# T-matrix methods for electromagnetic structured beams: A commented reference database for the period 2019-2023

Gérard Gouesbet

► To cite this version:

Gérard Gouesbet. T-matrix methods for electromagnetic structured beams: A commented reference database for the period 2019-2023. Journal of Quantitative Spectroscopy and Radiative Transfer, 2024, 328, pp.109015. 10.1016/j.jqsrt.2024.109015. hal-04574850

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Review

# T-matrix methods for electromagnetic structured beams: A commented reference database for the period 2019–2023

Gérard Gouesbet

CORIA-UMR 6614- Normandie Université, CNRS-Université et INSA de Rouen, Campus Universitaire du Madrillet, 76800, Saint-Etienne-du Rouvray, France

<figure><img src="image_3.png" /></figure>

## ARTICLE INFO

**Keywords:**

Generalized Lorenz-Mie theories

Extended boundary condition method

T-matrix formulations

Light and acoustical scatterings

Beam shape coefficients

Optical particle characterization

## ABSTRACT

The present paper pursues a fairly usual tradition basically devoted to reviews describing advances in some light scattering theories in which the illuminating beam is described using vector wave function expansions, with expansion coefficients known as beam shape coefficients. After the most recent review dealing with the period 2014–2018, the present review deals with the fiducial period 2019–2023. Relying on about 700 references, it discusses as well other light scattering approaches and applications to optical particle characterization techniques, therefore providing an overview of light scattering approaches and applications during half a decade.

## 1. Introduction

The present review is inscribed in a tradition of reviews regularly published since a few decades, commenting new advances in some light scattering theories (discussed below) and applications, namely published in 1994 [1], 2000 [2], 2009 [3,4], 2014 [5] and 2019 [6], see as well reports on the OPS/OPC/LIP conferences in the introduction to Section 9.

This review basically deals with light scattering theories (and applications) in the case when the scatterer is illuminated by a structured electromagnetic beam (specifically laser beams) which is described in terms of vector wave function expansions, with expansion coefficients known as Beam Shape Coefficients (BSCs). Such theories encompass Generalized Lorenz-Mie Theories (GLMTs), e.g. [7,8] and references therein dating back to 1982 [9], and Extended Boundary Condition Method (EBCM), e.g. [10,11], also called Null-Field Method [12]. GLMTs are rigorous analytical theories which are however limited to classes of particles which, thanks to their symmetry and morphological properties, allow one to use the method of separation of variables to solve Maxwell's equations, with the expression GLMT *stricto sensu* denoting the case when the particle is a homogeneous spherical particle. Conversely, EBCM is a semi-analytical method allowing one to deal with non-spherical particles, although it may become ill-conditioned in the cases of highly elongated or flattened particles [12–15]. Although EBCM is mostly used in the case of plane wave illumination (e.g. solar light), due to applications dealing with light scattering by airborne particles (on earth or elsewhere), it happened that EBCM has been used as well in the case of structured beams to which the present review is essentially devoted, e.g. [16–20] (for quotations outside of the

present fiducial period). EBCM has also been mentioned in Problems 13 and 14 of [21], which have been solved since then, see “Introduction” of [6], and has originally been used by using an Angular Spectrum Decomposition (ASD) rather than BSCs, e.g. [22,23] devoted to a review of electromagnetic scattering of arbitrary shaped beams by arbitrary shaped particles and to the relationship between ASD and BSCs. Acoustical fields will be considered as well due to analogies between acoustical and electromagnetic scatterings, in particular the fact that incident acoustical fields may be described by using acoustical BSCs.

Although the expression “T-matrix” has been recurrently (and erroneously) used as synonymous of EBCM, it happens that GLMTs may be viewed as T-matrix methods as well, particularly GLMT *stricto sensu*, e.g. [22,24]. A discussion of this issue may be found in the introduction of [6], with quotations from Doicu et al. [12], Bates and Wall [25], Hackman and Lim [26], Doicu and Wriedt [27], Nieminen et al. [15, 28], Zakharova et al. [29], Alonso et al. [30].

The references quoted in the present review have been, for most of them, obtained from citation alerts in such a way that issues related to GLMTs and EBCM for structured beams are certainly covered in a fairly exhaustive way. Other papers, less closely related to GLMTs and EBCM, but obtained as well from citation alerts, allow me to cover a broader scope related to other light scattering theories, and to applications, although I cannot claim exhaustibility for such topics which do not pertain to the main stream of this review. Papers prior to the fiducial period may be quoted as well for pedagogic skills, in particular to help the understanding of the story of such or such specific issue. It is then intended that the present review may be useful to experts in the field, but may provide as well a comprehensive inroad for the newcomers,

<footnote>E-mail address: gouesbet@coria.fr.</footnote>
although there may be some amount of bias generated by the limited knowledge and expertise of the author.

As a last comment, let us mention that a quoted paper is discussed with some details in such or such section (or subsection) although it could as well be eligible to be located in other sections (subsections). There is then a choice of location for extended comments in a given section (subsection). Papers still relevant to this section (subsection) but commented elsewhere may be reached by using keywords.

The paper is organized as follows. Section 2 is devoted to advances concerning the evaluation of BSCs. Section 3 discusses light scattering for various shapes and morphologies. Section 4 discusses light scattering for various beam shapes. Section 5 is devoted to mechanical effects of light. Section 6 deals with internal fields and resonances. Section 7 is devoted to photonic nanojets. Section 8 deals with the optical theorem. Section 9 is devoted to optical particle characterization techniques while, in particular, Section 10 deals with the use of neural networks/machine learning. Section 11 is devoted to alternative light theories. Section 12 deals with acoustical waves. Section 13 deals with quantum related issues. Section 14 is devoted to reviews and tutorials. Section 15 is devoted to miscellaneous issues while Section 16 is a conclusion.

## 2. Advances to the evaluation of beam shape coefficients

Sections 2.1 to 2.3 are devoted to BSCs in spherical coordinates while Section 2.4 is devoted to spheroidal coordinates, and Section 2.5 to miscellaneous issues. In spherical coordinates, BSCs are usually denoted $G_{n,TM}^m$ and $G_{n,TE}^m$ in which $n$ ranging from 1 to $\infty$ is the partial wave (multipole) order, $m$ is the azimuthal order, $TM$ stands for “Transverse Magnetic” and $TE$ for “Transverse Electric”. In spheroidal coordinates, they may be denoted $G_{n,TM}^m$ and $G_{n,TE}^m$ [31], although the physical meaning of $TM$ and $TE$ does not exactly identify with the one in spherical coordinates due to the fact that the GLMT for spheroids cannot rely on scalar Bromwich potentials originally used for spheres [32]. BSCs and beam shape distributions in circular and elliptical cylindrical coordinates are not considered in the present section, but see [33] for a review. Other papers devoted to BSCs can be reached by using “BSC” as a keyword. Papers devoted to quadrature techniques can be reached by using “quadrature” as a keyword. Some of them are discussed below in this section when comparing different methods of evaluations of BSCs, see end of Section 2.1.

### 2.1. Localized approximations

Various methods can be used to evaluate BSCs in spherical coordinates, namely quadratures, e.g. [34] to be completed by [35] demonstrating that we can actually define an infinite number of quadratures, localized approximations, e.g. [36,37], finite series, e.g. [38,39] and the use of an ASD, e.g. [40–42]. Up to recently, the most famous and celebrated method has been the use of a localized approximation (with variants), e.g. [43] for a review to be completed by [44–46]. A rigorous justification of localized approximations for so-called arbitrary shaped beams has been established by using what has been called the N-beam method [47]. Unfortunately, these so-called arbitrary shaped beams excluded (i) beams exhibiting an axicon angle, that is to say a term of the form $\exp(\pm ikz \cos \alpha)$ – for a propagation in the z-direction – in the description of the beam and/or (ii) exhibiting topological charges, that is to say a term of the form $\exp(i\ell\varphi)$, with $\ell$ being the topological charge and $\varphi$ the azimuthal angle – again for a propagation in the z-direction – in the description of the beam.

Using again the N-beam method, it has then been demonstrated that localized approximations for axicon beams were only valid for small axicon angles [48], while limitations of localized approximations due to topological charges have been introduced in [49]. As a result, several papers revisiting localized approximations have been published

namely [50] dealing with on-axis zeroth-order Bessel beam, [51] dealing with Bessel beams and radiation pressure forces, [52] dealing with Mathieu beams, [53] dealing with paraxial Laguerre-Gaussian beams freely propagating, [54] dealing with Laguerre-Gauss beams focused by a lens, [55] dealing with discrete superpositions of Bessel beams (frozen waves, see below).

In the fiducial period, these papers may be completed by Votto et al. [56] comparing three different methods (quadrature, localized approximation, finite series) of evaluations of BSCs of paraxial Laguerre-Gauss beams freely propagating, Valdivia et al. [57] dealing similarly with the comparisons of the same three different methods in the case of Bessel-Gauss beams. Other papers dealing with localized approximations may be reached by using “loca” as a keyword.

### 2.2. Finite series

Finite series method has been efficient, decades ago, to speed up computations of BSCs with respect to quadrature techniques (in the case when quadratures have to be evaluated numerically), see [38, 39]. It has afterward been virtually given up due to the success of the localized approximation techniques. However, limitation of these localized techniques, depicted in the previous subsection, provided a renewal of the interest for finite series. We then have to quote [58,59] dealing with paraxial Laguerre-Gauss beams freely propagating, [60, 61] dealing with Laguerre-Gauss beams focused by a lens, possibly with comparisons against other methods. Let us also note that the finite series techniques originally relied on particular choices of the angle $\theta$ in spherical coordinates, namely $\theta = 0$ and $\theta = \pi/2$, depending on whether $(n-m)$ is odd or even, in which $n$ is the partial wave order and $m$ the azimuthal order, using both the fields and the derivatives of the fields. A modified series technique in which $\theta$ is not necessarily specified to such special values has been designed in [62], with applications to linearly polarized Bessel beams. Finite series have also been used to express the BSCs of certain beams in terms of the BSCs of other beams, the latter having been evaluated using finite series. This approach has been used to express the BSCs of Hermite-Gaussian beams [63] and of Ince-Gaussian beams [64] through finite series Laguerre-Gaussian BSCs. Furthermore, Lin et al. [65] dealt with a radial quadrature method to formulate the BSCs of shaped beams, which is shown to be equivalent to the finite series method. As a by-product, this paper established finite series expansions of associated Legendre functions. For other papers dealing with the finite series technique, use “finite” as a keyword. This encompasses in particular [56,57] quoted in the previous subsection.

### 2.3. Blowing-ups

When dealing with the finite series technique for (paraxial) Gaussian beams, it has been observed, decades ago, that the BSCs blow-up when the order of the BSCs increases. This phenomenon could be attributed to a lack of numerical accuracy but it has been suggested that it actually corresponds to a genuine phenomenon, e.g. [8], pp. 163–170. The revival of the finite series summarized in the previous subsection sets again the issue on the stage, in particular after the observation of blowing-ups when dealing as well with paraxial Laguerre-Gauss beams freely propagating [56]. In [66], it is shown, in the case of paraxial Gaussian beams, relying on the properties of generalized Bessel polynomials, that blowing-ups are indeed genuine phenomena. This result is confirmed in [67], relying on the properties of both hypergeometric functions and generalized Bessel polynomials. Criteria for blowing-ups are furthermore established, leading to the conclusion (with variants) that blowing-ups occur for a critical order $n_c$ which is $O(1/s^2)$ in which $s$ is the beam confinement parameter of the Gaussian beam defined as $1/(kw_0)$ in which $k$ is the wavenumber and $w_0$ the beam waist radius. Finally Shen et al. [68] established a physical interpretation for blowing-ups, showing that they actually originate from the contribution of evanescent waves. Note that evanescent waves are discussed as well
by Wang et al. [69] who studied the spherical harmonics expansion of evanescent waves using an ASD for homogeneous and inhomogeneous plane waves, and for circular Gaussian beams. They found that the BSCs of the evanescent waves increase exponentially along the increasing partial wave number, corroborating the connection between evanescent waves and blowing-ups.

## 2.4. Advances in spheroidal coordinates

For this subsection, we must introduce the distinction between extrinsic and intrinsic methods to the evaluation of BSCs. According to [21], intrinsic methods evaluate BSCs of a GLMT posed in a certain coordinate system in terms of quantities pertaining to the same coordinate system while extrinsic methods evaluate BSCs of a GLMT posed in a certain coordinate system in terms of BSCs in a different coordinate system. When dealing with the GLMT for spheroids, BSCs have originally been evaluated using an extrinsic method, see [31] for a review. The development of intrinsic methods in spheroidal coordinates then became a desirable task, in particular with the hope of developing localized approximations in spheroidal coordinates, see problems 2 and 5 in [21]. A first approach was published in 2013 [70] which established a general framework of an intrinsic method to the evaluation of BSCs in spheroidal coordinates in the most general off-axis case, followed by [71] which dealt with the intrinsic evaluation of spheroidal BSCs in the case of an on-axis parallel configuration and by [72] which implemented the special case of a zeroth-order Bessel beam. In the fiducial period, this line of research has been pursued by [73] which dealt with an intrinsic method for the evaluation of BSCs in spheroidal coordinates for oblique illumination. This line of research has however not been yet achieved. In particular intrinsic localized approximations in spheroidal coordinates are not yet available.

For a review on intrinsic and extrinsic methods, dealing as well with cylindrical coordinates, see the introduction of [73]. The evaluation of spheroidal BSCs using either intrinsic or extrinsic methods is discussed as well in [74].

## 2.5. Miscellaneous

Instead of an ASD, Shen et al. [75] introduced a cylindrical spectrum decomposition (CSD) to evaluate BSCs of shaped beams in spherical coordinates, in a GLMT framework. The method first expands the incident field using a CSD and then expands the cylindrical waves of the CSD into spherical waves based on the relationship between cylindrical and spherical waves. This approach may be viewed as an extrinsic approach (as defined in the previous subsection). CSD is used as well by Shen et al. [76], with elliptical Gaussian beams, cosine Gauss beams and Hermite-Gauss beams as examples, allowing one to evaluate the BSCs with efficient one dimensional integrations. Shen and Yu [77] presented a new efficient quadrature method to the evaluation of BSCs relying on azimuthal and radial integrals, with applications to tilted plane waves and Gaussian beams. See as well Shen et al. [78] who, in a GLMT framework, displayed BSCs in terms of combinations of expansion coefficients of a scalar function, with applications to x-polarized Bessel beams and Gaussian beams.

# 3. Light scattering for various shapes and morphologies

Although the present review is intended to deal mainly with “arbitrary shaped beams” (particularly laser beams), a few papers dealing with plane waves will be nevertheless quoted for at least two reasons (i) Lorenz-Mie Theory (LMT) is actually a special case of GLMT using plane wave BSCs and (ii) in the general case of oblique illumination, plane wave scattering must be treated in a GLMT framework, with BSCs more complicated than the ones used in LMT e.g. [79] for oblique illumination, and [9] for BSCs in the LMT framework.

## 3.1. Spheres and spherical coordinates

Particles in spherical coordinates are homogeneous spheres (GLMT *stricto sensu*), but also multilayered spheres, e.g. [80–82], spheres with an eccentrically located spherical inclusion (sometimes referred to as the eccentric stratified sphere model), e.g. [83–85], and assemblies of spheres and aggregates, e.g. [19,86]. The case with several inclusions may have to be considered as well.

In the fiducial period, spheres are discussed by Yang et al. [87] in the case of a Bessel beam interacting with a charged sphere, with analytical expressions of BSCs derived in the GLMT framework using an ASD method, by Chrissoulidis et al. [88] dealing with the scattering of a corrugated plane wave by a sphere, using an extended Lorenz-Mie solution (which could possibly be treated in a GLMT framework as confirmed by a private communication with Chrissoulidis), by Orlov and Berskys [89] who discussed, in a GLMT framework, the scattering of gold and silicon spheres illuminated by electromagnetic fields which carry both longitudinal and transverse angular momentum (called pho-tonic wheels), with BSCs (denoted $A_{mn}$ and $B_{mn}$) evaluated by using a quadrature method under the form of one-dimensional integrals (see as well this paper below when discussing aggregates), and correctly stated that GLMT is a T-matrix method, by Tang et al. [90] dealing with the scattering of arbitrary-shaped optical polarized beams (vector Bessel and Airy beams) by a perfect electromagnetic conductor (PEMC) sphere in a GLMT framework (PEMC may be viewed as a man-made metamaterial, acting as a perfect reflector of electromagnetic waves, although losses may be introduced), with BSCs evaluated using an ASD, by Albashir et al. [91] who developed a code to calculate extinction and scattering cross-sections of spherical nanoparticles, relying on Mie theory, by Peng et al. [92] who studied the case of irregular spheres, in relation with the use of spherical calibrators which are not perfect and may then cause calibration errors, and proposed a method of estimation and compensation of these errors. Let us note as well the use of ASD to deal with the scattering by a sphere proposed as well by Kashter et al. [93], with a rebuttal given by Gouesbet et al. [42], already quoted above, and which has been an opportunity to review the uses of ASD to deal with light scattering.

The case of multilayered concentric dielectric spheres is discussed by Hudak et al. [94] who described a numerical iterative optimization algorithm to the design of optical scattering properties (in a plane wave context), by Rasskazov et al. [95], and [96], who presented a computer code for calculating near- and far-field electromagnetic properties of multilayered spheres, using transfer-matrices with close-form expressions, and in particular provided explicit expressions for the special case of lossless core and shell (still in a plane wave context), by Guo and Shen [97] who dealt with internal and external fields of a multilayered sphere in a GLMT framework, using an improved recurrence algorithm, by Li and Li [98] who proposed an efficient progressive methodology to deal with multilayered concentric nanoparticles, considering two adjacent shell layers at a time, with a marching procedure from the innermost to the outmost layer, completed by Li and Li [99]. See as well Lamberg et al. [100] dealing with the simulation of scattering from layered spheres with applications to corneal sensing, using a T-matrix method, and Li et al. [101] dealing with a study of the degree of linear polarization for a shell-core structure and coated super-spheroids.

For the case of sphere with inclusions, we have Chrissoulidis et al. [102] who dealt with the case of a sphere eccentrically placed in a dielectric sphere, with consideration as well of non-spherical bodies (for a plane wave illumination, but frequency-modulated, and with double amplitude and phase modulation), Eremenko et al. [103] who dealt with frequency domain simulations for electromagnetic resonant oscillations in a ball with eccentric spherical dielectric inclusion, using Hertz potentials (GLMT for eccentrically located inclusion could have been used, as agreed by the authors, following a private communication),

The case of aggregates is discussed by Han et al. [104] who discussed glistenings (namely fluid-filled micro vacuoles in intraocular
lens, supposed to be spherical) that cause light scattering and may affect the quality of vision, a problem which is viewed as a scattering by a single object made of an assembly of spheres, in the framework of a T-matrix formulation, relying on the expansion of the incident wave and of the scattered wave in terms of vector spherical wave functions, by Höller et al. [105] dealing with superstructures of gold nanospheres which offer augmented surface-enhanced Raman scattering activities beyond the limits of their individual building blocks, studying the optical properties of ideal spherical particles using Mie theory, and the optical properties of systems containing a finite number of coupling spheres being obtained using a generalized multipole Mie theory, referring to the GLMT for assemblies of spheres and aggregates and to an algorithm developed by Yu-Lin Xu, modified and extended to allow more efficient parallelized calculations [106], by Orlov and Berskys [89], already quoted above, dealing with a chiral cluster of three nanoparticles in the focal plane of a photonic wheel, by Haspel and Cohen [107] dealing with scattering by a collection of sparsely spaced spherical scatterers of Mie-scattering sizes, and by Thong and Yoon [108] using GLMT, Generalized Multiple-Particle Mie (GMM) solution, and geometrical optics, to study analytical far-field solutions of an aggregate of spheres, to be completed by another paper from the same authors [109] dealing with extinction cross-section curve of an electromagnetic scattering from aggregates, and more specifically with the inverse problem of deducing the aggregate configuration from the extinction cross-section curve, using GMM-GLMT approaches with a global optimizer, by Olmos-Trigo and Xavier Zambrana-Puyalto [110] who discussed the behavior of the helicity (defined as the projection of the total angular momentum onto the linear momentum) of the incoming beam when interacting with a lossy sphere, with a focus on the helicity inside the sphere, in comparison with helicity properties of the scattered wave, by Argentin et al. [111] who used the Rayleigh-Debye-Gans approximation to study Fractal Aggegates (RDG-DA) of soot particles under planar illumination, demonstrating that correction factors to the RDG-FA for forward scattering and the absorption cross-section are due to a competition between internal field hot-spots caused by point contact between the spherical monomers and a decrease of the field amplitude as the field propagates through the aggregate, and proposed a semi-empirical model intended to correct the RDG-FA theory, and by Arfan et al. [112] who analyzed the scattering of a Gaussian beam by a PEMC sphere in the framework of the GLMT, with BSCs evaluated by using a localized approximation.

