Abstract

Theoretical and experimental bases are given for measuring the complex forward-scattering amplitude of single particles through self-reference interferometry. Our analyses reveal the nondimensional parameters that primarily control the accuracy and resolution of the complex amplitude data. We propose a measurement protocol, Complex Amplitude Sensing version 1 (CAS-v1), for effectively utilizing self-reference interferometry as a universal tool for inline measurements of the complex forward-scattering amplitude of single sub- and super-micron particles suspended in a fluid flow. The CAS-v1 protocol will facilitate applications of self-reference interferometry to real-time particle measurements in the industrial, biomedical, and environmental sciences.

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

1. Introduction

The complex scattering amplitude $S(\theta )= {\rm{Re}}S + i{\rm{Im}}S = {{\;}}|S |{e^{i\Delta }}$ is defined by the amplitude $|S |$ and phase shift $\Delta $ of the scattered field observed at an angle $\theta $ relative to those of the plane-wave field incident to the scattering particle [13]. The complex $S(\theta )$ parameter reflects physical properties of the scattering particle such as complex refractive index, volume, shape, and orientation, and thus plays a fundamental role in almost all the technical methods for non-invasive optical particle characterization [46].

Light-scattering sensors are designed to measure the power of the scattered field ${\propto} {|{S(\theta )} |^2}$ at θ ≠ 0°, avoiding the direct optical beam of the incident field [7]. On the other hand, light-extinction sensors are designed to measure the interference power of the scattered and incident fields in the forward direction ${\propto} {\rm{Im}}S({0^\circ } )$ by monitoring the direct beam [7,8]. A simultaneous measurement of both ${\rm{Re}}S$ and ${\rm{Im}}S$ is not possible at optical frequencies without using specially designed interferometric methods. Some interferometric imaging microscopy techniques have been shown to be able to measure both the amplitude and phase of the forward-scattered field from each particle [9,10]. However, the necessity of acquiring and processing of the high-resolution microscopic image of each particle in these techniques would severely limits the data throughput, which could be a crucial factor for the monitoring of the temporal and spatial variations of particle concentrations in industrial, bio-medical, and environmental applications.

Several different types of optical interferometric techniques have been developed for phase-sensitive, high-throughput detection of the scattered field of single particles [1117]. Some of them are able to detect even single nanoparticles much smaller than the wavelength (with particle diameter ${d_{\rm{p}}}$ << 0.1 μm) thanks to the ${\propto} d_{\rm{p}}^3$ dependence of the interference signal power in contrast to the ${\propto} d_{\rm{p}}^6$ dependence of the scattering signal power [12,15,16]. In recent optics community, these techniques could also be regarded as particular instances of the interferometric scattering microscopy (iSCAT) method [1820].

To the author’s knowledge, the simultaneous measurement of both the amplitude and phase shift of the scattered field for single particles in a fluid flow has been achieved for the first time by Taubenblatt and Batchelder [13] by using an interferometric scheme based on Nomarski optics. Their measurement data [13] were equivalent to both ${\rm{Re}}S({0^\circ } )$ and ${\rm{Im}}S({0^\circ } )$ as defined here. Their scheme requires each scattering particle to move sequentially across both of the two adjacent focal spots of two orthogonally-polarized Gaussian beams [13]. This requirement demands the additional effort of implementing an ingenious technique into their scheme [13] for selecting only the detection signals for acceptable particle trajectories across the beams [cf. 13]. Bassini et al. [14] developed an experimental apparatus to measure both ${\rm{Re}}S({0^\circ } )$ and ${\rm{Im}}S({0^\circ } )$ (which are connected to the phase shift and extinction defined in their paper) of single particles through a Mach–Zehnder-type interferometer associated with heterodyne signal detection. Their interferometric scheme for single-particle detection requires precise control of a particle’s 3D position during measurement. For this reason, their experimental results were limited to immobilized particles deposited on the surface of a microscope slide [14]. Ignatovich and Novotny [15] proposed a simple scheme for interferometric detection of the backward-scattered field of single nanoparticles in a liquid flow. This scheme was not designed to separate the components of the amplitude and phase shift of the scattered field from the interferometric signals. Deutsch et al. [16] developed a dual phase interferometric scheme for separate detection of the amplitude and phase shift of the backward-scattered field of single nanoparticles in a liquid flow by extending the scheme of Ignatovich and Novotny [15]. In their scheme for interferometric detection of the backward-scattered field, the phase difference between the scattered and reference fields at the signal power detector varies as ∼$2k{z_{\rm{p}}}$ radians, where k and ${z_{\rm{p}}}$ denote the wavenumber and particle coordinate along the beam axis, respectively. For this reason, the width of the statistical ${z_{\rm{p}}}$-distribution of the particle trajectory in their scheme must be much smaller than the wavelength for quantitative signal detection. In their scheme [15,16], nanoscale flow channels (15 μm long with a 0.5 μm × 0.5 μm cross-section [cf. 17]) were employed to limit the width of the statistical ${z_{\rm{p}}}$ distribution to be comparable to the wavelength. Mitra et al. [17] introduced a heterodyne scheme to the backward-scattering interferometric scheme of Ignatovich and Novotny [15] to suppress the random error of signals attributable to the statistical ${z_{\rm{p}}}$ variation of the particle trajectory within the nanochannel.

Giglio and Potenza [21] proposed a self-reference homodyne interferometric scheme to measure both the amplitude and phase shift of the forward-scattered field, which are equivalent to ${\rm{Re}}S({0^\circ } )$ and ${\rm{Im}}S({0^\circ } )$, of single particles in a fluid flow. In their proposed scheme, single particles in a fluid flow are directed into a tightly focused Gaussian beam, and the amplitude and phase shift of the forward-scattered field are derived from the interferometric modulation of the cross-sectional power density distribution of the beam in the far forward region. Potenza et al. [22] embodied for the first time the self-reference interferometric scheme as a practical method for single particle measurement through formulations of theoretical signal models and experimental verification, and they designated it as the Single Particle Extinction and Scattering (SPES) method. Moteki [23] developed a modified version of the SPES instrument and an inverse model for unknown particle characterization using complex $S({0^\circ } )$ data. From a practical point of view, the self-reference interferometric scheme [21] has several remarkable advantages over the other interferometric schemes for inline scattered-field detection proposed so far [1117]:

  • 1. Interferometric signals in the self-reference scheme are nearly insensitive to vibrations and temperature changes in the environment thanks to the perfect common-path configuration of the reference and signal beams [21,22].
  • 2. The complex S measurements in the forward direction $\theta \approx 0^\circ $ can be advantageous for robust inference of the volume and complex refractive index of nonspherical particles because the orientation-dependent S variation is smaller at $\theta \approx 0^\circ $ than at any other value of $\theta $ [cf. 2,3,23].
  • 3. The interferometric signal from the self-reference homodyne scheme is less prone to errors caused by the ${z_{\rm{p}}}$ distribution of the particle trajectory as compared to conventional homodyne schemes for detecting the backward-scattered field.
  • 4. Thanks to the third advantage, the self-reference scheme is free from the requirement of using nanoscale flow channels that might induce near-field electromagnetic interactions between the scattering particle and the channel walls [cf. 2426].
  • 5. Thanks to the third advantage, the self-reference scheme increases our freedom in choosing the flow channel thickness that limits the upper limit of the target ${d_{\rm{p}}}$ range as well as the depth of the sensing volume.

Despite the importance of the third, fourth, and fifth advantages, the effects of the ${z_{\rm{p}}}$ distribution of the particle trajectory on the measurement results have neither been mentioned nor investigated in previous studies of the self-reference interferometric scheme [2123].

In this study, we update the theoretical framework of the self-reference interferometric scheme to be able to understand and predict the effects of the ${z_{\rm{p}}}$ distribution of the particle trajectory and other instrument parameters on the quality of the derived complex $S({0^\circ } )$ data. We also provide a theoretical tool to evaluate the applicability of plane-wave scattering theory in the self-reference interferometric scheme that actually uses a Gaussian beam to excite the scattered field from a particle at an arbitrary location in the beam.

In section 2, we develop a theoretical model of the signal waveforms and then provide a method to simulate a statistical distribution of waveform amplitudes that can be quantitatively compared with experimental data. In section 3, we discuss the major effects of the ${z_{\rm{p}}}$ distribution of the particle trajectory and other instrument parameters on measurement data. An inversion algorithm to derive complex $S({0^\circ } )$ values from waveform amplitude data is also presented in this section. In section 4, we discuss the applicability conditions of the proposed method. In section 5, we summarize our findings and suggest a protocol for quality-controlled $S({0^\circ } )$ measurements that uses the self-reference interferometric scheme. The proposed protocol is named Complex Amplitude Sensing version 1 (CAS-v1). This designation is intended to avoid confusion between the complex $S({0^\circ } )$ datasets derived from our CAS-v1 protocol and those from other procedures.

2. Theory

Successful predictions of the interferometric signals of the self-reference scheme under conditions of arbitrary ${z_{\rm{p}}}$ require a thorough reformulation of the SPES signal model [22,23] so as to take into account the higher order differences in both wavefront curvature and wavefront phase between the scattered field and incident beam field. Firstly, we derive an incident field model with a flow cell present (subsection 2.1). Next, a scattered field model is derived for each of the plane wave scattering theory and Gaussian beam scattering theory with the same order of approximation as the incident field model (subsection 2.2). Then, we combine these results to formulate the interference power density at an observation plane using a useful set of nondimensional parameters (subsection 2.3). Finally, a new model of signal waveforms is derived by analytically integrating the interference power density over the detector surface (subsection 2.4).

2.1 Incident beam field

In the optical system of the self-reference interferometer (Fig. 1), a focused Gaussian beam of TEM00 mode with vacuum wavelength $\lambda $ and waist spot size ${\omega _0}$ propagates in the + z direction in air, possibly in the presence of a glass flow cell. The trajectories of the individual particles traveling across the beam are assumed to be parallel to the x-axis. The origin of our laboratory coordinate system $O({xyz} )$ is defined to be the beam waist center location with the flow cell absent. The beam is incident to the surface of a multi-element or position-sensitive photodetector that monitors the power density distribution over the beam cross-section at $z = {z_{{\rm{pd}}}}$. Paraxial theory with scalar-field approximation is used in our description of the beam field assuming that the far-field vergence angle ${\theta _{{\rm{FF}}}} \equiv \lambda /({{n_{\rm{a}}}\pi {\omega_0}} )$ is less than ∼0.1 radian, where ${n_{\rm{a}}}$ is the refractive index of air. We assume here that the flow cell system consists of a pair of identical glass plates (each with thickness ${l_{\rm{g}}}$ and refractive index ${n_{\rm{g}}}$) and a plane parallel flow channel filled with a working fluid (with half-thickness ${l_{\rm{w}}}$ and refractive index ${n_{\rm{w}}}$). The thickness of the individual homogeneous layers in the flow cell system is assumed to be much greater than the wavelength (${l_{\rm{g}}},{{\;}}{l_{\rm{w}}} > > {{\;}}\lambda $). This condition is required for accurate prediction of the electric field within each layer using theoretical methods within the framework of geometric optics approximation. We assume that the flow cell position ${z_{\rm{f}}}$, which is defined as the center position of the flow channel (Fig. 1), is adjustable by an operator using a z-translation stage and that it has been coarsely adjusted such that the beam waist position ${z_0}$ is located within the flow channel. According to the ray-matrix method for Gaussian beam propagation [e.g., 27,28], the beam waist position ${z_0}$ is dependent on the flow cell position ${z_{\rm{f}}}$ through the relation ${z_0} ={-} ({{n_{\rm{w}}}/{n_{\rm{a}}} - 1} ){z_{\rm{f}}} + ({{n_{\rm{w}}}/{n_{\rm{a}}} - {n_{\rm{w}}}/{n_g}} ){l_{\rm{g}}} + ({{n_{\rm{w}}}/{n_{\rm{a}}} - 1} ){l_{\rm{w}}}.$ The beam waist position ${z_0}$ shifts with the flow cell position ${z_{\rm{f}}}$, which can be finely adjusted through a motor-driven z-translation stage on which the flow cell system is mounted. The range of ${z_{\rm{f}}}$ must be

$$\begin{array}{c}{\left( {1 - \frac{{{n_{\rm{a}}}}}{{{n_{\rm{g}}}}}} \right){l_{\rm{g}}} + \left( {1 - \frac{{2{n_{\rm{a}}}}}{{{n_{\rm{w}}}}}} \right){l_{\rm{w}}} \le {z_{\rm{f}}} \le \left( {1 - \frac{{{n_{\rm{a}}}}}{{{n_{\rm{g}}}}}} \right){l_{\rm{g}}} + {l_{\rm{w}}}} \end{array}$$
to ensure that ${z_0}$ is located within the flow channel. We define the optimal value of ${z_{\rm{f}}}$ as
$$\begin{array}{c}{{z_{{\rm{fo}}}} \equiv \left( {1 - \frac{{{n_{\rm{a}}}}}{{{n_{\rm{g}}}}}} \right){l_{\rm{g}}} + \left( {1 - \frac{{{n_{\rm{a}}}}}{{{n_{\rm{w}}}}}} \right){l_{\rm{w}}}} \end{array}$$
so that the beam waist is located at the center of the flow channel. It is not trivial to adjust the flow cell position ${z_{\rm{f}}}$ to be sufficiently close to the optimal position ${z_{{\rm{fo}}}}$ unless an observable indicator of ${z_{\rm{f}}} - {z_{{\rm{fo}}}}$ is available. Such an indicator will be introduced in section 3.3.

The electric field of the Gaussian beam at an arbitrary position on the surface of the detector ${\mathbf{r}} = {{\mathbf{r}}_{{\rm{pd}}}} \equiv ({x,y,{z_{{\rm{pd}}}}} )$ is given by

$$\begin{array}{c}{{E_{{\rm{inc}}}}({{{\mathbf{r}}_{{\rm{pd}}}}} )= {E_0}\frac{{{\omega _0}}}{{\omega ({{z_{{\rm{pd}}}}} )}}\exp\left[ { - \frac{{{x^2} + {y^2}}}{{{\omega^2}({{z_{{\rm{pd}}}}} )}}} \right]\exp\left[ {i{k_{\rm{a}}}\frac{{{x^2} + {y^2}}}{{2{R_{{\rm{beam}}}}({{z_{{\rm{pd}}}}} )}} + i\varphi ({{z_{{\rm{pd}}}},{z_0}} )- i\psi ({{z_{{\rm{pd}}}},{z_0}} )} \right],}\end{array}$$
where ${E_0}$ is the field amplitude at the center of the beam waist, $\omega (z )$ is the z-dependent spot size, ${R_{{\rm{beam}}}}(z )$ is the z-dependent radius of curvature of the wavefront, ${k_{\rm{a}}}$ is the wavenumber in air, $\varphi ({{z_{{\rm{pd}}}},{z_0}} )$ is the plane-wave phase shift during propagation through a layered medium from $z = {z_0}$ to $z = {z_{{\rm{pd}}}}$, and $\psi ({{z_{{\rm{pd}}}},{z_0}} )$ is the Gouy phase shift during propagation through that medium from $z = {z_0}$ to $z = {z_{{\rm{pd}}}}$. Time dependence ${e^{ - i\omega t}}$ is assumed for all fields. According to the ray-matrix method for Gaussian beam propagation [e.g., 27], ${R_{{\rm{beam}}}}({{z_{{\rm{pd}}}}} )$ and $\omega ({{z_{{\rm{pd}}}}} )$ are respectively given by ${R_{{\rm{beam}}}}({{z_{{\rm{pd}}}}} )= {z_{{\rm{pd\ast }}}} + z_{{\rm{Ra}}}^2/{z_{{\rm{pd\ast }}}}$ and $\omega ({{z_{{\rm{pd}}}}} )= {\omega _0}{[{1 + {{({{z_{{\rm{pd\ast }}}}/{z_{{\rm{Ra}}}}{{\;}}} )}^2}} ]^{1/2}}$, in which the Rayleigh range in air ${z_{{\rm{Ra}}}} \equiv {n_{\rm{a}}}\pi \omega _0^2/\lambda $ and the corrected detector position ${z_{{\rm{pd\ast }}}} \equiv {z_{{\rm{pd}}}} - \Delta {z_{{\rm{pdb}}}}$ with $\Delta {z_{{\rm{pdb}}}} \equiv 2[{({1 - {n_{\rm{a}}}/{n_{\rm{g}}}} ){l_{\rm{g}}} + ({1 - {n_{\rm{a}}}/{n_{\rm{w}}}} ){l_{\rm{w}}}} ]$ have been used. In evaluating Eq. (3) in later sections, we will use the fact that the Gouy phase shift of a Gaussian beam through a layered medium $({m = 1,2,\ldots } )$ is given by $\psi = {\rm{ta}}{{\rm{n}}^{ - 1}}\left( {\mathop \sum \limits_m \Delta {z_m}/{z_{{\rm{R}}m}}} \right),$ where $\Delta {z_m}$ and ${z_{{\rm{R}}m}}$ are the thickness and Rayleigh range in the $m$th layer, respectively, assuming the beam waist is located within the first layer [cf. 28]. The corresponding plane wave phase shift is simply given by $\varphi = \mathop \sum \limits_m {k_m}\Delta {z_m}$, where ${k_m}$ is the wavenumber in the $m$th layer.

2.2 Scattered field

In the framework of plane wave scattering theory, the complex S is solely determined from the particle’s physical properties and is independent of the beam parameters apart from the medium’s wavelength. For this reason, plane wave scattering theory is attractive as a theoretical framework in which a general-purpose protocol for complex S measurement is constructed. The use of exact Gaussian beam scattering theory is necessary to evaluate the validity of assuming plane wave scattering theory. We employ a definition of the complex scattering amplitude S with the physical dimension of [length] following [3,7,23]. In this subsection, the scattered field at $z = {z_{{\rm{pd}}}}$ is formulated within each of the frameworks of plane wave scattering theory and Gaussian beam scattering theory.

 figure: Fig. 1.

Fig. 1. Illustration of the optical system and the Gaussian beam around the particle-sensing region in the self-reference interferometric scheme. The laboratory coordinate system $O({xyz} )$ and some physical and coordinate variables are also defined in the figure. The wavefronts of the beam and scattered fields at the observation plane$\;z{{\;}} = {{\;}}{z_{{\rm{pd}}}}$ are also shown schematically.

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2.2.1 Plane wave scattering theory

Here we suppose that a scattering particle is located at ${\mathbf{r}} = {{\mathbf{r}}_{\rm{p}}} \equiv ({{x_{\rm{p}}},{y_{\rm{p}}},{z_{\rm{p}}}} )$ within a flow channel and is illuminated by a Gaussian beam (cf. Figure 1). As long as the maximum dimension of a scattering particle is sufficiently small compared to both the radius of the wavefront curvature ${R_{{\rm{beam}}}}({{z_{\rm{p}}}} ){{\;}}$and the spot size $\omega ({{z_{\rm{p}}}} )$ of the beam, we can treat the problem as if the scattering particle is excited by a plane wave localized around ${r_p}$. A simple geometric analysis shows that the angle $\theta $ subtended by the localized plane wave vector and the beam axis is at most ${\theta _{FF}}/\sqrt 2 $, which is less than ∼0.1/$\sqrt 2 $, within the region $\sqrt {{x^2} + {y^2}} \le {{\;}}\omega (z )$ for any${{\;}}z$. Here we denote S within the framework of plane wave scattering theory as ${S^{({{\rm{PW}}} )}}$. Because ${\theta _{FF}}$ is assumed to be less than ∼0.1 radian, we approximate ${S^{({{\rm{PW}}} )}}(\theta )$ by ${S^{({{\rm{PW}}} )}}({0^\circ } )$ and hereafter denote it as ${S^{({{\rm{PW}}} )}}$. This approximation is accurate unless the particle size is much larger than the wavelength.

