Beyond Mie Theory: Systematic Computation of Bulk ...

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

𝑁 cls = 1, 𝑎𝑖 = 300nm 𝑁 cls = 100, 𝑎𝑖 = 300nm 𝑁 cls = 100, 𝑎𝑖 = 500nm 𝑁 cls = 100, 𝑎𝑖 = 500nm 𝑁 cls = 100, 𝑎𝑖 = 500nm 𝑁 cls = 100, 𝑎𝑖 = 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 𝑁 cls and radii 𝑎𝑖 (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: • Computing methodologies→ Rendering.

Additional Key Words and Phrases: Radiative transfer, bulk scattering pa-rameters, wave optics

Authors’ 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 materials—such as marble, milk,wax, and human skin—are 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 𝜎t (aka. optical density), the scatteringcoefficient 𝜎s, and the phase function 𝑓p.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 • 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 clusters—or falling in thenear-field region of each other—are 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:

• Establishing a computational framework for modeling light scat-tering from clusters of particles (§4).

• Showing how radiative transfer parameters can be computednumerically based on our formulation (§5).

• Demonstrating how our technique can be applied to systemati-cally compute scattering parameters for a variety of participatingmedia (§6).

2 RELATED WORKRadiative transfer. Simulating the propagation of light in par-

ticipating media has been widely studied in graphics [Novák 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; Müller 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 Maxwell’sequations 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 Maxwell’s 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


Beyond Mie Theory: Systematic Computation of Bulk Scattering Parameters based on Microphysical Wave Optics • 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 Ström 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; Guillén 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; Guillén 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 frequency𝜔 is defined by theMaxwell curl equationsas

∇ × E(r) = i𝜔 ` (r)H(r),∇ × H(r) = −i𝜔 Y (r) E(r), (1)

where ∇ × . 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 :=

√−1 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

∇2 × E(r) − 𝑘 (r)2 E(r) = 0, (2)

where ∇2 = ∇ × ∇ , and 𝑘 (r) = 𝜔√Y (r)`0 is the medium’s wave

number at r. Note that the wave number 𝑘 has a dependence on thefrequency 𝜔 ; 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 disjointregion𝑉 , with potentially inhomogeneous permittivity Y2 (r). Underthis assumption, we can solve Equation (2) by expressing it as thevolume integral equation (see §S1 on the supplemental or §3.1 ofMishchenko’s 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 ∈ R3 Positionr̂ ∈ S2 Direction to r𝑟 ∈ R DistanceY (r) Permittivity` (r) Permeability𝜔 Wave angular frequency [s−1]

_ = 2𝜋𝜔−1 Wavelength [m]𝑘 (r) = 𝜔

√Y (r)` (r) Wavenumber at r

𝑚 (r) = 𝑘2 (r)/𝑘1 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 (r̂) Far-field angular distribution of the scattered radiation⇐⇒𝐺 Free-space dyadic Green’s function (5)⇐⇒𝑇 Dyad transition operator (9)

𝑔 (n̂, r) Planar field scalar propagator𝑉𝑖 Volume suspended by particle/cluster 𝑖

R𝑖 ∈ R3 Representative position of particle/cluster 𝑖R̂𝑖 𝑗 ∈ S2 Direction from R𝑗 to R𝑖𝑅𝑖 𝑗 ∈ R Distance from R𝑗 to R𝑖𝑁 cls Number of particles in a cluster

Esca𝑖

(r) Scattered field of r ∈ 𝑉𝑖 (8)E𝑖 (r) Exciting field in r ∈ 𝑉𝑖Eexc𝑖 𝑗

(r) Partial exciting field in r ∈ 𝑉𝑖 from particle 𝑗 (10)⇐⇒𝐴near𝑖

(n̂inc, r) Near-field scattering dyad of particle/cluster 𝑖 (21)⇐⇒𝐴𝑖 (n̂inc, n̂sca) Far-field scattering dyad of particle/cluster 𝑖 (24)

^t (n̂inc), ^s (n̂inc) Extinction (29) and scattering (30) cross-sections [m2]𝑓p (n̂inc, n̂sca) Phase function (31) [sr−1]

𝜌 Particles density [m−3]𝜎t (n̂inc), 𝜎s (n̂inc) Extinction (32) and scattering (33) coefficients [m−1]

inhomogeneities in the medium in the form of scatterers:

E(r) = Einc (r) + Esca (r) (3)

= Einc (r) + 𝑘21∫𝑉

[𝑚2 (r′) − 1]⇐⇒𝐺 (r, r′) · E(r′) dr′, (4)

with 𝑘1 the wave number at the hosting medium,𝑚(r) = 𝑘2 (r)/𝑘1the index of refraction of the interior regions 𝑉 with respect to thehosting medium, the operator . · . is the dot product1 and

⇐⇒𝐺 (r, r′)

the free-space dyadic Green’s function defined as:

⇐⇒𝐺 (r, r′) =

(⇐⇒𝐼 + 𝑘−21 ∇ ⊗ ∇

) exp(i𝑘1 |r − r′ |)4𝜋 |r − r′ | , (5)

where⇐⇒𝐼 is the identity dyad, and . ⊗ . denotes the dyadic product of

two vectors. Note that the derivative operator ∇ 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.with𝑚(r′) ≠ 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 . · . as the vector-vector, vector-dyadic and dyadic-dyadic dotproducts.



R̂𝑖 𝑗

rR𝑖 R𝑗

𝑅𝑖 𝑗

𝑉𝑖 𝑉𝑗

R𝑘R̂𝐶𝑘

R𝑖

R𝐶

𝑅𝐶𝑘

R̂𝑖𝑘

𝐶

R̂𝐶𝑘R𝐶

𝐶

Eexc𝐶𝑘

\ inc

𝜙 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 𝑅𝑖 𝑗 between two particles 𝑖 and 𝑗 is very large (i.e., 𝑅𝑖 𝑗 → ∞), neglecting the potential interactions betweenparticles. In our work (middle) we differentiate between near field scattering of particles within a small region in space (cluster𝐶 centered at R𝐶 ), and particles𝑘 on the far-field region of the cluster (distance 𝑅𝐶𝑘 → ∞). For large values of 𝑅𝐶𝑘 , the direction between particle 𝑘 and any particle 𝑗 ∈ 𝐶 is 𝑑𝑃𝑥𝑖𝑘 ≈ R̂𝐶𝑘 :Therefore, we can assume a planar exciting field Eexc

𝐶𝑘(r) on the whole cluster𝐶 from particle 𝑘 , with direction R̂𝐶𝑘 (right).

