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Abstract
Propagation of a coherent light beam through a random medium generates speckle patterns, in which some information of media and object is hidden. Speckles produced by particles smaller than wavelength are studied thoroughly, yet it is also essential to investigate speckles produced by larger particles. In this paper, the spatial intensity correlations of transmitted speckle patterns generated by large particles are studied theoretically and experimentally. A semi-empirical expression of spatial intensity correlation function of speckle patterns is derived based on Bethe-Salpeter equation, taking particle size and concentration into account. After performing experiments with various particle sizes and concentrations, we fit the theoretical expression to experimental results and determine the introduced parameters. We analyze the variation of spatial intensity correlation function with particle size and concentration. Theoretical analyses and experimental results given in this paper have potential applications in coherent imaging through random and disordered media.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Coherent light scattering by random media produces large intensity fluctuations called speckle patterns. It has been proved that speckle patterns are stochastic but with hidden information about the object to be imaged and the media. Statistical methods are always used to get the information [1–3]. There are theories and techniques analyzing the statistical properties of speckle patterns, and their relevance to the object and the random media [4–9].
Theories based on mesoscopic method are abundant on speckles produced by particles of much smaller size than illuminating wavelength, while those particles are called point scatterers [2,3,10,11]. For point scatterers, spatial and angular intensity correlations of speckle patterns have been thoroughly studied, and are divided into three parts: short-range correlations ($C_1$), long-range correlations ($C_2$) and infinite-range correlations($C_3$) [4,12–18]. The reflection-transmission correlations of speckle patterns have also been investigated plentifully [19–21]. There have been experimental methods to investigate the relations between speckle patterns and particles [6–8,22–25]. Scatterers include red cells, bacteria and monodisperse spherical particles of various sizes, while analyzing arguments includes speckle contrast, speckle spot size and fractal dimension.
Natural environments always contain a large amount of particles that have a wide range of sizes [3,26]. Light scattering in such media also generates speckles. Nevertheless, as shown above, theories based on mesoscopic methods have not discussed the speckle patterns for large particles yet. On the other hand, experimental reports for large particles have not given theoretical investigations, nor the specific theoretical expression of correlation function of speckle patterns. Here large particles stand for those much larger than illuminating wavelength. Thereby, the aim of this paper is to study the Spatial Intensity Correlation Function (SICF) of transmitted speckle patterns generated by monodisperse large particles of various sizes and concentrations, theoretically and experimentally.
This paper is organized as follows. In Section 2.1, we profile the multiple scattering theory and speckle theory based on mesoscopic method, and demonstrate the derivation of SICF for point scatterers. In Section 2.2, we calculate the first-order and second-order statistics of large particles, and derive a semi-empirical theoretical expression concerning particle size and concentration by analyzing the scattering principle for large particles. Experimental setup is shown in Section 3.1 with the calculation of SICF in Section 3.2. In Section 4, the experimental results of SICF are illustrated, and we fit the theoretical expression of SICF to that of experiments and analyze all the results. Section 5 gives the conclusion.
2. Theoretical analysis of speckle patterns produced by large particles
2.1 Multiple scattering and short-range correlations for point scatterers
We will profile mesoscopic theory for multiple scattering by point scatterers. Firstly, let us introduce the first-order and second-order dielectric correlation function. For point scatterers it reads [11]
Light scattering in random media is described by Dyson equation which reads [2,3]
in Fourier domain. $\mathbf {k}$ is frequency coordinate, and $\langle G(\mathbf {k}) \rangle$ represents average Green function. On the right-hand side of Eq. (2), $k_{\mathrm {eff}} = k_0 + \mathrm {i} \mathrm {Im}(k_{\mathrm {eff}})$ is the effective wavenumber with $k_0$ the wavenumber in host media, $\mathrm {i} = \sqrt {-1}$ and $\mathrm {Im} (\cdot )$ means imaginary parts. The imaginary parts is calculated theoretically by the dielectric correlation functions of all orders in the medium [2].Light fields correlations when scattering in random media are governed by Bethe-Salpeter (BS) equation. In ladder approximation and in the case of point scatterers, BS equation is simplified to [10,11]
Equation (3) is the BS equation in the ladder approximation, where the superscript $^*$ denotes complex conjugate and $\langle \cdot \rangle$ means ensemble average. $\langle G(\mathbf {r} , \mathbf {r}') \rangle$ represents average Green function given by Eq. (2), and $\langle E(\mathbf {r}) \rangle$ is the average field at position $\mathbf {r}$. $E(\mathbf {r}_0)$ corresponds to the incident field. $L(\mathbf {r}_1,\mathbf {r}_2)$ is the ladder propagator containing all the scattering sequences from $\mathbf {r}_1$ to $\mathbf {r}_2$.
