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Abstract
In this paper, the Indices of Polarimetric Purity (IPPs) [1-3] have been proposed to analyze the depolarization performances of mono-dispersion and poly-dispersion scattering systems. Here, we mainly investigate the influences of the particles’ density, Refractive Index (RI) of the medium, incident wavelengths, the mixing ratio of bi-dispersion scattering particles and particle-size distributions of poly-dispersion scattering system on the depolarization performances for the backscattering detection. For the mono-dispersion scattering system under same incident wavelength, if the relative RI ratios ($m$) increase linearly, the depolarization performances of the system will first weaken and then strengthen, and of course, the incident wavelength and density of scattering particles will also influence the depolarization performances of the scattering system. For the bi-dispersion scattering system, the proportion of small particles will be negatively correlated with the depolarization property of the dispersion system, and meanwhile, the particle-size distributions will also affect the depolarization performances greatly in the poly-dispersion scattering system. The results demonstrate that the IPPs can be used to describe the depolarization performances of dispersion systems effectively.
© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
As one of the intrinsic properties of light, in recent years, the polarization state of the light has attracted a lot of attentions due to its great application potentials in communications [4,5], navigation [6,7], detection [8] and imaging [9,10]. The polarization state of the light will change after the interaction with a specific material sample, and it will carry the information of the material, so the polarization property of light can be used to characterize the polarization properties of the target object [11]. In recent years, the incidences of human malignant tumors and other diseases have increased year by year, and early diagnosis is crucial to improve the survival rate and quality of life of patients. Relevant clinical research results show that more than 80% of the cancer at early stage originate from superficial epithelial tissue [12]. Compared with other imaging methods, optical polarization imaging can improve the imaging quality of shallow tissue greatly without the inhabitation the multiple scattered photons from deeper tissues, which is a potential auxiliary diagnostic method for the early stage cancer.
In general, the depolarization performances of a medium are influenced by many factors, such as shape, size, the number density, and Refractive Index (RI) (${n_s}$) of the scattering particles in the medium system. Biological tissue is composed of many cells with similar morphological structure and intercellular substances, and its components are complex, so the choice of the relevant optical models is crucial for evaluating the depolarization performances. In the 1990s, Anderson et al. first applied the “polarization difference” method to the detection of skin cancer [13], after which Jacques et al. obtained the “polarization degree” parameter [14]. Jaillon, Wang and Mehrubeoglu got the scattering coefficient, particle size and optical activity of biological tissues by analyzing the spatial distribution of the polarization state of the detecting light [15–17]. Anderson et al. found that polarization difference method could eliminate the influence of reflected light on images when applied to the diagnosis of skin diseases [13]. Subsequently Mourant et al. try to apply the polarization difference method to the differentiation of cancerous tissues [18]. However, the structure of biological tissue is complex, and various fibrous structures in the tissue will influence measurement of polarization parameters. Therefore, researchers still try to find polarization parameters corresponding to other specific microstructures [19]. In recent years, Mueller matrix (MM) [20–27] has attracted more and more attentions as a characterization method which can comprehensively reflect the polarization characteristics of media, and has been preliminarily used for the optical communication [20–22], polarization imaging [23,24], detection of cancerous tissues [26,27]. Since then, in order to solve the problems, in addition to unclear physical significance of MM elements and difficulty in extracting information, MM decomposition [28–30], Indices of Polarimetric Purity (IPPs) [1,2] and other methods have been proposed and applied in biomedical detections [31]. At present, the main problems in the biomedical applications of polarization imaging methods are how to analyze the obtained polarization data and whether there are better parameters to characterize the detected medium.
