arXiv:0705.0725v1 [physics.optics] 5 May 2007
Toward Full Spatio-Temporal Control on the Nanoscale
Maxim Durach,1Anastasia Rusina,1Keith Nelson,2and Mark I. Stockman1
1Department of Physics and Astronomy, Georgia State University, Atlanta, Georgia 30303, USA
2Department of Chemistry, MIT, Cambridge, MA 02139, USA
(Dated: February 5, 2008)
We introduce an approach to implement full coherent control on nanometer length scales. It is
based on spatio-temporal modulation of the surface plasmon polariton (SPP) fields at the thick edge
of a nanowedge. The SPP wavepackets propagating toward the sharp edge of this nanowedge are
compressed and adiabatically concentrated at a nanofocus, forming an ultrashort pulse of local fields.
The one-dimensional spatial profile and temporal waveform of this pulse are completely coherently
PACS numbers: 78.67.-n, 71.45.Gm, 42.65.Re, 73.20.Mf
Two novel areas of optics have recently attracted a
great deal of attention: nanooptics and ultrafast op-
tics. One of the most rapidly developing directions in
ultrafast optics is quantum control, in which coherent
superpositions of quantum states are created by exci-
tation radiation to control the quantum dynamics and
outcomes1,2,3,4. Of special interest are coherently con-
trolled ultrafast phenomena on the nanoscale where the
phase of the excitation waveform along with its polariza-
tion provides a functional degree of freedom to control
nanoscale distribution of energy5,6,7,8,9,10. Spatiotempo-
ral pulse shaping permits one to generate dynamically
predefined waveforms modulated both in frequency and
in space to focus ultrafast pulses in the required micro-
scopic spatial and femtosecond temporal domains11,12.
In this Letter, we propose and theoretically develop a
method of full coherent control on the nanoscale where a
spatiotemporally modulated pulse is launched in a graded
nanostructured system. Its propagation and adiabatic
concentration provide a possibility to focus the optical
energy in nanoscale spatial and femtosecond temporal re-
gions. The idea of adiabatic concentration13,14(see also
Ref. 15) is based on adiabatic following by the propa-
gating surface plasmon-polariton (SPP) wave of a plas-
monic waveguide, where the phase and group velocities
decrease toward a limit at which the propagating SPP
wave is adiabatically transformed into a standing surface
plasmon (SP) mode. This effect has been further devel-
oped theoretically16,17and observed experimentally.18
To illustrate the idea of this full coherent control, con-
sider first the adiabatic concentration of a plane SPP
wave propagating along a nanowedge of silver19, as shown
in Fig. 1(a); the theory is based on the Wentzel-Kramers-
Brillouin (WKB) or quasiclassical approximation, also
called the eikonal approximation in optics20as suggested
in Refs. 13,14. The propagation velocity of the SPP along
such a nanowedge is asymptotically proportional to its
thickness. Thus when a SPP approaches the sharp edge,
it slows down and asymptotically (in the ideal limit of
zero thickness at the apex) stops while the local fields
are increased and nano-concentrated.
Now consider a family of SPP rays (WKB trajecto-
ries) propagating from the thick side of a nanowedge,
FIG. 1: (a) Illustration of adiabatic concentration of energy
on the wedge. The distribution of local field intensity I in the
normal plane of propagation of SPPs (the yz plane). The in-
tensity in relative units is color coded with the color scale bar
shown to the right. (b) Trajectories of SPP rays propagating
from the thick to sharp edge of the wedge. The initial coor-
dinate is coded with color. The black curves indicate lines of
equal phase (SPP wave fronts).
as shown in Fig. 1(b). One possibility to launch such
SPPs is to have nanoscale inhomogeneities (nanoparti-
cles or nanoholes) at the thick edge of the wedge. Each
of the optical pulses from the spatiotemporal shaper is fo-
cused on the corresponding nanoparticle, scattering from
it and generating an SPP wave. This scattering is neces-
sary to impart large transverse momenta on SPPs, which
is required for their efficient transverse focusing. These
SPP waves propagate toward the sharp edge, adiabati-
cally slow down which increases the field amplitudes, and
constructively interfere as they converge at the nanofo-
The phases of the SPPs rays (i.e., the corresponding
wave fronts) are defined by the spatiotemporal modulator
in such a way that the rays converge to a reconfigurably
chosen point at the sharp edge of the nanowedge where
they acquire equal phases. When the SPPs propagate
along the rays, adiabatic concentration takes place be-
cause the SPP wavelength tends to zero proportionally
to the thickness of the wedge. This allows one to fo-
cus optical energy at a predefined nanofocus at the sharp
edge. The temporal structure of the generated SPPs can
be chosen in such a way that at the nanofocus the lo-
cal fields form an ultrashort pulse. Using spatiotemporal
modulation of the excitation field at the thick edge, one
can arbitrarily move the nanofocus along the sharp edge.