From the above considerations, the scattered field at ${\mathbf{r}} = {{\mathbf{r}}_{{\rm{pd}}}}$ is given by

$$\begin{array}{c}{{E_{{\rm{sca}}}}({{{\mathbf{r}}_{{\rm{pd}}}}} )= \frac{{{\rm{exp}}[{i\varphi ({{{\mathbf{r}}_{{\rm{pd}}}},{{\mathbf{r}}_{\rm{p}}}} )} ]}}{{{R_{{\rm{sca}}}}({{z_{{\rm{pd}}}}} )}}{S^{({{\rm{PW}}} )}}{E_{{\rm{inc}}}}({{{\mathbf{r}}_{\rm{p}}}} ),\;} \end{array}$$
where ${R_{{\rm{sca}}}}({\rm{z}} )$ is the z-dependent radius of curvature of the wavefront of the scattered field; $\varphi ({{{\mathbf{r}}_{{\rm{pd}}}},{{\mathbf{r}}_{\rm{p}}}} )$ is the phase shift of the scattered wave, which can be approximated as $\varphi ({{{\mathbf{r}}_{{\rm{pd}}}},{{\mathbf{r}}_{\rm{p}}}} )\approx \varphi ({{z_{{\rm{pd}}}},{z_{\rm{p}}}} )+ {k_a}[{{{({x - {x_{\rm{p}}}} )}^2} + {{({y - {y_{\rm{p}}}} )}^2}} ]/[{2{R_{{\rm{sca}}}}({{z_{{\rm{pd}}}}} )} ]$ and ${E_{{\rm{inc}}}}({{{\mathbf{r}}_{\rm{p}}}} )$ is the localized plane wave field incident to the scattering particle. According to the ray-matrix method for spherical wave propagation [e.g., 28], we have ${R_{{\rm{sca}}}}({{z_{{\rm{pd}}}}} )= {z_{{\rm{pd}}}} - \Delta {z_{{\rm{pds}}}}$ with $\Delta {z_{{\rm{pds}}}} \equiv ({1 - {n_{\rm{a}}}/{n_{\rm{w}}}} )({{z_{\rm{f}}} + {l_{\rm{w}}}} )+ ({1 - {n_{\rm{a}}}/{n_{\rm{g}}}} ){l_{\rm{w}}} + ({{n_{\rm{a}}}/{n_{\rm{w}}}} ){z_{\rm{p}}}.{{\;}}$The ${E_{{\rm{inc}}}}({{{\mathbf{r}}_{\rm{p}}}} )$ in Eq. (4) is explicitly given by
$${E_{{\rm{inc}}}}({{{\mathbf{r}}_{\rm{p}}}} )= {E_0}\frac{{{\omega _0}}}{{\omega ({{z_{\rm{p}}}} )}}{\rm{exp}}\left[ { - \frac{{x_{\rm{p}}^2 + y_{\rm{p}}^2}}{{{\omega^2}({{z_{\rm{p}}}} )}}} \right]{\rm{exp}}\left[ {i{k_{\rm{w}}}\frac{{x_{\rm{p}}^2 + y_{\rm{p}}^2}}{{2{R_{{\rm{beam}}}}({{z_{\rm{p}}}} )}} + i\varphi ({{z_{\rm{p}}},{z_0}} )- i\psi ({{z_{\rm{p}}},{z_0}} )} \right]. $$
where the radius of curvature of the wavefront ${R_{{\rm{beam}}}}({Z_p})$. and the spot size${{\;}}\omega ({{z_{\rm{p}}}} )$ are respectively given by ${R_{{\rm{beam}}}}({{z_{\rm{p}}}} )= {z_{{\rm{p\ast }}}} + z_{{\rm{Rw}}}^2/{z_{{\rm{p\ast }}}}$ and $\omega ({{z_{\rm{p}}}} )= {\omega _0}{[{1 + {{({{z_{{\rm{p\ast }}}}/{z_{{\rm{Rw}}}}{{\;}}} )}^2}} ]^{1/2}}$, in which ${z_{{\rm{Rw}}}}$ is the Rayleigh range in the working fluid, ${z_{{\rm{Rw}}}} \equiv {n_{\rm{w}}}\pi \omega _0^2/\lambda $ and ${z_{{\rm{p\ast }}}} \equiv {z_{\rm{p}}} - {z_0}$ is the axial position of the scattering particle relative to the beam waist position. A rigorous solution of the scattering of a plane wave by a spherical, homogeneous, isotropic particle is given by Lorenz–Mie Theory (LMT). We computed theoretical ${S^{({{\rm{PW}}} )}}$ values for single spherical particles using custom MATLAB code that implements published numerical algorithms for LMT [2,7].

2.2.2 Gaussian beam scattering theory

A rigorous description of the scattering of a Gaussian beam by a spherical, homogeneous, isotropic particle is given by the Generalized Lorenz–Mie Theory (GLMT) [29]. The GLMT was formulated using a coordinate system centered at the particle’s location ${{\mathbf{r}}_{\rm{p}}}$ [30]. For this reason, the complex S in the GLMT, which is denoted by ${S^{({{\rm{GB}}} )}}$ hereafter, takes into account the amplitude and phase shift of the incident field at ${{\mathbf{r}}_{\rm{p}}}$ relative to those at the center of the beam waist in addition to the particle’s physical properties. As a result, the formula for ${E_{{\rm{sca}}}}({{{\mathbf{r}}_{{\rm{pd}}}}} )$ corresponding to Eq. (4) is given by

$$\begin{array}{c}{{E_{{\rm{sca}}}}({{{\mathbf{r}}_{{\rm{pd}}}}} )= \frac{{{\rm{exp}}[{i\varphi ({{{\mathbf{r}}_{{\rm{pd}}}},{{\mathbf{r}}_{\rm{p}}}} )} ]}}{{{R_{{\rm{sca}}}}({{z_{{\rm{pd}}}}} )}}{S^{({{\rm{GB}}} )}}({{{\mathbf{r}}_{\rm{p}}}} ){E_0}\;,}\end{array}$$
in which the ${S^{({{\rm{GB}}} )}}({{{\mathbf{r}}_{\rm{p}}}} )$ parameter implicitly reflects the ${{\mathbf{r}}_{\rm{p}}}$-dependence of the local excitation field ${E_{{\rm{inc}}}}({{{\mathbf{r}}_{\rm{p}}}} )$ incident to the scattering particle. Theoretical${{\;}}{S^{({{\rm{GB}}} )}}({{{\mathbf{r}}_{\rm{p}}}} )$ values for single spherical particles are computed using custom MATLAB code that implements published numerical algorithms for calculating beam shape coefficients [30,31] according to a localized model for the ${L^ - }$-order Gaussian beam [32,33].

2.3 Signal power density

In this subsection, we develop the set of formulae needed for computing the optical power density [Wm−2] at ${{\mathbf{r}}_{{\rm{pd}}}}$:

$$\begin{array}{c}{{{|{{E_{{\rm{inc}}}}({{{\mathbf{r}}_{{\rm{pd}}}}} )+ {E_{{\rm{sca}}}}({{{\mathbf{r}}_{{\rm{pd}}}}} )} |}^2} = {{|{{E_{{\rm{inc}}}}({{{\mathbf{r}}_{{\rm{pd}}}}} )} |}^2} + {{|{{E_{{\rm{sca}}}}({{{\mathbf{r}}_{{\rm{pd}}}}} )} |}^2} + 2{\rm{Re}}[{{E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )} ],} \end{array}$$
where ${|{{E_{{\rm{inc}}}}} |^2}$, ${|{{E_{{\rm{sca}}}}} |^2},$ and $2{\rm{Re}}({{E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}} )$ are the unperturbed beam power density, scattered power density, and interference power density, respectively. For convenience, we use a set of scaled coordinates $({\xi ,\eta ,\zeta } )$ for the particle’s 3D position in a beam
$$\begin{array}{c}{({\xi ,\eta ,\zeta } )\equiv \left( {\frac{{{x_{\rm{p}}}}}{{\omega ({{z_{\rm{p}}}} )}},\frac{{{y_{\rm{p}}}}}{{\omega ({{z_{\rm{p}}}} )}},\frac{{{z_{{\rm{p\ast }}}}}}{{{z_{{\rm{Rw}}}}}}} \right).}\end{array}$$

We also use a set of scaled coordinates $({\tilde x,\tilde y} )\equiv ({x/\omega ({{z_{{\rm{pd}}}}} ),y/\omega ({{z_{{\rm{pd}}}}} )} )$ for the 2D position on the observation plane at $z = {z_{{\rm{pd}}}}$.

2.3.1 Plane wave scattering theory

Here we derive detailed expressions of ${|{{E_{{\rm{sca}}}}({{{\mathbf{r}}_{{\rm{pd}}}}} )} |^2}$ and ${E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )$ for plane wave scattering theory. We assume that the angular dependence of the scattered power density ${|{{E_{{\rm{sca}}}}({{{\mathbf{r}}_{{\rm{pd}}}}} )} |^2}$ can be ignored within the solid angle subtended by the surface of the photodetector. From Eqs. (4), (5), and (8), we obtain

$$\begin{array}{c}{{{|{{E_{{\rm{sca}}}}({{{\textbf{r}}_{{\rm{pd}}}}} )} |}^2} = \frac{1}{{z_{{\rm{pd}}}^2}}{{|{{S^{({{\rm{PW}}} )}}} |}^2}\left( {\frac{1}{{\omega_0^2}}\frac{{2{P_{{\rm{inc}}}}}}{\pi }} \right)\frac{1}{{1 + {\zeta ^2}}}\exp[{ - 2({{\xi^2} + {\eta^2}} )}} ],\end{array}$$
where ${P_{{\rm{inc}}}} = \pi {({{\omega_0}{E_0}} )^2}/2$ is the total power [W] of the Gaussian beam and an approximation $R_{{\rm{sca}}}^2({{z_{{\rm{pd}}}}} )\approx z_{{\rm{pd}}}^2$ have been used. From Eqs. (3)–(5) we can write ${E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )$ as
$$\begin{aligned}{E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )&= E_0^2\left[ {\frac{{\omega_0^2}}{{\omega ({{z_{{\rm{pd}}}}} )\omega ({{z_{\rm{p}}}} )}}} \right]\exp\left[ { - \frac{{{x^2} + {y^2}}}{{{\omega^2}({{z_{{\rm{pd}}}}} )}}} \right]\exp[{ - ({{\xi^2} + {\eta^2}} )} ]\frac{{{S^{({{\rm{PW}}} )}}}}{{{R_{{\rm{sca}}}}({{z_{{\rm{pd}}}}} )}}\\&\quad\times {\rm{exp}}\left\{ {i{k_{\rm{a}}}\left[ {\frac{{{{({x - {x_{\rm{p}}}} )}^2} + {{({y - {y_{\rm{p}}}} )}^2}}}{{2{R_{{\rm{sca}}}}({{z_{{\rm{pd}}}}} )}} - \frac{{{x^2} + {y^2}}}{{2{R_{{\rm{beam}}}}({{z_{{\rm{pd}}}}} )}}} \right]} \right\}{\rm{exp}}\left[ {i{k_{\rm{w}}}\frac{{x_{\rm{p}}^2 + y_{\rm{p}}^2}}{{2{R_{{\rm{beam}}}}({{z_{\rm{p}}}} )}}} \right]\\&\quad\times {\rm{exp}}\{{i[{\varphi ({{z_{{\rm{pd}}}},{z_{\rm{p}}}} )+ \varphi ({{z_{\rm{p}}},{z_0}} )- \varphi ({{z_{{\rm{pd}}}},{z_0}} )- \psi ({{z_{\rm{p}}},{z_0}} )+ \psi ({{z_{{\rm{pd}}}},{z_0}} )} ]} \},\end{aligned}$$
in which the three $\varphi $-terms cancel out. The two $\psi $-terms in Eq. (10) are given by $\psi ({{z_{\rm{p}}},{z_0}} )= {\rm{ta}}{{\rm{n}}^{ - 1}}\zeta $ and $\psi ({{z_{{\rm{pd}}}},{z_0}} )\approx \frac{\pi }{2} - {z_{{\rm{Ra}}}}/{z_{{\rm{pd}}}}$. Radii of curvature of scattered wavefront and beam wavefront are respectively approximated as$1/{R_{{\rm{sca}}}} \approx ({1 + \Delta {z_{{\rm{pds}}}}/{z_{{\rm{pd}}}}} )/{z_{{\rm{pd}}}}$and$1/{R_{{\rm{beam}}}} \approx ({1 + \Delta {z_{{\rm{pdb}}}}/{z_{{\rm{pd}}}} - {{({{z_{{\rm{Ra}}}}/{z_{{\rm{pd}}}}} )}^2}} )/{z_{{\rm{pd}}}}$ by neglecting the second and higher order terms of $\Delta {z_{{\rm{pds}}}}/{z_{{\rm{pd}}}}$ and $\Delta {z_{{\rm{pdb}}}}/{z_{{\rm{pd}}}}$. By using these approximation formulae and exact relationships $\Delta {z_{{\rm{pds}}}} - \Delta {z_{{\rm{pdb}}}} = ({{n_{\rm{a}}}/{n_{\rm{w}}}} ){z_{{\rm{p\ast }}}}$ and ${k_{\rm{w}}}({x_{\rm{p}}^2 + y_{\rm{p}}^2} )/[{2{R_{{\rm{beam}}}}({{z_{\rm{p}}}} )} ]= \zeta ({{\xi^2} + {\eta^2}} )$, the Eq. (10) is reduced to
$$\begin{aligned}{E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )&\approx \;iE_0^2\frac{{{z_{{\rm{Ra}}}}}}{{z_{{\rm{pd}}}^2}}{({1 + {\zeta^2}} )^{ - 1/2}}{S^{({{\rm{PW}}} )}}\exp[{ - ({{\xi^2} + {\eta^2}} )} ]\exp[{i\zeta ({{\xi^2} + {\eta^2}} )} ]\\ &\quad\times {\rm{exp}}({ - i\delta } ){\rm{exp}}[{i\delta ({1 + {\zeta^2}} )({{\xi^2} + {\eta^2}} )} ]{\rm{exp}}[{i({\zeta ({{{\tilde x}^2} + {{\tilde y}^2}} )- {\rm{ta}}{{\rm{n}}^{ - 1}}\zeta } )} ]\\&\quad\times {\rm{exp}}[{i\delta ({{{\tilde x}^2} + {{\tilde y}^2}} )} ]{\rm{exp}}\{{ - [{{{\tilde x}^2} + {{\tilde y}^2} + i2({1 + \varepsilon } ){{({1 + {\zeta^2}} )}^{1/2}}{{\;}}({\xi \tilde x + \eta \tilde y} )} ]} \},{{\;}}\end{aligned}$$
in which we have introduced nondimensional parameters $\varepsilon \equiv \Delta {z_{{\rm{pds}}}}/{z_{{\rm{pd}}}}$ and $\delta \equiv {z_{{\rm{Ra}}}}/{z_{{\rm{pd}}}}$ and used approximations ${R_{{\rm{sca}}}}({{z_{{\rm{pd}}}}} )\approx {z_{{\rm{pd}}}}$ and $\omega ({{z_{{\rm{pd}}}}} )\approx {\omega _0}{z_{{\rm{pd}}}}/{z_{{\rm{Ra}}}}$ for the phase-insensitive amplitude. We have ignored the second and higher order terms of $\varepsilon $ and $\delta $ in Eq. (11). The incorporations of both $\varepsilon $ and $\delta $ up to the first order in the ${E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )$ model are needed to make the approximation order with respect to every nondimensional parameter ($\varepsilon $, $\delta $, $\xi $, $\eta $, $\zeta $, $\tilde x$, $\tilde y$) consistent under their value ranges in our experiments.

We can evaluate the interference power density $2{\rm{Re}}[{{E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )} ]$ as a function of particle’ 3D position $({\xi ,\eta ,\zeta } )$ using Eq. (11). The expressions of ${|{{E_{{\rm{sca}}}}({{{\mathbf{r}}_{{\rm{pd}}}}} )} |^2}$ and ${E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )$ obtained here are equivalent to the corresponding expressions in previous SPES studies [22,23] when $\varepsilon = \delta = {{\;}}\zeta = 0$.

2.3.2 Gaussian beam scattering theory

Next we derive detailed expressions for ${|{{E_{{\rm{sca}}}}({{{\mathbf{r}}_{{\rm{pd}}}}} )} |^2}$ and ${E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )$ for the case of Gaussian beam scattering theory. As in the case of plane wave scattering theory, we assume that the scattered power density ${|{{E_{{\rm{sca}}}}({{{\mathbf{r}}_{{\rm{pd}}}}} )} |^2}$ is uniform over the photoelectric surface of the detector. From Eq. (6), we have

$$\begin{array}{c}{{{|{{E_{{\rm{sca}}}}({{{\mathbf{r}}_{{\rm{pd}}}}} )} |}^2} = \frac{1}{{R_{{\rm{sca}}}^2({{z_{{\rm{pd}}}}} )}}{{|{{S^{({{\rm{GB}}} )}}} |}^2}\left( {\frac{1}{{\omega_0^2}}\frac{{2{P_{{\rm{inc}}}}}}{\pi }} \right).}\end{array}$$

Using a similar procedure as the derivation of Eq. (11), we obtain from Eqs. (3) and (6) a succinct form of ${E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )$ formula:

$$\begin{aligned}{E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )&\approx \approx \;iE_0^2\frac{{{z_{{\rm{Ra}}}}}}{{z_{{\rm{pd}}}^2}}{S^{({{\rm{GB}}} )}}\exp({ - i\beta \zeta } )\exp({ - i\delta } )\exp[{i\delta ({1 + {\zeta^2}} )({{\xi^2} + {\eta^2}} )} ]\\&\quad\times {\rm{exp}}[{i\zeta ({{{\tilde x}^2} + {{\tilde y}^2}} )} ]{\rm{exp}}[{i\delta ({{{\tilde x}^2} + {{\tilde y}^2}} )} ]\\&\quad\times {\rm{exp}}\{{ - [{{{\tilde x}^2} + {{\tilde y}^2} + i2({1 + \varepsilon } ){{({1 + {\zeta^2}} )}^{1/2}}{{\;}}({\xi \tilde x + \eta \tilde y} )} ]} \}.{{\;}}\end{aligned}$$
where we have introduced the non-dimensional instrument parameter $\beta \equiv {k_{\rm{w}}}{z_{{\rm{Rw}}}}$.

The numerical results of these formulae for ${|{{E_{{\rm{sca}}}}({{{\mathbf{r}}_{{\rm{pd}}}}} )} |^2}$ and ${E_{{\rm{sca}}}}E_{{\rm{inc}}}^\ast ({{{\mathbf{r}}_{{\rm{pd}}}}} )$ approach to those for plane wave scattering theory when ${\omega _0}/{d_{\rm{p}}} \to \infty $. For actual ${\omega _0}/{d_{\rm{p}}}$ values, the accuracy of the plane wave formulae Eqs. (9) and (11) can be evaluated through comparisons with these Gaussian beam formulae.

2.4 Signal waveforms

The observed signal power [W] is an integration of the signal power density ${|{{E_{{\rm{sca}}}}({{{\mathbf{r}}_{{\rm{pd}}}}} )} |^2} + 2{\rm{Re}}[{{E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )} ]$ over the surface of the photodetector. A signal photodetector for self-reference interferometry should have at least two photosensitive segments along the particle moving direction (x-axis) to measure both asymmetric and symmetric components of the interference power density distribution with respect to optical axis that are required to determine both real and imaginary parts of $S({0^\circ } )$ [2123]. In addition, it is desirable to place at least two photosensitive segments along y-axis to estimate the lateral position of particle trajectory [2123], which is one of the mandatory parameters for single particle $S({0^\circ } )$ measurements whenever the lateral dimension of sample stream is greater than the beam spot size. Other requirements on the signal photodetector are superior uniformity of responsivity and small intersegment gaps over the photoelectric surface for photometric accuracy. To satisfy all these requirements, we employ a circular-shaped quadrant photodiode (QPD) of radius$\;a$ with negligibly small intersegment gap and placed it at $z = {z_{{\rm{pd}}}}$ with a particular orientation in the coordinate system $O({xyz} )$ (Fig. 2).

 figure: Fig. 2.

Fig. 2. Illustration of the photoelectric surface of a circular quadrant photodiode and the Gaussian beam spot at $z = {z_{{\rm{pd}}}}$ in the laboratory coordinate system $O({xyz} )$.

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In this QPD orientation, the two orthogonal, intersegment gaps in the QPD surface are oriented at 45° with respect to the x- and y-axes. For later convenience, we define the beam filling factor f as

$$\begin{array}{c}{f \equiv \frac{{\omega ({{z_{{\rm{pd}}}}} )}}{a} \approx \frac{{{\omega _0}{z_{{\rm{pd}}}}}}{{a{z_{{\rm{Ra}}}}}},}\end{array}$$
which was set to $f \approx 0.5$ in our experiments.

Because we can assume that $|{{E_{{\rm{sca}}}}({{{\mathbf{r}}_{{\rm{pd}}}}} )} |$ is uniform over the solid angle subtended by the QPD surface thanks to the small value of ${\theta _{{\rm{FF}}}}/f$, the scattered signal power density integrated over the whole QPD surface ${P_{{\rm{sca}}}}({{\rm{Tot}}} )$ is given by

$$\begin{array}{c}{{P_{{\rm{sca}}}}({{\rm{Tot}}} )\approx {{|{{E_{{\rm{sca}}}}({{{\mathbf{r}}_{{\rm{pd}}}}} )} |}^2}\pi {a^2}.\;}\end{array}$$

In contrast to the scattered power density ${|{{E_{{\rm{sca}}}}({{{\mathbf{r}}_{{\rm{pd}}}}} )} |^2}$, the surface integration of the interference power density $2{\rm{Re}}[{{E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )} ]$ requires great care because of its non-uniformity over the QPD surface. In the QPD configuration of Fig. 2, it is convenient to perform the surface integrals with respect to the polar coordinates ($\tilde r,\phi $), which are connected with the Cartesian coordinates ($\tilde x,\tilde y$) through the relation $({\tilde r{\rm{cos}}\phi ,\tilde r{\rm{sin}}\phi } )= ({\tilde x,\tilde y} )$. For evaluating the observation signals obtained by the QPD configuration of Fig. 2, we define the following four quantities: the surface integrals of $2{\rm{Re}}[{{E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )} ]$ over the whole QPD surface

$$\begin{array}{c}{{P_{{\rm{ext}}}}({{\rm{Tot}}} )= \mathop \int \limits_0^{2\pi } \left\{ {\mathop \int \limits_0^\infty 2{\rm{Re}}[{{E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )} ]{\omega^2}({{z_{{\rm{pd}}}}} )\tilde rd\tilde r} \right\}d\phi ,}\end{array}$$
the difference between the surface integrals of the two photodiode segments along the x-axis (A and C)
$$\begin{array}{c}{{P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )= - \mathop \int \limits_{ - \pi /4}^{\pi /4} \left\{ {\mathop \int \limits_{ - \infty }^\infty 2{\rm{Re}}[{{E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )} ]{\omega^2}({{z_{{\rm{pd}}}}} )\tilde rd\tilde r} \right\}d\phi ,\;}\end{array}$$
the difference between the surface integrals of the two photodiode segments along the y-axis (B and D)
$$\begin{array}{c}{{P_{{\rm{ext}}}}({{\rm{B}} - {\rm{D}}} )= - \mathop \int \limits_{\pi /4}^{3\pi /4} \left\{ {\mathop \int \limits_{ - \infty }^\infty 2{\rm{Re}}[{{E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )} ]{\omega^2}({{z_{{\rm{pd}}}}} )\tilde rd\tilde r} \right\}d\phi ,{{\;}}}\end{array}$$
and the summation of Eqs. (15) and (16)
$$\begin{array}{c}{{P_{{\rm{extsc}}}}({{\rm{Tot}}} )\equiv {P_{{\rm{ext}}}}({{\rm{Tot}}} )+ {P_{{\rm{sca}}}}({{\rm{Tot}}} )\;.\;}\end{array}$$

We can quantify the beam-power normalized signals ${P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )/{P_{{\rm{inc}}}}$, ${P_{{\rm{ext}}}}({{\rm{B}} - {\rm{D}}} )/{P_{{\rm{inc}}}}$, and ${P_{{\rm{extsc}}}}({{\rm{Tot}}} )/{P_{{\rm{inc}}}}$ by using the photocurrent signals from the QPD, as described in [23].

