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Accurate and fast modeling of scattering from random arrays of nanoparticles using the discrete dipole approximation and angular spectrum method

Maryam Baker, Weilin Liu, and Euan McLeod

Open Access Open Access

Abstract

Lens-free microscopes can utilize holographic reconstruction techniques to recover the image of an object from the digitally recorded superposition of an unperturbed plane wave and a wave scattered by the object. Image reconstruction most commonly relies on the scalar angular spectrum method (ASM). While fast, the scalar ASM can be inaccurate for nanoscale objects, either because of the scalar approximation, or more generally, because it only models field propagation and not light-matter interaction, including inter-particle coupling. Here we evaluate the accuracy of the scalar ASM when combined with three different light-matter interaction models for computing the far-field light scattered by random arrays of gold and polystyrene nanoparticles. Among the three models—a dipole-matched transmission model, an optical path length model, and a binary amplitude model—we find that which model is most accurate depends on the nanoparticle material and packing density. For polystyrene particles at any packing density, there is always at least one model with error below 20%, while for gold nanoparticles with 40% or 50% surface coverage, there are no models that can provide errors better than 30%. The ASM error is determined in comparison to a discrete dipole approximation model, which is more computationally efficient than other full-wave modeling techniques. The knowledge of when and how the ASM fails can serve as a first step toward improved resolution in lens-free reconstruction and can also be applied to other random nanoparticle array applications such as lens-based super-resolution imaging, sub-diffraction beam focusing, and biomolecular sensing.

© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. Introduction

Lens-free microscopes are competitive imaging devices due to their simple hardware and high space-bandwidth product, which provides high resolution across ultra-large fields of view [1]. In its simplest configuration, digital lens-free microscopy is resolution-limited by the pixel size of the image sensor [2]. Using pixel super-resolution, lens-free microscopes are able to approach and even slightly exceed half-pitch resolutions of $\lambda /2$, or $\sim$250 nm [3–5]. These pixel super-resolution techniques may involve source shifting [6,7], wavelength scanning [4], compensating for limited pixel active areas [3], and computational algorithms involving prior knowledge about the type of sample [8]. Lens-free microscopes have been applied to the imaging of pathology slides [9], waterborne parasites [10], cell dynamics [11,12], and nanoparticle sensing [13–15] with sufficient resolution and a much larger field of view than conventional microscopes at a fraction of the cost.

Improving system resolution is critical to further extending the boundaries of detection for lens-free microscopes. The impacts of coherence, separation distances, and noise on the resolution of lens-free reconstructions have previously been quantified [16]. However, the accuracy of the electromagnetic algorithms involved in holographic reconstruction when dealing with nanoscale targets has received relatively little attention, and could lead to further improvements in resolution. Applications that could benefit from resolutions significantly better than 250 nm include the detection of hemozoin crystals in malaria [17], subcellular imaging in general [12], quality control in nanomaterial synthesis [14], sensing nanoparticle-labeled biomarker molecules [18,19], or sensing ultrafine ($<100$ nm diameter) particulate matter air-pollutants, which have been linked to neurodegenerative diseases such as Alzheimer’s [20,21]. Even for nanoparticles with size $\ll \lambda /2$, which may never be resolvable in the far-field, improvements in reconstruction could lead to better estimation of the nanoparticle properties, such as size or refractive index [14,22,23].

The angular spectrum method (ASM) is popular for reconstructing images in digital holography by back-propagating light measured at the sensor plane to the object plane. The ASM relies on neither Fraunhofer nor Fresnel approximations and is computationally fast, due to the use of the fast Fourier transform (FFT) [24,25]. The field is decomposed into a superposition of plane waves propagating at different angles:

$$\textbf{E}(x,y,z) = \mathcal{F}^{{-}1}\left\{\mathcal{F}\{\textbf{E}(x,y,0)\}e^{iz\sqrt{k^2-k_x^2-k_y^2}}\right\},$$
with the Fourier transform defined as:
$$\mathcal{F}\{\textbf{E}(x,y,z)\} = \frac{1}{4\pi^2}\iint_{-\infty}^\infty \textbf{E}(x,y,z)e^{{-}i(k_xx+k_yy)}\;dxdy,$$
where $\textbf {E}(x,y,0)$ is the field to be propagated, $\textbf {E}(x,y,z)$ is the field propagated over some distance $z$, $k$ is the wavenumber, and $k_x$ and $k_y$ are the spatial frequency components along $x$ and $y$ respectively. Often, a scalar field approximation is used, in which case $\mathbf {E}\rightarrow E$.

For the ASM to be accurate in describing or reconstructing the scattering from nanoparticles, the amplitude and phase delay imposed on the transmitted field by the nanoparticles must be correctly included in the electric field being propagated by the ASM. While nanoscale descriptions have been studied for conduction gratings [26] and diffractive phase elements [27] and successfully implemented in ASM for optical element design [28], an accurate description of nanoscale objects for digital holography is still needed.

Here we compare the accuracy of three scalar ASM transmission functions that model the scattering of $\lambda = 610$ nm light from 30 nm ($\sim \lambda /20$), 60 nm ($\sim \lambda /10$), and 100 nm ($\sim \lambda /6$) gold and polystyrene nanoparticles. We study randomly assembled nanoparticle arrays since they have many applications in super-resolution imaging [29–33] and focusing [34–36], as well as in biological sensing [37–39]. Computational accuracies for different particle densities and sizes are determined through comparisons with the discrete dipole approximation (DDA), finite difference time domain (FDTD), and Mie theory.

