## Abstract

Monodisperse polystyrene microspheres are often utilized in optical phantoms since optical properties such as the scattering coefficient and the scattering phase function can be calculated using the Mie theory. However, the calculated values depend on the inherent physical parameters of the microspheres which include the size, refractive index, and solid content. These parameters are often provided only approximately or can be affected by long shelf times. We propose a simple method to obtain the values of these parameters by measuring the collimated transmission of polystyrene microsphere suspensions from which the wavelength-dependent scattering coefficient can be calculated using the Beer-Lambert law. Since a wavelength-dependent scattering coefficient of a single suspension is insufficient to uniquely derive the size, refractive index and solid content by the Mie theory, the crucial and novel step involves suspending the polystyrene microspheres in aqueous sucrose solutions with different sucrose concentrations that modulates the refractive index of the medium and yields several wavelength-dependent scattering coefficients. With the proposed method, we are able to obtain the refractive index within 0.2% in the wavelength range from 500 to 800 nm, the microsphere size to approximately 15 nm and solid content within 2% of their respective reference values.

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Optical phantoms are often utilized for calibration of optical systems [1] and validation of inverse models for estimation of optical properties [2]. While optical phantoms can come in liquid [3] or solid form [4], water-based suspensions of nearly monodisperse polystyrene microspheres are especially attractive since the optical properties, such as the scattering coefficient and scattering phase function, can be derived using the Mie theory [5]. Suspensions of polystyrene microspheres (SPMs) also come in standardized forms and exhibit significantly smaller batch-to-batch variations in comparison to some other liquid or solid phantoms. Furthermore, the optical properties of other phantoms have to be measured by a well-validated optical modality that is not readily available in every laboratory or institution [4]. These modalities can also introduce bias and uncertainty into the estimated values of optical properties.

Even though the optical properties of SPMs can be accurately derived from the Mie theory, the accuracy of calculated values still depends on the inherent physical properties of the microspheres. Among these properties is the size distribution of the microspheres, which is for commercially available SPMs usually modeled by a Gaussian distribution with supplier-provided mean and standard deviation of the diameter. In our experience, the supplier-provided mean often deviates from the true value and can thus adversely impact the derived scattering coefficient and scattering phase function. Uncertainties in the derived scattering coefficient and scattering phase function can also be introduced by the polystyrene refractive index, which varies quite significantly by more than 0.5% in the literature [6,7]. Finally, the accuracy of the derived scattering coefficient can also be affected by the number density of the microspheres in the SPMs. This information is commonly given by suppliers as the solid content of suspension, from which the number density can be calculated.

The main challenge in accurate utilization of the Mie theory for derivation
of optical properties is thus accurate characterization of the SPMs. While
methods such as scanning and transmission electron microscopy offer
characterization in terms of size, refractive index and solid content have
to be determined separately. For size distribution one can also utilize
dynamic light scattering. Extensive work has been done in the
characterization of aerosol particles, where size and refractive index are
the main unknowns [8–11]. However, the techniques are not
directly applicable to the field of SPMs (the suspensions have to be
atomized or nebulized) and equipment is not readily available in
laboratories that work in the field of biomedical optics. Similar drawback
can be inferred from advanced methods that include single particle
trapping and characterization in terms of size and refractive index [12–14]. A simpler and more
direct method for determination of the refractive index that is
independent of the size and shape of the particles is the immersion liquid
technique (ILT) [15,16]. This technique is based on
transmittance measurement of particles suspended in media with different
refractive indices. By observing the maximum of the transmission, the
refractive index can be deduced. Finally, Ma *et al*.
measured the complex refractive index of the SPM with a double-integrating
sphere setup by observing diffuse reflectance and transmittance [7]. In this case the size and the number
density of the SPMs had to be known *a priori*. A double
integrating sphere was also recently utilized by Postelmans *et
al*. for size distribution determination of SPMs [17]. However, in this case the refractive
index of the polystyrene microspheres had to be set to bulk polystyrene
[18].

The existing methods mostly provide either incomplete or complicated solutions to the problem of SPM characterization. In this regard, multiple methods would have to be utilized to separately yield the size, solid content and refractive index of polystyrene microspheres. Not only that some methods require demanding experimental equipment, using multiple methods to characterize SPMs is time and sample consuming.

