[1]Noah Flemens

[1]Jeffrey Moses

1]School of Applied and Engineering Physics, Cornell University, 142 Sciences
Drive, Ithaca, 14853, New York, United States

2]BAE Systems, P.O. Box 868, MER15-1813, Nashua, 03061, New Hampshire, United
States

# High quantum efficiency parametric amplification via hybridized nonlinear
optics

<EMAIL_ADDRESS>Dylan Heberle  Jiaoyang Zheng  Devin J. Dean  Connor Davis
Kevin Zawilski  Peter G. Schunemann<EMAIL_ADDRESS>[  [

###### Abstract

Parametric amplifiers have allowed breakthroughs in ultrafast, strong-field,
and high-energy density laser science and are an essential tool for extending
the frequency range of powerful emerging diode-pumped solid-state laser
technology. However, their impact is limited by inherently low quantum
efficiency due to nonuniform light extraction. Here we demonstrate a new type
of parametric amplifier based on hybridized nonlinear optics. Hybridization of
parametric amplification with idler second harmonic generation induces unusual
evolution dynamics for a fully parametric amplifier – with saturating rather
than cyclic gain – observed here for the first time. This allows highly
uniform light extraction enabling unprecedented efficiency for a lossless
amplifier with Gaussian-like intensity profiles – a 48-dB single-stage gain
with 68% quantum efficiency and 44% pump-to-signal energy conversion – a
several-fold improvement over the standard. Possessing both laser-like high
quantum efficiency and the advantages of thermal-loading free parametric
systems, this simple approach can be implemented widely and have significant
impact by increasing several-fold the power available for science and
industry.

Technology based on optical parametric amplification (OPA) and its variants,
developed over the past five decades akhmanov1965 ; Baumgartner79 ; DUBIETIS92
; Cerullo03 ; Fattahi2014 , has contributed greatly to a 21st-century
revolution in energetic picosecond and femtosecond pulsed sources of visible
through mid-infrared light. With each advance followed breakthroughs in
strong-field science, high-energy density science, and the nonlinear
spectroscopy of ultrafast molecular and materials phenomena Baltuska02 ;
Ledingham03 ; Nibbering05 ; Mourou2006 ; SALAMIN2006 ; Sansone2011 ; Danson15
. Unlike lasers, however, in which gain saturates and makes uniform extraction
of pump energy feasible, the gain dynamics of OPA are cyclic. A perpetual back
and forth energy exchange begins with a transfer of power from the pump to the
desired signal and idler waves followed by a return of power to the pump
Armstrong:62 . These dynamics are the root cause of low quantum efficiency
(defined as the ratio of emitted signal photons to incident pump photons) in
OPA, since they make uniform light extraction across the naturally bell-shaped
transverse beam and pulse profiles of pump lasers impractical.

Figure 1: Top: simulated pump-signal conversion dynamics of waves with
Gaussian intensity profiles for OPA (left) versus HPA (right), representing a
1D slice across the center of a 3D spatiotemporal wave at specific propagation
lengths. [Insets: photon exchange diagram and simulated (continuous-wave and
plane-wave) wave propagation dynamics at the peak of the spatiotemporal
profile.] In OPA, energy flow is cyclic, with cycle period dependent on the
local pump and seed intensities. Temporary extraction of pump photons via
conversion to the signal and idler begins at the profile center, working its
way outwards and extraction is always highly nonuniform, resulting in low
quantum efficiency integrated across the full 2+1D spatiotemporal extent.
Bottom: the evolving spatiotemporally integrated pump-to-signal energy
conversion efficiency and quantum efficiency is shown for OPA and HPA, with
circles corresponding to the specific lengths shown in the top panels. For
OPA, in this example, the maximum energy conversion efficiency (quantum
efficiency) peaks at 19% (30%) at 3.0 mm. This limitation to conversion
efficiency is caused by non-uniform light extraction (‘Extraction Limit
(OPA)’), and corresponds to a ring-shaped signal beam and spherical shell-
shaped spatiotemporal profile, a shape that is usually impractical for
applications. The maximum energy efficiency (quantum efficiency) achieved
while maintaining a bell-shaped profile (‘Practical Limit (OPA)’) peaks at
only 10% (16%) at 2.3 mm. In contrast, in HPA, energy flow has a quasi-
saturating behavior characterized by damped oscillations, with oscillatory
convergence of the pump field to the signal and idler SH fields. The
saturating gain allows nearly uniform pump light extraction that
asymptotically approaches 100% quantum efficiency (‘Quantum Defect Limit’).
For example, at 6.5 mm, $>$50% energy conversion ($>$78% quantum efficiency)
is achieved, or $\sim$5 times greater efficiency than the OPA practical limit.
At this length, the residual pump wave is characterized by low-amplitude
ripples that are most pronounced towards the profile edges, and the amplified
signal beam closely matches the profile of the incident pump. In all
simulations, the initial pump/seed pulse energy ratio is 105 and the quantum
defect, $\omega_{s}/\omega_{p}$, is 0.64. All simulation parameters are common
for OPA and HPA cases, the only difference being the condition of phase-
matched idler SHG included for HPA.

