Computational assessment of an effective-sphere model for characterizing colloidal fractal aggregates with holographic microscopy
Jerome Fung Samantha Hoang










Journal of Quantitative Spectroscopy and Radiative Transfer






























elsarticle-num




    1]Jerome Fungcor1jfung@ithaca.edu[1]Department of Physics & Astronomy, Ithaca College, 953 Danby Road, Ithaca, NY 148502]Samantha Hoangshoang@wellesley.edu[2]Department of Physics, Wellesley College, 106 Central Street, Wellesley, MA 02481[cor1]Corresponding author
        We perform simulations to evaluate a recent experimental technique for using in-line holographic microscopy and an effective-sphere
        model to measure the population-averaged fractal dimension D_f of an ensemble of colloidal fractal aggregates.
        In this technique, models based on Lorenz-Mie scattering by a uniform sphere are fit to digital holograms of a population of fractal aggregates to determine the effective refractive indices n_eff and effective radii a_eff of the aggregates.
        A scaling relationship between n_eff and a_eff based on the Maxwell Garnett effective-medium theory then determines D_f.
        Here we use a multisphere superposition code to calculate the exact holograms produced by aggregates with tunable fractal dimensions D_f.
        We show that n_eff and a_eff become less sensitive to the aggregate orientation as D_f increases.
        We also show that the Maxwell Garnett scaling relationship 
        correctly determines D_f to within 10.5% when multiple scattering is negligible and the population-averaged coefficient of determination ⟨ R^2⟩_p > 0.6, indicating that the holograms are well-described by the effective-sphere model. 
        
        fractal aggregates, digital holography, light scattering, T-matrix
        
        
        
        § INTRODUCTION
        
        Fractal-like aggregates of small particles commonly arise from both natural and artificial processes.
        Examples include soot <cit.>, protein aggregates formed by heating <cit.>, and aggregates of polystyrene spheres formed by salting out a suspension  <cit.>. 
        Experimental studies on fractal aggregates have frequently focused on determining their fractal dimension D_f, which characterizes their structure and strongly affects how they interact with light <cit.>.
        Direct imaging of individual aggregates using scanning or transmission electron microscopy <cit.> can allow detailed examination of their structure and the measurement of D_f.
        However, in applications such as the modeling of radiative transfer by an ensemble of particles, the fractal dimension of a population of aggregates, rather than the structure of a single aggregate, is the quantity of interest.
        Determining population-averaged fractal dimensions requires the time-consuming imaging of many aggregates; moreover, electron microscopy cannot be performed in situ.
        Macroscopic light scattering can determine fractal dimensions that are averaged over many aggregates. 
        However, bulk light scattering techniques can be affected by polydispersity, and interpreting the results may require strong assumptions to be made about the scatterers <cit.>.
        
        Recently, Wang and co-workers pioneered the use of in-line holographic microscopy for characterizing non-absorbing fractal aggregates <cit.>. 
        In holographic microscopy (also known as digital holographic microscopy or holographic video microscopy), light scattered from an aggregate of interest interferes with unscattered incident light to form an interference pattern, or hologram. 
        Rather than using reconstruction techniques to produce an image of the aggregate <cit.>, 
        Wang et al. analyzed holograms of fractal aggregates by fitting effective-sphere models based on Lorenz-Mie scattering
        by a uniform sphere. 
        From their fits, they determined each aggregate's effective radius a_eff and effective refractive index n_eff.
        Using the Maxwell Garnett effective-medium theory, Wang et al. showed that the relationship between a_eff and n_eff for aggregates of different sizes within a population depends on D_f. 
        Wang et al. applied this approach to colloidal aggregates formed in situ from polystyrene spheres, bovine pancreas insulin, and bovine serum albumin as they flowed through a microfluidic device.
        
        While the D_f values Wang et al. determined for their aggregates were consistent with their expectations, questions remain about the limits of validity of their approach.
        In this work, we computationally test their approach of using holographic microscopy and an effective-sphere model to characterize the fractal dimension D_f of a population of aggregates.
        We generate aggregates with a known, tunable D_f that are composed of monodisperse, non-absorbing spheres.
        We use an exact multisphere superposition code to calculate the holograms that those aggregates would propose.
        We then fit effective-sphere models to the exact holograms, determine n_eff and a_eff for each aggregate, and infer D_f from the distribution of n_eff and a_eff for a population of aggregates.
        We compare the values of D_f inferred from holographic microscopy to the input D_f values for aggregates composed of primary spheres of different sizes.
        Our results suggest criteria that may help experimenters determine whether D_f values obtained using the effective-sphere approach of Wang et al. are reliable.
        