2. Methods

Figure 1 shows the simulation geometry. We calculate the fields scattered by a plane wave incident on random nanoparticle arrays with densities ranging from a single particle to 80$\%$ lattice fill fraction, defined as the percentage of two-dimensional (2D) lattice sites occupied by a nanoparticle, over a $1.8$ $\mu$m $\times 1.8$ $\mu$m area. Particles are constrained to sit on square lattice sites so that particle centroids can be precisely located in the numerical angular spectrum method, which propagates a pixelated field. The particles in the array were placed on a discrete square grid to avoid position-related discretization errors when representing the fields as a matrix for ASM modeling. To mitigate variations in performance due to the randomness of the particle locations, different instances of random particle locations were generated, and the resulting simulation times and deviations between different models were averaged. The number of random instances required was determined by calculating the standard error of the mean. At least ten instances were performed and, if necessary, more were added until the standard error of the mean was less than $1\%$. Refractive indices of $n=0.21 + 3.272i$ for gold [40] and $n=1.5895$ for polystyrene [41,42] were used, both corresponding to $\lambda =610$ nm, which is the emission wavelength for europium chelate nanoparticles that we plan to use in future experiments. Transmitted incident and scattered intensities at a distance $z=15\lambda$ for $x$- and $y$-polarized incident plane waves were separately computed and then averaged to model unpolarized light. Based on this propagation distance and a half-width of $w=0.9$ $\mu$m for the array size, the Fresnel number is $w^2/(z \lambda ) = 0.145$, corresponding to the far-field.

 figure: Fig. 1.

Fig. 1. Simulation of scattering from random arrays of nanoparticles. (a) Simulation set-up. A uniform plane wave is incident in the $+z$ direction. The nanoparticles are unsupported (suspended in free space), but all lie in the $x$-$y$ plane. (b) An example of a random array of 10 gold 30-nm diameter nanoparticles, corresponding to 0.2182% lattice site fill fraction. (c) Comparison of ASM and DDA cross section of scattered fields generated by the nanoparticle array shown in (b).

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We primarily use DDA as the point of comparison to evaluate the ASM model accuracy because DDA has a good balance of accuracy and computational efficiency. We wrote our own DDA code since open source codes, e.g., OpenDDA and DDSCAT, use discrete Fourier transform methods, which require that dipoles be placed on a regular 3D lattice and computational times scale with the total lattice size. In our case, particles may be sparsely placed on a lattice and our code’s computational times ignore empty sites, scaling with the total number of dipoles rather than the total lattice size. To ensure the accuracy of our DDA code, we first validate it against both Mie scattering and FDTD simulations. Mie scattering is a good validation tool for single spherical particles of any size, but cannot be easily applied to arrays of particles with high surface coverage. Previously, Mie theory has been combined with multiple scattering theory to calculate the average reflectivity of randomly assembled nano-monolayers, but not for calculating full scattering patterns [43,44]. This multiple-scattering approach did not incorporate shadowing effects for large angles of incidence, making it only valid for low surface coverages (<25%). We use Mie theory to bound the error in approximating particles as single dipoles, which becomes less accurate as the size of the nanoparticle approaches the length scale of the incident wavelength [45]. FDTD simulations could be used to model any of our geometries, but the computational cost is too great to run all of the comparisons shown here. FDTD simulations were only run for four “edge” cases where the errors in DDA could be the greatest: a single 100 nm particle (both polystyrene and gold) and an array of 100 nm particles with the largest lattice site fill fraction tested here (80$\%$, 330 particles), again for polystyrene and gold.

ASM simulations were conducted for three transmission functions that modeled the light-matter interaction in different ways. The first transmission function is the dipole matched transmission (DMT) model, where the amplitude attenuation and phase delay imposed on the field by each particle were calculated to match the analytical far-field scattering from a single dipolar nanoparticle illuminated by a plane wave. This transmission model matches DDA for a single particle, but does not account for inter-particle coupling in the case of multiple particles. The second model is the optical path length (OPL) transmission function, which models the nanoparticles as small slabs of material that impose a phase change and attenuation on the incident field based on the material and thickness of the slab. This is based on the model commonly applied for Fourier optics modeling of thin lenses. The last model is the binary amplitude mask (BAM) transmission function where nanoparticles are modeled as an opaque mask. This is the simplest model, as it does not involve material parameters, and could be considered appropriate for materials with low-transmission in the visible range of wavelengths, such as thick metal films or metal films with induced surface plasmon polaritons. The DMT and OPL transmission functions were applied to both gold and polystyrene nanoparticles, while the BAM transmission function was applied only to gold nanoparticles. The ASM, Mie theory, and DDA simulations were done on a computer with a 2.2 GHz Quad-Core Intel Core i3 processor with 16 GB 1600 MHz DDR3 memory. DDA simulations modeling a nanoparticle as more than one dipole were done on a computer with two 2.29 GHz Intel Xeon Gold 5218 processors with 256 GB RAM. FDTD simulations were done on a computer with an Intel Xeon CPU E5-2660 v3 2.6 GHz processor with 256 GB of 2133 MT/s RDIMM memory.

2.1 Discrete dipole approximation

The DDA calculation was done in MATLAB by approximating each nanoparticle as a single dipole. For comparison, we also tested a model where the 100 nm gold nanoparticles were discretized by 81 dipoles, which reduced the error for a single nanoparticle when compared to FDTD simulations. We used $M=81$ dipoles because it satisfies the volume relation $M>\frac {4\pi }{3}|m|^3(ka)^3$, outlined by Draine and Flatau [46], which for the 100 nm gold particles ($a=50$ nm) is equivalent to $M>20$. When using DDA to model the scattering from gold nanoparticles, the convergence in terms of number of dipoles per particle is rather slow, and previous studies have found that nearly 1 million dipoles are needed to reduce the error significantly below 5% when calculating the scattering from 100 nm gold particles [47,48]. Such a large number of dipoles is computationally infeasible for us when there is a high percentage of filled lattice sites.

