Scattering from spheres made of time-varying and dispersive materials

G. Ptitcyn1,2,∗ A. G. Lamprianidis2,∗ T. Karamanos2,∗ V. S. Asadchy3, R. Alaee2,

M. Muller2, M. Albooyeh4, M. S. Mirmoosa1, S. Fan3, S. A. Tretyakov1, and C. Rockstuhl2,51Department of Electronics and Nanoengineering,

Aalto University, P.O. Box 15500, FI-00076 Aalto, Finland2 Institute of Theoretical Solid State Physics, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany

3Ginzton Laboratory and Department of Electrical Engineering,Stanford University, Stanford, California 94305, USA

4Mobix Labs Inc., 15420 Laguna Canyon, Irvine, California 92618, USA5 Institute of Nanotechnology, Karlsruhe Institute of Technology, 76131 Karlsruhe, Germany

Exploring the interaction of light with time-varying media is an intellectual challenge that, inaddition to fundamental aspects, provides a pathway to multiple promising applications. Timemodulation constitutes here a fundamental handle to control light on entirely different grounds.That holds particularly for complex systems simultaneously structured in space and time. However,a realistic description of time-varying materials requires considering their material dispersion. Thecombination thereof has barely been considered but is crucial since dispersion accompanies materialssuitable for dynamic modulation. As a canonical scattering problem from which many generalinsights can be obtained, we develop and apply a self-consistent analytical theory of light scatteringby a sphere made from a time-varying material exemplarily assumed to have a Lorentzian dispersion.We discuss the eigensolutions of Maxwell’s equations in the bulk and present a dedicated Mie theory.The proposed theory is verified with full-wave simulations. We disclose effects such as energy transferfrom the time-modulation subsystem to the electromagnetic field, amplifying carefully structuredincident fields. Since many phenomena can be studied on analytical grounds with our formalism,it will be indispensable when exploring electromagnetic phenomena in time-varying and spatiallystructured finite objects of other geometries.

I. INTRODUCTION

One of the most recent extensions of electromagnet-ics and optics is the concept of materials with time-varying properties. A time variation unlocks an addi-tional degree of freedom in electromagnetic systems thattremendously increases the possibilities for controllinglight-matter interactions [1–3]. Temporal material mod-ulations enable novel approaches to exceed conventionallimitations [4, 5] and design efficient systems that re-alize unconventional functionalities [6–10]. Historically,time-modulated structures were first studied when engi-neering radio-frequency antennas [11–13] to manipulatetheir bandwidth. In electronics, temporal modulations attwice the carrier frequency have been exploited in para-metric amplifiers since the 19th century. Time-varyingsystems found applications from microwaves to optics.They led to discoveries of many intriguing phenomenasuch as magnetless nonreciprocity [5, 6, 14–18], fre-quency conversion [19, 20], amplification [21–23], Dopplershift [24, 25], Fresnel drag [26], camouflage [27, 28],breaking antenna performance limits [29], temporal bire-fringence [30, 31], temporal photonic crystals [10, 32, 33],temporal discontinuities [34], power combiners [35], lightstopping and time reversal [36, 37], control of scatteringand radiation [38, 39], enhanced wireless power trans-fer [40], control of absorption [41], and more. However,the vast majority of prior contributions considered dis-

∗ These authors contributed equally.

persionless materials. The absence of dispersion is syn-onymous with the assumption of an instantaneous (in-ertialess) response. This assumption is, generally speak-ing, non-physical, approximately holds only for systemswith very small variations over the time and/or negligiblefrequency dispersion. Examples of dispersionless mate-rials whose properties can be modulated in time includelithium niobate (LiNBO3) [42] and silicon [43] (in theirtransparency frequency regions). But the modulationdepth of such dispersionless materials – a parameter thatdetermines the strength of the effects caused by temporalmodulations – is typically very low, being of the order of10−4 − 10−3 [44]. In contrast, material candidates thatallow large modulation depths of the permittivity, includ-ing electron plasmas [45] and aluminum-doped zinc [46]and indium tin oxides [34], are usually strongly dispersiveat the frequencies of interest (specifically, in the epsilon-near-zero region, where the modulation depth is large).To our knowledge, only a few recent papers tackled thisproblem and studied the influence of frequency dispersionin time-modulated materials [47–49].

One of the development routes considers time mod-ulation as an additional degree of freedom in spatiallymodulated structures such as metamaterials or meta-surfaces. Constituents of such devices are meta-atomswith finite sizes in all three spatial dimensions. To thebest of our knowledge, almost all previous contributionsconsidered time-varying structures that are infinite in atleast one spatial dimension, for instance, bulk media [50],slabs [48], and coatings of cylinders [20]. A couple of re-cent studies considered light scattering from finite-sized

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particles, a sphere and a conductive spherical shell, withtime-varying properties [51, 52]. However, accommodat-ing dispersion in such models remains a challenge, and,therefore, it was set aside.

The problem of light scattering by a time-invariantand dispersive sphere was solved a century ago by Gus-tav Mie [53]. An extension of this theory towards time-varying particles represents a solid initial step towardsthe design of time-varying metamaterials and metasur-faces. The study of light scattering from a sphere isalso very instructive since many basic phenomena can beexplored with analytical or semi-analytical calculations.The gained insights can be applied to understand andexplain the behavior of scatterers with a more compli-cated shape that require a full-wave numerical approachfor their full exploration. Therefore, exploring the caseof canonical objects, especially spheres, can be consid-ered to be at the heart of scattering theory. An insightthat one can obtain, for example, concerns the abilityof spheres to support scattering resonances where eitherelectric or magnetic multipole moments are driven intoresonance. During the last decade, passive metasurfacesmade from scatterers supporting such Mie resonanceshave demonstrated a variety of novel optical phenom-ena [54–56], and we envisage a substantial broadening ofpossible applications when time variations are consideredas an additional degree of freedom in these systems.

This paper extends the Mie theory to spheres madefrom a dispersive material with a periodically time-varying permittivity. The findings of this paper are,nevertheless, applicable to an arbitrary aperiodic mod-ulation in the limit of a very large period. First, weshow how the dispersion relation of the eigenmodes of ahomogeneous unbound dispersive medium transforms inthe presence of temporal material modulations and an-alyze band structures of frequency-dispersive time crys-tals. Next, we introduce a field Ansatz in spherical coor-dinates for solving the scattering problem. The expres-sions for the T-matrix elements of dynamic spheres arederived, and its power balance is analyzed. The analysisindicates a possibility to observe a negative absorptionin the system, i.e., transfer of energy from time-varyingmatter to photons. This effect happens for an incidentfield carefully chosen in both its spatial and spectral dis-tributions. Finally, we perform full-wave simulations oflight scattering by a dispersive sphere via a finite-elementtime-domain method and find excellent agreement be-tween the simulated and theoretical results. Based onthe developed theory, one can further extend and gen-eralize the analytical study of this paper towards struc-tures with lower spatial dimensions such as infinite slabsand cylinders. Alternatively, the insights that are gener-ated from these results can be useful to study arbitrarilyshaped objects using solely full-wave simulations.

ത𝐄inc(𝜔0)

ത𝑅e(𝑛𝜔m, 𝜔0)

ത𝐄sca(𝜔0 + 𝜔m)

ത𝐄sca(𝜔0)

ത𝐄sca(𝜔0 − 𝜔m)ത𝐄sca(𝜔0 + 𝑛𝜔m)…

𝜔m

FIG. 1. Illustration of the scattering of light by a spherecomposed of a time-varying and dispersive medium.

II. THEORETICAL ANALYSIS

In this section, we perform the electromagnetic anal-ysis of the canonical problem of scattering by a spherecomposed of a time-varying and dispersive medium, em-bedded in free space. The problem is illustrated in Fig. 1.The section is organized into four subsections. First, westudy the electromagnetic wave equation that governsthe electromagnetic fields inside spatially homogeneousbut time-varying and dispersive bulk media. Second, wediscuss the response function of a medium modeled bya Lorentz-type oscillator equation with a time-varyingbulk electron density. Then, we develop a generalizedMie theory that treats the scattering problem of homo-geneous spherical scatterers made of such time-varyingand dispersive media. Finally, we present expressions forobservable quantities such as the total scattered and ab-sorbed power by such a scattering system.