A superposition of such fields can render arbitrary spa-
tiotemporal modulation on the sharp edge, enabling one
to exert full control over nanoscale fields in space and
Turning to the theory, consider a nanofilm of metal
in an xy plane whose thickness d in the z direction is
adiabatically changing with the coordinate-vector ρ =
(x,y) in the plane of the nanofilm. Let εm= εm(ω) be
the dielectric permittivity of this metal nanofilm, and εd
be the permittivity of the embedding dielectric. Because
of the symmetry of the system, there are odd and even
(in the normal electric field) SPPs. It is the odd SPP that
is a slow-propagating, controllable mode. The dispersion
relation for this mode defining its effective index n(ρ) is
where k0= ω/c is the radiation wave vector in vacuum.
Let τ be a unit tangential vector to the SPP trajectory
(ray). It obeys a conventional equation of ray optics20
= ∇n − τ (τ∇n) ,(2)
where l is the length along the ray and ∇ = ∂/∂ρ.
Now let us consider a nanofilm shaped as a nanowedge
as in Fig. 1(b). In such a case, n = n(y), and these
trajectory equations simplify as
From these, it follows that nx ≡ τxn = const.
SPP wave vector, related to its momentum, is k(ρ) =
k0n(ρ)τ; this is the conservation of kx (the transverse
momentum). This allows one to obtain a closed solu-
tion for the ray. The tangent equation for the ray is
dx/dy = τx/τy, where τy=
get an explicit SPP trajectory (ray) equation as
?1 − n2
x/n2. From this, we
x − x0=
where ρ0 = (x0,y0) is the focal point where rays with
any nxconverge. To find the trajectories, as n(y) we use
the real part of effective index (1), as WKB suggests.
When the local thickness of the wedge is subwavelength
(k0d ≪ 1), the form of these trajectories can be found
analytically. Under these conditions, dispersion relation
(1) has an asymptotic solution
na= lnεm− εd
Substituting this into Eq. (4), we obtain explicit equa-
tions of trajectories,
x − x0−
+ y2=¯ n2
FIG. 2: (a) Phase (real part of eikonal Φ) acquired by a SPP
ray propagating between a point with coordinate x on the
thick edge and the nanofocus, displayed as a function of x.
The rays differ by frequencies that are color coded by the
vertical bar. (b) The same as (a) but for extinction of the ray
where ¯ na = na/(k0tanθ), and tanθ is the slope of the
wedge. Thus, each SPP ray is a segment of a circle whose
center is at a point given by x = x0+
and y = 0. This analytical result is in agreement with
Fig. 1 (b). If the nanofocus is at the sharp edge, i.e.,
y0= 0, then these circles do not intersect but touch and
are tangent to each other at the nanofocus point.
As an example we consider a silver19nanowedge illus-
trated in Fig. 1 (b) whose maximum thickness is dm= 30
nm and whose length (in the y direction) is L = 5 µm.
Trajectories calculated from Eq. (4) for ?ω = 2.5 eV are
shown by lines (color used only to guide eye); the nanofo-
cus is indicated by a bold red dot. The different trajec-
tories correspond to different values of nx in the range
0 ≤ nx ≤ n(L). In contrast to focusing by a conven-
tional lens, the SPP rays are progressively bent toward
the wedge slope direction.
The eikonal is found as an integral along the ray
?(¯ na/nx)2− y2
Consider rays emitted from the nanofocus [Fig. 1 (b)].
