In this study, a flexible tool to simulate the bulk optical properties of polydisperse spherical particles in an absorbing host medium is described. The generalized Mie solution for Maxwell’s equations is consulted to simulate the optical properties for a spherical particle in an absorbing host, while polydispersity of the particle systems is supported by discretization of the provided particle size distributions. The number of intervals is optimized automatically in an efficient iterative procedure. The developed tool is validated by simulating the bulk optical properties for two aqueous nanoparticle systems and an oil-in-water emulsion in the visible and near-infrared wavelength range, taking into account the representative particle sizes and refractive indices. The simulated bulk optical properties matched closely (R2 ≥ 0.899) with those obtained by reference measurements.
© 2014 Optical Society of America
Light transport studies are important in many fields, including atmospheric sciences, biomedical optics, food science and many more [1,2]. Propagation of electromagnetic radiation can be well described with the radiative transfer theory (RTT), taking into account the bulk optical properties (BOP) of the considered medium: bulk absorption coefficient µa, bulk scattering coefficient µs, reduced scattering coefficient µs’, and scattering phase function p(θ) or derived anisotropy factor g . The medium’s µa and µs’, and/or µs and g can indirectly be estimated from measurements in the time, frequency or continuous wave domain when combined with inverse light propagation models . However, many sample types need extensive sample preparation and a sufficient amount of sample is generally needed for this type of measurements. Additionally, many different samples need to be considered in order to determine the natural variation of the medium’s BOP, resulting in time-consuming experiments [3,4]. On the other hand, the BOP estimation routines are often vulnerable to instrumental noise. This is especially true for samples characterized with high absorption and/or low scattering as spectroscopic measurements on these samples usually suffer from low signal-to-noise ratios . Additionally, for the direct measurement of the scattering phase function, an accurate nephelometer is required and the single scattering condition has to be fulfilled by diluting or slicing the sample . For many media, mainly solids, with substantial scattering, it is very challenging to achieve this single scattering condition and, therefore, to measure the scattering phase function. As a result, the scattering phase function for biological tissues and many other scattering media is often approximated by the Henyey-Greenstein phase function that describes the angular scattering pattern in function of a single estimated or measured anisotropy factor . However, the derived angular function only provides a rough estimation of reality as simplicity often comes at the expense of a reduced accuracy [6,7]. Because of its simplicity, use of the Henyey-Greenstein phase function in RTT solutions is computationally faster and easier to interpret. Nevertheless, for more accurate simulations of light propagation through turbid media, a correct input for the scattering phase function is essential. This is especially true if spatial information on absorption, or spatial or angular information on reflectance, or transmittance for short source-detector distances is of interest. Accordingly, for many applications, it would be more interesting to simulate the correct µa, µs, p(θ) and/or g for homogeneous media or parts of media from the physical microstructure properties, rather than to measure and/or estimate them [7–11].
Many scattering media can be approximated as a suspension or colloid of spherical particles in a host medium [7,12–18]. The Mie solution to Maxwell’s equations describes the scattering of electromagnetic radiation by a single spherical and homogeneous particle in a host and, therefore, allows for the calculation of the BOP for such particulate systems. Taking into account the particle radius (r), the wavelength in vacuum (λ0) and the refractive index of both the particle (np = complex number) and the host medium (nm = real number), the Mie solution describes the electromagnetic field inside and outside the spherical scatterer in terms of a vector spherical harmonic expansion. Integration of the radiative fluxes over a concentric sphere much larger than the particle (far field observation) results in the scattering matrix, describing the relationship between the incident and scattered electric field components [19,20]. Important physical quantities can be obtained from these fields, including the particle’s high angular-resolution scattering phase function, anisotropy factor and scattering and absorption cross-sections. The latter two are the amount of energy removed from the incident field due to respectively scattering and absorption by a single particle. In the conventional Mie formulation, the spherical scattering particle is assumed to be embedded in a non-absorptive host medium [19,20]. For many applications of visible light in atmospheric science or the study of pigmented materials, this assumption holds . However, in spectroscopic applications, the near infrared (NIR) and infrared (IR) wavelengths can be more valuable, as in this wavelength-range molecules of interest generally have important overtone, combination and fundamental vibrations. As the molecules from the host medium usually absorb a significant amount of NIR and IR radiation, the absorptive host material will influence the scattering and absorption by the particle. Therefore, many researchers focused on the expansion of the Mie solution to also account for the effect of an absorbing host medium (nm = complex number) on the optical properties of the embedded particle [22–36]. It was found that ignoring the absorption index of the host medium results in significant errors if the absorption index is high (cfr. water in the near infrared and infrared region) and/or if the particles are large compared to a wavelength [23–26,28–30].
As long as the scattering events are independent, the µa and the µs for the particles in a monodisperse system can be easily calculated by respectively multiplying the particle’s absorption and scattering cross-sections, obtained through the Mie solution, with the particle density. The normalized phase function and anisotropy factor, on the other hand, are independent of the particle density and remain unchanged [29,30]. According to van de Hulst , a mutual distance between the particles of at least three times the particle radius, corresponding to a particle volume fraction of maximum 4.74%, should be sufficient to fulfill the independent scattering condition. A closed form solution, describing the effect of dependent scattering on the BOP for any combination of sphere size, wavelength, refractive indices of particle and medium, particle volume fraction and polydispersity index is, to our knowledge, not available . Consequently, simulations relying on the Mie solution are only valid in absence of dependent scattering.
