar fields from a static random medium and providing a short derivation of the reciprocity relation. In Sec.3, we employ the Markov-like approximation along the scattering axis to derive a concise expression for the scattered intensity in far zone. In Sec.4, we introduce two novel media to illustrate the above expression. In Sec.5, our results are summarized.

2. Scattering on a random medium

Let us suppose that a monochromatic plane wave of frequency $ω$ and amplitude $a$ is incident on a weak scatterer occupying a finite domain $D$ along the direction $\vec{s_{0}}$ ( $| \vec{s_{0} |} = 1$ ). The space-dependent part of the monochromatic plane wave field may then be expressed in the form

U^{(i)} (\vec{r}, ω) = a (ω) \exp (i k \vec{s_{0}} \cdot \vec{r}) .

where

\vec{r} = (x, y, z)

is the position vector of a field point,

k = ω / c

is the wave number and

c

is the speed of light in vacuum. In what follows, it should be noted that the angular frequency dependence of all the quantities of interest will be suppressed for brevity. According to the weak potential scattering theory, the spectral density of the scattered field in the far zone along direction

\vec{r} = r \vec{s}

(

| \vec{s |} = 1

) can be expressed within the first Born approximation as [1]

S^{(s)} (r \vec{s})= \frac{1}{r^{2}} S^{(i)} \tilde{C_{F}} [- k (\vec{s} - \vec{s_{0}}), k (\vec{s} - \vec{s_{0}})] .

Here,

S^{(i)} = | a |^{2}

is the spectral density of the incident plane wave and

\tilde{C_{F}}

denotes the six-dimensional Fourier transform

\tilde{C_{F}} (- \vec{K}, \vec{K}) = \int_{D} \int_{D} C_{F} (\vec{r_{1}^{'}}, \vec{r_{2}^{'}}) \exp [- i \vec{K} (\vec{r_{2}^{'}} - \vec{r_{1}^{'}})] d^{3} r_{1}^{'} d^{3} r_{2}^{'},

with

\vec{K} = k (\vec{s} - \vec{s_{0}})

being the momentum transfer vector and

C_{F}

being the spatial correlation function of the scattering potential:

C_{F} (\vec{r_{1}^{'}}, \vec{r_{2}^{'}}) = {〈 F^{*} (\vec{r_{1}^{'}}) F (\vec{r_{2}^{'}}) 〉}_{m},

where the angular brackets with subscript “

m

” indicate the ensemble average of the scattering medium and the asterisk denotes the complex conjugate. Further, the scattering potential

F (\vec{r^{'}})

is related to the refractive index of the medium by the formula [1]

F (\vec{r^{'}}) = \frac{1}{4 π} k^{2} [n^{2} (\vec{r^{'}}) - 1] .

The general structure of the Schell-model media which are routinely invoked can be expressed as [1]

C_{F} (\vec{r_{1}^{'}}, \vec{r_{2}^{'}}) = \sqrt{I_{F} (\vec{r_{1}^{'}}}) \sqrt{I_{F} (\vec{r_{2}^{'}}}) μ_{F} (\vec{r_{2}^{'}} - \vec{r_{1}^{'}}),

with

I_{F} (\vec{r^{'}}) = C_{F} (\vec{r^{'}}, \vec{r^{'}})

being the strength of the scattering potential and

μ_{F} (\vec{r_{2}^{'}} - \vec{r_{1}^{'}})

being the degree of spatial correlation, which depends on

\vec{r_{1}^{'}}

and

\vec{r_{2}^{'}}

only through the difference

\vec{r_{2}^{'}} - \vec{r_{1}^{'}}

. Just like any other correlation function, in order for

C_{F}

to be physically realizable, it suffices to prove that

C_{F}

obey non-negative definiteness conditions. The sufficient condition for genuine 2D correlation function has been derived in detail in [5] and the extension to 3D correlation function has been introduced in [3,6], which states that a valid

C_{F}

must have the following superposition form:

C_{F} (\vec{r_{1}^{'}}, \vec{r_{2}^{'}}) = \int p (\vec{v}) H_{0}^{*} (\vec{r_{1}^{'}}, \vec{v}) H_{0} (\vec{r_{2}^{'}}, \vec{v}) d^{3} v .