Computed from this equation for frequencies in the visi-
ble range, the phases of the SPPs at the thick edge of the
wedge (for y = L) are shown in Fig. 2 (a) as functions
of the coordinate x along the thick edge. The colors of
the rays correspond to the visual perception of the ray
frequencies. The gained phase dramatically increases to-
ward the blue spectral region, exhibiting a strong disper-
sion. The extinction for most of the frequencies except
for the blue edge, displayed in Fig. 2 (b), is not high.
Now consider the evolution of the field intensity along
a SPP ray. For certainty, let SPPs propagate along the
corresponding rays from the thick edge of the wedge to-
ward the nanofocus as shown in Fig. 1 (b). In the pro-
cess of such propagation, there will be concentration of
the SPP energy in all three directions (3d nanofocusing).
This phenomenon differs dramatically from what occurs
in conventional photonic ray optics.
To describe this nanofocusing, it is convenient to intro-
duce an orthogonal system of ray coordinates whose unit
vectors are τ (along the ray), η = (−τy,τx) (at the sur-
face normal to the ray), and ez(normal to the surface).
The concentration along the ray (in the τ direction) oc-
curs because the group velocity vg = [∂(k0n)/∂ω]−1of
SPP asymptotically tends to zero (for the antisymmetric
mode) for k0d → 0 as vg= v0gd where v0g= const.13,14
This contributes a factor A?= 1/?vg(d) to the ampli-
The compression of a SPP wave in the ez (vertical)
tude of an SPP wave.
direction is given by a factor of Az=
where W is the energy density of the mode. Substituting
a standard expression20for W, one obtains explicitly
1 +|n|2+ |κd|2
where κm= k0√n − εmand κd= k0√n − εd.
To obtain the compression factor A⊥for the η direc-
tion), we consider conservation of energy along the beam
of rays corresponding to slightly different values of nx.
Dividing this constant energy flux by the thickness of
this beam in the η direction, we arrive at
The ray amplitude thus contains the total factor which
describes the 3d adiabatic compression: A = A?A⊥Az.
Now consider the problem of coherent control. The
goal is to excite a spatiotemporal waveform at the thick
edge of the wedge in such a way that the propagating
SPP rays converge at an arbitrary nanofocus at the sharp
edge where an ultrashort pulse is formed. To solve this
problem, we use the idea of back-propagation or time-
reversal.21We generate rays at the nanofocus as an ultra-
short pulse containing just several oscillations of the light
field. Propagating these rays, we find amplitudes and
phases of the fields at the thick edge at each frequency
as given by the eikonal Φ(ρ). Then we complex con-
jugate the amplitudes of frequency components, which
corresponds to the time reversal. We also multiply these
amplitudes by exp(2ImΦ) which pre-compensates for the
losses. This provides the required phase and amplitude
modulation at the thick edge of the wedge.
We show an example of such calculations in Fig. 3.
Panel (a) displays the trajectories of SPPs calculated ac-
cording to Eq. (4). The trajectories for different frequen-
cies are displayed by colors corresponding to their visual
perception. There is a very significant spectral disper-
sion: trajectories with higher frequencies are much more
FIG. 3: (a) Trajectories (rays) of SPP packets propagating
from the thick edge to the nanofocus displayed in the xy
plane of the wedge. The frequencies of the individual rays
in a packet are indicated by color as coded by the bar at
the top. (b)-(d) Spatiotemporal modulation of the excitation
pulses at the thick edge of the wedge required for nanofocus-
ing. The temporal dependencies (waveforms) of the electric
field for the phase-modulated pulses for three points at the
thick edge boundary: two extreme points and one at the cen-
ter, as indicated, aligned with the corresponding x points at
panel (a). (e) The three excitation pulses of panels (b)-(d) (as
shown by their colors), superimposed to elucidate the phase
shifts, delays, and shape changes between these pulses. The
resulting ultrashort pulse at the nanofocus is shown by the
black line. The scale of the electric fields is arbitrary but
consistent throughout the figure.
curved. The spatial-frequency modulation that we have
found succeeds in bringing all these rays (with different
frequencies and emitted at different x points) to the same
nanofocus at the sharp edge.
The required waveforms at different x points of the
thick edge of the wedge are shown in Fig. 3 (b)-(d) where
the corresponding longitudinal electric fields are shown.
The waves emitted at large x, i.e., at points more distant
from the nanofocus, should be emitted significantly ear-
lier to pre-compensate for the longer propagation times.
