Coherently controlled femtosecond energy localization on nanoscale

Transcription

1 Appl. Phys. B 74 [Suppl.], S63 S67 (22) DOI:.7/s Applied Physics B Lasers and Optics m.i. stockman, s.v. faleev d.j. bergman 2 Coherently controlled femtosecond energy localiation on nanoscale Department of Physics and Astronomy, Georgia State University, Atlanta, GA 333, USA 2 School of Physics and Astronomy, Raymond and Beverly Sacler Faculty of Eact Sciences, Tel Aviv University, Tel Aviv, Israel Received: 7 September 2/ Revised version: 2 December 2 Published online: 2 June 22 Springer-Verlag 22 ABSTRACT We predict and quantitatively evaluate the unique possibility of concentrating the energy of an ultra-fast ecitation of a nanosystem in a small part of the whole system by means of coherent control (phase modulation of the eciting ultra-short pulse). Such concentration is due to dynamic properties of surface plasmons and leads to local fields enhanced by orders of magnitude. This effect eists for both engineered and random nanosystems. We also discuss possible applications. PACS p; 42.5.Md; n; 78.2.Bh Introduction There recently has been a tremendous interest in both ultra-fast (femtosecond and attosecond) laser-induced kinetics and in nanoscale properties of matter. Particular attention has been attracted by phenomena that are simultaneously nanoscale and ultra-fast; see e.g. [ ]. Fundamentally, nanosie eliminates effects of electromagnetic retardation and thus facilitates coherent ultra-fast kinetics. The entire electromagnetic dynamics in nanosystems is due to propagation of surface plasmons, while photons have much larger wavelengths to contribute to the nanolocaliation of energy (this is the underlying reason for the negligible effects of electromagnetic retardation). On the applied side, nanoscale design of optoelectronic devices is justified if their operating times are ultra-short to allow for ultra-fast computing and ultra-wide-band transmission of information. One of the key problems of ultra-fast/nanoscale physics is ultra-fast ecitation of a nanosystem where the transferred energy localies at a given site. Because the electromagnetic wavelength is on a much larger microscale, it is impossible to employ light-wave focusing for that purpose. On the other hand, the long-range dipole interaction causes ultra-fast transfer of an ecitation in nanostructures [], which brings about a redistribution of the ecitation energy across a nanosystem and may lead to delocaliation. For instance, consider a local ecitation of the system using a near-field scanning optical microscope (NSOM) or a nano-aperture. In this case, Fa: +-44/65-427; mstockman@gsu.edu the source of the ecitation is indeed well-localied, but this initially localied ecitation will spread over the nanosystem on the atto- to nanosecond scale due to the dipolar interaction between different parts of the nanosystem. Note that, additionally, the NSOM or nano-aperture spectral bandwidth may be insufficient to conduct the ultra-fast localied ecitation. To solve this problem, we propose to use phase modulation of an eciting femtosecond pulse as a functional degree of freedom to coherently control spatial-temporal distribution of the ecitation energy. This possibility eists due to the fact that polar ecitations (surface plasmons) in inhomogeneous nanosystems tend to be localied with their oscillation frequency (and, consequently, phase) correlated with position [, 2]. The pulse phase modulation will cause the eciting field to take energy away from surface plasmons localied in those parts of the system where the oscillations are out of phase with the driving pulse and move it, with time, to the surface-plasmon ecitations in other parts where such oscillations occur in phase with the driving pulse. Alternatively, one may think that the instantaneous frequency of the eciting pulse is changing, adiabatically passing through the resonance with a localied mode. As a result, this mode will initially be ecited and, later, as the pulse progresses, deecited and its ecitation energy moved to the net group of modes and further in the frequency domain in the same manner. Because the frequency and localiation in space for the eigenmodes of a nanosystem are correlated, it will lead to the flow of the ecitation energy in