# Phase-sensitive pump-probe measurement of the complex nonlinear
susceptibility of silicon across the direct band edge

C. D. Cruz  1 J. C. Stephenson  2 and J. K. Wahlstrand1  1Nanoscale Device
Characterization Division, National Institute of Standards and Technology,
Gaithersburg, MD 20899, USA
2Associate, Sensor Science Division, National Institute of Standards and
Technology, Gaithersburg, MD 20899, USA<EMAIL_ADDRESS>
###### Abstract

Pump-probe supercontinuum (SC) spectral interferometry in reflection is used
to measure time-dependent variations in the complex dielectric function ($n$,
$k$) over the visible wavelength range in bulk silicon. Changes in $n$
($\Delta n$) due to a near infrared pump pulse are derived from transient
fluctuations in the reflected SC probe amplitude, as is measured in
conventional pump-probe experiments. Additionally, modifications to $k$
($\Delta k$) are derived from phase shifts in the reflected SC. The ultrafast
response is attributed to two-photon absorption ($\Delta k$) and the optical
Kerr effect ($\Delta n$), enabling measurement of effective two-photon
($\beta$) and Kerr ($n_{2}$) coefficients. The onset of strong two-photon
absorption as the two-photon energy is tuned through the $E_{1}$ = 3.4 eV
direct band edge of silicon is observed for the first time to our knowledge.
These constitute the first measurements of the complex nonlinear response of a
bulk medium from reflectivity alone. The technique allows straightforward
spectroscopic measurements of the $\chi^{(3)}$ nonlinear response at the
surface of absorbing materials.

††journal: opticajournal††articletype: Research Article

##  1 Introduction

The optical Kerr effect and two-photon absorption play an increasingly
important role in integrated optics [1, 2] and laser processing of materials
[3, 4]. The most commonly used techniques for quantifying these aspects of the
third-order nonlinear susceptibility $\chi^{(3)}$ measure nonlinear changes in
the amplitude, distortion, or direction of an optical beam transmitted through
the sample under study [5, 6, 7]; however, transmission measurements in highly
absorbing bulk materials require preparation of thin samples. Conventional
pump-probe reflection measurements by themselves do not allow access to the
change in the full complex dielectric function (they can in combination with
transmission measurements, e.g. [8, 9]). _Phase-sensitive_ pump-probe
reflectivity measurements, in contrast, do allow measurement of the change in
the full complex dielectric function.

The complex reflection coefficient at normal incidence for a single interface
between vacuum and a material with complex refractive index $\tilde{n}\equiv
n+ik$ is

$r=\frac{\tilde{n}-1}{\tilde{n}+1}.$ (1)

Small perturbations of the complex refractive index, for example due to the
presence of a pump pulse, change the complex reflectivity according to

$\Delta
r\approx\frac{dr}{d\tilde{n}}\Delta\tilde{n}=\frac{2}{(\tilde{n}+1)^{2}}\Delta\tilde{n},$
(2)

where $\Delta\tilde{n}=\Delta n+i\Delta k$. A more experimentally accessible
quantity is the normalized change in reflectivity,

$\frac{\Delta r}{r}=\frac{2\Delta\tilde{n}}{\tilde{n}^{2}-1}.$ (3)

Ordinary pump-probe experiments measure pump-induced changes in the power
reflectivity $R\equiv|r|^{2}$, i.e. $\Delta R/R\approx 2\mathrm{Re}(\Delta
r/r)$. In dielectrics and many semiconductor materials, $n\gg k$ for photon
energies up to well above the band edge, and therefore $\tilde{n}^{2}-1\approx
n^{2}-1$ or $\Delta R/R\approx 4\Delta n/(n^{2}-1)$. A standard pump-probe
measurement is thus sensitive only to $\Delta n$ in these materials. A change
in the imaginary part of the refractive index $\Delta k$ mostly changes the
imaginary part of $\Delta r/r$, which changes the _phase_ of reflected light.
The pump-induced phase shift upon reflection is $\Delta\phi=\mathrm{Im}(\Delta
r/r)\approx 2\Delta k/(n^{2}-1)$.