In DDA, the total field at position $\mathbf {r}$ is calculated using the dyadic Green’s function, $\overleftrightarrow {\textbf {G}}$ and the dipole moment of each particle, $\mathbf {p}_n$ located at $\mathbf {r}_n$ [49]:

$$\textbf{E}(\textbf{r}) = \textbf{E}_0(\textbf{r}) + \omega^2\mu_0\mu_b\sum_{n=1}^{N} \overleftrightarrow{\textbf{G}}(\textbf{r},\textbf{r}_n)\textbf{p}_n,$$
$$\overleftrightarrow{\textbf{G}}(\textbf{r},\textbf{r}_n) = \frac{e^{ikR}}{4\pi R}\left[\left(1+\frac{ikR-1}{k^2R^2}\right)\overleftrightarrow{\textbf{I}}+\frac{3-3ikR-k^2R^2}{k^2R^2}\frac{\textbf{RR}}{R^2}\right],$$
where $\textbf {E}_0$ is the incident field, $\omega$ is the radial frequency of the incident light, $\mu _0$ is the free-space magnetic permeability, $\mu _b$ is the relative permeability of the background medium, $\overleftrightarrow {\textbf {I}}$ is a $3 \times 3$ identity matrix, $\mathbf {R} = \textbf {r}-\textbf {r}_n$, $R=|\mathbf {R}|$, and RR is an outer product. The dipole moment is given by,
$$\textbf{p}_n = \alpha(\omega)\textbf{E}(\textbf{r}_n)\varepsilon_b,$$
where $\alpha$ is the dipole polarizability and $\varepsilon _b$ is the background relative permittivity. The polarizability is modified from the polarizability derived from the Claussius-Mossotti relation $\alpha _{\mathrm {CM}}$ to account for the radiation reaction [46]:
$$\alpha = \frac{\alpha_{\mathrm{CM}}}{1-\frac{ik^3}{6\pi\varepsilon_0}\alpha_{\mathrm{CM}}},$$
$$\alpha_{\mathrm{CM}} = 3\varepsilon_0V\frac{\varepsilon-\varepsilon_b}{\varepsilon+2\varepsilon_b},$$
where $\varepsilon$ is the particle relative permittivity and $V$ is the particle volume. Note that inter-particle coupling is correctly handled, as the field exciting the $n^{\mathrm {th}}$ particle includes the contribution from the field scattered by the other $N-1$ particles. In this way, Eqs. (3) and (5) constitute a simultaneous set of equations where the dipole moments can be found when the incident light, polarizabilities, and all particle locations are known.

2.2 Mie theory

To evaluate the accuracy of the DDA and ASM simulations for single particles, comparisons were performed to Mie theory. A scattered field calculator was implemented in MATLAB using the Mie theory formalism in the Diffraction by a Conducting Sphere section of Principles of Optics by Born and Wolf [50]. The only limitation in the accuracy of the Mie theory for a single particle is the number of terms included in the infinite series [45]. For our simulations, we included successive terms in the series until the relative difference between the current and previous calculated field irradiances was less than $10^{-5}$, which only required four terms.

2.3 Finite difference time domain simulations

FDTD simulations were performed in FDTD Solutions (Lumerical Corp.). Unlike Mie theory, it works for an arbitrary number of randomly placed nanoparticles. Therefore, we can use it as a gold standard for both single nanoparticles and random nanoparticle arrays. To avoid diffraction artifacts caused by the boundaries of the simulation, we use the total-field scattered field (TFSF) scheme. The scattered field is recorded 0.5 $\mu$m away from the particle center over an area of 12 $\mu$m $\times$ 12 $\mu$m. Unlike the other simulation approaches, the FDTD simulations did not extend all the way to $z=15\lambda$ since a long propagation distance would be too computationally intensive for the required wide field of view. But if there is good agreement in the intermediate near-field, then there will also be good agreement in the far-field. To ensure the convergence of the FDTD simulation and conserve memory use, a two-scale mesh grid was used, with a fine mesh size of $\le 2$ nm enclosing the particles and a coarse mesh size throughout the domain of $\le 10$ nm. The stability factor used in all simulations is 0.99. Our convergence criteria for the FDTD simulations was the error between fields for decreasing mesh sizes. For our simulations, the error between the previous field computed from a coarser mesh and the current field computed with a finer mesh size was $3.7\times 10^{-6}\%$, which we determined was sufficient for convergence.

2.4 Angular spectrum method

The scattered far-field was computed based on Eq. (1), but with a scalar field assumption and FFTs:

$$\hat{E}(\xi, \eta) = \frac{\Delta x\Delta y}{(2\pi)^2} \sum_{u=1}^{N_x}\sum_{v=1}^{N_y}E((u-1)\Delta x, (v-1)\Delta y)\;\;e^{i2\pi\left(\frac{(u-1)(\xi-1)}{N_x}+\frac{(v-1)(\eta-1)}{N_y}\right)}$$
where $(u,v)$ are indices corresponding to discrete $(x,y)$ coordinates, $\Delta x$ and $\Delta y$ are the sampling pitches, $N_x$ and $N_y$ are the number of $x$ and $y$ values, and $(\xi ,\eta )$ are the frequency-space indices with sampling pitches $\Delta \xi = (2\pi )/(N_x \Delta x)$ and $\Delta \eta = (2\pi )/(N_y \Delta y)$. The use of FFTs requires the field to be sampled on a regular rectangular grid. The grid spacings $\Delta x$ and $\Delta y$ were chosen to equal the nanoparticle diameter in order to identify whether there are computationally-efficient ways that are still accurate in simulating light propagation through the structure, without having to discretize each particle into many voxels. The particle sizes are already significantly smaller than the wavelength, so the sampling resolution for wave propagation using ASM should be more than sufficient.

An $N_x \times N_y = 8001 \times 8001$ field was generated to model the electric field transmitted by the nanoparticle plane in ASM with values,

$$t_{uv} = \begin{cases} T & \textrm{if particle centered at (u,v)}\\ 1 & \textrm{otherwise} \end{cases}$$
corresponding to the transmission of a unit amplitude incident plane wave through the nanoparticle array, where $T$ was assigned corresponding to one of the three transmission functions described below. This electric field matrix $[t_{uv}]$ is used as the field $E$ in Eq. (8). The field was then propagated using the ASM and cropped to match the field size computed using DDA. This field size was chosen to reduce the numerical errors related to the discrete sampling of the free-space transfer function. Note that larger matrices can significantly increase computational cost in ASM.

