2021

[1]Matthew Yeung

[1]Philip D. Keathley

[1]Research Laboratory of Electronics, Massachusetts Institute of Technology,
77 Massachusetts Ave, Cambridge, 02139, MA, USA

2]Institute of Biophotonics, National Yang Ming Chiao Tung University, Linong
Street, Beitou District, Taipei City, 112304, Taiwan

3]IBM T.J. Watson Research Center, IBM Quantum, Yorktown Heights, 10598, NY,
USA

# Lightwave Electronic Harmonic Frequency Mixing

<EMAIL_ADDRESS>Lu-Ting Chou  Marco Turchetti  Shih-Hsuan Chia  Karl K.
Berggren<EMAIL_ADDRESS>*  [  [

###### Abstract

Frequency mixers are fundamental building blocks in many electronic systems.
They enable frequency conversion for signal detection and processing. While
conventional electronic frequency mixers operate in the GHz or at best in the
THz frequency range, a compact and scalable petahertz-scale electronic
frequency mixer would enable practical field-resolved optical signal
processing and readout without the need for nonlinear optical elements. We
used nanoscale antenna structures to experimentally demonstrate electronic
harmonic mixing beyond 0.350 PHz. Through this harmonic mixing process, we
demonstrate time-domain sampling of optical field waveforms spanning more than
one octave of bandwidth. The high nonlinearity of the devices enabled field-
resolved detection of spectral content outside of that contained within the
local oscillator, greatly extending the range of detectable frequencies
compared to conventional heterodyning techniques. We anticipate these devices
to provide compact, PHz-scale solid-state detection for integration with
emerging technologies such as compact frequency combs, optical waveform
synthesizers, and compact timing systems. Our work has important implications
for various fields such as chemistry, physics, material science, and biology,
where PHz signals exhibiting femtosecond-scale dynamics are of great interest
for spectroscopic analysis or imaging.

###### keywords:

integrated, lightwave electronics, optical, oscilloscope, ultrafast,
spectroscopy

##  1 Introduction

Field-resolved optical detection techniques are of paramount importance to the
advancement of ultrafast, light-based science and technology
krauszAttosecondMetrologyElectron2014 . By enabling the direct exploration of
light-matter interaction dynamics, field-resolved optical detection is being
increasingly used for the investigation of light-matter interactions in
biological systems pupezaFieldresolvedInfraredSpectroscopy2020 , charge
transfer dynamics in molecules and materials itoBuildupDephasingFloquet2023 ;
dienstbierTracingAttosecondElectron2023 ; jinUltrafastDynamicsVan2018 ;
reimannSubcycleObservationLightwavedriven2018 ;
schmidTunableNonintegerHighharmonic2021 ;
langerSymmetrycontrolledTemporalStructure2017 , near fields surrounding
metasurfaces mamaikinElectricfieldresolvedNearinfraredMicroscopy2022 , higher-
harmonic generation huangPolarizationControlIsolated2018 ;
wangRolesPhotocarrierDoping2017 ; schubertSubcycleControlTerahertz2014 ;
hohenleutnerRealtimeObservationInterfering2015 , and to inform the development
of novel light sources based on optical waveform synthesis
leskoSixoctaveOpticalFrequency2021 ;
steinleitnerSinglecycleInfraredWaveform2022 ;
wirthSynthesizedLightTransients2011 . Optical field sampling provides the most
direct field-resolved measurement of light-matter interaction dynamics in the
time-domain, but has traditionally been limited to the terahertz to mid-
infrared spectral regions (few to tens of terahertz) limiting its
applicability.

Significant efforts have been dedicated to extending optical field sampling
techniques to the petahertz-scale to provide time-domain optical field readout
without the need for spectral phase retrieval. Notable techniques include
electro-optic sampling (EOS)
ridenteElectroopticCharacterizationSynthesized2022 ;
ziminUltrabroadbandAllopticalSampling2022 ;
keiberElectroopticSamplingNearinfrared2016 , tunneling ionization with a
perturbation for the time-domain observation of an electric field (TIPTOE)
parkDirectSamplingLight2018 ; blochlSpatiotemporalSamplingNearpetahertz2022 ;
kimPetahertzOpticalOscilloscope2013 ; liuSingleshotMeasurementFewcycle2022 ,
and nonlinear photoconductive sampling in solids
sederbergAttosecondOptoelectronicField2020 ;
ziminPetahertzscaleNonlinearPhotoconductive2021 . Despite these efforts, such
optical field sampling techniques operating at PHz frequencies often
necessitate large pulse energies, undesirable for many materials borrego-
varillasAttosecondSpectroscopyInvestigation2022  and organisms
wangThreephotonNeuronalImaging2020 . Furthermore, these techniques typically
rely on few-cycle carrier-envelope phase-stable pulses, requiring bulky and
costly amplifiers operating at kHz repetition rates
herbstRecentAdvancesPetahertz2022 . As a result, the practicality of
petahertz-scale optical field sampling is severely limited. The ultimate
desire is for a simple, inexpensive, easily fabricated device that can sense
an optical field with sub-cycle time resolution: the optical equivalent of an
off-the-shelf electronic oscilloscope.

Recently, Bionta et al biontaOnchipSamplingOptical2021  experimentally
demonstrated an on-chip sub-optical-cycle optical-field-driven tunneling
device to measure degenerate (i.e. gate and signal having the same central
frequency) near-infrared fields down to 5 fJ using a Mach-Zender
interferometer (MZI) using carrier-envelope phase (CEP) stable pulses. While
degenerate sampling was demonstrated in Ref. biontaOnchipSamplingOptical2021 ,
non-degenerate (i.e. gate and signal having different frequency) upsampling
(i.e. higher frequencies are sampled using a lower frequency) or downsampling
(i.e. lower frequencies are sampled using a higher frequency) was only shown
in theory. Non-degenerate sampling would significantly broaden the available
detection bandwidth, potentially enabling detectors capable of sampling fields
spanning from the terahertz to the petahertz. This increased bandwidth would
allow for seamless amplitude- and phase-resolved characterization of nonlinear
processes of interest, such as solid-state higher harmonic generation
wangRolesPhotocarrierDoping2017 ; keathleyUncoveringExtremeNonlinear2023 ,
coherent Raman scattering marksNonlinearInterferometricVibrational2004 ;
kumarInvitedArticleComplex2018 , and multiphoton processes
alvesChallengesProgressProspects2022 ; wenFutureChallengesHybrid2019 ;
wangThreephotonNeuronalImaging2020 ; evansCoherentAntiStokesRaman2008 ,
without the need for nonlinear frequency conversion, spectral phase retrieval,
or a spectrally-overlapped local oscillator reference.

In this work, we explore non-degenerate time-domain sampling of optical field
waveforms using optical-field-driven tunneling from plasmonic nanoantennas. An
overview schematic is shown in Fig. 1. We show that the nanoantenna devices
can be conceptualized as petahertz-electronic harmonic frequency mixers as in
Fig. 1e and that the harmonic frequency mixing process enables sampling across
octaves of bandwidth without the requirement of absolute carrier-envelop phase
(CEP) stabilization or few-cycle pulse generation. As an experimental
demonstration, a non-CEP-stable, 10-cycle gate waveform having a central
frequency of 0.177 PHz (1690 nm) was used to sample its second harmonic
centered at 0.353 PHz (850 nm), further extending the known frequency limits
of electronic harmonic frequency mixing for optical metrology
evensonExtensionAbsoluteFrequency1972 . To demonstrate operation in the few-
cycle limit without CEP locking and to experimentally investigate the temporal
and spectral response of the mixing process and its dependence on the driving
waveform, we used a broadband, few-cycle supercontinuum as the signal while
driving the antennas with both a short (1.5-cycle) and long (10-cycle) gating
waveform. Accurate sampling was achieved in the few-cycle limit despite no CEP
stabilization of the driving laser source. In addition, we were able to
observe bandwidth reduction and map the entire fundamental pass band of the
sampling response under long-gate illumination when using the broadband
supercontinuum source as the signal.

Given their demonstrated ability to operate as petahertz-electronic harmonic
frequency mixers, we anticipate similar nanoantenna devices will be used as
basic building blocks for electromagnetic signal detection and processing at
optical frequencies.

Figure 1:  (a) A gate pulse illuminates the nanoantenna network and drives
sub-optical cycle electron emission. A small signal is introduced over a
variable delay. This small signal modulates the electron emission from the
nanoantennas leading to the optical-frequency mixing process. (b) A
representative scanning electron microscope image showing the nanoantennas.
(c) Finite-difference-time-domain (FDTD) simulation of the electric field
enhancement at the tip of a gold nanoantenna. (d) FDTD simulation of the field
enhancement and group delay imparted by the antenna response as a function of
frequency. Within the spectrum, we highlight the experimental frequencies used
with the gate at frequency 0.177 PHz and a higher frequency signal at 0.353
PHz, which corresponds to the second harmonic of the gate (SHG). Additionally,
the broadband supercontinuum (SCG) is utilized as a signal for studying the
bandwidth characteristics. (e) The devices can be conceptualized as electronic
harmonic frequency mixers (left schematic) with the gate serving as the local
oscillator (LO, with central frequency $f_{\text{LO}}$), and the signal as the
optical frequency input (OF, with central frequency $f_{\text{OF}}$). The
mixing process provides a current signal at baseband (intermediate frequency,
IF) for detection of harmonics of the local oscillator $kf_{\text{LO}}$ (right
plot). Here we measure the baseband response for field-resolved sampling of
the signal as a function of delay $\tau$.