space, allowing for the directed localiation of the entire ecitation energy at a given site of the nanosystem. We will confirm below this heuristic idea by quantitative computations demonstrating the possibility of this directed-energy concentration for the selected nanosystems, controlled by the phase modulation of the ecited ultra-short pulse. Coherent control has been recently used to spatially concentrate the energy of an ultra-short nonlinearly generated pulse in a given high harmonic [3]. Impressive results have been obtained by using coherent control to affect spatial movement of particles and polariation using continuouswave (cw) fields (see e.g. [4, 5] and references cited therein). The possibility has been shown to concentrate the energy of acoustic waves at a given time and site inside a region when these waves are generated by a laser-induced ecitation of the surface of that region [6]. Our effect is based on the same general idea of interference between different com-

2 S64 Applied Physics B Lasers and Optics ponents of the resonant eciting radiation, governed by the phase modulation. However, it is different, because it requires ultra-short pulses to preserve the coherence in time in contrast to [4, 5], but it is a linear effect unlike [3, 5], and it results in the concentration of energy in the space on the nanoscale and not on the micro- to macroscale as in [6]. 2 Analytical theory The local field potential ϕ(r, t) induced by the eternal potential ϕ (r, t) is determined by the retarded Green s function G r (r, r ; t) of the system, ϕ(r, t) = ϕ (r, t) ϕ (r, t ) 2 r 2 Gr (r, r ; t t ) d 3 r dt. We consider the nanosystem formed by an active component (e.g. a metal or a semiconductor) described by the relative dielectric function ε(ω), embedded in a supporting host. This Green s function has the spectral epansion in the coordinate frequency domain: G r (r, r ; ω) = ϕ i (r )ϕ i (r)s i / (s(ω) s i ). (2) i Here s(ω) = /[ ε(ω)] is the spectral parameter [7]. The material-independent eigenmodes (surface plasmons) ϕ n (r) and the corresponding eigenvalues s n are found from the generalied eigenproblem [8]: [ (θ(r) s n ) ] r r ϕ n(r) =, (3) where θ(r) = if r belongs to the nanosystem, and θ(r) = otherwise. These eigenmodes depend only on the geometry of the system, but not on its material composition. That composition and the light frequency enter the problem only via the spectral parameter s(ω). Such a separation of the geometric and material properties is characteristic of the spectral method [7] and provides great computational advantages: the differential eigenproblem (3) is solved only once for a given geometry and used in the solution (2) for all frequencies of the radiation and any materials of the system. And, last but not least in this respect, the spectral solution given by () and (2) proves to be eceptionally stable numerically, in particular because the analytical properties of the Green s function are eactly taken into account by the spectral epansion of (2). To completely define the eigenproblem, in addition to (3), the homogeneous Dirichlet conditions (ero of the potential) are imposed on the boundary planes perpendicular to the eternal field () direction, and the homogeneous Neumann conditions (ero of the normal electric field) on the other four boundary planes [7]. By using these and not the frequently used periodic boundary conditions, we avoid spurious effects of the interaction between the adjacent cells in the latter method. We also require θ(r) = at the boundaries. It follows from (3) and the boundary conditions that these eigenmodes (surface plasmons) are orthogonal and normalied [8]: ϕn (r) ϕ m (r) d 3 r = δ mn. (4) r r () 3 Numerical computations Numerically, we have resolved eigenproblem (3) using the finite-difference method to convert it to a matri generalied eigenproblem that is solved using LAPACK [9]. Because θ(r) is singular, we smooth its discontinuities using Gaussian filtering with an RMS width equal to the grid step sie. This does not affect the properties of the system at intermediate or large scales and actually makes the small-scale structure more realistic by eliminating sharp edges. After computing the Green s function (2) in the real-space and frequency domains, the temporal convolution in () has been effected