        
        
        
        § THEORY
        
        
         §.§ The fractal dimension of a fractal aggregate
        
        The geometry of a fractal-like aggregate is characterized by two parameters: the fractal dimension D_f and the fractal prefactor k_f. 
        An aggregate composed of N spheres each of radius a_0 (known as primary spheres) satisfies the geometric relationship
        
        N = k_f(R_g/a_0)^D_f.
        
        Here, R_g is the radius of gyration of the aggregate. In general, as D_f increases, aggregates become increasingly close-packed.
        A straight linear chain of particles has D_f=1, while a three-dimensional crystalline array of particles has D_f=3.
        We focus here on the characterization of D_f, as do Wang et al.<cit.>. 
        
        
        
        
         §.§ Holographic microscopy
        
        
        In in-line holographic microscopy, 
        the recorded hologram is an interference pattern formed by the scattered wave from an object of interest and the unscattered portion of the incident wave (Fig. <ref>).
        Following Lee and co-workers, we model the formation of a normalized digital hologram using <cit.>
        I_hol = 1 + 2 αRe{E⃗_s·ϵ̂} + α^2 | E⃗_s |^2.
        
        Here I_hol is the measured hologram intensity at a location on the detector, E⃗_s is the complex scattered electric field, ϵ̂ is a unit vector in the direction of the polarization of the incident field, α is a scaling parameter that depends on details of the optical system, and Re denotes the real part.
        The first term on the right side of Eq. (<ref>) is proportional to the intensity of the incident field prior to normalization, the second term encodes the amplitude and phase of the scattered wave, and the third term is proportional to the intensity of the scattered field. 
        A hologram thus contains information about the position of the scatterers (via the variation of the phase of E⃗_s on the detector) and about the optical properties of the scatterers (via the angular dependence of E⃗_s).
        Once E⃗_s can be calculated for an object, Eq. (<ref>) can be used to compute the hologram formed by that object.
        
        For dilute samples in which the scatterers are comparable in size to the wavelength of light and are well-separated, information about the scatterers can be retrieved from experimentally recorded holograms by fitting a computational model based on Eq. (<ref>). 
        Both the three-dimensional positions of the scatterers as well as their optical properties (such as their size and refractive index) can be determined.
        This has been demonstrated for isolated spheres <cit.>, clusters of spheres <cit.>, and cylinders <cit.>. 
        Here we only consider the optical properties of the scatterers.
        
        
        
         §.§ Modeling fractal aggregates using effective-medium theory
        
        Wang et al. use the Maxwell Garnett effective-medium theory <cit.> to predict n_eff and a_eff for fractal aggregates modeled as uniform spheres <cit.>. 
        They assume that an aggregate with radius of gyration R_g can be modeled as a uniform sphere of radius R_g. 
        While a sphere of radius R_g does not necessarily enclose the aggregate, particularly for low D_f aggregates,
        they define a volume fraction ϕ equal to the ratio between the volume of a sphere of radius R_g and the total volume of the N primary particles of radius a_0 in the aggregate:
        ϕ≡N·4/3π a_0^3/4/3π R_g^3.
        
        Using Eq. (<ref>) to eliminate N and assuming that R_g=a_eff leads to
        ϕ = k_f ( a_eff/a_0)^D_f-3.
        
        The Maxwell Garnett effective-medium theory relates the Lorentz-Lorenz factors L(n) of the refractive indices n_eff of the effective spheres and n_0 of the primary particles:
        
        L(n_eff) = ϕ L(n_0)
        
        where
        
        L(n) = n^2-n_med^2/n^2+2n_med^2.
        
        Here n_med is the refractive index of the surrounding medium.
        Combining Eqs. (<ref>) and (<ref>) leads to
        ln(  L(n_eff)/L(n_0)) = (D_f-3) ln( a_eff/a_0) + ln k_f.
        
        This is the scaling relationship between n_eff and a_eff from which Wang et al. determine D_f.
        
