Surface plasmons in graphene offer a compelling route to many useful photonic technologies1,2,3. As a plasmonic material, graphene offers several intriguing properties, such as excellent electro-optic tunability4, crystalline stability, large optical nonlinearities5 and extremely high electromagnetic field concentration6. As such, recent demonstrations of surface plasmon excitation in graphene using near-field scattering of infrared light7,8 have received intense interest. Here we present an all-optical plasmon coupling scheme which takes advantage of the intrinsic nonlinear optical response of graphene. Free-space, visible light pulses are used to generate surface plasmons in a planar graphene sheet using difference frequency wave mixing to match both the wavevector and energy of the surface wave. By carefully controlling the phase matching conditions, we show that one can excite surface plasmons with a defined wavevector and direction across a large frequency range, with an estimated photon efficiency in our experiments approaching 10−5.
Graphene has attracted significant interest recently as a unique optical material. In particular, it has been predicted and experimentally shown that graphene can support highly confined surface plasmons1,9, with electrically tunable dispersion7,8. Despite these promising discoveries, the burgeoning field of graphene plasmonics has some serious obstacles to overcome if it is to progress from the proof-of-principle stage. Problems arise due to the small wavelength of the surface plasmons, two orders of magnitude smaller than light of the same frequency. This has led to the development of specialized measurement techniques, most of which use infrared light and geometries with scattering resonances10,11,12 or near-field sources7,8 to excite graphene surface plasmons. However, the far-infrared regime, in which graphene plasmons are predicted to have long lifetimes, lacks developed sources and detectors compared to the visible regime. Alternative approaches, such as the manipulation of surface acoustic waves to couple to the graphene surface plasmons13,14, therefore hold promise.
Particularly desirable is the potential to excite a plasmon eigenstate with a singular energy, momentum and direction, vital for many future applications, including plasmonic circuits. In this respect, very recent progress has been made, with the development of carefully designed nanoantennas which can locally excite and direct surface plasmons in graphene11. Here, the combination of infrared source frequency and nanoantenna dimensions determine the frequency, wavevector and direction of the surface plasmons generated. In this letter, we investigate a competing approach that embodies many of these desirable aspects of directivity without requiring careful nanofabrication of antennas. This all-optical approach can access a distinctly broad frequency range, even down to the far infrared. We coherently excite surface plasmons using two visible frequency free-space beams via difference frequency generation (DFG), an effect which we monitor through changes in reflectance, and can tune the frequency and wavevector of the surface plasmon through careful adjustment of incident light sources. This potential to excite and detect plasmons purely with free-space optics, and at frequencies different from that of the plasmons themselves, has the potential to significantly expand the technological possibilities for graphene plasmonics.
The intrinsic nonlinear interactions of graphene with light are surprisingly large5,15,16,17,18. Moreover, large enhancements of nonlinear optical effects are predicted by the presence of highly confined plasmons in graphene19,20. It seems intuitive, then, to attempt the converse: to use the nonlinear interaction between optical fields to resonantly drive surface plasmons. Similar approaches have been demonstrated experimentally for thin metallic films21,22, and have been recently proposed for graphene, with various coupling schemes proposed for the difference frequency mixing of infrared light in a graphene film23 and in graphene-clad waveguide structures23,24. Similar in concept to that described in ref. 23, Fig. 1 shows our nonlinear coupling scheme illustrated on a dispersion diagram. By illuminating the graphene with two intense laser pulses with well-defined angles of incidence but different frequencies, labelled here fpump and fprobe, one can phase match both the frequency and wavevector, k, of the surface plasmon. This wave mixing process is a second-order nonlinear effect, normally forbidden in centro-symmetric crystals25, but possible in graphene because of the distinctively non-local, spatial character of the interaction20. The inset in Fig. 1 shows the experimental arrangement used (see Methods). The experiments are carried out in a non-collinear geometry, using two beams incident on the samples at angles θpump and θprobe providing sufficient in-plane momentum to match to the surface plasmon, as illustrated in Fig. 1. We measure the differential reflection of the probe beam, defined as ΔR/R = (R − R0)/R0, where R and R0 are the reflections with and without the presence of the pump pulse, respectively. The polarization of both incoming beams is set in the plane of incidence (transverse magnetic polarized). To isolate the nonlinear reflection signal, we vary the temporal overlap of the two pulses.
