Random light scattering by collections of ellipsoids

doi:10.1364/OE.20.029296

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Random light scattering by collections of ellipsoids

Zhangrong Mei and Olga Korotkova »View Author Affiliations

Optics Express, Vol. 20, Issue 28, pp. 29296-29307 (2012)
http://dx.doi.org/10.1364/OE.20.029296

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▸ Article Outline
– Abstract
1. Introduction
2. Scattering matrix theory for the collection of particles with different types
3. Spectral pair-scattering matrix for the collection of ellipsoids
4. Example of two correlated plane waves scattering on the collection of ellipsoids
5. Concluding remarks
– References and links

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Abstract

Theory of weak scattering of random optical fields from deterministic collections of particles with soft ellipsoidal scattering potentials of arbitrary shapes and orientations is developed. Far-field intensity distribution produced on scattering is shown to be influenced by source correlation properties as well as by a number, shapes and orientations of scatterers. The theory extends previous results on scattering from collections of spheres with soft Gaussian potentials and is applicable to analysis of a wide range of media including blood cells.

1. Introduction

In the theory of weak scattering of random light from collections of particles it is assumed, as a rule, that the scattering potentials are Gaussian functions of radial distance from the particles’ centers. To some extent such analytically convenient model approximates hard-edge spherical particles with radii coinciding with the variance of a Gaussian distribution [1

E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6(8), 1142–1149 (1989). [CrossRef]

–5

C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17(3), 1753–1765 (2009). [CrossRef] [PubMed]

]. For more precise approximation multi-Gaussian potentials may be employed [6

S. Sahin, G. Gbur, and O. Korotkova, “Scattering of light from particles with semisoft boundaries,” Opt. Lett. 36(20), 3957–3959 (2011). [CrossRef] [PubMed]

Recent advances in the bio-optical applications, involving measurements of scattered light statistics and reconstruction of physical properties of tissue constituencies require the extension of the scattering theory to the case of collections with ellipsoidal particles, having arbitrary shapes and orientations. Such development could be of interest for ektacytometry, a technique based on anomalous diffraction for quantifying the deformability of red blood cells [7

G. J. Streekstra, A. G. Hoekstra, E. J. Nijhof, and R. M. Heethaar, “Light scattering by red blood cells in ektacytometry: Fraunhofer versus anomalous diffraction,” Appl. Opt. 32(13), 2266–2272 (1993). [CrossRef] [PubMed]

–10

R. S. Brock, X. H. Hu, P. Yang, and J. Lu, “Evaluation of a parallel FDTD code and application to modeling of light scattering by deformed red blood cells,” Opt. Express 13(14), 5279–5292 (2005). [CrossRef] [PubMed]

On employing the recently introduced theory for scattering of scalar fields with arbitrary spatial, spectral and correlation properties from deterministic and random collections [11

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

,12

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056609 (2007). [CrossRef] [PubMed]

] we develop expressions for far-field second-order statistics of scattered illumination specifically in the case when the collection consists of ellipsoids. Such studies have been previously carried out for soft spheres in Refs [13

S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78(6), 063815 (2008). [CrossRef]

,14

S. Sahin and O. Korotkova, “Effect of the pair-structure factor of a particulate medium on scalar wave scattering in the first Born approximation,” Opt. Lett. 34(12), 1762–1764 (2009). [CrossRef] [PubMed]

]. Corresponding inverse problems, i.e. estimation of orientations and shapes of ellipsoids from scattered light spectral and intensity distributions is beyond the scope of this paper but can be approached in a similar fashion to the inverse problems involving spheres [15

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

–17

Z. Tong and O. Korotkova, “Method for tracing the position of an alien object embedded in a random particulate medium,” J. Opt. Soc. Am. A 28(8), 1595–1599 (2011). [CrossRef] [PubMed]

The paper is organized as follows: in section 2 the brief review of the scattering matrix theory is suggested; section 3 gives the derivation of the spectral pair-scattering matrix for deterministic collection of ellipsoids; in section 4 several examples are considered in which random light is scattered from ellipsoids with deterministic potentials but of different shapes and orientations; finally in section 5 the concluding remarks are given and the extension of this study to random collections of ellipsoids is outlined.

2. Scattering matrix theory for the collection of particles with different types

Let us consider a stochastic scalar wave field scattered by a weak random medium. The second-order correlation properties of incident field

U^{(i)} (r; ω)

at a pair of points

r_{1}

r_{2}

and angular frequency ω can be characterized by the cross-spectral density function [11

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

]

W^{(i)} (r_{1}, r_{2}; ω) = 〈 U^{(i) *} (r_{1}; ω) U^{(i)} (r_{2}; ω) 〉,

(1)

the asterisk denoting the complex and the angular brackets denoting the average of the statistical ensemble monochromatic realizations of the incident field. The angular correlation function of the incident field is defined as the four-dimensional Fourier transform of

W^{(i)}

\begin{array}{l} A^{(i)} (u_{1}, u_{2}; ω) = 〈 a^{(i) *} (u_{1}; ω) a^{(i)} (u_{2}; ω) 〉 \\ = \frac{k^{4}}{{(2 π)}^{4}} \int \int W^{(i)} (r_{1}, r_{2}; ω) \exp [- i (u_{1} \cdot r_{1} - u_{2} \cdot r_{2})] d^{2} r_{1} d^{2} r_{2}, \end{array}

(2)

where

a^{(i)} (u; ω)

is the spectral amplitude of the incident field

U^{(i)} (r; ω)

k = ω / c

is the wave number and

c

is the speed of light in vacuum,

u_{1}

and

u_{2}

are unit vectors and the integration is performed over the entire source plane.

