Incandescent temporal metamaterials

2025-04-24 0 0 8.44MB 43 页 10玖币

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J. Enrique V´azquez-Lozano∗and I˜nigo Liberal†

Department of Electrical, Electronic and Communications Engineering,

Institute of Smart Cities (ISC), Universidad P´ublica de Navarra (UPNA), 31006 Pamplona, Spain

(Dated: October 12, 2022)

Regarded as a promising alternative to spatially shaping matter, time-varying media can be

seized to control and manipulate wave phenomena, including thermal radiation. Here, based

upon the framework of macroscopic quantum electrodynamics, we elaborate a comprehensive

quantum theoretical formulation that lies the basis for investigating thermal emission eﬀects in

time-modulated media. Our theory unveils new physics brought about by time-varying media:

nontrivial correlations between thermal ﬂuctuating currents at diﬀerent frequencies and positions,

thermal radiation overcoming the black-body spectrum, and quantum vacuum ampliﬁcation eﬀects

at ﬁnite temperature. We illustrate how these features lead to striking phenomena and novel

thermal emitters, speciﬁcally, showing that the time-modulation releases strong ﬁeld ﬂuctuations

conﬁned within epsilon-near-zero (ENZ) bodies, and that, in turn, it enables a narrowband

(partially coherent) emission spanning the whole range of wavevectors, from near to far-ﬁeld regimes.

On the basis of the latest scientiﬁc and technological

breakthroughs in nanophotonics and material science,

the development of metamaterials have brought forth

an ideal playground for engineering innovative forms of

light-matter interactions [1]. In the quest for reaching

an increasing control over wave phenomena, a new

burgeoning approach consists in harnessing time as an

additional degree of freedom to be exploited [2]. This

revival of time-varying media [3], has in turn boosted the

discovery of new physics associated with time-dependent

optical phenomena, ultimately giving rise to the emerging

ﬁeld of temporal metamaterials [4,5].

Another research area where structuring matter and

shaping their optical properties is attracting a great deal

of attention is the engineering of thermal emission [6–8].

As a basic mechanism of heat transfer [9], whereby

an incandescent object at ﬁnite temperature emits

(thermal) light [10], thermal radiation is of fundamental

interest. Likewise, it is also the basis of multiple

technological applications including heat and energy

management [11,12], light sources [13], sensing [14],

and communications [15]. In sharp contrast to the

behavior of (non-thermal) light emanating from coherent

sources, thermal light is characterized for displaying a

broadband spectrum and isotropic ﬁeld distribution, as

well as for being unpolarized and temporally incoherent.

Due to these properties, the control and manipulation

of thermal ﬁelds has long been (and continues to

be) a challenging issue. Signiﬁcant eﬀorts have been

made to explore and stretch out the physical limits of

thermal radiation (imposed by Planck’s [16,17] and

Kirchhoﬀ ’s radiation laws [18,19]) by looking into the

distinctive features occurring at the nanoscale [20–23].

Practical implementations have mainly been based on

metamaterials [24], metasurfaces [25], photonic

crystals [26], or subwavelength structures such as

gratings [27], to name a few. In this sense, besides

aﬀording a better far-ﬁeld thermal emission performance

(e.g., via the so-called thermal extraction schemes [28]),

nanophotonic engineering has stimulated the unveiling of

a plethora of novel near-ﬁeld thermal eﬀects [29]. Akin

to customary coherent optical sources, the near-ﬁeld

radiation of thermal emitters greatly diﬀers from that in

the far-ﬁeld regime [30–32]. This is essentially due to the

existence of evanescent modes, which are dominant in

the near-ﬁeld and negligible in the far-ﬁeld [33]. Apart

from modifying the spectral distribution, the evanescent

contribution gives access to additional channels over the

frequency-wavevector (ω-k) space, thus strengthening

thermal ﬁelds by several orders of magnitude [27,30–32].

Just like in the ﬁeld of nanophotonics, it seems

natural to think that passing from spatially structured to

time-modulated materials could revolutionize the ﬁeld of

thermal engineering [see Fig. 1]. A clear example comes

through the grating structures [27], whose temporal

analog could similarly open new opportunities [34].

Moreover, the temporal dimension owns in itself

some fundamental attributes tied to the principle of

causality [35]. Nonetheless, the topic of thermal emission

in time-varying media is at a very incipient stage [36–38]

and the underlying physics is not fully understood as yet.