We define the beam-power normalized signals versus particle coordinate $\xi $ along each particle trajectory as the “signal waveforms”. From Eqs. (16)–(18), analytical formulae of the signal waveforms are derived as detailed in Appendices A and B. In the derivations, the assumption of $\eta = 0$ was employed to perform the surface integrals analytically. We show in later sections that the assumption of $\eta = 0$ does not introduce appreciable errors in our final results.

3. Experiment and theoretical interpretation

In this section, we provide experimental verification of the theory developed in section 2. After describing the experimental apparatus and samples, we demonstrate a practical method for optimizing the flow cell position according to the signal waveforms. Then we discuss the factors controlling the accuracy and resolution of the S measurement under the optimized flow cell position. After explaining the inversion algorithm for determining S from observed signal waveforms, experimental S data for sample particle size standards are compared with theoretical S values.

3.1 Apparatus and samples

The entire optical system of our experimental apparatus is shown schematically in Fig. 3. A red He-Ne laser with λ = 0.6328 μm, ∼2 mW beam power, and linear polarization (Thorlabs, HNL020LB) was used as the source of Gaussian beam. The TEM00 mode purity is at least 95% according to the manufacturer’s product specifications. An optical isolator was used to prevent laser instability due to back reflections. A polarization beam splitter (PBS) with a rotatable half waveplate (1/2 WP) was used to adjust the splitting power ratio between the signal beam and laser-power reference beam. The reference photodiode signal was subtracted from the ${P_{{\rm{extsc}}}}({{\rm{Tot}}} )/{P_{{\rm{inc}}}}$ signal to suppress the influence of laser-power noise from the ${P_{{\rm{extsc}}}}({{\rm{Tot}}} )/{P_{{\rm{inc}}}}$ signal waveform. To emphasize the importance of this laser noise cancelling operation in the experiments, the ${\rm{Tot}}$ signature for the signal is rendered as ${\rm{T}} - {\rm{R}}$ hereafter.

 figure: Fig. 3.

Fig. 3. Schematic diagram of our experimental apparatus.

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We chose one of the three beam expanders with 2×, 10×, or 20× magnification (Thorlabs, GBE2-A, GBE10-A, and GBE20-A, respectively) depending on the desired ${\omega _0}$ value within the range ∼2–20 μm. A diffraction-limited aspheric lens with 50-mm focal length (Thorlabs, AL2550G-A) was used to form the beam focus.

A custom-made fused-quartz glass flow cell (Japan Cell) was used to direct waterborne particles into the beam focal region. The mean flow velocity $\bar v$ was maintained at ∼0.2 m s−1 as driven by a peristaltic pump with Tygon tubing. The glass wall thickness and the channel half thickness of the flow cell were ${l_{\rm{g}}}$ = 1.5 mm and ${l_{\rm{w}}}$ = 25 μm, respectively (cf. Fig. 1). The refractive indices of the glass wall material (i.e., fused quartz) and the working fluid (i.e., water) at λ = 0.6328 μm were assumed to be ${n_{\rm{g}}}$ = 1.457 [34] and ${n_{\rm{w}}}$ = 1.33154 [35], respectively.

A circular-shaped quadrant photodiode (QPD) with radius a = 5 mm and intersegment gap = 0.01 mm (PIN-SPOT-9DMI, OSI optoelectronics) was placed at $z = {z_{{\rm{pd}}}}$ with the orientation illustrated in Fig. 2 for acquiring the three independent signal waveforms ${P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )/{P_{{\rm{inc}}}}$, ${P_{{\rm{ext}}}}({{\rm{B}} - {\rm{D}}} )/{P_{{\rm{inc}}}}$, and ${P_{{\rm{extsc}}}}({{\rm{T}} - {\rm{R}}} )/{P_{{\rm{inc}}}}$ simultaneously. A custom-made analog electronic circuit with a 3-dB cutoff frequency of ∼1 MHz was used to perform differential amplification of the photocurrent signals from each photodiode segments. A digitizer with 14-bit resolution and 2.5-MHz sampling rate (NI, PCI-6133) was used to import continuously the raw waveforms to a desktop computer. Individual single-particle signal waveforms were then extracted from the raw waveforms (up to ∼100 particles s−1) according to an appropriate set of triggering criteria. For closer comparison of experiment with theory, we picked only particle detection events with $|\eta |$ < ∼0.2 in which the peak amplitude of the $|{{P_{{\rm{ext}}}}({{\rm{B}} - {\rm{D}}} )/{P_{{\rm{inc}}}}} |$ waveform was less than 0.5 times the peak amplitude of the $|{{P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )/{P_{{\rm{inc}}}}} |$ waveform. The ${P_{{\rm{ext}}}}({{\rm{B}} - {\rm{D}}} )/{P_{{\rm{inc}}}}$ signal was used only for the event-filtering operation. The particle number concentration in sample water was adjusted to be less than ∼108 particles cm−3 to reduce the occurrence of waveform overlap between two successive detection events.

We used spherical polystyrene (PS) size standards (Thermo scientific, 3000 and 4000 Series) of 12 different nominal values of ${d_{\rm{p}}}$ from a range 0.3–5.0 μm to carry out the quantitative comparison of experimental results with theoretical values. The complex refractive index of PS at λ = 0.6328 μm was assumed to be 1.5854 + 6.1764×10−7i from the experimental data of Zhang et al. [36]. Spherical silica size standards (Thermo scientific, 8000 series) of 3 different nominal ${d_{\rm{p}}}$ values from a range ∼0.5–1.6 μm were also used to show example data for a different particle refractive index. The complex refractive index of the silica spheres at λ = 0.6328 μm was assumed to be 1.457 + 0i [34].

Quantitative comparison between measured and simulated signal waveforms requires accurate knowledge of the flow cell position ${z_{\rm{f}}}$ relative to the beam waist position ${z_{\rm{o}}}$. A practical method to optimize the flow cell position according to the observed signals is elaborated in subsection 3.3. We used a stepper-motor driven linear translation stage (Thorlabs, LNR25ZFS) for changing the ${z_{\rm{f}}}$ value with sufficient accuracy and repeatability (< ∼0.5 μm).

3.2 Signal waveforms

For each of ${P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )/{P_{{\rm{inc}}}}$ and ${P_{{\rm{extsc}}}}({{\rm{T}} - {\rm{R}}} )/{P_{{\rm{inc}}}}$, we define the waveform amplitude (WFA) and the waveform width (WFW) as illustrated in Fig. 4. The two-parameter pair (WFA, WFW) is regarded as a set of easily extractable data features from the observed signal waveforms.

 figure: Fig. 4.

Fig. 4. Diagram illustrating the definitions of waveform amplitude (WFA) and waveform width (WFW).

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The parameter WFA(A−C) is defined by $[{{P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )({{\xi_ + }} )- {{\;}}{P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )({{\xi_ - }} )} ]/2{P_{{\rm{inc}}}}$ in which ${\xi _ - }$ and ${\xi _ + }$ are the extreme points of ${P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )(\xi )$ in the negative and positive $\xi $ domains, respectively. The value of the WFA(A−C) can be either positive or negative. The WFW(A−C) is the distance between the two extreme points ${\xi _ + } - {\xi _ - }$. The WFA(T−R) is defined by the depth of the waveform valley $- {P_{{\rm{extsc}}}}({{\rm{T}} - {\rm{R}}} )({\xi = 0} )/{P_{{\rm{inc}}}}$, which is always positive. The WFW(T−R) is defined by the full-width at half maximum (FWHM) of the $- {P_{{\rm{extsc}}}}({{\rm{T}} - {\rm{R}}} )(\xi )/{P_{{\rm{inc}}}}$ waveform.

The WFAs and WFWs were calculated from the observed signal waveforms of each detection event. The WFAs and WFWs for single particles are used for various analyses including data inversion to estimate the complex S value. Furthermore, we can simulate the WFAs and WFWs for single particles according to the theoretical formulae given in Appendices A and B. In theoretical methods, we designate the WFAs at $\zeta $ = 0 as “ideal” WFAs without any $\zeta $-dependent artifacts.

A quantitative comparison between theory and experiment is possible only for an ensemble of detected particles rather than for a single detected particle, owing to the stochastic particle-to-particle variations in a moving trajectory across a beam. Appendix C describes the procedure for simulating the signal waveforms for an ensemble of detected particles.

In subsections 3.3–4, we show that both WFAs and WFWs are strongly dependent on the particle’s $\zeta $ coordinate and suggest a method to adjust the flow cell position ${z_{\rm{f}}}$ to the optimal value ${z_{{\rm{fo}}}}$. In section 3.5, we evaluate the validity of assuming plane wave scattering theory in our analyses and interpretations of WFA data. In section 3.6, we propose a procedure for determining the value of ${\omega _0}$ from the measured WFAs for test particle size standards. In section 3.7, we propose an inversion algorithm for estimating the ${\rm{Re}}S$ and ${\rm{Im}}S$ values from the measured WFAs.

3.3 Effects of the particle’s $\zeta $ coordinate on the WFA

Here, we demonstrate the significance of the effects of the $\zeta $ coordinate of the particle trajectory on the WFAs and WFWs, and clarify the necessity of optimizing the flow cell position ${z_{\rm{f}}}$ to minimize measurement errors. In this subsection, we assume ${l_w}/{z_{{\rm{Rw}}}}$ > ∼0.1 so that the observed (WFA, WFW) values appreciably change as functions of $\zeta $ within its range $- {l_w}/{z_{{\rm{Rw}}}} < $ $\zeta < + {l_w}/{z_{{\rm{Rw}}}}$ at the optimized condition ${z_f}$ = ${z_{f{\rm{o}}}}$ and also that they are functions of ${z_{\rm{f}}}$ within its adjustable range given by Eq. (1). In order to demonstrate the concepts through exemplary results, we present (WFA, WFW) datasets for PS samples with ${d_{\rm{p}}} = $ 0.803 μm obtained with the instrument conditions of ${\omega _0} = $ 3.34 μm and ${l_w}/{z_{{\rm{Rw}}}}$ = 0.339.

Figure 5 shows scatterplots of WFA(T−R) versus WFA(A−C) for ∼1000 particles at three different ${z_f}$ = ${z_{f{\rm{o}}}}$ values (−10, 0, and +10 μm). The shape of the cluster of the WFA data depends on the $\zeta $ coordinate of the particle trajectory, and therefore the mean and standard deviation of WFA change over the value range of $\zeta $. A deviation of ${z_{\rm{f}}}$ from the optimal ${z_{{\rm{fo}}}}$ results in a systematic deviation of the measured WFAs from the ideal WFAs because of the corresponding shift of the $\zeta $ distribution from the origin $\zeta $=0. The results imply that it is desirable to preset the flow cell position ${z_{\rm{f}}}$ such that $|{{z_{\rm{f}}} - {z_{{\rm{fo}}}}} |/{z_{{\rm{Rw}}}}$ < ∼0.1 to minimize systematic errors in the WFAs. It should be noted that, even at the optimal flow cell position ${z_{\rm{f}}} = {z_{{\rm{fo}}}}$, the mean WFAs could differ from the ideal WFAs at $\zeta $=0 because of the finite width of the $\zeta $ distribution and the non-symmetric spread of the data around the ideal WFAs (cf. Figs. 5(d)–(f)).

 figure: Fig. 5.

Fig. 5. Scatterplots of WFA(T−R) versus WFA(A−C) for ∼103 particles of PS samples with ${d_{\rm{p}}}$ = 0.803 μm obtained at ${\omega _0}$ = 3.34 μm (i.e.,$\;{l_w}/{z_{{\rm{Rw}}}}$ = 0.33). In each panel, the light blue open circle with error bars shows the mean and standard deviation of the single-particle WFA data (dots). The left, middle, and right columns show the experimental results, simulated results from plane wave scattering theory (PW), and simulated results from Gaussian beam scattering theory (GB), respectively. The upper row panels (a–c) show the results at$\;{z_{\rm{f}}} - {z_{{\rm{fo}}}}$ = −10 μm, the middle row panels (d–f) show the results at$\;{z_{\rm{f}}} - {z_{{\rm{fo}}}}$ = 0 μm, and the bottom row panels (g–i) show the results at$\;{z_{\rm{f}}} - {z_{{\rm{fo}}}}$ = +10 μm. The color scales indicate the $\zeta $ coordinate of the particle trajectory.

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Figure 6 shows scatterplots of WFW(T−R) versus WFA(T−R) for the same dataset shown in Fig. 5. In each panel of Fig. 6, WFW(T−R) varies mostly in accordance with the ${\tilde z_{\rm{p}}}$-dependent particle velocity (cf. Eqs. (C1)–(C3). On the WFA(T−R)–WFW(T−R) plane, the outline shape of the data cluster changed remarkably depending on the magnitude of $|{{z_{\rm{f}}} - {z_{{\rm{fo}}}}} |/{z_{{\rm{Rw}}}}$, with the data cluster having a “single line” shape when $|{{z_{\rm{f}}} - {z_{{\rm{fo}}}}} |/{z_{{\rm{Rw}}}}$ < ∼0.05 and a “horseshoe” shape otherwise, with the “horseshoe” broadening with increasing $|{{z_{\rm{f}}} - {z_{{\rm{fo}}}}} |/{z_{{\rm{Rw}}}}$. We found that this empirical rule was applicable to other PS sizes and other experimental conditions unless ${l_w}/{z_{{\rm{Rw}}}}$ << 1. When ${l_w}/{z_{{\rm{Rw}}}}$ > ∼0.1, the outline shape of the data cluster on the WFA(T−R)–WFW(T−R) plane is a precise indicator of $|{{z_{\rm{f}}} - {z_{{\rm{fo}}}}} |/{z_{{\rm{Rw}}}}$, which in practice is crucial for fine ${z_{\rm{f}}}$ adjustment around the optimal position ${z_{{\rm{fo}}}}$. Under the optimized ${z_{\rm{f}}}$ condition in which $|{{z_{\rm{f}}} - {z_{{\rm{fo}}}}} |/{z_{{\rm{Rw}}}} \approx $ 0, it is possible to further improve the accuracy and resolution of the WFA measurements by avoiding the larger $|\zeta |$ datapoints by thresholding the WFW(T−R) value. In our experiments, we did not impose such a WFW(T−R) threshold for the sake of simplicity.

 figure: Fig. 6.

Fig. 6. The same scatterplots as Fig. 5 but for WFW(T−R) versus WFA(T−R). The physical unit of the displayed WFW(T−R) values is 0.4 μs, which is the sampling time interval of the digitizer. The WFW(T−R) is inversely proportional to the sample flow rate, and thus its absolute value is unimportant here.

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When ${l_w}/{z_{{\rm{Rw}}}}$ << 1, the width of the $\zeta $-distribution of particles in the flow channel is too small to identify the $\zeta $-dependent variation of WFA(T−R) in the presence of other noise sources. Furthermore, WFA(T−R) does not appreciably change as a function of ${z_f}$ within the adjustable range (Eq. (1)). For this reason, once a coarse ${z_{\rm{f}}}$ adjustment has been made satisfying the condition of Eq. (1), further fine adjustment of ${z_{\rm{f}}}$ is not needed in practice.

3.4 Applicability of plane wave scattering theory

Here we test the applicability of plane wave scattering theory (PW) through comparison with exact Gaussian beam scattering theory (GB). The practical significance of this theoretical test can be realized from the fact that, under given experimental conditions and target particle size range, sufficient accuracy when using the PW assumption is implicitly assumed when we apply the data inversion algorithm designed for inferring the particle’s ${S^{({{\rm{PW}}} )}}$ from the measured WFAs.

Figure 7 displays the systematic errors of the simulated WFAs of a single PS sphere assuming PW relative to those assuming GB as a function of particle diameter ${d_p}$ for the two values of ${\omega _0}$ employed in our experiments (${\omega _0}$ = 3.34 and 17.3 μm). The figure also show the results for a single absorbing sphere (with imaginary part 0.05) for comparison. In the ${\omega _0}$ = 3.34 μm case, the systematic errors in WFAs using the PW assumption were always less than ±1% in the submicron size range ${d_{\rm{p}}}$< ∼1 μm. In the ${\omega _0}$ = 17.3 μm case, the corresponding systematic errors in WFAs were almost always less than ±1% in the particle size range of ${d_{\rm{p}}}$< ∼5 μm. In Fig. 7, the complex structures in the larger ${d_{\rm{p}}}$ regions for nonabsorbing sphere are supposed to be attributable to electromagnetic wave resonances within the sphere that depend on the incident beam shape. This hypothesis was supported by the fact that these complex features smoothed out as the imaginary part of particle’s refractive index increased.

 figure: Fig. 7.

Fig. 7. Errors in the simulated WFAs for a nonabsorbing (with refractive index of PS: 1.5854 + 6.1764×10−7i) and an absorbing (with refractive index of 1.5854 + 5×10−2i) single sphere assuming plane wave scattering theory (PW) relative to those assuming exact Gaussian beam scattering theory (GB) plotted versus particle diameter ${d_{\rm{p}}}$. Top row (a,b) and bottom row (c,d) show the results at ${\omega _0}$ = 3.34 μm and ${\omega _0}{{\;}}$ = 17.3 μm, respectively. Right column (b,d) shows the same plots as left column (a,c) but with 10 times magnification in vertical axes for visibility. The $\zeta $ coordinate of the particle trajectory was set to zero. Other input parameters were prescribed in accordance with our experimental conditions described in section 3.1.

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From these results, we assumed in our experiments that the systematic errors of using the PW assumption were almost always less than ± ∼1%, provided that the particle diameter ${d_{\rm{p}}}$ was smaller than $3{\omega _0}$. In this study, we define applicability criterion of the PW assumption as ${d_{\rm{p}}} < {{\;}}3{\omega _0}$. We denote ${S^{({{\rm{PW}}} )}}$ as S hereafter unless violating this applicability criterion of the PW assumption.

3.5 Estimating the spot size at the beam waist

An accurate prediction of the spot size at the beam waist ${\omega _0}$ is not trivial due to potential contributions from various sources of wavefront aberrations such as contributions of higher transverse modes in the output beam from the laser cavity [37]; misalignment, tilt, and surface roughness of the optical components [38]; and spherical aberration caused by focusing through a glass plate [39]. In this study, we obtained a plausible estimate of ${\omega _0}$ for each beam expander condition by numerically finding the best agreement between the simulated and experimental data for spherical PS-size standards. For this estimation, we find the minimum of the mean squared residual error (MSRE) by using the Golden section search method [cf. 40] as follows:

$$\begin{array}{c}{MSRE({{\omega_0}} )= \frac{1}{{{N_p}}}\mathop \sum \limits_{{i_p} = 1}^{{N_p}} \left\{ {{{\left[ {\frac{{{\langle\rm{WFA}}{{({{\rm{A}} - {\rm{C}}} )}_{{\rm{sim}},{i_p}}\rangle}}}{{{\langle\rm{WFA}}{{({{\rm{A}} - {\rm{C}}} )}_{{\rm{mea}},{i_p}}\rangle}}} - 1} \right]}^2} + {{\left[ {\frac{{{\langle\rm{WFA}}{{({{\rm{T}} - {\rm{R}}} )}_{{\rm{sim}},{i_p}}\rangle}}}{{{\langle\rm{WFA}}{{({{\rm{T}} - {\rm{R}}} )}_{{\rm{mea}},{i_p}}\rangle}}} - 1} \right]}^2}} \right\},}\end{array}$$
where the angle brackets denote the arithmetic mean of the simulated (subscript “sim”) or measured (subscript “mea”) single-particle datapoints. Here, we assume that the flow cell position ${z_{\rm{f}}}$ has already been optimized to be $|{{z_{\rm{f}}} - {z_{{\rm{fo}}}}} |/{z_{{\rm{Rw}}}} \approx $ 0 according to the procedure described in section 3.3. We used ∼103 datapoints for computing each arithmetic mean. Figure 8 summarizes the complete computation procedure for ${\omega _0}$ estimation.