For optical excitation, one expects optical nonlinearity arising due to saturable absorption caused by Pauli blocking of interband transitions26. A typical measurement of the temporal dynamics recorded for this process (λpump = 547 nm, λprobe = 615 nm) is shown by the black curve in Fig. 2. Note that we normalize the signal by the pump fluence, ϕ, to remove artefacts due to power variation27. The asymmetric line shape of the signal is due to the relaxation dynamics of the excited electrons cooling27,28, with temporal broadening caused by the spatial overlap of the non-collinear beam spots.
For non-degenerate pump and probe beams, in addition to (incoherent) saturable absorption effects, one can expect (coherent) wave mixing signals. This coherent contribution to the probe reflection is expected to be significantly enhanced when the difference frequency field generated by the pump and probe matches that of the graphene surface plasmons. This is analogous to that of a stimulated Raman process, corresponding to a transfer of energy from pump to probe pulses25 via the generation of surface plasmons. An example of the recorded temporal dynamics under such a resonant condition is presented in Fig. 2. Comparing the two curves in this figure, we see that the ‘non-resonant’ signal (that is, when one is not phase matching to plasmon excitation) gives rise to an asymmetric lineshape representative of carrier cooling dynamics. Under ‘resonant’ conditions (that is, when phase matching conditions are satisfied) we observe a fast additional contribution to the signal, giving rise to a more symmetric lineshape, as one would expect for a coherent signal. For certain experimental geometries and excitation fluences, signal enhancements of up to ×4 are observed (see Supplementary Fig. 1). It should be noted that, depending on efficiencies, it may be possible to isolate the coherent signal using a heterodyne detection scheme29, which could also allow detection of a plasmon in a different spatial position from where generated.
To observe the presence of the coherent signal, we vary the difference frequency from 0 to 60 THz to isolate any resonant, coherent conditions (see Methods). In this way, it is possible to interrogate a section of the surface plasmon dispersion, for example the region illustrated by the pink line in Fig. 1. By altering the experimental geometry, we investigate here three different regions of the dispersion diagram corresponding to (θpump = 55°, θprobe = 45°), (θpump = 50°, θprobe = 70°) and (θpump = 15°, θprobe = 125°).
Figure 3 shows the results of these three measurement geometries, superimposed on the surface plasmon dispersion (black line). The dispersion was calculated according to the model outlined in ref. 30, with the SiO2 substrate phonon frequencies as given and a Fermi energy of Ef = 0.5 eV . This Fermi energy is larger than the measured intrinsic doping of our graphene samples (see Methods for sample details), which we attribute to a significantly raised electron temperature expected under illumination by ultrafast pulses31,32 (see Supplementary Information, Supplementary Fig. 2). Hybridization with the substrate phonons leads to four branches30,33. The overlaid colour plots are placed on the diagram so that the maximum differential reflection signal achieved in each delay-scan corresponds to the difference frequency and wavevector of the data set.
Near the regions defined by the surface plasmon dispersion in graphene, we observe clear enhancement in the differential reflection. The assignment of the spectral features to surface plasmon excitation is further supported by the polarization dependence of the signal (see Supplementary Fig. 3). The observation of these resonant features over the incoherent background is also strongly dependent on the magnitudes of both pump and probe intensities (see Supplementary Figs 1 and 2). For larger difference frequencies, up to 150 THz, we do not observe any further resonance features in our spectra (see Supplementary Fig. 4).
The lower branch of the plasmon dispersion relation gives rise to the largest mixing signals for the low-wavevector phase matching (set-ups a, b in Fig. 3), while the upper branches give rise to the largest signals for the high-wavevector (set-up c in Fig. 3) region. Although we observe clear resonance features in all three of these experimental geometries, we also observe a change in sign of the signal between the low- (Fig. 3a, b) and high-wavevector (Fig. 3c) regions. The absolute differential reflectivity signal size also increases with increasing wavevector.