In this study, we present in detail a simple ILT-inspired method for characterization of SPMs that can be deployed in virtually every laboratory or institution since it requires only a collimated transmission setup. Our method is based on transmittance measurements of SPMs immersed in an aqueous sucrose solution with different mass concentrations of sucrose (also known as Brix values), where the reference measurement at each Brix value is the equivalent transparent aqueous sucrose solution. From there, the scattering coefficient is obtained through the Beer-Lambert law and fitted using the Mie theory derived scattering coefficient. Since our proposed method relies on the properties of aqueous solution of sucrose, we first provide the refractive index and density as a function of Brix. Secondly, we show the sensitivity of the scattering coefficient on the refractive index and size of the polystyrene microspheres. Thirdly, we discuss in detail the validity of the Beer-Lambert law which can be simply used to yield the scattering coefficient. Finally, we test the proposed method on several commercially available SPMs.

## 2. Materials and methods

#### 2.1 Mie theory and derived quantities

Polystyrene microspheres obtained from various suppliers are usually
not entirely monodisperse and thus the derived quantities used in this
study depend on the size distribution function
*n*(*d*), which is usually given by a
Gaussian model

*C*is the normalization constant defined as

*N*represents the total number of microspheres in a SPM volume

*V*resulting in number density $\tilde n$. The above integration limits are deliberately set to ± 5

*σ*, where the Gaussian distribution falls approximately to 0. For the values of $\bar d$ and

*σ*used in our manuscript, such definition avoids physically meaningless negative diameters and high diameters at which the available numerical implementation of the Mie theory becomes inaccurate [5,19]. Note that we can also define dimensionless probability distribution function from Eq. (2), $p(d )= n(d )/\tilde n$, with a unity integral. Suppliers usually provide the number density $\tilde n$ of the microspheres as solid content

*s*, which is defined as the ratio between the mass of the microspheres and the total volume of the sample SPM. The relationship between the number density $\tilde n$ and solid content

_{c}*s*is given as where ${\rho _{{\rm{poly}}}}$ is the density of the polystyrene (1.05 g/ml) and ${\bar V_{{\rm{one}}}}$ the average volume of one polystyrene microsphere defined as

_{c}*λ*is the wavelength of light in vacuum,

*σ*

_{sc}the scattering cross section given as a function of the dimensionless size parameter and relative refractive index. The wavelength dependent parameters

*n*

_{s}and

*n*

_{m}stand for the refractive indices of polystyrene microspheres and medium. The scattering cross section

*σ*

_{sc}for spherical scatterers can be obtained from Mie theory with numerical approaches readily available in the book by Bohren and Huffman [5] and implemented by Mätzler [19]. Figure 1(a) shows a few examples of the wavelength-dependent scattering coefficients for various diameters $\bar d$ of polystyrene microspheres suspended in water with

*σ*set to 3% of the nominal $\bar d$,

*n*

_{s}(

*λ*) taken from [6] and

*n*

_{m}(

*λ*) from [20].

#### 2.2 Transmittance and scattering coefficient measurement

Transmittance measurements were performed using 10 mm path length glass
cuvettes (OS 100-10-20, Hellma Analytics, Hellma GmbH & Co. KG,
Müllheim, Germany) and a cuvette holder (CUV-UV/VIS, Avantes,
Apeldoorn, The Netherlands). Light from a broadband tungsten-halogen
light source (SLS201/M, Thorlabs Inc., New Jersey, USA) was delivered
to the cuvette holder using a 200 µm optical fiber, where it
was collimated by a lens
(*f* = 8.7 mm CUV-UV/VIS,
Avantes). A second lens (*f* = 8.7
mm) was used to collect the light, which was then transmitted to a
spectrograph-based spectrometer (ImSpector V10E 400-1000 nm, Specim,
Spectral Imaging Ltd., Oulu, Finland) by a second 200 µm
optical fiber (Fig. 1(b)). The transmittance was defined as a ratio of the dark
corrected signal obtained through the cuvette containing the SPM and
the dark corrected signal through the same cuvette containing
transparent aqueous sucrose solution with the same medium refractive
index as the SPM. The wavelength range considered in our study spanned
from 500 to 800 nm.

