Abstract

Recently, the temporal control of graphene carrier density has emerged as a viable means to create various frequency shifting, modulation, and sensing photonic devices. Here we describe a general theoretical approach to calculate the graphene plasmon transformation after rapid changes of the Fermi level and carrier density. The approach is based on solving the Maxwell equations supplemented by the microscopic current equation. We derive formulas for the amplitudes of the transmitted and reflected plasmons after a rapid carrier density drop. The relation of these amplitudes and the Fourier transformed finite-difference time-domain (FDTD) fields is also established by introducing the concept of differential spectral transformation of wavepackets. The results of the analytical and FDTD approaches refute the claims of plasmon amplification under rapid carrier changes that appeared in recent theoretical studies. The presented theoretical and computational approaches form a basis of time-varying electromagnetics of graphene plasmonics.

© 2018 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

1. INTRODUCTION

The ability to control guided-mode propagation through the refractive index allows one to engineer various optoelectronic devices, for example, ultrafast modulators in telecommunications [1]. The refractive index can be changed, for example, by using an external electric field (electro-refraction) or by manipulating the carrier density (plasma-dispersion effect). The smallness of these effects in conventional materials stimulates researchers to look for novel materials with enhanced nonlinear properties. The appearance of graphene—a promising candidate for photonic applications in infrared and terahertz ranges—has attracted significant interest in both experimental studies and their theoretical understanding as reviewed recently [2,3].

Graphene sheets with finite carrier density are capable of supporting propagating modes—graphene plasmons [46]. Their propagation can be directed by creating various sheet corrugations [7], curved landscapes [8], or topological effects [9]. Graphene-based nanowaveguides hold promise as printed interconnects [10]. Strong nonlinearities in graphene can enable all-optical modulation [11,12]. Applying gate voltage one can also tune graphene conductivity and realize two-dimensional (2D) transformation optics using graphene [13], dynamically controlled devices [14], dynamically tunable and active hyperbolic metamaterials [15], and reconfigurable terahertz nanodevices [16]. The conductivity control is an apparent advantage of graphene over metal layers and metal interfaces [13] in addition to lower losses and stronger confinement [17], which can further be improved by making heterostructures [1820]. Depending on the characteristic time scale, the conductivity changes can be slow or fast compared to the plasmon period. In particular, the variation of graphene properties on a scale of a few 10s of femtoseconds was demonstrated [2123]. The instantaneous response of graphene to ultrafast optical fields, elucidating the role of hot carriers on sub-100 fs time scales, has also been recently studied [24]. In general, the control by external voltage in graphene extends other directions of nonlinear control of optical properties of 2D materials, for example, using terahertz fields for ultrafast optical modulation in semiconductor quantum wells [25,26].

The feasibility of controlling graphene properties in time necessitates the development of a theoretical understanding of plasmon transformation under time variations of graphene parameters, in particular, for rapid density changes. In general, the transformation of waves in time-varying media has a rather long history. Most of the studies considered the transformation of waves in bulk materials [2729]. The scattering of surface waves under rapid carrier injection was also investigated for a plasma half-space [30,31] and a plasma layer [32]. The consideration of wave scattering, either bulk or surface, requires a need for careful consideration of material equations for the medium, for example, plasma [33] or polar molecules [34].

The problem of temporal graphene plasmon transformation was addressed very recently in Refs. [35,36]. Among other regimes, these studies considered step-like variations of the Fermi level and calculated the amplitudes of the surface plasmons excited at the temporal discontinuity. A particularly noticeable prediction is the appearance of plasmon amplification. In Ref. [35], the plasmon amplification occurs if the Fermi level rapidly decreases. In contrast, in Ref. [36], the plasmon amplification was predicted for rapid density increases. In our opinion, there are at least two issues with these predictions. First, solving the same problem, these studies provide different transmission and reflection coefficients for the plasmons for step-like variations of carrier density; compare Eq. (10) in Ref. [35] and Eq. (7) in Ref. [36]. Second, and more fundamental, the appearance or disappearance of carriers under a rapid Fermi level variation cannot inject electromagnetic energy into the plasmon. Energy injection is possible under several scenarios: carriers recombine producing photons (or in this case, plasmons) or external forces perform work on the system (parametric processes). To achieve plasmon emission one essentially needs negative conductivity, or population inversion. Although it is possible to obtain in graphene [37,38], this was not present in the problem considered in Refs. [35,36]. Parametric processes are claimed to be responsible for the amplification in Ref. [36]. The energy of a graphene plasmon consists of the energy of the electromagnetic field and the kinetic energy of the moving carriers (current). Since the change of density neither increases the energy of the electromagnetic field nor the kinetic energy of the carriers, there is no parametric mechanism that can inject energy into the plasmons. A resolution of these paradoxes is important both for theoretical advancement of ultrafast graphene plasmonics as well as for creation of novel optoelectronic devices.

In this paper we solve the problem of graphene plasmon scattering under a rapid time variation of carrier density using the Maxwell equations and the microscopic current equation. Our model is not limited by the quasi-static plasmon approximation adopted in Refs. [35,36]. While our general approach is applicable to either decrease or increase, we consider explicitly only the case of rapid decrease. The small time scale of the changes also justifies the neglect of collisions, which we adopted here for the sake of shedding some light on energy redistribution after the temporal discontinuity. We investigate the frequencies of the transformed plasmons and derive formulas for their amplitudes. A comparison of the energies of the excited modes shows that the plasmon amplification does not exist at temporal discontinuities, which is in contrast to the predictions of Refs. [35,36]. We also study the plasmon scattering using finite-difference time-domain (FDTD) simulations and present a correct procedure to relate the FDTD Fourier transforms to the transmission and reflection coefficients for monochromatic waves. Finally, we discuss the culprits behind the predictions of plasmon amplification in Refs. [35,36].

2. ANALYTICAL SOLUTION OF THE SCATTERING ON TIME DISCONTINUITY

A. Dispersion and Energy of Graphene Plasmons for Constant Parameters

We take a surface plasmon at frequency ω propagating along a graphene sheet surrounded by a dielectric with permittivity ϵ; see Fig. 1(a). In all numerical results we set ϵ=1. The electromagnetic field of a transverse magnetic (TM) plasmon has three components {Hz,Ex,Ey}. The tangential Ex component can be written as

Ex(x,y,t)=E0eihxiωtϰ|y|,
where h is the propagation wavevector, ϰ is the decay constant, and E0 is the amplitude, which can be assumed to be real. The magnetic field at y=0+ is H0=iωE0/(cϰ). The equation for the graphene current in the Drude model is
jx=σEx,σ=icΩ/ω,
where Ω is the frequency parameter characterizing the graphene response. It conveniently includes the carrier concentration, temperature, band structure, etc. Specific values for Ω are not required since the solution will depend only on the ratio ω/Ω. For graphene, this parameter is ΩEfN, where Ef is the Fermi level and N is the electron density. For a regular electron gas, ΩN. In Eq. (2) we neglected collisions for the sake of simplicity. Using the standard boundary conditions across the current sheet, we arrive at the following dispersion for the TM plasmon:
h(ω)=ωcϵ1+ϵω24π2Ω2,ϰ(ω)=ϵω22πcΩ.
Figure 2(a) shows the phase nph=ch/ω and group indices ngr=cω/h obtained from Eq. (3). At small frequencies, ω/Ω1, the plasmon is weakly localized and both indices are close to 1. At large frequencies, ω/Ω1, they grow linearly with frequency and hϰ. In this regime (also known as the quasi-static, nonretarded, or plasmon approximation) the plasmon becomes strongly confined to the sheet [39].

 figure: Fig. 1.

Fig. 1. (a) Illustration of a surface plasmon propagating along a graphene sheet at t<0. (b) Time dependence of the graphene carrier density. (c) Dispersion diagram showing the frequency transformation of the initial plasmon when the carrier density decreases from N1 to N2. The lines labeled by 1 and 2 are the dispersion curves for the plasmons at densities N1 and N2, respectively. The shaded region shows the continuous spectrum for bulk waves; the hatched region defines the spectrum of the generated transient radiation. The dashed lines h=ϵ|ω|/c are the light lines.

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 figure: Fig. 2.

Fig. 2. (a) Phase nph and group ngr refractive indices for graphene plasmons. The dashed line shows the asymptotics nphω/(2πΩ). (b) Decomposition of the plasmon energy W0 into the electric WE, magnetic WH, and kinetic Wk parts.

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The time-averaged plasmon energy W0=Wf+Wk consists of the energies of its electric WE and magnetic WH fields (Wf=WE+WH) outside of the sheet and the kinetic energy Wk of the oscillating graphene carriers:

WE,H=E02ϵ4πϰ(h2ϰ2±1),Wk=|j0|2cΩ,
where j0=σE0 and we adopted the convention that the real field is twice the real part of the complex field. The energies obey WE=WH+Wk. Note that Wk depends only on the parameter Ω since it has to be uniquely determined by the coupling of the current and the electromagnetic field to enable the energy exchange between them.

The dependence of the relative energy components on frequency is shown in Fig. 2(b). With increasing ω/Ω, which corresponds to increasing plasmon confinement, the magnetic component decreases while the kinetic one increases.

B. Formulation of the Temporal Scattering Problem

We assume that initially, at t<0, there is a surface plasmon with properties described in Section 2.A propagating along a graphene sheet. We denote the parameters of the initial plasmon by subscript 1: ω1, Ω1, h1, and ϰ1. The amplitudes of Ex and Hz fields are E0 and H0, respectively. At t=0, the Fermi level rapidly decreases and, therefore, the carrier density drops from N1 to N2; see Fig. 1(b). This translates into a change from Ω1 to Ω2, which can be described by the jump parameter

γ=Ω2/Ω1=Ef2/Ef1<1
as in Refs. [35,36]. We are interested in finding the fields created by the temporal discontinuity defined by Eq. (5).