The ${\omega _0}$ values were estimated to be 1.92, 3.34, and 17.3 μm, when we used the beam expanders with magnifications 20×, 10×, and 2×, respectively (cf. Fig. 3).

 figure: Fig. 8.

Fig. 8. Procedure for estimating ${\omega _0}$ by using the WFA data for spherical PS size standards.

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3.6 Data inversion

Here we propose an inversion algorithm to determine the complex scattering amplitude S of individual detected particles from the WFA data. We assume that the WFA data have been obtained under the optimized condition $|{{z_{\rm{f}}} - {z_{{\rm{fo}}}}} |/{z_{{\rm{Rw}}}} \approx $ 0. Because the $\zeta $ coordinates of the detected particles distribute around $\zeta \approx $ 0, the observed signal waveforms of ${P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )/{P_{{\rm{inc}}}}$ and ${P_{{\rm{extsc}}}}({{\rm{T}} - {\rm{R}}} )/{P_{{\rm{inc}}}}$ can be represented by their theoretical formulae (cf. Appendix A) at $\zeta $ = 0 as follows:

$$\begin{array}{c}{{{\left. {\frac{{{P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )}}{{{P_{{\rm{inc}}}}}}} \right|}_{\eta = 0,{{\;\;}}\zeta = 0}} \approx \frac{4}{{{z_{{\rm{Ra}}}}}}\frac{{{e^{ - \frac{3}{2}{\xi ^2}}}}}{{\sqrt {2\pi } }}\left\{ {\left[ {2\varepsilon \xi + 2\sqrt 2 ({1 - 2\varepsilon {\xi^2}} )D\left( {\frac{\xi }{{\sqrt 2 }}} \right)} \right]{\rm{Re}}S - \xi \delta {\rm{Im}}S} \right\}\;\;}\end{array}$$
and
$${{{\left. {\frac{{{P_{{\rm{extsc}}}}({{\rm{T}} - {\rm{R}}} )}}{{{P_{{\rm{inc}}}}}}} \right|}_{\eta = 0,{{\;\;}}\zeta = 0}} \approx \frac{4}{{{z_{{\rm{Ra}}}}}}{e^{ - 2{\xi ^2}}}({ - 1 + 2\varepsilon {\xi^2}} )\textrm{Im}S + \frac{2}{{z_{{\rm{Ra}}}^2}}\frac{1}{{{f^2}}}{e^{ - 2{\xi ^2}}}{{|S |}^2},\;}$$
where D in Eq. (21) is the Dawson function. The WFAs of these theoretical waveforms, Eqs. (21) and (22), are given by
$$\begin{array}{c}{WFA({{\rm{A}} - {\rm{C}}} )= \frac{4}{{{z_{{\rm{Ra}}}}}}\frac{{{e^{ - \frac{3}{2}\xi _ + ^2}}}}{{\sqrt {2\pi } }}\left\{ {\left[ {2\varepsilon {\xi_ + } + 2\sqrt 2 ({1 - 2\varepsilon \xi_ +^2} )D\left( {\frac{{{\xi_ + }}}{{\sqrt 2 }}} \right)} \right]{\rm{Re}}S - \delta {\xi_ + }{\rm{Im}}S} \right\}\;}\end{array}$$
and
$$\begin{array}{c}{WFA({{\rm{T}} - {\rm{R}}} )= \frac{4}{{{z_{{\rm{Ra}}}}}}{\textrm{Im}}S - \frac{2}{{z_{{\rm{Ra}}}^2}}\frac{1}{{{f^2}}}{{|S |}^2},}\end{array}$$
where we have employed the symmetry of the ${P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )/{P_{{\rm{inc}}}}$ waveform, and only the positive extreme point ${\xi _ + }$ was used to describe the theoretical WFA(A−C).

We propose an inversion algorithm to determine the unknown $S{{\;}}$from the waveform data by regarding Eqs. (23)–(24) as a system of nonlinear equations with respect to ${\rm{Re}}S$ and ${\rm{Im}}S$ under a given set of WFAs. We designed a Newton-type iterative algorithm [cf. 40] to solve the system of nonlinear equations such that the $S$-dependent parameter ${\xi _ + }$ was also updated in accordance with the current estimate of S in each iteration. This inversion algorithm was designed to continuously map the statistical $\zeta $ distribution of the detected particles around $\zeta \approx $ 0 into a two-dimensional distribution of the derived ${\rm{Re}}S$ and ${\rm{Im}}S$ values around their true values. The entire computational procedure is shown in Fig. 9.

 figure: Fig. 9.

Fig. 9. Data inversion algorithm for estimating ${\rm{Re}}S$ and ${\rm{Im}}S$ from the WFA data.

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3.7 Accuracy and resolution

Here we present the complex S data of the spherical PS size standards derived through the inversion algorithm and compare the results for different conditions of ${\omega _0}$ and ${d_{\rm{p}}}$. In particular, we demonstrate that the accuracy and resolution of complex S data are both predominantly controlled by the nondimensional parameter ${l_{\rm{w}}}/{z_{{\rm{Rw}}}}$ unless ${l_{\rm{w}}}/{z_{{\rm{Rw}}}}$ << 1.

Figure 10 compares the histograms of the real and imaginary parts of the complex S data of the ${d_{\rm{p}}}$ = 0.401 μm sample between the two different conditions, ${\omega _0}$ = 1.92 μm (${l_{\rm{w}}}/{z_{{\rm{Rw}}}}$ = 1.02) and ${\omega _0}$ = 3.34 μm (${l_{\rm{w}}}/{z_{{\rm{Rw}}}}$ = 0.339). Each histogram plot exhibits a skewed statistical distribution with a single mode near the true value and a longer tail toward the left (i.e., negative skewness). The negative skewness of the real- and imaginary-$\;S$ distributions causes a negative shift of their mean values from their respective true values, and it therefore induces negative systematic errors in the real- and imaginary-$\;S$ measurements. Detailed analyses of the data inversion process using the simulated noiseless WFAs revealed that the negatively skewed real- and imaginary-$\;S$ probability distributions seen in Fig. 10 were the result of the nonlinear mapping from the symmetric $\zeta $ distribution within the $\zeta $ domain of $|\zeta |\le {l_{\rm{w}}}/{z_{{\rm{Rw}}}}$ (cf. Figs. 5(d)–(e)). The width of the S distribution is therefore predicted to increase with the width of the $\zeta $ distribution (that is, ${\sim} 2{l_{\rm{w}}}/{z_{{\rm{Rw}}}}$) in accordance with the experimental results shown in Fig. 10. In each panel of Fig. 10, the S distribution derived from the experimental WFAs was more or less broader than that from the simulated WFAs. This difference is attributable to the additional uncertainties in the experimental WFA data mostly due to the statistical $\eta $ distribution of the detected particles and the background noise in the acquired waveforms.

 figure: Fig. 10.

Fig. 10. Normalized occurrences of the ReS and ImS values derived through the inversion algorithm from the simulated (noiseless) and experimental WFA data of the spherical PS size standards with ${d_{\rm{p}}}\;$ = 0.401 μm obtained under the two different ${\omega _0}\;$conditions: (a, b) ${\omega _0}\;$ = 1.92 μm (${l_{\rm{w}}}/{z_{{\rm{Rw}}}}$ = 1.02) and (c, d) ${\omega _0}\;$ = 3.34 μm (${l_{\rm{w}}}/{z_{{\rm{Rw}}}}$ = 0.339). Theoretical S values for ${d_{\rm{p}}}\;$ = 401 ± 0.006 μm (mean ± expanded uncertainty (k = 2)) are also shown in each panel.

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Table 1 summarizes the systematic and random errors of the complex S data for spherical PS size standards obtained under all of the experimental conditions. The systematic and random errors for each sample were evaluated as

$$\begin{array}{c}{{\textrm{Systematic}}\;{\textrm{error}}\;{\textrm{in}}\;x \equiv \left|{\frac{\langle x \rangle}{{{x_{{\rm{true}}}}}}} \right|- 1\;}\end{array}$$
and
$$\begin{array}{c}{\textrm{Random}\;{\textrm{error}}\;{\textrm{in}}\;x \equiv \frac{{\sqrt {{{\langle({x - \langle x \rangle} )}^2}\rangle} }}{{|\langle x \rangle |}},}\end{array}$$
respectively, where x is either ${\rm{Re}}S$ or ${\rm{Im}}S$, ${x_{{\rm{true}}}}$ is the corresponding theoretical value at the nominal ${d_{\rm{p}}}$ for the sample, and angle brackets denote the arithmetic mean.

Tables Icon

Table 1. Systematic and random errors of (ReS, ImS) data for spherical PS size standards.

The theoretical systematic and random errors in the $({{\rm{Re}}S,{\rm{Im}}S} )$ data derived from the simulated noiseless WFAs listed in Table 1 suggest that the nondimensional parameter ${l_{\rm{w}}}/{z_{{\rm{Rw}}}}$ is a primary controlling factor of the accuracy and resolution of the complex S measurements. The sign and magnitude of the systematic errors in the $({{\rm{Re}}S,{\rm{Im}}S} )$ data derived from the measured WFAs were qualitatively in accordance with the theoretical predictions.

In the cases of ${\omega _0}$ = 1.92 and 3.34 μm, the measurement ${\rm{Re}}S$ systematic errors were more negative (by ∼ −5%) than the corresponding theoretical${{\;{\rm{Re}}}}S$ systematic errors, whereas the measurement ${\rm{Im}}S$ systematic errors were more positive (by ∼ +3–10%) than the theoretical ${\rm{Im}}S$ systematic errors. These discrepancies were likely due to wavefront aberrations of the Gaussian beam accumulated through the beam expander, lens, and flow cell. In comparison with these two cases, the smaller systematic discrepancies in the case of ${\omega _0}$ = 17.3 μm are likely due to the less serious magnitude of the wavefront aberrations thanks to the smaller beam vergence angle ${\theta _{{\rm{FF}}}}$.

Furthermore, the random errors in ${\rm{Re}}S$ and ${\rm{Im}}S$ derived from the measured WFAs were fairly in accordance with the theoretical predictions, besides the additional contributions from the background noise.

In Fig. 11, we plot the measured S data of spherical PS and silica size standards along with their theoretical S values on the complex plane in the cases of ${\omega _0}$ = 3.34 and 17.3 μm. The systematic and random errors in these measured S data are listed in Table 1. In the case of ${\omega _0}$ = 17.3 μm, agreement between the measurement and theory for each ${d_p}$ is excellent for both PS and silica samples. In the ${\omega _0}$ = 3.34 μm case, however, there remains a systematic discrepancy of the phase shift $\Delta $ between the measured and theoretical values for both PS and silica samples. The common $\Delta $ bias for every particle sample implies the presence of systematic wavefront aberrations of the Gaussian beam, as mentioned earlier. Quantitative analyses of the wavefront aberrations and their mitigation strategy are beyond the scope of this paper.

 figure: Fig. 11.

Fig. 11. Measured $({{\rm{Re}}S,{\rm{Im}}S} )$ data of spherical PS and silica size standards and their theoretical $({{\rm{Re}}S,{\rm{Im}}S} )$ values. Panels (a) and (c) respectively show the datasets obtained at ${\omega _0}\;$ = 3.34 and 17.3 μm. Panels (b) and (d), respectively, expand the subdomains of (a) and (c) indicated by the dotted rectangles. The scatterplots of small dots show the single-particle datapoints. Colors indicate the different samples. Each filled circle with error bars indicates the mean and standard deviation of the ∼103 single-particle data. Each triplet of open squares indicates the theoretical $({{\rm{Re}}S,{\rm{Im}}S} )$ values corresponding to the lower bound, center, and upper bound of ${d_{\rm{p}}}\;$within the expanded uncertainty range (k = 2). The solid and dashed curves show the theoretical S curves for spheres assuming refractive indices of PS (1.5854 + 6.1764×10−7i) and silica (1.457 + 0i), respectively.

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Our experimental results demonstrate that, apart from the instrument-dependent magnitudes of the wavefront aberrations and background noises, the accuracy and resolution of S measurements using the self-reference interferometer are both primarily determined through the nondimensional parameter ${l_{\rm{w}}}/{z_{{\rm{Rw}}}}$ that bounds the width of the statistical $\zeta $ distribution of the particle trajectory. It should be noted that we could make the magnitude of these $\zeta $-dependent errors smaller if we had filtered out the detection events with relatively large WFWs (cf. Figs. 6(d)–(f)). This WFW-based filtering should be effective in reducing the $\zeta $-dependent errors when ${l_{\rm{w}}}/{z_{{\rm{Rw}}}}$ < ∼1. Fortunately, the statistical distribution of the $\eta $ coordinate of the particle trajectory within our tolerance $|\eta |$ < ∼0.2 was of minor significance as a source of error in our S measurements.

4. Discussion

Our theoretical and experimental results suggest that the nondimensional parameter ${l_{\rm{w}}}/{z_{{\rm{Rw}}}}$ should be as small as possible. However, the allowed ${l_{\rm{w}}}/{z_{{\rm{Rw}}}}$ range is more or less strictly limited by the following experimental requirements:

  • 1. The flow channel thickness $2{l_{\rm{w}}}$ must be sufficiently larger (at least by several times) than the upper bound of the target ${d_{\rm{p}}}$ range to prevent clogging of the flow channel.
  • 2. The flow channel thickness $2{l_{\rm{w}}}$ must be much larger (by around 20 times) than the medium wavelength $\lambda /{n_{\rm{w}}}$ to make the near-field electromagnetic wave interactions between the inner walls and scattering particles within the channel negligible.
  • 3. The beam waist diameter $2{\omega _0}$ must not be considerably larger (no more than ∼30 times) than the lower bound of the target ${d_{\rm{p}}}$ range to keep the magnitude of the WFAs sufficiently greater than the background noise.

These three requirements should be kept in mind when designing a new measurement system for particles in liquids according to the proposed principles.

The proposed method for inline S measurements can also be applicable to airborne particles by directing them into the beam directly without using a flow cell, and setting ${n_{\rm{w}}}$ = ${n_{\rm{g}}}$ = 0 and assuming an appropriate flow velocity profile. The first and second experimental requirements do not apply in the case of direct airflow sampling.

In a previous study characterizing the performance of the SPES method [41], the observed change in the statistical distribution of the complex S data for nonspherical particles depending on the cross-sectional dimensions of the flow channel was qualitatively interpreted to be the result of the difference in randomness of particle orientations that would be affected by the shear profile in the channel. In future studies using the self-reference interferometric scheme for inline S measurements, we recommend including the effects of the parameter ${l_{\rm{w}}}/{z_{{\rm{Rw}}}}$ in interpretations of their data.

Although our experimental demonstration was limited to homogeneous, isotropic, spherical scatterers, the proposed method is also applicable to inhomogeneous, anisotropic, nonspherical scatters by regarding the measured S value for each particle as an observation at a particular particle’s 3D orientation. In case of our experiments, incident field is polarized along y-axis in the $O({xyz} )$ and thus the S parameter should be regarded as the ${S_{22}}$ element of the complex scattering amplitude matrix [cf. 23].

There seems to be no appreciable effects of radiation pressure or thermophoresis on particle trajectory in our experiments as we did not recognize any appreciable distortion of measured signal waveforms from the theoretical waveforms. These effects could be noticeable if we increase the power density of a focused beam. A lower power density is therefore recommended to avoid potential artifacts.

In larger target ${d_{\rm{p}}}$ range beyond ∼5 μm, it becomes harder to infer particle’s physical parameters from the complex $S({0^\circ } )$ data obtained at a visible wavelength due to the non-uniqueness of theoretical $S({0^\circ } )$ curves on a complex plane. In such large size parameter range $({ \gg 1} )$, inline digital holography [42,43] will be a suitable option for optical single particle characterizations.

5. Conclusions

The theoretical foundations and experimental demonstrations were presented for inline complex${{\;}}S({0^\circ } )$ measurements of single particles using the self-reference interferometric scheme, under the assumptions of the paraxial theory for Gaussian beam propagation through aplanatic optical systems and the plane wave scattering theory for predicting the scattered field. From our results, we propose a protocol for quality-controlled complex${{\;}}S({0^\circ } )$ measurements, the Complex Amplitude Sensing version 1 (CAS-v1), which consists of the following seven steps:

  • 1. Choose an appropriate set of experimental parameters${{\;}}\lambda $ and ${l_{\rm{w}}}$, and set a design value of ${\omega _0}$ depending on the target particle materials and target ${d_{\rm{p}}}$ range. For ${\omega _0}$ value, only a rough estimate (according to the optical system setup) is needed at this step.
  • 2. Coarsely adjust the flow cell position ${z_{\rm{f}}}$ to be within the proper ${z_{\rm{f}}}$ domain given by Eq. (1).
  • 3. Unless ${l_{\rm{w}}}/{z_{{\rm{Rw}}}}$ << 1, further adjust the flow cell position ${z_{\rm{f}}}$ such that $|{{z_{\rm{f}}} - {z_{{\rm{fo}}}}} |/{z_{{\rm{Rw}}}}$ << 1 by monitoring the shape of scatterplots of WFW(T−R) versus WFA(T−R) for spherical PS size standards as an observable indicator of $|{{z_{\rm{f}}} - {z_{{\rm{fo}}}}} |/{z_{{\rm{Rw}}}}$.
  • 4. If necessary, apply a threshold value of WFW(T−R) for event filtering to reduce the width of the $\zeta $ distribution of detected particles from the natural width $2{l_{\rm{w}}}/{z_{{\rm{Rw}}}}$.
  • 5. Determine the ${\omega _0}$ value according to the procedure shown in Fig. 8.
  • 6. Check the applicability of plane wave scattering theory through comparison with Gaussian beam scattering theory.
  • 7. Introduce the particle samples of interest and derive the complex${{\;}}S({0^\circ } )$ value from the measured WFA(A−C) and WFA(T−R) data for each detected particle through the inversion algorithm shown in Fig. 9.

The derived complex${{\;}}S({0^\circ } )$ data are useful for classification and quantification of physical properties of unknown particles as demonstrated by the number of applications of the SPES methods [4448]. For example, Potenza et al. [22] used a lookup table of Lorenz–Mie calculation results for estimating the refractive index and size of single spherical particles from measured${{\;}}S({0^\circ } )$ values. Villa et al. [41] proposed a method for physical interpretation of complex $S({0^\circ } )$ data for single, nonspherical particles according to anomalous diffraction theory [1]. Moteki [23] proposed a Bayesian inversion method for estimating the complex refractive index, shape, and volume-equivalent size distribution of each particle species from a cluster of complex ${{\;}}S({0^\circ } )$ datapoints. In such methods for inferring the physical properties of unknown particles from complex${{\;}}S({0^\circ } )$ data, the use of quality-controlled datasets obtained through the proposed CAS-v1 protocol will strengthen the robustness of the results.

