W. Forysiak, R. G. Flesch, 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: wladek@acms.arizona.edu
We introduce the dynamic nonlinear optical skin effect in which a pulse incident on a saturable absorbing interface is self-reflected from a moving absorption front. The motion of the front causes the self-reflected wave to be red-shifted by the Doppler effect, which in turn serves as an experimentally observable signature for the front propagation.
In the linear optical skin effect a pulse incident from air is reflected from a highly absorbing interface after penetrating only a fraction of a wavelength into the absorbing medium, this distance being the skin depth [1, 2]. The skin effect is therefore of fundamental importance in understanding the electrodynamics of pulse propagation at condensed matter interfaces, such as metals for field frequencies below the plasma frequency [2], and semiconductors with highly absorbing excitonic features [3]. In addition, it belongs to an important class of optical problems for which the notion of an electromagnetic field envelope varying slowly on the scale of a wavelength simply does not apply. The skin effect cannot be understood on the basis of envelope equations but is rather a consequence of Maxwell's equations for the interface.
In this Letter we introduce the dynamic nonlinear optical skin effect for pulses and elucidate the underlying physics. In the nonlinear skin effect a high intensity pulse is incident upon a nonlinear absorbing interface. Broadly speaking, saturation of the absorption allows the incident field to penetrate beyond the linear skin depth into the medium, and this causes an absorption front to propagate into the medium which separates the regions of low (saturated) and high (unsaturated) absorption. The front is excited by the incident pulse which is in turn reflected from the sharp absorption front, yielding a self-reflected pulse [4]. Thus the absorption front acts as a moving mirror from which the pulse is self-reflected, and the pulse suffers a red-shift due to the Doppler effect [5].
Continuous wave (cw) self-reflection from stationary absorption fronts for plane wave [4] and transverse Gaussian [7] fields incident at sharp and smooth [6] interfaces has been studied theoretically but not experimentally verified so far. In part, this is due to the extremely high absorption and strong saturation required for its manifestation, but the difficulty of obtaining good experimental signatures should not be overlooked. Here we explore the transient regime using the two-level Maxwell-Bloch equations. In particular, we show that moving fronts are excited by the incident pulse [8, 9] and that the self-reflected pulse bears clear spectral signatures due to the Doppler effect, which should be observable experimentally.
We consider the time-dependent propagation of a linearly polarized plane electromagnetic wave incident on a nonlinear medium composed of two-level systems. For propagation along the z-axis, and taking the electric field polarized along the x-axis, Maxwell's curl equations take the form [1, 2]
where 
. The specification of the problem is completed
with the constitutive relation 
,
where 
is the optical polarisation.
To elucidate the basic physics we
employ a two-level model to describe the optical response
with lower electronic state 
and upper state 
.
The Bloch equations are then (see for example Ref. [10])
where, 
is the off-diagonal density matrix element,
is the population difference between the lower
and upper states, 
is the transition frequency,
p is the dipole moment in the field direction,
and 
and 
are phenomenological damping
constants for the population and polarisation, respectively.
The polarization due to the atoms is then given by
, with N(z) the density of two-level
systems which varies along z in general.
Equations (1) and (2) are solved using a standard discretization scheme described by Yee [11] and the Bloch equations integrated in time using a fourth-order Runge-Kutta method. The initial condition for the field is
along with a similar expression for 
with 
.
Here 
is the peak input electric field, 
is the central pulse frequency, 
is the full-width at
the 
points of the pulse intensity profile in time units,
and 
is the position of the pulse center at t=0. The
nonlinear interface was imposed by tailoring the density profile
, with 
the longitudinal position of
the interface. The medium was initialized using 
, and n=1
in the medium. When initializing the field we ensured that the field protruded
negligibly into the nonlinear medium at t=0.
In the limit of cw fields, as previously studied by Roso-Franco [4], the self-reflected wave arises when the normalized parameters,
are both greater than unity. Physically, 
determines the
linear absorption per wavelength, 
,
and this quantity should be greater than unity for the linear skin
effect. For saturation of the absorption and self-reflection F>1
is required, since 
is the peak incident field intensity normalized
to the cw saturation intensity.
We consider the transient regime in which the incident pulse
width 
is much shorter than the population relaxation
time 
, but longer than the polarization dephasing time
, 
. For concreteness we adopt the
following specific parameters,
rads 
,
fs, 
ns, 
fs, 
Cm,
cm 
and 
m.