Here

\vec{v}

is the three-dimensional vector,

H_{0} (\vec{r^{'}}, \vec{v})

is an arbitrary function whose choice defines the type of correlations and

p (\vec{v})

must be non-negative for all values of its argument.

For Schell-model media, $H_{0} (\vec{r^{'}}, \vec{v})$ usually has the Fourier-like kernel [5], i.e.,

H_{0} (\vec{r^{'}}, \vec{v}) = τ (\vec{r^{'}}) \exp (- i \vec{r^{'}} \cdot \vec{v}),

where

τ (\vec{r^{'}})

is the amplitude of the random scattering potential. Then upon substituting from Eq. (8) into Eq. (7), we find that

C_{F} (\vec{r_{1}^{'}}, \vec{r_{2}^{'}}) = τ^{*} (\vec{r_{1}^{'}}) τ (\vec{r_{2}^{'}}) μ_{F} (\vec{r_{2}^{'}} - \vec{r_{1}^{'}}),

where

μ_{F} (\vec{r_{2}^{'}} - \vec{r_{1}^{'}}) = ∭ p (\vec{v}) \exp [- i \vec{v} \cdot (\vec{r_{2}^{'}} - \vec{r_{1}^{'}})] d^{3} v .

In the special situation when the scattering potential varies much slowly with position that over the effective width of $| μ_{F} |$ the function $τ (\vec{r^{'}})$ is essentially constant, in other words, $τ (\vec{r^{'}})$ is a slow function of $\vec{r^{'}}$ and $μ_{F} (\vec{r_{2}^{'}} - \vec{r_{1}^{'}})$ is a fast function of $\vec{r_{2}^{'}} - \vec{r_{1}^{'}}$ , then the correlation function of the scattering potential can be approximated as

C_{F} (\vec{r_{1}^{'}}, \vec{r_{2}^{'}}) = τ^{2} (\frac{\vec{r_{2}^{'}} + \vec{r_{1}^{'}}}{2}) μ_{F} (\vec{r_{2}^{'}} - \vec{r_{1}^{'}}) .

Such a model is well known as quasi-homogeneous media (also known as locally homogeneous media), which was introduced by Silverman in the literature [7]. Upon substituting from Eq. (11) and (3) into Eq. (2), and introducing the sum and difference coordinates:

\vec{r_{s}^{'}} = (\vec{r_{2}^{'}} + \vec{r_{1}^{'}}) / 2, \begin{matrix} \vec{r_{d}^{'}} = \vec{r_{2}^{'}} - \vec{r_{1}^{'}} \end{matrix},

we will obtain the reciprocity relation that the spectral density distribution of the scattered field in the far zone is proportional to the Fourier transform of the degree of potential correlation [1], i.e.

S^{(s)} (r \vec{s})= \frac{1}{r^{2}} S^{(i)} I_{F}^{D} \tilde{μ_{F}} [k (\vec{s} - \vec{s_{0}})],

where

I_{F}^{D} = \int_{D} τ^{2} (\vec{r_{s}^{'}}) d^{3} r_{s}^{'}

is the integrated value of the potential strength and

\tilde{μ_{F}} (\vec{K}) = \int_{D} μ_{F} (\vec{r_{d}^{'}}) \exp (- i \vec{K} \cdot \vec{r_{d}^{'}}) d^{3} r_{d}^{'}

is the three-dimensional Fourier transform of the degree of its correlation. As mentioned above,

p

and

μ_{F}

are the Fourier transform pair, predictably, the intensity distribution in the far field can be controlled by choosing a suitable non-negative function

p

.

3. Markov-like approximation

Now, we consider the layer-structured media whose scattering potential is delta correlated along the scattering axis (z-axis), that is to say, the degree of spatial correlation $μ_{F} (\vec{r_{d}^{'}})$ can be expressed as

μ_{F} (\vec{r_{d}^{'}}) = δ (z_{d}^{'}) μ_{F} (\vec{ρ_{d}^{'}}),

where

μ_{F} (\vec{ρ_{d}^{'}})

is the two-dimensional correlation function and

\vec{r_{d}^{'}} = (\vec{ρ_{d}^{'}}, z_{d}^{'})

