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Physica B 338 (2003) 361–365

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Femtosecond energy concentration in nanosystems:coherent control

Mark I. Stockmana,*, Sergey V. Faleeva,1, David J. Bergmanb

aDepartment of Physics and Astronomy, Georgia State University, Atlanta, Georgia 30303, USAbSchool of Physics and Astronomy, Raymond and Beverly Sackler Faculty of Exact Sciences, Tel Aviv University, Tel Aviv, 69978, Israel

Abstract

We describe the unique possibility of concentrating the energy of an ultrafast excitation of a nanosystem in a small

part of the whole system by means of coherent control (phase modulation of the exciting ultrashort pulse). Such

concentration is due to dynamic properties of surface plasmons and leads to local fields enhanced by orders of

magnitude. This effect exists for both ‘‘engineered’’ and random nanosystems. We also discuss possible applications.

r 2003 Elsevier B.V. All rights reserved.

PACS: 78.47.+p; 42.50.Md; 78.67.�n; 78.20.Bh

Keywords: Ultrafast excitation; Coherent control; Surface plasmons; Phase modulation; Local fields

There recently has been a tremendous interest inboth ultrafast (femtosecond and attosecond) laser-induced kinetics and in nanoscale properties ofmatter. Particular attention has been attracted byphenomena that are simultaneously nanoscale and

ultrafast, see, e.g., Refs. [1–10]. This interest is due,in particular, to the necessity to fabricate devicesthat are not only nanoscale in size, but alsoultrafast for the purposes of computing, informa-tion transfer and storage, etc.

One of the key problems of ultrafast/nanoscalephysics is ultrafast excitation of a nanosystemwhere the transferred energy localizes at a givensite. Because the electromagnetic wavelength is on

onding author.

ddress: mstockman@gsu.edu (M.I. Stockman).

ttp://www.phy-astr.gsu.edu/stockman.

address: Sandia National Laboratories, Livermore,

USA.

- see front matter r 2003 Elsevier B.V. All rights reserve

/j.physb.2003.08.021

a much larger microscale, it is impossible toemploy light-wave focusing for that purpose. Also,the long-range dipole interaction will cause ultra-fast transfer of excitation in nanostructures caus-ing delocalization. To solve this problem, we haveproposed to use phase modulation of an excitingfemtosecond pulse as a functional degree offreedom to coherently control spatial distributionof the excitation energy [11]. This possibility existsdue to the fact that polar excitations (surfaceplasmons) in inhomogeneous nanosystems tendto be localized with their oscillation frequency(and, consequently, phase) correlated withposition [12,13].

Coherent control was used recently to spectrallyconcentrate the energy of an ultrashort, nonli-nearly generated pulse in a certain high harmonic[14]. Impressive results were obtained by usingcoherent control to affect spatial movement ofparticles and polarization using continuous-wave

d.

ARTICLE IN PRESS

0 10 20 30x

0

10

20

30

z

0 10 20 30x

0

10

20

30

z

Fig. 1. Spatial distribution of smoothed density of two

nanosystems: V-shape (left panel) and random planar compo-

site (RPC) (right panel). Each unit on the coordinate axes

represents 2–10 nm:

V-shape

100 200 100 200-1

0

1

100 200 100 200-80-40

04080

100 200 100 200t (fs)

t (fs)

t (fs)t (fs)

t (fs)

t (fs)-80-40

04080

Positive Chirp(� = 0.3)(� = -0.3)

(e)

(f)

(d)

Negative Chirp

Ez

(t)

Ez

(t)

Ez

(t)

(0)

-1

0

1

-80-40

04080

-80-40

04080

(b)

(c)

(a)

Ez

(t)

Ez

(t)

Ez

(t)

(0)

Fig. 2. Time dependence of fields in a V-shaped nanostructure,

excited by pulses of duration T ¼ 50 fs; carrier frequency

_o0 ¼ 0:8 eV; and amplitude 1, with negative chirp ða ¼ �0:3Þ(left panel) and positive chirp ða ¼ 0:3Þ (right panel). Panels (a),

(d) show the exciting pulses; (b), (e) show the field at the

‘‘opening’’ of the V-shape (x ¼ 23; z ¼ 32); (c), (f) show the

field at the apex of this nanosystem (x ¼ 17; z ¼ 2). All fields

are plotted using the same units, where the maximum value of

the exciting field is set as 1.