As the parameters retrieved through the Mie solution are the result of summations over series expansions, the Mie solution cannot be integrated over a range of particle sizes. Hence, the BOP for polydisperse particle systems can only be retrieved after discretization of the particle size distribution (PSD), calculating the particle’s optical properties and number of particles for each fraction, and, finally, recombining them into the system’s BOP [29,30,38]. The number of intervals to be considered for an accurate final result, however, strongly depends on the range of the particle size, wavelength and refractive indices of particles and medium. This is because these properties determine the variation of the particle’s optical properties over its relevant size range . Moreover, illumination of the scattering particle by an electromagnetic wave excites the electrons in the molecules of scattering particles due to the interaction with the electric field of the incident wave. The dipole re-radiation pattern from the oscillating electrons in the molecules of scattering particles interferes both constructively and destructively in a radiation scattering pattern. Mainly because of these interference effects, particles scatter light in various directions with varying efficiencies and different particle sizes scatter particularly stronger or weaker. This phenomenon is referred to as Mie resonance . These interference effects depend on the aforementioned parameters and introduce variability in the particle’s optical properties for the relevant size range. However, the variability is highly unpredictable and can only be revealed after calculating the particle’s optical properties for a sufficient number of particle sizes in the range of interest. Therefore, it was suggested by Mishchenko  to increase the number of intervals until the required numerical accuracy is reached (optimal number of intervals). In many studies [7,12–18], the discretization approach is used in combination with number-frequency PSDs based on a power-law or lognormal probability density function (PDF). However, no criteria have been proposed on how to select the number of intervals to be considered. Moreover, for different PDF’s, defined by different parameters, and for different wavelengths (and consequently different refractive indices), the same number of intervals was usually considered, while the optimal number of intervals is expected to differ significantly. Other researchers simply used the exact same intervals that were the outcome of the considered technique to measure the particle size (typically electron microscopy, laser light scattering, dynamic light scattering, etc.) [39–41]. Another alternative is to use an extremely high number of intervals such that for each situation the required numerical accuracy is reached. However, this is computationally very inefficient.
The main objective of this study was the development of an algorithm that calculates the BOP for polydisperse particulated systems. The proposed method discretizes the PSD, while the optimal number of intervals is automatically determined by an efficient iterative procedure. The particle optical properties are calculated for each interval with the Mie solution and were recombined over all intervals to obtain the BOP of the sample. To account for the effect of an absorbing host on the absorption and scattering properties of the embedded particles, the generalized Mie solution for absorbing hosts was consulted. Finally, the simulation tool is validated on two aqueous nanoparticle systems and an oil-in-water emulsion in the visible and NIR wavelength range by comparing the simulated BOP with the ones measured in a double integrating sphere (DIS) and unscattered transmittance setup, which is considered to be the ‘golden standard’ for BOP estimation.
The novelty of the consulted approach lies in the automatic conversion of the BOP for polydisperse systems, while the user only needs to specify the required accuracy.
2. Materials and methods
2.1 Mie solution for a spherical particle embedded in an absorbing host medium
The expression of the electromagnetic field inside and outside a spherical scatterer with a vector spherical harmonic expansion, satisfying Maxwell’s equations, has been described in [19,20]. By applying the boundary conditions of the electric and magnetic fields at the sphere surface, the outgoing and internal spherical wave can be obtained in terms of the expansion coefficients (Mie coefficients).
In the far field (FF), the particle’s optical properties can be obtained by integrating the outgoing radiative fluxes (Poynting vector) over a sphere concentric to the particle, but whose radius is much larger than the particle’s radius. The properties of the scattering particle observed in this integrated sphere are, therefore, coupled to the absorption effect of the host medium in an inseparable manner. This is, however, no issue for the derivation of the normalized scattering phase function and the anisotropy factor as they are not related to the absolute scattering intensity. Accordingly, similar to the situation of a non-absorptive host medium, the two components of the scattered wave that are respectively parallel and perpendicular to the scattering plane can be obtained in the far-field from the scattering amplitudes. The optical properties derived from the FF asymptotic form can also be rescaled to the particle’s surface by removing the exponential absorption of the scattered wave due to the absorptive host medium between the particle’s surface and the observational point on the integrated sphere. The procedure to obtain this FF scattering and extinction cross-section for a spherical scatterer in an absorbing host medium was derived by Mundy et al.  and consulted in many other studies [24,26,29,30,42,43].
The inherent scattering and absorption properties of a spherical scatterer in an absorbing host medium can be calculated when the local Poynting vector is integrated in the near-field (NF), directly at the particle’s surface, as shown by Lebedev et al. . Similar analytical expressions for the NF extinction and scattering cross-sections were derived in [23,28]. The NF absorption and scattering cross-sections reduce to their corresponding FF counterparts if the absorption of the host medium is absent . Since the scattering phase function and anisotropy factor represent the angular distribution of the scattered energy at a sufficient distance from the sphere’s surface, it is convenient to only derive these parameters in the FF [24,29,30].