Within the accuracy of the first-order Born approximation the cross-spectral density of the total field (the sum of the incident field and the scattered field) is given by the expression [12

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056609 (2007). [CrossRef] [PubMed]

]

\begin{array}{l} W^{(t)} (r_{1}, r_{2}; ω) = \int \int \int \int M (u_{1}, u_{2}; {u^{'}}_{1}, {u^{'}}_{2}; ω) A^{(i)} ({u^{'}}_{1}, {u^{'}}_{2}; ω) \\ \times \exp [i (u_{2} \cdot r_{2} - u_{1} \cdot r_{1})] d^{2} u_{1 ⊥} d^{2} u_{2 ⊥} d^{2} {u^{'}}_{1 ⊥} d^{2} {u^{'}}_{2 ⊥}, \end{array}

(3)

where

M (u_{1}, u_{2}; {u^{'}}_{1}, {u^{'}}_{2}; ω) = S^{*} (u_{1}; {u^{'}}_{1}; ω) S (u_{2}; {u^{'}}_{2}; ω),

(4)

is the spectral pair-scattering matrix of random medium,

S (u, u^{'}; ω)

is the ordinary scattering matrix that describes the change in the amplitude of a plane wave incident along direction

u^{'}

and scattered along direction

u

u_{⊥} = (u_{x}, u_{y}, 0)

is the projection of

u

in the x-y plane. Within the validity of the first Born approximation the scattering matrix has simple relation with the scattering potential [18

M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999).

]

f (r; ω) = \frac{k^{2}}{4 π} [n^{2} (r; ω) - 1],

(5)

with

n (r; ω)

being the spatial and spectral refractive index distribution within the scatterer:

S (u_{2}; {u^{'}}_{2}; ω) = \tilde{f} [k (u - u'); ω]

(6)

The far-zone approximation of the cross-spectral density in Eq. (3) has the form [11

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

]

\begin{array}{l} W^{(t)} (r u_{1}, r u_{2}; ω) \approx \pm \frac{4 π^{2}}{k^{2} r^{2}} u_{1 z} u_{2 z} \int \int M (u_{1}, u_{2}; {u^{'}}_{1}, {u^{'}}_{2}; ω) \\ \times A^{(i)} ({u^{'}}_{1}, {u^{'}}_{2}; ω) d^{2} {u^{'}}_{1 ⊥} d^{2} {u^{'}}_{2 ⊥} . \end{array}

(7)

With the help of Eq. (7) we may at once determine the spectrum

S^{(t)} (r u; ω)

of the total field using the formula [10

]

S^{(t)} (r u; ω) = W^{(t)} (r u, r u; ω),

(8)

If the scattering medium is a deterministic collection of particles with L different types, the scattering potential of the whole collection is then given by the formula

F (r; ω) = \sum_{l = 1}^{L} \sum_{m = 1}^{M_{l}} f_{l} (r - r_{m}; ω),

(9)

where

r_{m}

is the center of the mth particle,

f_{l}

is the scattering potential of the scatterer of type l,

M_{l}

is the number of particle of type l.

Within the validity of the first Born approximation the pair-scattering matrix [see Eq. (4)] is related to the scattering potentials by the formula

M (u_{1}, u_{2}; {u^{'}}_{1}, {u^{'}}_{2}; ω) = \sum_{l 1 = 1}^{L} \sum_{l 2 = 1}^{L} \sum_{n = 1}^{M_{l 1}} \sum_{m = 1}^{M_{l 2}} \int \int f_{l 1}^{*} (r_{1} - r_{n}; ω) f_{l 2} (r_{2} - r_{m}; ω) e^{- i [K_{2} \cdot r_{2} - K_{1} \cdot r_{1}]} d^{3} r_{1} d^{3} r_{2},

(10)

where

K_{α} = k (u_{α} - {u^{'}}_{α})

(α = 1, 2)

are the momentum-transfer vectors and three-dimensional integration is carried over the scattering medium.

3. Spectral pair-scattering matrix for the collection of ellipsoids

Suppose that the scattering particles are ellipsoids centered at points

r_{n} = (x_{n}, y_{n}, z_{n})

oriented along the coordinate axes of a particle frame [ξ, η, ζ] (see Fig. 1 ), having three-dimensional (soft) Gaussian potentials

Fig. 1 Illustrating the notation relating to a single ellipsoid

f_{l} (r_{n}, ω) = G \exp {- [\frac{{(ξ - x_{n})}^{2}}{2 σ_{l x}^{2}} + \frac{{(η - y_{n})}^{2}}{2 σ_{l y}^{2}} + \frac{{(ζ - z_{n})}^{2}}{2 σ_{l z}^{2}}]} .