A. Unbounded time-varying media with frequencydispersion

We begin by studying the equations that governelectromagnetic waves inside source-free homogeneous,isotropic, linear, non-magnetic, with only electric dipo-lar polarization response, time-varying bulk media withtemporal dispersion. In such media, we consider thatMaxwell’s equations are coupled with the following con-stitutive relations where the displacement vector D andthe magnetic flux density B are given by

D(r, t) = ε0E(r, t) + P(r, t), (1a)

B(r, t) = µ0H(r, t), (1b)

where E and H are the electric and magnetic fields, re-spectively, ε0 is the electric permittivity of vacuum, andµ0 is the magnetic permeability of vacuum. The polar-

3

ization vector P(r, t) of the medium is given by [47, 57]

P(r, t) = ε0

+∞∫−∞

Re(t, t− τ) E(r, τ) dτ , (2)

where Re(t, t− τ) is the electric response function of thetime-varying, dispersive medium. The response func-tion Re(t, t − τ) expresses the polarization density Pat time t induced by an electric field impulse at timeτ . Equation (2) constitutes our Ansatz for the electric-field-driven polarization induced inside such a medium.It is important to note that this response function hasthe property Re(t, t − τ) = 0, for t ≤ τ , because ofcausality. Also, in the limiting case of non-time-varyingmedia, the response function becomes invariant with re-spect to its first argument t. Moreover, in the limit-ing case of dispersionless media with an instantaneousresponse, the dependency of Re(t, t − τ) on its secondargument t − τ is that of the Dirac delta distribution,Re(t, t − τ) = R′e(t)δ(t − τ). This last assumption ofdispersionless media with an instantaneous response isfound in several recent publications [7–9, 51], but consti-tutes generally a physical assumption that is valid onlyin limited and approximate cases.

By making use of the Fourier transforms of thequantities involved, adopting the convention X(t) =

1√2π

∫ +∞−∞ X(ω)e−iωtdω, we will switch from the above

time-space representation of the governing equations tothe corresponding frequency-space representation. Notethat bar sign represents frequency domain quantities inthe Fourier transform. Therefore, Maxwell’s equationsread [47, 48, 58]

∇ ·E(r, ω) = −+∞∫−∞

Re(ω − ω′, ω′)∇ ·E(r, ω′)dω′, (3a)

∇ ·H(r, ω) = 0, (3b)

∇×E(r, ω) = iωµ0H(r, ω), (3c)

∇×H(r, ω) = −iωε0E(r, ω)

− iωε0

+∞∫−∞

Re(ω − ω′, ω′)E(r, ω′)dω′. (3d)

The response function in frequency domain has been de-fined as the double Fourier transform Re(ω − ω′, ω′) =

12π

+∞∫∫−∞

Re(t, t− τ)ei(ωt−ω′τ)dtdτ . It gives the polarization

denisty P(ω) at frequency ω induced by an electric fieldimpulse at frequency ω′. Let us note that in the limitingcase of non-time-varying media the response function infrequency domain Re(ω − ω′, ω′) is a Dirac delta distri-bution with respect to its first argument and takes thefollowing form: Re(ω − ω′, ω′) = δ(ω − ω′)χ(ω), whereit collapses into the usual electric susceptibility tensor.Moreover, in the limiting case of dispersionless media

with instantaneous response, Re becomes invariant withrespect to its second argument.

We move on by combining the last two equations toobtain the wave equation for the electric field:

∇×∇×E(r, ω)

= k20(ω)

E(r, ω) +

+∞∫−∞

Re(ω − ω′, ω′)E(r, ω′)dω′

, (4)

where k0(ω) = ω√µ0ε0 = ω/c0 is the wavenumber of free

space. This wave equation can be simplified for disper-sionless media as reported, e.g., in Ref. 59.

Next, we calculate the eigenfunctions that solve thishomogeneous integro-differential equation, whose opera-tor is non-diagonal in terms of the frequency ω. Theseeigenfunctions are the fundamental solutions to thesesource-free Maxwell’s equations. They are of paramountimportance, as general solutions induced by an arbi-trary source can always be written as a superposition ofthese fundamental solutions weighted with suitable am-plitudes.

To find these eigenfunctions, we use the method of sep-aration of variables, and seek for solutions of the electricfield E(r, ω) that have the following form:

E(r, ω) =

∫A(κ)Sκ(ω)Fκ(r)dκ, (5)

where A(κ) is a complex amplitude. In this Ansatz, thedependency of the eigenfunctions on the spatial and fre-quency arguments is separated. By introducing the sep-aration constant κ2, we obtain the following set of cou-pled equations for the spatial part of the eigenfunctions,Fκ(r), and for the spectral part of the eigenfunctions,Sκ(ω):

∇×∇× Fκ(r) = κ2Fκ(r), (6a)

k20(ω)

Sκ(ω) +

+∞∫−∞

Re(ω − ω′, ω′)Sκ(ω′)dω′

= κ2Sκ(ω). (6b)

Both of the above equations constitute themselves eigen-value type of equations where κ2 is the common eigen-value. Fκ(r) is the corresponding eigenfunction of thedifferential operator of the first equation and Sκ(ω) isthe corresponding eigenfunction of the integral operatorof the second equation. The first equation (6a) for thespatial, vectorial profile of the eigenfunction Fκ(r), is anordinary monochromatic electromagnetic wave equationwith wavenumber κ. Depending on the coordinate sys-tem in which the solution is sought, the eigensolutionsof the equation could be a set of plane waves, cylindricalwaves, or spherical waves. As we wish to study electro-magnetic scattering of light by a sphere, we choose spher-ical waves as eigensolutions, since they allow to apply theinterface conditions needed. At this point we could have

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picked plane waves or cylindrical waves, and proceed fur-ther in a similar way, if we were to study the interactionof light with a slab or an infinite cylinder, respectively.

Such spherical-coordinate solutions to (6a) are knownas vector spherical harmonics (VSHs). Details on theirdefinition and properties can be found in Appendix A.

The VSHs are denoted here as F(ι)α,µνκ(r), where the in-

dex “(ι)” in the superscript takes the value “(1)” to referto regular VSHs finite at the origin r = 0, or the value“(3)” to refer to radiating VSHs that comply with the ra-diation condition at infinity. Hence, the former solutionsdescribe standing waves, whereas the latter solutions de-scribe scattered fields. Moreover, there is a set of foureigenvalues that characterize the VSHs. First, we havethe index α that takes the values M,N to refer to trans-verse electric (TE) or transverse magnetic (TM) waves,i.e., multipoles of magnetic or electric type, respectively.Then, apart from the wavenumber κ, we also have theeigenvalues µ and ν with µ being the angular momen-tum along the z-axis and ν being the multipolar order ofthe VSH.

It is important to note that the spatial eigenfunctionsFκ(r) are solenoidal for κ 6= 0, i.e., ∇·Fκ(r) = 0. For so-lutions with κ = 0, it is straightforward to show that themagnetic field becomes irrotational, i.e., ∇×∇×F0(r) =0. We then have a non-zero induced electric charge den-sity distribution since the divergence of F0(r) does nothave to be zero. The above observation follows from (3a),(3d) and (5)–(6b). Such implications that arise for κ = 0where the electric field ceases to be solenoidal are dis-regarded in the remainder of our analysis. This allowsexpanding the fields inside the time-varying scattereronly using the TE and TM spherical waves while avoid-ing the third multipolar family of longitudinal sphericalwaves [60–62].

Let us now focus on the eigenvalue equation (6b). It isnewly introduced by the time-variance of the medium andinvolves the spectral eigenfunction Sκ(ω). This eigen-value equation plays the role of a dispersion relation oftime-varying systems. The important thing to noticehere is that, due to the time-variance of the medium,the system is not translationally invariant in time, andwe encounter a coupling among different frequency com-ponents. Equation (6b) governs this spectral coupling ofelectromagnetic field harmonics inside the medium. Ingeneral, the equation has to be solved numerically byprojecting it onto a Hilbert space H of finite dimen-sions. This leads to a finite linear system of equationswhose eigenvalues κ2 and corresponding eigenfunctionsSκ(ω) can be calculated numerically. Two assumptionshave to be applied to make the system solvable.

First, we have to truncate the infinite spectrum intoa finite spectral window. This will always numericallycompromise the results. However, the truncation effectcan be made arbitrarily small if the spectral window issufficiently large compared to the spectral region of in-terest, since, usually, the spurious truncation effects willmainly affect the frequencies closer to the edges of the

truncated spectral window.Second, we need to discretize the frequency ω in (6b),

which implies a time-periodic modulation of the medium.So, once we have thatRe(t, t−τ) = Re(t+jTm, t−τ), withTm being the modulation period and j ∈ Z, the Fouriertransform of the response function becomes discrete:

Re(ω−ω′, ω′) =∑j δ(ω−ω′− jωm)R

′e(ω−ω′, ω′), with

ωm = 2π/Tm being the modulation frequency. Hence,for such a system with discrete translational symmetryin time, it is instructive to introduce a new eigenvalue,the Floquet frequency Ω. This eigenvalue takes valueswithin the frequency range [0, ωm).

