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A homemade Static Light scattering studies has been used to determine angle resolved scattered intensity for different polarization states of the incident laser light. Classical light scattering set ups are being used to study morphological aspects of scatterers using simple set ups using low power lasers. Red blood cells form rather interesting as well as a challenging system for scattering experiments. The scattering spectrometer consists of a scattering arm, a scattering turn table and collimating arm. Along with polarizers integrated in the collimating arm as well as scattering arms ensures collection of scattered flux with the required polarization state. This technique is being developed for its
*in vitro* studies using fresh red blood cells. A brief review of the theoretical models used for scattering from Red Blood Cells (RBC) has been discussed in the paper. Scattering pattern (scattering plots) as well as polar plots of scattered flux have been determined for different polarization state of the incident light. Insight into the orientation of major axis of particles can be inferred from the polar plots.

Static light scattering is a classical field the birth of which may be assigned to Rayleigh scattering and the explanation of blueness of sky [

The scattering in any direction is described by four amplitude functions, S_{1}, S_{2}, S_{3} and S_{4} all functions of θ and f which form a matrix S(θ, f) of four elements. The definition of S(θ, f) is

The corresponding intensity matrix F has 16 elements. By making θ = 0, we obtain the four complex matrix components S_{2}(0), S_{2}(0), S_{3}(0), S_{4}(0) forming the matrix S(0) of four elements. The electric field at a point P beyond the slab is

where q = 2pNlk^{−}^{2}.

In red blood cells, the size typically varies from 6 μ to 8 μ and has a biconcave disc shape. An ensemble of randomly oriented dilute Red Blood Cells (RBC) will therefore be well suited model system for scattering.

The spatial arrangement of particles comes to picture, when one takes into account the scattering from multiple particles. While single particle scattering takes into account the scattering from simple particles, multiple scattering has been always a mathematical difficulty. Experimentally, for multiple scattering events, it is difficult to convolute for different scattering events. With a perspective of colloidal interactions, it is well characterized by the pair potential existing between the particles which is well known from DLVO theory for colloids. Even though the nature of potential is not well known, the Derjaguin, Landau, Verwey and Overbook (DLVO) potential along with the Resealed Mean spherical approximation by Henson and Hayter provides a good model to explain the many particle interaction and stability of particle in a medium [

In the opposite regime, if the attractive forces dominate, the particles tent to become clusters. These clusters in the case of red blood cell (RBC) clusters arc called rouleaux [

The normal disc shaped red blood cells are shown in

Another situation is the sickle cell anemia in which the RBC takes the shape of sickle (see

A typical set-up for the static light scattering experiment consist of a laser beam illuminating a sample and a detector set up at scattering angle θ measuring the intensity

I(θ, t) of the scattered light [

Here E_{i}(r, t) is the electric field, E_{0} is the Amplitude and I denotes the incident light intensity. The polarization n_{i} is perpendicular to scattering plane. k_{i} is the incident light wave vector k_{i} = 2πn/λ_{i} and ω_{i} is the angular frequency which varies in time.

In a colloidal state, if the particles are charged to form ordering, and if we ignore multiple scattering, the scattered electric field may be written simply as a summation:

Here is ω_{i} written as ω because the scattering is elastic.

Light scattering from blood or of tissues are important in many medical applications. The light diffusion in tissues or cell suspension is important photodymanic laser therapy [

It is known that the scattering and absorption of light in blood are largely governed by the red blood cells. It is the refractive index of RBCs, as well as their size, shape and orientation that determine how light propagates. The size of an RBC is typically 6 - 10 wavelengths in the optical region and the full wave methods require fast computers with large RAM, when applied to such large objects. The contrast between the blood cell and the surrounding plasma is quite small. For scattering at a vacuum wavelength of 630 nm the typical values for the index of refraction is 1.40 for a blood cell and 1.35 for the plasma. Thus the blood cell forms a weakly scattering system. The different methods used to model RBC are described as follows:

1) FDTD method

2) The T-matrix method

3) Mie scattering

4) DDA method

5) The superposition approximation,

6) The Rytov approximation.

In general, the normal RBC shape is modeled as discocyte, i.e. an axially symmetric disk, slightly indented on the axis and having the mirror plane symmetry perpendicular to the axis. The information from these calculations is useful in the development of models of light propagation in whole blood containing many RBC.

