Dynamic light scattering (DLS) is a widely applied technique in multiple scientific and industrial fields for the size characterization of nanoscale objects in solution. While DLS is typically applied to characterize systems under static conditions, the emerging interest in using DLS on temporally evolving systems stimulates the latent need to improve the time resolution of measurements. Herein, we present a DLS microscopy setup (micro-DLS) that can accurately characterize the size of particles from autocorrelation functions built from sub-100 ms time windows, several orders of magnitude faster than previously reported. The system first registers the arrival time of the scattered photons using a time-correlated single photon counting module, which allows the construction of the autocorrelation function for size characterization based on a time window of freely chosen position and width. The setup could characterize both monomodal (60 or 220 nm polystyrene particles; PS) and multimodal size distributions (e.g., mixture of 20 nm LUDOX and 80 nm PS) with high accuracy in a sub-100 ms time window. Notably, the width of the size distribution became narrower as a shorter time window was used. This was attributed to the ability of the system to resolve the sub-ensemble of the broad size distribution, as the broad distribution could be reconstructed by accumulating the distribution obtained by consecutive 80 ms time windows. A DLS system with high temporal resolution will accelerate the expansion of its application toward systems that evolve as a function of time beyond its conventional use on static systems.

## I. INTRODUCTION

Characterization of the size of sub-micron and nano-objects in solution is a fundamentally important and ever-present need in many scientific and industrial fields. Among particle size analysis techniques^{1,2} (particle counting by imaging, scattering, and physical separation), light scattering methods have been applied for this purpose. Dynamic Light Scattering (DLS) has been used for many decades to measure the size of nano-objects in colloidal solutions. DLS finds applications in a diverse range of fields,^{3} including colloid and polymer sciences, as well as food chemistry,^{4} nanotechnology,^{5,6} nanomedicine,^{7} and biomedical sciences,^{8} where the size characterization of nano-objects in colloidal solutions is often essential. DLS is also consistently applied in industry for size characterization of colloids,^{9} for example, in the development of ceramics, cosmetics, food and beverages, pharmaceuticals, and pigments, among others.

DLS is based on the correlation of time dependent light intensity fluctuations of interfering wavefronts, originated by scattered light at individual colloidal particles undergoing Brownian motion.^{10} The diffusion coefficient of the particles in solution can be obtained from the decay of an autocorrelation function, which can be converted to its size. Large particles will diffuse slower than small particles and consequently show a slower decay of their autocorrelation curve. In its early stages, DLS was also known as quasi-elastic scattering, photon correlation spectroscopy, and intensity fluctuation spectroscopy.^{11} After the invention of the laser, the development of DLS bloomed in the late 1960s, and its solid foundation was established.^{11–15} A typical DLS setup is composed of a coherent light source and a photodetector, assembled in a configuration that optically couples them at the sample. The detector is connected to a data acquisition module that can achieve the high acquisition rates needed to capture fluctuations from particle dynamics. The wide applicability of DLS^{9} stems from its ability to non-destructively^{8} provide ensemble particle size estimates over a wide range of sizes, as well as its relatively quick and easy operation. However, conventional DLS is limited by multiple factors, such as sample concentration, the presence of pollutants (large particles), and the stationarity of conditions.^{11}

As a way to overcome some of these limitations, DLS can be implemented on an optical microscope. The core benefit of this technique, so-called “micro-DLS,” is its high spatial resolution, opening the possibility to probe the local properties of heterogeneous materials,^{16} which also improves DLS performance in the presence of pollutants. Another advantage is its ability to probe opaque samples (with high light-scattering and light-absorbing properties), a highly desired feature in fields of study where regular DLS is unable to perform measurements. There are multiple reports in the literature of light-scattering microscopy techniques^{17–19} as well as DLS-specific microscope designs.^{16,20,21} Hiroi and Shibayama^{22} were the first to propose a micro-DLS technique to effectively probe highly opaque and turbid samples by using a back-scattering arrangement, a confocal optical design, and partial heterodyne correction of the detected signal. The seminal work by Hiroi and Shibayama^{22} led to various applications of micro-DLS, such as the measurement of particle size distributions in turbid solutions,^{23} the aggregation states of carbon nanotubes,^{24} and the study of anomalous size distributions of silver nanocolloids.^{25} These studies further highlighted the power of micro-DLS in measuring particle size distributions on samples reaching high opacity conditions.