If a low number density SPM is investigated, the scattering coefficient
*µ*_{s} can be obtained from
transmittance through the Beer-Lambert law ${\mu _s} ={-} \ln
T/d$, where *T* is the
measured transmittance and *d* is the cuvette path
length. Since in general, the Beer-Lambert law returns the attenuation
coefficient, the validity of the attenuation coefficient being equal
to the scattering coefficient is discussed in the Section 3.3 using Monte Carlo
simulations.

#### 2.3 Parameter optimization procedure

Unknown parameters of the SPMs in our study involve the solid content
*s*_{c}, refractive index
*n*_{s}(*λ*) and diameter
*d* of the microspheres. The solid content usually
changes after a long shelf time due to potential water evaporation.
The refractive index
*n*_{s}(*λ*) and diameter
*d* of the microspheres are usually provided only
approximately by suppliers or in the literature.

We have modeled
*n*_{s}(*λ*) as a
one-term Sellmeier equation (*λ* specified in
meters)

In total, we are faced with a problem of optimizing 4 unknown
parameters (*s _{c}*,

*d*,

*a*,

*b*) by fitting the Mie theory predicted scattering coefficient to measurements. While spectral regularization over a wavelength range from 500 to 800 nm yields many measurement points, we show in Section 3.4 there is a significant crosstalk between the unknown parameters, if only a single SPM is used. We have found that this crosstalk can be alleviated, if the starting SPM is diluted in aqueous sucrose solutions with different Brix values thus changing the refractive index of the surrounding medium

*n*

_{m}. Therefore, the proposed cost function that is optimized can be written as

*i*-th SPM diluted by an aqueous sucrose solution. The Mie theory predicted scattering coefficient ${\mu _{\rm{s}}}$ (as defined in Eq. (5)) depends on the solid content

*s*

_{c}, diameter

*d*and refractive index of the medium

*n*

_{m}and polystyrene

*n*

_{s}. The medium refractive index further depends on Brix ${B_i}$ and wavelength ${\lambda _j}$, while polystyrene refractive index depends on parameters

*a*and

*b*of the Sellmeier equation, and wavelength ${\lambda _j}$.

Our procedure involves the optimization of parameters
(*s*_{c}, *d*,
*a*, *b*) using 31 wavelength points
uniformly distributed between 500 and 800 nm and five different SPMs
with medium Brix values of 0.0, 20.0, 30.0, 50.0 and 60.0%.
Optimization was performed using Trust Region Reflective algorithm
with lower boundaries set to (a = 1.0,
b = 1.0 m^{2},
*d* = 0.1 µm,
*s*_{c} = 0.0 g/ml) and
upper boundaries to (a = 4.0,
b = 4.0 m^{2},
*d* = 2.0 µm,
*s*_{c} = 1.0 g/ml). In
Sect. 3.4, 16 initial values
sets were utilized which were varied ± 30%
from their nominal true values. In Sect. 3.5, a single set of initial values was used
(a = 2.0, b = 3.0
m^{2}, *d* = 0.1
µm, *s*_{c} = 0.01
g/ml).

#### 2.4 Sample preparation for transmittance measurements

As pointed out in the previous section, our optimization procedure utilizes five different dilutions of original SPMs with aqueous sucrose solution yielding different medium Brix values. In this way, the medium refractive index is varied to regularize the optimization. Since the transmittance of each resulting SPM is measured against a transparent reference with the same Brix value (to match the refractive index), each SPM requires a matching transparent aqueous sucrose solution without the scatterers.

For this study, we have diluted the original SPMs by knowing the
approximate solid content, diameter and polystyrene refractive index
beforehand. This prior knowledge was used to obtain scattering
coefficients of final suspensions that are within the validity of the
Beer-Lambert law (i.e., in the order of 1-4 cm^{−1}),
which we discuss in Section 3.3. Although at least some approximate prior information is
often available for SPMs from the suppliers, we later also describe a
mixing procedure, when the specifications of the SPM are unknown.

