# The role of heat generation and fluid flow in plasmon-enhanced reduction-
oxidation reactions

Ieng-Wai Un  [<EMAIL_ADDRESS>Yonatan Sivan  [

###### Abstract

Recently, we have shown that thermal effects play a crucial role in speeding
up the rate of bond-dissociation reactions. This was done by applying a simple
temperature-shifted Arrhenius Law to the experimental data, corroborated with
detailed account of the heat diffusion occurring within the relevant samples
and identification of errors in the temperature measurements. Here, we provide
three important extensions of our previous studies. First, we analyze thermal
effects in reduction-oxidation (redox) reactions, where charge transfer is an
integral part of the reaction. Second, we analyze not only the spatial
distribution of the temperature, but also its temporal dynamics. Third, we
also model the fluid convection and stirring. An analysis of two exemplary
experimental studies allows us to show that thermal effects can explain the
experimental data in one of experiment (Baumberg and coworkers), but not in
the other (Jain and coworkers), showing that redox reactions are not
necessarily driven by non-thermal charge carriers.

###### keywords:

heat generation, fluid convection, plasmon-assisted photocatalysis, reduction-
oxidation reactions

SECE-BGU] School of Electrical and Computer Engineering, Ben-Gurion University
of the Negev, Ben-Gurion University SECE-BGU] School of Electrical and
Computer Engineering, Ben-Gurion University of the Negev, Ben-Gurion
University IR,NMR,UV

##  1 Introduction

The idea of using illuminated metallic surfaces to enhance the yield of
chemical reactions (aka plasmon-assisted photocatalysis) was first proposed by
Nitzan and Brus in 1981 1 and experimentally implemented two years later by
Chen and Osgood 2. Then, after several decades of additional demonstrations
(see e.g., Refs 3, 4, 5, 6, 7, 8), this line of research has rapidly gained
popularity due to several high impact publications (see, e.g. Refs 9, 10, 11
for some recent reviews). The main motivation behind this growing interest was
the claim that the reaction rate can be increased due to the excitation of
high energy non-thermal electrons in the metal (aka “hot” electrons), which
then tunnel out of the metal, and provide the necessary energy for the
reactants to allow them to be converted into the products more efficiently.

However, the claims in some of the more famous papers on the topic 12, 13, 14,
15, 16 were shown to be based on a series of technical errors (including
improper temperature measurements, improper data normalization etc., see
discussion in Refs. 17, 18, 19, 20), and instead, a purely thermal mechanism
was shown to be able to explain the experimental data quite convincingly 17,
19, 21. In particular, a shifted Arrhenius Law for the reaction rate,
$R\sim\exp\left(-\mathcal{E}_{a}/k_{B}(T({\bf r})+aI_{\textrm{inc}}\right)$
whereby the temperature of the system was corrected for the illumination-
induced heating was shown to provide an excellent fit to the published data,
essentially with no fit parameters. This result was corroborated with the
first ever complete calculation of the steady-state electron non-equilibrium
in the metal 22, a consequent Fermi golden-rule argument 23 that implied on
the unlikeliness of nonthermal electrons to cause the catalysis, and by
detailed thermal simulations where the dynamics of the heat generated from
each of the nanoparticles (NPs) in the system was properly modelled 19.

In a consequent paper 24, it was shown that the tedious modelling of the
contributions of each of the heated NPs in the system can be accurately
replaced by an effective medium approximation whereby the penetration depth of
the electromagnetic fields was calculated from the NP density and absorption
cross-section, and consequently used to describe the heat source. This
approach also enabled accounting for the exact reactor geometry, thus,
facilitating a quantitative comparison to experimental data. This series of
work thus established thermal simulations as a necessary tool in
investigations of the underlying physics of plasmon-assisted photocatalysis
experiments.

The bottom line of the thermal modelling was that when attempting to
quantitatively separate thermal and non-thermal effects in plasmom-assisted
photocatalysis experiments, one has to overcome a conceptual difficulty - the
thermocatalysis control experiments must reproduce the exact spatially non-
uniform temperature profile induced by the illumination, otherwise, when
subtracting the thermocatalysis rate from the photocatalysis rate (e.g., as in
Refs. 16, 25), any difference between the temperature distributions in an
inaccurate control and the corresponding photocatalysis experiment is bound to
be incorrectly interpreted as “hot” electron action. This issue is
particularly important because the Arrhenius Law shows that the reaction rate
has an exponential sensitivity to the temperature distribution.

Detailed measurements and/or calculations of the temperature distribution in
the studied samples indeed constituted a central role in several recent
demonstrations of non-thermal effects in plasmon-assisted photocatalysis 26,
25, 27, 28. However, while the simple thermal calculations as done so far were
sufficient for relatively simple scenarios, they may not be sufficient to
account for more complicated ones. Those include, in particular, high
intensity illumination which invokes nonlinear thermo-optic effects 29, 30,
31, 32, charge transport (in particular, in electrochemistry experiments 33,
34, 27, 35, 36, 37), or in the presence of strong gas flows, liquid stirring
etc.. In this context, there are several claims in the literature that the
latter can effectively eliminate thermal gradients, and thus, make
thermocatalysis control experiments based on external heating far more similar
to the generally more complex temperature distribution in photocatalysis
experiments. As noted above, this can simplify the distinction between thermal
and non-thermal effects, however, somewhat peculiarly, to the best of our
knowledge, no estimates nor proper numerical simulations of the effectiveness
of gas flow, convection or liquid stirring were previously presented. In Ref.
19 we offered some simplistic estimates which showed that gas flow is not
expected to be significant in removing temperature gradients; however, one may
ask whether the situation changes in a liquid environment, especially in the
presence of rapid external stirring which can be stronger than gas flow.

In order to quantify these effects, in the current manuscript we add to the
(validated Ref. 24) effective medium modelling used previously some more
advanced effects such as gas flow and liquid convection/stirring via the
proper fluid mechanics equations. We reproduce numerically the temperature
distribution in two experiments performed in a liquid environment where
natural convection and forced stirring occurred. This specific choice of
experiments allowed us also to address another common claim, namely, the
validity of the thermal analysis to a wider variety of chemical reactions.
Indeed, all the chemical reactions we previously identified as likely to be
driven by thermal effects in Refs. 19, 17 were bond-dissociation type.
However, a different type of reactions, namely, reduction-oxidation (redox)
reactions are, by definition, associated with charge transfer, so that they
are frequently associated with the optical generation of “hot” electrons or
electron-hole pairs 38.

The manuscript is organized as follows. In Section 2 we introduce our improved
thermal model along with the fluid dynamics equations. In Section 3 we
describe the two experiments, the corresponding temperature gradients and
temporal dynamics, and draw conclusions regarding the effectiveness of the
convection/stirring as well as the mechanism responsible for the faster
reactions reported. Potentially counter-intuitively, we show that the results
of the first experiment can be explained just with a thermal theory (i.e., the
shifted-Arrhenius model), while the results of the latter cannot.