###  1.1 Principle of Sampling via Harmonic Mixing

Figure 2:  Simulations of the electron emission and harmonic mixing process.
(a) Electron emission calculated using the Fowler-Nordheim tunneling rate
(solid orange curves) driven by a 4-cycle Gaussian pulse (black-dashed line
envelope, solid blue line field waveform) with a center frequency
($\omega_{0}$) of 0.177 PHz (1690 nm) with CEP at 0, $\pi/2$, $\pi$, and
$3\pi/2$. (b) CEP-dependent frequency response and the corresponding relative
phase using a 4-cycle Gaussian pulse with a center frequency of 0.177 PHz. (c)
CEP of the sampled field using a 10-cycle gate and 2-cycle signal that are
both 0.177 PHz where the CEP is linearly ramped. The blue, orange, and yellow
lines correspond to the CEP of the gate, signal, and sampled electric field
waveform, respectively. (d) The CEP of the sampled field when the 10-cycle
gate is 0.177 PHz and the 2-cycle signal is the second harmonic of the gate at
0.353 PHz. (e) The CEP of the sampled field when the 10-cycle gate is 0.177
PHz and the 2-cycle signal is the third harmonic of the gate at 0.528 PHz. (f)
The CEP of the sampled field when the 10-cycle gate is 0.177 PHz and the
2-cycle signal is the fourth harmonic of the gate pulse at 0.704 PHz.

Our approach shares similarities with the TIPTOE method
parkDirectSamplingLight2018 ; choTemporalCharacterizationFemtosecond2019 ,
with the distinction that we used Fowler-Nordheim (FN) electron emission from
the nanoantenna surface as our nonlinear medium. This provides several key
advantages. First, the field enhancement and reduced work function of the
antenna emitter (here taken to be gold) increase the field sensitivity by
orders of magnitude relative to a gas-based medium, allowing for the use of
relatively low-energy optical driving and sampling pulses. Second, the
geometry of the antenna can be changed to enable both full and half-wave
rectification for control over the electronic frequency response of the
devices ritzkowskyEngineeringFrequencyResponse2022 . Third, the sub-wavelength
nature of the devices removes the need for phase-matching and potentially
provides spatial resolution.

The sub-optical-cycle tunneling current is driven from the nanoantenna to a
collector as seen in Fig. 2a. Note that for this work we always assume an
asymmetric, diode-like configuration with a single antenna facing a flat
collecting wire which provides a half-wave rectified response. A small
additional signal (to be sampled), is superimposed and varied in time relative
to the gate pulse. The time-averaged charge transferred across the nanoantenna
gap is measured as a function of this delay. The measured current is

$I(\tau)=q\int_{-T_{rep}/2}^{T_{rep}/2}\Gamma[E_{\mathrm{gate}}(t-\tau)+E_{\mathrm{signal}}(t)]\,dt$
(1)

where $\Gamma$ is the Fowler-Nordheim equation. Provided that
$E_{\mathrm{signal}}$ is sufficiently small, such that the electric field
perturbation seen by the nanoantenna can be assumed to be linear and
$E_{\mathrm{gate}}$ can be Taylor-expanded to the first order. This resulting
expression is given by

$I(\tau)\approx
q\int_{-T_{rep}/2}^{T_{rep}/2}\Gamma[E_{\mathrm{gate}}(t-\tau)]+[\frac{d\Gamma}{dE}|_{E_{\mathrm{gate}}(t-\tau)}\cdot
E_{\mathrm{signal}}(t)]\,dt$ (2)

In this equation, the integral of the second term corresponds to the sampled
electric field waveform. This integral represents the small-signal cross-
correlation between $\frac{d\Gamma}{dE}|_{E_{\mathrm{gate}}(t-\tau)}$ and
$E_{\mathrm{signal}}(t)$ which we denote as $I_{cc}(\tau)$. Since this
expression represents a cross-correlation, in general, the Fourier-transformed
expression can be written as
$\tilde{I}_{cc}(\omega)\approx\mathcal{F}[\frac{d\Gamma}{dE}|_{E_{\mathrm{gate}}}]^{*}\cdot
E_{\mathrm{signal}}(\omega)$ where
$\tilde{H}(\omega)=\mathcal{F}[\frac{d\Gamma}{dE}|_{E_{\mathrm{gate}}}]^{*}$
is the full complex frequency response of the detector, which is plotted in
Fig. 2b.

Due to the high nonlinearity of the FN tunneling rate $\Gamma$ and rectified
current response, $\tilde{H}(\omega)$ contains integer harmonics of
$E_{\mathrm{gate}}$. As shown in Fig. 2b, integer harmonic frequency bands
appear in the sampling response outside of the gate spectrum centered around
$\omega_{0}$. The appearance of these harmonics is a result of mixing between
higher-order components of the electronic frequency response with higher-order
signal fields. Thus, these devices can be conceptualized as electronic
harmonic optical-frequency mixers (see left panel of Fig. 1b). Due to the
frequency mixing process, higher-order frequency components of the signal are
shifted to the baseband (see right panel of Fig. 1b), which enables highly
broadband, phase-resolved readout of optical waveforms through standard
electronic detection, and without the need for separate wave mixing using
nonlinear optical elements which introduce phase-matching constraints and
require larger driving pulse energies
ziminUltrabroadbandAllopticalSampling2022 . We emphasize that while one can
certainly obtain a mixing response via conventional heterodyning and
homodyning using an $E^{2}$ detector (e.g. a photodiode), $E^{2}$ nonlinearity
without rectification is insufficient to measure a pulse’s own phase function
(i.e. only spectral intensity and relative phase differences are provided) or
any higher-order frequencies.

Conceptualizing the devices as electronic optical frequency mixers also aids
in describing important properties of their response. In particular, it allows
a transparent explanation of why absolute CEP locking of the gate and signal
pulse is not a requirement for time-domain sampling, even for the case of
signals comprised of higher-order harmonics, provided that they originate from
the same laser source. To illustrate why, consider the case of perfectly
sinusoidal gate and signal functions, where the signal is a harmonic of the
gate. In this case, the gate becomes analogous to the local oscillator input
of the mixer at central frequency $f_{\text{LO}}$, and the harmonic signal of
the optical frequency input at central frequency
$f_{\text{OF}}=kf_{\text{LO}}$ as represented in Fig. 1e. We can then
represent $\frac{d\Gamma}{dE}|_{E_{\mathrm{gate}}(t-\tau)}$ as

$\frac{d\Gamma}{dE}|_{E_{\mathrm{gate}}(t-\tau)}=h_{0}+\frac{1}{2}\bigg{(}\sum_{n=1}^{\infty}\tilde{h}_{n}e^{in\varphi}e^{i2\pi
nf_{\text{LO}}(t-\tau)}+\text{c.c.}\bigg{)}$ (3)

where $\varphi$ represents the absolute phase shift of the gate (LO),
analogous to the CEP for the case of a pulsed gate. Likewise, the signal is
represented by

$E_{\text{signal}}(t)=\frac{1}{2}\tilde{a}_{k}e^{ik\varphi+\Delta\varphi}e^{i2\pi
kf_{\text{LO}}t}+\text{c.c.}$ (4)

where $\Delta\varphi$ represents any remaining phase difference in addition to
$k\varphi$. The DC output response is then formed by the multiplication of
conjugate and non-conjugate coefficients of $\tilde{h}_{k}$ and
$\tilde{a}_{k}$ and is found to be

$I_{\text{cc}}(\tau)=\frac{1}{4}\tilde{h}_{k}^{*}\tilde{a}_{k}e^{i2\pi
kf_{\text{LO}}\tau}e^{i\Delta\varphi}+\text{c.c.}$ (5)

Note that since the signal and current responses are both phase-locked to the
local oscillator (gate), the absolute phase terms $\varphi$ always cancel and
the harmonic mixing response is not sensitive to any fluctuations of the
absolute phase $\varphi$.

These behaviors translate directly to the case of pulsed gate and signal
inputs. A finite envelope of the gate pulse leads to a broadening of the
harmonic pass bands. When shifting the carrier-envelope phase of the gate, the
phase of each passband in $\tilde{H}(\omega)$ increases as integer steps with
the harmonic number as shown in Fig. 2b. These shifts are equal and opposite
to those of optically-generated harmonic pulses meaning that so long as the
source of harmonics and the nonlinear optical radiation originate from the
same driving laser the sampled response does not depend on the absolute
carrier-envelope phase.