While there is a vast amount of literature using DLS and micro-DLS techniques, most of their applications have been in the size characterization of nano-objects in steady state. Typical DLS measurements require time ranges from seconds to minutes to obtain an autocorrelation curve for size determination. Yet, there has been an emerging interest in using DLS to follow systems that evolve over time. For example, “fast dynamic light scattering (FDLS)”^{26} was developed to study the formation of calcium carbonate *in situ* at a time resolution of 20 s. Recently, Gowayed *et al.* used DLS to study liquid droplets of an aqueous glycine solution prepared by a CW laser^{27} by following the change in the size distribution of glycine clusters as a function of irradiation times at a time resolution of 2 min. More recently, a numerical analysis methodology was proposed to extract particle size distributions of non-stationary DLS signals and was applied to transient signals measured at a time resolution as short as 10 s.^{28} This trend of studying transient systems by applying DLS techniques in a time-resolved manner sparks the latent need for the improvement of DLS and micro-DLS in their temporal resolution.

Here, we report a micro-DLS implementation with post-data-processing capabilities that can retrieve monomodal and multimodal size distributions at a time resolution as short as 40 ms. Our inspiration came from the single molecule fluorescence spectroscopy field, where the photon arrival times are often registered using a time correlated single photon counting (TCSPC) device. The data can then be post-processed by choosing specific time ranges to access photophysical and chemical information that varies over time.^{29,30} While commercial DLS setups output the autocorrelation curve from the measurement directly, the advantage of post-processing the data from DLS measurements has started to be recognized and adopted in DLS lately. For example, Malm and Corbett^{31} showed that it can improve the precision and accuracy of measured size distributions when a statistical analysis is made on a series of autocorrelation curves with shorter acquisition times, compared with an analysis on a single autocorrelation curve built over the whole acquisition time. Hiroi *et al.*^{32} recorded photon arrival times as data instead of autocorrelation curves and post-constructed autocorrelation curves for arbitrary time windows over the course of the acquired intensity transients. They demonstrated that they could reject the contribution of pollutants such as large particles and extract the size distribution of dominant particles in their sample. While these works aimed to improve the size precision by taking advantage of post-processing data, we consider the time-resolving ability and flexibility of choosing the time range of interests as boosters of DLS capabilities and an asset to following dynamic chemical processes *in situ*. In a very recent development, Liénard *et al.*^{28} reported a method to follow the particle size distribution of a system under non-equilibrium by analyzing consecutive and overlapping time windows extracted from a single measurement at a time resolution as short as 10 s. The present work focuses on demonstrating the ability of micro-DLS to measure monomodal and multimodal size distributions at as fast as 40 ms time resolution, paving the way for time resolved *in situ* applications.

## II. SYSTEM DESIGN AND METHODOLOGY

### A. Optical assembly and data acquisition

The setup was home-built based on an inverted optical microscope (Olympus IX73) (Fig. 1). A 532 nm continuous wave laser (Laser Quantum, Opus 532) was expanded by a telescope to slightly overfill the back aperture of a microscope objective [Olympus, 20×, numerical aperture (NA) 0.4, air]. The beam was reflected by a dichroic beamsplitter (AHF analysentechnik AG, F73-512 Shortpass Dichroic) and focused on the sample by the objective (spot size estimated as ∼2 *µ*m diameter in the lateral direction and ∼8 *µ*m in the vertical direction), then Rayleigh scattering was collected by the same objective and sent toward the detector. The power of the laser after the objective was set between 160 and 16 *µ*W, depending on the sample conditions. The scattering signal was spatially filtered by a 25 *µ*m pinhole placed at the focal plane of the tube lens (confocal detection), and a single photon counting module [avalanche photodiode (APD), Laser components GmbH, Count® Blue] was coupled to the pinhole plane by a relay lens (Thorlabs, 75 mm biconvex, LB4330). The position of the APD was shifted in the *z* axis from the exact focal position to slightly overfill the APD detector. This greatly reduced the detection of signals due to multiple scatterings. A 532 nm bandpass filter (AHF, F94-532 HC laser clean-up maxline) was placed before the detector to only allow the detection of light from Rayleigh scattering, followed by a neutral density filter (Thorlabs, ND10A) to attenuate the signal intensity. Care was taken to fine-tune the alignment of the confocal system (pinhole and APD placement) by maximizing the detected intensity in the *x*, *y* planes while maintaining a slight overfill of the detector by shifting the *z* position. Confocal alignment is important for micro-DLS since we need to remove the effect of multiple scattering and maximize the detection of single scattering at the detection volume (i.e., maximizing the autocorrelation amplitude at a given point inside the sample). Data acquisition was made with a multichannel event timer and time-correlated single photon counting (TCSPC) device (PicoQuant, MultiHarp 150) that transmits information to the PC, where an acquisition program developed in LabView stores the data as a list of time tagged photon arrival events. An external synchronization (sync) signal of 80 MHz, generated by a photodiode coupled to fs laser (Spectra-Physics, Tsunami) pulses, was used as a time-tag reference for the TCSPC acquisition.

