Abstract

We study the problem of a temporal discontinuity in the permittivity of an unbounded medium with Lorentzian dispersion. More specifically, we tackle the situation in which a monochromatic plane wave forward-traveling in a (generally lossy) Lorentzian-like medium scatters from the temporal interface that results from an instantaneous and homogeneous abrupt temporal change in its plasma frequency (while keeping its resonance frequency constant). In order to achieve momentum preservation across the temporal discontinuity, we show how, unlike in the well-known problem of a nondispersive discontinuity, the second-order nature of the dielectric function now gives rise to two shifted frequencies. As a consequence, whereas in the nondispersive scenario the continuity of the electric displacement D and the magnetic induction B suffices to find the amplitude of the new forward and backward wave, we now need two extra temporal boundary conditions. That is, two forward and two backward plane waves are now instantaneously generated in response to a forward-only plane wave. We also include a transmission-line equivalent with lumped circuit elements that describes the dispersive time-discontinuous scenario under consideration.

© 2021 Chinese Laser Press

1. INTRODUCTION

In the past few years, time-variant metamaterials/metasurfaces have become a hot research topic within the photonics community, given their potential to boost the degree of manipulation of light–matter interactions achieved by their time-invariant predecessors. The latter, through the subwavelength space modulation of the electric and/or magnetic response [1], allow for alluring possibilities in the way light is controlled and thus enable a vast range of interesting phenomena and promising applications from strengthened nonlinearities [2] and ϵ-near-zero (ENZ) propagation [3,4] to artificial Faraday rotation [5] and optically driven topological states [6]. On the other hand, an externally induced time modulation in some of the properties of these engineered structures largely broadens the degree of harnessing of light manipulation, in which case we have a time-varying metamaterial. This spatiotemporal modulation is the supporting platform of such fascinating effects as magnetless nonreciprocity [7] or time reversal [8], just to name a few. In this regard, the research on active metasurfaces has gained a lot of momentum in the past few years [912].

One avenue to induce this temporal variation is the time modulation of a medium’s dielectric function, e.g., electro-optically. In Ref. [13], a nonstationary interface was reported from plasma ionization by a high-power electromagnetic pulse. The problem of wave propagation in an unbounded medium with a rapid change—and, to a lesser extent, a slab with sinusoidal time variation—in its constitutive parameters was first theoretically studied in Ref. [14] for the case of nondispersive permittivity and/or permeability. These nondispersive step transients, further explored in Refs. [15,16], effectively produce a “time interface”: based on the continuity of D and B, an instantaneous frequency shift occurs to accommodate the new permittivity while preserving the wave momentum, and a forward wave and a backward wave arise whose amplitudes are quantified by what can be seen as the temporal dual Fresnel coefficients. These step-like discontinuities were later analyzed, e.g.,  in a half-space [17] and a dielectric layer [18]. Moreover, Refs. [19,20] addressed the adiabatic frequency conversion of optical pulses going through slabs with arbitrarily time-varying refractive index, while Refs. [21,22] reported wave solutions for a smooth or arbitrary transition of the refractive index, respectively. Wave propagation undergoing periodic temporal inhomogeneities of the permittivity has also been investigated in a half-space [23], a slab [2429], or a space-time-periodic (traveling-wave modulation) medium [3033]: time-periodic variations exhibit frequency-periodic band-structured dispersion relations that include wave vector gaps [26], dual bandgaps of space-periodic media. As shown in Refs. [24,25,29,3437], this time-Floquet modulation can be harnessed to achieve parametric amplifiers.

Nonetheless, most of the aforementioned works consider nondispersive susceptibilities only (excepting Refs. [17,18], where a plasma is parameterized with a nonstationary electron density, and Ref. [23], where the time-varying parameter is conductivity). In Ref. [38], on the contrary, closed-form Green’s functions are obtained for pulsed excitations within spatially homogeneous media with abrupt or gradual temporal changes, either without dispersion or considering a cold ionized lossless plasma described with Drude dispersion. Very recently, the question of time-varying dispersion has been studied from different angles, namely, a transmission line [39] and a meta-atom [40] with time-modulated reactive loads, and the analysis of the instantaneous radiation of nonharmonic dipole moments [41] and nonstationary Drude–Lorentz polarizabilities [42].

In the present work, we assume an initial plane wave at t<0 with frequency ω and bring in the effects of Lorentzian dispersion when considering a step-like change in the plasma frequency ωp with otherwise constant resonance frequency ω0. Unlike in Refs. [1416], this abrupt change gives rise to two shifted frequencies (in the simplified lossless case, a lower frequency ω1<ω0 and an upper frequency ω2>ω0) that bear the following interpretation when ω0 is considerably larger than ω: while ω1 reflects in essence the change in permittivity similarly to the nondispersive case, ω2 characterizes a wave of a different nature, viz. one that has a negligible magnetic component; the medium at ω2 thus possesses ENZ characteristics.

We begin by defining in Section 2 the differential equation describing the Lorentzian-like dielectric response characterizing our time-varying medium to further derive the initial conditions across the temporal change in ωp at t=0. This transition is perceived as abruptly varying the volumetric density of ω0-resonating dipoles N=N(t), our control parameter. As a starting point, we mainly look into the case where this number changes from zero to a specified value N+. From the differential equation relating the polarization vector P to the electric field E, we show that E, P, and dPdt are all continuous across the temporal discontinuity at t=0. In Section 3 we use preservation of momentum to analytically find ω1 and ω2 and also give a detailed numerical account for the evolution of the frequency split over time when the transition is gradual rather than abrupt. A dynamic analysis toward a full-wave solution is developed in Section 4 for a lossless scenario. The approach is first based, in Section 4.A, on the scattering-parameter model from Ref. [16]. It is further substantiated—and confirmed—in Section 4.B by a Laplace-transform-based first-principles solution to the amplitudes for the forward and backward propagating constituents at ω1 and ω2; this comprehensive development also recovers ω1 and ω2. Furthermore, we developed a finite-difference time-domain (FDTD) [43] solver whose simulation results perfectly agree with our analytical predictions. In Section 5, we show how this one-dimensional spatial problem may be likened to a transmission line equivalent that is relatively simple to use. Further phenomena related to losses are described in Section 6. Finally, conclusions are drawn in Section 7.

2. TIME-VARYING LORENTZIAN DISPERSION: INITIAL CONDITIONS

Let us consider, for t<0, an x-polarized electric field plane wave traveling in the +z direction and oscillating at a purely real frequency ω in an unbounded dispersive medium (for simplicity, we will assume it lossless for now) whose electric polarization charge P responds to the electric field E following a susceptibility χ that can be described in the frequency domain by a Lorentzian resonance centered at ω0, such that

P(ω)=ϵ0χ(ω)E(ω)=ϵ0ωp2ω02ω2E(ω),
where ωp=qeNϵ0me is the plasma frequency, and N is the volumetric density of polarizable atoms, with me and qe the electron’s mass and charge, respectively. In the time domain, this relation adopts the form of the following second-order differential equation:
d2P(t)dt2+ω02P(t)=ϵ0ωp2E(t),
which can be also written as the convolution P(t)=ϵ0χ(t)*tE(t), where χ(t)=ωp2ω0sin(ω0t)U(t) is the system’s impulse response, U(t) is the step function, and *t denotes the linear time-invariant (LTI) convolution operation with respect to t.

Now, let us allow N—and thus ωp—to be time dependent and consider a scenario where it abruptly changes—instantaneously and homogeneously—at t=0 as N(t)=N+(N+N)U(t), with N and N+, some arbitrary positive constants. After defining A(t)=ωp2(t), Eq. (2) becomes

d2P(t)dt2+ω02P(t)=ϵ0A(t)E(t),
where we have indicated that the resonance frequency ω0 is time invariant. Moreover, despite our linear system now being time variant (LTV), one can still use the convolution operator and write P(t)=ϵ0χn(t)*t[A(t)E(t)] [44], where we have defined the normalized impulse response χn(t)=1ω0sin(ω0t)U(t), or, in the frequency domain,
P(ω)=ϵ012πA(ω)*ωE(ω)ω02ω2,
with χn(ω)=1ω02ω2. In short, the dielectric response to an impulse applied at time τ is only a function of N(t=τ) and not of N(t>τ): intuitively, any new dipoles brought into the medium after the electric-field impulse at t=τ simply have no excitation to respond to; mathematically, this can be traced back to the invariance of the coefficients in the left-hand side of Eq. (3) and gives us one key piece of information: regardless of the step-function discontinuity in N(t), P(t) is continuous [note that χn(t=0)=0], and so is dP(t)dt (only a spike in E would determine otherwise). In the more general framework of LTV systems, the response observed at time t due to an impulse at time τ can in this case be recast as h(t,τ)=ϵ0A(τ)χn(tτ), which allows us to write
P(t)=th(t,τ)E(τ)dτ.

Importantly, the depicted situation differs from the model assumed in Ref. [42], where h(t,τ)=ϵ0A(t)χn(tτ). Formally, our continuity of both P(t) and dPdt across t=0 can be substantiated as follows. Applying the one-sided Laplace transform L{f(t)}=f˜(s), defined over the temporal interval t:[0,), to Eq. (3), and solving for P˜(s), we have

P˜(s)=ϵ0L{A(t)E(t)}+sP(0)+dPdt(0)s2+ω02,
where, e.g.,  P(0) stands for P(t=0). A direct application of the initial value theorem (IVT) [45]
P(0+)=limssP˜(s)
provides the continuity condition for P:
P(0+)=P(0).

Similarly, for dPdt,

dPdt(0+)=lims[s2P˜(s)sP(0+)].

However, by virtue of Eq. (8) and substituting Eq. (6) with the understanding that limsL{A(t)E(t)}=0, we find that dPdt is continuous as well:

dPdt(0+)=lims[s2P˜(s)sP(0)]=dPdt(0).

Finally, the Laplace-domain polarization emerges when N=0 as

P˜(s)=ϵ0L{A(t)E(t)}s2+ω02,
which immediately connects with Eq. (4).

3. KINEMATICS: PRESERVATION OF MOMENTUM

The existence of dispersion does not change the fact that, as dictated by electromagnetic momentum conservation, the new waves arising after the temporal boundary must be shifted in frequency with respect to ω, as shown in Refs. [1416] for a nondispersive scenario. Our initial wave oscillating at ω has a wavenumber k(ω) so, after the temporal jump, the supported new frequencies ω+ will be those that satisfy the equality k(ω)=k+(ω+). This leads, when there is no magnetic response, to the transcendental equation ωcϵ(ω)=ω+cϵ+(ω+), which in our lossless case can be written, when ϵ=1 and thus the relative dielectric permittivity ϵ(ω)=1+χ(ω), as

ω1+ωp2ω02ω2=ω+1+ωp+2ω02ω+2.

Squaring both sides of Eq. (12) leads to a quartic polynomial equation in ω+ whose four roots determine the new frequencies for t>0:

ω+=±K±K24ω02ω2(ω2ω02)(ω2ω02ωp2)2(ω2ω02),
for which we will denote ±ω1 and ±ω2, with
K=ω2(ω2+ωp+2ωp2)ω02(ω02+ωp+2).

For definiteness, we will choose K+ for ±ω1 and K for ±ω2 such that ω1<ω2 (ω2<ω1) when |ω|<ω0 (|ω|<ω02+ωp2): note that only in the interval ω0<|ω|<ω02+ωp2 of anomalous dispersion, which we will not address and for which ϵ<0, do we get complex solutions—more precisely, purely real (imaginary) ω1 (ω2). Specializing Eq. (12) to the case ωp=0 (the medium is vacuum for t<0), we have the following characteristic equation for ω+:

ω+4(ω2+ω02+ωp+2)ω+2+ω02ω2=0,
and Eq. (13) reduces to
ω+=±ω2+ω02+ωp+2±(ω2+ω02+ωp+2)24ω02ω22.

In order to illustrate how the frequencies evolve from ω to (ω1,ω2), let us for a moment assume that N(t)=N+(N+N)[1+tanh(Rt)]/2, with R some constant describing the transition rate [in the limit R, we have [1+tanh(Rt)]/2U(t)]. In Fig. 1(a), we consider a transition from vacuum (N=0) to a Lorentzian-like medium with N+ chosen such that ε+(ω)=4, and with ω0=2ω. As soon as N+>0, ω splits into the pair (ω1,ω2)=(ω,ω0). This is understood once we make ωp=0 in Eq. (12) and take the limit ωp+0: in addition to the trivial solution ϵ1(ω1=ω)=1, we also have ϵ2(ω2=ω0)=(ωω0)2, which in this case is equal to 0.25. Physically, this is just the manifestation of the natural frequency ω0 of the newly added oscillators, which may surface depending on the boundary conditions.

 figure: Fig. 1.

Fig. 1. (a) Temporal evolution of ωl and ϵl (l=1,2) as ωp(t) transitions from ωp=0 (vacuum) to ωp+ such that ϵ+(ω)=4 with ω0=2ω, resulting in ωp+=3ω. (b) Dispersion diagram k+ versus ω+, showing the two solutions for ω+ that achieve momentum conservation. The dashed green line represents ω0.

Download Full Size | PPT Slide | PDF

Now, we could think of “quantizing” tanh(Rt) and consider the entire continuous transition NN+ as a succession of infinitesimal step-function-like temporal discontinuities. By doing this, we next go on by applying Eq. (12) twice in our second temporal jump, with both ω itself and ω0 as the input frequencies: it turns out that ω1(t) and ω2(t) are interrelated such that they both give rise to the same pair of output frequencies, so, notably, there is a pair (ω1(t),ω2(t)) [blue and red solid lines in Fig. 1(a)]. This interrelation shows up in that ω1(t)ω2(t)=ϵωω0 or, alternatively—from the mentioned transcendental equation ωl(t)=ϵϵl(t)ω, with l=1,2 and ϵl=Δϵ+(ωl)ϵ1(t)ϵ2(t)=ϵ(ωω0)2, allowing us to further write ω1(t)=ϵ2(t)ω0 and ω2(t)=ϵ1(t)ω0. Of course, by making R, our original ω is instantaneously split into the final values of (ω1,ω2), whereas making R finite alters the dynamics of the problem: we have a transient and thus the amplitudes of the final forward and backward waves will be different. In addition, Fig. 1(b) shows the graphical match of momentum from the dispersion diagram of our Lorentzian when the blue solid line Re(k+) crosses the dashed black line k.