M.I. Stockman et al. / Physica B 338 (2003) 361–365362

(CW) fields (see, e.g., Refs. [15,16] and referencescited therein). The possibility was shown tospatially concentrate the energy of laser-generatedacoustic waves [17]. Our effect is based on a similargeneral idea of interference between differentcomponents of the resonant exciting radiation,governed by the phase modulation. However, it isdifferent, because it requires ultrashort pulses incontrast to Refs. [15,16], and it is a linear effectunlike Refs. [14,16]. Also, it results in a nanoscalespatial concentration of energy, in contrast to themacro- to microscale concentration in Ref. [17].

The local field potential jðr; tÞ; induced by theexternal potential j0ðr; tÞ; is determined by theretarded Green’s function Grðr; r0; tÞ of the system

jðr; tÞ ¼j0ðr; tÞ

�Z

j0ðr0; t0Þ

q2

qr02Grðr; r0; t � t0Þ d3r0 dt0: ð1Þ

We consider the nanosystem formed by an activecomponent (e.g., a metal or semiconductor),described by the relative dielectric function eðoÞ;embedded in a dielectric host. This Green’sfunction is computed as a spectral expansion inthe coordinate-frequency domain:

Grðr; r0;oÞ ¼X

i

jiðr0ÞjiðrÞsi=ðsðoÞ � siÞ; ð2Þ

where sðoÞ ¼ 1=½1 � eðoÞ� is the spectral parameter[18]. The material-independent eigenmodes (sur-face plasmons) jiðrÞ; and the correspondingeigenvalues si; are found by numerically solving ageneralized Dirichlet–Neumann eigenproblem ona grid, as described in Ref. [19].

We study two planar systems, an ‘‘engineered’’V-shape with and without defects (vacancies) anda random planar composite (RPC). An RPC canbe made, in practice, by partially covering asmooth dielectric surface by a metal (or semicon-ductor), forming what is often called a semicon-tinuous metal-dielectric film. The geometry ofthese systems after smoothing is shown in Fig. 1.We have computer-generated them by positioningcubes of size 2 � 2 � 2 on the central xz plane. Thematerial dielectric function is set as that of silver[20] in a host whose dielectric constant is eh ¼ 2:0:The exciting field j0ðr; tÞ is a z-polarized chirped

pulse with Gaussian envelope and duration T

j0ðr; tÞ ¼ � z expf�io0ðt � T=2Þ

� ½1 þ aðt � T=2Þ=T �

� ð32Þ ½ðt � T=2Þ=T �2g þ c:c:; ð3Þ

where a is a dimensionless phase-modulationparameter.

Starting with the V-shape, we show in Figs. 2(a)and (d) two excitation pulses, with identical

ARTICLE IN PRESS

V-shape; Negative Chirp (� = -0.3)

xz-80

-40

0

40

80

Ez

xz-80

-40

0

40

80

Ex

xz-80

-40

0

40

80

Ez

xz-80

-40

0

40

80

Ez

(a) (b)

(c) (d)

t = 118 fs

t = 80.3 fst = 57.3 fs

t = 89.7 fs

Fig. 3. Spatial distributions of local electric fields for V-shape

and negative chirp ða ¼ �0:3Þ at various instants of time. The

largest field component is displayed for each instant. Panel (a)

corresponds to the instant of maximum field exhibited in

Fig. 2(b), and panel (c) corresponds similarly to Fig. 2(c).

M.I. Stockman et al. / Physica B 338 (2003) 361–365 363

spectral composition and envelopes, but withdifferent phase modulation ða ¼ 70:3Þ: these pulsesare time reversed (phase conjugated) with respectto each other. Another excitation pulse (notshown) had the identical envelope and the samemean (carrier) frequency o0; but no phasemodulation ða ¼ 0Þ: Comparing the responses tothese pulses allows one to distinguish the effects ofthe phase control proper (i.e., the time dependenceof pulse phases) from those of the spectral

composition that is also changed by the phasemodulation.

