beyond mie theory: systematic computation of bulk
TRANSCRIPT
Beyond Mie Theory: Systematic Computation of Bulk ScatteringParameters based on Microphysical Wave Optics
YU GUO, University of California, Irvine, USAADRIAN JARABO, Universidad de Zaragoza - I3A, SpainSHUANG ZHAO, University of California, Irvine, USA
๐ cls = 1, ๐๐ = 300nm ๐ cls = 100, ๐๐ = 300nm ๐ cls = 100, ๐๐ = 500nm ๐ cls = 100, ๐๐ = 500nm ๐ cls = 100, ๐๐ = 500nm ๐ cls = 100, ๐๐ = 500nmIsotropic Isotropic Isotropic Isotropic Anisotropic Postively correlated_ = 700nm _ = 700nm _ = 700nm Multi-spectral _ = 700nm _ = 400nm
Fig. 1. We introduce a new technique to compute bulk scattering parameters (i.e., the extinction and scattering coefficients as well as the single-scatteringphase function) in a systematic fashion. By considering wave optical effects and particle (scatterer) interactions at the microscopic level, our technique enjoysthe generality of supporting a wide range of media (e.g., isotropic, anisotropic, and correlated). In this figure, we show renderings of thin slabs lit with asmall area light from behind (top). Additionally, we show visualizations of the corresponding particle distributions (middle) as well as per-cluster particlecounts ๐ cls and radii ๐๐ (bottom).
Light scattering in participating media and translucent materials is typi-cally modeled using the radiative transfer theory. Under the assumptionof independent scattering between particles, it utilizes several bulk scat-tering parameters to statistically characterize light-matter interactions atthe macroscale. To calculate these parameters based on microscale materialproperties, the Lorenz-Mie theory has been considered the gold standard. Inthis paper, we present a generalized framework capable of systematicallyand rigorously computing bulk scattering parameters beyond the far-fieldassumption of Lorenz-Mie theory. Our technique accounts for microscalewave-optics effects such as diffraction and interference as well as interac-tions between nearby particles. Our framework is general, can be plugged inany renderer supporting Lorenz-Mie scattering, and allows arbitrary packingrates and particles correlation; we demonstrate this generality by comput-ing bulk scattering parameters for a wide range of materials, includinganisotropic and correlated media.
CCS Concepts: โข Computing methodologiesโ Rendering.
Additional Key Words and Phrases: Radiative transfer, bulk scattering pa-rameters, wave optics
Authorsโ addresses: Yu Guo, [email protected], University of California, Irvine, USA;Adrian Jarabo, [email protected], Universidad de Zaragoza - I3A, Spain; Shuang Zhao,[email protected], University of California, Irvine, USA.
1 INTRODUCTIONParticipating media and translucent materialsโsuch as marble, milk,wax, and human skinโare ubiquitous in the real world. These ma-terials allow light to penetrate their surfaces and scatter in theinterior. In computational optics and computer graphics, how lightinteracts with participating media and translucent materials is typ-ically modeled using the radiative transfer theory (RTT). Underthis formulation, a participating medium consists of microscopicparticles (scatterers) randomly dispersed in some homogeneous em-bedding medium. After entering a translucent material, light travelsin straight lines in the embedding medium and occasionally col-lides with a particle and gets redirected into a new direction. Tocapture the macroscopic behavior of light, the RTT uses a statisticaldescription of the particles (the medium bulk parameters), namelythe extinction coefficient ๐t (aka. optical density), the scatteringcoefficient ๐s, and the phase function ๐p.While purely phenomenological in origin, the RTT has been
demonstrated a corollary of Maxwell equations, under the assump-tion of far-field or independent scattering [Mishchenko 2002]. There-fore, these optical bulk parameters can be obtained from first prin-ciples, using e.g., Lorenz-Mie theory [van der Hulst 1981; Frisvad
ACM Trans. Graph., Vol. 1, No. 1, Article . Publication date: September 2021.
2 โข Guo, Jarabo, and Zhao
et al. 2007]. However, although very successful in practice, this the-ory neglects the interactions occurring between particles in theirnear-field, including wave-optics effects such as diffraction andinterference with neighbor particles. Consequently, Lorenz-Mie the-ory is largely limited to isotropic media with relatively low packingrates. Examples of particles arranged as clustersโor falling in thenear-field region of each otherโare widespread in nature: Fromdense media where the particles density and packing rate is large,to spatially-correlated media such as clouds or biological structureswhere microscopic scatterers form clusters.
Previously, the classical radiative transfer theory has been gener-alized to handle materials with (statistically) organized microstruc-tures. Anisotropic media [Jakob et al. 2010], for instance, have bulkscattering parameters with stronger directional dependency com-pared to isotropic media. Additionally, media comprised of particleswith correlated locations can exhibit non-exponential transmittanceand characteristic scattering profiles [Bitterli et al. 2018; Jarabo et al.2018]. Although several empirical models have been proposed tomodel these media, these models work on the macro-scale directly,they are still based on the very same far-field assumption of Lorenz-Mie scattering, and lack the generality to capture wave-optics ormulti-spectral effects. Therefore, techniques capable of computingthe bulk optical parameters of a material, based on its microscopicproperties, have been lacking.
In this paper, we bridge this gap by introducing a new techniqueto systematically and rigorously compute the bulk scattering pa-rameters. The elementary building block of our technique is particleclusters in which individual particles follow user-specified distribu-tions. Within a cluster, we consider full near-field light transporteffects; Between clusters, on the contrary, we use a far-field ap-proximation to allow efficient modeling of macroscopic level lighttransport.
Our formulation is derived from first principles of light transport(i.e., Maxwell electromagnetism) and reduces to the Lorenz-Mietheory in the special case of single-particle scatterers. Based onthis formulation, we demonstrate how the bulk parameters can becomputed numerically. Using our technique, we systematically gen-erate radiative transfer optical parameters capturing multi-spectral,anisotropic, and correlated scattering effects for particles with arbi-trary distributions (Figure 1).Concretely, our contributions include:
โข Establishing a computational framework for modeling light scat-tering from clusters of particles (ยง4).
โข Showing how radiative transfer parameters can be computednumerically based on our formulation (ยง5).
โข Demonstrating how our technique can be applied to systemati-cally compute scattering parameters for a variety of participatingmedia (ยง6).
2 RELATED WORKRadiative transfer. Simulating the propagation of light in par-
ticipating media has been widely studied in graphics [Novรกk et al.2018], building upon the radiative transfer equation (RTE), intro-duced 125 years ago by von Lommel [1889] (see [Mishchenko 2013]
for a historical perspective). This scalar radiative formulation hasbeen extended in graphics accounting for anisotropic [Jakob et al.2010], refractive [Ament et al. 2014], bispectral [Gutierrez et al.2008], or spatially-correlated media [Jarabo et al. 2018; Bitterli et al.2018]. All these works assume a radiometric light transport model,establishing no connections with the electromagnetic behaviourgoverning light transport. From a wave-optics perspective, a fewworks have generalized light transport in media to account for wave-based properties, including polarized light transport [Wilkie et al.2001; Jarabo and Arellano 2018], or coherence [Bar et al. 2019]. Thislast work is of special relevance, given that it was able to simulatepurely wave-based phenomena such as speckle or coherent back-scattering on top of a radiative model. All these works build on theassumption of the far-field approximation and independent scat-tering, which largely simplifies computations. A notable exceptionis the near-field model proposed by Bar et al. [2020], that rendersspeckle statistics in the near-field zone of the camera, although itstill considers independent far-field scattering between particles.In contrast, in this work we explicitly relate the radiometric lighttransport modeled by the RTE with physics-based optics based onelectromagnetism, and generalize the independent scattering ap-proximation to account clusters of particles in the near field.
Modeling scattering in media. The phase function models the aver-age scattering distribution at a light interaction with the medium. Acommon approach is to use simple phenomenological models, suchas the Henyey-Greenstein phase function [Henyey and Greenstein1941] or mixtures of von Mishes-Fisher distributions [Gkioulekaset al. 2013], as well as other functions modeling the scattering ofidealized anistropic particles [Zhao et al. 2011; Heitz et al. 2015];however, these methods lack an explicit relationship with the un-derlying microscopic material properties. Under the assumptionof geometric optics, several works have proposed to precomputethe phase functions of more complex particles for granular materi-als [Meng et al. 2015; Mรผller et al. 2016] or cloth fibers [Aliaga et al.2017] using explicit path tracing, by neglecting wave effects. A morerigorous phase function is based on the Lorenz-Mie theory [van derHulst 1981], which provides closed-form solutions for the Maxwellโsequations for spherical particles [Jackel and Walter 1997; Frisvadet al. 2007]. Sadeghi et al. [2012] generalized the Lorenz-Mie the-ory to larger non-spherical particles in the context of accuratelymodeling rainbows. To avoid the expensive sum series of the Lorenz-Mie theory, Guo et al. [2021] proposed to use the geometric opticsapproximation [Glantschnig and Chen 1981], which gives a goodapproximation to Lorenz-Mie theory for larger particles at signifi-cantly lower cost. All these approaches provide accurate rigoroussolutions to the far-field scattering of disperse particles.Beyond Lorenz-Mie, several exact rigorous solutions have been
proposed for computing electromagnetic scattering of particles inmedia, including the finite elements method (FEM), the finite dif-ference time domain (FDTD) method, or the boundary elementsmethod (BEM) [Wu and Tsai 1977], which solve the Maxwellโs equa-tions for arbitrary shapes. Xia et al. [2020] proposed using BEM foraccurately precomputing the far-field scattering of individual fibers.Unfortunately these methods are very slow as the number of par-ticles increases, limiting its applicability to individual elements in
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Beyond Mie Theory: Systematic Computation of Bulk Scattering Parameters based on Microphysical Wave Optics โข 3
problems with reduced dimensionality. The T-matrix method [Wa-terman 1965] generalizes the Lorentz-Mie theory to particles ofarbitrary shape in both the near- and far-fields, with the only as-sumption of the computed field being outside a sphere surroundingthe particles. This method was later extended to clusters of multipleparticles [Peterson and Strรถm 1973; Mackowski and Mishchenko2011]. We leverage the T-matrix method for computing the scatter-ing of groups of particles.