(11)

The variances

σ_{l x}^{2}

σ_{l y}^{2}

and

σ_{l z}^{2}

are taken to be independent of position but, in general, may depend on the frequency. The orientation of the particle frame [ξ, η, ζ] relative to the laboratory frame [x, y, z] is defined by three counter-clockwise rotation angles

α_{l}

β_{l}

and

γ_{l}

as shown in Fig. 2 . Starting with the laboratory frame, we obtain the particle frame by rotation α of the y-z plane around the x axis followed by rotation β of the x'-z' plane around the y' axis and by rotation γ of the x”-y” plane around the z” axis. The coordinates in the particle frame [ξ, η, ζ] are related to the laboratory frame[x, y, z] as

Fig. 2 Illustrating three basic rotations defining the orientation of the particle frame [ξ, η, ζ] relative to the laboratory frame [x, y, z].

\begin{array}{l} (\begin{matrix} ξ \\ η \\ ζ \end{matrix}) = R (\begin{matrix} x \\ y \\ z \end{matrix}) = (\begin{matrix} A_{l_11} & A_{l_12} & A_{l_13} \\ A_{l_21} & A_{l_22} & A_{l_23} \\ A_{l_31} & A_{l_32} & A_{l_33} \end{matrix}) (\begin{matrix} x \\ y \\ z \end{matrix}) \\ = (\begin{matrix} \cos β_{l} \cos γ_{l} & \cos α_{l} \sin γ_{l} + \sin α_{l} \sin β_{l} \cos γ_{l} & \sin α_{l} \sin γ_{l} - \cos α_{l} \sin β_{l} \cos γ_{l} \\ - \cos β_{l} \sin γ_{l} & \cos α_{l} \cos γ_{l} - \sin α_{l} \sin β_{l} \sin γ_{l} & \sin α_{l} \cos γ_{l} + \cos α_{l} \sin β_{l} \sin γ_{l} \\ \sin β_{l} & - \sin α_{l} \cos β_{l} & \cos α_{l} \cos β_{l} \end{matrix}) \\ \times (\begin{matrix} x \\ y \\ z \end{matrix}) . \end{array}

(12)

On substituting from Eq. (12) into Eq. (11), we obtain the following expression for the scattering potential of a particle

\begin{array}{l} f_{l} (r_{n}, ω) = G \exp {- [B_{l}^{(1)} {(x - x_{n})}^{2} + B_{l}^{(2)} (^{y - y_{n}) 2} + B_{l}^{(3)} (^{z - z_{n}) 2} \\ + 2 B_{l}^{(4)} (x - x_{n}) (y - y_{n}) + 2 B_{l}^{(5)} (x - x_{n}) (z - z_{n}) + 2 B_{l}^{(6)} (y - y_{n}) (z - z_{n})]}, \end{array}

(13)

where

B_{l}^{(1)} = A_{l_11}^{2} / (2 σ_{l x}^{2}) + A_{l_21}^{2} / (2 σ_{l y}^{2}) + A_{l_31}^{2} / (2 σ_{l z}^{2}),

(14a)

B_{l}^{(2)} = A_{l_12}^{2} / (2 σ_{l x}^{2}) + A_{l_22}^{2} / (2 σ_{l y}^{2}) + A_{l_32}^{2} / (2 σ_{l z}^{2}),

(14b)

B_{l}^{(3)} = A_{l_13}^{2} / (2 σ_{l x}^{2}) + A_{l_23}^{2} / (2 σ_{l y}^{2}) + A_{l_33}^{2} / (2 σ_{l z}^{2}),

(14c)

B_{l}^{(4)} = A_{l_11} A_{l_12} / (2 σ_{l x}^{2}) + A_{l_21} A_{l_22} / (2 σ_{l y}^{2}) + A_{l_31} A_{l_32} / (2 σ_{l z}^{2}),

(14d)

B_{l}^{(5)} = A_{l_11} A_{l_13} / (2 σ_{l x}^{2}) + A_{l_21} A_{l_23} / (2 σ_{l y}^{2}) + A_{l_31} A_{l_33} / (2 σ_{l z}^{2}),

(14e)

B_{l}^{(6)} = A_{l_12} A_{l_13} / (2 σ_{l x}^{2}) + A_{l_22} A_{l_23} / (2 σ_{l y}^{2}) + A_{l_32} A_{l_33} / (2 σ_{l z}^{2}) .

