W. Forysiak, J. V. Moloney
Arizona Center for Mathematical Sciences
Department of Mathematics
University of Arizona
Tucson, AZ 85721
E. M. Wright
Optical Sciences Center
University of Arizona
Tucson, AZ 85721
Email: jml@acms.arizona.edu
We calculate the spatio-temporal evolution of intense, femtosecond pulses incident on a saturable absorbing interface in the regime of self-reflection, using the FD-TD computational method. The pulses induce a curved, moving absorption front in the nonlinear medium, which acts as a transient focusing mirror for the reflected pulse energy.
The propagation of an intense optical field incident from air on a saturable absorber can give rise to a self-reflected wave, if the absorption is sufficiently large [1]. The reflected wave arises at a wavelength-scale, spatial transition between the saturated and unsaturated regions of the absorber [2], akin to the skin effect at a linear boundary, and is self-reflected in the sense that the transition region is induced by the incident field itself. The self-reflected wave first predicted for a cw incident plane wave field[1], was subsequently extended to consideration of a cw incident beam with a Gaussian transverse profile [3]. In this latter case, the emergence of a halo was predicted in the far field at high intensities. Recently, the FD-TD method [4, 5] was used to study the dynamics of self-reflection under pulsed excitation, and to predict an intensity dependent Doppler shift in the reflected pulse [6].
In this Letter, we include transverse variations, in order to study the spatio-temporal dynamics of self-reflection. The FD-TD method discretises the differential form of Maxwell's equations directly and allows one to determine the evolution of an optical field in a nonlinear medium, subject to the given constitutive relations between the electric field and polarisation (see e.g. [7, 8]), and without recourse to the SVEA. We examine pulse shaping in the near-field, close to the absorber boundary, and predict a new nonlinear focusing effect, which we attribute to the formation of a transient, focusing mirror in the absorber. As the incident pulse impinges on and strongly saturates the absorber, it excites a moving reflection front [6] which is shaped according to the transverse profile of the incident pulse. If the incident pulse transverse profile is bell-shaped, so too is the resulting mirror, and the reflected pulse is focused according to the waist and intensity of the incident pulse. In addition to being re-shaped, the reflected pulse is spectrally broadened and red-shifted due to the Doppler effect at the moving mirror [6].
We consider the time-dependent propagation of a 2-D transverse
electric (TE) polarised pulse, in which the electric field is polarized
along the y-axis and also assumed uniform along that axis,
. Then
the Maxwell's equations for the electric
and magnetic field quantities, 
, 
and 
, are,
where the z-axis is the propagation direction, and the x-axis is the
transverse direction. The nonlinear optical response of the saturable
absorber is included using a two-level model via the constitutive
relation, 
, where the macroscopic polarisation,
, is determined by the Bloch
equations [9],
Here, N is the density, 
is the off-diagonal density
matrix element, 
is the population difference,
is the transition frequency, p is the dipole moment,
and 
and 
are the population and
polarization damping constants.
The conditions for self-reflection of a continuous plane-wave incident
field require that the normalised parameters,
, and,
,
are greater than unity, with 
the peak input field.
Physically, this requires the
linear absorption to be large on a wavelength scale, and the incident
field to be strong enough to saturate the absorption [2].
In the case of ultrashort pulses, we also require that the
incident pulse duration 
is greater than the polarization decay time
but less than the population decay time, 
[6].
To meet these conditions for the sub-100 fs pulses to which we were
restricted by computational resources, we adopted the following medium
parameters for illustrative purposes: 
ns, 
fs,
rads 
( 
nm),
Cm and 
cm 
.
For these values a normalised
field strength of F=1 corresponds to an electric field strength
of 
V/m.
The linear complex refractive index of the absorbing
medium is given by,
,
where 
is the normalised
detuning. For the assumed parameter values 
, and 
has a substantial imaginary contribution so that the linear
skin effect is to be expected at the absorber boundary.
The nonlinear Maxwell's equations were numerically integrated using Yee's
second-order FD-TD scheme [4] for the field updates
and a fourth-order Runge-Kutta method for the medium updates.
On the driving face (z=0) of the computational domain, the
incident pulse was defined to be Gaussian-shaped in space and time.
Periodic boundary conditions were imposed at the transverse boundaries
( 
) which were placed far enough away for the optical
intensity to be reduced by more than 50dB compared with pulse center.
The medium was initialised uniformly with 
and n=1 for
.
Figures 1 and 2 show results from a pair of sample calculations for
incident pulses with peak fields of (a) 
V/m,
and (b) 
V/m. The computational domain was
m ( 
), discretised onto a
numerical grid, with the saturable absorber interface
situated at 
m. The incident pulse was 80fs
long (FWHM of the electric field), with a beam waist of
m (1/e half-width of the field), and was initialised
such that its peak entered the computational domain at z=0 when
t=120fs. The temporal snapshots are taken after reflection
from the interface (at t = 770 ps) when the pulses are travelling
back towards z=0. In Fig. 1 only the extracted field envelopes are
shown, since the carrier is not easily discernible. The corresponding
spectra in Fig. 2 were obtained from the full electric field data.