Appendix A : signal waveforms for plane wave scattering theory

Here, the analytical signal waveforms are given for plane wave scattering theory. From Eqs. (9), (14), and (15), the normalized scattered power ${P_{{\rm{sca}}}}({{\rm{Tot}}} )/{P_{{\rm{inc}}}}$ along the particle trajectory at $\eta = 0$ is given by

$$\begin{array}{c}{{{\left. {\frac{{{P_{{\rm{sca}}}}({{\rm{Tot}}} )}}{{{P_{{\rm{inc}}}}}}} \right|}_{\eta = 0}} = \frac{2}{{z_{{\rm{Ra}}}^2}}\frac{{{e^{ - 2{\xi ^2}}}}}{{1 + {\zeta ^2}}}\frac{1}{{{f^2}}}{{|{{S^{({{\rm{PW}}} )}}} |}^2}.}\end{array}$$

We will use Eq. (A1) to derive the signal waveform ${P_{{\rm{extsc}}}}({{\rm{Tot}}} )/{P_{{\rm{inc}}}}$ through the relation Eq. (19). From Eq. (11), ${E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )$ at $\eta = 0$ can be written as

$$\begin{aligned}{E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )&\approx {{\;}}iE_0^2\frac{{{z_{{\rm{Ra}}}}}}{{z_{{\rm{pd}}}^2}}{({1 + {\zeta^2}} )^{ - 1/2}}{S^{({{\rm{PW}}} )}}{\rm{exp}}({ - {\xi^2}} ){\rm{exp}}({i\zeta {\xi^2}} )\\&\quad\times {\rm{exp}}({ - i\delta } ){\rm{exp}}[{i\delta ({1 + {\zeta^2}} ){\xi^2}} ]{\rm{exp}}[{i({\zeta {{\tilde r}^2} - {\rm{ta}}{{\rm{n}}^{ - 1}}\zeta } )} ]\\&\quad\times {\rm{exp}}({i\delta {{\tilde r}^2}} ){\rm{exp}}\{{ - [{{{\tilde r}^2} + i2({1 + \varepsilon } ){{({1 + {\zeta^2}} )}^{1/2}}{{\;}}\xi \tilde r{\rm{cos}}\phi } ]} \}.\end{aligned}$$

In order to make the integrations of Eq. (A2) with respect to $\tilde r$ and $\phi $ analytically tractable, we approximate the $\tilde r$-dependent terms, ${\rm{exp}}({i\delta {{\tilde r}^2}} )$ and ${\rm{exp}}[{i({\zeta {{\tilde r}^2} - {\rm{ta}}{{\rm{n}}^{ - 1}}\zeta } )} ]$, by truncated Taylor series expressions of

$$\begin{array}{c}{\exp({i\delta {{\tilde r}^2}} )\approx 1 + i{{\tilde r}^2}\delta }\end{array}$$
and
$$\begin{array}{c}{\exp[{i({\zeta {{\tilde r}^2} - {\rm{ta}}{{\rm{n}}^{ - 1}}\zeta } )} ]\approx 1 + i({{{\tilde r}^2} - 1} )\zeta + \left( { - \frac{{{{\tilde r}^4}}}{2} + {{\tilde r}^2} - \frac{1}{2}} \right){\zeta ^2} + i\left( { - \frac{{{{\tilde r}^6}}}{6} + \frac{{{{\tilde r}^4}}}{2} - \frac{{{{\tilde r}^2}}}{2} + \frac{1}{2}} \right){\zeta ^3},\;}\end{array}$$
respectively, assuming $\delta < < 1$ and $|\zeta |< \sim 1$. We intentionally combined the $\tilde r$-dependent ${\rm{exp}}({i\zeta {{\tilde r}^2}} )$ and $\tilde r$-independent ${\rm{exp}}({ - i{\rm{ta}}{{\rm{n}}^{ - 1}}\zeta } )$ prior to the Taylor series expansion to emphasize the importance of unifying the truncation order with respect to $\zeta $ for these mutually cancelling terms. From Eqs. (16)–(17) and (A2)-(A4), the waveform formulae of ${P_{{\rm{ext}}}}({{\rm{Tot}}} )/{P_{{\rm{inc}}}}$ and ${P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )/{P_{{\rm{inc}}}}$ are given by
$$\begin{array}{c}{{{\left. {\frac{{{P_{{\rm{ext}}}}({\rm{\varSigma }} )}}{{{P_{{\rm{inc}}}}}}} \right|}_{\eta = 0}} = \frac{4}{{{z_{{\rm{Ra}}}}}}\frac{{{e^{ - {\xi ^2}}}}}{{\sqrt {1 + {\zeta ^2}} }}{\textrm{Re}}\left\{ {\frac{i}{\pi }{\rm{exp}}({ - i\delta } ){\rm{exp}}[{i\delta ({1 + {\zeta^2}} ){\xi^2}} ]{\rm{exp}}({i\zeta {\xi^2}} ){S^{({{\rm{PW}}} )}}U({\rm{\varSigma }} )} \right\},}\end{array}$$
in which $U({\rm{\varSigma }} )$ with ${\rm{\varSigma }} = {\rm{Tot\;or\;A}} - {\rm{C}}$ is the analytically integrable functions
$$\begin{array}{c}{U({{\rm{Tot}}} )\equiv \mathop \int \limits_0^{2\pi } \mathop \int \limits_0^\infty {\rm{exp}}\{{ - [{{{\tilde r}^2} + i2({1 + \varepsilon } ){{({1 + {\zeta^2}} )}^{1/2}}{{\;}}\xi \tilde r{\rm{cos}}\phi } ]} \}Q({{{\tilde r}^2},\delta ,\zeta } )\tilde rd\tilde rd\phi ,}\end{array}$$
and
$$\begin{array}{c}{U({{\rm{A}} - {\rm{C}}} )\equiv - \mathop \int \limits_{ - \pi /4}^{\pi /4} \mathop \int \limits_{ - \infty }^\infty {\rm{exp}}\{{ - [{{{\tilde r}^2} + i2({1 + \varepsilon } ){{({1 + {\zeta^2}} )}^{1/2}}{{\;}}\xi \tilde r{\rm{cos}}\phi } ]} \}Q({{{\tilde r}^2},\delta ,\zeta } )\tilde rd\tilde rd\phi ,}\end{array}$$
respectively, where $Q({{{\tilde r}^2},\delta ,\zeta } )\equiv {\rm{Eq}}.\;({{\rm{A}}3} )\times {\rm{Eq}}.\;({{\rm{A}}4} )$. In the actual computation of Eq. (A5), we also use the following approximations
$$\begin{array}{c}{\exp({ - i\delta } )\approx 1 - i\delta ,\;\;\exp[{i\delta ({1 + {\zeta^2}} ){\xi^2}} ]\approx 1 + i\delta ({1 + {\zeta^2}} ){\xi ^2},\;\;}\end{array}$$
and
$$\begin{array}{c}{\exp({i\zeta {\xi^2}} )\approx 1 + i{\xi ^2}\zeta - \frac{1}{2}{\xi ^4}{\zeta ^2} - i\frac{1}{6}{\xi ^6}{\zeta ^3},} \end{array}$$
respectively, assuming $\delta < < 1$ and $|\zeta |< \sim 1$. Analytical formulae for the waveforms ${P_{{\rm{ext}}}}({{\rm{Tot}}} )/{P_{{\rm{inc}}}}$ and ${P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )/{P_{{\rm{inc}}}}$ after performing the integrals in Eqs. (A6) and (A7) are available from [49] as a MATEMATICA notebook file. The signal waveform ${P_{{\rm{extsc}}}}({{\rm{Tot}}} )/{P_{{\rm{inc}}}}$ is given by summing the ${P_{{\rm{ext}}}}({{\rm{Tot}}} )/{P_{{\rm{inc}}}}$ waveform of Eq. (A5) with Eq. (A6) and the ${P_{{\rm{sca}}}}({{\rm{Tot}}} )/{P_{{\rm{inc}}}}$ waveform of Eq. (A1).

In the particular case of $\varepsilon = \delta = {{\;}}\zeta = 0$, the waveforms of ${P_{{\rm{ext}}}}({{\rm{Tot}}} )/{P_{{\rm{inc}}}}$ and ${P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )/{P_{{\rm{inc}}}}$ are reduced to

$$\begin{array}{c}{{{\left. {\frac{{{P_{{\rm{ext}}}}({{\rm{Tot}}} )}}{{{P_{{\rm{inc}}}}}}} \right|}_{\eta = 0,{{\;\;}}\zeta = 0,\varepsilon = 0,\delta = 0}} = - \left( {\frac{4}{{{z_{{\rm{Ra}}}}}}} \right){e^{ - 2{\xi ^2}}}\textrm{Im}{S^{({{\rm{PW}}} )}}\;}\end{array}$$
and
$$\begin{array}{c}{{{\left. {\frac{{{P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )}}{{{P_{{\rm{inc}}}}}}} \right|}_{\eta = 0,{{\;\;}}\zeta = 0,\varepsilon = 0,\delta = 0}} = - \left( {\frac{4}{{{z_{{\rm{Ra}}}}}}} \right){e^{ - 2{\xi ^2}}}erfi\left( {\frac{\xi }{{\sqrt 2 }}} \right){\textrm{Re}}{S^{({{\rm{PW}}} )}},} \end{array}$$
respectively, each of which is mathematically identical to the corresponding waveform in the SPES method (cf. Eqs. (21) and (22) of Moteki [23] at $\eta = 0$).

It should also be noted that the ${P_{{\rm{ext}}}}({{\rm{B}} - {\rm{D}}} )/{P_{{\rm{inc}}}}$ waveform along the $\eta $ coordinate at $\xi = 0$ is simply given by interchanging $\xi $ and $\eta $ in the ${P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )/{P_{{\rm{inc}}}}$ waveform.

Appendix B : signal waveforms for the Gaussian beam scattering theory

Here, the analytical signal waveforms are given for Gaussian beam scattering theory. From Eqs. (12), (14), and (15), we have

$$\begin{array}{c} {{{\left. {\frac{{{P_{{\rm{sca}}}}({{\rm{Tot}}} )}}{{{P_{{\rm{inc}}}}}}} \right|}_{\eta = 0}} = \frac{2}{{z_{{\rm{Ra}}}^2}}\frac{1}{{{f^2}}}{{|{{S^{({{\rm{GB}}} )}}} |}^2}.\;}\end{array}$$

From Eq. (13), ${E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )$ at $\eta = 0$ can be written as

$$\begin{array}{c}{{E_{{\rm{sca}}}}E_{{\rm{inc}}}^{\rm{\ast }}({{{\mathbf{r}}_{{\rm{pd}}}}} )\approx \;iE_0^2\frac{{{z_{{\rm{Ra}}}}}}{{z_{{\rm{pd}}}^2}}{S^{({{\rm{GB}}} )}}\exp({ - i\beta \zeta } )\exp({ - i\delta } )\;\exp[{i\delta ({1 + {\zeta^2}} ){\xi^2}} ]}\\{ \times \exp({i\zeta {{\tilde r}^2}} )\exp({i\delta {{\tilde r}^2}} )\exp\{{ - [{{{\tilde r}^2} + i2({1 + \varepsilon } ){{({1 + {\zeta^2}} )}^{1/2}}{{\;}}\xi \tilde r{\rm{cos}}\phi } ]} \}.}\end{array}$$

In order to make the analytical integration of Eq. (B2) with respect to $\tilde r$ and $\phi $ tractable, we approximate the $\tilde r$-dependent term ${\rm{exp}}({i\zeta {{\tilde r}^2}} )$ by a truncated Taylor series

$$\begin{array}{c}{\exp({i\zeta {{\tilde r}^2}} )\approx 1 + i{{\tilde r}^2}\zeta - \frac{1}{2}{{\tilde r}^4}{\zeta ^2} - i\frac{1}{6}{{\tilde r}^6}{\zeta ^3}\;\;}\end{array}$$
and also approximate the $\tilde r$-dependent term ${\rm{exp}}({i\delta {{\tilde r}^2}} )$ by Eq. (A3). From Eqs. (16)–(17) and Eqs. (B2)–(B3), the waveform formulae of ${P_{{\rm{ext}}}}({{\rm{Tot}}} )/{P_{{\rm{inc}}}}$ and ${P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )/{P_{{\rm{inc}}}}$ are given by
$$\begin{array}{c}{{{\left. {\frac{{{P_{{\rm{ext}}}}({\rm{\varSigma }} )}}{{{P_{{\rm{inc}}}}}}} \right|}_{\eta = 0}} = \frac{4}{{{z_{{\rm{Ra}}}}}}{\textrm{Re}}\left\{ {\frac{i}{\pi }{\rm{exp}}({ - i\delta } ){\rm{exp}}[{i\delta ({1 + {\zeta^2}} ){\xi^2}} ]{\rm{exp}}({ - i\beta \zeta } ){S^{({{\rm{GB}}} )}}V({\rm{\varSigma }} )} \right\},}\end{array}$$
in which $V({\rm{\varSigma }} )$ with ${\rm{\varSigma }} = {\rm{Tot\;or\;A}} - {\rm{C}}$ is an integrable function defined by
$$\begin{array}{c} {V({{\rm{Tot}}} )\equiv \mathop \int \limits_0^{2\pi } \mathop \int \limits_0^\infty {\rm{exp}}\{{ - [{{{\tilde r}^2} + i2({1 + \varepsilon } ){{({1 + {\zeta^2}} )}^{1/2}}{{\;}}\xi \tilde r{\rm{cos}}\phi } ]} \}R({{{\tilde r}^2},\delta ,\zeta } )\tilde rd\tilde rd\phi \;\;} \end{array}$$
and
$$\begin{array}{c}{V({{\rm{A}} - {\rm{C}}} )\equiv - \mathop \int \limits_{ - \pi /4}^{\pi /4} \mathop \int \limits_{ - \infty }^\infty {\rm{exp}}\{{ - [{{{\tilde r}^2} + i2({1 + \varepsilon } ){{({1 + {\zeta^2}} )}^{1/2}}{{\;}}\xi \tilde r{\rm{cos}}\phi } ]} \}R({{{\tilde r}^2},\delta ,\zeta } )\tilde rd\tilde rd\phi ,\;} \end{array}$$
respectively, where $R({{{\tilde r}^2},\delta ,\zeta } )\equiv {\rm{Eq}}.{{\;}}({{\rm{A}}3} )\times {\rm{Eq}}.{{\;}}({{\rm{B}}3} )$. In the actual computation of Eq. (B4), we use the truncated Taylor series approximations of Eq. (A8). The final analytical formulae of Eq. (B4) for the waveforms ${P_{{\rm{ext}}}}({{\rm{Tot}}} )/{P_{{\rm{inc}}}}$ and ${P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )/{P_{{\rm{inc}}}}$ are available from [49]. The signal waveform ${P_{{\rm{extsc}}}}({{\rm{Tot}}} )/{P_{{\rm{inc}}}}$ is given by summing the ${P_{{\rm{ext}}}}({{\rm{Tot}}} )/{P_{{\rm{inc}}}}$ waveform of Eq. (B4) with Eq. (B5) and the ${P_{{\rm{sca}}}}({{\rm{Tot}}} )/{P_{{\rm{inc}}}}$ waveform of Eq. (B1).

In the particular case of $\varepsilon = \delta = {{\;}}\zeta = 0$, the waveforms of ${P_{{\rm{ext}}}}({{\rm{Tot}}} )/{P_{{\rm{inc}}}}$ and ${P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )/{P_{{\rm{inc}}}}$ are reduced to

$$\begin{array}{c}{{{\left. {\frac{{{P_{{\rm{ext}}}}({{\rm{Tot}}} )}}{{{P_{{\rm{inc}}}}}}} \right|}_{\eta = 0,{{\;\;}}\zeta = 0,\varepsilon = 0,\delta = 0}} = - \left( {\frac{4}{{{z_{{\rm{Ra}}}}}}} \right){e^{ - {\xi ^2}}}\textrm{Im}{S^{({{\rm{GB}}} )}}\;\;} \end{array}$$
and
$$\begin{array}{c} {{{\left. {\frac{{{P_{{\rm{ext}}}}({{\rm{A}} - {\rm{C}}} )}}{{{P_{{\rm{inc}}}}}}} \right|}_{\eta = 0,{{\;\;}}\zeta = 0,\varepsilon = 0,\delta = 0}} = - \left( {\frac{4}{{{z_{{\rm{Ra}}}}}}} \right){e^{ - {\xi ^2}}}erfi\left( {\frac{\xi }{{\sqrt 2 }}} \right){\textrm{Re}}{S^{({{\rm{GB}}} )}},\;} \end{array}$$
respectively. The waveform formulae Eqs. (B7)–(B8) are analogous to their plane wave counterparts Eqs. (A10)–(A11), except for the amplitude-scaling factor ${\rm{exp}}({ - {\xi^2}} )$ of the incident field which has been incorporated into the ${S^{({{\rm{GB}}} )}}$ parameter.

Appendix C : data simulation procedure

Here, the computational procedure is given for simulating an ensemble of signal waveforms of detected particles flowing through a plane-parallel channel (cf. Fig. 1).

We define the scaled z coordinate of particle ${\tilde z_{\rm{p}}} \in ({ - 1,{{\;}}1} )$ by the following equation:

$$\begin{array}{c}{{z_{\rm{p}}} = {l_{\rm{w}}}{{\tilde z}_{\rm{p}}} + {z_{\rm{f}}},}\end{array}$$
in which ${z_{\rm{p}}}$ and ${z_{\rm{f}}}$ are the z coordinates of the particle and the flow cell, respectively, and ${l_{\rm{w}}}$ is the half thickness of the flow channel (cf. Fig. 1). Assuming a laminar flow condition within a plane-parallel channel [cf. 50], the ${\tilde z_{\rm{p}}}$-dependent particle velocity toward the positive x-direction is given by
$$\begin{array}{c}{v({{{\tilde z}_{\rm{p}}}} )= \frac{3}{2}\bar v({1 - \tilde z_{\rm{p}}^2} ),\;}\end{array}$$
where $\bar v$ denotes the mean $v({{{\tilde z}_{\rm{p}}}} )$ within the domain $- 1 \le {\tilde z_{\rm{p}}} \le 1$. From the particle velocity and the beam spot size at $z = {z_{\rm{p}}}$, the $\xi $ coordinate of the particle can be written as a function of time as
$$\begin{array}{c}{\xi (t )= \frac{1}{{\omega ({{z_{\rm{p}}}} )}}v({{{\tilde z}_{\rm{p}}}} )t.}\end{array}$$

Once we have a ${\tilde z_{\rm{p}}}$ value, we can compute signal waveforms as functions of time according to $\xi (t )$ and the waveform formulae given in the Appendix A or B.

Next, we explain the algorithm for acquiring random ${\tilde z_{\rm{p}}}$ samples from a particular probability density function according to the flow velocity profile Eq. (C2). If we assume that the particle number concentration is random but uniform within the flow channel, the probability density function of ${\tilde z_{\rm{p}}}$ occurrence is given by

$$\begin{array}{c}{g({{{\tilde z}_{\rm{p}}}} )= \frac{3}{4}({1 - \tilde z_{\rm{p}}^2} ),\;}\end{array}$$
in which $g({{{\tilde z}_{\rm{p}}}} )$ satisfies the normalization condition $\mathop \int \limits_{ - 1}^1 g({{{\tilde z}_{\rm{p}}}} )d{\tilde z_{\rm{p}}} = 1$. Here, we use the inverse transform method [cf. 40] to pick random samples of ${\tilde z_{\rm{p}}}$ from its probability density function $g({{{\tilde z}_{\rm{p}}}} )$. In this method, we construct the cumulative distribution function of $g({{{\tilde z}_{\rm{p}}}} )$
$$\begin{array}{c}{G({{{\tilde z}_{\rm{p}}}} )\equiv \mathop \int \limits_{ - 1}^{{{\tilde z}_{\rm{p}}}} g(z )dz = \frac{1}{4}({ - \tilde z_{\rm{p}}^3 + 3{{\tilde z}_{\rm{p}}} + 2} ),} \end{array}$$
and then compute a numerical solution of the following equation with respect to ${\tilde z_{\rm{p}}}$:
$$\begin{array}{c}{\;G({{{\tilde z}_{\rm{p}}}} )= u[{0,1} ],} \end{array}$$
in which $u[{0,1} ]$ denotes the random variable uniformly distributed on the interval $[{0,1} ]$.

Funding

Ministry of Education, Culture, Sports, Science and Technology (JPMXD1300000000); Environmental Restoration and Conservation Agency (JPMEERF20172003, JPMEERF20202003); Japan Society for the Promotion of Science (JP19H04236, JP19H04259, JP19H05699, JP19KK0289).

Acknowledgements

The author thanks Drs. A. Yoshida, S. Ohata, J. Schwarz, and D. Murphy for valuable information and comments motivating this research.

Disclosures

The author declares no conflict of interest.