### B. Data processing and analysis

#### 1. Window slicing and autocorrelation

^{10}$g2\tau $ is defined as

^{33}This correlation algorithm calculates the ACF by using the photon arrival information within the region of interest, and the problem is reduced to an event-counting task for discrete predefined bins. This algorithm avoids the construction of time dependent intensity vectors, and by using a multitau scheme of logarithmically spaced bin groups, the ACF can be calculated over a wide range of lag times while reducing the computation cost. In our analysis, we used logarithmically spaced bins with 32 points per decade, ranging from

*μ*s to ms (1 × 10

^{−6}to 1 × 10

^{−2}s).

#### 2. Reconstruction of decay time distributions

*τ*and with a characteristic decay constant

*Γ*,

*Γ*, the diffusion coefficient

*D*can be obtained from the following equation:

*q*is the wavevector.

*g*

_{2}(

*τ*), it is convenient to define a model to which it can be directly fitted. Assuming a homodyne Gaussian process, this is normally achieved by employing the Siegert relation, where

*β*is the coherence factor (a setup dependent quantity),

^{34}Under this condition, the amplitude and decay constants obtained from the Siegert relation [Eq. (4)] will be systematically shifted from their real values. These so-called “apparent measured values” can be corrected using partial heterodyne theory.

^{22}The second order ACF,

*g*

_{2}(

*τ*), can be written using the amplitude

*A*and apparent decay constant

*Γ*

_{a},

*D*

_{a}, can be found in the following set of equations:

*q*is the wavevector,

*n*is the refractive index,

*λ*is the wavelength of the incident light, and

*θ*is the scattering angle.

^{35,36}However, for systems where complex distributions are expected (highly polydisperse and multimodal), more advanced methods should be used. The aim in this last case is to reconstruct the distribution of time decay constants, which can be seen as an inversion problem for Eq. (8). This inversion problem can be reduced to the ill-posed problem of finding the solution of a Fredholm integral, or more specifically, an inverse Laplace transform. Historically, different numerical approaches have been applied to obtain valid solutions to this problem. Among the most common are the constrained regularization method for data inversion (CONTIN)

^{37}and non-negative least squares (NNLS). Other more advanced methods include regularization and statistical treatments to improve the quality of the inverted distributions.

^{28}For this work, we adopted the NNLS method with Tikhonov regularization,

^{38}a widely studied inversion method. The inversion was implemented in Python by using a package based on the NNLS algorithm by Lawson and Hanson.

^{39}

*g*

_{2}(

*τ*). This way, we avoid having to numerically convert $g1\tau $ to $g2\tau $ by taking the square root of $g2\tau $, which would amplify the noise in the experimental ACF curve and could have a negative effect on the numerical inversion method. As it has been shown in a recent study,

^{28}if we discretize Eq. (8) and apply the Siegert relation [Eq. (4)] by squaring the first-order autocorrelation

*g*

_{1}(

*τ*),

*g*

_{2}(

*τ*) as

*b*

_{i}components of the distribution. By applying the numerical inversion method to $g2\tau $, one can then obtain a good approximation of the squared distribution function.