To further illustrate this absolute phase cancellation for the case of a
pulsed gate and signal, we performed time-domain current cross-correlation
(using Eqn. 1) simulations using a 10-cycle Gaussian pulse at 0.177 PHz as the
gate and a 2-cycle signal at frequencies of various integer harmonic orders
(details can be found in the Methods section). To investigate the effect of
CEP on the sampled electric field waveform, we simultaneously increased the
CEP of both the gate and signal pulses linearly from 0 to 2$\pi$ in individual
simulations. We tracked the resulting sampling response for each simulation
and found that, regardless of the CEP of the gate and signal pulses, the
sampled electric field always had a CEP of 0, as shown in Fig. 2c. Next, we
kept the same gate conditions (10-cycle Gaussian at 0.177 PHz) and changed the
signal pulse to a two-cycle SHG, THG, and 4HG pulse, with the signal CEP being
2, 3, and 4 times the gate CEP, respectively. This resulted in a sampled
electric field with a CEP of 0, as illustrated in Fig. 2d, e, f. These results
can also be seen in the phase in the frequency domain in Fig. 2b, where the
phase is constant.

##  2 Results and discussion

In this section, we present the results of the three types of measurements
performed. First, we describe the degenerate optical field sampling
experiments using 10-cycle near-infrared gate and signal waveforms both having
a central frequency of 0.177 PHz as shown in Fig. 3 to illustrate the
capability of multicycle sampling without the need for CEP stabilization. We
next discuss the demonstration of non-degenerate optical-field upsampling to
demonstrate the capability of harmonic frequency mixing. We did this by using
a 10-cycle 0.177 PHz gate to sample a signal pulse of a frequency of 0.353 PHz
that was generated through the second harmonic generation of the gate
waveform. Finally, we demonstrate the sampling performance without CEP
stabilization in the few-cycle limit and compare the frequency properties of
the sampling response when gated using both a short, 1.5-cycle gate, and a
long, 10-cycle gate. To do this we used a broadband, self-compressed
supercontinuum (SC) source with continuous spectral coverage from 0.13 to 0.35
PHz as the signal.

Figure 3:  Degenerate optical field sampling of a 10-cycle 0.177 PHz pulse.
(a) The sampled electric field and (b) the corresponding frequency-domain
intensity of the sampled electric field compared to a commercial spectrometer
and the simulated frequency response. The extracted group delay from the
sampled optical field is compared to the group delay retrieved from FROG.

###  2.1 Degenerate Sampling

To perform degenerate optical field sampling with a 10-cycle (57 fs) 0.177 PHz
(1690 nm) pulse (degenerate sampling of a 63-cycle pulse is also shown in SI
Fig. 13), a network of triangular nanoantennas on fused silica substrates was
illuminated with a commercial optical parametric amplifier, with varying gap
sizes of 20-50 nm between the triangle apex and collector wire as depicted in
Fig. 1b, c. The sampling was carried out using a gate and signal pulse pair
generated with an MZI, with the pulse duration determined through frequency-
resolved optical gating (FROG) measurements (See SI Fig. 11 for the
measurement schematic, Fig. 12 for FROG).

We first confirmed that the devices were operated in the optical-field
emission regime through the analysis of current scaling with intensity by
measuring the output current as a function of the incident gate pulse energy
(see SI Fig. 10). It is worth noting that we observed a gradual decline in
performance over time in most of our devices, resulting in the need to
increase the input pulse energy to achieve the same current as the device was
used over longer periods of time. Similar effects have been noted and studied
in past work and were attributed to small structural changes in the antennas
under illumination resulting in reduced intensity enhancement
yangLightPhaseDetection2020 .

Having confirmed operation in the optical-field emission regime, we then
illuminated the antennas with the signal and gate pulse energies at 4.4 pJ (24
V/$\mu$m) and 5.4 nJ (0.85 V/nm), respectively. Note that the small-signal
gain from the high nonlinearity of the optical-field emission response enables
the sampling of signal pulse energies on the order of 1000$\times$ smaller
than those of the gate. The sampled electric field is shown in Fig. 3a. The
measured optical period was 5.6 fs which matched the expected value for a
frequency of 0.177 PHz. The pulse of the sampled field was 57 fs full-width at
half maximum (FWHM), in close agreement with the 58 fs FWHM pulse duration
retrieved FROG measurement (see SI Fig. 12c). The accuracy of the sampled
field was further verified through spectral intensity and group delay
analysis. In Fig. 3b the squared modulus of the Fourier transformation of the
sampled pulse (solid blue line) is shown to agree well with the spectrum as
measured using a commercial grating-based indium gallium arsenide spectrometer
(dashed blue line). Also, the extracted group delay of the sampled pulse
(solid orange line) is shown to be concave up and agrees well with the group
delay retrieved from the FROG measurement (dashed orange line).

We note that in these comparisons, we have ignored the impact of the antenna
response. To justify this, we simulated the electromagnetic response of the
nanoantenna based on the experimentally fabricated geometry. The field
enhancement and phase are plotted as a function of frequency in Fig. 1c, d.
The results of the simulation show a nearly constant amplitude and group delay
from 0.168 PHz to 0.191 PHz (1570 nm to 1784 nm), corresponding to the
spectral extent of the signal pulse. The simulated field enhancement factor
ranges from 17 to 22, and the group delay from -4.6$\times 10^{-3}$ to
-7.4$\times 10^{-3}$ fs within this range, leading to negligible reshaping of
the pulse.

###  2.2 Non-degenerate Sampling

Figure 4:  (a) Non-degenerate optical field sampling of a
$\omega_{\mathrm{signal}}$ = 0.353 PHz signal using a lower-frequency gate,
$\omega_{\mathrm{gate}}$ = 0.177 PHz. (b) The corresponding frequency-domain
intensity of the sampled electric field compared to a commercial spectrometer
and the simulated frequency response (at 0.353 PHz, the relative intensity is
2.3x lower than 0.177 PHz). The extracted group delay from the sampled optical
field (orange solid curve) is compared to the group delay retrieved from FROG
(dashed orange curve).

In the previous section, we discussed degenerate sampling where the frequency
of the gate matched that of the signal. Here, we demonstrate non-degenerate
up-sampling of the second harmonic signal via harmonic mixing using the
fundamental ($\omega$ = 0.177 PHz) as the gate.

We performed the non-degenerate up-sampling by adding lenses and a type 1
beta-barium borate ($\beta$-BaB2O4, BBO) crystal for generating the SHG (0.353
PHz) signal (see SI Fig. 14 for measurement schematic). We illuminated the
antennas on the device using 23 pJ (49 V/$\mu$m) for the SHG signal (after
filtering the fundamental) and 4.9 nJ (0.80 V/nm) as the gate. Shown in Fig.
4a is the sampled SHG using the 10-cycle 0.177 PHz as a gate. The measured
optical period was 2.8 fs matching the expected optical period for a center
frequency of 0.353 PHz.

To verify the accuracy of the sampled field, we performed FROG and use a
commercial grating-based silicon spectrometer as a reference in the same way
as for the case of degenerate sampling. In Fig. 4b we compare the
corresponding squared modulus of the Fourier-transformed sampled fields (solid
blue line) to the spectral intensity measured using the spectrometer (dashed
blue line). We can see in the lower-frequency portion of the frequency domain
spectrum that no residual fundamental (0.177 PHz) was observed in the
measurement. The pulse duration from our FROG measurement was found to be 49
fs FWHM which matches almost perfectly with the sampled pulse duration of 48
fs FWHM (see supplementary information Fig. 15 for a comparison of FROG and
sampled fields). The group delay of the sampled trace (solid orange curve) is
concave down and matches relatively well with the FROG-retrieved group delay
(dashed orange curve).

As before, we ignored the amplitude and phase response of the nanoantenna.
Again looking at the spectral field enhancement and group delay of the antenna
response as shown in Fig. 1d, we see that over the spectral range of the
signal from 0.333 PHz and 0.370 PHz (810 nm and 900 nm), the field enhancement
amplitude and group delay were relatively constant, ranging from 8 to 12 and
from -8.4 $\times 10^{-3}$ fs radians to -2.6 $\times 10^{-2}$ fs,
respectively. As for the degenerate case, this leads to negligible pulse
reshaping in the time domain.

There are several aspects of this measurement that are important to emphasize.
First, given that the second harmonic generation used to generate the signal
was phase-locked to the second harmonic of the sampling response as noted in
Section 1, there was no need for CEP stabilization of the gate. Second, the
nonlinearity of the mixing process enabled the phase-resolved, interferometric
pulse readout without the need for optical generation of a spectrally-
overlapped local oscillator as would typically be required for all-optical
homodyning or heterodyning techniques
ziminUltrabroadbandAllopticalSampling2022 . Third is that the direct cross-
correlation output was an accurate representation of the signal pulse as
retrieved using FROG without the need for any post-analysis, such as phase
retrieval, despite the fact that the signal pulse was appreciably shorter than
the gate (48 fs FWHM signal duration versus 60 fs FWHM gate pulse duration),
and of a significantly higher carrier frequency.