4. DYNAMICS: PLANE WAVE(S) IN A TIME-VARYING LORENTZIAN MEDIUM

A. Temporal-Interface Scattering Coefficients

In order to find the electromagnetic fields after the temporal discontinuity at t=0, we need to solve the wave equation subject to the temporal boundary conditions (BCs), including those stated at the end of Section 2. One can find in the literature [1416] that, in a nondispersive medium, it suffices to consider temporal continuity for both D and B, which ensures that magnetic and electric fields H and E remain bounded, respectively: D(z,t=0+)=D(z,t=0) and B(z,t=0+)=B(z,t=0). This latter condition obviously becomes H(z,t=0+)=H(z,t=0) when magnetism is not present. In our case these two still apply, but two extra BCs are needed to determine the amplitudes of the forward and backward waves for both frequencies (ω1 and ω2): we can now use the fact—remarked upon in Section 2—that P(0+)=P(0) and dP(0+)dt=P(0)dt, where, e.g.,  P(0±) stands for P(z,t=0±) to reduce notation. Importantly, the joint continuities of D and P lead to the continuity of E: these three conditions are linearly dependent, so we choose to discard P.

If we adopt the eiωt time-harmonic convention and use k=k=k+, our initial forward waves can be written as [note that, in order to simplify notation, E stands for E(z,t<0)], e.g.,  

E=eiωteikz,
H=ϵη0eiωteikz,
D=ϵ0ϵeiωteikz,
dPdt=iωϵ0(ϵ1)eiωteikz.

Let us now see the complex space-time harmonic dependencies from a different perspective and adopt the space-harmonic complex dependence eikz, in which case forward and backward waves will be described by eiωt and eiωt, respectively. For t>0, the fields can be expressed as

E+=eikzl=12(fleiωlt+bleiωlt),
H+=eikz1η0l=12ϵl(fleiωltbleiωlt),
D+=eikzϵ0l=12ϵl(fleiωlt+bleiωlt),
dP+dt=eikziϵ0l=12ωl(ϵl1)(fleiωltbleiωlt),
where the unknowns fl and bl represent the amplitudes of the forward and backward electric field waves oscillating at frequency ωl. Enforcing the time continuity of these four waves at t=0 leads—after some straightforward simplifications, replacing ωl=ϵϵlω, and using the BC for H to simplify the BC for dPdt—to the following system of equations:
[1111ϵ1ϵ1ϵ2ϵ2ϵ1ϵ1ϵ2ϵ21ϵ11ϵ11ϵ21ϵ2][f1b1f2b2]=[1ϵϵ1ϵ],
which gives us the closed-form solution to the unknown amplitudes:
[f1,b1]=ϵ2ϵ2ϵ(ϵ2ϵ1)(ϵ±ϵ1),
[f2,b2]=ϵϵ12ϵ(ϵ2ϵ1)(ϵ±ϵ2),
where + () gives the forward fl (backward bl) amplitude. A set of analogous equations expressed only in terms of frequencies can be found in Appendix A.

In Fig. 2(a) we show the temporal evolution of the electromagnetic waves at z=λ/16 around the temporal jump (at t=0, indicated with black dashed lines) that results from Eqs. (12)–(20) when we consider the transition of Fig. 1 (the results obtained from FDTD simulations—marked with circles—when N(t) follows the previously mentioned tanh(Rt) profile perfectly converge to these results as we make R larger. Here we use R=105/T, with T=2πω).

 figure: Fig. 2.

Fig. 2. (a) Electromagnetic waves versus time at z=λ/16 for a transition from ϵ(ω)=1 (vacuum) to ϵ+(ω)=4, with ω0=2ω (the solid lines are analytical results, while the circular markers represent numerical FDTD simulations). (b1) New frequencies ωi [and ϵ+(ωi)] for t>0 and (b2) wave amplitude coefficients versus ω0/ω, considering ωp+=3ω [ϵ+(ω)1 as ω0/ω]. Panels (c1), (c2) are the same, but with ϵ+(ω)=4 [ωp+χ+(ω)ω0 as ω0/ω].

Download Full Size | PPT Slide | PDF

Although less practical from a mathematical standpoint than the unilateral Laplace transform (see Appendix D) in this case, perhaps taking the Fourier transform (FT{}) of E(t) over the whole time interval <t<+ helps reveal the transient nature of our discontinuity. Taking z=0 for simplicity, from Eqs. (17a) and (18a), the spectrum of E(z=0,t) becomes

E(z=0,ω)=FT{E(z=0,t)U(t)+E+(z=0,t)U(t)}=iωωl=12(iflωωl+iblω+ωl),
which shows nonzero spectral content over the entire <ω<+ range, commensurate to the fact that an abrupt change of the medium’s properties gives rise to operational frequencies that extend to infinity.

1. Approximations for ω0ω

Now, let us ask ourselves what happens when ω0 increases, in which case we have to consider two different scenarios. In Figs. 2(b1) and 2(b2), we keep ωp+ fixed with the value that makes ϵ+(ω)=4 when ω0=2ω and increase the ratio ω0/ω: as this ratio tends to , we have ϵ1(ω1ω)1 and ϵ2(ω2ω0)0 [Fig. 2(b1)], which makes f11 [see Fig. 2(b2)]. Noting that l(fl+bl)=0, this means the initial plane wave is not altered by the temporal discontinuity, as one would expect from the fact that, given that the new medium is effectively transparent at ω, no transfer of energy should take place between ω and ω0.

On the contrary, in Figs. 2(c1) and 2(c2) we consider that ωp+ varies such that ϵ+(ω)=4 regardless of ω0/ω [see constant black dashed line in Fig. 2(c1)]: by making ω0/ω, we now have ωp+ and

ϵ1[ω1ϵϵ+(ω)ω]ϵ+(ω),
ϵ2[ω2ϵ+(ω)ω0]ϵϵ+(ω)(ωω0)20,
which transform Eq. (20a) into [f1,b1]=ϵ2ϵ1(ϵ±ϵ1), i.e., the exact same expressions of the nondispersive scenario [1416]. Nonetheless, we now also have f2=b2=ϵ1ϵ2ϵ10 [in this precise example f2=b2=f1, as depicted in Fig. 2(c2)]: one thus has to wonder how to connect this solution including oscillations at ω2 with the nondispersive situation and first realize that the new medium is effectively ENZ [3,4]. Substituting ϵ20 into Eq. (18) we see that H+(ω2)0 [and D+(ω2)0], making the Poynting vector at ω2 tend to zero and all of the power purely reactive. We must recognize, however, that as soon as we allow for some infinitesimally small loss (see Section 6), as required by our Lorentzian in order to become physical, ω2 becomes purely imaginary and its components immediately vanish (more details can be found in Appendix C). Noting that, when there is no dispersion, E and P are discontinuous across the temporal boundary—which entails a change of electromagnetic energy density—our suddenly vanishing ω2 components are nothing but the dispersive manifestation of this behavior.

B. First-Principles Approach: Use of Laplace Transform

From Maxwell’s equations with a general polarization vector,

×E=μ0Ht,
×H=ϵ0Et+Pt+J,
one can derive the pertinent wave equation
××E=μ0ϵ02Et2μ02Pt2μ0Jt.

Transforming into the Laplace domain, taking into account Eqs. (8) and (10) and restricting ourselves to ωp=0,

1μ0××E˜(r,s)=ϵ0[s2E˜(r,s)sE(r,0)E(r,0)t][s2P˜(r,s)+sJ˜(r,s)J(r,0)].

Now combine Eq. (25) with the constitutive relation Eq. (11) to obtain

1μ0××E˜(r,s)+ϵ0s2(s2+ω02+ωp+2)s2+ω02E˜(r,s)=ϵ0[sE(r,0)+E(r,0)t]sJ˜(r,s)+J(r,0).

Let us take the one dimensional reduction of Eq. (24) with E=x^E(z,t). In view of preservation of momentum we take k=k=ωμ0ε0 throughout. Also, =ik, so from

(k2+μ0ϵ02t2)E=μ02Pt2μ0Jt,
Eq. (26) becomes
ϵ0(ω2+s2s2+ω02+ωp+2s2+ω02)E˜(z,s)=ϵ0[sE(z,0)+Et(z,0)]sJ˜(z,s)+J(z,0),
or
E˜(z,s)=(s2+ω02)ϵ0[sE(z,0)+Et(z,0)]sJ˜(z,s)+J(z,0)ϵ0(ω2s2+ω2ω02+s4+s2ω02+s2ωp+2).

The denominator of Eq. (29) can be factored as

s4+(ω2+ω02+ωp+2)s2+ω2ω02=(s2s12)(s2s22)=(s2+ω12)(s2+ω22)
with sl=±iωl. Note the agreement with the kinematic characteristic Eq. (15).

For t<0, the electric field is given as E(z,t<0)=cos(ωtkz). At the time t=0,

E(z,t=0)=cos(kz),
Et(z,t=0)=ωsin(kz).

We are now able to rewrite Eq. (29) in the form

E˜(z,s)=(s2+ω02)sE(z,0)+Et(z,0)(s2+ω12)(s2+ω22)(s2+ω02)sJ˜(z,s)J(z,0)ϵ0(s2+ω12)(s2+ω22).

An inverse transform of Eq. (32) for the source-free case yields

E(z,t)=E1++E1+E2++E2,
where
E1±=ω02ω12ω22ω1212(1±ωω1)cos(ω1tkz),
E2±=ω02ω22ω22ω1212(1±ωω2)cos(ω2tkz),
which are the same exact expressions that result from keeping the real part of Eq. (18a), with [fl,bl] from Eq. (20) [or, more directly, Eq. (A2)] simplified through ϵ=1. Under the approximations of Eq. (22), the latter results simplify to
E1±1±ϵ22ϵ2cos(ω1tkz),
E2±ϵ212ϵ2(1±1ω0ωϵ2)cos(ω2tkz).

A corresponding approximation for the magnetic field is then

H1±±1±ϵ22ϵ21η0cos(ω1tkz),H2±±ϵ212ϵ2(1±ωω0ϵ2)ωω0ϵ21η0cos(ω2tkz)0.

5. TRANSMISSION-LINE MODEL

The time-varying Lorentzian response described in Eq. (3) can be viewed as the (polarization) charge response to an applied voltage across a series time-varying LC circuit and thus rewritten as

L(t)d2P(t)dt2+1C(t)P(t)=E(t),
with distributed shunt inductance L(t)=1ϵ0ωp2(t) and per-unit-length series capacitance C(t)=1ω02L(t)=ϵ0[ωp(t)ω0]2, in units of [H/m] and [F/m], respectively {note that L(t) is dimensionally different from the per-unit-length series inductance that models μ0, in [H/m]. A shunt inductance can, e.g.,  characterize a thin aperture in the transverse wall of a waveguide [46]. Distributed (or lumped) series capacitors and shunt inductors can also model negative permeabilities and permittivities, respectively [47,48]}. Two important facts must be pointed out here: (i) ω0 is kept constant, and (ii) there is no dL(t)dtdP(t)dt term. Accordingly, considering that, for t<0, our dispersive medium is modeled as a transmission line with an LC branch, we can think of our sudden change in ωp(t) as the connection of a new LpCp branch in parallel as shown in Fig. 3, with Lp=LL+LL+ and Cp=1ω02Lp=C+C (note that both Lp and Cp will be non-Foster when ωp+<ωp, and note also that disconnecting the LC branch will turn the medium into vacuum. These aspects will be discussed in our future study); when ωp=0 (vacuum), L= and C=0, and thus Lp=L+ and Cp=C+. Now the inductor Lp forbids a discontinuity in Pp that would generate a spike of polarization current dPpdt across the new branch: Pp(0+)=Pp(0)=0 [vCp(0+)=vCp(0)=0]. Besides, there can be no discontinuity in the magnetic flux linkage Φp=LpdPpdt—the fact that we use the term “magnetic” should not give rise to confusion: we are using inductors to model the dispersive behavior of the dielectric function, but there is no magnetism involved; more specifically, in the picture of a mass-spring oscillator, the inductor represents mass and is therefore related to (mechanical) momentum and kinetic energy, whereas the capacitor models the spring constant and is related to potential energy. A further argument is that, unlike ΦL0, ΦL and ΦLp are related to dHdz, not to H, and hence vL and vLp are related to E, not to dEdz—that would lead to a spike in vLp:dPp(0+)dt=dPp(0)dt=0. Therefore, we finally have d2Pp(0+)dt2=E(0+)Lp=E(0)Lp. Toward the end of Section 2, continuity conditions for P and dPdt were derived from a functional-analysis point of view of our LTI system’s response; as it is clear that the voltages and currents across the LC branch are also continuous, we have now arrived, from a circuital perspective, to the same continuity conditions.

6. LOSSY CASE

If we introduce loss into our time-varying Lorentzian medium, Eqs. (3) and (37) must be extended as

d2P(t)dt2+γdP(t)dt+ω02P(t)=ϵ0ωp2(t)E(t),
L(t)d2P(t)dt2+R(t)dP(t)dt+1C(t)P(t)=E(t),
with resistivity R(t)=γL(t), in [Ω·m]. For conciseness, we will not write down here the lengthy expressions of the complex frequencies that enforce k+=k [Fig. 4(a) shows the complex-frequency dispersion diagram for ϵ+(ω)=40.1i with ω0=2ω] according to
ω1+ωp2ω02ω2+iωγ=ω+1+ωp+2ω02ω+2+iω+γ,
but it is worth pointing out that three different scenarios open up. We will now restrict the discussion to the particular case where ωp=0 (further insights will be presented in an upcoming study). If, starting from γ=0, we gradually increase loss, a positive imaginary part—note that, given that we are adopting the eiωt convention, Im(ω+)>0 represents frequencies that are damped—begins to show up in the two pairs of solutions from Eq. (13) [this is seen in Fig. 4(b), depicting the variation of these complex frequencies with γ/ω], so we have two distinct pairs of complex frequencies (complex conjugate pairs in the Laplace transform s plane): ±ωlr+iωli, with ωlr and ωli real and positive (ωlf=+ωlr+iωli and ωlb=ωlr+iωli will therefore describe forward- and backward-propagating evanescent waves, respectively). Each pair can then be seen as the two characteristic roots of the natural response of some underdamped resistor-inductor-capacitor (RLC) oscillator, and the forward and backward waves for frequency l will have the form eωlitcos(ωlrt±kz+φ), φ being a phase term. If we define the s-plane frequencies sl=iωlrωli, the electromagnetic waves for t>0 can be described as
E+=eikzl=12(fleslt+blesl*t),
H+=eikz1η0l=12(ϵlflesltϵl*blesl*t),
D+=eikzϵ0l=12(ϵlfleslt+ϵl*blesl*t),
dP+dt=eikzϵ0l=12(slχlfleslt+sl*χl*blesl*t),
and thereby the unknown amplitudes can be calculated as
[1111ϵ1ϵ1*ϵ2ϵ2*ϵ1ϵ1*ϵ2ϵ2*s1χ1s1*χ1*s2χ2s2*χ2*][f1b1f2b2]=[1ϵϵsχ],
where s=iω. This character of the waves, decaying with t but not with z, is clearly seen in Figs. 5(b) and 5(c), which depicts the underdamped scenario associated with γ=0.5ω.
 figure: Fig. 3.