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8. M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Extinction and the optical theorem. Part I. Single particles,” J. Opt. Soc. Am. A 25(7), 1504–1513 (2008). [CrossRef]  

9. F. Charriere, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett. 31(2), 178–180 (2006). [CrossRef]  

10. S. Khadir, D. Andren, P. C. Chaumet, S. Monneret, N. Bonod, M. Kall, A. Sentenac, and G. Baffou, “Full optical characterization of single nanoparticles using quantitative phase imaging,” Optica 7(3), 243–248 (2020). [CrossRef]  

11. D. R. Pettit and T. W. Peterson, “Coherent Detection of Scattered-Light from Sub-Micron Aerosols,” Aerosol Sci. Technol. 2(3), 351–368 (1982). [CrossRef]  

12. J. S. Batchelder and M. A. Taubenblatt, “Interferometric Detection of Forward Scattered-Light from Small Particles,” Appl. Phys. Lett. 55(3), 215–217 (1989). [CrossRef]  

13. M. A. Taubenblatt and J. S. Batchelder, “Measurement of the Size and Refractive-Index of a Small Particle Using the Complex Forward-Scattered Electromagnetic-Field,” Appl. Opt. 30(33), 4972–4979 (1991). [CrossRef]  

14. A. Bassini, M. Menchise, S. Musazzi, E. Paganini, and U. Perini, “Interferometric system for precise submicrometer particle sizing,” Appl. Opt. 36(31), 8121–8127 (1997). [CrossRef]  

15. F. V. Ignatovich and L. Novotny, “Real-time and background-free detection of nanoscale particles,” Phys. Rev. Lett. 96(1), 013901 (2006). [CrossRef]  

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18. K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy,” Phys. Rev. Lett. 93(3), 037401 (2004). [CrossRef]  

19. J. Ortega-Arroyo and P. Kukura, “Interferometric scattering microscopy (iSCAT): new frontiers in ultrafast and ultrasensitive optical microscopy,” Phys. Chem. Chem. Phys. 14(45), 15625–15636 (2012). [CrossRef]  

20. R. W. Taylor and V. Sandoghdar, “Interferometric Scattering Microscopy: Seeing Single Nanoparticles and Molecules via Rayleigh Scattering,” Nano Lett. 19(8), 4827–4835 (2019). [CrossRef]  

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22. M. A. C. Potenza, T. Sanvito, and A. Pullia, “Measuring the complex field scattered by single submicron particles,” AIP Adv. 5(11), 117222 (2015). [CrossRef]  

23. N. Moteki, “Capabilities and limitations of the single-particle extinction and scattering method for estimating the complex refractive index and size-distribution of spherical and non-spherical submicron particles,” J. Quant. Spectrosc. Radiat. Transfer 243, 106811 (2020). [CrossRef]  

24. G. Videen, “Light-Scattering from a Sphere on or near a Surface,” J. Opt. Soc. Am. A 8(3), 483–489 (1991). [CrossRef]  

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26. M. Paulus, P. Cay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green's tensor for stratified media,” Phys. Rev. E 62(4), 5797–5807 (2000). [CrossRef]  

27. A. Gerrard and J. M. Burch, Matrix methods in optics (John Wiley & Sons, 1975).

28. A. E. Siegman, Lasers (University Science Books, 1986).

29. G. Gouesbet, B. Maheu, and G. Grehan, “Light-Scattering from a Sphere Arbitrarily Located in a Gaussian-Beam, Using a Bromwich Formulation,” J. Opt. Soc. Am. A 5(9), 1427–1443 (1988). [CrossRef]  

30. J. A. Lock and G. Gouesbet, “Generalized Lorenz-Mie theory and applications,” J. Quant. Spectrosc. Radiat. Transfer 110(11), 800–807 (2009). [CrossRef]  

31. J. A. Lock, “Contribution of High-Order Rainbows to the Scattering of a Gaussian Laser-Beam by a Spherical-Particle,” J. Opt. Soc. Am. A 10(4), 693–706 (1993). [CrossRef]  

32. J. A. Lock and G. Gouesbet, “Rigorous Justification of the Localized Approximation to the Beam-Shape Coefficients in Generalized Lorenz-Mie Theory .1. On-Axis Beams,” J. Opt. Soc. Am. A 11(9), 2503–2515 (1994). [CrossRef]  

33. G. Gouesbet and J. A. Lock, “Rigorous Justification of the Localized Approximation to the Beam-Shape Coefficients in Generalized Lorenz-Mie Theory .2. Off-Axis Beams,” J. Opt. Soc. Am. A 11(9), 2516–2525 (1994). [CrossRef]  

34. W. S. Rodney and R. J. Spindler, “Index of Refraction of Fused Quartz Glass for Ultraviolet, Visible, and Infrared Wavelengths,” J. Opt. Soc. Am. 44(9), 677–679 (1954). [CrossRef]  

35. G. M. Hale and M. R. Querry, “Optical-Constants of Water in 200-Nm to 200-Mum Wavelength Region,” Appl. Opt. 12(3), 555–563 (1973). [CrossRef]  

36. X. N. Zhang, J. Qiu, X. C. Li, J. M. Zhao, and L. H. Liu, “Complex refractive indices measurements of polymers in visible and near-infrared bands,” Appl. Opt. 59(8), 2337–2344 (2020). [CrossRef]  

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41. S. Villa, T. Sanvito, B. Paroli, A. Pullia, B. Delmonte, and M. A. C. Potenza, “Measuring shape and size of micrometric particles from the analysis of the forward scattered field,” J. Appl. Phys. 119(22), 224901 (2016). [CrossRef]  

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44. M. A. C. Potenza, T. Sanvito, and A. Pullia, “Accurate sizing of ceria oxide nanoparticles in slurries by the analysis of the optical forward-scattered field,” J. Nanopart. Res. 17(2), 110–118 (2015). [CrossRef]  

45. M. A. C. Potenza, S. Albani, B. Delmonte, S. Villa, T. Sanvito, B. Paroli, A. Pullia, G. Baccolo, N. Mahowald, and V. Maggi, “Shape and size constraints on dust optical properties from the Dome C ice core, Antarctica,” Sci. Rep. 6(1), 1–9 (2016). [CrossRef]  

46. M. A. C. Potenza, Z. Krpetic, T. Sanvito, Q. Cai, M. Monopoli, J. M. de Araujo, C. Cella, L. Boselli, V. Castagnola, P. Milani, and K. A. Dawson, “Detecting the shape of anisotropic gold nanoparticles in dispersion with single particle extinction and scattering,” Nanoscale 9(8), 2778–2784 (2017). [CrossRef]  

47. M. F. Simonsen, L. Cremonesi, G. Baccolo, S. Bosch, B. Delmonte, T. Erhardt, H. A. Kjaer, M. Potenza, A. Svensson, and P. Vallelonga, “Particle shape accounts for instrumental discrepancy in ice core dust size distributions,” Clim. Past 14(5), 601–608 (2018). [CrossRef]  

48. L. Cremonesi, C. Minnai, F. Ferri, A. Parola, B. Paroli, T. Sanvito, and M. A. C. Potenza, “Light extinction and scattering from aggregates composed of submicron particles,” J. Nanopart. Res. 22(11), 344–417 (2020). [CrossRef]  

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References

  • View by:

  1. H. C. van de Hulst, Light scattering by small particles (John Wiley & Sons, 1957).
  2. C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, 1983).
  3. M. I. Mishchenko, Electromagnetic scattering by particles and particle groups: an introduction (Cambridge University, 2014).
  4. A. R. Jones, “Light scattering for particle characterization,” Prog. Energy Combust. Sci. 25(1), 1–53 (1999).
    [Crossref]
  5. R. L. Xu, “Light scattering: A review of particle characterization applications,” Particuology 18, 11–21 (2015).
    [Crossref]
  6. R. A. V. and M and A. Yurkin, “Single-particle characterization by elastic light scattering,” Laser Photonics Rev. 15(2), 2000368 (2021).
    [Crossref]
  7. M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University, 2002).
  8. M. J. Berg, C. M. Sorensen, and A. Chakrabarti, “Extinction and the optical theorem. Part I. Single particles,” J. Opt. Soc. Am. A 25(7), 1504–1513 (2008).
    [Crossref]
  9. F. Charriere, A. Marian, F. Montfort, J. Kuehn, T. Colomb, E. Cuche, P. Marquet, and C. Depeursinge, “Cell refractive index tomography by digital holographic microscopy,” Opt. Lett. 31(2), 178–180 (2006).
    [Crossref]
  10. S. Khadir, D. Andren, P. C. Chaumet, S. Monneret, N. Bonod, M. Kall, A. Sentenac, and G. Baffou, “Full optical characterization of single nanoparticles using quantitative phase imaging,” Optica 7(3), 243–248 (2020).
    [Crossref]
  11. D. R. Pettit and T. W. Peterson, “Coherent Detection of Scattered-Light from Sub-Micron Aerosols,” Aerosol Sci. Technol. 2(3), 351–368 (1982).
    [Crossref]
  12. J. S. Batchelder and M. A. Taubenblatt, “Interferometric Detection of Forward Scattered-Light from Small Particles,” Appl. Phys. Lett. 55(3), 215–217 (1989).
    [Crossref]
  13. M. A. Taubenblatt and J. S. Batchelder, “Measurement of the Size and Refractive-Index of a Small Particle Using the Complex Forward-Scattered Electromagnetic-Field,” Appl. Opt. 30(33), 4972–4979 (1991).
    [Crossref]
  14. A. Bassini, M. Menchise, S. Musazzi, E. Paganini, and U. Perini, “Interferometric system for precise submicrometer particle sizing,” Appl. Opt. 36(31), 8121–8127 (1997).
    [Crossref]
  15. F. V. Ignatovich and L. Novotny, “Real-time and background-free detection of nanoscale particles,” Phys. Rev. Lett. 96(1), 013901 (2006).
    [Crossref]
  16. B. Deutsch, R. Beams, and L. Novotny, “Nanoparticle detection using dual-phase interferometry,” Appl. Opt. 49(26), 4921–4925 (2010).
    [Crossref]
  17. A. Mitra, B. Deutsch, F. Ignatovich, C. Dykes, and L. Novotny, “Nano-optofluidic Detection of Single Viruses and Nanoparticles,” ACS Nano 4(3), 1305–1312 (2010).
    [Crossref]
  18. K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy,” Phys. Rev. Lett. 93(3), 037401 (2004).
    [Crossref]
  19. J. Ortega-Arroyo and P. Kukura, “Interferometric scattering microscopy (iSCAT): new frontiers in ultrafast and ultrasensitive optical microscopy,” Phys. Chem. Chem. Phys. 14(45), 15625–15636 (2012).
    [Crossref]
  20. R. W. Taylor and V. Sandoghdar, “Interferometric Scattering Microscopy: Seeing Single Nanoparticles and Molecules via Rayleigh Scattering,” Nano Lett. 19(8), 4827–4835 (2019).
    [Crossref]
  21. M. Giglio and M. A. C. Potenza, “A method of measuring properties of particles and corresponding apparatus,” International Patent No: WO2006137090A1 (2006).
  22. M. A. C. Potenza, T. Sanvito, and A. Pullia, “Measuring the complex field scattered by single submicron particles,” AIP Adv. 5(11), 117222 (2015).
    [Crossref]
  23. N. Moteki, “Capabilities and limitations of the single-particle extinction and scattering method for estimating the complex refractive index and size-distribution of spherical and non-spherical submicron particles,” J. Quant. Spectrosc. Radiat. Transfer 243, 106811 (2020).
    [Crossref]
  24. G. Videen, “Light-Scattering from a Sphere on or near a Surface,” J. Opt. Soc. Am. A 8(3), 483–489 (1991).
    [Crossref]
  25. G. Videen, “Light-Scattering from a Sphere Behind a Surface,” J. Opt. Soc. Am. A 10(1), 110–117 (1993).
    [Crossref]
  26. M. Paulus, P. Cay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green's tensor for stratified media,” Phys. Rev. E 62(4), 5797–5807 (2000).
    [Crossref]
  27. A. Gerrard and J. M. Burch, Matrix methods in optics (John Wiley & Sons, 1975).
  28. A. E. Siegman, Lasers (University Science Books, 1986).
  29. G. Gouesbet, B. Maheu, and G. Grehan, “Light-Scattering from a Sphere Arbitrarily Located in a Gaussian-Beam, Using a Bromwich Formulation,” J. Opt. Soc. Am. A 5(9), 1427–1443 (1988).
    [Crossref]
  30. J. A. Lock and G. Gouesbet, “Generalized Lorenz-Mie theory and applications,” J. Quant. Spectrosc. Radiat. Transfer 110(11), 800–807 (2009).
    [Crossref]
  31. J. A. Lock, “Contribution of High-Order Rainbows to the Scattering of a Gaussian Laser-Beam by a Spherical-Particle,” J. Opt. Soc. Am. A 10(4), 693–706 (1993).
    [Crossref]
  32. J. A. Lock and G. Gouesbet, “Rigorous Justification of the Localized Approximation to the Beam-Shape Coefficients in Generalized Lorenz-Mie Theory .1. On-Axis Beams,” J. Opt. Soc. Am. A 11(9), 2503–2515 (1994).
    [Crossref]
  33. G. Gouesbet and J. A. Lock, “Rigorous Justification of the Localized Approximation to the Beam-Shape Coefficients in Generalized Lorenz-Mie Theory .2. Off-Axis Beams,” J. Opt. Soc. Am. A 11(9), 2516–2525 (1994).
    [Crossref]
  34. W. S. Rodney and R. J. Spindler, “Index of Refraction of Fused Quartz Glass for Ultraviolet, Visible, and Infrared Wavelengths,” J. Opt. Soc. Am. 44(9), 677–679 (1954).
    [Crossref]
  35. G. M. Hale and M. R. Querry, “Optical-Constants of Water in 200-Nm to 200-Mum Wavelength Region,” Appl. Opt. 12(3), 555–563 (1973).
    [Crossref]
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2021 (1)

R. A. V. and M and A. Yurkin, “Single-particle characterization by elastic light scattering,” Laser Photonics Rev. 15(2), 2000368 (2021).
[Crossref]

2020 (4)

S. Khadir, D. Andren, P. C. Chaumet, S. Monneret, N. Bonod, M. Kall, A. Sentenac, and G. Baffou, “Full optical characterization of single nanoparticles using quantitative phase imaging,” Optica 7(3), 243–248 (2020).
[Crossref]

N. Moteki, “Capabilities and limitations of the single-particle extinction and scattering method for estimating the complex refractive index and size-distribution of spherical and non-spherical submicron particles,” J. Quant. Spectrosc. Radiat. Transfer 243, 106811 (2020).
[Crossref]

X. N. Zhang, J. Qiu, X. C. Li, J. M. Zhao, and L. H. Liu, “Complex refractive indices measurements of polymers in visible and near-infrared bands,” Appl. Opt. 59(8), 2337–2344 (2020).
[Crossref]

L. Cremonesi, C. Minnai, F. Ferri, A. Parola, B. Paroli, T. Sanvito, and M. A. C. Potenza, “Light extinction and scattering from aggregates composed of submicron particles,” J. Nanopart. Res. 22(11), 344–417 (2020).
[Crossref]

2019 (1)

R. W. Taylor and V. Sandoghdar, “Interferometric Scattering Microscopy: Seeing Single Nanoparticles and Molecules via Rayleigh Scattering,” Nano Lett. 19(8), 4827–4835 (2019).
[Crossref]

2018 (1)

M. F. Simonsen, L. Cremonesi, G. Baccolo, S. Bosch, B. Delmonte, T. Erhardt, H. A. Kjaer, M. Potenza, A. Svensson, and P. Vallelonga, “Particle shape accounts for instrumental discrepancy in ice core dust size distributions,” Clim. Past 14(5), 601–608 (2018).
[Crossref]

2017 (3)

M. A. C. Potenza, Z. Krpetic, T. Sanvito, Q. Cai, M. Monopoli, J. M. de Araujo, C. Cella, L. Boselli, V. Castagnola, P. Milani, and K. A. Dawson, “Detecting the shape of anisotropic gold nanoparticles in dispersion with single particle extinction and scattering,” Nanoscale 9(8), 2778–2784 (2017).
[Crossref]

M. J. Berg, N. R. Subedi, and P. A. Anderson, “Measuring extinction with digital holography: nonspherical particles and experimental validation,” Opt. Lett. 42(5), 1011–1014 (2017).
[Crossref]

M. J. Berg, Y. W. Heinson, O. Kemppinen, and S. Holler, “Solving the inverse problem for coarse-mode aerosol particle morphology with digital holography,” Sci. Rep. 7(1), 1–9 (2017).
[Crossref]

2016 (2)

S. Villa, T. Sanvito, B. Paroli, A. Pullia, B. Delmonte, and M. A. C. Potenza, “Measuring shape and size of micrometric particles from the analysis of the forward scattered field,” J. Appl. Phys. 119(22), 224901 (2016).
[Crossref]

M. A. C. Potenza, S. Albani, B. Delmonte, S. Villa, T. Sanvito, B. Paroli, A. Pullia, G. Baccolo, N. Mahowald, and V. Maggi, “Shape and size constraints on dust optical properties from the Dome C ice core, Antarctica,” Sci. Rep. 6(1), 1–9 (2016).
[Crossref]

2015 (3)

M. A. C. Potenza, T. Sanvito, and A. Pullia, “Accurate sizing of ceria oxide nanoparticles in slurries by the analysis of the optical forward-scattered field,” J. Nanopart. Res. 17(2), 110–118 (2015).
[Crossref]

M. A. C. Potenza, T. Sanvito, and A. Pullia, “Measuring the complex field scattered by single submicron particles,” AIP Adv. 5(11), 117222 (2015).
[Crossref]

R. L. Xu, “Light scattering: A review of particle characterization applications,” Particuology 18, 11–21 (2015).
[Crossref]

2012 (1)

J. Ortega-Arroyo and P. Kukura, “Interferometric scattering microscopy (iSCAT): new frontiers in ultrafast and ultrasensitive optical microscopy,” Phys. Chem. Chem. Phys. 14(45), 15625–15636 (2012).
[Crossref]

2010 (2)

B. Deutsch, R. Beams, and L. Novotny, “Nanoparticle detection using dual-phase interferometry,” Appl. Opt. 49(26), 4921–4925 (2010).
[Crossref]

A. Mitra, B. Deutsch, F. Ignatovich, C. Dykes, and L. Novotny, “Nano-optofluidic Detection of Single Viruses and Nanoparticles,” ACS Nano 4(3), 1305–1312 (2010).
[Crossref]

2009 (1)

J. A. Lock and G. Gouesbet, “Generalized Lorenz-Mie theory and applications,” J. Quant. Spectrosc. Radiat. Transfer 110(11), 800–807 (2009).
[Crossref]

2008 (1)

2006 (2)

2004 (1)

K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy,” Phys. Rev. Lett. 93(3), 037401 (2004).
[Crossref]

2000 (1)

M. Paulus, P. Cay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green's tensor for stratified media,” Phys. Rev. E 62(4), 5797–5807 (2000).
[Crossref]

1999 (1)

A. R. Jones, “Light scattering for particle characterization,” Prog. Energy Combust. Sci. 25(1), 1–53 (1999).
[Crossref]

1997 (2)

1994 (2)

1993 (2)

1991 (2)

1989 (1)

J. S. Batchelder and M. A. Taubenblatt, “Interferometric Detection of Forward Scattered-Light from Small Particles,” Appl. Phys. Lett. 55(3), 215–217 (1989).
[Crossref]

1988 (1)

1987 (1)

1982 (1)

D. R. Pettit and T. W. Peterson, “Coherent Detection of Scattered-Light from Sub-Micron Aerosols,” Aerosol Sci. Technol. 2(3), 351–368 (1982).
[Crossref]

1973 (1)

1966 (1)

1954 (1)

Albani, S.

M. A. C. Potenza, S. Albani, B. Delmonte, S. Villa, T. Sanvito, B. Paroli, A. Pullia, G. Baccolo, N. Mahowald, and V. Maggi, “Shape and size constraints on dust optical properties from the Dome C ice core, Antarctica,” Sci. Rep. 6(1), 1–9 (2016).
[Crossref]

and M, R. A. V.

R. A. V. and M and A. Yurkin, “Single-particle characterization by elastic light scattering,” Laser Photonics Rev. 15(2), 2000368 (2021).
[Crossref]

Anderson, P. A.

Andren, D.

Baccolo, G.

M. F. Simonsen, L. Cremonesi, G. Baccolo, S. Bosch, B. Delmonte, T. Erhardt, H. A. Kjaer, M. Potenza, A. Svensson, and P. Vallelonga, “Particle shape accounts for instrumental discrepancy in ice core dust size distributions,” Clim. Past 14(5), 601–608 (2018).
[Crossref]

M. A. C. Potenza, S. Albani, B. Delmonte, S. Villa, T. Sanvito, B. Paroli, A. Pullia, G. Baccolo, N. Mahowald, and V. Maggi, “Shape and size constraints on dust optical properties from the Dome C ice core, Antarctica,” Sci. Rep. 6(1), 1–9 (2016).
[Crossref]

Baffou, G.

Bassini, A.

Batchelder, J. S.

M. A. Taubenblatt and J. S. Batchelder, “Measurement of the Size and Refractive-Index of a Small Particle Using the Complex Forward-Scattered Electromagnetic-Field,” Appl. Opt. 30(33), 4972–4979 (1991).
[Crossref]

J. S. Batchelder and M. A. Taubenblatt, “Interferometric Detection of Forward Scattered-Light from Small Particles,” Appl. Phys. Lett. 55(3), 215–217 (1989).
[Crossref]

Beams, R.

Berg, M. J.

Bird, R. B.

R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport phenomena, revised 2nd edition (John Wiley & Sons, 2007).

Bohren, C. F.

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, 1983).

Bonod, N.

Bosch, S.

M. F. Simonsen, L. Cremonesi, G. Baccolo, S. Bosch, B. Delmonte, T. Erhardt, H. A. Kjaer, M. Potenza, A. Svensson, and P. Vallelonga, “Particle shape accounts for instrumental discrepancy in ice core dust size distributions,” Clim. Past 14(5), 601–608 (2018).
[Crossref]

Boselli, L.

M. A. C. Potenza, Z. Krpetic, T. Sanvito, Q. Cai, M. Monopoli, J. M. de Araujo, C. Cella, L. Boselli, V. Castagnola, P. Milani, and K. A. Dawson, “Detecting the shape of anisotropic gold nanoparticles in dispersion with single particle extinction and scattering,” Nanoscale 9(8), 2778–2784 (2017).
[Crossref]

Braat, J.

Burch, J. M.

A. Gerrard and J. M. Burch, Matrix methods in optics (John Wiley & Sons, 1975).

Cai, Q.

M. A. C. Potenza, Z. Krpetic, T. Sanvito, Q. Cai, M. Monopoli, J. M. de Araujo, C. Cella, L. Boselli, V. Castagnola, P. Milani, and K. A. Dawson, “Detecting the shape of anisotropic gold nanoparticles in dispersion with single particle extinction and scattering,” Nanoscale 9(8), 2778–2784 (2017).
[Crossref]

Castagnola, V.

M. A. C. Potenza, Z. Krpetic, T. Sanvito, Q. Cai, M. Monopoli, J. M. de Araujo, C. Cella, L. Boselli, V. Castagnola, P. Milani, and K. A. Dawson, “Detecting the shape of anisotropic gold nanoparticles in dispersion with single particle extinction and scattering,” Nanoscale 9(8), 2778–2784 (2017).
[Crossref]

Cay-Balmaz, P.

M. Paulus, P. Cay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green's tensor for stratified media,” Phys. Rev. E 62(4), 5797–5807 (2000).
[Crossref]

Cella, C.