Upon this ground, here we put forward a quantum

formalism to address near- and far-ﬁeld thermal emission

in time-varying media. Noteworthily, this theoretical

formulation would also be extensible to purely quantum

phenomena (e.g., Casimir forces [39] and vacuum

ampliﬁcation eﬀects [40–42]) at ﬁnite temperature. Our

formalism allows us to unveil new thermal physics

associated with time-modulated materials, including

ﬂuctuating currents with non-local (space and frequency)

correlations, and far-ﬁeld thermal emission beyond

the black-body spectrum. In turn, these properties

lead to novel thermal phenomena empowered by the

time-modulation, such as the releasing of ﬂuctuations

trapped within a material body, or the narrowband

near-to-far ﬁeld thermal linking.

arXiv:2210.05565v1 [physics.optics] 11 Oct 2022

Ideal black body

aSpatially structured material

bTime-modulated material

enhanced control

Thermal emission

Photon frequency

Thermal emission

Photon frequency

Thermal emission

Photon frequency

time

non-local correlations

beyond black body

dyn. vacuum effects

narrowband spectra

directional emission

polarized propagation

broadband spectra

omnidirectional fields

unpolarized radiation

new physics

FIG. 1. Breakthroughs in thermal emission. a, According to Planck’s law, the broadband emission spectrum (tied

to isotropic and unpolarized thermal radiation) of a black body in thermal equilibrium only depends on the temperature.

b, Thermal emission can be controlled by structuring the matter, enabling narrowband, directive, and polarized radiation.

c, Modulating temporally the optical properties of a medium enables more sophisticated ways to obtain similar eﬀects, yielding

the emergence of new physics, such as non-local correlations, overcoming the black-body spectrum, or dynamical vacuum eﬀects.

A SEMICLASSICAL APPROACH TO THERMAL

RADIATION FROM FLUCTUATING CURRENTS

Theoretical modeling of thermal emission is

typically carried out within the framework of

ﬂuctuational electrodynamics [43]. According to

this semiclassical approach, the emergence of thermal

radiation emanating from a body at temperature

Tcan be understood as a result of the radiation

emitted by the ﬂuctuating electromagnetic (EM)

currents (mathematically characterized by means

of the current density correlations) [see Fig. 2].

The theoretical cornerstone of this formalism is the

ﬂuctuation-dissipation theorem (FDT) [44–46], which

provides with a relationship between the correlations

of ﬂuctuating systems and the linear response of the

system associated to its dissipative features:

hj∗(ρ;ω)·j(ρ0;ω0)ith =4πε0ε00(ρ, ω)~ω2

Θ(ω, T )δ[ω−ω0]δ[ρ−ρ0],(1)

where the brackets h···ith denote a thermal ensemble

average, Θ(ω, T )=[e~ω/(kBT)−1]−1, and ε(r, ω) =

ε0(r, ω)+iε00(r, ω) is the lossy and dispersive permittivity

of the body. A very important feature of Eq. (1) is that

thermally ﬂuctuating currents at diﬀerent frequencies

and positions are uncorrelated, describing the stochastic

nature of thermal ﬁelds.

Once the ﬂuctuating currents are known, the spectral

energy density, S(r;ω) = hE∗(r;ω)·E(r;ω)ith, at a

given position, r, and frequency, ω, can be directly

found from the connection between ﬁelds and currents

E(r;ω) = iωµ0Rd3ρG(r,ρ, ω)·j(ρ;ω) via the dyadic

Green’s function of the body, G(r,ρ, ω) [see Fig. 2].

Fluctuational electrodynamics is a very successful

theory that has made possible breakthrough advances in

engineering thermal ﬁelds with photonic nanostructures.

At the same time, it is a semiclassical theory that does

not allow for the simultaneous modeling of quantum

vacuum and thermal ﬂuctuations, particularly in their

interaction with dynamical systems. In addition, its

extension and applicability to time-varying media is not

rigorously justiﬁed [36,37]. In the following, from a

ﬁrst-principles approach, we introduce a full-quantum

formalism to thermal emission in time-varying media,

which enables the calculation of quantum and thermally

ﬂuctuating current correlations, and thermal ﬁelds,

without the need of any additional assumptions beyond

those implicitly set in the Hamiltonian of the system.

FIG. 2. Thermal emission from EM ﬂuctuations.