Equation (6b) can now take the following discreteform:

k20(Ωj)

[Sκ(Ωj) +

NΩ∑l=1

R′e(Ωj − Ωl,Ωl)Sκ(Ωl)

]= κ2(Ω)Sκ(Ωj), (7a)

where

Ωj = Ω + (j + j0)ωm (7b)

with j = 1, 2, . . . , NΩ and j0 being an integer that wechose appropriately for the truncated spectral window ofinterest. NΩ is the total number of frequencies of thediscretized and truncated spectrum.

We see that the Floquet frequency Ω constitutes anextra eigenvalue of our system. It characterizes an in-finite periodic comb of frequencies passing through thefrequency Ω. Equation (7b) gives the frequencies of sucha spectral comb within an arbitrarily truncated spectralwindow. Due to the medium’s periodic time modula-tion, only the frequencies contained in each such spectralcomb are mutually coupled, constituting an independentsystem. Therefore, for each Floquet frequency Ω, equa-tion (7a), repeated for all values of the index j, forms alinear system of equations that can be written in matrixform as

K(Ω) · ~Sκ(Ω) = κ2(Ω) ~Sκ(Ω), (8)

where we have defined the vector:

~Sκ(Ω) = [Sκ(Ω1) · · · Sκ(ΩNΩ)]T, (9)

and the matrices:

K(Ω) = k20(Ω) ·

[I + Re(Ω)

], (10)

k0(Ω) = diag [k0(Ω1) · · · k0(ΩNΩ)] , (11)

with I being the identity matrix and with the j-th-row-,l-th-column-element of the matrix Re(Ω) being equal to

R′e(Ωj − Ωl,Ωl).Consequently, in the eigenvalue equation (8), we end

up with a matrix K(Ω) of finite dimensions NΩ × NΩ,whose NΩ eigenvalues κ2

i (Ω) and corresponding eigen-

vectors ~Sκi(Ω) can be calculated numerically for each

5

Floquet frequency Ω [63]. The eigenvalues and the cor-responding eigenvectors are enumerated by the indexi = 1, . . . , NΩ.

Finally, the expansion of the fields in (5) can now takethe following form within the Hilbert space H of finitedimensions that we constructed for the case of such aperiodically modulated, time-varying medium:

E(r, ω) =

∫ ωm

0

NΩ∑i,j=1

Ai(Ω)δ(ω − Ωj)Sκi(Ωj)Fκi(r)dΩ,

(12)

with Ai(Ω) being complex amplitudes. The above equa-tion constitutes our general Ansatz for the expansionof fields inside periodically modulated, time-varying me-dia [64].

B. The response function of time-varying media

In this subsection, we will consider a simple case ofa time-varying medium and derive its response functionRe(ω − ω′, ω′). We consider a polarizable medium inwhich there exist bound polarizable electrons of a singlespecies that live inside the potential well of a Lorentzianharmonic oscillator. Let p(r, t) be the induced dipolemoment of a single bound electron driven by the electricfield. It shall obey the following differential equation ofmotion: [

∂2

∂t2+ γn

∂

∂t+ ω2

n

]p(r, t) =

e2

meE(r, t), (13)

where γn is the damping factor of the oscillator, ωn isits resonance frequency, and e,me are the charge and theeffective mass of an electron, respectively. The solutionto the above differential equation is given by the followingconvolution integral:

p(r, t) =

∫ +∞

−∞αe(t− τ)E(r, τ) dτ, (14a)

where

αe(t)=1√

ω2n −

γ2n

4

e2

meH(t)e−

γn2 tsin

(t

√ω2n −

γ2n

4

), (14b)

is the electric polarizability kernel of the Lorentzian os-cillator and H(t) is the Heaviside step function [47].

Now, let us consider that the bulk electron density ofthese electron species, N(t), gets modulated in time. Thepolarization density of the medium shall then be given bythe following equation:

P(r, t) =

∫ +∞

−∞αe(t− τ)N(τ)E(r, τ) dτ. (15)

This equation implies that the electric field at each mo-ment τ excites only the available number of electrons in

unit volume N(τ) [65]. Moreover, the model assumesthat the electrons oscillate inside a Lorentzian potentialwell that remains invariant even if the bulk electron den-sity varies in time. However, one would expect, for exam-ple, that the damping factor becomes larger with increas-ing bulk electron density, due to a higher rate of electroncollisions. In this work we will avoid such considerationsfor simplicity.

It is straightforward to show that the polarization den-sity obeys then the following differential equation:[

∂2

∂t2+ γn

∂

∂t+ ω2

n

]P(r, t) =

e2

meN(t)E(r, t). (16)

Such a model for the polarization density has alreadybeen reported in Refs. 49 and 66.

Furthermore, comparing the above equation with theAnsatz of Eq. (2), we get that the response function ofsuch a medium is equal to

Re(t, t− τ) =αe(t− τ)N(τ)

ε0, (17)

which, in frequency domain, takes the following form:

Re(ω − ω′, ω′) =1√2π

e2

meε0

N(ω − ω′)ω2n − ω2 − iγnω

. (18)

A more general type of response function could be givenby a superposition of such Lorentz harmonic oscillatorsand an additional Drude term in order to account fordifferent electron species [67].

In general, for a periodically modulated bulk elec-tron density N(t), with the modulation frequency ωm,we have N(ω − ω′) =

∑j Njδ(ω − ω′ − jωm), with

j ∈ Z and Nj being complex coefficients. For exam-ple, for our numerical demonstration in the next sec-tion, we will consider the particular case where the bulkelectron density is harmonically modulated in time ac-cording to N(t) = N0 [1 +Mscos(ωmt)], where N0 isthe bulk electron density of the unmodulated medium,Ms is the modulation strength, taking values from 0to 1, and ωm is the modulation frequency. In such acase, we find that N(ω − ω′) =

√2πN0δ(ω − ω′) +

Ms

√2πN0 [δ(ω − ω′ + ωm) + δ(ω − ω′ − ωm)] /2. We

will use the above formulas in the next section to nu-merically solve the eigenvalue problem of Eq. (8).

C. Scattering by time-varying spheres withfrequency dispersion

Let us move on to the particular problem of electro-magnetic scattering by a sphere composed of a time-varying and dispersive material. The sphere has a radiusR, is embedded in free space and centered at the originof the coordinate system. We begin by expanding theincident field in the following series of regular VSHs [68]:

Einc

(r, ω) =∑νµ,α

Aincα,µν(ω)F

(1)α,µνk0

(r), (19)

6

where the free-space wavenumber k0(ω) is a function offrequency. More details about the above expansion of theincident field can be found in Appendix B. Accordingly,the scattered field can be expanded in the following seriesof radiating VSHs [69]:

Esca

(r, ω) =∑νµ,α

Ascaα,µν(ω)F

(3)α,µνk0

(r). (20)

Finally, according to the Ansatz of (12), the field inducedinside the sphere is expanded over the following series ofregular VSHs:

Eind

(r, ω) =∫ ωm

0

NΩ∑i,j=1

∑νµ,α

Aindα,µνi(Ω)δ(ω − Ωj)Sκi(Ωj)F

(1)α,µνκi(r)dΩ.

(21)

Let us highlight here that, in the formula above, theeigenvalues κi [70] and the frequencies Ωj , are functionsof the Floquet frequency Ω. This dependence is droppedin our notation here for simplicity but implicitly alwaysassumed. In comparison to the two previous expansionsof the fields in free space [Eqs. (19,20)], one can see herehow our Ansatz for the fields inside the time-varyingsphere [Eq. (21)] changes according to Eq. (12). Dueto the time variance, there is no unique wavenumber cor-responding to each frequency anymore. Instead, we havea bunch of wavenumbers corresponding to each comb offrequencies characterized by the Floquet frequency Ω.

The series expansions of the respective magnetic fieldscan be taken by making use of the Maxwell-Faradayequation (3c), together with the following property of

VSHs: ∇× F(ι)α,µνκ(r) = κF

(ι)β,µνκ(r), where β 6= α.

Now, solving this electromagnetic scattering prob-lem within the defined finite-dimensional Hilbert spaceH means calculating the unknown complex amplitudesAscaα,µν(ω), Aind

α,µνi(Ω) given the amplitudes Aincα,µν(ω).