##  2 Model

Previously in Refs. 17, 19, calculations of the temperature distribution in
plasmon-assisted photocatalysis systems relied on a straightforward solution
of the heat equation accounting for the multiple absorbing metal NPs as
nanometric heat sources within an effective medium model for the light
absorption. Specifically, one solved

$c_{p}\rho_{0}\dfrac{\partial T({\bf r},t)}{\partial
t}=\nabla\cdot\left(\kappa\nabla T\right)+p_{\textrm{abs}}({\bf r},t),$ (1)

where $c_{p}$ is the heat capacity at constant pressure, $\rho_{0}$ is the
(constant) mass density of the fluid, $T$ is the temperature and the term
$\nabla\cdot(\kappa\nabla T)$ accounts for heat diffusion/conduction.
Typically, homogenized values of $c_{p}$, $\rho_{0}$ and $\kappa$ for the host
19 were used; this means that the metal parameters were ignored due to the
small volume fraction they occupy. Finally, $p_{\textrm{abs}}$ is the heat
source density induced by the light absorption; it has been shown that by
applying the effective medium approximation for the electromagnetic
properties, it is well described by 24

$\displaystyle p_{\textrm{abs}}({\bf
r},t)=\left(I_{\textrm{inc}}(\varrho)/\delta_{\textrm{skin}}\right)\textrm{exp}(-|z|/\delta_{\textrm{skin}}),$
(2)

where $\varrho$ is the distance from the propagation optical axis, $|z|$ is
the distance along the propagation direction of the incident beam from the
spot area, $I_{\textrm{inc}}(\varrho)$ is the intensity profile of the
incident beam and $1/\delta_{\textrm{skin}}$ is the absorption coefficient
experienced by the incident beam; it can either be obtained by the measurement
directly (see e.g., Section 3.1), or can be related to the NP density
$n_{\textrm{p}}$ (number per volume) and the absorption cross-section
$\sigma_{\textrm{abs}}$ by using the effective medium theory 19, 24 (see e.g.,
Section 3.2)

$\displaystyle 1/\delta_{\textrm{skin}}=n_{p}\sigma_{\textrm{abs}}.$ (3)

In the current study, in order to model more complicated scenarios in which
the temperature dynamics is accompanied by significant fluid density and
velocity dynamics, we add to the heat equation a term responsible for heat
convection and determine the mass velocity and density via the Navier-Stokes
equation (5) coupled with the continuity equation (6), namely,

$\displaystyle c_{p}\rho({\bf r},t)\left[\dfrac{\partial T({\bf
r},t)}{\partial t}+({\bf u}({\bf r},t)\cdot\nabla)T\right]$
$\displaystyle=\nabla\cdot\left(\kappa\nabla T\right)+p_{\textrm{abs}}({\bf
r},t).$ (4) $\displaystyle\rho\dfrac{\partial{\bf u}}{\partial t}+\rho({\bf
u}\cdot\nabla){\bf u}$ $\displaystyle=-\nabla p+\rho_{0}{\bf
g}+(\rho-\rho_{0}){\bf g}+\nabla\left[\mu\left(\nabla{\bf u}+(\nabla{\bf
u})^{\intercal}\right)\right],$ (5)
$\displaystyle\dfrac{\partial\rho}{\partial t}+\nabla\cdot\left(\rho{\bf
u}\right)$ $\displaystyle=0.$ (6)

As well known, these equations ensure conservation of energy, momentum and
mass in the fluid. Here, the unknown variables ${\bf u}$, $p$, $\rho$ and $T$
are, respectively, the flow velocity, the pressure, fluid density and
temperature. As before, $\rho_{0}$ is the (constant and uniform) fluid density
at the ambient conditions. Since Eq. (4)-(6) contain five equations in terms
of six unknown flow-field variables, we also require the equation-of-state to
relate the fluid density to the pressure and temperature. For liquids, the
fluid density can be assumed to be pressure-independent due to the small
compressibility of most liquids. This provides the necessary sixth equation.

On the right hand side of Eq. (5), the terms $-\nabla p$ and $\rho{\bf g}$
are, respectively, the pressure forces and the gravity forces. These two terms
nearly cancel each other for liquids. However, the variations of density with
temperature which arise in the gravity force give rise to a buoyancy force
lifting the fluid. This force is represented by $(\rho-\rho_{0}){\bf g}$ and
is responsible for the natural convection in the fluid. The last term
$\nabla\left[\mu\left(\nabla{\bf u}+(\nabla{\bf u})^{\intercal}\right)\right]$
represents the viscous forces ($\mu$ is the viscosity). This is usually small
when only the natural convection in the fluid is considered.

In addition, for plasmon-assisted photocatalysis in liquid phase, since
typical experiments were performed at atmospheric pressure, the additional
pressure at a depth of a few cm of liquid (a few hundreds Pa) has a negligible
effect on the material parameters; $\mu$, $c_{p}$ and $\kappa$ are, thus,
assumed to be pressure-independent. However, the temperature dependence of
these parameters is non-negligible.

The above assumptions are suitable for liquid hosts. One can also use the
model equations (4)-(6) to calculate the temperature distribution in samples
in which the catalysts are nanostructured solids and the reactants are gases
(e.g., as in Refs. 13, 15, 16, 14). In these photocatalysis experiments, the
catalyst samples are usually put in a reaction chamber consisting of inlet and
outlet channels between which gases flow in the chamber. When modeling these
systems, we have to take into account the pressure-dependence of the gas
density via the equation of state due to the high compressibility of gas. To
include the gas flow, one should use the inlet boundary condition and specify
the gas velocity in the inlet as $\left(=\dfrac{\textrm{volumetric flow
rate}}{\textrm{inlet section area}}\right)$ via a Dirichlet boundary
condition. Simple estimates of the power flow due to convection performed in
Ref. 19 showed that such gas flows can have, at most, a moderate quantitative
effect on the temperature distribution in the system. More accurate modelling
requires knowledge (or computation) of the gas flow velocity inside the
sample; this information is not usually available, and is hard to compute.

##  3 Representative Analysis

In this Section, we apply the model described above to two representative
experiments. In contrast to a variety of experimental studies that aim to
mimic practical conditions for catalysis, these two studies chose a simple
configuration that facilitates the isolation of the underlying physical
mechanism. In particular, in both experiments the heat source was highly
localized, such that the spatio-temporal dynamics of heat occurred in a
homogeneous environment, and thermal gradients and dynamics were pronounced.

###  3.1 Huang et al. [J. Baumberg’s group, 39]

First, we look at the results of Huang et al. 39 who studied dithionite-
mediated redox reactions in a water suspension of Au NP aggregates. The
aggregates were estimated to consist of a random arrangement of 15 NPs on
average, each $\sim 60$nm in size. The suspension was put in a 1 cm $\times$
0.4 cm $\times$ 2.5 cm cuvette which was heated using a $785$ nm spatially-
narrow ($\sim 110\ \mu$m diameter) 125 mW CW beam. The presence of the Au NPs
enhances not only the reaction, but also the Raman signal from the redox
reporter molecules, allowing for surface enhanced Raman spectroscopy (SERS) to
be used to monitor the rate of the reaction (i.e., the density of both
reactants and products) taking place in the illuminated region, see Fig. 1(a).
No reaction takes place in the absence of the metal NPs.