For these parameters 
so that the linear skin effect is
expected at low input intensities. We have numerically verified that this is
indeed the case, and the input pulse suffers minimal distortion in
profile or spectrum upon reflection.
Figures 1(a) and 1(b) show an example of the calculated pulses at two different
times for a peak input field of 
V/m. Although our
calculations employ the full field, we display only the envelope obtained from
joining the peaks as shown by the solid line, since it is not possible
to resolve the carrier in the plots. The field strength is associated with the
scale shown on the left-hand-side of the plots. In Fig. 1(a) for t=1.68 ps
the peak of the input pulse has not yet reached the interface at
m, but one can
clearly see the leading edge of the pulse is penetrating only a short distance
into the interface, as expected for the skin effect. The dashed line in
Fig. 1(a), which is associated with the right-hand scale, is the local
wavelength for the field. This is determined numerically by calculating
the local wavenumber K via 
, where a prime
signifies a z-derivative, and converting to wavelength. In Fig. 1(a) the
local wavelength remains constant at the input value 
nm.
In contrast, Fig. 1(b) shows the field profile at a later
time t=2.75 ps following reflection from the interface (for times between
the results shown in Figs. 1(a) and 1(b),
the field profile shows strong ringing due to interference
between the incident and reflected fields). The reflected pulse has developed
a double-peaked structure (solid line), and become
significantly chirped (dashed line). In particular, the central
portion of the pulse has a peak local wavelength of 990 nm, a
significant red-shift. This red-shift is also evident in the
reflected pulse spectrum (solid line) shown in Fig. 1(c) corresponding
to Fig. 1(b), along with considerable spectral broadening and modulation
(the input spectrum is shown by the dashed line and is associated with the
left-hand scale).
The results shown in Fig. 1 are typical of what we observe in
our simulations in the nonlinear regime, namely, distortion of the
reflected field profile and significant spectral modulation and associated
red-shift. To expose the physics underlying these phenomena we show
in Fig. 2(a) and 2(b) the spatial distribution of the full field and
the population difference 
at various times
corresponding to the results shown in Fig. 1.
Figure 2(a) shows that the field penetrates progressively further into
the interface as the absorption is saturated, as can be seen by comparing the
field profiles at t=1.61 ps (solid line) and t=1.95 ps (dashed line) or t=2.28 ps
(single-dash-dot line) (at t=2.62 ps the field has been mostly refelcted).
At t=1.61 ps (solid line) Fig. 2(b) shows that
n=0 before the interface signifying zero absorption, and n=1 beyond
the interface signifying large absorption due to the two-level systems.
At later times after the input pulse has penetrated into the interface,
the population difference is depleted and the absorption front
is seen to propagate into the nonlinear medium. The propagating
absorption front maintains a sharp wavelength scale
transition region so that the linear skin effect still occurs but
now from a moving absorption front. Thus the self-reflected field
must suffer a red-shift due to the Doppler-effect, akin to reflection
from a mirror moving away from a source [5].
To validate this physical picture we have determined the
absorption front velocity from the numerical simulation in Fig. 2(b),
and the result is shown in Fig. 2(c). After initially
accelerating the front reaches a maximum velocity of 
before decelerating back to zero velocity. The maximum wavelength shift
of the reflected pulse due to the Doppler effect is then
for 
[5],
or 
nm for
the free-space wavelength nm used here. Thus, based
on the Doppler effect we expect a maximum local wavelength of
nm, in good agreement with the numerical results
in Fig. 1(b). The Doppler effect upon reflection from the moving
absorption front can therefore explain the magnitude of the observed
pulse wavelength chirp.
We are now in a position to further explain the physics underlying
the pulse profiles and spectra in Fig. 1: As the leading edge of the
input field penetrates into the medium the absorption front accelerates
and the local wavelength of the reflected field increases, and
on the trailing edge of the pulse the absorption front decelerates
and the local wavelength decreases. This explains the initial
rise and then decrease in the wavelength chirp in Fig. 1(b) (dashed line).
Note that the field-profile in Fig. 1(b) (solid line)
exhibits a minimum at the same point that the local wavelength peaks,
and this begs a physical explanation.
A graph of the cw intensity reflectivity of the linear interface
[12], for the same parameter values used to generate Figs 1
and 2, shows that the reflectivity decreases, relative to the pulse
center wavelength nm, for wavelengths red detuned
from the resonance.