2.4.1 Transmission function $\#$1: dipole matched transmission (DMT)

The DMT function was derived by equating the analytical far-field DDA and ASM solutions for a single particle. The derivation is given in the Appendix, resulting in:

$$T_\mathrm{DMT} = 1+\frac{ik\alpha}{2\varepsilon_0 \Delta x \Delta y},$$

2.4.2 Transmission function $\#$2: optical path length (OPL)

The OPL transmission function applies an attenuation and phase delay at a nanoparticle location corresponding to the nanoparticle’s complex refractive index and thickness. This model is based off of the phase delay induced by a lens due its shape and refractive index as shown in Fig. 2(a) [51]:

$$t_l(x,y) = \exp{[ik\Delta_0 n_b]}\exp \left[ik(n-n_b)\Delta(x,y)\right],$$
where $\Delta _0$ is the center thickness of the lens and $\Delta (x,y)$ is the varying thickness due to the radius of curvature along the lens. Since the nanoparticle diameter $2a$ is equal to the pixel pitch in our discretized ASM, the spherical nanoparticle does not physically fill the entire cubiod pixel volume. To correct for this effect, we use an effective pixel thickness $\bar {h}$ equal to the spatially-averaged optical path length through a spherical nanoparticle (see Fig. 2(b) and (c)):
$$\bar{h} = \frac{\int_{{-}a}^a \int_{{-}a}^a {2\operatorname{Re} [\sqrt{a^2-x^2-y^2}]dxdy}}{\Delta x \Delta y}=\frac{\pi}{3}a,$$
where we have used $\Delta x \Delta y = (2a)^2$. Dropping the constant phase factor from Eq. (11), the resulting transmission function is,
$$T_\mathrm{OPL} = e^{\frac{i 2\pi}{\lambda}\bar{h}(n - n_b)}.$$

2.4.3 Transmission function $\#$3: binary amplitude mask (BAM)

The binary amplitude mask (BAM) models nanoparticles as an opaque mask, and is only considered for the gold nanoparticle case:

$$T_\mathrm{BAM} = 0.$$
While this model may be reasonable for a continuous gold film that is thicker than the skin depth ($\sim$30 nm at this wavelength) and has high reflectivity, it is less accurate for films of only a few tens of nanometers in thickness [52], or isolated particles. Here, we test if there is any range of parameters where the BAM approximation is valid.

 figure: Fig. 2.

Fig. 2. Geometrical considerations for the OPL transmission function. All panels show the optical thickness of a $y$-$z$ cross section at the plane $x=0$. (a) A lens. (b) A spherical particle. (c) A square pixel representing a matrix element in the ASM. The value of $\bar {h}$ is chosen according to Eq. (12) such that the optical path lengths averaged over the area of the pixel in the $x$-$y$ plane are the same in (b) and (c).

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3. Results and discussion

3.1 Validation of DDA results against the Mie theory and FDTD

To gauge the accuracy of the different ASM transmission functions, we will compare the results to DDA simulations. First, however, we validate the DDA simulation by comparing to Mie scattering for single particles and FDTD simulations for a few representative cases. A comparison of Mie scattering to DDA and to the ASM methods is shown in Fig. 3, where the percent error with respect to Mie scattering is defined as,

$$\Delta I_{\mathrm{Mie}} = 100 \times \sqrt{\frac{\sum\sum (I_{\mathrm{Mie}}-\tilde{I})^2}{\sum\sum(I_{\mathrm{Mie}}-1)^2}},$$
where the summations are run over all points in the $x$ and $y$ directions in the imaging plane, $I_{\mathrm {Mie}}=|\mathbf {E}_{\mathrm {Mie}}|^2$, and $\tilde {I}$ is a normalized intensity, either for DDA or for ASM, depending on the comparison being made:
$$\tilde{I} = |\tilde{E}|^2 = \left|(|E|-1)\sqrt{\frac{\sum\sum(|\mathbf{E}_{\mathrm{Mie}}|-1)^2}{\sum\sum(|E|-1)^2}}+1\right|^2$$
By subtracting the background and normalizing the total power across the full field of view before calculating the error, the metric in Eq. (15) effectively calculates the pattern error between the two methods.

The range of $\theta$ plotted in Fig. 3, $10^\circ \le \theta \le 65^\circ$, corresponds to sensor plane areas from $3\,\mathrm {\mu m} \times 3\, \mathrm {\mu m}$ to $40\, \mathrm {\mu m} \times 40\, \mathrm {\mu m}$. When evaluating the accuracy of DDA, for polystyrene particles, $\Delta I_{\mathrm {Mie}}<1.05\%$ over all angles, while for gold, $\Delta I_{\mathrm {Mie}}<10\%$. Overall, the agreement between DDA and Mie scattering remains reasonable for all particle sizes, and so any deviations between ASM and DDA greater than these errors between DDA and Mie scattering can be considered as significant errors in the ASM model.

As a further confirmation of the validity of our DDA implementation, we compared our results to those of FDTD simulations, which, unlike Mie scattering, can handle large numbers of potentially coupled particles. These results are shown in Fig. 4. Due to burdensome computational times and memory requirements, we only compare the most error-prone “edge” cases of greatest particle coverage (80% lattice site fill fraction) with the largest particles (100 nm diameter) as well as single particle simulations, which can be validated against Mie scattering to ensure the accuracy of the FDTD framework (Fig. 4(a)). For 100 nm polystyrene particles, the error between DDA and both validating methods, Mie theory and FDTD, was less than 10%. For gold, errors are $\sim 2\times$ larger. In an attempt to reduce the error, we discretized the gold nanoparticle into 81 dipoles. For a single particle, this significantly reduced the error between DDA and FDTD; however for the array of 330 particles (80% lattice site fill fraction), the error worsened (Fig. 4(b)). This might imply that more than 81 dipoles are needed to more accurately model each nanoparticle for the 330 particle case. In some applications, up to a million dipoles per particle can be required, depending on the desired level of accuracy [47,48]. Due to computational limits, we were not able to model more than 81 dipoles per particle across 330 nanoparticles. With a deviation of less than 22% between DDA and the validating models for the least accurate simulations, the comparisons of ASM against DDA are further validated to this level of accuracy.

 figure: Fig. 3.