        
        
        
        § SIMULATION METHODS
        
        
         §.§ Generation of fractal aggregates
        
        
        In this work, we consider fractal aggregates with the D_f and k_f values shown in Table <ref>.
        We generate aggregates using the algorithms of Filippov et al.<cit.> as implemented in the software package FLAGE <cit.>.
        There are two main approaches for generating fractal aggregate geometries in which both the number of particles N and the fractal dimension D_f are tunable.
        One approach, known as particle-cluster aggregation or the sequential algorithm, involves adding particles one at a time to a growing aggregate such that Eq. (<ref>) is always satisfied <cit.>. 
        In the other approach, cluster-cluster aggregation, larger aggregates are created by repeatedly merging smaller sub-clusters in a manner that satisfies Eq. (<ref>) <cit.>.
        While either approach generates aggregates whose radius of gyration is well-described by Eq. (<ref>), the cluster-cluster approach tends to produce clusters whose pair correlation function scales as expected <cit.>.
        We therefore use a cluster-cluster aggregation algorithm to generate all aggregates except for those with D_f=1.1, for which we use a particle-cluster algorithm. 
        Aggregates with D_f=1.1 are so nearly linear that very few cluster-cluster mergers are possible, which causes the cluster-cluster algorithm to seldom converge. 
        
        
        
        
        
         §.§ Calculation of exact holograms
        
        
        Throughout our work, except in Sec. <ref>, we assume that the primary particles are monodisperse, non-absorbing spheres whose refractive index is n_0 = 1.59, and we assume that the spheres are surrounded by a medium with refractive index n_med = 1.33. 
        These values correspond to polystyrene spheres in water, one of the experimental systems studied by Wang et al.
        We consider three different values for the radius a_0 of the primary particles: 5, 20, and 80.
        We also assume that the aggregates are illuminated by incident light with a vacuum wavelength of λ_0=447 that is linearly polarized in the horizontal direction.
        We assume that our detector is an array of 180×180 pixels with a pixel size of 0.1. 
        We place all aggregates so that their centers of mass are located 30 from the detector along the optical axis and are centered on the detector. 
        While a microscope objective lens usually images holograms onto a camera (typically with a pixel size of ∼10) in experiments, we do not explicitly model the lens except for its effect on the pixel size.
        To model the aggregates being randomly tumbled as in the experiments of Wang et al., we randomly rotate aggregates about their centers of mass using an algorithm due to Brannon <cit.>. 
        Two unit vectors uniformly distributed on the unit sphere are generated and then orthogonalized using the Graham-Schmidt procedure. The cross product of the resulting vectors then yields a third orthonormal vector to define a rotation matrix.
        
        In order to calculate the hologram formed by a given aggregate using Eq. (<ref>), it is necessary to calculate the electric field scattered by the aggregate.
        We use the multisphere superposition code SCSMFO1B by Mackowski and Mishchenko <cit.>.
        This code calculates the exact scattered field (up to series truncation) from a cluster of arbitrary, non-overlapping spheres as a series expansion in
        vector spherical harmonics. 
        The SCSMFO1B code is wrapped into the open-source package HoloPy (version 3.2.1), which allows for convenient generation and analysis of holograms <cit.>.
        
        
        
        
        
         §.§ Fitting efffective-sphere models
        
        
        We also use HoloPy to fit effective-sphere models to exact holograms. We use the Levenberg-Marquardt solver  (as wrapped in HoloPy) to
        fit a model based on the Lorenz-Mie scattering solution <cit.>. 
        In fitting the model, we vary the particle refractive index n_eff, the particle radius a_eff, 
        the x, y, and z coordinates of the sphere center, and the scaling parameter α that regulates the amplitude of the scattered electric field.
        Nonlinear least squares algorithms (like the Levenberg-Marquardt algorithm) require initial guesses for the parameter values. 
        We provide initial guesses for n_eff and a_eff using the values that are predicted by Eq. (<ref>). 
        If necessary, the need to provide initial guesses could be avoided by using Bayesian inference or machine learning techniques <cit.>.
        We reject any fits where the minimizer does not return parameter values that differ from the initial guess values, which would indicate that the minimizer did not converge.
        
        We assess the goodness of fit of the effective-sphere models to the exact holograms using the R^2 coefficient of determination, a dimensionless parameter given by <cit.>
        R^2 = 1 - ∑(I_holo - I_model)^2/∑ (I_holo - I̅_holo)^2.
        
        Here, I_holo denotes the pixel values of an exact hologram, I_model denotes the pixel values of the best-fit model, I̅_holo denotes the mean value of the exact hologram pixels, and the sums are taken over all hologram pixels. 
        When the model matches the exact hologram, R^2=1.
        