###  2.3 Supercontinuum Sampling with Long and Short Gates

Figure 5:  (a) Sampling measurements of the 1.5-cycle supercontinuum using the
short-pulse gate (blue curve) and long-pulse gate (orange curve) along with a
simulation of $I_{\text{cc}}(\tau)$ using the experimentally sampled 1.5-cycle
signal and 10-cycle gate as inputs (yellow curve). (b) Comparison between the
short-gate-sampled supercontinuum and the FROG retrieved pulse envelope. (c)
Normalized intensity spectra of the supercontinuum when measured using the
short-pulse gate (solid blue curve); through a commercial spectrometer (solid
green curve); a full simulation of the sampling spectrum (solid purple curve)
incorporating the simulated supercontinuum spectrum, the calculated short-gate
sampling response ${|\tilde{H}_{\text{SG}}(\omega)|}^{2}$ (dashed blue curve),
and the intensity enhancement from the nanoantenna (solid light blue curve);
the measured long-gate-sampled spectrum (solid orange curve); the calculation
of the long-gate sampling response ${|\tilde{H}_{\text{LG}}(\omega)|}^{2}$
(dashed orange curve); and the simulated long-gate-sampled cross-correlation
response (solid yellow curve).

To demonstrate that CEP-stabilization is not required even in the few-cycle
limit, we measured the sampling response using a non-CEP-stable 1.5-cycle
pulse as the gate and signal. To further explore the spectral and temporal
properties of the sampling response (and thus the electronic emission
response), and investigate how these properties are influenced by the choice
of gate pulse, we then measured the sampling response using the 10-cycle pulse
as the gate while keeping the 1.5-cycle pulse as the signal.

To start we used a compressed, 1.5-cycle supercontinuum source with a central
frequency near 0.17 PHz having broad and continuous spectral coverage from
below 0.13 to beyond 0.25 PHz as the gate and signal pulse. To generate the
1.5-cycle pulse, we used a commercially available large-mode-area photonic
crystal fiber to generate a nearly transform-limited duration supercontinuum
pulse at the fiber output based on soliton self-compression. The self-
compression process significantly influences the direct output duration, which
was predicted to be 6 fs based on generalized nonlinear Schrödinger (NLSE)
simulation results. Additional details regarding the simulation and
supercontinuum generation (SCG) process can be found in the methods section
for further reference. To validate these findings, we conducted FROG
measurements, which yielded a retrieved pulse duration of 9 fs (Fig. 5b and SI
Fig. 18 for further details). Furthermore, using intensity autocorrelation
(IAC) measurements, we observed an autocorrelation trace width of 13 fs FWHM.
Accounting for a deconvolution factor of 1.54 and assuming a hyperbolic secant
(sech) pulse shape, this corresponds to an estimated pulse duration of 8.4 fs
FWHM (refer to SI Fig. 19).

The short-gate-sampled trace was measured using the supercontinuum pulse as
both gate and signal and is shown in Fig. 5a, with its spectrum shown in Fig.
5c (blue curves). A gate pulse energy of 0.6 nJ (1.1 V/nm) was used for the
gate pulse and a pulse energy of 80 fJ (37 V/$\mu$m) was used for the signal
(see SI Fig. 17 for measurement schematic). The temporal shape of the short-
gate-sampled trace is in good agreement with the retrieved FROG trace as shown
in Fig. 5b (for more details see SI Fig. 18), with a central pulse duration of
8.5 fs FWHM (compare to 9 fs and 8.4 fs for the FROG and IAC results,
respectively). Moreover, the measurement of the short-gate sampled trace using
a non-CEP-stable laser highlights the mixing effect discussed earlier in
Section 1. Since the CEP undergoes stochastic drift, one would anticipate
substantial variations in the emitted current and resultant sampling response
from pulse-to-pulse. However, these variations are averaged away as each
sampling trace corresponds to integration over 1.8 $\times 10^{7}$ pulses, and
the signal waveform is reproduced containing only the relative CEP difference
as discussed in Section 1. Our results demonstrate that the measurement is
insensitive to shifts in the absolute CEP of the driving waveform in the few-
cycle limit so long as the gate and signal exhibit relative CEP stability.

In Fig. 5b, we see that the spectrum of the short-gate-sampled trace spans
from just below 0.16 PHz to roughly 0.24 PHz. Near the central spectral peak,
structure is observed with several peaks and more prominent dips at 0.167,
0.175, and 0.180 PHz. To understand the response, we compare the spectrum to
the supercontinuum spectrum measured with a spectrometer (green curve), as
well as a full simulation of the spectral response (purple curve) that
incorporates our NLSE simulation result $\tilde{E}_{\text{NLSE}}(\omega)$, the
sampling response $\tilde{H}_{\text{SG}}(\omega)$ (dashed blue curve) and the
calculated antenna response
$\tilde{H}_{\text{pl}}(\omega)=\tilde{E}_{\text{tip}}(\omega)/\tilde{E}_{\text{inc}}(\omega)$,
where $\tilde{E}_{\text{tip}}$ is the excited field at the tip and
$\tilde{E}_{\text{inc}}$ the field incident on the antenna. The intensity
enhancement at the antenna tip ${|\tilde{H}_{\text{pl}}(\omega)|}^{2}$ is
shown as solid light blue curve for reference. We note that the NLSE
simulation describes the details of the measured supercontinuum spectrum very
accurately, with the only major difference being that the spectral structure
near the center of the pulse is more pronounced as in our sampling result (for
direct comparison, see SI Fig. 21).

The spectral response of the short-gate-sampled trace generally agrees well
with the full simulation. While the spectral dips near 0.17 PHz are more
pronounced in the sampled trace spectrum, their frequency locations align well
with the NLSE simulation result. Here $\tilde{H}_{\text{SG}}(\omega)$ was
calculated assuming a perfect sech pulse with a peak field of 28 V/nm,
corresponding to a field enhancement of 30. We recognize that for the measured
short-gate-sampled trace, the spectral cutoff at lower frequencies is a bit
sharper than expected. We attribute this discrepancy to two possible causes:
(1) our limited scan range resulting in an underestimation of long-wavelength
spectral components; and (2) weaker focusing of the long-wavelength components
onto the nanoantenna detectors.

To study the impact of the gate pulse on the sampling response, we next
compare the case of the few-cycle gate to that using the 10-cycle pulse with a
central frequency of 0.177 PHz as the gate while leaving the signal pulse
fixed. As touched on in Section 2, the bandwidth of the integer harmonic peaks
in $\tilde{H}(\omega)$ can be adjusted by varying the cycle count of the gate
pulse. When the cycle count decreases (i.e. the pulse duration is shorter),
the frequency bandwidth of the pulse expands, leading to a wider bandwidth of
the integer harmonic peaks in $\tilde{H}(\omega)$. Conversely, increasing the
cycle count narrows the bandwidth of the integer harmonic peaks. The field
enhancement, nonlinearity of the emission response, and gate pulse waveform
all contribute to impacting the precise shape of $\tilde{H}(\omega)$. By
maintaining the same broadband signal while altering the gate pulse waveforms,
we can examine precisely how $\tilde{H}(\omega)$ impacts the resultant time
and frequency content of the underlying electronic response, and how well our
emission model captures this electronic response.

The long-gate-sampled trace is shown in Fig. 5a with its spectrum in Fig. 5c
(solid orange curves). For the measurements shown, we used a 6.6 nJ (0.94
V/nm) gate and a 2 pJ (52 V/$\mu$m) signal pulse. Note that the peak field
strengths of the gate and signal are comparable to those of the short-gate-
sampled case when accounting for the increased pulse duration.

Due to the longer gate pulse, a longer train of sub-cycle current bursts were
generated causing the long-gate-sampled trace to have an increased pulse
duration. Along with the increased sampling response duration, we also see the
expected bandwidth reduction in the spectral response shown in Fig. 5c. While
the central frequencies are faithfully retrieved, with a similar structure to
the short-gate-sampled result, NLSE, and measured spectrum, there is a sharp
dropoff moving towards higher and lower frequencies. We observe dips at 0.156,
0.186, and 0.190 PHz in the spectrum which is due to a combination of the gate
pulse shape and localized fields at the tip. The dashed orange curve shows the
sampling response ${|\tilde{H}_{\text{LG}}(\omega)|}^{2}$ of the long gate
with a peak field of 30 V/nm and field enhancement of 32. Through the
simulation, we estimate the window of emitted current from the 10-cycle gate
pulse to be roughly 32 fs under the conditions in the measurement, still
significantly shorter than the original pulse duration of roughly 60 fs. We
emphasize that operating at lower peak field values or with materials having
higher work function could further shorten this duration and increase the
bandwidth response due to larger nonlinearities, at the cost of lower current
yields and thus lower signal-to-noise ratios.

