Journal of Spectroscopy

Volume 2019, Article ID 3406319, 8 pages

https://doi.org/10.1155/2019/3406319

## Polystyrene Microsphere Optical Properties by Kubelka–Munk and Diffusion Approximation with a Single Integrating Sphere System: A Comparative Study

Correspondence should be addressed to Ali Shahin; moc.oohay@88nihahs.ila

Received 25 September 2019; Revised 26 October 2019; Accepted 14 November 2019; Published 1 December 2019

Academic Editor: Daniel Cozzolino

Copyright © 2019 Ali Shahin et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

The optical properties of 1 *μ*m polystyrene in the wavelength range of 500–750 nm were estimated by using a white light spectrophotometric transmittance spectroscopy and a single integrating sphere system. To retrieve the optical characteristics, two analytical methods, namely, diffusion approximation and Kubelka–Munk were used, and then their results were compared with Mie theory calculations. The correspondence of the Kubelka–Munk scattering coefficient with Mie was obvious, and relative errors varied between 6.73% and 2.66% whereas errors varied between 6.87% and 3.62% for diffusion theory. Both analytical methods demonstrated the absorption property of polystyrene over the abovementioned wavelength range. Although absorption coefficient turned out to be much lower than scattering, constructing a realistic optical phantom requires taking into account absorption property of polystyrene. Complex refractive index of polystyrene based on these two methods was determined. Inverse Mie algorithm with scattering coefficient was also used to retrieve the real part of refractive index and absorption coefficient for calculating the imaginary part of refractive index. The relative errors of the real part did not exceed 2.6%, and the imaginary part was in consistence with the prior works. Finally, the presented results confirm the validity of diffusion theory with a single integrating sphere system.

#### 1. Introduction

Optical characterization techniques require both an experimental setup to measure the radiometric characteristics and a light propagation method to extract the macroscopic optical properties. Generally, these techniques are categorized as direct and indirect methods [1, 2]. The direct method can be performed only ex vivo, and the samples used in this technique are thin enough so that the multiple scattering events can be negligible, but the advantage of this method depends on simple exponential equations to extract the optical properties [2]. In contrast, indirect methods can be implemented ex vivo and in vivo with thick samples [2–5]. Unfortunately, the estimation of the optical properties via these methods is based on more complicated mathematical principles and analytical and numerical solutions of the radiative transfer equation (RTE) [1–4]. Kubelka–Munk may be considered as the simplest analytical method that has been used for optical characterization, but the drawback of this method is limited to the case of highly scattering medium, and the sample’s thickness should exceed the transport length. Also, this model does not take into account the jump in the refractive index between the sample and surrounding medium [6, 7]. Recently, this approach has been modified to overcome the refractive index mismatch between the sample and holder, and the thickness does not exceed the transport length [7]. The present method has been investigated with a single integrating sphere system over visible and N-IR wavelength range on biological tissues and optical phantoms [7, 8]. On the other hand, diffusion approximation has been considered as an approximate solution of the radiative transfer equation for semi-infinite and high diffusive medium for collimated incident light [9, 10]. That has been used as a quantitative method to retrieve the optical properties of several media using diffuse reflectance spectra which supplies a correlation of measurements with some constraints, i.e., a highly diffusive medium with a large distance between the semi-infinite sample and source. Accordingly, diffusion approximation has been used in optical characterization that has been in agreement to theoretical models such as Mie theory and the Monte Carlo method [11, 12]. Also, this approach has been used with an integrating sphere method in vivo with a large medium to simulate the boundary conditions of this model [13, 14].

On the other hand, polystyrene microsphere has been widely used as a tissue-simulating phantom component to mimic scattering property of a certain tissue. The ability to estimate its optical characteristics accurately via Mie theory calculation that provides a level of validation does not exist in any other optical phantom component. However, most prior research groups deal with this material as a pure scattering without taking into account absorption property and retrieved scattering coefficient by using Mie theory [15, 16]. Thus, precise optical characterization of these materials has an enormous effect on constructing ideal phantoms [15].

The aim of this research is to investigate the validity of diffusion approximation with a single integrating sphere system compared with analytical and theoretical approaches. Thus, albedo, scattering coefficient, extinction coefficient, and complex refractive index of a 1 *μ*m polystyrene microsphere (07310-15, Polybead, Polysciences, USA) were determined by using diffusion approximation and Kubelka–Munk based on radiometric measurements using a single integrating sphere system and collimated transmission spectroscopy. Then, Mie theory calculation was implemented to compare its scattering coefficient with the two models’ results which was considered as a standard method for a spherical particle like polystyrene. Finally, complex refractive index of a 1 *μ*m polystyrene was determined in the range of wavelength 500–750 nm based on these two models’ scattering coefficient and inverse Mie theory calculation.