For other papers devoted to the scattering by spheres and the use of spherical coordinates, use “sphere”, “spherical”, “bead”, “drop”, “bubble”, “aggregate”, “cluster”, “inclusion”, “core-shell”, “Mie”, “LMT” as keywords.

### 3.2. Spheroids and spheroidal coordinates

For this subsection, we shall be content to ask the reader to gather papers devoted to the issue of spheroids and spheroidal particles by using “spheroid”, “spheroidal”, as keywords. The case of ellipsoids is furthermore discussed by [113] to be later commented.

### 3.3. Cylinders and cylindrical coordinates

Mitri [114] discussed optical scattering, absorption and extinction cross-sections, and energy efficiencies of a cylindrical material, with arbitrary geometrical cross-sections, exhibiting circular dichroism in arbitrary-shaped monochromatic light sheets. The theoretical approach relies on the use of partial wave series expansions in cylindrical coordinates and of the Poynting theorem. An emphasis is stressed on mode conversions between TM and TE components of the scattered field. Examples are provided in the case of a circular PEMC cylinder in air illuminated by circularly polarized Airy light sheets. Loh [115] studied Stokes and anti-Stokes quantum fields from a mesoscopic microparticle with cylindrical geometry. Khoobjou et al. [116] dealt with an optimal

design for decreasing optical loss and dispersion in photonic crystal fibers (specifically simple hexagonal photonic crystal fibers, photonic crystal fibers with elliptical holes and with elliptical and stellar holes), using GLMT to investigate scattering, absorption and extinction cross-sections allowing one to obtain dispersion, and absorption of energy of the nanostructures under study, with BSCs computed by using the triple quadrature technique [34]. Chen et al. [117] dealt with the study of Gaussian beam propagation through a biaxial anisotropic circular cylinder, using cylindrical vector wave expansions, following the GLMT scheme, with BSCs evaluated by using a localized approximation. Zhang et al. [118] studied Gaussian beam scattering by an infinite cylinder with a spherical inclusion, using both cylindrical and spherical vector wave functions, and GLMT addition theorems, with the Gaussian beam described by using a localized approximation. Next, Chen et al. [119] dealt with a study of electromagnetic beam propagation through a gyrotropic anisotropic circular cylinder, with a theoretical structure akin to a GLMT although, as stated by the authors, they used a projection method which relies on “an explicit description of the incident beam”, meaning that the evaluation of BSCs (specifically cylindrical BSCs) is not necessarily required. A GLMT for cylinders is however recovered if the incident fields are expanded over cylindrical vector wave functions. Otherwise, the projection method takes the form of a semi-analytical method on which integrals related to incident fields have to be evaluated numerically. Various kinds of illuminating beams are considered, namely Gaussian beams, Hermite-Gaussian beams and zeroth-order Bessel beam. In particular, Gaussian beams are described by using a third-order Davis beam formulation [120–122], with a good agreement with results using a localized beam model. Mitri [123] studied the scattering of a dielectric cylinder in light-sheets of arbitrary wavefronts and polarizations, using a GLMT-like approach in terms of BSCs.

Other papers relevant to the present subsection may be found by using “cylinder”, “cylindrical”, “fiber” as keywords.

### 3.4. Slabs

Slabs are considered by Zhang et al. [124] who proposed a general theoretical scheme to investigate the propagation of a focused electromagnetic beam in a uniaxial anisotropic slab, using cylindrical vector wave functions and a SIEM method with surface integration (from a private communication), applied to the case of Gaussian beams – For SIEM and akin numerical techniques, see [125] and papers quoted in subsection 8.5 of [6] –, Hermite-Gaussian beams and doughnut mode beams, Lu et al. [126] comparing the behavior of differently polarized Bessel vortex beams through a uniaxial anisotropic slab, using transformations of spherical vector wave functions in terms of cylindrical vector wave functions, Liu et al. [127] dealing with the reflection and transmission of a Bessel vortex beam by a stratified uniaxial anisotropic beam, using cylindrical vector wave functions, and closed form expressions of expansion coefficients, with the example of a three-layered slab, Yan et al. [128] dealing with electromagnetic wave propagation through a chiral slab, using a projection method already mentioned above, and discussing its relationship with GLMT, applying their approach to Gaussian beams, Hermite-Gaussian beams, and radial doughnut mode beams, and Sigel [129] dealing with light propagation in a layered refractive index profile.

### 3.5. Chiral particles

Su et al. [130] dealt with internal and near-surface fields for a chiral sphere under arbitrary laser beam illumination (applied to Gaussian beams, Hermite-Gaussian beams, doughnut mode, Laguerre-Gaussian beams, and Bessel beams), using again the semi-analytical projection method, with results in agreement with those from the GLMT. Li and Feng [131] investigated the scattering by a pair of chiral spheres in an elliptical Gaussian beam. Bai et al. [132] dealt with the scattering
of chiral spheres in the GLMT framework, under illumination by dual Gaussian beams, with BSCs evaluated using a localized approximation. Olmos-Trigo et al. [133] used a GLMT framework to search for objects that yield maximum electromagnetic chirality in their emitted wavefields. Other papers relevant to the present subsection may be found by using “chiral” as a keyword.

### 3.6. Last words

As last words in this Section 3, let us mention the case of cuboids discussed by [134], see subsection of “photonic jets”, and cases when particles are made from metamaterials which can be reached by using “metamaterial” as a keyword. An exotic case is the one of Janus particles discussed by Gao et al. [135]. To reach other papers related to this section, use “non-spherical” as a keyword.

## 4. Light scattering for various beam shapes

### 4.1. Gaussian beams

Gaussian beams have been, for a long time, the most common kind of beams considered in various fields, particularly optical particle characterization, and the development of GLMTs has then indeed been focused on Gaussian beams, e.g. [7,136–138], although GLMT is actually a genuine arbitrary beam theory [139,140]. Examples in the fiducial period are by Khosravi et al. [141] who simulated Gaussian electromagnetic wave interaction and its effects on the dynamics of metallic nanospheres, computing average forces and torques, Khosravi et al. [142] dealing again with forces and torques under Gaussian beam illumination, with BSCs evaluated in a GLMT framework, Khosravi et al. [143] using again the GLMT framework to study the temperature of plasma nanospheres produced by Gaussian illumination, with the purpose of increasing the arsenal of methods available to fight against cancer cells, providing an alternative to surgical and chemical therapies, and Gao et al. [144] who, based on the equivalent medium theory, discussed the scattering characteristics of terahertz Gaussian beams by moist media, numerical results showing that the extinction coefficient of particles is mainly affected by humidity, and that the scattering efficiency is affected by both temperature and humidity, relying on the GLMT with BSCs of the Gaussian beam expressed by using a localized approximation.

For other papers dealing with Gaussian beams, see [145–147], and [148], to be discussed in the sequel, and many other papers which can be reached by using “Gauss” as a keyword.

### 4.2. Light sheets

Gaussian beams pertain to a class of radially symmetrical beams characterized by a beam waist radius $w_0$. This symmetry may be broken leading to Gaussian laser sheets characterized by two different beam waist radii, say $w_{0x}$ and $w_{0y}$. Earlier investigations of Gaussian laser sheets in a GLMT framework may be found in [149–151], and [152]. Such beams may be called as well elliptical Gaussian beams, e.g. [76, 131]. The concept of laser sheets is however more general. An example is provided by Zhang et al. [153] who studied the scattering of a non-paraxial Bessel light-sheet by a sphere of arbitrary size in the GLMT framework, to be completed by [154] who again studied the scattering of non-paraxial Bessel pincer light-sheets depending on two parameters (namely, the order $l$ of the Bessel beam and a scaling parameter $\alpha_0$) by spherical scatterers, in a GLMT framework using the ASD and expressing the BSCs in terms of one-dimensional integrals. Another example concerns the case of Airy light-sheets, which can be reached by using “Airy” as a keyword. For other papers devoted to light-sheets, see [155] and many other papers which may be reached by using “sheet” as a keyword.

### 4.3. Bessel beams

After the original works by Durnin [156], Durning et al. [157], and Mishra [158]. Bessel beams are the most paradigmatic example of nondiffracting beams, e.g. [159], and Recami et al. [160] in a book edited by Hernandez-Fugueroa et al. [161], and see as well Rondon [162] for the use of the terminology “localized waves”. They indeed possess the property of propagating without modifying their transverse intensity profile. In other words, they are “tube-like” and do not exhibit any waist, then contrasting with the most familiar case of Gaussian beams. They furthermore have the property of self-healing, i.e. they are reconstructed in their integrity after interaction with an obstacle. This property can be well understood if we know that the ASD of Bessel beams is made from plane waves all having the same tilted angle with respect to the direction of propagation. Therefore, after an obstacle, the beam is reconstructed from portions of the plane wave components which have not been affected by the scattering phenomenon. They possess an axicon angle and, apart of the case of zeroth-order Bessel beams, they exhibit as well a topological charge. In the last case, they are then vortex beams, with a null singularity on the axis, excepted in the cases – largely ignored – when the topological charge is $\pm 2$, e.g. [163–165], see subsection 3.4 in [6] in which beams with topological charges are named helical beams. Furthermore, the existence of the axicon angle implies that the velocity along the direction of propagation of the beam is $c \cos \alpha$, and can therefore been made – in principle – as small as possible. It is however a general property of laser beams that they propagate (even in vacuum) slower than the speed of light [166]. We furthermore recall that they pertain to the kind of beams which may be inaccurately described by localized approximations (Section 2.1).

Examples of works devoted to Bessel beams in the fiducial period are by Qu et al. [167] dealing with the scattering of aerosols by a high-order Bessel beam for multimedia information transmission in atmosphere, combining GLMT and GMM, by Cui et al. [168] who studied the local dynamical characteristics of Bessel beams upon reflection near the Brewster angle, by Zhang and Shen [169] who compared standard elliptical Bessel beams (EBB) and refracted circular Bessel beams (CBB) scattering by a spherical particle, using the ASD and expressing the BSCs as single quadratures or in closed forms (EBB and CBB of this paper may however be diffracting), by Glukhova and Yurkin [170,171] who proposed a general classification of all Bessel types based on the superposition of transverse Hertz vector potentials, generalized the formalism of the Mueller matrices to arbitrary Bessel beams accounting for their vorticity, and implemented these beams in the ADDA code (dealing with the discrete dipole approximation DDA) using GLMT to provide reference data, simulating the scattering of zeroth-order Bessel beam by a homogeneous or coated sphere, by Bai et al. [172] who studied the interactions between cluster chiral nanoparticles and a high-order Bessel beam, in the GLMT-GMM framework, by Dong et al. [173] dealing with the propagation of Bessel beams in the atmospheric environment, with scattering by raindrop particles simulated using GLMT and the evaluation of BSCs using a localized approximation (specifically a version named the Integral Localized Approximation, ILA), by Wang et al. [174] who relied on GLMT to study the scattering of metal particles under Bessel beam illumination, with BSCs provided in closed forms, by Bai et al. [175] who studied the interactions between collective multilayered cells and off-axis high-order Bessel beams, in a GLMT framework, with BSCs provided in closed forms, by Arfan et al. [176] dealing with the scattering for a PEMC sphere under Bessel beam illumination, in a GLMT framework, and by Chafiq et al. [177] who dealt with the calculation of BSCs for various nondiffracting beams (cosine beams, Bessel beams, and parabolic beams) relying on the use of an ASD and of a Whittaker integral. For other papers dealing with nondiffracting beams, use “nondiffracting” as a keyword. For other papers dealing with Bessel beams, see as well Mun et al. [178], and use “Bessel” as a keyword.
## 4.4. Mathieu beams

Mathieu beams pertain to the class of nondiffracting beams [179, 180]. In contrast with Bessel beams, these Mathieu beams exhibit an elliptical feature which have been used for instance by Lopez-Marisca et al. [181] and by Kartashov et al. [182]. They exhibit as well an axicon angle and a topological charge. We already quote such beams, specifically zeroth-order Mathieu beams, when dealing with the issue of localized approximations [52]. In the fiducial period, as an example, Mathieu beams have been discussed by El Mechate et al. [183] who introduced a new index in the description of the beams, allowing one to control the angular spectrum of the beams to change their symmetries, without modifying their nondiffracting property. Other papers devoted to Mathieu beams, in particular acoustical Mathieu beams, can be reached by using “Mathieu” as a keyword.

## 4.5. Frozen waves

Frozen waves pertain as well to the class of nondiffracting beams. They are obtained by superpositions of Bessel beams and allow one to modulate the beam along the longitudinal axis but also and simultaneously along a transverse direction, effectively providing intensity patterns of arbitrary shapes. Within this framework, Ambrosio [184] dealt with scalar millimeter-structured beams allowing one to choose a two-dimensional spatial intensity pattern dependent upon the longitudinal coordinates, and derived analytical expressions of BSCs using GLMT, with relevance to the manipulation guidance and/or trapping of micro- and millimeter-sized particles, Ambrosio et al. [185] dealt with the evaluation, using three distinct approaches in a GLMT framework, of BSCs of continuous frozen waves, built from the integral superposition of Bessel beams over their longitudinal wavenumber, while Ambrosio [186] dealt with zeroth-order continuous frozen waves, in the GLMT framework, for different polarizations. See as well Ambrosio et al. [187] for discrete frozen waves, Ambrosio et al. [55] already quoted above, and other papers to be reached by using “frozen” as a keyword.

## 4.6. Bessel-Gauss beams

Bessel beams have the disadvantage of carrying an infinite amount of energy, implying that they are not physically realizable. This defect is repaired by using Bessel-Gauss beams which are finite energy, physically realizable wave fields, resulting from the apodization of Bessel beams by Gaussian functions. Beside Valdivia et al. [57] and Shen et al. [41] already quoted above, who discussed the connection between the ASD, the quadrature method, and a localized approximation of GLMT, in the case of vector Bessel-Gauss beams, we have papers by Shi et al. [188] who dealt with the case when the scatterers are formed by a spherical marine aerosol in the GLMT framework, with different polarizations, with BSCs evaluated using an ASD in terms of one-dimensional integrals, and Shi et al. [189] dealing with the scattering of dusty medium (uniform charged spheres and non-uniform concentric layered spheres) interacting with a Bessel-Gauss beam using GLMT and ASD, for instance with the aim to study the transmission of such beams during sandstorms. More generally, Chafiq and Belafhal [190] dealt with nondiffracting beams made physical (and diffracting) by using a Gaussian envelope, under the name of “pseudo-diffracting beams” (pNDBs) and, beside Bessel-Gauss beams, studied as well cosine-Gauss beams, Mathieu-Gauss beams, parabolic-Gauss beams and Lommel-Gauss beams, which are investigated under a paraxial approximation. They specifically calculate the Fourier transform of pNDBs in the focal plane of a converging thin lens, and examined the propagation in free space after the focal plane using the Huygens-Fresnel diffraction integral. Other papers on the issue may be reached by using “Bessel-Gauss” as a keyword.

## 4.7. Laguerre-Gauss beams

Laguerre-Gauss beams are helical beams (having topological charges) without exhibiting any axicon angle, which carry angular momenta of two kinds, namely a spin angular momentum associated with the polarization state of the beam and an orbital angular momentum related to the spatial variations of the field. We already encountered above Laguerre-Gauss beams either freely propagating, or focused by a lens. Beside, Dong et al. [191] dealt with a propagation model of partially coherent Laguerre-Gauss beams in turbulent atmosphere after scattering from a Gaussian rough surface, Arfan et al. [192] studied the scattering of a Laguerre-Gauss beam from a spherical marine aerosol particle using GLMT, Sun et al. [193] dealt with the scattering of electrically large arbitrarily shaped targets (specifically spheres, almonds, blunt cones, and blade models) illuminated by an off-axis Laguerre-Gauss beam, with a comparison with the on-axis case, using a physical optics approach which is a high-frequency approximation technique. For regular scatterers, GLMT may be used to the sake of comparisons. Also, Liu et al. [194] focused their interest on the evaluation of BSCs of Laguerre-Gauss beams using different methods in a GLMT framework (using ASD expressions approximated in the form of series or of single integrals, and a localized approximation). In this study, some BSCs using finite series are found to blow-up for high order partial waves, a phenomenon attributed to the loss of significant decimals in processing the series (but see blowing-up discussion in Section 2.3). For other papers on the issue, see [178], and use “Laguerre” as a keyword.

## 4.8. Pulses and laser pulses

We recall that GLMTs under laser pulse illuminations have been discussed by Gouesbet and Gréhan [195], Mées et al. [196–200], Han et al. [201], and Mées et al. [202]. Returning to the fiducial period, we have Panetta et al. [203] who dealt with the decay phase of the interaction of a femtosecond pulse with particles, focusing on how energy initially trapped within a particle is released over time after interaction with an incident Gaussian pulse, Zhang et al. [204] dealing with the optical breakdown, a photoionization process leading to a laser-induced plasma, during femtosecond laser propagation in water-cloud droplets, with an agreement between experiments and predictions, invoking as well GLMT to deny any disturbance of their results in the case of two droplets with respect to the single droplet case, Chaussonnet and Bardet [205] who investigated the scattering of an ultrashort pulse, viewed as a plane wave with a Gaussian temporal envelope, by a spherical air bubble by means of the LMT and the Debye expansion, relying as well on GLMT, to be completed by Chaussonnet et al. [206] who dealt with the modeling of multiple scattering transient of an ultrashort laser pulse by spherical particles (namely a cloud of bubbles in water) using the LMT, the RTE (Radiative Transfer Equation) and Monte-Carlo simulations, Tribelsky and Miroshnichenko [207] who introduced two models to investigate unusual counterintuitive dynamics of scattering by incident pulses, with application to the case of dynamic light scattering of a square-envelope pulse by an infinite circular cylinder in the vicinity of destructive interferences at Fano resonances. Other papers relevant to the issue are available by using “pulse” as a keyword.

