1. Introduction

At present, optical diagnostic techniques are essential experimental tools employed in a large and diverse set of research and industrial applications. Based on measurements of scattering intensity and light extinction, a number of these techniques are able to determine the parameters of scattering particles, such as size, shape and concentration. Examples of applications are numerous: In meteorology, the measurement of atmospheric constituents is performed by analyzing the backscattering light signal via the LIDAR (LIght Detection And Ranging) technique [1]. In oceanography, marine picoplankton are characterized from the laser light scattered by the illuminated particles [2]. The optical radiation is actively used in various bio-medical applications as a non-invasive diagnostic tool as well as to treat a number of diseases [3]. During the past two decades, the investigation of laser light propagation in skin tissues and the human brain has been particularly extensive [3–4, 9]. Finally, in Combustion Engineering a variety of laser based diagnostics have been developed over more than three decades in order to determine the physical properties of fuel droplets from atomizing sprays [5]. These optical measurements are of fundamental importance in the improvement of the combustion efficiency and in the reduction of pollutant emissions from modern internal combustion engines and gas turbines.

The major limitation of the current existing laser techniques for particle characterization is commonly related to attenuation and multiple scattering (schematically represented in Fig.1) of the source radiation. Under optically thick conditions, such phenomena can introduce substantial errors in the measurements of size and concentration. The magnitude of error varies with position, in a manner dependent on the source, detector, and medium geometries.

Quantifying and reducing attenuation and multiple scattering effects requires knowledge of the radiative transfer of optical energy through the medium. In the field of laser diagnostics the migration of photons is generally described by the radiative transfer equation (RTE), specifying a balance of energy between the incident, outgoing, absorbing, and scattering radiation propagated through the medium. The RTE is given in Eq.(1).

 

Fig. 1. Schematic illustration of attenuation (a) and multiple scattering (b). The incident radiation is attenuated when it traverses the turbid medium due to both scattering and absorption. Depending on the position along the laser line-of-sight, particles are not illuminated in the same conditions. Multiple scattering, so-called “Extraneous light”, is detected after being multiply scattered by a number of particles (b).

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where t is time, r⃗ is the vector position, s⃗ is the incident direction of propagation, f(s⃗′, s⃗) is the scattering phase function derived from the appropriate scattering theory (e.g. Lorentz-Mie or Rayleigh-Gans theory), dΩ′ is the solid angle spanning s⃗′ and c is the speed of the light in the surrounding medium. The RTE can be summarized as follows: the change of radiance along a line of sight (Eq.(1) term (a)), corresponds to the loss of radiance due to the extinction of incident light (Eq.(1) term (b)) plus the amount of radiance that is scattered from all other directions s⃗′, into the incident direction s⃗ (Eq.(1) term (c)) The total extinction represented by Eq.(1) term (b) equals the radiance lost due to scattering of the incident light in all other directions, minus the radiance that is absorbed at each light-droplet interaction. RTE is applicable for a wide range of turbid media; however the analytical solutions are only available in rather simple circumstances where assumptions and simplifications are introduced to reduce the equation to a more tractable form. Since there are no analytical solutions available to the transport equation in realistic cases, numerical techniques have been developed and utilized. The most versatile and widely used numerical solution is based on the statistical Monte Carlo (MC) technique [6].

The MC technique for photon migration is principally applied to biomedical applications [7–9]. However, the method has also been employed to a variety of other situations [10–12]. Owing to the large improvements in computer performance, modern MC codes allow the production of realistic simulations and the development of new MC codes is an active research topic [12–13]. Recently the method has been employed for the complex case of light scattering within inhomogeneous polydisperse turbid media such as spray systems [15–16]. MC simulations are primarily employed to solve radiative problems within the intermediate scattering regimes where no approximation applies. Depending on the optical thickness, the scattering of light within a turbid medium can be classified into 3 regimes:

- In the single scattering regime the average number of scattering events ≤1 and the non-scattered ballistic photons are dominant. For off-axis detection, the single scattering approximation which assumes that photons have experienced only one scattering event prior arriving to the detector applies.

- The intermediate single-to-multiple scattering regime operates when the average number of scattering events is between 2 and 9. In this regime, one dominant scattering order is clearly defined. No approximation can be made under such a regime.