by the fast Fourier transform method. We use 496 frequency or temporal points. The rank of the eigenproblem is 892, corresponding to the space dimensions in, y and of (all sies are in the grid steps). We have verified by testing that this number of spatial and time frequency points allows us to achieve sufficient computational precision. We study two planar systems, an engineered V-shape and a random planar composite (RPC). In practice, a RPC can be made by partially covering a smooth, usually dielectric, surface by another material, usually a metal, forming what is often called a semicontinuous metal dielectric film. The geometry of these systems after smoothing is shown in Fig. a and b, respectively. We have computer-generated them by positioning cubes on the central y plane. The eciting pulse ϕ (r, t) is -polaried with a Gaussian envelope and duration T: { ϕ (r, t) = ep iω (t T/2) [ + α (t T/2) /T ] (3/2) [ (t T/2) /T ] } 2 + c. c., (5) whereα is a dimensionless phase-modulation parameter. Note that, due to the nanosie of the system, the wave electric field is uniform on the system scale. Therefore, the dependence of the potential in (5) on the coordinate vector r ={, y, } is linear in the coordinate. Starting with the V-shape, we show in Fig. 2a and d two ecitation pulses, with identical spectral composition and envelope, but with different phase modulation (α =±.3): these pulses are time-reversed (phase-conjugated) with respect to each other. Another ecitation pulse (not shown) had the identical envelope and the same mean (carrier) frequency ω, but no phase modulation (α = ). Comparing the 3 2 a 3 2 ) FIGURE Spatial distribution of smoothed density of two nanosystems: V-shape (left) and random planar composite (RPC) (right). Each unit on the coordinate aes represents 2 to nm b

3 STOCKMAN et al. Coherently controlled femtosecond energy localiation on nanoscale S65 E +/ +t/ E +t/ 4 4 E +t/ 4 4 Negative Chirp ( α =. 3 ) 2 t+fs/ 2 t+fs/ 2 t+fs/ E +/ +t/ Positive Chirp ( α =. 3 ) 2 t+fs/ E +t/ 4 4 E +t/ 4 4 (e) 2 t+fs/ (f) 2 t+fs/ FIGURE 2 Time dependence of fields in a V-shaped nanostructure, ecited by pulses of duration T = 5 fs, carrier frequency hω =.8eV, and amplitude, with negative chirp (α =.3) (left) and positive chirp (α =.3) (right). Panels, show the eciting pulses;, (e) show the field at the opening of the V-shape ( = 23, = 32);, (f) show the field at the ape of this nanosystem ( = 7, = 2). All fields are plotted using the same units, where the maimum value of the eciting field is set as responses to these pulses allows one to distinguish the effects of the phase control proper (i.e. the time dependence of pulse phases) from those of the spectral composition that is also changed by the phase modulation. In Fig. 2, the panels, (e), and, (f) show the temporal dynamics of the local fields at two characteristic sites of the system: the opening and the ape of the V-shape. It is evident that even with identical power spectra and envelopes, the phase modulation dramatically affects the temporal behavior: for the negative chirp, oscillations at the opening of the V-shape (panel ) first grow and then decay rapidly due to an unfavorable phase relation established with the driving pulse. This is analogous to the phenomenon of adiabatic passage through a resonance. By contrast, oscillations at the ape (panel ) evolve later and last longer. Consequently, the field oscillations in different parts of the system do not overlap in time, which suggests that a localied wave of the oscillating polariation is compelled to propagate from the opening to the ape of the nanostructure. In marked contrast to this behavior, for the positively chirped pulse the oscillations at the opening (Fig. 2e) and the ape (Fig. 2f) of the V-shape essentially overlap in time, indicating that there is no spatial localiation of the ecitation energy. In the absence of phase modulation (data not shown), the temporal dynamics is radically different from both of these cases: the local fields then closely reproduce the shape of the ecitation pulse, ecept for a tail of free-induction decay, a behavior similar to the response of noble-metal nanospheres [5]. The spatial distributions of the local fields at different instants for the case of negative chirp are shown in Fig. 3. The distribution in panel for t = 57.3 fs (near the time of the ecitation-pulse maimum) is peaked at the V-shape opening. At the end of the driving pulse (t =.3 fs, panel ), the ecitation energy has sharply concentrated at the ape of this nanosystem, where the maimum field enhanced almost by two orders of magnitude (with respect to the 4 E 4 4 E 4 Phase Modulation: Negative Chirp ( α =. 