3.2 Foldy-Lax EquationsWe now consider a medium filled with 𝑁 finite discrete particleswith volume 𝑉𝑖 and index of refraction𝑚𝑖 (r). Considering an inci-dent E-field Einc (r), we can rewrite Equation (4) as

E(r) = Einc (r) +∫R3

𝑈 (r′)⇐⇒𝐺 (r, r′) · E(r′) dr′, (6)

where⇐⇒𝐺 (r, r′) is the dyadic Green’s function (5), and 𝑈 (r) the po-

tential function given by

𝑈 (r) =𝑁∑𝑖=1

𝑈𝑖 (r) with 𝑈𝑖 (r) ={0, (r ∉ 𝑉𝑖 )𝑘21 [𝑚

2𝑖(r) − 1] . (r ∈ 𝑉𝑖 )

(7)

By combining Equations (6) and (7), we can express the field at anyposition r ∈ R3 following the so-called Foldy-Lax equation [Foldy1945; Lax 1951] as

E(r) = Einc (r) +𝑁∑𝑖=1

=:Esca𝑖

(r)︷︸︸︷∫𝑉𝑖

⇐⇒𝐺 (r, r′) ·

∫𝑉𝑖

⇐⇒𝑇𝑖 (r′, r′′) · E𝑖 (r′′) dr′′ dr′,

(8)with Esca

𝑖(r) and E𝑖 (r) the scattered and partial field of particle 𝑖 , and

⇐⇒𝑇𝑖 (r, r′) the dyad transition operator for particle 𝑖 defined as [Tsanget al. 1985]

⇐⇒𝑇𝑖 (r, r′) = 𝑈𝑖 (r) 𝛿 (r − r′)

⇐⇒𝐼

+𝑈𝑖 (r)∫𝑉𝑖

⇐⇒𝐺 (r, r′′) ·

⇐⇒𝑇𝑖 (r′′, r′) dr′′,

(9)

with 𝛿 (𝑥) the Dirac delta. The partial field at particle 𝑖 is defined asE𝑖 (r) = Einc (r) + ∑𝑁

𝑗 (≠𝑖)=1 Eexc𝑖 𝑗

(r), where the partial exciting fieldEexc𝑖 𝑗

(r) from particles 𝑗 to 𝑖 is

Eexc𝑖 𝑗 (r) =∫𝑉𝑗

⇐⇒𝐺 (r, r′) ·

∫𝑉𝑗

⇐⇒𝑇𝑗 (r′, r′′) · E𝑗 (r′′) dr′′ dr′, (10)

with r ∈ 𝑉𝑖 . Note that the scattered and exciting fields for par-ticle 𝑗 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 𝑗 on particle 𝑖 .However, if the distance 𝑅𝑖 𝑗 := ∥R𝑖 − R𝑗 ∥ between particles (withR𝑖 denoting the center of particle 𝑖) is large, we can approximatethe propagation distance between any point r ∈ 𝑉𝑖 and r′ ∈ 𝑉𝑗 as

∥r − r′∥ ≈ 𝑅𝑖 𝑗 + (R̂𝑖 𝑗 · Δr) − (R̂𝑖 𝑗 · Δr′), (11)

with R̂𝑖 𝑗 := (R𝑖−R𝑗 )/𝑅𝑖 𝑗 , Δr := r−R𝑖 and Δr′ := r′ −R𝑗 (see Figure 2,left). With this approximation, we can now express Eexc

𝑖 𝑗(r) for

a point r ∈ 𝑉𝑖 using its far-field approximation (see §S3 in thesupplemental for the derivation), as

Eexc𝑖 𝑗 (r) =exp(i𝑘1 𝑅𝑖 𝑗 )

𝑅𝑖 𝑗𝑔(R̂𝑖 𝑗 ,Δr) Eexc1𝑖 𝑗 (R̂𝑖 𝑗 ), (12)

with r ∈ 𝑉𝑖 a point in particle 𝑖 , 𝑔(n̂, r) = exp(i𝑘1n̂ · r), and Eexc1𝑖 𝑗 thefar-field exciting field from particle 𝑗 to particle 𝑖 defined as

Eexc1𝑖 𝑗 (R̂𝑖 𝑗 ) = (4𝜋)−1 (⇐⇒𝐼 − R̂𝑖 𝑗 ⊗ R̂𝑖 𝑗 ) (13)

·∫𝑉𝑗

𝑔(−R̂𝑖 𝑗 ,Δr′)∫𝑉𝑗

⇐⇒𝑇𝑗 (r′, r′′) · E𝑗 (r′′) dr′′ dr′.

The dyad (⇐⇒𝐼 − R̂𝑖 𝑗 ⊗ R̂𝑖 𝑗 ) ensures a transverse planar field, which

allows to solely characterize Eexc1𝑖 𝑗 (R̂𝑖 𝑗 ) by the propagation directionR̂𝑖 𝑗 . In order for Equation (12) to be valid, the distance 𝑅𝑖 𝑗 needs tohold the far-field criteria, which relates the 𝑅𝑖 𝑗 with the radius ofthe particle 𝑎 𝑗 following the inequality [Mishchenko et al. 2006]:

𝑘1𝑅𝑖 𝑗 ≫ max(1,𝑘21𝑎

2𝑗

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



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 medium’s bulk scattering parameters, which we willdiscuss in §4.1.

The two forms of computing the exciting field from particle 𝑗 to 𝑖[Equations (10) and (12)] suggest that we can consider two subsetsof particles 𝑗 depending on their distance with respect to the pointof interest r: One set of 𝑁near particles in the near field and anotherset of 𝑁far particles in the far field. With that, we can now calculatethe exciting field in particle 𝑖 as

E𝑖 (r) = Einc (r) +𝑁near∑

𝑗 (≠𝑖)=1Eexc𝑖 𝑗 (r) +

𝑁far∑𝑘=1

Eexc𝑖𝑘

(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 𝑘 do not have neighbor particles in their respectivenear field region. More formally, we now consider a cluster𝐶 of 𝑁𝐶

particles, where all particles 𝑖 ∈ 𝐶 are in their respective near-fieldregion, and that the particles of the cluster have a bounding spherecentered at R𝐶 with radius 𝑎𝐶 (see Figure 2, middle).