- The multiple scattering regime is defined when the average number of scattering events is greater or equal to 10. In this regime, the relative amount of each scattering order tends to be equal and no dominant scattering order is apparent. The diffusion approximation can be applied in this regime.

By definition, the distance of light propagation between two scattering or absorbing events corresponds to the free path length l_fp. The mean free path length, which is the average distance between two light-particle interactions, is inversely proportional to the extinction coefficient, such that: $\overline{l_{\mathrm{fp}}}=1⁄{\mu }_e$ where µ_e=µ_s+µ_a. Here, µ_s is the scattering coefficient and µ_a is the absorption coefficient. The optical depth (OD) can be calculated by dividing the total length, l, traversed by a light beam, by the mean free path length: $OD=l⁄\overline{l_{\mathrm{fp}}}=l·{\mu }_e$. The optical depth provides an estimation of the average number of times that photons have interacted with the scattering particles, prior to exiting a medium of length l. The classification of each scattering regime can be performed from the value of the optical depth as presented in Table 1.

Table 1. Classification of the scattering regimes as a function of the optical depth. The intermediate scattering regime is the most complex. Neither the single scattering assumption nor the diffusion approximation applies in this regime.

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In the work presented here, scattering of light in the intermediate regime is investigated both experimentally and via MC simulation. The two-dimensional distribution of light intensity for the forward and side scattering detection is analyzed. The effects of scattering phase function, optical depth, and detection acceptance angle are highlighted. An analysis of the signal contributions made by individual scattering orders is also provided in this article. An empirical approach to include multiple scattering within the Beer-Lambert law is proposed based on the simulated and experimental results. This extrapolation is deduced from the numerical calculations of the scattering orders’ contributions to the detected light intensity. This article is Part I of a complete study of light scattering in turbid media. In Part II [17], complementary information including both two-dimensional light intensity distributions and time-resolved analysis of individual scattering orders will be presented, based on further numerical MC results.

2. Description of the experimental setup

The experiment is based on the transmission of a laser beam through a homogeneous monodisperse scattering medium of known optical properties and the detection of the intensity profiles of the scattered light. The incident laser light is produced by a Spectra-Physics Tsunami Ti:Sapphire mode-locked laser. Light pulses with ~80 fs duration (FWHM ~11 nm centered at ~800 nm) and 10 nJ energy were transmitted through a 10 mm×10 mm×45 mm optical glass cell containing a suspension of polystyrene spheres immersed in distilled water (see Fig. 1). Several solutions are prepared by varying the concentration and the size of the polystyrene spheres. The number density of spheres in each cell was adjusted to provide optical depths of 2, 5 and 10 for two cases of sphere diameter, D, equal to 1 µm, and 5 µm. The initial concentrations of polystyrene spheres were diluted with distilled water and the value of the optical depth was verified from the Beer-Lambert law by measuring the transmission of the incident light for each sample with a photodiode and a lock-in amplifier. A smaller cell (5 mm wide instead of 10 mm) was employed in order to check the concentration of polystyrene spheres at the highest optical depth, corresponding to OD=10. These measurements were used to quantify the optical depth with a maximum error of 5%. The refractive index of the scattering particles was determined from the experimental results published by Ma et al. [18]. According to these authors, for the case of polystyrene spheres illuminated at 800 nm, the refractive index, n, is 1.578–0.0007i. The light intensity scattered from the cell was detected using an Andor iXon DV887 Electron Multiplying CCD camera. This camera has single photon detection capability without an image intensifier due to its multiplication gain feature. At 800 nm, the quantum efficiency reaches 70% when using thermoelectric cooling set to -20°C. The intensity profiles of the scattered light were detected on both the front and side face of the cell. A 10×10 mm surface is imaged onto 200×200 CCD pixels, resulting in an image resolution of 50 µm. Two F/#’s, equal to 1.8 and 5.6 were successively employed. The focal distance, f, of the camera lens equals 10 cm. The distance, L, between the surface of the sample and the camera lens was equal to 152 mm. From these parameters, the detection acceptance angle of the collection optics is equal to θ_a=8.5° for F/#=1.8 and θ_a=1.5° for F/#=5.6. Due to the high pulse repetition rate (82 MHz) and the long detection aperture time (0.015s), the laser source is perceived by the camera as a continuous wave laser source.