Fig. 3. Errors in single particle scattering calculations with respect to Mie theory. The angle $\theta$ describes the field of view at the scattering plane, as indicated in Fig. 1(a). The purple curve in (d) corresponds to the error for a 100 nm gold nanoparticle discretized by 81 dipoles.

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 figure: Fig. 4.

Fig. 4. Comparison to FDTD simulations for 100 nm diameter particles. In (b), 330 particles corresponds to an 80$\%$ fill fraction of the available lattice sites. The purple bar corresponds to 100 nm gold nanoparticle(s) discretized by 81 dipoles. The FDTD error, $\Delta I$, is defined analogously to that of $\Delta I_{\mathrm {Mie}}$ in Eq. (15).

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3.2 Transmission function $\#$1: dipole matched transmission

Some examples of scattered field calculations using the DMT model are shown in Fig. 5. The insets show how the deviation between this model and DDA increases as more particles are added to the simulation. Increasing deviation is expected as the DMT model does not account for inter-particle coupling. Similarly, deviations are greater for gold (top row of Fig. 5) than polystyrene (bottom row) as plasmonic effects, such as localized surface plasmon resonance, can enhance inter-particle coupling. Another interesting aspect of the fields in Fig. 5 is the bright spot in the center of each field, where one might intuitively expect a shadow instead. We believe this is an example of the well-known Arago spot.

 figure: Fig. 5.

Fig. 5. Scattering patterns. The interference between a unit amplitude incident plane wave and scattered light (in-line hologram) is calculated at the imaging plane $z=15\lambda$ for an off-center single particle, $\sim$1800 particles, and $\sim$3600 particles. The images show field magnitudes ($|E|$) calculated using the DMT model. The insets show the field cross sections along the $x$-axis for the DMT ASM (orange) and DDA (blue) calculations. As more 30 nm particles are added, the deviation between the DMT and ASM models increases.

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A compilation of all ASM to DDA comparison results is shown in Fig. 6. The percent deviation of the ASM scattered field with respect to the DDA reference field is defined as,

$$\Delta I = 100 \times \sqrt{\frac{\sum\sum (I_{\mathrm{DDA}}-\tilde{I}_{\mathrm{ASM }})^2}{\sum\sum(I_{\mathrm{DDA}}-1)^2}},$$
where an ASM field normalized against the DDA result is defined analogously to the field normalized against Mie theory defined in Eq. (16):
$$\tilde{I}_{\mathrm{ASM}} = |\tilde{E}_{\mathrm{ASM}}|^2 = \left|(|E_{\mathrm{ASM}}|-1)\sqrt{\frac{\sum\sum(|\mathbf{E}_{\mathrm{DDA}}|-1)^2}{\sum\sum(|E_{\mathrm{ASM}}|-1)^2}}+1\right|^2$$

 figure: Fig. 6.

Fig. 6. Deviation in ASM models with respect to DDA. Results for all particle sizes and both materials are presented. Each data point represents the mean deviation from enough different instances of random arrays generated using the same parameters to reach a standard error of the mean $<1\%$ of the mean. Symbol colors and shapes have the same meaning in each panel. The deviation between DDA and Mie scattering for single particles is also shown for reference.

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Of the two ASM models tested on polystyrene, DMT was more accurate than OPL for low and modest lattice site fill fractions. Even for larger lattice site fill fractions of polystyrene particles, the deviation with respect to DDA remained less than 25%. For gold, the DMT model was more accurate than the other ASM models for all lattice site fill fractions for 30 nm particles as well as for fill fractions below 50% for 60 nm and a lattice site fill fractions below 40% for 100 nm gold particles modeled as a single dipole and lattice site fill fractions below 10% (10 nanoparticles) for 100 nm gold particles discretized by 81 dipoles. The gold nanoparticles scatter more light between particles than the polystyrene nanoparticles do, resulting in more error in the DMT model as more particles are added and as the particles get larger. This demonstrates the importance of accounting for scattered field contributions from inter-particle interactions.

To ensure that our conclusions were not tied to a particular instance of a randomly-generated mask, we calculated the standard error of the mean for each case and added trials of new randomly-generated masks to reduce the standard error of the mean to below 1%. The standard deviation of the error between the DDA and DMT simulations over the various random configurations was less than 1.2% for all polystyrene particle simulations. For gold particles, the standard deviation of the error was generally less than 10%, with the exception of the 100 nm gold particles discretized by 81 dipoles, where the 2.42% lattice fill fraction exhibited $\sigma = 24\%$, and the 10% lattice fill fraction exhibited $\sigma = 12\%$. The largest deviations between two random instances of the array occur when one instance has ample spacing between all particles while another instance has particles touching each other.

Another source of error of the DMT transmission function is the small-angle cosine approximation (see appendix Eq. (25)). The relative error in this approximation exceeds 1% for $\theta > 38^\circ$. The field of view in the simulations run here corresponds to $\theta = 33^\circ$, which is within the range of accuracy of the approximation, as shown by the relatively constant levels of deviation in Fig. 3 for $\theta < 38^\circ$. As expected, the deviation increases rapidly as $\theta$ increases beyond 38$^\circ$ for both gold and polystyrene. While the deviation is about 4$\%$ or less for polystyrene up to 50$^\circ$, the approximation breaks down sooner for gold, perhaps due to increased side-scattering.

We also compared a polarized DDA model (Eq. (20)) with the DMT model, which remains unchanged whether or not the field described in Eq. (22) is polarized. For a single 100 nm gold nanoparticle, the average error between the methods is $8\%$, and for the $80\%$ fill fraction case the average error between the two models is 111$\%$. The primary difference between these two models in the case of polarized light is the decay of the scattered field along the direction of polarization, which is correctly captured by the DDA model, but not by the DMT model.