Since both the incident field Einc (r) and the exciting field Eexc𝐶𝑘

(r)from particle 𝑘 are in the far-field region, we can assume both fieldsto be planar waves defined as

Einc (r) = Einc0 exp(i𝑘1n̂ · Δr) = Einc0 𝑔(n̂,Δr), (16)

Eexc𝐶𝑘

(r) = Eexc0𝐶𝑘 exp(i𝑘1R̂𝐶𝑘 · Δr) = Eexc0𝐶𝑘 𝑔(R̂𝐶𝑘 ,Δr), (17)

with Einc0 the amplitude of the planar incident field, n̂ its direction,and Δr = r − R𝐶 . Equivalently, Eexc0𝐶𝑘 =

exp(i𝑘1 𝑅𝐶𝑘 )𝑅𝐶𝑘

Eexc1𝐶𝑘 (R̂𝐶𝑘 ) isthe amplitude of the exciting field at 𝐶 from particle 𝑘 , and R̂𝐶𝑘 itsdirection.Now, let us slightly abuse the dot product notation, remove the

dependency on the spatial dependency on each term, and use (𝜑1 •𝜑2) =

∫𝜑1 (𝑥) 𝜑2 (𝑥) d𝑥 for scalar-valued functions 𝜑1 and 𝜑2. From

the far-field assumptions, plugging Equation (15) into the definitionof the scattered field from particle 𝑖 ∈ 𝐶 in Equation (8) (with𝑁near = 𝑁 cls) yields

Esca𝑖 (r) =⇐⇒𝐺 •

⇐⇒𝑇𝑖 • E𝑖

=⇐⇒𝐺 •

⇐⇒𝑇𝑖 •

Einc +𝑁far∑𝑘=1

Eexc𝐶𝑘

+𝑁 cls∑

𝑗 (≠𝑖)=1Eexc𝑖 𝑗

.(18)

By recursively expanding Eexc𝑖 𝑗

and some algebraic operations (seethe supplemental for the full derivation), this results into

Esca𝑖 (r) = E0 ·⇐⇒𝐺 •

⇐⇒𝑇𝑖 •

[𝑔(n̂) +

𝑁 cls∑𝑗 (≠𝑖)=1

[...]𝑔 (n̂)𝑗

](19)

+𝑁far∑𝑘=1

Eexc0𝐶 𝑗

⇐⇒𝐺 •

⇐⇒𝑇𝑖 •

[𝑔(R̂𝐶𝑘 ) +


[...]𝑔 (R̂𝐶𝑘 )𝑗

] ,where the domain of integration in the spatial domain of𝑔(n̂inc,Δr′)is Δr′ = r′ − R𝐶 , and "[...]𝜑

𝑙" term represents the recursivity as

[...]𝜑𝑗=

⇐⇒𝐺 •

⇐⇒𝑇𝑗 •

𝜑 +𝑁 cls∑

𝑙 (≠𝑗)=1[...]𝜑

𝑙

. (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 𝐶 . We can thus define the scattering dyad

⇐⇒𝐴near𝑖

(n̂inc, r)relating a unit-amplitude planar incident field at particle 𝑖 fromdirection n̂inc with the scattered field at point r as

⇐⇒𝐴near𝑖 (n̂inc, r) =

⇐⇒𝐺 •

⇐⇒𝑇𝑖 •

[𝑔(n̂inc) +


[...]𝑔 (n̂inc)

𝑗

]. (21)

By considering constant Einc0 and Eexc0𝐶𝑘 for the whole cluster 𝐶 , wecan compute the cluster’s scattering dyad as

⇐⇒𝐴near𝐶 (n̂inc, r) =

𝑁𝐶∑𝑖=1

⇐⇒𝐴near𝑖 (n̂inc, r), (22)

which defines the scattered field for a unit-amplitude incoming pla-nar field in a scene consisting of the particles forming cluster 𝐶 . Inpractice, the scattering dyad

⇐⇒𝐴near𝐶

(n̂inc, r) can be computed numeri-cally using standard methods from computational electromagnetics(see §5 for more details).

Far-field approximation. Equation (21) represents the generalform of the scattering dyad for particle 𝑖 , which results into a five-dimensional function. Assuming that r is in the far-field region of aparticle 𝑖 ∈ 𝐶 , 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 𝑖 as

Esca𝑖 (r) ≈ 𝑒 i𝑘1𝑅𝑖

𝑅𝑖

(⇐⇒𝐴𝑖 (n̂, R̂𝑖 ) · Einc0 +

𝑁far∑𝑘=1

⇐⇒𝐴𝑖 (R̂𝐶𝑘 , R̂𝑖 ) · Eexc0𝐶𝑘

), (23)

with 𝑅𝑖 := |r − R𝑖 | and R̂𝑖 := r−R𝑖/𝑅𝑖 , and

⇐⇒𝐴𝑖 (n̂inc, n̂sca) = (

⇐⇒𝐼 − R̂𝑖 ⊗ R̂𝑖 ) ·

𝑔(−n̂sca)4𝜋 •

⇐⇒𝑇𝑖

•[𝑔(n̂inc) +

𝑁near∑𝑗 (≠𝑖)=1

[...]𝑔 (n̂inc)

𝑗

].

(24)



300nm 600nm 900nm

Fig. 3. Comparison against Lorenz-Mie theory: We compare our methodwith clusters containing a single particle (i.e., 𝑁 cls = 1) against a ref-erence solution based on Lorenz-Mie theory for three different particleradii 𝑎𝑖 ∈ {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 is𝑚 = 1.5 + 0.1i.

Finally, since R̂𝑖 ≈ R̂𝐶 for all particles 𝑖 ∈ 𝐶 , we can approximatethe far-field scattered field of cluster 𝐶 as

Esca𝐶 (r) = 𝑒 i𝑘1𝑅𝐶

𝑅𝐶

(⇐⇒𝐴𝐶 (n̂, R̂𝐶 ) · E0 +

𝑁far∑𝑘=1

⇐⇒𝐴𝐶 (R̂𝐶𝑘 , R̂𝐶 ) · Eexc0𝐶𝑘

), (25)

where⇐⇒𝐴𝐶 (n̂inc, n̂sca) =

𝑁𝐶∑𝑖=1

⇐⇒𝐴𝑖 (n̂inc, n̂sca), (26)

is the far-field scattering dyad of cluster 𝐶 .Thus, by grouping the individual particles into 𝑁 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) +𝑁 cls∑𝐶 𝑗=1

Esca𝐶 𝑗(r), (27)

with Esca𝐶 𝑗

(r) the scattered field at cluster 𝐶 𝑗 .