        
        
        
        § RESULTS AND DISCUSSION
        
        
         §.§ Exact hologram calculations and effective-sphere model fits
        
        
        Figure <ref> shows renderings, exact holograms, best-fit effective-sphere models, and residuals for three aggregates with different fractal dimensions each consisting of N=300 primary spheres.
        The renderings show how the aggregates are nearly linear when D_f=1.3 but are much more tightly packed when D_f = 2.5.
        The holograms of these three aggregates are well-described by the effective-sphere model, as indicated by the values of R^2 > 0.9. 
        The residuals show subtler differences between the exact and effective-sphere holograms. 
        In particular, the elongated nature of the aggregate with D_f=1.3 in Figure <ref>(a) leads to the diagonal bands in the residuals.
        
        
        
        
         §.§ Orientation dependence of n_eff and a_eff
        
        In the experiments of Wang et al., the aggregates tumble as they flow through a microfluidic channel <cit.>. 
        Each aggregate is typically imaged in several orientations, and the resulting values of n_eff and a_eff are averaged together. 
        It is therefore worthwhile to assess how sensitive n_eff and a_eff are to the aggregate orientation.
        
        To study this question, we generate one aggregate with N=300 primary spheres of radius a_0 = 5 for each of the fractal dimensions in Table <ref>. 
        We calculate exact holograms for 800 randomly-chosen orientations for each aggregate, determine n_eff and a_eff from each hologram, and compile the resulting distributions of n_eff and a_eff in Figures <ref> and <ref>.
        
        
        
        We make two principal observations about Figures <ref> and <ref>.
        First, n_eff and a_eff are anti-correlated. This is consistent with experimental observations for single spheres  <cit.>. 
        This anti-correlation results from the product of  the sphere radius and the sphere refractive index frequently appearing in the Lorenz-Mie solution.
        Second, as D_f increases, the distributions of n_eff and a_eff generally become narrower. 
        While the marginal distributions are not always monomodal (such as the a_eff distribution for D_f=2.3 in Fig. <ref>(b)), it is still possible to calculate normalized sample standard deviations (Fig. <ref>). 
        For n_eff, we normalize the  sample standard deviation Δ n_eff by the difference between the sample mean n̅_eff and the medium index n_med.
        For a_eff, we normalize the sample standard deviation Δ a_eff by the sample mean a̅_eff.
        The normalized sample standard deviations decrease as D_f increases because the morphology of the aggregates becomes increasingly compact.
        For elongated low-D_f aggregates, the size of the geometric projection of the aggregate onto the hologram plane (which is correlated with a_eff) strongly depends on the orientation of the aggregate relative to the optical axis.
        For more compact high-D_f aggregates, the projections are approximately the same no matter how the aggregates are oriented relative to the optical axis. 
        Consequently, a_eff and n_eff are much less sensitive to the aggregate orientation.
        
        
        
         §.§ Determination of D_f using holographic microscopy
        
        We next consider whether the values of D_f determined using holographic microscopy and the scaling relation between n_eff and a_eff [Eq. (<ref>)] are consistent with the known input values.
        For each of the fractal dimensions in Table <ref>, we calculate exact holograms for 225 aggregates consisting of between N=60 and N=600 primary particles.
        We calculate holograms for five random orientations of each aggregate.
        We then determine n_eff and a_eff for each hologram by fitting an effective-sphere model.
        Then, following Wang et al., we determine the fractal dimension by fitting Eq. (<ref>) to the orientation-averaged values of n_eff and a_eff for each aggregate.
        For clarity, we henceforth use D_f,i to denote input fractal dimensions and D_f,h to denote fractal dimensions determined from fitting Eq. (<ref>).
        
        
        
        Figure <ref> shows orientation-averaged values of n_eff and a_eff for aggregates composed of primary spheres with radius a_0 = 5.
        The color of each dot indicates ⟨ R^2⟩, the orientation-averaged value of the R^2 coefficient of determination.
        The solid lines are best fits of the scaling relation Eq. (<ref>) to the points, and the fractal dimensions D_f,h determined from those fits are shown.
        The uncertainties in D_f,h are determined from the fits of Eq. (<ref>) but are underestimates since they neglect the spread in the values of a_eff. 
        