To understand the origin of these coupling behaviours, we have developed a simple theoretical model that captures the salient features of this nonlinear reflection and generation of plasmons. We briefly summarize the model in the Methods (full details are presented in the Supplementary Information). In Fig. 4, we plot the modelled differential probe reflectance, normalized by fluence, for the simplified case of continuous plane-wave pump and probe beams. While this simple model ignores the non-equilibrium nature of the excitation, as observed in experiment, we show below that it is sufficient to describe some of the salient features of our results. Similar to Fig. 3, the differential reflectance is plotted versus difference frequency and in-plane wavevector. It can be seen that the simulation qualitatively produces the main features of Fig. 3. In particular, the change in the sign of differential reflectance at the Brewster angle is clearly observed, as is the enhancement of the signal when the difference frequency and wavevector align with the plasmon dispersion relation.
The model also reproduces some of the main features arising from different coupling efficiencies to different bands (Fig. 4). Generally, the highest coupling efficiency occurs for the dispersion regions which are most ‘plasmon-like’ in origin (see also Supplementary Fig. 7). This is most obvious comparing the data in Figs 3c and 4c, where the coupling to the upper bands is much stronger than the lowest band in both model and experiment. For lower-wavevector cases, the coupling to the highest band is overestimated in the model compared to the experiment. This could possibly be caused by frequency-dependent losses in the graphene sheet unaccounted for in our simple model. The model also reproduces the increasing absolute signal strength with increasing wavevector observed in experiment, which is a consequence of both larger changes in the reflection coefficient for a corresponding change in absorption for higher angles, and due to spatial dispersion20 in the signal. Indeed, it can be shown that the magnitude of enhancement is proportional to the square of the plasmon quality factor, Q2 (see Supplementary Information), in agreement with predictions from ref. 23.
In addition to the surface plasmon resonance conditions, for the highest-wavevector region in Fig. 3c there is an additional resonant enhancement found in experiment at low frequencies (<3 THz) that is not reproduced in our model (Fig. 4c). The position of this peak lies within the expected region of intraband transitions in graphene, indicated by the dotted line in Fig. 3. This feature is also largely independent of polarization, unlike the enhancements we attribute to surface plasmon coupling (see Supplementary Fig. 3).
In principle, the reflection/transmission expressions obtained (see Methods, equation (1)) can be inverted to allow an experimental determination of the nonlinear conductivity σ(2), given transmission or reflection data. This is difficult in the present set-up, in part given the broad bandwidth of the pulses, uncertainty over some system parameters, and difficulty of investigating a large number of angles to quantify possible wavevector and frequency dependence of σ(2). However, as an estimate, we take the simplest possible model, in which the effective nonlinear susceptibility χ(2) is independent of frequency and wavevector. This corresponds to a nonlinear conductivity function obeying σ(2)(ω) = i|σ(2)(ωprobe)|(ω/ωprobe), where the value at the (fixed) probe frequency represents a single fitting parameter. We find that a value of |σ(2)(ωprobe)|≍2.4 × 10−12 A m V −2 produces the same peak signal as observed in Fig. 3b. It should be emphasized that this represents a rather conservative estimate of σ(2)(ω). In particular, the mobility of μ ≍ 2,000 cm2 V−1 s−1 corresponds to a plasmon linewidth of γ = 2π × 1.6 THz that is narrower than the measurement bandwidth, indicating that only a fraction of the pulse can efficiently excite plasmons. Reducing the measurement bandwidth could therefore give rise to greater coupling to the surface plasmons, while also reducing the effects of non-equilibrium carriers on the measurements. While a comparison to a bulk nonlinear crystal is not directly meaningful, it is nonetheless interesting to note that a bulk nonlinear crystal with the thickness t ≍ 0.3 nm of a graphene layer would require a nonlinear susceptibility of χ(2) ∼ |σ(2)(ωprobe)|/(ɛ0ωprobet) ∼ 3 × 10−7 m V −1 to produce the equivalent in-plane nonlinear currents. This value is approximately three orders of magnitude larger than in GaAs.
Finally, from the inferred value of σ(2) and the input beam parameters, our model enables us to estimate the conversion efficiency η of pump photons to plasmons for our experimental pulse intensities (see Supplementary Information). We find a value of η ≍ 6 × 10−6, while noting that the actual conversion could be significantly higher with narrow pulses, again as the estimated value of σ(2) does not account for the large pulse bandwidth. We note that this experimentally obtained value of η is of the same order as predicted in ref. 23, once adjusted for our experimental parameters.