3.3 Transmission function $\#$2: optical path length transmission function

The OPL transmission function was the second-best ASM model after the DMT function for most cases, and performed better than the DMT function at high lattice site fill fractions. For polystyrene, the percent deviation with respect to DDA was lower than 15% for all particle sizes, as can be seen in Fig. 6. The OPL transmission function is a reasonable approximation for dielectric particles as scattering is low between particles and can be neglected in some cases. However the OPL transmission function is not a good approximation for gold as it does not accurately describe the scattering from metallic particles, and the error with respect to DDA is $>80\%$ for all particle sizes and lattice site fill fractions except for the 30-80% lattice site fill fractions for the 100 nm nanoparticles discretized by 81 dipoles.

The deviation for polystyrene has an unusual behavior as it first decreases while particles are added and then increases (Fig. 6(a–c)). The initial decrease in deviation could be due to effective “smoothening” of the surface consisting of the array of nanoparticles, such that the array can be more accurately described using low spatial frequency components. As the OPL transmission function is based off of a model for macroscopic thin lenses, it works best for smooth, slowly-varying surfaces. As more particles are added, gaps between particles become filled, leading to a smoother surface. However, at very high particle densities $>60\%$, errors due to inter-particle coupling can take over, leading to an increase in error. Note that as we are simulating spherical particles, we never generate a continuous smooth film in the DDA model.

Notably, the Arago spot is much weaker for a single gold particle in the OPL (and BAM) model than the (more accurate) DMT model, as seen in Fig. 7. This demonstrates a need for a more sophisticated description of the diffraction effects for gold nanoparticles than can be provided by the OPL and BAM models.

 figure: Fig. 7.

Fig. 7. Comparison of Arago’s spot between the different ASM models. The same colorbar applies panels (a)–(c). The agreement with DDA is significantly better for the DMT model than for the other ASM models.

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3.4 Transmission function $\#$3: binary amplitude mask transmission function

The BAM transmission function was only applied to gold to test the high opacity approximation, which was expected to apply to larger particles filling a larger percentage of the array area. The smallest particle size, 30 nm, is on the order of the skin depth [52], so some transmission would be expected even for a complete film. While the deviation did decrease as the particle size and lattice site fill fraction increased, as can be seen in the bottom row of Fig. 6, there remains significant error in the BAM model for all cases tested here.

3.5 Simulation times

Each ASM simulation took between 11 and 13 seconds to run, while the DDA simulations were 150–685 s for 30 nm particles, 10–40 s for 60 nm particles, and 1–5 s for 100 nm particles when modeling the 100 nm particle as a single dipole and between $1.2\times 10^3$–$1.2\times 10^4$ s when modeling the 100 nm particles as 81 dipoles. For the single dipole approximation, the DDA time decreased as the particle size increased because fewer large particles are used to fill the 1.8 $\mathrm {\mu m}$ x 1.8 $\mathrm {\mu m}$ window. The relative time efficiency for ASM calculations is notable for 30 and 60 nm particles. Although the 100 nm DDA simulations were fastest, they were the DDA solutions with the highest potential error due to approximating particles on the length scale of the incident light as a single dipole [53]. The Mie simulations of the field from a single polystyrene or gold nanoparticle took 17–144 s depending on the particle size. The FDTD simulation time depends on the fine mesh area. For the single particle cases, the fine mesh only enclosed one particle, and the simulation took $6\times 10^4$–$7.2\times 10^4$ s to run. For the arrays with 80% lattice site fill fraction, the fine mesh area needed to enclose all the nanoparticles, so the simulation was slower, taking $1.3\times 10^5$–$1.64\times 10^5$ s.

4. Conclusion

A summary guide as to which simulations are accurate and fast is provided in Table 1. The ASM models were considered accurate when the error was less than 30% with respect to the DDA simulation. The Mie and FDTD methods are considered inherently accurate in situations where they can be used, considering computational resources and the number of particles. Overall, the most accurate model for light-matter interaction using the ASM was the DMT function, which maintained its accuracy for 30 nm, 60 nm, and 100 nm polystyrene nanoparticles of large densities and for the same sizes of gold nanoparticles of medium to low densities. The transmission function approximations all have large inaccuracies for large densities of large gold nanoparticles. ASM computations are significantly faster than DDA for 30 nm and 60 nm nanoparticles and faster than Mie and FDTD for single particles of all sizes, particularly for large observation planes.

Tables Icon

Table 1. Summary of accuracy and speed of the different investigated scattering models, which ranged up to 80% lattice site fill fraction. If the model is not applicable over the entire experimental domain, the upper limit on the number of particles is specified in parentheses. The models are considered accurate if the deviation is less than 30$\%$ and considered fast if the simulations take 30 seconds or less. $N_{81}$ corresponds to particles discretized with 81 dipoles each, while $N_{1}$ corresponds to particles modeled as single dipoles.

View Table

Mie theory and FDTD simulations were used for validation and to estimate the uncertainty in the DDA calculations. The error between DDA and Mie was less than 2% for all particle sizes except for the 100 nm gold particles, which had error less than 10%. These errors correspond to uncertainties in ASM error when using DDA as a reference, as the DDA itself may be inaccurate to this level. There are some combinations of particle material, size, lattice site fill fraction, and ASM models where the deviation between DDA and ASM are $<1\%$ (smaller than the DDA uncertainty), and other parameter combinations where the deviations are significantly greater than the DDA uncertainty. This demonstrates the importance of choosing an accurate scalar ASM light-matter interaction model, considering the type of particle and surface coverage density. Notably, for gold particles with 40% or 50% lattice site fill fractions, there are no ASM methods with accuracy better than $30\%$, suggesting an alternative light scattering and propagation algorithm is required for those cases.

We expect these results to prove useful in applications where the scattered field for large observation planes or from large numbers of dielectric particles is desired. Beyond the reconstruction of lensfree images with extreme resolution, these applications might include optimization for metamaterial devices and sub-diffraction limited beam focusing.