Firstly, we prepared transparent solutions without the scatterers by
mixing a specified quantity of deionized water and sucrose (BioXtra
≥99.5%, S7903-1KG, Sigma-Aldrich Inc.) to yield the
target Brix value for five solutions given in Table 1, along with the required masses
of water (*m*_{H2O}) and sucrose
(*m*_{sucrose}). A vital step in the sample
preparation is thorough mixing that results in optically uniform
samples. Mixing proved to be particularly important and somewhat
challenging for the samples with high Brix values, since the density
and viscosity of the liquid increases significantly. These samples
were mixed with a magnetic stirrer (ARE Heating Magnetic Stirrer, VELP
Scientifica, Usmate, Italy) for roughly 10 minutes and sonicated in an
ultrasonic bath (Ultra 8020, James Products Europe Limited, England)
to eliminate any bubbles that formed during the mixing step.

Secondly, we prepared diluted SPMs in a two-step procedure to conserve
with the amount of sucrose and to yield a smaller quantity of the
final sample for easier mixing. In the first step, a pre-calculated
volume of the original SPM (*V*_{orig}) was
added to the volume of pure deionized water
(*V*_{addH2O}). After thorough mixing with a
magnetic stirrer, the second step involved extraction of a smaller
volume of the diluted SPM (*V*_{ext}) to which
we added the same amount of sucrose
(*m*_{sucrose}) as in the corresponding
transparent reference solutions, to obtain the same medium refractive
index. The volumes are provided in Table 2 for all microsphere sizes that were tested
experimentally in this study (obtained from microParticles GmbH,
Berlin, Germany). Note that the preparation of sucrose-free sample 1
follows a simplified procedure that does not require addition of
sucrose in the second step.

Often the specifications, e.g. solid content
*s _{c}*, of SPMs are unknown, especially after
a long shelf time or after unrecorded dilutions. In this case, prior
knowledge cannot be used to calculate the required volumes and masses
of the original SPM, deionized water and sucrose. However, the
following procedure can be used to yield the final scattering
coefficient that is still within the validity of the Beer-Lambert law
and introduces practically negligible errors in terms of mixtures.

Firstly, a small but easily measureable mass of original SPM is
extracted and mixed with a required mass of sucrose to yield the
target Brix value. The required mass of sucrose is calculated by using
the total mass of the original SPM and disregarding the volume
occupied by the scatterers. Note that in this case we make a crude
approximation that sucrose can dissolve in the whole volume of the
original SPM. Since for high solid contents the available aqueous
volume is significantly smaller, we actually obtain a larger Brix
value. Next, we separately mix a transparent aqueous sucrose solution
with the true target Brix value. Subsequently, we add the solution to
the SPM until the transmittance corresponds to the target scattering
coefficient, where the Beer-Lambert law holds (i.e., in the order of
1-4 cm^{−1}). As a reference in the transmittance
measurements, we use the transparent aqueous solution of sucrose.
Finally, all the masses and dilutions should be monitored and recorded
so that they can be traced back to the original SPM. The above
procedure should be repeated for five target Brix values as given in
Table 1.

While the above procedure initiates with a wrong Brix value in the SPM, it should be noted that upon dilution with the transparent aqueous sucrose solution with the correct Brix value, the final Brix value virtually coincides with the correct Brix value. This is a reasonable assumption since the final solid content of the diluted SPM is so small that can be neglected without introducing significant errors. For example, when the solid content of the original SPM is large, the error of the initial Brix value is also large. However, since the solid content is large, the SPM will have to be significantly diluted by the transparent equivalent, which will finally yield a suspension with a Brix value that is virtually the same as that of the transparent aqueous sucrose solution. Likewise, if the solid content of the original SPM is small, then the error of the initial Brix value is also small. By addition of the transparent equivalent, the error of the Brix value is further reduced.

Since possible errors of the Brix value directly translate to the
errors of medium refractive index, we have evaluated the errors in
terms of different solid contents (between 0.001 and 0.1 g/ml) of the
original SPM and different target Brix values (between 0.0 and
60.0%). Figure 2
shows the maximum relative error of the medium refractive index
*n*_{med} calculated across different solid
contents and displayed as a function of the Brix value. It can be seen
that the relative error of the medium refractive index never exceeds
0.05% and significantly decreases with increasing diameter of
the microspheres. For 0.0% Brix value the error is 0%,
since dilution is done with deionized water and the medium refractive
index does not change from the initial one.

Once the scattering coefficients of five polystyrene suspensions with
different Brix values are measured, they can be fitted using
Eq. (7) by
optimizing the parameters of the original SPM, where the solid content
*s _{c}* can be traced back through the recorded
masses and dilutions of each suspension (i.e., all of the final SPMs
share the same original SPM).