 

Fig. 2. Configuration of the experimental set-up: Two optical paths are independently considered for either the forward or the side detection.

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For each measurement, 10 images were recorded and averaged. An illustration of the experiment is shown in Fig. 2. For each measurement, an appropriate selection of several Neutral Density Filters (NDF) was used in order to detect the maximum number of photons on the CCD chip while avoiding saturation. The Full Width at Half Maximum (FWHM) of the initial laser beam intensity profile was measured along the vertical axis Z, at X=5 mm and was found to be equal to d_a=2.55 mm when θ_a=8.5° and d_b=2.61 mm when θ_b=1.5°.

3. Description of the Monte Carlo simulation

The implementation of MC simulation for the propagation of light radiation through turbid media has been treated by a large body of literature. Accurate description of the technique can be found in [5–8] and [13–16]. The fundamental steps of the MC simulation are as follows:

Photons enter the simulated scattering medium from an initial position with an incident direction of propagation. The free path length, l, before each light-particle interaction is derived from the Beer-Lambert law and is calculated as a function of the extinction coefficient µ_e using a random number ξ uniformly distributed between 0 and 1: l=-ln(ξ)/µ_e. The extinction coefficient is deduced such that: µ_e=N·σ_e where N is the number density and σ_e is the extinction cross section of the scattering particles. At each particle interaction, photons can be either absorbed or scattered. If the particles are non-absorbing, the extinction coefficient is then equal to the scattering coefficient and the albedo Λ(Λ=µ_s/(µ_s+µ_a) is equal to one. In the MC technique, independent scattering is assumed requiring a distance between individual particles greater than three times the radius of the particles [19]. The MC model treats light as a collection of distinct entities, and as a consequence, interference phenomena are neglected in the simulation. This requires a random distribution of the scattering particles and the absence of periodic structures within the turbid medium. After a scattering event, the photon’s new direction is selected based on a random number and the Cumulative Probability Density Function (CPDF) calculated from the appropriate scattering phase function f. The scattering phase function is defined as a function of the properties of the scattering particles. Lorentz-Mie, Rayleigh-Gans [19–20] or Henyey-Greenstein [21] phase functions are typically employed. The polar scattering angle θ_s defined between 0 and π is calculated from the inverse CPDF of f by: θ_s=CPDF^-1(ξ), where ξ is once again a random number generated between 0 and 1. The resolution of the sampled scattering angle, θ_s, equals 0.1°. The azimuthal scattering angle, φ_s, is uniformly distributed between 0 and 2π such that the scattered radiation is assumed to be independent to the orientation of the scattering particle with respect to the direction of the incident radiation (valid for spherical particles). When a new direction of propagation is defined, the position of the next scattering point is calculated again and the process is repeated until the photon is either absorbed or exits the medium at a boundary. The optimum number of photons employed in the simulation depends on the desired accuracy and on the detector characteristics. The final direction of propagation, the final position, the number of scatters, and the total path length are calculated for each light entity. If the conditions of detection are met (e.g. photon lies within the field of view of the detector with its trajectory within the acceptance angle), such data are written to disk. The process is repeated for a sufficiently large amount of photons such that the distribution of the light intensity impinging on the detector is accurately represented.

 

Fig. 3. Simulation configuration: The laser source S is modeled from the experimental image matrix (200×200 pixels). Photons are sent from S into a scattering single cubic volume (L=10 mm) containing scattering polystyrene spheres suspended in distilled water.

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For the data presented here, the experimental cell corresponds to a cubic volume of 10 mm length. The laser wavelength is assumed to be monochromatic and equal to 800 nm. The laser light is assumed to be unpolarized and the polarization state is not tracked. The intensity profile of the source, S, is modeled from the experimental matrix of the incident laser beam as illustrated in Fig. 3. This matrix is obtained from the EM-CCD camera by imaging the surface of a cell containing only distilled water (without polystyrene spheres). Using this technique, the exact experimental source is considered and any irregularity in the laser beam profile is accounted for within the modeling, allowing a more realistic MC simulation. Computed photons are recorded at the exit position, provided detection conditions are met. This implies that the angle between the vector normal to the detection face (front face or side face) and the vector direction of the photons must be within the acceptance angle, θ_a (see Fig. 3). The absorption of 800 nm light within the polystyrene sphere solution is negligible, so a real (non-absorbing) index of refraction is assumed for the scatterers, such that n=1.578-0.0i. The same assumption applies for the surrounding medium composed of distilled water, where n=1.33-0.0i. The resulting Lorentz-Mie scattering phase function of a single polystyrene sphere is illustrated in Fig. 3, for the two diameters D=1 and 5 µm. For each simulation, 3 billion photons are sent through the scattering medium and the resultant computational time is ~7.5 hours at optical depth OD=2 and ~25 hours at OD=10, when using a modern Intel(R) Core(TM) 2 CPU 6600 @ 2.40GHz processor. The relative speed of computation is then on the order of 9 µs/photon (at OD=2) and 30 µs/photon (at OD=10).