Schematic representation of the ﬂuctuating density currents

moving inside a hot body. The solid red arrows represent

two particular electric currents which are locally correlated,

releasing so, via the corresponding Green’s function (dashed

red arrows), the emission of thermal ﬁelds out the material.

A QUANTUM APPROACH TO THERMAL

EMISSION IN TIME-VARYING MEDIA

To establish a general theoretical framework

for extending the study of thermal emission and

the behavior of ﬂuctuating thermal currents and

ﬁelds in time-varying media, we make use of

macroscopic quantum electrodynamics [47,48].

Within this framework, instead of bare photons,

one actually deals with elementary excitations,

namely, EM ﬁeld-matter coupled states modeled

by a continuum of harmonic oscillators [49]. These

modes are described by polaritonic operators,

f(r, ωf;t), which, in the Heisenberg picture,

obey the equal-time commutation relations:

[ˆ

f(r, ωf;t),ˆ

f(r0, ω0

f;t)] = [ˆ

f†(r, ωf;t),ˆ

f†(r0, ω0

f;t)] = 0,

and [ˆ

f(r, ωf;t),ˆ

f†(r0, ω0

f;t)] = ˆ

Iδ[r−r0]δ[ωf−ω0

f],

where ˆ

Iis the identity operator.

To describe the dynamical behavior of the time-varying

quantum system we assume a perturbative approach with

a Hamiltonian given by ˆ

H=ˆ

H0+ˆ

HT[41,47–50],

where ˆ

H0represents the macroscopic body without

time-modulation,

H0=Zd3rZ+∞

dωf~ωfˆ

f†(r, ωf;t)·ˆ

f(r, ωf;t),(2)

while ˆ

HTaccounts for the perturbation describing the

changes induced by the time-modulation,

HT=−Zd3rˆ

P(r;t)·ˆ

E(r;t).(3)

Here, the polarization ﬁeld operator, given by ˆ

P(r;t)≡

0dτ∆χ(r, t, τ )ˆ

E(r;τ), is tied to the time-varying

susceptibility of the medium ∆χ(r, t, τ ) [41], and the

electric ﬁeld operator ˆ

E(r;t) = ˆ

E(+)(r;t) + ˆ

E(−)(r;t),

whose positive-frequency component reads as,

E(+)(r;t) = Z∞

dωfZd3ρGE(r,ρ, ωf)·ˆ

f(ρ, ωf;t),(4)

with GE(r,ρ, ωf)= ip~ε00(ρ, ωf)/πε0(ωf/c)2G(r,ρ, ωf)

being the response function characterizing the

background medium, G(r,ρ, ωf) the dyadic Green’s

function for the unmodulated system, and noticing

that ˆ

E(−)(r;t)=[ˆ

E(+)(r;t)]†. Similar perturbation

Hamiltonians are adopted for modeling other nonlinear

quantum processes [51,52]. Moreover, it is implicitly

assumed that the susceptibility function, ∆χ, is small

enough so that it could be regarded as a perturbation

to the background structure, and does not signiﬁcantly

aﬀect to the quantization procedure [52].

The emission spectrum, both in the far- and

near-ﬁelds, is given by [53–57]

S(r;ω) = hˆ

E(+)(r;ω)†·ˆ

E(+)(r;ω)ith ,(5)

where ˆ

E(+)(r;ω) = Lω[ˆ

E(r, t)] is the Laplace’s

transform of the electric ﬁeld operator. By solving

the Heisenberg equations of motion for the polaritonic

operators, i~∂tˆ

O= [ ˆ

O,ˆ

H], performing an integral in the

complex frequency plane, and rearranging the terms [50],

E(+)(r;ω) can be compactly written as follows:

E(+)(r;ω) = iωµ0Zd3ρG(r,ρ, ω)·ˆ

j(ρ;ω),(6)

with the current density operator,

j(ρ;ω)=2ωhpπ~ε0ε00(ρ, ω)ˆ

f0−iLω[∆˜χ(ρ, t)ˆ

E(ρ;t)]i,(7)

where ˆ

f0≡ˆ

f(r, ω;t= 0). The ﬁrst term in

Eq. (7) corresponds to the currents associated with

a system without time-modulation, while the second

term represents the currents excited due to the

time-modulation of the permittivity. In order to

obtain Eq. (7), we have conducted a sharp but routine

assumption, whereby the time-varying susceptibility

exhibits a modulation which actually is local in time,

i.e., ∆χ(r, t, τ) = ∆ ˜χ(r, t)δ[t−τ] [3,4,41,58,59].