Lastly, to test the validity of our measurements and accuracy of our sampling
response simulations, we took our experimentally sampled 10-cycle gate pulse
from Fig. 3a and the short-gate-sampled supercontinuum from Fig. 5a and used
these to calculate the current cross-correlation $I_{\text{cc}}(\tau)$ using
Eqn. 1. We accounted for a peak gate field strength of 30 V/nm for the
10-cycle gate pulse and a signal to gate peak field ratio of 0.03. The
simulated $I_{\text{cc}}(\tau)$ is shown in Fig. 5a, and its corresponding
spectral intensity is shown in Fig. 5c (solid yellow curves). While certain
details of the temporal tails of the calculated result are different in
comparison to the long-gate-sampled measurement, the central lobe is
faithfully reproduced. The calculated response has a central lobe centered
around t = 0 fs of duration 20 fs FWHM, whereas the measured long-gate-sampled
supercontinuum has a central lobe duration of 18 fs FWHM. The calculated
spectral response also reproduces the bandwidth limits and key features
observed in the measurement, indicating that our simplified Fowler-Nordheim
model provides a reasonable estimate of the underlying current response.

##  3 Concluding Remarks and Outlook

In this study, we used nanoantenna networks to demonstrate a broadband, on-
chip electronic optical frequency harmonic mixer using optical-field-driven
tunneling. We showed how the harmonic frequency mixing process enables
accurate optical field sampling of waveforms having central frequencies
greater than that of the gate waveform using a commercial laser without the
need for a few-cycle laser source or carrier-envelope phase locking. In
comparison to wave mixing in crystals, the optical-field-driven tunneling
mechanism provides access to higher-order nonlinearities, and thus larger
mixing bandwidths, while eliminating the need for phase-matching or a separate
photodetection element. Similar devices could be used to create compact and
sensitive sampling optical oscilloscopes with bandwidths spanning multiple
octaves. We anticipate that such optical field oscilloscopes will provide
needed time-domain detection tools that will help accelerate the development
of ultrafast source technologies (e.g. compact frequency combs and optical
waveform synthesizers), and enable new approaches to the investigation of
nonlinear light-matter interactions. Beyond field sampling, electronic
optical-frequency harmonic mixers could also be incorporated as fundamental
components within future lightwave electronic systems for petahertz-scale
communication and computation.

##  4 Methods

###  4.1 Nanofabrication

We started with 1 cm $\times$ 1 cm fused silica pieces (MTI Corp.) and cleaned
them using piranha for 10 minutes prior to use. For the nanoantenna array
fabrication, we spin-coated polymethyl methacrylate A2 (Microchem) at 2,750
revolutions per minute and baked at
$180\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for 2 minutes. Afterward,
DisCharge H2O X2 (DisChem Inc.) was spun at 3,000 revolutions per minute so
that charging did not occur during the electron beam lithography write. The
electron beam lithography was performed at 125 keV with a dose ranging from
4000-6000 $\mu$C/cm2 with proximity effect correction. Development of the
exposed polymethyl methacrylate samples was done at
$0\text{\,}\mathrm{\SIUnitSymbolCelsius}$ in a solution of 3:1 2-propanol to
methyl isobutyl ketone for 50 seconds. Electron beam evaporation was performed
at 2$\times$$10^{-6}$ Torr where we first deposited a 2 nm titanium adhesion
layer, then 20 nm of gold. Lift-off was performed by submerging the samples in
a $65\text{\,}\mathrm{\SIUnitSymbolCelsius}$ solution of N-methyl pyrrolidone
(Microchem) for one hour.

Contacts were made to the nanoantenna using photolithography. We spin coat
nLOF 2035 at 3,000 revolutions per minute, then bake the resist at
$110\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for 90 seconds. The exposure was
performed using a maskless aligner with a wavelength of 375 nm and at a dose
of 300 mJ/cm2. After exposure, we do a post-exposure bake at
$110\text{\,}\mathrm{\SIUnitSymbolCelsius}$ for 90 seconds, then develop for
90 seconds in AZ726. We then use electron beam lithography to deposit 10 nm of
a titanium adhesion layer and 50 nm of gold. Liftoff was performed at room
temperature in a solution of acetone for at least 6 hours. The samples were
ashed for 30 seconds, then mounted on a printed circuit board (PCB), and wire
bonded.

###  4.2 Laser and measurement Methods

We used a LightConversion optical parametric amplifier pumped by a Yb:KGW
laser pulse picked to 1 MHz (Cronus 3P) for our experiments. The idler output
was compressed using a prism pair. The samples on the PCB were used as-is in
ambient conditions and the output was connected to a trans-impedance amplifier
with a gain of 1 V/nA (FEMTO). We measured the signal pulse-induced current
through a lock-in measurement referenced by chopping the signal arm at $\sim$
277 Hz. The x- and y-channels of the lock-in are output to an oscilloscope
(Keysight) with a sampling rate of at least 25-50 kSa/s. Prior to sampling
measurements, we illuminated the device with the gate pulse and ensured that
the photocurrent remained constant for at least several minutes. To temporally
control the delay between the signal and gate pulse, we placed the gate pulse
on a closed-loop piezo stage (Piezosystem Jena) with $\pm$ 14 nm (0.05 fs)
repeatability.

In every measurement, we used neutral density (ND) filters to control the
power of the signal and gate pulses. After recombining the two pulses, we
placed a linear film polarizer with an extinction ratio of $\geq 10^{5}$
(across all wavelengths) to ensure the two beams were horizontally polarized.
Once collinearity was ensured between the gate and signal arm, both the signal
and gate pulses were focused onto the nanoantenna chip through a reflective
objective (Ealing). See SI Fig. 11 for the schematic.

For frequency doubling, we used a type 1, 1.5 mm thick beta-barium borate
($\beta$-BaB2O4, BBO) crystal with $\theta$=24 $\phi$=90 to generate the
second harmonic (SHG) of $\omega_{gate}$ = 0.177 PHz. The SHG (0.353 PHz) is
filtered using a 0.207 PHz (1450 nm) high-frequency pass filter with ND 2 at
0.177 PHz along with a broadband achromatic half waveplate used to rotate the
SHG polarization from vertical to horizontal, to match the gate pulse. Note
that the integration time was crucial for accurate measurement of the higher
frequencies.

For the non-degenerate supercontinuum measurement using a 10-cycle
$\omega_{gate}$ = 0.177 PHz, we ensured that the phase induced by the optics
was equivalent to when the degenerately sampled supercontinuum was measured.
We also utilize the same device for degenerate and non-degenerate sampling.

For frequency-resolved optical gating (FROG) measurements, we used the RANA
approach (see Ref. jafari100ReliableAlgorithm2019 ) as a robust retrieval
algorithm. The MATLAB code is available on the Trebino Group website (see
trebinoMATLABCodeAvailable ).

We padded the retrieved waveforms with zeros before taking an FFT to improve
spectral resolution. This was justified as the spacing between consecutive
pulses in time in the experiment is much larger (one microsecond) than the
time window of the retrieved waveforms.

###  4.3 Electromagnetic, Sampling, and Supercontinuum Generation Simulation

For the field enhancement simulations, we used open-source finite-difference
time-domain package (FDTD) PyMeep oskooiMeepFlexibleFreesoftware2010 . For the
gold nanoantenna and silicon oxide substrate, we used the standard materials
library included in the Python package. For setting up the FDTD conditions, we
used periodic boundary conditions in the nanoantenna on a silicon oxide plane
with perfectly matched layers in the direction of propagation of the plane-
wave source to prevent multi-reflections affecting the simulation. To obtain
realistic field enhancements, we ensured that the apex of the triangle had a
radius of curvature of 10 nm, which we extracted through scanning electron
microscopy of our devices.

In our sampling simulations, the Fowler-Nordheim equation, expressed as
$\Gamma=\alpha\phi^{-1}E^{2}\exp(-\beta\phi^{3/2}/E)$, was utilized. Here,
$\alpha$ is 1.54$\times 10^{-6}$ A eV V-2, $\beta$ is 6.83 eV-3/2V nm-1,
$\phi$ represents the work function (taken as 5.1 eV), and E denotes the
electric field. With this equation, we numerically calculated the current
cross-correlation by

$I_{CC}(\tau)\propto\int\Gamma(E_{\mathrm{gate}}(t-\tau)+E_{\mathrm{signal}}(t))\,dt$

using a ratio of signal to the gate of 0.03.

For the simulated frequency responses $\tilde{H}(\omega)$ shown in Fig. 5c, we
used a sech centered at 0.17 PHz with a pulse duration of 8.5 fs with the same
center frequency as the measured SCG and with a field enhancement of
30$\times$ for the short-gate-sampled SCG. For the long-gate-sampled SCG we
used a 0.17 PHz Gaussian with a pulse duration of 60 fs and a field
enhancement of 32$\times$.