Fig. 3. Transmission-line equivalence of our unbounded time-varying dispersive medium. At t=0, the switch is closed, effectively connecting our series tank circuit LpCp.

Download Full Size | PPT Slide | PDF

 figure: Fig. 4.

Fig. 4. (a) Dispersion diagram in the complex k- and ω-planes when ϵ+(ω)=40.1i, which results in ωp+3ω and γ=0.1ω when ω0=2ω. Conservation of momentum is achieved when the surfaces [ω+,Re[k+(ω+)]/k] and [ω+,Im[k+(ω+)]/k] simultaneously intersect the Re(k+)/k=1 and Im(k+)/k=0 planes (gray color), respectively. These intersection curves are marked in black and give rise to four complex frequencies that, in this example, form two complex-conjugate pairs in the s plane [the plotted region Re(ω+)>0 only includes one complex frequency per pair]. (b) Evolution of the four complex frequencies versus γ, with the other parameters fixed [ω0 and ωp+ from panel (a)]: at γ=7.01ω we reach a critical point, and the pair of complex frequencies linked to ω0 (ω2f and ω2b) splits into two purely imaginary frequencies for which ϵ becomes purely real and negative; the latter is seen in panel (c). Note that, in this overdamped region (γ>7.01ω), the notation ω2f and ω2b is not strictly rigorous: this pair simply becomes ω2 and ω3, and the associated waves represent non-propagating damping.

Download Full Size | PPT Slide | PDF

 figure: Fig. 5.

Fig. 5. (a) Electromagnetic waves versus time at z=λ/16 when ω0 and ωp+ are taken from Fig. 4 and γ=0.5ω (numerical FDTD simulations are marked with circles): we have ω2f and ω2b, corresponding to an underdamped scenario. (b) Separate components of E(z=0,t>0). (c) Two snapshots of the separate components of E+ [the solid and dashed lines represent E(z,t=0+) and E(z,t=T/32), respectively], showing forward and backward propagation for both ω1 and ω2.

Download Full Size | PPT Slide | PDF

If we keep increasing γ, we will reach a critical point (γ=7.01ω in Fig. 4) at which the second pair of complex frequencies collapses into the same purely imaginary frequency +iω2i, so one can think of this pair as the two equal characteristic roots of some critically damped RLC oscillator. Propagation for +iω2i is obviously forbidden, with ϵ2 purely real and negative [see Fig. 4(c)], and the waves will have the form eω2itcos(kz+φ). Also, assuming ω<ω0, in general we have |ϵ2|1. Further, if γ is increased beyond the point of critical damping, +iω2i is split into two different purely imaginary frequencies, as corresponds to an overdamped RLC oscillator, which we will denote ω2 and ω3 (the retrieval of the temporal-interface scattering coefficients is described in Appendix B). This time-decaying non-propagating nature associated with ω2 and ω3 is illustrated in the overdamped scenario of Fig. 6 (γ=7.3ω); see red and green plots in Figs. 6(b) and 6(c). Finally, Fig. 7 depicts the evolution of the scattering coefficients with γ/ω and how f2+b2 (x2+x3 after the critical point) remains bounded, despite these coefficients separately diverging.

 figure: Fig. 6.

Fig. 6. Same as Fig. 5 but with γ=7.3ω, yielding an overdamped regime with purely imaginary ω2 and ω3 describing oscillations that do not propagate. This is seen in panel (c): red (E2) and green (E3) curves.

Download Full Size | PPT Slide | PDF

 figure: Fig. 7.

Fig. 7. Real and imaginary parts of the complex amplitude coefficients versus γ/ω. As ω2fω2b near the point of critical damping γ=7.01ω, f2 and b2 tend to diverge but with opposite signs [panel (b)], keeping f2+b2 (or else x2+x3) bounded.

Download Full Size | PPT Slide | PDF

Incidentally, only when ωp=0 do we have s1 and s1* (and s2 and s2* in the underdamped case). In general, for ωp<ωp+, the characteristic roots s1f and s1b will approximately, but not exactly, form a complex conjugate pair. As a consequence, ϵ+*(is1f)ϵ+(is1b), meaning that forward and backward waves do not propagate in the very same medium.

7. CONCLUSION

We investigate the “reflection/transmission” of a monochromatic plane wave at a dispersive temporal boundary, substantiated as a step-like change in the plasma frequency of a Lorentz-type dielectric function, and we present a transmission-line equivalent modeling this transition. The fact that two frequencies rather than one, each with forward and backward propagating constituents, are instantaneously generated after the transition is in line with the second-order nature of the dispersion in the medium. When we omit loss, we can still connect this behavior with the well-known dispersionless case and show how, as ω0/ω increases, the lower frequency ω1 tends to the dispersionless solution, whereas the upper frequency ω2, linked to ω0, presents a markedly different phenomenon: not only does the medium acquire ENZ character at ω2, but also the forward and backward waves’ amplitudes tend to converge, effectively constituting a standing wave along z which, in the limit of negligible loss, almost instantaneously fades out. Importantly, one can see from the mathematics developed that the described analogy, exemplified in this work for a transition from free space, also holds for the inverse transition to free space, or any other transition for that matter. In an upcoming study, the issue of power storage/conveyance and conversion will be addressed in depth, but it is already evident from the above discussion that, in the ω0/ω limit, no power propagates at ω2.

APPENDIX A: SCATTERING COEFFICIENTS FOR THE LOSSLESS SCENARIO IN TERMS OF ω

We can substitute ϵl=ωωlϵ in Eq. (19) to arrive at a set of equations expressed only in terms of frequencies:

[1111ωω1ωω1ωω2ωω2(ωω1)2(ωω1)2(ωω2)2(ωω2)2ω1ωω1ωω2ωω2ω][f1b1f2b2]=[1111].

Note that, as the elements of the right-hand side are all equal, this system is perfectly conditioned for numerical solving. The expressions for the unknown amplitudes in Eq. (20) thus have the alternative form

[f1,b1]=12ω22ω2ω22ω12ω1ω2(ω1±ω),
[f2,b2]=12ω2ω12ω22ω12ω2ω2(ω2±ω).

APPENDIX B: SCATTERING COEFFICIENTS IN A LOSSY OVERDAMPED SCENARIO

Given that we now have purely imaginary ω2 and ω3, which describe no propagation, the coefficients f2 and b2 are replaced with x2 and x3. The matrix system of equations becomes

[1111ϵ1ϵ1*ϵ2ϵ3ϵ1ϵ1*ϵ2ϵ3s1χ1s1*χ1*s2χ2s3χ3][f1b1x2x3]=[1ϵϵsχ].

APPENDIX C: ADDING A SMALL LOSS WHEN ω0

We saw in Section 4.A in the main text [see Figs. 2(c1) and 2(c2)] how, for a given prescribed value of (lossless) ϵ+(ω), taking the limit ω0/ω leads to a situation that is equivalent to the well-known problem of a temporal interface in a nondispersive medium, except for the fact that we now have additional forward and backward oscillations at ω2—for which the medium is ENZ (ϵ20)—with nonzero amplitudes f2=b2=ϵ1ϵ2ϵ1. We also stated how adding an infinitesimally small amount of loss would lead to instantaneously vanishing ω2 components, thereby drawing an exact correspondence with the nondispersive scenario. Let us see this behavior in more detail with the numerical example of Fig. 8.

 figure: Fig. 8.

Fig. 8. (a1) Normalized real and imaginary parts of the complex frequencies ω2f (ω2) and ω2b (ω3) versus ω0χi/ω, for χi=103 (lines) and χi=104 (markers). (a2) Normalized dielectric functions ϵ2f (ϵ2) and ϵ2b (ϵ3). (b) Complex amplitude coefficients f2 (x2) and b2 (x3). (c) E2f (E3) and E2b (E3) versus normalized time, for z=0 and several ω0/ω ratios; note how the black and red plots, corresponding to the underdamped region where f2=b2, are superimposed.

Download Full Size | PPT Slide | PDF

Figure 8(a1) shows the real and imaginary parts of the complex frequencies ω2f and ω2b, which form a complex-conjugate pair in the s plane when ω0/ω is smaller than the point of critical damping (see Section 6). Beyond the critical point, this pair becomes purely imaginary: denoting χ+(ω) by χriχi, it can be shown that, as ω0/ω, we have γχiχrω02ω and ω3iγi, together with ω2i(χr+1)χrχiω when, additionally, χi0 (ω3 and ω2 replace ω2b and ω2f, respectively, in the overdamped region). Consequently, ϵ3 and ϵ2, purely real and negative, behave in the limit as ϵ30 and ϵ2[χi(χr+1)χr]2ϵ [see Fig. 8(a1)]. Finally, we have x30 and x2f2+b2=ϵ1ϵϵ1 [Fig. 8(b)].

For a given ω0/ω ratio, ω3 (ω2) is directly (inversely) proportional to χi, which means, in principle, that the oscillation will die out faster (slower) as we increase loss. However, x30 in the limit ω0/ω, so all we care about is ω2, which is indeed bounded by i(χr+1)χrχiω. That is, the larger the loss, the slower the non-oscillatory damping, which is perfectly consistent with intuition: we need loss to be infinitesimally small in order to make non-oscillatory damping instantaneous right after the temporal discontinuity; this is better understood by noting that, in our circuital analogy, L0 in the limit ω0/ω, so the RC time constant dictates the decay rate [note that this is the opposite of the underdamped regime, where the damped oscillation from the pair (ω2f,ω2b) will die out faster as we increase loss]. This is illustrated in Fig. 8(c), where the electric fields decay one order of magnitude faster for χi=104 (markers) than for χi=103 (lines).

Moreover, note that the normalized frequencies ω2fχi/ω and ω2bχi/ω do not depend on χi when plotted versus ω0χi/ω—as shown in Fig. 8(a1), where lines and markers represent different values of χi—very much like the normalized dielectric functions in Fig. 8(a2) and the amplitude coefficients in Fig. 8(b). Interestingly, we know from Section 6 that, at the critical point, both f2 and b2 diverge, though with bounded f2+b2: not only do we now observe this behavior, but also f2+b2 remains constant [see magenta and orange plots in Fig. 8(b)].

Finally, in Fig. 9 we show how, in the limit of χi0 and ω0, the resulting waves, though continuous, converge to the well-known solution of a nondispersive medium undergoing a step-like change in its dielectric function [1416], with discontinuous E and P [black and red lines in Fig. 9(a), respectively].

 figure: Fig. 9.

Fig. 9. (a) Electromagnetic waves versus time at z=λ/16 for the transition from vacuum to ϵ+(ω)=4iχi, considering ω0=104ω and several values of χi specified in panel (b); all the curves are virtually the same and overlap with the chosen t-axis scale, converging to the solution of the nondispersive lossless case. (b) Zoomed-in view of the transition of panel (a), both for underdamped (solid lines) and overdamped (dashed lines) scenarios.

Download Full Size | PPT Slide | PDF

APPENDIX D: SATISFACTION OF PARSEVAL’S THEOREM

We herein show how one can still find a form of the Parseval–Plancherel theorem [49,50]—also known as Rayleigh’s energy theorem [51]—that is satisfied by our infinite-energy double-sided signals. Assuming a lossless Lorentzian, E+(t) in Eq. (17a) will have infinite energy, and yet we can consider some positive real number σ such that eσtE+(t) is Lebesgue square-integrable [52]: eσtE+(t)L2(0,). Direct application of Parseval’s theorem for finite-energy signals results in

0e2σt|E+(t)|2dt=12π|FT{eσtE+(t)U(t)}(ω)|2dω=12π|Lr{E+(t)}(σ+iω)|2dω,
with Lr{} the unilateral (right-sided) Laplace transform. Similar considerations allow us to write, for the left-sided signal E,
0e2σt|E(t)|2dt=12π|FT{eσtE(t)U(t)}(ω)|2dω=12π|Ll{E(t)}(σ+iω)|2dω.

Finally, we can write, for our double-sided signal E(t)=EU(t)+E+U(t), the energy equality

e2σ|t||E(t)|2dt=12π|FT{e|σ|tE(t)}(ω)|2dω=12π|Ll{E(t)}(σ+iω)+Lr{E(t)}(σ+iω)|2dω,
where (Ll+Lr){E(t)}(s) represents the bilateral Laplace transform of E(t), whose region of convergence (ROC) is given in this case by |Re(s)|<σ.

Funding

Office of Naval Research (N00014-16-1-2029).

Disclosures

The authors declare no conflicts of interest.

Data Availability

Data that support the findings of this study are available in the main text and the appendices. Additional information is available from the corresponding author upon reasonable request.