C. Frequencies of Scattered Waves

The scattering results in the creation of two plasmons and transient bulk radiation going to y±. One plasmon propagates in the +x direction (transmitted) and the other propagates in the x direction (reflected). Their frequencies can be found using the invariance of the spatial dependence eih1x set by the initial plasmon and their dispersion relation at Ω2, as illustrated in Fig. 1(c). The initial dispersion at Ω1 defines the relation between h1 and ω1. After the density changes, the dispersion for Ω2 gives two frequencies ±ω2 that correspond to the fixed h1. One frequency describes the transmitted plasmon, the other—reflected. For γ<1, the plasmons will be frequency downshifted while the bulk radiation will always be frequency upshifted. We will analyze the scattering at several values of ω1/Ω1 corresponding to various degrees of plasmon localization; see the dispersion curves in Fig. 2(a).

Figure 3(a) shows the change in the relative plasmon frequency ω2/ω1 as a function of Ω2/Ω1. For all values of ω1/Ω1, the frequency decreases with decreasing Ω2/Ω1; in the quasi-static regime ω1/Ω11, ω2/ω1γ, as in Ref. [35]. It is quite interesting that as the carrier density N2 decreases, the ratio ω2/Ω2 increases, making the excited plasmon more and more confined; see Fig. 3(b).

 figure: Fig. 3.

Fig. 3. Frequency ω2 of the excited plasmon relative (a) to the initial frequency ω1 and (b) to the frequency parameter Ω2 as a function of Ω2/Ω1.

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D. Amplitudes of Scattered Waves

To find the scattered fields eih1x, we use the Maxwell equations (in Gaussian units)

ih1EyExy=1cHzt,
Hzy=ϵcExt+4πcjxδ(y),
ih1Hz=ϵcEyt,
where δ(y) is the delta function describing the sheet current. Integrating Eq. (6) over the infinitely small jump time, we obtain continuity of the fields. Equations 6(a)6(c) should be supplemented by a material equation describing the current with a corresponding initial condition at t=0+. It is reasonable to assume that the decrease of carrier density (basically, the carrier removal) leaves the velocity of the remaining carriers unchanged. This gives a sudden reduction of current at t=0. We can represent the current at t<0 as two components: j1=j1(1)+j1(2), where j1(1) disappears after the jump and j1(2) remains after the jump. These components are j1(1,2)=E0icΩ1(1,2)/ω1, where Ω1(1)+Ω1(2)=Ω1 and Ω1(2)=Ω2. This gives
jx(t=0+)jx(t=0)=j1(2)j1(1)+j1(2)=Ω1(2)Ω1(1)+Ω1(2)=Ω2Ω1.
Thus, the current at t>0 obeys the time-domain Drude equation with the derived initial condition:
jxt=cΩ2Ex,jx(t=0+)=Ω2Ω1jx(t=0).
The fields and current at t=0 are given by the initial plasmon.

To solve Eqs. (6) and (8) with the specified initial conditions, we use the Laplace transform technique [30,32]. From the system of linear equations, we express the transform E˜x(x,y,p) of Ex(x,y,t), where p is the Laplace variable, as

E˜x(x,y,p)=Ex(x,y,0)p+iω1+A(p)eih1xϰ(p)|y|,
where ϰ(p)=h12+ϵp2/c2 and the coefficient A(p) defines the free-wave part of the solution:
A(p)=iϰp(Ω2Ω1)E0ω1Ω2(p+iω1)D(p),D(p)=ϰ(p)+ϵp22πcΩ2.
We can now apply the inverse Laplace transform to Eq. (10) to find the field Ex(x,y,t) in the time domain. By closing the integration path, we represent Ex(x,y,t) in terms of contributions from the poles of Eq. (9) and from the integrals along the branch cuts of ϰ(p). The residues at the poles give the amplitudes of the transmitted and reflected plasmons. The integrals along the branch cuts describe the transient processes.

Evaluating the residues at p=iω2, which is the solution of D(p)=0, we obtain the amplitudes of the transmitted and reflected plasmons:

Ext,r=πcϰ2ω22(Ω2Ω1)ω1(ω1ω2)(ϵω22+2π2Ω22)E0.
Defining ξ=ω2ϰ1/(ω1ϰ2), we can express the transmission and reflection coefficients for Ex and Hz fields using Eq. (11):
tE=Ext/E0,rE=Exr/E0,
tH=Hzt/H0=ξtE,rH=Hzr/H0=ξrE.

E. Energy Balance

The energy balance not only gives us physical insight into the temporal scattering but also provides justification for the solution. In the absence of collisions, the energy of the initial plasmon W0 should transform into the energy of the excited (transmitted Wt and reflected Wr) plasmons and transient bulk radiation Wb. Also, the removed carriers had some kinetic energy, which can be considered as loss Wl. The energy balance becomes

W0=Wt+Wr+Wb+Wl.
The plasmon energies can be directly calculated using Eqs. (4) and (11). The loss is, apparently, Wl=(1Ω2/Ω1)Wk. The bulk energy Wb can be calculated from the branch cut integrals and represented in terms of its angular density wb(θ) using
Wb=02πdθwb(θ),wb(θ)=c2h14π2|A(iω(θ))|2cos2θ,
where ω(θ)=ch1/(ϵcosθ) is the frequency of the wave component propagating at an angle θ with respect to +x. We confirmed numerically that Eq. (14) holds with accuracy exceeding 1012. This adds credibility to our results.

The results for several values of ω1/Ω1 are shown in Fig. 4. In general, the plasmon transformation depends strongly not only on the jump Ω2/Ω1 but also on the initial condition ω1/Ω1. The lost energy Wl increases linearly with decreasing Ω2/Ω1. Furthermore, the loss becomes higher with increasing confinement of the initial plasmon or ω1/Ω1 as expected from the energy distribution in the initial plasmon; see Fig. 2(b). In contrast, the transient radiation Wb decreases with increasing confinement. The transmitted plasmon energy Wt always decreases with decreasing Ω2/Ω1. The reflected plasmon energy Wr increases with decreasing Ω2/Ω1. Interestingly, the excitation of the plasmons takes place even when the carrier density becomes arbitrarily small but finite. This process is especially pronounced when the initial plasmon is strongly confined ω1/Ω1=15. In this case, when most of the existing carriers disappear, the electromagnetic energy, which is Wf/W0=0.5, seems to drive the remaining carriers so that both transmitted and reflected plasmons are excited almost equally, Wt/W0Wr/W00.25.

 figure: Fig. 4.

Fig. 4. Energy distribution after temporal scattering as a function of Ω2/Ω1 for (a) ω1/Ω1=15 and (b) ω1/Ω1=1.

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The transient radiation propagates from the graphene sheet, and its angular distribution is described by Eq. (15). Figure 5 shows typical angular distributions. The weakly localized plasmon, ω1/Ω1=1, produces a rather narrow radiation peak at small angle. The strongly localized plasmon, ω1/Ω1=15, emits almost vertically in a very broad angular range.

 figure: Fig. 5.

Fig. 5. Far-field angular distribution wb(θ)/W0 for Ω2/Ω1=0.5 and several values of ω1/Ω1.

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F. Temporal Scattering in Quasi-Static Approximation

Since the quasi-static approximation is often used, it is instructive to derive the corresponding transmission and reflection coefficients without using the general formulas in Eqs. (11)–(13). It was shown in Section 2.A that in this limit the plasmon energy consists of the electric and kinetic parts. The magnetic field is rather small |Hz/Ex|=2πΩ/(ϵω)1. According to Fig. 4(a), the temporal discontinuity produces scattered (transmitted and reflected) plasmons and very weak bulk radiation. Also, a significant part of the initial plasmon energy is taken away by the removed carriers. Thus, we can write the electric field of the scattered plasmons at t>0 as

Ex(x,y,t)=E0(tEeiω2t+rEeiω2t)eih1xϰ2|y|.
The current can be obtained from Eq. (16) using the Drude model. To find tE and rE in Eq. (16), we have to apply the boundary conditions that relate the dominant components of the modes: electric field and surface current. The continuity of Ex and current jump in Eq. (8) give
tE=(1+γ)/2,rE=(1γ)/2.
Note that the derived formulas do not rely on the continuity of Hz since Hz is rather small in the quasi-static approximation. However, Hz can still be obtained from Eq. (17). Using Eq. (13) with ξγ gives
tH=γ(1+γ)/2,rH=γ(1γ)/2.
Our derived quasi-static results in Eq. (18) agree well with the general formulas in Eqs. (11)–(13) for ω1/Ω11, as seen from Fig. 6.

 figure: Fig. 6.

Fig. 6. Comparison of the plasmon transmission tH and reflection rH coefficients obtained here (general and quasi-static results) and available from the literature for ω1/Ω1=15. Equation (11) is the general formula and Eq. (18) is the quasi-static formula derived here. Equation (19) is from Ref. [35]. Equation (20) is from Ref. [36].

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G. Analysis of Transmission and Reflection Coefficients

Here we compare our results with those of Refs. [35,36] where the scattering of a graphene plasmon on a temporal discontinuity was considered in the quasi-static approximation, i.e., for a strongly confined plasmon. In Ref. [35], matching the field distributions of the incident plasmon and excited (reflected and transmitted) plasmons using the continuity of Hz and its time derivative resulted in, see Eq. (10) in Ref. [35],

|tH|=(1+γ)/(2γ),|rH|=|1γ|/(2γ).
Reference [36] developed a propagator matrix using the continuity of Hz and Ex and obtained, see Eq. (7) in Ref. [36],
tH=(1+γ)/2,rH=(1γ)/2.
Apparently, Eqs. (19) and (20) are quite different considering that they were obtained without any restrictions on the value of the jump parameter γ given by Eq. (5). One can immediately see from Eq. (19) that tH>1 for any γ<1, i.e., Ω2<Ω1. Since the mode profile was assumed to be constant due to the strong confinement, the energy of the transmitted plasmon exceeds that of the incident. This was pointed out and commented on in Ref. [35] as some energy injection into the system. However, it is a puzzling result because of the lack of any energy source. Indeed, a decrease of the energy level removes some carriers but by no means injects energy into the system. In contrast to Eq. (19), Eq. (20) gives tH>1 for γ>1, i.e., Ω2>Ω1. The corresponding gain was attributed in Ref. [36] to a parametric process. However, carrier injection simply adds carriers that initially do not form any current and therefore cannot contribute to the plasmon kinetic energy. The formulas in Eq. (20) also have a different problem due to their asymptotic values at γ0: tH,rH1/2. From Eq. (13) this means that the electric field of the exited plasmon diverges: tE, rE since ξω2/Ω20; see Fig. 3(b). In fact, the magnetic Hz component of the excited plasmons should vanish since in the limit Ω2/Ω10 we obtain extremely well-confined plasmons, which have no magnetic field; see also Fig. 2(b).