We simulated the spectral broadening using a 1690-nm and 60-fs Gaussian pulse
to evaluate the coupled pulse energy and fiber length for optimal spectral
broadening and temporal compression based on a 12.2 $\mu$m endlessly single
mode, large mode area photonic crystal fiber (NKT Photonics). The simulation
solved a generalized nonlinear Schrödinger (NLSE) equation based on the split-
step method dudleySupercontinuumGenerationOptical2010 , which considered the
fiber dispersion, self-phase modulation, self-steepening, and the Raman
response. We then experimentally controlled the fiber length to be 4 mm,
coupling 100-nJ of pulse energy, and the corresponding measured spectrum was
compared to the simulation using the same parameter.

Data and selected code used can be found in the following repositories: Data
and selected code: https://github.com/qnngroup/manu-HarmonicMixer.git FDTD of
the nanoantenna, example simulation code is provided,
https://github.com/qnngroup/pymeep-nanoantenna-simulator

## Supplementary information

###  4.4 Limits of sampling a short signal pulse with a long gate pulse

Figure 6: Simulated sampling of a single cycle $\omega_{\mathrm{signal}}$ =
0.353 PHz and varied cycle $\omega_{\mathrm{gate}}$ = 0.177 PHz. (a) The
sampled time-domain electric fields with the single cycle
$\omega_{\mathrm{signal}}$ as a reference. (b) The corresponding gate-
dependent frequency-domain response for varied cycle count (dashed lines) and
the sampled response (solid lines).

Illuminating the nanoantennas with a gate pulse generates sub-optical-cycle
current bursts, creating a comb-like frequency response with peaks centered at
higher harmonics of the fundamental, with the bandwidth of these harmonics
being dependent on the cycle count of the gate pulse. The device’s limitations
are determined by the gate frequency and pulse duration, with the latter being
directly related to time resolution due to the electron burst’s dependence on
the optical period. To illustrate this, we performed the following simulations
using a Gaussian 1-, 5-, and 10-cycle $\omega_{\mathrm{gate}}$ = 0.177 PHz and
a single cycle $\omega_{\mathrm{signal}}$ = 0.353 PHz. Shown in Fig. 6a from
top to bottom is a 1-cycle signal (2.9 fs full-width at half maximum, FWHM of
the absolute modulus squared of the electric field) as a reference and the
corresponding sampled electric field using a 1-, 5- and 10-cycle gate. When
sampling is performed, the CEP of the signal does not change due to the
harmonic mixing process as described in the main text.

When we use a 1-cycle gate to measure a 1-cycle frequency-doubled signal, the
sampled pulse duration is 2.9 fs, which is identical to the pulse duration of
the signal. The corresponding Fourier-transformed amplitude is also shown in
Fig. 6b and has a frequency bandwidth of $\Delta$f=0.125 PHz. As the cycle
count of the gate pulse is increased to 5 cycles, it is seen that some of the
frequencies are distorted in the time domain, resulting in a non-symmetric
electric field waveform as shown in Fig. 6a, however, the sampled pulse
duration is still 2.9 fs (see SI Fig. 7). From the corresponding Fourier-
transformed amplitude shown in Fig. 6b, we observe the expected frequency
response from a 5-cycle gate, which has a bandwidth of 0.08 PHz, and the
sampled spectrum is attenuated at the valleys centered at 0.263 PHz and 0.441
PHz. Finally, when the gate consists of 10 cycles, the non-symmetric electric
field becomes even more distorted, resulting in a sampled pulse duration of
13.4 fs. The frequency response with a 10-cycle gate has a bandwidth of 0.04
PHz and causes the previous valleys, which were observed with a 5-cycle gate,
to approach 0, resulting in a further broadening of the sampled electric
field.

Figure 7:  The sampled pulse duration when a 1-cycle 850 nm signal and varied
cycle 1690 nm gate is used

In Fig. 7, we show the difference in pulse duration, which is defined as the
FWHM of the absolute modulus squared of the electric field. When a 1-cycle
1690 nm gate is used to sample a 1-cycle 850 nm signal, the pulse duration of
the sample field is identical. When the 1-cycle gate is changed to a 5-cycle
gate, the pulse sampled pulse duration is nearly the same but has two side
lobes at $\pm$ 5.5 fs which have a normalized intensity that is 0.29. Lastly,
when the gate cycle count is increased to 10, several additional side lobes
appear at $\pm$ 5.5 fs, $\pm$ 11.4 fs, and $\pm$ 17.2 fs with normalized
intensities of 0.73, 0.29, and 0.05, respectively.

Figure 8:  The sampling response as a function of gate frequency. We use a
10-cycle Gaussian pulse (a) with a carrier frequency of 0.353 PHz and 0.177
PHz to illustrate the frequency bandwidth dependence when the gate pulse
changes. (b) The corresponding time domain electron burst for the two gate
wavelengths used. (c) The frequency response of the electron burst for the two
gate wavelengths used and their corresponding envelope if a 1-cycle pulse was
used (dashed lines)

Shown in Fig. 8a are two 10-cycle Gaussian pulses used as
$\omega_{\mathrm{gate}}$, one with a carrier frequency corresponding to 0.353
PHz and another with 0.177 PHz. The corresponding Fowler-Nordheim electron
emission rate for the two Gaussian pulses is shown in Fig. 8b. Since there are
several cycles, there are several bursts of electrons that are generated. The
inset displays the main burst at t=0, which is modulated by the signal pulse,
enabling small signal sampling. The current burst full-width at half maximum
(FWHM) determines the time resolution and is 0.28 fs for
$\omega_{\mathrm{gate}}$ = 0.353 PHz and 0.55 fs for $\omega_{gate}$ = 0.177
PHz. Lastly, the frequency response given the 10-cycle
$\omega_{\mathrm{gate}}$ is shown in Fig. 8c. Since we are using a 10-cycle
pulse, there are multiple electron bursts in time, resulting in a reduction in
the frequency bandwidth response. If we were using a single-cycle gate pulse,
the frequency domain response would be the envelope shown in the black dashed
lines.

Figure 9:  (a) Illustration on the electronic comb generation in time. (b) The
effect of the electronic comb with varied field enhancement (FE) using a
5-cycle $\omega_{\mathrm{gate}}$ = 0.177 PHz optical field. The corresponding
sub-optical-cycle electron emission burst is shown in (b) and the
corresponding Fourier transform is shown in (c) as the field enhancement is
increased.

Shown in Fig. 9a is the gate-dependent electron emission in time. The electron
emission can be seen as the multiplication of a comb in time with the peaks
separated depending on the device geometry. With an asymmetric design, the
spacing is the optical period of the gate frequency. Mathematically, it can be
described as the following:

$\begin{array}[]{cc}h(t)=f(t)\cdot g(t)\\\ k(t)=h(t)\times i(t)\end{array}$
(6)

In time, an electronic comb (f) is generated and the amplitude of the comb
follows an envelope function (g), resulting in h and is purely dependent on
the gate-dependent localized electric fields at the tip. Then, h is convolved
with the electron emission rate envelope with finite bandwidth (i) which is
directly related to the optical period of the gate frequency.

Shown in Fig. 9b is the electron emission rate for 3 different field
enhancements. When the field enhancement is small, the main electron emission
burst centered at t = 0 plays the largest role in perturbative sampling. As
the field enhancement increases by a factor of 10 (equivalently an intensity
enhancement of 1000), the side electron emission bursts start to increase in
amplitude and the smaller amplitude sub-optical-cycle oscillations in the
electric field start to contribute to electron emission.

###  4.5 Supplementary information regarding the experiments

Figure 10:  Current output of the gold nanoantenna network with a new device
and with a device after performing measurements for $>$4 hours using the
10-cycle $\omega$ = 0.177 PHz.

There are numerous papers demonstrating that such nanoantenna devices are
operating in the optical field emission regime using 10s to 100s of pJ pulse
energies with few-cycle pulses. To confirm our devices are operating in the
Fowler-Nordheim regime with the 10-cycle $\omega$ = 0.177 PHz pulse, we
measured the current output versus peak field shown in Fig. 10 with a typical
device that has never been used for sampling, denoted as ”New Device”, and a
device which we used for more than 4 hours of illumination (denoted as ”Used
Device ($>$4 hr)”). We fit the low-pulse-energy section of the new device
current vs peak field curve to the conventional multiphoton photoemission
form, $E_{MP}\propto|F_{0}|^{N}$, where $F_{0}$ is the incident peak field.
Using a gold nanostructure, we take the work function to be 5.1 eV with the
incident photon energy to be 0.729 eV, so we expect N = 7, and experimentally,
we find N = 7.

The crossover from multiphoton photoemission to optical-field emission occurs
at roughly 0.285 V/nm as shown in Fig. 10. We also show that after more than 4
hours of performing measurements on a device, the response slightly degrades
and the field emission response settles into one that is well described by the
Fowler-Nordheim tunneling rate as shown by the right curve in Fig. 10. The
measured photocurrent and nonlinearity after long-term operation were
sufficient to continue using the device for optical field sampling.

####  4.5.1 Degenerate Sampling

Figure 11:  A schematic of the experimental setup for the measurement of
$\omega$ = 0.177 PHz. The laser light was split using a beamsplitter (BS). One
arm has a delay stage and was used as the gate pulse, while the signal arm was
chopped and neutral density filters were used to attenuate the signal.
Eventually, the two pulses are recombined using an identical beamsplitter
before being sent to a reflective objective where they are focused onto the
nanoantenna devices.