REFERENCES

1. N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations (Wiley, 2006).

2. T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009). [CrossRef]  

3. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97, 157403 (2006). [CrossRef]  

4. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008). [CrossRef]  

5. T. Kodera, D. L. Sounas, and C. Caloz, “Artificial faraday rotation using a ring metamaterial structure without static magnetic field,” Appl. Phys. Lett. 99, 031114 (2011). [CrossRef]  

6. M. A. Gorlach, X. Ni, D. A. Smirnova, D. Korobkin, D. Zhirihin, A. P. Slobozhanyuk, P. A. Belov, A. Alù, and A. B. Khanikaev, “Far-field probing of leaky topological states in all-dielectric metasurfaces,” Nat. Commun. 9, 909 (2018). [CrossRef]  

7. Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009). [CrossRef]  

8. V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016). [CrossRef]  

9. A. M. Shaltout, V. M. Shalaev, and M. L. Brongersma, “Spatiotemporal light control with active metasurfaces,” Science 364, eaat3100 (2019). [CrossRef]  

10. K. Chen, Y. Feng, F. Monticone, J. Zhao, B. Zhu, T. Jiang, L. Zhang, Y. Kim, X. Ding, S. Zhang, A. Alù, and C.-W. Qiu, “A reconfigurable active Huygens’ metalens,” Adv. Mater. 29, 1606422 (2017). [CrossRef]  

11. K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018). [CrossRef]  

12. L. Zhang, X. Q. Chen, R. W. Shao, J. Y. Dai, Q. Cheng, G. Castaldi, V. Galdi, and T. J. Cui, “Breaking reciprocity with space-time-coding digital metasurfaces,” Adv. Mater. 31, 1904069 (2019). [CrossRef]  

13. E. Yablonovitch, “Self-phase modulation of light in a laser-breakdown plasma,” Phys. Rev. Lett. 32, 1101–1104 (1974). [CrossRef]  

14. F. R. Morgenthaler, “Velocity modulation of electromagnetic waves,” IRE Trans. Microw. Theory Tech. 6, 167–172 (1958). [CrossRef]  

15. Y. Xiao, D. N. Maywar, and G. P. Agrawal, “Reflection and transmission of electromagnetic waves at a temporal boundary,” Opt. Lett. 39, 574–577 (2014). [CrossRef]  

16. C. Caloz and Z. Deck-Léger, “Spacetime metamaterials-part ii: theory and applications,” IEEE Trans. Antennas Propag. 68, 1583–1598 (2020). [CrossRef]  

17. R. L. Fante, “Transmission of electromagnetic waves into time-varying media,” IEEE Trans. Antennas Propag. 19, 417–424 (1971). [CrossRef]  

18. R. L. Fante, “On the propagation of electromagnetic waves through a time-varying dielectric layer,” Appl. Sci. Res. 27, 341–354 (1973). [CrossRef]  

19. Y. Xiao, G. P. Agrawal, and D. N. Maywar, “Spectral and temporal changes of optical pulses propagating through time-varying linear media,” Opt. Lett. 36, 505–507 (2011). [CrossRef]  

20. Y. Xiao, D. N. Maywar, and G. P. Agrawal, “Optical pulse propagation in dynamic Fabry–Perot resonators,” J. Opt. Soc. Am. B 28, 1685–1692 (2011). [CrossRef]  

21. A. G. Hayrapetyan, J. B. Gãtte, K. K. Grigoryan, S. Fritzsche, and R. G. Petrosyan, “Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media,” J. Quant. Spectrosc. Radiat. Transfer 178, 158–166 (2016). [CrossRef]  

22. M. Chegnizadeh, K. Mehrany, and M. Memarian, “General solution to wave propagation in media undergoing arbitrary transient or periodic temporal variations of permittivity,” J. Opt. Soc. Am. B 35, 2923–2932 (2018). [CrossRef]  

23. F. A. Harfoush and A. Taflove, “Scattering of electromagnetic waves by a material half-space with a time-varying conductivity,” IEEE Trans. Antennas Propag. 39, 898–906 (1991). [CrossRef]  

24. D. Holberg and K. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14, 183–194 (1966). [CrossRef]  

25. D. E. Holberg and K. S. Kunz, “Parametric properties of dielectric slabs with large permittivity modulation,” Radio Sci. 3, 273–286 (1968). [CrossRef]  

26. J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009). [CrossRef]  

27. J. R. Zurita-Sánchez, J. H. Abundis-Patiño, and P. Halevi, “Pulse propagation through a slab with time-periodic dielectric function ε(t),” Opt. Express 20, 5586–5600 (2012). [CrossRef]  

28. J. S. Martnez-Romero and P. Halevi, “Standing waves with infinite group velocity in a temporally periodic medium,” Phys. Rev. A 96, 063831 (2017). [CrossRef]  

29. J. S. Martnez-Romero and P. Halevi, “Parametric resonances in a temporal photonic crystal slab,” Phys. Rev. A 98, 053852 (2018). [CrossRef]  

30. J. Simon, “Action of a progressive disturbance on a guided electromagnetic wave,” IRE Trans. Microw. Theory Tech. 8, 18–29 (1960). [CrossRef]  

31. A. A. Oliner and A. Hessel, “Wave propagation in a medium with a progressive sinusoidal disturbance,” IRE Trans. Microw. Theory Tech. 9, 337–343 (1961). [CrossRef]  

32. R. S. Chu and T. Tamir, “Wave propagation and dispersion in space-time periodic media,” Proc. Inst. Electr. Eng. 119, 797–806 (1972). [CrossRef]  

33. R. L. Fante, “Optical propagation in space–time-modulated media using many-space-scale perturbation theory,” J. Opt. Soc. Am. 62, 1052–1060 (1972). [CrossRef]  

34. T. T. Koutserimpas and R. Fleury, “Electromagnetic waves in a time periodic medium with step-varying refractive index,” IEEE Trans. Antennas Propag. 66, 5300–5307 (2018). [CrossRef]  

35. T. T. Koutserimpas, A. Alù, and R. Fleury, “Parametric amplification and bidirectional invisibility in PT-symmetric time-Floquet systems,” Phys. Rev. A 97, 013839 (2018). [CrossRef]  

36. T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-Hermitian time-Floquet systems,” Phys. Rev. Lett. 120, 087401 (2018). [CrossRef]  

37. T. T. Koutserimpas and R. Fleury, “Electromagnetic fields in a time-varying medium: exceptional points and operator symmetries,” IEEE Trans. Antennas Propag. 68, 6717–6724 (2020). [CrossRef]  

38. L. Felsen and G. Whitman, “Wave propagation in time-varying media,” IEEE Trans. Antennas Propag. 18, 242–253 (1970). [CrossRef]  

39. M. Mirmoosa, G. Ptitcyn, V. Asadchy, and S. Tretyakov, “Time-varying reactive elements for extreme accumulation of electromagnetic energy,” Phys. Rev. Appl. 11, 014024 (2019). [CrossRef]  

40. G. Ptitcyn, M. S. Mirmoosa, and S. A. Tretyakov, “Time-modulated meta-atoms,” Phys. Rev. Res. 1, 023014 (2019). [CrossRef]  

41. M. S. Mirmoosa, G. A. Ptitcyn, R. Fleury, and S. A. Tretyakov, “Instantaneous radiation from time-varying electric and magnetic dipoles,” Phys. Rev. A 102, 013503 (2020). [CrossRef]  

42. M. S. Mirmoosa, T. T. Koutserimpas, G. A. Ptitcyn, S. A. Tretyakov, and R. Fleury, “Dipole polarizability of time-varying particles,” arXiv:2002.12297 (2020).

43. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

44. D. M. Sols and N. Engheta, “Functional analysis of the polarization response in linear time-varying media: a generalization of the Kramers-Kronig relations,” Phys. Rev. B 103, 144303 (2021). [CrossRef]  

45. A. V. Oppenheim, A. S. Willsky, and H. Nawab, Signals and Systems (Pearson Education, 1996).

46. D. M. Pozar, Microwave Engineering (Wiley, 2011).

47. G. Eleftheriades, A. Iyer, and P. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microw. Theory Tech. 50, 2702–2712 (2002). [CrossRef]  

48. N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005). [CrossRef]  

49. M.-A. Parseval, “Mémoire sur les séries et sur l’intégration complète d’une équation aux différences partielles linéaires du second ordre, à coefficients constants,” Mém. prés. par divers savants, Acad. des Sciences, Paris 1, 638–648 (1806).

50. M. Plancherel and M. Leffler, “Contribution à l’étude de la représentation d’une fonction arbitraire par des intégrales définies,” Rendiconti del Circolo Matematico di Palermo (1884-1940) 30, 289–335 (1910). [CrossRef]  

51. L. Rayleigh, “LIII. On the character of the complete radiation at a given temperature,” London, Edinburgh, Dublin Philos. Mag. J. Sci. 27, 460–469 (1889). [CrossRef]  

52. H. Lebesgue, “Intégrale, longueur, aire,” Ann. Mat. Pura Appl. (1898-1922) 7, 231–359 (1902). [CrossRef]  

References

  • View by:

  1. N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations (Wiley, 2006).
  2. T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
    [Crossref]
  3. M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97, 157403 (2006).
    [Crossref]
  4. B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
    [Crossref]
  5. T. Kodera, D. L. Sounas, and C. Caloz, “Artificial faraday rotation using a ring metamaterial structure without static magnetic field,” Appl. Phys. Lett. 99, 031114 (2011).
    [Crossref]
  6. M. A. Gorlach, X. Ni, D. A. Smirnova, D. Korobkin, D. Zhirihin, A. P. Slobozhanyuk, P. A. Belov, A. Alù, and A. B. Khanikaev, “Far-field probing of leaky topological states in all-dielectric metasurfaces,” Nat. Commun. 9, 909 (2018).
    [Crossref]
  7. Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
    [Crossref]
  8. V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
    [Crossref]
  9. A. M. Shaltout, V. M. Shalaev, and M. L. Brongersma, “Spatiotemporal light control with active metasurfaces,” Science 364, eaat3100 (2019).
    [Crossref]
  10. K. Chen, Y. Feng, F. Monticone, J. Zhao, B. Zhu, T. Jiang, L. Zhang, Y. Kim, X. Ding, S. Zhang, A. Alù, and C.-W. Qiu, “A reconfigurable active Huygens’ metalens,” Adv. Mater. 29, 1606422 (2017).
    [Crossref]
  11. K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
    [Crossref]
  12. L. Zhang, X. Q. Chen, R. W. Shao, J. Y. Dai, Q. Cheng, G. Castaldi, V. Galdi, and T. J. Cui, “Breaking reciprocity with space-time-coding digital metasurfaces,” Adv. Mater. 31, 1904069 (2019).
    [Crossref]
  13. E. Yablonovitch, “Self-phase modulation of light in a laser-breakdown plasma,” Phys. Rev. Lett. 32, 1101–1104 (1974).
    [Crossref]
  14. F. R. Morgenthaler, “Velocity modulation of electromagnetic waves,” IRE Trans. Microw. Theory Tech. 6, 167–172 (1958).
    [Crossref]
  15. Y. Xiao, D. N. Maywar, and G. P. Agrawal, “Reflection and transmission of electromagnetic waves at a temporal boundary,” Opt. Lett. 39, 574–577 (2014).
    [Crossref]
  16. C. Caloz and Z. Deck-Léger, “Spacetime metamaterials-part ii: theory and applications,” IEEE Trans. Antennas Propag. 68, 1583–1598 (2020).
    [Crossref]
  17. R. L. Fante, “Transmission of electromagnetic waves into time-varying media,” IEEE Trans. Antennas Propag. 19, 417–424 (1971).
    [Crossref]
  18. R. L. Fante, “On the propagation of electromagnetic waves through a time-varying dielectric layer,” Appl. Sci. Res. 27, 341–354 (1973).
    [Crossref]
  19. Y. Xiao, G. P. Agrawal, and D. N. Maywar, “Spectral and temporal changes of optical pulses propagating through time-varying linear media,” Opt. Lett. 36, 505–507 (2011).
    [Crossref]
  20. Y. Xiao, D. N. Maywar, and G. P. Agrawal, “Optical pulse propagation in dynamic Fabry–Perot resonators,” J. Opt. Soc. Am. B 28, 1685–1692 (2011).
    [Crossref]
  21. A. G. Hayrapetyan, J. B. Gãtte, K. K. Grigoryan, S. Fritzsche, and R. G. Petrosyan, “Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media,” J. Quant. Spectrosc. Radiat. Transfer 178, 158–166 (2016).
    [Crossref]
  22. M. Chegnizadeh, K. Mehrany, and M. Memarian, “General solution to wave propagation in media undergoing arbitrary transient or periodic temporal variations of permittivity,” J. Opt. Soc. Am. B 35, 2923–2932 (2018).
    [Crossref]
  23. F. A. Harfoush and A. Taflove, “Scattering of electromagnetic waves by a material half-space with a time-varying conductivity,” IEEE Trans. Antennas Propag. 39, 898–906 (1991).
    [Crossref]
  24. D. Holberg and K. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14, 183–194 (1966).
    [Crossref]
  25. D. E. Holberg and K. S. Kunz, “Parametric properties of dielectric slabs with large permittivity modulation,” Radio Sci. 3, 273–286 (1968).
    [Crossref]
  26. J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009).
    [Crossref]
  27. J. R. Zurita-Sánchez, J. H. Abundis-Patiño, and P. Halevi, “Pulse propagation through a slab with time-periodic dielectric function ε(t),” Opt. Express 20, 5586–5600 (2012).
    [Crossref]
  28. J. S. Martnez-Romero and P. Halevi, “Standing waves with infinite group velocity in a temporally periodic medium,” Phys. Rev. A 96, 063831 (2017).
    [Crossref]
  29. J. S. Martnez-Romero and P. Halevi, “Parametric resonances in a temporal photonic crystal slab,” Phys. Rev. A 98, 053852 (2018).
    [Crossref]
  30. J. Simon, “Action of a progressive disturbance on a guided electromagnetic wave,” IRE Trans. Microw. Theory Tech. 8, 18–29 (1960).
    [Crossref]
  31. A. A. Oliner and A. Hessel, “Wave propagation in a medium with a progressive sinusoidal disturbance,” IRE Trans. Microw. Theory Tech. 9, 337–343 (1961).
    [Crossref]
  32. R. S. Chu and T. Tamir, “Wave propagation and dispersion in space-time periodic media,” Proc. Inst. Electr. Eng. 119, 797–806 (1972).
    [Crossref]
  33. R. L. Fante, “Optical propagation in space–time-modulated media using many-space-scale perturbation theory,” J. Opt. Soc. Am. 62, 1052–1060 (1972).
    [Crossref]
  34. T. T. Koutserimpas and R. Fleury, “Electromagnetic waves in a time periodic medium with step-varying refractive index,” IEEE Trans. Antennas Propag. 66, 5300–5307 (2018).
    [Crossref]
  35. T. T. Koutserimpas, A. Alù, and R. Fleury, “Parametric amplification and bidirectional invisibility in PT-symmetric time-Floquet systems,” Phys. Rev. A 97, 013839 (2018).
    [Crossref]
  36. T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-Hermitian time-Floquet systems,” Phys. Rev. Lett. 120, 087401 (2018).
    [Crossref]
  37. T. T. Koutserimpas and R. Fleury, “Electromagnetic fields in a time-varying medium: exceptional points and operator symmetries,” IEEE Trans. Antennas Propag. 68, 6717–6724 (2020).
    [Crossref]
  38. L. Felsen and G. Whitman, “Wave propagation in time-varying media,” IEEE Trans. Antennas Propag. 18, 242–253 (1970).
    [Crossref]
  39. M. Mirmoosa, G. Ptitcyn, V. Asadchy, and S. Tretyakov, “Time-varying reactive elements for extreme accumulation of electromagnetic energy,” Phys. Rev. Appl. 11, 014024 (2019).
    [Crossref]
  40. G. Ptitcyn, M. S. Mirmoosa, and S. A. Tretyakov, “Time-modulated meta-atoms,” Phys. Rev. Res. 1, 023014 (2019).
    [Crossref]
  41. M. S. Mirmoosa, G. A. Ptitcyn, R. Fleury, and S. A. Tretyakov, “Instantaneous radiation from time-varying electric and magnetic dipoles,” Phys. Rev. A 102, 013503 (2020).
    [Crossref]
  42. M. S. Mirmoosa, T. T. Koutserimpas, G. A. Ptitcyn, S. A. Tretyakov, and R. Fleury, “Dipole polarizability of time-varying particles,” arXiv:2002.12297 (2020).
  43. A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).
  44. D. M. Sols and N. Engheta, “Functional analysis of the polarization response in linear time-varying media: a generalization of the Kramers-Kronig relations,” Phys. Rev. B 103, 144303 (2021).
    [Crossref]
  45. A. V. Oppenheim, A. S. Willsky, and H. Nawab, Signals and Systems (Pearson Education, 1996).
  46. D. M. Pozar, Microwave Engineering (Wiley, 2011).
  47. G. Eleftheriades, A. Iyer, and P. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microw. Theory Tech. 50, 2702–2712 (2002).
    [Crossref]
  48. N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).
    [Crossref]
  49. M.-A. Parseval, “Mémoire sur les séries et sur l’intégration complète d’une équation aux différences partielles linéaires du second ordre, à coefficients constants,” Mém. prés. par divers savants, Acad. des Sciences, Paris 1, 638–648 (1806).
  50. M. Plancherel and M. Leffler, “Contribution à l’étude de la représentation d’une fonction arbitraire par des intégrales définies,” Rendiconti del Circolo Matematico di Palermo (1884-1940) 30, 289–335 (1910).
    [Crossref]
  51. L. Rayleigh, “LIII. On the character of the complete radiation at a given temperature,” London, Edinburgh, Dublin Philos. Mag. J. Sci. 27, 460–469 (1889).
    [Crossref]
  52. H. Lebesgue, “Intégrale, longueur, aire,” Ann. Mat. Pura Appl. (1898-1922) 7, 231–359 (1902).
    [Crossref]