## 4.9. Last words

As last words in this section, let us comment other kinds of beams, starting with Hermite-Gaussian beams, already quoted in Section 2.2, see Votto et al. [63], and with other papers which can be reached by using the word “Hermite” as a keyword, pursuing with Ince-Gaussian beams, already quoted as well in Section 2.2, see [64]. Gaussian vortex beams (i.e. Gaussian beams modulated by a helical term) are considered by Shi et al. [208], dealing with aerosol scattering and vortex beam
transmission in hazy atmospheres, and relying on GLMT. They in partic-
ular show that the attenuation of the Gaussian vortex beams is smaller
than that of the traditional Gaussian beam in propagating through
aerosol particles. Lu et al. [209] used the ASD to derive the BSCs of
circular Airy beams (which may in part be viewed as Bessel-like beams
because they exhibit a nearly nondiffractive property) by spherical
particles (both Rayleigh and Mie particles) in a GLMT framework,
in relation with abrupt autofocusing property and the generation of
optical forces and particle trapping, allowing the “intriguing” transport
behavior of Mie particle along curved and straight trajectories and
therefore transport along a circumventing route. Circular Airy beams,
with an arbitrary conical angle, are discussed as well by Lu et al. [210]
who studied their propagation relying on the ASD, with a closed-form
expression of the angular spectrum derived by using the stationary
phase method. We may be as well interested by another example
of arbitrary shaped beams, namely beams emitted by antennas and
interacting with objects, such as in Zhou et al. [211] who dealt with
simulations validated by experiments using a human torso phantom.
Airy beams as Airy light-sheets may furthermore be reached by using
the word “Airy” as a keyword, similarly for Lommel beams with
“Lommel” as a keyword. For doughnut beams, use “doughnut” as a
keyword.

**5. Mechanical effects of light**

5.1. Optical forces and torques

Theoretical studies of mechanical effects of light in a GLMT frame-
work have been published in a few papers, namely [7,136], [139,
140,212–214]. The last paper dealt with both forces and torques but
expressions for torques are given without any demonstration, such a
demonstration in the GLMT framework retrieved and slightly modified
from Polaert’s thesis is presented in [8], pp. 298–309. In these papers,
a first categorization of optical forces was introduced in terms of a di-
chotomy between mixing forces (involving couplings between incident
and scattered fields) and recoil forces (involving only scattered fields).
However, strange as it may be, no further systematic investigation
of the categorizations of optical forces has been pursued, except for
Lock [215] who dealt with a second level categorization between
gradient and scattering forces, but in the restricted case of on-axis
Gaussian beams in the weak confinement limit. For completeness, let
us note that a thorough review on GLMTs and mechanical effects of
laser light, for the period 1985–2018, has been provided in [74], on the
occasion of Arthur Ashkin’s receipt of the 2018 Nobel prize in physics
for his pioneering work in optical levitation and manipulation.

A systematic investigation of categorizations of optical forces in
the GLMT framework has been carried out in the fiducial period. It
started with the study of the scattering of Rayleigh particles (more
generally with electric dipoles defined only by its first electric Mie
coefficient *a*₁) illuminated by Bessel beams, both for longitudinal [216]
and transverse [217] components. These studies reveal the existence of
unexpected forces, called axicon forces (because it was believed that
they were typical of axicon beams), later renamed non-standard forces
when it became evident that they have a more general significance.
These works may be completed by [218], dealing with so-called axicon
terms associated with gradient forces, whose exact status has been
revealed later [219], and by [220] who thoroughly investigated optical
forces produced by on-axis Bessel beams. Thanks to a simpler on-axis
framework, forces obtained in the general off-axis case were then better
understood, including non-standard forces who existence was then
confirmed. The GLMT restricted to dipoles is furthermore discussed
for larger classes of beams, namely beams (i) called non-dark on-axis
axisymmetric beams of the first kind [221], including zeroth-order
circularly symmetric Bessel beams and localized models of Gaussian
beams (restricted or not restricted to the weak confinement limit), and
(ii) called non-dark axisymmetric beams of the second kind, and dark

axisymmetric beams (Bessel beams for special polarizations, Lommel
beams, Bessel-Gauss beams, Laguerre-Gauss beams) [222], involving as
well a mini-review to the best of the current understanding.

A question at stake was then to examine the relationship between
the GLMT restricted to electric dipoles and the dipole theory of forces.
After numerical comparisons in the case of Bessel beams, both for lon-
gitudinal [223] and transverse optical force components [224] which
produced excellent agreements, up to 1000 and 3000 decimal places
respectively, it has been formally demonstrated for arbitrary shaped
beams [225,226] that the GLMT restricted to electric dipoles indeed
agreed with the dipole theory of forces, but for an important formal
(not numerical) difference, namely while the dipole theory of forces
is expressed in terms of the total electric field (i.e. encompassing all
multipoles), the GLMT restricted to electric dipoles leads to the same
results by considering a few low-order multipoles. In other words, the
dipole theory of forces implicitly contains an infinite number of partial
waves whose contributions are exactly zero. Another result is that non-
standard forces appeared as being components of forces called spin-curl
forces, without having been up to this moment clearly identified.

These studies have afterward been completed for the cases of mag-
netodielectric particles (defined by their electric and magnetic Mie
coefficients *a*₁ and *b*₁ respectively) in the GLMT framework [227] with
comparisons against the dipole theory of forces [228], of quadrupoles
[229,230], and of arbitrary sized particles [231]. A summary of this
story is available from Gouesbet and Ambrosio [232]. As a whole, it
has been found that forces may be sorted along a partition relying on a
three-level categorization (mixing and recoil forces, gradient and non-
gradient forces, scattering and non-standard forces) and on a two-level
decomposition (K-forces with K being an integer ranging from 1 to ∞,
and electric/magnetic/magnetoelectric forces).

Other papers in the fiducial period for the year 2019 are by Zhang
et al. [233] who experimentally confirmed that radiation pressure is
similar to mechanical pressure and can also improve the luminescence
performance of perovskite quantum dots, showing furthermore that
operating by means of radiation pressure is more convenient than that
of mechanical pressure, radiation pressure being obtained by using a
400 nm femtosecond laser, by Zaza et al. [234] who demonstrated
that surfactant-free silicon nanoparticles of a predefined and narrow
size range can be selectively immobilized on a substrate by opti-
cal printing from a polystyrene colloidal suspension, with theoretical
analysis performed in a GLMT framework, by Rui et al. [113] who
calculated optical forces and torques exerted on Rayleigh particles with
spherical/ellipsoidal shapes in connection with optical trapping effects
and the manipulation of movement behaviors of trapped nanoparticles,
by Brzobohaty et al. [235,236] who dealt with self-arrangement of
wave-length-size dielectric particles to form a colloidal waveguide, due
to optical binding (i.e. force interactions between particles mediated by
scattered light), with Gaussian beam illumination and use of the GLMT
with a localized approximation, to model a dual-beam trap, or a multi-
sphere Mie theory, as well as FDTD (Finite-Difference Time-Domain)
method, by Gong et al. [237] who studied the radiation forces on re-
dundant particles (affecting the performance of optical surfaces) by an
incident Gaussian beam, in a GLMT framework, by Zhang et al. [238]
who studied optical trapping forces, discussed in terms of gradient and
scattering forces, exerted by a focused Bessel-Gauss correlated Schell-
model on Rayleigh dielectric spheres, by Alinezhad and Reihani [239]
who studied optical tweezers produced by a tightly focused Gaussian
beam, in a GLMT framework with BSCs evaluated by using the ASD, and
exhibited reasonable agreement between experiments and theoretical
results, by Ju et al. [240] who investigated radiation forces on a three-
layered particle in a focused Gaussian beam, in a GLMT framework with
BSCs evaluated by using a localized approximation, by Lin et al. [241]
who studied oscillations of microparticles embedded in viscoelastic hy-
drogels induced by harmonically modulated optical radiation pressure,
with a comparison between experiments and GLMT calculations, by
Rafferty et al. [242] who studied the optical deformation of single
aerosol droplets, using a cavity-enhanced Raman scattering, with GLMT used to calculate optical stresses, and with a discussion of the excitation of whispering-gallery modes, by Herranen et al. [243] who presented numerical T-matrix methods for modeling the dynamics of arbitrarily shaped particles (specifically cubic, ellipsoidal, Gaussian random ellipsoidal particles) trapped with optical tweezers, using Laguerre-Gaussian modes, by Yu et al. [244] providing an approach to fully decomposing an optical force into conservative and nonconservative components, by Ha et al. [245] who studied the effect of polarization on optical micromanipulation in a hexagonal lattice formed by three equiamplitude plane waves that have their wave vectors lying equiangularly in a plane, leading to trapping versus detrapping phenomena tunable by tayloring the polarization of the incident beams, by Ge et al. [246] who carried out a numerical study using FDTD and Maxwell stress tensor of optical trapping properties of nanoparticles on metallic film with periodic circular holes, by Zheng et al. [247] dealing with the decomposition of optical forces into gradient and scattering parts, by Mitri [248] dealing with radiation forces and torques (per-length) exerted on perfect electrically conducting (PEC) cylinders of corrugated and elliptical geometrical cross-sections (under plane wave illumination), by Mitri [249] who dealt with optical radiation force on an electrically conducting elliptical cylinder having a smooth or ribbed surface, using however cylindrical Bessel functions instead of Mathieu functions usually used to deal with elliptical coordinates, leading to a loss of “analyticity” due to the fact that cylindrical wave functions are not orthogonal to elliptical surfaces (private communication kindly acknowledged), by Mitri [250,251] who dealt with the optical radiation force expression for a PEMC cylinder of circular cross-section illuminated by plane quasi-standing, standing, or progressive waves, in order to examine the possibility of cylinder trapping, using rigorous partial wave expansions in cylindrical coordinates, by Mitri [252] dealing with optical forces exerted on a cylinder of circular cross-section in an electric line source illumination, using partial wave expansions in cylindrical coordinates.

For the year 2020, we have papers by Mitri [253] dealing with forces and torques for cylindrical particles of arbitrary geometrical cross-sections exhibiting a circular dichroism, illuminated by light-sheets, in a GLMT-framework for the evaluation of incident wave BSCs, completed by [254] with a stronger focus on optical TM-TE conversion for a long PEMC circular cylinder, by Vaippully et al. [255] dealing with torques generated on spherical birefringent microspheres with a cylindrical outershape compared to that for a spherical outershape, in the Rayleigh and Mie regimes, by Bai et al. [256] who investigated the stability and dynamics of chiral nanoparticles in lateral optical binding induced by a high-order Bessel beam described in terms of closed-forms BSCs in the GLMT-framework, by Mitri [257] who dealt with exact expressions for radiation force components (per-length) exerted on PEC elliptical cylinders illuminated by Gaussian light sheets, with scattering coefficients derived using a semi-analytical procedure which requires a single numerical integration, by Mitri [258] who studied the oscillatory dynamic radiation force in optical heterodyning, caused by a heating effect in the case of interference/mixing of two electromagnetic plane waves driven at different slightly frequencies illuminating a circular geometric cross-section cylinder, by Mitri et al. [259] dealing on electromagnetic radiation force on a PEMC sphere of arbitrary size, illuminated by linearly-polarized plane progressive waves, with a plausible extension based on GLMT, by Chen et al. [260] who studied transverse optical forces on particles associated with the breaking of electromagnetic symmetries, relying on GLMT, by Vennes and Preston [261] who studied the optical deformation of homogenous and core-shell spherical particles illuminated by arbitrary shaped electromagnetic beams, with the incident beams (Gaussian beams and zeroth-order Bessel beams) and the optical stress described in the GLMT-framework, BSCs of Gaussian beams being obtained using a localized approximation and BSCs of zeroth-order Bessel beams being available in closed-forms, by Ohwal et al. [262] who dealt with the establishment of an optical

trapping curve for the prediction of trapping parameters, considering gradient forces and viscous drag forces, for polystyrene small-sized particles in the Rayleigh regime flowing in a water channel, and Gaussian beam illumination, by Lin et al. [263] using the detection by an optical coherence tomography imaging system of bead motions illuminated by a photonic light sheet beam generating radiation pressure for acceleration of the beads, by Devi and De [264] using GLMT with a localized approximation, and dipole approximation, to study the reversal of optical forces on nanoparticles in a nonlinear optical trap, incorporating third-order optical nonlinearity, and comparing pulsed and cw excitations (see as well Yadav et al. [265] on this issue), by Yadav et al. [266] studying optical trapping efficiency for silver nanoparticles using dipole approximation and GLMT with a localized approximation for on-axis particles, under Gaussian beam illumination, including higher-order optical nonlinear effects, and observation of Fano resonance effects, together with a study for pulsed excitation, by Song et al. [267] studying the optical torque exerted on an absorptive dielectric sphere of arbitrary size illuminated by a linearly-polarized Airy light-sheet, relying on the GLMT and the ASD method, by Devi et al. [268] presenting a comparison study for dielectric and metallic nanoparticles in nonlinear traps, using GLMT with BSCs evaluated using a localized approximation, by Song et al. [269] dealing with the optical radiation force on a lossless dielectric sphere of arbitrary size illuminated by a linearly polarized Airy light sheet, based on the GLMT and the ASD method, with results compared to those obtained using the dipole approximation method in which the gradient force has also been computed for a Rayleigh sphere, by Xu et al. [270] who showed that Kerker interference, the interplay between electric and magnetic dipoles induced in nanoparticles, permits the intensity gradient to exert a nonconservative force (in contrast with the fact that intensity-gradient force is conventionally considered conservative), in the case of a standard optical trap created with linearly or elliptically polarized Gaussian beam, by Zheng et al. [271] who provided a general formulation for computing the optical gradient and scattering forces on a spherical chiral particle illuminated by generic monochromatic fields, by Shi et al. [272] dealing with optical binding and lateral forces on a pair of chiral particles in linearly polarized plane waves, using full wave simulation and GLMT, with a focus on the dipole approximation, and by Ha et al. [273] dealing with the decomposition of optical forces on a spherical particle illuminated by a Bessel beam into conservative and non conservative parts.

For the year 2021, we have papers by Tian et al. [274] estimating radiative forcing efficiency of spherical dust aerosol based on direct satellite observations, using LMT and T-matrix to calculate the optical properties of the particles, with case studies over the Sahara and Taklimakan Desert, by Neves et al. [275] who reported analytical expressions for BSCs in cylindrical and rectangular metallic waveguides, for particle radius ranging from the Rayleigh regime to large microparticles, with the aim of sorting particles in optical chromatography using optical forces, by Donato et al. [276] devoted to photonic force microscopy (PFM) where an optically trapped probe in liquid is scanned over a sample (possibly in a three-dimensional environment) to achieve both morphology characterization and force sensing detection through its thermal fluctuations, i.e. by recording an interference pattern which fluctuates due to the fluctuations of the position of the particle induced by Brownian motion, with illumination by cylindrical vector beams, and forces calculated by EBCM, reducing to GLMT for spherical particles, by expanding the focused illuminating beams with an ASD and expressing the BSCs of each individual plane wave component of the spectrum (which is indeed a correct way to use ASD, see subsection 2.6 of [277]), by Shahabadi et al. [278] who studied the optical trapping of anti-reflection core-shell microspheres by regular Gaussian beams and other structured beams including radially polarized Gaussian, petal and hard-aperture truncated circular Airy beams, with GLMT used to calculate the scattered fields from the trapped particles, by De Angelis et al. [279] who carried out simulation
of optical forces by a microstructured continuous superposition of first-order non paraxial Bessel beams (micrometer nondiffractive frozen waves), having both orbit and spin angular momentum, on absorptive Rayleigh particles, allowing the particles to follow circular or elliptical trajectories, by [280] who experimentally and theoretically studied the revolution of a trapped particle in counter-propagating dual-beam optical tweezers under different air pressures, exhibiting the possibility of trapping micro-sized particles in microgravity environment showing great potential in space applications, using fixed-step Runge-Kutta method for the analysis of dynamical behaviors, and radiation forces evaluated using a T-matrix method (namely EBCM, according to a private communication). PEMC spheres are considered as well in Mitri [281] dealing with the optical trapping of a PEMC sphere using nonparaxial Gaussian single-beam tweezers, with on-axis BSCs obtained from a complex source-point method, compared to standard solutions using the principle of localization, with a particular attention paid to the Rayleigh regime, see as well Mitri et al. [282] dealing again with forces exerted on PEMC and AEMC (absorptive electromagnetic conductor) spheres, using a GLMT-like approach based on multipole series expansions. Other papers are by Li et al. [283] who carried out a numerical analysis of optical force in the Rayleigh regime, investigated by comparison with the GLMT, in the case of a spherical particle trapped in a Gaussian beam, with a complementary discussion in terms of gradient and scattering forces, by Soleimani and Zakery [284] who used multi-beam optical tweezers to create optical lattices and manipulate microparticles, decomposing optical forces into gradient and scattering parts for both Mie particles and Rayleigh particles, and writing that “unfortunately, decomposition of the force into gradient and scattering forces for Mie particles by GLMT has not been reported”, a statement which is no more true as discussed at the beginning of the present section, by Wu et al. [285] who studied optical pulling force arising from nonparaxial accelerating beams, relying on GLMT and Maxwell’s stress tensor (mind however that the effects of Maxwell stress tensor are already implemented in GLMT), by Li et al. [286] dealing with the 3D calibration of microsphere position in optical tweezers, using FDTD and scalar diffraction theorem, under illumination by a focused Gaussian beam, with a good agreement between simulations and experiments, by Bai et al. [287] dealing with the optical binding force of stratified chiral particles illuminated by Bessel beams described by BSCs, using a GLMT framework, by Devi and De [288] who presented interesting comparative studies of various theories for laser trapping under femtosecond pulsed excitation, however containing erroneous statements corrected in [289], by Rafferty and Preston [290] who performed a theoretical and experimental investigation of trapping positions of weakly absorbing spherical particles in a dual-beam optical trap, using Gaussian beams, with theoretical calculations using the GLMT, by Zhang et al. [291] who carried out a numerical analysis of optical trapping force affected by lens misalignments, in the geometrical optics regime, by Xiu et al. [292] dealing with lateral optical forces on spherical particles with illumination by circularly polarized plane waves, using the “generalized Lorenz-Mie method”.