For degenerate sampling of 0.177 PHz (1690 nm) as shown in SI. Fig. 11, we
used an MZI to generate a pulse pair. We used ND filters to attenuate the gate
and signal pulses and place a chopper in the signal arm as a reference for the
measurement of the current-induced change when the small signal perturbed the
sub-optical cycle gate. After the pulse pair was recombined at the second
beamsplitter, we used a linear polarizer with extinction coefficients greater
than $10^{5}$ at 1690 nm.

Figure 12:  The frequency-resolved optical gating (FROG) (a) measured and (b)
retrieved spectrograms. (c) A comparison of the squared modulus of sampled
optical fields and the retrieved pulse versus time. (d) A comparison of the
squared modulus of the Fourier transformed sampled fields, retrieved pulse
versus frequency, a spectrometer reference, group delay from the sampled
optical fields, and the retrieved group delay.

We performed frequency-resolved optical gating (FROG) measurements of the
pulses in time to provide a reference to compare the sampled fields against.
In Fig. 12a, b we show the measured and retrieved spectrograms of the 10-cycle
1690 nm pulse used in the measurements. The degenerate sampled fields in Fig.
12c show relatively good agreement in the time domain with the FROG result. In
both traces a small side lobe is present in time near 65 fs. The retrieved
pulse duration was 58 fs FWHM and the sampled pulse duration was 57 fs FWHM.
Lastly, the frequency domain comparison between the sampled intensity,
retrieved intensity, and the grating-based spectrometer is shown in Fig. 12d.
The sampled group delay and retrieved group delay show very good agreement.

Figure 13:  Degenerate optical field sampling of a 63-cycle (218 fs) 0.291 PHz
pulse. (a) The corresponding current vs peak field before and after sampling.
(b) The degenerately sampled field with varying x-axis limits of 1760 fs, 400
fs, and 10 fs. The frequency-resolved optical gating (FROG) (c) measured and
(d) retrieved spectrograms. (e) A comparison of the squared modulus of sampled
optical fields and the retrieved pulse versus time. (f) A comparison of the
squared modulus of the Fourier transformed sampled fields, retrieved pulse
versus frequency, a spectrometer reference, group delay from the sampled
optical fields, and the retrieved group delay.

We also performed degenerate field sampling using a 63-cycle pulse with a
center frequency of 0.291 PHz (1030 nm) using an air-cooled 6-watt
LightConversion Carbide after characterizing the pulse using SHG FROG. To
begin our measurement, we employed the built-in pulse picker to reduce the
repetition rate down to 500 kHz from 1 MHz. We then directed the output of the
laser through the MZI with a closed-loop linear piezo stage delay line
(Smaract SLC2445-S with MCS2) with $\pm$ 40 nm ($\pm$ 0.13 fs) repeatability.
Utilizing a pristine device, we measured the current as a function of the gate
pulse energy. We proceeded with sampling once the total photocurrent reached
approximately 20 pA (blue line in Fig. 13a), which is equivalent to 1.4 nJ
(0.328 V/nm) for this specific device measured. After sampling, we remeasured
the current as a function of the gate pulse energy and observed degradation
(orange curve Fig. 13a) associated with long pulse durations and highly
localized fields at the nanoantenna tips. We tried to obtain 20 pA of current
after sampling, however, this specific device rapidly degraded at 2.3 nJ.
Using a signal pulse energy of 0.02 nJ (38.6 V/um), we degenerately sampled
the pulse as shown in Fig. 13b.

In the top panel (1.76 ps x-axis range) of Fig. 13b, at $\pm$ 500 fs, small
side lobes are seen. As we continue zooming in until the 10 fs x-axis range,
the measured optical period is 3.4 fs and the expected optical period for a
center frequency of 291 PHz is 3.4 fs. It is clear that the technique can be
generalized beyond 10-cycle pulses, however, since the devices were not
operating in the tunneling regime for these measurements, these results have
been placed in the supplemental information and require future studies to
better understand the mechanism.

To verify the sampled field, we compared our measured results with SHG FROG.
As seen in Fig. 13c, d is the measured and retrieved spectrograms of the
63-cycle 1030 nm pulse used in the measurements. The degenerate sampled fields
in Fig. 13e show relatively good agreement in the time domain with the FROG
result demonstrating that there is a small side lobe in time at $\pm$500 fs.
The retrieved pulse duration is 218 fs and the sampled pulse duration is 216
fs. Lastly, shown in Fig. 13f is the frequency domain comparison between the
sampled intensity, retrieved intensity, and the grating-based spectrometer.
The sampled group delay and retrieved group delay are also shown and show
reasonable agreement.

####  4.5.2 Non-degenerate Sampling

Figure 14:  A schematic of the experimental setup for the measurement of the
$\omega_{\mathrm{gate}}$ = 0.177 PHz and $\omega_{\mathrm{signal}}$ = SHG of
0.177 PHz. The laser light first passes through a telescope and is split using
a broadband beamsplitter (BS). One arm has a delay stage and was used as the
gate pulse, while the chopped signal arm has a lens to focus onto the
nonlinear crystal used to double the frequency and another lens was used to
collimate the SHG. To control the SHG power, an ND filter was placed before
the BBO and a 0.207 PHz (1450 nm) high-frequency pass filter (ND 2 at 0.177
PHz) was used to attenuate the residual $\omega$ = 0.177 PHz and a broadband
achromatic waveplate (WP) was used to rotate the SHG polarization from
vertical to horizontal. The two pulses are recombined using an identical
broadband beamsplitter before being sent to a reflective objective where they
are focused onto the nanoantenna devices.

For non-degenerate sampling of 0.353 PHz (850 nm) using a 0.177 PHz (1690 nm)
gate, we modified the previously shown setup by adding a telescope for the
1690 nm gate to control the beam spot size before focusing on the
nanoantennas. This modification was performed since when the SHG was generated
and collimated using the type-1 1.5 mm thick BBO the beam spot was smaller in
size before entering the objective. We also used a shortpass filter which was
ND2 at wavelengths longer than 1450 nm as well as an achromatic half waveplate
to rotate the SHG such that its polarization matched the gate pulse
polarization. After the second beamsplitter, the pulse pair passed through a
linear polarizer which had an extinction ratio that is $>10^{5}$ at 1690 nm
and $>10^{6}$ at 850 nm.

Figure 15:  The frequency-resolved optical gating (FROG) (a) measured and (b)
retrieved spectrograms. (c) A comparison of the squared modulus of sampled
optical fields and the retrieved pulse versus time. (d) A comparison of the
squared modulus of the Fourier transformed sampled fields, retrieved pulse
versus frequency, a spectrometer reference, group delay from the sampled
optical fields, and the retrieved group delay.

We performed FROG measurements on the SHG, and the measured and retrieved
spectrograms of the SHG of the 10-cycle 1690 nm pulse used in the measurements
are shown in Fig. 15a, b. The non-degenerate sampled fields in Fig. 15c show
relatively good agreement in the time domain with the FROG result. The
retrieved pulse duration was 49 fs FWHM and the sampled pulse duration was 48
fs FWHM. Lastly, shown in Fig. 15d is the frequency domain comparison between
the sampled intensity, retrieved intensity, and the grating-based
spectrometer. The sampled group delay and retrieved group delay also show very
good agreement.

Figure 16:  Downsampling using a supercontinuum pumped by a 78.4 MHz Er Fiber
Laser. (a) The experimental schematic for the measurement. (b) Degenerate
sampling of the dispersive wave. (c) Non-degenerate sampling of the soliton
using the dispersive wave. (d) The corresponding Fourier transformed sampled
fields.

To demonstrate non-degenerate downsampling and to show that lower frequencies
that are not integer harmonics can still be sampled, we used a separate few-
cycle supercontinuum source with details that can be found in
putnamFewcycleCarrierEnvelopephasestable2019 . After supercontinuum
generation, an SF10 prism pair was used to spatially disperse the beam. After
the prism pair, the Er pump was spatially filtered, and the soliton and
dispersive wave contributions were spatially separated. The higher-frequency
dispersive wave portion of the supercontinuum was returned through the prism
pair for compression and the removal of spatial chirp and was used as the
gate. We note that the lower-frequency soliton, which was used as the signal,
was not returned through the prism pair leaving some residual spatial chirp.

For the gate pulse, 19 pJ (0.21 V/nm at focus) of the 3-cycle (12 fs)
dispersive wave (0.250 PHz, 1200 nm) was used. For the signal, 30 pJ (0.15
V/nm at focus) of the 11-cycle (65 fs) soliton (0.166 PHz, 1600 nm) was used.
The sampling measurement configuration is shown in detail in Fig. 16a.
Degenerate sampling of the dispersive wave was performed and is shown in Fig.
16b. Next, we performed non-degenerate downsampling of the soliton and
measured an optical period of approximately 6.0 fs as expected as Fig. 16c.
Lastly, the normalized Fourier-transformed time-domain sampled waveforms are
shown in Fig. 16d. This emphasizes the ability of the nonlinear mixing process
to also sample lower frequency contributions at non-integer harmonics in the
few-cycle regime without CEP locking.