3 ) t 57.3 fs t.3 fs t 89.7 fs 4 E 4 4 FIGURE 3 Spatial distributions of local electric fields for V-shape and negative chirp (α =.3) at various instants of time. The largest field component is displayed for each instant. Panel corresponds to the instant of maimum field ehibited in Fig. 2b and panel corresponds similarly to Fig. 2c. Panel shows the instant of the overall maimum of the local field, and panel illustrates the long-time behavior. Panels,, are for the instances separated by time intervals on order of the optical period, while panel shows the long-time behavior 4 E t 8 fs driving field) develops (with the normal () polariation). Within two oscillations (t = 89.7 fs, panel ), the maimum field is equally enhanced and concentrated at the ape, but in the parallel () polariation. Note that the local energy density, proportional to the square of the local field, has its maimum enhanced by four orders of magnitude. This energy concentration persists long after the ecitation pulse is over (see panel at t = 8 fs). To summarie, for a comparatively long period (t 2 fs), almost the entire ecitation energy is concentrated at one point, at the ape of the V-shaped nanostructure, where the local fields are greatly enhanced and the oscillations persist in the form of free induction long after the driving pulse is over. The energy concentration is very similar to the ninth-wave effect [] with one principal distinction: here it is due to the phase modulation of the eciting pulse, while in [] it is a result of the randomness of the system. The local fields at the V-shape for the case of positive chirp, shown in Fig. 4, behave quite differently. Their maimum is reached earlier, between t = 58.5 and 68.3 fs. In sharp contrast to the case of negative chirp, the energy is never concentrated for a significant period of time at one location: it transfers from the ape (t = 58.5 fs, panel ) to the opening of the V-shape (t = 64.3 fs, panel ), and back to the ape (t = 68.3 fs, panel ) during a period of a few oscillations of the field. This behavior is also suggested by the field oscillations overlapping in time in Fig. 3e and f. By the end of the driving pulse (t = 98.4 fs, see Fig. 4d), the local fields are delocalied and significantly decayed with respect to their maimum, dramatically distinct from the negative-chirp case for the corresponding time interval (cf. Fig. 3c and d). For an unchirped pulse (α = ) (data not shown), there is no pronounced spatial dynamics or energy localiation. The local field pulse is spread out in time to the duration of 2 fs

4 S66 Applied Physics B Lasers and Optics 4 E 4 4 E 4 Phase Modulation: Positive Chirp ( α =. 3 ) t 58.5 fs t 64.3 fs 4 t 68.3 fs E 4 4 FIGURE 4 Spatial distributions of local electric fields for V-shape and positive chirp (α =.3) at various instants of time. The largest field component is displayed for each instant 4 E t 98.4 fs and is significantly smaller in amplitude than for the case of the chirped pulses, reaching only the magnitude of 4.We have thus demonstrated some nontrivial effects of the coherent control: (i) the phase modulation significantly affects the dynamics of the spatial distribution of local fields in nanostructures, allowing one to induce energy localiation; (ii) the specific time sequence of the phases is important, and not only the spectral composition of the ecitation pulse. The other systems considered, RPCs, are random and possess chaotic eigenmodes [8]. For a RPC, the temporal behavior of the eciting pulse and the local fields at the points of local and global maima are displayed in Fig. 5. In the absence of the phase modulation (Fig. 5a c), the temporal response of the composite is similar to that of metal nanoparticles in [5]. The kinetics for the case of phase modulation (see Fig. 5d f) resembles that for the V-shape (cf. Fig. 2). The phase modulation (positive chirp in this case) causes the field pulse at the