This simpliﬁes the mathematical treatment, and allows

us to use the aforementioned equal-time commutation

relationships. Despite this particularization, it should

be noted that the formalism is completely general, and

a time-modulation with an arbitrary form would be

feasible with the proper adoption of time-dependent

commutation relations [47].

Equations (5)–(7) provide a quantum framework

for the computation of thermal emission spectra that

is conceptually similar to the semiclassical treatment

sketched above: ﬂuctuating EM currents give rise to

ﬂuctuating thermal ﬁelds by means of the corresponding

propagator (the dyadic Green’s function) through the

entire medium. Notwithstanding, the use of a quantum

formulation generalizes the semiclassical treatment,

enabling the evaluation of thermal currents and ﬁelds in

time-varying media, the calculation of purely quantum

phenomena such as vacuum ampliﬁcation eﬀects, and,

ultimately, a sound, self-consistent, and systematic

formulation that directly arises from the assumptions on

the Hamiltonian of the system, without the need of any

semiclassical additions to the theory.

FLUCTUATING CURRENTS IN TIME-VARYING

MACROSCOPIC BODIES

Next, we use this formalism to obtain a general form

of the ﬂuctuating currents excited in a macroscopic

body whose permittivity is modulated in time, providing

an extension to usual forms justiﬁed through the

ﬂuctuation-dissipation theorem. To this end, we ﬁrst

note that Eq. (7) is an implicit equation, where the

current density operator is deﬁned as a function of

TABLE I. Fluctuation-dissipation theorem for time-varying systems: First and second-order current density correlations.

hˆ

j†

0(ρ;ω)·ˆ

j1(ρ0;ω0)ith =ω0hµ0

πiZV

d3ρ00 Zdω00ω00∆ ˜χ(ρ0, ω0−ω00)G(ρ0,ρ00, ω00)hˆ

j†

0(ρ;ω)·ˆ

j0(ρ00;ω00)ith;

hˆ

j†

0(ρ;ω)·ˆ

j2(ρ0;ω0)ith =ω0hµ0

πiZV

d3ρ00 Zdω00ω00∆ ˜χ(ρ0, ω0−ω00)G(ρ0,ρ00, ω00)hˆ

j†

0(ρ;ω)·ˆ

j1(ρ00;ω00)ith

hˆ

j†

1(ρ;ω)·ˆ

j1(ρ0;ω0)ith =ωω0hµ0

πi2ZZV

d3˜

ρd3˜

ρ0ZZ d˜ωd˜ω0˜ω˜ω0∆ ˜χ∗(ρ, ω −˜ω)∆˜χ(ρ0, ω0−˜ω0)

hG∗(ρ,˜

ρ,˜ω)G(ρ0,˜

ρ0,˜ω0)hˆ

j†

0(˜

ρ; ˜ω)·ˆ

j0(˜

ρ0; ˜ω0)ith +G(ρ,˜

ρ,˜ω)G∗(ρ0,˜

ρ0,˜ω0)hˆ

j0(˜

ρ; ˜ω)·ˆ

j†

0(˜

ρ0; ˜ω0)ithi

the electric ﬁeld operator, which is itself generated

by the current density operator. This fact makes a

clear signature of the sharply intertwined dynamic of

the system. At any rate, such an equation may be

solved iteratively leading to a solution in the form of a

series expansion of current density operators at diﬀerent

orders [50]:

j(r;ω) =

∞

n=0

jn(r;ω).(8)

Accordingly, the ﬁrst three elements of the series can

be explicitly written as [50]:

j0=ωp4π~ε0ε00(r, ω)ˆ

f(r, ω;t= 0); (9a)

j1∝Zd3ρ0

Zdω0ω0∆˜χ(r, ω−ω0)G(r,ρ0, ω0)ˆ

j0(ρ0;ω0)

+Zd3ρ0

Zdω0ω0∆˜χ(r, ω−ω0)G∗(r,ρ0, ω0)ˆ

j†

0(ρ0;ω0); (9b)

j2∝Zd3ρ0

Zdω0ω0∆˜χ(r, ω−ω0)G(r,ρ0, ω0)ˆ

j1(ρ0;ω0).(9c)

These expressions for the current density operators have

a clear physical meaning: the current density operators

of successive orders result from the ﬁelds generated

by the preceding ones. Speciﬁcally, a current density

operator of order nat position ρ0and frequency ω0,

i.e., ˆ

jn(ρ0;ω0), generates a ﬁeld at position rvia

the propagation of the Green’s function G(r,ρ0, ω0).