We would like to adopt diagrammatic approach to demonstrate Eq. (3). The diagrammatic rules are as follows: [3]
(1) A solid line corresponds to the trajectory of a field.
(2) A dashed line corresponds to the trajectory of a conjugated field.
(3) A circle corresponds to one point (for point scatterers it also means one single particle).
As is shown in Fig. 1, two fields get into the medium and both reach $\mathbf {r}_2$. Ladder propagator $L(\mathbf {r}_1,\mathbf {r}_2)$ takes the two fields to propagate from $\mathbf {r}_2$ to $\mathbf {r}_1$ parallel. And every twin scattering events occur at the same point. Two fields are limited to identical trajectories, so that they keep their correlations. Otherwise they will lose correlations and fall into the first term of Eq. (3). Finally two fields separate in $\mathbf {r}_1$ and reach $\mathbf {r},\mathbf {r}'$ respectively.
Ladder propagator contains all possible scattering sequences connecting $\mathbf {r}_2$ to $\mathbf {r}_1$. It varies smoothly with $\mathbf {r}_1$ so we may as well take it out from the integration of Eq. (3). Then we get [3]
Intensity correlations are more cared about in practice since it is easier to measure. It is theoretically derived from field correlations. Three types of intensity correlations have been given in Appendix A. Substituting Eq. (4) into Eq. (22) leads to the expression of $C_1$ [2–4,17]
Equation (5) gives the expression of short-range ($C_1$) spatial correlation function for point scatterers. On the other hand, angular correlations lead to the well-known memory effect [5]. We will focus on spatial correlations in the following. Equation (5) is independent of the size of particles, hence only applicable to point scatterers. We will see in Section 4 that correlation function varies with size and concentration for large particles.
The other types of correlations, whether for point scatterers or for large particles, will not be discussed in this paper since $C_1$ correlations dominate in transmission speckles (see Appendix A for more details) [3,27].
2.2 Spatial intensity correlation function of speckle patterns for large particles
It is also necessary to start with the dielectric statistics when analyzing fields correlations for large particles. We begin with a single homogeneous spherical particle of radius $R_0$ and dielectric constant $\epsilon _p$, embedded in a homogeneous host medium of dielectric constant $\epsilon _0$, with dielectric function $\epsilon (\mathbf {r})$ in whole space satisfying [28]
Taking Fourier transform of Eq. (7) we get [29]
In a group of identical particles, we assume the position of each particle is stochastic in whole space and any two particles have no correlations. By definition, the correlation function $B \left ( \mathbf {r} - \mathbf {r}' \right )$ of $\delta \epsilon (\mathbf {r})$ is given by [3]
The Fourier transform of $B\left ( \mathbf {r} - \mathbf {r}' \right )$ is, according to Eq. (9), given by
Substitute Eq. (8) into Eq. (10), we get
Equations (7) and (11) represents the first-order and second-order statistics of particles. It is seen that, $B \left ( \mathbf {r} - \mathbf {r}' \right )$ tends to be a Dirac delta function if $R_0$ approaches zero, which corresponds to point scatterers given by Eq. (1).
We would like to take diagrammatic approach to demonstrate light field correlations in large particles. The diagrammatic rules are as follows:
(1) A solid line corresponds to the trajectory of a field.