#### 2.5 Refractive index and density of aqueous sucrose solution

The refractive index of aqueous sucrose solution is given in the literature mainly at the wavelength of 589.3 nm. Since the proposed method requires the medium refractive index at multiple wavelengths, we have utilized a commercial refractometer (Abbemat MW, Anton Paar GmbH, Graz, Austria) with which we were able to measure the refractive index at eight wavelengths including 436.3, 480.3, 514.2, 533.1, 546.1 589.3, 633.7 and 657.2 nm and at two temperatures, namely 20.0 and 25.0°C.

The density of the sucrose solution was measured at 25°C using a 50 ml certified pycnometer (ISOLAB Laborgeräte GmbH, Eschau, Germany).

#### 2.6 Monte Carlo simulations

To investigate the effects of multiple scattering and anisotropy factor on the validity of the Beer-Lambert law, we have modeled light propagation through a glass cuvette with a custom-developed OpenCL implementation of the Monte Carlo method. The measurement setting was modeled as a multi-layer structure with source-cuvette and cuvette-detector distance set to 10 mm, cuvette wall thickness to 1.25 mm and cuvette path length to 10 mm. The refractive index of the medium surrounding the cuvette was set to 1.0, the refractive index of the cuvette was modeled as that of quartz [21] and the refractive index of the interior was modeled according to the Brix value of the sucrose solution. The source beam and the detector diameter were set to 4.0 mm with a beam divergence and detection acceptance angle of 0.66 degrees. Transmittance was defined as a ratio between the number of detected photon packets obtained through a glass cuvette filled with the SPM and with the reference transparent aqueous sucrose solution. For each simulation we have propagated 500 million photon packets. The simulations were performed on a personal computer (i7, 32 GB RAM, Nvidia GTX 1070).

## 3. Results and discussion

#### 3.1 Refractive index and density of aqueous sucrose solution

The refractive index and density of aqueous sucrose solution had to be measured to ensure accurate calculations of the scattering coefficient and solid content of the SPMs given a certain Brix value.

Tables 3 and 4 show the refractive index of the
aqueous sucrose solution at two temperatures 20.0 and 25.0°C,
and multiple wavelengths. The corresponding graphs of the aqueous
sucrose solution refractive index as a function of wavelength and Brix
value are presented in Fig. 3(a). While the refractive index was measured in the narrow
range from 436.4 to 657.2 nm, it can be extrapolated up to 850 nm
using a fitted three term Sellmeier equation without introducing
significant errors. Extrapolation was tested on the refractive index
of water measured by Daimon *et al*. [20], by fitting the three term
Sellmeier equation to 8 wavelength points between 436.4 and 657.2 nm.
Subsequently, the relative error between the true measured and fitted
values was calculated in the range from 450 to 850 nm. The maximum
error due to the extrapolation was located at 850 nm and amounted to
merely 0.002% or at the 5^{th} decimal place in the
refractive index. Since we expect that the aqueous suspension of
sucrose exhibits similar wavelength dependence of the refractive
index, the extrapolation should not pose a problem.

Table 5 and Fig. 3(b) show the dependence of aqueous sucrose solution density on the Brix value. The density was measured at 7 points (red circles), to which a third order polynomial was fitted (shown with navy blue line in Fig. 3(b)).

#### 3.2 Influence of the polystyrene refractive index and diameter deviations on the Mie theory derived reduced scattering coefficient

One of the main optical properties of turbid phantoms is the reduced
scattering coefficient *µ*_{s}’.
Advantage of the SPM-based optical phantoms over other alternatives is
that the optical properties such as the reduced scattering coefficient
can be derived using Mie theory. However, the calculated quantities
significantly depend on the optical and physical properties provided
for the SPM. Since the polystyrene refractive index given within the
literature can vary by more than 0.5%, it is important to study
these dependencies. Figure 4(a) depicts the literature provided polystyrene refractive
indices and the red region demonstrates the relative deviation between
two commonly cited values. Although the relative difference of
0.5% might seem insignificant, such discrepancies in the
polystyrene refractive index result in much higher relative
differences of more than 5% in the derived reduced scattering
coefficient for 1-µm-sized spheres (Fig. 4(b)). In other words, the relative
error that propagates from the polystyrene refractive index to the
reduced scattering coefficient is amplified approximately by a factor
of 10. For example, the blue region in Fig. 4(a) corresponds
to ± 0.2% relative deviation of the
polystyrene refractive index and results
in ± 2% relative deviation in the derived
reduced scattering coefficient for 1 µm sized spheres.