The particle’s absorption and scattering cross-sections obtained from the FF and NF approaches were already thoroughly compared for both absorptive and non-absorptive host media [24,29,30]. Although no differences were found for applications where the host medium was non-absorbing, the differences between FF and NF cross-sections increased if absorption by the host medium increased and/or if the particle size increased relative to a wavelength. Therefore, it is important to define the apparent absorption and scattering cross-sections that are consistent with the RTT . Since the scattered wave is only a relevant quantity for the RTT in the FF, the apparent scattering cross-section should be derived in the far field and rescaled to the particle’s surface as illustrated in [24,44]. According to Fu and Sun , the NF absorption cross-section includes the absorption by the particle itself, but not the non-exponential absorption of the scattered intensity by the host medium near the surface of the particle. For example, the evanescent waves related to the total external and internal reflections at the particle’s surface propagate to the far field along the tangential direction of the particle surface. As a result, the exponential attenuation along the radial direction would always underestimate the absorption of the scattered radiation in the host medium . As this non-exponential absorption is related to the presence of the particle, it should be added to the NF absorption to obtain the apparent absorption cross-section of the particle. The non-exponential absorption of the scattered radiation is actually the difference between the NF and FF scattering cross-section. Consequently, the real extinction of incident radiation that matches the RTT is the same as the NF extinction . A clear overview on how to calculate the particle’s FF, NF and apparent optical properties can be found in .
2.2 Discretization for polydisperse suspensions
As stated earlier, for systems with monodisperse, independently scattering particles, the bulk absorption and bulk scattering coefficients can be obtained by multiplying the related cross-sections with the particle density (number of particles per volume unit), while the normalized phase function and anisotropy factor remain the same [29,30]. For a polydisperse system, however, the absorption and scattering cross-sections, phase functions and anisotropy factors of the different particles (of the same material) depend on the particle radius. To calculate the BOP for such a system, the particles have to be divided in subgroups where each subgroup represents the particles with a radius that falls within a predefined interval j. Accordingly, the Mie solution can be consulted to calculate the apparent absorption (σs) and scattering cross-sections (σa), the anisotropy factor (g) and the scattering phase function (p(θ)) for the mean particle radius of each interval (rj). These optical properties can be combined with the particle density (number of particles per volume unit) for each subgroup (F(rj)) to obtain the BOP :
Where α represents the absorption coefficient of the pure host medium , whilesymbolizes the absorption due to the particles only. Because of the presence of the particles there is a displacement effect on the host medium’s absorption. To correct for this, Modest  introduced, the particle’s absorption cross-section if the particle was of the same material as the host medium (np = nm) [Eq. (1)]. The scattering phase function for a polydisperse system [Eq. (1)] is obtained by summing the non-normalized scattering phase function (σsp(θ)) for the different subgroups and normalizing with µs . A similar approach is followed for the anisotropy factor [Eq. (1)].
As most polydisperse particle systems have a smooth PSD, a higher number of considered intervals (J), and consequently a smaller interval size, would result in a more accurate estimation of the system’s BOP. However, two similar particles with nearly the same radius have highly correlated optical properties. Therefore, an optimal number of intervals exists, satisfying the desired numerical accuracy for the BOP without computational overload. In this study, an iterative procedure is proposed that systematically increases the number of intervals, while monitoring the convergence of the obtained BOP. The normalized error εi(x) [Eq. (2)] is introduced as the absolute value of the relative difference between a bulk optical property calculated in the current cycle of the iteration (xi) and the same bulk optical property in the next cycle (xi + 1):
The normalized error can be monitored for BOP µa, µs, g,and p(θ). As εi(p(θ)) is a vector, it is more convenient to monitor its mean and maximum. For systems with a relatively low concentration of scattering particles, µa is dominated by α. Hence, it is more convenient to monitor, rather than εi(µa) [Figs. 1(b) and 2(b)-2(f)]. The iterative process of further discretization continues until the BOP have converged, i.e. when the normalized errors for the BOP are below a predefined threshold value (e.g. 0.01%). This threshold can be specified by the user for each BOP individually and relates directly to the maximum relative error allowed on the simulated value for the BOP of interest.
It might be of interest to consider more and smaller intervals in the particle size regions where the variability on the particle density and/or the particle’s optical properties is larger. However, as the latter strongly depends on the conditions (wavelength and refractive indices) and therefore is unknown, equally sized intervals are considered here. To simplify the iterative procedure (see next paragraph), the two intervals at the beginning and the end of the PSD are, however, half in size. The cycle of the iterative procedure is indicated with i [Fig. 1(b)]. The number of considered intervals Ji for cycle i is given by:
The radius for interval j in cycle i to be used for the calculation of the interval’s optical properties is the center of that interval and is calculated as follows:
The minimum (rmin) and maximum considered particle radius (rmax) are respectively the lower and upper boundary of the measured PSD, or can be calculated for predefined PDF’s for a given population-percentage that has to be included, or can be specified by the user.