This can be done by imposing the following interface con-ditions on the surface of the sphere:

r×[E

ind(r, ω)−E

sca(r, ω)−E

inc(r, ω)

]∣∣∣r=R

= 0, (22a)

r×[H

ind(r, ω)−H

sca(r, ω)−H

inc(r, ω)

]∣∣∣r=R

= 0, (22b)

that enforce the continuity of the tangential fields accord-ing to Maxwell’s equations. Here, we need to make useof the following orthogonality property of the VSHs [71]:∮SR

[r× F

(ι)α,µνκ(r)

]· F(ι′)

α′,−µ′ν′κ′(r)ds

(−1)µ+δαNR2 z(ι′)α′,ν(κ′R)

= δα′βδµ′µδν′ν z(ι)α,ν(κR), (23)

where integration is done over the spherical surface SRof radius R, upon which we need to enforce the aboveinterface conditions. δij is the Kronecker delta, β 6= α,

and z(ι)α,ν(x) is the generalized spherical Bessel function

defined in Appendix A. Finally, by substituting Eqs. (19-21) into Eqs. (22a,22b) and by making use of Eq. (23),we end up with the following inhomogeneous system ofequations to be solved:

∑i

Aindα,µνi(Ω)Sκi(Ωj)z

(1)α,ν(κiR) = Asca

α,µν(Ωj)z(3)α,ν(xj) +Ainc

α,µν(Ωj)z(1)α,ν(xj), (24a)

∑i

κiAindα,µνi(Ω)Sκi(Ωj)z

(1)β,ν(κiR) = k0(Ωj)

[Ascaα,µν(Ωj)z

(3)β,ν(xj) +Ainc

α,µν(Ωj)z(1)β,ν(xj)

], (24b)

where xj = k0(Ωj)R. Let us note that the last two equa-tions are equivalent to Eqs. (22,23) of Ref. 51. By intro-ducing the following definitions of column vectors

~Aincα,µν(Ω) =

[Aincα,µν(Ω1) · · · Ainc

α,µν(ΩNΩ)]T, (25a)

~Ascaα,µν(Ω) =

[Ascaα,µν(Ω1) · · · Asca

α,µν(ΩNΩ)]T, (25b)

~Aindα,µν(Ω) =

[Aindα,µν1(Ω) · · · Aind

α,µνNΩ(Ω)]T, (25c)

and matrices

S(Ω)=[~Sκ1

(Ω) · · · ~SκNΩ(Ω)], (26a)

κ(Ω)=diag[κ1(Ω) · · · κNΩ(Ω)] , (26b)

Z(ι)α,ν(Ω)=diag

[z(ι)α,ν(κ1(Ω)R) · · · z(ι)

α,ν(κNΩ(Ω)R)

], (26c)

ˆZ(ι)α,ν(Ω)=diag

[z(ι)α,ν(k0(Ω1)R) · · · z(ι)

α,ν(k0(ΩNΩ)R)], (26d)

together with the definitions in Eqs. (9,11), we canrewrite the above set of equations in the following matrixform:

7[S · Z(1)

α,ν − ˆZ

(3)α,ν

S · κ · Z(1)β,ν −k0 · ˆ

Z(3)β,ν

]·

[~Aindα,µν~Ascaα,µν

]=

[ˆZ

(1)α,ν 0

0 k0 · ˆZ

(1)β,ν

]·

[~Aincα,µν~Aincα,µν

], (27)

where 0 is a matrix with dimensions NΩ×NΩ filled withzeros. In the above equation, the dependencies on theFloquet frequency Ω were dropped for simplicity. Let usintroduce now the following T-matrix

Tα,ν(Ω) =

[T11α,ν T12

α,ν

T21α,ν T22

α,ν

]

=

[S · Z(1)

α,ν − ˆZ

(3)α,ν

k−10 · S · κ · Z

(1)β,ν −

ˆZ

(3)β,ν

]−1

·

[ˆZ

(1)α,ν 0

0ˆZ

(1)β,ν

]. (28)

By introducing also the following two T-matrices withdimensions NΩ ×NΩ:

Tindα,ν(Ω) = T11

α,ν + T12α,ν , (29a)

Tscaα,ν(Ω) = T21

α,ν + T22α,ν , (29b)

we finally end up with the following expressions for thecomplex amplitudes of the induced and scattered fieldsas functions of the incident amplitudes

~Aindα,µν(Ω) = Tind

α,ν(Ω) · ~Aincα,µν(Ω),

~Ascaα,µν(Ω) = Tsca

α,ν(Ω) · ~Aincα,µν(Ω).

(30a)

(30b)

The last two equations solve the scattering problem thatwe studied.

Finally, let us discuss some important symmetry prop-erties of the above T-matrices, defined by the spatio-temporal symmetries of the corresponding scattering sys-tem that they represent. First of all, due to the fact thatour scattering system is time-varying, we end up havinga T-matrix that is non-diagonal with respect to the fre-quency ω. This property implies an inelastic scatteringprocess. In fact, for the specific case of a time-modulatedsystem with discrete translational symmetry over time,i.e., with a modulation period Tm, according to the Flo-quet theorem, we get a T-matrix that is block diagonalover frequency ω, with each block involving a comb of fre-quencies characterized by the Floquet frequency Ω anda period of ωm = 2π/Tm. This is the sole change thatthe structure of the T-matrix undergoes due to the intro-duced time-variance of the scattering system. The spatialsymmetries of the system of the spherical scatterer con-tinue to be exactly the same as in the stationary case.Since such a scattering system is rotationally invariantwith respect to the z-axis, we have a T-matrix that isdiagonal with respect to the eigenvalue µ, the angularmomentum along the z-axis. Actually, rotational invari-ance of the system along an arbitrary axis, due to itsspherical symmetry, implies also a T-matrix that is diag-onal with respect to the multipolar order ν. Moreover,due to the point inversion invariance of such a scattering

system, we end up having a T-matrix that is diagonalwith respect to the eigenvalue α, since the TE and TMVSHs with a fixed multipolar order ν (mod 2) have anopposite parity symmetry. Scatterers of different, non-spherical geometry, would generally break those spatialsymmmetry properties of their T-matrices.

D. Observable scattering quantities

In this subsection we will provide expressions forthe scattered and absorbed energy by spherical time-modulated scatterers. Following Eq. (5.18) of Ref. 68,as well as our conventions for the Fourier transforms ofthe fields and the above expressions for the incident andscattered fields in terms of series of VSHs, as presentedin Eqs. (19,20), we can get the following expressions forthe total scattered energy W sca:

W sca =

∫ ωm

0

NΩ∑j=1

P sca(Ωj) dΩ

=

∫ ωm

0

NΩ∑j=1

∑νµ,α

∣∣Ascaα,µν(Ωj)

∣∣2Z0k2

0(Ωj)dΩ, (31a)

and for the total absorbed energy W abs:

W abs =

∫ ωm

0

NΩ∑j=1

P abs(Ωj) dΩ

= −∫ ωm

0

NΩ∑j=1

∑νµ,α

<[Aincα,µν(Ωj)

]∗Ascaα,µν(Ωj)

Z0k2

0(Ωj)dΩ,

(31b)

where P sca(ω), P abs(ω) are the total scattered andabsorbed powers, respectively, and Z0 is the waveimpedance of free space.

We can also reach alternative expressions for the totalscattered and absorbed energies by performing a singularvalue decomposition [72] of the following matrices:

k−10 (Ω) · Tsca

α,ν(Ω) · k0(Ω)

= Uα,ν(Ω) · Σα,ν(Ω) · V†α,ν(Ω), (32)

where Σα,ν(Ω) are diagonal matrices that contain the sin-

gular values σα,νs(Ω) of the decomposition, and Uα,ν(Ω),

Vα,ν(Ω) are matrices whose columns contain the cor-respondent left- and right-singular vectors, ~uα,νs(Ω),~vα,νs(Ω), respectively. The right- and left-singular vec-tors contain the incident and scattered multipolar spectra

8

of the singular modes of the time-varying scattering sys-tem. By expanding the following vectors on the full basisof the right-singular vectors:

k−10 (Ω) · ~Ainc

α,µν(Ω) =

NΩ∑s=1

S incα,µνs(Ω)~vα,νs(Ω), (33)

where S incα,µνs(Ω) = ~v†α,νs(Ω) · ~Ainc

α,µν(Ω) are complex co-efficients, we can arrive to the following alternative ex-pressions for the total scattered and absorbed energies:

W sca =1

Z0

∫ ωm

0

∑νµ,α

NΩ∑s=1

σ2α,νs

∣∣S incα,µνs

∣∣2 dΩ, (34a)

W abs = − 1

Z0

∫ ωm

0

dΩ∑νµ,α

NΩ∑s=1

σα,νs∣∣S incα,µνs

∣∣2 [σα,νs + <~v†α,νs · ~uα,νs

]+

NΩ∑s,s′=1s′ 6=s

σα,νs′<[S incα,µνs

]∗ S incα,µνs′~v

†α,νs · ~uα,νs′

,

(34b)

where we dropped the dependence of the quantities onthe Floquet frequency Ω for simplicity. The sum in thelast row of the last equation for the absorbed energy cor-responds to couplings among the singular modes.