Having discussed the issues with Eqs. (19) and (20), let us study how our results, given by Eqs. (11)–(13), compare with them; see Fig. 6. One can clearly see that our results agree neither with Eq. (19) nor with Eq. (20). In contrast to increasing tH with decreasing γ obtained in Ref. [35], our tH decreases discarding the claim of gain appearance; see also Fig. 4(a). Moreover, in contrast to rHtH1/2 at Ω2/Ω10, our rH and tH vanish as expected. In general, it is quite possible to obtain |tE,H|>1 since the coefficients define only the field values at y=0. Indeed, our simulations show that, for example, for ω1/Ω1=1 we obtain |tE,H|>1 at γ=0.5. However, due to the change of the mode profile, the plasmon energy always decreases.

Where is the origin of mistakes in Refs. [35,36]? Both of them considered the problem explicitly in the quasi-static approximation. In this approximation, the modes are defined by their electric fields and surface currents. Thus, one should use the boundary conditions for these components. However, the behavior of the microscopic current at the temporal discontinuity, see Eq. (7), was not employed in Refs. [35,36] as a necessary condition to assess the plasmon transformation.

3. EXTRACTING TRANSMISSION FROM FDTD SIMULATIONS WITH TEMPORAL DISCONTINUITIES

While analytical results can be derived in some limited geometries, numerical simulations are required in most practical situations related to device modeling. From this point of view, it is instructive to compare the analytical results with FDTD simulations. We applied an in-house developed 2D FDTD code [40] to simulate the temporal scattering of a graphene plasmon. Initially, a plasmon wavepacket propagating in the +x direction along the graphene sheet is created; see Fig. 7, frame t=0. The current discontinuity at t=0, see Eq. (7), gives rise to its scattering. Figure 7, frame t=0.15/Ω1shows a typical distribution of Ex(x,y) after the jump when some transient radiation escapes from the sheet region. Subsequently, one observes the separation of the field into the transmitted and reflected plasmon wavepackets; see Fig. 7, frame t=7/Ω1.

 figure: Fig. 7.

Fig. 7. Snapshots of the electric field distribution Ex(x,y,t) produced by a plasmon wavepacket with central frequency ω1/Ω1=10 at three time moments t=0, 0.15/Ω1, 7/Ω1. Initially the wavepacket propagates along the graphene sheet at y=0 in the +x direction (frame t=0). Soon after the carrier density jump Ω2/Ω1=0.2 at t=0, transient processes take place (frame t=0.15/Ω1). Finally, two propagating plasmons are formed (frame t=7/Ω1).

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The spectra of the electric and magnetic fields of the transmitted plasmon were computed at some specific points near the sheet using the running Fourier transform and plotted together with the incident spectra in Fig. 8(a). According to the analytical dependence of the frequency conversion ω2(ω1) shown in Fig. 8(b), the components at ω1/Ω1=15 should be transformed to ω2/Ω1=11.8; this agrees well with the simulation result in Fig. 8(a). Focusing on amplitudes, according to Eqs. (11)–(13), see also Fig. 6, at ω1/Ω1=15 and Ω2/Ω1=0.6 we should have tE=Ext/E0=0.924 and tH=Hzt/H0=0.704. None of these is seen in Fig. 8(a). Instead, Fig. 8(a) shows that the spectrum of the electric field of the transmitted plasmon is higher than of the incident; the magnetic spectrum is only slightly lower. Apparently, the textbook procedure of simply dividing the spectra does not work. Furthermore, the transmitted spectra are visibly narrower than the incident spectra. In fact, to find the transmission coefficient from the FDTD results one needs not only to apply the spectral shift but also the spectral compression or expansion.

 figure: Fig. 8.

Fig. 8. Obtaining the transmission coefficients tE,H from the FDTD simulations. (a) Absolute value of the incident Exi(ω), Hzi(ω) and transmitted Ext(ω), Hzt(ω) field spectra for Ω2/Ω1=0.6. (b) Analytically calculated frequency ω2 of the excited plasmon as a function of the frequency ω1 of the incident plasmon. (c) Differential spectral transormation f=dω2/dω1, which is the derivative of ω2(ω1) shown in frame (b). (d) Comparison of the transmission coefficients obtained numerically from the FDTD simulations and analytically using Eqs. (11)–(13).

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Let us define the magnetic field spectrum of the incident plasmon wavepacket by Hzi(ω) and the transmitted spectrum by Hzt(ω). How can we find the transmission coefficient for a monochromatic plasmon at some fixed ω1 within the spectral range of Hzi(ω)? After a temporal jump, a monochromatic plasmon at ω1 will create a plasmon at ω2. Let us consider a small spectral interval Δω1 of the wavepacket spectrum. The corresponding spectral components will be transformed into Δω2 after the temporal discontinuity according to

Hzt(ω2)Δω2=tHHzi(ω1)Δω1,
where tH is the required transformation coefficient. Thus, the following procedure should be used to find tH:
tH=f·Hzt(ω2)Hzi(ω1),f=dω2(ω1)dω1,
where we defined the differential spectral transformation coefficient f. The same procedure can be used for tE, rE or rH. Without temporal discontinuity or for frequency shifts ω2=ω1+const we obtain Δω2=Δω1 and f=1.

Thus, according to derived Eq. (22) the transmission coefficients tH or tE for monocromatic waves can be found from the wavepacket spectra by taking the ratio of the spectral components at corresponding frequencies and multiplying by the spectral transformation factor f. Figure 8(c) shows the spectral transformation factor calculated analytically based on the frequency dependence ω2(ω1) shown in Fig. 8(b). In this specific case, this factor differs significantly from 1 and therefore plays a crucial role in extracting the transmission coefficients from the FDTD simulations. Figure 8(d) compares the analytical results obtained from Eq. (11) and the numerical results calculated using Eq. (22). Accounting for the spectral transformation allowed us to match perfectly the analytical results in the whole spectral range of the incident wavepacket. We also note that Ref. [35] presented the spectra obtained from time-domain finite element method (FEM) modeling. However, the transmission was evaluated as the ratio of the peak heights. This may have led to the erroneous estimation of the transmission coefficient since the value of the spectral transformation was not taken into account.

The described procedure can also be applied to quasi-monochromatic wavepackets if the transmission or reflection coefficients at the central frequency, rather than a finite range, are needed. This allows one to integrate Eq. (21) over the frequencies and pull tH out of the integral

dωHzt(ω)=tHdωHzi(ω).
In practice, the complex spectrum may show oscillations that correspond to the delay of the wavepacket so it is easier to deal with absolute values. Since the tE,H vary rather weakly over the spectral ranges in Fig. 8(d), we can evaluate them by applying this quasi-monochromatic procedure, which requires finding the areas under the spectra in Fig. 8(a). The area of each peak is proportional to the product of its width and height. For example, we obtain |tH|=14.2·1.92/(15.4·2.50)0.71 and |tE|=46·1.92/(38·2.50)0.93, which agree well with results shown in Fig. 8(d). Similar estimates can be done for |rE,H|. Thus, to find the transmission and reflection coefficients for monochromatic waves in a nonstationary case using FDTD simulations we need to evaluate the areas under the spectal peaks, not the heights as in a stationary case.

4. CONCLUSION

To conclude, the temporal scattering of a graphene plasmon by a rapid carrier density change was considered. By solving the Maxwell equations supplemented by the material equation for the sheet current using the Laplace transform technique we obtained general formulas for the temporal transmission and reflection coefficients as well as for the transient bulk radiation in the case of carrier density decrease. A procedure to derive the coefficients in the quasi-static limit using the boundary conditions for the electric field and current was proposed and its results were verified by a comparison with the general formulas. It was shown that the energy of the excited plasmons and bulk radiation is smaller than the initial plasmon energy by the amount of the kinetic energy of the removed carriers. This refutes the claim of plasmon amplification in Ref. [35]. The presence of carrier collisions, which are neglected in this study, will lead to the damping of the excited surface plasmons but should not significantly change the energy balance immediately after the rapid density change.

When new carriers are created, one has to solve separately for the dynamics of the existing and newly created carriers since they have different initial velocities. Among other things, this leads to the appearance of the so-called free-streaming mode, which consists of a self-consistent distribution of direct current and magnetic field [41]. This mode takes some energy of the initial plasmon. Thus, a rapid increase of carrier density always reduces the energy, in contrast to the predictions of Ref. [36].

The temporal scattering was also modeled using the FDTD approach. The procedure to find the temporal transmission and reflection coefficients was developed and its results agreed with the analytical ones. It was shown that finding the temporal transmission and reflection coefficients from the FDTD simulations of wavepacket propagation requires not only finding the Fourier transforms of the wavepackets but also accounting for their differential spectral compression or expansion. Besides scattering, rapid changes in graphene, as in plasma layers [42], can create a viable mechanism of launching graphene plasmons using optical beams without using various corrugations or coupling prisms.

Funding

Ministry of Education and Science of the Russian Federation (Minobrnauka) (3.3854.2017/4.6).