Recently submilliradian pump-probe phase measurements have become possible,
enabling transient phase spectroscopy [10, 11, 12]. Here we employ a phase-
sensitive pump-probe scheme to measure the wavelength-dependent transient
nonlinear response of silicon at very short time delay, where the pump pulse
overlaps the probe pulse. Using Eq. (3), we extract effective nonlinear
coefficients as a function of wavelength and compare to previous results and
calculations of the two-photon absorption spectrum in silicon. We use this to
measure the two-photon absorption coefficient in bulk Si for two-photon energy
(defined as $\hbar\omega_{1}+\hbar\omega_{2}$ for excitation at two
frequencies $\omega_{1}$ and $\omega_{2}$) well above the direct band edge
(3.4 eV), which requires the photon energy of at least one excitation beam to
be above the indirect band edge (1.1 eV), at which Si is highly absorbing. The
direct two-photon absorption regime is, with the exception of a few data
points around a two-photon energy of 4 eV [13], unexplored in Si. These
measurements on Si are of fundamental interest, but we believe the larger
impact of this technique will be its ability to measure inhomogeneous flakes
and thin films on opaque substrates.

##  2 Experiment

Figure 1: Phase-sensitive pump-probe reflectivity measurements.

The experimental scheme based on pump-probe spectral interferometry is
depicted in Fig. 1. We use supercontinuum spectral interferometry [14]; many
aspects of the specific experimental apparatus are described in detail in Ref.
[12]. The light source is a Yb:KGW laser (Light Conversion Carbide [15]) that
pumps an optical parametric amplifier (OPA, Light Conversion Orpheus [15]).
The laser is operated at 3 kHz repetition rate. A supercontinuum (SC) probe
beam is generated in water by some of the 1.03 $\mu$m output from the laser.
The SC is split into two pulses with a delay $\tau_{pr}$ using a Michelson
interferometer. A wavelength tunable pump beam is generated using the OPA. The
pump and probe beams are combined using a dichroic mirror and focused on the
sample with a microscope objective (numerical aperture 0.42). The central part
of the input beam is normally incident on the sample. Reflected light is
sampled with a beamsplitter placed between the objective and the dichroic
mirror and is then sent to a spectrometer. Probe spectra spanning 450-680 nm
are collected by a CCD camera at 3 kHz and the pump beam is chopped at 750 Hz.
Captured spectra are sorted and averaged into “pump on” and “pump off”
spectra. The probe spectra have interference fringes spaced by $1/\tau_{pr}$,
and Fourier analysis of these fringes allows extraction of the probe frequency
dependent phase and amplitude [16].

The achievable delay between probe pulses $\tau_{pr}$ has an upper limit set
by the resolution of the spectrometer. In a pump-probe experiment this sets
the upper limit on the temporal measurement window, because spectral
interferometry requires a reference pulse with an unperturbed spectral phase.
While there are techniques to avoid this limitation for long delay phase
spectroscopy [10, 12], here we are exclusively concerned with the short time
response of the sample, during and a few hundred femtoseconds after the pump
pulse, so we used the maximum time delay possible for our spectrometer
resolution, approximately 1 ps.

The supercontinuum pulses are heavily chirped by propagation through the water
cell and subsequent optics, with low frequencies arriving first. For a given
pump-probe time delay, the approximately 200 fs long pump pulse therefore
overlaps with a small range of probe frequencies. The extracted phase and
amplitude as a function of pump-probe delay and probe optical frequency are
shown in Fig. 2 for a (100) oriented silicon wafer with parallel pump and
probe polarization along the (011) crystal direction. A pump central
wavelength of 1190 nm was used. Figure 2a shows the pump-induced phase shift
$\Delta\phi(\omega_{p},\tau_{ep})=\mathrm{Im}[\Delta
r(\omega_{p},\tau_{ep})/r]$, where $\omega_{p}/(2\pi)$ is the probe frequency
and $\tau_{ep}$ is the time delay between the pump pulse and the 570 THz (525
nm) component of the probe pulse. Figure 2b shows the pump-induced normalized
field amplitude change $\Delta A(\omega_{p},\tau_{ep})=\mathrm{Re}[\Delta
r(\omega_{p},\tau_{ep})/r]$ (note that $\Delta A$ is a normalized amplitude
change and not a change in absorbance). The pump-probe delay is varied over 10
ps in steps of 500 fs. To calibrate the pump and probe beam fluence and
optimize spatial overlap, we image the sample plane using a second CCD camera.
The width (full width at half maximum) of the probe (pump) beam at the sample
surface is 3 $\mu$m (14 $\mu$m).

Figure 2: Phase-sensitive pump-probe reflectivity spectrum in bulk silicon as
a function of pump-probe time delay (with respect to the 570 THz component of
the probe pulse) for a 1190 nm pump pulse at 9 mJ/cm2 incident fluence. (a)
Extracted pump-induced phase change $\Delta\phi(\omega_{p},\tau_{ep})$. (b)
Extracted pump-induced amplitude change $\Delta A(\omega_{p},\tau_{ep})$.