Figure 1, top left, illustrates this well-known problem. Since the OPA gain
cycle period lengthens with decreasing local pump intensity, pump energy is
only temporarily extracted at each location starting inside and moving
outwards along the transverse profile. The limit to signal conversion
efficiency imposed by this non-uniform pump extraction is several times
smaller than the limit due to the quantum defect, i.e. the ratio of signal and
pump photon energies (Fig. 1, bottom). Deepening the problem, maximum
extraction occurs at an amplifier length at which significant back-conversion
of signal power at the beam center results in a ring-shaped beam. If a bell-
shaped signal beam is desired, as is usually the case for applications, the
practical conversion limit can be up to another factor of two smaller.
Generally, for Gaussian spatiotemporal intensity profiles, the quantum
efficiency ranges from 10-30% depending on the amplifier’s gain. The pump-to-
signal energy conversion efficiency – further reduced by the size of the
quantum defect – typically ranges from single-percent to $\sim$20%.

Low quantum efficiency means the great majority of pump photons go unused.
This inefficiency is of particular significance today with the advent of
record-high average power femtosecond and picosecond diode-pumped solid-state
(DPSS) lasers Jauregui2013 ; Fattahi2014 ; Sincore18 ; Saraceno2019 , recently
exceeding ten kilowatts Muller:20 , that can be used to pump OPA. It remains a
challenge to efficiently convert that power to ultrafast sources across the
visible and infrared spectrum where they can enable new and more cost-
effective applications for science, medicine, and industry, and in the mid-
infrared transmission windows to advance ranging and sensing technologies. Our
present work addresses this barrier for laser science and applications
resulting from poor OPA efficiency. We demonstrate a way to fundamentally
alter the nonlinear dynamics of OPA to enable uniform pump energy extraction
and thus, high quantum efficiency, via a hybridization of three-wave mixing
processes in an ordinary OPA crystal. The result is a first solution that
possesses both high quantum efficiency through laser-like spatiotemporally
uniform pump extraction and the advantages of an OPA such as the absence of
thermal loading, which is essential for scaling OPA with the high average
powers of emerging DPSS pump lasers.

Historically, several methods for boosting pump light extraction have been
demonstrated in bulk-crystal based parametric amplifiers, though none are
widely employed. 74% net quantum efficiency in a sub-Joule scale, nanosecond
chirped pulse power amplifier with 10 dB gain has been achieved via top-hat
beam and pulse shaping Bagnoud:05 . This is the simplest use of conformal
profiles Begishev:90 ; Moses:11 ; Cao:18 ; Mackonis:20 ; Fischer:21 , a
general technique for synchronizing OPA conversion cycles across the
spatiotemporal transverse extent of the interacting waves by pre-amplifier
beam and pulse shaping. However, the difficulty in making the shaping process
itself efficient, especially for femtosecond and picosecond pulses, has
limited the use of this practice.