(14f)

On substituting from Eq. (13) into Eq. (10), we find that that, within the accuracy of the first Born approximation, the pair scattering matrix for the collection of ellipsoids takes form

\begin{array}{l} M (u_{1}, u_{2}; {u^{'}}_{1}, {u^{'}}_{2}; ω) = \sum_{l 1 = 1}^{L} \sum_{l 2 = 1}^{L} \frac{G^{2} π^{3}}{\sqrt{B_{l 1}^{(1)} C_{l 1}^{(1)} C_{l 1}^{(3)} B_{l 2}^{(1)} C_{l 2}^{(1)} C_{l 2}^{(3)}}} \exp [- (\frac{K_{1 X}^{2}}{4 B_{l 1}^{(1)}} + \frac{K_{2 X}^{2}}{4 B_{l 2}^{(1)}})] \\ \times \exp [- (\frac{K_{1 Y}^{2}}{4 C_{l 1}^{(1)}} + \frac{K_{2 Y}^{2}}{4 C_{l 2}^{(1)}})] \exp [- (\frac{K_{1 Z}^{2}}{4 C_{l 1}^{(3)}} + \frac{K_{2 Z}^{2}}{4 C_{l 2}^{(3)}})] \\ \times \sum_{n = 1}^{M_{l 1}} \sum_{m = 1}^{M_{l 2}} e^{- i [K_{2} \cdot r_{2 m} - K_{1} \cdot r_{1 n}]}, \end{array}

(15)

where for

j = 1, 2

C_{l j}^{(1)} = B_{l j}^{(2)} - {(B_{l j}^{(4)})}^{2} / B_{l j}^{(1)},

(16a)

C_{l j}^{(2)} = B_{l j}^{(6)} - B_{l j}^{(4)} B_{l j}^{(5)} / B_{l j}^{(1)},

(16b)

C_{l j}^{(3)} = B_{l j}^{(3)} - {(B_{l j}^{(5)})}^{2} / B_{l j}^{(1)} - {(C_{l j}^{(2)})}^{2} / C_{l j}^{(1)},

(16c)

K_{j X} = K_{j x},

(16d)

K_{j Y} = K_{j y} - \frac{B_{l j}^{(4)}}{B_{l j}^{(1)}} K_{j x},

(16e)

K_{j Z} = K_{j z} - \frac{B_{l j}^{(5)}}{B_{l j}^{(1)}} K_{j x} - \frac{C_{l j}^{(2)}}{C_{l j}^{(1)}} (K_{j y} - \frac{B_{l j}^{(4)}}{B_{l j}^{(1)}} K_{j x}) .

(16f)

If all particles in the collection are identical ellipsoids, i.e., each particle has the same rotation angles

α

β

and

γ

, and, hence, the rotation matrix

R

in Eq. (12) is a constant, Eq. (15) reduces to

\begin{array}{l} M (u_{1}, u_{2}; {u^{'}}_{1}, {u^{'}}_{2}; ω) = \frac{G^{2} π^{3}}{B^{(1)} C^{(1)} C^{(3)}} \exp [- \frac{1}{4 B^{(1)}} (K_{1 X}^{2} + K_{2 X}^{2})] \\ \times \exp [- \frac{1}{4 C^{(1)}} (K_{1 Y}^{2} + K_{2 Y}^{2})] \exp [- \frac{1}{4 C^{(3)}} (K_{1 Z}^{2} + K_{2 Z}^{2})] \\ \times \sum_{n = 1}^{M} \sum_{m = 1}^{M} e^{- i [K_{2} \cdot r_{2 m} - K_{1} \cdot r_{1 n}]} . \end{array}

(17)

When

α = β = γ = 0

and

σ_{x} = σ_{y} = σ_{z} = σ

, Eq. (17) reduces to the case which all particles in the collection are identical spheres, the pair scattering matrix has the form [12

O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056609 (2007). [CrossRef] [PubMed]

]

M (u_{1}, u_{2}; {u^{'}}_{1}, {u^{'}}_{2}; ω) = G^{2} {(2 π)}^{2} σ^{6} \exp [- σ^{2} (K_{1}^{2} + K_{2}^{2}) / 2] \sum_{n = 1}^{M} \sum_{m = 1}^{M} e^{- i [K_{2} \cdot r_{2 m} - K_{1} \cdot r_{1 n}]} .

(18)

4. Example of two correlated plane waves scattering on the collection of ellipsoids

As an illustrative example, we consider an incident field

U^{(i)}

consisting of two mutually correlated homogeneous plane waves propagation along directions

{u^{'}}_{1}

and

{u^{'}}_{2}

, scattered by a collection of ellipsoids with different shapes and orientations (see Fig. 3 ).

Fig. 3 Notation relating to scattering of two correlated plane waves by a collection of ellipsoids with different shapes and orientations.

The spectral amplitude

a^{(i)} (u^{'}, ω)

of the incident field has the following form

a^{(i)} (u^{'}, ω) = a^{(i)} ({u^{'}}_{1}, ω) δ^{(2)} (u^{'} - {u^{'}}_{1}) + a^{(i)} ({u^{'}}_{2}, ω) δ^{(2)} (u^{'} - {u^{'}}_{2}),

(19)