        For 1.1 ≤ D_f,i≤ 1.9, we find excellent agreement between the data points for each aggregate geometry and Eq. (<ref>). 
        For these values of D_f,i, the agreement between D_f,h and D_f,i is 4.5% or better.
        ⟨ R^2⟩ generally decreases as a_eff increases. 
        This is particularly noticeable for the relatively linear aggregates with D_f,i=1.1 in Fig. <ref>(a). 
        For large, nearly-linear aggregates, the agreement between the exact holograms and an effective-sphere model can be poorer when the long axis of the aggregate is nearly perpendicular to the optical axis.
        
        For 2.1≤ D_f,i≤ 2.5, however, the agreement between the shape of the distribution of points and Eq. (<ref>) is less good [Fig. <ref>(f)–(h)]. 
        For the smallest clusters (with the lowest a_eff), the values of L(n_eff)/L(n_0) lie above the prediction of Eq. (<ref>).
        The points with the lowest a_eff values skew the fits of Eq. (<ref>) and affect the determination of D_f,h. 
        For the aggregates with D_f,i=2.5, the value of D_f,h is nearly 20% lower than the input value.
        We attribute this behavior to multiple scattering, a claim we explore further in Sec. <ref>.
        
        
        
        
        We observe similar results for aggregates composed of primary spheres with radius a_0=20 (Fig. <ref>).
        For aggregates with D_f,i≥1.3, D_f,h agrees with D_f,i to 10.5% or better.
        In addition, we do not observe significant deviations from the scaling predicted by Eq. (<ref>) for small aggregates with D_f,i≥ 2.1.
        However, at low D_f,i, the quality of the effective-sphere hologram fits becomes poor, as indicated by the values of ⟨ R^2⟩≪ 1. 
        For D_f,i=1.1 [Fig. <ref>(a)], the quality of the fits is so poor that Eq. (<ref>) does not correctly determine the fractal dimension.
        This is because low-D_f (and hence, highly elongated) aggregates composed of 20-nm-radius spheres have a size approaching or exceeding the wavelength of light.
        Consequently, the exact holograms of these aggregates exhibit pronounced deviations from axisymmetry. 
        Figure <ref> shows holograms of an aggregate with D_f,i=1.3 consisting of N=240 spheres.
        While the central bright spot in the exact hologram is well-captured by the effective-sphere fit, the exact hologram exhibits pronounced diagonal bands due to interference between the waves scattered by different spheres within the aggregate <cit.>.
        
        
        
        
        For aggregates composed of primary spheres of radius a_0=80, the discrepancies between the exact holograms and the effective-sphere holograms are so large for low-D_f,i aggregates that most of the effective-sphere fits do not successfully converge.
        For this primary particle size, we therefore only consider aggregates with 1.5 ≤ D_f,i≤ 2.5 (Fig. <ref>).
        Even at D_f,i=1.5, all the values of ⟨ R^2⟩ are smaller than 0.6.
        Only for D_f,i≥ 2.1 do the fits of Eq. (<ref>) determine D_f,h to within 10% of the input values.
        
        
        
        For each of the primary particle radii we have considered, the scaling relationship in Eq. (<ref>) accurately determines the fractal dimension for some but not all of the values of D_f,i. 
        In order to draw conclusions about the conditions under which Eq. (<ref>) is useful, we summarize our results on D_f,h in Fig. <ref>.
        Here we show the fractional difference between D_f,h and D_f,i as a function of ⟨ R^2⟩_p, the population-averaged value of R^2 for all hologram fits at a given D_f,i (e.g., all of the points within one of the panels of Fig. <ref>, <ref>, and <ref>).
        Several features are apparent in Fig. <ref>.
        The points inside the orange oval have both a small fractional difference (⪅ 10%) between D_f,h and D_f,i as well as high values (>0.6) of the population-averaged coefficient of determination ⟨ R^2⟩_p.
        There are also four points (corresponding to low-D_f,i aggregates composed of 20- or 80-nm-radius primary particles) for which the quality of the fits is poor: ⟨ R^2⟩_p< 0.5.
        For two of these points, there is a large (>20%) discrepancy between D_f,h and D_f,i.
        While for the other two points, the discrepancy between D_f,h and D_f,i is less than 4%, these results suggest that Eq. (<ref>) is unreliable when ⟨ R^2⟩_p ≲ 0.6.
        