Roughly speaking, at such a position, the action of the

ﬁeld over the time-varying susceptibility, ∆ ˜χ(r, ω −ω0),

generates a higher-order current at frequency ω, i.e.,

jn+1(r, ω). As we will show, the interplay between

sources of diﬀerent order result in nontrivial correlations

between ﬂuctuating currents at diﬀerent frequencies and

points of space. Moreover, a crucial aspect of the

current density operators is that they mix creation, ˆ

j†

and annihilation, ˆ

jn, polaritonic operators, akin to

Bogoliubov (or squeezing) transformations [60]. The

diﬀerence between creation and annihilation operators,

not present in semiclassical treatments, allows for the

prediction of dynamical vacuum eﬀects.

Subsequently, the correlation between ﬂuctuating

currents at diﬀerent frequencies and points of space can

also be written in a series form:

hˆ

j†(r;ω)·ˆ

j(r0;ω0)ith =X

l,m hˆ

j†

l(r;ω)·ˆ

jm(r0;ω0)ith.(10)

With this in mind, the corresponding nth-order

correlations can be obtained from a direct evaluation

of hˆ

j†

l(r;ω)·ˆ

jm(r0;ω0)ith, where l+m=n,

and h···ith ≡Tr[··· ˆρth], with ˆρth being the

thermal density operator that yields the thermal

ﬁelds at a given temperature T[47,48,52].

As expected, the zeroth-order correlation of the current

density, hˆ

j†

0·ˆ

j0ith, coincides with the original version of

the FDT given in Eq. (1). In other words, our quantum

formalism correctly recovers the semiclassical case for a

steady (non-time-modulated) system. At the same time,

it generalizes this result via the higher-order correlations.

Indeed, proceeding iteratively, one can ﬁnd closed form

expressions for such higher-order contributions to the

current density correlations [50]. The corresponding

results for the ﬁrst and second-order contributions are

presented in the Table I.

Comparing this result with the original form of

the FDT for stationary systems, one may realize

that, even truncating the expansion at the second

order, time-varying media bring new physics to

ﬂuctuational electrodynamics. First feature concerns

to the breakdown of locality, since higher-order

ﬂuctuating currents are correlated at diﬀerent

frequencies and position of space. For conventional

(non-time-modulated) thermal emitters, the correlations

are local both in position and frequency. This becomes

evident at a glance from the involvement of the Dirac

delta functions [see Eq. (1)], and is physically understood

as a consequence of the random nature of the thermal

ﬁelds. However, higher-order terms include integrals

over frequencies and positions. In this manner, the

time-modulation enables the possibility of correlating (or

connecting) diﬀerent frequencies appearing at diﬀerent

locations of the material system, thus underscoring its

non-local character and, consequently, the potential to

enhance the coherence of thermal ﬁelds.

There is an additional feature that aﬀect to the

photon distribution. Indeed, in the hˆ

j†

1·ˆ

j1ith term,

the black-body spectrum, Θ(ω, T ), appears inside

a frequency integral. For conventional thermal

emitters, the spectrum of thermal radiation is given by

Ireal(ω, T ) = α(ω)IBB(ω, T ), where α(ω) = (ω)≤1

is the spectral absorptivity (related to the emissivity

by the Kirchhoﬀ’s radiation law), and IBB ∝Θ(ω, T ),

refers to the black-body emission spectrum. Herein,

Θ(ω, T ) acts as a ﬁxed frequency window that

ultimately sets an upper limit for the radiative heat

transfer. Thus, customary spatial-like nanophotonic

engineering of emission spectra have so far been limited

to the control of the optical absorptivity (or the

emissivity) of materials [61,62], within the limits

imposed by the black-body spectrum [see Fig. 1(b)].

By contrast, our analysis reveals that time-modulated

thermal emitters can aﬀect to the black-body’s photon

distribution, thus making tunable the accessible window

of frequencies [see Fig. 1(c)]. This suggests an additional

and unprecedented way to engineer the density of states.

