1. Introduction

Sixty years since the identification of nonlinear three-wave mixing as a means for amplification of light [1–4], optical parametric amplification (OPA) continues to be a major research area. More recently, a rise in research activity was fueled by the emergence of Ti:sapphire and Yb-doped solid-state laser amplifiers that can be used as powerful OPA pump sources in the near-infrared, giving rise to significant advances to meet the needs of nonlinear spectroscopy, nonlinear imaging, strong-field and relativistic optics applications [5–12]. Throughout the wide frequency range of sources, new records are regularly set for pulse duration, intensity, pulse energy and average power. However, poor amplifier efficiency is usually a limiting factor.

Poor efficiency is the result of two problems. First is the energy lost to an undesired wave due to the splitting of pump energy between signal and idler photons (energy lost $= \omega _{\textrm {idler}}/\omega _{\textrm {pump}}$ or $\omega _{\textrm {signal}}/\omega _{\textrm {pump}}$, depending on whether the seeded signal or unseeded idler wave is desired) – the so-called quantum defect. This problem is especially severe when a transfer of energy to a much lower frequency is required.

Second is the problem of spatiotemporally asynchronous conversion of pump photons, which is usually discussed in terms of an intensity-dependent exponential gain coefficient, $G \propto \textrm {exp}(gz)$, where $g$ is proportional to the square-root of the local pump intensity, which varies in time and space. However, the root of this problem is the cyclic flow of energy between waves – known as conversion-back-conversion cycles – and the cycle period’s dependence on the pump intensity (Fig. 1) – a problem identified at least four decades ago and investigated in several works [13–15]. For conversion of all available pump photons to the signal and idler frequencies, all spatiotemporal coordinates of interacting waves must share a common length at which the peak of the conversion cycle occurs, since only one crystal length can be chosen. This cannot occur for free-space OPA with bell-shaped beam and pulse profiles. The maximum photon conversion efficiency can be estimated analytically via the exact Jacobi elliptic function solutions of the process (neglecting dispersion and diffraction) [15]. When each pump profile is Gaussian and the signal wavelength is seeded uniformly in space and time (as if by a much longer seed pulse and wider seed beam), only 20% of pump photons are converted for an amplifier with a peak gain of 100. When the peak gain is $10^5$, the number drops to 10%. These poor numbers, combined with the quantum defect, underlie the many published reports of energy conversion from pump to signal in the 5-20% range for $\omega _s>\omega _i$ and sub-percent to few percent when $\omega _s\ll \omega _i$. Similar efficiencies are obtained for the unseeded wave (i.e., the idler). The roughly one to two order of magnitude drop in energy from pump to signal and idler beams greatly increases the cost of OPA systems while limiting their power and thus the scientific or industrial applications made accessible by them.

 

Fig. 1. In OPA, conversion of pump to signal and idler occurs cyclically (a) with pump depletion (b) having a strong dependence on local pump intensity due to the variation in cycle period. (c) At the optimal crystal length for a Gaussian spatiotemporal pump intensity profile (gray dashed line in (b)), the profile wings remain mostly undepleted in both space and time, a severe source of OPA inefficiency. Similarly, the signal gain coefficient (d) depends on the pump-intensity-dependent conversion cycle period, leading to gain narrowing in the signal and idler spatiotemporal profiles that reflects the inefficient extraction of power from the pump (e).

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To mitigate the problem, pulse and beam shaping have been proposed. Uniform pump and seed profiles eliminate the asynchronous spatiotemporal conversion problem, and flattop-profile pump shaping in one or more transverse dimensions has been implemented successfully in some systems, e.g., [16,17]. Idealized conformal profiles [14,15] have been proposed for maximizing energy conversion and amplification bandwidth, as well as passive methods for achieving conformal-like profiles, such as cascaded extraction [18] and pump pulse reshaping caused by spatiotemporal variations in impedance matching within an enhancement cavity [19,20]. For most applications, however, the pump shaping is a challenge, either due to implementation complexity, or because the pulse shaping itself imparts too significant a loss to be practical for boosting efficiency.

Alternatively, recent works have focused on forcing evolution dynamics with suppressed back-conversion. Adiabatic frequency conversion [21,22], which uses the nonlinear optical analog of rapid adiabatic passage via a swept phase-matching condition, is one such method. Adiabatic frequency conversion can achieve full photon population inversion in the presence of a strong wave in sum- and difference-frequency generation and four-wave mixing Bragg scattering [23–27]. However, in OPA, where the aim is to deplete the strong pump wave, there is still a bandwidth-efficiency trade-off [28,29]. Thus, while efficiency gains have been observed in adiabatic OPA processes, the technique is used more widely to extend bandwidth [22,30–35]. Recently, dissipative variants of OPA were proposed – named quasi-parametric amplification (QPA) or dissipative OPA – in which the introduction of material losses to the idler wave or spatial separation of the idler damps out conversion cycles by preventing the recombination of signal and idler photons that returns energy to the pump [36–40]. In one such work, the addition of a samarium dopant to a bulk yttrium calcium oxyborate crystal allowed 41% energy conversion efficiency from pump to signal [36], with conversion cycles largely damped across the majority of the spatiotemporal profile, thus making the asynchronicity of the conversion cycle irrelevant. These promising findings provide hope for the development of widely applicable solutions to the spatiotemporally inhomogeneous conversion problem in OPA based on back-conversion suppression, and thus for achieving energy efficiency approaching the quantum defect limit.