####  4.5.3 Supercontinuum Sampling

For the supercontinuum measurements discussed in the main text, we pumped a
commercially available endlessly single-mode, large-mode-area photonic crystal
fiber with roughly 100 nJ of pulse energy to generate a supercontinuum
spanning 0.13 to 0.35 PHz. This supercontinuum was then split and recombined
using an identical pair of beamsplitters. Each arm had an ND filter with the
same thickness of glass to control the pulse energy focused onto the
nanoantennas. After the pulses were recombined, the pulse pair was passed
through the same linear polarizer as in prior measurements.

Figure 17:  A schematic of the experimental setup for the measurement of the
supercontinuum generated using $\omega$ = 0.177 PHz. The laser light was split
using a beamsplitter (BS1). One arm had a delay stage and was used as the gate
pulse, while the signal arm was chopped and neutral density filters were used
to attenuate the signal. Eventually, the two pulses were recombined using an
identical beamsplitter before passing the linear polarizer and sent to a
reflective objective where they were focused onto the nanoantenna devices.
Figure 18:  The frequency-resolved optical gating (FROG) (a) measured and (b)
retrieved spectrograms. (c) A comparison of the squared modulus of sampled
optical fields and the retrieved pulse versus time. (d) A comparison of the
squared modulus of the Fourier transformed sampled fields, retrieved pulse
versus frequency, a spectrometer reference, group delay from the sampled
optical fields, and the retrieved group delay.

We performed FROG measurements on the SCG as shown in Fig. 18. Fig. 18a and b
are the measured and retrieved spectrograms of the supercontinuum pulse used
in the measurements. The degenerate sampled fields in Fig. 18c show relatively
good agreement with the pulses retrieved from the FROG measurements in the
time domain. The retrieved pulse duration was 9 fs FWHM and the sampled pulse
duration was 8.5 fs FWHM. Lastly, shown in Fig. 18d is the frequency domain
comparison between the sampled intensity, retrieved intensity, simulated
intensity spectrum, and the grating-based spectrometer. The sampled group
delay and retrieved group delay are also shown and show very good agreement.

Figure 19:  The measured and reconstructed interferometric autocorrelation of
the supercontinuum source.

For the interferometric autocorrelation measurements (IAC) (SI Fig. 19), we
focused the light onto a 40 $\mu$m thick BBO and detected the IAC using a
silicon photodiode after passing through a linear polarizer. We measured a
duration of 13 fs FWHM, which corresponds to an 8.4-fs pulse width considering
a deconvolution factor of 1.54 assuming a sech pulse shape. To validate the
field sampling, we compared the measured IAC trace to the reconstructed IAC
trace obtained using the sampled field shown in Fig. 5. The two traces share
comparable features, both having a sub-two-cycle component in the center part
of the IAC trace along with pronounced side lobes. The second-order
dispersion-induced chirp was not observed to be significant in either trace.
The IAC traces both show significant side lobes, which are explained by the
long tails observed in the time domain of the sampled waveform. These time-
domain tails were also observed in the simulation of the nonlinear pulse
propagation. They are primarily induced by third-order dispersion accumulated
during spectral broadening between normal and abnormal dispersion regimes as
well as complex higher-order dispersion during nonlinear compression.

Figure 20:  A schematic of the experimental setup for the measurement of the
supercontinuum generated using $\omega$ = 0.177 PHz with a 10-cycle gate
pulse. The laser light was split using a beamsplitter (BS), such that one path
is for the gate and the other path is for the supercontinuum signal pulse. The
gate pulse was directed through a delay stage to enable temporal control of
the two pulses. Eventually, the 10-cycle gate and supercontinuum signal were
recombined and passed through a linear polarizer before being focused onto the
nanoantennas.

We split the gate pulse into two arms such that one can be used to generate
the exact continuum used for the degenerate sampling of the supercontinuum. We
add a telescope on the gate arm to control the beam spot size difference
before being focused on the nanoantennas. We kept the same ND filter and
transmitted the beam through the same beamsplitter to ensure the pulses
accumulates nearly the same phase as the degenerate supercontinuum sampling.
After the two pulses recombine at BS2, they passed through the same linear
polarizer as used before.

Figure 21:  (a) Simulated supercontinuum in frequency with the simulated
frequency response, and gold antenna intensity enhancement (right y-axis). (b)
The corresponding time-domain SCG with and without interacting with the gold
nanoantenna.

We used the simulated supercontinuum fields in frequency and added the
electromagnetically simulated gold field enhancement amplitude and phase to
understand how the gold nanoantenna modified the supercontinuum. We add the
gold field enhancement amplitude and phase into the frequency spectrum and
normalized the squared modulus of the corresponding frequency spectrum which
is shown in Fig. 21a. After incorporating the gold antenna response, we also
looked at how the simulated frequency response plays a role in the simulated
frequency spectrum. We observed that there is a reduction of intensity which
is mainly attributed to the gate-dependent frequency response. Also, it can be
seen that where the sampled (blue) and gold antenna intensity enhancement
(teal) curves intersect at 0.217 PHz, the sampled field starts to increase and
is related to the intensity enhancement from the nanoantenna. Afterward, we
took the inverse Fourier transform of the simulated frequency spectrum and
normalized the squared modulus of the time-domain spectrum to understand how
the gold nanoantenna affects the pulse duration which is shown in Fig. 21b. We
observe that the gold nanoantenna does not significantly modify the
supercontinuum in frequency or time since the devices are being operated away
from their resonant wavelength.

###  4.6 Regarding Alternative Materials

Figure 22:  FDTD simulation of the gold nanoantenna and platinum nanoantenna
as a reference for a non-plasmonic metal. In (a) is the field enhancement and
phase as a function of frequency while (b) is in wavelength

As these devices are based on resonant nanoantennas, we performed simulations
using platinum, a metal that is not plasmonic, to demonstrate other
nanoantenna materials could be used. It can be observed that toward DC
frequencies (Fig. 22a), the field enhancement converges. It can be easily seen
in Fig. 22b, where in the long wavelength field enhancement of the gold and
platinum antennas converge, which is a characteristic of metal nanoantennas.

## Acknowledgments

We also graciously thank Prof. Sixian You for the use of the LightConversion
Cronus 3P system used in this work. This work was carried out in part through
the use of MIT.nano. We also graciously thank LightConversion for providing
the 6-Watt Carbide used for testing degenerate sampling of a 63-cycle pulse.
We would like to thank Matteo Castellani and Adina Bechofer for their comments
and suggestions on the manuscript before submission. Finally, we would like to
acknowledge the many helpful discussion and conversations with Felix
Ritzkowsky regarding this work and the underlying topics.

This work was supported by the Air Force Office of Scientific Research under
award number FA9550-18-1-0436. M. Y. Acknowledges support from the National
Science Foundation Graduate Research Fellowship Program, Grant No. 1745302.
L.-T. C acknowledges financial support from the Ministry of Education (R.O.C)
for the Overseas Internship Program directed by Prof. Shih-Hsuan Chia at
National Yang Ming Chiao Tung University and the National Science and
Technology Council (R.O.C) for the doctoral fellowship program. S.-H. C
acknowledges the support from the Young Scholar Fellowship Program by the
National Science and Technology Council (R.O.C) under award number NSTC
112-2636-E-A49-004. The opinions and views expressed in this publication are
from the authors and not necessarily from the National Science Foundation, the
Air Force Office of Scientific Research, the Ministry of Education, or the
National Science and Technology Council.

## Declarations

  * •

Funding: This work was supported by the Air Force Office of Scientific
Research under award number FA9550-18-1-0436.

  * •

Conflict of interest/Competing interests: The authors do not declare any
competing interests.

  * •

Availability of data and materials: Experimental data is provided on a public
repository https://github.com/qnngroup/manu-HarmonicMixer.git

  * •

Code availability: Code used in this manuscript is provided on a public
repository https://github.com/qnngroup/manu-HarmonicMixer.git. For the FDTD of
the nanoantenna, an example simulation code is provided,
https://github.com/qnngroup/pymeep-nanoantenna-simulator

  * •

Authors’ contributions: M. Y., L. T. C., and P. D. K. conceived the
experiments. M. Y. designed the experiments along with contributions from L.
T. C. M. Y. performed the data analysis with input and contributions from P.
D. K. and L. T. C. M. Y. performed the sampling and electromagnetic
simulations with input from P. D. K. L. T. C. performed the retrieval and
supercontinuum generation simulations with input from S. H. C. M. Y. and M. T.
performed the device fabrication. M. Y. wrote the initial draft of the
manuscript, with contributions in revision from K. K. B., L. T. C., and P. D.
K. Input was provided from all authors. The project was supervised by K. K. B.
and P. D. K.

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