REFERENCES

1. G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4, 518–526 (2010). [CrossRef]  

2. F. J. García de Abajo, “Special issue “2D materials for nanophotonics”,” ACS Photon. 4, 2959–2961 (2017). [CrossRef]  

3. Q. Guo, C. Li, B. Deng, S. Yuan, F. Guinea, and F. Xia, “Infrared nanophotonics based on graphene plasmonics,” ACS Photon. 4, 2989–2999 (2017). [CrossRef]  

4. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6, 749–758 (2012). [CrossRef]  

5. J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012). [CrossRef]  

6. Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012). [CrossRef]  

7. T. M. Slipchenko, M. L. Nesterov, R. Hillenbrand, A. Y. Nikitin, and L. Martín-Moreno, “Graphene plasmon reflection by corrugations,” ACS Photon. 4, 3081–3088 (2017). [CrossRef]  

8. D. Smirnova, S. H. Mousavi, Z. Wang, Y. S. Kivshar, and A. B. Khanikaev, “Trapping and guiding surface plasmons in curved graphene landscapes,” ACS Photon. 3, 875–880 (2016). [CrossRef]  

9. D. A. Kuzmin, I. V. Bychkov, V. G. Shavrov, and V. V. Temnov, “Plasmonics of magnetic and topological graphene-based nanostructures,” Nanophotonics 7, 597–611 (2018). [CrossRef]  

10. G. Lovat, P. Burghignoli, and R. Araneo, “Low-frequency dominant-mode propagation in spatially dispersive graphene nanowaveguides,” IEEE Trans. Electromagn. Compat. 55, 328–333 (2013). [CrossRef]  

11. W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14, 955–959 (2014). [CrossRef]  

12. J. D. Cox and F. J. García de Abajo, “Transient nonlinear plasmonics in nanostructured graphene,” Optica 5, 429–433 (2018). [CrossRef]  

13. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011). [CrossRef]  

14. T.-T. Kim, H.-D. Kim, R. Zhao, S. S. Oh, T. Ha, D. S. Chung, Y. H. Lee, B. Min, and S. Zhang, “Electrically tunable slow light using graphene metamaterials,” ACS Photon. 5, 1800–1807 (2018). [CrossRef]  

15. J. S. T. Smalley, F. Vallini, X. Zhang, and Y. Fainman, “Dynamically tunable and active hyperbolic metamaterials,” Adv. Opt. Photon. 10, 354–408 (2018). [CrossRef]  

16. P.-Y. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-based plasmonic platform for reconfigurable terahertz nanodevices,” ACS Photon. 1, 647–654 (2014). [CrossRef]  

17. M. Jablan, M. Soljačić, and H. Buljan, “Plasmons in graphene: fundamental properties and potential applications,” Proc. IEEE 101, 1689–1704 (2013). [CrossRef]  

18. A. V. Kretinin, Y. Cao, J. S. Tu, G. L. Yu, R. Jalil, K. S. Novoselov, S. J. Haigh, A. Gholinia, A. Mishchenko, M. Lozada, T. Georgiou, C. R. Woods, F. Withers, P. Blake, G. Eda, A. Wirsig, C. Hucho, K. Watanabe, T. Taniguchi, A. K. Geim, and R. V. Gorbachev, “Electronic properties of graphene encapsulated with different two-dimensional atomic crystals,” Nano Lett. 14, 3270–3276 (2014). [CrossRef]  

19. A. Woessner, M. B. Lundeberg, Y. Gao, P. A.-G. A. Principi, M. Carrega, K. Watanabe, M. P. T. Taniguchi, G. Vignale, J. Hone, R. Hillenbrand, and F. H. L. Koppens, “Highly confined low-loss plasmons in graphene-boron nitride heterostructures,” Nat. Mater. 14, 421–425 (2015). [CrossRef]  

20. J. D. Caldwell and K. S. Novoselov, “Mid-infrared nanophotonics,” Nat. Mater. 14, 364–366 (2015). [CrossRef]  

21. I. Gierz, J. C. Petersen, M. Mitrano, C. Cacho, I. C. E. Turcu, E. Springate, A. Stöhr, A. Köhler, U. Starke, and A. Cavalleri, “Snapshots of non-equilibrium Dirac carrier distributions in graphene,” Nat. Mater. 12, 1119–1124 (2013). [CrossRef]  

22. S. Ulstrup, J. C. Johannsen, F. Cilento, J. A. Miwa, A. Crepaldi, M. Zacchigna, C. Cacho, R. Chapman, E. Springate, S. Mammadov, F. Fromm, C. Raidel, T. Seyller, F. Parmigiani, M. Grioni, P. D. C. King, and P. Hofmann, “Ultrafast dynamics of massive Dirac fermions in bilayer graphene,” Phys. Rev. Lett. 112, 257401 (2014). [CrossRef]  

23. M. Trushin, A. Grupp, G. Soavi, A. Budweg, D. De Fazio, U. Sassi, A. Lombardo, A. C. Ferrari, W. Belzig, A. Leitenstorfer, and D. Brida, “Ultrafast pseudospin dynamics in graphene,” Phys. Rev. B 92, 165429 (2015). [CrossRef]  

24. M. Baudisch, A. Marini, J. D. Cox, T. Zhu, F. Silva, S. Teichmann, M. Massicotte, F. Koppens, L. S. Levitov, F. J. García de Abajo, and J. Biegert, “Ultrafast nonlinear optical response of Dirac fermions in graphene,” Nat. Commun. 9, 1018 (2018). [CrossRef]  

25. A. V. Maslov and D. S. Citrin, “Numerical calculation of the terahertz field-induced changes in the optical absorption in quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8, 457–463 (2002). [CrossRef]  

26. S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310, 651–653 (2005). [CrossRef]  

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

28. D. K. Kalluri, Electromagnetics of Time Varying Complex Media: Frequency and Polarization Transformer, 2nd ed. (CRC Press, 2010).

29. K. Qu, Q. Jia, M. R. Edwards, and N. J. Fisch, “Theory of electromagnetic wave frequency upconversion in dynamic media,” Phys. Rev. E 98, 023202 (2018). [CrossRef]  

30. M. I. Bakunov and S. N. Zhukov, “Conversion of a surface electromagnetic wave at the boundary of a time-varying plasma,” Plasma Phys. Rep. 22, 649–658 (1996). [CrossRef]  

31. M. I. Bakunov, A. V. Maslov, and S. N. Zhukov, “Time-dependent scattering of a standing surface plasmon by rapid ionization in a semiconductor,” Opt. Lett. 25, 926–928 (2000). [CrossRef]  

32. M. I. Bakunov, A. V. Maslov, and S. N. Zhukov, “Scattering of a surface plasmon polariton by rapid plasma creation in a semiconductor slab,” J. Opt. Soc. Am. B 16, 1942–1950 (1999). [CrossRef]  

33. M. I. Bakunov and A. V. Maslov, “Reflection and transmission of electromagnetic waves at a temporal boundary: comment,” Opt. Lett. 39, 6029 (2014). [CrossRef]  

34. K. Huang and T. Hong, “Dielectric polarization and electric displacement in polar-molecule reactions,” J. Phys. Chem. A 119, 8898–8902 (2015). [CrossRef]  

35. G. A. Menendez and B. Maes, “Time reflection and refraction of graphene plasmons at a temporal discontinuity,” Opt. Lett. 42, 5006–5009 (2017). [CrossRef]  

36. J. Wilson, F. Santosa, M. Min, and T. Low, “Temporal control of graphene plasmons,” Phys. Rev. B 98, 081411 (2018). [CrossRef]  

37. S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85, 035443 (2012). [CrossRef]  

38. T. Li, L. Luo, M. Hupalo, J. Zhang, M. C. Tringides, J. Schmalian, and J. Wang, “Femtosecond population inversion and stimulated emission of dense Dirac fermions in graphene,” Phys. Rev. Lett. 108, 167401 (2012). [CrossRef]  

39. Y. V. Bludov, A. Ferreira, N. M. R. Peres, and M. I. Vasilevskiy, “A primer on surface plasmon-polaritons in graphene,” Int. J. Mod. Phys. B 27, 1341001 (2013). [CrossRef]  

40. A. V. Maslov, “Levitation and propulsion of a Mie-resonance particle by a surface plasmon,” Opt. Lett. 42, 3327–3330 (2017). [CrossRef]  

41. M. I. Bakunov and A. V. Maslov, “Trapping of an electromagnetic wave by the boundary of a time-varying plasma,” Phys. Rev. E 57, 5978–5987 (1998). [CrossRef]  

42. M. I. Bakunov and A. V. Maslov, “Trapping of electromagnetic wave by nonstationary plasma layer,” Phys. Rev. Lett. 79, 4585–4588 (1997). [CrossRef]  