(2) A dashed line corresponds to the trajectory of a conjugated field.
(3) A small circle corresponds to one point.
(4) A large dotted circle corresponds to one single particle.
Correlations of two fields come from identical trajectories for point scatterers. For large particles however, two fields follow parallel but no longer identical trajectories, as shown in Fig. 2. Every twin scattering events happen at two close positions instead of a certain point. The ladder propagator is now related to four points, which is $L(\mathbf {r}_1,\mathbf {r}'_1,\mathbf {r}_2,\mathbf {r}'_2)$ [3]. It takes the two fields to propagate from $(\mathbf {r}_2,\mathbf {r}'_2)$ to $(\mathbf {r}_1,\mathbf {r}'_1)$ parallel. Two fields are no longer at the same point in every twin scattering events. They separate at $\mathbf {r}_1,\mathbf {r}'_1$ and finally reach $\mathbf {r}$, $\mathbf {r}'$ respectively.
BS equation in this case is expressed by [11]
In this situation, the ladder propagator cannot be solved exactly, and all scattering sequences contribute to correlations of fields. Also the exact form of correlation function can no longer be derived from Bethe-Salpeter equation. However, an assumption could be made to get a semi-empirical expression of SICF.
Field correlations are related to all the dielectric correlation functions, as well as propagation of each field, as shown in Fig. 2. Even the propagation before $(\mathbf {r}_1,\mathbf {r}'_1)$ counts. Thereby, we assume that $L(\mathbf {r}_1,\mathbf {r}'_1,\mathbf {r}_2,\mathbf {r}'_2)$ could be divided into three parts
Substituting Eqs. (13) and (14) into Eq. (12), and taking $L_p \left ( \mathbf {r}_1,\mathbf {r}_2 \right )$ out from the integration yield
Unlike point scatterers, the term $L_p \left ( \mathbf {r}_1,\mathbf {r}_2 \right )$ has effect on the field correlation function. The effect could be included in average Green function, as $L_p \left ( \mathbf {r}_1,\mathbf {r}_2 \right )$ is taken out. In this case, the Green function in Eq. (15) consists of both the actual value and the contribution from ladder propagator. Thus we introduce the equivalent wavenumber, and replace $k_{\mathrm {eff}}$ in Green function with ${k_{\mathrm {eq}}}$ based on the above analysis.
Substituting Eq. (4) into Eq. (22), and performing the integration in Fourier domain, we get
Equation (16) has given a semi-empirical expression for SICF. There are two arguments, $\mu$ and $k_{\mathrm {eq}}$, to be determined by fitting. $\mu$ is the proportion of the dielectric correlations in ladder propagator to the function $B(\mathbf {r}-\mathbf {r}')$, in terms of spatial scale. $k_{\mathrm {eq}}$ consists of two part. One is the value of effective wavenumber $k_{\mathrm {eff}}$. The other is the contribution from ladder propagator. We will fit Eq. (16) to experimental results in Section 4. Following the form of $k_{\mathrm {eff}}$ for point scatterers, we express $k_{\mathrm {{eq}}}$ by
with $\mathrm {Im} \left ( k_{\mathrm {{eq}}} \right )$ means the imaginary part of $k_{\mathrm {{eq}}}$.3. Measurement of spatial intensity correlations of speckle pattern
3.1 Experimental setup
The experimental setup is shown in Fig. 3. A 532nm laser beam (100-mW solid-state laser) is expanded as the incident wave. The particle suspension is held in a transparent container with thickness of 1cm. An imaging system composed of a 20$\times$ microscope objective, a 300-mm convex lens and a CMOS camera (Daheng MER2-231-41U3M) respectively, is placed behind the suspension to record transmitted speckle patterns. This imaging system is used to enlarge speckle patterns, since the average size of speckle spot in this experiment is smaller than or comparable to the pixel size of camera [21]. The actual magnification of the optical system is also related to the location of objective and lens. In this experiment, we use microscope calibration ruler to determine the actual magnification. And the object plane is in the internal surface of the wall of container, the one that is closer to the microscope objective.