δ^{(2)} (u)

being the spherical Dirac

δ

function. On substituting from Eq. (19) into Eq. (2), the angular correlation function of the incident field can be expressed as the following form

\begin{array}{l} A^{(i)} ({u^{'}}_{1}, {u^{'}}_{2}; ω) = a ({u^{'}}_{1}, {u^{'}}_{1}; ω) δ^{(2)} (u^{'} - {u^{'}}_{1}) δ^{(2)} (u^{'} - {u^{'}}_{1}) \\ + a ({u^{'}}_{2}, {u^{'}}_{2}; ω) δ^{(2)} (u^{'} - {u^{'}}_{2}) δ^{(2)} (u^{'} - {u^{'}}_{2}) \\ + a ({u^{'}}_{1}, {u^{'}}_{2}; ω) δ^{(2)} (u^{'} - {u^{'}}_{1}) δ^{(2)} (u^{'} - {u^{'}}_{2}) \\ + a ({u^{'}}_{2}, {u^{'}}_{1}; ω) δ^{(2)} (u^{'} - {u^{'}}_{2}) δ^{(2)} (u^{'} - {u^{'}}_{1}), \end{array}

(20)

where

a ({u^{'}}_{p}, {u^{'}}_{q}; ω) = 〈 a^{(i) *} ({u^{'}}_{p}; ω) a^{(i)} ({u^{'}}_{q}; ω) 〉 (p, q = 1, 2),

(21)

is the angular correlation function of the incident field. This model incident field while covering the simplest case helps to obtain insight into the process and results of interaction between particulate collections and any random light field. If a random field has a more involved structure, it can always be broken into elementary modes in Eq. (20).

Suppose now that

a ({u^{'}}_{p}, {u^{'}}_{q}; ω)

has Gaussian form, i.e., that

a ({u^{'}}_{p}, {u^{'}}_{q}; ω) = a_{p q} e^{(- k^{2} Δ^{2} / 2) {({u^{'}}_{p} - {u^{'}}_{q})}^{2}} (p, q = 1, 2),

(22)

where

a_{p q}

and

Δ

depend, in general, on frequency

ω

. On substituting from Eq. (20) with the help Eq. (22) into Eq. (7), the cross-spectral density of the total field in the far zone can be expressed as

\begin{array}{l} W^{(t)} (r u_{1}, r u_{2}; ω) = \frac{4 π^{2}}{k^{2} r^{2}} u_{1 z} u_{2 z} {a_{11} M (u_{1}, u_{2}; {u^{'}}_{1}, {u^{'}}_{1}; ω) \\ + a_{22} M (u_{1}, u_{2}; {u^{'}}_{2}, {u^{'}}_{2}; ω) + e^{(- k^{2} Δ^{2} / 2) {({u^{'}}_{1} - {u^{'}}_{2})}^{2}} a_{12} \\ \times [M (u_{1}, u_{2}; {u^{'}}_{1}, {u^{'}}_{2}; ω) + M (u_{1}, u_{2}; {u^{'}}_{2}, {u^{'}}_{1}; ω)] . \end{array}

(23)

Using Eqs. (15) and (23), we obtain for the spectral density of the total far field [see Eq. (8)] along the direction specified by unit vector u the expression

\begin{array}{l} S^{(t)} (r u; ω) = \frac{4 G^{2} π^{5} u_{z}^{2}}{k^{2} r^{2}} {a_{11} \sum_{l 1 = 1}^{L} \sum_{l 2 = 1}^{L} \sqrt{\frac{1}{B_{l 1}^{(1)} C_{l 1}^{(1)} C_{l 1}^{(3)} B_{l 2}^{(1)} C_{l 2}^{(1)} C_{l 2}^{(3)}}} \exp [- (\frac{1}{4 B_{l 1}^{(1)}} + \frac{1}{4 B_{l 2}^{(1)}}) K_{1 X}^{2}] \\ \times \exp [- (\frac{1}{4 C_{l 1}^{(1)}} + \frac{1}{4 C_{l 2}^{(1)}}) K_{1 Y}^{2}] \exp [- (\frac{1}{4 C_{l 1}^{(3)}} + \frac{1}{4 C_{l 2}^{(3)}}) K_{1 Z}^{2}] \sum_{n = 1}^{M_{l 1}} \sum_{m = 1}^{M_{l 2}} e^{- i [K_{1} \cdot (r_{2 m} - \cdot r_{1 n})]} \\ + a_{22} \sum_{l 1 = 1}^{L} \sum_{l 2 = 1}^{L} \sqrt{\frac{1}{B_{l 1}^{(1)} C_{l 1}^{(1)} C_{l 1}^{(3)} B_{l 2}^{(1)} C_{l 2}^{(1)} C_{l 2}^{(3)}}} \exp [- (\frac{1}{4 B_{l 1}^{(1)}} + \frac{1}{4 B_{l 2}^{(1)}}) K_{2 X}^{2}] \\ \times \exp [- (\frac{1}{4 C_{l 1}^{(1)}} + \frac{1}{4 C_{l 2}^{(1)}}) K_{2 Y}^{2}] \exp [- (\frac{1}{4 C_{l 1}^{(3)}} + \frac{1}{4 C_{l 2}^{(3)}}) K_{2 Z}^{2}] \sum_{n = 1}^{M_{l 1}} \sum_{m = 1}^{M_{l 2}} e^{- i [K_{2} \cdot (r_{2 m} - \cdot r_{1 n})]} \\ + e^{- \frac{k^{2} Δ^{2} {({u^{'}}_{2} - {u^{'}}_{1})}^{2}}{2}} Re [a_{12} \sum_{l 1 = 1}^{L} \sum_{l 2 = 1}^{L} \sqrt{\frac{1}{B_{l 1}^{(1)} C_{l 1}^{(1)} C_{l 1}^{(3)} B_{l 2}^{(1)} C_{l 2}^{(1)} C_{l 2}^{(3)}}} \exp [- (\frac{K_{1 X}^{2}}{4 B_{l 1}^{(1)}} + \frac{K_{2 X}^{2}}{4 B_{l 2}^{(1)}})] \\ \times \exp [- (\frac{K_{1 Y}^{2}}{4 C_{l 1}^{(1)}} + \frac{K_{2 Y}^{2}}{4 C_{l 2}^{(1)}})] \exp [- (\frac{K_{1 Z}^{2}}{4 C_{l 1}^{(3)}} + \frac{K_{2 Z}^{2}}{4 C_{l 2}^{(3)}})] \sum_{n = 1}^{M_{l 1}} \sum_{m = 1}^{M_{l 2}} e^{- i [K_{2} \cdot r_{2 m} - K_{1} \cdot r_{1 n}]}]} . \end{array}