4.1 Relationship with the Radiative Transfer Theory

The scattering dyad⇐⇒𝐴𝐶 (n̂inc, n̂sca) given by Equation (26) models

how a particles cluster 𝐶 scatters a planar unit-amplitude incidentfield from direction n̂inc towards direction n̂sca 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 𝑝 (𝑅𝑖 , b𝑖 ), with 𝑅𝑖 and b𝑖denoting, respectively, the position and properties (e.g., shape, size,index of refraction...) of a particle 𝑖; and (iii) We are interested onthe average field ⟨E(r)⟩.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

⇐⇒𝐴(n̂inc, n̂sca). Let us first assume that the

distribution of particle properties b𝑖 are independent of the particles

position, and compute the average scattering dyad ⟨⇐⇒𝐴(n̂inc, n̂sca)⟩ =∫

Ω

⇐⇒𝐴𝑖 (n̂inc, n̂sca)𝑝 (b𝑖 ) db𝑖 . 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 �̂� inc and �̂�inc of the incidentfield as shown in Figure 2 (right), and equivalently for the scat-tered field �̂� sca and �̂�

sca, we can compute the polarized scatteringcomponents 𝑺\ and 𝑺𝜙 from the average cluster’s scattering dyad⟨⇐⇒𝐴𝐶 (n̂inc, n̂sca)⟩ as

𝑺\ (n̂inc, n̂sca) = �̂�sca · ⟨

⇐⇒𝐴𝐶 (n̂inc, n̂sca)⟩ · �̂�

inc,

𝑺𝜙 (n̂inc, n̂sca) = �̂�sca · ⟨

⇐⇒𝐴𝐶 (n̂inc, n̂sca)⟩ · �̂�

inc. (28)

Then, based on the two scattering components 𝑺\ and 𝑺𝜙 , we canobtain the optical parameters of the medium as

^t (n̂inc) = 4𝜋ℜ[𝑺 (n̂inc, n̂inc)

𝑘2𝑖

], (29)

^s (n̂inc) =∫S2

|𝑺\ (n̂inc, n̂sca) |2 + |𝑺𝜙 (n̂inc, n̂sca) |2

2𝑘21dn̂sca,

(30)

𝑓p (n̂inc, n̂sca) =|𝑺\ (n̂inc, n̂sca) |2 + |𝑺𝜙 (n̂inc, n̂sca) |2

2𝑘21^s, (31)

with 𝑺 (n̂inc, n̂inc) = 𝑺𝜙 (n̂inc, n̂inc) = 𝑺\ (n̂inc, n̂inc), ℜ[𝑥] returningthe real part of a complex number 𝑥 , and S2 the unit sphere ofdirections. Lastly, assuming a uniform distribution of clusters, wecan compute the extinction and scattering coefficients as

𝜎t (n̂inc) = ^t (n̂inc)𝜌

⟨𝑁 cls⟩, (32)

𝜎s (n̂inc) = ^s (n̂inc)𝜌

⟨𝑁 cls⟩, (33)

with 𝜌 the number of particles per differential volume, and ⟨𝑁 cls⟩ theaverage number of particles per cluster. Note that the optical prop-erties defined in Equations (29)–(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 [Nováket al. 2018] build on the assumption of independent scattering—thatis, particles are in their respective far-field region. It is easy to verifythat this is a special case of Equation (15) with 𝑁 cls = 1, causing thescattering dyad

⇐⇒𝐴𝐶 of Equation (26) to reduce to

⇐⇒𝐴𝐶 (n̂inc, n̂sca) =

⇐⇒𝐴𝑖 (n̂inc, n̂sca) =

𝑔(n̂sca) ·⇐⇒𝑇𝑖 · 𝑔(n̂inc)4𝜋 , (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



Fig. 4. Comparison against Lorenz-Mie theory: We compare the extinctionand scattering cross sections computed with our method for𝑁 cls = 1 againstthe results obtained using Lorenz-Mie theory. As in Figure 3, our resultsshow perfect agreement.

for⇐⇒𝐴𝑖 (n̂inc, n̂sca) for spheres and cylinders, while numerical solu-

tions of⇐⇒𝐴𝑖 (n̂inc, n̂sca) 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 dyad⇐⇒𝐴𝐶 (n̂inc, n̂sca) of Equation (26), which in turn determines the bulkscattering parameters following Equations (29)–(33). These bulkscattering parameters can be directly used in any renderer support-ing participating media [Novák et al. 2018] using tabulated phasefunction and cross sections.Computing

⇐⇒𝐴𝐶 (n̂inc, n̂sca) essentially boils down to solving the

time-harmonic Maxwell equations for an incident unit-amplitudeplanar field with direction n̂inc. While several different methods ex-ist for that purpose (see §16 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 𝑁 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 ⟨

⇐⇒𝐴𝐶 (n̂inc, n̂sca)⟩, 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 cluster’s bounding sphere). As

we will demonstrate in §6, 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 (§6.1). Then, we apply our technique described in §4and §5 to compute bulk scattering parameters for a wide range ofparticipating media (§6.2).

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 𝜎t, 𝜎s are directionally independent,and the single-scattering phase function 𝑓p 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



Sparse Intermediate Dense 𝑎𝑖=400nm 𝑎𝑖=500nm 𝑎𝑖=600nm 𝑁 cls = 20 𝑁 cls = 100 𝑁 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 𝑁 cls = 100 particles, and each particle has radius 𝑎𝑖 = 500nm. For each phase function, we vary: (a) The distance between particleswithin the cluster; (b) The particle size 𝑎𝑖 ; and (c) The number of particles 𝑁 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 (𝑎𝑖= 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 𝑎𝑖=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 𝑁 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 𝜎t, 𝜎s to be directionally dependent, and have



𝑁 cls = 1 𝑁 cls = 50 𝑁 cls = 100 𝑁 cls = 500

𝑎𝑖=300n

m𝑎𝑖=400n

m𝑎𝑖=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 𝑎𝑖 and per-cluster particle count 𝑁 cls.

full 4D phase functions 𝑓p. 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 𝑁 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 × 360. Due to the symmetry of the disc, theresulting phase function 𝑓p is three-dimensional, and we tabulatedit with the resolution 90 × 180 × 360.