2021 (1)

D. M. Sols and N. Engheta, “Functional analysis of the polarization response in linear time-varying media: a generalization of the Kramers-Kronig relations,” Phys. Rev. B 103, 144303 (2021).
[Crossref]

2020 (3)

T. T. Koutserimpas and R. Fleury, “Electromagnetic fields in a time-varying medium: exceptional points and operator symmetries,” IEEE Trans. Antennas Propag. 68, 6717–6724 (2020).
[Crossref]

M. S. Mirmoosa, G. A. Ptitcyn, R. Fleury, and S. A. Tretyakov, “Instantaneous radiation from time-varying electric and magnetic dipoles,” Phys. Rev. A 102, 013503 (2020).
[Crossref]

C. Caloz and Z. Deck-Léger, “Spacetime metamaterials-part ii: theory and applications,” IEEE Trans. Antennas Propag. 68, 1583–1598 (2020).
[Crossref]

2019 (4)

L. Zhang, X. Q. Chen, R. W. Shao, J. Y. Dai, Q. Cheng, G. Castaldi, V. Galdi, and T. J. Cui, “Breaking reciprocity with space-time-coding digital metasurfaces,” Adv. Mater. 31, 1904069 (2019).
[Crossref]

A. M. Shaltout, V. M. Shalaev, and M. L. Brongersma, “Spatiotemporal light control with active metasurfaces,” Science 364, eaat3100 (2019).
[Crossref]

M. Mirmoosa, G. Ptitcyn, V. Asadchy, and S. Tretyakov, “Time-varying reactive elements for extreme accumulation of electromagnetic energy,” Phys. Rev. Appl. 11, 014024 (2019).
[Crossref]

G. Ptitcyn, M. S. Mirmoosa, and S. A. Tretyakov, “Time-modulated meta-atoms,” Phys. Rev. Res. 1, 023014 (2019).
[Crossref]

2018 (7)

M. A. Gorlach, X. Ni, D. A. Smirnova, D. Korobkin, D. Zhirihin, A. P. Slobozhanyuk, P. A. Belov, A. Alù, and A. B. Khanikaev, “Far-field probing of leaky topological states in all-dielectric metasurfaces,” Nat. Commun. 9, 909 (2018).
[Crossref]

K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
[Crossref]

M. Chegnizadeh, K. Mehrany, and M. Memarian, “General solution to wave propagation in media undergoing arbitrary transient or periodic temporal variations of permittivity,” J. Opt. Soc. Am. B 35, 2923–2932 (2018).
[Crossref]

J. S. Martnez-Romero and P. Halevi, “Parametric resonances in a temporal photonic crystal slab,” Phys. Rev. A 98, 053852 (2018).
[Crossref]

T. T. Koutserimpas and R. Fleury, “Electromagnetic waves in a time periodic medium with step-varying refractive index,” IEEE Trans. Antennas Propag. 66, 5300–5307 (2018).
[Crossref]

T. T. Koutserimpas, A. Alù, and R. Fleury, “Parametric amplification and bidirectional invisibility in PT-symmetric time-Floquet systems,” Phys. Rev. A 97, 013839 (2018).
[Crossref]

T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-Hermitian time-Floquet systems,” Phys. Rev. Lett. 120, 087401 (2018).
[Crossref]

2017 (2)

J. S. Martnez-Romero and P. Halevi, “Standing waves with infinite group velocity in a temporally periodic medium,” Phys. Rev. A 96, 063831 (2017).
[Crossref]

K. Chen, Y. Feng, F. Monticone, J. Zhao, B. Zhu, T. Jiang, L. Zhang, Y. Kim, X. Ding, S. Zhang, A. Alù, and C.-W. Qiu, “A reconfigurable active Huygens’ metalens,” Adv. Mater. 29, 1606422 (2017).
[Crossref]

2016 (2)

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

A. G. Hayrapetyan, J. B. Gãtte, K. K. Grigoryan, S. Fritzsche, and R. G. Petrosyan, “Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media,” J. Quant. Spectrosc. Radiat. Transfer 178, 158–166 (2016).
[Crossref]

2014 (1)

2012 (1)

2011 (3)

2009 (3)

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[Crossref]

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
[Crossref]

J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009).
[Crossref]

2008 (1)

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[Crossref]

2006 (1)

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97, 157403 (2006).
[Crossref]

2005 (1)

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).
[Crossref]

2002 (1)

G. Eleftheriades, A. Iyer, and P. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microw. Theory Tech. 50, 2702–2712 (2002).
[Crossref]

1991 (1)

F. A. Harfoush and A. Taflove, “Scattering of electromagnetic waves by a material half-space with a time-varying conductivity,” IEEE Trans. Antennas Propag. 39, 898–906 (1991).
[Crossref]

1974 (1)

E. Yablonovitch, “Self-phase modulation of light in a laser-breakdown plasma,” Phys. Rev. Lett. 32, 1101–1104 (1974).
[Crossref]

1973 (1)

R. L. Fante, “On the propagation of electromagnetic waves through a time-varying dielectric layer,” Appl. Sci. Res. 27, 341–354 (1973).
[Crossref]

1972 (2)

R. S. Chu and T. Tamir, “Wave propagation and dispersion in space-time periodic media,” Proc. Inst. Electr. Eng. 119, 797–806 (1972).
[Crossref]

R. L. Fante, “Optical propagation in space–time-modulated media using many-space-scale perturbation theory,” J. Opt. Soc. Am. 62, 1052–1060 (1972).
[Crossref]

1971 (1)

R. L. Fante, “Transmission of electromagnetic waves into time-varying media,” IEEE Trans. Antennas Propag. 19, 417–424 (1971).
[Crossref]

1970 (1)

L. Felsen and G. Whitman, “Wave propagation in time-varying media,” IEEE Trans. Antennas Propag. 18, 242–253 (1970).
[Crossref]

1968 (1)

D. E. Holberg and K. S. Kunz, “Parametric properties of dielectric slabs with large permittivity modulation,” Radio Sci. 3, 273–286 (1968).
[Crossref]

1966 (1)

D. Holberg and K. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14, 183–194 (1966).
[Crossref]

1961 (1)

A. A. Oliner and A. Hessel, “Wave propagation in a medium with a progressive sinusoidal disturbance,” IRE Trans. Microw. Theory Tech. 9, 337–343 (1961).
[Crossref]

1960 (1)

J. Simon, “Action of a progressive disturbance on a guided electromagnetic wave,” IRE Trans. Microw. Theory Tech. 8, 18–29 (1960).
[Crossref]

1958 (1)

F. R. Morgenthaler, “Velocity modulation of electromagnetic waves,” IRE Trans. Microw. Theory Tech. 6, 167–172 (1958).
[Crossref]

1910 (1)

M. Plancherel and M. Leffler, “Contribution à l’étude de la représentation d’une fonction arbitraire par des intégrales définies,” Rendiconti del Circolo Matematico di Palermo (1884-1940) 30, 289–335 (1910).
[Crossref]

1902 (1)

H. Lebesgue, “Intégrale, longueur, aire,” Ann. Mat. Pura Appl. (1898-1922) 7, 231–359 (1902).
[Crossref]

1889 (1)

L. Rayleigh, “LIII. On the character of the complete radiation at a given temperature,” London, Edinburgh, Dublin Philos. Mag. J. Sci. 27, 460–469 (1889).
[Crossref]

1806 (1)

M.-A. Parseval, “Mémoire sur les séries et sur l’intégration complète d’une équation aux différences partielles linéaires du second ordre, à coefficients constants,” Mém. prés. par divers savants, Acad. des Sciences, Paris 1, 638–648 (1806).

Abundis-Patiño, J. H.

Agrawal, G. P.

Alù, A.

T. T. Koutserimpas, A. Alù, and R. Fleury, “Parametric amplification and bidirectional invisibility in PT-symmetric time-Floquet systems,” Phys. Rev. A 97, 013839 (2018).
[Crossref]

M. A. Gorlach, X. Ni, D. A. Smirnova, D. Korobkin, D. Zhirihin, A. P. Slobozhanyuk, P. A. Belov, A. Alù, and A. B. Khanikaev, “Far-field probing of leaky topological states in all-dielectric metasurfaces,” Nat. Commun. 9, 909 (2018).
[Crossref]

K. Chen, Y. Feng, F. Monticone, J. Zhao, B. Zhu, T. Jiang, L. Zhang, Y. Kim, X. Ding, S. Zhang, A. Alù, and C.-W. Qiu, “A reconfigurable active Huygens’ metalens,” Adv. Mater. 29, 1606422 (2017).
[Crossref]

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[Crossref]

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).
[Crossref]

Asadchy, V.

M. Mirmoosa, G. Ptitcyn, V. Asadchy, and S. Tretyakov, “Time-varying reactive elements for extreme accumulation of electromagnetic energy,” Phys. Rev. Appl. 11, 014024 (2019).
[Crossref]

Bacot, V.

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

Belov, P. A.

M. A. Gorlach, X. Ni, D. A. Smirnova, D. Korobkin, D. Zhirihin, A. P. Slobozhanyuk, P. A. Belov, A. Alù, and A. B. Khanikaev, “Far-field probing of leaky topological states in all-dielectric metasurfaces,” Nat. Commun. 9, 909 (2018).
[Crossref]

Brongersma, M. L.

A. M. Shaltout, V. M. Shalaev, and M. L. Brongersma, “Spatiotemporal light control with active metasurfaces,” Science 364, eaat3100 (2019).
[Crossref]

Caloz, C.

C. Caloz and Z. Deck-Léger, “Spacetime metamaterials-part ii: theory and applications,” IEEE Trans. Antennas Propag. 68, 1583–1598 (2020).
[Crossref]

T. Kodera, D. L. Sounas, and C. Caloz, “Artificial faraday rotation using a ring metamaterial structure without static magnetic field,” Appl. Phys. Lett. 99, 031114 (2011).
[Crossref]

Castaldi, G.

L. Zhang, X. Q. Chen, R. W. Shao, J. Y. Dai, Q. Cheng, G. Castaldi, V. Galdi, and T. J. Cui, “Breaking reciprocity with space-time-coding digital metasurfaces,” Adv. Mater. 31, 1904069 (2019).
[Crossref]

Cervantes-González, J. C.

J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009).
[Crossref]

Chegnizadeh, M.

Chen, K.

K. Chen, Y. Feng, F. Monticone, J. Zhao, B. Zhu, T. Jiang, L. Zhang, Y. Kim, X. Ding, S. Zhang, A. Alù, and C.-W. Qiu, “A reconfigurable active Huygens’ metalens,” Adv. Mater. 29, 1606422 (2017).
[Crossref]

Chen, X. Q.

L. Zhang, X. Q. Chen, R. W. Shao, J. Y. Dai, Q. Cheng, G. Castaldi, V. Galdi, and T. J. Cui, “Breaking reciprocity with space-time-coding digital metasurfaces,” Adv. Mater. 31, 1904069 (2019).
[Crossref]

Cheng, Q.

L. Zhang, X. Q. Chen, R. W. Shao, J. Y. Dai, Q. Cheng, G. Castaldi, V. Galdi, and T. J. Cui, “Breaking reciprocity with space-time-coding digital metasurfaces,” Adv. Mater. 31, 1904069 (2019).
[Crossref]

Chu, R. S.

R. S. Chu and T. Tamir, “Wave propagation and dispersion in space-time periodic media,” Proc. Inst. Electr. Eng. 119, 797–806 (1972).
[Crossref]

Cui, T. J.

L. Zhang, X. Q. Chen, R. W. Shao, J. Y. Dai, Q. Cheng, G. Castaldi, V. Galdi, and T. J. Cui, “Breaking reciprocity with space-time-coding digital metasurfaces,” Adv. Mater. 31, 1904069 (2019).
[Crossref]

Dai, J. Y.

L. Zhang, X. Q. Chen, R. W. Shao, J. Y. Dai, Q. Cheng, G. Castaldi, V. Galdi, and T. J. Cui, “Breaking reciprocity with space-time-coding digital metasurfaces,” Adv. Mater. 31, 1904069 (2019).
[Crossref]

Deck-Léger, Z.

C. Caloz and Z. Deck-Léger, “Spacetime metamaterials-part ii: theory and applications,” IEEE Trans. Antennas Propag. 68, 1583–1598 (2020).
[Crossref]

Ding, X.

K. Chen, Y. Feng, F. Monticone, J. Zhao, B. Zhu, T. Jiang, L. Zhang, Y. Kim, X. Ding, S. Zhang, A. Alù, and C.-W. Qiu, “A reconfigurable active Huygens’ metalens,” Adv. Mater. 29, 1606422 (2017).
[Crossref]

Eddi, A.

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

Edwards, B.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[Crossref]

Eleftheriades, G.

G. Eleftheriades, A. Iyer, and P. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microw. Theory Tech. 50, 2702–2712 (2002).
[Crossref]

Engheta, N.