M. A. C. Potenza, Z. Krpetic, T. Sanvito, Q. Cai, M. Monopoli, J. M. de Araujo, C. Cella, L. Boselli, V. Castagnola, P. Milani, and K. A. Dawson, “Detecting the shape of anisotropic gold nanoparticles in dispersion with single particle extinction and scattering,” Nanoscale 9(8), 2778–2784 (2017).
[Crossref]

Chakrabarti, A.

Charriere, F.

Chaumet, P. C.

Colomb, T.

Cremonesi, L.

L. Cremonesi, C. Minnai, F. Ferri, A. Parola, B. Paroli, T. Sanvito, and M. A. C. Potenza, “Light extinction and scattering from aggregates composed of submicron particles,” J. Nanopart. Res. 22(11), 344–417 (2020).
[Crossref]

M. F. Simonsen, L. Cremonesi, G. Baccolo, S. Bosch, B. Delmonte, T. Erhardt, H. A. Kjaer, M. Potenza, A. Svensson, and P. Vallelonga, “Particle shape accounts for instrumental discrepancy in ice core dust size distributions,” Clim. Past 14(5), 601–608 (2018).
[Crossref]

Cuche, E.

Dawson, K. A.

M. A. C. Potenza, Z. Krpetic, T. Sanvito, Q. Cai, M. Monopoli, J. M. de Araujo, C. Cella, L. Boselli, V. Castagnola, P. Milani, and K. A. Dawson, “Detecting the shape of anisotropic gold nanoparticles in dispersion with single particle extinction and scattering,” Nanoscale 9(8), 2778–2784 (2017).
[Crossref]

de Araujo, J. M.

M. A. C. Potenza, Z. Krpetic, T. Sanvito, Q. Cai, M. Monopoli, J. M. de Araujo, C. Cella, L. Boselli, V. Castagnola, P. Milani, and K. A. Dawson, “Detecting the shape of anisotropic gold nanoparticles in dispersion with single particle extinction and scattering,” Nanoscale 9(8), 2778–2784 (2017).
[Crossref]

Delmonte, B.

M. F. Simonsen, L. Cremonesi, G. Baccolo, S. Bosch, B. Delmonte, T. Erhardt, H. A. Kjaer, M. Potenza, A. Svensson, and P. Vallelonga, “Particle shape accounts for instrumental discrepancy in ice core dust size distributions,” Clim. Past 14(5), 601–608 (2018).
[Crossref]

S. Villa, T. Sanvito, B. Paroli, A. Pullia, B. Delmonte, and M. A. C. Potenza, “Measuring shape and size of micrometric particles from the analysis of the forward scattered field,” J. Appl. Phys. 119(22), 224901 (2016).
[Crossref]

M. A. C. Potenza, S. Albani, B. Delmonte, S. Villa, T. Sanvito, B. Paroli, A. Pullia, G. Baccolo, N. Mahowald, and V. Maggi, “Shape and size constraints on dust optical properties from the Dome C ice core, Antarctica,” Sci. Rep. 6(1), 1–9 (2016).
[Crossref]

Depeursinge, C.

Deutsch, B.

B. Deutsch, R. Beams, and L. Novotny, “Nanoparticle detection using dual-phase interferometry,” Appl. Opt. 49(26), 4921–4925 (2010).
[Crossref]

A. Mitra, B. Deutsch, F. Ignatovich, C. Dykes, and L. Novotny, “Nano-optofluidic Detection of Single Viruses and Nanoparticles,” ACS Nano 4(3), 1305–1312 (2010).
[Crossref]

Dykes, C.

A. Mitra, B. Deutsch, F. Ignatovich, C. Dykes, and L. Novotny, “Nano-optofluidic Detection of Single Viruses and Nanoparticles,” ACS Nano 4(3), 1305–1312 (2010).
[Crossref]

Erhardt, T.

M. F. Simonsen, L. Cremonesi, G. Baccolo, S. Bosch, B. Delmonte, T. Erhardt, H. A. Kjaer, M. Potenza, A. Svensson, and P. Vallelonga, “Particle shape accounts for instrumental discrepancy in ice core dust size distributions,” Clim. Past 14(5), 601–608 (2018).
[Crossref]

Ferri, F.

L. Cremonesi, C. Minnai, F. Ferri, A. Parola, B. Paroli, T. Sanvito, and M. A. C. Potenza, “Light extinction and scattering from aggregates composed of submicron particles,” J. Nanopart. Res. 22(11), 344–417 (2020).
[Crossref]

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipies, 3rd edition (Cambridge University, 2007).

Gerrard, A.

A. Gerrard and J. M. Burch, Matrix methods in optics (John Wiley & Sons, 1975).

Giglio, M.

M. Giglio and M. A. C. Potenza, “A method of measuring properties of particles and corresponding apparatus,” International Patent No: WO2006137090A1 (2006).

Gouesbet, G.

Grehan, G.

Hale, G. M.

Hanson, S. G.

Heinson, Y. W.

M. J. Berg, Y. W. Heinson, O. Kemppinen, and S. Holler, “Solving the inverse problem for coarse-mode aerosol particle morphology with digital holography,” Sci. Rep. 7(1), 1–9 (2017).
[Crossref]

Holler, S.

M. J. Berg, Y. W. Heinson, O. Kemppinen, and S. Holler, “Solving the inverse problem for coarse-mode aerosol particle morphology with digital holography,” Sci. Rep. 7(1), 1–9 (2017).
[Crossref]

Huffman, D. R.

C. F. Bohren and D. R. Huffman, Absorption and scattering of light by small particles (John Wiley & Sons, 1983).

Ignatovich, F.

A. Mitra, B. Deutsch, F. Ignatovich, C. Dykes, and L. Novotny, “Nano-optofluidic Detection of Single Viruses and Nanoparticles,” ACS Nano 4(3), 1305–1312 (2010).
[Crossref]

Ignatovich, F. V.

F. V. Ignatovich and L. Novotny, “Real-time and background-free detection of nanoscale particles,” Phys. Rev. Lett. 96(1), 013901 (2006).
[Crossref]

Jones, A. R.

A. R. Jones, “Light scattering for particle characterization,” Prog. Energy Combust. Sci. 25(1), 1–53 (1999).
[Crossref]

Kalkbrenner, T.

K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy,” Phys. Rev. Lett. 93(3), 037401 (2004).
[Crossref]

Kall, M.

Kemppinen, O.

M. J. Berg, Y. W. Heinson, O. Kemppinen, and S. Holler, “Solving the inverse problem for coarse-mode aerosol particle morphology with digital holography,” Sci. Rep. 7(1), 1–9 (2017).
[Crossref]

Khadir, S.

Kjaer, H. A.

M. F. Simonsen, L. Cremonesi, G. Baccolo, S. Bosch, B. Delmonte, T. Erhardt, H. A. Kjaer, M. Potenza, A. Svensson, and P. Vallelonga, “Particle shape accounts for instrumental discrepancy in ice core dust size distributions,” Clim. Past 14(5), 601–608 (2018).
[Crossref]

Kogelnik, H.

Krpetic, Z.

M. A. C. Potenza, Z. Krpetic, T. Sanvito, Q. Cai, M. Monopoli, J. M. de Araujo, C. Cella, L. Boselli, V. Castagnola, P. Milani, and K. A. Dawson, “Detecting the shape of anisotropic gold nanoparticles in dispersion with single particle extinction and scattering,” Nanoscale 9(8), 2778–2784 (2017).
[Crossref]

Kuehn, J.

Kukura, P.

J. Ortega-Arroyo and P. Kukura, “Interferometric scattering microscopy (iSCAT): new frontiers in ultrafast and ultrasensitive optical microscopy,” Phys. Chem. Chem. Phys. 14(45), 15625–15636 (2012).
[Crossref]

Lacis, A. A.

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University, 2002).

Li, T.

Li, X. C.

Lightfoot, E. N.

R. B. Bird, W. E. Stewart, and E. N. Lightfoot, Transport phenomena, revised 2nd edition (John Wiley & Sons, 2007).

Lindfors, K.

K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy,” Phys. Rev. Lett. 93(3), 037401 (2004).
[Crossref]

Liu, L. H.

Lock, J. A.

Maggi, V.

M. A. C. Potenza, S. Albani, B. Delmonte, S. Villa, T. Sanvito, B. Paroli, A. Pullia, G. Baccolo, N. Mahowald, and V. Maggi, “Shape and size constraints on dust optical properties from the Dome C ice core, Antarctica,” Sci. Rep. 6(1), 1–9 (2016).
[Crossref]

Maheu, B.

Mahowald, N.

M. A. C. Potenza, S. Albani, B. Delmonte, S. Villa, T. Sanvito, B. Paroli, A. Pullia, G. Baccolo, N. Mahowald, and V. Maggi, “Shape and size constraints on dust optical properties from the Dome C ice core, Antarctica,” Sci. Rep. 6(1), 1–9 (2016).
[Crossref]

Marian, A.

Marquet, P.

Martin, O. J. F.

M. Paulus, P. Cay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green's tensor for stratified media,” Phys. Rev. E 62(4), 5797–5807 (2000).
[Crossref]

Menchise, M.

Milani, P.

M. A. C. Potenza, Z. Krpetic, T. Sanvito, Q. Cai, M. Monopoli, J. M. de Araujo, C. Cella, L. Boselli, V. Castagnola, P. Milani, and K. A. Dawson, “Detecting the shape of anisotropic gold nanoparticles in dispersion with single particle extinction and scattering,” Nanoscale 9(8), 2778–2784 (2017).
[Crossref]

Minnai, C.

L. Cremonesi, C. Minnai, F. Ferri, A. Parola, B. Paroli, T. Sanvito, and M. A. C. Potenza, “Light extinction and scattering from aggregates composed of submicron particles,” J. Nanopart. Res. 22(11), 344–417 (2020).
[Crossref]

Mishchenko, M. I.

M. I. Mishchenko, Electromagnetic scattering by particles and particle groups: an introduction (Cambridge University, 2014).

M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University, 2002).

Mitra, A.

A. Mitra, B. Deutsch, F. Ignatovich, C. Dykes, and L. Novotny, “Nano-optofluidic Detection of Single Viruses and Nanoparticles,” ACS Nano 4(3), 1305–1312 (2010).
[Crossref]

Monneret, S.

Monopoli, M.

M. A. C. Potenza, Z. Krpetic, T. Sanvito, Q. Cai, M. Monopoli, J. M. de Araujo, C. Cella, L. Boselli, V. Castagnola, P. Milani, and K. A. Dawson, “Detecting the shape of anisotropic gold nanoparticles in dispersion with single particle extinction and scattering,” Nanoscale 9(8), 2778–2784 (2017).
[Crossref]

Montfort, F.

Moteki, N.

N. Moteki, “Capabilities and limitations of the single-particle extinction and scattering method for estimating the complex refractive index and size-distribution of spherical and non-spherical submicron particles,” J. Quant. Spectrosc. Radiat. Transfer 243, 106811 (2020).
[Crossref]

N. Moteki, “Analytical formulae of signal waveforms for self-reference interferometry with CAS-v1 protocol (Zenodo),” https://doi.org/10.5281/zenodo.4643041 , (2021).

Musazzi, S.

Novotny, L.

B. Deutsch, R. Beams, and L. Novotny, “Nanoparticle detection using dual-phase interferometry,” Appl. Opt. 49(26), 4921–4925 (2010).
[Crossref]

A. Mitra, B. Deutsch, F. Ignatovich, C. Dykes, and L. Novotny, “Nano-optofluidic Detection of Single Viruses and Nanoparticles,” ACS Nano 4(3), 1305–1312 (2010).
[Crossref]

F. V. Ignatovich and L. Novotny, “Real-time and background-free detection of nanoscale particles,” Phys. Rev. Lett. 96(1), 013901 (2006).
[Crossref]

Ortega-Arroyo, J.

J. Ortega-Arroyo and P. Kukura, “Interferometric scattering microscopy (iSCAT): new frontiers in ultrafast and ultrasensitive optical microscopy,” Phys. Chem. Chem. Phys. 14(45), 15625–15636 (2012).
[Crossref]

Paganini, E.

Parola, A.

L. Cremonesi, C. Minnai, F. Ferri, A. Parola, B. Paroli, T. Sanvito, and M. A. C. Potenza, “Light extinction and scattering from aggregates composed of submicron particles,” J. Nanopart. Res. 22(11), 344–417 (2020).
[Crossref]

Paroli, B.

L. Cremonesi, C. Minnai, F. Ferri, A. Parola, B. Paroli, T. Sanvito, and M. A. C. Potenza, “Light extinction and scattering from aggregates composed of submicron particles,” J. Nanopart. Res. 22(11), 344–417 (2020).
[Crossref]

M. A. C. Potenza, S. Albani, B. Delmonte, S. Villa, T. Sanvito, B. Paroli, A. Pullia, G. Baccolo, N. Mahowald, and V. Maggi, “Shape and size constraints on dust optical properties from the Dome C ice core, Antarctica,” Sci. Rep. 6(1), 1–9 (2016).
[Crossref]

S. Villa, T. Sanvito, B. Paroli, A. Pullia, B. Delmonte, and M. A. C. Potenza, “Measuring shape and size of micrometric particles from the analysis of the forward scattered field,” J. Appl. Phys. 119(22), 224901 (2016).
[Crossref]

Paulus, M.

M. Paulus, P. Cay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green's tensor for stratified media,” Phys. Rev. E 62(4), 5797–5807 (2000).
[Crossref]

Perini, U.

Peterson, T. W.

D. R. Pettit and T. W. Peterson, “Coherent Detection of Scattered-Light from Sub-Micron Aerosols,” Aerosol Sci. Technol. 2(3), 351–368 (1982).
[Crossref]

Pettit, D. R.

D. R. Pettit and T. W. Peterson, “Coherent Detection of Scattered-Light from Sub-Micron Aerosols,” Aerosol Sci. Technol. 2(3), 351–368 (1982).
[Crossref]

Potenza, M.

M. F. Simonsen, L. Cremonesi, G. Baccolo, S. Bosch, B. Delmonte, T. Erhardt, H. A. Kjaer, M. Potenza, A. Svensson, and P. Vallelonga, “Particle shape accounts for instrumental discrepancy in ice core dust size distributions,” Clim. Past 14(5), 601–608 (2018).
[Crossref]

Potenza, M. A. C.

L. Cremonesi, C. Minnai, F. Ferri, A. Parola, B. Paroli, T. Sanvito, and M. A. C. Potenza, “Light extinction and scattering from aggregates composed of submicron particles,” J. Nanopart. Res. 22(11), 344–417 (2020).
[Crossref]

M. A. C. Potenza, Z. Krpetic, T. Sanvito, Q. Cai, M. Monopoli, J. M. de Araujo, C. Cella, L. Boselli, V. Castagnola, P. Milani, and K. A. Dawson, “Detecting the shape of anisotropic gold nanoparticles in dispersion with single particle extinction and scattering,” Nanoscale 9(8), 2778–2784 (2017).
[Crossref]

S. Villa, T. Sanvito, B. Paroli, A. Pullia, B. Delmonte, and M. A. C. Potenza, “Measuring shape and size of micrometric particles from the analysis of the forward scattered field,” J. Appl. Phys. 119(22), 224901 (2016).
[Crossref]

M. A. C. Potenza, S. Albani, B. Delmonte, S. Villa, T. Sanvito, B. Paroli, A. Pullia, G. Baccolo, N. Mahowald, and V. Maggi, “Shape and size constraints on dust optical properties from the Dome C ice core, Antarctica,” Sci. Rep. 6(1), 1–9 (2016).
[Crossref]

M. A. C. Potenza, T. Sanvito, and A. Pullia, “Accurate sizing of ceria oxide nanoparticles in slurries by the analysis of the optical forward-scattered field,” J. Nanopart. Res. 17(2), 110–118 (2015).
[Crossref]

M. A. C. Potenza, T. Sanvito, and A. Pullia, “Measuring the complex field scattered by single submicron particles,” AIP Adv. 5(11), 117222 (2015).
[Crossref]

M. Giglio and M. A. C. Potenza, “A method of measuring properties of particles and corresponding apparatus,” International Patent No: WO2006137090A1 (2006).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipies, 3rd edition (Cambridge University, 2007).

Pullia, A.

S. Villa, T. Sanvito, B. Paroli, A. Pullia, B. Delmonte, and M. A. C. Potenza, “Measuring shape and size of micrometric particles from the analysis of the forward scattered field,” J. Appl. Phys. 119(22), 224901 (2016).
[Crossref]

M. A. C. Potenza, S. Albani, B. Delmonte, S. Villa, T. Sanvito, B. Paroli, A. Pullia, G. Baccolo, N. Mahowald, and V. Maggi, “Shape and size constraints on dust optical properties from the Dome C ice core, Antarctica,” Sci. Rep. 6(1), 1–9 (2016).
[Crossref]

M. A. C. Potenza, T. Sanvito, and A. Pullia, “Accurate sizing of ceria oxide nanoparticles in slurries by the analysis of the optical forward-scattered field,” J. Nanopart. Res. 17(2), 110–118 (2015).
[Crossref]

M. A. C. Potenza, T. Sanvito, and A. Pullia, “Measuring the complex field scattered by single submicron particles,” AIP Adv. 5(11), 117222 (2015).
[Crossref]

Qiu, J.

Querry, M. R.

Rodney, W. S.

Sandoghdar, V.

R. W. Taylor and V. Sandoghdar, “Interferometric Scattering Microscopy: Seeing Single Nanoparticles and Molecules via Rayleigh Scattering,” Nano Lett. 19(8), 4827–4835 (2019).
[Crossref]

K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy,” Phys. Rev. Lett. 93(3), 037401 (2004).
[Crossref]

Sanvito, T.

L. Cremonesi, C. Minnai, F. Ferri, A. Parola, B. Paroli, T. Sanvito, and M. A. C. Potenza, “Light extinction and scattering from aggregates composed of submicron particles,” J. Nanopart. Res. 22(11), 344–417 (2020).
[Crossref]

M. A. C. Potenza, Z. Krpetic, T. Sanvito, Q. Cai, M. Monopoli, J. M. de Araujo, C. Cella, L. Boselli, V. Castagnola, P. Milani, and K. A. Dawson, “Detecting the shape of anisotropic gold nanoparticles in dispersion with single particle extinction and scattering,” Nanoscale 9(8), 2778–2784 (2017).
[Crossref]

S. Villa, T. Sanvito, B. Paroli, A. Pullia, B. Delmonte, and M. A. C. Potenza, “Measuring shape and size of micrometric particles from the analysis of the forward scattered field,” J. Appl. Phys. 119(22), 224901 (2016).
[Crossref]

M. A. C. Potenza, S. Albani, B. Delmonte, S. Villa, T. Sanvito, B. Paroli, A. Pullia, G. Baccolo, N. Mahowald, and V. Maggi, “Shape and size constraints on dust optical properties from the Dome C ice core, Antarctica,” Sci. Rep. 6(1), 1–9 (2016).
[Crossref]

M. A. C. Potenza, T. Sanvito, and A. Pullia, “Accurate sizing of ceria oxide nanoparticles in slurries by the analysis of the optical forward-scattered field,” J. Nanopart. Res. 17(2), 110–118 (2015).
[Crossref]

M. A. C. Potenza, T. Sanvito, and A. Pullia, “Measuring the complex field scattered by single submicron particles,” AIP Adv. 5(11), 117222 (2015).
[Crossref]

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[Crossref]

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K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy,” Phys. Rev. Lett. 93(3), 037401 (2004).
[Crossref]

Subedi, N. R.

Svensson, A.

M. F. Simonsen, L. Cremonesi, G. Baccolo, S. Bosch, B. Delmonte, T. Erhardt, H. A. Kjaer, M. Potenza, A. Svensson, and P. Vallelonga, “Particle shape accounts for instrumental discrepancy in ice core dust size distributions,” Clim. Past 14(5), 601–608 (2018).
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R. W. Taylor and V. Sandoghdar, “Interferometric Scattering Microscopy: Seeing Single Nanoparticles and Molecules via Rayleigh Scattering,” Nano Lett. 19(8), 4827–4835 (2019).
[Crossref]

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W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipies, 3rd edition (Cambridge University, 2007).

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M. I. Mishchenko, L. D. Travis, and A. A. Lacis, Scattering, absorption, and emission of light by small particles (Cambridge University, 2002).

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M. F. Simonsen, L. Cremonesi, G. Baccolo, S. Bosch, B. Delmonte, T. Erhardt, H. A. Kjaer, M. Potenza, A. Svensson, and P. Vallelonga, “Particle shape accounts for instrumental discrepancy in ice core dust size distributions,” Clim. Past 14(5), 601–608 (2018).
[Crossref]

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H. C. van de Hulst, Light scattering by small particles (John Wiley & Sons, 1957).

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W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipies, 3rd edition (Cambridge University, 2007).

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S. Villa, T. Sanvito, B. Paroli, A. Pullia, B. Delmonte, and M. A. C. Potenza, “Measuring shape and size of micrometric particles from the analysis of the forward scattered field,” J. Appl. Phys. 119(22), 224901 (2016).
[Crossref]

M. A. C. Potenza, S. Albani, B. Delmonte, S. Villa, T. Sanvito, B. Paroli, A. Pullia, G. Baccolo, N. Mahowald, and V. Maggi, “Shape and size constraints on dust optical properties from the Dome C ice core, Antarctica,” Sci. Rep. 6(1), 1–9 (2016).
[Crossref]

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R. L. Xu, “Light scattering: A review of particle characterization applications,” Particuology 18, 11–21 (2015).
[Crossref]

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R. A. V. and M and A. Yurkin, “Single-particle characterization by elastic light scattering,” Laser Photonics Rev. 15(2), 2000368 (2021).
[Crossref]

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Zhao, J. M.