*b*

_{i}×

*Γ*

_{a}(

*i*)] matrix

*K*, composed of discrete exponential decays, as described by Eq. (10) and ranging from the minimum to the maximum lag times of the autocorrelation. In our analysis, we chose to upsample the base distribution, so we generated three times

*b*

_{i}number of exponential decays describing the autocorrelation curve. The matrix

*K*and vector $g2\tau $ can be used to obtain the distribution

*G*(

*Γ*

_{a}) by applying a NNLS optimization to a Tikhonov regularization problem,

*α*is the regularization parameter, and it should be adjusted to avoid overfitting and to control the smoothness and broadness of the distribution. For our analysis, we chose the

*L*-curve method

^{40,41}to find the optimum

*α*value, where $Kx\u2212y22$ and $x22$ are plotted for a range of

*α*and the elbow of the

*L*curve represents the optimum value. Since this leads to an iterative process, care should be taken in choosing an optimum range of

*α*, to reduce computation times while preserving optimum solutions. For our analysis, we used a range of 400 logarithmically spaced

*α*values, ranging from 10

^{−4}to 10

^{−2}. Once the distribution of time decay constants is obtained, it can be transformed into a distribution of apparent diffusion coefficients

*D*

_{a}by using Eqs. (6) and (7).

#### 3. Partial heterodyne analysis and calculation of real size distribution

*D*

_{a}measured on micro-DLS is systematically shifted from that of real

*D*values due to a constant electric field (e.g., laser scattered at the glass–liquid interface) on top of the fluctuating electric field from the scattering. Hiroi and Shibayama

^{22}showed that the amplitude of the second-order ACF [$g2\tau \u22121$] varies at different focal depths, and it in fact decreases as the focus gets closer to the glass interface due to the larger contribution of a time-independent field from the reflection at the glass below the sample. The authors showed

^{22,23}that the amplitude

*A*of the ACF [$g2\tau \u22121$] changes with the intensity of the reflected (

*I*

_{r}) and scattered light (

*I*

_{s}),

*D*

_{a}, is also altered by the introduction of the static field, and they used the partial heterodyne method

^{42}to calculate the diffusion coefficient,

*D*, from

*D*

_{a},

*A*was used to obtain a distribution of diffusion coefficients

*D*from the

*D*

_{a}distribution using Eq. (13). To ensure that the partial-heterodyne treatment works, we followed the procedure described by Hiroi and Shibayama

^{22,23}to reproduce the linear trend between

*D*

_{a}and

*A*at various focal depths. On our setup, the linear trend was observed up to a depth of 100

*µ*m over the glass interface. The value

*A*did not change after this depth, and therefore the measurements were performed below this value. Assuming our system is composed of diffusing spheres, the hydrodynamic radius

*R*

_{h}of the particles can be calculated using the Stokes–Einstein equation,

*k*

_{B}is the Boltzmann constant,

*T*is the temperature, and

*η*is the viscosity of the medium.

#### 4. Log-normal fitting of size distribution

^{43}

*s*is the particle diameter,

*s*

_{0}is the central peak position,

*H*is the full width at half maximum, and

*ρ*is the skewness parameter.

## III. EXPERIMENTAL EVALUATION OF THE SYSTEM

### A. Monomodal and multimodal size distributions

We first evaluated the ability of our system to correctly characterize the size distributions of samples containing particles with monomodal size distributions. For this purpose, we used polystyrene particles of 60 nm (Spherotech, Inc., catalog number: PP-008-10, lot number: AH01) at 0.1 wt. % and 220 nm (Spherotech, Inc., catalog number: PP-025-10, lot number: AH01) at 5 wt. %. Early works on micro-DLS^{16} addressed the possibility that the small observation volume could be an issue, possibly bringing the system over the limit of a gaussian process due to significant fluctuations of the average number of particles when the concentration is too low.^{34} To avoid this issue, the sample concentration should be high enough so that the average number of particles in the scattering volume is kept constant over time. A signal with strong and frequent spikes could point toward the detection of single scatterers and not the interaction between them. The concentration of polystyrene (PS) particles was chosen based on the suggestions of previous reports^{22} so that it is high enough to show the expected signal behavior without sudden intensity jumps. ∼200 *µ*l of the colloids were placed in a glass vial (4 ml), the bottom of which was cut, and a glass coverslip was glued to it. Laser power was adjusted to 160 *µ*W at the sample for 60 nm and 16 *µ*W for 220 nm colloids. Figures 3(a) and 3(b) show how the binned photon counts evolve over time, showing the expected random fluctuations around a mean value; no prominent spikes were observed.