Wave optics in surface scattering. Inspired on the vast backgroundon electromagnetic surface scattering in optics (see [Frisvad et al.2020] for a general survey), several works in graphics have takeninto account relevant wave effects including diffraction-aware BS-DFs [He et al. 1991; Stam 1999; Cuypers et al. 2012; Dong et al. 2015;Holzschuch and Pacanowski 2017; Toisoul and Ghosh 2017; Werneret al. 2017; Yan et al. 2018], goniochromatic patterns due to thin-layerinterference [Smits and Meyer 1992; Gondek et al. 1994; Belcour andBarla 2017; Guillรฉn et al. 2020], or birefringence [Steinberg 2019].These works assume single scattering, with no interaction betweendifferent particles with a few exceptions that assume full incoher-ence after single scattering [Falster et al. 2020; Guillรฉn et al. 2020].Notably, Moravec [1981] andMusbach et al. [2013] computed the fullelectromagnetic surface scattering by solving the wave propagationusing the FDTD method.
3 PRELIMINARIESWe now briefly revisit the basics on first principles of (classical)light transport theory based on Maxwell electromagnetism. Table 1summarizes the symbols used along the paper.
3.1 Electromagnetic ScatteringThe propagation of a time-harmonic monochromatic electromag-netic field with frequency๐ is defined by theMaxwell curl equationsas
โ ร E(r) = i๐ ` (r)H(r),โ ร H(r) = โi๐ Y (r) E(r), (1)
where โ ร . is the curl operator; E(r) and H(r) indicate, respec-tively, the (vector-valued) electric and magnetic fields at r; ` (r) andY (r) denote the (scalar-valued) magnetic permeability and electricpermittivity at r, respectively; and i :=
โโ1 is the imaginary unit.
Assuming a non-magnetic medium satisfying ` (r) = `0 with `0being the magnetic permeability of a vacuum, Equation (1) reducesto the electric field wave equation
โ2 ร E(r) โ ๐ (r)2 E(r) = 0, (2)
where โ2 = โ ร โ , and ๐ (r) = ๐โY (r)`0 is the mediumโs wave
number at r. Note that the wave number ๐ has a dependence on thefrequency ๐ ; in the following we omit such dependence for brevity.
We now assume an infinite homogeneous isotropic medium withpermittivity Y1, filled with scatterers bounded by a finite disjointregion๐ , with potentially inhomogeneous permittivity Y2 (r). Underthis assumption, we can solve Equation (2) by expressing it as thevolume integral equation (see ยงS1 on the supplemental or ยง3.1 ofMishchenkoโs work [2006] for a step-by-step derivation) as the sumof the incident field Einc (r) and the scattered field Esca (r) due to
Table 1. Symbols used along the paper.
Symbol Definition
r โ R3 Positionrฬ โ S2 Direction to r๐ โ R DistanceY (r) Permittivity` (r) Permeability๐ Wave angular frequency [sโ1]
_ = 2๐๐โ1 Wavelength [m]๐ (r) = ๐
โY (r)` (r) Wavenumber at r
๐ (r) = ๐2 (r)/๐1 Relative refractive index at rH(r) Magnetic field at rE(r) Electric field at r (4)
Einc (r) Incident electric field rEsca (r) Scattered electric field at r (4)E0 Amplitude of a planar electric field
Esca1 (rฬ) Far-field angular distribution of the scattered radiationโโ๐บ Free-space dyadic Greenโs function (5)โโ๐ Dyad transition operator (9)
๐ (nฬ, r) Planar field scalar propagator๐๐ Volume suspended by particle/cluster ๐
R๐ โ R3 Representative position of particle/cluster ๐Rฬ๐ ๐ โ S2 Direction from R๐ to R๐๐ ๐ ๐ โ R Distance from R๐ to R๐๐ cls Number of particles in a cluster
Esca๐
(r) Scattered field of r โ ๐๐ (8)E๐ (r) Exciting field in r โ ๐๐Eexc๐ ๐
(r) Partial exciting field in r โ ๐๐ from particle ๐ (10)โโ๐ดnear๐
(nฬinc, r) Near-field scattering dyad of particle/cluster ๐ (21)โโ๐ด๐ (nฬinc, nฬsca) Far-field scattering dyad of particle/cluster ๐ (24)
^t (nฬinc), ^s (nฬinc) Extinction (29) and scattering (30) cross-sections [m2]๐p (nฬinc, nฬsca) Phase function (31) [srโ1]
๐ Particles density [mโ3]๐t (nฬinc), ๐s (nฬinc) Extinction (32) and scattering (33) coefficients [mโ1]
inhomogeneities in the medium in the form of scatterers:
E(r) = Einc (r) + Esca (r) (3)
= Einc (r) + ๐21โซ๐
[๐2 (rโฒ) โ 1]โโ๐บ (r, rโฒ) ยท E(rโฒ) drโฒ, (4)
with ๐1 the wave number at the hosting medium,๐(r) = ๐2 (r)/๐1the index of refraction of the interior regions ๐ with respect to thehosting medium, the operator . ยท . is the dot product1 and
โโ๐บ (r, rโฒ)
the free-space dyadic Greenโs function defined as:
โโ๐บ (r, rโฒ) =
(โโ๐ผ + ๐โ21 โ โ โ
) exp(i๐1 |r โ rโฒ |)4๐ |r โ rโฒ | , (5)
whereโโ๐ผ is the identity dyad, and . โ . denotes the dyadic product of
two vectors. Note that the derivative operator โ applies over r. Intu-itively, Equation (4) models the scattering field as the superpositionof the spherical wavelets resulting from a change of permittivity (i.e.with๐(rโฒ) โ 1). Note also the recursive nature of Equation (4); wewill deal with this recursivity in the following section, computingEsca (r) as a function of the incident field Einc (r).
1In the paper we use . ยท . as the vector-vector, vector-dyadic and dyadic-dyadic dotproducts.
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4 โข Guo, Jarabo, and Zhao
Rฬ๐ ๐
rR๐ R๐
๐ ๐ ๐
๐๐ ๐๐
R๐Rฬ๐ถ๐
R๐
R๐ถ
๐ ๐ถ๐
Rฬ๐๐
๐ถ
Rฬ๐ถ๐R๐ถ
๐ถ
Eexc๐ถ๐
\ inc
๐ inc
Fig. 2. Schematical representation of the particles scattering geometry. Previous methods, including Lorenz-Mie theory, assume independent scattering ofparticles (left), assuming that the distance ๐ ๐ ๐ between two particles ๐ and ๐ is very large (i.e., ๐ ๐ ๐ โ โ), neglecting the potential interactions betweenparticles. In our work (middle) we differentiate between near field scattering of particles within a small region in space (cluster๐ถ centered at R๐ถ ), and particles๐ on the far-field region of the cluster (distance ๐ ๐ถ๐ โ โ). For large values of ๐ ๐ถ๐ , the direction between particle ๐ and any particle ๐ โ ๐ถ is ๐๐๐ฅ๐๐ โ Rฬ๐ถ๐ :Therefore, we can assume a planar exciting field Eexc
๐ถ๐(r) on the whole cluster๐ถ from particle ๐ , with direction Rฬ๐ถ๐ (right).
3.2 Foldy-Lax EquationsWe now consider a medium filled with ๐ finite discrete particleswith volume ๐๐ and index of refraction๐๐ (r). Considering an inci-dent E-field Einc (r), we can rewrite Equation (4) as
E(r) = Einc (r) +โซR3
๐ (rโฒ)โโ๐บ (r, rโฒ) ยท E(rโฒ) drโฒ, (6)
whereโโ๐บ (r, rโฒ) is the dyadic Greenโs function (5), and ๐ (r) the po-
tential function given by
๐ (r) =๐โ๐=1
๐๐ (r) with ๐๐ (r) ={0, (r โ ๐๐ )๐21 [๐
2๐(r) โ 1] . (r โ ๐๐ )
(7)
By combining Equations (6) and (7), we can express the field at anyposition r โ R3 following the so-called Foldy-Lax equation [Foldy1945; Lax 1951] as
E(r) = Einc (r) +๐โ๐=1
=:Esca๐
(r)๏ธท ๏ธธ๏ธธ ๏ธทโซ๐๐
โโ๐บ (r, rโฒ) ยท
โซ๐๐
โโ๐๐ (rโฒ, rโฒโฒ) ยท E๐ (rโฒโฒ) drโฒโฒ drโฒ,
(8)with Esca
๐(r) and E๐ (r) the scattered and partial field of particle ๐ , and
โโ๐๐ (r, rโฒ) the dyad transition operator for particle ๐ defined as [Tsanget al. 1985]
โโ๐๐ (r, rโฒ) = ๐๐ (r) ๐ฟ (r โ rโฒ)
โโ๐ผ
+๐๐ (r)โซ๐๐
โโ๐บ (r, rโฒโฒ) ยท
โโ๐๐ (rโฒโฒ, rโฒ) drโฒโฒ,
(9)
with ๐ฟ (๐ฅ) the Dirac delta. The partial field at particle ๐ is defined asE๐ (r) = Einc (r) + โ๐
๐ (โ ๐)=1 Eexc๐ ๐
(r), where the partial exciting fieldEexc๐ ๐
(r) from particles ๐ to ๐ is
Eexc๐ ๐ (r) =โซ๐๐
โโ๐บ (r, rโฒ) ยท
โซ๐๐
โโ๐๐ (rโฒ, rโฒโฒ) ยท E๐ (rโฒโฒ) drโฒโฒ drโฒ, (10)
with r โ ๐๐ . Note that the scattered and exciting fields for par-ticle ๐ have essentially the same form. As shown by Mishchenko[2002], the Foldy-Lax equation (8) solves exactly the volume integralequation (4) for multiple arbitrary particles in the medium, without
any assumptions on their composition or packing rate, beyond theassumption of a homogeneous hosting medium.