The vector with the radius boundaries (bounds) for each interval is given by:
The upper bound of the jth interval is equal to the lower bound of the j + 1th interval. The bounds are used to calculate the vector with the particle density (number of particles per volume unit) for each interval or subgroup (F(ri,j)) as follows:
Where CND is the cumulative number-frequency distribution. This is the number of particles per volume unit with a radius smaller than or equal to the given bound. The CND is calculated from the volume fraction of the scattering particles in combination with the given number- or volume-frequency PSDs (normal, log-normal power law, gamma, bimodial, …) defined as a PDF, or in terms of discrete size ranges (measured PSDs). For the latter case, interpolation of the CND is done with a monotone piecewise cubic interpolation technique . Due to the construction proposed in Eqs. (3)-(5), all the radii for cycle i, except the two utmost ones, will return in cycle i + 1:
Therefore, the Mie calculations from cycle i can be recycled in cycle i + 1, saving nearly 50% in computational effort. The entire computational procedure, as well as the monitoring of the convergence of the BOP [Eq. (2)] has been programmed in Matlab (version 7.10, The Mathworks Inc., Massachusetts, USA), and consults the parallel computing toolbox.
In Fig. 1, a screenshot of a media file (Media 1) is presented. Media 1 shows an animation of the BOP convergence with increasing discretization of the PSD. In Fig. 1(a), a normal PDF for the volume-frequency PSD is discretized into equally sized (except the two utmost) intervals. In Fig. 1(b) the decrease of the normalized errors with increasing number of considered intervals is illustrated. Note that the BOP for cycle i + 1 need to be simulated to calculate the normalized errors for cycle i. Additionally, if the relevant BOP of cycle i converged (normalized errors < threshold), the BOP for cycle i + 1 are further considered as they should be the most accurate. In general, the scattering phase function converges in the latest stage, indicated by the maximum and mean normalized error [Fig. 1(b)]. The convergence of the BOP is displayed in Figs. 1(c) and 1(d). In Fig. 1(c), the convergence of, µs and g follow the same trend, but this is not generally true (Media 1). The scattering phase function [Fig. 1(d)] for a single particle size (cycle 1) shows a ripple structure. However, as the number of considered intervals increases, the ripple structure of p(θ) is smoothed out. As it takes some cycles to get a smooth p(θ), the number of intervals needed to obtain a converged scattering phase function is larger than for the other BOP.
In Fig. 2, four different normal PDF’s for the volume-frequency PSD are considered and the convergence of the simulated BOP is illustrated with the normalized errors decreasing with increasing number of intervals considered. Again, it is demonstrated that, compared to the other BOP, a higher number of considered intervals is needed for the scattering phase function to converge. This is generally true if the thresholds for all BOP are set equal. The main conclusion is, however, that the optimal number of intervals strongly depends on the mean particle size, polydispersity and the BOP of interest. It should be noted that in Fig. 2, the wavelength and refractive indices of particles and medium were kept constant, while they also have a significant impact (results not shown).
The automatic convergence, as described in this manuscript, will always result in the optimal number of considered intervals, while the user only has to specify the threshold for the normalized error on the simulated BOP. Generally, this normalized error will decrease with increasing number of intervals. However, occasionally [Figs. 2(b) and 2(d)], there might be a small increase, depending on the complexity of the PSD and the particle’s optical properties in that range. Therefore, it is advised to monitor the convergence of all BOP, rather than a single one, and change the threshold for each BOP as a function of the required accuracy.
2.3 Validation on silica nanoparticle suspensions and an oil-in-water emulsion
The BOP simulation approach described in this manuscript has been validated for two aqueous nanoparticle suspensions and an oil-in-water emulsion in the visible and NIR wavelength range. The simulated BOP were compared with the ones obtained from DIS and unscattered transmittance measurements, which are considered to be the ‘golden standard’ method for BOP estimation .
According to the manufacturer, the two silica nanoparticle suspensions (Sicastar, Micromod, Germany) had a particle size (diameter) of respectively 300 and 800 nm with a polydispersity index below 0.2 and a silica nanoparticle concentration of 50 mg/ml (2 g/ml density). The actual volume-frequency PSD was measured with dynamic light scattering (Zetasizer Nano ZS, Malvern Instruments, UK) and the complex refractive indices for silica and water in the Vis/NIR were adopted from respectively [48,49] and . For the measurement of the BOP, the silica suspensions were diluted with 1/3rd water to obtain 1.67% (v/v) silica nanoparticles and fulfill the independent scattering condition . To prevent particle aggregation, the suspensions were vortexed and sonicated for 10 minutes before PSD, DIS and unscattered transmittance measurements were performed.
The consulted Intralipid 20% (batch 10FH1726, expiring date 07/2014, Fresenius Kabi, Germany) is a stabile emulsion of soybean oil in water with mainly oil droplets smaller than 500 nm, acting as spherical scattering particles in the Vis/NIR [18,40,52]. The volume-frequency PSD of the scattering particles was measured with laser light scattering (Mastersizer 2000, Malvern Instruments, UK) and the complex refractive index for soybean oil in the Vis/NIR range was obtained from [4,5]. The volume concentration of the scattering particles in pure intralipid is 22.7% . To prevent dependent scattering in DIS and unscattered transmittance measurements, the intralipid was diluted 8 times with water to obtain a volume fraction of 1.82% of scattering particles. The diluted intralipid was thoroughly stirred before PSD, DIS and unscattered transmittance measurements were carried out.
Accordingly, the measured volume-frequency PSDs and volume fractions of the scattering particles were combined to calculate the cumulative number-frequency distributions (CND), which are consulted in Eq. (6). The CND and the complex refractive indices for scattering particles and host medium, obtained from literature, were loaded in the simulation tool and the BOP for the suspensions and emulsion were calculated for the 500 – 2500 nm wavelength range in steps of 1 nm. The discretization was stopped when the normalized error εi(x) for each BOP was below 0.01%.