III. NUMERICAL STUDY AND DISCUSSION

In this section, we will demonstrate and discuss nu-merical results based on the theoretical approach thatwe developed in the previous section. The section is di-vided into three subsections. In the first subsection, wepresent our results regarding the bulk media dynamics oftime-varying and dispersive media and discuss their mainelectromagnetic properties. In the second subsection, westudy the properties of the scattering system of a homo-geneous spherical scatterer composed of a time-varyingand dispersive medium. Properties of the T-matrix char-acterizing the scattering system are presented. We alsohighlight the prototypical ability of this system to act asan active element by transferring energy from the exter-nal modulation of the medium to the radiated electro-magnetic field, resulting in negative electromagnetic ab-sorption. In the last subsection, we numerically comparethe developed semi-analytical approach to full-wave opti-cal simulations, highlighting the accuracy and efficiencyof our method.

A. Bulk media dynamics

We begin by considering a dispersive medium de-scribed by a single Lorentz-type oscillator with natu-

ral resonance frequency ωn. The medium’s electric sus-ceptibility χ(ω), in the non-time-varying case, is plot-ted in Fig. 2(a). The considered bulk electron densityis N0 = 11ω2

nmeε0/e2, and the damping factor of the

oscillator is γn = ωn/8.

Then, we study the effect of the introduced time-variance on such dispersive medium. To do that, weassume that the bulk electron density of the Lorentz os-cillator varies harmonically in time according to N(t) =N0 [1 +Ms cos(ωmt)] and the corresponding responsefunction of the medium is given by Eq. (18) accordingto the analysis of Sec. II B. We use this time modulationto study the eigenvalues, i.e., the supported wavenumbers

κ(Ω), and the corresponding eigenvectors ~Sκ(Ω), i.e., thecorresponding spectral eigenfunctions, of such a systemas described by Eq. (8).

With the purpose to ease the understanding, let us dis-cuss initially what happens in the limiting case of verysmall modulation strengths Ms → 0. In analogy to pe-riodically spatially modulated materials, this case wouldcorrespond to an empty-lattice approximation, i.e., prac-tically there is no modulation but the periodicity is stillintroduced. This assumption allows us to work with theanalytical dispersion relation while observing the onsetof a band structure. This approach facilitates the under-standing of the further results.

In this case, the integral operator in Eq. (6b) remainspredominantly diagonal, with very small off-diagonalterms. This property indicates a weak spectral cou-pling among frequencies. This means that the spectraleigenfunctions Sκ(ω) tend to delta distributions. Theytend to associate a unique wavenumber κ to each fre-quency ω, as is the case for the usual dispersion re-lation of non-time-varying media. Thus, we see thatκ(ω) →

√[1 + χ(ω)]ω2/c20, as Ms → 0. We show in

Fig. 2(b) how the band structure of such a system isformed by folding the wavenumbers κ(ω) into the fun-damental spectral band. This spectral band ranges fromthe zero frequency to the modulation frequency of themedium ωm, encompassing in this manner all the Flo-quet frequencies Ω. The blue dashed lines show the bandstructure formed by folding the solid blue line within thefundamental spectral band that corresponds to a mod-ulation frequency ωm = ωn/2. The folding takes placeperiodically in frequencies denoted by the dashed purplelines.

We illustrate in Fig. 2(c) the truncated discrete spec-trum of the response of a system periodically modulatedwith a frequency ωm and excited, also, by a periodic ex-citation. ωp = ωm/Np, with Np ∈ N, is the frequencythat corresponds to the superperiod of the combined pe-riodicities of the modulation and excitation. We can seethat the spectrum can be separated into a set of Np finitecombs of frequencies with a periodicity of ωm. Each suchcomb of frequencies, indicated by a different color, corre-sponds to a different Floquet frequency Ω, which variesbetween zero and ωm. Due to the periodic time modu-lation of the medium, there is only coupling among the

9

FIG. 2. Bulk media dynamics: (a) Plot of the electric susceptibility of an unmodulated medium with dispersion correspondingto a single Lorentz oscillator. (b) Illustration of the folding of the band structure (dashed blue line) of the medium as we startmodulating the bulk electron density of the Lorentz oscillator with frequency ωm, in the limit of weak modulation strengthMs. Solid lines represent the dispersion relation of the unmodulated medium. The dashed purple lines indicate the foldingfrequencies. (c) Illustration of the truncated, discrete spectrum of the response of a periodically modulated system excited by aperiodic illumination. Spectral combs, characterized by the Floquet frequencies Ω, constitute independent systems of coupledfrequencies and are illustrated with different colors. Due to the periodic time-modulation of the system, there is coupling amongthe frequencies of each such spectral comb. (d) Plot of the band structure diagram of a time-modulated medium with strongmodulation strength. The opening of a band gap can be observed. (e) Plot of the spectral content of the eigenmodes |Sκi(Ωj)|that correspond to the system of a single comb of frequencies. We can observe the spectral coupling that is introduced by thestrong time modulation of the medium. (f) Plot of the eigenvalues, i.e., the wavenumbers κi, that correspond to the eigenmodespresented in (e).

frequencies of each such spectral comb. Spectral combsof different Floquet frequencies Ω constitute independentsystems and do not couple to each other. For numericalreasons, we have to truncate the spectrum. Here, a trun-cated spectral window of NΩ bands is illustrated, whichcorresponds, also, to the number of frequencies of eachsuch truncated spectral comb. The actual distributionof power among the frequencies of each comb can be de-

composed over the full basis of eigenvectors ~Sκi(Ω).

In general, what changes with the introduced periodictime modulation of the medium, is that now we have a setof wavenumbers κi(Ω) associated with each Floquet fre-quency Ω, and with each wavenumber corresponding toa different, generally broadband, spectrum, given by the

corresponding eigenvectors ~Sκi(Ω). We solve the eigen-value equation (8) and plot in Fig. 2(d) the band struc-ture, i.e., the set of eigenvalues/wavenumbers that cor-respond to each Floquet frequency Ω, i.e., to each spec-tral comb. For the pertinent case we consider a modu-lation strength MS = 0.9 and a modulation frequencyωm = ωn/10. The color of the line in this figure encodes

the imaginary part of the eigenvalues. Generally, posi-tive and negative imaginary values correspond to spectraleigenmodes with predominant spectral content over pos-itive and negative frequencies, respectively. Let us notethat the graph focuses on the region of eigenvalues witha small real part. We can see that, due to strong enoughtime modulation of the medium, we introduce a band gapin the lower band of the band structure [7, 10, 22]. Thisis not possible for low modulation strengths as the effectis indicative of strong spectral coupling.

In Figs. 2(e,f) we plot, respectively, the eigenvec-tors and the corresponding eigenvalues of the subsystemwith Floquet frequency Ω = ωm/200. The eigenmodesare ordered with respect to ascending eigenmode cen-tral frequency, which is defined as the following sum:∑j Ωj |Sκi(Ωj)|

2. We can see in Fig. 2(e) that each

eigenmode has a different spectral content distributedover the frequencies of the spectral comb characterizedby the particular Floquet frequency Ω. The spectral sup-port becomes wider for eigenmodes that support high fre-quencies, whereas for the eigenmodes that predominantlysupport the frequencies Ωj/ωn ≈ 0,±1 it becomes mini-

10

mally narrow. As discussed previously, the matrix withthe eigenvectors plotted in Fig. 2(e) shall approach theidentity matrix in the limit of MS → 0. The degree ofnon-diagonality of the matrix S(Ω) is indicative of thestrength of the spectral coupling within the time-varyingsystem. In Fig. 2(f), we plot the sorted eigenvalues,i.e., the wavenumbers associated with the correspond-ing eigenvectors as sorted in Fig. 2(e). Let us note theresemblance of Fig. 2(f) with the unmodulated case il-lustrated in Fig. 2(b). The sorted wavenumbers of thestrongly-modulated case are quite similar to those of theunmodulated case. However, we still have quite signifi-cant deviations, as it is indicated by the presence of theopen band gap in Fig. 2(d).