Transient changes in the phase and amplitude are observed at a probe optical
frequency that depends on $\tau_{ep}$ due to the probe chirp. At a given pump-
probe time delay, the closely spaced probe pulses ($\sim 1$ ps) are both
affected by the pump pulse at different wavelengths. When the pump pulse
affects the first probe pulse but not the second, the phase shift induced by
the pump has a particular sign. When the pump pulse affects the second probe
pulse but not the first, the sign of the pump-induced phase shift is flipped.
In contrast, the pump’s effect on the amplitude does not flip sign depending
on which pulse is affected.

The effect of the pump pulse on each of the probe pulses can be well separated
within a 500 fs time window. In spectral interferometry, reliable extraction
of the frequency-dependent phase shift of a pulse requires that the reference
pulse be unperturbed at the frequencies where the pulse of interest is
perturbed by the pump pulse. Often it is possible to arrange for the reference
pulse to not be affected at all by the pump pulse. Our scheme, where both
pulses are affected by the pump pulse, is analogous to (spatial) folded
wavefront interferometry, where a beam that propagates through a locally
perturbed medium is split and interfered with itself so that an unperturbed
region acts as a reference for a phase measurement of the perturbed region
(e.g. [17]).

In both phase and amplitude we observe transient spikes that correspond to the
pump pulse overlapping with the two probe pulses. To analyze the data, we draw
on previous single-shot supercontinuum spectral interferometry (SSSI) based
measurements in gases [14, 18], where knowledge of the probe pulse’s spectral
phase $\phi_{p}(\omega)$ was used to calculate the intrapulse-time-dependent
phase shift and amplitude change from frequency domain data. The time domain
change in complex reflectivity is found from the measurements using

$\frac{\Delta r(t)}{r}=\frac{\mathcal{F}^{-1}[(A_{p}(\omega)+\Delta
A(\omega))e^{i[\phi_{p}(\omega)+\Delta\phi(\omega)]}]}{\mathcal{F}^{-1}[A_{p}(\omega)e^{i\phi_{p}(\omega)}]}-1,$
(4)

where $\mathcal{F}^{-1}$ indicates the inverse Fourier transform,
$A_{p}(\omega)$ is the probe spectral amplitude, $\phi_{p}$ is the probe
spectral phase, $\Delta A(\omega)$ is the change in probe spectral amplitude,
and $\Delta\phi$ is the change in probe spectral phase. This is a
straightforward generalization of SSSI analysis [14, 19], which in previous
work could assume a pure phase shift in the time domain, to include both a
phase shift and amplitude change in the time domain. The quantities $\Delta
A(\omega)$ and $\Delta\phi(\omega)$ (shown in Fig. 2) are found from the
spectral interferometry analysis described above. The quantity $A_{p}(\omega)$
is also found directly from the probe interferograms captured by the CCD
camera. We find the last quantity, the probe spectral phase
$\phi_{p}(\omega)$, in the usual way by tracking the frequency of the maximum
transient phase shift as a function of time delay $\tau_{ep}$, which is
related to $\phi_{p}(\omega)$ via $\tau_{ep}=\phi_{p}^{\prime}(\omega)$ [14,
19]. The probe spectral phase is well fit by a polynomial
$\phi_{p}(\omega)=\phi_{2}(\omega-\omega_{0})^{2}+\phi_{3}(\omega-\omega_{0})^{3}$.

##  3 Results and discussion

For each pump-probe time delay (i.e. each row in the subplots shown in Fig.
2), we use Eq. (4) to translate the frequency domain phase and amplitude
changes to the intrapulse time dependent complex reflectivity change $\Delta
r(t)/r=\Delta A(t)+i\Delta\phi(t)$. We truncate the data to include the
response of only one probe pulse, which results in a $\sim 500$ fs measurement
window in $t$ centered on the time at which the pump pulse overlaps the probe
pulse. Examples of extracted time dependent data are shown in Fig. 3. Note
that the time axis in Fig. 3 is the delay within the probe pulse $t$ with
respect to a the center of the pump pulse, not the pump-probe delay
$\tau_{ep}$, the vertical axis in Fig. 2.

Figure 3: Time domain changes found from data (points) and fits (lines) to Eq.
(5) in bulk silicon. (a) Time-domain phase change $\Delta\phi(t)$ and (b)
Time-domain normalized amplitude change $\Delta A(t)$ pumped at 1190 nm and
probed at 492 nm.