Another crucial aspect of the hˆ

j†

1·ˆ

j1ith term, is

that it contains anti-normally ordered correlations

hˆ

j0·ˆ

j†

0ith [63,64], indicating so the eventual occurrence

of vacuum ampliﬁcations eﬀects [42]. In fact, this term

is the dominant contribution in the zero-temperature

limit (T→0). In this sense, going beyond

a semiclassical approach, our quantum formalism

would allow for unifying quantum photon production

eﬀects (such as the dynamical Casimir eﬀect [40,41],

parametric ampliﬁcation, ...) and thermal emission

processes. In turn, it paves the way to study vacuum

ampliﬁcation eﬀects at a ﬁnite temperature, which might

be required for the analysis of realistic experimental

conﬁgurations. Finally, our quantum formalism for

ﬂuctuating currents sets the basis for calculating

quantum vacuum forces in time-varying media, such as

Casimir forces [39] and quantum friction [65–70].

NEW THERMAL EMISSION EFFECTS IN

TIME-VARYING MEDIA

Thus far we have sketched out the theoretical model

of thermally ﬂuctuating currents and their correlations

in time-varying media. Next, we illustrate some of

the main consequences of such a time-modulation in

connection with the emergence of new thermal emission

eﬀects. To this end, we revisit a historical example that

helped initiating the ﬁeld of nanophotonic engineering

of thermal emission [30–32]. It consists of a silicon

carbide (SiC) substrate (z < 0), in contact with

vacuum (z > 0) [see Fig. 3]. The frequency-dependent

permittivity of SiC is described by a Drude-Lorentz

model, so that ε(ω) = ε∞(ω2

L−ω2−iγω)/(ω2

T−ω2−iγω),

where ε∞= 6.7, ωL= 29.1 THz, ωT= 23.8 THz,

and γ= 0.14 THz [31], stand, respectively, for the

high-frequency-limit permittivity, the longitudinal and

Thermal emission

SiC

SPhPs

Fluctuating

currents

Vacuum

FIG. 3. Thermal emission from a semi-inﬁnite planar

slab of SiC subjected to a harmonic time-modulation.

Schematic depiction of a semi-inﬁnite slab of SiC at

temperature Twith a random distribution of ﬂuctuating

currents moving inside. The horizontal axis represents the

time, so that each section displays a diﬀerent susceptibility.

transverse optical phonon frequencies, and the damping

factor (or characteristic collision frequency). Because

of such frequency dispersion, a SiC substrate supports

nontrivial far and near-ﬁeld thermal ﬂuctuations,

including the thermal excitation of surface phonon

polaritons. Here, instead on introducing a grating to

outcouple near-ﬁeld thermal waves [27], we consider

a time-harmonic modulation of the susceptibility:

∆˜χ(r, t) = ε0δχ sin Ωt. As we will show, it leads to new

thermal wave phenomena, associated with the release

of strong epsilon-near-zero (ENZ) thermal ﬂuctuations

trapped within the material body.

Evaluating the dyadic Green’s functions

One of the major technical diﬃculties in evaluating

thermal emission from a time-modulated system lies

in computing the product of multiple dyadic Green’s

functions, which represent the interaction between

ﬂuctuating currents and the radiated ﬁelds. Due to

the translational symmetry of the proposed system,

one can take advantage of an angular spectrum

representation [33,55], whereby the dyadic Green’s

function is expressed as a superposition of plane waves:

G(r,ρ, ω) = k2

4π2RRVd2κkˆ

G(κx, κy;ω|z, ρz)eik0κkρk,

where k0=ω/c,κk= (κx, κy,0), and ρk=

(ρx, ρy,0). This formalism provides with valuable

physical intuition by separating the modes into

propagating and evanescent, from the character of

their wavevector. Indeed, for each half-space, ki=

k0(κx, κy,pεi(ω)κz,i), so kz,i =k0˜

kz,i =k0pεi(ω)κz,i,

where κz,i =p1−κ2

R/εi(ω), and κ2

R=κ2

x+κ2

y[32].

Thus, for lossless media (i.e., those where εi(ω) is real),

it is possible to set the usual correspondence of κ2

R≤εi

and κ2

R> εi, with propagating and evanescent modes.

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IncandescenttemporalmetamaterialsJ.EnriqueVazquez-LozanoandI~nigoLiberalyDepartmentofElectrical,ElectronicandCommunicationsEngineering,InstituteofSmartCities(ISC),UniversidadPublicadeNavarra(UPNA),31006Pamplona,Spain(Dated:October12,2022)Regardedasapromisingalternativetospatiallyshapingmatter,tim...

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Incandescent temporal metamaterials

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