Far-field Foldy-Lax Equations. Equation (10) defines the exactexciting field resulting from the scattering by particle ๐ on particle ๐ .However, if the distance ๐ ๐ ๐ := โฅR๐ โ R๐ โฅ between particles (withR๐ denoting the center of particle ๐) is large, we can approximatethe propagation distance between any point r โ ๐๐ and rโฒ โ ๐๐ as
โฅr โ rโฒโฅ โ ๐ ๐ ๐ + (Rฬ๐ ๐ ยท ฮr) โ (Rฬ๐ ๐ ยท ฮrโฒ), (11)
with Rฬ๐ ๐ := (R๐โR๐ )/๐ ๐ ๐ , ฮr := rโR๐ and ฮrโฒ := rโฒ โR๐ (see Figure 2,left). With this approximation, we can now express Eexc
๐ ๐(r) for
a point r โ ๐๐ using its far-field approximation (see ยงS3 in thesupplemental for the derivation), as
Eexc๐ ๐ (r) =exp(i๐1 ๐ ๐ ๐ )
๐ ๐ ๐๐(Rฬ๐ ๐ ,ฮr) Eexc1๐ ๐ (Rฬ๐ ๐ ), (12)
with r โ ๐๐ a point in particle ๐ , ๐(nฬ, r) = exp(i๐1nฬ ยท r), and Eexc1๐ ๐ thefar-field exciting field from particle ๐ to particle ๐ defined as
Eexc1๐ ๐ (Rฬ๐ ๐ ) = (4๐)โ1 (โโ๐ผ โ Rฬ๐ ๐ โ Rฬ๐ ๐ ) (13)
ยทโซ๐๐
๐(โRฬ๐ ๐ ,ฮrโฒ)โซ๐๐
โโ๐๐ (rโฒ, rโฒโฒ) ยท E๐ (rโฒโฒ) drโฒโฒ drโฒ.
The dyad (โโ๐ผ โ Rฬ๐ ๐ โ Rฬ๐ ๐ ) ensures a transverse planar field, which
allows to solely characterize Eexc1๐ ๐ (Rฬ๐ ๐ ) by the propagation directionRฬ๐ ๐ . In order for Equation (12) to be valid, the distance ๐ ๐ ๐ needs tohold the far-field criteria, which relates the ๐ ๐ ๐ with the radius ofthe particle ๐ ๐ following the inequality [Mishchenko et al. 2006]:
๐1๐ ๐ ๐ โซ max(1,๐21๐
2๐
2
). (14)
This far-field assumption is both the basis for the Lorenz-Mie the-ory [van der Hulst 1981] (to model electromagnetic scattering fromsmall spherical particles) and, as shown by Mishchenko [2002], atthe core of the radiative transfer theory.
In the following, we relax the assumption of near-field scatteringand compute the Foldy-Lax equations for clusters of particles forboth the near- and far-field regions. Then, we use them to compute
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Beyond Mie Theory: Systematic Computation of Bulk Scattering Parameters based on Microphysical Wave Optics โข 5
the scatteringmatrix to be used in the RTE to efficiently approximatelight transport between clusters of particles.
4 SCATTERING FROM CLUSTERS OF PARTICLESIn this section, we present our main theoretical result: the far-fieldapproximated scattering dyad relating a field incoming at a particle,which will be shown in Equation (24). This dyad can then be usedto compute a mediumโs bulk scattering parameters, which we willdiscuss in ยง4.1.
The two forms of computing the exciting field from particle ๐ to ๐[Equations (10) and (12)] suggest that we can consider two subsetsof particles ๐ depending on their distance with respect to the pointof interest r: One set of ๐near particles in the near field and anotherset of ๐far particles in the far field. With that, we can now calculatethe exciting field in particle ๐ as
E๐ (r) = Einc (r) +๐nearโ
๐ (โ ๐)=1Eexc๐ ๐ (r) +
๐farโ๐=1
Eexc๐๐
(r) . (15)
In what follows, we derive the far-field Foldy-Lax equations forgroups of particles where a cluster of these particles are in theirrespective near-field region, while the other elements in the systemare in the far field. For the simplicity of our derivations, we considera single far-field incident field in the cluster, and assume that thefar-field particles ๐ do not have neighbor particles in their respectivenear field region. More formally, we now consider a cluster๐ถ of ๐๐ถ
particles, where all particles ๐ โ ๐ถ are in their respective near-fieldregion, and that the particles of the cluster have a bounding spherecentered at R๐ถ with radius ๐๐ถ (see Figure 2, middle).
Since both the incident field Einc (r) and the exciting field Eexc๐ถ๐
(r)from particle ๐ are in the far-field region, we can assume both fieldsto be planar waves defined as
Einc (r) = Einc0 exp(i๐1nฬ ยท ฮr) = Einc0 ๐(nฬ,ฮr), (16)
Eexc๐ถ๐
(r) = Eexc0๐ถ๐ exp(i๐1Rฬ๐ถ๐ ยท ฮr) = Eexc0๐ถ๐ ๐(Rฬ๐ถ๐ ,ฮr), (17)
with Einc0 the amplitude of the planar incident field, nฬ its direction,and ฮr = r โ R๐ถ . Equivalently, Eexc0๐ถ๐ =
exp(i๐1 ๐ ๐ถ๐ )๐ ๐ถ๐
Eexc1๐ถ๐ (Rฬ๐ถ๐ ) isthe amplitude of the exciting field at ๐ถ from particle ๐ , and Rฬ๐ถ๐ itsdirection.Now, let us slightly abuse the dot product notation, remove the
dependency on the spatial dependency on each term, and use (๐1 โข๐2) =
โซ๐1 (๐ฅ) ๐2 (๐ฅ) d๐ฅ for scalar-valued functions ๐1 and ๐2. From
the far-field assumptions, plugging Equation (15) into the definitionof the scattered field from particle ๐ โ ๐ถ in Equation (8) (with๐near = ๐ cls) yields
Esca๐ (r) =โโ๐บ โข
โโ๐๐ โข E๐
=โโ๐บ โข
โโ๐๐ โข
Einc +๐farโ๐=1
Eexc๐ถ๐
+๐ clsโ
๐ (โ ๐)=1Eexc๐ ๐
.(18)
By recursively expanding Eexc๐ ๐
and some algebraic operations (seethe supplemental for the full derivation), this results into
Esca๐ (r) = E0 ยทโโ๐บ โข
โโ๐๐ โข
[๐(nฬ) +
๐ clsโ๐ (โ ๐)=1
[...]๐ (nฬ)๐
](19)
+๐farโ๐=1
Eexc0๐ถ ๐
โโ๐บ โข
โโ๐๐ โข
[๐(Rฬ๐ถ๐ ) +
๐ clsโ๐ (โ ๐)=1
[...]๐ (Rฬ๐ถ๐ )๐
] ,where the domain of integration in the spatial domain of๐(nฬinc,ฮrโฒ)is ฮrโฒ = rโฒ โ R๐ถ , and "[...]๐
๐" term represents the recursivity as
[...]๐๐=
โโ๐บ โข
โโ๐๐ โข
๐ +๐ clsโ
๐ (โ ๐)=1[...]๐
๐
. (20)
Note this recursivity is similar to the one appearing in the renderingequation [Kajiya 1986]. Each element in the sum in Equation (19)above is the result of the amplitude of the far-field incident or ex-citing fields, and a series that encode all the near-field scattering inthe cluster ๐ถ . We can thus define the scattering dyad
โโ๐ดnear๐
(nฬinc, r)relating a unit-amplitude planar incident field at particle ๐ fromdirection nฬinc with the scattered field at point r as
โโ๐ดnear๐ (nฬinc, r) =
โโ๐บ โข
โโ๐๐ โข
[๐(nฬinc) +
๐ clsโ๐ (โ ๐)=1
[...]๐ (nฬinc)
๐
]. (21)
By considering constant Einc0 and Eexc0๐ถ๐ for the whole cluster ๐ถ , wecan compute the clusterโs scattering dyad as
โโ๐ดnear๐ถ (nฬinc, r) =
๐๐ถโ๐=1
โโ๐ดnear๐ (nฬinc, r), (22)
which defines the scattered field for a unit-amplitude incoming pla-nar field in a scene consisting of the particles forming cluster ๐ถ . Inpractice, the scattering dyad
โโ๐ดnear๐ถ
(nฬinc, r) can be computed numeri-cally using standard methods from computational electromagnetics(see ยง5 for more details).
Far-field approximation. Equation (21) represents the generalform of the scattering dyad for particle ๐ , which results into a five-dimensional function. Assuming that r is in the far-field region of aparticle ๐ โ ๐ถ , by using the far-field approximation of the scatteredor exciting field (12) (we refer to the supplemental document for thederivation), we get the scattered field by particle ๐ as
Esca๐ (r) โ ๐ i๐1๐ ๐
๐ ๐
(โโ๐ด๐ (nฬ, Rฬ๐ ) ยท Einc0 +
๐farโ๐=1
โโ๐ด๐ (Rฬ๐ถ๐ , Rฬ๐ ) ยท Eexc0๐ถ๐
), (23)
with ๐ ๐ := |r โ R๐ | and Rฬ๐ := rโR๐/๐ ๐ , and
โโ๐ด๐ (nฬinc, nฬsca) = (
โโ๐ผ โ Rฬ๐ โ Rฬ๐ ) ยท
๐(โnฬsca)4๐ โข
โโ๐๐
โข[๐(nฬinc) +
๐nearโ๐ (โ ๐)=1
[...]๐ (nฬinc)
๐
].
(24)
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6 โข Guo, Jarabo, and Zhao
300nm 600nm 900nm
Fig. 3. Comparison against Lorenz-Mie theory: We compare our methodwith clusters containing a single particle (i.e., ๐ cls = 1) against a ref-erence solution based on Lorenz-Mie theory for three different particleradii ๐๐ โ {300nm, 600nm, 900nm}. As expected, for a single particle ourmethod reduces to the same results as Lorenz-Mie theory. The wavelengthis _ = 600nm, while the refractive index of the particle is๐ = 1.5 + 0.1i.