Following the analysis methodology described earlier, we computed $g2\tau \u22121$ from a time window width of 180 s [Figs. 3(c) and 3(d)] and reconstructed the size distribution functions [Figs. 3(e) and 3(f)]. We report the peak center *s*_{0} and full width half maximum (FWHM) after fitting the distribution to a log-normal function [Eq. (14)]. The reconstructed distribution for 60 nm particles reported *s*_{0} = 69 nm and FWHM = 54 nm, while the distribution for 220 nm particles reported *s*_{0} = 264 nm and FWHM = 226 nm. The same samples were also measured in a commercial DLS setup (Malvern Panalytical Ltd., Zetasizer Nano ZS) by accumulating six measurements of 30 s, and the distributions were retrieved from the instrument software for comparison. From fitting a log-normal function to the commercial DLS data, the values *s*_{0} = 75 nm and FWHM = 51 nm for the 60 nm particles and *s*_{0} = 300 nm and FWHM = 303 nm for the 220 nm particles were obtained. Skewness parameters are also reported in Fig. 3 as a reference for the level of skewness in each case. Both peak center and FWHM values gave results close to the specifications provided by the company as well as the values obtained in the reference measurement on the commercial DLS setup.

Next, we evaluated the capacity of the method to resolve multimodal size distributions by measuring the mixtures of two sizes of polystyrene particles [60 nm (0.1 wt. %) and 220 nm (5 wt. %)]. Data were acquired by using a laser power of 160 *µ*W at the sample, and the autocorrelation was constructed from a time window width of 180 s. Figures 4(a)–4(c) show the reconstructed size distributions from the mixtures of 60 and 220 nm particles at three different volume ratios: 100:1, 10:1, and 1:1. All three distributions show two predominant peaks reflecting the size distribution of each polystyrene solution. The ratio of peak amplitudes qualitatively reflects the concentration change for each of the mixtures. After fitting the individual peaks of the distributions to a log-normal function, we can see that the peak attributed to 60 nm particles does not significantly diverge from its nominal value, showing *s*_{0} values of 69, 66, and 68 nm [panels (a)–(c)]. Similarly, the peak attributed to 220 nm particles shows *s*_{0} values of 267, 254, and 259 nm.

### B. Evaluation of time window widths

We measured three separate samples containing 60 nm (0.1 wt. %) polystyrene particles, 220 nm (5 wt. %) polystyrene particles, as well as a 1:40 (volume ratio) mixture of 20 nm (50 wt. %) LUDOX (Signa-Aldrich, LUDOX TM-50 colloidal silica) particles and 80 nm (0.24 wt. %) polystyrene particles, for a total time span of 180 s. Following our analysis methodology, we sliced the main data into smaller time windows, ranging from 0 to *T*, with a fixed starting time. A complete analysis was run for a representative range of *T* values from 20 ms to 180 s. To avoid overfitting the noise components, the minimum autocorrelation lag time was cut to 2 × 10^{−5} s for the 60 nm particle sample, 1.5 × 10^{−5} s for the 220 nm particle sample, and 3 × 10^{−6} for the 20–80 nm particle mixture. Figures 5, 6, and 8 show the results of the analysis for each sample. A global summary of the results is represented as panel (a) in each figure, where each vertical trace represents a reconstructed size distribution at a specific window width *T*, and the distribution amplitude is represented as the shading. The size distributions at representative window widths *T* (the log-normal fit results as the inset) are shown on panels (b)–(e), and their corresponding autocorrelations on panels (f)–(i).

For 60 nm (0.1 wt. %) polystyrene particles, we noticed that the size distribution became narrower as the time window width *T* decreased [Fig. 5(a)], while the peak position remained around similar values. Figures 5(b)–5(e) show the size distributions reconstructed at various *T*. The results shown at *T* = 1 s [Fig. 5(b)],*T* = 300 ms [Fig. 5(c)], and *T* = 80 ms [Fig. 5(d)] show that the method could successfully characterize the size of particles, even though *T* = 1 s has been considered an extremely short measurement in conventional DLS.^{31} In fact, the quality of ACF remained similar between these *T* values [Figs. 5(f)–5(h)]. For the lower values of *T* (the minimum studied value was *T* = 20 ms), the size distribution randomly splits into a bimodal distribution or greatly shifts relative to the expected distribution. For example, the quality of the ACF at *T* = 20 ms [Fig. 5(i)] is not as good as that in Figs. 5(f)–5(h), mainly because the signal-to-noise ratio decreases as there are less events to correlate. We conclude that *T* = 80 ms was the limit where the reconstruction of a monomodal distribution was sensible based on the data obtained under this specific experimental condition.