Samples scattering light are polystyrene microspheres dispersed in deionized water, for the density of polystyrene is close to water and the microspheres will suspend for much longer time than recording period. For each set, we homogenize the suspension before recording. Use of deionized water prevents the possibility of particle aggregation caused by electrostatic interaction. Particles in suspensions are moving continuously, resulting in speckle pattern changing in time. And hence we set exposure time to be $10^{-4}\mathrm {s}$ to make sure that each speckle pattern corresponds to one single realization of suspension.
To investigate the variation of speckles with particle size and concentration, we used suspensions of five particle sizes, and different concentrations for each size. Particle size mentioned here is double the $R_0$. For every set of given size and given concentration, we recorded 1000 speckle patterns. Figure 4 shows one single speckle pattern recorded in this experiment.
Fig. 4. A sample of speckle pattern recorded, for particles of 10$\mathrm {\mu }\textrm {m}$ size, and volume concentration $c_p = 3.13 \times 10^{-3}$.
3.2 Calculation of spatial intensity correlation function (SICF)
Speckle patterns recorded in experiments are composed of three parts, diffuse intensity $\delta I$, ballistic intensity $\langle I \rangle$ and the unwanted pattern. The unwanted pattern stems from interference effect in optical system when coherent light is used. Ballistic intensity is the average intensity contributing uniformly to intensity correlation, and thereby is always neglected. Diffuse intensity changes in time but the other two do not, which means it is possible to extract the diffuse intensity from the whole speckle.
We denote a measured speckle pattern by $I_m (\mathbf {r})$, and express it by
where $I_i (\mathbf {r})$ stands for unwanted pattern, and $I (\mathbf {r}) = \delta I + \langle I \rangle$ is the ideal speckle pattern.The correlation function $C\left ( \Delta \mathbf {r} \right )$ for an ensemble of speckle pattern $I (\mathbf {r})$ is introduced by Eq. (20) in Appendix A. For experiments, $\mathbf {r}=(x,y)$ is a transverse position in the speckle image, $\Delta \mathbf {r}= \left | \mathbf {r}-\mathbf {r}' \right |$ stands for a transverse shift between two points, and $\delta I = I - \langle I \rangle$ indicates statistical fluctuation of $I$, with $\langle \cdot \rangle$ the ensemble average. To continue, it is found that $\langle I_m (\mathbf {r}) \rangle = I_i (\mathbf {r}) + \langle I (\mathbf {r}) \rangle$, and then the normalization of $\langle \delta I(\mathbf {r}) \delta I(\mathbf {r}') \rangle$ is directly $C\left ( \Delta \mathbf {r} \right )$ according to Appendix A. Thus we get [21]
Speckles recorded in this experiment is isotropic and $C\left ( \Delta \mathbf {r} \right )$ is only related to absolute value of distance, so we may replace $\Delta \mathbf {r}$ by $\Delta r$. By Eq. (19), SICF could be calculated from recorded speckle patterns.
4. Results and discussions
Values of the scattering mean free path $l_s$ and transport mean free path $l_t$ of the samples used in experiments are given in Table 1 and Table 2, respectively. Sample 1-5 in both tables are distinguished by concentration from each other. For sizes of 5$\mathrm {\mu }\textrm {m}$, 10$\mathrm {\mu }\textrm {m}$ and 20$\mathrm {\mu }\textrm {m}$, five concentrations are 0.989‰, 1.318‰, 1.758‰,2.344‰and 3.125‰. For sizes of 30$\mathrm {\mu }\textrm {m}$ and 50$\mathrm {\mu }\textrm {m}$, concentrations are 1.832‰, 2.094‰, 2.393‰,2.734‰and 3.125‰. Values of $l_s$ and $l_t$ for spherical particles can be calculated using Mie theories [26].