B. The scattering system of a time-varying anddispersive sphere

Next, we will study the properties of the scatteringsystem of a homogeneous spherical scatterer made of thetime-varying and dispersive medium that was studied inthe previous subsection. We consider the same materialdispersion as the one used in Fig. 2(a), and we considerthat the bulk electron density of the material is againmodulated harmonically with a modulation frequency ofωm = ωn/10 and a modulation strength of Ms = 0.9, asit was the case in the example illustrated in Figs. 2(d-f).We consider the radius of the sphere R to be equal to thefree-space wavelength that corresponds to the resonancefrequency of the Lorentz oscillator, i.e., R = 2πc0/ωn.

In Fig. 3(a) we plot the Mie coefficients (up to thequadrupolar order) that correspond to such a spherein the non-time-varying case of Ms = 0. At very lowfrequencies, we observe the Rayleigh scattering regionwhere the Mie coefficients diminish in amplitude. Atlow frequencies, away from the resonance frequency ofthe Lorentz oscillator where the material losses are max-imized, we note the appearance of sharp and denselypacked multipolar resonances. On the other hand, atfrequencies larger than the resonance frequency of theLorentz oscillator, we have a material with negative di-electric permittivity [see Fig. 2(a)] that demonstratesmodes of much lower quality factors that are spectrallywell-separated.

Next, we introduce time modulation to the materialfrom which the sphere is made. We plot in Figs. 3(b-e) the absolute values of the elements of T-matrices ofsuch a time-modulated sphere given by Eq. (29b). Theresults are plotted in a logarithmic scale for multipolesup to the quadrupolar order. We combine the calculatedresults for the T-matrices of all the Floquet frequenciesin a single plot. There, the x-axis corresponds to thefrequency ωinc of the incident multipolar excitation. They-axis corresponds to the scattered band order of radiat-ing multipoles. The output frequency ωsca of the radiatedmultipole at a scattering band order bo is given by theformula ωsca = ωinc+boωm. A zero scattering band order,

FIG. 3. The scattering system of a time-varying and dis-persive sphere: (a) Plot of the absolute value of the Miecoefficients that correspond to a sphere made of a disper-sive medium without time modulation. (b-e) Plots of theT-matrix elements of a sphere with introduced strong timemodulation, for different multipolar orders. Time modulationleads to an inelastic scattering process where there is spectralcoupling among different input and output frequencies givenby ωsca = ωinc + boωm.

i.e., bo = 0, means that the frequencies of the incidentand scattered multipoles are the same. Therefore, in thelimit of low modulation strengths Ms → 0, we shall havea predominant response solely at the zero scattering bandorder, bo = 0. The color of the plots encodes the ampli-tude of a radiated multipole at frequency ωsca once thesphere is excited by a single multipole of unit amplitudeat frequency ωinc. The white lines in Figs. 3(b-e) denotean output frequency being zero, i.e., ωsca = 0.

There are several interesting features to be observed inFigs. 3(b-e). Most importantly, we see that, due to thetime modulation of the sphere’s material, a monochro-matic excitation gives rise to a polychromatic response.This implies an inelastic scattering process. The general

11

FIG. 4. Singular value decomposition of the matrices of the scattering system revealing the presence of singular dipolar modesthat demonstrate negative absorption, i.e., a transfer of energy from the time-varying matter to the photons in an inelasticscattering process. An optimized excitation of the system can give birth to such modes. (a,b) Plots of the ratio of the absorbedand scattered powers under the system’s excitation with its singular modes. The presence of singular modes with negativeabsorption can be observed. Plots of the output (b,e) and input (c,f) spectra of the left- and right-singular vectors of thesingular modes that demonstrate maximally negative absorption for each Floquet frequency Ω.

condition for a resonant response is that the sphere is atresonance simultaneously both at the input and outputfrequencies ωinc and ωsca. This, of course, happens pre-dominantly when the input and output frequencies coin-cide. However, there are several other cases where sucha resonant inelastic scattering process takes place. Forexample, we can see that there is a strong response alonglines parallel to the white ones, where we have a con-stant output frequency ωsca that shall be associated witha sharp multipolar resonance supported by the spherethere. Such sharp resonances have a significant spec-tral echo predominantly in negative scattering band or-ders, with the response, though, weakening as the differ-ence between the input and output frequencies increases.Moreover, we also observe the appearance of sharper fea-tures with an even stronger response along those spectrallines. We can associate these features with the simulta-neous presence of sharp multipolar resonances at the re-spective input frequencies, leading to enhanced double-resonant effects. Furthermore, there is a beating patternalong those spectral lines. The periodicity thereof is re-lated to the modulation frequency ωm and it indicatesa spectral echo of a multipolar resonance at the inputfrequency. Another interesting feature is that even somecoupling between input and output frequencies of oppo-site sign can be observed. This may lead to interestingphenomena such as parametric amplification [73, 74] andnon-reciprocity [6]. Finally, we observe that for low inputfrequencies, the response of the sphere is weak, especiallyfor larger multipolar orders, since in this case the opticalsize of the sphere is small.

As we highlighted already, there is an inelastic scatter-

ing process when we introduce a time modulation of thescatterer. It implies that the photons interact with thetime-varying matter and exchange energy. This makesus wonder whether it is possible to create an active ele-ment out of such a time-varying scatterer that extractsenergy from the time-varying matter and provides it tothe photons. Therefore, we search for the possibility ofusing our scattering system to realize negative total ab-sorption, even though the dispersive model of the Lorentzoscillator that we employ is rather lossy around the res-onant frequency of the oscillator. Such an observationhas already been reported in Ref. 51 for a lossless systemwithout material dispersion.

To this end, we perform a singular value decompo-sition of the T-matrices that correspond to each spec-tral comb with a specific Floquet frequency. The de-composition is given by Eq. (32). The total scatteredand absorbed energies by our scattering system are thengiven by Eqs. (34a,34b). Then, we excite our scatter-ing system with each right-singular vector of the de-composed matrices, i.e., we consider excitations withS incα,µνs(Ω) = δαα′δµµ′δνν′δss′δ(Ω − Ω′) sweeping the val-

ues of α′, s′,Ω′, with µ′ being arbitrary and ν′ fixed to 1as we focus on the dipolar response of the system. Forall such excitations of our system, we observe the sign ofthe absorbed power. Exciting the system with a singleright-singular vector means that we excite only a singlespectral comb of some Floquet frequency, with a particu-lar spectral distribution of the power over the frequenciesof the comb. Simultaneously, our excitation consists ofa single incoming multipole (dipole). Therefore, it is aquite special excitation, not only spectrally but also spa-

12

tially. It corresponds to an excitation with a particularangular spectrum of plane waves that comprise such anincoming multipole. An arbitrary excitation of the sys-tem can be decomposed into this basis of right-singularvectors. Exciting our system, though, with a single singu-lar mode enables us to ignore the inter-modal couplingsdue to the terms in the third row of Eq. (34b).

We consider as a scattering system the same spherethat we studied before in this subsection, and we plotour results in Fig. 4. The singular modes are orderedin a descending order of their respective singular values,i.e., in a descending order of total scattered power, as itis implied by Eq. (34a). In Figs. 4(a,d), we observe thatfor many of the spectral combs with varying Floquet fre-quency Ω, we can have singular modes that demonstratesignificantly negative values of absorbed power, i.e., asignificant transfer of energy from the time-varying mat-ter to the photons of the electromagnetic field duringthe inelastic scattering process. Such modes can be ex-cited only with the particular illumination of the corre-sponding right-singular vectors. The spectral content ofthe right-singular vectors, ~vα,1s0(Ω), which correspondto singular modes that demonstrate a maximally neg-ative absorption (indicated with the index s0), is plot-ted in logarithmic scale and for each Floquet frequencyin Figs. 4(c,f). The black-colored columns of the fig-ure indicate an absence of a singular mode with nega-tive absorption for that particular Floquet frequency. InFigs. 4(b,e) we plot the norm of the elements of the cor-responding left-singular vectors ~uα,1s0(Ω), i.e., the spec-tral content of the scattered fields once the system getsexcited by the corresponding right-singular vectors. Weobserve that the input and output spectra of the singularmodes that demonstrate negative absorption are charac-terized by a spectral distribution of power predominantlyover the low frequencies where the material losses due todispersion are low. Due to the presence of a lossy spec-tral region and the size of the considered sphere, we donot find any singular mode with negative absorption forthe quadrupolar modes. It would only become possiblefor larger sizes of the sphere. It is rather remarkable thatnegative absorption can be achieved even in the presenceof strong material losses once we optimize the system’sexcitation. Finally, let us note that the presence of thethird row of Eq. (34b), corresponding to inter-modal cou-plings, allows for the possibility of attaining negative ab-sorption under other excitation schemes as well, that in-volve, in general, a superposition of such singular modes.

