## 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.

© 2012 OSA

## 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 [1–5]. For more precise approximation multi-Gaussian potentials may be employed [6].

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 [7–10].

On employing the recently introduced theory for scattering of scalar fields with arbitrary spatial, spectral and correlation properties from deterministic and random collections [11,12] 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 [13,14]. 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 [15–17].

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 ${U}^{(i)}(r;\omega )$ at a pair of points ${r}_{1}$, ${r}_{2}$ and angular frequency *ω* can be characterized by the cross-spectral density function [11]

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]

*x-y*plane. Within the validity of the first Born approximation the scattering matrix has simple relation with the scattering potential [18]with $n(r;\omega )$being the spatial and spectral refractive index distribution within the scatterer:

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

With the help of Eq. (7) we may at once determine the spectrum ${S}^{(t)}(ru;\omega )$ of the total field using the formula [10]

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

*m*th 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

## 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

The variances ${\sigma}_{lx}^{2}$, ${\sigma}_{ly}^{2}$ and ${\sigma}_{lz}^{2}$ 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 ${\alpha}_{l}$, ${\beta}_{l}$ and ${\gamma}_{l}$ 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

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

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

If all particles in the collection are identical ellipsoids, i.e., each particle has the same rotation angles $\alpha $, $\beta $and $\gamma $, and, hence, the rotation matrix $R$ in Eq. (12) is a constant, Eq. (15) reduces to

When $\alpha =\beta =\gamma =0$and ${\sigma}_{x}={\sigma}_{y}={\sigma}_{z}=\sigma $, Eq. (17) reduces to the case which all particles in the collection are identical spheres, the pair scattering matrix has the form [12]

## 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 ${{u}^{\prime}}_{1}$ and ${{u}^{\prime}}_{2}$, scattered by a collection of ellipsoids with different shapes and orientations (see Fig. 3 ).

The spectral amplitude ${a}^{(i)}({u}^{\prime},\omega )$ of the incident field has the following form

Suppose now that $a({{u}^{\prime}}_{p},{{u}^{\prime}}_{q};\omega )$has Gaussian form, i.e., that

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

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:: $\lambda =0.6328\times {10}^{-6}\text{m}$, ${a}_{1}=0.6{e}^{i\pi /7}$, ${a}_{2}=0.9{e}^{i\pi /6}$, ${{\theta}^{\prime}}_{1}=-\pi /4$, ${{\varphi}^{\prime}}_{1}=-\pi /3$, ${{\theta}^{\prime}}_{2}=\pi /6$, ${{\varphi}^{\prime}}_{1}=\pi /5$, $k\Delta =1$, $d=1$. The angles $\theta $ and $\varphi $ are the polar and the azimuthal angles of the unit vector **u** in spherical coordinates, i.e., ${u}_{x}=\mathrm{cos}\theta \mathrm{cos}\varphi $, ${u}_{y}=\mathrm{cos}\theta \mathrm{sin}\varphi $, ${u}_{z}=\mathrm{sin}\theta $.

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.

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 $k{\sigma}_{z}$ 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.

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.

## 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]. 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).

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