Initially, the cuvette was subject to a hot water bath whose temperature was
externally controlled. It was then observed that products were detected for
heating of the sample by 8 K or more above room temperature. This enabled
Huang et al. to conclude that if under illumination the sample temperature
remained lower than that threshold, but reactants can be detected, then, the
reaction is likely to be driven by high energy non-thermal electrons.

A rough estimate of the heat generation and transfer to the suspension from
each single aggregate then allowed Huang et al. to predict that the (steady-
state) temperature rise of a single aggregate with respect to its environment
may reach $1-2$ K. In addition, they performed an estimate of the overall
cuvette temperature rise which relied on the assumption of infinitely fast
heat diffusion (such that the total heat generated in the sample spreads
evenly in its volume). This gave a slightly higher temperature of $\sim 4$ K
for the average temperature in the sample (due to the accumulated heating from
the multiple aggregates). With these conservative estimates and in light of
the difficulties to measure the temperature distribution at high accuracy, the
authors were unable to ascribe clearly the many effects observed to non-
thermal carriers.

Here, we employ the model (4)-(6) to improve upon the original heating
estimates. Our motivation to do so is the fact that since the heat diffusion
has a finite strength, it is reasonable to assume that some level of non-
uniformity exists in the suspension, such that the temperature of the
illuminated (and probed) region may be higher than its surroundings. Thus, in
order to determine the actual level of temperature non-uniformity, we first
perform a standard heat transfer simulation (neglecting the natural convection
in water, namely, we solve Eq. (4) by assuming that ${\bf u}=0$, but
accounting for heat diffusion in water) with a heat source Eq. (2) in which
the transverse intensity profile has a Gaussian shape, namely,
$I_{\textrm{inc}}(\varrho)=\left(P_{\textrm{inc}}/(\pi\varrho_{b}^{2})\right)\exp\left(-\varrho^{2}/\varrho_{b}^{2}\right)$,
where $P_{\textrm{inc}}$ is the illumination power and $\varrho_{b}$ is the
beam spot radius. We note, however, that the 3D simulation of fluid flow and
the temperature dynamics of the water in a rectangular cuvette requires a
large computing time. To obtain a quick result without compromising accuracy,
we replaced the rectangular cuvette used in the experiment by a cylindrical
one, such that the height and the cross-section of the model are,
respectively, equal to the height and the cross-section of the actual cuvette
used in the experiments, see Figure 1(a). This allows us to simplify the 3D
model into a 2D model using an axisymmetric geometry. As the following result
show, the cuvette size and shape have a negligble effect on the dynamics in
this case, as the cuvette is sufficiently large.

The analysis below is enabled by the extraction of a key parameter from the
experimental data, namely, the absorption coefficient
$1/\delta_{\textrm{skin}}$ (3) which is itself deduced from the attenuation
measurement. Specifically, the reported $83\%$ attenuation per $cm$
corresponds to a penetration depth of $\delta_{skin}\approx 5.6$ mm ($\sim
20\%$ of the total thickness). We set the bottom of the cuvette to a
temperature of $20^{\circ}$C; the top surface of the water and the outer
surfaces of the cuvette dissipate heat via natural convective flow of air
driven by the temperature difference between the cuvette and its surrounding.

Initially, the system is at the room temperature. Then, the laser is turned on
for 140 seconds. The simulation (see Figure 1(b)-(c)) shows a strongly non-
uniform temperature profile occurring because the beam is much smaller than
the cuvette in which the liquid suspension was. In particular, the temperature
in the region close to the illuminated NPs grows by about $27$ K, while
regions further away hardly heat up at all.

These hydrostatic results imply that the highly non-uniform temperature
distribution induces a density difference between the hot and cold regions
thereby leading to buoyancy-driven flow such that the illuminated region cools
down. For this reason, we further improve the temperature calculations by
modeling both heat diffusion and convection as well as simulating the fluid
flow via Eqs. (4)-(6). This simulation shows that the temperature non-
uniformity induced by the local heating is hardly affected by the natural
convection, see Figure 1(b). In particular, the temperature rise at the
illumination region decreases by only a few K due to the natural convection,
see Figure 1(c). The lower part of the sample is even less affected, because
its bottom is set to room temperature.

Thus, since the temperature of the illuminated (hence, monitored) region is
much higher than the $8$ K threshold, we conclude that the reaction in Ref. 39
was likely driven by the heat generated in the sample (rather than by
generation of high energy non-thermal carriers).

This conclusion is further supported by additional evidence that the
temperature dynamics that emerges from our simulations is consistent with the
experimentally-measured product dynamics. Such an analysis was attempted in
Ref. 40, yet, was applied to heating of a single NP by a single pulse; thus,
much like Ref. 14, it overlooked cumulative thermal effects and reached an
invalid conclusion about the negligible role played by thermal effects in Ref.
40. In contrast, the simulation results (see inset of Figure 1(c)) which look
at the macroscopic (i.e., cumulative) heat dynamics show that the temperature
of the illuminated region increases rapidly by $\sim 20$ K within the first
second of the illumination (before settling on a much slower heating rate). To
see that, it is insightful first to understand the origin of the time scale
for the temperature rise in our simulations. This is done by approximating the
heat source in the experiment as an infinitely small heat source turned on
abruptly, namely, $P_{\textrm{inc}}\delta^{(3)}({\bf r})H(t)$. The ensuing
heat dynamics is given by

$\displaystyle\Delta T({\bf r},t)\approx\Delta T({\bf
r},\infty)\left[1-\textrm{erf}\left(\dfrac{r}{\sqrt{4Dt}}\right)\right],$ (7)

where $\Delta T({\bf r},\infty)$ is the “steady-state” temperature rise (i.e.,
at long times, assuming that the laser was not turned off),
$D=\kappa/c_{p}\rho_{0}$ is the (effective) thermal diffusivity,
$\textrm{erf}(x)$ is the error function and $H(t)$ is the Heaviside step
function.

As shown in the inset of Fig. 1(c), within roughly 1 second, the temperature
rise reaches about $80\%$ of its “steady-state” value before slowing down
significantly.

This is the time scale necessary for the build-up of the temperature gradient
for which the outgoing heat flux becomes comparable to the illumination-
induced heat generation. In that sense, this dynamics is independent of the
geometry of the cuvette, i.e., it is determined only by the beam size and the
host diffusivity. Then, based on the estimate $\textrm{erf}(x)\sim x$, we need
to find the time that satisfies $\dfrac{r}{\sqrt{4Dt}}\sim 0.2$. Using the
water diffusivity ($\sim 0.15\cdot 10^{-6}m^{2}/s$) and the beam radius, we
get that the temperature rises within the time scale of $\sim 0.5$ s.