In Figure 11 (b), we visualize slices of the computed single-scatteringphase function 𝑓p with two incident directions n̂inc. 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. (b–d) 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.



𝑚 = 1.2 + 0i 𝑚 = 1.3 + 0i 𝑚 = 1.4 + 0i 𝑚 = 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 𝑚 = 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.

𝑁 cls 𝑓p res. timeRegular (Fig. 6) 1–500 180 × 360 3–16sMulti-spectral (Fig. 8) 100 180 × 360 × 50 35mVarying particle sizes (Fig. 10) 100 180 × 360 7–108sAnisotropic (Fig. 12) 100 180 × 360 × 90 13mCorrelated (Fig. 13) 100 180 × 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 with𝑁 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



Forward Backward

(a) Incident direction (b) Phase function slice

Fig. 11. Visualizations of slices 𝑓p (n̂inc, ·) of a phase function for two inci-dent directions n̂inc 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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x, 700nm y, 700nm z, 700nm

x, multi. y, multi. z, multi.

Fig. 12. Renderings of homogeneous Lucymodels with the same anisotropicmedium as in Figure 11. The medium’s orientation—which determinesthe axis of the disk—is aligned respectively with the 𝑥-, 𝑦-, and 𝑧-axisin the three columns , leading to distinctive appearances . We show single-wavelength (_ = 700nm) renderings on the top and multi-spectral ones onthe bottom.

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(a) Negatively correlated particles (b) Positively correlated particles

(a1) unc. clusters (a2) neg. clusters (b1) unc. clusters (b2) pos. clusters

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http://www.philiplaven.com/mieplot.htm

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] (§S1), and its formulation for a medium with multipleparticles embedded using the Foldy-Lax equations [Foldy 1945; Lax1951](§S2) and their far-field approximations (§S3.

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 frequency𝜔 is defined by theMaxwell curl equationsas

∇ × E(r) = i𝜔` (r)H(r),∇ × H(r) = −i𝜔Y (r)E(r), (S.1)

with ∇× . 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 =

√−1.

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

∇2E(r) = i𝜔` (r)∇ × H(r)= −i2𝜔2`0Y (r)E(r), (S.2)

with ∇2 = ∇ × ∇, which by arithmetic reordering reduces to theelectric field wave equation

∇2 × E(r) − 𝑘 (r)2E(r) = 0, (S.3)

where 𝑘 (r) = 𝜔√Y (r)`0 is the medium’s wave number at r. Note

that the wave number 𝑘 has a dependence on the frequency 𝜔 ; 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 𝑉0, and the finite disjointregion occupìed by the scatterers𝑉 , so that𝑉

⋃𝑉0 = R3. Under this

Authors’ 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

∇2 × E(r) − 𝑘21E(r) = 0, r ∈ 𝑉0, (S.4)

∇2 × E(r) − 𝑘2 (r)2E(r) = 0, r ∈ 𝑉 , (S.5)

with 𝑘1 the constant wave number at the hosting medium, and𝑘2 (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

∇2 × E(r) − 𝑘21E(r) = 𝑈 (r) E(r), (S.6)

with𝑈 (r) = 𝑘21 [𝑚2 (r) − 1] the potential function at r, and𝑚(r) =

𝑘2 (r)/𝑘1 the index of refraction at r. It is trivial to verify that forr ∈ 𝑉0 in the hosting medium𝑚(r) = 1, then the potential function𝑈 (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

⇐⇒𝐺 (r, r′) that satisfies the impulse response of the linear system

in Equation (S.3), modeled as

∇2 ×⇐⇒𝐺 (r, r′) − 𝑘21

⇐⇒𝐺 (r, r′) =

⇐⇒𝐼 𝛿 (r − r′), (S.8)

where⇐⇒𝐼 is the identity dyad, and 𝛿 (·) is the Dirac delta function.

Note that the derivatives are with respect to r. Multiplying bothsides of the differential equation by 𝑈 (r) E(r), and integrating bothsides with respect to r′ over the entire space R3, we get

(∇2 ×

⇐⇒𝐼 − 𝑘21

⇐⇒𝐼

) Esca (r)︷︸︸︷∫R3

𝑈 (r′)⇐⇒𝐺 (r, r′) · E(r′) dr′ = 𝑈 (r) Esca (r), (S.9)

with . · . the dyadic-vector dot-product. Since the potential function𝑈 (r) vanishes everywhere outside 𝑉 , we can express the scatteredfield Esca (r) as an integral on the space occupied by scatterers 𝑉



only, as

Esca (r) =∫𝑉

𝑈 (r)⇐⇒𝐺 (r, r′) · E(r′) dr′

= 𝑘21

∫𝑉

[𝑚2 (r′) − 1]⇐⇒𝐺 (r, r′) · E(r′) dr′. (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

⇐⇒𝐺 (r, r′) =

(⇐⇒𝐼 + 𝑘−21 ∇ ⊗ ∇

) exp(i𝑘1 |r − r′ |)4𝜋 |r − r′ | , (S.11)

where . ⊗ . denotes the dyadic product of two vectors, and thederivative operator ∇ 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) +∫𝑉

𝑘21 [𝑚2 (r′) − 1]︸︷︷︸𝑈 (r′)

⇐⇒𝐺 (r, r′) · E(r′) dr′. (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. with𝑚(r′) ≠ 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 𝑁 finite discrete particles withvolume 𝑉𝑖 and index of refraction 𝑚𝑖 (r). We can now define thepotential function𝑈𝑖 (r) for each particle 𝑖 as

𝑈𝑖 (r) ={

0, r ∉ 𝑉𝑖𝑘21 [𝑚

2𝑖(r) − 1] r ∈ 𝑉𝑖 ,

(S.13)

with the total potential function 𝑈 in Equation (S.12) defined as𝑈 (r) = ∑𝑁

𝑖=1𝑈𝑖 (r). By combining Equations (S.12) and (S.13), wecan express the field at any position r ∈ R3 following the so-calledFoldy-Lax equation [Foldy 1945; Lax 1951] as