In our previous researches [20–26], we have studied quantitatively the depolarization behaviors of incident light in mono-dispersion and poly-dispersion systems, which have been measured by degree of polarization (DoP) [26]. In fact, the significance of IPPs relies on the fact that each of them is sensitive to specific depolarization mechanisms. Here, we quantitatively describe the depolarization performances of the medium based on the MM obtained by Monte Carlo (MC) simulation. The four eigenvalues of the 4×4 covariance matrix can be obtained, from which the IPPs of the medium can also be calculated. Meanwhile, the Degree of Polarimetric Purity, also called the Depolarization Index (${P_\Delta }$) [27] of turbid media can be deduced and calculated, which can accurately describe the depolarization performances of scattering environment. Here, we investigate the influences of different parameters’ changes in mono-dispersion, bi-dispersion and poly-dispersion systems on the depolarization ability. For backward scattering, in a mono-dispersion system, simulation results show that the increasing density of scattering particles will lead to the increasing depolarization ability of the medium. Meanwhile, the relative RI ratio and incident wavelength also affect the depolarization performance of the system. In a poly-dispersion system, the simulation results confirm that with increasing proportion of small particles, the depolarization property of the medium decreases for the backscattering detection. The depolarization performances of the scattering systems depend on the mean values and standard deviations of the scattering particle size distributions greatly. For some special biological tissues, we can simply use mono-dispersion system and poly-dispersion system to simulate its scattering processes, which has important guiding significance for early cancer diagnosis.
2. Theoretical background
2.1 Monte Carlo method
The transmission of light waves in scattering systems is a complex process. For recognizing and analyzing them, researchers have established different radiation transmission models and various analytical methods. Among them, MC method which can be used to study statistical problems, has been widely used with the development of computers. Kattawar and Plass were the first to calculate the status of polarization of light using MC method [32]. Since then many papers have been published on MC models and their applications. In fact, MC method, also known as statistical simulation method, is a very important numerical calculation method guided by probability statistics theory, that is, the method of using random numbers to solve many calculation problems. In 2005, Ramella-Roman's team compared three different MC procedures [33] (Meridian plane MC, Euler MC and Quaternion MC), and the polarization states of scattered light after three kinds of processing had been compared. Here, we chose Meridian plane MC to apply to the commonly used MATLAB simulation platform. MC is applied to the transmission of light wave in scattering system, because it is simple and flexible, and most importantly, the simulated results agree with the experimental results well [34,35].
Here, we simulated the transmitting process of 106 photons into the scattering medium, during which the photons would interact with the scattering particles and experience scattering again and again until they reached to a backward detector. The collision process of emitted photons and scattered particles is simplified to simulate the interaction between light and media. Then, the single scattering distance of the photon in the scattering medium can be determined by the following formula:
where $\xi $ is a random number between 0 and 1, ${u_e}$ is the extinction coefficient of the transmission or reflection media, ${u_s}$ and ${u_a}$ represent the scattering and absorbing coefficients, respectively, and all of these parameters can be derived from Mueller's theory.In general, the medium systems we studied include mono-dispersion systems and bi-dispersion systems. Mono-dispersion system has one size of scattering particles, and bi-dispersion system has two different sizes of scattering particles. In a bi-dispersion system, the new ${u_{e\textrm{ - }total}}$ can be expressed as [36]
where ${x_1}$, ${x_2}$ represent the relative volume fraction of the first and second types of particles in a bi-dispersion media system respectively, and ${x_1} + {x_2} = 1$, ${u_{e1}}$ and ${u_{e2}}$ are the extinction coefficients of the two particles respectively.The process of MC simulation involves the probability problem of collision between the photons and particles for a bi-dispersion medium, and the collision probability (${p_k}$) depends on sizes and the number densities of certain type of particles inside the medium [36].