location of a local maimum to be suppressed in magnitude and shortened in duration (see panel (e)). At the position of the global maimum (Fig. 5f), the chirp leads to the noticeable delay of the response relative to the field pulse at the local maimum position and significant (by two orders of magnitude) enhancement of the maimum field with respect to the driving pulse. The spatial distributions of the local fields for a RPC in the case of phase modulation (α =.3) are shown in Fig. 6. Similar to the V-shape, there is pronounced temporal dynamics of the local fields. At the beginning (t = 46.8 fs, panel ), the field distribution is delocalied. By the end of the pulse, the local fields are concentrated in a few hot spots and enhanced by two orders of magnitude compared to the eciting field amplitude (see panel ), similar to the ninth-wave effect [8]. For phase-modulation-free (unchirped) ecitation of RPC, illustrated in Fig. 7, there is dramatic dynamics of local fields: as time progresses, the ecitation initially distributed over the entire system (Fig. 7a) concentrates in a very narrow area (Fig. 7b). This is eactly the ninth-wave effect as introduced in [], which is due to the randomness of the system and consequent chaos of its eigenmodes. Note that this effect is obtained here without the benefit of the dipole approimation, from the solution of the differential equation boundary problem. Importantly, as a comparison with Fig. 6 shows, the ninth-wave effect (the spatial concentration of the ecitation energy) takes place both in the presence and absence of chirp, but the site of the energy concentration depends on the phase modulation, i.e. is coherently controllable. For the negative chirp, α =.3 (data not shown), the kinetics also ehibits Phase Modulation: Positive Chirp ( α =. 3 ) t 46.8 fs t 68.4 fs E +/ No Chirp +t/ ( α = ) 2 t+fs/ E +t/ 2 t+fs/ E +t/ 2 t+fs/ E +/ +t/ Positive Chirp ( α =. 3 ) 2 t+fs/ E +t/ (e) 2 t+fs/ E +t/ (f) 2 t+fs/ FIGURE 5 For a RPC, temporal behavior of the eciting pulses (duration T = 5 fs and carrier frequency hω =.9 ev) for the cases of no phase modulation (α = ) (left) and phase modulation with α =.3 (right). Graphs and are for the eciting pulses, and (e) are for the sites of the local maima, and and (f) are for the overall maima of the electric fields E FIGURE 6 Local electric fields at a RPC in the presence of phase modulation (α =.3) at the times indicated. Only the maimum component of the field is displayed at each moment. In panel is plotted the time of the maimum field in Fig. 5e, and panel similarly corresponds to Fig. 5f E E No Phase Modulation ( α = ) t 76. fs t 92.7 fs E FIGURE 7 Local electric fields at a RPC in the absence of phase modulation (α = ) at the times indicated. In panel is plotted the time of the maimum field in Fig. 5b, and panel similarly corresponds to Fig. 5c

5 STOCKMAN et al. Coherently controlled femtosecond energy localiation on nanoscale S67 the concentration of energy, but differs significantly from both the cases of α =.3 and α =. 4 Concluding remarks To briefly summarie the major results obtained: we predicted and quantitatively investigated the unique possibility to coherently control, on the nanoscale, the spatialtemporal concentration of energy transferred to a nanosystem by a femtosecond pulse. The phase modulation of the eciting pulse provides a degree of freedom necessary to compel the nanosystem to concentrate its ecitation energy at a certain point within a certain time interval. We showed that the ecitation energy localiation in nanosystems depends on the phase modulation, on the spectral composition of the eciting pulse, and, for a given spectral composition, on the specific time dependence of the pulse phase. For a V-shaped nanosystem, the negative-chirped Gaussian-envelope 5-fs pulse drives a traveling wave of surface plasmons, resulting in a concentration of the ecitation energy at the tip of the V-shape (Fig. 3). This energy concentration is similar to the ninth-wave effect [], but with the distinction that it is caused by the pulse phase modulation and not by the randomness of the system. In contrast, for the time-reversed, positively chirped pulse, the ecitation energy is never concentrated at one site of the system, but rather transfers forth and back across its entire