In this work, we introduce a new back-conversion suppression approach for solving the OPA conversion efficiency problem that can work in ordinary bulk nonlinear media (Fig. 2). We find that the nonlinear evolution dynamics resulting from simultaneously phase-matched OPA and idler second harmonic generation (SHG) are those of a damped nonlinear oscillator (Fig. 3). This closed, conservative system, which we call second harmonic amplification (SHA), mimics the damped oscillatory behavior and convergence to a static steady state characteristic of OPA with linear loss (such as QPA). However, as the system is fully parametric there is no loss: all of the energy remains in coherent optical fields at the end of the device. To demonstrate the close similarity of dynamical behavior between SHA and QPA and their relationship to conventional OPA, we further introduce a Duffing oscillator model of parametric amplification that unifies the description of all three processes under a generalized theoretical framework.

 

Fig. 2. The second harmonic amplification (SHA) concept, consisting of simultaneous OPA and SHG of the idler. (a) Photon energy exchange diagram. (b) Schematic of the desired energy exchange within a device phase matched for both processes.

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Fig. 3. (a) Evolution dynamics of SHA. The idler is converted to its SH during OPA gain saturation, damping signal back-conversion and eventually converting all pump energy to signal and idler SH fields, with the evolution dynamics of a damped oscillator. The quantum defect, $\omega _s/\omega _p$ (dashed line), sets a conversion efficiency upper limit. (b, d) The existence of conversion cycle damping regardless of local pump intensity enables (c, e) spatiotemporally uniform pump depletion at a suitable length (dashed line in (b, d)).

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To demonstrate the promise of the SHA approach to strongly boost OPA efficiency, we perform spatiotemporal pulse propagation analyses in implementations relevant to modern ultrafast laser systems with ordinary Gaussian intensity profiles: birefringent phase matching in cadmium silicon phosphide (CSP) is predicted to allow conversion of a 1-ps, 2-$\mu$m laser to 3-$\mu$m with 55% energy efficiency, corresponding to 80% pump depletion after 50 dB gain, and a readily manufacturable lithium niobate (LNB) quasi-phase matching (QPM) structure can amplify 180-fs, 1.63 $\mu$m signal pulses with 68% depletion of a 1-$\mu$m pump resulting in 44% energy conversion to the signal. In both cases, net unidirectional energy flow over the majority of the spatiotemporal extent of the interacting waves is observed, eliminating spatial and temporal gain narrowing, and with a several-fold improvement in conversion efficiency compared to the performance of conventional OPA. The SHA process thus appears to offer a solution to the longstanding problem of OPA inefficiency due to spatiotemporally inhomogeneous conversion for many modern ultrafast laser systems. Finally, phase-matching bandwidth and other practical considerations are discussed.

2. Theoretical analysis

2.1 Second harmonic amplification (SHA) wave-mixing system

We use the naming convention of ‘idler’ to mean the unseeded wave in OPA. Incorporating idler SHG with OPA results in coupled evolution equations for four nondegenerate fields,

When $|\Delta k_{SHG}|\gg 0$, the SHG process cannot deplete energy from the OPA system significantly and the system dynamics are quantitatively similar to those of ordinary OPA (Fig. 1(a)), with cyclical conversion. However, numerical integration of the four coupled equations when $\Delta k_{SHG}\cong 0$ results in qualitatively different monochromatic, plane-wave evolution dynamics involving the mixing of all four waves (Fig. 3(a)). We may reason that the initial behavior, i.e., before the onset of signal gain saturation takes place, is well approximated as pure OPA. This can be seen through examination of the idler evolution equation, Eq. (1c). For the case of $\Delta k_{OPA} = \Delta k_{SHG} = 0$, the change to the idler wave is equally weighted by a gain term, $A_p A_s^*$, and a loss term, $A_{2i} A_i^*$. However, for the initial conditions of an optical parametric amplifier, $|A_{p,0}| \gg |A_{s,0}|$ and $A_{i,0} = A_{2i,0} = 0$, gain must strongly exceed loss until significant pump depletion occurs and thus $|A_p| \approx |A_s| \approx |A_i|$. These initial dynamics are illustrated in Fig. 3(a), with little observed deviation from ordinary OPA evolution through the first conversion peak.

Significant idler SH intensity appears shortly before full pump depletion, at which point qualitative changes to the behavior compared to conventional OPA are observed in the form of damped oscillations of energy between the pump, signal and idler, with a step-wise monotonic displacement of intensity to the idler SH. We refer to this hybrid process as SHA. The pump and idler fields reduce asymptotically to zero intensity while the signal field asymptotically approaches its quantum-defect-limited intensity (i.e., the creation of one signal photon for each annihilated pump photon). The SH intensity asymptotically approaches $(\omega _{2i}/2\omega _p)=(\omega _{i}/\omega _p)$ times the initial pump intensity (i.e., the creation of one idler SH photon for every two annihilated pump photons). A reversal of the process is not observed. We note that similar behavior was observed in self-doubling OPO when investigated numerically in single-pass operation, where in contrast the investigation studied the case of the signal (seeded wave), rather than the idler, experiencing simultaneous SHG [42].