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[Crossref]

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M. A. C. Potenza, T. Sanvito, and A. Pullia, “Measuring the complex field scattered by single submicron particles,” AIP Adv. 5(11), 117222 (2015).
[Crossref]

Appl. Opt. (7)

Appl. Phys. Lett. (1)

J. S. Batchelder and M. A. Taubenblatt, “Interferometric Detection of Forward Scattered-Light from Small Particles,” Appl. Phys. Lett. 55(3), 215–217 (1989).
[Crossref]

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M. F. Simonsen, L. Cremonesi, G. Baccolo, S. Bosch, B. Delmonte, T. Erhardt, H. A. Kjaer, M. Potenza, A. Svensson, and P. Vallelonga, “Particle shape accounts for instrumental discrepancy in ice core dust size distributions,” Clim. Past 14(5), 601–608 (2018).
[Crossref]

J. Appl. Phys. (1)

S. Villa, T. Sanvito, B. Paroli, A. Pullia, B. Delmonte, and M. A. C. Potenza, “Measuring shape and size of micrometric particles from the analysis of the forward scattered field,” J. Appl. Phys. 119(22), 224901 (2016).
[Crossref]

J. Nanopart. Res. (2)

M. A. C. Potenza, T. Sanvito, and A. Pullia, “Accurate sizing of ceria oxide nanoparticles in slurries by the analysis of the optical forward-scattered field,” J. Nanopart. Res. 17(2), 110–118 (2015).
[Crossref]

L. Cremonesi, C. Minnai, F. Ferri, A. Parola, B. Paroli, T. Sanvito, and M. A. C. Potenza, “Light extinction and scattering from aggregates composed of submicron particles,” J. Nanopart. Res. 22(11), 344–417 (2020).
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R. A. V. and M and A. Yurkin, “Single-particle characterization by elastic light scattering,” Laser Photonics Rev. 15(2), 2000368 (2021).
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Nano Lett. (1)

R. W. Taylor and V. Sandoghdar, “Interferometric Scattering Microscopy: Seeing Single Nanoparticles and Molecules via Rayleigh Scattering,” Nano Lett. 19(8), 4827–4835 (2019).
[Crossref]

Nanoscale (1)

M. A. C. Potenza, Z. Krpetic, T. Sanvito, Q. Cai, M. Monopoli, J. M. de Araujo, C. Cella, L. Boselli, V. Castagnola, P. Milani, and K. A. Dawson, “Detecting the shape of anisotropic gold nanoparticles in dispersion with single particle extinction and scattering,” Nanoscale 9(8), 2778–2784 (2017).
[Crossref]

Opt. Lett. (2)

Optica (1)

Particuology (1)

R. L. Xu, “Light scattering: A review of particle characterization applications,” Particuology 18, 11–21 (2015).
[Crossref]

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J. Ortega-Arroyo and P. Kukura, “Interferometric scattering microscopy (iSCAT): new frontiers in ultrafast and ultrasensitive optical microscopy,” Phys. Chem. Chem. Phys. 14(45), 15625–15636 (2012).
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Phys. Rev. E (1)

M. Paulus, P. Cay-Balmaz, and O. J. F. Martin, “Accurate and efficient computation of the Green's tensor for stratified media,” Phys. Rev. E 62(4), 5797–5807 (2000).
[Crossref]

Phys. Rev. Lett. (2)

K. Lindfors, T. Kalkbrenner, P. Stoller, and V. Sandoghdar, “Detection and spectroscopy of gold nanoparticles using supercontinuum white light confocal microscopy,” Phys. Rev. Lett. 93(3), 037401 (2004).
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M. A. C. Potenza, S. Albani, B. Delmonte, S. Villa, T. Sanvito, B. Paroli, A. Pullia, G. Baccolo, N. Mahowald, and V. Maggi, “Shape and size constraints on dust optical properties from the Dome C ice core, Antarctica,” Sci. Rep. 6(1), 1–9 (2016).
[Crossref]

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W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical recipies, 3rd edition (Cambridge University, 2007).

N. Moteki, “Analytical formulae of signal waveforms for self-reference interferometry with CAS-v1 protocol (Zenodo),” https://doi.org/10.5281/zenodo.4643041 , (2021).

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H. C. van de Hulst, Light scattering by small particles (John Wiley & Sons, 1957).

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M. I. Mishchenko, Electromagnetic scattering by particles and particle groups: an introduction (Cambridge University, 2014).

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A. E. Siegman, Lasers (University Science Books, 1986).

M. Giglio and M. A. C. Potenza, “A method of measuring properties of particles and corresponding apparatus,” International Patent No: WO2006137090A1 (2006).

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Figures (11)

Fig. 1.
Fig. 1. Illustration of the optical system and the Gaussian beam around the particle-sensing region in the self-reference interferometric scheme. The laboratory coordinate system $O({xyz} )$ and some physical and coordinate variables are also defined in the figure. The wavefronts of the beam and scattered fields at the observation plane $\;z{{\;}} = {{\;}}{z_{{\rm{pd}}}}$ are also shown schematically.
Fig. 2.
Fig. 2. Illustration of the photoelectric surface of a circular quadrant photodiode and the Gaussian beam spot at $z = {z_{{\rm{pd}}}}$ in the laboratory coordinate system $O({xyz} )$ .
Fig. 3.
Fig. 3. Schematic diagram of our experimental apparatus.
Fig. 4.
Fig. 4. Diagram illustrating the definitions of waveform amplitude (WFA) and waveform width (WFW).
Fig. 5.
Fig. 5. Scatterplots of WFA(T−R) versus WFA(A−C) for ∼103 particles of PS samples with ${d_{\rm{p}}}$  = 0.803 μm obtained at ${\omega _0}$  = 3.34 μm (i.e., $\;{l_w}/{z_{{\rm{Rw}}}}$  = 0.33). In each panel, the light blue open circle with error bars shows the mean and standard deviation of the single-particle WFA data (dots). The left, middle, and right columns show the experimental results, simulated results from plane wave scattering theory (PW), and simulated results from Gaussian beam scattering theory (GB), respectively. The upper row panels (a–c) show the results at $\;{z_{\rm{f}}} - {z_{{\rm{fo}}}}$  = −10 μm, the middle row panels (d–f) show the results at $\;{z_{\rm{f}}} - {z_{{\rm{fo}}}}$  = 0 μm, and the bottom row panels (g–i) show the results at $\;{z_{\rm{f}}} - {z_{{\rm{fo}}}}$  = +10 μm. The color scales indicate the $\zeta $ coordinate of the particle trajectory.
Fig. 6.
Fig. 6. The same scatterplots as Fig. 5 but for WFW(T−R) versus WFA(T−R). The physical unit of the displayed WFW(T−R) values is 0.4 μs, which is the sampling time interval of the digitizer. The WFW(T−R) is inversely proportional to the sample flow rate, and thus its absolute value is unimportant here.
Fig. 7.
Fig. 7. Errors in the simulated WFAs for a nonabsorbing (with refractive index of PS: 1.5854 + 6.1764×10−7i) and an absorbing (with refractive index of 1.5854 + 5×10−2i) single sphere assuming plane wave scattering theory (PW) relative to those assuming exact Gaussian beam scattering theory (GB) plotted versus particle diameter ${d_{\rm{p}}}$ . Top row (a,b) and bottom row (c,d) show the results at ${\omega _0}$  = 3.34 μm and ${\omega _0}{{\;}}$  = 17.3 μm, respectively. Right column (b,d) shows the same plots as left column (a,c) but with 10 times magnification in vertical axes for visibility. The $\zeta $ coordinate of the particle trajectory was set to zero. Other input parameters were prescribed in accordance with our experimental conditions described in section 3.1.
Fig. 8.
Fig. 8. Procedure for estimating ${\omega _0}$ by using the WFA data for spherical PS size standards.
Fig. 9.
Fig. 9. Data inversion algorithm for estimating ${\rm{Re}}S$ and ${\rm{Im}}S$ from the WFA data.
Fig. 10.
Fig. 10. Normalized occurrences of the ReS and ImS values derived through the inversion algorithm from the simulated (noiseless) and experimental WFA data of the spherical PS size standards with ${d_{\rm{p}}}\;$  = 0.401 μm obtained under the two different ${\omega _0}\;$ conditions: (a, b) ${\omega _0}\;$  = 1.92 μm ( ${l_{\rm{w}}}/{z_{{\rm{Rw}}}}$  = 1.02) and (c, d) ${\omega _0}\;$  = 3.34 μm ( ${l_{\rm{w}}}/{z_{{\rm{Rw}}}}$  = 0.339). Theoretical S values for ${d_{\rm{p}}}\;$  = 401 ± 0.006 μm (mean ± expanded uncertainty (k = 2)) are also shown in each panel.
Fig. 11.
Fig. 11. Measured $({{\rm{Re}}S,{\rm{Im}}S} )$ data of spherical PS and silica size standards and their theoretical $({{\rm{Re}}S,{\rm{Im}}S} )$ values. Panels (a) and (c) respectively show the datasets obtained at ${\omega _0}\;$  = 3.34 and 17.3 μm. Panels (b) and (d), respectively, expand the subdomains of (a) and (c) indicated by the dotted rectangles. The scatterplots of small dots show the single-particle datapoints. Colors indicate the different samples. Each filled circle with error bars indicates the mean and standard deviation of the ∼103 single-particle data. Each triplet of open squares indicates the theoretical $({{\rm{Re}}S,{\rm{Im}}S} )$ values corresponding to the lower bound, center, and upper bound of ${d_{\rm{p}}}\;$ within the expanded uncertainty range (k = 2). The solid and dashed curves show the theoretical S curves for spheres assuming refractive indices of PS (1.5854 + 6.1764×10−7i) and silica (1.457 + 0i), respectively.

Tables (1)

Tables Icon

Table 1. Systematic and random errors of (ReS, ImS) data for spherical PS size standards.

Equations (51)

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( 1 n a n g ) l g + ( 1 2 n a n w ) l w z f ( 1 n a n g ) l g + l w
z f o ( 1 n a n g ) l g + ( 1 n a n w ) l w
E i n c ( r p d ) = E 0 ω 0 ω ( z p d ) exp [ x 2 + y 2 ω 2 ( z p d ) ] exp [ i k a x 2 + y 2 2 R b e a m ( z p d ) + i φ ( z p d , z 0 ) i ψ ( z p d , z 0 ) ] ,
E s c a ( r p d ) = e x p [ i φ ( r p d , r p ) ] R s c a ( z p d ) S ( P W ) E i n c ( r p ) ,
E i n c ( r p ) = E 0 ω 0 ω ( z p ) e x p [ x p 2 + y p 2 ω 2 ( z p ) ] e x p [ i k w x p 2 + y p 2 2 R b e a m ( z p ) + i φ ( z p , z 0 ) i ψ ( z p , z 0 ) ] .
E s c a ( r p d ) = e x p [ i φ ( r p d , r p ) ] R s c a ( z p d ) S ( G B ) ( r p ) E 0 ,
| E i n c ( r p d ) + E s c a ( r p d ) | 2 = | E i n c ( r p d ) | 2 + | E s c a ( r p d ) | 2 + 2 R e [ E s c a E i n c ( r p d ) ] ,
( ξ , η , ζ ) ( x p ω ( z p ) , y p ω ( z p ) , z p z R w ) .
| E s c a ( r p d ) | 2 = 1 z p d 2 | S ( P W ) | 2 ( 1 ω 0 2 2 P i n c π ) 1 1 + ζ 2 exp [ 2 ( ξ 2 + η 2 ) ] ,
E s c a E i n c ( r p d ) = E 0 2 [ ω 0 2 ω ( z p d ) ω ( z p ) ] exp [ x 2 + y 2 ω 2 ( z p d ) ] exp [ ( ξ 2 + η 2 ) ] S ( P W ) R s c a ( z p d ) × e x p { i k a [ ( x x p ) 2 + ( y y p ) 2 2 R s c a ( z p d ) x 2 + y 2 2 R b e a m ( z p d ) ] } e x p [ i k w x p 2 + y p 2 2 R b e a m ( z p ) ] × e x p { i [ φ ( z p d , z p ) + φ ( z p , z 0 ) φ ( z p d , z 0 ) ψ ( z p , z 0 ) + ψ ( z p d , z 0 ) ] } ,
E s c a E i n c ( r p d ) i E 0 2 z R a z p d 2 ( 1 + ζ 2 ) 1 / 2 S ( P W ) exp [ ( ξ 2 + η 2 ) ] exp [ i ζ ( ξ 2 + η 2 ) ] × e x p ( i δ ) e x p [ i δ ( 1 + ζ 2 ) ( ξ 2 + η 2 ) ] e x p [ i ( ζ ( x ~ 2 + y ~ 2 ) t a n 1 ζ ) ] × e x p [ i δ ( x ~ 2 + y ~ 2 ) ] e x p { [ x ~ 2 + y ~ 2 + i 2 ( 1 + ε ) ( 1 + ζ 2 ) 1 / 2 ( ξ x ~ + η y ~ ) ] } ,
| E s c a ( r p d ) | 2 = 1 R s c a 2 ( z p d ) | S ( G B ) | 2 ( 1 ω 0 2 2 P i n c π ) .
E s c a E i n c ( r p d ) ≈≈ i E 0 2 z R a z p d 2 S ( G B ) exp ( i β ζ ) exp ( i δ ) exp [ i δ ( 1 + ζ 2 ) ( ξ 2 + η 2 ) ] × e x p [ i ζ ( x ~ 2 + y ~ 2 ) ] e x p [ i δ ( x ~ 2 + y ~ 2 ) ] × e x p { [ x ~ 2 + y ~ 2 + i 2 ( 1 + ε ) ( 1 + ζ 2 ) 1 / 2 ( ξ x ~ + η y ~ ) ] } .
f ω ( z p d ) a ω 0 z p d a z R a ,
P s c a ( T o t ) | E s c a ( r p d ) | 2 π a 2 .
P e x t ( T o t ) = 0 2 π { 0 2 R e [ E s c a E i n c ( r p d ) ] ω 2 ( z p d ) r ~ d r ~ } d ϕ ,
P e x t ( A C ) = π / 4 π / 4 { 2 R e [ E s c a E i n c ( r p d ) ] ω 2 ( z p d ) r ~ d r ~ } d ϕ ,
P e x t ( B D ) = π / 4 3 π / 4 { 2 R e [ E s c a E i n c ( r p d ) ] ω 2 ( z p d ) r ~ d r ~ } d ϕ ,
P e x t s c ( T o t ) P e x t ( T o t ) + P s c a ( T o t ) .
M S R E ( ω 0 ) = 1 N p i p = 1 N p { [ W F A ( A C ) s i m , i p W F A ( A C ) m e a , i p 1 ] 2 + [ W F A ( T R ) s i m , i p W F A ( T R ) m e a , i p 1 ] 2 } ,
P e x t ( A C ) P i n c | η = 0 , ζ = 0 4 z R a e 3 2 ξ 2 2 π { [ 2 ε ξ + 2 2 ( 1 2 ε ξ 2 ) D ( ξ 2 ) ] R e S ξ δ I m S }
P e x t s c ( T R ) P i n c | η = 0 , ζ = 0 4 z R a e 2 ξ 2 ( 1 + 2 ε ξ 2 ) Im S + 2 z R a 2 1 f 2 e 2 ξ 2 | S | 2 ,
W F A ( A C ) = 4 z R a e 3 2 ξ + 2 2 π { [ 2 ε ξ + + 2 2 ( 1 2 ε ξ + 2 ) D ( ξ + 2 ) ] R e S δ ξ + I m S }
W F A ( T R ) = 4 z R a Im S 2 z R a 2 1 f 2 | S | 2 ,
Systematic error in x | x x t r u e | 1
Random error in x ( x x ) 2 | x | ,
P s c a ( T o t ) P i n c | η = 0 = 2 z R a 2 e 2 ξ 2 1 + ζ 2 1 f 2 | S ( P W ) | 2 .
E s c a E i n c ( r p d ) i E 0 2 z R a z p d 2 ( 1 + ζ 2 ) 1 / 2 S ( P W ) e x p ( ξ 2 ) e x p ( i ζ ξ 2 ) × e x p ( i δ ) e x p [ i δ ( 1 + ζ 2 ) ξ 2 ] e x p [ i ( ζ r ~ 2 t a n 1 ζ ) ] × e x p ( i δ r ~ 2 ) e x p { [ r ~ 2 + i 2 ( 1 + ε ) ( 1 + ζ 2 ) 1 / 2 ξ r ~ c o s ϕ ] } .
exp ( i δ r ~ 2 ) 1 + i r ~ 2 δ
exp [ i ( ζ r ~ 2 t a n 1 ζ ) ] 1 + i ( r ~ 2 1 ) ζ + ( r ~ 4 2 + r ~ 2 1 2 ) ζ 2 + i ( r ~ 6 6 + r ~ 4 2 r ~ 2 2 + 1 2 ) ζ 3 ,
P e x t ( Σ ) P i n c | η = 0 = 4 z R a e ξ 2 1 + ζ 2 Re { i π e x p ( i δ ) e x p [ i δ ( 1 + ζ 2 ) ξ 2 ] e x p ( i ζ ξ 2 ) S ( P W ) U ( Σ ) } ,
U ( T o t ) 0 2 π 0 e x p { [ r ~ 2 + i 2 ( 1 + ε ) ( 1 + ζ 2 ) 1 / 2 ξ r ~ c o s ϕ ] } Q ( r ~ 2 , δ , ζ ) r ~ d r ~ d ϕ ,
U ( A C ) π / 4 π / 4 e x p { [ r ~ 2 + i 2 ( 1 + ε ) ( 1 + ζ 2 ) 1 / 2 ξ r ~ c o s ϕ ] } Q ( r ~ 2 , δ , ζ ) r ~ d r ~ d ϕ ,
exp ( i δ ) 1 i δ , exp [ i δ ( 1 + ζ 2 ) ξ 2 ] 1 + i δ ( 1 + ζ 2 ) ξ 2 ,
exp ( i ζ ξ 2 ) 1 + i ξ 2 ζ 1 2 ξ 4 ζ 2 i 1 6 ξ 6 ζ 3 ,
P e x t ( T o t ) P i n c | η = 0 , ζ = 0 , ε = 0 , δ = 0 = ( 4 z R a ) e 2 ξ 2 Im S ( P W )
P e x t ( A C ) P i n c | η = 0 , ζ = 0 , ε = 0 , δ = 0 = ( 4 z R a ) e 2 ξ 2 e r f i ( ξ 2 ) Re S ( P W ) ,
P s c a ( T o t ) P i n c | η = 0 = 2 z R a 2 1 f 2 | S ( G B ) | 2 .
E s c a E i n c ( r p d ) i E 0 2 z R a z p d 2 S ( G B ) exp ( i β ζ ) exp ( i δ ) exp [ i δ ( 1 + ζ 2 ) ξ 2 ] × exp ( i ζ r ~ 2 ) exp ( i δ r ~ 2 ) exp { [ r ~ 2 + i 2 ( 1 + ε ) ( 1 + ζ 2 ) 1 / 2 ξ r ~ c o s ϕ ] } .
exp ( i ζ r ~ 2 ) 1 + i r ~ 2 ζ 1 2 r ~ 4 ζ 2 i 1 6 r ~ 6 ζ 3
P e x t ( Σ ) P i n c | η = 0 = 4 z R a Re { i π e x p ( i δ ) e x p [ i δ ( 1 + ζ 2 ) ξ 2 ] e x p ( i β ζ ) S ( G B ) V ( Σ ) } ,
V ( T o t ) 0 2 π 0 e x p { [ r ~ 2 + i 2 ( 1 + ε ) ( 1 + ζ 2 ) 1 / 2 ξ r ~ c o s ϕ ] } R ( r ~ 2 , δ , ζ ) r ~ d r ~ d ϕ
V ( A C ) π / 4 π / 4 e x p { [ r ~ 2 + i 2 ( 1 + ε ) ( 1 + ζ 2 ) 1 / 2 ξ r ~ c o s ϕ ] } R ( r ~ 2 , δ , ζ ) r ~ d r ~ d ϕ ,
P e x t ( T o t ) P i n c | η = 0 , ζ = 0 , ε = 0 , δ = 0 = ( 4 z R a ) e ξ 2 Im S ( G B )
P e x t ( A C ) P i n c | η = 0 , ζ = 0 , ε = 0 , δ = 0 = ( 4 z R a ) e ξ 2 e r f i ( ξ 2 ) Re S ( G B ) ,
z p = l w z ~ p + z f ,
v ( z ~ p ) = 3 2 v ¯ ( 1 z ~ p 2 ) ,
ξ ( t ) = 1 ω ( z p ) v ( z ~ p ) t .
g ( z ~ p ) = 3 4 ( 1 z ~ p 2 ) ,
G ( z ~ p ) 1 z ~ p g ( z ) d z = 1 4 ( z ~ p 3 + 3 z ~ p + 2 ) ,
G ( z ~ p ) = u [ 0 , 1 ] ,

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