References

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  1. G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4, 518–526 (2010).
    [Crossref]
  2. F. J. García de Abajo, “Special issue “2D materials for nanophotonics”,” ACS Photon. 4, 2959–2961 (2017).
    [Crossref]
  3. Q. Guo, C. Li, B. Deng, S. Yuan, F. Guinea, and F. Xia, “Infrared nanophotonics based on graphene plasmonics,” ACS Photon. 4, 2989–2999 (2017).
    [Crossref]
  4. A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6, 749–758 (2012).
    [Crossref]
  5. J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
    [Crossref]
  6. Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).
    [Crossref]
  7. T. M. Slipchenko, M. L. Nesterov, R. Hillenbrand, A. Y. Nikitin, and L. Martín-Moreno, “Graphene plasmon reflection by corrugations,” ACS Photon. 4, 3081–3088 (2017).
    [Crossref]
  8. D. Smirnova, S. H. Mousavi, Z. Wang, Y. S. Kivshar, and A. B. Khanikaev, “Trapping and guiding surface plasmons in curved graphene landscapes,” ACS Photon. 3, 875–880 (2016).
    [Crossref]
  9. D. A. Kuzmin, I. V. Bychkov, V. G. Shavrov, and V. V. Temnov, “Plasmonics of magnetic and topological graphene-based nanostructures,” Nanophotonics 7, 597–611 (2018).
    [Crossref]
  10. G. Lovat, P. Burghignoli, and R. Araneo, “Low-frequency dominant-mode propagation in spatially dispersive graphene nanowaveguides,” IEEE Trans. Electromagn. Compat. 55, 328–333 (2013).
    [Crossref]
  11. W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14, 955–959 (2014).
    [Crossref]
  12. J. D. Cox and F. J. García de Abajo, “Transient nonlinear plasmonics in nanostructured graphene,” Optica 5, 429–433 (2018).
    [Crossref]
  13. A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
    [Crossref]
  14. T.-T. Kim, H.-D. Kim, R. Zhao, S. S. Oh, T. Ha, D. S. Chung, Y. H. Lee, B. Min, and S. Zhang, “Electrically tunable slow light using graphene metamaterials,” ACS Photon. 5, 1800–1807 (2018).
    [Crossref]
  15. J. S. T. Smalley, F. Vallini, X. Zhang, and Y. Fainman, “Dynamically tunable and active hyperbolic metamaterials,” Adv. Opt. Photon. 10, 354–408 (2018).
    [Crossref]
  16. P.-Y. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-based plasmonic platform for reconfigurable terahertz nanodevices,” ACS Photon. 1, 647–654 (2014).
    [Crossref]
  17. M. Jablan, M. Soljačić, and H. Buljan, “Plasmons in graphene: fundamental properties and potential applications,” Proc. IEEE 101, 1689–1704 (2013).
    [Crossref]
  18. A. V. Kretinin, Y. Cao, J. S. Tu, G. L. Yu, R. Jalil, K. S. Novoselov, S. J. Haigh, A. Gholinia, A. Mishchenko, M. Lozada, T. Georgiou, C. R. Woods, F. Withers, P. Blake, G. Eda, A. Wirsig, C. Hucho, K. Watanabe, T. Taniguchi, A. K. Geim, and R. V. Gorbachev, “Electronic properties of graphene encapsulated with different two-dimensional atomic crystals,” Nano Lett. 14, 3270–3276 (2014).
    [Crossref]
  19. A. Woessner, M. B. Lundeberg, Y. Gao, P. A.-G. A. Principi, M. Carrega, K. Watanabe, M. P. T. Taniguchi, G. Vignale, J. Hone, R. Hillenbrand, and F. H. L. Koppens, “Highly confined low-loss plasmons in graphene-boron nitride heterostructures,” Nat. Mater. 14, 421–425 (2015).
    [Crossref]
  20. J. D. Caldwell and K. S. Novoselov, “Mid-infrared nanophotonics,” Nat. Mater. 14, 364–366 (2015).
    [Crossref]
  21. I. Gierz, J. C. Petersen, M. Mitrano, C. Cacho, I. C. E. Turcu, E. Springate, A. Stöhr, A. Köhler, U. Starke, and A. Cavalleri, “Snapshots of non-equilibrium Dirac carrier distributions in graphene,” Nat. Mater. 12, 1119–1124 (2013).
    [Crossref]
  22. S. Ulstrup, J. C. Johannsen, F. Cilento, J. A. Miwa, A. Crepaldi, M. Zacchigna, C. Cacho, R. Chapman, E. Springate, S. Mammadov, F. Fromm, C. Raidel, T. Seyller, F. Parmigiani, M. Grioni, P. D. C. King, and P. Hofmann, “Ultrafast dynamics of massive Dirac fermions in bilayer graphene,” Phys. Rev. Lett. 112, 257401 (2014).
    [Crossref]
  23. M. Trushin, A. Grupp, G. Soavi, A. Budweg, D. De Fazio, U. Sassi, A. Lombardo, A. C. Ferrari, W. Belzig, A. Leitenstorfer, and D. Brida, “Ultrafast pseudospin dynamics in graphene,” Phys. Rev. B 92, 165429 (2015).
    [Crossref]
  24. M. Baudisch, A. Marini, J. D. Cox, T. Zhu, F. Silva, S. Teichmann, M. Massicotte, F. Koppens, L. S. Levitov, F. J. García de Abajo, and J. Biegert, “Ultrafast nonlinear optical response of Dirac fermions in graphene,” Nat. Commun. 9, 1018 (2018).
    [Crossref]
  25. A. V. Maslov and D. S. Citrin, “Numerical calculation of the terahertz field-induced changes in the optical absorption in quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8, 457–463 (2002).
    [Crossref]
  26. S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310, 651–653 (2005).
    [Crossref]
  27. F. D. Morgenthaler, “Velocity modulation of electromagnetic waves,” IRE Trans. Microwave Theory Tech. 6, 167–172 (1958).
    [Crossref]
  28. D. K. Kalluri, Electromagnetics of Time Varying Complex Media: Frequency and Polarization Transformer, 2nd ed. (CRC Press, 2010).
  29. K. Qu, Q. Jia, M. R. Edwards, and N. J. Fisch, “Theory of electromagnetic wave frequency upconversion in dynamic media,” Phys. Rev. E 98, 023202 (2018).
    [Crossref]
  30. M. I. Bakunov and S. N. Zhukov, “Conversion of a surface electromagnetic wave at the boundary of a time-varying plasma,” Plasma Phys. Rep. 22, 649–658 (1996).
    [Crossref]
  31. M. I. Bakunov, A. V. Maslov, and S. N. Zhukov, “Time-dependent scattering of a standing surface plasmon by rapid ionization in a semiconductor,” Opt. Lett. 25, 926–928 (2000).
    [Crossref]
  32. M. I. Bakunov, A. V. Maslov, and S. N. Zhukov, “Scattering of a surface plasmon polariton by rapid plasma creation in a semiconductor slab,” J. Opt. Soc. Am. B 16, 1942–1950 (1999).
    [Crossref]
  33. M. I. Bakunov and A. V. Maslov, “Reflection and transmission of electromagnetic waves at a temporal boundary: comment,” Opt. Lett. 39, 6029 (2014).
    [Crossref]
  34. K. Huang and T. Hong, “Dielectric polarization and electric displacement in polar-molecule reactions,” J. Phys. Chem. A 119, 8898–8902 (2015).
    [Crossref]
  35. G. A. Menendez and B. Maes, “Time reflection and refraction of graphene plasmons at a temporal discontinuity,” Opt. Lett. 42, 5006–5009 (2017).
    [Crossref]
  36. J. Wilson, F. Santosa, M. Min, and T. Low, “Temporal control of graphene plasmons,” Phys. Rev. B 98, 081411 (2018).
    [Crossref]
  37. S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85, 035443 (2012).
    [Crossref]
  38. T. Li, L. Luo, M. Hupalo, J. Zhang, M. C. Tringides, J. Schmalian, and J. Wang, “Femtosecond population inversion and stimulated emission of dense Dirac fermions in graphene,” Phys. Rev. Lett. 108, 167401 (2012).
    [Crossref]
  39. Y. V. Bludov, A. Ferreira, N. M. R. Peres, and M. I. Vasilevskiy, “A primer on surface plasmon-polaritons in graphene,” Int. J. Mod. Phys. B 27, 1341001 (2013).
    [Crossref]
  40. A. V. Maslov, “Levitation and propulsion of a Mie-resonance particle by a surface plasmon,” Opt. Lett. 42, 3327–3330 (2017).
    [Crossref]
  41. M. I. Bakunov and A. V. Maslov, “Trapping of an electromagnetic wave by the boundary of a time-varying plasma,” Phys. Rev. E 57, 5978–5987 (1998).
    [Crossref]
  42. M. I. Bakunov and A. V. Maslov, “Trapping of electromagnetic wave by nonstationary plasma layer,” Phys. Rev. Lett. 79, 4585–4588 (1997).
    [Crossref]

2018 (7)

D. A. Kuzmin, I. V. Bychkov, V. G. Shavrov, and V. V. Temnov, “Plasmonics of magnetic and topological graphene-based nanostructures,” Nanophotonics 7, 597–611 (2018).
[Crossref]

J. D. Cox and F. J. García de Abajo, “Transient nonlinear plasmonics in nanostructured graphene,” Optica 5, 429–433 (2018).
[Crossref]

T.-T. Kim, H.-D. Kim, R. Zhao, S. S. Oh, T. Ha, D. S. Chung, Y. H. Lee, B. Min, and S. Zhang, “Electrically tunable slow light using graphene metamaterials,” ACS Photon. 5, 1800–1807 (2018).
[Crossref]

J. S. T. Smalley, F. Vallini, X. Zhang, and Y. Fainman, “Dynamically tunable and active hyperbolic metamaterials,” Adv. Opt. Photon. 10, 354–408 (2018).
[Crossref]

M. Baudisch, A. Marini, J. D. Cox, T. Zhu, F. Silva, S. Teichmann, M. Massicotte, F. Koppens, L. S. Levitov, F. J. García de Abajo, and J. Biegert, “Ultrafast nonlinear optical response of Dirac fermions in graphene,” Nat. Commun. 9, 1018 (2018).
[Crossref]

K. Qu, Q. Jia, M. R. Edwards, and N. J. Fisch, “Theory of electromagnetic wave frequency upconversion in dynamic media,” Phys. Rev. E 98, 023202 (2018).
[Crossref]

J. Wilson, F. Santosa, M. Min, and T. Low, “Temporal control of graphene plasmons,” Phys. Rev. B 98, 081411 (2018).
[Crossref]

2017 (5)

A. V. Maslov, “Levitation and propulsion of a Mie-resonance particle by a surface plasmon,” Opt. Lett. 42, 3327–3330 (2017).
[Crossref]

G. A. Menendez and B. Maes, “Time reflection and refraction of graphene plasmons at a temporal discontinuity,” Opt. Lett. 42, 5006–5009 (2017).
[Crossref]