This is so because even
though the absorption decreases the skin depth increases, thus allowing
more path length in the medium over which absorption of the field can occur
(the situation is more complicated for blue-detuning but that does not
concern us here). Thus the red-shifted peak portion of the reflected pulse
in Fig. 1(b) experiences a lower reflectivity than the wings, giving rise to
the double-peaked reflected field. More quantitatively,
for the maximum red-shifted wavelength of nm the reflectivity
is reduced to 2%. This can also be intuited by realizing that a red-shift of
nm corresponds to 4.6 linewidths ( ), and a
significant reduction in absorption and reflection is to be expected.
Furthermore, this physical picture correctly indicates that
for lower input fields the Doppler shift is reduced, in which case the
differential reflection coefficient between the wings and center of the
pulse need not be as large as in Fig. 1(b). In this case the reflected
pulse can be single peaked, though still red-shifted.
An experimentally measurable signature of the double-peaked
reflected field in Fig. 1(b) is the modulated spectrum in
Fig. 1(c). The reflected field is composed of two peaks with a
spacing 
m. Treating these as point sources
in z, we expect a modulation in the wavelength spectrum with a
period 
m, which agrees
reasonably with the modulation period in Fig. 1(c). In the case
that the reflected field is single-peaked the reflected spectrum
is red-shifted but with no modulation.
While the slowly-varying envelope approximation (SVEA)
[4, 12], cannot capture
the physics of the evolution of the self-reflected wave,
we can employ it to demonstrate front propagation in the medium away
from the interface. In particular, we introduce the field and polarization
envelopes via the definitions
and
, where
and
,
to obtain the usual Maxwell-Bloch equations in the SVEA from
Eqs. (1) and (2) [10]. For the case
considered here with ,
we may ignore the population relaxation, and in the limit of small 
adiabatically eliminate the polarization. We then obtain the pair of
coupled equations
where 
. These equations admit travelling wave
solutions depending on the variable 
, where v is the
front velocity [8]. Assuming a pulse profile
with 
and 
, a constant, and a corresponding
bleaching of the two-level system inversion from the ground-state to
transparency, 
and 
, we find the
front solution
where 
is the characteristic spatial width of the
travelling wave front, and the front velocity is given by
[8, 9].
For the parameters above, this gives v/c=0.026, which is smaller than the
value reported above due to neglect of the self-reflected wave in the present
calculation.
Thus we have analytic confirmation that absorption fronts can propagate
in the nonlinear medium, and we identify these with the fronts seen
in our numerical simulations. Numerical integration
of the full SVEA equations also clearly shows front propagation for
these parameters, but with a damped, oscillatory front profile.
The nonlinear skin effect appears
over a much broader range of parameters than employed here.
As in the cw case we require the dimensionless parameters
and F to be larger than unity so that there is a linear skin
effect and a high enough level of nonlinear absorption saturation.
In particular, we note that especially for longer pulses, F can be
substantially smaller than the value of F=320 in the example above,
though the front velocities and spectra are correspondingly reduced.
For the pulse duration and material relaxation times, we require
to obtain propagating absorption fronts, where
the condition 
is imposed to avoid the regime of
self-induced transparency (the input pulse area above
is 
) [13].
Given these restrictions, exitonic resonances in quantum well materials
are prime candidates for the observation of this effect,
where high absorptions and suitable relaxation times are available [3].
It remains to see whether the high levels of saturation can
be achieved, and if under these condition, simple two-level
polarisation dynamics can be realised.
In conclusion, we have introduced the dynamic nonlinear optical skin effect for reflection of pulses from a highly absorbing interface. This new basic effect for the electrodynamics of interfaces combines the concepts of self-r eflected waves [4] and front propagation, and is also a prime example of a nonlinear optical phenomenon where the SVEA fails and the full Maxwell equations must be employed. We have shown that the nonlinear optical skin effect arises from moving absorption fronts so that the red-shifting and spectral modulation of the reflected pulse are clear experimental signatures of the effect.
We thank Dr. Robert Indik for constructive discussions and suggestions. Ewan Wright would like to thank Profs. Ray Chiao, Alex Kaplan, John McCullen, Pierre Meystre, and Dr. John Garrison for their insightful remarks concerning this work. The authors also wish to thank the Arizona Center for Mathematical Sciences (ACMS) for support. ACMS is sponsored by AFOSR contracts F49620-94-1-0463 and F49620-94-1-0051. E. M. Wright is partially supported by the Joint Services Optical Program.