.This assumption can be viewed as a counterpart of the well-known Markov approximation [8] and is first studied by Olga, the interested reader should consult reference [4] for a detailed statement. As a matter of fact, we can treat the adjacent layers of the medium as statistically independent if it is synthesized layer by layer. In this way, the function

p (\vec{v})

reduces to

p (v_{x}, v_{y}, 0)

and Eq. (10) becomes

\begin{array}{l} μ_{F} (\vec{r_{d}^{'}}) = \iint p (\vec{v}) e x p (- i \vec{v} \cdot \vec{ρ_{d}^{'}}) d^{2} v \int \exp (- i v_{z} \cdot z_{d}^{'}) d v_{z} \\ \begin{matrix} = 2 π \end{matrix} δ (z_{d}^{'}) \iint p (\vec{v}) e x p (- i \vec{v} \cdot \vec{ρ_{d}^{'}}) d^{2} v . \end{array}

From Eq. (16) it follows that

μ_{F} (\vec{ρ_{d}^{'}}) = 2 π \iint p (\vec{v}) e x p (- i \vec{v} \cdot \vec{ρ_{d}^{'}}) d^{2} v .

Upon substituting from Eqs. (16) and (18) into Eq. (15),we find that

\begin{array}{l} \tilde{μ_{F}} (\vec{K}) = \int_{D} μ_{F} (\vec{ρ_{d}^{'}}) \exp (- i \vec{K_{ρ}} \cdot \vec{ρ_{d}^{'}}) d^{2} ρ_{d}^{'} \int δ (z_{d}^{'}) \exp (- i K_{z} \cdot z_{d}^{'}) d z_{d}^{'} \\ \begin{matrix} = 2 π \int p ( \end{matrix} \vec{v}) e x p (- i \vec{v} \cdot \vec{ρ_{d}^{'}}) \exp (- i \vec{K_{ρ}} \cdot \vec{ρ_{d}^{'}}) d^{2} ρ_{d}^{'} d^{2} v \\ \begin{matrix} = {(2 π)}^{3} \end{matrix} p (- \vec{K_{ρ}}), \end{array}

where

\vec{K} = (\vec{K_{ρ}}, K_{z})

is the momentum transfer vector which has been mentioned above and

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

is the two-dimensional vector. Therefore, the spectral density distribution for field scattered by quasi-homogeneous media in the far zone will take the following form:

S^{(s)} (r \vec{s})= \frac{{(2 π)}^{3}}{r^{2}} S^{(i)} I_{F}^{D} p (- \vec{K_{ρ}}) .

Equation (20) is the most valuable result in this paper, which provides a convenient way of designing the weak scattering media for controllable intensity patterns. What should be noted is that the average strength of the potential is not a determining factor in relation to the spectral density distribution in far zone, it only affects the result as the proportionality factor. Without loss of generality, we take

τ (\vec{r^{'}}) = e x p (- \frac{r^{'}^{2}}{2 σ^{2}}),

which corresponds to a soft-edge spherical potential [9]. Here,

σ

is its typical width. Then upon substituting from Eq. (21) into Eq. (14), we obtain the integrated value of the potential strength

I_{F}^{D} = π^{3/2} σ^{3} .

4. Elementary examples

Some simple examples can help to illustrate the previous results. As an example, let us first consider the weak medium which can scatter light forming frame-like intensity profiles with Cartesian symmetry. In this case, we may choose $p (\vec{v})$ in the form

p (\vec{v}) = p_{o} (\vec{v}) - p_{i} (\vec{v}),

where the outer and inner distributions are:

p_{o} (\vec{v}) = \frac{A}{C_{L}^{2}} \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp (- \frac{m δ_{o x}^{2} v_{x}^{2}}{2}) \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp (- \frac{m δ_{o y}^{2} v_{y}^{2}}{2}),

p_{i} (\vec{v}) = \frac{A}{C_{L}^{2}} \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp (- \frac{m δ_{i x}^{2} v_{x}^{2}}{2}) \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp (- \frac{m δ_{i y}^{2} v_{y}^{2}}{2}),

C_{L} = \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix}),

(\begin{matrix} M \\ m \end{matrix})

stands for the binomial coefficient,

C_{L}

is the normalization factor and the non-negative value of parameter

A

will be discussed below. We should note that on the basic of Markov-like approximation, the function

p (\vec{v})

has reduced to

p (v_{x}, v_{y}, 0)