In the simulations the refractive index for the RBC is set to n_{1} = 1.406 and the refractive index of the surrounding blood plasma is set to n_{2} = 1.345. The absorption is neglected in both regions. The model of the disk-like normal RBC used in the simulations is defined in the references [^{3}. The membrane of RBC has a negligible influence on the scattered field, and hence the RBC models do not include the membrane or any other internal structure.

cross section around the z axis.

where k = n_{2}w/c_{0} is the wave number for the plasma and c_{0} is the velocity of light in vacuum. The RBC is rotated around the y―direction in order to change the angle of incidence. The rotation angle of the axis of symmetry relative to the z-direction is denoted θ_{i}.

The scattering probability was calculated as a function of the zenith scattering angle θ_{s} This scattering probability, P(θ_{s}), was calculated by numerical integration of the differential scattering cross section, σ_{diff}(θ_{s}, f_{s}), over all azimuthal angles fξ[0, 2π]:

The differential cross section is defined as the time averages of the Poynting vector of the scattered and incident fields, respectively. Furthermore, r is the radial unit vector, E(r, θ, f) is the scattered electric field, H*(r, θ, f) is the complex conjugate of the

where

Corresponding magnetic field. and n_{0} = 120 πΩ is the wave impedance of vacuum.

The FDTD algorithm was originally proposed by K. S. Yee in 1966 [

Because of the finite computational domain, the values of the fields on the boundaries must be defined so that the solution region appears to extend infinitely in all directions. With no truncation conditions, the scattered waves will be artificially reflected at the boundaries leading to inaccurate results. The perfectly matched layer (PML) ABC suggested by Berenger [

The field formulation forms the basis of this formulation which considers the scattering patterns and computes the corresponding total field/scattered fields. The computational grid are created for computations and is divided into two regions. The total field region encloses the scattering particles. whereas in the scattered field region, only the scattered field matrix are stored. At the boundary between the two regions, special boundary conditions are needed to connect the fields in different regions. Here, the incident field is either added or subtracted from the total field. The conditions used for the simulations in the paper are described in detail in Ref [

The FDTD is inherently a near-field method. To determine the far-field scattering pattern, the near-field data is transformed to the far-field by the near-field to far-field (NFFF) transformation. The details of the NFFF technique can be found in [

This method as known as null field method was formulated by Waterman [

Mie scattering is celebrated a theory formulated by Gustav Mie [

The Discrete Dipole Approximation (DDA) is closely related to the method of moments. The original derivation of the method is less rigorous and does not rely on the volume integral equation. The principle of the method is as follows: The scattering volume is divided into N parts, each part is small enough to be represented by a dipole moment. The induced dipole moment of each volume element is equal to the electric field in the volume multiplied by the polarizability of the volume. The electric field is a superposition of the fields from the sources external to the object and the electric fields from the sources inside the object. The field from the external sources is the incident plane wave and hence the electric field in a volume j is given by

where in the term _{j} from a dipole P(r_{k}). It can be found in basic textbooks in electromagnetic theory. In iteration method i.e. the conjugate gradient method is applied to solve the equations. More details can be found in the references [

The superposition approximation is based on the assumption that multiple scattering effects are small within the cell. The scattering object may be divided into several parts each part being a scattering objects. The multiple scattering effects between the objects are neglected. The advantage is that the volume of the scattering objects is reduced implying a reduction of the CPU-time and the required RAM of the computer. In the example a blood cell was divided in two halves through the yz-plane. The far-field pattern E_{f} in yz-plane for each half was calculated by the FDTD method. Then the two far-fields were added and compared with the same far-field from the whole blood cell. The patterns agree very well. The far fields are the same except for the phase shifts. The superposition approximation facilitates the calculation of far-fields for one RBC and then sum up for phase shifts to get the far fields for the RBCs. The far field approximation is thus calculated using the equation:

where r is the translational vector of RBC n relative to the origin.

The Rytop approximation is a frequently used method in tomography [_{1} for the object and n_{2} for the surrounding medium. The approximation assumes that when the wave passes the object, the phases of two waves are shifted while its amplitude and polarization are unaltered. The wave is assumed to travel as straight rays parallel to z-axis, Let d(x, y) be the total distance that ray travel inside scattering object. If z = z_{1} is a plane behind the object, the total electric field in that plane may be computed as

Thus the phase is shifted to an angle

where S is the plane z = z_{1}. All reflection of the wave is neglected. The approximation gives far field amplitude that is only accurate for the angles θ_{s} < π/2. The code for the far field amplitude is very simple and short and for that reason this method is alternative to far more time and memory requiring full wave methods.