Figures 1(a) and 2(a) show that for low incident pulse intensities the
self-reflected pulse, though diminished in amplitude, is unchanged
spatially or spectrally. The pulse energy is reduced during the
absorptive reflection, and the calculated reflectivity of 
compares favourably with the predicted steady-state linear value
of 
from Fresnel's law and the expression for the complex
refractive index above.
The pulse waist remains unchanged, evolving almost imperceptibly on
the time scale of the computation, because of its long diffraction
length ( 
m).
In this case, therefore, since the pulse is too weak to significantly
saturate the absorber, the physical properties of the interface remain
unmodified and the pulse is partially reflected, according to the
linear skin effect, as if from a flat mirror.
In contrast, Figs. 1(b) and 2(b) show that the high intensity pulse
undergoes significant nonlinear spatial and spectral
reshaping at the interface.
Figure 1(b) shows the pulse profile after it has propagated away from
the interface, close to its focal point, where the pulse waist
(at maximum ) is reduced to 2.2 m, a considerable
reduction in spot size in comparison to the linear case in Fig. 1(a).
At this point, the
peak electric field is actually fractionally greater than the peak
incident field, despite the decreased reflectivity of 
.
The reflectivity is reduced compared to the linear case because
the high power pulse reduces the medium absorption and as a result
the pulse penetrates further into the interface.
The spectral broadening that results from the nonlinear
saturation of the absorber is clearly evident in Fig. 2(b) in
comparison to Fig. 2(a). The longitudinal spectral broadening
(in 
) and red-shifting of the reflected field are due to the
self-reflection of the incident pulse from an absorption front which
initially accelarates and then deccelarates into the
saturable medium, which in turn causes the reflected light to be
red-shifted and chirped.
These phenomena are captured in the first-order approximation
for the front-velocity v [6]
,
where R is the linear reflectivity of the interface. Thus, the
higher the local intensity in the pulse 
the higher the
front velocity, and the higher the Doppler red-shift of that
locally self-reflected portion of the pulse. Thus, due to the
distribution of intensities in the pulse, the reflected spectrum
is chirped and suffers a net red-shift.
The expression for the front velocity also captures the source of the transverse
spectral broadening in Fig. 2(b) (in 
): The local intensity
also varies transversely which gives rise to transverse
spectral chirping of the self-reflected field. In the spatial
domain this translates to the fact that the self-reflected field
acquires a phase curvature, which then causes the self-reflected
field to focus transversely in front of the interface as in Fig.1(b).
In other terms, for a bell-shaped incident field profile the
front velocity will be higher at the beam center than at the
beam wings, which causes the transverse profile of the absorbing
front to be curved into the medium. Thus, the self-reflection
occurs from a curved moving mirror which produces focusing of the
reflected pulse.
The formation of a curved focusing reflection front is shown in Fig.
3, where the spatial distribution of the population difference, n,
is plotted after its creation by the leading edge of the pulse.
Once this curved mirror is formed, the rest of the pulse is reflected towards the x=0 axis,
leading to the tight focus seen in Fig. 1(a). The spectral width
of the broadest feature in Fig. 2(b) accurately reflects the inverse
incident pulsewidth. Furthermore, we note the signature of the
ordinary (unfocused) plane-wave reflection at the leading edge of the
pulse in Fig. 1(a), on a timescale of approximately 
, during
which the mirror formation takes place.
The dependence of this novel self-reflection phenomenon on incident
beam waist and intensity has been studied numerically.
We observe that the focus gets closer to the interface with increasing
intensity and decreasing pulse waist. This effect could be
exploited as a writable and erasable focusing/deflecting mirror,
with the virtue that is has no mechanical parts.
For example, an intense and
broad switching pulse, could write a curved mirror to direct a
subsequent packet of low power pulses. The mirror would decay
on a timescale of the inversion decay ( 
) time.
In conclusion, using the FD-TD method, we have calculated the near-field, spatio-temporal dynamics of intense femtosecond optical pulses self-reflected from a saturable, highly absorbing interface. The transverse profile of the incident field induces a curved, moving saturation front in the absorber, which acts as a transient focusing mirror for the incident pulse energy.
Acknowledgement: Effort sponsored by the Air Force Office of Scientific Research, Air Force Materiel Command, USAF, under grant numbers AFOSR-94-1-0051, AFOSR-94-1-0144DEF and AFOSR-94-1-0463. Ewan Wright was also supported by AFOSR contract F49620-94-1-0343, and partially by the Joint Services Optical Program.