Appendix

The DMT function is derived by equating the analytical DDA and ASM solutions for a single particle in the Rayleigh scattering regime. The interference of a unit-amplitude $x$-polarized plane wave and the scattered far-field terms ($kR\gg 1$) of the dyadic Green’s function (Eq. (4)) at some observation plane a distance $z$ away from a single dipole placed at the origin so that $\mathbf {R}=\mathbf {r}$ is:

$$\begin{aligned}\textbf{E}_{\mathrm{DDA},x} &= \textbf{E}_{\mathrm{inc},x}+\textbf{E}_{\mathrm{scatt},x}\\ &= e^{ikz}\mathbf{\hat{n}}_x + \omega^2\mu_0\mu\frac{e^{ikr}}{4\pi r}\left(\overleftrightarrow{\textbf{I}}-\frac{\textbf{rr}}{r^2}\right)\textbf{p}\\ &= e^{ikz} \mathbf{\hat{n}}_x + A \frac{e^{ikR}}{r^3} \left( r^2 \overleftrightarrow{\mathbf{I}}-\mathbf{rr} \right) \mathbf{\hat{n}}_x \end{aligned}$$
where $\mathbf {p}$ has been evaluated using Eq. (5) and $A = \varepsilon _b \alpha \omega ^2 \mu _0 \mu /(4\pi )$. Assuming a nonmagentic background medium and $|A|/r \ll 1$, the irradiance detected at some distance $z$ away from the nanoparticle for an $x$-polarized plane wave becomes:
$$I_{\mathrm{DDA},x} = \frac{cn_b\varepsilon_0}{2}|\mathbf{E}_{\mathrm{DDA},x}|^2 = \frac{cn_b\varepsilon_0}{2} \left(1+\frac{2(y^2+z^2)}{r^3}[A'\cos(k(r-z))-A^{\prime\prime}\sin(k(r-z))]\right),$$
where $A' = \operatorname {Re}(A)$ and $A'' = \operatorname {Im}(A)$, $c$ is the speed of light, and $n_b$ is the refractive index of the background. Combining this result with the same calculation for an incident $y$-polarized plane wave, the irradiance for a single nanoparticle illuminated by an unpolarized plane wave is:
$$I_\mathrm{DDA} = \frac{1}{2}(I_x+I_y) = \frac{cn_b\varepsilon_0}{2} \left( 1+\frac{r^2+z^2}{r^3}\left[A' \cos(k(r-z))-A^{\prime\prime}\sin(k(r-z))\right]\right)$$
Using the scalar ASM, the electric field corresponding to a unit amplitude plane wave interacting with a single dipolar nanoparticle located at the origin is:
$$E_\mathrm{ASM}(x,y,0) = 1+B \delta(x,y),$$
where $B$ describes the amplitude and phase of the interaction between the scattering dipole and the incident light. In the DMT model, $B$ is found analytically by equating the irradiance predicted by DDA (Eq. (21)) with that predicted by the ASM. Applying the ASM (Eq. (1)) to Eq. (22),
$$\begin{aligned} E_{\mathrm{ASM}}(x,y,z) &= e^{ikz} + B \mathcal{F}^{{-}1}\left\{\left(\frac{1}{4\pi^2}\right) e^{iz\sqrt{k^2-k_x^2-k_y^2}}\right\}\\ &= e^{ikz}+B\frac{z}{2\pi}e^{ikr}\frac{1-ikr}{r^3}, \end{aligned}$$
where we have used the partial derivative of the Weyl identity [54] with respect to $z$ to compute the inverse Fourier transform.

Assuming $kr \gg 1$ and $(|B|zk)/(2\pi r^2) \ll 1$, as would be the case in the far-field, the irradiance computed from Eq. (23) is:

$$I_{ASM} = \frac{cn_b\varepsilon_0}{2} \left(1+\frac{kz}{\pi r^2}[B^{\prime\prime} \cos (k(r-z))+B'\sin (k(r-z))]\right),$$
where $B' = \operatorname {Re}(B)$ and $B'' = \operatorname {Im}(B)$. To reconcile the prefactors in Eqs. (21) and (24), we introduce a small-angle approximation for $\cos \theta =z/r$, which we apply to the prefactor in Eq. (21):
$$\frac{r^2+z^2}{r^3} = \frac{z}{r^2}\left(\frac{1}{\cos \theta}+\cos \theta \right) \approx \frac{2z}{r^2}$$
The error of this approximation is less than $1\%$ for $\theta < 38 ^{\circ }$. Reconciling Eqs. (21), (24), and (25), solving for $B$, and substituting into Eq. (22), we find the dipole-matched transmission function:
$$T_\mathrm{DMT} = 1+\frac{B}{\Delta x \Delta y} = 1+\frac{ik\alpha}{2\varepsilon_0 \Delta x \Delta y},$$
where we have mapped the continuous delta function in Eq. (22) to a single-pixel value corresponding to the notation in Eq. (9).

Funding

National Science Foundation (ECCS-1807590).

Disclosures

EM: University of California (P).

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

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Figures (7)
Fig. 1.
Fig. 1. Simulation of scattering from random arrays of nanoparticles. (a) Simulation set-up. A uniform plane wave is incident in the $+z$ direction. The nanoparticles are unsupported (suspended in free space), but all lie in the $x$-$y$ plane. (b) An example of a random array of 10 gold 30-nm diameter nanoparticles, corresponding to 0.2182% lattice site fill fraction. (c) Comparison of ASM and DDA cross section of scattered fields generated by the nanoparticle array shown in (b).

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Fig. 2.
Fig. 2. Geometrical considerations for the OPL transmission function. All panels show the optical thickness of a $y$-$z$ cross section at the plane $x=0$. (a) A lens. (b) A spherical particle. (c) A square pixel representing a matrix element in the ASM. The value of $\bar {h}$ is chosen according to Eq. (12) such that the optical path lengths averaged over the area of the pixel in the $x$-$y$ plane are the same in (b) and (c).

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Fig. 3.
Fig. 3. Errors in single particle scattering calculations with respect to Mie theory. The angle $\theta$ describes the field of view at the scattering plane, as indicated in Fig. 1(a). The purple curve in (d) corresponds to the error for a 100 nm gold nanoparticle discretized by 81 dipoles.