T. M. Slipchenko, M. L. Nesterov, R. Hillenbrand, A. Y. Nikitin, and L. Martín-Moreno, “Graphene plasmon reflection by corrugations,” ACS Photon. 4, 3081–3088 (2017).
[Crossref]

F. J. García de Abajo, “Special issue “2D materials for nanophotonics”,” ACS Photon. 4, 2959–2961 (2017).
[Crossref]

Q. Guo, C. Li, B. Deng, S. Yuan, F. Guinea, and F. Xia, “Infrared nanophotonics based on graphene plasmonics,” ACS Photon. 4, 2989–2999 (2017).
[Crossref]

2016 (1)

D. Smirnova, S. H. Mousavi, Z. Wang, Y. S. Kivshar, and A. B. Khanikaev, “Trapping and guiding surface plasmons in curved graphene landscapes,” ACS Photon. 3, 875–880 (2016).
[Crossref]

2015 (4)

K. Huang and T. Hong, “Dielectric polarization and electric displacement in polar-molecule reactions,” J. Phys. Chem. A 119, 8898–8902 (2015).
[Crossref]

M. Trushin, A. Grupp, G. Soavi, A. Budweg, D. De Fazio, U. Sassi, A. Lombardo, A. C. Ferrari, W. Belzig, A. Leitenstorfer, and D. Brida, “Ultrafast pseudospin dynamics in graphene,” Phys. Rev. B 92, 165429 (2015).
[Crossref]

A. Woessner, M. B. Lundeberg, Y. Gao, P. A.-G. A. Principi, M. Carrega, K. Watanabe, M. P. T. Taniguchi, G. Vignale, J. Hone, R. Hillenbrand, and F. H. L. Koppens, “Highly confined low-loss plasmons in graphene-boron nitride heterostructures,” Nat. Mater. 14, 421–425 (2015).
[Crossref]

J. D. Caldwell and K. S. Novoselov, “Mid-infrared nanophotonics,” Nat. Mater. 14, 364–366 (2015).
[Crossref]

2014 (5)

A. V. Kretinin, Y. Cao, J. S. Tu, G. L. Yu, R. Jalil, K. S. Novoselov, S. J. Haigh, A. Gholinia, A. Mishchenko, M. Lozada, T. Georgiou, C. R. Woods, F. Withers, P. Blake, G. Eda, A. Wirsig, C. Hucho, K. Watanabe, T. Taniguchi, A. K. Geim, and R. V. Gorbachev, “Electronic properties of graphene encapsulated with different two-dimensional atomic crystals,” Nano Lett. 14, 3270–3276 (2014).
[Crossref]

S. Ulstrup, J. C. Johannsen, F. Cilento, J. A. Miwa, A. Crepaldi, M. Zacchigna, C. Cacho, R. Chapman, E. Springate, S. Mammadov, F. Fromm, C. Raidel, T. Seyller, F. Parmigiani, M. Grioni, P. D. C. King, and P. Hofmann, “Ultrafast dynamics of massive Dirac fermions in bilayer graphene,” Phys. Rev. Lett. 112, 257401 (2014).
[Crossref]

M. I. Bakunov and A. V. Maslov, “Reflection and transmission of electromagnetic waves at a temporal boundary: comment,” Opt. Lett. 39, 6029 (2014).
[Crossref]

P.-Y. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-based plasmonic platform for reconfigurable terahertz nanodevices,” ACS Photon. 1, 647–654 (2014).
[Crossref]

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14, 955–959 (2014).
[Crossref]

2013 (4)

M. Jablan, M. Soljačić, and H. Buljan, “Plasmons in graphene: fundamental properties and potential applications,” Proc. IEEE 101, 1689–1704 (2013).
[Crossref]

G. Lovat, P. Burghignoli, and R. Araneo, “Low-frequency dominant-mode propagation in spatially dispersive graphene nanowaveguides,” IEEE Trans. Electromagn. Compat. 55, 328–333 (2013).
[Crossref]

I. Gierz, J. C. Petersen, M. Mitrano, C. Cacho, I. C. E. Turcu, E. Springate, A. Stöhr, A. Köhler, U. Starke, and A. Cavalleri, “Snapshots of non-equilibrium Dirac carrier distributions in graphene,” Nat. Mater. 12, 1119–1124 (2013).
[Crossref]

Y. V. Bludov, A. Ferreira, N. M. R. Peres, and M. I. Vasilevskiy, “A primer on surface plasmon-polaritons in graphene,” Int. J. Mod. Phys. B 27, 1341001 (2013).
[Crossref]

2012 (5)

S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85, 035443 (2012).
[Crossref]

T. Li, L. Luo, M. Hupalo, J. Zhang, M. C. Tringides, J. Schmalian, and J. Wang, “Femtosecond population inversion and stimulated emission of dense Dirac fermions in graphene,” Phys. Rev. Lett. 108, 167401 (2012).
[Crossref]

A. N. Grigorenko, M. Polini, and K. S. Novoselov, “Graphene plasmonics,” Nat. Photonics 6, 749–758 (2012).
[Crossref]

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
[Crossref]

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).
[Crossref]

2011 (1)

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
[Crossref]

2010 (1)

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4, 518–526 (2010).
[Crossref]

2005 (1)

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310, 651–653 (2005).
[Crossref]

2002 (1)

A. V. Maslov and D. S. Citrin, “Numerical calculation of the terahertz field-induced changes in the optical absorption in quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8, 457–463 (2002).
[Crossref]

2000 (1)

1999 (1)

1998 (1)

M. I. Bakunov and A. V. Maslov, “Trapping of an electromagnetic wave by the boundary of a time-varying plasma,” Phys. Rev. E 57, 5978–5987 (1998).
[Crossref]

1997 (1)

M. I. Bakunov and A. V. Maslov, “Trapping of electromagnetic wave by nonstationary plasma layer,” Phys. Rev. Lett. 79, 4585–4588 (1997).
[Crossref]

1996 (1)

M. I. Bakunov and S. N. Zhukov, “Conversion of a surface electromagnetic wave at the boundary of a time-varying plasma,” Plasma Phys. Rep. 22, 649–658 (1996).
[Crossref]

1958 (1)

F. D. Morgenthaler, “Velocity modulation of electromagnetic waves,” IRE Trans. Microwave Theory Tech. 6, 167–172 (1958).
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Akinwande, D.

P.-Y. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-based plasmonic platform for reconfigurable terahertz nanodevices,” ACS Photon. 1, 647–654 (2014).
[Crossref]

Alonso-González, P.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
[Crossref]

Alù, A.

P.-Y. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-based plasmonic platform for reconfigurable terahertz nanodevices,” ACS Photon. 1, 647–654 (2014).
[Crossref]

Andreev, G. O.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).
[Crossref]

Araneo, R.

G. Lovat, P. Burghignoli, and R. Araneo, “Low-frequency dominant-mode propagation in spatially dispersive graphene nanowaveguides,” IEEE Trans. Electromagn. Compat. 55, 328–333 (2013).
[Crossref]

Badioli, M.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
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Bakunov, M. I.

M. I. Bakunov and A. V. Maslov, “Reflection and transmission of electromagnetic waves at a temporal boundary: comment,” Opt. Lett. 39, 6029 (2014).
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M. I. Bakunov, A. V. Maslov, and S. N. Zhukov, “Time-dependent scattering of a standing surface plasmon by rapid ionization in a semiconductor,” Opt. Lett. 25, 926–928 (2000).
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M. I. Bakunov, A. V. Maslov, and S. N. Zhukov, “Scattering of a surface plasmon polariton by rapid plasma creation in a semiconductor slab,” J. Opt. Soc. Am. B 16, 1942–1950 (1999).
[Crossref]

M. I. Bakunov and A. V. Maslov, “Trapping of an electromagnetic wave by the boundary of a time-varying plasma,” Phys. Rev. E 57, 5978–5987 (1998).
[Crossref]

M. I. Bakunov and A. V. Maslov, “Trapping of electromagnetic wave by nonstationary plasma layer,” Phys. Rev. Lett. 79, 4585–4588 (1997).
[Crossref]

M. I. Bakunov and S. N. Zhukov, “Conversion of a surface electromagnetic wave at the boundary of a time-varying plasma,” Plasma Phys. Rep. 22, 649–658 (1996).
[Crossref]

Bao, J.

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14, 955–959 (2014).
[Crossref]

Bao, W.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).
[Crossref]

Basov, D. N.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).
[Crossref]

Baudisch, M.

M. Baudisch, A. Marini, J. D. Cox, T. Zhu, F. Silva, S. Teichmann, M. Massicotte, F. Koppens, L. S. Levitov, F. J. García de Abajo, and J. Biegert, “Ultrafast nonlinear optical response of Dirac fermions in graphene,” Nat. Commun. 9, 1018 (2018).
[Crossref]

Belzig, W.

M. Trushin, A. Grupp, G. Soavi, A. Budweg, D. De Fazio, U. Sassi, A. Lombardo, A. C. Ferrari, W. Belzig, A. Leitenstorfer, and D. Brida, “Ultrafast pseudospin dynamics in graphene,” Phys. Rev. B 92, 165429 (2015).
[Crossref]

Biegert, J.

M. Baudisch, A. Marini, J. D. Cox, T. Zhu, F. Silva, S. Teichmann, M. Massicotte, F. Koppens, L. S. Levitov, F. J. García de Abajo, and J. Biegert, “Ultrafast nonlinear optical response of Dirac fermions in graphene,” Nat. Commun. 9, 1018 (2018).
[Crossref]

Birkedal, V.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310, 651–653 (2005).
[Crossref]

Blake, P.

A. V. Kretinin, Y. Cao, J. S. Tu, G. L. Yu, R. Jalil, K. S. Novoselov, S. J. Haigh, A. Gholinia, A. Mishchenko, M. Lozada, T. Georgiou, C. R. Woods, F. Withers, P. Blake, G. Eda, A. Wirsig, C. Hucho, K. Watanabe, T. Taniguchi, A. K. Geim, and R. V. Gorbachev, “Electronic properties of graphene encapsulated with different two-dimensional atomic crystals,” Nano Lett. 14, 3270–3276 (2014).
[Crossref]

Bludov, Y. V.