Finally, since Rฬ๐ โ Rฬ๐ถ for all particles ๐ โ ๐ถ , we can approximatethe far-field scattered field of cluster ๐ถ as
Esca๐ถ (r) = ๐ i๐1๐ ๐ถ
๐ ๐ถ
(โโ๐ด๐ถ (nฬ, Rฬ๐ถ ) ยท E0 +
๐farโ๐=1
โโ๐ด๐ถ (Rฬ๐ถ๐ , Rฬ๐ถ ) ยท Eexc0๐ถ๐
), (25)
whereโโ๐ด๐ถ (nฬinc, nฬsca) =
๐๐ถโ๐=1
โโ๐ด๐ (nฬinc, nฬsca), (26)
is the far-field scattering dyad of cluster ๐ถ .Thus, by grouping the individual particles into ๐ cls near-field
clusters, and assuming that all clusters and observation point rlay in their respective far field, we can approximate the Foldy-Laxequation (8) as
E(r) = Einc (r) +๐ clsโ๐ถ ๐=1
Esca๐ถ ๐(r), (27)
with Esca๐ถ ๐
(r) the scattered field at cluster ๐ถ ๐ .
4.1 Relationship with the Radiative Transfer Theory
The scattering dyadโโ๐ด๐ถ (nฬinc, nฬsca) given by Equation (26) models
how a particles cluster ๐ถ scatters a planar unit-amplitude incidentfield from direction nฬinc towards direction nฬsca in the far-field region.However, for rendering we are generally interested on the averagefield intensity (i.e., radiance).As shown by Mishchenko [2002], the radiative transfer equa-
tion (RTE) directly derives from the far-field Foldy-Lax equationsunder three additional assumptions: (i) The amount of coherentbackscattering is negligible; (ii) The particles are randomly dis-tributed according to some distribution ๐ (๐ ๐ , b๐ ), with ๐ ๐ and b๐denoting, respectively, the position and properties (e.g., shape, size,index of refraction...) of a particle ๐; and (iii) We are interested onthe average field โจE(r)โฉ.Following these assumptions, and after a lengthy derivation,
Mishchenko demonstrates that the bulk scattering properties canbe obtained from the far-field Foldy-Lax form, and in particularfrom the scattering dyad
โโ๐ด(nฬinc, nฬsca). Let us first assume that the
distribution of particle properties b๐ are independent of the particles
position, and compute the average scattering dyad โจโโ๐ด(nฬinc, nฬsca)โฉ =โซ
ฮฉ
โโ๐ด๐ (nฬinc, nฬsca)๐ (b๐ ) db๐ . Then, note that the Foldy-Lax equation
for clusters of particles (27), we derived above has the same form asthe original Foldy-Lax equation (8). Thus, by the same derivationfollowed by Mishchenko we get to an equivalent RTE based on thescattering dyad of clusters.
Computing the scattering parameters. By taking the vectors of theparallel and perpendicular polarization ๏ฟฝฬ๏ฟฝ inc and ๏ฟฝฬ๏ฟฝinc of the incidentfield as shown in Figure 2 (right), and equivalently for the scat-tered field ๏ฟฝฬ๏ฟฝ sca and ๏ฟฝฬ๏ฟฝ
sca, we can compute the polarized scatteringcomponents ๐บ\ and ๐บ๐ from the average clusterโs scattering dyadโจโโ๐ด๐ถ (nฬinc, nฬsca)โฉ as
๐บ\ (nฬinc, nฬsca) = ๏ฟฝฬ๏ฟฝsca ยท โจ
โโ๐ด๐ถ (nฬinc, nฬsca)โฉ ยท ๏ฟฝฬ๏ฟฝ
inc,
๐บ๐ (nฬinc, nฬsca) = ๏ฟฝฬ๏ฟฝsca ยท โจ
โโ๐ด๐ถ (nฬinc, nฬsca)โฉ ยท ๏ฟฝฬ๏ฟฝ
inc. (28)
Then, based on the two scattering components ๐บ\ and ๐บ๐ , we canobtain the optical parameters of the medium as
^t (nฬinc) = 4๐โ[๐บ (nฬinc, nฬinc)
๐2๐
], (29)
^s (nฬinc) =โซS2
|๐บ\ (nฬinc, nฬsca) |2 + |๐บ๐ (nฬinc, nฬsca) |2
2๐21dnฬsca,
(30)
๐p (nฬinc, nฬsca) =|๐บ\ (nฬinc, nฬsca) |2 + |๐บ๐ (nฬinc, nฬsca) |2
2๐21^s, (31)
with ๐บ (nฬinc, nฬinc) = ๐บ๐ (nฬinc, nฬinc) = ๐บ\ (nฬinc, nฬinc), โ[๐ฅ] returningthe real part of a complex number ๐ฅ , and S2 the unit sphere ofdirections. Lastly, assuming a uniform distribution of clusters, wecan compute the extinction and scattering coefficients as
๐t (nฬinc) = ^t (nฬinc)๐
โจ๐ clsโฉ, (32)
๐s (nฬinc) = ^s (nฬinc)๐
โจ๐ clsโฉ, (33)
with ๐ the number of particles per differential volume, and โจ๐ clsโฉ theaverage number of particles per cluster. Note that the optical prop-erties defined in Equations (29)โ(33) are directionally dependent, sothey are general and can represent both isotropic and anisotropicmedia.
4.2 Relationship with Independent ScatteringMost previous works rendering light transport in media [Novรกket al. 2018] build on the assumption of independent scatteringโthatis, particles are in their respective far-field region. It is easy to verifythat this is a special case of Equation (15) with ๐ cls = 1, causing thescattering dyad
โโ๐ด๐ถ of Equation (26) to reduce to
โโ๐ด๐ถ (nฬinc, nฬsca) =
โโ๐ด๐ (nฬinc, nฬsca) =
๐(nฬsca) ยทโโ๐๐ ยท ๐(nฬinc)4๐ , (34)
which encodes the scattered field in the far-field region of a particlewhen excited by an incident unit-amplitude planar field. The Lorenz-Mie theory [van der Hulst 1981] provides closed-form expressions
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Beyond Mie Theory: Systematic Computation of Bulk Scattering Parameters based on Microphysical Wave Optics โข 7
Fig. 4. Comparison against Lorenz-Mie theory: We compare the extinctionand scattering cross sections computed with our method for๐ cls = 1 againstthe results obtained using Lorenz-Mie theory. As in Figure 3, our resultsshow perfect agreement.
forโโ๐ด๐ (nฬinc, nฬsca) for spheres and cylinders, while numerical solu-
tions ofโโ๐ด๐ (nฬinc, nฬsca) have been proposed for scatterers of arbitrary
shapes via, for example, the T-matrix method [Waterman 1965], ormore recently based on the BEM for cylindrical fibers [Xia et al.2020]. Our work is therefore a generalization of these works toparticles in the near field.
5 COMPUTING THE BULK SCATTERING PARAMETERSWe now detail our numerical computations of the scattering dyadโโ๐ด๐ถ (nฬinc, nฬsca) of Equation (26), which in turn determines the bulkscattering parameters following Equations (29)โ(33). These bulkscattering parameters can be directly used in any renderer support-ing participating media [Novรกk et al. 2018] using tabulated phasefunction and cross sections.Computing
โโ๐ด๐ถ (nฬinc, nฬsca) essentially boils down to solving the
time-harmonic Maxwell equations for an incident unit-amplitudeplanar field with direction nฬinc. While several different methods ex-ist for that purpose (see ยง16 of [Mishchenko 2014] for an overview),we opt for the superposition T-matrix method [Mackowski andMishchenko 1996] that has been demonstrated efficient for mod-erately large ๐ cls, can handle scatterers with arbitrary geometry,and is based on the principles of the Foldy-Lax equations, making itparticularly appealing for our work.In practice, we use the open-source CUDA-based CELES solver
[Egel et al. 2017], which implements the superposition T-matrixmethod proposed by Mackowski and Mishchenko [2011] for spheri-cal or randomly rotated particles. In our implementation, we focuson clusters of spherical particles. Since the Lorenz-Mie theory alsoassumes spherical particles, this allows us to directly compare ourresults with those computed using the Lorenz-Mie theory (see Fig-ures 3 and 4). Note that the T-matrix method does not introduceassumptions on the size of particles but, similar to Lorenz-Mie the-ory, larger particles result in more expensive computations.To compute the average scattering dyad โจ
โโ๐ด๐ถ (nฬinc, nฬsca)โฉ, we
average the scattered field of several random realizations of theclusters (each of which obtained by randomly sampling the po-sition of the particles inside the clusterโs bounding sphere). As
we will demonstrate in ยง6, we use a wide array of distributionsincluding particles uniformly distributed over the volume of thecluster, positively-correlated particles following Shaw et al. [2002],negatively-correlated particles using Poisson sampling of the sphere,and anisotropic distributions by uniformly sampling the particleson a oriented 2D disk.Lastly, we represent the resulting phase function as well as the
extinction and scattering cross sections as tabulated (i.e., piecewiseconstant) functions that can be used for rendering.
6 EXPERIMENTSIn this section, we first validate our technique by comparing bulkscattering parameters computed with our method and the Lorenz-Mie theory (ยง6.1). Then, we apply our technique described in ยง4and ยง5 to compute bulk scattering parameters for a wide range ofparticipating media (ยง6.2).
6.1 ValidationTo validate our technique, we compare computed bulk scatteringparameters provided by our implementation and MiePlot [Laven2011], a free software based on the Lorenz-Mie theory. We focuson the configuration where a cluster contains only one (spherical)particle as this is a fundamental assumption of the Lorenz-Mietheory.
In Figure 3, we visualize computed single-scattering phase func-tions at the wavelength 600 nm with three particle radii (300, 600,and 900 nm). We set the refractive index of the particle to 1.5 + 0.1i.Additionally, we show in Figure 4 the corresponding extinction andscattering cross sections ^t and ^s given by Equations (29) and (30),respectively. In all these examples, our computed scattering param-eters match those predicted by the Lorenz-Mie theory perfectly.