They should also have different amplitudes due to the
differences in A. Finally, there is clearly a negative chirp
(gradual decrease of frequency with time). This is due
to the fact that the higher frequency components prop-
agate more slowly and therefore must be emitted earlier
to form a coherent ultrashort pulse at the nanofocus.
In Fig. 3 (e) we display together all three of the rep-
resentative waveforms at the thick edge to demonstrate
their relative amplitudes and positions in time. The pulse
at the extreme point in x (shown by blue) has the longest
way to propagate and therefore is the most advanced in
time. The pulse in the middle point (shown by green) is
intermediate, and the pulse at the center (x = 0, shown
by red) is last. One can notice also a counterintuitive fea-
ture: the waves propagating over longer trajectories are
smaller in amplitude though one may expect the oppo-
site to compensate for the larger losses. The explanation
of this fact is that the losses are actually insignificant for
the frequencies present in these waveforms. What deter-
mines the relative magnitudes of these waveforms is the
coefficient A⊥ of the transverse concentration, which is
much higher for the peripheral trajectories than for the
Figure 3 (e) also shows the resulting ultrashort pulse
in the nanofocus. This is a transform-limited, Gaussian
pulse. The propagation along the rays completely com-
pensates the initial phase and amplitude modulation, ex-
actly as intended. As a result, the corresponding electric
field of the waveform is increased by a factor of 100. Tak-
ing the other component of the electric field and the mag-
netic field into account, the corresponding increase of the
energy density is by a factor ∼ 104with respect to that
of the SPPs at the thick edge.
Consider the efficiency of the energy transfer to the
nanoscale. This is primarily determined by the cross sec-
tion σSPPfor scattering of photons into SPPs. For in-
stance, for a metal sphere of radius R at the surface of
the wedge, one can obtain an estimate σSPP∼ R6/(d3
where λ is the reduced photon wavelength.
R ∼ dm, we estimate σSPP∼ 3 nm2. Assuming opti-
cal focusing into a spot of ∼ 300 nm radius, this yields
the energy efficiency of conversion to the nanoscale of
∼ 10−3. Taking into account the adiabatic concentration
of energy by a factor of 104, the optical field intensity
at the nanofocus is enhanced by one order of magnitude
with respect to that of the incoming optical wave.
The criterion of applicability of the WKB approxima-
tion is ∂k−1/∂y ≪ 1. Substituting k = k0n and Eq.
(5), we obtain a condition dm/(naL) << 1. This condi-
tion is satisfied everywhere including the nanofocus since
na ∼ 1 and dm ≪ L for adiabatic grading. The min-
imum possible size of the wavepacket at the nanofocus
in the direction of propagation, ∆x, is limited by the lo-
cal SPP wavelength: ∆x ∼ 2π/k ≈ 2πdf/na, where df
is the wedge thickness at the nanofocus. The minimum
transverse size a (waist) of the SPP beam at the nanofo-
cus can be calculated as the radius of the first Fresnel
zone: a = π/kx ≥ π/(k0nx). Because nx is constant
along a trajectory, one can substitute its value at the
thick edge (the launch site), where from Eq. (5) we ob-
tain nx≈ n = na/dm. This results in a ≈ πdm/na; thus
a is on order of the maximum thickness of the wedge,
which is assumed also to be on the nanoscale.
To briefly conclude, we have proposed and theoretically
investigated an approach to full coherent control of spa-
tiotemporal energy localization on the nanoscale. From
the thick edge of a plasmonic metal nanowedge, SPPs
are launched, whose phases and amplitudes are indepen-
dently modulated for each constituent frequency of the
spectrum and at each spatial point of the excitation. This
pre-modulates the departing SPP wave packets in such a
way that they reach the required point at the sharp edge
of the nanowedge in phase, with equal amplitudes form-
ing a nanofocus where an ultrashort pulse with required
temporal shape is generated. This system constitutes
a “nanoplasmonic portal” connecting the incident light
field, whose features are shaped on the microscale, with
the required point or features at the nanoscale.
This work was supported by grants from the Chemi-
cal Sciences, Biosciences and Geosciences Division of the
Office of Basic Energy Sciences, Office of Science, U.S.
Department of Energy, a grant CHE-0507147 from NSF,
and a grant from the US-Israel BSF.
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