Alternatively, it has been shown that idler loss during amplification can
inhibit the sum-frequency generation process responsible for back-conversion
of the amplified signal wave to the pump, thereby allowing more uniform
extraction of photons from the pump. Jeys demonstrated an eight-pass
parametric generator with the idler absorbed between passes by a roof prism
Jeys:96 . 80% pump photon depletion was achieved, though the net quantum
efficiency was limited to 45% due to significant optical reflective losses.
More recently, several works have explored concurrent idler loss during
amplification by means of material absorption or radiation Ma:15 ; El-
Ganainy:15 ; Zhong:16 ; Ma:17 . The OPA evolution dynamics in the presence of
idler loss were found to exhibit damped power oscillations with a gradual
increase in pump-to-signal energy conversion efficiency, an indication of the
system’s non-conservative (non-parametric) nature. These quasi-saturating gain
dynamics allowed vastly improved pump extraction, with an impressive 70%
internal quantum efficiency achieved with $\sim$20 dB gain in a YCOB crystal
doped with Sm3+ to induce idler absorption Ma:15 . An example of the ‘gain via
loss’ paradigm Perego2018 ; Feng2017 , improved conversion efficiency comes at
the expense of loss of the idler power as well as the resulting thermal
loading. Additionally, the approach requires the engineering of new,
application-appropriate lossy nonlinear crystals, limiting its applicability.

### Hybridized parametric amplification

Recently, several of us proposed a hybridized parametric amplification (HPA)
scheme that employs concurrent idler self-frequency doubling to alter the
nonlinear dynamics of OPA. It was discovered that displacement of idler
photons by second harmonic generation (SHG) to a coherent copropagating idler
second harmonic (SH) wave shares a general dynamical form with the case of OPA
undergoing concurrent idler linear absorption: that of a damped nonlinear
(Duffing) oscillator Flemens:21 . These dynamics allow uniform pump energy
extraction from spatiotemporally bell-shaped beams (Fig. 1, top right),
approximating the gain saturation that is inherent to a laser amplifier but
keeping advantages unique to fully parametric (lossless) amplification, such
as phase-matching limited gain band tunability and no thermal loading.
Capitalizing on this behavior, our numerics predict that HPA allows pump-to-
signal conversion efficiency approaching full quantum efficiency – exceeding
the practical OPA limit by severalfold (Fig. 1, bottom) – while producing an
output beam profile closely matching that of the bell-shaped pump, and without
any transfer of power to the gain medium. This may be a key enabling feature
for high-average-power and high-energy applications that could enable new
breakthroughs in ultrafast laser science. In the following, we describe an
experimental demonstration of this principle in a single-stage, 48-dB mid-
infrared amplifier with 68% quantum efficiency and in an ordinary OPA bulk
crystal, with efficiency far exceeding its ordinary OPA counterpart. Moreover,
we clearly observe the novel evolution dynamics enabling the improved
performance, with experimental data closely matching a theoretical
spatiotemporal propagation analysis.

### Experimental design and setup

For this proof-of-principle demonstration, we chose a signal wavelength
relevant for applications in the mid-wave infrared atmospheric passband at
3390 nm, and a pump wavelength of 2190 nm, within or not far from holmium-,
thulium-, and chromium-doped DPSS ultrafast laser emission bands. For this
wavelength range, CdSiP2 (CSP) is a convenient OPA crystal, with high
nonlinear coefficient, 2.45-eV band edge and good transparency out to 6.5
$\mu$m, and favorable Type-I OPA phase-matching when pumped near 2 $\mu$m
Sanchez:14 ; Liang:17 . A previous analysis of non-collinear birefringent
phase-matching for simultaneous Type-I OPA and Type-I idler SHG found broad
applicability to common OPA crystals and pump wavelengths, with wide signal
tunability Dean2021Apr . Fig. 2, top extends this analysis to CSP at our
choice of pump wavelength, and finds a range of signal wavelengths over which
the joint phase-matching angles can be satisfied. At a signal wavelength of
3390 nm, the phase-matching conditions support a near-collinear wave-vector
geometry ($\theta_{p},\theta_{s},\theta_{i}\simeq 44\deg$), though we note
that Poynting-vector non-collinearity and therefore spatial walkoff occurs due
to the involvement of two waves with extraordinary polarization (pump and
idler SH). Fig. 2, bottom illustrates that this Type-I birefringent phase-
matching scheme is more broadly applicable to common OPA crystals and common
DPSS pump laser wavelengths and allows tuning over a wide signal wavelength
range with small non-collinear angles.