This temperature dynamics matches with the experimental observation that the
characteristic bands of the products appeared in the SERS emission after a few
seconds of illumination. Due to the instantaneous nature of the SERS
measurement technique, the limitations discussed in Ref. 20 can be alleviated
such that the growth of the reaction rate can be directly ascribed to heat. To
our knowledge, this is the first successful attempt to distinguish thermal
from nonthermal effects in plasmon-assisted photocatalysis by the time
dynamics, as originally suggested by Baffou et al. in Ref. 20. Indeed, the
(rather unique ) uniformity of the sample and the large and relatively simple
reactor geometry (with respect to the illumination beam) in this experiment
enables the simple analysis above. Nevertheless, Raman signal dynamics of a
few second time scales was observed in various other plasmon-assisted
photocatalysis experiments which involved a more complicated reactor geometry,
see e.g., Refs. 41, 42. Although an estimate of the heat dynamics is far more
complex in these cases, it is more than reasonable to associate the faster
reaction rates to heating, especially since simple estimates we made for these
cases imply that the thermal dynamics may be on the few-second time scale or
even faster.

A few-second time scale dynamics was also observed in many other experiments
where the signal was not monitored instantaneously, but rather, using a mass
spectrometer. While this prevents one from drawing firm conclusions about the
underlying physical mechanism (see further discussion in Ref. 20), the slow
dynamics in these cases is likely due to thermal effects, especially in those
experiments that were shown to be thermally-driven by other means (such as
thermal calculations, and fits to Arrhenius curves; e.g., Refs. 43, 14, 13,
44, 16, 45, 46). Similarly, few-second time scale dynamics was also observed
in various photoelectrochemistry experiments 47, 34, 48, 49, 50, 51. While the
heat dynamics analysis is applicable to these works too, it is not possible
yet to draw firm conclusions before the theory is complemented by a proper
study of the charge dynamics.

Figure 1: (Color online) (a) Schematic of the simulation domain. A laser beam
of wavelength 785 nm is focused at the water–air interface from the top of the
open cylindrical glass cuvette. (b) The temperature distribution at $t=140$ s.
(c) The temperature distribution along the cylindrical axis (blue solid line)
and along the radius at the water–air interface (orange solid line) at $t=140$
s. The inset shows the temperature dynamics at the laser spot center. The
dotted symbols represent the corresponding temperature distribution (and
temperature dynamics in the inset) when the free convection of water is
neglected.

###  3.2 Yu et al. [P. K. Jain’s group, 52]

In contrast, the experiment of Yu et al. 52 cannot be explained within our
simple thermal model shifted-Arrhenius Law, because our thermal calculations
predict a much lower reaction rate than what was actually observed
experimentally. To see this explicitly, we perform the thermal calculations
for the configuration studied in Ref. 52. Specifically, the sample in Ref. 52
consisted of a $3.5$ cm long cuvette with a circular cross-section of radius
of $1$ cm, see Fig. 2(a). The bottom of the cuvette was attached to an
external heater, and was complemented with a small rotor that was used to stir
the $10$ cc water volume. A rectangular cotton patch was attached to the
cuvette wall, and was doped with $\sim 2\times 10^{13}$ Au NPs of $\sim 12$ nm
diameter. The patch was illuminated by a continuous-wave laser beam of spot
size $\sim$ 1 cm2 at a wavelength of $532$ nm and incident power of $1$ W. A
thermocouple was inserted at the center of the cuvette, half way into its
depth to monitor the temperature.

Figure 2: (Color online) (a) Schematic of the simulation domain. A continuous-
wave laser beam is illuminating the side wall of an open cylindrical glass
cuvette. A 2D cross-section of the steady-state temperature distribution in
the cuvette for (b) $\Omega=0$ rpm (without stirring) and (c) $\Omega=120$
rpm.

The thermal distribution in this system was chosen as a case study in Ref. 24
in order to demonstrate explicitly the dependence of the temperature
distribution and total reaction rate on the parameters of the system. These
calculations relied on the formalism described in Ref. 19 which consisted of a
summation of the heat diffusing from a large ensemble of absorbing metal NPs;
the calculation was also validated using an effective medium approximation in
Ref. 24.

In the current work, in order to model the experimental conditions more
faithfully, we simulate the heat transfer and convection of the liquid heated
by the illumination via Eqs. (4)-(6). To account for the stirring, we separate
the calculation domain into two parts, a rotating part and a stationary part.
The rotating part contains the magnetic stir bar itself 111The specific stir
bar geometry we used may not have been identical to the one used in the
experiment, yet, this is not expected to have any significant effect on the
results.  and the water in its vicinity, while the stationary part contains
all other domains outside the rotating part. In the rotating part, we
reformulate Eqs. (4)-(6) in a rotating frame but in terms of the velocity flow
${\bf u}$ and temperature $T$ in the stationary frame, namely,

$\displaystyle c_{p}\rho\left[\dfrac{\partial}{\partial t}+\left({\bf
u}-\bm{\Omega}\times{\bf r}\right)\cdot\nabla\right]T$
$\displaystyle=\nabla\cdot\left(\kappa\nabla T\right)+p_{\textrm{abs}}.$ (8)
$\displaystyle\rho\left[\dfrac{\partial}{\partial t}+\left({\bf
u}-\bm{\Omega}\times{\bf r}\right)\cdot\nabla\right]{\bf u}$
$\displaystyle=-\nabla p+\rho_{0}{\bf g}+(\rho-\rho_{0}){\bf
g}+\nabla\left[\mu\left(\nabla{\bf u}+(\nabla{\bf
u})^{\intercal}\right)\right]$ (9)
$\displaystyle\left[\dfrac{\partial}{\partial t}-\left(\bm{\Omega}\times{\bf
r}\right)\cdot\nabla\right]\rho+\nabla\cdot\left(\rho{\bf u}\right)$
$\displaystyle=0,$ (10)

where $\bm{\Omega}=\Omega{\hat{\bf z}}$ is the rotation vector with $\Omega$
being the angular speed. In the stationary part, the ordinary heat transfer
and Navier-Stokes equations (4)-(6) are solved. For the temperature, we set a
continuity condition at the boundary between the rotating and the stationary
parts. At the bottom of the cuvette, we set the boundary condition to the
controlled temperature, effectively assuming that the cuvette is in perfect
contact with the hotplate surface. At the top and outer surfaces, we use a
convective heat flux boundary condition driven by the temperature difference
between the cuvette and the surrounding. For the flow velocity, we apply the
continuity condition to couple the flow velocity on the boundary between the
rotating and stationary parts. In the stationary part, no-slip condition is
set for the fluid flow at the interface between the liquid and the cuvette
while a slip condition is set at the water-air interface.

When the liquid is not stirred, the temperature distribution within the sample
is clearly non-uniform, see Fig. 2(b). In particular, the temperature near the
cotton patch is $\sim 30^{\circ}$C higher than the average temperature in the
bulk of the liquid and higher than the measured temperature.

The intuitive expectation is that the stirring would reduce these gradients
within the sample. Indeed, as shown in Fig. 2(c) for stirring speed of 120
rpm, the temperature variation in the bulk of the liquid is much smaller.
However, since the heat source (namely, the cotton patch) is positioned on the
wall of cuvette, where the liquid flow vanishes (due to the no-slip boundary
condition), the heat gradients in the vicinity of the heat source are not
reduced as efficiently by the stirring as in the bulk of the sample. Indeed,
Fig. 3(a) shows that the azimuthal flow velocity (parallel to the cuvette
wall) forms the well-known boundary layer 53, 54 whereby the flow velocity
drops linearly towards the cuvette wall. As a result, heat drainage away from
the heat source position is not as effective as in other parts of the sample.
In that sense, our results (Fig. 3(a)) imply that stirring could have been
most effective were the heat source positioned half way between the wall and
cuvette center, or simply spread evenly within the cuvette volume.