For the year 2022, we have papers by Chen et al. [148] who theoretically and experimentally investigated the optical trapping of polystyrene microspheres in optical tweezers under Gaussian beam illumination, using EBCM, with a disagreement between experimental measurements and theoretical results, attributed to a slowing effect (representing an exciting problem which would certainly be better understood in a GLMT framework if the beam was correctly described), by Zhang et al. [293] who demonstrated an enhanced transverse optical gradient force (up to two orders of magnitude) acting on Rayleigh core-shell particles immersed in a simple optical field formed by two linearly polarized plane waves, relying on GLMT (viewed as a numerical technique (!)) and on an analytical analysis, by Nalupurackal et al. [294] studying the simultaneous optical trapping and magnetic micromanipulation of ferromagnetic iron-doped upconversion hexagonally shaped microparticles, and also using GLMT for spherical particles, by Chen

et al. [295] dealing with the enhancement of performance of a counter-propagating dual-beam optical trap with asymmetric configurations, by Gao et al. [135] dealing with optical trapping of Janus particles composed of two independent hemispheres with different refractive indices, both from experimental and theoretical points of view (using a T-matrix approach, i.e. EBCM), by Devi et al. [296] dealing with femtosecond pulsed excitation to enhance the efficiency of optical trapping of nanoparticles, including nanomaterials having negative indices, over cw excitations, using GLMT for spherical particles and localized approximation to evaluate the BSCs, and dipole approximation, but erroneously or at least ambiguously concluded that GLMT using localized approximation is not valid for arbitrarily sized particles, by Kincaid et al. [297] who studied optical forces and time of flight measurements on microspheres inside hollow core photonic crystal fibers (HC-PCFs) using a rigorous treatment (GLMT) with BSCs, allowing one in particular to predict Mie resonances, shown to be superior to the use of ray optics, by Yan et al. [298] who studied axial optical forces and torques on a multilayered graphene-coated gold nanosphere with illumination by a vector Bessel beam, using a GLMT framework with the BSCs given by the ASD and the multipole expansion, by Edelstein et al. [299] who calculated the optical force exerted on a spherical magneto-optical sphere illuminated by a plane wave, with magneto-optical gradient forces and magneto-optical extinction forces, as well as scattering and extinction cross-sections, presented in an analytical form, by Tang et al. [300] who studied the scattering, force and torque, of an arbitrary-shaped optical polarized beam by a PEMC sphere, in a GLMT framework with BSCs attained in a previous work using the multipole expansion method, by Wu et al. [301] who studied the optical multiple-particle trapping of particles based on a rotationally-symmetric power-exponent-phase vortex beam, with an experiment showing that the number of captured particles is equal to the topological charge, by Andrew et al. [302] who dealt with optical micro micromachines which have the potential to improve the capabilities of optical tweezers by amplifying forces and allowing for indirect handling and probing of specimens, and designed and tested an unsupported microlever, suitable for general-purpose optical tweezer studies, and demonstrated stable trapping performance, by Yu et al. [303] who dealt with focused structured light fields which can exhibit multiple focal points in the longitudinal direction, suggesting that they are suitable for longitudinal optical trapping and manipulation, by Chen et al. [304] who theoretically and numerically demonstrated that Fano-like resonances can induce a left-handed optical torque on a dipolar plasmonic core-shell nanoparticle in the interference optical field composed of two linearly polarized plane waves, and derived an analytical expression of the optical torque up to electric quadrupole terms, and by Chen et al. [305] who dealt with “optical matter”, i.e. a class of materials formed by self-organization of particle into ordered structures, and devised an analytical Markov state model to understand and control the dynamics and metastable states of an optical matter system, relying on a Langevin dynamics approach and on GMM theory. We also have papers by Song et al. [306] who studied the optical forces and spin torques on a magneto-dielectric Mie sphere under illumination by an Airy light-sheet, using GLMT, by Zhang et al. [307] who used as well GLMT to analyze the effects of auto-focusing and self-bending Bessel light-sheets on illuminated dielectric spheres, by Joshi and Bowling [308] and Joshi et al. [309] dealing with the motion of beads in an optical tweezers, relying on Mie scattering theory to compute the laser beam force and Stokes law to compute the drag force, including a random force that is produced by Brownian motion, by Tang et al. [310] who dealt with the optical radiation forces acting on a homogeneous and lossless dielectric spherical particle for trapping, by a polarized Airy beam using GLMT, with BSCs derived with the help of ASD, revealing that dielectric spheres can be trapped in the main caustics of the beam or its sidelobes and guided along parabolic trajectories, by Alinezhad et al. [311] theoretically and experimentally showing that a dry objective can be utilized for optimal trapping of small polystyrene beads, with optical forces calculated using GLMT, by
Yadav et al. [312] who presented theoretical studies on nonlinear laser trapping of metal/dielectric core/shell nanoparticles, and discussed the effect of optical nonlinearity and of Fano resonances under on-axis focused femtosecond pulsed excitation, using GLMT with BSCs evaluated using a localized approximation.

For the year 2023, we have papers by Zhou et al. [313] who explored the potential application of a superimposed Hermite-Gaussian correlated Schell-model beam with off-axis multi-vortices in optical trapping of Rayleigh particles, with the focused beam intensity and radiation forces, discussed in terms of scattering and gradient forces, derived in detail with an acknowledgment of the existence of non-standard forces, by Tang et al. [314] who studied forces and torques exerted on an AEMC sphere by arbitrary-shaped optical polarized beams, in the framework of the GLMT, by Oliveira et al. [315] who proposed the use of the semiconductor polyaniline for the preparation of spherical-shaped microparticles to serve as handles in optical tweezers experiments, concluding that for absorbing beads the geometric optics model does not provide a good approximation for the trap stiffness when the particle radius is of the order of the wave-length and that more sophisticated models are then needed, such as GLMT, by Savelyev and Karpeev [316] who calculated and studied the three-dimensional microstructures used in the formation of optical traps on the optical axis in a near diffraction zone, using the FDTD method, by Li et al. [317] who comprehensively studied the optical torque caused by a Bessel beam on a charged sphere of arbitrary size, in the GLMT framework, by Mandal et al. [318] dealing with the optical force and resulting trajectory of a nanoparticle encapsulated in a nanotube, using LMT for plane wave illumination, by Zhou et al. [319] who studied the electromagnetic fields of an autofocusing vector Pearcey beam and its optical force and curved trajectory transport effects exerted on a microsphere, based on GLMT and the ASD to derive the BSCs, by Ma et al. [320] who studied bipolar optical forces on a dipolar nanoparticle, particularly thermal induced switching between optical pulling and optical pushing, under Bessel beam illumination using GLMT, by Zeng et al. [321] who studied a programmable and high-quality patterning of fullerene optoelectronic devices with complex aggregate nanostructures, for a contamination-free laser printing approach, with optical trapping forces under higher-order Laguerre-Gaussian beam illumination and Gaussian beam illumination, evaluated using the on-axis GLMT or FDTD, and the use as well of a Rayleigh model, by Bai et al. [322] dealing with GLMT to study the radiation forces exerted on chiral spheres by dual Gaussian laser beams, with a quadrature method to evaluate the BSCs, by Sadafi et al. [323] who dealt with orbital angular momentum and polarization states of highly focused vector vortex beams which can be engineered to selectively excite desired multipoles in nanoparticles, making them ideal candidates for complex optical applications, and published a GLMT based platform to compute optical forces on particles placed on a plasmonic substrate, achieving 3D trapping regions for highly absorptive and high-refractive-index particles, by Tiwari et al. [324] who used optical tweezers to trap and configure spherical micro-robots so that they are attached to a single stationary waveguide, allowing for the enhancement of bio-inspired optical resonators, with experimental results verified by using FDTD-based simulations, achieving good agreement, by Guzatov [325] who provided an analytical solution to study the light pressure of the Bessel beam on a spherical particle, and by Lu et al. [326] who studied the optical trapping performance of circular Pearcey beams on Mie particles, relying on the GLMT for the theoretical analysis and using optical tweezers experiments to confirm the theoretical conclusion that the use of circular Pearcey beams leads to an enhanced trapping performance for larger Mie particles, with also the conclusion that these beams performed better than the more conventional circular Airy beams with the same focusing length.

## 5.2. Photophoretic forces

With Ambrosio [327], we begin to deal with the examination of other kinds of forces, which can have an amplitude orders of magnitude greater than gradient and scattering optical forces, namely photophoretic forces exerted on low-loss particles of the order of a few micrometers under plane wave or uniform illumination. Such thermal forces, which can be attractive or repulsive, arise from the uneven heating of the particle surface under illumination over a hemisphere. Different regimes may be considered (i) a free molecular regime when the Knudsen number $Kn>>1$, i.e. when the particle is much smaller than the mean free path of the gas molecules, so that photophoretic forces can be investigated using kinetic theory of gases, (ii) a slip-flow regime for $Kn<1$, and (iii) a continuous regime when $Kn<<1$. The applied force then depends on the asymmetry factor which involves an integration of the normalized absorbed light intensity over the volume of the particle. The author considered particles with arbitrary index of refraction and analytically investigated photophoretic forces in terms of the asymmetry factor, for both slip flow and free molecular regimes. He made use of the Mie theory for lossy and homogeneous spheres with arbitrary size parameters, and obtained the photophoretic forces as analytic closed-form solutions in the case of low-loss spherical particles under plane wave illumination. Examples of the behavior of the asymmetric factor for micro-spheres with distinct materials are provided and discussed, for all optical regimes, i.e. for Rayleigh, Mie and geometrical regimes. Next, Ambrosio [328] dealt with the extension of the previous paper to the study of longitudinal photophoretic forces for both slip flow and free molecular regimes beyond the case of plane waves, for arbitrary-index spheres of arbitrary size parameter, illuminated by on-axis axisymmetric beams of the first kind using GLMT, with examples of calculations in the case of Gaussian and zeroth-order Bessel beams. Next, Ambrosio et al. [329] established analytic and closed-form expressions for the photophoretic forces exerted by arbitrary shaped beams on homogeneous and low-loss spherical particles in both the free molecular regime and slip flow regime, with the aid of GLMT to expand the internal electromagnetic fields, while Sun et al. [330] dealt as well with the prediction of longitudinal and transverse photophoretic forces exerted on an absorptive homogeneous sphere in a gaseous medium for the slip-flow regime, under Bessel beam illumination using GLMT, for both on-axis and off-axis illuminations.

Photophoretic forces are studied as well by Wang et al. [331] who carried out the theoretical prediction of photophoretic forces on a dielectric sphere in gaseous media, illuminated by a circularly symmetric on-axis high-order Bessel beam, and derived an analytic closed-form formula for the asymmetry factor, within the framework of GLMT, focusing exclusively on the electromagnetic part of the problem, by Mitri [332] who presented generalized mathematical expressions for the longitudinal and transverse photophoresis asymmetry factors for a light-absorptive magnetodielectric circular cylinder of arbitrary asymmetry factor and permeability, illuminated by an arbitrarily shaped polarized light-sheet, relying on the GLMT for light sheets to the evaluation of BSCs, by Mitri [333] dealing with the optical Magnus effect in the photophoresis of a spinning absorptive dielectric circular cylinder in air illuminated by axisymmetric wave fields such as plane waves, by Moura et al. [334] who demonstrated that the optical trapping and manipulation of spherical-shaped semiconductor (germanium) particles (stable trapping, oscillatory dynamics, or repelling) can be strongly modulated and controlled using optical tweezer based on a highly focused Bessel beam, and considered gradient, scattering, and photophoretic forces to carry out the analysis.

Finally, when dealing with photophoretic forces, specifically for the evaluation of photophoretic asymmetry factors, and likely for other problems as well, classes of integrals appear which involve particular products of Bessel functions of the first kind with complex argument and integer orders, and also products of derivatives of such Bessel
function. Such integrals have been carried out using numerical calculations, e.g. in problems involving the illumination of lossy infinite cylinders, either in isolation or close to conducting corner spaces or planar boundaries, by plane waves or light-sheets [332,335,336]. To speed up computations, Ambrosio et al. [337] demonstrated that these integrals can actually be resolved analytically. Similarly, Ambrosio et al. [338] established closed-form expressions for integrals of products between (Ricatti)-Bessel functions and derivatives of (Ricatti)-Bessel functions of arbitrary kind, integer order and complex argument.

Other papers devoted to mechanical effects of light may be found by using “force”, “torque”, “trap”, “tweezer” as keywords.

## 6. Internal fields, resonances and WGMs/MDRS, and plasmons

Whispery-Gallery Modes (WGMS) and Morphological-Dependent Resonances (MDRs) are resonances related to internal fields in particles generated by an incident illumination. Although these two words are usually considered as synonymous, there are circumstances where the two kinds of phenomena must be distinguished, namely for instance the case of a sphere with an internal eccentric spherical inclusion. “WGMs” may then be strictly reserved to resonant modes associated with waves creeping below the surface of the host sphere, namely those which are “whispering” like in St Paul’s in London or in the Temple of Heaven in Beijing. The resonant modes which interact with the internal sphere may then be called MDRs without being WGMs. This case has been investigated both in the framework of ray optics Hamiltonian chaotic dynamics [339,340] and in the GLMT framework [84,85].

In the fiducial period, we have papers by Kolwas [341] who dealt with the properties of plasmons (electromagnetic excitations coupled to coherent electron charge density oscillations on a metal/dielectric interface), and the control of their decay dynamics in spherical metal nanoparticles, which may be relevant to GLMT insofar as shapes and spectral widths of the metal nanoparticle spectra may be studied on the ground of its formalism, by Sun et al. [342] who dealt with optical resonance scattering of a dielectric sphere of arbitrary size illuminated by polarized Airy beams in a GLMT framework, with BSCs obtained with a multipole expansion method, by Abad-Arredondo [343] who dealt with fluorescence triggered by radioactive β decay in optimized hyperbolic cavities, reported a 100-fold amplification in periodic structures, and revealed that enhancement of up to 10 000-fold is possible because of the hybridization and spectral overlap of WGMs and localized-plasmon modes, with possible applications to nuclear-optical imaging, the use of highly energetic gamma rays, Cerenkov luminescence, and visible and near-infrared fluorescence, by Wang et al. [344] dealing with assembly-induced microwave band resonances in gold nanoparticles and demonstrated the effectiveness of an ultrathin and flexible localized surface plasmon prepared by layer-by-layer assembly of gold nanoparticles, in a study comprising both simulation and experiments, by Zambrana-Puyalto et al. [345] who studied the physical mechanisms governing the coupling of a tangential beam into a WGM of a microsphere, in relation with the symmetry properties of the illuminating light, using GLMT – see as well Zambrana-Puyalto et al. [146] for a related topic –, by Koushki [346] who studied the effect of conjugation with organic molecules on the surface plasmon resonance of gold nanoparticles and application in optical biosensing, on the basis of a precise theoretical model to simulate the ultraviolet-visible absorption spectrum of a colloid containing gold nanoparticles in the presence of different surfactants, by Marhaba and Samad [347] who described the plasmonic coupling effect on the Localized Surface Plasmon Resonance (LSPR) of Platinum dimer nanoparticles, investigating the optical extinction spectra over a broad range (from the visible to the near infra-red), with measurements performed using a numerical method based on a finite element method which permits the determination of both the interparticle distance and the polarization effects on the LSPR, by Song et al. [348] who studied optical resonances and rainbow scattering of an electromagnetic nondiffracting Airy light sheet by a

lossless homogeneous dielectric sphere of arbitrary size, with calculations done in the GLMT framework, with BSCs expressed in terms of single quadratures, by Tiwari et al. [349] who reported the theoretical investigations on WGMs of dielectric hollow core–shell microspheres illuminated by plane waves and focused Gaussian beams, using GLMT, by D’Ambrosio et al. [350] who simulated and experimentally investigated the resonance spectrum and the scattered light field of WGMs in a spherical microresonator made with a liquid droplet excited with a laser beam, starting from a GLMT based model, with agreement between theory and experiments, by Luo et al. [351] who studied germanium material exhibiting a strong electric-type resonance due to its semi-metal properties, which can be seen in the visible and near-infrared spectral range, with calculations carried out using the FDTD method, by Sun et al. [352] who focused on the morphology-dependent scattering properties of spherical/spheroidal nanoparticles which the GLMT can analytically describe, and dealt at the same time with other complicated shapes, with a Left-handed Circularly Polarized (LCP) light illuminating nanoparticles with three kinds of morphologies (spheres, oblate and prolate spheroids), in order to optimize the scattering intensity and to deal with the spin Hall effect which refers to the transverse displacement of scattered light due to spin–orbit interaction, by overlapping the electric and magnetic dipolar modes, by Hoang et al. [353] who emphasized the usefulness and convenience of using multipole expansion coefficients in designing Mie resonances, and then discussed single Mie resonances supported by spheres and pillars, collective Mie resonances supported by linear nanochains of the spheres and pillars, Fesbach resonances resulting from the unique structure of three optically coupled nanochains and Feshbach theory of overlapping resonances for analyzing and optimizing photonic crystal defect cavities, all these discussions being based on the exact solutions to the Maxwell’s equations, by Sedeh et al. [354] who dealt with the role of the helical wavefronts of Laguerre-Gaussian beams in the excitation and suppression of higher-order resonant modes inside all-dielectric meta-atoms of various shapes (nanocuboids, nanocylinders), aspect ratios, and orientations, with scattered fields described up to the electric octupole moment term, with the case of Gaussian beam illumination as a reference, by Kaur et al. [355] who demonstrated MDRs with the help of an encased optical nano-antenna which generated a tightly focused beam with a spot size of ~3µm, inside an optical fiber tip cavity, with a barium titanate microsphere placed besides the optical axis inside the cavity, using in particular the GLMT with the Van de Hulst’s localization principle (related to the optical approximation), by Sarita et al. [356] who studied the interactions of gold and silver spherical nanoparticles of a wide range of radius size with illumination from plane waves and from two different tightly focused beams, for a wide range of wavelengths, leading to intensity enhancement in the near-field due to localized surface plasmons, using GLMT, by Esmaili-Dastjerdipour and Shahshahani [357] who theoretically studied a fiber optic sensor based on surface plasmon resonance with a coating of nanocomposite layers, by Minin et al. [358] who demonstrated that, by introducing an air cavity, it is possible to increase the intensity of the electromagnetic fields at the poles of a sphere by an order of magnitude, thanks to higher-order Mie resonances, by Tang et al. [359] who studied the controllable adjustment of the distance between metal nanoparticles and quantum dots by controlling the thickness of SiO₂ shell of Ag@SiO₂ nanoparticles, in relation with the unique properties of metal nanoparticles which can enhance the fluorescence emission intensity of quantum dots through the local surface plasmon resonances, and by Stollberger et al. [360] who presented the first experimental analysis of the influence of single, micrometer-sized aerosol particles, on the reflectivity of a resonant Fabry–Perot etalon, together with a theoretical model capable of describing and predicting the particle influence on the transfer function of the etalon.

For other papers devoted to the issue of this Section, see papers by Mun et al. [178], by Swetha et al. [361,362], and use “resonance”, “resonant”, “MDR”, “WGM”, “plasmon”, “whispering” and “internal” as keywords.
# 7. Photonic nanojets

We consider in this section a specific class of *scattered* beam shapes, namely photonic nanojets (PNJs) which may be defined as narrow, high-intensity, beams emerging from the shadow-side of an illuminated particle and which propagate over a distance longer than the wavelength, simultaneously maintaining a sub-wavelength full-width at half maximum (FWHM). The issue of PNJs has been considered in the GLMT framework, first in 2014 [363], in the case of spheroidal particles under Gaussian beam illumination, following many previous works such as by Chen et al. [364], Itagi and Challener [365], Cui et al. [366], Valdivia-Valero and Nieto-Vesperinas [367], to name a few.