A number of experimental works have observed signal or idler SHG simultaneously phase matched during OPA (often as an undesired, parasitic process, though sometimes with the goal of obtaining the upconverted wave, see, e.g., [44] for some examples), but without investigating the four-wave dynamics occurring beyond the first conversion peak. Here we observe that the dynamics of SHA beyond the first conversion half-cycle enable a solution to the spatiotemoporal inhomogeneous conversion problem of OPA. Independence of the qualitative dynamics to the local initial pump intensity can be seen in Fig. 3(b); at lower intensity, the dynamics are merely slowed. As in QPA, efficiency optimization can now be carried out in a different way from the conventional approach. Rather than choosing a crystal length where the greatest fraction of the spatiotemporal energy distribution is close to the first conversion peak, a crystal length is chosen at which the conversion oscillations have been damped over the majority of the spatiotemporal extent (Fig. 3(c)), thus allowing nearly full pump depletion and conversion.

2.2 Damped Duffing oscillator universal model of parametric amplification

Further insight into the nonlinear evolution dynamics can be gained by nondimensionalizing Eqs. (1):

The fraction of photons in the $j$th field relative to the total number of initial photons, which we refer to as the fractional photon number of the $j$th field, is given by $n_j \equiv \left | u_j \right |^2$. There are two independent Manley-Rowe equations describing conservation of fractional photon number,

 

Fig. 4. Fractional photon number exchange dynamics under perfect phase matching conditions and corresponding photon mixing diagrams for OPA processes involving the displacement of idler photons. (a) SHA: OPA with simultaneous idler SHG. (b) QPA: OPA with simultaneous idler linear absorption. All cases: $\gamma _0=0.35$, $n_{p,0}=0.9$. $n_d$ is fractional photon number of displaced idler photons.

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Additional insight into the dynamics can be obtained by examining a single-field equation for the pump obtained from Eqs. (2) by use of the Manley-Rowe equations. Setting $\Delta _{OPA}=\Delta _{SHG}=0$, we differentiate Eq. (2a) and combine with Eqs. (2b), (2c), (3a) and (3b) to find:

Equation (4) for the pump field is the force equation of a damped, undriven Duffing oscillator. The first term is a linear restoring force that decreases as the number of displaced idler photons, $n_d(\zeta )=2n_{2i}(\zeta )$, grows. The second term acts as a nonlinear softening of the restoring force. Under the constraints of OPA initial conditions and the Manley-Rowe equations, the sum of the first two terms must always be negative (i.e., the force is always restoring). The third term results in damping given by the coefficient $\gamma (\zeta )=\gamma _0 \sqrt {n_{d}(\zeta )/2} =\gamma _0 \sqrt {n_{2i}(\zeta )}=\gamma _0 u_{2i}$, which grows monotonically from zero as idler photons are unidirectionally displaced to the idler SH field. While $u_i$ may switch signs between conversion cycles, $u_{2i}$ does not since it grows in proportion to $u_i^2$ (Eq. (2d), with $\Delta _{SHG}=0$). Thus, $\gamma (\zeta )$ never switches sign, meaning the oscillator is always damped and never experiences gain.

We note, when $\gamma (\zeta )=0$ (i.e., when there is no SHG), Eq. (4) corresponds to the case of conventional OPA, and is the undamped, undriven cubic Duffing equation, $d_{\zeta }^2 u_p = -(1+n_{p,0}) u_p + 2 u_p^3$, which is well known to have Jacobi elliptic function solutions [45], the known solutions to OPA [3]. In terms of the Duffing equation, Eq. (4), the system parameter $\gamma _0$ can now be understood as a damping parameter. Since the damping coefficient of the Duffing equation, $\gamma (\zeta )$, is dependent on the number of photons in the idler SH field (an evolving quantity), one might not expect evolution with the standard characteristics of a damped oscillator. However, since the fractional idler SH photon number monotonically approaches a steady-state value, $n_{2i}(\infty ) = \lim _{\zeta \to \infty } n_{2i}(\zeta ) = n_{p,0}/2$, the damping coefficient also monotonically approaches a steady-state value related to $\gamma _0$ by $\gamma (\infty ) = \lim _{\zeta \to \infty } \gamma (\zeta ) = \gamma _0\sqrt {n_{2i}(\infty )} = \gamma _0\sqrt {n_{p,0}/2} \simeq \gamma _0/\sqrt {2}$. Indeed, Fig. 5, which shows a numerical solution of Eqs. (2) for $\gamma (\infty )=$ 0, 0.2, 1.0, and 4.0, clearly depicts the four regimes of a damped oscillator: undamped, underdamped, critically damped, and overdamped, respectively. By virtue of $\gamma _0 =\Gamma _{SHG}/2\Gamma _{OPA}$, and thus $\gamma (\infty ) \simeq \gamma _0/\sqrt {2} = \Gamma _{SHG}/2\sqrt {2}\Gamma _{OPA}$, these values of $\gamma (\infty )$ also correspond to no SHG, stronger OPA, stronger SHG, and much stronger SHG, respectively. For each value of $\gamma (\infty )$, the dynamics of $n_p(\zeta )$ are shown for three initial values of $n_{s,0}$ corresponding to 10, 30, and 50 dB photon gain. The effect of a decreased initial fractional signal photon number (higher gain) is primarily a delay of the onset of the oscillatory dynamics. Once they begin, the oscillations have nearly identical amplitude, frequency, and decay rate. Notably, by Eq. (3a), in each case the signal fractional photon number asymptotically approaches $1$, meaning every pump photon produces a signal photon, equivalent to the full quantum-defect-limited conversion efficiency in energy, $I_s = I_{s,0} + (\omega _s/\omega _p)I_{p,0}$.