As a result, the radial gradients of the temperature are nearly eliminated at
stirring speeds of 120 rpm in most of the volume, yet, they persist near the
cuvette wall, see Fig. 3(b)-(c). More specifically, the maximal temperature
(occurring on the cotton patch) is reduced from $\sim 80^{\circ}$C (no
stirring) to $\sim 63^{\circ}$C (120 rpm). Similarly, the radial temperature
gradient is reduced from $\sim 27^{\circ}$C to $\sim 10^{\circ}$C in a
monotonic yet nonlinear fashion (see also Fig. 3(d)). Importantly, this shows
that the thermocouple position in the experiment causes its reading to
underestimate the temperature on the patch where the limiting step of the
reaction is expected to occur.

In the experiment itself, a much higher rotation speed was applied 55.
Unfortunately, it is difficult to predict how much would the gradients be
further reduced with such an additional increase of the stirring speed.
Indeed, 3D maps of the liquid flow (not shown) imply on the development of
turbulent flow, an effect which will cause the convective heat transfer to be
substantially enhanced by the intense mixing of the fluid 56. In this case, a
much more sophisticated numerical framework compared to the one employed here
is required for the correct treatment of the turbulent convection heat
transfer.

Nevertheless, the authors of Ref. 52 were careful enough to account for this
potential complication. In particular, in order to demonstrate that the
product generation in their experiment does not originate from simply a
photothermal effect, the authors performed thermal control experiments where
the stirred liquid in the cuvette was heated up externally in the dark whereby
the thermocouple temperature reached $70^{\circ}$C 57. Under this condition,
they reported that negligible product generation was observed.

Figure 3: (Color online) (a) 1D cross-section (marked by the red solid line in
Fig. 2(a)) of the steady-state azimuthal velocity $u_{\theta}$ for various
angular speeds of the magnetic stir bar (30 rpm, red dotted dash line; 60 rpm,
green dotted line; 90 rpm, orange dashed line; 120 rpm, blue solid line). (b)
and (c) 1D cross-sections of the steady-state temperature distribution along
$x$-direction (at $z=1.75$ cm; marked by the red solid line in Fig. 2(a)) and
along the $z$-direction (at $x=-1$ cm; marked by the blue solid line in Fig.
2(a)), respectively, for the same angular speeds of the magnetic stir bar in
(a); the case of 0 rpm is added (magenta dots). The shaded region in (c)
indicates the NP-coated cotton substrate. (d) The color dots represent the
maximum temperature on the cotton patch for various angular speeds of the
magnetic stir bar. The black dots represent the averaged temperature at the
thermocouple, as appropriate for its high thermal conductivity. The error bars
represent the temperature variation along the thermocouple.The black dashed
lines in (b), (c) and (d) represent the temperature 70∘C measured by the
thermocouple in the control experiments 57.

Thus, while it is unclear how much the gradients associated with the boundary
layer would be further reduced at higher stirring speeds, already at the
rotation speeds we could reliably simulate it seems that the temperature of
the NPs is lower than in the $70^{\circ}$ control experiment. Thus, since the
thermocatalysis experiment yielded no reaction, thermal effects seem to be
incapable of explaining the experimental observation of the reactants under
illumination, and in fact, optically-induced non-thermal electrons seem to be
responsible for the reaction rate in its entirety; a similar situation was
demonstrated in Ref. 28. In fact, together with the quantitative predictions
of the number of nonthermal electrons generated in the metal 22, 23, with a
quantification of the number of nonthermal electrons that participated in
their reaction and with reactant diffusion dynamics (needed to estimate the
arrival time of the reactants to the gas chromatograph), Jain’s team can
quantify the tunnelling rate of electrons out of the metal.

This experimental approach shows how to bypass the need to match the
temperature distributions of the photocatalysis and thermocatalysis
experiments to enable a correct identification of the underlying mechanism
responsible for the catalysis 19. However, in other cases where the thermal
control was performed under more severe conditions compared to that of the
photocatalysis case (see e.g., Ref. 45), the reaction rate was not negligible
so that one cannot quantify the relative importance of non-thermal electrons
to the reaction rate. In that sense, the conclusions of Ref. 45 are
unfortunately invalid.

A possible reason for the significant non-thermal electron action in this
study is the presence of hole scavengers, which effectively prevent the
electron-hole pairs generated by the plasmonic decay from recombining. In
bond-dissociation reactions (where no hole-scavengers were present, e.g. Refs.
16, 15, 14) the electron-hole pairs generated from plasmon decay recombine or
thermalize at femto-second timescale 22. However, when a hole-scavenger is
present, the holes generated in the plasmon decay process are quickly filled
by electrons from the hole-scavenger molecules in the solution. The high-
energy non-thermal electrons, having their place taken by electrons from the
hole scavengers, are trapped at high-energy states for much longer,
sufficiently long so they can participate in the redox reaction. Thus, our
thermal calculations re-affirm the conclusions in Ref. 52, and support a
similar approach adopted earlier 58, 59, 60, 61, 28, 36.

##  4 Summary and outlook

The thermal simulations performed in this manuscript demonstrate again that
detailed thermal calculations can enable the correct identification of the
underlying physical mechanism responsible for the speeding up of a certain
chemical reaction by revealing the steady-state spatial distribution of the
temperature as before 17, 19, 18, 21, and here also by looking at the
temperature dynamics.

The next challenge in understanding the underlying physics/chemistry in such
systems and quantifying it is to introduce aspects of the electron dynamics
into the quantitative thermal model, and potentially even introduce a
quantitative model of chemical interactions, electron transfer between the NPs
and the chemical moieties and possibly reaction dynamics. This will require
quantitative calculations of “hot” electrons (as in Ref. 22, 23) as well as of
their tunnelling (as e.g., in Refs. 62, 63, 64) and the chemical reaction
dynamics itself.

The formulation presented here will also enable the quantitative study of
electrochemical reactions, which attain growing attention in the context of
plasmon-assisted photocatalysis, see, e.g. Refs. 33, 34, 65, 35, 27, 37. We
are hopeful that similar analysis can be applied to future experiments, in
pursuit for the correct interpretation of the mechanism underlying the
speeding up of the chemical reaction.

We wish to thank J. Huang, J. J. Baumberg and P. K. Jain for sharing their raw
data and for many useful discussions; we also thank Y. Dubi for many useful
conversations and Y. Green and Y. Feldman for their guidance on fluid
mechanics simulations. We acknowledge funding from the Ben-Gurion University
VP office for “multi-disciplinary research”.