In the fiducial period, we have papers by Wang et al. [368] who dealt with a numerical study of tunable PNJs generated by biocompatible hydrogel core-shell microspheres under plane wave illumination, for surface-enhanced Raman scattering applications, and demonstrated that the Raman signals in the case of adding the core-shell microspheres in the system can be enhanced about 20 times in water and 100 times in air as compared to the case in which no core-shell microspheres are present, with simulations carried out by using FDTD runs from a commercial software, by Chen et al. [369] who studied how the polarizations and amplitude profiles of the incident beam affect the shape, size, and location of the PNJs generated from illuminated microspheres, mainly under illumination by plane waves, but as well by Gaussian beam illumination, simulations being again carried out by using the FDTD, by Uenohara et al. [370] who attempted to control the PNJ intensity distribution flexibly by adjusting the intensity distribution of two incident laser beams, a Gaussian beam and a radially polarized beam, still using FDTD simulations, by Wei et al. [371] who dealt with PNJs generated by a dielectric ellipsoidal particle illuminated by a plane wave, using the open source DDSCAT (relying on the discrete dipole approximation DDA) for the calculations, by Liu et al. [372] who dealt with photonic hooks (subwavelength curved beams) based on the physical principles of photonic jets, shaping photonic hook and controlling the bending direction of the hook, via well-controlled illumination of finite-size graded-index micro-ellipsoid (or oblate spheroids), using FDTD simulations, by Dorofeev et al. [134] who examined the role of the tangential electric field component to the terahertz jet and hook formation by dielectric cuboids and spheres surrounded by ideally conducting screens, a work which is aimed at elucidation of photonic jet formation mechanisms in the presence of a metal screen whose plane is parallel to the incident wave front and encompasses the dielectric object without gaps, by Lin et al. [263] who studied the characteristics of photonic jets generated by a dielectric sphere illuminated by a Gaussian beam, within the framework of GLMT with BSCs evaluated by using the modified localized approximation, with a special attention paid to the influence of the beam waist radius and beam focal center location on the key properties (maximal intensity, focal distance, longitudinal waist and FWHM) of the photonic jets, by Mandal and Dantham [373] who reported a theoretical investigation on short and elongated PNJs generated by single solid/hollow core-shell microparticles illuminated with focused Gaussian beams, using a GLMT approach, with the electric field enhancement values of the PNJs upon focused Gaussian beam illumination being compared to those obtained from the plane wave illumination, by Zhu et al. [374] who used the FDTD method to study the properties of photonic jets generated by a dielectric spheroid with Bessel beam excitation, by Huang et al. [375] dealing with the characteristics of photonic jets generated by a dielectric sphere illuminated by a Gaussian beam, in the GLMT framework, by Cheng and Shen [376] who studied curved photonic jets (i.e. photonic hooks) produced from a spherical particle illuminated by a Bessel-Gaussian beam, reporting curved hollow photonic jets produced by a spherical particle under the illumination of a circular high-order Bessel-Gaussian beam, using a GLMT-like framework in which the BSCs are obtained by a double quadrature technique which is reduced to a single quadrature technique, by Jia et al. [377] who reported on

PNJs generated by a spheroidal particle illuminated by a vector Bessel beam, using a DDA approach, by Yousefi et al. [378] dealing with PNJ generation by spheres under converging and diverging beams, and their properties including the effective focal length and the FWHM, using a FDTD software package, with comparisons against the case of plane wave illuminations and [379] who presented the photonic nanojet as a phenomenon in a structured light generator system that is implemented to modify the source focal spot size and emission angle, under illumination of a microlens array and spheroids with different aspect ratios, with prolate, spherical, and oblate geometries, by a focused Gaussian beam, using FDTD, by Zhang and Lin [380] who discussed the design of a novel device for PNJs (in Chinese), by Mandal et al. [381] who reported a theoretical investigation of PNJs produced by single dielectric microspheres illuminated by focused broadband radiation (polychromatic light) from a Halogen lamp, a supercontinuum source, a light-emitting diode, and a Hg arc lamp, using a GLMT-like approach with BSCs evaluated using a localized approximation, by Gasparic et al. [382] dealing with the PNJ produced by a Gaussian beam illuminating a low refractive index microsphere in air, within the framework of GLMT, with the calculations carried out for a huge number of parameters, by Wu et al. [383] dealing with a polarization-sensitive photonic jet of a dielectric sphere excited by a zeroth-order Bessel beam, in the framework of GLMT, with BSCs expressed in closed-forms, by Joshi and Dantham [384] who dealt with PNJ-assisted fluorescence microscopic technique which is helpful in enhancing fluorescence signals of various samples at ultralow concentration, with a study of all relevant parameters under Gaussian beam illumination, by Chen et al. [385] who studied a photonic jet generated by illuminating a dielectric scatterer with symmetric four Airy beams, by Karamehmedovic and Glückstadt [386] who demonstrated numerically the feasibility of axial and angular control of the position of a PNJ by lossless phase-only modulation of a fixed Gaussian beam illuminating a fixed 2D circular homogeneous dielectric micro-lens, showing as well that their phase-only modality can be used to calibrate and improve the confinement of PNJ generation, and by Tang et al. [387] who dealt with the scattering of a radially polarized Bessel beam by a PEMC sphere, relying on the GLMT, with an emphasis on the photonic nanojet (and the “bottle beam”) generated by the interaction between the PEMC sphere and the illuminating beam. For other papers on the issue, use “jet” as a keyword.

# 8. Optical theorem

The optical theorem considered in this section (not to be confused with the Schelkunoff optical/equivalence theorem) relates the forward scattering amplitude to the total extinction cross-section of the scatterer. It soon appeared by the beginning of the eighties that the classical version of the electromagnetic optical theorem was not valid in the GLMT framework. This is the starting point of studies which finally ended with a complete understanding of the situation, depicted in [166], obtained by invoking the ASD. As a summary, the optical theorem is valid for each plane wave in the spectrum, but is not valid for the whole beam because the forward directions of the plane wave components do not correspond to the forward direction of the beam. By the way, the same argument allows one to understand that the average velocity of a structured beam is smaller, even in vacuum, than the speed of light. Such a failure of the optical theorem is not restricted to vectorial beams such as electromagnetic beams, but occurs as well for scalar beams. All this is reviewed in a recent paper dealing with the failures of the optical theorem in electromagnetism, acoustics and quantum mechanics [388], in particular discussing a long story from the beginning of the eighties when the failure of the optical theorem in the GLMT framework has been discovered up to the understanding of the deep and very simple reason for this failure, understood in 2014 and published in 2016 [166].
In the fiducial period, we have papers by Rondon and Lee [389] who dealt with a generalized optical theorem for Rayleigh far-field scattering approximation, for any nondiffracting beam, with results illustrated in the case of a Bessel beam, by Rondon [162] who presented a general derivation for the optical theorem, with results concerning plane waves, and two different illuminating probes, and by Zhang et al. [390] who studied a generalized optical theorem in the case of an infinite cylinder embedded in a weakly absorbing host, under plane wave illumination. Also, dealing with acoustical waves instead of electromagnetic waves, Zhang [391] proposed a generalized optical theorem for an arbitrary incident field in the context of acoustic waves, more generally applicable to any scalar wave. See as well the review by Markel [392] and use “optical theorem” as a keyword.

## 9. Optical particle characterization

Originally, the motivation to build GLMTs, particularly GLMT stricto sensu to begin with, has been the field of Optical Particle Characterization (OPC), as described by [277,393]. This field has been the motivation for the creation of the OPS (Optical Particle Sizing) conference in Rouen, France in 1987, e.g. [394], thereafter reformatted to OPC, and eventually to LIP (Laser-light and Interactions with Particles) as an OPC follow-up, again in Rouen, France in 2012, e.g. [395], afterward in Marseille, France, in 2014 [396], Xi'An, China, in 2016 [397], College Station, U.S.A. in 2018 [398], and Warsaw, Poland, in 2020, but postponed due to the covid pandemic, although there has been a JQSRT special issue [399,400], and in Warsaw, Poland, 2022 [401].

### 9.1. Phase-Doppler technique

The first genuine applications of GLMT have been devoted to the phase-Doppler technique [81,402–405]. The issue is less prolific to-day insofar as the phase-Doppler technique and its relationship with GLMT is concerned because it became very mature but, for a few papers in the fiducial period, see Azarskov and Dyvnych [406] who described a laser Doppler velocimeter (an ancestor of phase-Doppler technique) using a powerful laser diode allowing one to build a device for measuring two vector components of the true airspeed of heavy unmanned aerial vehicles, Dafsari et al. [407] who used phase Doppler anemometry to experimentally study the effect of a refractive index change on droplet size measurement according to the fuel temperature, using specifically commercial aviation light fuels, confirming that the change in the refractive index with fuel temperature has a great influence on the droplet size measurements, and proposed a method of calculating the refractive index according to the fuel temperature change, Akinyemi et al. [408] who used a Phase-Doppler Particle Analyzer (PDPA) in a study devoted to the effect of the air-to-liquid mass ratio on the internal flow and near-field spray characteristics of a two-phase swirl burst injector, Zhang et al. [409] who used an in-situ laser Doppler current probe, with a compact dual wavelength (491 nm and 532 nm) diode pumped solid state laser as the light source, to achieve the simultaneous measurements of micro-scale subsurface current speeds and the characterization of micron-particles, and Bardera et al. [410] who dealt with a laser Doppler velocimeter for measurements of the flow around the wind sensors of a planetary exploration vehicle, carried out in a wind tunnel to simulate the very low Reynolds number of Mars atmosphere. For other papers on the issue, use “Doppler” as a keyword.

### 9.2. Rainbow and rainbow refractometry

In the fiducial period, we have papers by Zhou et al. [411] who analyzed a rainbow refractometer in order to extract the temperature gradient profile inside heating droplets, relying on the GLMT for multilayered spheres to simulate the rainbow pattern of a radially

inhomogeneous heating droplet in the case of a Gaussian beam illumination, investigating the influences of the layer number of the multilayered sphere, beam waist radius of the Gaussian beam, and location of the droplet within the beam, on the rainbow pattern, and showing that the influence of the beam waist radius should be taken into account if the beam waist radius of the beam is smaller than the radius of the droplet, by Li et al. [412] who dealt with the measurement of transient evaporation of a pre-heated monodisperse ethanol droplet stream injected into ambient air with phase rainbow refractometry and high-speed microscopic shadowgraphy, focusing on the interaction among evaporating droplets, by Cao et al. [413] who dealt with the characterization of refractive index and size of a spherical drop by using Gaussian beam scattering in the secondary rainbow region, with the waist radius of the beam comparable with the drop radius, with calculations performed using GLMT and Debye series, by Li et al. [414] who dealt with colloidal droplets, namely droplets containing dispersed insoluble particles, and with the simultaneous measurement of refractive index, host droplet size and colloid concentration of suspensions, using rainbow refractometry, with concentration measurements obtained from intensity attenuation, relying on the separate identification of the *p* = 0 and *p* = 2 rays in the rainbow signal, see as well Li et al. [415] for the study of colloidal droplets, by Kanjanasaku et al. [416] who measured the size and the refractive index of quasi-monodispersed ethane droplets and water droplets in high pressure conditions, in the pressure range 1–50 bars, close to the ones of the critical point, using global rainbow refractometry, by Tufaile and Tufaile [417] who explored the halo formation based on the concept of rainbow ray formation and caustics in parallel with the studies of dynamical systems, such as particles bouncing in an open billiard, in relation with experiments involving a laser beam injected in a glass cylinder viewed as an open billiard, by Li et al. [418] who extended conventional optical configurations of rainbow refractometry set-up for point measurements or line measurements to allow also the particle position in a plane to be determined, leading to a technique designed as planar rainbow refractometry, by Wang et al. [419] dealing with spherical droplet characterization based on the simulated secondary rainbows with simultaneous measurements of droplet size, size distribution, refractive index and temperature, relying on GLMT and the use of focused Gaussian beams to achieve high spatial resolution in denser droplet sprays, see as well [147], by Ni et al. [420] who measured droplet size and temperature under different conditions, to investigate the evaporation process of ethanol droplets under ambient temperature, by analyzing rainbow patterns in the vicinity of the rainbow angle, see as well [421], by Wang et al. [422] who employed vector-ray tracing to calculate Möbius shifts of the third-order and the fourth-order rainbows for a spheroidal droplet, by Wu et al. [423] who proposed a refractometry technique for long range measurements at distances up to 1650 mm of transparent and spherical droplet refractive index and size, under the form of both a synthetic aperture standard rainbow refractometry for refractive index and size of mono-dispersed droplets, and a synthetic aperture global rainbow refractometry for spray measurement, by Li et al. [424] who studied a method for scattering angle calibration in the rainbow region using a droplet stream, viewed as an emerging technique operating at backscatter angles and characterizing droplets or complex droplets in terms of size and refractive index, by Wang et al. [425] who proposed a synthetic aperture rainbow refractometry by synthetizing rainbow signals of the same droplet with dual-wavelength laser beams, in order to increase the aperture of rainbow refractometry, allowing one to deal with long distance and small droplet measurements, by Ni et al. [426] who dealt with the evaporation behavior of the ethanol and heptane binary droplets using both experimental and simulation methods, using in particular a rainbow technique, by Chiewudomrat et al. [427] who extended the rainbow technique in order to allow for the characterization of internal gradients, by Li et al. [428] who recorded incomplete rainbow signals from droplets with varying index refractive index or at long working
distances, which are usually considered as invalid and rejected, and studied their capability of achieving correct measurements in terms of feasibility and accuracy, by Li et al. [429] who evaluated by simulations and experiments the effect of signal-noise-ratio (SNR) on the inversion accuracy of the global rainbow signals of droplets with three typical size distributions (normal, log-normal and bimodal normal). For other papers on the issues of the present subsection, use “rainbow” as a keyword.

## 9.3. Holography

In the fiducial period, we have papers by Middleton et al. [430] who used holographic particle characterization to measure the sizes and compositions of individual colloidal particles dispersed in fluid media, rapidly amassing statistics on the distributions of these properties, even for complex heterogeneous dispersions, by Berdeu et al. [431] who proposed a hologram reconstruction algorithm based on alternating a model fitting step and a regularized inversion step, applying this algorithm in the context of fluid mechanics, where holograms of evaporating droplets were analyzed allowing one to jointly reconstruct the droplet and the surrounding vapor wake, theoretically describing the wave diffracted by the spherical droplet using a Mie model (Lorenz-Mie theory extended to radially inhomogeneous spheres), while the vapor wake is described by a phase transmittance plane whose diffracted wave is propagated according to the Rayleigh-Sommerfeld theory, by Kasimberg et al. [432] who used Holographic Video Microscopy (HVM), in which particles flowing through a microfluidic channel are illuminated with a collimated laser beam, to study protein aggregation and demonstrate its capability to quantitatively distinguish protein aggregates and silicon oil in the presence of varying amounts of surfactants, showing that it is possible to directly measure the size distributions of protein aggregates and silicon oil present in a sample, and emphasizing that HVM is a particle-resolved technology that is able to distinguish multiple different particle species in a single sample even when they are at the same size, by Wu et al. [433] who dealt with dual-view tomographic particle holography, combining tomography and holography, as a promising approach to yield accurate 3D measurement of particle fields, with a standard model of particle hologram which has been developed in the framework of GLMT to model and investigate the light scattering effects on three configurations of dual-view tomographic particle holography, that is, dual-beam dual-view Gabor inline holography, single beam dual-view combing Gabor inline holography with off-axis scattering holography, and single beam dual-view off-axis scattering holography, by Alexander et al. [434] who dealt with precise measurements in digital holographic microscopy by modeling the optical train, allowing the motion of spherical and non-spherical colloidal particles and biological organisms to be tracked with nanometer-scale precision and their optical properties to be inferred particle by particle, and discussed as well how machine learning might play a role in future advances, by Altman and Grier [435] who interpreted holographic molecular binding assays, which used holographic video microscopy to directly detect molecules binding to the surfaces of micrometer-scaled colloidal beads by monitoring associated changes in the light scattering properties of beads, with holograms of individual spheres being analyzed against a fitting to a model based on the Lorenz-Mie theory, relying on an effective medium theory in which coated spheres are treated as being homogeneous and comparing the effective medium theory results with those of the discrete dipole approximation – For a complementary study devoted to the direct detection of antibodies binding to colloidal spheres, see as well Snyder et al. [436] –, by Coëtmellec et al. [437] who revisited Luneberg integrals which allow one to describe the propagation of a vectorial electromagnetic field and, together with a new approach of computations based on Zernike polynomials, lead to the simulation of holograms illustrated in the case of the digital in-line holography with an opaque disk, by Song et al. [438] who applied

digital holography technology in a quantitative measurement of the density distribution of a low refractive index transparent substance (e.g. the vapor of suspended droplets), assuming that the droplet inner and outer gradients could be analyzed and inversed with an appropriate light scattering model under certain conditions, possibly using the GLMT for multilayered spheres, by Odete et al. [439] who discussed a label-free viability assay using holographic video microscopy, more specifically Total Holographic Characterization (THC) which is a single-particle characterization technology that determines the size and index of refraction of individual particles using the Lorenz-Mie theory of light scattering and, measuring the viability of *Saccharomyces cerevisiae* yeast in the presence of various concentrations of isopropanol as a function of time, demonstrated that THC can effectively distinguish living and dead yeast cells by the index of refraction of individual cells, see as well Boltyanski et al. [440] dealing as well with THC for a complementary work concerning *Saccharomyces cerevisiae* yeasts, by Abdulali et al. [441] who used quantitative analysis of holographic microscopy data to precisely and rapidly measure the diameter and refractive index of individual colloidal spheres, applying this technique to inhomogeneous or aspherical particles to retrieve measured diameter and refractive index which represent properties of an effective sphere enclosing each particle, and showing that, when applied to populations of fractal aggregates, it is possible to measure the fractal dimensions of an individual fractal cluster by probing how its effective diameter and refractive index changes as it undergoes rotational diffusion, producing in summary a fast and effective method to assess the average structural properties of individual colloidal fractal aggregates from sequences of holograms recorded at multiple angles, see as well Abdulali et al. [442] for a related work dealing as well with effective sphere analysis of fractal aggregates, by Brault et al. [443] who presented a new method to achieve autofocus in digital holography microscopy, a method which is based on the insertion of calibrated objects into a sample placed on a slide, by Belaid et al. [444] who described tele-holography, a new concept for lossless compression and transmission of inline digital holograms based on wavelet transform, reducing drastically the bit rate of the hologram transmission flow, with the theoretical scheme being supported by experiments, by Brault et al. [445] who, in the context of digital in-line holographic microscopy, described an unsupervised methodology to estimate the aberration of an optical microscopy system from a single hologram, the method being based on the Inverse Problems Approach reconstructions of holograms of spherical objects, with the forward model being based on a Lorenz-Mie model distorted by optical aberrations described by Zernike polynomials, by Brault et al. [446] who dealt with the problem of detecting the weak signal of a known pattern superimposed to similar or more contrasted patterns, which is a problem which can generally be relevant to the field of imaging, and then proposed a fast-detection algorithm applicable to various robust loss functions evaluated on simulated and experimental in-line holograms, by Onofri and Lamadie [447] who dealt with the numerical analysis of Gabor holograms using Lorenz-Mie theory, Debye series, and the principle of localized approximation, by Onofri and Lamadie [448] again who introduced a set of Matlab routines and a graphical user interface to model holograms of homogeneous coated and multilayered particles, based on Lorenz-Mie theory, by Olivier et al. [449] who analyzed and studied the effects of some parameters (wrong magnification, partial coherence of the illumination, optical aberrations) using simulations and experiments on calibrated spherical objects and a rigorous Lorenz-Mie model to evaluate the relevance and requirements of model refinements, in the framework of inverse problem approaches in in-line configuration of digital holographic microscopy, by Grier et al. [450] who demonstrated label-free holographic assays for antibodies and for antigenic proteins from pathogenic viruses, including SARS-CoV-2 and H1N1, by Altman and Grier [451] who reported holographic particle tracking measurements of diffusion and sedimentation of colloidal particles between parallel horizontal walls, by Noury et al. [452] using three-dimensional (3D) phase maps in a
digital inline holography system to measure the coordinates of tiny
particles, previously applied to spherical particles, and extended here
to the case of an infinite cylindrical filament to determine the optimal
focus plane, with a practical implementation to the investigation of
the dynamics of beads-on-a-string structures under an air flow current,
structures which are characterized by several quasi-spherical beads
(i.e. drops) linked by a quasi-cylindrical filament (i.e. thread), by
Snyder and Grier [453] who dealt with aberrations introduced by real-
world imaging (holography) conditions, and identified a previously
overlooked source of spherical aberration as a significant source of
errors, by Brault et al. [454] dealing as well with aberrations in digital
in-line holographic microscopy, first by model fitting calibration using
transparent beads inserted in the sample, a step which estimates the
aberrations of the optical system as a function of the lateral position in
the field of view and at each wavelength, second by using a regularized
inverse problem approach to reconstruct the transmittance and the
phase of objects of interest, and applied the method to the case of
bacteria imaging in Gram-stained blood smears, and by Fedorov and
Mironov [455] who, in a study relevant to holography, dealt with the
interaction between an illuminating wave and a spherical particle in the
framework of the GLMT. For other papers on the issue of holography,
use “holo” as a keyword.