6.2 Main ResultsWe now demonstrate the versatility of our technique by computingbulk scattering parameters for a range of participating media. In allcases, we set the cluster size to roughly the same order of magnitudeof the coherence area of sunlight. This allows us to assume anincident planar field. Further, we assume that particles outside thecluster might receive different incident field. Then, light scatteringoutside the cluster is assumed to be sufficiently far away, followingthe central assumption of RTT.By default, we set the refractive indices of the particles and the
embedding media to 1.33 + 0i and 1, respectively. Please see Table 2for the performance statistics of our experiments.
Isotropic media. In computer graphics, volumetric light transporteffects are typically simulated using isotropic mediawhere the extinc-tion and scattering coefficients ๐t, ๐s are directionally independent,and the single-scattering phase function ๐p is formulated as a 1Dfunction on the angle between the incident and scattered directions.
Our technique can produce bulk scattering parameters for isotropicmedia using particles distributed in radially symmetric densities.We conduct a few ablation studies to demonstrate how different par-ticle arrangements in a cluster affect the resulting parameters. Weuse a wavelength of 700 nm for all these studies and represent the
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8 โข Guo, Jarabo, and Zhao
Sparse Intermediate Dense ๐๐=400nm ๐๐=500nm ๐๐=600nm ๐ cls = 20 ๐ cls = 100 ๐ cls = 500
(a) Varying particle spacing (b) Varying particle radii (c) Varying particle counts
Fig. 5. Comparison of the resulting (normalized) phase function for different cluster parameters, for a planar incident field at _ = 700nm. Unless mentionedotherwise, the clusters have ๐ cls = 100 particles, and each particle has radius ๐๐ = 500nm. For each phase function, we vary: (a) The distance between particleswithin the cluster; (b) The particle size ๐๐ ; and (c) The number of particles ๐ cls. We visualize all phase functions in logarithmic scale to better show theirlow-magnitude regions.
1D phase functions as tabulated (i.e., piecewise constant) functionsusing 180 equal-sized bins.In our first study, we use a cluster of 100 particles with radii
500 nm. Then, we vary the distances between particles (by usingbounding spheres with different sizes and distributing particlesuniformly in these spheres). As shown in Figure 5 (a), the closer theparticles are to each other, the more forward the resulting phasefunction is. This is expected: With sparsely distributed particles, itis simpler for light to pass straightly through.Our second ablation study examines the effect of particle size.
With 100 uniformly distributed particles, we apply our technique tothree particle sizes (๐๐= 400, 500, and 600 nm). As shown in Figure 5(b), as we increase the particles radius, the phase function becomesmore forward and increases its frequency. This agrees with thebehaviour of single particles predicted by Lorenz-Mie theory.In our third study, we vary the number of particles in a cluster
while keeping the particle size fixed to ๐๐=500 nm. Figure 5 (c) showsthat as we increase the number of particles, the phase function getsmore forward and of higher-frequency, in a behaviour somewhatcorrelated with the particles size. This is the result of the increasingnumber of diffractive elements on the cluster, that instead of makingscattering more diffuse (as predicted by geometric optics) increasesits forward frequency.Lastly, we show in Figure 6 monochrome renderings using bulk
scattering parameters obtained with varying combinations of parti-cle count and radius.
Multi-spectral results. Since our technique is derived using mi-crophysical wave optics, it allows systematic generation of multi-spectral parameters based on a single (monochrome) configurationof particle cluster.To demonstrate this, we use a configuration of 100 uniformly
distributed particles (per cluster) with radius 500 nm and compute
bulk scattering parameters at 50 wavelengths ranging from 400 nmto 700 nm. In Figure 7, we visualize the computed phase functionsat five wavelengths as well as multi-spectral renderings of a back-lit thin slab. The smooth changes in scattering parameters acrosswavelength have resulted in a characteristic rainbow-like effect.When using the single-particle configuration (with identical overallparticle density per unit volume), the rainbow effect is missing.
Figure 8 shows renderings of the Lucy model using these scatter-ing parameters.
Varying particle refractive indices. We show in Figure 9 how therefractive index of the particles affects the final appearance. In thisexample, all four media is formed by clusters of 100 particles withradii 500 nm. We keep the imaginary part of refractive index to 0and vary the real part from 1.2 to 1.5. Increasing the refractive indexleads to a stronger backward scattering, which makes the renderedobject less transparent.
Varying particle sizes. Our technique supports clusters comprisedof particles with varying sizes. In Figure 10, we illustrate how varia-tions of particle sizes affects macro-scale object appearance. Specifi-cally, on the top of this figure, we show bulk phase functions of fourisotropic media generated using our method with ๐ cls = 100 anduniformly distributed particles. Further, the particle sizes per clusterfollow normal distributions with the mean 300 nm and standarddeviations varying from 20 nm to 200 nm.The bottom of Figure 10 shows renderings of the Lucy model
using the four media. We can see that, when the variation in particlesizes increases, the object tends to appear overall more opaque (i.e.,with lower light transmition).
Anisotropic media. Anisotropic media allow the extinction andscattering coefficients ๐t, ๐s to be directionally dependent, and have
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Beyond Mie Theory: Systematic Computation of Bulk Scattering Parameters based on Microphysical Wave Optics โข 9
๐ cls = 1 ๐ cls = 50 ๐ cls = 100 ๐ cls = 500
๐๐=300n
m๐๐=400n
m๐๐=500n
m
Fig. 6. Renderings of homogeneous Lucy models at _ = 700nm. The bulkscattering parameters are computed using our method with different com-binations of particle radius ๐๐ and per-cluster particle count ๐ cls.
full 4D phase functions ๐p. Previously, although the scattering param-eters of anisotropic media can be devised based on the microflakemodels [Jakob et al. 2010; Heitz et al. 2015], equivalences of theLorenz-Mie theory, to our knowledge, have been lacking.
By using anisotropic particle distributions, our technique can gen-erate bulk scattering parameters for anisotropic media. To demon-strate this, we use a configuration where the cluster contains ๐ cls =100 particles following an anisotropic Gaussian distribution, as il-lustrated in Figure 11 (a). We tabulate the extinction and scatteringcross sections using the latitude-longitude parameterization witha resolution of 180 ร 360. Due to the symmetry of the disc, theresulting phase function ๐p is three-dimensional, and we tabulatedit with the resolution 90 ร 180 ร 360.
In Figure 11 (b), we visualize slices of the computed single-scatteringphase function ๐p with two incident directions nฬinc. In Figure 12, weshow renderings of the Lucy model with three (spatially invariant)orientations.
Ours
Single-particle(a) Phase function (b) Thin-slab rendering
Fig. 7. Multi-spectral results: (a) visualizations of phase functions; (b)corresponding multi-spectral renderings of a thin slab lit by a small arealight from behind. Results on the top are generated using a cluster of 100particles with radii 500nm. Results on the bottom are obtained using aconventional single-particle setting. We used identical particle counts perdifferential volume for both configurations.
(a) Multi. (b) 400nm (c) 550nm (d) 700nm
Fig. 8. (a) Multi-spectral rendering of a homogeneous Lucy model usingidentical bulk scattering parameters as the top row of Figure 7. (bโd) Mono-chrome renderings of the same model at three wavelengths.
Correlated particles. In Figure 13, we demonstrate the effect ofparticles correlation within the cluster, by analyzing particles dis-tributed using both negative (Poisson sampled) and positive cor-relation [Jarabo et al. 2018]. We compare the effect of introducingmicroscopic correlation on media where the clusters position is it-self correlated, compared with uniformly distributed particles insidethe clusters. These two levels of correlation have significant effecton the final appearance of the translucent materials.
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10 โข Guo, Jarabo, and Zhao
๐ = 1.2 + 0i ๐ = 1.3 + 0i ๐ = 1.4 + 0i ๐ = 1.5 + 0i
Fig. 9. Effect of the refractive index of the particles. The top of this figurevisualizes the bulk phase functions of clusters of 100 particles with radii500nm. The refractive index of the particles range from ๐ = 1.2 + 0i to1.5 + 0i suspended in the vacuum. The bottom figures show renderings ofthe Lucy model for media with each refractive index.
Table 2. Performance statistics for our simulation. The numbers are col-lected using a workstation equipped with an Intel i7-6800K six-core CPUand an Nvidia GTX 1080 GPU. Timings are given for each random realizationof the particles within the cluster; to compute the average scattering dyadwe average 50 realizations.
๐ cls ๐p res. timeRegular (Fig. 6) 1โ500 180 ร 360 3โ16sMulti-spectral (Fig. 8) 100 180 ร 360 ร 50 35mVarying particle sizes (Fig. 10) 100 180 ร 360 7โ108sAnisotropic (Fig. 12) 100 180 ร 360 ร 90 13mCorrelated (Fig. 13) 100 180 ร 360 98s
7 DISCUSSION AND CONCLUSIONLimitations and future work. While taking into account the effect
of the near-field on clusters, our work is still based on the RTT.Therefore it relies on the far-field approximation to represent ascattering dyad useful for rendering. Therefore, while we can han-dle near- and far-field scattering, we cannot accurately model thescattering in the intermediate region, which we treat as the far field.Using more accurate representations, that capture the effects at suchmid-field region could further enhance the generality of our theoryand, thus, is an interesting future topic. This would however requireexploring an alternative light transport framework beyond the RTT.Recent light transport models tracking light coherence [Steinberg
N(300, 20) nm N(300, 60) nm N(300, 100) nm N(300, 200) nm
Fig. 10. Our technique supports clusters comprised of particles with vary-ing sizes. The top of this figure visualizes bulk phase functions of four mediagenerated with๐ cls = 100 and uniformly distributed particles. Further, sizesof particles in each cluster are normally distributed with the same mean(300nm) but varying standard deviations (20nm, 60nm, 100nm, and 200nm).The bottom of this figure shows renderings of the Lucy model made of thefour media, respectively.
and Yan 2021] are a promising framework for modeling such mid-field scattering.