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# Front-induced transitions control THz waves

Aidan W. Schiff-Kearn  Department of Physics and Centre for the Physics of
Materials, McGill University, Montreal, H3A 2T8 Canada  these authors
contributed equally to this work  Lauren Gingras  Department of Physics and
Centre for the Physics of Materials, McGill University, Montreal, H3A 2T8
Canada  Department of Physics, University of Ottawa, Ottawa, K1N 6N5 Canada
these authors contributed equally to this work  Simon Bernier  Department of
Physics and Centre for the Physics of Materials, McGill University, Montreal,
H3A 2T8 Canada  Nima Chamanara  Department of Physics and Centre for the
Physics of Materials, McGill University, Montreal, H3A 2T8 Canada  Kartiek
Agarwal  Department of Physics and Centre for the Physics of Materials, McGill
University, Montreal, H3A 2T8 Canada  Jean-Michel Ménard  Department of
Physics, University of Ottawa, Ottawa, K1N 6N5 Canada  David G. Cooke
Department of Physics and Centre for the Physics of Materials, McGill
University, Montreal, H3A 2T8 Canada

###### Abstract

Relativistically moving dielectric perturbations can be used to manipulate
light in new and exciting ways beyond the capabilities of traditional
nonlinear optics. Adiabatic interaction with the moving front modulates the
wave simultaneously in both space and time, and manifests a front-induced
transition in both wave vector and frequency yielding exotic effects including
non-reciprocity and time-reversal [1, 2, 3, 4, 5, 6, 7]. Here, we introduce a
technique called SLIPSTREAM, Spacetime Light-Induced Photonic STRucturEs for
Advanced Manipulation. The technique is based on the creation of relativistic
fronts in a semiconductor-filled planar waveguide by photoexcitation of mobile
charge carriers. Here we demonstrate the capabilities of SLIPSTREAM for novel
manipulation of THz light pulses through relativistic front-induced
transitions. In the sub-luminal front velocity regime, we generate temporally
stretched THz waveforms, with a quasi-static field lasting for several
picoseconds tunable with the front interaction distance. In the super-luminal
regime, the carrier front outpaces the THz pulse and a time-reversal operation
is performed via a front-induced intra-band transition. We anticipate our
platform will be a versatile tool for future applications in the THz spectral
band requiring direct and advanced control of light at the sub-cycle level.

Light in the terahertz (THz) band, already immensely important for
spectroscopy and imaging applications [8], has been touted as a frontier for
next generation high-speed wireless communications [9, 10, 11]. As with other
spectral bands, the advancement of future THz technologies will be done in
lockstep with the development of new methods for controlling the spatial and
temporal properties of THz light. While significant advances have been made on
controlling the spatial evolution of THz waves with metasurfaces and wavefront
engineering [12, 13, 14, 15], methods for actively controlling the temporal
properties of THz light on the level of the electric field cycle are typically
only possible through self-induced temporal phase modulation using intense THz
pulses [16, 17, 18, 19, 20, 21, 22, 23, 24]. Only recently an arbitrary THz
pulse shaper was introduced, employing a hybrid digital wave synthesis
technique using light-induced structures in a waveguide [25]. Such planar
waveguide approaches have the benefit of being amenable to chip-scale
integration [26].
Alternative approaches to nonlinear optics, however, draw on the concept of
time-varying linear media introduced first by Morgenthaler [27, 3, 7]. These
ideas were tested in experiments involving plasma physics via the rapid
ionization of gases [28] or sudden carrier injection through optical
excitation in semiconductors [29]. Temporal modulation is attractive for
wavelength conversion as it does not require nonlinear media or strong field
interactions. Subsequently, several studies have emerged, focused on
temporally modulated photonic crystals [30, 31] and cavities [32, 33] for
wavelength conversion applications. The non-adiabatic modulation of meta-atoms
was recently demonstrated as a means to achieve wavelength conversion in the
THz band [34]. In addition, sub-cycle slicing using intense optical pulses
illuminating semiconductor surfaces is also capable of pulse narrowing and
spectral expansion in the THz band [35, 36].
Simultaneous control over the wavevector and frequency of light through
combined spatial and temporal modulation can produce a variety of exotic
photonic operations [37, 38, 1]. These spacetime, front-induced transitions
can be used to time-reverse a light pulse,[2, 39, 40], for example, which has
important applications in dispersion compensation [41] and imaging through
random scattering media [42]. THz interactions with moving fronts were first
investigated in pioneering experiments using photoexcitation in a
semiconductor to create an overdense, reflective plasma interaction that is
counter-propagating with the incoming THz wave. Considerable spectral
broadening was observed via relativistic Doppler shifts [43, 44, 45]. Here we
introduce SLIPSTREAM for exploring front-induced transitions in the THz band
and use it to demonstrate two unconventional photonic operations: the temporal
stretching and time-reversal of THz pulses.

Figure 1: a, Schematic of the SLIPSTREAM platform, generating and manipulating
THz pulses using the injection of a NIR pulse with a tilted intensity profile
into a Si-filled PPWG. b, Symbolic construction of a THz plateau pulse by the
space-time-delayed, ultrafast switching of static Schottky currents built into
the top and bottom interfaces, shown in successive snapshots in time. c,
Measured THz transients typical of the sub-luminal, luminal and super-luminal
regimes of front velocity for a maximal induced photocarrier density of about
$5\times 10^{16}$ cm3. d, Integrated spectral power, peaking for the luminal
case due to synchronicity with the emission, maximizing interaction with the
moving front.

The SLIPSTREAM technique is based on the same compact platform for spatio-
temporal manipulation we have previously introduced, capable of performing a
range of photonic operations including pulse steering [46], mode control [47],
spectral shaping[48], arbitrary pulse shaping [25] and phase control [49]. As
shown schematically in Fig. 1a, a parallel plate waveguide (PPWG) supporting
dispersionless, low loss, TEM THz propagation is formed by coating both sides
of a $d=50$-$\mu$m-thick high resistivity float zone silicon ($\rho>10,000$
$\Omega$-cm) slab with optically transparent and conducting indium tin oxide
(sheet resistance of 1 $\Omega$/sq). The transparency of the ITO coatings
allows optical excitation of the silicon through the top plate with a near-
infrared (1035 nm, 190 fs, 30 $\mu$J) pump pulse derived from a Yb:KGW
amplified femtosecond laser system (Light Conversion, PHAROS). This excitation
is close to the indirect band gap of Si, with pump penetration depths
exceeding 100 $\mu$m, ensuring a homogeneous excitation across the Si slab.
The pump pulse has its pulse front tilted using a diffraction grating,
producing a linear gradient in the pump arrival time along the waveguide
propagation axis. In this manner, as the pump pulse front illuminates the
silicon, a moving front of photoinduced carrier density is created within the
PPWG. A portion of the NIR beam is split off to a CMOS camera to record the
pump intensity spatial profile.
The photoexcitation of a moving front results in the emission of a phase-
locked THz pulse within the waveguide that propagates in the TEM mode. Since
no electrical bias is applied and the plates are grounded, the sole source of
THz emission is due to transient currents created by built-in Schottky fields,
oppositely oriented at the metal-semiconductor interfaces, upon arrival of the
pump pulse at the top and bottom waveguide plates, as in Fig. 1b.
Photoinjecting free carriers within the band bending regions at these
interfaces leads to the transient currents $J_{\text{top}}(t,z)$ and
$J_{\text{bot}}(t-\zeta,z)$ directed normal to the waveguide plates and
temporally separated by the transit time of the NIR pump pulse $\zeta$. These
delayed and transient currents emit THz light in the same manner as an Auston
switch and similar to a continuous version of a segmented dc-to-ac radiation
converter [50, 51, 52, 53, 54, 55, 56, 57]. Once coupled out of the PPWG, the
THz emission is detected by free-space electro-optic sampling in a 5-mm-thick
(110) GaP crystal.