(24)

Next, we illustrate the typical behavior of the far-field spectral density calculated from Eq. (24). Unless specified in the captions, the incident waves and the particle parameters are chosen as follows::

λ = 0.6328 \times 10^{- 6} m

a_{1} = 0.6 e^{i π / 7}

a_{2} = 0.9 e^{i π / 6}

{θ^{'}}_{1} = - π / 4

{ϕ^{'}}_{1} = - π / 3

{θ^{'}}_{2} = π / 6

{ϕ^{'}}_{1} = π / 5

k Δ = 1

d = 1

. The angles

θ

and

ϕ

are the polar and the azimuthal angles of the unit vector u in spherical coordinates, i.e.,

u_{x} = \cos θ \cos ϕ

u_{y} = \cos θ \sin ϕ

u_{z} = \sin θ

In Fig. 4 the selected collections later used in Figs. 5 -7 are presented where the ellipsoids are located in the same plane. Figure 5 shows the behavior of the spectral density of the far field produced by scattering of two correlated plane waves. It can be seen from Fig. 5 that with the increase of the number of particles the interference effects gradually disappear and the spectral density distribution becomes more pronounced around two centers corresponding to directions of the incident waves. This result can be used for the detection of the particle number density of the scattering medium.

Fig. 4 Notation of the position of the ellipsoid particles,

R = 2 k^{- 1}

Fig. 5 Contours of the spectral density of the far field, produced by scattering of two correlated plane waves on different number of ellipsoid particles with Gaussian potential. The parameters were taken as:

α = β = γ = π / 4

k σ_{x} = k σ_{y} = 1

k σ_{z} = 1.5

(a) one ellipsoid particle; (b) three ellipsoid particles; (c) five ellipsoid particles; (d) nine ellipsoid particles.

Fig. 7 Contours of the spectral density of the far field, produced by scattering of two correlated plane waves on five identical ellipsoid particles for different orientation (position shown in Fig. 4c). The parameters were taken as:

k σ_{x} = 1

k σ_{y} = 1

k σ_{z} = 1.5

β = γ = 0

, (a)

α = 0

; (b)

α = π / 6

; (c)

α = π / 4

; (d)

α = π / 3

In Fig. 6 the spectral density of the far field produced by scattering of two correlated plane waves on five identical ellipsoid particles with different values of parameter

k σ_{z}

is shown (particles’ positions in this case corresponds to Fig. 4(c)). With the size increase, the interference are diminished and the intensity distribution becomes more concentrated around the two maxima. This relation provides a tool for determination of the typical particles’ deformation.

Fig. 6 Contours of the spectral density of the far field, produced by scattering of two correlated plane waves on five identical ellipsoid particles with different parameter

k σ_{z}

(position shown in Fig. 4c). The parameters were taken as:

α = β = γ = π / 4

k σ_{x} = k σ_{y} = 1

, (a)

k σ_{z} = k σ_{y}

; (b)

k σ_{z} = 1.3 k σ_{y}

; (c)

k σ_{z} = 1.6 k σ_{y}

; (d)

k σ_{z} = 1.9 k σ_{y}

Figure 7 illustrates typical variation of the spectral density distribution of the far field produced by scattering of two correlated plane waves on five identical ellipsoidal particles (see Fig. 4(c)) for different orientation angles. As the orientation angle increases, the interference effects disappear and the distribution also becomes more concentrate. This result can be used for detecting of average particles’ deflection.

Figures 8 and 9 show the spectral density of the field produced by scattering of two correlated plane waves on collections of differently orientated ellipsoids shown as Figs. 10 and 11 . The orientation of the far field significantly changes and becomes similar to that for the ellipsoids in collections with large number of particles. Estimation of the preferential orientation may be based on this dependence.