Figure 2: Top: Joint Type-I OPA and Type-I idler SHG birefringent phase-
matching diagram (showing the angles of wave-vectors relative to a uniaxial
crystal’s optical axis) and calculated angles in CSP at the experimental pump
wavelength of 2190 nm. Generally, two non-collinear configurations are
possible across the signal tuning range, indicated by
$(\theta_{p,1},\theta_{s,1})$ and $(\theta_{p,2},\theta_{s,2})$, which share a
common $\theta_{i}$ that is fixed at the SHG phase-matching angle. A wide
signal wavelength tuning range exists below 3.5 µm and above 4.2 µm with non-
collinear angles of up to a few degrees. At the experimental signal wavelength
of 3390 nm, a solution exists with near-collinear wave-vector geometry. The
bottom panel illustrates a wide signal tuning range for other combinations of
common bulk OPA media and common DPSS pump laser wavelengths. Ranges indicate
joint Type-I OPA and Type-I idler SHG phase matching for pump-signal angles
limited to 4 $\deg$.

Experimental implementation of HPA requires no significant modifications to
the simple apparatus of an OPA device. The only additional consideration in
our experiment was ensuring that all idler frequencies are phase matched for
SHG. This was achieved by insertion of a spectral bandpass filter in the
signal seed path. To satisfy the phase-matching condition at near-collinear
angles, we used a Rochon prism to spatially combine the orthogonally polarized
pump (up to 100 $\mu$J/pulse at 10 kHz pulse rate) and signal seed (0.7
nJ/pulse) beams before passing through a 6-mm CSP crystal. The beam sizes
($\sim$1-mm 1/e2 diameter) and pulse durations ($\sim$1-ps FWHM, slightly
chirped, transform-limited duration of $\sim$600 fs) were chosen to
approximately match, ensuring the full spatiotemporal extent of the pump was
overlapped with signal light. Within the device, transfer of pump photons to
the signal and idler is accompanied by frequency doubling of the idler (at
6186 nm) to its SH (at 3093 nm). We monitored pump, signal, and idler SH beams
at the device exit as the pump power was increased to its maximum. Since the
idler SHG process is efficient and the exit face of the CSP crystal was anti-
reflection (AR) coated for 2.0-4.8 $\mu$m, monitoring the low-power residual
idler beam was impractical. Measured spectra are shown in Extended Data Fig. 1
alongside corresponding simulated spectra and temporal profiles.

Figure 3: Top: pump-to-signal energy conversion efficiency comparing the HPA
experiment (blue squares) to simulated HPA (black circles) and the equivalent
OPA without SHG (red circles). Error bars indicate the standard deviation of
signal powers measured by a thermopile-based power meter over 1 minute.
Bottom: measured signal and idler SH beam profiles with increasing applied
pump power.

## Results

### High quantum efficiency via uniform light extraction

The pump-to-signal energy conversion efficiency internal to the device as a
function of input pump power is shown in Fig. 3, top, along with the results
of spatiotemporal wave propagation simulations using the model and framework
of Flemens:21 . The experimentally measured (blue squares) and simulated
(black circles) data for the HPA process are in good agreement, with a clear
trend of monotonically increasing efficiency. This behavior is as expected
given the dynamics of HPA, which inhibit signal back-conversion and provide
saturating gain at all spatiotemporal coordinates of the amplifier. Fig. 3,
bottom corroborates this behavior, illustrating a signal beam profile (imaged
at the crystal exit plane) that expands in size with increasing pump power,
becoming roughly equal in size to the incident pump beam as the pump light
extraction becomes more uniform. The slightly lower measured performance at
higher powers of the HPA system compared to simulations might be explained by
the observed onset of idler SHG reversal at the beam center in the presence of
spatial walk-off (an effect not captured by the simulations). Idler SHG
reversal is a consequence of imperfect phase matching across the full
transverse wave-vector and spectral content of the fields. Since for higher
pump powers, idler SHG begins earlier in the crystal, decoherence due to phase
mismatch is more pronounced. In other words, idler SHG decoherence limits the
device length, and thus greater efficiency requires shorter length and higher
pump intensity. In our experiment, an observed spatial asymmetry of the idler
SH indicates significant Poynting vector walk-off (Fig. 3 bottom), which would
have a more pronounced effect for smaller beam size. Thus, our choices of beam
size, pump intensity, and crystal length represents an optimal balance between
decoherence and spatial walk-off effects.