 

Fig. 5. Pump fractional photon number evolution obtained by numerical solution of Eqs. (2) under various damping conditions, $\gamma (\infty )= \gamma _0\sqrt {n_{p,0}/2}$, and initial fractional signal photon number, $n_{s,0}$. These can be understood in terms of the Duffing equation model as the damping regimes of an anharmonic oscillator: (a) undamped: $\gamma (\infty )=0$ (no SHG), (b) underdamped: $\gamma (\infty )=0.2$ (stronger OPA), (c) critically damped: $\gamma (\infty )=1$ (stronger SHG), and (d) overdamped: $\gamma (\infty )=4$ (much stronger SHG). $n_{s,0}$ = 0.1, 0.001, and 0.00001 correspond to a maximum signal photon gain of 10 (solid), 30 (dashed), and 50 dB (dotted), respectively.

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For a practical amplifier device, regimes near critical damping (Fig. 5(c)) are ideal from a design perspective as oscillations are avoided and convergence to full pump depletion occurs at the shortest length. We note, when indices of refraction $\textrm {n}_p \approx \textrm {n}_s \approx \textrm {n}_i \approx \textrm {n}_{2i}$ (as is typical for four off-resonant fields) the damping parameter $\gamma (\infty )$ reduces to $\approx \sqrt {\omega _i \omega _{2i}/2\omega _p \omega _s}$. Thus, critical damping occurs when the signal and idler frequencies are related by the ratio $\omega _i/\omega _s \approx (1+\sqrt {5})/2$, the golden ratio. We note, in this regime, $\omega _{2i} > \omega _p$, providing a route to efficient upconversion of a laser through generation of the idler SH.

As SHA is a conservative, closed nonlinear system, its close resemblance to a dissipative system is remarkable, and begs comparison of SHA to OPA with actual loss, i.e. QPA, where irreversible removal of idler photons is achieved by linear absorption [36,40] rather than by SHG. This is modeled by replacement of Eq. (2c) with

Table 1. Parameters of the Duffing oscillator equation, Eq. (4), for each process. $n_d(\zeta )$ is the fractional photon number displaced from the idler field by loss (QPA) or SHG (SHA) and $\gamma (\zeta )$ is a damping parameter.

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While QPA and SHA exhibit close dynamical behavior, a key physical distinction is that an incoherent linear loss mechanism prevents coupling of the lost energy back to the system in QPA, while the idler SH field of SHA coherently drives the idler field throughout propagation and the irreversibility of power flow from idler to idler SH is a consequence of the coherent dynamics at degeneracy. This phenomenon enables the conservative SHA system to have a loss-like channel – mimicking a heat bath – through its full nonlinear evolution. These distinctly different physical damping mechanisms have different practical consequences. In QPA, idler photons are dissipated by material absorption and lost as heat, while in SHA, a nonlinear polarizability converts them to a coherent copropagating wave. At the end of the medium, this wave can be separated from the amplified signal by a beamsplitter, allowing complete removal and possible reuse of its energy.

3. Spatiotemporal propagation analysis

We employed a spatiotemporal propagation analysis for coupled signal, pump, idler, and idler SH waves to predict the success of two realistic amplifier designs based on SHA. Propagation equations for cw fields of finite spatial width including diffraction and Poynting vector beam walk-off were first employed to identify a range of beam sizes where these spatial propagation effects play a negligible role in the field evolution (see Appendix 1). The four coupled equations for femtosecond pulse propagation were then solved for beam sizes within this range for each spatial coordinate independently, using the exact material dispersion given by Sellmeier equations. This allowed us to perform a full spatiotemporal analysis of all non-negligible propagation effects consistent with a collinear geometry. Nonlinear polarization terms beyond quadratic order were not included. (See Appendix 2 for propagation equations.)

To reflect the bell-shaped profiles of real lasers, in all simulations, the initial pump and signal intensity profiles were 1st-order Gaussian in both temporal and spatial extent. The two spatial beam dimensions were set equal and rotational symmetry about the propagation axis is assumed. The peak intensity of the pump was chosen to be below the reported damage threshold of each nonlinear medium. Pulse durations are given in FWHM and beam radius in $1/e^2$ dimension.

3.1 Birefringent phase matching in CSP

We begin by investigating SHA achieved via birefringent phase matching, a scheme relevant for high-energy amplification in a bulk medium. Our analysis uses CSP, a material relevant to mid-infrared applications of OPA, and a 2.05-$\mu$m pump wavelength relevant to Ho- and Tm-doped solid-state gain media. Our analysis predicts that a high-energy 2-$\mu$m picosecond laser can be downconverted to 3.0 $\mu$m with less than a factor of two loss in energy, a severalfold efficiency improvement compared to standard OPA.