E(r) = Einc (r) +𝑁∑𝑖=1

∫𝑉𝑖

⇐⇒𝐺 (r, r′) ·

∫𝑉𝑖

⇐⇒𝑇𝑖 (r′, r′′) · E𝑖 (r′′) dr′′ dr′

= Einc (r) +𝑁∑𝑖=1

Esca𝑖 (r), (S.14)

with E𝑖 (r) = Einc (r) + ∑𝑁𝑗 (≠𝑖)=1 E

exc𝑖 𝑗

(r), where the partial excitingfield Eexc

𝑖 𝑗(r) from particles 𝑗 to 𝑖 and Esca

𝑖(r) the scattered field

from particle 𝑖 . Note that we overload the dot-product operatoraccounting for the dyad-dyad case. The dyad transition operator

⇐⇒𝑇𝑖 (r, r′) for particle 𝑖 defined as [Tsang et al. 1985]

⇐⇒𝑇𝑖 (r, r′) = 𝑈𝑖 (r)𝛿 (r − r′)

⇐⇒𝐼

+𝑈𝑖 (r)∫𝑉𝑖

⇐⇒𝐺 (r, r′′) ·

⇐⇒𝑇𝑖 (r′′, r′) dr′′, (S.15)

with 𝛿 (𝑥) the Dirac delta,⇐⇒𝐼 the identity dyad. The partial exciting

field Eexc𝑖 𝑗

(r) is defined as

Eexc𝑖 𝑗 (r) =∫𝑉𝑗

⇐⇒𝐺 (r, r′) ·

∫𝑉𝑗

⇐⇒𝑇𝑗 (r′, r′′) · E𝑗 (r′′) dr′′ dr′, (S.16)

with r ∈ 𝑉𝑖 . Note that the exciting field Eexc𝑖 𝑗

(r) has essentially thesame form as the scattered field Esca

𝑗(r) from particle 𝑗 . 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 𝑗 on particle 𝑖 . However, if the distance between par-ticles 𝑅𝑖 𝑗 = |R𝑖 −R𝑗 |, with R𝑖 the origin of particle 𝑖 , is large, so that𝑘1𝑅𝑖 𝑗 ≫ 1, we can approximate the propagation distance betweenpoints r ∈ 𝑉𝑖 and r′ ∈ 𝑉𝑗 as |r−r′ | ≈ 𝑅𝑖 𝑗 +(R̂𝑖 𝑗 ·Δr)−(R̂𝑖 𝑗 ·Δr′), withR̂𝑖 𝑗 =

R𝑖−R𝑗

𝑅𝑖 𝑗, Δr = r−R𝑖 and Δr′ = r′−R𝑗 . With this approximation,

and after some algebraic operations, we can now approximate thedyadic Green’s function as

⇐⇒𝐺 (r, r′) ≈ (

⇐⇒𝐼−R̂𝑖 𝑗 ⊗ R̂𝑖 𝑗 )

exp(i𝑘1𝑅𝑖 𝑗 )4𝜋𝑅𝑖 𝑗

𝑔(R̂𝑖 𝑗 ,Δr)𝑔(−R̂𝑖 𝑗 ,Δr′), (S.17)

with 𝑔(n̂, r) = exp(i𝑘1n̂ · r). With this approximation, we can nowexpress Eexc

𝑖 𝑗(r) for a point r ∈ 𝑉𝑖 using its far-field approximation,

as

Eexc𝑖 𝑗 (r) =exp(i𝑘1 𝑅𝑖 𝑗 )

𝑅𝑖 𝑗𝑔(R̂𝑖 𝑗 ,Δr) Eexc1𝑖 𝑗 (R̂𝑖 𝑗 ), (S.18)

with r ∈ 𝑉𝑖 a point in particle 𝑖 , and Eexc1𝑖 𝑗 the far-field exciting fieldfrom particle 𝑗 to particle 𝑖 defined as

Eexc1𝑖 𝑗 (R̂𝑖 𝑗 ) =(⇐⇒𝐼 − R̂𝑖 𝑗 ⊗ R̂𝑖 𝑗 )

4𝜋·∫𝑉𝑗

𝑔(−R̂𝑖 𝑗 ,Δr′)∫𝑉𝑗

⇐⇒𝑇𝑗 (r′, r′′)·E𝑗 (r′′) dr′′ dr′.

(S.19)The dyad (

⇐⇒𝐼 − R̂𝑖 𝑗 ⊗ R̂𝑖 𝑗 ) to ensure a transverse planar field, which

allows to solely characterize Eexc1𝑖 𝑗 (R̂𝑖 𝑗 ) by the propagation directionR̂𝑖 𝑗 . In order to Equation (S.19) to be valid, the distance 𝑅𝑖 𝑗 needsto hold the far-field criteria, which relates the 𝑅𝑖 𝑗 with the radius ofthe particle 𝑎 𝑗 following the inequality [Mishchenko et al. 2006]

𝑘1𝑅𝑖 𝑗 ≫ max

(1,𝑘21𝑎

2𝑗

2

). (S.20)

The two forms of computing the exciting field from particle 𝑗

to 𝑖 (Equations (S.16) and (S.19)) suggest that we can consider twosubsets of particles 𝑗 depending on their distance with respect tothe point of interest r: One set of 𝑁near particles in the near field


Supplementary MaterialBeyond Mie Theory: Systematic Computation of Bulk Scattering Parameters based on Microphysical Wave Optics • 3

and another set of 𝑁far particles in the far field. With that, we cannow the exciting field in particle 𝑖 as

E𝑖 (r) = Einc (r) +𝑁near∑

𝑗 (≠𝑖)=1Eexc𝑖 𝑗 (r) +

𝑁far∑𝑘=1

Eexc𝑖𝑘

(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 𝑘 in the far fieldregion of the cluster of particles. More formally, let us now considera cluster 𝐶 of 𝑁𝐶 particles, where all particles 𝑗 ∈ 𝐶 are in theirrespective near-field region, and that the particles of the cluster arebounded on a sphere centered at R𝐶 with radius 𝑎𝐶 .Since both the incident field Einc (r) and the exciting field Eexc

𝐶𝑘from particle 𝑘 are in the far-field region, we can assume that bothfields are planar waves defined as

Einc (r) = Einc0 exp(i𝑘1n̂ · Δr) = Einc0 𝑔(n̂,Δr), (S.22)

Eexc𝐶𝑘

(r) = Eexc0𝐶𝑘 exp(i𝑘1R̂𝐶𝑘 · Δr) = Eexc0𝐶𝑘 𝑔(R̂𝐶𝑘 ,Δr) (S.23)

with Einc0 and Eexc0𝐶𝑘 =exp(i𝑘1 𝑅𝐶𝑘 )

𝑅𝐶𝑘Eexc1𝐶𝑘 (R̂𝐶𝑘 ) (S.19) the amplitude

of the planar incident field and the exciting field from particle 𝑘respectively, n̂ and R̂𝐶𝑘 the propagation direction of the each field,and Δr = r − R𝐶 .