C. Numerical performance of the developedalgorithm in comparison to a full-wave solver

Our numerical analysis of the problem of scatteringby a time-varying and dispersive sphere was verified byfull-wave simulations performed in Comsol Multiphysicswith the finite element time-domain method. To solidifythe comparison and study the efficiency of the developed

semi-analytical approach, we have considered two differ-ent sets of simulations. We again adopt the Lorentz os-cillator model with time-modulated bulk electron densityto account for material dispersion and time modulation.Equation (16) is embedded inside the full-wave numeri-cal solver. We name the first set as “slow modulation,weak dispersion” since we consider a relatively slow butstill strong modulation of the medium of the scatterer.The scatterer is excited in a spectral window character-ized by weak material dispersion, i.e., far away from theresonance of the Lorentz oscillator. On the other hand,the second set of simulations also considers a strong mod-ulation amplitude, but now with a fast modulation fre-quency. Moreover, the sphere is excited in a spectralwindow centered around the resonance frequency of theLorentz oscillator, where we encounter maximal disper-sion. Hence, we name this second set of simulations as“fast modulation, strong dispersion”.

For the first set of simulations, we consider a Lorentzmodel with damping factor γn = ωn/8 and bulk electrondensity N0 = 11ω2

nmeε0/e2. This material can be con-

sidered dispersionless far away from its natural frequencyand, therefore, we simulate the excitation [see Eq. (B1)]of the sphere with a Gaussian pulse of unit amplitudeE0 = 1 V/m and temporal width T0 = 2.9 × 2π/ωnthat is centered at the frequency ω0 = 0.3ωn. The pulseis polarized along the x-axis and propagates along the+z-direction [75]. The pulse is temporally centered att0 = 8T0. The material is modulated with frequencyωm = ωn/15 and modulation strength Ms = 0.9. Wechoose a sphere radius of R = 7.095 c0/ωn to ensure a sig-nificant scattering response. This combination of excita-tion, modulation parameters, and sphere dimensions pro-vides a rich scattering spectrum. Locating the sphere atthe origin of the coordinate system, we compare the fieldsat two arbitrarily chosen spatial points: behind [pointA(0,0,1.43R)] and above [point B(1.43R,0,0)] the sphere.Since the electric fields’ x-component is the strongest atpoint A and the magnetic fields’ y-component is strongat point B, we present them in time domain in Fig. 5(a).The respective norms of the fields are presented in fre-quency domain in Fig. 5(b). The semi-analytical resultsobtained from the developed theory are plotted with solidyellow lines. The numerical results from Comsol are plot-ted with dashed red lines. The highlighted light-blue areadenotes the temporal/spectral region where 99% of theenergy of the incident pulse resides. We observe thatthe results obtained by the numerical simulations almostideally match the semi-analytical results.

In Fig. 5(c), we plot the total power scattered by thesphere [see Eq. (31a)], i.e., the power that we calculate byintegrating the power flux of the Poynting vector of thescattered field over a spherical shell that surrounds thescatterer. To perform a multipolar decomposition of thescattered fields numerically, we measure the fields overa distribution of points located over a spherical shell ofradius 1.43R. We use a surface integral (see Eq. (5.175)from Ref. 68) to extract from the Comsol simulations

13

FIG. 5. Comparison of analytical and numerical results from a full-wave time-domain solver, for two sets of simulations: (a-d)“slow modulation, weak dispersion” and (e-h) “fast modulation, strong dispersion”. (a,e) Plots of the electric (at points Aand A’, respectively) and magnetic (at points B and B’, respectively) fields in the time domain. (b,f) Plots of the Fouriertransform of the fields in (a,e). The highlighted light-blue regions indicate the temporal/spectral windows where 99% of theenergy of the incident Gaussian pulse resides. (c,g) Plots of the normalized total scattered power together with the individualmultipolar contributions. Solid lines show the analytical results, whereas cross-markers indicate the numerical results of thefull-wave simulation. (d,h) Plots of the probability distribution of the logarithmic relative error between the fields calculatedanalytically and numerically. These plots involve the error statistics among a considered distribution of points over a sphericalshell surrounding the scatterer, and, also, over a broadband spectral window.

the multipolar amplitudes of the scattered field Ascaα,µν(ω)

and compare them with the ones that we obtain analyt-ically. The scattered power spectra are normalized tothe spectral peak of the total power flux of the incidentfield passing through the geometrical cross-section of thescatterer. The values of the total scattered power areplotted with a black solid line. The individual multi-polar contributions (up to the quadrupolar order) areplotted with colored solid lines. We use cross-markers toplot the numerical results obtained from Comsol. Again,we see a perfect agreement between the semi-analyticaland the numerical results. Finally, in Fig. 5(d), we plotthe probability distribution of the logarithmic relative er-ror between the analytically and numerically calculatedfields over the points of the previously considered spher-ical shell surrounding the scatterer and over a spectralwindow between the frequencies [0.1ωn, 0.93ωn], wherethe signals are strong. The graph indicates a relative er-ror distribution predominantly within the range of 1%and 10%.

The second set of simulations considers a less lossymaterial for which the Lorentz model parameters readγn = ωn/120 and N0 = 1.12ω2

nmeε0/e2. To study the

effects of strong dispersion, we excite the scatterer atthe resonance frequency (i.e., ω0 = ωn). To capture arich frequency spectrum, we choose the pulse width T0 =1.934 × 2π/ωn and the sphere radius R = 1.824 c0/ωn.The pulse is again temporally centered at t0 = 8T0. Themodulation strength of the material is considered againto be Ms = 0.9. In contrast, the modulation frequency isnow ωm = ωn/2, which corresponds to a relatively highmodulation speed. Figures 5(e-h) are the counterpartsto Figs. 5(a-d), but now for the case of the second setof simulations. The only difference is that in Figs. 5(e,f)the observation points are located at A′(0,0,2.432R) andB′(2.432R,0,0). As for the Figs. 5(g,h), we have theobservation points located over a spherical shell of ra-dius 2.432R, and the spectral window considered forthe statistics of Fig. 5(h) being between the frequencies[0.827ωn, 1.172ωn]. Again, we can observe that for thesecond set of simulations characterized by fast modula-tion and strong dispersion, the simulation and analyticalresults are in almost perfect agreement.

In Appendix C we present results for a comparativestudy between two different material models: one ac-counting for temporal dispersion, as it was the case with

14

the results presented in this subsection, and the otherignoring it. Our results there propound the appreciationof the importance of taking into account the temporaldispersion.

Finally, let us highlight that the numerical simulationsare computationally considerably more demanding thanthe presented semi-analytical approach. While the Com-sol simulations for the first setup lasted for 12 days re-quiring 110 gigabytes of RAM, and for the second setupthey lasted for 5 days requiring 43 gigabytes of RAM,the semi-analytical algorithm uses 2 gigabytes of RAMto calculate T-matrices and only needs approximately 15seconds for both setups.

IV. CONCLUSION

To summarize, we have presented an analytical modelthat describes light scattering on spheres made of disper-sive and time-varying media. First, we comprehensivelystudied the propagation of electromagnetic waves in un-bounded time-varying media with frequency dispersion.We then applied this theory to treat the problem of lightscattering by spheres composed of such media. In con-trast to other approaches for theoretical investigations oftime-modulated structures, the developed route consid-ers spatially confined scatterers, incorporates frequencydispersion, and allows an arbitrary modulation speed andamplitude. In addition to that, we verify our findings us-ing full-wave simulations.

This study can be considered referential since it treatssuch a canonical object as a sphere. It makes an essen-tial initial step towards a general understanding of allkinds of scattering effects in time-varying structures. Ithas the crucial advantage that it can be used to study allkinds of effects considering a simple shaped object suchas a sphere in a short amount of time and with mini-mal computational resources. The understanding bornefrom these investigations provides the language to discuss

more elaborate systems that are no longer feasible for ananalytical treatment but require a numerical full-wavesimulation to capture all the details. In the past, theanalytical solution of the canonical problem of scatteringby a stationary sphere was one of the key cornerstonesin the development of the theory of light scattering, andwe hope that this extension of the theory to time-varyingcanonical scatterers will serve the same important pur-pose.

A further extension of this study can include a properdescription of absorption with the associated dispersion.Such a study would be crucial for providing insightsinto parametric amplification for realistic systems. Al-ternatively, this study can be further extended towardsnon-spherical geometries using, for instance, an analyt-ical solution for other simple structures, such as slabsor cylinders, or employing simulations for more compli-cated three-dimensional structures. In addition to that,one can consider arrays of time-varying particles.