For 220 nm (5 wt. %) polystyrene particles (Fig. 6), the results are qualitatively similar to those for 60 nm particles (Fig. 5). We observed a similar trend in the size distribution, becoming narrower as *T* decreased [Fig. 6(a)], while the peak position remained around similar values. Figures 6(b)–6(d) show the reconstructed size distributions at the *T* values of 1 s, 300 ms, and 80 ms, respectively. The variation of the peak center values for 220 nm (*s*_{0} = 300, 243, and 253 nm) was larger than that for 60 nm, although all these values remain in the expected size range. Despite the larger variations in *s*_{0}, it appears from the analysis shown in Fig. 6(a) that a monomodal distribution could be reconstructed until *T* = 40 ms. The narrowing behavior of the distribution as the time window *T* decreases is more significant for 220 nm (FWHM varied from 116 nm at *T* = 1 s to 33 nm at *T* = 80 ms) than what was observed for 60 nm (FWHM varied from 38 nm at *T* = 1 s to 24 nm at *T* = 80 ms). We performed further analysis to understand why we observe smaller FWHM at lower *T* values.

We hypothesized that the variation in FWHM might be due to the ability to probe the sub-ensemble within a broad size distribution at a short time window width. Although samples are monodisperse, there is still some distribution of particle diameters [Figs. 3(e) and 3(f)]. When the window size *T* becomes smaller, we may be undercovering this heterogeneity of the particle size that is normally hidden by averaging over a long measurement time. The initial time to define the time window was fixed at 0 s for the analysis shown in Figs. 5 and 6. Here, we constructed distributions from consecutive 80 ms windows over a 180 s time span (i.e., the first span: 0–80 ms, the second span: 81–160 ms, and the last span: 179.92–180 s), normalized their amplitude to one, and accumulated them. Figure 7(a) (60 nm) and Fig. 7(c) (220 nm) show several examples of the size distribution constructed from a single 80 ms window at different times, overlayed on the size distribution constructed from a 180 s window. The FWHM of the size distributions at *T* = 80 ms is smaller than that at *T* = 180 s, and the contrast is more significant for 220 nm particles [Fig. 7(c)]. Despite the difference in FWHM, the peak positions from the *T* = 80 ms analysis are distributed within the range of the size distribution from *T* = 180 s. Figure 7(b) (60 nm) and Fig. 7(d) (220 nm) show the accumulated size distributions from consecutive 80 ms windows over a 180 s time, overlayed on the size distribution from a 180 s window. For the 60 nm sample, the size distribution obtained by accumulating consecutive single 80 ms windows is similar to that obtained from a 180 s window [Fig. 7(b)]. On the other hand, for the 220 nm sample, a shift in the peak position was observed, although both size distributions remain qualitatively similar. Remarkably, the size distribution obtained by accumulating consecutive singles of 80 ms resulted in the peak value being closer to the specification given by the supplier. In addition, for both samples, the FWHM was narrower when the size distribution was obtained by accumulating consecutive single 80 ms windows. Similar results were reported by previous research,^{31} where the accuracy and precision of conventional DLS measurements were improved by averaging multiple sub-measurements compared with a single long measurement. These results prove our hypothesis that the variation of the peak position and smaller FWHM at the decreased *T* value were due to the capability of our system to probe the sub-ensemble of the sample heterogeneity. Because the size distribution is larger for the 220 nm sample, the effect of probing the sub-ensemble appears to be more significant. This analysis further confirms our claim that our method can characterize the size distribution of monodisperse nanoparticles with a time window *T* as short as 40 ms.