etent within a few oscillations of the fields. In the case of a random planar composite, the energy concentration occurs spontaneously for an unmodulated Gaussian pulse, manifesting the ninth-wave effect. However, when a phase modulation is introduced, the energy concentrates at a completely different location. Thus the energyconcentration ( ninth-wave ) effect is coherently controllable. This effect of coherently controlled energy localiation can be detected in eperiments with nanometer spatial resolution, the most straightforward of which would invoke only time-integrated measurements, but no temporal resolution. In particular, the concentration of energy can be recorded by positioning a local probe (say a dye molecule or a cluster) at a given site of the system using the techniques of scanning probe microscopy. This effect will produce enhanced integral radiation from such a probe (fluorescence, Raman scattering, etc.) that is maimied when the phase modulation of the eciting pulse concentrates the energy at the probe position. This enhancement will be most pronounced in the case of nonlinear responses, such as two-photon-ecited fluorescence or the higher-order Raman effect. Phase-dependent photodamage of the probe, or even of the nanosystem itself, recorded by postradiation electron or scanning probe microscopy, would be another indication of the effect. A time-resolved observation of radiation from the local probe would give richer information, but is much more difficult to perform. The coherently controlled ultra-fast energy localiation in nanosystems, introduced in this paper, can have applications in different fields that require directed nanosie-selective ecitation or modification. One such application could be ultrafast computing using nanosystems, where the present effect can be used to energie and control processes on the nanoscale for femtosecond times. Ultra-fast site-selective photomodification of nanosystems may be another field of application. ACKNOWLEDGEMENTS This work was supported in part by the Chemical Sciences, Biosciences and Geosciences Division of the Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy. Partial support for the work of DJB was provided by grants from the US Israel Binational Science Foundation and the Israel Science Foundation. MIS is grateful to S.T. Manson for useful comments, to M.Yu. Ivanov and H. Rabit for valuable suggestions, and to M.M. Murnane and H.C. Kapteyn for stimulating discussions. REFERENCES S. Link, C. Burda, M.B. Mohamed, B. Nikoobacht, M.A. El-Sayed: Phys. Rev. B 6, 686 (2) 2 B. Lamprecht, G. Schider, R.T. Lechner, H. Ditlbachter, J.R. Krenn, A. Leitner, F.R. Aussenegg: Phys. Rev. Lett. 84, 472 (2) 3 T. Klar, M. Perner, S. Grosse, G. von Plessen, W. Spirkl, J. Feldman: Phys. Rev. Lett., 4249 (998) 4 J. Lehmann, M. Mershdorf, W. Pfeifer, A. Thon, S. Voll, C. Gerber: Phys. Rev. Lett. 85, 292 (2) 5 B. Lamprecht, J.R. Krenn, A. Leitner, F.R. Aussenegg: Phys. Rev. Lett. 83, 442 (999) 6 J.-H. Klein-Wiele, P. Simon, H.-G. Rubahn: Phys. Rev. Lett., 45 (998) 7 F. Stiet, J. Bosbach, T. Wenel, T. Vartanyan, A. Goldmann, F. Träger: Phys. Rev. Lett. 84, 5644 (2) 8 G. Kulcsár, D. AlMawlawi, F.W. Budnik, P.R. Herman, M. Moskovits, L. Zhao, R.S. Marjoribanks: Phys. Rev. Lett. 84, 549 (2) 9 M. Perner, S. Gresillon, J. Mär, G. von Plessen, J. Feldmann, J. Porstendorfer, K.-J. Berg, G. Berg: Phys. Rev. Lett. 85, 792 (2) M.I. Stockman: Phys. Rev. Lett. 84, (2); Phys. Rev. B 62, 494 (2) M.I. Stockman: Phys. Rev. Lett. 79, 4562 (997); Phys. Rev. E 56, 6494 (997) 2 M.I. Stockman, L.N. Pandey, T.F. George: Phys. Rev. B 53, 283 (996) 3 R. Bartels, S. Bachus, E. Zeek, L. Misoguti, G. Vdovin, I.P. Christov, M.M. Murnane, H.C. Kapteyn: Nature 46, 64 (2) 4 B.K. Dey, M. Shapiro, P. Brumer: Phys. Rev. Lett. 85, 325 (2) 5 R.D.R. Bhat, J.E. Sipe: Phys. Rev. Lett. 85, 5432 (2) 6 Y.S.Kim,M.Tadi,H.Rabit,A.Askar,J.B.McManus:Phys.Rev.B5, (994) 7 D.J. Bergman, D. Stroud: In Solid State Physics, Vol. 46, ed. by H. Ehrenreich, D. Turnbull (Academic, Boston, MA 992) p M.I. Stockman, S.V. Faleev, D.J. Bergman: Phys. Rev. Lett. 87, 67 4 (2) 9 E. Anderson, Z. Bai, C. Bischof, J. Demmel, J. Dongarra, J. Du Cro, A. Greenbaum, S. Hammarling, A. McKenney, S. Ostrouchov, D. Sorensen: LAPACK Users Guide, 2nd edn. (SIAM, Philadelphia, PA 995)