ACKNOWLEDGMENTS

We acknowledge support by the German ResearchFoundation through Germany’s Excellence Strategy viathe Excellence Cluster 3D Matter Made to Order (EXC-2082/1 - 390761711). A. G. L. acknowledges supportfrom the Max Planck School of Photonics, which is sup-ported by BMBF, Max Planck Society, and FraunhoferSociety and from the Karlsruhe School of Optics andPhotonics (KSOP). T. K. acknowledges support from theAlexander von Humboldt Foundation through the Hum-boldt Research Fellowship for postdoctoral researchers.R. A. acknowledges support from the Alexander vonHumboldt Foundation through the Feodor Lynen (Re-turn) Research Fellowship. G. P. acknowledges partialfunding from the Academy of Finland (project 330260).V. A. and S. F. acknowledge the support of a MURIproject from the U.S. Air Force of Office of Scientific Re-search (Grant No FA9550-21-1-0244).

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[63] Let us note that at the limit of Tm → ∞, j0 → −∞,NΩ →∞, we get the general case of time-varying mediathat are not periodically modulated, where the discreteset of eigenvalues κ2

i (Ω) ends up being a continuum ofeigenvalues κ in the complex plane, and the correspond-ing eigenvectors ~S(κi)(Ω) end up being the original spec-tral eigenfunctions Sκ(ω).

[64] Note that the integral here over the Floquet frequency Ω

is defined as limω′m→ωm

∫ ω′m0

dΩ, but we keep it like this for

brevity.[65] Let us note that this model implies the assumption that

the excited electrons decay as time tends to infinity ac-cording to the electric polarizability kernel. Ref. 47 pro-poses another model for the response function of themedium that is given by the formula Re(t, t − τ) =αe(t − τ)N(t)/ε0. We would like to highlight that, eventhough there may be various ways to model the responsefunction of the time-varying medium depending on theparticular physical considerations that one would need toadopt, any kind of such phenomenological model can bedirectly embedded in an identical way inside the rest ofthe theoretical framework that is developed in this sec-tion.

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h(1)ν (−x) = (−1)νh

(2)ν (x).

[70] It should be noted that the eigenvalues that we calcu-late numerically are κ2

i (Ω). For this expansion of thefields, we select the principle branch of the square rootκi(Ω) = +

√κ2i (Ω). This κi(Ω) is the wavenumber that

we use in the regular VSHs of the expansion. The choiceof the branch of the square root here does not playa role, since for regular VSHs we have the symmetry

F(1)α,µν−κi(r) = (−1)νF

(1)α,µνκi(r), which follows from the

respective symmetry of spherical Bessel functions. There-fore, picking the other branch of the square root simplyleads to an equivalent representation.

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Appendix A: Vector Spherical Harmonics (VSHs)

In this section, we will provide the basic definitions ofthe VSHs that play a central role in our theoretical anal-ysis in the main text. We begin with defining the scalar

spherical harmonics, ψ(ι)µνκ(r), i.e., the spherical solutions

of the scalar monochromatic Helmholtz equation. Theirspatial representation in the spherical coordinate systemis given by the formula [60]:

ψ(ι)µνκ(r) = γµν z

(ι)M,ν (κr) Pµν (cosθ)eiµφ, (A1)

where κ is the wavenumber of the monochromatic waveequation, ν(ν + 1) is the eigenvalue of the squared totalangular momentum operator J2 with ν taking integervalues 1, 2, . . . and µ is the eigenvalue of the projection

of the total angular momentum operator along the z-

axis Jz taking integer values −ν, . . . , 0, . . . , ν. z(ι)M,ν (x)

denotes the spherical Bessel (ι = 1) and Hankel (ι =3) functions of the first kind of order ν. Pµν (x) are theassociated Legendre functions of the first kind and γµν =√

(2ν+1)(ν−µ)!4πν(ν+1)(ν+µ)! are normalization coefficients.

By following Ref. 60, we can construct a full-set ofdivergence-free vector spherical harmonics based on theabove scalar spherical harmonics. We will use the symbolα to denote the TE (α = M) and TM (α = N) eigen-waves. α can be associated with the eigenvalue of theparity operator, of which the VSHs are eigenstates. The

spatial representation of such VSHs, F(ι)α,µνκ(r), is given

by the formulas below [60, 68]:

F(ι)M,µνκ(r) , ∇×

[rψ(ι)

µνκ(r)]

= iz(ι)M,ν(κr)fM,µν(r), (A2a)

F(ι)N,µνκ(r) ,

1

κ∇× F

(ι)M,µνκ(r)

= rν(ν + 1)

κrψ(ι)µνκ(r) + z

(ι)N,ν(κr)fN,µν(r), (A2b)

where:

fM,µν(r) = γµν

[θτ (1)µν (θ) + iφτ (2)

µν (θ)]eiµφ, (A3a)

fN,µν(r) = γµν

[θτ (2)µν (θ) + iφτ (1)

µν (θ)]eiµφ, (A3b)

τ (1)µν (θ) = µ

Pµν (cosθ)

sinθ, (A3c)

τ (2)µν (θ) =

∂Pµν (cosθ)

∂θ, (A3d)

and z(ι)N,ν(x) is defined as:

z(ι)N,ν(x) =

1

x

∂

∂x[x z

(ι)M,ν(x)]. (A3e)

Appendix B: Expansion of the incident field in seriesof VSHs

In this section, we are discussing the coefficientsAincα,µν(ω) that expand the incident field in series of VSHs.

Explicit expressions for these coefficients, for the cases ofincoming focused beams or even single propagating planewaves can be found in Ref. 76. Moreover, the translationaddition theorem of the radiating VSHs [see Eq. (C.68)in Ref. 68] can provide expressions for the coefficients forthe case of multipolar emitters placed in the vicinity ofthe scatterer.

In this work, for the numerical demonstration of ouralgorithm, we have been considering an excitation by anx-polarized plane wave propagating along the z-axis andhaving a Gaussian pulse envelope with width T0 and car-rier frequency ω0. Its representation in time domain isgiven by the formula

Einc(r, t)=E0xe− (t−t0−z/c0)2

2T20 cos[ω0(t− t0 − z/c0)], (B1)

17

whereas its representation in the frequency domain isgiven by

Einc

(r, ω) = x eiωz/c0

× E0T0

2

[e−

T20 (ω0−ω)2

2 + e−T2

0 (ω0+ω)2

2

]eiωt0 , (B2)

where t0 is a time delay and c0 is the speed of light infree space. Expanding the plane wave xeiωz/c0 in a se-ries of VSHs around the origin of the coordinate system[see Eq. (19) here and Eq. (8) in Ref. 76], we finally getthe following expression for the incident spherical ampli-tudes:

Aincα,µν(ω) = E0T0π iν+1γ−1ν

[δµ,1 + (−1)δαN δµ,−1

][e−

T20 (ω0−ω)2

2 + e−T2

0 (ω0+ω)2

2

]eiωt0 , (B3)

where δij is the Kronecker delta. Let us note thatthe above equation holds true only for positive frequen-cies ω. For the negative frequencies we can make useof the following symmetry property that satisfies thecondition for real fields in time domain: Ainc

α,µν(−ω) =

(−1)ν+µ+δαN[Aincα,−µν(ω)

]∗.

Appendix C: The compromise effects of ignoringtemporal dispersion

In this section, we study the impact of ignoring tem-poral dispersion in our material model on the scatter-ing response from the spheres. The results are shown inFig. 6, where the same cases as previously studied areconsidered with the notable difference that the “Analyti-cal” results correspond here to a material model with notemporal dispersion. Specifically, the electric responsefunction of the time-varying medium, instead of beinggiven by Eq. (18), as it was the case in Fig. 5, here it isgiven by:

Re(ω − ω′, ω′) =1√2π

e2

meε0

N(ω − ω′)ω2n − ω2

0 − iγnω0. (C1)

This model considers a vanishing dispersion across allfrequencies, i.e. a material constant that takes the valueequal to what we had previously at the frequency ω0, thecentral frequency of the incident pulse. Also, let us clarifythat the “Numerical” results here are the same as thoseof Fig. 5, i.e., they correspond to full-wave simulationsthat account for temporal dispersion.

We can observe that disregarding temporal dispersionin our material model can lead to significant deviations.The deviations are more pronounced in the “strong dis-persion” case, where we excite our sphere within a spec-tral region where the time-varying medium is charac-terized by strong temporal dispersion. The importanceof accounting for material temporal dispersion is high-lighted by our results here.

18

FIG. 6. Same as Fig. 5 but with the “Analytical” results being based on a material model that ignores temporal dispersion.The effects of the compromise of ignoring temporal dispersion are clearly visible, especially in the ”strong dispersion” case.