9.4. Imaging, microscopy and diffractometry

GLMT has been early applied to imaging systems, see [456]. In the fiducial period, we have papers by Wu et al. [457] who dealt with dual-beam interferometric particle imaging for size measurement of opaque metal droplet, a method in which the studied metal droplet is illuminated by two laser beams from different views, generating an interference pattern in the far-field from which it is possible to derive the size of the droplet, this technique being furthermore compared with digital inline holography leading to a highly consistent agreement, by Shanblatt et al. [458] who demonstrated a fast, flexible, and accurate paraxial wave propagation model to serve as a forward model for propagation-based X-ray phase contrast imaging in parallel-beam or cone-beam geometry, well suited for propagation-based X-ray phase contrast tomographic reconstructions, furthermore capable of modeling arbitrary objects, including those that are strongly multi-scattering, with simulation studies which were conducted to compare the model to other forward models in the X-ray regime, such as the Mie and full-wave Rytov solutions, by Shan et al. [459] using a forward diffrac- tometry set-up to measure PSDs (Particle Size Distributions) of small particles (similar to a Malvern diffractometry commercial system) and discussing the associated inversion problem, and also by [460] dealing as well with an inversion problem using the artificial fish swarm algo- rithm, by Sukhanovskii et al. [461] who discussed the drawbacks of two measurement techniques, namely IPI (interferometric particle imaging) and GPT (glare point technique) for droplet sizing, applied for measure- ment in nozzle (dense) sprays, showing that reconstructed droplet size distribution strongly depends on parameters of the optical system and post-processing such as laser intensity, focal ratio, brightness threshold and others, by Liu et al. [462] who investigated microsphere-assisted super-resolution imaging, relying on photonic nanojet which is sim- ulated by FDTD and software super-resolution imaging experiments, which are carried out by using silica microspheres and glass spheres un- der different illumination modes, by Stilgoe et al. [463] who dealt with microscope images of strongly scattering objects via vectorial transfer matrices, and presented an accurate and efficient T-matrix approach for calculating the image of a mesoscopic particle that is captured using a high-numerical-aperture objective lens, calculated images being able to accurately replicate experimental observations, a study which is relevant in particular to holographic optical tweezers apparatus, with a complementary use of ray optics and a study of the effect of resonances, and the possibility of calculating forces using the particle properties extracted from the image, by Stepanov et al. [464] who

dealt with the measurements of the size of liquid droplets in a spray
for fuel injector using two methods, namely Interferometric Particle
Imaging (IPI) and a Glare Point Technique (GPT), the combination
of these methods allowing one to significantly expand the range of
measured droplet sizes, although the results obtained by the IPI and
GPT methods begin to diverge with an increase in particle density, by
Qieni et al. [465] who presented a method for simultaneously inferring
both the refractive index and the size of a particle with an extended
IPI technique, with opposite two-sheet illumination at dual scatter-
ing angles, with experiments on polystyrene microspheres and water
droplets suggesting that the method is promising for many relevant
applications, such as fuel combustion and environmental monitoring,
whose theory can rely on Debye series, geometric optics approach and
glare points analysis, by Lin et al. [466] who developed three con-
figurations of dual-beam interferometric particle imaging for opaque
metal droplet size measurement, with signals respectively sampled from
backward scattering, forward scattering, and hybrid-direction scatter-
ing regions, with a simultaneous recording of digital inline holograms
of the same metal droplets, leading to a consistent agreement between
the results of all these techniques, and a comparison as well of two
theoretical approaches, namely LMT and geometrical optics, by Jobert
et al. [467] who generated and analyzed images based on lens-free
imaging of light scattering signatures with a holed image sensor, in
order to retrieve particle size and refractive index, relying on LMT and
classical radiometric calculations, with experiments carried out using
calibrated laboratory spherical particles, by Dohet-Erlay et al. [468]
who proceeded to an analysis of three-dimensional objects in quan-
titative phase contrast microscopy, specifically studying the validity
of a planar equation using numerically simulated images (holograms)
of spherical objects under illumination by a polarized plane wave,
and using the LMT to compute the resulting optical field, by Swetha
et al. [361,362] dealing with electron microscopy to characterize shape
and size of Ag nanoparticles or to detect silver nanoparticles, by Wu
et al. [469] who used a high-speed stereoscopic shadowgraph system
generating images, which is applied to investigate the 3D splashing
dynamics of silicone oil dropping on liquid films quantitatively and
allows one to determine secondary droplet diameter, number and mass
fraction, and to reconstruct the 3D coordinates of these droplets and
their trajectories, by Garcia-Magariño et al. [470] who dealt with the
Interferometric Laser Imaging for Droplet Sizing (ILIDS) method (also
called IPI in other papers) for long range instantaneous measurements
of transparent spherical particles, relying on glare point properties
which can be analyzed using a geometric optics approach, relevant to
the planned characterization of the spatial size distribution of simulated
supercooled large droplets in a windtunnel, by Xiong et al. [471] who
used a laser sizer to investigate the characteristics of jet flash atomiza-
tion evaporation under a depressurized condition, using a large beam
allowing one to use the classical LMT to analyze the data, by Farhat
et al. [472] who dealt with the characterization of bubble dynamics by
imaging, especially upon operating with high flowrates, using a high-
speed camera and a LED light source, by Cannaday et al. [473] who
used a mobile angular scattering microscope for the characterization
of organelle size distribution, allowing one to determine information
about the health of the biological sample, the stability and accuracy of
the system having been investigated by measuring the scattering from
polystyrene beads with mean sizes of 5 and 1.75 µm with narrow size
distributions, with the use of a GLMT model to obtain diameter size
estimates, by Denneulin et al. [474] who proposed an unsupervised
regularized inversion method for reconstruction of the 3D refractive
index map of a sample in tomographic diffractive microscopy (as-
sociated with hologram acquisition), with use of a LMT scattering
model, by Jüngst et al. [475] who developed image analysis routines
to investigate droplet disruption (puffing, micro-explosions) in spray-
flame synthesis of metal oxides, using droplet shadowgraph imaging
with high spatiotemporal resolution on a high-speed camera, by Sax
et al. [476] using IPI in configurations which are not restricted to the
use of forward-scattering angles, but may be extended to other scatter-
ing angles, especially in the back-scatter region, with the identification
of suitable scattering angles both for droplets and bubbles, by Garcia
et al. [477] who investigated the planar LIF/Mie droplet sizing tech-
nique, also called Planar Droplet Sizing (PDS), which is a laser imaging
method which combines the detection of Lorenz-Mie scattering and
Laser-Induced-Fluorescence (LIF) emission from droplets illuminated
with a laser beam, from both an experimental and theoretical point of
view, allowing one measurements of the Sauter Mean Diameter (SMD)
over a wide field using a calibration procedure based on phase-Doppler
anemometer, by Putz et al. [478], who dealt with images which are
recorded using the elastically side-scattered light of spherical micro-
particles (beads) in an imaging flow cytometer, and with often seen
two or more bright spots rather than an image resembling the outline
of the particle, and demonstrated that these spots are in fact glare
points, using the LMT for the theoretical approach, a paper which
might be read together with the one by Brittain IV et al. [479] who
proposed a flow cytometer with enhanced light-scatter sensitivity for
the analysis of biological nanoparticles, allowing one to fully resolve
70 nm polystyrene and 98.6 nm silica beads, viruses down to 81 nm
and extracellular vesicles at least as small as 65 nm, and finally by
Grillini et al. [480] who dealt with the comparison of several imaging
models for spectral unmixing in oil painting, although the authors
mentioned that they “decided to not include in our study models that
explicitly solve the radiative transfer equation, although some might
very relevant for future works, for example, the Kulbelka-Munk or the
four-flux model”.

A special mention concerns the use of commercial particle size
analysers such as by Miraji et al. [481] who used a Malvern ana-
lyzer to study PSDs of particles in the framework of a study devoted
to nutritional quality and in vitro digestion of immature rice-based
processed products, by Kang et al. [482] using as well a Malvern
system to measure grain sizes in the context of a study devoted to
the increasing effective moisture during the Holocene in a semi-arid
region of Central Asia, see as well [483,484] for similar studies, next
by Zhao et al. [485] who used Mie theory for grain-size analysis,
conducted with an American Microtrac 83500 laser particle sizer with
a large size-range, using both laser diffractometry and dynamic light
scattering, in the framework of a study devoted to loess soils, by Lee
et al. [486] using a Malvern laser diffractometer to measure mean
particle size powders, during a study devoted to the effects of deep
cryogenic treatment for tungsten carbide-iron cemented carbide, by
Loutati et al. [487] who used a laser scattering particle size analyzer
(Horiba LA-950 V2 with a 650 nm and a 405 nm laser source) to
determine the PSD of dry milded powders, with data analyzed using
the LMT, by Trejo et al. [488] who presented a study to analyze the
performance of three commercial particle counter sensors with similar
characteristics, by An et al. [489] who used as well a Horiba sizer to
measure the PSD of cells to study the effect of cell size on the mechanics
of strawberry fruit tissue, by Beck-Broichsitter et al. [490] who used a
Malvern laser diffractometer to describe the spatial varying PSD along
aggregate surfaces, with data analyzed using the LMT, see as well Zhang
et al. [491] who evaluated particle-size-distribution-based homogeneity
of laboratory mycotoxin samples using laser diffraction particle size
analysis, by Rulko et al. [492] who reported on the characterization of
solid volume fraction as a function of height in gravity-driven particle
curtains using optical techniques and simulations performed with a
discrete element method model, in which solid volume fraction was
obtained from experimentally measured transmittances which have
been found sensitive to the optical properties of the particles through
Monte-Carlo ray tracing, both experimental and computational tests
being performed with 1 mm and 2 mm particles, whose size distri-
butions have been analyzed using image processing techniques, by Li
et al. [493] dealing as well with laser diffraction for grain-size analysis
and unraveling source-to-sink dust transport in Central and East Asia.

For other papers related to the issue, use “microscop”, “imag”,
“diffr” as keywords.

9.5. Angular scattering measurements

A somewhat obvious way to measure particle sizes is to record
angular scattering of particles (most usually spherical) and to use LMT
or GLMT to deal with an inverse problem. We then have papers by
Rao et al. [494] who theoretically and experimentally dealt with the
angular distribution of the intensity of light scattered by a spherical
air bubble in the oceanic medium which is distinctive near the critical
scattering angle region, when light interacts with a bubble with a
refractive index less than its surrounding medium, with LMT as the
theoretical approach, by Cannaday et al. [495] who relied on an angu-
larly resolved elastic light scattering which is an established technique
for probing the average size of organelles in biological tissues and
cellular ensembles, and investigated the effect of angular range on the
stability and accuracy of Mie theory fits to scattering from organelle-
sized beads and from single cells, as well as the effect that angular
range has on the uncertainty using computational methods. They obtain
experimental data from single mammalian cells, and performed simu-
lations using GLMT with a linearly polarized excitation beam, for three
different size distributions, namely a narrow log-normal distribution
of polystyrene beads, a broad log-normal distribution of polystyrene
beads, and a mitochondria-dominated distribution available from the
literature. Also, Assmann et al. [496] used a wide-angle scattering
system in which elastically scattered light is collected by an ellipsoidal
mirror and imaged onto a CCD-camera over a wide angular range, to
measure droplet sizes and PSD in flames, relying on LMT and on the
measurement of characteristic maxima in the scattering pattern and
on the measured scattering intensities. For other relevant papers, use
“angular” and “multi-angle” as keywords.

9.6. Extinction method

Another somewhat obvious method (in principle) is to retrieve
particle properties from the extinction of light propagating in a cloud
of such particles. This technique basically analyzes the variations in
the transmittance of a collimated polychromatic light beam passing
through a medium containing an ensemble of particles. Examples are
by Onofri et al. [497] who reported a novel microscale approach to
characterize the PSD and absolute concentration of colloidal suspen-
sions, relying on three detection channels to record simultaneous light
extinction spectra, with experiments carried out on particle standards
with sizes ranging from 30 nm to 0.5 µm and volume concentrations
of 1 to 1000 ppm, clearly demonstrating the value and potential of
the proposed method, by Skidanenko et al. [498] who dealt with
the computation of optical extinction spectra for nanoparticles with
different concentrations of metals and different architectures (core–
shell, inverse core–shell, or alloy) and demonstrated that the use of
an optical spectrum fitting technique to the spectra of monodisperse
noninteracting bimetallic nanoparticles with a predetermined structure
appeared to be effective for the determination of the internal structure
of nanoparticles, except large nanoparticles with a radius of more than
60 nm containing less than about 25% silver atoms, computations
of optical extinction spectra being carried out with a multisphere T-
matrix (likely EBCM) program, by Jiang et al. [499] who studied the
influence of particle properties on measuring a low-particulate-mass
concentration by using a white-cell light extinction method, dealing
with several mineral and ash aerosols with different size distributions,
shapes and compositions, to illustrate the efficiency of the system for
particles having different optical properties, and provided a discussion
of the extinction properties of spherical particles and of non-spherical
particles, including the GLMT for spheroids, to be completed by Jiang
et al. [500] who measured particle size and concentration of non-
spherical particles by combined light extinction and scattering method,
with simulations of scattering properties of prolate spheroids relying
on EBCM. Another paper is Erenen et al. [501] to measure PSDs,
more specifically concentration and volumetric PSDs of colloids and
nanofluids, with a particular attention paid to the issues of sensitivity to complex refractive index uncertainties and to noise. For a given wavelength of the polychromatic illuminating beam, and under some restrictions such as the absence of dependent scattering – produced by near-field optical interactions between particles – or of multiple scattering effects, the light extinction for homogeneous, isotropic, non-magnetic spheres, is analytically modeled by the LMT under the form of an expression incorporating a normalized PSD function in number, from which it is possible to retrieve a PSD in weight if the particle density is known. This requires a careful inversion procedure which has been selected by Erenen et al. [501] as being the Tikhonov regularization procedure also known as penalized least square method. Experiments are carried out using polystyrene latex beads commonly employed for the calibration of light extinction based instruments to generate monomodal or bimodal colloidal suspensions in water. These beads were illuminated by a highly stabilized UV light source while the transmitted light was recorded by a spectrometer which was highly sensitive in the UV range and which offered a high signal-to-noise ratio, before being fed to a laptop for data inversion (these light extinction experiments have been complemented by Dynamic Light Scattering experiments). Another paper is by Dey and Hesselbarth [502] who provided a detailed study by theory, simulation and measurement of the electromagnetic scattering from alumina ceramic spheres, covering a frequency range from 50 to 220 GHz, in order to simultaneously determine both the particle electrical and geometrical attributes, relying on extinction cross-sections.

See as well Chaâbani et al. [503] below for “extinction”, and use the word “extinction” for other papers.

## 9.7. Time-shift technique

Li et al. [504] dealt with a time-shift technique using a highly focused Gaussian beam to separate different scattering orders during the passage of a drop/particle through the beam, typically detecting light in the near backscatter direction. The authors then investigated the possibility of using the time-shift technique to carry out measurements in the case of a drop with an embedded particle, and therefore dealt with the light scattering from drops with an inclusion. The theoretical analysis relied on a ray optics technique, allowing the signal generation arising from the drop passing through a plane wave of a focused beam to be simulated. Validations of the ray tracing code are carried out by comparisons with the case of pure liquid drops in the Lorenz-Mie framework. It is concluded that, with suitable signal detection and modifications to the signal processing, the time-shift technique should be able to unequivocally detect whether a drop contains a spherical particle or not. For time-shift technique, see also Li et al. [505] who is devoted to the development and application of a Monte-Carlo ray-tracing model to simulate light scattering when a colloid suspension droplet (spherical particle containing multiple spherical inclusions), passes through a highly focused Gaussian laser sheet, relevant to the use of the time-shift technique for particle sizing, the incident laser sheet being treated as a large number of polarized light rays and the scattering from the embedded inclusions being computed using the LMT, leading to a good agreement between simulated and measured time-shift signals, indicating that the polarized Monte-Carlo code reliably captures the light scattering when a colloid suspension droplet passes through a highly focused Gaussian beam, and again Li et al. [506] dealing with a focus on the estimation of the colloidal nanoparticle concentration within drops. Also, Schaefer et al. [507] used the time-shift technique to deal with spray measurements, together with the use of a Debye decomposition or of geometrical optics, using the fact that the time shift between different scattering mode peaks in the signal either on one detector or between detectors will be linearly proportional to the diameter of the drop and inversely proportional to its velocity, and introduced a novel signal processing approach, based on cross-correlation functions between four detector signals, for estimating size and velocity from time-shift signals, despite lower signal-to-noise ratios arising due to unfavorable measurement conditions or due to the complex structure of drops.

## 9.8. Dynamic light scattering

This subsection is devoted to Dynamic Light Scattering (DLS) which is based on the analysis of the intensity fluctuations of the scattered light, caused by the Brownian motion of the dispersed particles, and requires the measurement of the autocorrelation function of the fluctuations. It may be called as well Diffusion Broadening Spectroscopy although this terminology has seemingly became obsolete, or Photon Correlation Spectroscopy, e.g. [508] and reference therein. In the fiducial period, examples of paper are by Derkachov et al. [509] who investigated single micrometer-sized droplets containing spherical nanoparticles, in evaporation in an electrodynamic quadrupole trap, obtaining exponentially decaying autocorrelation functions for the light scattered by evaporating microdroplets as well as microaggregates of nanoparticles formed at the end of the evaporation process, using three laser wavelengths, different sizes of silica and titania nanoparticles and their concentrations in different dispersion media (liquids), and registering autocorrelation function at the evaporation stage of microdroplets and at the final stage of evolution corresponding to microaggregates, with a discussion of the influence of MDRs on the signals at the initial stage of the microdroplet temporal evolutions, and by Chaâbani et al. [503] who used DLS to analyze colloidal heterostructures, together with scanning electron microscope measurements, and observations of extinction spectra, with FDTD simulations allowing one to better understand scattering results. Relying on a similar basic phenomenon, namely Brownian motion, Xiong et al. [510] relied on the principle that, because the diffusivity of particles undergoing the Brownian motion is inversely proportional to their sizes, the size distribution of submicron particles can be estimated by tracking their movement. They then used a Particle Tracking Analysis (PTA), associated with Monte-Carlo simulations, to demonstrate that, in principle, PTA can be used to size natural oceanic particles, and afterward conducted a series of lab experiments using microbeads of NIST-traceable sizes to evaluate the performance of Viewsizer 3000, a PTA-based commercial instrument. For other papers on the issue, use “Dynamic” or “dynamic” as keywords.

## 9.9. N-flux methods, multiple scattering and radiative transfer

An efficient method to deal with multiple scattering, and radiative transfer, analysis is the use of a N-flux method, more particularly 2-flux and 4-flux methods, such as described by [511–515], while another one is the use of Monte-Carlo methods [516]. In the fiducial period, we have papers by Vargas et al. [517] who dealt with the measurements of scattering and absorption cross-sections of light diffusing materials retrieved from reflectance and transmittance spectra of collimated radiation, using N-flux radiative transfer models, specifically the MLG-four-flux radiative model (Maheu-Letoulouzan-Gouesbet), considering a system consisting of $TiO_2$ particles dispersed in a matrix and incorporating a comparison with Lorenz-Mie evaluations, by Duveiller et al. [518] who dealt with reflectance and transmittance of flowable dental resin composite predicted by the two-flux Kubelka-Munk model, with Saunder correction, and by the four-flux model which looks preferable when the sample under study becomes more scattering, to be completed by Duveiller et al. [519] who dealt with the improvement of the predictive performances of the two-flux model applied to translucent dental resin composites, by Rio et al. [520] who studied the influence of crystalline phases on optical characteristics of a glass-ceramic in the visible range via simulations by the four-flux method, with the aid of a transfer matrix formalism, to anticipate the transmittance and reflectance of an object for different set of parameters, the impact of each aforementioned parameter on the color rendering being separately discussed and simulations being confronted to experimental data to assert their validity, by Yang et al. [521] who used a four-flux method in the framework of a study devoted to shielding radiation damage to metallic turbine blades, by Vargas et al. [522] who dealt with a model
to substrate contributions of scattered radiation from measured direct transmittance and specular reflectance by light diffusing materials, using both the LMT and four-flux models, by Haratoka et al. [523] who dealt with the design of thermochromic glazing windows, the radiative transfer through glazings being solved using four flux models to predict the optical properties of the glazing, by Mishra et al. [524] who analyzed the applicability of the Kubelka-Munk model and of the four-flux model to study radiative cooling paints and solar thermal absorbers, over the entire wavelength range from visible to IR, by Zhu et al. [525] who investigated the impact of the particle size of dispersion phase in polymer blends on near-infrared diffuse reflectance spectra, using Mie scatter calculations, and Monte-Carlo simulation. See as well Barrios and Alvarez for works related to four-flux models [526].