Figure 2: a, Carrier density profiles in the waveguide for different beam
clippings of a sub-luminal front at $v_{f}=0.86$ $c/n_{\text{Si}}$, varying
either its termination (top) or start position (bottom). b, Corresponding
measured (solid and filled) and simulated (dashed) THz plateau pulses.

The emitted THz electric field transients, shown in Fig. 1c, depend critically
on the pulse front velocity, $v_{f}$, with respect to the THz TEM mode phase
velocity $c/n_{\text{Si}}$ with refractive index $n_{\text{Si}}=3.418$ [58].
The front velocity is continuously tunable from the sub-luminal to super-
luminal regime with the femtosecond pump pulse front tilt angle. In the sub-
luminal regime, $v_{f}<c/n_{\text{Si}}$, a THz plateau pulse is created with a
stretch of quasi-dc electric field bookended by unipolar structure, shown in
the top panel of Fig. 1c. The emitted THz waves escape the excitation front
which produces a superposition of linearly delayed single cycle pulses along
the propagation axis. The result is a temporally stretched THz pulse whose
duration is tunable with front velocity and total propagation distance. At the
luminal condition $v_{f}=c/n_{\text{Si}}$, the emitted THz pulse is of single
cycle duration, as all emission points add in phase along the entire
propagation axis. This produces the maximum amplitude THz emission as shown in
the integrated spectral power in Fig. 1d. Tuning the front to the super-
luminal regime $v_{f}=1.5$ $c/n_{\text{Si}}$, the pulse character remains
single cycle but loses some high frequency components as the front overtakes
the THz wave and it experiences losses in the plasma, here with a plasma
frequency $\omega_{p}/2\pi\approx 1$ THz.
In the sub-luminal regime, the duration of the THz field plateau is tunable
with propagation length as shown in Fig. 2. Truncated carrier density
profiles, calibrated from camera images and power measurements, are given in
Fig. 2a with the subsequently measured THz pulses plotted in Fig. 2b. The
positions of two sharp knife edges in the beam map directly to start
$t_{\text{start}}$ and stop times $t_{\text{end}}$ of the THz pulse,
corresponding to the measured pulse arising from the leading and trailing edge
of the carrier density front, respectively. Sliding either knife edge by
$\Delta z$ and measuring the pulse duration change $\Delta t$ allows a precise
calibration of the front velocity through

$v_{f}=\frac{1}{\Delta t/\Delta z+n/c},$ (1)

which in this case yields a sub-luminal velocity of $v_{f}=0.86$
$c/n_{\text{Si}}$. The amplitude of the plateau follows the Gaussian profile
of the NIR beam due to the linearity of field emission with the local current
density.

Figure 3: a, Plasma dispersion relation $\beta$ projected onto the unexcited
TEM mode wavenumber $\beta_{0}$ (horizontal black line) for excitation
conditions: (1 - light blue line) $\omega_{2}/\beta_{2}<v_{f}$, (2 - cyan
line) $\omega_{2}/\beta_{2}=v_{f}$ and (3 - green line)
$\omega_{2}/\beta_{2}>v_{f}$. The dashed line is the front velocity
$v_{f}=1.0035$ $c/n_{\text{Si}}$. The left inset shows that only forward
intra-band scattering is possible at weak excitation (blue line), whereas the
right inset shows that a time-reversal scattering event
($-\beta_{1}\rightarrow\beta_{2}$, empty dot to solid circle) becomes phase
matched for highly photoexcited (red line) dispersion conditions. b, Real
space pictures that follow the evolution of initially-forward (pulse A) and
initially-backward (pulse B) propagating THz waves for plasma conditions
induced below and above critical fluence. c, Measured single cycle THz pulses
generated by a slightly super-luminal front for indicated levels of pump
fluence. The pulses on the left display time-reversal at $t=2.5$ ps and, on
the right, a chirped pulse builds up about 10 ps later from low-frequency
waves surviving the plasma. d, Amplitude spectra of only the pulses displaying
time-reversal in c for all fluences explored, categorized above critical
fluence (and below in the inset). e, Integrated spectral power of the time-
reversed THz pulses (squares) and those arising 10 ps later (circles).

Two-dimensional finite-difference time-domain (2D-FDTD) simulations were
performed to further understand the influence of the moving front on the THz
pulse waveforms. A perfectly conducting layer was chosen to bound the domain
and Schottky fields were placed on the top and bottom extending into the
silicon by the Debye length (about 1 $\mu$m). The dispersion of photoexcited
silicon was modelled assuming a Drude form with a scattering time of 0.1 ps
and carrier densities governed by the profiles in Fig. 2a, reaching a maximum
of $10^{17}$ cm-3 [59]. The electric field of the emitted THz wave coupled to
the TEM mode was extracted at a fixed position within the waveguide just
outside the illuminated region and a filter function was applied in order to
approximate the experimental losses from the waveguide and detection response
function [60]. The simulated THz pulses, shown in Fig. 2b agree well with the
detected THz waveforms, qualitatively reproducing the variations in the
plateau amplitudes.
Setting $v_{f}>c/n_{\text{Si}}$ forces the emitted THz waves to interact with
the moving front over the entire propagation distance, bringing about front-
induced transitions. Deep in the super-luminal regime, the underdense plasma
rapidly overtakes the THz wave and the pulse experiences both absorption and
dispersion much the same as it would for transiting a static dispersive plasma
[49], as we have verified for $v_{f}=1.5$ $c/n_{\text{Si}}$ conditions. For
$v_{f}$ very close but still exceeding $c/n_{\text{Si}}$, however, the front
interacts with the THz emission for an extended interaction length without
being consumed by the absorptive plasma. The wave therefore exists within the
plasma skin depth, which is about $150$ $\mu$m at $f=0.5$ THz and which spans
a single wavelength in the photoexcited silicon, supporting single cycle
pulses.
Figure 3 demonstrates the concept of a front-induced transition, which is
explained in more detail in ref. [1]. For dispersionless ($\omega_{p}\tau\ll
1$) and underdense ($\omega_{p}<\omega$) plasmas where hard reflections can be
ignored, the photonic band structure of the photoexcited silicon is
approximately linear with a phase velocity given by

$\frac{\omega_{2}}{\beta_{2}}\approx\frac{c}{n_{\text{Si}}}\biggl{(}1+\frac{\omega_{p}^{2}\tau^{2}}{2}\biggr{)}.$
(2)