It consists of homemade light scattering Spectrometer in which a laboratory available spectrometer has been modified for acting as the scattering setup. The set up is a classical light scattering setup with a 1) Collimating Arm, 2) Scattering Turn Table, 3) Scattering Arm.

This has been shown in

Collimating Arm consists of a pinhole tube sufficiently wide for the laser beam to pass through and is made up of a aluminum. Following this is a lens and a polarizer system. The polarizer can be rotated in the collimating arm, so as to get required polarization state for the incident laser beam. A separate circular polarizer (quarter wave plate) can be inserted whenever one needs circular polarized light. By setting the polarizer and the circular polarizer one can get circular or a elliptical polarized light.

Scattering Turn

Scattering Arm consists of a pin hole which collects the scattered flux from the scattering cell. The scattered light flux is collected in the same plane as the incident light flux. The scattered light is collected by the lens and iris diaphragm, and further the scattered light is fed to a photo diode. The iris can be opened or closed, as on our will. In between the lens and the iris, a polarizer is again introduced so as to get vertical-vertical (VV) for horizontal-horizontal (HH) or vertical-horizontal (VH) polarization state. The laser used here is a 2 mV Helium Neon laser with 632.5 nm wavelength. The full photograph of experimental setup is shown in

Sample preparation has been done in the university health centre. Human blood (5 ml) was injected out into a centrifuging tube and was citrated with sodium citrate solution. This was rotated at 3000 rpm. Blood plasma from this has been removed, and 1 ml of Red Blood Cell (RBC) has been injected out to a separate tube. This has been diluted (as required) to make a very dilute system. It has been insured that the RBC cells do not aggregate. This is done by using a buffer as the diluting liquid. The pH has been kept at a value around 7.0. Glass cell of about 8 mm outer diameter has been fabricated for this work. The scattering data is typically collected between angles (θ) of 20˚ to 130˚ with an increment of 5˚. Since the cells remain alive approximately for 3 hours, the experiments are to be finished before three hours. The photograph of the set up is shown in

Angle resolved scattered Intensity have been determined and plots have been plotted between Scattered Intensity Vs Angle for

1) Unpolarized Incident light

2) Vertically Incident polarized light

3) Circularly Incident polarized light

Polar plots are plotted by first fixing an angle (θ = 30˚, 60˚, 90˚, 120˚) and then varying the Azimuthal angle (φ) by rotating the Analyzer in the scattering arm.

re 4.10 and 4.11 shows the similar polar plots for the angle (θ = 90˚, 120˚). These two plots show increase scattered flux for azimuthal angle (φ = 30˚) which is different from the polar plot for θ = 30˚ and 60˚. Polar plots using Incident circularly polarized light for θ = 30˚, 60˚, 90˚, 120˚ have been plotted in

Polar plots using vertically polarized Incident light have been plotted in

The scattered intensity Vs angle plots show similar variation. This nature of variation is not similar to that have seen in literature. Nevertheless variation should be compared with theoretical calculations to get a good insight into the morphological conditions of scatteres as well as its orientation.

The present investigation is carried out to study the scattering properties of RBC cells with different polarized laser light as individual scatterrers. As it is evident from literature, RBC can form clusters or aggregates known as rouleuax. However, the present study is focused on single particle scattering from RBC which may be modeled as “disc shaped” particles. They can be studied as Mie scatterers. Nevertheless, the theory is formulated basically for spherical (larger) particles. RBC being discs can orient in many directions and to get insight into this phenomenon, we have plotted the polar plots. Polar plots with unpolarized, circular polarized and linear polarized incident light have been plotted. Scattered intensity Vs Angle shows similar angular variations for all polarization work.

Similar variation for polarization variations is seen in literature [

The paper reports the fabrication of indigenously developed scattering spectrometer and it reports angle resolved scattered intensity plots. Also the Polar Plots have been plotted to show the significance of orientations. A quest of Ideal single particle scatters is underlying such experiment and theoretical studies.

Joseph, D. and Kumar, A. (2016) Static Laser Light Scattering Studies from Red Blood Cells. Optics and Photonics Journal, 6, 237-260. http://dx.doi.org/10.4236/opj.2016.610025