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Fig. 4.
Fig. 4. Comparison to FDTD simulations for 100 nm diameter particles. In (b), 330 particles corresponds to an 80$\%$ fill fraction of the available lattice sites. The purple bar corresponds to 100 nm gold nanoparticle(s) discretized by 81 dipoles. The FDTD error, $\Delta I$, is defined analogously to that of $\Delta I_{\mathrm {Mie}}$ in Eq. (15).

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Fig. 5.
Fig. 5. Scattering patterns. The interference between a unit amplitude incident plane wave and scattered light (in-line hologram) is calculated at the imaging plane $z=15\lambda$ for an off-center single particle, $\sim$1800 particles, and $\sim$3600 particles. The images show field magnitudes ($|E|$) calculated using the DMT model. The insets show the field cross sections along the $x$-axis for the DMT ASM (orange) and DDA (blue) calculations. As more 30 nm particles are added, the deviation between the DMT and ASM models increases.

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Fig. 6.
Fig. 6. Deviation in ASM models with respect to DDA. Results for all particle sizes and both materials are presented. Each data point represents the mean deviation from enough different instances of random arrays generated using the same parameters to reach a standard error of the mean $<1\%$ of the mean. Symbol colors and shapes have the same meaning in each panel. The deviation between DDA and Mie scattering for single particles is also shown for reference.

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Fig. 7.
Fig. 7. Comparison of Arago’s spot between the different ASM models. The same colorbar applies panels (a)–(c). The agreement with DDA is significantly better for the DMT model than for the other ASM models.

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Tables (1)

Tables Icon

Table 1. Summary of accuracy and speed of the different investigated scattering models, which ranged up to 80% lattice site fill fraction. If the model is not applicable over the entire experimental domain, the upper limit on the number of particles is specified in parentheses. The models are considered accurate if the deviation is less than 30 % and considered fast if the simulations take 30 seconds or less. N 81 corresponds to particles discretized with 81 dipoles each, while N 1 corresponds to particles modeled as single dipoles.

View Table

Equations (26)

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E ( x , y , z ) = F − 1 { F { E ( x , y , 0 ) } e i z k 2 − k x 2 − k y 2 } ,
F { E ( x , y , z ) } = 1 4 π 2 ∬ − ∞ ∞ E ( x , y , z ) e − i ( k x x + k y y ) d x d y ,
E ( r ) = E 0 ( r ) + ω 2 μ 0 μ b ∑ n = 1 N G ↔ ( r , r n ) p n ,
G ↔ ( r , r n ) = e i k R 4 π R [ ( 1 + i k R − 1 k 2 R 2 ) I ↔ + 3 − 3 i k R − k 2 R 2 k 2 R 2 RR R 2 ] ,
p n = α ( ω ) E ( r n ) ε b ,
α = α C M 1 − i k 3 6 π ε 0 α C M ,
α C M = 3 ε 0 V ε − ε b ε + 2 ε b ,
E ^ ( ξ , η ) = Δ x Δ y ( 2 π ) 2 ∑ u = 1 N x ∑ v = 1 N y E ( ( u − 1 ) Δ x , ( v − 1 ) Δ y ) e i 2 π ( ( u − 1 ) ( ξ − 1 ) N x + ( v − 1 ) ( η − 1 ) N y )
t u v = { T if particle centered at (u,v) 1 otherwise
T D M T = 1 + i k α 2 ε 0 Δ x Δ y ,
t l ( x , y ) = exp ⁡ [ i k Δ 0 n b ] exp ⁡ [ i k ( n − n b ) Δ ( x , y ) ] ,
h ¯ = ∫ − a a ∫ − a a 2 Re ⁡ [ a 2 − x 2 − y 2 ] d x d y Δ x Δ y = π 3 a ,
T O P L = e i 2 π λ h ¯ ( n − n b ) .
T B A M = 0.
Δ I M i e = 100 × ∑ ∑ ( I M i e − I ~ ) 2 ∑ ∑ ( I M i e − 1 ) 2 ,
I ~ = | E ~ | 2 = | ( | E | − 1 ) ∑ ∑ ( | E M i e | − 1 ) 2 ∑ ∑ ( | E | − 1 ) 2 + 1 | 2
Δ I = 100 × ∑ ∑ ( I D D A − I ~ A S M ) 2 ∑ ∑ ( I D D A − 1 ) 2 ,
I ~ A S M = | E ~ A S M | 2 = | ( | E A S M | − 1 ) ∑ ∑ ( | E D D A | − 1 ) 2 ∑ ∑ ( | E A S M | − 1 ) 2 + 1 | 2
E D D A , x = E i n c , x + E s c a t t , x = e i k z n ^ x + ω 2 μ 0 μ e i k r 4 π r ( I ↔ − rr r 2 ) p = e i k z n ^ x + A e i k R r 3 ( r 2 I ↔ − r r ) n ^ x
I D D A , x = c n b ε 0 2 | E D D A , x | 2 = c n b ε 0 2 ( 1 + 2 ( y 2 + z 2 ) r 3 [ A ′ cos ⁡ ( k ( r − z ) ) − A ′ ′ sin ⁡ ( k ( r − z ) ) ] ) ,
I D D A = 1 2 ( I x + I y ) = c n b ε 0 2 ( 1 + r 2 + z 2 r 3 [ A ′ cos ⁡ ( k ( r − z ) ) − A ′ ′ sin ⁡ ( k ( r − z ) ) ] )
E A S M ( x , y , 0 ) = 1 + B δ ( x , y ) ,
E A S M ( x , y , z ) = e i k z + B F − 1 { ( 1 4 π 2 ) e i z k 2 − k x 2 − k y 2 } = e i k z + B z 2 π e i k r 1 − i k r r 3 ,
I A S M = c n b ε 0 2 ( 1 + k z π r 2 [ B ′ ′ cos ⁡ ( k ( r − z ) ) + B ′ sin ⁡ ( k ( r − z ) ) ] ) ,
r 2 + z 2 r 3 = z r 2 ( 1 cos ⁡ θ + cos ⁡ θ ) ≈ 2 z r 2
T D M T = 1 + B Δ x Δ y = 1 + i k α 2 ε 0 Δ x Δ y ,
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