SHA requires simultaneous phase matching of OPA and idler SHG processes, or $\Delta k_{OPA}=\Delta k_{SHG} \approx 0$. CSP is uniaxial with broad phase-matching bandwidth for Type-I OPA pumped at 2.05 $\mu$m [47,48]. At a crystal orientation given by $\theta =44.8^\circ$, simultaneous phase matching occurs at a 3.0-$\mu$m signal, corresponding to a 6.5-$\mu$m idler and 3.25-$\mu$m idler SH (Fig. 6). While $\Delta k_{SHG}$ varies more rapidly with frequency than $\Delta k_{OPA}$, our analysis predicts sufficient bandwidth to support efficient SHA for pulses down to $\sim$1 ps.

 

Fig. 6. Type-I birefringent phase-matching curves for simultaneous OPA (blue) and idler SHG (orange) processes in CSP with tuning angle $\theta =44.8^\circ$ and a 2.05-$\mu$m pump. Simultaneous phase matching occurs for a 3.0-$\mu$m signal, corresponding to a 6.5-$\mu$m idler and 3.25-$\mu$m idler SH. (Sellmeier coefficients taken from [46].)

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Figure 7 compares SHA (left column) with standard OPA (right column) for high-gain (50-dB) amplification of a 1-ps, 2-nJ, 2-mm radius, 3.0-$\mu$m signal wave in a few millimeters of CSP, pumped by an 870-$\mu$J, 2.05-$\mu$m wavelength pump of the same duration and 1-mm radius, corresponding to a peak intensity (50 GW/cm$^2$) below the damage threshold of CSP. These parameters correspond to $\gamma _0=0.27$, putting the SHA system in the underdamped regime. In SHA (Figs. 7(a) and (e)), the evolution of energy exchange over the full spatiotemporal profile of the waves displays an asymptotic approach to full conversion, with damped conversion cycles as typified by underdamping. Full damping at the spatiotemporal profile center for over 1 mm of propagation allows depletion of the spatiotemporal wings to catch up before any significant back-conversion occurs at the center. Back-conversion remains suppressed just past the optimal length of 2.5 mm, where 80% pump depletion (Fig. 7(c)) and 55% energy conversion to the signal (Fig. 7(g)) is observed with >10$^5$ gain in a single stage. The remaining 45% of the energy is split between the idler at 6.5 $\mu$m (3%), the idler SH at 3.2 $\mu$m (22%), and the unconverted pump (20%). The effect of temporal walk-off, resulting in a slight temporal asymmetry, is observed but does not significantly interfere with device performance. Figure 8 shows the SHA seed, amplified signal, and idler SH spectra and temporal profiles. The amplified signal takes on the initial pump profile with 1.0-ps duration, i.e., almost no temporal gain narrowing is observed – a result of the near-uniform spatiotemporal conversion. In contrast, for conventional OPA (Figs. 7(b),(d),(f),(h)), only 12% pump depletion and 8% energy conversion to the signal can be achieved (at the optimal length of 0.7 mm, corresponding to approximately one half-period of the conversion-back-conversion cycle at the spatiotemporal peak). Accordingly, significant spatiotemporal gain narrowing of the amplified conventional OPA signal is observed. The conversion efficiency of SHA is $\sim 7$ times greater than that of conventional OPA in this example.

 

Fig. 7. Spatiotemporal evolution of SHA (left) vs. conventional OPA (right) in CSP with birefringent phase matching. (a,b) Pump depletion dynamics, with optimum length for conversion efficiency indicated (dashed line). (c,d) Intensity profile of the residual pump at the optimum length. (e,f) Signal amplification dynamics. (g,h) Signal intensity profile at the optimum length. SHA results in 55% pump-to-signal energy conversion vs. 8% for conventional OPA.

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Fig. 8. Spatially integrated (a) spectra and (b) pulse profiles of amplified signal and idler SH in the CSP SHA device.

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In Figs. 7(a) and (e), we observe an eventual reversal of energy flow in SHA that returns energy to the pump field. Since pulsed laser beams possess a distribution of wave-vectors and frequencies, some phase mismatch is inherent in any real application. Figure 9 shows the intensity dynamics at the center of each pulse ($(x,t)=(0,0)$). At $z \gtrsim$ 2 mm, energy can be observed returning from idler SH to idler field and, subsequently, from signal and idler to pump field. These modified dynamics are almost exactly matched by numerical integration of Eqs. (2) setting either $\Delta k_{SHG}$ or $\Delta k_{OPA}$ equal to $\pm$0.033 mm$^{-1}$ (dotted curves), indicating a nonzero average wave-vector mismatch magnitude.

 

Fig. 9. Intensity evolution of the four fields at $t=x=0$ showing eventual back-conversion. Dotted lines: corresponding solution of Eqs. (2) for $(\Delta k_{SHG})\Delta k_{OPA} = (-)$0.033 mm$^{-1}$.

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3.2 Superlattice quasi-phase matching in LNB

While the introduction of a noncollinear geometry could be used to tune the signal frequency of the birefringent phase matching approach above, QPM offers additional flexibility as well as Type-0 geometries with large nonlinear coefficient and the absence of spatial walk-off. In this example, our analysis predicts that a 1-$\mu$m pump laser can amplify chirped 180-fs transform-limited (TL) pulses at 1.63 $\mu$m in OPCPA configuration in a LNB superlattice QPM device, producing millijoule output with > 50 dB gain and an energy conversion efficiency of 44$\%$ – a five-fold efficiency increase compared to standard OPA – while simultaneously producing a chirped 100-fs TL idler SH pulse at 1.40-$\mu$m with 21$\%$ conversion efficiency.