. In addition, the above formula has been previously used to model random sources for optical frames [10]. Since the function

p (\vec{v})

must be non-negative, we must set

δ_{i x} > δ_{o x}

and

δ_{i y} > δ_{o y}

.Then upon substituting from Eq. (23) into Eq. (18), we obtain the two-dimensional spatial correlation coefficient of the scattering potential:

\begin{array}{l} μ_{F} (\vec{ρ_{d}^{'}}) = \frac{A}{C_{L}^{2} δ_{o x} δ_{o y}} \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix}) \exp (- \frac{x_{d}^{'}^{2}}{2 m δ_{o x}^{2}}) \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix}) \exp (- \frac{y_{d}^{'}^{2}}{2 m δ_{o y}^{2}}) \\ \begin{matrix} \begin{matrix} \begin{matrix}  \end{matrix} \end{matrix} & - \end{matrix} \frac{A}{C_{L}^{2} δ_{i x} δ_{i y}} \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix}) \exp (- \frac{x_{d}^{'}^{2}}{2 m δ_{i x}^{2}}) \sum_{m = 1}^{M} \frac{{(- 1)}^{m - 1}}{\sqrt{m}} (\begin{matrix} M \\ m \end{matrix}) \exp (- \frac{y_{d}^{'}^{2}}{2 m δ_{i y}^{2}}), \end{array}

where

\vec{ρ_{d}^{'}} = (x_{d}^{'}, y_{d}^{'})

,

x_{d}^{'} = x_{2}^{'} - x_{1}^{'}

and

y_{d}^{'} = y_{2}^{'} - y_{1}^{'} .

Due to the fact that the degree of correlation is one at the coinciding arguments [11], i.e.,

μ_{F} (0) = 1

, one finds that the coefficient

A

must take the form:

A = {(\frac{1}{δ_{o x} δ_{o y}} - \frac{1}{δ_{i x} δ_{i y}})}^{- 1} .

Upon substituting from Eqs. (22), (23) and (28) into Eq. (20), we obtain the far-field intensity distribution

\begin{array}{l} S^{(s)} (r \vec{s}) = \frac{8 π^{9 / 2} σ^{3} S^{(i)}}{r^{2} C_{L}^{2}} (^{\frac{1}{δ_{o x} δ_{o y}} - \frac{1}{δ_{i x} δ_{i y}}) - 1} \\ \begin{matrix}  \end{matrix} \times {\sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp [- \frac{m δ_{o x}^{2} k^{2} {(s_{x} - s_{0 x})}^{2}}{2}] \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp (- \frac{m δ_{o y}^{2} k^{2} {(s_{y} - s_{0 y})}^{2}}{2}) \\ \begin{matrix}  \end{matrix} - \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp (- \frac{m δ_{i x}^{2} k^{2} {(s_{x} - s_{0 x})}^{2}}{2}) \sum_{m = 1}^{M} {(- 1)}^{m - 1} (\begin{matrix} M \\ m \end{matrix}) \exp (- \frac{m δ_{i y}^{2} k^{2} {(s_{y} - s_{0 y})}^{2}}{2})} . \end{array}

Figures 1 and 2 present the far-field scattered spectral density as functions of the two-dimensional coordinates of direction vector $\vec{s}$ .Here we assume that the scatterer is illuminated with the normal incidence plane wave, i.e., $\vec{s_{0}} = (0, 0, 1)$ . As is shown in Figs. 1 and 2, in control of the outer and inner correlation coefficients, the far-field scattered intensity can be adjusted to form square frames (corresponding to Fig. 1(b)) or rectangular frames (corresponding to Fig. 2(b)) when the value of $M$ is high. Once the value of $M$ is reduced to one, then we will obtain the density plots of circular ring (corresponding to Fig. 1(a)) or elliptical ring (corresponding to Fig. 2(a)).