Our current implementation requires precomputing the bulk opti-cal properties of the media. This limits the applicability of our workto media with homogeneous particle statistical properties. Find-ing faster approximations for our scattering functions, in the samespirit as the geometric optics approximation for Lorenz-Mie the-ory [Glantschnig and Chen 1981], is an interesting future research.An efficient analytic approximation would also be very useful forfitting real-world measurements as well as in inverse scattering ap-plications, which are now limited by the expensive precomputation.Finally, while our theory is fully general in terms of particles
shape, composition, and distribution, our implementation is cur-rently limited in practice to clusters of spherical particles. Allowingarbitrary particle shapes by using an alternative implementation ofthe T-matrix method would further improve the versatility of ourtechnique.
Conclusion. In this paper, we introduce a new technique to sys-tematically compute bulk scattering parameters for participatingmedia. Built upon first principles of light transport (i.e., Maxwell
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Beyond Mie Theory: Systematic Computation of Bulk Scattering Parameters based on Microphysical Wave Optics โข 11
Forward Backward
(a) Incident direction (b) Phase function slice
Fig. 11. Visualizations of slices ๐p (nฬinc, ยท) of a phase function for two inci-dent directions nฬinc at _ = 700nm. This phase function is computed using aconfiguration where 100 particles with radii 300nm follow an anisotropicGaussian distribution.
electromagnetism), our technique models a translucent materialas clusters of particles randomly distributed in embedding media.Our work generalizes the widely-used Lorenz-Mie theory for rig-orously deriving optical properties of scattering media, and can bereadily used in any radiative-based light transport simulator. Wehave demonstrated the significant effects of departing from the un-derlying assumptions of Lorenz-Mie theory, and the versatility formodeling a wide range of participating media by modifying thearrangement of particles within each cluster, including isotropic,anisotropic, and correlated media.
ACKNOWLEDGMENTSWe thank the anonymous reviewers for their comments and sugges-tions. Yu and Shuang are partially supported by NSF grant 1813553.Adrian is partially supported by the European Research Council(ERC) under the EU Horizon 2020 research and innovation pro-gramme (project CHAMELEON, grant No 682080), the EU MSCA-ITN programme (project PRIME, grant No 956585) and the SpanishMinistry of Science and Innovation (project PID2019-105004GB-I00).
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ACM Trans. Graph., Vol. 1, No. 1, Article . Publication date: September 2021.
12 โข Guo, Jarabo, and Zhao
(a) Negatively correlated particles (b) Positively correlated particles
(a1) unc. clusters (a2) neg. clusters (b1) unc. clusters (b2) pos. clusters
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ACM Trans. Graph., Vol. 1, No. 1, Article . Publication date: September 2021.
Supplementary MaterialBeyond Mie Theory: Systematic Computation of Bulk ScatteringParameters based on Microphysical Wave Optics
YU GUO, University of California, Irvine, USAADRIAN JARABO, Universidad de Zaragoza - I3A, SpainSHUANG ZHAO, University of California, Irvine, USA
In this document we derive the far-field scattered field of clustersof particles from the Foldy-Lax equations. For completeness andselfcontainedness, we start by reviewing time-harmonic electro-magnetics following the derivations described by Mishchenko etal. [2006] (ยงS1), and its formulation for a medium with multipleparticles embedded using the Foldy-Lax equations [Foldy 1945; Lax1951](ยงS2) and their far-field approximations (ยงS3.
From these, we later derive the scattering dyad encoding theresponse of a cluster of particles in the far field, which later can beused to compute the (radiative) optical properties of a scatteringmedium.
S1 ELECTROMAGNETIC SCATTERINGThe propagation of a time-harmonic monochromatic electromag-netic field with frequency๐ is defined by theMaxwell curl equationsas
โ ร E(r) = i๐` (r)H(r),โ ร H(r) = โi๐Y (r)E(r), (S.1)
with โร . the curl operator, E(r) and H(r) the electric and magneticfield at r respectively, ` (r) and Y (r) the magnetic permeability andelectric permittivity at r respectively, and i =
โโ1.
By assuming a non-magnetic medium (i.e. ` (r) = `0, with `0 themagnetic permeability of a vacuum), and taking the curl on the firstline in Equation (S.1) we get
โ2E(r) = i๐` (r)โ ร H(r)= โi2๐2`0Y (r)E(r), (S.2)
with โ2 = โ ร โ, which by arithmetic reordering reduces to theelectric field wave equation
โ2 ร E(r) โ ๐ (r)2E(r) = 0, (S.3)
where ๐ (r) = ๐โY (r)`0 is the mediumโs wave number at r. Note
that the wave number ๐ has a dependence on the frequency ๐ ; inthe following we omit such dependence for brevity.Let us now assume an infinite homogeneous isotropic medium
with permittivity Y1, filled with scatterers with potentially inhomo-geneous permittivity Y2 (r). This separates the space in two differentregions: The surrounding infinite region ๐0, and the finite disjointregion occupรฌed by the scatterers๐ , so that๐
โ๐0 = R3. Under this
Authorsโ addresses: Yu Guo, [email protected], University of California, Irvine, USA;Adrian Jarabo, [email protected], Universidad de Zaragoza - I3A, Spain; Shuang Zhao,[email protected], University of California, Irvine, USA.
configuration, we can express Equation (S.3) as two different waveequations
โ2 ร E(r) โ ๐21E(r) = 0, r โ ๐0, (S.4)
โ2 ร E(r) โ ๐2 (r)2E(r) = 0, r โ ๐ , (S.5)
with ๐1 the constant wave number at the hosting medium, and๐2 (r) the potentially inhomogeneous wave number at the scatter-ers. Equations (S.4) and (S.5) can be expressed together in a singleinhomogeneous differential equation as
โ2 ร E(r) โ ๐21E(r) = ๐ (r) E(r), (S.6)
with๐ (r) = ๐21 [๐2 (r) โ 1] the potential function at r, and๐(r) =
๐2 (r)/๐1 the index of refraction at r. It is trivial to verify that forr โ ๐0 in the hosting medium๐(r) = 1, then the potential function๐ (r) vanishes, and Equation (S.6) reduces to Equation (S.4).
Solving the inhomogeneous linear differential equation describedin Equation (S.6) results in two terms: The contribution of the in-cident field Einc (r), which is the sole contribution in the case ofa homogeneous medium, and the scattered field Esca (r) resultingof introducing inhomogeneities (i.e. scatterers) in the embeddingmedium, as
E(r) = Einc (r) + Esca (r) . (S.7)
The fist part trivially satisfies Equation (S.4) for the incidentfield Einc (r). In order to compute the scattered field Esca (r), weenforce energy conservation by computing a solution that vanishesat large distances. We introduce the free-space dyadic Green func-tion
โโ๐บ (r, rโฒ) that satisfies the impulse response of the linear system
in Equation (S.3), modeled as
โ2 รโโ๐บ (r, rโฒ) โ ๐21
โโ๐บ (r, rโฒ) =
โโ๐ผ ๐ฟ (r โ rโฒ), (S.8)
whereโโ๐ผ is the identity dyad, and ๐ฟ (ยท) is the Dirac delta function.
Note that the derivatives are with respect to r. Multiplying bothsides of the differential equation by ๐ (r) E(r), and integrating bothsides with respect to rโฒ over the entire space R3, we get
(โ2 ร
โโ๐ผ โ ๐21
โโ๐ผ
) Esca (r)๏ธท ๏ธธ๏ธธ ๏ธทโซR3
๐ (rโฒ)โโ๐บ (r, rโฒ) ยท E(rโฒ) drโฒ = ๐ (r) Esca (r), (S.9)
with . ยท . the dyadic-vector dot-product. Since the potential function๐ (r) vanishes everywhere outside ๐ , we can express the scatteredfield Esca (r) as an integral on the space occupied by scatterers ๐
ACM Trans. Graph., Vol. 1, No. 1, Article . Publication date: September 2021.
2 โข Guo, Jarabo, and Zhao
only, as
Esca (r) =โซ๐
๐ (r)โโ๐บ (r, rโฒ) ยท E(rโฒ) drโฒ
= ๐21
โซ๐
[๐2 (rโฒ) โ 1]โโ๐บ (r, rโฒ) ยท E(rโฒ) drโฒ. (S.10)
Now, the only term missing for computing Esca (r) is the Greenfunction that solves Equation (S.8), which has a well-known solutionas
โโ๐บ (r, rโฒ) =
(โโ๐ผ + ๐โ21 โ โ โ
) exp(i๐1 |r โ rโฒ |)4๐ |r โ rโฒ | , (S.11)
where . โ . denotes the dyadic product of two vectors, and thederivative operator โ applies over r.Finally, by plugin Equation (S.10) into Equation (S.7) we get the
volume integral equation [Mishchenko et al. 2006, Sec.3.1] that solvesthe Maxwell equations (S.1) as the sum of the incident field Einc (r)and the scattered field Esca (r) due to inhomogeneities in themediumin the form of scatterers:
E(r) = Einc (r) + Esca (r)
= Einc (r) +โซ๐
๐21 [๐2 (rโฒ) โ 1]๏ธธ ๏ธท๏ธท ๏ธธ๐ (rโฒ)
โโ๐บ (r, rโฒ) ยท E(rโฒ) drโฒ. (S.12)
Intuitively, Equation (S.12) models the scattering field as the su-perposition of the spherical wavelets resulting from a change ofpermitivitty (i.e. with๐(rโฒ) โ 1). This is a general equation thatsolves the Maxwell equations for non-magnetic media in arbitrarysetups. Note also the recursive nature of Equation (S.12); we will dealwith this recursivity in the following section, computing Esca (r) asa function of the incident field Einc (r).