The samples were loaded in a 1 mm pathlength cuvette for measurement of the total reflectance and the total and unscattered transmittance from 500 to 2250 nm in steps of 50 nm . The BOP were estimated from these measurements through an inverse adding-doubling routine . Measurements were repeated 3 times and both the average and standard deviation of the measured BOP were calculated [Figs. 4-5].
3.1 Mie solution for a spherical particle embedded in an absorbing host medium
In Fig. 3, the simulated apparent optical properties for a spherical silica particle in ‘water’ at 633 nm for 4 different values for the imaginary part of the medium’s refractive index are shown. As silica is a very weak absorber at 633 nm (Im(np) = 8.5*10−8), the absorption cross-section in an non-absorbing host (red) is very small. However, as the absorption by the host medium increases, the non-exponential absorption of scattered radiation at the particle’s surface increases, resulting in an increase of the particle’s apparent absorption cross-section [Fig. 3(a)]. For this particular case, absorption by the host medium reduces the overall scattering cross-section and results in more forward scattering (higher g), especially for larger particles [Figs. 3(b) and 3(c)]. Moreover, the Mie resonances seemed to be reduced due to absorption by the host medium. More examples and a thorough interpretation of simulated optical properties for spherical particles in an absorbing host medium can be found in [23,24,28–30].
3.2 Validation on silica nanoparticle suspensions
In Fig. 4 the simulated and measured bulk scattering properties (µs, g and µs’) for the two considered nanoparticle suspensions are shown. The simulated and measured µa spectra are nearly the same (R2 ≥ 0.964). Moreover, as there is no visible difference with the µa spectrum of the water fraction (98.3% v/v) itself , they are not shown. The normalized cumulative volume-frequency PSDs in Fig. 4(a) illustrate the difference in particle size and polydispersity between the two nanoparticle systems. The suspension with larger particles (800 nm diameter) results in a higher µs and g spectrum, compared to the one with smaller particles (300 nm diameter) [Figs. 4(b) and 4(c)]. As the effects on µs and g partially compensate each other, the µs’ spectra for the two suspensions are closer to each other, with the larger nanoparticles producing a more flat µs’ spectrum [Fig. 4(d)].
The simulated bulk scattering properties correspond very well with the measured ones [Figs. 4(b)-4(d) and Table 1], especially at wavelengths where the standard deviation (error bars) on the measurements is small. For wavelengths above 1700 nm, the standard deviation on the measured bulk scattering properties increases, while the correlation with neighboring wavelengths decreases. The uncertainty on the measurements at these wavelengths is caused by the very high absorption by water in combination with low scattering [Fig. 4(b)], resulting in diffuse reflectance values close to the noise level. Therefore, noise dominates the measured scattering properties in the wavelength range above 1700 nm . This is especially true for the small nanoparticles, as the scattering is even lower compared to the large nanoparticles. This is probably the main reason for the mismatch (R2 = 0.604) between the measured and simulated anisotropy factor for the 300 nm nanoparticles and wavelengths above 1700 nm. For wavelengths below 1700 nm, a much better match was found (R2 = 0.802). At around 1450 nm, there is a slight disagreement between the measured and simulated µs’ for the 800 nm nanoparticles [Fig. 4(d)]. As there is no plausible reason for a dip in the measured µs’ at this water absorption peak, it is most likely that there is little cross-talk in the separation between µs’ and µa from the DIS measurements.
3.3 Intralipid emulsion
The same procedure as for the nanoparticle suspensions was followed for a dilution of intralipid [Fig. 5]. Because the BOP and the PSDs for Intralipid 20% emulsions have been widely studied, and are constant and stable, results from literature were added for comparison [18,40,52]. The normalized cumulative volume-frequency PSD for intralipid, as measured in this study with laser light scattering (Mastersizer), is shown in Fig. 5(a) and agrees well with the transmission electron microscopy (TEM) results reported by van Staveren et al. .
The simulated bulk scattering properties based on the PSD measurement from this study match very closely with the measured BOP (R2 for µa, µs, g and µs’ of respectively 0.979, 0.992, 0.941 and 0.987) and with the results reported in . This indicates the reliability of the consulted measurements and simulation approach. Small disagreements between measured and simulated BOP can possibly be attributed to errors in the measurement of the BOP and/or PSDs and to slightly incorrect refractive indices for the particles and the host medium which were retrieved from literature. The PSDs reported by Michels et al. , estimated from nephelometry measurements, and Kodach et al. , from TEM, respectively under- and overestimate the particle size when compared to the results from this study. Consequently, also the simulated bulk scattering properties (µs, g and µs’), simulated from those PSDs, are respectively under- and overestimated compared to the BOP measured in this study and . The reason for this is unclear as the very small batch-to-batch variability and high stability in scattering properties of Intralipid 20% has been widely demonstrated [54–56].