Fig. 1 Illustration of the far-field scattered spectral density distribution for selected values of the parameters $M$ , namely, (a) $M = 1$ , (b) $M = 40$ . The other parameters are $k δ_{o x} = k δ_{o y} = 20, k δ_{i x} = k δ_{i y} = 40$

Download Full Size | PDF

Fig. 2 Illustration of the far-field scattered spectral density distribution for selected values of the parameters $M$ , namely, (a) $M = 1$ , (b) $M = 40$ . The other parameters are $k δ_{o x} = 10, k δ_{o y} = 20, k δ_{i x} = 15, k δ_{i y} = 30.$

Download Full Size | PDF

For the second example, we will discuss the weak medium which can scatter light forming intensity patterns with azimuthal dependence in the far zone. The original idea of a distribution that have arbitrary dependence on the azimuthal variable belongs to Fei and Olga and is used to construct random sources [12]. In order to design the suitable medium, here, we choose $p (\vec{v})$ in the form:

p (\vec{v}) = B \exp (- δ^{2} v^{2}) \cos^{2} (\frac{n φ}{2}),

where

\vec{v} = (v, φ)

is the position vector in the polar coordinate system,

n

is an integer,

δ

is a constant and the non-negative value of parameter

B

will be discussed below. The reader who has interest may consult with Ref [12]. for detailed derivation of the formulas. Upon substituting from Eq. (30) into Eq. (18), we can then obtain, after tedious integration (cf. Ref [12].), that

\begin{array}{l} μ_{F} (\vec{ρ_{d}^{'}}) = \frac{π^{2} B}{δ^{2}} \exp (- \frac{ρ_{d}^{'}^{2}}{4 δ^{2}}) + (^{- i) n} \cos (n φ_{d}^{'}) \frac{π^{5 / 2} ρ_{d}^{'} B}{4 δ^{3}} \exp (- \frac{ρ_{d}^{'}^{2}}{8 δ^{2}}) \\ \begin{matrix} \times [I_{(n - 1) / 2} (\frac{ρ_{d}^{'}^{2}}{8 δ^{2}}) - I_{(n + 1) / 2} (\frac{ρ_{d}^{'}^{2}}{8 δ^{2}})] \end{matrix}, \end{array}

where

ρ_{d}^{'} = \sqrt{x_{d}^{'}^{2} + y_{d}^{'}^{2}},

φ_{d}^{'} = \arctan (y_{d}^{'} / x_{d}^{'})

with

x_{d}^{'} = x_{2}^{'} - x_{1}^{'}

and

y_{d}^{'} = y_{2}^{'} - y_{1}^{'}

and

I_{n}

denotes a modified Bessel function of order

n

. As previously stated, the degree of correlation is one at the coinciding arguments [11], then the coefficient

B

must take the form:

B = \frac{δ^{2}}{π^{2}} .

Upon substituting from Eqs. (22), (30) and (32) into Eq. (20), we obtain the far-field intensity distribution

S^{(s)} (r \vec{s})= \frac{4 π^{5 / 2} σ^{3} S^{(i)} δ^{2}}{r^{2}} e x p (- K_{ρ}^{2} δ^{2}) \cos^{2} [n (φ_{K} + π) / 2],

where

K_{ρ} = k \sqrt{{(s_{x} - s_{0 x})}^{2} + {(s_{y} - s_{o y})}^{2}}

and

φ_{K} = arc \tan (\frac{s_{y} - s_{0 y}}{s_{x} - s_{0 x}})

.

The illustration of the far-field scattered spectral density distribution for $n = 3$ and $n = 4$ is shown in Fig. 3. In this simulation, we still assume that the incident plane wave is along z-axis, i.e., $\vec{s_{0}} = (0, 0, 1)$ . As is shown in Fig. 3, the far-field scattered intensity patterns with azimuthal dependence will be formed as long as the two-dimensional spatial correlation coefficient of the scattering potential satisfies the condition of Eq. (31). In the special case that the parameter $n$ reduces to zero, we find that the desired medium will scatter light forming the circular intensity patterns, which is shown in Fig. 4.

Fig. 3 Illustration of the far-field scattered spectral density distribution for selected values of the parameters $n$ , namely, (a) $n = 3$ , (b) $n = 4$ . In addition, $k δ = 40$ .

Download Full Size | PDF

Fig. 4 Illustration of the far-field scattered spectral density distribution for the parameters $n = 0$ . The other parameters are the same as in Fig. 3.