Because of the short measurement window, the experiment only captures the
transient spike when the pump and probe pulses overlap in time and the
cumulative signal directly following the pump pulse. The shape of the temporal
response depends on the pump and probe wavelengths, but in all cases is well
fit by the expression

$\frac{\Delta r(t)}{r}=\Delta
A(t)+i\Delta\phi(t)=A_{T}\exp(-t^{2}/\tau_{T}^{2}).$ (5)

Fits are shown as black solid lines in Fig. 3. In the following text we refer
to the term proportional to $A_{T}$ as the transient response, which is
proportional to the approximately Gaussian irradiance profile
$I_{e}(t)=I_{0}e^{-t^{2}/\tau_{T}^{2}}$ of our pump pulse. The time duration
parameter $\tau_{T}$ extracted from the fits is approximately 120 fs (the full
width at half maximum is 200 fs), which agrees with the pulse duration we
measure from the temporal dependence of the phase shift in transmission in a
thin fused silica sample.

A transient response of the probe amplitude ($\Delta n$ in reflection when
$n\gg k$) that follows the pump irradiance envelope can be modeled by an
effective Kerr response [20],

$\Delta n(t)=2n_{2}I_{e}(t),$ (6)

where $n_{2}$ is the Kerr coefficient for one beam acting on itself. Note that
this expression includes a factor of 2 that is caused by “weak wave
retardation” [22, 23, 21], Similarly, a change in the probe phase ($\Delta k$
in reflection when $n\gg k$) can be modeled by a two-photon absorption
coefficient,

$\Delta
k(t)=\frac{\lambda}{4\pi}\Delta\alpha(t)=\frac{\lambda}{4\pi}\left[2\beta
I_{e}(t)\right],$ (7)

where $\beta$ is the two-photon absorption coefficient and $\lambda$ is the
probe wavelength. The dependence of $A_{T}$ (phase and amplitude) on pump
power (proportional to fluence) is shown in Fig. 4a for 1190 nm pump
wavelength and is found to be linear within the measurement uncertainty, which
is consistent with the expressions above.

Figure 4: (a) Power dependence of the transient phase shift and amplitude
change in Si. The error bars are derived from the measurement noise. (b)
Polarization dependent $\Delta\phi(t)$. For both plots, the pump wavelength is
1190 nm and probe wavelength is 457 nm.

We also compared the response with pump and probe polarized parallel and
perpendicular to one another. Time-dependent phase traces are shown in Fig. 4b
for 1190 nm pump wavelength. The transient change in both phase and amplitude
(latter not shown) is smaller for perpendicular polarization. In
semiconductors, the primary source of an optical nonlinearity that follows the
pump irradiance envelope is the bound electronic nonlinearity (a true
$\chi^{(3)}$), which is responsible for the optical Kerr effect and two-photon
absorption [24]. The approximate factor of 3 change with polarization is
consistent with expectations for the third-order susceptibility for a cubic
crystal [24].

The pump irradiance falls as $I(z)=I_{0}e^{-\alpha z}$ within the material
because of absorption, and $\Delta n$ and $\Delta k$ drop accordingly. In an
appendix in Ref. [25], Sabbah and Riffe show that if the length scale $\delta$
of changes to the optical properties near the surface is much greater than the
probe observation depth $d_{\mathrm{obs}}=\lambda/(4\pi n)$, the change of
optical constants below the surface can be neglected and the Fresnel formula
for reflection, Eq. (3), can be used. For the pump and probe wavelengths used,
$\delta/d_{\mathrm{obs}}>500$ putting us firmly within this limit.
Ellipsometry measurements indicated the presence of an approximately $10$ nm
thick native oxide layer. The Kerr coefficient of SiO2 is approximately
$3\times 10^{-16}$ cm2/W [26], and double passing this thin layer is predicted
to produce a peak phase shift $\Delta\phi=8\pi n_{2}IL/\lambda\approx 3\times
10^{-5}$ radians, much smaller than the phase shift we observe.