There is also a study by Rebolledo-Oyarce et al. [527] who studied the growth of microalgae depending on the color and amount of light received, the study of the light transfer and its effect on the growth of *D. terciolecta* being compared with three models, namely (i) Beer-Lambert law (ii) Two-flux approximation and (iii) radiative transfer equation, and by Ribeiro et al. [528] who used LMT, and radiative transfer through both Monte-Carlo and two-flux models, to investigate experimental and theoretical coupled approaches for the analysis of radiative transfer in photoreactors containing particulate media, with the case study of TiO₂ powders for photocatalytic reactions. Radiative transfer is investigated as well by Diao et al. [529] who dealt with a comparison of the influence of Martian and Earth's atmospheric environments on terahertz band electromagnetic waves, in which absorption coefficients are calculated based on the theory of atmospheric radiative transfer and a database, together with the LMT to study transmission characteristics of terahertz waves with different wavelengths, by Membre et al. [530] who, dealing with the spectral variability of the optical properties of pictorial layers with applications on the field of cultural heritage, presented the study of optical properties of pictorial layers as seen in the case of paintings and illuminated manuscripts, namely the influence of morphological parameters on reflectance spectra using both hyperspectral measurements on physical samples and reflectance spectra calculated using an optical model based on the four-flux model, by Huang et al. [531] who discussed opaque thermal barrier materials for thermal shielding of turbine blades, with a theoretical analysis relying on the evaluation of reflectances and transmittances in the spirit of flux models of radiation, by Yu et al. [532] who dealt with an assay providing on-site quantification of tetrahydrocannanibol in oral fluid, relying on a light detection method in which light absorption could be detected, together with the use of the 2-flux Kubelka-Munk theory of light propagation in turbid media, by Wang and Niklasson [533] who presented a revised Kubelka-Munk two-flux radiative transfer model to determine the absorption and backscattering coefficients of composites containing Au nanospheres from experimental transmittance and reflectance spectra, by Niklasson et al. [534] who presented a method to obtain absorption and backscattering coefficients of a light scattering material and applied it to the technologically important case of sunscreen lotions, modeling the light scattering properties by two-flux theory and inverting spectrophotometric total transmittance and reflectance measurements to obtain backscattering and absorption coefficient spectra, by Eymard et al. [535] who applied a four flux optical model on total and diffuse transmittance and reflectance measurements of a Thermoplastic PolyOlefin (TPO) to extract values of refractive index, absorption, scattering coefficients and anisotropy coefficients of scattered light, by Aziz et al. [536] who presented the strategy of high-temperature photon suppression, based on a ceramic composite consisting of a second component with a smaller refractive index and a controlled particle size, using LMT, measurements of reflectances and transmittances of the composite, and the use of a two-flux model to calculate a diffusive transmittance, by Gautheron et al. [537] who investigated the impact of translucency on light angular distribution and corresponding internal reflectances by the means of the radiative transfer equation, which is indeed more rigorous than the 2-flux

Kubelka-Munk model or even than the 4-flux (but more demanding), by Wang et al. [538] who discussed the thermal radiative properties of zirconium oxide films in the near-infrared wavelengths, with a hybrid approach combining the Kubelka-Munk and discrete ordinate methods used to invert radiative properties, by Barrios et al. [539] who studied the intrinsic and extrinsic scattering and absorption coefficients obtained by inversion of spectrophotometric transmittance and reflectance data, in a work devoted to Polymer Dispersed Liquid Crystal (PDLC) that switch from a translucent milky appearance (off state) to a transparent clear appearance (on state), relying on the use of two-flux and four-flux models, and obtaining good fits although the average crossing parameter was found to be inconsistent with fitted parameters, by Duveiller et al. [540] dealing with the objective to evaluate the prediction accuracy of the Kubelka-Munk reflectance theory and other more innovative two-flux and four-flux methods for predicting reflectance and transmittance factor of w flowable dental resine composites of various thicknesses, and by Huang et al. [541] who used a modified Kubelka-Munk four-flux method to calculate the reflectance spectrum of a composite film with translucent substrate containing uniformly dispersed pigment particles.

Independently of N-flux models, we have a paper by Jönsson et al. [542] dealing with multiple scattering phenomena relying on online accelerated Monte-Carlo simulations of light transport through scattering media, in which billions of “photon” packets are traced through simulated cubic volumes, and acceleration being achieved by the use of general-purpose computing on graphics processing units, reducing the computation time by a factor up to 200 in comparison with a single central processing unit thread, and even up to 800 by using four graphic cards, relying on LMT to generate the scattering phase functions from spherical particles, by Zeng et al. [543] who used the radiative transfer equation by a discrete ordinate method in bubbling fluidized beds and circulating fluidized beads, using as well Mie theory, by Wang et al. [544] who used as well a Monte-Carlo approach to study impaired visibility resulting from rainfall contributing greatly to the occurrence of traffic accidents, and presented a numerical simulation approach to analyze the extent to which the coupling of spray and raindrops reduces visibility. See also Chaussonnet et al. [206] dealing with multiple scattering.

We also have a paper by Mishchenko and Dlugach [545] who studied multiple scattering of light by particles embedded in an absorbing host medium using a recently developed single-scattering and vector radiative-transfer methodology directly based on the Maxwell equations, and using a generalized radiative transfer equation, in a configuration relying on a simple scattering model in the form of particles randomly and sparsely distributed throughout a plane-parallel layer or a homogeneous absorbing medium surrounded by a vacuum, with the specific cases of particles in absorbing gaseous atmosphere, and air bubbles in water, to be complemented with Mishchenko and Yurkin [546] dealing with the issue of co- and counter-propagating wave effects in an absorbing medium. Beyond multiple scattering, we may deal with dependent scattering as in Cao et al. [547] who studied the radiative heat transfer in splash and dilution zones of the oxy-fuel combustion in a pressurized circulating fluidized bed when dependent-scattering is considered, relying on Mie theory and three different models.

For other papers relevant to the issues of this subsection, use “flux”, “multiple”, “radiative”, and “Monte-Carlo” as keywords.

## 9.10. Miscellaneous

In this subsection, I list papers by Zambrana-Puyalto et al. [548] who used an optical set-up to measure the vortex circular dichroism signal from cylindrical nanoholes illuminated with an incident circularly polarized paraxial Laguerre-Gauss beam, with all simulations carried out in a Mie-framework, by Zhang et al. [549] who developed a sensor in which dust particles illuminated by a laser beam generate a scattered
light which is converted into pulse signal by a photodiode, this pulse
signal being proportional to dust concentration, by Hou et al. [550]
who carried out a study on backward scattering characteristics and
detection of submicron particles under illumination by a Gaussian
beam, relying on GLMT and on a fairly biunivoque correspondence
between backscatter light energy and particle diameter in the submi-
cronic range. More precisely, the backscattering light energy has a
"fairly linear" relationship with micro-oscillations on the logarithmic
coordinate as the particle size changes.

We also have papers by Stone et al. [551] who relied on the fact that
light reflected under laser sheet illumination of particles is correlated
with the optical particle concentration to deal with the modeling
and a validation approach for predicting particle concentrations, more
precisely a dimensionless number defined as Concentration Rank Factor
(CRF), of airborne dust during the filling process of cylindrical silos,
with direct comparison between experiments and Computational Fluid
Dynamics (CFD) computations, by Moteki [552] who measured the
complex forward-scattering amplitude of single sub- and super-particles
suspended in a fluid flow by self-reference interferometry, a technique
in which a focused Gaussian beam is incident on particles and the
optical power density is obtained from the combination of incident
and scattered fields, the scattered field being theoretically described
in the framework of plane scattering theory and as well by GLMT with
BSCs evaluated according to a localized model, by Semenov et al. [553]
who developed a particulate matter sensor which operates as a particle
counter and employs interferometric detection of the coherent light
scattered by fine dust moving particles, relying on the interference
between illuminating reference beam and scattered light, by Cinar
et al. [554] who focused on the comparability of analytical ultracen-
trifuge instruments, dynamic light scattering, field-flow fractionation,
and multi-angle light scattering methods in terms of the scope of the
information to be obtained, i.e. primarily hydrodynamic sizes, by Odelli
et al. [555] who, in the framework of a study devoted to the interaction
between fish skin gelatin and pea protein at air-water interface after
ultrasound treatment, evaluated the size of bubbles using LMT from the
value of the basckscattered light depending on particle concentration,
particle diameter, and refracted index of the dispersed and continuous
phases, by Zanini and Sada [556] who dealt with a comprehensive
description of all the optical phenomena occurring when light interacts
with a moving dispersed phase in a constrained environment such as
a real microfluidic channel, requiring to identify fingerprints in the
detected optical signal that are correlated with the shape and content
type of the dispersed phase from those connected to uncertainties of
the optical detection system and/or instabilities in the microfluidic
apparatus leading to dispersed phase size distribution, and to quantify
how diffraction, transmission, absorbance and reflection contribute to
the overall response in the detected intensity after light-matter interac-
tion, by Liu et al. [557] who presented an approach to determine the
location of the liquid-to-solid phase transition and the volume fraction
and number density of particles of both phases within a gas phase
reactor, where freshly nucleated particles are solid and solidify during
growth and cooling, using spectrally-resolved line-of-sight attenuation
measurements applied to silicon nanoparticles in a microwave plasma
reactor which is furthermore analyzed using Direct Numerical Sim-
ulation (DNS), by Lasa-Alonso et al. [558] who proposed a method
to determine the scattering cross-section and global polarization state
of radiation through measurements of two Stokes parameters at an
angle of choice in far-field, the method being applied to cylindrically
symmetric samples whose optical response is well described by a single
multipolar order, and by Yao et al. [559] who developed a single-
particle optical counter with polarization detection for the real-time
measurement of the optical size and depolarization ratio of atmospheric
particles, using two detection angles on the chamber which are 60
° for forward scattering detection and 120° for backward detection,
the apparatus having been calibrated using a commercially produced
polystyrene latex sphere.

10. Machine learning

The use of machine learning (neural networks) is a promising approach for OPC whose applications are only at the beginning of what we can predict to become a main stream. An “in-principle” idea concerning the characterization of spherical particles illuminated by Gaussian beams might be as follows: (i) feed a neural network with a huge number of direct scattering calculations using GLMT and (ii) provide to the network experimental data which would then be used to solve the inverse problem and retrieve particle properties from the data. In this spirit, Cao et al. [560] presented a hybrid approach using machine learning and genetic algorithm to obtain the geometrical and optical parameters of a sphere from its angular scattering pattern in a light sheet in which (i) scattering patterns related to different parameters are first generated by numerically solving Mie scattering based on angular spectrum theory and then (ii) multilayer perception neural network is employed to roughly estimate the parameter, while genetic algorithm is adopted to retrieve the precise value, while the discussion of the method used to generate direct scattering patterns to be inverted are discussed by Zhai and Cao [155].

Another study is by Altman and Grier [561] who dealt with the use of CATCH (Characterizing and Tracking Colloids Holographically), i.e. a machine-learning system for analyzing the properties of colloidal dispersions from holographic microscopy images, with the implementation being validated with simulated data and also through experimental measurements on model colloidal dispersions, illustrating the utility of this machine-learning approach for measuring the concentrations of colloidal dispersions, for characterizing particles in heterogeneous dispersions, and for measuring single-particle dynamics, to be completed by Altman and Grier [562] who dealt with holographic particle characterization using in-line holography video microscopy to track and characterize individual colloidal particles dispersed in their native fluid media, the information encoded in a hologram being extracted by fitting a generative model based on the LMT of light scattering, with the aid of the end-to-end neural-network solution of CATCH.

We also have Stumpf et al. [563] dealing with an imaging technique for determining the volume fraction of each liquid component in two-component droplets, in which multiple images of the same droplet at different times are used, and the processing of these images being supported by a machine learning algorithm which is taught using synthetically generated images and validated on droplets with known mixture fractions placed in an acoustic levitator, and the application of the technique being demonstrated by measuring the volume fraction in splashed secondary droplets following the impact of a drop of one liquid onto a film of a different liquid, Tang et al. [564] dealing with a neural network for the inverse scattering of charged spheres illuminated by plane waves, with parameters such as the charge and radius of the spherical particle being used as the input of the neural network, and the corresponding far field scattering intensities being derived by the GLMT, Zhao et al. [565] who dealt with the inverse problem of designing a monochromatic spatially-structured light field that recovers the pre-specified profile of optical force exerted on a particle, which requires high dimensional optimization and involves lengthy calculations, attacked by using a back-propagation algorithm which is facilitated by a currently available machine learning framework, with two illustrative examples in which the inverse designed structured light fields reproduce, respectively, a pre-defined spatial pattern of optical force and negative longitudinal optical force in a transversely trapping area, Liu et al. [566] in which machine learning has been applied to angular laser-light scattering patterns obtained from numerical simulation, with full Maxwell's equations numerically solved by the FDTD method, in the case of a plane excitation source, probing cells with three wavelengths (blue, green and red) and different directions of scattered light collection (forward, side and backward), in order to determine the optimum conditions to distinguish cell properties, Taei et al. [567] who presented a method to simplify the setup using
unsupervised machine learning technique, allowing one to accurately measure particle diameter with a precision of 0.1 µm and a refractive index of 0.001 using only a laser and a camera without the need for complex alignment of components, and Cioffi et al. [568] dealing with neural network-based image classification widely used in life science applications, here implementing the open-set concept on scattering snapshots of living cells to distinguish between unknown and known cell classes, targeting four different known monoblast cell classes and a single tumoral unknown monoblast cell line, also investigating the influence on experimental sample errors and optimized neural network hyperparameters to obtain a high unknown cell class detection accuracy.

Other papers may be reached by using “machine”, “network” or “artificial” as keywords.

# 11. Alternative light scattering approaches

## 11.1. Extended boundary condition method for structured beams

As already mentioned in the introduction, EBCM is mostly used in the case of plane wave illumination, and as quoted there, the number of papers for structured beams is fairly scarce. It is however the light scattering approach which is usually quoted as *the* T-matrix approach, although this extension from EBCM to T-matrix is somewhat misleading, particularly for the newcomer. One important common point between EBCM (in spherical coordinates) and GLMT *stricto sensu* is that they both may use an expansion of the incident fields in terms of vector spherical wave functions, i.e. they then both use BSCs encoding of the beams, see [569,570] for a more refined discussion.

In the fiducial period, we have papers by Vennes et al. [571] in which EBCM and GLMT are combined according to prescriptions given in [22,24], and BSCs for Gaussian beams evaluated using the localization principle, in a study, both theoretical and experimental, devoted to cavity-enhanced Raman scattering from optically deformed single aqueous aerosol microdroplets in a dual-beam optical trap, in which resonances/WGMs are excited, calculations leading to a good agreement with measured Raman spectra and correctly predicting the splitting of WGMs in measured spectra, by Liu and Mischenko [572] who used the numerically exact “T-matrix” method to model light scattering and absorption by aged smoke aerosols at lidar wavelength ranging from 355 to 1064 nm assuming the aerosols to be smooth spheroids or Chebyshev particles, by Han et al. [573] dealing with the analysis of power transmission through intraocular lens with glis- tenings and whitening by “T-matrix” (presumably EBCM), Vavilin and Fernadez-Corbaton [574] dealing with the polychromatic T-matrix (that provides the complete description of the linear interaction be- tween the electromagnetic field and a given object), which affords the direct treatment of the interaction of electromagnetic pulses with ob- jects, and by Saito and Yang, [575] who used the Invariant-Imbedding T-matrix method and an improved geometric optics method to compute the single-scattering properties of smooth and roughened hexagonal column ice crystals over a wide size parameter range, directed to- ward a better understanding of the impact of surface roughness on backscattering properties, by Sun et al. [576] who used the semi- analytical Invariant-Imbedding T-matrix and physical-geometric optics method to study the scattering properties of non-spherical particles (spheroids, cylinders, hexagonal prisms) with verification by using the finite-difference method. See as well [243] already quoted.

## 11.2. Ray optics

Ray optics is valid when the size parameter of the scatterer is large enough. For a discussion on this issue, the reader may refer to Ungut et al. [577] dealing with a quantitative comparison between geometrical (ray) optics and wave optics. In this subsection, and others

below in the same section, GLMT may be used to provide reference data which are compared with ray optics computations.

In the fiducial period, we have papers by Yu et al. [578] who pro- posed a model, which can be applied to shaped beam illumination, to compute the salient optical caustic partitions occurring in the primary rainbow for oblate spheroidal drops, by Bin et al. [579] who studied the characteristics of laser scattering by droplets, based on geometrical optics, and taking into account the distribution of raindrop sizes and shapes, by Duan et al. [580] who dealt with the vectorial complex ray model (VCRM) for simulation and experimental validation of three- dimensional light scattering by a real liquid jet, see as well Duan et al. [581] who used again the VCRM to study the light scattering by an infinite cylinder of arbitrarily smooth cross section, and performed a numerical study on the scattering patterns of composite elliptical cylinders related to the appearance of natural rainbows, by Bredikhin and Kazakov [582] and Bredikhin [583] who presented an algorithm for describing focusing properties of a transparent sphere within the framework of the geometrical optics using a Gaussian beam approxi- mation that gives a good description of the field in the region of lens caustic, with comparisons made against GLMT concerning the spatial distributions of photonic jet relative intensity formed in the vicinity of a polystyrene sphere placed in water and illuminated by a Gaussian beam, with the conclusion that GLMT provides a refined description of oscillatory behaviors of intensities, by Lock [584] who used ray optics to study the imaging through a homogeneous circular cylinder, focusing on the role of virtual caustics, rainbow glare points, and image fragmentation, by Guo et al. [585] who considered the scattering of light in participating media composed of sparsely and randomly distributed discrete particles, whose size is expected to range from the scale of the wavelength to the scale several orders of magnitude greater than the wavelength, resorting to geometrical optics approximation, together with a practical Monte-Carlo solution to solve the transfer of energy in discrete participating media, and comparisons between geo- metrical approximation and LMT, by Wang et al. [586] who developed a vector ray-tracing model to compute the optical caustics associated with the primary rainbow for an oblate spheroidal drop illuminated by a Gaussian beam, by Lee et al. [587] who relied on Mie scattering and Monte-Carlo ray tracing method to study the spectral transmission behavior of visible light in the seawater, by Malik and Boruah [588] who proposed an augmented ray-optics model for computation of the optical force on spherical microscopic particles due to a beam with an arbitrary complex amplitude profile, the accuracy of the augmented model being demonstrated by comparing numerically with the existing model and with the experimental observations considering a focused vortex beam, by Duan et al. [589] who dealt with the detailed descrip- tion of a numerical ray model implementation allowing the simulation of the 3D scattering field by a non-spherical particle, with application to the rainbow scattering of spheroidal drops.

For other papers on the issue, use “geometric” and “ray” as key- words.

## 11.3. Dipole approximation and DDA

While ray optics is valid in the case of large particles (with respect to the wavelength), the dipole approximation is conversely valid for small particles (with respect to the wavelength), with the Rayleigh approximation as a special case of the electric dipole approximation. DDA (discrete dipole approximation) is a numerical technique in which large particles are viewed as a collection of elementary dipole particles.