Table 1. Scattering mean free path $l_s$ of the samples
Table 2. Transport mean free path $l_t$ of the samples
Figure 5 shows experimental results of SICF for large particles, with (a) for different particle size but identical volume concentration, and (b)-(f) for different volume concentration of each particle size.
Fig. 5. SICF of experimental results: (a) different particle sizes, with $c_p = 3.13 \times 10^{-3}$, (b)-(f) different volume concentrations for each size, with 5$\mathrm {\mu }\textrm {m}$, 10$\mathrm {\mu }\textrm {m}$, 20$\mathrm {\mu }\textrm {m}$, 30$\mathrm {\mu }\textrm {m}$, 50$\mathrm {\mu }\textrm {m}$, respectively. The numbers with ‰ in legends represent volume concentration.
From Fig. 5(a), for particles of 5$\mathrm {\mu }\textrm {m}$ and 10$\mathrm {\mu }\textrm {m}$ there is a primary peak around $\Delta r = 0$ where the SICF declines rapidly. The rest three curves have also peaks around $\Delta r = 0$ but decline smoothly with $\Delta r$ becoming larger, and as particle size increases this smooth region gets wider. From Fig. 5(b)-(f) it could be seen that as volume concentration of particles increases, SICF gets lower. As a result, particle size determines the width of the smooth region of SICF, while particle concentration determines the height of the smooth region of SICF.
To compare the theoretical expression to experimental results, we fit Eq. (16) by least squares method to experimental results for each term. Corresponding results of theoretical SICF fitted are shown in Fig. 6. In Fig. 6(a) we show fitted theoretical SICFs for different particle sizes but identical concentration, while in Fig. 5(b)-(f) we show both theoretical and experimental SICFs for each particle size. It is seen that theoretical expressions are in consistent with experimental results after being fitted. Larger particles generate wider SICF, and for particles much larger than wavelength, there is also a smooth region for each.
Fig. 6. Fitted theoretical SICFs: (a) different particle sizes, with volume concentration $c_p = 3.13 \times 10^{-3}$, (b)-(f) both theoretical and experimental results for each particle size in (a).
For the same particle size, SICF given theoretically varies also with particle concentration as is shown in Fig. 7. The smooth region of SICF has similar width yet different heights at the bottom for different concentration. Consequently, size of particle determines the width of the smooth region and concentration determines the height where to get into the smooth region, which is consistent to experimental results.
Fig. 7. Fitted theoretical SICFs: (a) different particle concentrations, with particle size of 30$\mathrm {\mu }\textrm {m}$, (b)-(f) both theoretical and experimental results for each concentration in (a). The numbers with ‰ in legends represent volume concentration.
In addition, it is found that there are oscillations on the scale of wavelength. These oscillations spread at the bottom of each primary peak. And it is seen that the oscillations of fitting results basically synchronize with the oscillations of experimental results.
Fitting results of $\mathrm {Im} \left ( k_{\mathrm {{eq}}} \right )$ and $\mu$ are plotted in Fig. 8(a) and (b), respectively. It has been demonstrated in Section 2.2 that, $k_{\mathrm {{eq}}}$ characterize the equivalent wavenumber. This equivalent one consists both the actual value of $k_{\mathrm {eff}}$ and the contribution of ladder propagator. It is seen from Fig. 8(a) that for each particle size, as volume concentration increases $\mathrm {Im} \left ( k_{\mathrm {{eq}}} \right )$ becomes smaller. The coefficient $\mu$ given by Eq. (14) contains the effect of all the scattering events. From Fig. 8(b), the value of $\mu$ fluctuates slightly around 2, which means that the principle of scattering and correlation of each particle size is similar. As a result, $\mathrm {Im} \left ( k_{\mathrm {{eq}}} \right )$ in Eq. (16) decreases with particle concentration increasing, causing the smooth region of SICF getting higher, while $\displaystyle \frac {R_0}{\mu }$ increases with particle size increasing, causing the smooth region of SICF getting wider.