where $k$ equals 1, 2. We use a computer to randomly generate a number between 0 and 1. If the generated random number locates between 0 and ${p_1}$, the photon packet will interact with the first type of particles. Accordingly, if the generated random number exceed ${p_1}$, the photon packet will interact with the second type of particles.2.2 The IPPs of material media
According to the concept of parallel decomposition of Stokes vectors, we can consider the emitting light as a convex linear combination of several incoherent totally polarized states [37]. Since we are studying passive linear optical systems, it can be used as a parallel combination of several pure elements [3]. This model can be justified because there are incoherent scattering processes in the depolarizing optical systems, and the fluctuations of the secondary harmonic in the medium are much slower than the incident wave, so these fluctuations are incoherent [3,38]. The Mueller-Jones matrix can be used to describe a pure non-depolarizing deterministic system, in which all of the completely polarized incident light will result in the completely polarized emitting light. Because of the biunivocal relation between MM and the coherent matrix H(M), any parallel decomposition expressed in terms of H(M) can be directly translated into the corresponding expression in terms of MM, and vice versa [37]. Since H is a positive semi-definite Hermitian matrix, it can be diagonalized through a characteristic transformation. In other words, a general MM of the system can be synthesized as the incoherent sum of four Mueller-Jones matrixs (as shown in Fig. 1) [1,3,37–43].
Fig. 1. The schematic of decomposition from MM to Mueller-Jones matrices: p1M1, p2M2, p3M3 and p4M4 represent respective incoherent components of the optical system.
The Hermitian matrix H was first introduced by S. R. Cloude [44] and R. Simon [45]. Thereafter, the relationship between the Hermitian matrix H and its corresponding Mueller matrix MM can be expressed as follows:
3. Mono-dispersion scattering system
For the non-invasive backward detection of the early diagnosis based on the polarization, the polarization characteristics of the measured tissues should be analyzed to determine whether cancer has occurred. Biological tissue is a very complex system, in which light scattering is produced by a large number kinds of macromolecules, organelles, small water droplets and so on. To simplify the simulation process, we simulated simple tissues, such as skin basal tissues and colon tissues. These tissues have no fibrous structure and are composed of nucleus, organelle and cytochylema, where polarization measurements showed no anisotropy [28,46]. Therefore, we can use the sphere scattering model to simulate these tissues. The model is an infinite plate medium, and the process of biological tissue carcinogenesis can be described by setting the parameters of the scatterers and the medium surroundings the scatterers. The parameters of the scatterers include size, refractive index (RI), scattering coefficient, and so on. The parameters of the medium around the scatterer include RI, absorptivity and so on. Here, the ratio between the RI of scatterer and the RI of surrounding medium is called relative RI ratio, which can be denoted by $m$. Based on previous knowledge, single-layer models can be used to simulate specific biological tissues [47]. The system model is shown in Fig. 2. We first investigate the variation of polarization characteristics due to the parameter variation of mono-dispersion medium, and then consider the case of bi-dispersion systems.
3.1 The influence of particles’ density
Based on previous studies about biological tissue [48], the simulation parameters have been set appropriately. Here, we set up homogeneous mono-dispersion system, in which the medium depth, the radius of the scatterer, and the particle density are set as 1 mm, 1 µm, and 2.0 * 10−5∼3.0 * 10−4/µm3 respectively, and the used RIs of particle and background media are 1.45 and 1.33, respectively. In the simulations, by changing the number of particles, we have investigated the influences of density of particles on the polarization characteristics of the scattering system at 632.8nm. The IPPs of the corresponding scattering system under different densities of particles can be obtained. From Eq. (10), we can calculate and obtain polarization purity (${P_\Delta }$), and the results are shown in Fig. 3.
Fig. 3. Cumulative ${P_\Delta }$ and $\Delta$ of a medium, for backward scattered, as a function of the density of particles.
As shown in Fig. 3, the relationship between the density of particles and polarization purity (${P_\Delta }$) is inversely proportional. It can numerically characterize the depolarization capability of the scattering system. To verify the correctness of the results, the Mueller matrices can also be decomposed by Lu-Chipman decomposition [29] to obtain the depolarization capability of the medium, which can be expressed as,
where ${M_\Delta }$ is a depolarizer, ${M_R}$ is a pure (non-depolarizing) retarder, and ${M_D}$ is a pure diattenuator. Another commonly used depolarization metric is the so-called depolarization power ($\Delta$), which is given by [29,43] where $\Delta \textrm{ = }1$ corresponds to an ideal depolarizer, and $\Delta \textrm{ = 0}$ is associated with a non-depolarizing media.As shown in Fig. 3, we draw the calculation results $\Delta$ as a line graph, which are consistent with our new characterization by IPPs, but the magnitude of ${P_\Delta }$ varies more greatly. With increasing the density of particles, the probability of a photon colliding with the particles increases accordingly, in which photons will undergo more scattering events before reaching to the backward detector. The scattering effect obtained by Mie scattering theory increases gradually, so the depolarization performance of the system will increase accordingly.