Diameter uncertainty is much less amplified than the uncertainty of the refractive index. Figure 5 shows the reduced scattering coefficient for 1 µm SPM with the blue region representing the ± 2% relative deviation in the nominal diameter. It can be seen that the relative deviation in the reduced scattering coefficient is also approximately 2% (red region), so the errors are propagated without amplification.

Based on these results, the relative error of the refractive index affects the reduced scattering coefficient much more than the diameter. Therefore, we aim to determine the refractive index within ± 0.2% of the reference value, while the diameter roughly within ± 2% of the reference diameter.

#### 3.3 Validity of the Beer-Lambert law

The Beer-Lambert law relates the transmittance to the attenuation coefficient, which in non-absorbing purely scattering media is the scattering coefficient. However, the validity of the Beer-Lambert law can be broken by multiple scattering and high anisotropy factor of the scatterers. In the first case, the signal due to multiple scattering is not as highly attenuated as one would expect, since some of the light can be scattered back into the light path. In the second case, the anisotropy factor can hinder the validity due to increasing the amount of forward scattered light, which cannot be discriminated from unscattered light.

Figure 6(a) shows the
relationship between the scattering coefficients
(*µ*_{s}) estimated by the Beer-Lambert
law from the Monte Carlo simulated transmittance and the true input
scattering coefficient at a wavelength of 600 nm. Different colors
represent different anisotropy factors as a result of polystyrene
microsphere diameter variation from 0.01 to 3 µm submerged in
an aqueous sucrose solution with Brix values varying from 0.0 to
60.0% (see the colorbar). We can clearly observe that the
estimated scattering coefficient starts to deviate from the true
scattering coefficients at high values. This difference is also more
pronounced for high anisotropy factors. Both observations agree with
the fact that at high scattering coefficients multiple scattering
events start contributing to the transmittance signal and for high
anisotropy factors more intense forward scattering is contributing to
the transmittance signal. The absolute value of the relative deviation
of the estimated scattering coefficient from the true scattering
coefficient is presented in Fig. 6(b). We can observe that the relative deviation
is less than 2% for true input scattering coefficient under 4
cm^{−1}.

Consequently, a scattering coefficient of approximately 2.5
cm^{−1} at 633 nm was enforced in our experiments,
which ensured that the scattering coefficient across the utilized
spectrum from 500 to 800 nm would not significantly exceed 4
cm^{−1}. In the experimental setting the target
transmittance for a 10 mm cuvette would therefore lie around
8.2%. It should be noted that while using small scattering
coefficients even below 1 cm^{−1} could prove useful in
terms of Beer-Lambert law validity, the relative error that propagates
into the scattering coefficient from the transmittance measurement
increases significantly when transmittance approaches 100%,
therefore the measured scattering coefficient should lie between 1 and
4 cm^{−1}.

In Fig. 6(c), which shows
the dependence of the anisotropy factor on the sphere diameter for
three different Brix values of the solution, we can see that the
anisotropy rises with the sphere diameter and Brix value. However, as
given in Fig. 6(b), the
relative deviation of the estimated scattering coefficient does not
exceed 2% for sphere diameters up to 3 µm and for true
input scattering coefficients under 4 cm^{−1}.

#### 3.4 Scattering coefficient fitting ambiguity

The most straightforward approach in the optimization of four unknown
parameters (*s _{c}*,

*d*,

*a*and

*b*) would be to fit the Mie theory predicted scattering coefficient to a single measured spectrum

*µ*

_{s}(

*λ*), i.e. choosing only one term in the sum over

*B*, Eq. (7). In this way, only a single dilution of the original SPM would be required. However, to show that several dilutions with different

_{i}*B*values are required, we firstly investigated the uniqueness of the optimization process when using a single and five different

_{i}*B*values by setting the initial values of optimized parameters to ± 30% from their true values. The optimization process was performed on the wavelength-dependent scattering coefficient

_{i}*µ*

_{s}(

*λ*) obtained from the Monte Carlo simulated transmittance through a 10 mm cuvette containing 1-µm-sized SPM to circumvent potential errors that would be introduced by the measurements but still account for the characteristics of the experimental setup. Since 4 unknown parameters were optimized, 16 initial value sets were tested.