The Mie solution for a spherical particle in an absorbing host medium has been a research topic for many years. However, a consensus on how to calculate the apparent particle optical properties that are relevant quantities for the RTT was only recently reached . This resulted in the development of computational codes for the calculation of the Mie coefficients, the scattering matrix and the derived particle optical properties (absorption and scattering cross-sections, anisotropy factor and scattering phase functions) [22,24,30,32]. These codes were consulted in the developed simulation tool. Furthermore, Mishchenko derived the RTT for the case of particles in an absorbing host medium directly from Maxwell’s equations, without an intermediate stop at the particle’s optical properties [33–35]. Nevertheless, up till now, no computational code had been developed to calculate the quantities that enter the RTT, similar to the BOP described in this manuscript. The developed algorithm allows for the comparison of the different approaches and gives more insight into the effect of an absorbing host on the scattering properties of a particle.
The validation of the proposed simulation tool for two nanoparticle suspensions and an oil-in-water emulsion, indicated that the simulations match very well with the measurements if the measurement’s signal-to-noise ratio is sufficient. Moreover, in situations where measurements suffer from noise, the simulations still perform well and give results which physically make sense. The latter is not always true for BOP derived from measurements which are susceptible to noise [Fig. 4(c)]. In addition, simulations of the BOP, following the proposed approach, make it straightforward to examine the effect of microstructure changes (refractive indices, particle size, polydispersity, …) on the BOP. Moreover, a simulation results in an accurate scattering phase function with arbitrary resolution. Such a discretized phase function can be combined with the RTT to obtain a more accurate simulation of light propagation, especially for short source-detector distances [9–11]. On the other hand, a simulation of the BOP can only be as accurate as the inputs that are provided (refractive indices, PSD) and the validity of the independent scattering sphere criteria. For this reason, the proposed tool is only valid for spherical scattering particles with a volume fraction less than 2% [16,37,51,57–59]. For non-spherical particles, the simulated µa, and µs for equivalent spheres will be sufficiently accurate. However, the anisotropy factor g and scattering phase function p(θ) can differ significantly . The T-matrix method for arbitrarily shaped and oriented particles could in these cases be a good alternative for the Mie solution [60,61]. However, it introduces extra degrees of freedom which might import additional sources of error. Furthermore, dependent scattering can be (partially) accounted for by introducing the static structure factor in the RTT [51,62]. Further development, practical implementation and validation of theories is, however, important to increase the accuracy and promote generalization in light propagation modelling.
The development of computational codes to simulate the BOP, starting from physical microstructure information, is important to promote light propagation modelling in turbid media. Moreover, inversion of these codes could even be more interesting as it would clear the way for extracting microstructural information from relatively simple and cost-efficient optical measurements .
A flexible tool to simulate the bulk optical properties for polydisperse spherical particles in an absorbing host medium has been elaborated. The generalization of the Mie solution accounts for the effect of an absorbing medium on the scattering properties of the embedded particles. Discretization of the particle size distribution allowed for the simulation of the bulk optical properties for polydisperse particle systems. Moreover, an efficient iterative procedure was demonstrated, resulting in an automatic optimization of the number of intervals and convergence for the bulk optical properties of interest. Following this approach, the bulk optical properties for two aqueous nanoparticle systems and an oil-in-water emulsion in the Vis/NIR wavelength range were simulated, starting from measured particle size distributions and refractive indices found in literature. The simulated results matched very closely with the reference bulk optical properties (R2 ≥ 0.899), estimated from double integrating sphere and unscattered transmittance measurements. Moreover, this validation showed that the simulations resulted in robust predictions with a physically meaningful outcome, even in situations where reference measurements are meaningless because of significant noise interference.
Ben Aernouts is funded as Ph. D. fellow of the Research Foundation-Flanders (FWO, grant 11A4813N). Rodrigo Watté, Robbe Van Beers and Filip Delport are funded by the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Flanders, respectively grants 101552, 131777 and 120584). The authors gratefully acknowledge IWT-Flanders for the financial support through the GlucoSens project (SB-090053). Gratitude is also expressed to Qiang Fu (UW), Wenbo Sun (NASA), Michael Mishchenko (NASA), Laurent Pilon (UCLA), Ping Yang (TAMU) and Jeppe Revall Frisvad (DTU) for insightful comments and discussions.
References and links
1. A. Torricelli, L. Spinelli, M. Vanoli, M. Leitner, A. Nemeth, N. D. T. Nguyen, B. M. Nicolaï, and W. Saeys, “Optical Coherence Tomography (OCT), Space-resolved Reflectance Spectroscopy (SRS) and Time-resolved Reflectance Spectroscopy (TRS): Principles and Applications to Food Microstructures,” in Food Microstructures: Microscopy, Measurement and Modelling, V. Morris and K. Groves, eds., 1st ed. (WoodHead Publishing, 2013), p. 480.
2. V. V. Tuchin, Tissue Optics: Light Scattering Methods and Instruments for Medical Diagnosis, 2nd ed. (SPIE Press, 2007), p. 840.