It should be noted that other sources could contribute to a response that
follows a 200 fs pulse intensity profile. These include the lattice response
(which we expect to be small, considering the relative size of the coherent
phonon signal in silicon observed with shorter pulses [25]) and ultrafast
carrier relaxation. However some caution is advised concerning the origin of
the transient response, because the pump-probe signal when the pulses overlap
in time has potential contributions from coherent diffraction effects [27,
28]. These cause transients in both amplitude and phase [21] that depend on
details like the pulse chirp. In the experiments described here, the pump and
probe beams are nondegenerate, which massively suppresses these two-beam
interference effects because the grating created by the pump and probe beams
is time dependent and averages to zero [28, 21]. For nondegenerate pulses as
used here, the only remaining contribution from two-beam effects is from bound
electronic $\chi^{(3)}$, which is unique in that it responds at the optical
frequency. Two-beam effects for a true $\chi^{(3)}$ are responsible for the
factor of 2 in Eqs. (6,7) [21, 20].

##  4 Extraction of nonlinear coefficients

We used the previously measured complex refractive index of Si [29] to
translate from peak phase shift and amplitude change to $\Delta n$ and $\Delta
k$ [Eq. (3)] and used Eqs. (6,7) to find effective nonlinear coefficients
$n_{2}$ and $\beta$. We emphasize that the $n_{2}$ and $\beta$ coefficients in
Eqs. (6,7) are nondegenerate coefficients that depend on both pump wavelength
and probe wavelength and can be different [30] from the degenerate
coefficients accessed by one-beam experiments [5, 31]. Extracted coefficients
are plotted (red points for $\beta$ and black points for $n_{2}$) as a
function of probe wavelength and two-photon energy (sum of pump photon energy
and probe photon energy) in Fig. 5.

Figure 5: Spectral dependence of measured coefficients (red squares: $\beta$,
left axis; black circles: $n_{2}$, right axis) in Si shown as a function of
probe wavelength (bottom axis) and two-photon energy (top axis) at pump
wavelengths 1190 nm. The solid line is a calculation of two-photon absorption
using a $\mathbf{k}\cdot\mathbf{p}$ model (only including direct electronic
transitions). The measured value $\beta$ (degenerate nonlinearity) from Ref.
[31] is shown as a blue star.

###  4.1 Uncertainty

As in most nonlinear optics measurements, the uncertainty in our measurement
of the nonlinear coefficients is largely due to uncertainty in the laser
irradiance. In the case of this experiment, we are concerned with the average
pump irradiance within the probe spot, which depends on the pulse energy, the
spatial beam profile at the sample surface, the temporal profile of the pump
pulse, and the overlap between pump and probe spots. The pump spot at the
sample surface was imaged using an InGaAs CCD camera. A resolution target was
used to calibrate the magnification of the image. The ratio of the summed
(dark count subtracted) image to the peak was found and combined with the
average power measurement to calculate the peak pump fluence of 9 mJ/cm2. When
setting up the measurement, we peaked up the signal size by adjusting the pump
spot position on the sample. Finally, for the pulse duration we use the
parameter $\tau_{T}=120$ fs (200 fs full width at half maximum) from the fit
of the time domain phase shift. We estimate the total fractional uncertainty
in the peak pump irradiance as 30%, with inputs from the average power (10%),
the pump spot area (25%), the pump-probe overlap (10%), and the pulse duration
(10%). This contributes to a 30% fractional uncertainty in each point in Fig.
5.

In contrast with many other techniques, where the signal-to-noise ratio can be
quite high, here the noise in the measurement of $\Delta\phi$ and $\Delta A$
is also a sizable fraction of the uncertainty budget for some data points,
particularly at high probe photon energy. For each data point in Fig. 5 we
added (in quadrature) the statistical uncertainty in the peak phase or
amplitude from the fit to the 30% uncertainty due to irradiance.

The absolute uncertainty in each nonlinear coefficient is at least 30%, but it
is important to note that the relative uncertainty between points in the
spectra shown in Fig. 5 is smaller because each measurement shares the same
pump irradiance. Thus the uncertainty in pump spot size and pulse duration
does not contribute, only the uncertainty from pump-probe overlap (due to
probe spatial chirp produced by chromatic aberration) and the phase
measurement. The spectroscopic information in the probe wavelength dependent
nonlinear coefficients is therefore of relatively higher quality than the
absolute coefficients. In future work we hope to perform new measurements with
an improved estimation of the pump fluence and report coefficients with
reduced uncertainty.