The maximum internal pump-to-signal energy conversion efficiency achieved is
44%, corresponding to 41 µJ of measured energy per signal pulse at 100 µJ
incident pump pulse energy due to imperfect AR coatings (4% reflections at
each face), and with 18 µJ of generated idler SH. These results correspond to
an internal quantum efficiency of 68%, corroborated by a separate analysis of
the measured depletion of the pump power (see Supplementary Section 1), and a
48-dB signal gain. This represents a new standard for lossless, thermal-
loading free parametric amplification, based on the novel underlying nonlinear
dynamics of the HPA approach. The efficiency far exceeds that of standard OPA
(Fig. 3, red circles), which peaks at 18% at 140 mW of pump power with a
donut-shaped beam due to significant back-conversion at the beam center.
Achieving a comparable amount of signal energy as our HPA result in a single
OPA stage while maintaining a bell-shaped amplified beam shape would require
4.5 times the pump power and would only be 9% efficient, according to
simulation.

Figure 4: (a) Depleted experimental pump beam profiles imaged to CSP crystal
end facet and normalized to input pump energy, and corresponding normalized
simulated output pump beam profiles from (b) HPA and (c) OPA. [Full datasets
for (a) and (b) are provided in Supplementary Video 1.]

### Observation of novel nonlinear dynamics

By examining the residual pump beam profile as a function of incident pump
power, we can also clearly discern the novel nonlinear dynamics that allow
uniform pump extraction in HPA. Fig. 4a shows the measured residual pump beam
at the output of the CSP device normalized in each frame to the input pump
power. Depletion starts from the center of the pump beam and moves outward
until a significant amount of energy is extracted by the signal beam. As the
damped oscillatory dynamics of HPA occur at each local transverse coordinate
but with asynchronous oscillation period, as predicted in Fig. 1, a concentric
ring pattern is formed across the beam that diminishes with increasing pump
power. The measured profiles exhibit a remarkable match to the HPA simulations
(Fig 4b), and highly contrast with simulations of the equivalent OPA device
(Fig. 4c). As seen, the undamped oscillations in pump extraction
characteristic of OPA also lead to the formation of concentric rings, but
these do not diminish with increasing pump power, preventing uniform
extraction across the pump beam at any power and resulting in low quantum
efficiency. Thus, we clearly observe the hybridized nonlinear optical system
dynamics leading to uniform pump extraction predicted in Flemens:21  for the
first time, and confirm the highly contrasting pump extraction dynamics of HPA
and OPA.

## Discussion

Since the HPA implementation demonstrated here is no less practical to
implement than standard OPA, working for unshaped beams, without idler loss
and thermal loading, and with standard bulk crystals, the technique offers a
solution that could be easily and widely adopted for the efficient frequency
down-conversion of modern high-power DPSS lasers. Simultaneous OPA and idler
SHG Type-I birefringent phase matching will generally be possible across
common OPA crystals and DPSS laser wavelengths for pulses hundreds of
femtoseconds and longer in duration, making it relevant to amplifiers based on
Yb-doped, Ho-doped, and many other gain media. This means HPA could be
employed to efficiently extend ultrafast lasers that can produce high average
power and/or high pulse energy to the large spectral ranges where suitable
gain media do not exist. For many applications, HPA may prove to be even more
simple and practical to implement than OPA, as it allows high efficiency to be
achieved even at high gain (48 dB in this demonstration). Thus, HPA can also
obviate the common practice of dividing available pump power into high-gain
pre-amplifier stages and a low-gain power amplifier. This technique is
employed to marginally improve overall efficiency since the practical OPA
efficiency limit increases with lower gain. Elimination of the need for two or
more amplifier stages cuts losses and could prove to be an important
simplification to parametric amplifier architecture, allowing technologies
that are more robust, portable and cost-effective.