Fig. 8 Contours of the spectral density of the far field, produced by scattering of two correlated plane waves on five different orientating ellipsoid particles shown as Fig. 10. The parameters were taken as:

k σ_{x} = 1

k σ_{y} = 1

k σ_{z} = 1.5

Fig. 9 Contours of the spectral density of the far field, produced by scattering of two correlated plane waves on three layers different orientating ellipsoid particles shown as Fig. 11. The parameters were taken as:

k σ_{x} = 1

k σ_{y} = 1

k σ_{z} = 1.5

Fig. 10 Illustrating the collection of different orientating ellipsoid particles, (a) Five particles in the same orientation

α = β = γ = 0

; (b) the rotation angle

α = π / 3

of a particle at the center; (c) the rotation angle

α = π / 3

of three particles in the x-direction; (d) the rotation angle

α = π / 3

of all five particles.

Fig. 11 Illustrating the collection of three layers different orientating ellipsoid particles, (a) all particles in the same orientation

α = β = γ = 0

; (b) the rotation angle

α = π / 3

of a layer of particles at

z = d + R

; (c) the rotation angle

α = π / 3

of two layers of particles at

z = d + R

and

z = d

; (d) the rotation angle

α = π / 3

of all particles.

5. Concluding remarks

Based on the scattering matrix approach and first Born approximation the theory for scalar light scattering by deterministic collections of ellipsoids with arbitrarily shapes and orientations is developed. A numerical example relating to intensity distribution produced on scattering of two mutually correlated plane waves from a collection of several ellipsoids with various locations, number, shapes and orientations has been discussed in detail. The results show that the relation between all physical properties of particles and scattered intensity pattern is immediately evident. The presented here theory may help in developing applications for obtaining number density and deformation structure information from the observed intensity pattern for some soft particles, such as, for instance, the blood cells.

As a final remark, we note that on passing from deterministic to random collections of ellipsoids, which would be a natural continuation of this work, it suffices to replace the pair-scattering matrix in Eq. (4) by its counterpart, which takes into account the correlation properties of individual ellipsoids. In collections consisting of ellipsoids with different shapes and orientations both self-correlations among members within the same subgroup and cross-correlations among the members of different subgroups must be taken into account [19

Z. Tong and O. Korotkova, “Pair-structure matrix of random collections of particles: Implications for light scattering,” Opt. Commun. 284(24), 5598–5600 (2011). [CrossRef]

]. It is a much more complex description compared to that for soft spheres which only can be discriminated by size. Ellipsoids of different subgroups can have different shapes and orientations, leading to five more degrees of freedom to be taken into account.

Acknowledgments

Z Mei's research is supported by the National Natural Science Foundation of China (NSFC) (11247004) and Zhejiang Provincial Natural Science Foundation of China (Y6100605). O. Korotkova's research is supported by US ONR (N00189-12-T-0136) and US US AFOSR (FA9550-12-1-0449).

References and links

1.	E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A 6(8), 1142–1149 (1989). [CrossRef]
2.	E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media: errata,” J. Opt. Soc. Am. A 7(1), 173–173 (1990). [CrossRef]
3.	D. G. Fischer and E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A 11(3), 1128–1132 (1994). [CrossRef]
4.	A. Dogariu and E. Wolf, “Spectral changes produced by static scattering on a system of particles,” Opt. Lett. 23(17), 1340–1342 (1998). [CrossRef] [PubMed]
5.	C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express 17(3), 1753–1765 (2009). [CrossRef] [PubMed]
6.	S. Sahin, G. Gbur, and O. Korotkova, “Scattering of light from particles with semisoft boundaries,” Opt. Lett. 36(20), 3957–3959 (2011). [CrossRef] [PubMed]
7.	G. J. Streekstra, A. G. Hoekstra, E. J. Nijhof, and R. M. Heethaar, “Light scattering by red blood cells in ektacytometry: Fraunhofer versus anomalous diffraction,” Appl. Opt. 32(13), 2266–2272 (1993). [CrossRef] [PubMed]
8.	G. J. Streekstra, A. G. Hoekstra, and R. M. Heethaar, “Anomalous diffraction by arbitrarily oriented ellipsoids: applications in ektacytometry,” Appl. Opt. 33(31), 7288–7296 (1994). [CrossRef] [PubMed]
9.	B. K. Wilson, M. R. Behrend, M. P. Horning, and M. C. Hegg, “Detection of malarial byproduct hemozoin utilizing its unique scattering properties,” Opt. Express 19(13), 12190–12196 (2011). [CrossRef] [PubMed]
10.	R. S. Brock, X. H. Hu, P. Yang, and J. Lu, “Evaluation of a parallel FDTD code and application to modeling of light scattering by deformed red blood cells,” Opt. Express 13(14), 5279–5292 (2005). [CrossRef] [PubMed]
11.	E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).
12.	O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 75(5), 056609 (2007). [CrossRef] [PubMed]
13.	S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A 78(6), 063815 (2008). [CrossRef]
14.	S. Sahin and O. Korotkova, “Effect of the pair-structure factor of a particulate medium on scalar wave scattering in the first Born approximation,” Opt. Lett. 34(12), 1762–1764 (2009). [CrossRef] [PubMed]
15.	D. G. Fischer and E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A 11(3), 1128–1135 (1994). [CrossRef]
16.	D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett. 32(24), 3483–3485 (2007). [CrossRef] [PubMed]
17.	Z. Tong and O. Korotkova, “Method for tracing the position of an alien object embedded in a random particulate medium,” J. Opt. Soc. Am. A 28(8), 1595–1599 (2011). [CrossRef] [PubMed]
18.	M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999).
19.	Z. Tong and O. Korotkova, “Pair-structure matrix of random collections of particles: Implications for light scattering,” Opt. Commun. 284(24), 5598–5600 (2011). [CrossRef]