The relative deviation of the estimated *s _{c}*,

*d*and

*n*

_{s}(based on the parameters

*a*and

*b*) from their respective true values is collected in Fig. 7 for the case of single and five different

*B*values in the optimization process. It can be observed that when using only a single dilution (Fig. 7(a)), the optimized values for many initial values significantly differ from their respective true values. This would suggest that the optimization process gets trapped in a local minimum. In contrast, when using five different

_{i}*B*values (Fig. 7(b)) the optimization process always ends in the global minimum, where the optimized values are virtually the same as the true ones. Although a single or five

_{i}*B*dilutions are investigated in our study, we have found that three

_{i}*B*dilutions are already adequate for reliably locating the global minimum.

_{i}Figure 8(a) outlines an
example from the above optimization test that uses a single
*B _{i}* dilution. The scattering coefficient
seems to nicely fit although the optimized values
(

*s*= 0.0325 g/ml,

_{c}*d*= 0.661 µm,

*n*

_{s}= 1.714) deviate significantly from the true values (

*s*= 0.05 g/ml,

_{c}*d*= 1.00 µm,

*n*

_{s}= 1.587). Using these optimized values to represent the scattering coefficient of five

*B*dilutions, one can observe that only one of the five spectra is fitted well (Fig. 8(b)). Therefore, in the case of a single

_{i}*B*dilution, the optimization process would stop, while in the case of five

_{i}*B*dilutions, the optimization process would continue until all five scattering coefficients are fitted well and the estimated values do not deviate significantly from their respective true values.

_{i}The above results suggest that when using only a single
*B _{i}* dilution in the optimization process,
different estimated parameters can yield the same scattering
coefficient. Therefore, the parameters experience significant
crosstalk that is eliminated when using several

*B*dilutions in the optimization process. To further demonstrate why a single

_{i}*B*dilution would be insufficient to discriminate between

_{i}*s*,

_{c}*d*and

*n*

_{s}, Fig. 9(a) shows the effect of the diameter

*d*and Fig. 9(b) the effect of the polystyrene refractive index

*n*

_{s}on the scattering cross section of nominally 1 µm microspheres. The scattering coefficient is obtained by multiplying the scattering cross section and the number density of the scatterers which is directly connected to the solid content

*s*(Eq. (3)). Therefore, changing the three parameters leads to multiplicative changes in the scattering coefficient and is thus difficult to discriminate when using only a single

_{c}*B*dilution.

_{i}#### 3.5 Refractive index, size and solid content of SPMs

The proposed optimization method was tested on three different SPMs with mean sphere diameters equal to 0.425, 0.721 and 1.05 µm (further details in Table 5). To assess the repeatability of measurements, the 1.05 µm SPM was measured twice within 7 days.

Figure 10 shows the
estimated polystyrene refractive index as a function of wavelength
according to the optimized values of parameters *a* and
*b* for different particles. Blue line represents the
polystyrene refractive index as measured by Nikolov *et
al*. [6] and the green
region represents ± 0.2% relative
deviation. It can be observed that for all sizes, the measured values
of the polystyrene refractive index fall within
the ± 0.2% region. Moreover, the
refractive index of the 1.05 µm microspheres from the same SPM
overlaps, which suggests good measurement repeatability. Although
likely not the sole reason, the discrepancies between the three
different SPMs could be due to batch-to-batch variations in the
microspheres synthesis process as also suggested by Jones *et
al*. [12]. The
discrepancies could also be a consequence of the random uncertainties
during the measurement procedure, which propagate into the
transmittance and subsequently affect the scattering coefficients. To
this end, however, further investigations showed that deliberate
random perturbations of the scattering coefficients even up to
5% have not significantly changed the estimated refractive
indices in Fig. 10.