        There is one other significant feature in Fig. <ref>. For aggregates with D_f,i=2.5 composed of 5-nm-radius spheres, even though ⟨ R^2⟩_p = 0.9992, the fractional discrepancy between D_f,i and D_f,h is 19.3%.
        This case is shown in Fig. <ref>(h).
        We next consider the origin of this discrepancy, which we attribute to the breakdown of the Maxwell Garnett effective-medium theory for high-D_f aggregates composed of small primary spheres due to multiple scattering.
        
        
        
         §.§ Breakdown of effective-medium theory at high D_f
        
        
        The results for aggregates with D_f,i≥2.1 composed of 5-nm-radius primary particles in Fig. <ref>
        deviate significantly from the scaling predictions of Eq. (<ref>) for the smallest values of a_eff.
        Theoretical considerations along with additional computations suggest that these deviations are due to the breakdown of the Maxwell Garnett effective-medium theory used to derive Eq. (<ref>) as a result of multiple scattering.
        
        The Maxwell Garnett theory assumes that an optical medium (here, the combination of a fractal aggregate and the medium surrounding it) can be modeled as consisting of dipolar inclusions within a homogeneous background.
        The volume fraction ϕ of the inclusions is assumed to be low, and the density of inclusions is assumed to be constant.
        As Wang et al. point out, one problem with applying the Maxwell Garnett theory to fractal aggregates is that the particles are not dispersed uniformly within an aggregate: there are more particles near the center of the aggregate than towards the edges <cit.>.
        (We note, as do Wang et al., that a more realistic effective-sphere model for a fractal aggregate could incorporate a radially-varying refractive index, as in the work of Khlebtsov <cit.>.)
        More importantly, all the inclusions are assumed to experience the same electric field, which requires that multiple scattering be negligible.
        
        A scaling argument due to Lu and Sorensen <cit.> suggests that multiple scattering may be non-negligible for low-N aggregates composed of small primary spheres in the Rayleigh limit, for which ka_0≪1.
        In this limit, each sphere scatters like a dipole.
        At distances r from a dipole such that kr<1, the near-field term in the scattered electric field (which is proportional to (kr)^-3) dominates over the usual far-field term (which is proportional to (kr)^-1).
        For aggregates small enough that kr_ij<1 for every interparticle distance r_ij within the aggregate, every primary particle is within
        the near field of every other particle.
        Lu and Sorensen consider the amplitude of the double-scattered electric field from an aggregate under this assumption and show that the ratio of the double-scattered to single-scattered fields is proportional to N^1-3/D. 
        Crucially, this ratio increases as N decreases. 
        This explains why we observe deviations from Eq. (<ref>) for low-N aggregates with D_f,i≥ 2.1 when a_0 = 5.
        (When a_0=5, ka_0 = 0.093, and for an aggregate with D_f,i=2.5 composed of N=100 spheres, kR_g = 0.587.)
        We do not observe similar deviations from Eq. (<ref>) for aggregates with a_0=5 and D_f,i≤1.9 or aggregates of any D_f,i composed of larger primary particles since it is not the case that every primary particle in such aggregates is in the near field of every other particle.
        
        Further evidence suggesting that multiple scattering is significant for high-D_f,i aggregates of 5-nm-radius particles comes from simulations in which we decrease the effects of multiple scattering by decreasing the refractive index of the primary particles.
        Figure <ref> shows n_eff and a_eff for aggregates with D_f,i=2.5 and a_0=5, but with n_0 = 1.356. 
        Here, the refractive index difference n_0 - n_med is an order of magnitude smaller than for the polystyrene spheres (n_0=1.59) previously considered. 
        While the aggregate geometries and orientations in Fig. <ref> are identical to those in Fig. <ref>(h),
        the agreement between the data and Eq. (<ref>) is much better, and D_f,h is determined with an accuracy of 0.4%.
        
        
        
         §.§ Effective-sphere approach for digital holography of fractal aggregates
        
        Why does the effective-sphere approach for analyzing digital holograms of fractal aggregates succeed when it does?
        The scaling approach of Sorensen <cit.> is a useful way to consider this question.
        Sorensen argues that there are two important regimes in the functional behavior of the structure factor S(q), which is proportional to the scattered intensity, for scattering by a fractal aggregate of fractal dimension D_f and radius of gyration R_g:
        
        S(q) =  
        1     qR_g < 1 
        
        C(qR_g)^-D_f    qR_g > 1.
        