Another possible significant advantage of HPA is illustrated by Fig. 3, top,
which shows a significant noise reduction as the amplifier is pushed into the
gain saturation regime at high pump powers. In OPA, gain non-uniformly
saturates and reverses to loss at the practical OPA limit, thus preventing a
strong reduction in noise at peak efficiency and causing increased
fluctuations in transverse profiles (as well as in the power spectrum in
optical parametric chirped pulse amplification (OPCPA)) Manzoni:11 ; Feng:21 .
In contrast, in HPA, gain saturation takes place at all transverse coordinates
simultaneously. This and the suppression of back-conversion produces an
observed clamping effect on noise at high conversion efficiency as the signal
transverse profile asymptotically grows, approaching the form of the incident
pump. This may also help to increase the pulse contrast, as recently predicted
for OPA with linearly absorbed idler Hu2020 .

The amplified signal bandwidth obtained at full efficiency corresponds to a
340-fs transform limited pulse duration. The simple bulk-crystal birefringent
phase matching approach taken here is not likely to offer, for any given
application, a few-cycle phase-matching bandwidth for both OPA and SHG
processes simultaneously in available crystals. However, we propose several
plausible routes to high-quantum-efficiency few-cycle pulse amplification via
HPA. For example, HPA could potentially accommodate both few-cycle pump and
signal pulses via a dual-chirp scheme with matching pump and signal chirp
rates. Such a configuration, employed previously for spectral compression of
the idler in difference frequency generation Luo:06 , can sharply reduce the
bandwidth of the idler pulse. A hybridized parametric amplifier using this
configuration would then require a broad phase-matching bandwidth for the OPA
process only since the idler SHG process would be narrowband, thus allowing
compatibility with standard noncollinear OPA phase-matching angles employed
for few-cycle amplification Cerullo03 . This approach is naturally suited to
femtosecond pump lasers such as Ti:sapphire amplifiers or DPSS lasers with
hollow-core fiber post compression. Fourier-domain OPA Schmidt2014 , in which
an angularly dispersed signal overlaps with a narrowband pump pulse at the
image plane of a diffraction grating, could potentially accommodate HPA in
few-cycle amplifiers that employ a relatively narrowband pump pulse, since
locally both the pump and signal are spectrally narrow in the amplifier
medium. Fourier-domain HPA could therefore potentially serve as a high-
efficiency alternative to few-cycle OPCPA pumped by picosecond lasers with the
Fourier-domain OPA capacity for allowing spatially engineered nonlinear
crystals at the Fourier plane Leblanc:20 , a route to the possible
accommodation of an octave-spanning signal bandwidth. Quasi-phase matching can
be used to suit the needs of many potential applications that do not require
large crystal apertures, using superlattice and other poling approaches
Chou:99 ; Liu:02 ; Norton:03 ; Lifshitz:05 ; Flemens:21 .

An especially intriguing possibility is to make use of the significant energy
displaced to the co-propagating idler SH field to pump a subsequent signal
amplification stage, which is practical when the idler SH frequency is greater
than the signal frequency. This was proposed in Flemens:21  as a potential
route for overcoming the quantum defect limit of OPA, i.e., for producing a
double-stage quantum efficiency above 100%, meaning more than one signal
photon is produced for every incident pump photon. Cascading HPA stages in
this way could have significant impact for atmospheric laser science
applications by improving the efficiency of long-wave infrared generation
pumped by near-infrared lasers, where the quantum defect is on the order of
ten percent and typical OPA energy efficiency is at the single to few percent
level, making it impractical for many applications.

While OPA has had a tremendous impact on ultrafast laser science and
technologies, it has perennially suffered from low quantum efficiency. The new
concept of parametric amplification employing hybridized nonlinear optics
overcomes this issue with a novel combination of positive attributes found in
ultrafast lasers and parametric amplifiers, including gain saturation without
back-conversion and thus uniform light extraction even at high gain, a tunable
gain band, and an absence of thermal loading. Through a capacity for high
efficiency at a high single-pass gain, it can also be used to simplify OPA-
based architecture. HPA thus sets a new quantum efficiency standard for
parametric amplification while possessing the attributes required to handle
high average power and broadly extend the frequency range of emerging DPSS
ultrafast pump laser technology.