OCIS Codes
(290.5850) Scattering : Scattering, particles
(290.5825) Scattering : Scattering theory

ToC Category:
Scattering

History
Original Manuscript: November 12, 2012
Manuscript Accepted: December 10, 2012
Published: December 17, 2012

Virtual Issues
Vol. 8, Iss. 1 Virtual Journal for Biomedical Optics

Citation
Zhangrong Mei and Olga Korotkova, "Random light scattering by collections of ellipsoids," Opt. Express 20, 29296-29307 (2012)
http://www.opticsinfobase.org/vjbo/abstract.cfm?URI=oe-20-28-29296

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References

E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media,” J. Opt. Soc. Am. A6(8), 1142–1149 (1989). [CrossRef]
E. Wolf, J. T. Foley, and F. Gori, “Frequency shifts of spectral lines produced by scattering from spatially random media: errata,” J. Opt. Soc. Am. A7(1), 173–173 (1990). [CrossRef]
D. G. Fischer and E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A11(3), 1128–1132 (1994). [CrossRef]
A. Dogariu and E. Wolf, “Spectral changes produced by static scattering on a system of particles,” Opt. Lett.23(17), 1340–1342 (1998). [CrossRef] [PubMed]
C. Zhao, Y. Cai, X. Lu, and H. T. Eyyuboğlu, “Radiation force of coherent and partially coherent flat-topped beams on a Rayleigh particle,” Opt. Express17(3), 1753–1765 (2009). [CrossRef] [PubMed]
S. Sahin, G. Gbur, and O. Korotkova, “Scattering of light from particles with semisoft boundaries,” Opt. Lett.36(20), 3957–3959 (2011). [CrossRef] [PubMed]
G. J. Streekstra, A. G. Hoekstra, E. J. Nijhof, and R. M. Heethaar, “Light scattering by red blood cells in ektacytometry: Fraunhofer versus anomalous diffraction,” Appl. Opt.32(13), 2266–2272 (1993). [CrossRef] [PubMed]
G. J. Streekstra, A. G. Hoekstra, and R. M. Heethaar, “Anomalous diffraction by arbitrarily oriented ellipsoids: applications in ektacytometry,” Appl. Opt.33(31), 7288–7296 (1994). [CrossRef] [PubMed]
B. K. Wilson, M. R. Behrend, M. P. Horning, and M. C. Hegg, “Detection of malarial byproduct hemozoin utilizing its unique scattering properties,” Opt. Express19(13), 12190–12196 (2011). [CrossRef] [PubMed]
R. S. Brock, X. H. Hu, P. Yang, and J. Lu, “Evaluation of a parallel FDTD code and application to modeling of light scattering by deformed red blood cells,” Opt. Express13(14), 5279–5292 (2005). [CrossRef] [PubMed]
E. Wolf, Introduction to the Theory of Coherence and Polarization of Light (Cambridge University Press, Cambridge, 2007).
O. Korotkova and E. Wolf, “Scattering matrix theory for stochastic scalar fields,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.75(5), 056609 (2007). [CrossRef] [PubMed]
S. Sahin and O. Korotkova, “Scattering of scalar light fields from collections of particles,” Phys. Rev. A78(6), 063815 (2008). [CrossRef]
S. Sahin and O. Korotkova, “Effect of the pair-structure factor of a particulate medium on scalar wave scattering in the first Born approximation,” Opt. Lett.34(12), 1762–1764 (2009). [CrossRef] [PubMed]
D. G. Fischer and E. Wolf, “Inverse problems with quasi-homogeneous random media,” J. Opt. Soc. Am. A11(3), 1128–1135 (1994). [CrossRef]
D. Zhao, O. Korotkova, and E. Wolf, “Application of correlation-induced spectral changes to inverse scattering,” Opt. Lett.32(24), 3483–3485 (2007). [CrossRef] [PubMed]
Z. Tong and O. Korotkova, “Method for tracing the position of an alien object embedded in a random particulate medium,” J. Opt. Soc. Am. A28(8), 1595–1599 (2011). [CrossRef] [PubMed]
M. Born and E. Wolf, Principles of Optics, 7th ed. (Cambridge University Press, Cambridge, 1999).
Z. Tong and O. Korotkova, “Pair-structure matrix of random collections of particles: Implications for light scattering,” Opt. Commun.284(24), 5598–5600 (2011). [CrossRef]

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