3. E. Zamora-Rojas, B. Aernouts, A. Garrido-Varo, W. Saeys, D. Pérez-Marín, and J. E. Guerrero-Ginel, “Optical properties of pig skin epidermis and dermis estimated with double integrating spheres measurements,” Innov. Food Sci. Emerg. Technol. 20, 343–349 (2013). [CrossRef]
4. E. Zamora-Rojas, B. Aernouts, A. Garrido-Varo, D. Pérez-Marín, J. E. Guerrero-Ginel, and W. Saeys, “Double integrating sphere measurements for estimating optical properties of pig subcutaneous adipose tissue,” Innov. Food Sci. Emerg. Technol. 19, 218–226 (2013). [CrossRef]
5. B. Aernouts, E. Zamora-Rojas, R. Van Beers, R. Watté, L. Wang, M. Tsuta, J. Lammertyn, and W. Saeys, “Supercontinuum laser based optical characterization of Intralipid phantoms in the 500-2250 nm range,” Opt. Express 21(26), 32450–32467 (2013). [CrossRef] [PubMed]
6. H. Berberoglu and L. Pilon, “Experimental measurements of the radiation characteristics of Anabaena variabilis ATCC 29413-U and Rhodobacter sphaeroides ATCC 49419,” Int. J. Hydrogen Energy 32(18), 4772–4785 (2007). [CrossRef]
7. S. Sharma and S. Banerjee, “Role of approximate phase functions in Monte Carlo simulation of light propagation in tissues,” J. Opt. A, Pure Appl. Opt. 5(3), 294–302 (2003). [CrossRef]
8. S. C. Kanick, V. Krishnaswamy, U. A. Gamm, H. J. Sterenborg, D. J. Robinson, A. Amelink, and B. W. Pogue, “Scattering phase function spectrum makes reflectance spectrum measured from Intralipid phantoms and tissue sensitive to the device detection geometry,” Biomed. Opt. Express 3(5), 1086–1100 (2012). [CrossRef] [PubMed]
9. R. Watté, Department of Biosystems, KU Leuven, B. Aernouts, and W. Saeys, are preparing a manuscript to be called, “Fpf-MC: A multilayer Monte Carlo method with free phase function choice,” (n.d.).
10. R. Watté, B. Aernouts, and W. Saeys, “A multilayer Monte Carlo method with free phase function choice,” Proc. SPIE 8429, 84290S (2012). [CrossRef]
11. A. Kienle, F. K. Forster, and R. Hibst, “Influence of the phase function on determination of the optical properties of biological tissue by spatially resolved reflectance,” Opt. Lett. 26(20), 1571–1573 (2001). [CrossRef] [PubMed]
12. A. Bhandari, B. Hamre, Ø. Frette, K. Stamnes, and J. J. Stamnes, “Modeling optical properties of human skin using Mie theory for particles with different size distributions and refractive indices,” Opt. Express 19(15), 14549–14567 (2011). [CrossRef] [PubMed]
13. B. Gélébart, E. Tinet, J. M. Tualle, and S. Avrillier, “Phase function simulation in tissue phantoms : a fractal approach,” Pure Appl. Opt. 5(4), 377–388 (1996). [CrossRef]
14. S. K. Sharma, S. Banerjee, and M. K. Yadav, “Light propagation in a fractal tissue model: a critical study of the phase function,” J. Opt. A, Pure Appl. Opt. 9(1), 49–55 (2007). [CrossRef]
17. R. Wang, “Modelling optical properties of soft tissue by fractal distribution of scatterers,” J. Mod. Opt. 47(1), 103–120 (2000). [CrossRef]
19. C. F. Bohren and D. R. Huffman, Absorption and Scattering of Light by Small Particles (John Wiley and Sons, 1983), p. 530.
20. H. C. van de Hulst, Light Scattering by Small Particles (John Wiley and Sons, 1957), p. 470.
21. K. Liou, An Introduction to Atmospheric Radiation, 2nd ed. (Academic Press, 2002), p. 583.
22. J. R. Frisvad, N. J. Christensen, and H. W. Jensen, “Computing the scattering properties of participating media using Lorenz-Mie theory,” ACM SIGGRAPH 2007 Pap. - SIGGRAPH 26(3), 60 (2007). [CrossRef]
24. Q. Fu and W. Sun, “Apparent optical properties of spherical particles in absorbing medium,” J. Quant. Spectrosc. Radiat. Transf. 100(1-3), 137–142 (2006). [CrossRef]
25. W. C. Mundy, J. A. Roux, and A. M. Smith, “Mie scattering by spheres in an absorbing medium,” J. Opt. Soc. Am. 64(12), 1593–1597 (1974). [CrossRef]
26. P. Chýlek, “Light scattering by small particles in an absorbing medium,” J. Opt. Soc. Am. 67(4), 561–563 (1977). [CrossRef]
27. A. Lebedev, M. Gartz, U. Kreibig, and O. Stenzel, “Optical extinction by spherical particles in an absorbing medium: application to composite absorbing films,” Eur. Phys. J. D 6, 365–373 (1999).
30. P. Yang, B.-C. Gao, W. J. Wiscombe, M. I. Mishchenko, S. E. Platnick, H.-L. Huang, B. A. Baum, Y. X. Hu, D. M. Winker, S. C. Tsay, and S. K. Park, “Inherent and apparent scattering properties of coated or uncoated spheres embedded in an absorbing host medium,” Appl. Opt. 41(15), 2740–2759 (2002). [CrossRef] [PubMed]
32. W. Sun, N. G. Loeb, and Q. Fu, “Light scattering by coated sphere immersed in absorbing medium: a comparison between the FDTD and analytic solutions,” J. Quant. Spectrosc. Radiat. Transf. 83(3-4), 483–492 (2004). [CrossRef]
34. M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium. 1. Foldy-Lax equations, order-of-scattering expansion, and coherent field,” Opt. Express 16(3), 2288–2301 (2008). [CrossRef] [PubMed]
35. M. I. Mishchenko, “Multiple scattering by particles embedded in an absorbing medium. 2. Radiative transfer equation,” J. Quant. Spectrosc. Radiat. Transf. 109(14), 2386–2390 (2008). [CrossRef]
36. R. Ceolato, N. Riviere, M. Berg, and B. Biscans, “Electromagnetic scattering from aggregates embedded in absorbing media,” in Progress In Electromagnetics Research Symposium Proceedings,Taipei (2013), pp. 717–721.