## Methods

### Pump and Seed Generation

The 2190 nm HPA pump and 3390 nm HPA seed sources were each derived from a
common home-built noncollinear 10-fs near-IR OPCPA system spanning 680-950 nm.
The OPCPA is white-light seeded and pumped by a 1.03 µm, 10-kHz, commercial
picosecond Yb:YAG amplifier (Amphos) frequency doubled to 515 nm. To generate
the mid-IR wavelengths used for this experiment, we employed adiabatic
difference frequency converters (ADFC) based on aperiodically poled lithium
niobate, similar to that used in Ref. Krogen2017Apr , designed by our group
and manufactured by HC Photonics. For the pump beam line, we used a 4-f pulse
shaper (Phasetech) to select a 0.5-nm band at 700.8 nm from the near-IR OPCPA
front end, which was subsequently converted to 2190 nm in a 2-cm LiNbO3 ADFC
pumped by the 1.03-µm Yb:YAG amplifier. This light was then amplified by OPA
to 100 µJ (1 W average power) using a 1030-nm pumped 2-mm periodically poled
LiNbO3 crystal (HC Photonics) pre-amplifier stage and 3-mm BiBO (Castech)
power amplifier stage. The OPA pump source is tunable via selection of the
near-IR band via the pulse shaper. For the seed beam line, we used a bandpass
filter (Andover) to select a fixed 790 $\pm$ 0.5 nm band from the near-IR
OPCPA front end, which was converted to 0.7-nJ, 3.39-µm pulses in a parallel
ADFC stage.

### HPA Stage

The HPA experiment was performed using a 6-mm CSP crystal grown by BAE
Systems. The input and output faces of the crystal were AR coated (TwinStar)
for 2.0-4.8 µm with 4% reflection losses on each face measured for the pump
and signal wavelengths. The pump beam path incorporated a translation stage to
fine tune the temporal overlap of the beams. A Rochon prism (RPM10, Thorlabs),
which converts incident orthogonally polarized beams to a collinear output,
was used to achieve spatial overlap of the pump and seed beams. The Rochon
prism additionally served as a control for pump power through tuning of an
upstream half-wave plate. The beams were then reduced in size to approximately
1-mm 1/e2 diameter and recollimated in a 4x shrinking magnification telescope.
Two steering mirrors after the telescope were used to align the common beam
path into the CSP crystal.

### Diagnostics

The input pump and seed energies were measured with a thermopile average power
and energy sensor (Ophir 3A). At the output of the CSP crystal, reflections of
the beams from a CaF2 beam sampler were imaged with a CaF2 lens in a 2F
configuration to a microbolometer camera (DataRay WinCamD-IR-BB). A Wollaston
prism was used to spatially separate the beams on the camera by polarization.
The transmission through the beam sampler was used to measure the power of
each beam with the thermopile sensor. Spectral filters (Thorlabs 2250$\pm$500
nm, Thorlabs 3500$\pm$500 nm, and Andover 3050$\pm$100 nm) were used to
isolate each beam for spatial profile and power measurements. The crystal
angle, signal alignment, and pump timing were adjusted to simultaneously
optimize the beam shape and power of the signal beam. For the signal
conversion efficiency measurements, the beam sampler was removed and the
3500$\pm$500 nm filter was used to isolate the signal.