###  4.2 Comparison to previous work and theory

For pump wavelength 1190 nm (1.04 eV), the two-photon energy range covers the
transition between indirect (phonon-assisted) and direct two-photon absorption
(the direct edge is near 3.4 eV). We find that $\beta$ begins to rise around
3.3 eV, which is similar to the onset of the corresponding one-photon
absorption edge [32, 29]. Our spectral resolution is limited by the pump
bandwidth to about 0.01 eV, which is narrower than any features we observe. We
consider the observed two-photon energy dependence to be evidence that we are
measuring two-photon absorption, since there is nothing special about the 500
nm probe wavelength in the linear optical properties of silicon that would
otherwise explain the rapid increase of the transient phase shift with
increasing probe photon energy. Measurements at other pump wavelengths (not
shown) also indicated an onset of strong two-photon absorption near 3.3 eV. We
find $\beta\approx 6.5$ cm/GW for two-photon energy around 3.1 eV, which is
higher than Z-scan measurements at 3.0 eV two-photon energy by Bristow _et
al._ [31], shown as a blue star, and other experimental and theoretical work
on indirect two-photon absorption in silicon [33, 34, 35]. Our measurement of
$n_{2}\approx 1.1\times 10^{-13}$ cm2/W near 3.1 eV is also higher than the
Bristow _et al._ value of $4.5\times 10^{-14}$ cm2/W for their highest
reported two-photon energy of 2.4 eV. The larger values we measure are likely
explained by the fact that we measure the nondegenerate susceptibility, which
can differ from the degenerate susceptibility [30] that was measured in Ref.
[31].

In direct band gap semiconductors such as GaAs and ZnSe, the spectral
dependence of $\beta$ and $n_{2}$ has been shown to be very consistent with an
analytical model based on parabolic valence and conduction bands [36]. Unlike
GaAs and ZnSe, and even other indirect semiconductors like Ge, the direct band
edge in Si at 3.4 eV is not associated with a conduction band minimum at the
center of the Brillouin zone ($\Gamma$) aligned with the valence band maximum
at $\Gamma$. Instead the lowest conduction band is concave down in reciprocal
space, and the onset of absorption with increasing photon energy occurs at
wavevectors well away from the zone center. We therefore compared the
measurements to a calculation of the two-photon absorption spectrum using a
30-band $\mathbf{k}\cdot\mathbf{p}$ model [37, 38]. We use the band energy and
coupling parameters provided by Richard _et al._ [37] and calculate two-photon
absorption spectra using the approach of Rioux and Sipe [38]. The calculation
was modified [39] to calculate nondegenerate two-photon absorption spectra,
which can result in an enhanced nonlinear response [30, 40]. The two-photon
absorption spectrum $\beta$ calculated is shown as a solid line in Fig. 5. We
find very good agreement considering that the theory does not include indirect
two-photon absorption.

##  5 Conclusion

In summary, we have demonstrated that the complex nonlinear susceptibility in
a highly absorbing bulk medium can be measured through phase-sensitive pump-
probe reflectivity spectroscopy. We hope that our measurement will stimulate
theoretical work on direct two-photon absorption in silicon. An important
benefit of phase-sensitive pump-probe reflectivity is its ability to measure
absolute nonlinearities at surfaces with a small spot size. This could be used
to characterize small samples such as exfoliated flakes and materials that are
spatially inhomogeneous.

The technique we demonstrated here has important advantages for characterizing
the nonlinear response. In the transmission geometry, going from the measured
phase shift to the change in refractive index requires knowledge of the
effective interaction length within the sample. In reflectivity all that is
needed to go from the measurement to the nonlinear change in complex
refractive index is the _linear_ complex refractive index, which can be easily
found through ellipsometry. Additionally, our analysis does not rely on any
assumptions about the 3D beam profile as it propagates through the sample,
only the irradiance profile at a single plane (the sample surface). In Z-scan,
the spatiotemporal irradiance profile can be difficult to characterize and
result in systematic error. Thus phase-sensitive pump-probe reflectivity,
particularly if its sensitivity can be improved, may offer advantages for
transparent samples as well.

Finally, these results on the nonlinear response of the silicon surface will
support future work on optical nonlinearity measurements of transparent films
on silicon substrates. Such films form the basis of many integrated optics
platforms and this technique, combined with precise knowledge of the nonlinear
response of the silicon surface, could be adapted to characterize the optical
nonlinearity of films.

Funding NIST Scientific & Technical Research Services (STRS)

Acknowledgments The authors thank Aaron Katzenmeyer for assistance with
ellipsometry measurements, Ethan Swagel for help implementing the 30-band
$\mathbf{k}\cdot\mathbf{p}$ model, and Valeria Viteri-Pflucker for early
experimental contributions.

Disclosures The authors declare no conflicts of interest.

Data availability Data underlying the results presented in this paper are not
publicly available at this time but may be obtained from the authors upon
reasonable request.

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