In a previous study, an aperiodically poled LNB device for simultaneous QPM of OPO and SHG was designed for the purpose of self-doubling OPO [43] using a numerical, iterative approach. Numerous other approaches exist for the design of multiple-process QPM, e.g., [44,49–52]. Here we employ the approach of [49], which for the phase matching of two simultaneous processes can have the simple form of a superlattice with inverted 50%-duty-cycle multi-domain stacks (Fig. 10(a)), where $\Lambda _{\pm }= 4\pi /(\Delta k_{OPA} \pm \Delta k_{SHG})$. We further use a rounding approach: each domain wall of the max$(\Lambda _{+},\Lambda _{-})$ periodic lattice is rounded to the nearest domain wall of the min$(\Lambda _{+},\Lambda _{-})$ periodic lattice. This final step ensures a QPM structure that only has one manufacturable domain size of width min$(\Lambda _{+},\Lambda _{-})/2$ even for noncommensurate values of $\Lambda _{\pm }$, where there are occasionally repeated positive or negative domains.

 

Fig. 10. (a) Superlattice QPM device schematic. (b) The Fourier transform of a LNB device consisting uniformly of 16.6-$\mu$m domains reveals sharp Bragg peaks at $\Delta k_{OPA}$ and $\Delta k_{SHG}$.

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We designed a superlattice QPM device for SHA with a 1.03 $\mu$m pump and 1.63 $\mu$m signal in Mg:LiNbO$_3$ in Type-0 configuration ($eee$). Simultaneous SHG doubles the 2.8 $\mu$m idler to a 1.4 $\mu$m idler SH. The device is generated from superimposed noncommensurate poling periods $\Lambda _{+}$ = 33.2 $\mu$m and $\Lambda _{-}$ = 425 $\mu$m with rounding as described above to eliminate small domains, resulting in a structure consisting uniformly of 16.6-$\mu$m domains of alternating sign of $d_{\textrm {eff}}$ and some repeated positive or negative domains. The spatial Fourier transform of its domain poling function (Fig. 10(b)) has sharp Bragg peaks with nearly equal magnitude at the exact values of $\Delta k_{OPA}=$ 0.204/$\mu$m and $\Delta k_{SHG} =$ 0.174/$\mu$m.

Numerical integration of the SHA wave-mixing dynamics in this structure illustrates efficient amplification of femtosecond signal pulses (Fig. 11). We modeled 50-dB amplification in an OPCPA configuration with a 5.5-ps, 2.7-mJ, 30-GW/cm$^2$ pump and a 120-fs, 8-nJ, 31-nm FWHM seed at 1.63 $\mu$m chirped to 5.6 ps. These chirped pulse durations mitigated the effects of group-velocity walk-off. Figures 11(a) and (c) shows clear underdamped SHA dynamics. At 7.4 mm, nearly homogeneous depletion of the pump in space and time is observed (Fig. 11(b)), resulting in 68% pump energy depletion with 44% going to the signal (1.2 mJ, Fig. 11(d)) with a 180-fs FWHM TL pulse duration and 21% going to a 1.40-$\mu$m, 100-fs TL idler SH (Figs. 11(e) and (f)). For comparison, a standard periodically poled QPM structure optimized for conventional OPA (not shown) results in 15% pump energy depletion, with only 9% of the pump energy going to the signal – a 5-fold decrease.

 

Fig. 11. Spatiotemporal evolution of a LNB superlattice QPM device employing SHA to efficiently amplify femtosecond near-IR pulses in OPCPA configuration. (a,c) Pump and signal dynamics, indicating spatiotemporally homogeneous amplification at a length of 7.4 mm, resulting in 44% pump-to-signal energy conversion. (b,d) Pump and signal profiles at 7.4 mm. (e) Spectra indicating 50-dB signal amplification. (f) TL pulse durations of 180 fs and 100 fs for amplified signal and idler SH, respectively.

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4. Further discussion

A detailed study of the phase matching of SHA in common OPA materials is beyond the scope of this article, but here we discuss several important considerations. In the example of birefringent phase matching above (Section 3.1), we observed efficient amplification of a 1-ps pulse. This example was chosen to make use of the full phase-matching bandwidth available for the chosen pump and signal wavelengths in a collinear geometry. The phase-matching constraints for simultaneous OPA and SHG are naturally greater than those of either process alone; as was the case in this example (Fig. 6), the two processes have greatly different gradients $d \Delta k/d \lambda _s$, and the larger gradient sets the bandwidth of the joint process. Thus, e.g., even if the OPA phase-matching bandwidth is broad, a narrow SHG phase-matching bandwidth can lead to a narrow SHA bandwidth. In the example of QPM in LNB (Section 3.2), both $d \Delta k_{OPA}/d \lambda _s$ and $d \Delta k_{SHG}/d \lambda _s$ were small enough at the crossing point of the phase-matching curves to allow for efficient amplification of sub-200 fs pulses.