In the fiducial period, we then have papers by Zhang et al. [590] who investigated the scattering of an Airy light-sheet by a non-spherical particle, namely a cube, dealing with the calculation of the internal and near-surface fields of the cube, using DDA, by Raschpichler et al. [591] who experimentally and theoretically studied surface properties and porosity of silica particles determined by wide-angle soft X-ray scatter- ing, for various incident photon energies, modeling the experimental
results using (i) scattering and absorption of X-rays by discrete polarizable dipoles (ii) conventional Mie theory to model a continuous medium, which is supplemented by second-order Mie scattering originating from individual pores and (iii) DDA applied to particles, by Glukhova and Yurkin [592,593] who considered the simulation of scattering of the high-order vector Bessel beam by arbitrary shaped particles in the framework of the DDA, with the open-source ADDA package, dealing as well with a classification of the various types of high-order vector Bessel beams, by Hui et al. [594] who reported an implementation of typical structured light beams including the Hermite-Gaussian beams, Laguerre-Gaussian beams, Bessel beams and Airy beams, in the DDA for scattering problems of small particles, by Mai et al. [595] who studied a spherical particle model and ten non-spherical particle models to describe the scattering properties of snow for potential use in precipitation estimation from spaceborne dual-frequency precipitation radar, single scattering properties of non-spherical snow particles being computed using DDA, while those of spherical particles are determined using Mie theory, and by Arnoldus [596] who considered the scattering of a small conducting sphere, for which only the induced electric and magnetic dipole moments contribute to the emitted radiation. Finally, Kocifaj et al. [597], in the framework of a study devoted to the pollution by night sky brightness, used and compared several approaches, namely using LMT, Henyey-Greenstein formalism, Rayleigh-Gans approximation, Rayleigh theory, and multiple scattering models, including as well comparisons with respect to observational data.

For other papers on the issue, use “Rayleigh”, “dipol”, “DDA” as keywords.

## 11.4. Debye series

Ray optics acts by relying on the use of various ray paths from a geometrical point of view. The Debye series method relies on wave optics but succeeds to select different waves corresponding to various ray paths. In other words, Debye series is the wavy analog of the geometrical ray optics. It has been introduced in the GLMT context in [598].

In the fiducial period, we have papers by Zhuang et al. [599] who used Debye series expansion for light scattering by a charged sphere, which is verified by comparing the obtained results by those from the LMT (for neutral spheres), for the calculation of far-field intensity, and absorption and extinction efficiencies of charged spherical particles illuminated by a plane wave, and the analysis of the rainbow produced by charged particles, by Liu [600] who discussed the disagreement observed when the Debye theory, calculating the far-field scattered light intensity of bubbles under plane wave illumination, provides quite different results from the ones calculated by the LMT, due to the fact that Debye coefficients converge very slowly for bubbles, therefore accordingly revised the Debye approach, and on account of the localization principle connected the Debye series of different orders with the geometric rays incident at different positions. Finally, after Bi et al. [601] who used Debye series to deal with the depolarization of nearly spherical particles, Bi and Gouesbet [602] proposed a new formulation of the Debye series based on the Riccati-differential equations to compute electromagnetic wave scattering by non-spherical particles, expanding the T-matrix in terms of the Debye series. In this expansion, the zeroth-order term corresponds to a combination of diffraction and external reflection which is given by unity minus the external reflection matrix. The higher-order terms are generated from the transmission matrix from the medium to the particle, the internal reflection matrix within the particle and the transmission matrix from the particle to the medium. After validation of the scheme in the case of a homogeneous sphere, representative results were given in the case of spheroids.

For other papers devoted to Debye series, use “Debye” as a keyword.

## 11.5. Finite element methods

Finite Element Methods (FEMs) are numerical methods in which the surface of the scatterer is discretized, e.g. with triangular patches, or the whole volume of the scatterer is discretized, e.g. with tetrahedral elements. See Wan and Tang [603] who proposed an accurate and efficient numerical simulation method, well suited for parallel computation, based on a dual-primal finite element tearing and interconnecting and Maxwell stress tensor to calculate the optical force and potential for trapping metal nanoparticles, in which using fully decoupled subdomain problems, a large-scale complex problem is converted into a series of small-scale simple problems, with the examples of silver nanocapsule, gold nanosphere dimer, and gold truncated cone dimer, Golharani et al. [604] who used the FEM to calculate field equations and the amplitude of scattered waves, in the case of plasma-covered long cylindrical non-isotropic dielectric lenses under the presence of plane monochromatic waves, see as well [605] dealing with a similar problem, Sahoo et al. [606] who dealt with three-dimensional modeling and numerical simulations based on a FEM, to estimate the scattering cross-section of single-sprouted nanostructures, under plane wave illumination, with experiments to study dark field microscopic images of nanoparticles and the plasmonic properties of single Au-Ag bimetallic nanoparticles.

For other papers on FEM, use “finite” as a keyword.

## 11.6. Finite-difference time-domain methods

FDTD methods pertain as well to the field of numerical techniques. As in fluid mechanics, it relies on the use of a grid allowing one to discretize Maxwell's equations in space, leading to finite-difference equations which are thereafter solved. Using such a technique, Lian et al. [607] dealt with the study of sub-wavelength artificial photonic structures which can be introduced to tailor and modulate the spectrum of materials, simulating optical absorption under different conditions in the case of SiO₂/Cu/ITPO arrays with surface plasmon resonances, from the near-infrared short-wave band to the near-infrared middle-wave band, Wu et al. [608] expressed arbitrary shaped beams by using an approximation in terms of an ASD, each plane wave in the ASD sample being thereafter injected into a FDTD method, and then specifically generated an arbitrary order Bessel beam with arbitrary incidence, to compute backscattering radar cross-sections in the case of spheres and spheroids, with validations carried out using comparisons against GLMT computations, Ding et al. [609] studied the optical properties of multi-particle phosphors together with a combined Monte-Carlo ray tracing combined method, both theoretically and experimentally, Savelyev [610] studied the diffraction of Gaussian beams and Laguerre-Gaussian modes on subwavelengths optical 3D microstructures with variable relief heights, with a FDTD method, Wang et al. [611] adopted a transmission filter based on Guide Mode Resonance (GMR) of all-dielectric metasurface as an example of metasurfaces for image sensors, with numerical calculations based on the FDTD method to study the relationship between the robustness of the metasurfaces and the optical resonant modes, and Liu et al. [612] employed a fluffy nanosphere with feature-shaped edges as a three-dimensional model system to study the localized light confinement effect and analyze the inherent mechanisms for strong light absorption properties, using FDTD method to compare the optical propagation and light field distribution around and inside a single nanosphere and GLMT as well since the spherical size is close to the incident wavelength.

For other papers on FDTD, use “FDTD” and “finite-difference” as keywords.

## 11.7. Miscellaneous

See Chen et al. [613] who dealt with shaped beam scattering by an object with a uniaxial anisotropic inclusion, the problem being
solved in a spherical basis by using the boundary conditions (with surface integrals numerically solved), method of moment technique and Schelkunoff's equivalence theorem which leads to a system of linear equations for the expansion coefficients of the scattered and internal fields when the description of the incident field is known. Examples are given for Gaussian beams, zeroth-order Bessel beams and Hertzian electric dipole radiations striking a spheroid with a uniaxial anisotropic spheroid inclusion and a circular cylinder with a uniaxial anisotropic circular cylinder inclusion. An important issue is that the technique used avoids to expand the illuminating beam in terms of vector wave functions, and instead relies on their explicit expressions, although the scattered and internal fields are expressed indeed in terms of vector wave functions. Comparisons are carried out with GLMT (with BSCs evaluated using a localized approximation) in the case of spheroids with spheroid dielectric inclusions, and with EBCM in the case of uniaxial anisotropic spheroids.

## 12. Acoustical waves

As already discussed in a previous review [6], analogies between electromagnetic and scalar waves, in particular acoustical waves, make the present section compulsory. In the fiducial period, we then have Bredikhin and Kazakov [582] who dealt with the conversion of pulsed laser radiation into acoustic radiation, produced by exciting polystyrene transparent spheres having a colloidal single-layer coating with laser radiation (λ = 1.064 µm), generating sound pulses in the liquid in the approximately 0.2–4 MHz range, Li et al. [614] who derived an expression for the axial acoustic radiation force on a spherical particle on the axis of a zeroth-order Mathieu sound field propagating in an ideal fluid, with the consideration of different kinds of particles including rigid spheres, fluid spheres and elastic spheres, Wu et al. [615] who dealt with a large-scale multiplexed weak reflector array fabricated with a femtosecond laser for a fiber-optic quasi-distributed acoustic sensing system which can be used to measure and locate acoustic signals along a fiber of several kilometers with large-scale monitoring applications such as pipeline monitoring, reservoir monitoring and safeguarding purposes, and demonstrated the fabrication of a prototype of a weak reflector array which has a length of 9.8 km, a reflector interval of 10 m, a reflectivity around –42 dB and a transmission loss of 0.34 dB/km, scattering losses being generated by point shape and line shape according to Mie-like processes, Pessôa and Neves [616] who presented a solution to the acoustic scattering and radiation force problem by a fluidic sphere, based on multipole expansion coefficients for a general acoustic field, and which is applied to two common acoustic fields: stationary waves and arbitrary located Bessel beams of orders 0 and 1, the behavior of the acoustic force and scattering being in agreement with the Rayleigh, Mie-like and geometric scattering regimes, showing in particular results for force in the stationary pressure waves, typical for acoustic levitation trap which exhibit nodes where small particles can be trapped by a restoring acoustic force, Schweitzer et al. [617] who studied the spray drying of colloidal suspensions, with experiments conducted on single suspension droplets trapped in an acoustic field as well as on droplets in a mini-spray dryer, Rondon [618] who studied the fundamental properties for the spin and orbital angular momentum using acoustic evanescent Bessel beams, and then explored the possible connections among optics, electromagnetic and sound waves and whether the acoustic spin density can be of interest by studying remarked similarities among evanescent waves, Wang et al. [619] who studied robust acoustic pulling forces using chiral surface waves, showing that long-range and robust acoustic pulling can be achieved by using a pair of one-way chiral surface waves supported on the interface between two phononic crystals composed of spinning cylinders with equal but opposite spinning velocities embedded in water, providing a new mechanism for the acoustic manipulation apart from the levitation, trapping and binding approaches, Ospina et al. [620] who demonstrated by means of a full three-dimensional numerical analysis and

a thorough experimental study that the acoustic force, and thus the particle's behavior, critically depends on its size, in the case of acoustic levitation and trapping, with the case study of expanded polystyrene particles levitated in air with an ultrasonic frequency of 40 kHz, using a theoretical model based on the GLMT, Farhat et al. [621] who considered the acoustic scattering of cylindrical objects in spinning motion around the axis of rotational symmetry, and investigated the radiation force and torque induced by various incident signals, e.g. cylindrical diverging and converging beams as well as quasi-Gaussian beams of different orders, Martin [622] who investigated time-harmonic acoustic waves in a compressible inviscid fluid, specifically various superpositions of Bessel beams called quasi-Gaussian beams, under the form of expansions as infinite series of regular spherical wave functions, discussing as well the case when the incident wavefield does not solve the Helmholtz equation, e.g. a paraxial Gaussian beam, Pessôa and Neves [623] who described an experimental set-up for a standing wave ultrasound trap for air microbubbles in oil, together with a model for the finite acoustic beam using the ASD technique for the acoustical BSCs, reconstructing the pressure field and acoustic scattering using the GLMT framework, which was validated using a FEM simulation, Li et al. [624] who evaluated the expansion coefficients of an off-axial Gaussian beam using the localized approximation and the translational addition theorem for spherical wave functions, then deriving the three-dimensional acoustic radiation force on an elastic sphere positioned in the beam, Li and Zhang [625] who derived expressions for calculating the three-dimensional acoustic radiation force on a multilayer microsphere (eukaryotic cell) positioned arbitrarily in a Gaussian beam, the model being expressed in terms of BSCs which are evaluated using an on-axis localized approximation while, for off-axis configurations, the BSCs are calculated by the localized approximation method together with the translational addition theorem of spherical wave functions, the theoretical model being validated by comparisons to FEM results, and Li and Zhang [626] dealing, theoretically and experimentally with acoustic radiation force of a rigid sphere arbitrarily positioned in a zeroth-order Mathieu beam, with expansion coefficients obtained by using the addition theorem of the Bessel functions.

Analogies between acoustical and electromagnetic scatterings are so strong that some authors may use the electromagnetic terminology to characterize acoustical phenomena, such as Wang et al. [619] speaking of “Lorenz-Mie theory for acoustic force” in an Appendix or Pessôa and Neves [623] speaking of a “generalized Lorenz-Mie theory framework”. One of the most striking facts is that the acoustical BSCs which encode the structure of the incident acoustical field may be evaluated with the arsenal which has been already developed for electromagnetic beams, notwithstanding obvious translations. A systematic effort for the transfer of the BSC-arsenal from electromagnetic waves to acoustical wave is then currently under development. For finite series, see Ambrosio and Gouesbet [627] dealing with acoustic arbitrary-order Bessel beams and field reconstructions, Ambrosio and Gouesbet [628] dealing with on-axis Bessel, Laguerre-Gauss and Gaussian beams, including a discussion of a modified version of the finite series method presented above in the context of optical fields. Also, Ambrosio and Gouesbet [629] focused on the case of acoustical Bessel beams, comparing a localized approximation procedure and quadrature results, with discussion of a version known as the integral localized approximation, showing that the localized approximations can serve as an accurate method to investigate physical properties of interest in acoustic scattering for Bessel beams with small axicon angles and topological charges. Localized approximations for on-axis Gaussian beams are discussed in Gouesbet and Ambrosio [630] in which Gaussian beams are taken as special cases of Laguerre-Gauss beams and in Gouesbet and Ambrosio [631] where they are described in the framework of a scheme of approximations due to Davis in the electromagnetic framework [120], [122,632], adapted to the case of acoustical waves. The case of off-axis Gaussian beams taken as special cases of off-axis Laguerre-Gauss beams is discussed in Gouesbet and Ambrosio [633]. It is expected that the case of “arbitrary
shaped beams” might be processed in a N-beam method similar to the one developed for electromagnetic beams, not only in spherical coordinates [47], but also in circular cylindrical coordinates [634] or in elliptical cylindrical coordinates [635], with limitations for beams with axicon angles [48] or having topological charges [49], as is the case for electromagnetic beams, this last sentence being as well a proposal for a research program devoted to an acoustical GLMT-like framework.

## 13. Quantum related issues

GLMT has been used to show the failure of optical theorem in quantum mechanics [636] or to study [199] two-photon absorption and fluorescence in a micro-cavity under double pulse illumination. More generally, since GLMT can compute the fields inside a particle, it is expected that studies of quantum phenomena in cavity should find a benefice in the use of GLMT. Although this idea may be misleading, there has been below a few papers dealing with quantum related issues, in which GLMT is or may be relevant according to references cited in the papers.

Andrews et al. [637] published a tutorial on Quantum Electrodynamics (QED) which is a key framework for developing the theory of light-matter interactions in modern optics and photonics, with representations of quantized light fully embracing the concept of photons, discussing in particular a most prevalent *semiclassical theory* in which light is treated classically, and matter is treated by methods based on quantum mechanics, and mentioning GLMT in a subsection devoted to optical binding where it is acknowledged that, for particles larger than the wavelength of the input light, it is the classical Lorenz-Mie formalism that should be applied.

Wang et al. [638] dealt with underwater quantum key distribution which is thought to be important for modern underwater communications in an unsecure environment, relevant in order to guarantee secure underwater communication between submarines and enhance safety for critical network nodes, and proposed a scheme for strengthening the security of an underwater system in terms of the secret key rate and the maximal transmission depth. One issue concerns the transmittance efficiency, and therefore the attenuation coefficient which is affected by absorption and scattering dependent, in the present case, on laser light scattering theories, although in the analysis considered in the paper the security analysis is based on pure seawater.

Tareeva et al. [639] discussed two-photon excited luminescence in polyethylene and polytetrafluoroethylene, excitation being produced by radiations with different wavelengths and indeed detected two-photon excited luminescence in polymers at low intensity of exciting light, and also registered luminescence in Teflon in the blue range under excitation by ruby laser pulses.

Davydov and Zlydneva [640] carried out a study on the simulation of the photon wave function to explain the Young's experiment and prospects for its use in quantum cryptography, dealing with the still controverted issue of the interpretation of quantum mechanics. According to the authors, for a photon wave packet, one can count the mathematical expectation (mean value) of coordinate in which the photon is detected, to establish the mean travel speed of the packet in space. It was then discovered that this mean speed is noticeably less of light speed. This is in agreement with the classical demonstration that the speed of laser light is smaller than the speed of light, even in vacuum [166].

Zhang et al. [641] dealt with a performance analysis of continuous-variable measurement-device-independent quantum key distribution under diverse weather conditions, in particular having in mind the case of battlefields. There, when implementing the quantum key distribution protocol in atmospheric channel, the quantum states are inevitably affected by atmospheric turbulence, rain, fog, and other factors of natural environment, mainly from the absorption and scattering of light. The paper then studied the effect of rain and fog of a quantum key

distribution protocol according to Mie scattering and a beam deflection model.

For other papers relevant to the issue, see Abad-Arredondo [343] already quoted for fluorescence and $\beta$ decay, and use “quantum” and "Raman" as keywords.

## 14. Reviews and tutorials

Angelsky et al. [642] reviewed the basic concepts and manifestations of the singular and structured light fields, dealing with the deep intrinsic relations between the singularities and the rotational phenomena in light, involving essentially the dynamical aspects of light fields and their interactions with matter. Some keywords and key expressions relevant to the review are as follows: topological nature of singularities, optical vortices and circular vortex beams, generation and diagnostics of optical singularities (sources, sinks, saddles, centres, stable and unstable spiral points), mechanical action of structured light fields on material objects, canonical spin-orbital decomposition of electromagnetic momentum.

Gouesbet et al. [632] reviewed diverging and converging schemes to the description of electromagnetic Gaussian beams. Indeed Gaussian beams are the most celebrated and used kind of laser beams. Their description beyond paraxial regimes has a long and venerable history, culminating maybe with the building of a scheme of approximations which can be named the Davis scheme of approximations [120] whose convergence has been considered as granted. Strange as it may be, a paper by Wang and Webb [643] demonstrated that, actually, the Davis scheme is divergent. This quite unexpected result has been dramatically overlooked. This has been the motivation for the present review dealing with diverging and converging schemes of approximations to describe fundamental electromagnetic Gaussian beams, one of the new results obtained is that a scheme of approximations known as the improved standard scheme, introduced more than two decades ago, is diverging as well. These divergences are the results of the behavior of asymptotic series similar to the ones encountered in quantum electrodynamics. They do not prevent to obtain correct results if only the first terms of the involved series are preserved.

Ren and Shen [644] reviewed the scattering of shaped beams by large particles, dealing mainly with various GLMTs and a discussion of the methods usable to the evaluation of BSCs, including the ASD method, while Gouesbet and Ambrosio [33] focused on cylindrical particles and reviewed the GLMTs for cylinders (with circular or elliptical cross-sections), with a particular interest devoted to the localized approximations in circular and elliptical cylindrical coordinates.

Bunea and Glückstad [645] reviewed the issue of optical trapping with a focus on the optical trapping of biological samples. This review encompassed the basic concepts of optical trapping (trapping spherical particles, anisotropic particles, laser light propagation modes, generation of single and multiple optical traps) and the applications for biological samples (single molecule studies, Raman analysis, microrheology measurements, particle and cell sorting, subcellular manipulation using optical methods, as well as the applications to measurements in complex biological media). Strategies for improving optical trapping in biological samples are listed and discussed, such as using advanced beam-shaping or “sculpting” the object.

Polimeno et al. [646] provided a review of optical forces in the T-matrix formalism. The authors provided a background on the theory of optical trapping and optomechanics, a discussion of the electromagnetic theory and of the T-matrix formalism, going on with applications to model particles and with forces and torques in the T-Matrix formalism, ending with a discussion of size scaling in optical trapping of nanowires. In this paper, “T-matrix” is viewed as synonymous with Extended Boundary Condition Method (EBCM) in contrast with another point of view shared by several authors, in which EBCM is only one of the T-matrix methods available, e.g. a discussion on this issue in the introduction of this paper.