The two insets of Fig. 3a show that increasing the plasma frequency from low
(blue) to high (red) adds curvature to the band, flattening at low THz
frequencies. Front-induced transitions on a dispersion diagram, from an
initial state $(\omega_{1},\beta_{1})$ to a final state
$(\omega_{2},\beta_{2})$, must obey the phase continuity relation
$\omega_{1}t-z_{f}\beta_{1}=\omega_{2}t-z_{f}\beta_{2}$ where $z_{f}=v_{f}t$.
The ratio of frequency change to wavenumber change is equal to the front
velocity, i.e.
$\Delta\omega/\Delta\beta=(\omega_{2}-\omega_{1})/(\beta_{2}-\beta_{1})=v_{f}$.
Thus the allowed transitions are tunable with the dispersion of the
perturbation via the optical excitation power controlling $\omega_{p}$ and the
pump tilt angle controlling $v_{f}$. The main diagram in Fig. 3a shows the
dispersion relative to unexcited silicon for three possible perturbations with
increasing $\omega_{p}$ labelled by 1, 2 and 3. The dispersion of the plasma
is modelled by a Drude response with a fixed scattering time $\tau$ of 0.1 ps
and a variable plasma frequency $\omega_{p}$ set by the pump fluence. In each
plot, the phase continuity condition is denoted by the dashed black line where
a slightly super-luminal front velocity of $v_{f}=1.0035$ $c/n_{\text{Si}}$
has been chosen for reasons that will be explained below.
The Schottky THz emission mechanism produces an outgoing spherical
electromagnetic wave containing components travelling towards the detector
(positive $\beta$) as well as those pointed backward into the plasma front
(negative $\beta$). For a super-luminal front, the emission is immediately
swept up by the plasma so only intra-band front-induced transitions may take
place. Considering that only positive wavevectors will be detected, three
scenarios are indicated in Fig. 3a: (1 - blue line)
$\omega_{2}/\beta_{2}<v_{f}$, where transitions involving solely positive
wavenumbers can occur. For high fluences (3 - green line)
$\omega_{2}/\beta_{2}>v_{f}$, transitions from backward propagating initial
states to forward propagating states within the plasma become possible. In
this scenario, the backward wave undergoes a time-reversal as the spatial
distribution remains unchanged but the group velocity inverts sign. Higher
plasma frequencies forbid intra-band scattering at lower frequencies until
time-reversal scattering from $-\beta_{1}$ to $+\beta_{2}$ takes over. Here,
in the intermediate regime, the special case (2 - cyan line)
$\omega_{2}/\beta_{2}=v_{f}$ prohibits all forward initial states from
undergoing transitions. The plasma frequency that satisfies this dispersion
condition allows equation 2 to be solved for the front velocity.
Figure 3b illustrates in real space a picture of the pulse time-reversal we
obtain through control over the plasma dispersion in the regime where $v_{f}$
is slightly super-luminal. At low fluence, i.e. scenario 1, the forward
propagating THz component (pulse A) can be projected up to higher frequency
states whereas its backwards counterpart (pulse B) is undetected, lost to the
plasma. When the front stops at the edge of the photoexcited region, the
blueshifted forward THz wave is ejected. In scenario 3, the optical fluence is
tuned above a critical value $F_{c}$ for which the slope of the photoexcited
band matches the front velocity. The additional curvature in the plasma
dispersion prohibits fast forward waves of type A from undergoing intra-band
transitions. Instead, the highly dispersive front prevents their passage and
they recede back to the plasma skin depth (pulse A′), where they are made to
co-travel with the front. Finally, in the same regime above critical fluence,
the initially-backwards wave (pulse B) is able to find transitions to final
states with positive $\beta$, reversing its group velocity through intra-band
reflection.
Figure 3c shows the influence of tuning the plasma dispersion curve on the
detected THz waveforms for a slightly super-luminal velocity front. For these
results we utilize a PPWG composed of ITO/Si/Au layering, where 5-10 nm of Ti
is used to adhere 150 nm of Au to the bottom layer. Varying the fluence of the
NIR pump pulse, we observe on the left single cycle THz pulses as expected,
however for an intermediate fluence (scenario 2), we see a near extinction of
the pulse followed by a $\pi$ phase inversion at higher fluences. This
extinction marks the crossover from forward intra-band scattering to a time-
reversal scattering event. The right plot in Fig. 3c highlights the
development at increasing fluence of a second low-frequency THz pulse emerging
roughly 10 ps later, corresponding to the type A′ pulses that have been held
back in the low-frequency plasma skin before ejection.
Next we take a frequency-domain view of our experiment, first by analyzing the
evolution of measured type B pulses as we alter their dielectric environment.
Amplitude spectra obtained from numerical fast-Fourier transform of these
pulses are shown in Fig. 3d, revealing how the final states populate as the
plasma frequency is swept through the scenarios 1, 2 and 3 described in Fig.
3a. A steep super-Gaussian filter has been applied in order to window the
pulses. Each spectrum displays a high-frequency shoulder due to waveguide
loss. As $\omega_{p}$ is decreased from its highest value, the spectra
acquired between scenarios 3 to 2 maintain a smooth broadband shape while
reducing in amplitude, consistent with front-induced transitions occurring
across the entire detection bandwidth. However, in the inset, between
scenarios 2 to 1 a spectral dip suddenly appears and develops near 0.4 THz. As
seen in Fig. 3a, weak excitation produces dispersion that is predominantly
linear, i.e. well-approximated by equation 2, and we therefore expect this dip
to arise from interference between co-propagating frequency states populated
inside and just outside the moving plasma front, both travelling in the
neighbourhood of $c/n_{\text{Si}}$. In Fig. 3e, we verify the distinct
behaviour of the pulses as the plasma frequency of the excitation is tuned
through the 3 identified scenarios. Type A′ pulses follow a continuous
decrease in integrated power, whereas as type A evolves into type B a minimum
occurs at scenario 2 which marks the crossover between forward intra-band
scattering to time-reversing front-induced transitions followed by a
resurgence and true nought.
In summary, we have demonstrated that the SLIPSTREAM platform enables the
interaction between THz light and relativistic moving fronts of photoinduced
dielectric modulation. In the sub-luminal regime, the relativistic front
generates a structured THz pulse displaying a quasi-static electric field
profile via the extraction of energy from built-in Schottky fields at metal-
semiconductor boundaries. The temporal profile of the electric field can be
tailored by the intensity distribution of the optical photoexcitation. This
capability allows, for example, applications where long, tunable THz pulses
may act as an ultrafast optical bias [61, 62]. In addition, we showed that
when the front velocity is tuned just above luminal in the silicon, the
fluence dependence of the dispersion of the carrier plasma is crucial in
determining the front-induced transitions that may occur. We observed time-
reversal of a THz pulse as the photoinduced dispersion of the dielectric
perturbation was optically tuned relative to the front velocity. This work
establishes our platform as a practical avenue to explore the interaction of
light with moving dielectric fronts. The ability to apply localized dielectric
modulation by optical means on a silicon chip has significant implications for
investigating, for example, the relativistic Doppler effect [44, 45, 63],
optical horizon analogues [64] and experimentally novel effects associated
with the motion of shocklike dielectric modulations travelling in photonic
crystal landscapes [65].
The authors wish to acknowledge financial support from the Natural Sciences
and Engineering Research Council of Canada (NSERC), Fonds de Recherche du
Qu$\acute{\text{e}}$bec Nature et Technologies (FRQNT) and the Canadian
Foundation for Innovation (CFI). We also gratefully acknowledge useful
discussions with E. Yablonovich.

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