Y. V. Bludov, A. Ferreira, N. M. R. Peres, and M. I. Vasilevskiy, “A primer on surface plasmon-polaritons in graphene,” Int. J. Mod. Phys. B 27, 1341001 (2013).
[Crossref]

Boubanga-Tombet, S.

S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85, 035443 (2012).
[Crossref]

Brida, D.

M. Trushin, A. Grupp, G. Soavi, A. Budweg, D. De Fazio, U. Sassi, A. Lombardo, A. C. Ferrari, W. Belzig, A. Leitenstorfer, and D. Brida, “Ultrafast pseudospin dynamics in graphene,” Phys. Rev. B 92, 165429 (2015).
[Crossref]

Budweg, A.

M. Trushin, A. Grupp, G. Soavi, A. Budweg, D. De Fazio, U. Sassi, A. Lombardo, A. C. Ferrari, W. Belzig, A. Leitenstorfer, and D. Brida, “Ultrafast pseudospin dynamics in graphene,” Phys. Rev. B 92, 165429 (2015).
[Crossref]

Buljan, H.

M. Jablan, M. Soljačić, and H. Buljan, “Plasmons in graphene: fundamental properties and potential applications,” Proc. IEEE 101, 1689–1704 (2013).
[Crossref]

Burghignoli, P.

G. Lovat, P. Burghignoli, and R. Araneo, “Low-frequency dominant-mode propagation in spatially dispersive graphene nanowaveguides,” IEEE Trans. Electromagn. Compat. 55, 328–333 (2013).
[Crossref]

Bychkov, I. V.

D. A. Kuzmin, I. V. Bychkov, V. G. Shavrov, and V. V. Temnov, “Plasmonics of magnetic and topological graphene-based nanostructures,” Nanophotonics 7, 597–611 (2018).
[Crossref]

Cacho, C.

S. Ulstrup, J. C. Johannsen, F. Cilento, J. A. Miwa, A. Crepaldi, M. Zacchigna, C. Cacho, R. Chapman, E. Springate, S. Mammadov, F. Fromm, C. Raidel, T. Seyller, F. Parmigiani, M. Grioni, P. D. C. King, and P. Hofmann, “Ultrafast dynamics of massive Dirac fermions in bilayer graphene,” Phys. Rev. Lett. 112, 257401 (2014).
[Crossref]

I. Gierz, J. C. Petersen, M. Mitrano, C. Cacho, I. C. E. Turcu, E. Springate, A. Stöhr, A. Köhler, U. Starke, and A. Cavalleri, “Snapshots of non-equilibrium Dirac carrier distributions in graphene,” Nat. Mater. 12, 1119–1124 (2013).
[Crossref]

Caldwell, J. D.

J. D. Caldwell and K. S. Novoselov, “Mid-infrared nanophotonics,” Nat. Mater. 14, 364–366 (2015).
[Crossref]

Camara, N.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
[Crossref]

Cao, Y.

A. V. Kretinin, Y. Cao, J. S. Tu, G. L. Yu, R. Jalil, K. S. Novoselov, S. J. Haigh, A. Gholinia, A. Mishchenko, M. Lozada, T. Georgiou, C. R. Woods, F. Withers, P. Blake, G. Eda, A. Wirsig, C. Hucho, K. Watanabe, T. Taniguchi, A. K. Geim, and R. V. Gorbachev, “Electronic properties of graphene encapsulated with different two-dimensional atomic crystals,” Nano Lett. 14, 3270–3276 (2014).
[Crossref]

Carrega, M.

A. Woessner, M. B. Lundeberg, Y. Gao, P. A.-G. A. Principi, M. Carrega, K. Watanabe, M. P. T. Taniguchi, G. Vignale, J. Hone, R. Hillenbrand, and F. H. L. Koppens, “Highly confined low-loss plasmons in graphene-boron nitride heterostructures,” Nat. Mater. 14, 421–425 (2015).
[Crossref]

Carter, S. G.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310, 651–653 (2005).
[Crossref]

Cavalleri, A.

I. Gierz, J. C. Petersen, M. Mitrano, C. Cacho, I. C. E. Turcu, E. Springate, A. Stöhr, A. Köhler, U. Starke, and A. Cavalleri, “Snapshots of non-equilibrium Dirac carrier distributions in graphene,” Nat. Mater. 12, 1119–1124 (2013).
[Crossref]

Centeno, A.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
[Crossref]

Chan, S.

S. Boubanga-Tombet, S. Chan, T. Watanabe, A. Satou, V. Ryzhii, and T. Otsuji, “Ultrafast carrier dynamics and terahertz emission in optically pumped graphene at room temperature,” Phys. Rev. B 85, 035443 (2012).
[Crossref]

Chapman, R.

S. Ulstrup, J. C. Johannsen, F. Cilento, J. A. Miwa, A. Crepaldi, M. Zacchigna, C. Cacho, R. Chapman, E. Springate, S. Mammadov, F. Fromm, C. Raidel, T. Seyller, F. Parmigiani, M. Grioni, P. D. C. King, and P. Hofmann, “Ultrafast dynamics of massive Dirac fermions in bilayer graphene,” Phys. Rev. Lett. 112, 257401 (2014).
[Crossref]

Chen, B.

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14, 955–959 (2014).
[Crossref]

Chen, J.

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
[Crossref]

Chen, P.-Y.

P.-Y. Chen, H. Huang, D. Akinwande, and A. Alù, “Graphene-based plasmonic platform for reconfigurable terahertz nanodevices,” ACS Photon. 1, 647–654 (2014).
[Crossref]

Chung, D. S.

T.-T. Kim, H.-D. Kim, R. Zhao, S. S. Oh, T. Ha, D. S. Chung, Y. H. Lee, B. Min, and S. Zhang, “Electrically tunable slow light using graphene metamaterials,” ACS Photon. 5, 1800–1807 (2018).
[Crossref]

Cilento, F.

S. Ulstrup, J. C. Johannsen, F. Cilento, J. A. Miwa, A. Crepaldi, M. Zacchigna, C. Cacho, R. Chapman, E. Springate, S. Mammadov, F. Fromm, C. Raidel, T. Seyller, F. Parmigiani, M. Grioni, P. D. C. King, and P. Hofmann, “Ultrafast dynamics of massive Dirac fermions in bilayer graphene,” Phys. Rev. Lett. 112, 257401 (2014).
[Crossref]

Citrin, D. S.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310, 651–653 (2005).
[Crossref]

A. V. Maslov and D. S. Citrin, “Numerical calculation of the terahertz field-induced changes in the optical absorption in quantum wells,” IEEE J. Sel. Top. Quantum Electron. 8, 457–463 (2002).
[Crossref]

Coldren, L. A.

S. G. Carter, V. Birkedal, C. S. Wang, L. A. Coldren, A. V. Maslov, D. S. Citrin, and M. S. Sherwin, “Quantum coherence in an optical modulator,” Science 310, 651–653 (2005).
[Crossref]

Cox, J. D.

M. Baudisch, A. Marini, J. D. Cox, T. Zhu, F. Silva, S. Teichmann, M. Massicotte, F. Koppens, L. S. Levitov, F. J. García de Abajo, and J. Biegert, “Ultrafast nonlinear optical response of Dirac fermions in graphene,” Nat. Commun. 9, 1018 (2018).
[Crossref]

J. D. Cox and F. J. García de Abajo, “Transient nonlinear plasmonics in nanostructured graphene,” Optica 5, 429–433 (2018).
[Crossref]

Crepaldi, A.

S. Ulstrup, J. C. Johannsen, F. Cilento, J. A. Miwa, A. Crepaldi, M. Zacchigna, C. Cacho, R. Chapman, E. Springate, S. Mammadov, F. Fromm, C. Raidel, T. Seyller, F. Parmigiani, M. Grioni, P. D. C. King, and P. Hofmann, “Ultrafast dynamics of massive Dirac fermions in bilayer graphene,” Phys. Rev. Lett. 112, 257401 (2014).
[Crossref]

De Fazio, D.

M. Trushin, A. Grupp, G. Soavi, A. Budweg, D. De Fazio, U. Sassi, A. Lombardo, A. C. Ferrari, W. Belzig, A. Leitenstorfer, and D. Brida, “Ultrafast pseudospin dynamics in graphene,” Phys. Rev. B 92, 165429 (2015).
[Crossref]

Deng, B.

Q. Guo, C. Li, B. Deng, S. Yuan, F. Guinea, and F. Xia, “Infrared nanophotonics based on graphene plasmonics,” ACS Photon. 4, 2989–2999 (2017).
[Crossref]

Dominguez, G.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).
[Crossref]

Eda, G.

A. V. Kretinin, Y. Cao, J. S. Tu, G. L. Yu, R. Jalil, K. S. Novoselov, S. J. Haigh, A. Gholinia, A. Mishchenko, M. Lozada, T. Georgiou, C. R. Woods, F. Withers, P. Blake, G. Eda, A. Wirsig, C. Hucho, K. Watanabe, T. Taniguchi, A. K. Geim, and R. V. Gorbachev, “Electronic properties of graphene encapsulated with different two-dimensional atomic crystals,” Nano Lett. 14, 3270–3276 (2014).
[Crossref]

Edwards, M. R.

K. Qu, Q. Jia, M. R. Edwards, and N. J. Fisch, “Theory of electromagnetic wave frequency upconversion in dynamic media,” Phys. Rev. E 98, 023202 (2018).
[Crossref]

Engheta, N.

A. Vakil and N. Engheta, “Transformation optics using graphene,” Science 332, 1291–1294 (2011).
[Crossref]

Fainman, Y.

Fang, W.