### Numerical Model

The 2D spatial plus 1D temporal HPA and OPA simulation results reported in
Figs. 3 and 4 employed a 1D temporal grid numerically propagated along the
crystal axis using a Fourier split-step approach that captured the full
dispersion series of CSP (using Sellmeier equations reported in
ZAWILSKI20101127 ) and a unidirectional propagation model of nonlinear wave
mixing (as described in Flemens:21 ) between signal, idler, idler SH, and pump
fields by quadratic-order electric polarizability. This captured OPA between
pump, signal and idler fields, and frequency doubling of the idler field to
the idler SH. In addition, to capture possible parasitic wave mixing, we
included two additional fields and nonlinear processes: SHG of the signal to a
signal SH field, and OPA between signal and idler SH to a field at their
difference frequency. To capture 3D spatiotemoporal dynamics, we defined the
initial transverse field profiles of the pump and signal to be perfect
Gaussians in space and time. All other fields were initialized to zero.
Simulations captured collinear propagation through a CSP crystal with theta
angle of 43.9 degrees.

Fields were sampled in space by a 62 by 62 grid, and the temporal pulse
propagation dynamics for each spatial grid point were computed individually by
the Fourier split-step method described above. Thus, linear temporal effects
(including group-velocity walk-off, group-velocity dispersion, and higher
order dispersion) were accounted for, but diffraction and spatial walk-off
were not. This was justified by separately carried out simulations including
diffraction and spatial walk-off that indicated only minor features due to
these effects, and which was corroborated by the excellent match between
experimental and numerical data found in our study, as shown in Figs. 3, 4,
and Supplementary Fig. 1. As noted, however, small discrepancies between
experimental and numerical data are likely due to minor Poynting-vector walk-
off of the extraordinary-polarized pump and idler SH beams from the ordinary-
polarized signal and idler beams, which was not captured in the simulations.
The pump and signal durations were chosen based on the filter bandwidths in
the experiment resulting in transform limited pulse durations of 1.2 ps for
the 2.2 µm pump and 0.9 ps for the 3.39 µm signal. The pump and signal beam
diameters were 1.0 and 1.2 mm, respectively. The signal seed energy matched
that of the experiment at 0.7 nJ and the pump energy was varied over the range
used in the experiment.

In order to produce the data shown in Fig. 4, the field intensities were
integrated over the temporal dimension for each spatial grid point. The data
shown in Fig. 3 were generated by additionally integrating over both spatial
dimensions.

### Calculation of Non-collinear Phase-Matching Angles

Non-collinear phase matching data reported in Fig. 2, bottom, were determined
from Sellmeier equations (LiNbO3: Zelmon:97 ; BBO: Tamosauskas:18 ; DKDP:
Kirby:87 ; CSP: ZAWILSKI20101127 ), using the approach of Ref. Dean2021Apr .

## Extended Data

Figure 1: (Extended Data) Top: Spectral measurements for the incident seed and
pump (at 1W power), and the emitted amplified signal and idler SH beams (after
48-dB signal gain), all normalized to 1. Corresponding simulated spectra
(middle) and temporal intensity profiles (bottom).

Extended Data Figure 1 Top: Spectral measurements for the incident seed and
pump (at 1W power), and the emitted amplified signal and idler SH beams (after
48-dB signal gain), all normalized to 1. Corresponding simulated spectra
(middle) and temporal intensity profiles (bottom).

## Supplementary Information

Supplementary Fig. 1, Section 1, and Video 1.

## Declarations

### Funding

This material is based upon work primarily supported by the National Science
Foundation (NSF) under Grant No. ECCS-1944653. Additional support is
acknowledged for the development of specific experimental infrastructure used
for this work: by the U.S. Department of Energy (DOE), Office of Science,
Basic Energy Sciences (BES), under Award #DE-SC0020141 (seed laser
infrastructure and pulse diagnostics), by the Department of the Navy, Office
of Naval Research under Grant No. N00014-19-1-2592 (pump laser development),
and by the Cornell Center for Materials Research with funding from the NSF
MRSEC program (DMR-1719875) (pump laser and propagation simulation code
development).

### Authors’ contributions

J.M. supervised the study. N.F. and J.M. conceived and planned the
experimental and numerical studies. N.F. performed the numerical analysis.
N.F. performed the measurements and analysed the data with assistance from
C.D. N.F. and J.Z. developed the pump laser source. D.H. and N.F. developed
the seed laser source. D.J.D. calculated noncollinear phase matching angles.
K.Z. and P.G.S. manufactured the CSP crystal. All the authors discussed the
results and contributed to writing the manuscript.

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