S2 FOLDY-LAX EQUATIONSLet us consider a medium filled with ๐ finite discrete particles withvolume ๐๐ and index of refraction ๐๐ (r). We can now define thepotential function๐๐ (r) for each particle ๐ as
๐๐ (r) ={
0, r โ ๐๐๐21 [๐
2๐(r) โ 1] r โ ๐๐ ,
(S.13)
with the total potential function ๐ in Equation (S.12) defined as๐ (r) = โ๐
๐=1๐๐ (r). By combining Equations (S.12) and (S.13), wecan express the field at any position r โ R3 following the so-calledFoldy-Lax equation [Foldy 1945; Lax 1951] as
E(r) = Einc (r) +๐โ๐=1
โซ๐๐
โโ๐บ (r, rโฒ) ยท
โซ๐๐
โโ๐๐ (rโฒ, rโฒโฒ) ยท E๐ (rโฒโฒ) drโฒโฒ drโฒ
= Einc (r) +๐โ๐=1
Esca๐ (r), (S.14)
with E๐ (r) = Einc (r) + โ๐๐ (โ ๐)=1 E
exc๐ ๐
(r), where the partial excitingfield Eexc
๐ ๐(r) from particles ๐ to ๐ and Esca
๐(r) the scattered field
from particle ๐ . Note that we overload the dot-product operatoraccounting for the dyad-dyad case. The dyad transition operator
โโ๐๐ (r, rโฒ) for particle ๐ defined as [Tsang et al. 1985]
โโ๐๐ (r, rโฒ) = ๐๐ (r)๐ฟ (r โ rโฒ)
โโ๐ผ
+๐๐ (r)โซ๐๐
โโ๐บ (r, rโฒโฒ) ยท
โโ๐๐ (rโฒโฒ, rโฒ) drโฒโฒ, (S.15)
with ๐ฟ (๐ฅ) the Dirac delta,โโ๐ผ the identity dyad. The partial exciting
field Eexc๐ ๐
(r) is defined as
Eexc๐ ๐ (r) =โซ๐๐
โโ๐บ (r, rโฒ) ยท
โซ๐๐
โโ๐๐ (rโฒ, rโฒโฒ) ยท E๐ (rโฒโฒ) drโฒโฒ drโฒ, (S.16)
with r โ ๐๐ . Note that the exciting field Eexc๐ ๐
(r) has essentially thesame form as the scattered field Esca
๐(r) from particle ๐ . As shown
by Mishchenko [2002], the Foldy-Lax equations (S.14) solve exactlythe volume integral equation (S.12) for multiple arbitrary particlesin the medium without any assumptions on their composition orpacking rate, beyond the assumption of a homogeneous hostingmedium.
S3 FAR-FIELD FOLDY-LAX EQUATIONSEquation (S.16) define the exact exciting field resulting from scatter-ing by particle ๐ on particle ๐ . However, if the distance between par-ticles ๐ ๐ ๐ = |R๐ โR๐ |, with R๐ the origin of particle ๐ , is large, so that๐1๐ ๐ ๐ โซ 1, we can approximate the propagation distance betweenpoints r โ ๐๐ and rโฒ โ ๐๐ as |rโrโฒ | โ ๐ ๐ ๐ +(Rฬ๐ ๐ ยทฮr)โ(Rฬ๐ ๐ ยทฮrโฒ), withRฬ๐ ๐ =
R๐โR๐
๐ ๐ ๐, ฮr = rโR๐ and ฮrโฒ = rโฒโR๐ . With this approximation,
and after some algebraic operations, we can now approximate thedyadic Greenโs function as
โโ๐บ (r, rโฒ) โ (
โโ๐ผโRฬ๐ ๐ โ Rฬ๐ ๐ )
exp(i๐1๐ ๐ ๐ )4๐๐ ๐ ๐
๐(Rฬ๐ ๐ ,ฮr)๐(โRฬ๐ ๐ ,ฮrโฒ), (S.17)
with ๐(nฬ, r) = exp(i๐1nฬ ยท r). With this approximation, we can nowexpress Eexc
๐ ๐(r) for a point r โ ๐๐ using its far-field approximation,
as
Eexc๐ ๐ (r) =exp(i๐1 ๐ ๐ ๐ )
๐ ๐ ๐๐(Rฬ๐ ๐ ,ฮr) Eexc1๐ ๐ (Rฬ๐ ๐ ), (S.18)
with r โ ๐๐ a point in particle ๐ , and Eexc1๐ ๐ the far-field exciting fieldfrom particle ๐ to particle ๐ defined as
Eexc1๐ ๐ (Rฬ๐ ๐ ) =(โโ๐ผ โ Rฬ๐ ๐ โ Rฬ๐ ๐ )
4๐ยทโซ๐๐
๐(โRฬ๐ ๐ ,ฮrโฒ)โซ๐๐
โโ๐๐ (rโฒ, rโฒโฒ)ยทE๐ (rโฒโฒ) drโฒโฒ drโฒ.
(S.19)The dyad (
โโ๐ผ โ Rฬ๐ ๐ โ Rฬ๐ ๐ ) to ensure a transverse planar field, which
allows to solely characterize Eexc1๐ ๐ (Rฬ๐ ๐ ) by the propagation directionRฬ๐ ๐ . In order to Equation (S.19) to be valid, the distance ๐ ๐ ๐ needsto hold the far-field criteria, which relates the ๐ ๐ ๐ with the radius ofthe particle ๐ ๐ following the inequality [Mishchenko et al. 2006]
๐1๐ ๐ ๐ โซ max
(1,๐21๐
2๐
2
). (S.20)
The two forms of computing the exciting field from particle ๐
to ๐ (Equations (S.16) and (S.19)) suggest that we can consider twosubsets of particles ๐ depending on their distance with respect tothe point of interest r: One set of ๐near particles in the near field
ACM Trans. Graph., Vol. 1, No. 1, Article . Publication date: September 2021.
Supplementary MaterialBeyond Mie Theory: Systematic Computation of Bulk Scattering Parameters based on Microphysical Wave Optics โข 3
and another set of ๐far particles in the far field. With that, we cannow the exciting field in particle ๐ as
E๐ (r) = Einc (r) +๐nearโ
๐ (โ ๐)=1Eexc๐ ๐ (r) +
๐farโ๐=1
Eexc๐๐
(r) . (S.21)
In the following, we will use this as motivation for defining theexciting field on a particle from a group of particles in the far field.
S4 FAR-FIELD FOLDY-LAX EQUATIONS FOR CLUSTERSOF PARTICLES
Here we derive the far-field Foldy-Lax equations for groups of par-ticles where the a cluster of these particles are in their respectivenear-field region, while the other elements in the system are inthe far field. For simplicity in our derivations, we consider a singlefar-field incident field, as well as single particle ๐ in the far fieldregion of the cluster of particles. More formally, let us now considera cluster ๐ถ of ๐๐ถ particles, where all particles ๐ โ ๐ถ are in theirrespective near-field region, and that the particles of the cluster arebounded on a sphere centered at R๐ถ with radius ๐๐ถ .Since both the incident field Einc (r) and the exciting field Eexc
๐ถ๐from particle ๐ are in the far-field region, we can assume that bothfields are planar waves defined as
Einc (r) = Einc0 exp(i๐1nฬ ยท ฮr) = Einc0 ๐(nฬ,ฮr), (S.22)
Eexc๐ถ๐
(r) = Eexc0๐ถ๐ exp(i๐1Rฬ๐ถ๐ ยท ฮr) = Eexc0๐ถ๐ ๐(Rฬ๐ถ๐ ,ฮr) (S.23)
with Einc0 and Eexc0๐ถ๐ =exp(i๐1 ๐ ๐ถ๐ )
๐ ๐ถ๐Eexc1๐ถ๐ (Rฬ๐ถ๐ ) (S.19) the amplitude
of the planar incident field and the exciting field from particle ๐respectively, nฬ and Rฬ๐ถ๐ the propagation direction of the each field,and ฮr = r โ R๐ถ .
Now, let us slightly abuse the dot product notation defining (๐1 โข๐2) =
โซ๐1 (๐ฅ) ยท ๐2 (๐ฅ) d๐ฅ , and remove the spatial dependency on
each term. By the planar incident field assumption, and pluggingEquation (S.21) into the definition of the scattered field from particle๐ โ ๐ถ (S.14), we get
Esca๐ (r) =โโ๐บ โข
โโ๐๐ โข E๐ (S.24)
=โโ๐บ โข
โโ๐๐ โข
Einc +๐farโ๐=1
Eexc๐ถ๐
+๐nearโ
๐ (โ ๐)=1Eexc๐ ๐
.By recursively expanding Eexc
๐ ๐, Equation (S.24) becomes
Esca๐ (r) =โโ๐บ โข
โโ๐๐ โข
[Einc +
๐farโ๐=1
Eexc๐ถ๐
(S.25)
+๐nearโ
๐ (โ ๐)=1
โโ๐บ โข
โโ๐๐ โข
[Einc +
๐farโ๐=1
Eexc๐ถ๐
+๐nearโ
๐ (โ ๐)=1[...]๐
] ],
where the "[...]๐ " term represents the recursivity as
[...]๐ =โโ๐บ โข
โโ๐๐ โข
[Einc +
๐farโ๐=1
Eexc๐ถ๐
+๐nearโ
๐ (โ ๐)=1[...]๐
]. (S.26)
By reordering Equation (S.25) we get
Esca๐ (r) =โโ๐บ โข
โโ๐๐ โข
[Einc +
๐nearโ๐ (โ ๐)=1
โโ๐บ โข
โโ๐๐ โข
[Einc +
๐nearโ๐ (โ ๐)=1
[...]Einc
๐
] ](S.27)
+๐farโ๐=1
โโ๐บ โข
โโ๐๐ โข
[Eexc๐ถ๐
+๐nearโ
๐ (โ ๐)=1
โโ๐บ โข
โโ๐๐ โข
[Eexc๐ถ๐
+๐nearโ
๐ (โ ๐)=1[...]E
exc๐ถ๐
๐
] ] ,where "[...]๐
๐" is similar to Equation (S.26), with form
[...]๐๐=
โโ๐๐ โข
โโ๐บ โข
[๐ +
๐nearโ๐ (โ ๐)=1
[...]๐๐]. (S.28)
Finally, by exploiting Equations (S.22) and (S.23), and contractingthe recursion, we transform Equation (S.27) into
Esca๐ (r) =โโ๐บ โข
โโ๐๐ โข
[๐(nฬ) +
๐nearโ๐ (โ ๐)=1
[...]๐ (nฬ)๐
]ยท Einc0 (S.29)
+๐farโ๐=1
โโ๐บ โข
โโ๐๐ โข
[๐(Rฬ๐ถ๐ ) +
๐nearโ๐ (โ ๐)=1
[...]๐ (Rฬ๐ถ๐ )๐
]ยท Eexc0๐ถ๐
.Note that each element in the sum in the equation above is the resultof the amplitude of the far-field incident or exciting fields, and aseries that encode all the near-field scattering in the cluster ๐ถ . Wecan thus define the scattering dyad
โโ๐ดnear๐
(nฬinc, r) relating a fieldincoming at particle ๐ from direction nฬinc with the field at point r as
โโ๐ดnear๐ (nฬinc, r) =
โโ๐บ โข
โโ๐๐ โข
[๐(nฬinc) +
๐nearโ๐ (โ ๐)=1
[...]๐ (nฬinc)
๐
]. (S.30)
Trivially, following our assumption of constant Einc0 and Eexc0๐ถ๐ forthe whole cluster ๐ถ , we can compute the clusterโs scattering dyadas:
โโ๐ดnear๐ถ (nฬinc, r) =
๐๐ถโ๐=1
โโ๐ดnear๐ (nฬinc, r) . (S.31)
The scattering dyadโโ๐ดnear๐ถ
(nฬinc, r) solves the scattering field for aunit-amplitude incoming planar field in a scene consisting of theparticles forming cluster๐ถ , and can be computed using any methodfrom computational electromagnetics.