Download Full Size | PDF

5. Discussion and conclusion

In summary, we have created a mathematical model for one to design the novel weak scattering media with controllable intensity patterns based on the Markov-like approximation. This method provides a convenient way for one to steer the scattered intensity. It is to be noted that the Markov-like approximation is reasonable if the medium is synthesized layer by layer. Then the question may arise that why we have to introduce the Markov-like approximation? In fact, in use of the Markov-like approximation, we have modified the reciprocity relation in another form, i.e., now the far-zone intensity distribution of the scattered plane wave is proportional to the 2D instead of 3D spatial Fourier transform of the correlation coefficient of the scattering potential. This relation is consistent with the well-known reciprocity relation occurring in radiation from planar quasi-homogeneous sources [11], which has been used to model a great deal of random sources [10,12–15]. In this way, the structure of the random sources may be applied to the random media we designed and we indeed do so in this paper.

In principle, the far field intensity is contributed by the scattering light and the un-scattering light together. Thus the scattered intensity steered here can be challenging to be separated in practice. An area for future study is to solve the problem how to separate the incident wave and the scattering wave efficiently.

In the end, we would like to point out that the random media with desired correlation functions may be experimentally produced with help of the 3D printing or sequences of liquid crystal light modulators [3,4].

Funding

National Science Foundation of China (61077012); National Natural Science Foundation of China (NSFC) (61675098); Ph.D. Programs Foundation of Ministry of Education of China (20123219110021); Innovation fund for the shanghai aerospace science and technology (SAST) (201350).

Acknowledgments

We gratefully acknowledge the reviewers for their useful comments and suggestions.

References and links

1. E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University, 2007).

2. D. G. Fischer and E. Wolf, “Inverse problems with quasi-homogeneous media,” J. Opt. Soc. Am. A 11(3), 1128–1135 (1994). [CrossRef]

3. O. Korotkova, “Design of weak scattering media for controllable light scattering,” Opt. Lett. 40(2), 284–287 (2015). [CrossRef] [PubMed]

4. O. Korotkova, “Can a sphere scatter light producing rectangular intensity patterns?” Opt. Lett. 40(8), 1709–1712 (2015). [CrossRef] [PubMed]

5. F. Gori and M. Santarsiero, “Devising genuine spatial correlation functions,” Opt. Lett. 32(24), 3531–3533 (2007). [CrossRef] [PubMed]

6. J. Li and O. Korotkova, “Scattering of light from a stationary nonuniformly correlated medium,” Opt. Lett. 41(11), 2616–2619 (2016). [CrossRef] [PubMed]

7. R. A. Silverman and F. Ursell, “Scattering of plane waves by locally homogeneous dielectric noise,” Proc. Camb. Philos. Soc. 54(4), 530–537 (1958). [CrossRef]

8. L. C. Andrews and R. L. Phillips, Laser Beam Propagation through Random Media (SPIE, 2005).

9. O. Korotkova, S. Sahin, and E. Shchepakina, “Light scattering by three-dimensional objects with semi-hard boundaries,” J. Opt. Soc. Am. A 31(8), 1782–1787 (2014). [CrossRef] [PubMed]

10. O. Korotkova and E. Shchepakina, “Random sources for optical frames,” Opt. Express 22(9), 10622–10633 (2014). [CrossRef] [PubMed]

11. L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

12. F. Wang and O. Korotkova, “Random sources for beams with azimuthal intensity variation,” Opt. Lett. 41(3), 516–519 (2016). [CrossRef] [PubMed]

13. Z. Mei, D. Zhao, O. Korotkova, and Y. Mao, “Gaussian Schell-model arrays,” Opt. Lett. 40(23), 5662–5665 (2015). [CrossRef] [PubMed]

14. S. Sahin and O. Korotkova, “Light sources generating far fields with tunable flat profiles,” Opt. Lett. 37(14), 2970–2972 (2012). [CrossRef] [PubMed]

15. C. Liang, F. Wang, X. Liu, Y. Cai, and O. Korotkova, “Experimental generation of Cosine-Gaussian-correlated Schell-model beams with rectangular symmetry,” Opt. Lett. 39(4), 769–772 (2014). [CrossRef] [PubMed]

Tunable scattering intensity with prescribed weak media

Abstract

1. Introduction

2. Scattering on a random medium

3. Markov-like approximation

4. Elementary examples

5. Discussion and conclusion

Funding

Acknowledgments

References and links

Cited By

Figures (4)

Equations (33)

Optics Express