#### 2. Materials and Methods

##### 2.1. Diffusion Approximation Method

Diffusion equation is a differential equation for fluence rate which can be derived from a radiative transfer equation depending on some assumptions as a highly homogenous scattering property for a semi-infinite medium. For a collimated beam incident normally on a surface and taking into account the refractive index mismatch, the diffuse reflectance from a semi-infinite medium can be expressed as a function of a reduced albedo and an internal diffuse reflectance *r*, as follows [10, 13]:where is the reduced albedo and *k* = (1 + *r*)/(1 − *r*) is a factor related to internal diffuse reflectance *r* which is given by [10, 13]where is the relative refractive index, is the polystyrene refractive index, and is the quartz cuvette refractive index [10, 13].

##### 2.2. Kubelka–Munk Model

Kubelka–Munk theory is an analytical solution of a radiative transfer equation (RTE) based on some presumptions. The simplicity of the Kubelka–Munk method has been the main reason for its widespread uses [6, 7]. This theory introduced special coefficients: *K* is a Kubelka–Munk absorption coefficient and *S* is a Kubelka–Munk scattering coefficient, which are related to macroscopic optical coefficients as follows [1, 6]:

This model has been studied and tested extensively by several research groups on chemical substances as well as biological tissues. Recently, a modified approach has been developed and implemented on tissues and optical phantom components with a single integrating sphere system [7]. The modified method takes into account the mismatch in the refractive index between sample and holder by using a Fresnel transmission coefficient expressed as follows [7]:where *n* is the refractive index of the sample. Scattering and absorption coefficients can be retrieved by measuring diffuse transmission and reflection spectra which are given by these correlations [7]:where is the diffuse transmittance, is the diffuse reflectance, is the collimated transmittance, and *d* is the sample thickness. Since a small concentration of polystyrene was used, the refractive index can be considered as a refractive index of the solvent.

##### 2.3. Mie Theory Calculation

Mie theory is an analytical solution of Maxwell’s equations for the scattering of electromagnetic radiation by a single spherical particle. It provides an exact solution for the scattering and the anisotropy coefficients of perfect spheres [17]. Matzler computer program was used to calculate Mie scattering parameters, i.e., scattering efficiency and anisotropy coefficient that were used to extract the scattering coefficient and anisotropy factor [18, 19]. This method requires particle diameter, fraction, and wavelength of radiation in the vacuum besides refractive index of the particle and refractive index of the solvent [18, 19]. A pure polystyrene sample was an aqueous suspension (07310-15, Polybead, Polysciences, USA), and the fraction of 1 *μ*m polystyrene particles was 2.5% (w/v). In addition, refractive indices of polystyrene as a scattering and water as a host material were found from previous work [20].

##### 2.4. Spectrophotometric Transmission Spectroscopy

A broadband, fiber-based spectrophotometric transmission setup was arranged. That consists of a broadband halogen-tungsten light source (HL-2000-HP-FHSA, Ocean Optics Inc., FL), a fiber-coupled cuvette holder with two collimating lenses (CUV-ATT-DA, Avantes Inc., Netherlands), and a USB portable spectrometer (USB4000 FL, Ocean Optics Inc.). The extinction coefficient of polystyrene samples can be measured within a broad wavelength range from 500–750 nm. The sample was illuminated with a collimated white light beam. Thus, the transmission intensity may be approximated by the Beer–Lambert Law [8, 9]:where is the extinction coefficient, is the intensity of light passed through the studied sample, is the reference intensity, *z* is the optical path length (thickness of cuvette), *c* is the concentration, and is the collimated transmittance [8, 9].

##### 2.5. Integrating Sphere System

A broadband single integrating sphere (819C-IS-5.3, Newport, USA) system was used. A schematic drawing of diffuse reflection measurement using a single integrating sphere system is shown in Figure 1(a). Diffuse transmittance measurement via an integrating sphere can be seen from Figure 1(b). Light from a tungsten-halogen lamp (HL-2000-HP-FHSA, Ocean Optics Inc., FL) was delivered to the sample of interest through a convex lens with a 10 cm focal length and a 1.5 mm pinhole, and the diffuse light emanated from the sample is guided through an optical fiber to the spectrometer (USB4000 FL, Ocean Optics, USA).