In Fig. 2, the panels (b), (e), and (c), (f) show thetemporal dynamics of the local fields at twocharacteristic sites of the system: the opening andthe apex of the V-shape. It is evident that even withidentical power spectra and envelopes, the phasemodulation dramatically affects the temporal be-havior: For the negative chirp, oscillations at theopening of the V-shape (panel (b)) first grow andthen decay rapidly due to an unfavorable phaserelation with the driving pulse. This is analogous tothe phenomenon of adiabatic passage through aresonance. By contrast, oscillations at the apex(panel (c)) evolve later and last longer. Conse-quently, the field oscillations in different parts ofthe system do not overlap in time, which suggeststhat a localized wave of the oscillating polarizationis compelled to propagate from the opening to theapex of the nanostructure. In marked contrast tothis behavior, for the positively chirped pulse, theoscillations at the opening (Fig. 2(e)) and apex(Fig. 2(f)) of the V-shape essentially overlap in time,indicating that there is no spatial localization of theexcitation energy. In the absence of phase modula-tion (data not shown), the temporal dynamics isradically different from both of these cases: Thelocal fields then closely reproduce the shape of theexcitation pulse, except for a tail of free-inductiondecay, a behavior similar to the response of noble-metal nanospheres [5].

The spatial distributions of the local fields atdifferent instants for the case of negative chirp areshown in Fig. 3. The distribution in panel (a) fort ¼ 57:3 fs (near the time of the excitation pulsemaximum) is peaked at the V-shape ‘‘opening’’. Atthe end of the driving pulse (t ¼ 80:3 fs; panel (b)),the excitation energy has sharply concentrated at

the apex of this nanosystem, where the maximumfield enhanced almost by two orders of magnitude(with respect to the driving field) develops (withthe normal ðxÞ polarization). Within two oscilla-tions (t ¼ 89:7 fs; panel (b)), the maximum field isequally enhanced and concentrated at the apex,but in the parallel ðzÞ polarization. This energyconcentration persists long after the excitationpulse is over (see panel (d) at t ¼ 118 fs). Tosummarize, for a comparatively long period

ðtE802120 fsÞ; almost the entire excitation energy

is concentrated at one point, at the apex 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. This energy concentration is very similar tothe ‘‘ninth wave’’ effect [10] with one principaldistinction: Here it is due to the phase modulationof the exciting pulse, while in Ref. [10] it is a resultof the randomness of the system.

The local fields at the V-shape for the case ofpositive chirp, shown in Fig. 4, behave quitedifferently. Their maximum is reached earlier,between t ¼ 58:5 and 68:3 fs: In sharp contrast tothe case of negative chirp, the energy is neverconcentrated for a significant period of time at onelocation: it transfers from the apex (t ¼ 58:5 fs;

ARTICLE IN PRESS

V-shape; Positive Chirp (� = 0.3)

xz-80

-40

0

40

80

Ez

xz-80

-40

0

40

80

Ez

xz-80

-40

0

40

80

Ex

t = 98.4 fst = 68.3 fs

t = 64.3 fst = 58.5 fs

xz-80

-40

0

40

80

Ez

(a) (b)

(c) (d)

Fig. 4. Spatial distributions of local electric fields for V-shape

and positive chirp ða ¼ 0:3Þ at various instants of time. The

largest field component is displayed for each instant.

t = 68.3 fs t = 92.6 fs

xz-100

-50

0

50

100

Ex

Ex

xz-100

-50

0

50

100α = 0.3 α = 0

(a) (b)

Fig. 5. Local electric fields in RPC at the indicated instants of

their overall (global) maxima: (a) The case of positive chirp

ða ¼ 0:3Þ; (b) no phase modulation ða ¼ 0:0Þ:

Single-Vacancy V-shape; Negative Chirp (� = -0.3)

t = 59.5 fs

xz-80

-40

0

40

80

Ez

t = 80.1 fs

xz-80

-40

0

40

80

Ey

t = 90.1 fs

xz-80

-40

0

40

80

Ex

t = 102 fs

xz-80

-40

0

40

80

Ey

(a) (b)

(c) (d)

Fig. 6. Same as in Fig. 3, but for V-shape with a vacancy.

M.I. Stockman et al. / Physica B 338 (2003) 361–365364

panel (a)) to the opening of the V-shape(t ¼ 64:3 fs; panel (b)), and back to the apex(t ¼ 68:3 fs; panel (c)) during a period of a fewoscillations of the field.

We have thus demonstrated some nontrivialeffects of the coherent control: (i) the phasemodulation significantly affects the dynamics ofthe spatial distribution of local fields in nanos-tructures, allowing one to induce energy localiza-tion. (ii) the specific time sequence of the phases isimportant, and not only the spectral compositionof the excitation pulse.