3.2 The influences of incident wavelength and relative RI ratio
From the above results, we know that the density of particles plays an important role in the depolarization performance of scattering medium. Here we synthetically consider the depolarization performance of medium with a wide range of incident wavelength of λ (400<λ<700) and relative RI ratios of m (1.01 <$m$ < 1.20). We consider the effect of incident wavelength and relative RI ratios variation on depolarization performance in scattering regime, in which we set the particle size as r = 1 µm, the number of particles as 2.0 * 10−5/µm3 and the ranges of incident wavelength as 400∼700 nm and the relative RI ratios as 1.01∼1.20, respectively. The two influencing factors for the depolarization performances of the scattering medium have been demonstrated in Fig. 4 by ${P_\Delta }$, in which we can observe that in each same incident wavelength, ${P_\Delta }$ increases firstly and then decreases with increasing the relative RI ratios. However, ${P_\Delta }$ reaches the maximum value at different RIs. In other words, as depicted in Fig. 4, with increasing the incident wavelength, the maximum value of ${P_\Delta }$ will move to the higher relative RI ratios.
Fig. 4. Cumulative ${P_\Delta }$ of a medium, for backward scattered, as a function of wavelength and relative RI ratios.
According to theoretical analysis, at the beginning, when the relative RI ratios is very small, the scattering effect is also small, in which a lot of photons will be directly transmitted, and few photons reach to the backward detection surface, so the ${P_\Delta }$ is small. With increasing relative RI ratios, more photons will experience scattering and reach to the backward direction, so the ${P_\Delta }$ tends to increase. When the relative RI ratios are increased to a certain extent, the photons will experience more collisions and are more likely to reach the backward detector due to the increase of scattering effect. So the polarization property of the photon decreases, and then ${P_\Delta }$ tends to decrease again. The depolarization performance of system decreases firstly and then increases with the increasing relative RI ratios. Why does the trend shift as the wavelength increases? This is because the larger the wavelength is and the smaller the corresponding particle size parameters is, so the greater the probability of backscattering is.
4. Poly-dispersion scattering system
4.1 The effect of mixed ratio
The above results demonstrate that the particle size parameters have great influences on the depolarization characteristics of the scattering medium system. All the above works are aimed at the pure mono-dispersion system to approximate the propagation of light inside a realistic medium. In order to simulate biological tissue properly, a mixed scattering medium model with different-size scattering particles need be established. Medium containing different-size particles should demonstrate different depolarization behaviors. With this purpose, the relationship between depolarization behavior and particle size in the medium has been investigated and discussed. Here, we investigate the depolarization performance of a medium with a mixture of two sizes of particles. In order to approximate the scattering effect of the mixed medium of nucleus and smaller organelle, we choose a mixed medium with two-size radii of r = 0.05 µm and r = 3 µm. The incident wavelength is set as 632.8 nm, and the relative RI ratio is selected as $m$≈1.09 in the medium, which is typical for biological tissues with ${n_s}$≈1.45 (organelles and nucleus) and ${n_b}$≈1.33 (cytoplasm). The densities of large particle and small particle are set as 2.0*10−4 /µm3 and 8.0*10−2 /µm3, respectively, and the depth of the medium is set as 28 µm.