 

Fig. 4. Polar scattering phase function (logarithmic scale) for a single polystyrene sphere of refractive index n=1.578-0.0i suspended in a solution of distilled water of refractive index n=1.33-0.0i. In (a) D=1 µm and in (b) D=5 µm.

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4. Results and comparison for the forward scattering detection

4.1. Polystyrene spheres of 1 µm diameter:

The first set of comparisons shows the forward scattering detection with polystyrene spheres of 1 µm diameter. The 2D intensity distribution is shown on the front face for the large detection acceptance angle θ_a=8.5° in Fig. 5(a). By increasing the optical depth from OD=2 to OD=10, the light intensity transmitted through the scattering sample is reduced and the laser beam profile diffuses.

At OD=2, the amount of light crossing the sample reaches a maximum value of 19% of the initial intensity. This result is found both experimentally and in the simulation. The Beer-Lambert law predicts a lower transmission of 13.5% in the same conditions. At OD=5, the simulated results differ somewhat from the experimental results (the maximum light transmission equals 2.2% experimentally versus 2.8% for the simulation) and the differences with the Beer-Lambert calculation (I_f/I_i=0.67%) become significant. It is also seen that the laser beam starts to diffuse and its FWHM is 1.28 times wider than its original value. At OD=10, the laser beam is now highly diffused with a FWHM equal to 1.85 d_a. The maximum transmission corresponds to 0.18% experimentally, 0.52% with the simulation and the Beer-Lambert law predicts only 0.0045%.

 

Fig. 5. Comparison between the front face experimental and simulated images at detection acceptance angle θ_a=8.5° in (a) and θ_a=1.5° in (b). Solutions of polystyrene spheres of 1 µm diameter are considered at optical depths OD=2, OD=5 and OD=10 in (a) and for OD=10 only in (b). The intensity scale of the images corresponds to the final light intensity, I_f, detected per pixel divided by the maximum value of the incident light intensity I_i. A comparison of the intensity profile along the vertical axis at X=5 mm is also shown on the right side of the figure. The solid line corresponds to the experimental results and the circles are the results from simulation.

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Such results demonstrate the considerable differences between the Beer-Lambert calculations, which neglect the effects of multiple scattering, and the results from both the experiment and from the MC simulation.

By reducing the detection acceptance angle to θ_a=1.5° (see Fig. 5 (b)), it is observed that the detected light intensity decreases while the incident laser beam profile tends to diffuse less. The FWHM is now 1.61 larger from its initial width and the light transmission equals ~0.025%. It is also apparent that at smaller detection acceptance angles, the number of photons detected is significantly reduced, resulting to a deterioration of the spatial resolution in the MC image. This is an artifact of the number of photons used in the computation. These results show that the amount of light intensity detected (experimentally and via simulation) differs considerably from that calculated by the Beer-Lambert formula. These divergences increase when increasing OD and for large θ_a. Discrepancies between the experimental and simulated results occur principally at OD=10.

 

Fig. 6. Probability distribution of scattering orders, P(n), at various optical depths, for the polystyrene spheres of 1 µm diameter. Recorded photons exit the scattering medium through the front face, within the indicated acceptance angle θ_a.

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The relative amount of light intensity calculated for each scattering order from n=0 to n=10 is plotted in Fig. 6. At scattering order n=0, photons cross the scattering sample without being scattered. This non-scattered light, generally termed ballistic light, corresponds to the ballistic photons, which traverse the scattering medium without encountering any scattering or absorption events. The influence of the optical depth and of the detection acceptance angle on the contribution P(0) of the ballistic photons can be observed by comparing Fig. 6(a) and 6(b).

