48
M. I. Stockman
[228, 229]. The field produced by them is a coherent superposition of waves with
different frequencies whose amplitudes and phases can arbitrarily vary in space and
with frequency. This modulation can be chosen so that all the frequency components
converge at the same focal spot at the same time forming an ultrashort pulse of the
nanolocalized optical fields.
As an example we consider a silver [32] nanowedge illustrated in Fig. 1.20 whose
maximum thickness is d m = 30 nm, the minimum thickness is d f = 4 nm, and whose
length (in the y direction) is L = 5 µm. Trajectories calculated by the WentzelKramers-Brillouin (WKB) method in Ref. [210] for ω = 2.5 eV are shown by
lines (color used only to guide eye); the nanofocus is indicated by a bold red dot.
In contrast to focusing by a conventional lens, the SPP rays are progressively bent
toward the wedge slope direction.
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 timereversal [220, 221, 235]. We generate rays at the nanofocus as an ultrashort pulse
containing just several oscillations of the optical field. Propagating these rays, we
find amplitudes and phases of the fields at the thick edge at each frequency as given
by the complex propagation phase (eikonal) Φ(ρ), where ρ is a 2-d coordinate vector
in the plane of the wedge. Then we complex conjugate the amplitudes of frequency
components, which corresponds to the time reversal. We also multiply these amplitudes by exp(2Im Φ), which pre-compensates for the Ohmic 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. 1.21. Panel (a) displays the
trajectories of SPPs calculated [210] by the WKB method. The trajectories for different frequencies are displayed by colors corresponding to their visual perception.
There is a very significant spectral dispersion: trajectories with higher frequencies
are much more 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. 1.21b–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 earlier to pre-compensate for the longer propagation times.
They should also have different amplitudes due to the differences in the adiabatic
compression along the different rays. Finally, there is clearly a negative chirp (gradual
decrease of frequency with time). This is due to the fact that the higher frequency
components propagate more slowly and therefore must be emitted earlier to form a
coherent ultrashort pulse at the nanofocus.
In Fig. 1.21e we display together all three of the representative 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
M. I. Stockman
[228, 229]. The field produced by them is a coherent superposition of waves with
different frequencies whose amplitudes and phases can arbitrarily vary in space and
with frequency. This modulation can be chosen so that all the frequency components
converge at the same focal spot at the same time forming an ultrashort pulse of the
nanolocalized optical fields.
As an example we consider a silver [32] nanowedge illustrated in Fig. 1.20 whose
maximum thickness is d m = 30 nm, the minimum thickness is d f = 4 nm, and whose
length (in the y direction) is L = 5 µm. Trajectories calculated by the WentzelKramers-Brillouin (WKB) method in Ref. [210] for ω = 2.5 eV are shown by
lines (color used only to guide eye); the nanofocus is indicated by a bold red dot.
In contrast to focusing by a conventional lens, the SPP rays are progressively bent
toward the wedge slope direction.
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 timereversal [220, 221, 235]. We generate rays at the nanofocus as an ultrashort pulse
containing just several oscillations of the optical field. Propagating these rays, we
find amplitudes and phases of the fields at the thick edge at each frequency as given
by the complex propagation phase (eikonal) Φ(ρ), where ρ is a 2-d coordinate vector
in the plane of the wedge. Then we complex conjugate the amplitudes of frequency
components, which corresponds to the time reversal. We also multiply these amplitudes by exp(2Im Φ), which pre-compensates for the Ohmic 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. 1.21. Panel (a) displays the
trajectories of SPPs calculated [210] by the WKB method. The trajectories for different frequencies are displayed by colors corresponding to their visual perception.
There is a very significant spectral dispersion: trajectories with higher frequencies
are much more 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. 1.21b–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 earlier to pre-compensate for the longer propagation times.
They should also have different amplitudes due to the differences in the adiabatic
compression along the different rays. Finally, there is clearly a negative chirp (gradual
decrease of frequency with time). This is due to the fact that the higher frequency
components propagate more slowly and therefore must be emitted earlier to form a
coherent ultrashort pulse at the nanofocus.
In Fig. 1.21e we display together all three of the representative 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