D. M. Sols and N. Engheta, “Functional analysis of the polarization response in linear time-varying media: a generalization of the Kramers-Kronig relations,” Phys. Rev. B 103, 144303 (2021).
[Crossref]

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[Crossref]

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97, 157403 (2006).
[Crossref]

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).
[Crossref]

N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations (Wiley, 2006).

Fan, S.

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
[Crossref]

Fante, R. L.

R. L. Fante, “On the propagation of electromagnetic waves through a time-varying dielectric layer,” Appl. Sci. Res. 27, 341–354 (1973).
[Crossref]

R. L. Fante, “Optical propagation in space–time-modulated media using many-space-scale perturbation theory,” J. Opt. Soc. Am. 62, 1052–1060 (1972).
[Crossref]

R. L. Fante, “Transmission of electromagnetic waves into time-varying media,” IEEE Trans. Antennas Propag. 19, 417–424 (1971).
[Crossref]

Felsen, L.

L. Felsen and G. Whitman, “Wave propagation in time-varying media,” IEEE Trans. Antennas Propag. 18, 242–253 (1970).
[Crossref]

Feng, Y.

K. Chen, Y. Feng, F. Monticone, J. Zhao, B. Zhu, T. Jiang, L. Zhang, Y. Kim, X. Ding, S. Zhang, A. Alù, and C.-W. Qiu, “A reconfigurable active Huygens’ metalens,” Adv. Mater. 29, 1606422 (2017).
[Crossref]

Fink, M.

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

Fleury, R.

T. T. Koutserimpas and R. Fleury, “Electromagnetic fields in a time-varying medium: exceptional points and operator symmetries,” IEEE Trans. Antennas Propag. 68, 6717–6724 (2020).
[Crossref]

M. S. Mirmoosa, G. A. Ptitcyn, R. Fleury, and S. A. Tretyakov, “Instantaneous radiation from time-varying electric and magnetic dipoles,” Phys. Rev. A 102, 013503 (2020).
[Crossref]

T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-Hermitian time-Floquet systems,” Phys. Rev. Lett. 120, 087401 (2018).
[Crossref]

T. T. Koutserimpas, A. Alù, and R. Fleury, “Parametric amplification and bidirectional invisibility in PT-symmetric time-Floquet systems,” Phys. Rev. A 97, 013839 (2018).
[Crossref]

T. T. Koutserimpas and R. Fleury, “Electromagnetic waves in a time periodic medium with step-varying refractive index,” IEEE Trans. Antennas Propag. 66, 5300–5307 (2018).
[Crossref]

M. S. Mirmoosa, T. T. Koutserimpas, G. A. Ptitcyn, S. A. Tretyakov, and R. Fleury, “Dipole polarizability of time-varying particles,” arXiv:2002.12297 (2020).

Fort, E.

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

Fritzsche, S.

A. G. Hayrapetyan, J. B. Gãtte, K. K. Grigoryan, S. Fritzsche, and R. G. Petrosyan, “Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media,” J. Quant. Spectrosc. Radiat. Transfer 178, 158–166 (2016).
[Crossref]

Galdi, V.

L. Zhang, X. Q. Chen, R. W. Shao, J. Y. Dai, Q. Cheng, G. Castaldi, V. Galdi, and T. J. Cui, “Breaking reciprocity with space-time-coding digital metasurfaces,” Adv. Mater. 31, 1904069 (2019).
[Crossref]

Gãtte, J. B.

A. G. Hayrapetyan, J. B. Gãtte, K. K. Grigoryan, S. Fritzsche, and R. G. Petrosyan, “Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media,” J. Quant. Spectrosc. Radiat. Transfer 178, 158–166 (2016).
[Crossref]

Gorlach, M. A.

M. A. Gorlach, X. Ni, D. A. Smirnova, D. Korobkin, D. Zhirihin, A. P. Slobozhanyuk, P. A. Belov, A. Alù, and A. B. Khanikaev, “Far-field probing of leaky topological states in all-dielectric metasurfaces,” Nat. Commun. 9, 909 (2018).
[Crossref]

Grigoryan, K. K.

A. G. Hayrapetyan, J. B. Gãtte, K. K. Grigoryan, S. Fritzsche, and R. G. Petrosyan, “Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media,” J. Quant. Spectrosc. Radiat. Transfer 178, 158–166 (2016).
[Crossref]

Hagness, S. C.

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Halevi, P.

J. S. Martnez-Romero and P. Halevi, “Parametric resonances in a temporal photonic crystal slab,” Phys. Rev. A 98, 053852 (2018).
[Crossref]

J. S. Martnez-Romero and P. Halevi, “Standing waves with infinite group velocity in a temporally periodic medium,” Phys. Rev. A 96, 063831 (2017).
[Crossref]

J. R. Zurita-Sánchez, J. H. Abundis-Patiño, and P. Halevi, “Pulse propagation through a slab with time-periodic dielectric function ε(t),” Opt. Express 20, 5586–5600 (2012).
[Crossref]

J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009).
[Crossref]

Harfoush, F. A.

F. A. Harfoush and A. Taflove, “Scattering of electromagnetic waves by a material half-space with a time-varying conductivity,” IEEE Trans. Antennas Propag. 39, 898–906 (1991).
[Crossref]

Hayrapetyan, A. G.

A. G. Hayrapetyan, J. B. Gãtte, K. K. Grigoryan, S. Fritzsche, and R. G. Petrosyan, “Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media,” J. Quant. Spectrosc. Radiat. Transfer 178, 158–166 (2016).
[Crossref]

Hessel, A.

A. A. Oliner and A. Hessel, “Wave propagation in a medium with a progressive sinusoidal disturbance,” IRE Trans. Microw. Theory Tech. 9, 337–343 (1961).
[Crossref]

Holberg, D.

D. Holberg and K. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14, 183–194 (1966).
[Crossref]

Holberg, D. E.

D. E. Holberg and K. S. Kunz, “Parametric properties of dielectric slabs with large permittivity modulation,” Radio Sci. 3, 273–286 (1968).
[Crossref]

Iyer, A.

G. Eleftheriades, A. Iyer, and P. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microw. Theory Tech. 50, 2702–2712 (2002).
[Crossref]

Jeon, W.

K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
[Crossref]

Jiang, T.

K. Chen, Y. Feng, F. Monticone, J. Zhao, B. Zhu, T. Jiang, L. Zhang, Y. Kim, X. Ding, S. Zhang, A. Alù, and C.-W. Qiu, “A reconfigurable active Huygens’ metalens,” Adv. Mater. 29, 1606422 (2017).
[Crossref]

Kang, B.

K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
[Crossref]

Khanikaev, A. B.

M. A. Gorlach, X. Ni, D. A. Smirnova, D. Korobkin, D. Zhirihin, A. P. Slobozhanyuk, P. A. Belov, A. Alù, and A. B. Khanikaev, “Far-field probing of leaky topological states in all-dielectric metasurfaces,” Nat. Commun. 9, 909 (2018).
[Crossref]

Kim, Y.

K. Chen, Y. Feng, F. Monticone, J. Zhao, B. Zhu, T. Jiang, L. Zhang, Y. Kim, X. Ding, S. Zhang, A. Alù, and C.-W. Qiu, “A reconfigurable active Huygens’ metalens,” Adv. Mater. 29, 1606422 (2017).
[Crossref]

Kodera, T.

T. Kodera, D. L. Sounas, and C. Caloz, “Artificial faraday rotation using a ring metamaterial structure without static magnetic field,” Appl. Phys. Lett. 99, 031114 (2011).
[Crossref]

Korobkin, D.

M. A. Gorlach, X. Ni, D. A. Smirnova, D. Korobkin, D. Zhirihin, A. P. Slobozhanyuk, P. A. Belov, A. Alù, and A. B. Khanikaev, “Far-field probing of leaky topological states in all-dielectric metasurfaces,” Nat. Commun. 9, 909 (2018).
[Crossref]

Koutserimpas, T. T.

T. T. Koutserimpas and R. Fleury, “Electromagnetic fields in a time-varying medium: exceptional points and operator symmetries,” IEEE Trans. Antennas Propag. 68, 6717–6724 (2020).
[Crossref]

T. T. Koutserimpas, A. Alù, and R. Fleury, “Parametric amplification and bidirectional invisibility in PT-symmetric time-Floquet systems,” Phys. Rev. A 97, 013839 (2018).
[Crossref]

T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-Hermitian time-Floquet systems,” Phys. Rev. Lett. 120, 087401 (2018).
[Crossref]

T. T. Koutserimpas and R. Fleury, “Electromagnetic waves in a time periodic medium with step-varying refractive index,” IEEE Trans. Antennas Propag. 66, 5300–5307 (2018).
[Crossref]

M. S. Mirmoosa, T. T. Koutserimpas, G. A. Ptitcyn, S. A. Tretyakov, and R. Fleury, “Dipole polarizability of time-varying particles,” arXiv:2002.12297 (2020).

Kremer, P.

G. Eleftheriades, A. Iyer, and P. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microw. Theory Tech. 50, 2702–2712 (2002).
[Crossref]

Kunz, K.

D. Holberg and K. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14, 183–194 (1966).
[Crossref]

Kunz, K. S.

D. E. Holberg and K. S. Kunz, “Parametric properties of dielectric slabs with large permittivity modulation,” Radio Sci. 3, 273–286 (1968).
[Crossref]

Kuramochi, E.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[Crossref]

Labousse, M.

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

Lebesgue, H.

H. Lebesgue, “Intégrale, longueur, aire,” Ann. Mat. Pura Appl. (1898-1922) 7, 231–359 (1902).
[Crossref]

Lee, K.

K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
[Crossref]

Leffler, M.

M. Plancherel and M. Leffler, “Contribution à l’étude de la représentation d’une fonction arbitraire par des intégrales définies,” Rendiconti del Circolo Matematico di Palermo (1884-1940) 30, 289–335 (1910).
[Crossref]

Martnez-Romero, J. S.

J. S. Martnez-Romero and P. Halevi, “Parametric resonances in a temporal photonic crystal slab,” Phys. Rev. A 98, 053852 (2018).
[Crossref]

J. S. Martnez-Romero and P. Halevi, “Standing waves with infinite group velocity in a temporally periodic medium,” Phys. Rev. A 96, 063831 (2017).
[Crossref]

Maywar, D. N.

Mehrany, K.

Memarian, M.

Min, B.

K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
[Crossref]

Mirmoosa, M.

M. Mirmoosa, G. Ptitcyn, V. Asadchy, and S. Tretyakov, “Time-varying reactive elements for extreme accumulation of electromagnetic energy,” Phys. Rev. Appl. 11, 014024 (2019).
[Crossref]

Mirmoosa, M. S.

M. S. Mirmoosa, G. A. Ptitcyn, R. Fleury, and S. A. Tretyakov, “Instantaneous radiation from time-varying electric and magnetic dipoles,” Phys. Rev. A 102, 013503 (2020).
[Crossref]

G. Ptitcyn, M. S. Mirmoosa, and S. A. Tretyakov, “Time-modulated meta-atoms,” Phys. Rev. Res. 1, 023014 (2019).
[Crossref]

M. S. Mirmoosa, T. T. Koutserimpas, G. A. Ptitcyn, S. A. Tretyakov, and R. Fleury, “Dipole polarizability of time-varying particles,” arXiv:2002.12297 (2020).

Monticone, F.

K. Chen, Y. Feng, F. Monticone, J. Zhao, B. Zhu, T. Jiang, L. Zhang, Y. Kim, X. Ding, S. Zhang, A. Alù, and C.-W. Qiu, “A reconfigurable active Huygens’ metalens,” Adv. Mater. 29, 1606422 (2017).
[Crossref]

Morgenthaler, F. R.

F. R. Morgenthaler, “Velocity modulation of electromagnetic waves,” IRE Trans. Microw. Theory Tech. 6, 167–172 (1958).
[Crossref]

Nawab, H.

A. V. Oppenheim, A. S. Willsky, and H. Nawab, Signals and Systems (Pearson Education, 1996).

Ni, X.

M. A. Gorlach, X. Ni, D. A. Smirnova, D. Korobkin, D. Zhirihin, A. P. Slobozhanyuk, P. A. Belov, A. Alù, and A. B. Khanikaev, “Far-field probing of leaky topological states in all-dielectric metasurfaces,” Nat. Commun. 9, 909 (2018).
[Crossref]

Notomi, M.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[Crossref]

Oliner, A. A.

A. A. Oliner and A. Hessel, “Wave propagation in a medium with a progressive sinusoidal disturbance,” IRE Trans. Microw. Theory Tech. 9, 337–343 (1961).
[Crossref]

Oppenheim, A. V.

A. V. Oppenheim, A. S. Willsky, and H. Nawab, Signals and Systems (Pearson Education, 1996).

Park, J.

K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
[Crossref]

Parseval, M.-A.

M.-A. Parseval, “Mémoire sur les séries et sur l’intégration complète d’une équation aux différences partielles linéaires du second ordre, à coefficients constants,” Mém. prés. par divers savants, Acad. des Sciences, Paris 1, 638–648 (1806).

Petrosyan, R. G.

A. G. Hayrapetyan, J. B. Gãtte, K. K. Grigoryan, S. Fritzsche, and R. G. Petrosyan, “Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media,” J. Quant. Spectrosc. Radiat. Transfer 178, 158–166 (2016).
[Crossref]

Plancherel, M.

M. Plancherel and M. Leffler, “Contribution à l’étude de la représentation d’une fonction arbitraire par des intégrales définies,” Rendiconti del Circolo Matematico di Palermo (1884-1940) 30, 289–335 (1910).
[Crossref]

Pozar, D. M.

D. M. Pozar, Microwave Engineering (Wiley, 2011).

Ptitcyn, G.

G. Ptitcyn, M. S. Mirmoosa, and S. A. Tretyakov, “Time-modulated meta-atoms,” Phys. Rev. Res. 1, 023014 (2019).
[Crossref]

M. Mirmoosa, G. Ptitcyn, V. Asadchy, and S. Tretyakov, “Time-varying reactive elements for extreme accumulation of electromagnetic energy,” Phys. Rev. Appl. 11, 014024 (2019).
[Crossref]

Ptitcyn, G. A.

M. S. Mirmoosa, G. A. Ptitcyn, R. Fleury, and S. A. Tretyakov, “Instantaneous radiation from time-varying electric and magnetic dipoles,” Phys. Rev. A 102, 013503 (2020).
[Crossref]

M. S. Mirmoosa, T. T. Koutserimpas, G. A. Ptitcyn, S. A. Tretyakov, and R. Fleury, “Dipole polarizability of time-varying particles,” arXiv:2002.12297 (2020).