Moreover, by using three IPPs magnitudes (P1, P2 and P3) as a coordinate system, a new representation of depolarization performance, the so called Purity-Space [1], can be obtained. Figure 5 shows the effect of the proportion of small particles on the resulted IPPs and ${P_\Delta }$ of the medium. According to the special meaning of each point in the purity space, we know that at “O” point, all IPPs equal to 0, representing the completely depolarized medium, and at “C” point, all IPPs equal to 1, representing the pure medium, namely the absolutely undepolarized medium. When the spots are closer to point “C”, it represents the depolarization performance of the medium is smaller. On the contrary, when the spots are closer to point “O”, it represents the depolarization ability is stronger [1]. As depicted in Fig. 5(a), the colors of the dots change from blue to red, representing the proportion of small particles in the system, in which the red means the smallest and the blue means the largest. As the proportion of small particles in the system increases, the point in the purity space gradually approaches to point “C”, it indicates that the depolarization ability of the media is gradually decreasing. Then we calculate the corresponding ${P_\Delta }$, as demonstrated in Fig. 5(b), from which we can know when the proportion of small particles increases, the value of ${P_\Delta }$ increases. In other words, the depolarization performance of the medium is decreasing. This is because that when the number of small particles increases gradually, more photons with large scattering angles are received in the backward detection, leading to a depolarization weakening. This trend is also consistent with the depolarization characteristics of tissue, which is weakened in the process of cancer [31].
Fig. 5. (a) P1, P2, P3 of a medium varying with the proportion of small particles in the purity space. (b) Cumulative ${P_\Delta }$ of a medium, for backward scattering, as a function of the proportion of small particles.
4.2 The effect of particle-size distribution
In above descriptions, the depolarization performances of mixed scattering system with two kinds of particles is discussed. In fact, the particle sizes in the real tissue will follow a certain distributions, which has a certain impacts on the depolarization performances of the scattering system. Lognormal distribution is suitable for all random processes and can well reflect the distribution characteristics of scattered particles in actual tissues. Therefore, in our Poly-dispersion scattering system, we set the distribution of particles sizes subject to lognormal distribution, which can be expressed as [49]
Here, we set that the poly-dispersion scattering medium contains twelve kinds of particles with different diameters that have the same RI, in which twelve sizes of the scattering particles is determined by each mean value according to logarithmic normal distribution with a certain standard deviations. For example, in our poly-dispersion scattering medium system with the mean value of $v\textrm{ = 2}\textrm{.0}$ µm, if the standard deviation of the logarithmic normal distribution is selected as 0.5µm, we can choose twelve sizes of the scattering particles as 0.400µm, 0.914µm, 1.429µm, 1.944µm, 2.459µm, 2.973µm, 3.488µm, 4.003µm, 4.518µm, 5.033µm, 5.547µm and 6.062µm, respectively. Meanwhile, the concrete number densities for each size particle can be also determined. Of course, if the standard deviation are changed, the sizes and number densities of the twelve particles will also change accordingly, as shown in Fig. 6.
Fig. 6. The number density of twelve-size particle distributions with mean diameter value of $v\textrm{ = 2}\textrm{.0}$ µm and standard deviation of 0.1 µm, 0.5 µm and 0.9 µm respectively.