5. Conclusions
In conclusion, transmitted speckle patterns generated by multiple scattering by large particles have been studied theoretically and experimentally. We have demonstrated the principles of coherent light scattering by large particles using diagrammatic representation, and derived a semi-empirical expression of short-range ($C_1$) spatial intensity correlation function. This expression takes particle size and concentration into account. For each given particle size and concentration in experiments, we fitted the theoretical expression by least squares method to experimental results . The expression given in this paper is consistent with experimental results, with respect to the variation of SICF with particle size and concentration.
Experiments performed and theoretical analysis have shown that in an optically thick medium containing large particles, spatial intensity correlations of speckle patterns vary with particle size and concentration. Particularly, apart from a primary peak, we have observed a smooth region in each spatial intensity correlation function, which is different from the function of small particles. These investigations introduce new analytical methods to study multiple scattering, and new types of speckle patterns. With a priori knowledge of speckle patterns, one can extract the information of particles and objects to be imaged. Results may find applications in coherent imaging through biological tissues, natural turbid water and atmosphere.
Appendix A: Intensity correlations: $C_1$, $C_2$ and $C_3$
Field correlations have been shown in Fig. 1. In this appendix, we show intensity correlations. Intensity correlation function of a speckle pattern is defined by [3]
There are three types of correlations in $C \left ( \mathbf {r}_1, \mathbf {r}_2 \right )$: short-range correlations ($C_1$), long-range correlations ($C_2$) and infinite-range correlations($C_3$). We use diagrams to show the geometry of each type, as is shown in Fig. 9. All diagrams are under the condition of point scatterers, and diagrammatic rules are the same as Fig. 1 only we use two colors to distinguish [27].
Fig. 9. Diagrammatic representation of contributions of three types to intensity correlations: (a) short-range correlations ($C_1$), (b) long-range correlations ($C_2$), (c) infinite-range correlations ($C_3$).
It could be seen from Fig. 9 that $C_1$ only concerns separation of fields, while $C_2$ and $C_3$ concern one and two exchanges of two pairs of fields. If there are there or more exchanges, it also belongs to $C_3$. The calculation of contribution of each geometry has to include the probability of exchange of fields. This probability in weak-scattering regime is much less than unity. So $C_2$ and $C_3$ have much less contributions to $C \left ( \mathbf {r}_1, \mathbf {r}_2 \right )$ than $C_1$. More details have been discussed in [3,27].
If there are no field correlations, the term $\langle E_1 E^*_1 E_2 E^*_2 \rangle$ would reduce to $\langle \left |E_1\right |^2 \rangle \langle \left |E_2\right |^2 \rangle$, where $\left | \cdot \right |$ means modulus. Otherwise there will be other contributions. The red lines in Fig. 9(a) is independent of the blue lines, which means that the extra contribution in $\langle E_1 E^*_1 E_2 E^*_2 \rangle$ could be written as $\langle E_1 E^*_2 \rangle \langle E^*_1 E_2 \rangle$. Thus, $\langle E_1 E^*_1 E_2 E^*_2 \rangle = \langle \left |E_1\right |^2 \rangle \langle \left |E_2\right |^2 \rangle + \langle E_1 E^*_2 \rangle \langle E^*_1 E_2 \rangle$ in Fig. 9(a), and the fields of $C_1$ are Gaussian random variables. Since it is not the same case for $C_2$ and $C_3$, the fields of $C_2$ and $C_3$ are non-Gaussian. Then accordingly we get
In experiment, the camera records intensity instead of field. It is useful to simplify Eq. (22) to make it accessible in data processing. As the fields for $C_1$ correlations are Gaussian, the intensity in this case obeys Rayleigh statistics [1]. Further, when $\mathbf {r}_1 = \mathbf {r}_2$ the intensity correlation function $C \left ( \mathbf {r}_1, \mathbf {r}_2 \right )$ is equal to unity. Then, to calculate Eq. (22) in experiment, only the normalization of $\langle \delta I\left ( \mathbf {r}_1\right ) \delta I\left ( \mathbf {r}_2\right ) \rangle$ needs to be calculated.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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