Qiu, C.-W.

K. Chen, Y. Feng, F. Monticone, J. Zhao, B. Zhu, T. Jiang, L. Zhang, Y. Kim, X. Ding, S. Zhang, A. Alù, and C.-W. Qiu, “A reconfigurable active Huygens’ metalens,” Adv. Mater. 29, 1606422 (2017).
[Crossref]

Rayleigh, L.

L. Rayleigh, “LIII. On the character of the complete radiation at a given temperature,” London, Edinburgh, Dublin Philos. Mag. J. Sci. 27, 460–469 (1889).
[Crossref]

Rotermund, F.

K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
[Crossref]

Salandrino, A.

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).
[Crossref]

Shalaev, V. M.

A. M. Shaltout, V. M. Shalaev, and M. L. Brongersma, “Spatiotemporal light control with active metasurfaces,” Science 364, eaat3100 (2019).
[Crossref]

Shaltout, A. M.

A. M. Shaltout, V. M. Shalaev, and M. L. Brongersma, “Spatiotemporal light control with active metasurfaces,” Science 364, eaat3100 (2019).
[Crossref]

Shao, R. W.

L. Zhang, X. Q. Chen, R. W. Shao, J. Y. Dai, Q. Cheng, G. Castaldi, V. Galdi, and T. J. Cui, “Breaking reciprocity with space-time-coding digital metasurfaces,” Adv. Mater. 31, 1904069 (2019).
[Crossref]

Silveirinha, M.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[Crossref]

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97, 157403 (2006).
[Crossref]

Simon, J.

J. Simon, “Action of a progressive disturbance on a guided electromagnetic wave,” IRE Trans. Microw. Theory Tech. 8, 18–29 (1960).
[Crossref]

Slobozhanyuk, A. P.

M. A. Gorlach, X. Ni, D. A. Smirnova, D. Korobkin, D. Zhirihin, A. P. Slobozhanyuk, P. A. Belov, A. Alù, and A. B. Khanikaev, “Far-field probing of leaky topological states in all-dielectric metasurfaces,” Nat. Commun. 9, 909 (2018).
[Crossref]

Smirnova, D. A.

M. A. Gorlach, X. Ni, D. A. Smirnova, D. Korobkin, D. Zhirihin, A. P. Slobozhanyuk, P. A. Belov, A. Alù, and A. B. Khanikaev, “Far-field probing of leaky topological states in all-dielectric metasurfaces,” Nat. Commun. 9, 909 (2018).
[Crossref]

Sols, D. M.

D. M. Sols and N. Engheta, “Functional analysis of the polarization response in linear time-varying media: a generalization of the Kramers-Kronig relations,” Phys. Rev. B 103, 144303 (2021).
[Crossref]

Son, J.

K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
[Crossref]

Sounas, D. L.

T. Kodera, D. L. Sounas, and C. Caloz, “Artificial faraday rotation using a ring metamaterial structure without static magnetic field,” Appl. Phys. Lett. 99, 031114 (2011).
[Crossref]

Taflove, A.

F. A. Harfoush and A. Taflove, “Scattering of electromagnetic waves by a material half-space with a time-varying conductivity,” IEEE Trans. Antennas Propag. 39, 898–906 (1991).
[Crossref]

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

Tamir, T.

R. S. Chu and T. Tamir, “Wave propagation and dispersion in space-time periodic media,” Proc. Inst. Electr. Eng. 119, 797–806 (1972).
[Crossref]

Tanabe, T.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[Crossref]

Taniyama, H.

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[Crossref]

Tretyakov, S.

M. Mirmoosa, G. Ptitcyn, V. Asadchy, and S. Tretyakov, “Time-varying reactive elements for extreme accumulation of electromagnetic energy,” Phys. Rev. Appl. 11, 014024 (2019).
[Crossref]

Tretyakov, S. A.

M. S. Mirmoosa, G. A. Ptitcyn, R. Fleury, and S. A. Tretyakov, “Instantaneous radiation from time-varying electric and magnetic dipoles,” Phys. Rev. A 102, 013503 (2020).
[Crossref]

G. Ptitcyn, M. S. Mirmoosa, and S. A. Tretyakov, “Time-modulated meta-atoms,” Phys. Rev. Res. 1, 023014 (2019).
[Crossref]

M. S. Mirmoosa, T. T. Koutserimpas, G. A. Ptitcyn, S. A. Tretyakov, and R. Fleury, “Dipole polarizability of time-varying particles,” arXiv:2002.12297 (2020).

Whitman, G.

L. Felsen and G. Whitman, “Wave propagation in time-varying media,” IEEE Trans. Antennas Propag. 18, 242–253 (1970).
[Crossref]

Willsky, A. S.

A. V. Oppenheim, A. S. Willsky, and H. Nawab, Signals and Systems (Pearson Education, 1996).

Xiao, Y.

Yablonovitch, E.

E. Yablonovitch, “Self-phase modulation of light in a laser-breakdown plasma,” Phys. Rev. Lett. 32, 1101–1104 (1974).
[Crossref]

Young, M. E.

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[Crossref]

Yu, Z.

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
[Crossref]

Zhang, L.

L. Zhang, X. Q. Chen, R. W. Shao, J. Y. Dai, Q. Cheng, G. Castaldi, V. Galdi, and T. J. Cui, “Breaking reciprocity with space-time-coding digital metasurfaces,” Adv. Mater. 31, 1904069 (2019).
[Crossref]

K. Chen, Y. Feng, F. Monticone, J. Zhao, B. Zhu, T. Jiang, L. Zhang, Y. Kim, X. Ding, S. Zhang, A. Alù, and C.-W. Qiu, “A reconfigurable active Huygens’ metalens,” Adv. Mater. 29, 1606422 (2017).
[Crossref]

Zhang, S.

K. Chen, Y. Feng, F. Monticone, J. Zhao, B. Zhu, T. Jiang, L. Zhang, Y. Kim, X. Ding, S. Zhang, A. Alù, and C.-W. Qiu, “A reconfigurable active Huygens’ metalens,” Adv. Mater. 29, 1606422 (2017).
[Crossref]

Zhao, J.

K. Chen, Y. Feng, F. Monticone, J. Zhao, B. Zhu, T. Jiang, L. Zhang, Y. Kim, X. Ding, S. Zhang, A. Alù, and C.-W. Qiu, “A reconfigurable active Huygens’ metalens,” Adv. Mater. 29, 1606422 (2017).
[Crossref]

Zhirihin, D.

M. A. Gorlach, X. Ni, D. A. Smirnova, D. Korobkin, D. Zhirihin, A. P. Slobozhanyuk, P. A. Belov, A. Alù, and A. B. Khanikaev, “Far-field probing of leaky topological states in all-dielectric metasurfaces,” Nat. Commun. 9, 909 (2018).
[Crossref]

Zhu, B.

K. Chen, Y. Feng, F. Monticone, J. Zhao, B. Zhu, T. Jiang, L. Zhang, Y. Kim, X. Ding, S. Zhang, A. Alù, and C.-W. Qiu, “A reconfigurable active Huygens’ metalens,” Adv. Mater. 29, 1606422 (2017).
[Crossref]

Ziolkowski, R. W.

N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations (Wiley, 2006).

Zurita-Sánchez, J. R.

J. R. Zurita-Sánchez, J. H. Abundis-Patiño, and P. Halevi, “Pulse propagation through a slab with time-periodic dielectric function ε(t),” Opt. Express 20, 5586–5600 (2012).
[Crossref]

J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009).
[Crossref]

Adv. Mater. (2)

K. Chen, Y. Feng, F. Monticone, J. Zhao, B. Zhu, T. Jiang, L. Zhang, Y. Kim, X. Ding, S. Zhang, A. Alù, and C.-W. Qiu, “A reconfigurable active Huygens’ metalens,” Adv. Mater. 29, 1606422 (2017).
[Crossref]

L. Zhang, X. Q. Chen, R. W. Shao, J. Y. Dai, Q. Cheng, G. Castaldi, V. Galdi, and T. J. Cui, “Breaking reciprocity with space-time-coding digital metasurfaces,” Adv. Mater. 31, 1904069 (2019).
[Crossref]

Ann. Mat. Pura Appl. (1898-1922) (1)

H. Lebesgue, “Intégrale, longueur, aire,” Ann. Mat. Pura Appl. (1898-1922) 7, 231–359 (1902).
[Crossref]

Appl. Phys. Lett. (1)

T. Kodera, D. L. Sounas, and C. Caloz, “Artificial faraday rotation using a ring metamaterial structure without static magnetic field,” Appl. Phys. Lett. 99, 031114 (2011).
[Crossref]

Appl. Sci. Res. (1)

R. L. Fante, “On the propagation of electromagnetic waves through a time-varying dielectric layer,” Appl. Sci. Res. 27, 341–354 (1973).
[Crossref]

IEEE Trans. Antennas Propag. (7)

C. Caloz and Z. Deck-Léger, “Spacetime metamaterials-part ii: theory and applications,” IEEE Trans. Antennas Propag. 68, 1583–1598 (2020).
[Crossref]

R. L. Fante, “Transmission of electromagnetic waves into time-varying media,” IEEE Trans. Antennas Propag. 19, 417–424 (1971).
[Crossref]

F. A. Harfoush and A. Taflove, “Scattering of electromagnetic waves by a material half-space with a time-varying conductivity,” IEEE Trans. Antennas Propag. 39, 898–906 (1991).
[Crossref]

D. Holberg and K. Kunz, “Parametric properties of fields in a slab of time-varying permittivity,” IEEE Trans. Antennas Propag. 14, 183–194 (1966).
[Crossref]

T. T. Koutserimpas and R. Fleury, “Electromagnetic fields in a time-varying medium: exceptional points and operator symmetries,” IEEE Trans. Antennas Propag. 68, 6717–6724 (2020).
[Crossref]

L. Felsen and G. Whitman, “Wave propagation in time-varying media,” IEEE Trans. Antennas Propag. 18, 242–253 (1970).
[Crossref]

T. T. Koutserimpas and R. Fleury, “Electromagnetic waves in a time periodic medium with step-varying refractive index,” IEEE Trans. Antennas Propag. 66, 5300–5307 (2018).
[Crossref]

IEEE Trans. Microw. Theory Tech. (1)

G. Eleftheriades, A. Iyer, and P. Kremer, “Planar negative refractive index media using periodically L-C loaded transmission lines,” IEEE Trans. Microw. Theory Tech. 50, 2702–2712 (2002).
[Crossref]

IRE Trans. Microw. Theory Tech. (3)

J. Simon, “Action of a progressive disturbance on a guided electromagnetic wave,” IRE Trans. Microw. Theory Tech. 8, 18–29 (1960).
[Crossref]

A. A. Oliner and A. Hessel, “Wave propagation in a medium with a progressive sinusoidal disturbance,” IRE Trans. Microw. Theory Tech. 9, 337–343 (1961).
[Crossref]

F. R. Morgenthaler, “Velocity modulation of electromagnetic waves,” IRE Trans. Microw. Theory Tech. 6, 167–172 (1958).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. B (2)

J. Quant. Spectrosc. Radiat. Transfer (1)

A. G. Hayrapetyan, J. B. Gãtte, K. K. Grigoryan, S. Fritzsche, and R. G. Petrosyan, “Electromagnetic wave propagation in spatially homogeneous yet smoothly time-varying dielectric media,” J. Quant. Spectrosc. Radiat. Transfer 178, 158–166 (2016).
[Crossref]

London, Edinburgh, Dublin Philos. Mag. J. Sci. (1)

L. Rayleigh, “LIII. On the character of the complete radiation at a given temperature,” London, Edinburgh, Dublin Philos. Mag. J. Sci. 27, 460–469 (1889).
[Crossref]

Mém. prés. par divers savants, Acad. des Sciences, Paris (1)

M.-A. Parseval, “Mémoire sur les séries et sur l’intégration complète d’une équation aux différences partielles linéaires du second ordre, à coefficients constants,” Mém. prés. par divers savants, Acad. des Sciences, Paris 1, 638–648 (1806).

Nat. Commun. (1)

M. A. Gorlach, X. Ni, D. A. Smirnova, D. Korobkin, D. Zhirihin, A. P. Slobozhanyuk, P. A. Belov, A. Alù, and A. B. Khanikaev, “Far-field probing of leaky topological states in all-dielectric metasurfaces,” Nat. Commun. 9, 909 (2018).
[Crossref]

Nat. Photonics (2)

Z. Yu and S. Fan, “Complete optical isolation created by indirect interband photonic transitions,” Nat. Photonics 3, 91–94 (2009).
[Crossref]

K. Lee, J. Son, J. Park, B. Kang, W. Jeon, F. Rotermund, and B. Min, “Linear frequency conversion via sudden merging of meta-atoms in time-variant metasurfaces,” Nat. Photonics 12, 765–773 (2018).
[Crossref]

Nat. Phys. (1)

V. Bacot, M. Labousse, A. Eddi, M. Fink, and E. Fort, “Time reversal and holography with spacetime transformations,” Nat. Phys. 12, 972–977 (2016).
[Crossref]

Opt. Express (1)

Opt. Lett. (2)

Phys. Rev. A (5)

J. S. Martnez-Romero and P. Halevi, “Standing waves with infinite group velocity in a temporally periodic medium,” Phys. Rev. A 96, 063831 (2017).
[Crossref]

J. S. Martnez-Romero and P. Halevi, “Parametric resonances in a temporal photonic crystal slab,” Phys. Rev. A 98, 053852 (2018).
[Crossref]

J. R. Zurita-Sánchez, P. Halevi, and J. C. Cervantes-González, “Reflection and transmission of a wave incident on a slab with a time-periodic dielectric function ε(t),” Phys. Rev. A 79, 053821 (2009).
[Crossref]

T. T. Koutserimpas, A. Alù, and R. Fleury, “Parametric amplification and bidirectional invisibility in PT-symmetric time-Floquet systems,” Phys. Rev. A 97, 013839 (2018).
[Crossref]

M. S. Mirmoosa, G. A. Ptitcyn, R. Fleury, and S. A. Tretyakov, “Instantaneous radiation from time-varying electric and magnetic dipoles,” Phys. Rev. A 102, 013503 (2020).
[Crossref]

Phys. Rev. Appl. (1)

M. Mirmoosa, G. Ptitcyn, V. Asadchy, and S. Tretyakov, “Time-varying reactive elements for extreme accumulation of electromagnetic energy,” Phys. Rev. Appl. 11, 014024 (2019).
[Crossref]

Phys. Rev. B (1)

D. M. Sols and N. Engheta, “Functional analysis of the polarization response in linear time-varying media: a generalization of the Kramers-Kronig relations,” Phys. Rev. B 103, 144303 (2021).
[Crossref]