W. Li, B. Chen, C. Meng, W. Fang, Y. Xiao, X. Li, Z. Hu, Y. Xu, L. Tong, H. Wang, W. Liu, J. Bao, and Y. R. Shen, “Ultrafast all-optical graphene modulator,” Nano Lett. 14, 955–959 (2014).
[Crossref]

Fei, Z.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).
[Crossref]

Ferrari, A. C.

M. Trushin, A. Grupp, G. Soavi, A. Budweg, D. De Fazio, U. Sassi, A. Lombardo, A. C. Ferrari, W. Belzig, A. Leitenstorfer, and D. Brida, “Ultrafast pseudospin dynamics in graphene,” Phys. Rev. B 92, 165429 (2015).
[Crossref]

Ferreira, A.

Y. V. Bludov, A. Ferreira, N. M. R. Peres, and M. I. Vasilevskiy, “A primer on surface plasmon-polaritons in graphene,” Int. J. Mod. Phys. B 27, 1341001 (2013).
[Crossref]

Fisch, N. J.

K. Qu, Q. Jia, M. R. Edwards, and N. J. Fisch, “Theory of electromagnetic wave frequency upconversion in dynamic media,” Phys. Rev. E 98, 023202 (2018).
[Crossref]

Fogler, M. M.

Z. Fei, A. S. Rodin, G. O. Andreev, W. Bao, A. S. McLeod, M. Wagner, L. M. Zhang, Z. Zhao, M. Thiemens, G. Dominguez, M. M. Fogler, A. H. C. Neto, C. N. Lau, F. Keilmann, and D. N. Basov, “Gate-tuning of graphene plasmons revealed by infrared nano-imaging,” Nature 487, 82–85 (2012).
[Crossref]

Fromm, F.

S. Ulstrup, J. C. Johannsen, F. Cilento, J. A. Miwa, A. Crepaldi, M. Zacchigna, C. Cacho, R. Chapman, E. Springate, S. Mammadov, F. Fromm, C. Raidel, T. Seyller, F. Parmigiani, M. Grioni, P. D. C. King, and P. Hofmann, “Ultrafast dynamics of massive Dirac fermions in bilayer graphene,” Phys. Rev. Lett. 112, 257401 (2014).
[Crossref]

Gao, Y.

A. Woessner, M. B. Lundeberg, Y. Gao, P. A.-G. A. Principi, M. Carrega, K. Watanabe, M. P. T. Taniguchi, G. Vignale, J. Hone, R. Hillenbrand, and F. H. L. Koppens, “Highly confined low-loss plasmons in graphene-boron nitride heterostructures,” Nat. Mater. 14, 421–425 (2015).
[Crossref]

García de Abajo, F. J.

J. D. Cox and F. J. García de Abajo, “Transient nonlinear plasmonics in nanostructured graphene,” Optica 5, 429–433 (2018).
[Crossref]

M. Baudisch, A. Marini, J. D. Cox, T. Zhu, F. Silva, S. Teichmann, M. Massicotte, F. Koppens, L. S. Levitov, F. J. García de Abajo, and J. Biegert, “Ultrafast nonlinear optical response of Dirac fermions in graphene,” Nat. Commun. 9, 1018 (2018).
[Crossref]

F. J. García de Abajo, “Special issue “2D materials for nanophotonics”,” ACS Photon. 4, 2959–2961 (2017).
[Crossref]

J. Chen, M. Badioli, P. Alonso-González, S. Thongrattanasiri, F. Huth, J. Osmond, M. Spasenović, A. Centeno, A. Pesquera, N. Camara, F. J. García de Abajo, R. Hillenbrand, and F. H. L. Koppens, “Optical nano-imaging of gate-tunable graphene plasmons,” Nature 487, 77–81 (2012).
[Crossref]

Gardes, F. Y.

G. T. Reed, G. Mashanovich, F. Y. Gardes, and D. J. Thomson, “Silicon optical modulators,” Nat. Photonics 4, 518–526 (2010).
[Crossref]

Geim, A. K.

A. V. Kretinin, Y. Cao, J. S. Tu, G. L. Yu, R. Jalil, K. S. Novoselov, S. J. Haigh, A. Gholinia, A. Mishchenko, M. Lozada, T. Georgiou, C. R. Woods, F. Withers, P. Blake, G. Eda, A. Wirsig, C. Hucho, K. Watanabe, T. Taniguchi, A. K. Geim, and R. V. Gorbachev, “Electronic properties of graphene encapsulated with different two-dimensional atomic crystals,” Nano Lett. 14, 3270–3276 (2014).
[Crossref]

Georgiou, T.

A. V. Kretinin, Y. Cao, J. S. Tu, G. L. Yu, R. Jalil, K. S. Novoselov, S. J. Haigh, A. Gholinia, A. Mishchenko, M. Lozada, T. Georgiou, C. R. Woods, F. Withers, P. Blake, G. Eda, A. Wirsig, C. Hucho, K. Watanabe, T. Taniguchi, A. K. Geim, and R. V. Gorbachev, “Electronic properties of graphene encapsulated with different two-dimensional atomic crystals,” Nano Lett. 14, 3270–3276 (2014).
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Figures (8)

Fig. 1.
Fig. 1. (a) Illustration of a surface plasmon propagating along a graphene sheet at t<0. (b) Time dependence of the graphene carrier density. (c) Dispersion diagram showing the frequency transformation of the initial plasmon when the carrier density decreases from N1 to N2. The lines labeled by 1 and 2 are the dispersion curves for the plasmons at densities N1 and N2, respectively. The shaded region shows the continuous spectrum for bulk waves; the hatched region defines the spectrum of the generated transient radiation. The dashed lines h=ϵ|ω|/c are the light lines.
Fig. 2.
Fig. 2. (a) Phase nph and group ngr refractive indices for graphene plasmons. The dashed line shows the asymptotics nphω/(2πΩ). (b) Decomposition of the plasmon energy W0 into the electric WE, magnetic WH, and kinetic Wk parts.
Fig. 3.
Fig. 3. Frequency ω2 of the excited plasmon relative (a) to the initial frequency ω1 and (b) to the frequency parameter Ω2 as a function of Ω2/Ω1.
Fig. 4.
Fig. 4. Energy distribution after temporal scattering as a function of Ω2/Ω1 for (a) ω1/Ω1=15 and (b) ω1/Ω1=1.
Fig. 5.
Fig. 5. Far-field angular distribution wb(θ)/W0 for Ω2/Ω1=0.5 and several values of ω1/Ω1.
Fig. 6.
Fig. 6. Comparison of the plasmon transmission tH and reflection rH coefficients obtained here (general and quasi-static results) and available from the literature for ω1/Ω1=15. Equation (11) is the general formula and Eq. (18) is the quasi-static formula derived here. Equation (19) is from Ref. [35]. Equation (20) is from Ref. [36].
Fig. 7.
Fig. 7. Snapshots of the electric field distribution Ex(x,y,t) produced by a plasmon wavepacket with central frequency ω1/Ω1=10 at three time moments t=0, 0.15/Ω1, 7/Ω1. Initially the wavepacket propagates along the graphene sheet at y=0 in the +x direction (frame t=0). Soon after the carrier density jump Ω2/Ω1=0.2 at t=0, transient processes take place (frame t=0.15/Ω1). Finally, two propagating plasmons are formed (frame t=7/Ω1).
Fig. 8.
Fig. 8. Obtaining the transmission coefficients tE,H from the FDTD simulations. (a) Absolute value of the incident Exi(ω), Hzi(ω) and transmitted Ext(ω), Hzt(ω) field spectra for Ω2/Ω1=0.6. (b) Analytically calculated frequency ω2 of the excited plasmon as a function of the frequency ω1 of the incident plasmon. (c) Differential spectral transormation f=dω2/dω1, which is the derivative of ω2(ω1) shown in frame (b). (d) Comparison of the transmission coefficients obtained numerically from the FDTD simulations and analytically using Eqs. (11)–(13).

Equations (25)

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Ex(x,y,t)=E0eihxiωtϰ|y|,
jx=σEx,σ=icΩ/ω,
h(ω)=ωcϵ1+ϵω24π2Ω2,ϰ(ω)=ϵω22πcΩ.
WE,H=E02ϵ4πϰ(h2ϰ2±1),Wk=|j0|2cΩ,
γ=Ω2/Ω1=Ef2/Ef1<1
ih1EyExy=1cHzt,
Hzy=ϵcExt+4πcjxδ(y),
ih1Hz=ϵcEyt,
jx(t=0+)jx(t=0)=j1(2)j1(1)+j1(2)=Ω1(2)Ω1(1)+Ω1(2)=Ω2Ω1.
jxt=cΩ2Ex,jx(t=0+)=Ω2Ω1jx(t=0).
E˜x(x,y,p)=Ex(x,y,0)p+iω1+A(p)eih1xϰ(p)|y|,
A(p)=iϰp(Ω2Ω1)E0ω1Ω2(p+iω1)D(p),D(p)=ϰ(p)+ϵp22πcΩ2.
Ext,r=πcϰ2ω22(Ω2Ω1)ω1(ω1ω2)(ϵω22+2π2Ω22)E0.
tE=Ext/E0,rE=Exr/E0,
tH=Hzt/H0=ξtE,rH=Hzr/H0=ξrE.
W0=Wt+Wr+Wb+Wl.
Wb=02πdθwb(θ),wb(θ)=c2h14π2|A(iω(θ))|2cos2θ,
Ex(x,y,t)=E0(tEeiω2t+rEeiω2t)eih1xϰ2|y|.
tE=(1+γ)/2,rE=(1γ)/2.
tH=γ(1+γ)/2,rH=γ(1γ)/2.
|tH|=(1+γ)/(2γ),|rH|=|1γ|/(2γ).
tH=(1+γ)/2,rH=(1γ)/2.
Hzt(ω2)Δω2=tHHzi(ω1)Δω1,
tH=f·Hzt(ω2)Hzi(ω1),f=dω2(ω1)dω1,
dωHzt(ω)=tHdωHzi(ω).

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