At the acceptance angle θ_a=8.5° P(0) equals 65%, 18% and 0, 6% for the respective optical depths OD=2, OD=5 and OD=10. By reducing the detection acceptance angle to θ_a=1.5°, the amount of multiply scattered light detected is reduced and P(0) increases significantly, reaching 98%, 87% and 16% (for OD=2, OD=5 and OD=10 respectively). At the same time, the contribution of the high scattering orders (n≥4) increases smoothly with increasing OD while the contribution of the low scattering orders (n≤3) is reduced abruptly.

4.2. Polystyrene spheres of 5 µm diameter:

The second set of comparisons is based on forward scattering detection with polystyrene spheres of 5 µm diameter. The intensity distribution profile is shown on the front face for the large detection acceptance angle θa=8.5° in Fig. 7(a). At OD=2 the light transmission reaches a maximum value of ~36%. This is observed both experimentally and via simulation.

 

Fig. 7. Comparison between the front face experimental and simulated images at detection acceptance angle θ_a=8.5° in (a) and θ_a=1.5° in (b). Solutions of polystyrene spheres of 5 µm diameter are considered at optical depths OD=2, OD=5 and OD=10 in (a) and for OD=10 only in (b). The intensity scale of the images corresponds to the final light intensity, I_f, detected per pixel divided by the maximum value of the incident light intensity I_i. A comparison of the intensity profile along the vertical axis at X=5 mm is also shown on the right side of the figure. The solid line corresponds to the experimental results and the circles are the results from simulation.

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At OD=5, this maximum equals 7% experimentally and 9% via simulation. Finally, at OD=10, I_f/I_i equals 0.9% experimentally and 1.4% via simulation.

These results show that the laser beam does not diffuse as much as it does for spheres of 1 µm diameter. The FWHM of the laser beam intensity profile in this case is 1.02 d_a, 1.10 d_a and 1.25 d_a for the respective OD=2, OD=5 and OD=10. For the small detection acceptance angle θ_a=1.5° and for OD=10, Fig. 7(b) shows both the FWHM 1.13 is times greater than the initial width, and the number of photons detected is reduced by a factor of ~10. However, contrary to the previous case with the 1 µm particles, the resultant statistics are sufficient for good spatial resolution in the MC image. The scattering phase function of the 5 µm polystyrene spheres is characterized by a significant forward scattering lobe, as shown in Fig. 3(b). This scattering feature is responsible for the increase of the light intensity scattered and multiply scattered in the forward direction. As a result, divergences with the Beer-Lambert calculations increase; whereas the broadening effect of the incident laser beam is reduced.

 

Fig. 8. Probability distribution of scattering orders, P(n), at various optical depths, for the polystyrene spheres of 5 µm diameter. Recorded photons exit the scattering medium through the front face, within the indicated acceptance angle θ_a.

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The amount of light intensity detected for each scattering order from n=0 to n=10 is plotted in Fig. 8. For θ_a=8.5°, the contribution of higher scattering orders is important even for the low optical depth OD=2. By decreasing θ_a from 8.5° to 1.5°, the detection of ballistic photons is once again largely improved: From 31.6% to 80.1% at OD=2, from 5.6% to 43.2% at OD=5 and from 0.23% to 4.9% at OD=10.

For large detection acceptance angle, Fig. 6 and Fig. 8 show that the dominant scattering order is transferred from one order to another when increasing OD. Conversely, at the lower detection acceptance angle, the dominant scattering order is n=0 for low optical depth. At higher optical depths each scattering order tends to have an equal contribution.

While the experimental and simulated results are qualitatively similar in all cases, some quantitative differences do exist. For both cases of polystyrene sphere diameters, the amount of detected light intensity is larger for the simulated data than for the measured data when applying θ_a=8.5°. Conversely, when θ_a=1.5°, the detected intensity is smaller for the simulated data. These differences are due to several factors. First, in the experiment an amount of light is reflected by the walls of the glass sample within the turbid medium. These internal reflection phenomena are not calculated in the present simulation. Additionally, computed photons are recorded only if the angle between the vector normal to the detection face and the direction of the exiting photons is within the acceptance angle. These selection criteria approximate the light collection efficiency of a real system, but do not account for the spatial filtering effects of the actual detector and imaging optics. A ray-tracing model which simulates photon propagation from the exit plane, through the collection optics, to the detector is required for better agreement between the experimental and simulated results. Finally, errors in the measurement of the experimental optical depth (estimated to a maximum of 5%) contribute to the observed quantitative differences. Despite these factors, the experimental and simulated transmission intensities are of the same order of magnitude under all conditions. By plotting normalized intensity, as shown on the right side of Fig. 5 and Fig. 7, it is apparent that the simulated profile remains true to the experimental results in all cases.