Now, let us slightly abuse the dot product notation defining (𝜑1 •𝜑2) =

∫𝜑1 (𝑥) · 𝜑2 (𝑥) d𝑥 , 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 particle𝑖 ∈ 𝐶 (S.14), we get


⇐⇒𝑇𝑖 • E𝑖 (S.24)

=⇐⇒𝐺 •

⇐⇒𝑇𝑖 •

Einc +𝑁far∑𝑘=1

Eexc𝐶𝑘

+𝑁near∑

𝑗 (≠𝑖)=1Eexc𝑖 𝑗

.By recursively expanding Eexc

𝑖 𝑗, Equation (S.24) becomes


⇐⇒𝑇𝑖 •

[Einc +

𝑁far∑𝑘=1

Eexc𝐶𝑘

(S.25)

+𝑁near∑

𝑗 (≠𝑖)=1

⇐⇒𝐺 •

⇐⇒𝑇𝑗 •

[Einc +

𝑁far∑𝑘=1

Eexc𝐶𝑘

+𝑁near∑

𝑙 (≠𝑗)=1[...]𝑙

] ],

where the "[...]𝑙 " term represents the recursivity as

[...]𝑙 =⇐⇒𝐺 •

⇐⇒𝑇𝑙 •

[Einc +

𝑁far∑𝑘=1

Eexc𝐶𝑘

+𝑁near∑

𝑚 (≠𝑙)=1[...]𝑚

]. (S.26)

By reordering Equation (S.25) we get


⇐⇒𝑇𝑖 •

[Einc +


⇐⇒𝐺 •

⇐⇒𝑇𝑗 •

[Einc +

𝑁near∑𝑙 (≠𝑗)=1

[...]Einc

𝑙

] ](S.27)

+𝑁far∑𝑘=1

⇐⇒𝐺 •

⇐⇒𝑇𝑖 •

[Eexc𝐶𝑘

+𝑁near∑

𝑗 (≠𝑖)=1

⇐⇒𝐺 •

⇐⇒𝑇𝑗 •

[Eexc𝐶𝑘

+𝑁near∑

𝑙 (≠𝑗)=1[...]E

exc𝐶𝑘

𝑙

] ] ,where "[...]𝜑

𝑙" is similar to Equation (S.26), with form

[...]𝜑𝑙=

⇐⇒𝑇𝑙 •

⇐⇒𝐺 •

[𝜑 +

𝑁near∑𝑚 (≠𝑙)=1

[...]𝜑𝑚]. (S.28)

Finally, by exploiting Equations (S.22) and (S.23), and contractingthe recursion, we transform Equation (S.27) into


⇐⇒𝑇𝑖 •

[𝑔(n̂) +


[...]𝑔 (n̂)𝑗

]· Einc0 (S.29)

+𝑁far∑𝑘=1

⇐⇒𝐺 •

⇐⇒𝑇𝑖 •

[𝑔(R̂𝐶𝑘 ) +


[...]𝑔 (R̂𝐶𝑘 )𝑗

]· Eexc0𝐶𝑘

.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 𝐶 . Wecan thus define the scattering dyad

⇐⇒𝐴near𝑖

(n̂inc, r) relating a fieldincoming at particle 𝑖 from direction n̂inc with the field at point r as

⇐⇒𝐴near𝑖 (n̂inc, r) =

⇐⇒𝐺 •

⇐⇒𝑇𝑖 •

[𝑔(n̂inc) +


[...]𝑔 (n̂inc)

𝑗

]. (S.30)

Trivially, following our assumption of constant Einc0 and Eexc0𝐶𝑘 forthe whole cluster 𝐶 , we can compute the cluster’s scattering dyadas:

⇐⇒𝐴near𝐶 (n̂inc, r) =

𝑁𝐶∑𝑖=1

⇐⇒𝐴near𝑖 (n̂inc, r) . (S.31)

The scattering dyad⇐⇒𝐴near𝐶

(n̂inc, r) solves the scattering field for aunit-amplitude incoming planar field in a scene consisting of theparticles forming cluster𝐶 , and can be computed using any methodfrom computational electromagnetics.

Far-field approximation. Equation (S.30) represents the generalform of the scattering dyad for particle 𝑖 , which results into a five-dimensional function. Assuming that r is in the far-field region ofa particle 𝑖 ∈ 𝐶 , and by using the far-field approximation of theGreen’s function (S.18), Equation (S.24) becomes

Esca𝑖 (r) ≈ (⇐⇒𝐼 − R̂𝑖 ⊗ R̂𝑖 )

exp(i𝑘1𝑅𝑖 )4𝜋𝑅𝑖

· 𝑔(−R̂𝑖 ) •⇐⇒𝑇𝑖 · E𝑖 , (S.32)

with 𝑅𝑖 = |r − R𝑖 | and R̂𝑖 =r−R𝑖𝑅𝑖

. Note that the term 𝑔(R̂𝑖 ,Δr)inEquation (S.18) vanishes for a single particle, since |Δr| = 0 andtherefore 𝑔(R̂𝑖 ,Δr) = 1.



Now, using the definition of the scattered fieldE𝑖 in Equation (S.21),and expanding Eexc following Equation (S.24), and expanding Eexc

𝑖 𝑗

following Equation (S.25) we get

Esca𝑖 (r) = (⇐⇒𝐼 − R̂𝑖 ⊗ R̂𝑖 )

exp(i𝑘1𝑅𝑖 )4𝜋𝑅𝑖

· 𝑔(−R̂𝑖 ) •⇐⇒𝑇𝑖 •

[Einc +

𝑁far∑𝑘=1

Eexc𝐶𝑘

+𝑁near∑

𝑗 (≠𝑖)=1

⇐⇒𝐺 •

⇐⇒𝑇𝑗 •

[Einc +

𝑁far∑𝑘=1

Eexc𝐶𝑘

+𝑁near∑

𝑙 (≠𝑗)=1[...]𝑙

] ], (S.33)

with "[...]𝑙 " 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

𝐶𝑘respectively,

and exploit the far field assumption to put Einc and Eexc𝐶𝑘

in theirplanar field form [Equations (S.22) and (S.23)], as

Esca𝑖 (r) ≈ 𝑒 i𝑘1𝑅𝑖

𝑅𝑖

(⇐⇒𝐴𝑖 (n̂, R̂𝑖 ) · Einc0 +

𝑁far∑𝑘=1


), (S.34)

with⇐⇒𝐴𝑖 (n̂inc, n̂sca) the far-field scattering dyad relating incident and

scattered directions n̂inc and n̂sca as

⇐⇒𝐴𝑖 (n̂inc, n̂sca) = (

⇐⇒𝐼 − R̂𝑖 ⊗ R̂𝑖 ) ·

𝑔(−n̂sca)4𝜋

•⇐⇒𝑇𝑖

•[𝑔(n̂inc) +


[...]𝑔 (n̂inc)

𝑗

].