The other systems considered, RPCs, are ran-dom and possess chaotic eigenmodes [19]. InFig. 5, we display two spatial distributions of thelocal fields for the cases of positive and zero chirpin the same RPC, at the instants when the overallmaximum fields are achieved. In each case, there isa significant concentration of the excitation

energy: the local fields form ‘‘hot spots’’ wherethey are enhanced by nearly two orders ofmagnitude. In the case of a positive chirp (panel(a)), the distribution differs dramatically from thatof the zero chirp (panel (b)), and both are verydifferent from the case of negative chirp (data notshown). Thus the locations of the hot spots can becoherently controlled: They are strongly depen-dent on the phase modulation, both the spectralcomposition and specific time dependence of theexcitation phase. Note that the concentration ofenergy at hot spots, even for a nonchirped pulse(Fig. 5(b)), is exactly the ‘‘ninth wave’’ effect asintroduced in Ref. [10], which is due to therandomness of the system causing chaos of itseigenmodes [12] and giant fluctuations of the localfields [21]. This effect is confirmed here, withoutrelying on the dipole approximation, by solvingthe differential equation boundary problem.

Since it is impossible to fabricate an idealnanostructure, there is a principal question of thestability of these results with respect to and theinfluence of their structural defects. For thisreason, we have studied a V-shape with a vacancyat the place of one of the constituent cubes. Forsuch a nanostructure, a sequence of the local fielddistributions is shown in Fig. 6 for the case ofnegative phase modulation (cf. Fig. 3 for thesystem without a defect). The general conclusion isthat the fields are quite stable with respect to this

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M.I. Stockman et al. / Physica B 338 (2003) 361–365 365

structural defect. Only slight accumulation ofenergy at the site of the defect and decrease of itsconcentration at the tip is seen in Fig. 6.

To briefly summarize the major results obtained:We predicted and quantitatively investigated theunique possibility to coherently control, on thenanoscale, the spatial concentration of energytransferred to a nanosystem by a femtosecondpulse. The phase modulation of the exciting pulseprovides a degree of freedom necessary to compelthe nanosystem to concentrate its excitationenergy at a certain point. We showed that the

excitation energy localization in nanosystems de-

pends on the phase modulation, both on the spectral

composition of the exciting pulse and, for a given

spectral composition, on the specific time depen-

dence of the pulse phase. In the case of a randomplanar composite (RPC), the energy concentrationoccurs spontaneously for the unmodulated Gaus-sian pulse, manifesting the ‘‘ninth wave’’ effect[10]. However, when a phase modulation isintroduced, the energy concentrates at a comple-tely different location.

This effect of phase-controlled energy localiza-tion can be detected in experiments withnanometer spatial resolution, the most straightfor-ward of which would invoke only time-integratedmeasurements, but no temporal resolution. Inparticular, the concentration of energy can berecorded by positioning a local probe (say, a dyemolecule, or a cluster) at a given site of the systemusing the techniques of scanning probe micro-scopy. There will be an enhancement of integral

nonlinear responses, such as two-photon-excitedfluorescence, or higher-order Raman effect. Aphase-dependent photo-damage of the probe, oreven of the nanosystem itself, recorded by post-radiation electron or scanning-probe microscopy,would be another indication of the effect. A time-resolved observation of radiation from the localprobe would give richer information, but is muchmore difficult to perform.

The coherently controlled ultrafast energy loca-lization in nanosystems, introduced in this paper,can have applications in different fields thatrequire directed nanosize-selective excitation ormodification. One such application could beultrafast computing using nanosystems, where

the present effect can be used to energize andcontrol processes on the nanoscale for femtose-cond times. Ultrafast site-selective photomodifica-tion of nanosystems may be another field ofapplication.

Acknowledgements

This work was supported by the ChemicalSciences, Biosciences and Geosciences Division ofthe Office of Basic Energy Sciences, Office ofScience, US Department of Energy. Partial sup-port for the work of D.J.B was provided by grantsfrom the US–Israel Binational Science Foundationand the Israel Science Foundation. M.I.S isgrateful to Y. Braiman, M.Yu. Ivanov, T.Kobayashi, and S.T. Manson for stimulatingdiscussions.

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