        Here C is a proportionality constant and q is the scattering wavenumber, defined by q ≡ (4 π n_med/λ) sin(θ/2), where θ is the scattering angle.
        The key idea is that near the forward direction (small q), the scattered intensity is insensitive to D_f, while at larger angles the angular dependence encodes D_f.
        
        Our results suggest that the effective-sphere approach succeeds only when qR_g ⪅ 1.
        In our simulations, the maximum value of q (at the corners of the detector) is given by q_max = 7.45^-1.
        For given values of a_0, D_f, k_f, and N, we can determine R_g using Eq. (<ref>).
        For aggregates with a_0=5, q_maxR_g is generally of order 1 or smaller. 
        For instance, for an aggregate of 400 spheres with a_0 = 5, D_f = 1.5, and k_f=1.4, q_maxR_g = 1.62.
        But when a_0 = 20, q_maxR_g = 6.46 for the same aggregate, and when a_0=80, q_maxR_g = 25.9.
        As we have seen, for aggregates composed of larger primary particles where q_maxR_g ≫ 1, the quality of the effective-sphere fits tends to be poor.
        The poor quality of the fits primarily arises from the lack of axial symmetry in the holograms, but Eq. (<ref>) suggests that the angular dependence of the scattered intensity (which affects the envelope of the hologram fringe pattern) is also inconsistent with scattering by a uniform sphere.
        In the Rayleigh-Gans approximation for scattering by a sphere, the envelope of the scattered intensity scales as q^-4, in contrast to the q^-D_f scaling for a fractal aggregate <cit.>.
        
        Because the scattered intensity does not depend on D_f when qR_g<1, in the approach analyzed here, D_f cannot be determined from a single hologram of an aggregate.
        But since holograms capture the scattered field at small scattering angles (including the forward direction), it follows from the optical theorem that holograms probe the extinction cross section of scatterers <cit.>.
        For the non-absorbing scatterers considered here, the extinction cross section is equal to the scattering cross section.
        The data in Figs. <ref>, <ref>, and <ref> thus encode how the scattering cross section scales with the size of an aggregate.
        Within the Rayleigh-Gans approximation, the scattering cross section for fractal aggregates scales with D_f and R_g<cit.>.
        This dependence enables us to retrieve D_f.
        
        
        
        § CONCLUSIONS
        
        It is relatively straightforward to incorporate holographic microscopy into an existing optical setup, and the effective-sphere approach for characterizing fractal aggregates introduced by Wang et al. may be of interest to other experimenters.
        We have evaluated this approach by computing exact holograms of fractal aggregates using a multisphere superposition code.
        Our simulations suggest criteria that may help experimenters assess whether or not the fractal dimensions they determine using this approach are reliable to within ∼10% or better.
        First, the effects of multiple scattering should be negligible.
        We have shown how the effects of multiple scattering may be observed in deviations from the Maxwell Garnett scaling relationship [Eq. (<ref>)] for aggregates for which kR_g⪅ 1.
        Second, as shown in Fig. <ref>, the population-averaged values of R^2 should be sufficiently large (⟨ R^2⟩_p ⪆ 0.6) to indicate that the effective-sphere models reliably describe the holograms.
        These criteria are useful because estimates of kR_g and ⟨ R^2⟩_p can be determined from the experimental holograms alone.
        No a priori knowledge of the radius and refractive index of the primary particles is necessary; in many experimental applications (such as the protein aggregates considered by Wang et al.), the optical properties of the primary particles are unknown.
        
        Several further extensions of this work are possible. First, in this work we have only carefully analyzed the determination of the population-averaged values of D_f. While Wang et al. assumed that the fractal prefactor k_f=1, it should in principle be possible to determine the fractal prefactor k_f using Eq. (<ref>) as well.
        Second, our simulations assume that every aggregate in a population has the same D_f and k_f, and that the primary particles are monodisperse.
        These assumptions are unlikely to be satisfied for naturally-occurring aggregates.
        Future work might thus explore the effects of particle polydispersity and variations in D_f.
        Finally, this work has only considered aggregates of optically non-absorbing particles. 
        We intend to extend these results to absorbing particles in the future, as well as to non-absorbing primary particles with other refractive indices.
        Nonetheless, our work confirms the findings of Wang and co-workers that holographic microscopy is an easy-to-use and effective tool for characterizing fractal aggregates.
        
        
        
        
        § ACKNOWLEDGEMENTS
        
        S. H. was supported by the Wellesley College IBM Faculty Research Fund for Science and Mathematics.