4.3. Extrapolation of the Beer-Lambert transmission to large detection acceptance angle and high optical depth:

The final transmitted light intensity, I_f, equals the sum of the non-scattered light intensity, I_b, from the ballistic photons, plus the light intensity, I_ms, from scattered and multiply scattered photons:

Assuming that I_ms is related to I_b by a coefficient k we have:

Here, k corresponds to the contribution of scattered and multiply scattered light over the contribution of non-scattered light. As described previously, P(0) is the contribution of ballistic photons and the contribution of scattered and multiply scattered photons equals P(tot)-P(0). For a normalized distribution where P(tot)=1, the multiply scattered photon contribution becomes, 1-P(0), and the coefficient k equals:

From Eq.(3) it follows:

By definition, the Beer-Lambert law describes the exponential reduction of the incident light intensity I_i along a line-of-sight as a function of the optical depth. In highly scattering environments, some photons which are scattered away from the optical axis undergo a succession of scattering events and are eventually redirected along the original path of the incident light. Experimentally, the amount of multiply-scattered light detected increases with the detection acceptance angle. The contribution of I_ms is not considered in the Beer-Lambert law, which applies only to the number of ballistic photons crossing the scattering sample, such that:

By including Eq.(6) in Eq.(5) we obtain:

Here, the final light intensity calculated from the conventional Beer-Lambert law is increased by the factor 1/P(0) which is the inverse probability density of the ballistic photon contribution.

Table 2. Deduction of the factor 1/P(0) from the Monte Carlo calculations. The results from the extrapolated Beer-Lambert transmission, given in Eq.(8), are compared with both the simulated results and the experiment.

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The results from the extrapolated Beer-Lambert law are compared in Table 1 with the experimental and simulated results for the transmission of the laser beam at various optical depths and detection acceptance angles. Good agreement is found between the simulated and experimental results for the three optical depth OD=2, OD=5 and OD=10.

 

Fig. 9. Factor 1/P(0) as a function of the optical depth. Each value of 1/P(0) is deduced from the MC simulation and is related to the amount of multiply scattered light intensity detected. The red and black lines are the best fit of the calculated data.

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The factor 1/P(0) is plotted in Fig.9 as a function of OD for the 1 µm and 5 µm sphere diameters and for the detections acceptance angles θ_a=8.5° and θ_a=1.5°. The simulation data shows that 1/P(0) increases exponentially with α·OD^β. This increase becomes more apparent at large detection acceptance angles and for scattering particles which exhibit a dominant forward scattering lobe (larger particles). From these results, the Beer-Lambert relation can be modified and written such that the light intensity from multiply scattered photons is considered:

The coefficients α and β are related to the detection acceptance angle and to the particle diameter D, and the term [-OD+α·OD^β] is always negative implying α·OD^β<OD.

5. Results and comparison for the side scattering detection

The first set of comparisons in this section is based on the detection of the scattered light on the side face for polystyrene spheres of 1 µm diameter (Fig. 10). The second set of comparison concerns the polystyrene spheres of 5 µm diameter (Fig. 11). In both cases, the 2D light intensity distribution is shown on the side face for the large detection acceptance angle θ_a=8.5° at optical depths OD=2, OD=5 and OD=10 (Fig. 10(a) and Fig. 11(a)). The small detection acceptance angle θ_a=1.5° is considered only for OD=10 (Fig. 10(b) and Fig. 11(b)). The intensity profile along the vertical axis at Y=5 mm is also displayed in these figures.

By increasing the optical depth from OD=2 to OD=10, the distance of photon penetration along the incident direction is reduced. Also, the circular cross-section of the incident laser beam becomes wider due to an increased number of scattering events within the cell. At equal OD it is seen that the broadening of the laser beam operates more efficiently for the 1 µm particles; whereas, for the 5 µm particles, photons tend to penetrate further within the cell. Furthermore, the light intensity detected is larger for the 1 µm particles than for the 5 µm particles. These effect are due to the larger forward scattering characteristics of the 5 µm spheres and to the larger amount of light scattered at ~90° for the 1 µm particles (see the respective scattering phase functions illustrated in Fig.4).