Then we investigate the depolarization performances of the twelve-size poly-dispersion scattering system with different deviations. As shown in Figs. 7(a)–7(b), we depict the backward IPPs for the twelve-size poly-dispersion scattering system with different mean diameters and standard deviations of $\sigma \textrm{ = }0.01$ µm, $\sigma \textrm{ = }1.05$ µm respectively. The mean diameters of particles increased from 0.1µm to 2.1µm. We can observe that as the mean values increases, the point in the purity space gets closer and closer to point “O”, for both of the standard deviations of $\sigma \textrm{ = }0.01$ µm and $\sigma \textrm{ = }1.05$ µm. It means that the difference in standard deviation will not change the overall trend of the depolarization performances. In general, the larger the particle, the higher probability of the photon scattering forward, and the photons will undergo more scattering times before they reach to the backward detector, which will cause a decrease in the DoP of the outgoing light. As depicted in Figs. 7(a)–7(b), when the standard deviation is very small, the depolarization performances of the scattering system changes more with increasing mean values of particles’ radius. It may be due to that more dispersive particle distribution will result in the average effect. We also explore the depolarization ability of the scattering system varying with the standard deviations under different mean values, in which we set the mean values of particles’ radius as $v\textrm{ = }0.1$µm and $v\textrm{ = }3.0$ µm, corresponding to Rayleigh scattering system and Mie scattering system respectively. As shown in Table 1, with increasing the standard deviation, the IPPs of the system gets smaller for both of the systems with mean values of $v\textrm{ = }0.1$ µm and $v\textrm{ = }3.0$ µm. This phenomenon indicates that in poly-dispersion scattering system, depolarization performances and standard deviations of system with the same mean value are positively correlated. When the standard deviation is larger, and the system contains a larger range of particles sizes, collisions between the launched photons and particles will be more random, and the depolarization effect will increase too. However, for the poly-dispersion system with smaller mean size particles, for different standard deviations, the changing ranges for three numbers (P1, P2, P3) of IPPs are relative larger. Comparing Table 1 with Table 2, the IPPs of scattering system with mean value $v\textrm{ = }3.0$ µm is smaller than the IPPs of system with mean value $v\textrm{ = }0.1$ µm, which agree well with the above results. Above results illustrate that the depolarization performances of the poly-dispersion scattering system depend on the mean values of particles’ radius and standard deviations of the scattering particle size distributions greatly, and most importantly, the depolarization performances of the scattering system can be characterized clearly by the IPPs effectively.
Fig. 7. P1, P2, P3 of a medium varying with different mean values for standard deviation of $\sigma \textrm{ = }0.01$ µm (a) and $\sigma \textrm{ = }1.05$ µm (b) in the purity space.
Table 1. IPPs and ${P_\Delta }$ of the poly-dispersion scattering system with mean value of $v\textrm{ = 0}\textrm{.1}$ µm (standard deviations range: 0∼0.65).
Table 2. IPPs and ${P_\Delta }$ of the poly-dispersion scattering system with mean value of $v\textrm{ = 3}\textrm{.0}$ µm (standard deviations range: 0∼0.65).
5. Conclusion
In this paper, based on the IPPs [1], we have mainly investigated the depolarization performances of the medium by the incident lights propagating in the mono-dispersing and poly-dispersing systems, in which the depolarization ability of the medium is represented numerically by ${P_\Delta }$ [27]. For a mono-dispersing system, our simulated results show that the depolarization performances of the medium are related to incident light wavelength, density of particles and relative RI ratios. The depolarization behavior of the medium is positively related to the density of particles, and the incident wavelength is negatively correlated with the depolarization performances of the scattering system. For the influence of relative RI ratios on depolarization performance of scattering regime, there is a tendency of increasing firstly and then decreasing. For a two-size particles’ media system, the results demonstrate that the evolution of depolarization performance of the bi-dispersing system is greatly influenced by the volume fraction of small particles in the medium. With increasing the proportion of small particles, the depolarization ability of the medium in the backward detection decreases gradually. For the poly-dispersing scattering system, by the coordinate system in Purity Space, we have also investigated their depolarization performances, from which we can obtain that the mean values and standard deviations of the scattering particle size distributions have great influences on the depolarization effect. The above results demonstrate that the IPPs can be used to describe the depolarization performances of dispersion systems effectively. And the IPPs is also expected to be efficiently used in other polarization technologies, such as the polarization detection, polarization imaging, polarization communications and so on.
Funding
National Natural Science Foundation of China (11804073, 61775050); Natural Science Foundation of Anhui Province (1808085MF188, 1808085QA21); Fundamental Research Funds for the Central Universities (JZ2018HGBZ0309, JZ2018HGTB0240, PA2019GDZC0098); Anhui Key Laboratory of Polarization Imaging Detection Technology (2018-KFJJ-02).
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