Phys. Rev. Lett. (6)

N. Engheta, A. Salandrino, and A. Alù, “Circuit elements at optical frequencies: nanoinductors, nanocapacitors, and nanoresistors,” Phys. Rev. Lett. 95, 095504 (2005).
[Crossref]

T. T. Koutserimpas and R. Fleury, “Nonreciprocal gain in non-Hermitian time-Floquet systems,” Phys. Rev. Lett. 120, 087401 (2018).
[Crossref]

E. Yablonovitch, “Self-phase modulation of light in a laser-breakdown plasma,” Phys. Rev. Lett. 32, 1101–1104 (1974).
[Crossref]

T. Tanabe, M. Notomi, H. Taniyama, and E. Kuramochi, “Dynamic release of trapped light from an ultrahigh-Q nanocavity via adiabatic frequency tuning,” Phys. Rev. Lett. 102, 043907 (2009).
[Crossref]

M. Silveirinha and N. Engheta, “Tunneling of electromagnetic energy through subwavelength channels and bends using ε-near-zero materials,” Phys. Rev. Lett. 97, 157403 (2006).
[Crossref]

B. Edwards, A. Alù, M. E. Young, M. Silveirinha, and N. Engheta, “Experimental verification of epsilon-near-zero metamaterial coupling and energy squeezing using a microwave waveguide,” Phys. Rev. Lett. 100, 033903 (2008).
[Crossref]

Phys. Rev. Res. (1)

G. Ptitcyn, M. S. Mirmoosa, and S. A. Tretyakov, “Time-modulated meta-atoms,” Phys. Rev. Res. 1, 023014 (2019).
[Crossref]

Proc. Inst. Electr. Eng. (1)

R. S. Chu and T. Tamir, “Wave propagation and dispersion in space-time periodic media,” Proc. Inst. Electr. Eng. 119, 797–806 (1972).
[Crossref]

Radio Sci. (1)

D. E. Holberg and K. S. Kunz, “Parametric properties of dielectric slabs with large permittivity modulation,” Radio Sci. 3, 273–286 (1968).
[Crossref]

Rendiconti del Circolo Matematico di Palermo (1884-1940) (1)

M. Plancherel and M. Leffler, “Contribution à l’étude de la représentation d’une fonction arbitraire par des intégrales définies,” Rendiconti del Circolo Matematico di Palermo (1884-1940) 30, 289–335 (1910).
[Crossref]

Science (1)

A. M. Shaltout, V. M. Shalaev, and M. L. Brongersma, “Spatiotemporal light control with active metasurfaces,” Science 364, eaat3100 (2019).
[Crossref]

Other (5)

N. Engheta and R. W. Ziolkowski, Metamaterials: Physics and Engineering Explorations (Wiley, 2006).

M. S. Mirmoosa, T. T. Koutserimpas, G. A. Ptitcyn, S. A. Tretyakov, and R. Fleury, “Dipole polarizability of time-varying particles,” arXiv:2002.12297 (2020).

A. Taflove and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method (Artech House, 2005).

A. V. Oppenheim, A. S. Willsky, and H. Nawab, Signals and Systems (Pearson Education, 1996).

D. M. Pozar, Microwave Engineering (Wiley, 2011).

Data Availability

Data that support the findings of this study are available in the main text and the appendices. Additional information is available from the corresponding author upon reasonable request.

Cited By

Optica participates in Crossref's Cited-By Linking service. Citing articles from Optica Publishing Group journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (9)

Fig. 1.
Fig. 1. (a) Temporal evolution of ωl and ϵl (l=1,2) as ωp(t) transitions from ωp=0 (vacuum) to ωp+ such that ϵ+(ω)=4 with ω0=2ω, resulting in ωp+=3ω. (b) Dispersion diagram k+ versus ω+, showing the two solutions for ω+ that achieve momentum conservation. The dashed green line represents ω0.
Fig. 2.
Fig. 2. (a) Electromagnetic waves versus time at z=λ/16 for a transition from ϵ(ω)=1 (vacuum) to ϵ+(ω)=4, with ω0=2ω (the solid lines are analytical results, while the circular markers represent numerical FDTD simulations). (b1) New frequencies ωi [and ϵ+(ωi)] for t>0 and (b2) wave amplitude coefficients versus ω0/ω, considering ωp+=3ω [ϵ+(ω)1 as ω0/ω]. Panels (c1), (c2) are the same, but with ϵ+(ω)=4 [ωp+χ+(ω)ω0 as ω0/ω].
Fig. 3.
Fig. 3. Transmission-line equivalence of our unbounded time-varying dispersive medium. At t=0, the switch is closed, effectively connecting our series tank circuit LpCp.
Fig. 4.
Fig. 4. (a) Dispersion diagram in the complex k- and ω-planes when ϵ+(ω)=40.1i, which results in ωp+3ω and γ=0.1ω when ω0=2ω. Conservation of momentum is achieved when the surfaces [ω+,Re[k+(ω+)]/k] and [ω+,Im[k+(ω+)]/k] simultaneously intersect the Re(k+)/k=1 and Im(k+)/k=0 planes (gray color), respectively. These intersection curves are marked in black and give rise to four complex frequencies that, in this example, form two complex-conjugate pairs in the s plane [the plotted region Re(ω+)>0 only includes one complex frequency per pair]. (b) Evolution of the four complex frequencies versus γ, with the other parameters fixed [ω0 and ωp+ from panel (a)]: at γ=7.01ω we reach a critical point, and the pair of complex frequencies linked to ω0 (ω2f and ω2b) splits into two purely imaginary frequencies for which ϵ becomes purely real and negative; the latter is seen in panel (c). Note that, in this overdamped region (γ>7.01ω), the notation ω2f and ω2b is not strictly rigorous: this pair simply becomes ω2 and ω3, and the associated waves represent non-propagating damping.
Fig. 5.
Fig. 5. (a) Electromagnetic waves versus time at z=λ/16 when ω0 and ωp+ are taken from Fig. 4 and γ=0.5ω (numerical FDTD simulations are marked with circles): we have ω2f and ω2b, corresponding to an underdamped scenario. (b) Separate components of E(z=0,t>0). (c) Two snapshots of the separate components of E+ [the solid and dashed lines represent E(z,t=0+) and E(z,t=T/32), respectively], showing forward and backward propagation for both ω1 and ω2.
Fig. 6.
Fig. 6. Same as Fig. 5 but with γ=7.3ω, yielding an overdamped regime with purely imaginary ω2 and ω3 describing oscillations that do not propagate. This is seen in panel (c): red (E2) and green (E3) curves.
Fig. 7.
Fig. 7. Real and imaginary parts of the complex amplitude coefficients versus γ/ω. As ω2fω2b near the point of critical damping γ=7.01ω, f2 and b2 tend to diverge but with opposite signs [panel (b)], keeping f2+b2 (or else x2+x3) bounded.
Fig. 8.
Fig. 8. (a1) Normalized real and imaginary parts of the complex frequencies ω2f (ω2) and ω2b (ω3) versus ω0χi/ω, for χi=103 (lines) and χi=104 (markers). (a2) Normalized dielectric functions ϵ2f (ϵ2) and ϵ2b (ϵ3). (b) Complex amplitude coefficients f2 (x2) and b2 (x3). (c) E2f (E3) and E2b (E3) versus normalized time, for z=0 and several ω0/ω ratios; note how the black and red plots, corresponding to the underdamped region where f2=b2, are superimposed.
Fig. 9.
Fig. 9. (a) Electromagnetic waves versus time at z=λ/16 for the transition from vacuum to ϵ+(ω)=4iχi, considering ω0=104ω and several values of χi specified in panel (b); all the curves are virtually the same and overlap with the chosen t-axis scale, converging to the solution of the nondispersive lossless case. (b) Zoomed-in view of the transition of panel (a), both for underdamped (solid lines) and overdamped (dashed lines) scenarios.

Equations (64)

Equations on this page are rendered with MathJax. Learn more.

P(ω)=ϵ0χ(ω)E(ω)=ϵ0ωp2ω02ω2E(ω),
d2P(t)dt2+ω02P(t)=ϵ0ωp2E(t),
d2P(t)dt2+ω02P(t)=ϵ0A(t)E(t),
P(ω)=ϵ012πA(ω)*ωE(ω)ω02ω2,
P(t)=th(t,τ)E(τ)dτ.
P˜(s)=ϵ0L{A(t)E(t)}+sP(0)+dPdt(0)s2+ω02,
P(0+)=limssP˜(s)
P(0+)=P(0).
dPdt(0+)=lims[s2P˜(s)sP(0+)].
dPdt(0+)=lims[s2P˜(s)sP(0)]=dPdt(0).
P˜(s)=ϵ0L{A(t)E(t)}s2+ω02,
ω1+ωp2ω02ω2=ω+1+ωp+2ω02ω+2.
ω+=±K±K24ω02ω2(ω2ω02)(ω2ω02ωp2)2(ω2ω02),
K=ω2(ω2+ωp+2ωp2)ω02(ω02+ωp+2).
ω+4(ω2+ω02+ωp+2)ω+2+ω02ω2=0,
ω+=±ω2+ω02+ωp+2±(ω2+ω02+ωp+2)24ω02ω22.
E=eiωteikz,
H=ϵη0eiωteikz,
D=ϵ0ϵeiωteikz,
dPdt=iωϵ0(ϵ1)eiωteikz.
E+=eikzl=12(fleiωlt+bleiωlt),
H+=eikz1η0l=12ϵl(fleiωltbleiωlt),
D+=eikzϵ0l=12ϵl(fleiωlt+bleiωlt),
dP+dt=eikziϵ0l=12ωl(ϵl1)(fleiωltbleiωlt),
[1111ϵ1ϵ1ϵ2ϵ2ϵ1ϵ1ϵ2ϵ21ϵ11ϵ11ϵ21ϵ2][f1b1f2b2]=[1ϵϵ1ϵ],
[f1,b1]=ϵ2ϵ2ϵ(ϵ2ϵ1)(ϵ±ϵ1),
[f2,b2]=ϵϵ12ϵ(ϵ2ϵ1)(ϵ±ϵ2),
E(z=0,ω)=FT{E(z=0,t)U(t)+E+(z=0,t)U(t)}=iωωl=12(iflωωl+iblω+ωl),
ϵ1[ω1ϵϵ+(ω)ω]ϵ+(ω),
ϵ2[ω2ϵ+(ω)ω0]ϵϵ+(ω)(ωω0)20,
×E=μ0Ht,
×H=ϵ0Et+Pt+J,
××E=μ0ϵ02Et2μ02Pt2μ0Jt.
1μ0××E˜(r,s)=ϵ0[s2E˜(r,s)sE(r,0)E(r,0)t][s2P˜(r,s)+sJ˜(r,s)J(r,0)].
1μ0××E˜(r,s)+ϵ0s2(s2+ω02+ωp+2)s2+ω02E˜(r,s)=ϵ0[sE(r,0)+E(r,0)t]sJ˜(r,s)+J(r,0).
(k2+μ0ϵ02t2)E=μ02Pt2μ0Jt,
ϵ0(ω2+s2s2+ω02+ωp+2s2+ω02)E˜(z,s)=ϵ0[sE(z,0)+Et(z,0)]sJ˜(z,s)+J(z,0),
E˜(z,s)=(s2+ω02)ϵ0[sE(z,0)+Et(z,0)]sJ˜(z,s)+J(z,0)ϵ0(ω2s2+ω2ω02+s4+s2ω02+s2ωp+2).
s4+(ω2+ω02+ωp+2)s2+ω2ω02=(s2s12)(s2s22)=(s2+ω12)(s2+ω22)
E(z,t=0)=cos(kz),
Et(z,t=0)=ωsin(kz).
E˜(z,s)=(s2+ω02)sE(z,0)+Et(z,0)(s2+ω12)(s2+ω22)(s2+ω02)sJ˜(z,s)J(z,0)ϵ0(s2+ω12)(s2+ω22).
E(z,t)=E1++E1+E2++E2,
E1±=ω02ω12ω22ω1212(1±ωω1)cos(ω1tkz),
E2±=ω02ω22ω22ω1212(1±ωω2)cos(ω2tkz),
E1±1±ϵ22ϵ2cos(ω1tkz),
E2±ϵ212ϵ2(1±1ω0ωϵ2)cos(ω2tkz).
H1±±1±ϵ22ϵ21η0cos(ω1tkz),H2±±ϵ212ϵ2(1±ωω0ϵ2)ωω0ϵ21η0cos(ω2tkz)0.
L(t)d2P(t)dt2+1C(t)P(t)=E(t),
d2P(t)dt2+γdP(t)dt+ω02P(t)=ϵ0ωp2(t)E(t),
L(t)d2P(t)dt2+R(t)dP(t)dt+1C(t)P(t)=E(t),
ω1+ωp2ω02ω2+iωγ=ω+1+ωp+2ω02ω+2+iω+γ,
E+=eikzl=12(fleslt+blesl*t),
H+=eikz1η0l=12(ϵlflesltϵl*blesl*t),
D+=eikzϵ0l=12(ϵlfleslt+ϵl*blesl*t),
dP+dt=eikzϵ0l=12(slχlfleslt+sl*χl*blesl*t),
[1111ϵ1ϵ1*ϵ2ϵ2*ϵ1ϵ1*ϵ2ϵ2*s1χ1s1*χ1*s2χ2s2*χ2*][f1b1f2b2]=[1ϵϵsχ],
[1111ωω1ωω1ωω2ωω2(ωω1)2(ωω1)2(ωω2)2(ωω2)2ω1ωω1ωω2ωω2ω][f1b1f2b2]=[1111].
[f1,b1]=12ω22ω2ω22ω12ω1ω2(ω1±ω),
[f2,b2]=12ω2ω12ω22ω12ω2ω2(ω2±ω).
[1111ϵ1ϵ1*ϵ2ϵ3ϵ1ϵ1*ϵ2ϵ3s1χ1s1*χ1*s2χ2s3χ3][f1b1x2x3]=[1ϵϵsχ].
0e2σt|E+(t)|2dt=12π|FT{eσtE+(t)U(t)}(ω)|2dω=12π|Lr{E+(t)}(σ+iω)|2dω,
0e2σt|E(t)|2dt=12π|FT{eσtE(t)U(t)}(ω)|2dω=12π|Ll{E(t)}(σ+iω)|2dω.
e2σ|t||E(t)|2dt=12π|FT{e|σ|tE(t)}(ω)|2dω=12π|Ll{E(t)}(σ+iω)+Lr{E(t)}(σ+iω)|2dω,