Finally, we examined the effect of time window width *T* on the mixture of 20 nm (50 wt. %) silica particles and 80 nm (0.24 wt. %) polystyrene particles (1:40 volume ratio). As described earlier, the starting time was fixed at 0, and *T* was varied for this analysis. Figure 8(a) clearly shows that we could reconstruct the bimodal distribution at a wide range of *T* values. As was the case for monodisperse samples, small variations of the peak position and the narrowing of the FWHM were observed as the *T* value decreased. This is attributed to the capability of probing the sub-ensemble at a smaller window width. Figures 8(b)–8(d) show the reconstructed size distributions at the *T* values of 1 s, 220 ms, and 80 ms, respectively. These distributions fall within the size distribution constructed from *T* = 180 s [Fig. 8(a), the vertical trace at the very right]. Below *T* = 80 ms, the peak from 20 nm particles disappeared, and the size distribution became monomodal with the peak attributed to 80 nm particles [e.g., *T* = 20 ms shown in Fig. 8(e)]. We concluded that the limit of the time window width was *T* = 80 ms for the characterization of multimodal distributions. Notably, the time resolution that could reconstruct monomodal and multimodal distributions was found to be similar.

## IV. CONCLUSION

We implemented a micro-DLS system with time resolved capability based on the post-processing of TCSPC data using an algorithm adopted from the single molecule fluorescence spectroscopy field. The analysis software was built so that one can choose the window width and its position arbitrarily to reconstruct the size distribution. We demonstrated the ability to reconstruct the particle size distributions of known samples, both monomodal and multimodal. Analysis of reconstructed size distributions from different time window widths showed that our implementation could reliably reconstruct size distributions from time windows as short as 80 ms for 60 nm and 40 ms for 220 nm polystyrene samples. Even for multimodal size distributions, the expected size distribution could be obtained at similar time window widths. Remarkably, when the window width is sufficiently small, the reconstructed size distribution reflects the sub-ensemble of the broad size distribution from a large time window width. The capability of obtaining the sub-ensemble information that constitutes the ensemble average is a valuable feature that is recognized in the field of single molecule spectroscopy. The time window width required to characterize the size distribution reported here is several orders of magnitude lower than what was previously reported for DLS applications. Our micro-DLS implementation allows the measurement of size distributions with high temporal resolution, paving the way for its application to study a wide range of non-stationary systems that evolve over time.

## ACKNOWLEDGMENTS

The authors would like to acknowledge the University of Geneva for funding and supporting this project.

## AUTHOR DECLARATIONS

### Conflict of Interest

O.U. and T.B.M.A. have Patent EP23173928.5 pending.

### Author Contributions

T.B.M.A. introduced the original idea and motivation. T.B.M.A., J.B., and O.U. built the setup. J.B. provided the initial data and assisted in the refinement of the experimental conditions. O.U. and N.B. refined the setup, performed the exploratory analysis and main measurements under the supervision of T.B.M.A. and J.B.. O.U., developed the analysis methodology and software, and performed the final data analysis. All authors participated in drafting the manuscript.

**Oscar Urquidi**: Data curation (lead); Formal analysis (lead); Methodology (lead); Software (lead); Visualization (lead); Writing – original draft (equal); Writing – review & editing (equal). **Natercia Barbosa**: Data curation (supporting); Formal analysis (supporting); Methodology (supporting); Validation (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). **Johanna Brazard**: Data curation (supporting); Formal analysis (supporting); Investigation (supporting); Methodology (supporting); Project administration (supporting); Software (supporting); Supervision (supporting); Validation (supporting); Writing – original draft (supporting); Writing – review & editing (supporting). **Takuji B. M. Adachi**: Conceptualization (lead); Data curation (supporting); Formal analysis (supporting); Funding acquisition (lead); Investigation (supporting); Methodology (supporting); Project administration (lead); Resources (lead); Software (supporting); Supervision (lead); Validation (lead); Visualization (equal); Writing – original draft (equal); Writing – review & editing (equal).

## DATA AVAILABILITY

The data that support the findings of this study are openly available in the Yareta depository at http://doi.org/10.26037/yareta:xptnb342p5fwvaqn5ud6eexfyi.

## REFERENCES

*Dynamic Light Scattering: Applications of Photon Correlation Spectroscopy*

*Encyclopedia of Analytical Chemistry*

_{3}nucleation and microgravity effect

*n*th-order intensity correlation functions

*Solutions of Ill-Posed Problems*

*Solving Least Squares Problems*