To conclude, by carefully manipulating the phase matching conditions, we show that one can generate surface plasmons with a defined wavevector and an efficiency approaching 10−5 for the pulse intensities used. This efficiency by no means represents a fundamental limit, and we believe that it could in principle be pushed towards a 10−2 level with future adjustments, such as increasing the surface plasmon Q factor from ∼5 to ∼30 with lattice-matched hBN substrates34, equalizing the intensities of the pump and probe beams (see Supplementary Fig. 1), or the use of narrower bandwidth pulses. Moreover, in principle, our approach could be extended to higher or lower frequencies, regions that are generally hard to access using present approaches2,35.
An identical pair of optical parametric amplifiers (OPAs), pumped by an amplified femtosecond laser system, generate the 100 fs-pulses at a repetition rate of 1 kHz. The wavelengths of the two OPAs are selected independently, and the beams are directed to the sample. The incident beams are weakly focused on the sample using 30 cm focal length lenses, giving rise to a very small uncertainty in angle ∼0.017 rad, and a similarly negligible uncertainty for the in-plane wavevectors. Sets of half-wave plates and polarizers determine both the average power and polarization, with the polarization set such that the electric vector of the light is in the plane of incidence (transverse magnetic polarized). The pump pulse fluence, ϕ, used is typically in the range ϕ ∼ 0.1 − 0.2 mJ cm−2, with a pump spot size on the sample of ∼300 μm radius. This pump fluence is an order of magnitude less than the photo-modification threshold for graphene36, and the probe fluence is typically two orders of magnitude smaller still.
To obtain difference frequencies from 0 to 60 THz, the pump wavelength is varied from 615 to 545 nm, with the probe wavelength set at 615 nm.
We record the differential reflection of the probe beam, defined as ΔR/R = (R − R0)/R0, where R and R0 are the reflections with and without the presence of the pump pulse, respectively. This differential reflection is recorded using a set of photo-balance diodes. To isolate the nonlinear reflection signal, we vary the temporal overlap of the two pulses using a motorized delay stage. Note that there is no appreciable signal from the quartz substrate (see Supplementary Fig. 8).
Samples for our experiments are fabricated from commercially grown CVD graphene on copper foil (Graphene Supermarket). Transfer to quartz substrates was performed in house by means of a standard metal etching and float technique using ammonium persulphate to etch the copper and PMMA as a support structure. Combined resistance and Raman spectroscopy37 give an estimated mobility of the samples of around 2,000 cm2 V−1 s−1 and a natural Fermi energy of ∼300 meV. Raman imaging indicates that the graphene is nominally single layer, with ∼80% coverage of the substrate.
In general, equations for the electromagnetic boundary conditions at the air–graphene–substrate interface relate the wavevector- and frequency-dependent reflection and transmission coefficients r(k, ω), t(k, ω) to the graphene current density J(k, ω). The current density, on the other hand, can be written in terms of the electric field using conductivity functions, which allows the equations to be solved in terms of fields alone. Nonlinear contributions imply that J(k, ω) depends on fields at other wavevectors and frequencies, which couple the various reflection and transmission coefficients together. For a second-order conductivity σ(2), we find that the probe transmission depends on the pump via with an analogous equation for the pump transmission (expressions for r are more involved but are directly related to t, see Supplementary Methods). Here tprobe(L) is the linear transmission coefficient, A is a function of linear optical properties and beam angles with, for notational simplicity, dependencies on k, ω being implicit.
To model the differential reflectivity of the probe as shown in Fig. 4, the probe wavelength and pump angles are fixed at 615 nm and 50°, respectively, and the pump and probe intensities are chosen to be 10 and 0.1 W μm−2, to closely correspond to the configuration in Fig. 3b.
This research has been supported by the European Commission (FP7-ICT-2013-613024-GRASP) and EPSRC fellowship (EP/K041215/1). D.E.C. also acknowledges support from Fundacio Privada Cellex Barcelona and the Ramon y Cajal programme. The authors would like to thank J. García de Abajo and M. Jablan for useful discussions, and N. Cole for technical assistance.
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Second order optical nonlinearity of graphene due to electric quadrupole and magnetic dipole effects
Scientific Reports (2017)