Contrary to the forward scattering detection case, the amount of light intensity detected on the side face increases with the optical depth. This increase seems to be “linearly” related to the value of OD. Thus, the optimum qualitative and quantitative comparison between the MC and experimental results are observed at OD=10; whereas, discrepancies occur principally at OD=2. Similarly, when reducing the acceptance angle to θ_a=1.5°, the number of detected photons is considerably reduced and the statistics are not sufficient to provide highly resolved 2D simulated images. It is observed experimentally, for both particle diameter cases, that the reduction of the detection acceptance angle from 8.5° to 1.5° leads to a reduction of light intensity by a factor of ~10.. Furthermore, the light intensity profile is found to be larger at θ_a=1.5° than at θ_a=8.5° with a broadening of the FWHM measured along the vertical axis at Y=5 mm. These phenomena are related to the low Signal to Noise Ratio and to the significant effects of the background light contribution when detecting with a small acceptance angle. Without these noise effects, it is assumed that the relative distribution of light intensity on the side face of the scattering sample would remain fairly equal for both cases of detection. Thus, it is deduced that the detection acceptance angle does not affect the relative distribution of light intensity for the side scattering detection.

The amount of light intensity detected for each scattering order from n=0 to n=10 is plotted in Fig. 12. Contrary to the forward scattering detection, it is seen here that the detection acceptance angle and the scattering phase function both have minor influence on the contribution of each scattering order. At equal OD, Fig. 12(a), (b), (c) and (d) show similar contributions. For the cases considered here, it is notable that the value of OD consistently remains close or equal to the dominant scattering order. It is deduced from these results that the optical depth is the only parameter that influences the contribution of the scattering orders. The value of the detection acceptance is insignificant on both the distribution of light intensity by scattering order and the relative spatial light intensity distribution. A reduction of θ_a leads only to a reduction of light intensity.

 

Fig. 10. Comparison between the side face experimental and simulated images at detection acceptance angle θ_a=8.5° in (a) and θ_a=1.5° in (b). Solutions of polystyrene spheres of 1 µm diameter are considered at optical depths OD=2, OD=5 and OD=10 in (a) and for OD=10 only in (b). The intensity scale of the images corresponds to the final light intensity, I_f, detected per pixel divided by the maximum value of the incident light intensity I_i. A comparison of the intensity profile along the vertical axis at Y=5 mm is also shown on the right side of the figure. The solid line corresponds to the experimental results and the circles are the results from simulation.

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Fig. 11. Comparison between the side face experimental and simulated images at detection acceptance angle θ_a=8.5° in (a) and θ_a=1.5° in (b). Solutions of polystyrene spheres of 5 µm diameter are considered at optical depths OD=2, OD=5 and OD=10 in (a) and for OD=10 only in (b). The intensity scale of the images corresponds to the final light intensity, I_f, detected per pixel divided by the maximum value of the incident light intensity I_i. A comparison of the intensity profile along the vertical axis at Y=5 mm is also shown on the right side of the figure. The solid line corresponds to the experimental results and the circles are the results from simulation.

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Fig. 12. Probability distribution of scattering orders, P(n), at various optical depths. (a)–(b) correspond to the 1 µm polystyrene spheres and (c)–(d) are for the 5 µm particles. Recorded photons exit the scattering medium through the side face, within θ_a.

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6. Conclusion

The propagation of a laser beam within different scattering solutions of monodisperse polystyrene spheres suspended in distilled water has been studied. Various configurations have been investigated by changing the properties of the scattering medium, (various concentration and size of particles) and of the collection optics (forward or side detection, different detection acceptance angles). In all cases, the spatial distribution of the light intensity deduced from the MC simulations, is found to be very close to the one measured experimentally. A computational analysis of the scattering orders contribution has revealed different specific features between the forward and side scattering detection. An extrapolation of the Beer-Lambert law for including the multiply scattered light intensity has been suggested. Finally, the capability of the MC model to accurately analyze the effects of multiple scattering on highly spatially resolved images has been demonstrated.