Specification of the phase-matching bandwidth for SHA in terms of system parameters is less straightforward than for OPA. As seen in Fig. 9 (dotted lines), a nonzero wave-vector mismatch in either the OPA or SHG process causes an eventual reversal of the conversion dynamics, with conversion-back-conversion oscillations growing until the system returns a significant portion of the energy to the pump. For efficient conversion in SHA, the goal is for every frequency to experience fully damped pump-signal conversion cycles. Thus, an essential question is whether a return of power to the pump wave due to wave-vector mismatch sets in before the conversion cycles are fully damped. Figures 12(a) and (b) shows the dependence of the length at which a reversal of the dynamics occur on $\Delta k_{OPA}$ and $\Delta k_{SHG}$, respectively. In either case, if $\Delta k$ is small enough, a window in $z$-space exists where conversion cycles are fully damped, allowing efficient SHA. For the particular case shown in Fig. 12, $|\Delta k| =$ 1.0 mm$^{-1}$ appears to be the marginal value. Thus, one would expect that as long as all involved frequencies possess $\Delta k_{OPA}, \Delta k_{SHG} <$ 1.0 mm$^{-1}$, a length exists at which all frequencies experience fully damped SHA, and mostly uniform spatiotemporal pump depletion can still take place.

 

Fig. 12. Effects of small wave-vector mismatch (in mm$^{-1}$) on SHA for $\gamma _0=0.35$ and $\Gamma _{OPA}=13$ mm$^{-1}$. (a) Nonzero $\Delta k_{SHG}$ leads to the eventual return of conversion-back-conversion cycles and regrowth of the pump wave. (b) A nearly identical effect is seen for nonzero $\Delta k_{OPA}$. (c) Simultaneous wave-vector mismatch of the two processes can enhance or partially offset the regrowth of the pump, depending on the relative signs of $\Delta k$.

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However, notably, the quantitative effects of small nonzero $|\Delta k_{SHG}|$ and $|\Delta k_{OPA}|$ are nearly identical, and Fig. 12(c) shows that one may even be used to partially offset the other. The dynamical behavior associated with nonzero $\Delta k_{SHG}$ is intuitive: $\Delta k_{SHG}z=\pi$ reverses the displacement of idler photons to the idler SH, turning idler loss into idler gain in Eq. (1c) and thus causing oscillations to grow rather than to decay. The similar effect on the dynamics due to nonzero $\Delta k_{OPA}$, however, is less intuitive, and requires additional study. We note that the offsetting behavior of two wave-vector mismatches common to a single field has been noted earlier in the context of cascaded third-order processes [53]. Further investigation of the offsetting behavior in SHA also requires further study.

Collinear birefringent phase matching of idler SHG requires a specific material orientation, e.g., the theta angle for a uniaxial crystal, and so choice of an idler wavelength fixes the theta angle. Once this is fixed, there is only one free parameter available for phase matching the OPA process: the pump wavelength or signal wavelength, since they are related by the OPA requirement $\omega _p - \omega _s = \omega _i$. As a result, for any pump wavelength and given nonlinear medium, in a collinear geometry there is likely only one signal wavelength (and corresponding idler) where $\Delta k_{SHG} = \Delta k_{OPA} = 0$. However, the use of a non-collinear angle between pump and signal should enable signal-wavelength tunable SHA for a given pump wavelength through the additional degree of freedom for phase matching. A superlattice QPM structure with Bragg peaks at both $\Delta k_{SHG}$ and $\Delta k_{OPA}$, as used in Section 3.2, can also provide arbitrary choice of signal and pump wavelengths within a range. Thus, in practice, non-collinear birefringent phase-matching and QPM allow some flexibility of choice of the signal wavelength that is required for a particular application. Alternatively, if the shortest possible duration is desired, one might select the signal wavelength that has the broadest phase-matching bandwidth. This approach allowed us to obtain a TL bandwidth supporting a sub-200-fs pulse by QPM in Section 3.2. It is unclear whether the phase-matching of SHA for few-cycle pulse applications will be feasible. However, the joint use of QPM and noncollinear angle is one possible avenue for investigation.

With phase-matching bandwidths supporting amplification of pulses down to sub-ps duration, SHA is clearly relevant to the down-conversion of many high-energy and high-average-power solid-state lasers. In the investigated CSP device, for example, a picosecond 2-$\mu$m laser is converted to a 3-$\mu$m amplified signal with >50% energy efficiency. SHA therefore provides a route to efficiently translate the frequency of picosecond solid-state lasers. As it works in ordinary bulk nonlinear media without absorption, SHA appears particularly well suited for high energy and high average power applications. As noted earlier, SHA can also up-convert the pump wave to a fractional harmonic of the pump frequency, producing an idler SH with frequency $1 < \omega _{2i}/\omega _p < 2$, when $\omega _i > \omega _s$. SHA is therefore also relevant to the up-conversion of ultrafast solid state lasers, which, if a frequency other than $2\omega _p$ is desired, normally requires an inefficient two-stage device consisting of sequential OPA and SHG (in either order). Thus, SHA can be expected to allow a significantly improved up-conversion efficiency.

For high efficiency, we note it is essential to seed the amplifier only with wavelengths that are appropriately phase-matched for both OPA and SHG processes. For example, if part of the seeded bandwidth is phase-matched for OPA only, this component of the pulse will compete for gain and experience ordinary OPA dynamics that lead to spatiotemporally inhomogeneous and inefficient amplification. Furthermore, to enable full depletion of the pump wave, its entire spatiotemporal extent should be overlapped with adequate seed field. For this reason, seed intensity profile sizes should be equal to or larger than the co-propagating pump profiles, as was the case in each of the device investigations of Section 3.