The fitted scattering coefficients as modeled by the Mie theory (red
dashed lines) and the measured scattering coefficients (navy blue) are
presented in Fig. 11 for
all the three test SPMs utilized in this study. One can observe that
for all Brix dilutions (different lines within each graph) and
microsphere sizes (a-d), the match is excellent. The red region
represents the relative root-mean-square (RMS) error calculated
between the fitted and measured scattering coefficients across the
different Brix dilutions for each test case. For all except the last
case (Fig. 11(d)), the
relative RMS error is mainly below 1%, while for the 0.425
µm microspheres, the RMS error seems to be closer to 2%.
It should be noted that the measured scattering coefficients do not
cross the target 2.5 cm^{−1} at 633 nm as set out in
Section 3.3. This is because
the SPM dilutions were made by assuming that the values of the solid
content and diameter as specified by the supplier are completely true.
Due to small differences in the two parameters and also due to errors
during the mixing procedure, the final scattering coefficient slightly
deviated from the target value. However, the masses of constituents
were precisely recorded during the dilution and subsequently traced
back to the original SPM.

Finally, the estimated values of remaining parameters
(*s _{c}* and

*d*) are given in Table 6 along with the supplier data. In addition, we also determined the

*s*by evaporating a sample of the original SPM and then measuring the mass of the residue. The relative deviation of the estimated parameters is given in parenthesis and is for

_{c}*s*calculated from the measured mass of the residue, while for the diameter

_{c}*d*from the supplier data. The relative deviation of the

*s*never exceeds 2% and follows the reference values quite nicely. However, the diameter seems to deviate slightly more from the supplier-provided values.

_{c}In our experience, the supplier-provided diameter often deviates from the true value and we have come across this problem often when dealing with SPM-based optical phantoms. Especially, when the optical phantoms were used with optical fiber probes that included small source-detector separations [22,23], where the acquired reflectance spectrum exhibits oscillations that are due to the scattering phase function dependence on the wavelength. The signature of these oscillations also significantly depends on the diameter of the microspheres. In Fig. 12, we present reflectance spectra of 1.05 µm SPM as measured with an optical fiber probe with a source-detector separation of 220 µm. Additionally, we also show Monte Carlo simulated reflectance spectra using Mie theory derived scattering phase functions for four different diameters (1.007, 1.021, 1.05 and 1.07 µm) and with the polystyrene refractive index estimated by the proposed method. It can be seen that the reflectance spectra simulated for 1.007 and 1.021 µm SPMs seem to fit best the measured reflectance, with 1.021 µm being the optimal. Figure 12 indicates that the supplier-provided value of 1.05 µm might not be accurate and could just be an estimate provided as a result of the controlled manufacturing process. Furthermore, both the proposed method and the measured reflectance with an optical fiber probe suggest that the true diameter is significantly below 1.05 µm. Since the simulated reflectance of a 1.021 µm microsphere suspensions seems to best fit the measurements, the estimate of our method is off by only –1.4% and not by 4.2%, as would be inferred from the supplier-provided data in Table 6.

## 4. Conclusion

Characterization of SPMs in terms of refractive index, size and solid content is important for accurate calculation of optical properties of SPM-based turbid phantoms. For this purpose, we have proposed a simple method that is based on collimated transmittance measurements of SPMs diluted by different sucrose solutions. We have shown that such measurements allow simultaneous estimation of the refractive index, size and solid content. Thus, sucrose-based dilution is the key to accurate estimation and to the best of our knowledge, it has not been utilized before. With the proposed method we were able to obtain the refractive index within 0.2%, the microsphere size to approximately 15 nm and solid content within 2% of their respective reference values. The obtained values nicely fall within the requirements set out in the Section 3.2, where the refractive index should be known within 0.2% and size within 2% of their reference values to yield the scattering coefficient within 2%.

Further work on the proposed method includes characterization of the region of validity especially in terms of the size of the polystyrene microspheres. Up to this point, the proposed method was tested for microspheres between 0.425 and 1.05 µm, although according to the validity of the Beer-Lambert law, the method should work without any modifications for polystyrene microspheres up to 3.0 µm. Additionally, the proposed method should be investigated on monodisperse microspheres of other materials such as silica, which are also commonly used for scattering in turbid phantoms. Finally, we believe that the determination of refractive index, size and solid content could be further improved by incorporating additional information in terms of reflectance measured with optical fiber probes at small source-detector separations as was shown in Fig. 12.

## Funding

Javna Agencija za Raziskovalno Dejavnost RS (J2-7211, J2-8173, P2-0232).

## Disclosures

The authors declare no conflicts of interest.

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