Far-field approximation. Equation (S.30) represents the generalform of the scattering dyad for particle ๐ , which results into a five-dimensional function. Assuming that r is in the far-field region ofa particle ๐ โ ๐ถ , and by using the far-field approximation of theGreenโs function (S.18), Equation (S.24) becomes
Esca๐ (r) โ (โโ๐ผ โ Rฬ๐ โ Rฬ๐ )
exp(i๐1๐ ๐ )4๐๐ ๐
ยท ๐(โRฬ๐ ) โขโโ๐๐ ยท E๐ , (S.32)
with ๐ ๐ = |r โ R๐ | and Rฬ๐ =rโR๐๐ ๐
. Note that the term ๐(Rฬ๐ ,ฮr)inEquation (S.18) vanishes for a single particle, since |ฮr| = 0 andtherefore ๐(Rฬ๐ ,ฮr) = 1.
ACM Trans. Graph., Vol. 1, No. 1, Article . Publication date: September 2021.
4 โข Guo, Jarabo, and Zhao
Now, using the definition of the scattered fieldE๐ in Equation (S.21),and expanding Eexc following Equation (S.24), and expanding Eexc
๐ ๐
following Equation (S.25) we get
Esca๐ (r) = (โโ๐ผ โ Rฬ๐ โ Rฬ๐ )
exp(i๐1๐ ๐ )4๐๐ ๐
ยท ๐(โRฬ๐ ) โขโโ๐๐ โข
[Einc +
๐farโ๐=1
Eexc๐ถ๐
+๐nearโ
๐ (โ ๐)=1
โโ๐บ โข
โโ๐๐ โข
[Einc +
๐farโ๐=1
Eexc๐ถ๐
+๐nearโ
๐ (โ ๐)=1[...]๐
] ], (S.33)
with "[...]๐ " representing the recursivity (S.26). Following Equa-tions (S.27) and (S.29) we reorder the equation to separate the con-tribution of the incident Einc and exciting fields Eexc
๐ถ๐respectively,
and exploit the far field assumption to put Einc and Eexc๐ถ๐
in theirplanar field form [Equations (S.22) and (S.23)], as
Esca๐ (r) โ ๐ i๐1๐ ๐
๐ ๐
(โโ๐ด๐ (nฬ, Rฬ๐ ) ยท Einc0 +
๐farโ๐=1
โโ๐ด๐ (Rฬ๐ถ๐ , Rฬ๐ ) ยท Eexc0๐ถ๐
), (S.34)
withโโ๐ด๐ (nฬinc, nฬsca) the far-field scattering dyad relating incident and
scattered directions nฬinc and nฬsca as
โโ๐ด๐ (nฬinc, nฬsca) = (
โโ๐ผ โ Rฬ๐ โ Rฬ๐ ) ยท
๐(โnฬsca)4๐
โขโโ๐๐
โข[๐(nฬinc) +
๐nearโ๐ (โ ๐)=1
[...]๐ (nฬinc)
๐
].
(S.35)
Finally, since Rฬ๐ โ Rฬ๐ถ for all particles ๐ โ ๐ถ we can approximatethe far-field scattered field of cluster ๐ถ as
Esca๐ถ
(r) =๐๐ถโ๐=1
Esca๐ (r)
=
๐๐ถโ๐=1
๐ i๐1๐ ๐
๐ ๐
(โโ๐ด๐ (nฬ, Rฬ๐ ) ยท E0 +
๐farโ๐=1
โโ๐ด๐ (Rฬ๐ถ๐ , Rฬ๐ ) ยท Eexc0๐ถ๐
),
โ ๐ i๐1๐ ๐ถ
๐ ๐ถ
( ๐๐ถโ๐=1
โโ๐ด๐ (nฬ, Rฬ๐ถ ) ยท E0 +
๐farโ๐=1
๐๐ถโ๐=1
โโ๐ด๐ (Rฬ๐ถ๐ , Rฬ๐ถ ) ยท Eexc0๐ถ๐
)=๐ i๐1๐ ๐ถ
๐ ๐ถ
(โโ๐ด๐ถ (nฬ, Rฬ๐ถ ) ยท E0 +
๐farโ๐=1
โโ๐ด๐ถ (Rฬ๐ถ๐ , Rฬ๐ถ ) ยท Eexc0๐ถ๐
),
(S.36)
withโโ๐ด๐ถ (nฬinc, nฬsca) =
โ๐๐ถ
๐=1โโ๐ด๐ (nฬinc, nฬsca) the far-field scattering
dyad of cluster ๐ถ .
Computing the far-field exciting field. Let us know compute thefar-field exciting field Eexc
๐๐ถfrom a cluster ๐ถ to a particle ๐ placed
in the far-field region of ๐ถ . By plugging Equation (S.21) into Equa-tion (S.19), and under the assumption of far-field incident fields(Equations (S.22) and (S.23)) we get the exciting field from a particle
๐ โ ๐ถ over particle ๐ as:
Eexc๐๐
(r) โ (โโ๐ผ โ Rฬ๐๐ โ Rฬ๐๐ ) ยท
๐ i๐1 (๐ ๐๐+Rฬ๐๐ ยทฮr)
4๐๐ ๐๐๐(Rฬ๐๐ ) โข
โโ๐๐
โข[Einc +
๐farโ๐โฒ=1
Eexc๐๐โฒ +
๐nearโ๐ (โ ๐)=1
Eexc๐ ๐
]. (S.37)
This equation has the same form as Equation (S.34), and thus wecan express it using the far-field scattering dyad defined in Equa-tion (S.35) as
Eexc๐๐
(r) = ๐ i๐1 (๐ ๐๐+Rฬ๐๐ ยทฮr)
๐ ๐๐
(โโ๐ด๐ (nฬ, Rฬ๐ ) ยท Einc0 +
๐farโ๐โฒ=1
โโ๐ด๐ (Rฬ๐ถ๐โฒ, Rฬ๐ ) ยท Eexc0๐ถ๐โฒ
),
(S.38)
which by summing the exciting field of all particles ๐ โ ๐ถ andfollowing the far-field approximation (Rฬ๐๐ โ Rฬ๐๐ถ ,โ๐ โ ๐ถ we get
Eexc๐๐ถ
(r) โ ๐ i๐1 (๐ ๐๐ถ+Rฬ๐๐ถ ยทฮr)
๐ ๐๐ถ
(โโ๐ด๐ถ (nฬ, Rฬ๐ถ )ยทEinc0 +
๐farโ๐โฒ=1
โโ๐ด๐ถ (Rฬ๐ถ๐โฒ, Rฬ๐ถ )ยทEexc0๐ถ๐โฒ
).
(S.39)Finally, if particle ๐ is itself contained in the near-field of a clusterof particles ๐ถ1, then it is trivial to compute the exciting field fromcluster ๐ถ to ๐ถ1 as
Eexc๐ถ1๐ถ
(r) =๐๐ถ1โ๐=1
Eexc๐๐ถ
(r). (S.40)
Thus, by grouping the individual particles into ๐ cls near-fieldclusters, and assuming that all clusters and observation point rlay in their respective far field, we can approximate the Foldy-Laxequation (S.14) as
E(r) = Einc (r) +๐ clsโ๐ถ ๐=1
Esca๐ถ ๐
(r), (S.41)
with Esca๐ถ ๐
(r) defined by pluging Equation (S.40) into Equation (S.36)as
Esca๐ถ ๐
(r) = ๐i๐1๐ ๐ถ๐
๐ ๐ถ ๐
(โโ๐ด๐ถ ๐
(nฬ, Rฬ๐ถ ๐) ยท Einc0 (S.42)
+๐ clsโ
๐ถ๐ (โ ๐ถ ๐ )=1
โโ๐ด๐ถ ๐
(Rฬ๐ถ ๐๐ถ๐, Rฬ๐ถ ๐
) ยท Eexc0๐ถ ๐๐ถ๐
), (S.43)
with Eexc0๐ถ ๐๐ถ๐the amplitude of the far-field exciting field from cluster
๐ถ๐ to cluster ๐ถ ๐ .
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287.Michael I Mishchenko. 2002. Vector radiative transfer equation for arbitrarily shaped
and arbitrarily oriented particles: a microphysical derivation from statistical elec-tromagnetics. Applied optics 41, 33 (2002), 7114โ7134.
Michael I Mishchenko, Larry D Travis, and Andrew A Lacis. 2006. Multiple scattering oflight by particles: radiative transfer and coherent backscattering. Cambridge UniversityPress.
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ACM Trans. Graph., Vol. 1, No. 1, Article . Publication date: September 2021.