(S.35)

Finally, since R̂𝑖 ≈ R̂𝐶 for all particles 𝑖 ∈ 𝐶 we can approximatethe far-field scattered field of cluster 𝐶 as

Esca𝐶

(r) =𝑁𝐶∑𝑖=1

Esca𝑖 (r)

=

𝑁𝐶∑𝑖=1

𝑒 i𝑘1𝑅𝑖

𝑅𝑖

(⇐⇒𝐴𝑖 (n̂, R̂𝑖 ) · E0 +

𝑁far∑𝑘=1


),

≈ 𝑒 i𝑘1𝑅𝐶

𝑅𝐶

( 𝑁𝐶∑𝑖=1

⇐⇒𝐴𝑖 (n̂, R̂𝐶 ) · E0 +

𝑁far∑𝑘=1

𝑁𝐶∑𝑖=1

⇐⇒𝐴𝑖 (R̂𝐶𝑘 , R̂𝐶 ) · Eexc0𝐶𝑘

)=𝑒 i𝑘1𝑅𝐶

𝑅𝐶

(⇐⇒𝐴𝐶 (n̂, R̂𝐶 ) · E0 +

𝑁far∑𝑘=1

⇐⇒𝐴𝐶 (R̂𝐶𝑘 , R̂𝐶 ) · Eexc0𝐶𝑘

),

(S.36)

with⇐⇒𝐴𝐶 (n̂inc, n̂sca) =

∑𝑁𝐶

𝑖=1⇐⇒𝐴𝑖 (n̂inc, n̂sca) the far-field scattering

dyad of cluster 𝐶 .

Computing the far-field exciting field. Let us know compute thefar-field exciting field Eexc

𝑘𝐶from a cluster 𝐶 to a particle 𝑘 placed

in the far-field region of 𝐶 . 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

𝑖 ∈ 𝐶 over particle 𝑗 as:

Eexc𝑘𝑖

(r) ≈ (⇐⇒𝐼 − R̂𝑘𝑖 ⊗ R̂𝑘𝑖 ) ·

𝑒 i𝑘1 (𝑅𝑘𝑖+R̂𝑘𝑖 ·Δr)

4𝜋𝑅𝑘𝑖𝑔(R̂𝑘𝑖 ) •

⇐⇒𝑇𝑗

•[Einc +

𝑁far∑𝑘′=1

Eexc𝑖𝑘′ +


Eexc𝑖 𝑗

]. (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

Eexc𝑘𝑖

(r) = 𝑒 i𝑘1 (𝑅𝑘𝑖+R̂𝑘𝑖 ·Δr)

𝑅𝑘𝑖

(⇐⇒𝐴𝑖 (n̂, R̂𝑖 ) · Einc0 +

𝑁far∑𝑘′=1

⇐⇒𝐴𝑖 (R̂𝐶𝑘′, R̂𝑖 ) · Eexc0𝐶𝑘′

),

(S.38)

which by summing the exciting field of all particles 𝑖 ∈ 𝐶 andfollowing the far-field approximation (R̂𝑘𝑖 ≈ R̂𝑘𝐶 ,∀𝑖 ∈ 𝐶 we get

Eexc𝑘𝐶

(r) ≈ 𝑒 i𝑘1 (𝑅𝑘𝐶+R̂𝑘𝐶 ·Δr)

𝑅𝑘𝐶

(⇐⇒𝐴𝐶 (n̂, R̂𝐶 )·Einc0 +

𝑁far∑𝑘′=1

⇐⇒𝐴𝐶 (R̂𝐶𝑘′, R̂𝐶 )·Eexc0𝐶𝑘′

).

(S.39)Finally, if particle 𝑘 is itself contained in the near-field of a clusterof particles 𝐶1, then it is trivial to compute the exciting field fromcluster 𝐶 to 𝐶1 as

Eexc𝐶1𝐶

(r) =𝑁𝐶1∑𝑘=1

Eexc𝑘𝐶

(r). (S.40)

Thus, by grouping the individual particles into 𝑁 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) +𝑁 cls∑𝐶 𝑗=1

Esca𝐶 𝑗

(r), (S.41)

with Esca𝐶 𝑗

(r) defined by pluging Equation (S.40) into Equation (S.36)as

Esca𝐶 𝑗

(r) = 𝑒i𝑘1𝑅𝐶𝑗

𝑅𝐶 𝑗

(⇐⇒𝐴𝐶 𝑗

(n̂, R̂𝐶 𝑗) · Einc0 (S.42)

+𝑁 cls∑

𝐶𝑘 (≠𝐶 𝑗 )=1

⇐⇒𝐴𝐶 𝑗

(R̂𝐶 𝑗𝐶𝑘, R̂𝐶 𝑗

) · Eexc0𝐶 𝑗𝐶𝑘

), (S.43)

with Eexc0𝐶 𝑗𝐶𝑘the amplitude of the far-field exciting field from cluster

𝐶𝑘 to cluster 𝐶 𝑗 .

REFERENCESLeslie L Foldy. 1945. The multiple scattering of waves. I. General theory of isotropic

scattering by randomly distributed scatterers. Physical review 67, 3-4 (1945), 107.Melvin Lax. 1951. Multiple scattering of waves. Reviews of Modern Physics 23, 4 (1951),

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), 7114–7134.

Michael I Mishchenko, Larry D Travis, and Andrew A Lacis. 2006. Multiple scattering oflight by particles: radiative transfer and coherent backscattering. Cambridge UniversityPress.

Leung Tsang, Jin Au Kong, and Robert T Shin. 1985. Theory of microwave remote sensing.John Wiley & Sons.


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