38. M. Mishchenko, J. Dlugach, E. G. Yanovitskij, and N. T. Zakharova, “Bidirectional reflectance of flat, optically thick particulate layers: an efficient radiative transfer solution and applications to snow and soil surfaces,” J. Quant. Spectrosc. Radiat. Transf. 63(2-6), 409–432 (1999). [CrossRef]
39. B. Cletus, R. Künnemeyer, P. Martinsen, A. McGlone, and R. Jordan, “Characterizing liquid turbid media by frequency-domain photon-migration spectroscopy,” J. Biomed. Opt. 14(2), 024041 (2009). [CrossRef] [PubMed]
40. V. M. Kodach, D. J. Faber, J. van Marle, T. G. van Leeuwen, and J. Kalkman, “Determination of the scattering anisotropy with optical coherence tomography,” Opt. Express 19(7), 6131–6140 (2011). [CrossRef] [PubMed]
42. C. F. Bohren and D. P. Gilra, “Extinction by a spherical particle in an absorbing medium,” J. Colloid Interface Sci. 72(2), 215–221 (1979). [CrossRef]
43. M. Quinten and J. Rostalski, “Lorenz-Mie theory for spheres immersed in an absorbing host medium,” Part. Part. Syst. Charact. 13(2), 89–96 (1996). [CrossRef]
44. M. I. Mishchenko, “Vector radiative transfer equation for arbitrarily shaped and arbitrarily oriented particles: a microphysical derivation from statistical electromagnetics,” Appl. Opt. 41(33), 7114–7134 (2002). [CrossRef] [PubMed]
45. B. Aernouts, R. Watté, J. Lammertyn, and W. Saeys, “A flexible tool for simulating the bulk optical properties of polydisperse suspensions of spherical particles in an absorbing host medium,” Proc. SPIE 8429, 84290R (2012). [CrossRef]
46. M. Modest, Radiative Heat Transfer, 2nd ed. (Academic Press, 2003), p. 822.
47. F. Fritsch and R. Carlson, “Monotone piecewise cubic interpolation,” SIAM J. Numer. Anal. 17(2), 238–246 (1980). [CrossRef]
48. M. Khashan and A. Nassif, “Dispersion of the optical constants of quartz and polymethyl methacrylate glasses in a wide spectral range: 0.2–3 μm,” Opt. Commun. 188(1-4), 129–139 (2001). [CrossRef]
49. I. Malitson, “Interspecimen comparison of the refractive index of fused silica,” J. Opt. Soc. Am. 55(10), 1205–1209 (1965). [CrossRef]
51. M. I. Mishchenko, D. H. Goldstein, J. Chowdhary, and A. Lompado, “Radiative transfer theory verified by controlled laboratory experiments,” Opt. Lett. 38(18), 3522–3525 (2013). [CrossRef] [PubMed]
52. H. J. van Staveren, C. J. Moes, J. van Marie, S. A. Prahl, and M. J. van Gemert, “Light scattering in Intralipid-10% in the wavelength range of 400-1100 nm,” Appl. Opt. 30(31), 4507–4514 (1991). [CrossRef] [PubMed]
53. S. A. Prahl, “Everything I think you should know about inverse adding-doubling,” http://omlc.ogi.edu/software/iad/iad-3-9-10.zip.
54. B. Cletus, R. Künnemeyer, P. Martinsen, and V. A. McGlone, “Temperature-dependent optical properties of Intralipid measured with frequency-domain photon-migration spectroscopy,” J. Biomed. Opt. 15(1), 017003 (2010). [CrossRef] [PubMed]
57. A. Ishimaru and Y. Kuga, “Attenuation constant of a coherent field in a dense distribution of particles,” J. Opt. Soc. Am. 72(10), 1317–1320 (1982). [CrossRef]
59. V. Twersky, “Low-frequency scattering by mixtures of correlated nonspherical particles,” J. Acoust. Soc. Am. 84(1), 409–415 (1988). [CrossRef]
60. D. W. Mackowski and M. I. Mishchenko, “Calculation of the T matrix and the scattering matrix for ensembles of spheres,” J. Opt. Soc. Am. A 13(11), 2266–2278 (1996). [CrossRef]
62. V. D. Nguyen, D. J. Faber, E. van der Pol, T. G. van Leeuwen, and J. Kalkman, “Dependent and multiple scattering in transmission and backscattering optical coherence tomography,” Opt. Express 21(24), 29145–29156 (2013). [CrossRef] [PubMed]
63. L. Wang, X. Sun, and F. Li, “Generalized eikonal approximation for fast retrieval of particle size distribution in spectral extinction technique,” Appl. Opt. 51(15), 2997–3005 (2012). [CrossRef] [PubMed]