We note that for devices operating in the underdamped regime, a consequence of a finite propagation length is that the signal’s spatiotemporal profile will exhibit a small modulation that grows in amplitude toward the edge of the profile, as observed in Figs. 7 and 11. This modulation period is roughly 2-5 times smaller than the 1/e$^2$ radius of the beam for the two devices studied here and will lead to the higher spatial frequency content of the signal beam diffracting away at roughly 4-25 times the rate of diffraction for the main Gaussian beam in the absence of relay imaging.

On account of SHA’s high photon conversion efficiency and the availability of the coherent idler SH at the output of the device, one might envision using cascaded SHA stages to obtain a pump-to-signal energy conversion efficiency that reaches or even exceeds the OPA quantum defect limit (i.e., an efficiency $\geq \omega _s/\omega _p$), as the substantial power in the idler SH field could be reused in a second SHA stage to further amplify the signal. This scheme, depicted in Fig. 13, works if $\omega _{2i}>\omega _s$ (equivalent to $\omega _s < (2/3)\omega _p$). Defining $\eta _i$ as the pump photon depletion efficiency for the i$^{th}$ stage (and assuming there are no residual idler photons), the overall energy conversion efficiency of pump to signal in this scheme is $(\eta _1^2+\eta _1 \eta _2/2)(\omega _s/\omega _p)$. Thus, for example, it is possible for the pump-to-signal energy conversion efficiency to equal the quantum defect limited efficiency for a per-stage photon conversion efficiency $\eta _1 = \eta _2 = 81.6\%$. A maximum energy conversion efficiency at 150% above the quantum defect is theoretically possible for $\eta _1 = \eta _2 = 100\%$. In the example of Section 3.1, the pump photon depletion efficiency is 80% and only 3% of the incident pump energy is lost to the idler wave, and is thus close to but not quite at the efficiency threshold needed for exceeding the quantum limit if repeated in an equally efficient subsequent stage where the idler SH is resused to pump the signal. This scheme therefore presents an intriguing future direction for achieving ultra-efficient parametric amplification, and may be especially useful for mid-wave and long-wave mid-infrared amplifiers pumped by near-infrared lasers, which possess a low quantum defect and are therefore very inefficient.

 

Fig. 13. Cascaded SHA stage scheme potentially capable of exceeding the quantum-limited OPA efficiency, $\omega _s/\omega _p$, for a high enough single-stage photon conversion efficiency. The idler SH from the first stage is reused to amplify the signal in the second stage, such that $\omega ^{'}_p = \omega _{2i}$ and $\omega ^{'}_s=\omega _s$. Each stage would incorporate an appropriate material and phase-matching scheme for SHA.

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In summary, we have found the evolution dynamics of hybridized OPA and idler SHG – a process we have termed ‘second harmonic amplification’ – to be suited to overcome one of the major problems limiting the conversion efficiency of parametric amplifiers: the spatiotemporally inhomogeneous conversion that is a result of the intensity-dependent conversion-back-conversion cycles that typify the evolution dynamics of conventional OPA. The dynamics of SHA are those of a damped anharmonic oscillator, describable by a Duffing oscillator model that also describes the dynamics of conventional OPA and OPA with linear idler loss (as due to absorption). Our analysis thus unifies the description of OPA and its variants that incorporate the displacement of idler photons – including the recently demonstrated QPA approach [36,40] – and may be useful for the analysis and design of many parametric amplifier systems.

Moreover, two device examples provided and investigated by a numerical analysis indicate a high expected pump-to-signal energy conversion efficiency (as high as 55% in one example) in common bulk nonlinear media used for OPA and for the bell-shaped pump beam and pulse profiles commonly obtained from pump lasers. Our examples included devices pumped by picosecond lasers at 1-$\mu$m and 2-$\mu$m wavelength that produce amplified signal pulses and accompanying idler SH pulses with picosecond and $<$ 200-fs TL duration. SHA is thus predicted to be a concept for back-conversion suppression and high efficiency in OPA with relevance to many applications, and may be especially useful for extending the reach and lowering the cost of high-power OPA-based laser systems. SHA might also be used for efficient upconversion of a pump laser to a selectable fractional harmonic frequency. Further exploration of the birefringent phase-matching range of common materials for SHA, including noncollinear geometry and temperature tuning approaches, and multi-process QPM techniques as well, may lead to wide applicability. An initial analysis of amplifier bandwidth and the dynamics of phase-mismatched SHA was given, but a more expansive analysis – including an investigation of bandwidths approaching the few-cycle limit – will be important for the possible expansion of the SHA approach to the few-cycle regime. As the provided amplifier examples reflect realistic ultrafast laser systems and manufacturable devices, we expect an experimental investigation of our findings will be forthcoming.

Appendix 1: evaluation of diffraction and beam walk-off in SHA

We performed an initial analysis of beam propagation effects in order to find the smallest beam size where diffraction and Poynting vector walk-off are negligible for the SHA device considered in our study. Monochromatic fields at frequencies $\omega _{j,0}$ for signal, pump, idler, and idler SH waves with 1D spatial Gaussian beam profiles were propagated in the plane of Poynting vector walk-off (spatial coordinate x) using the four coupled pulse propagation equations for OPA and idler SHG (shown in the spatial Fourier domain) with diffraction and Poynting vector walk-off:

(6a)$$d_{z}E_s(k_x) = i \frac{\omega_{s,0} d_{\textrm{eff}}}{n_s c} \mathcal{F} \{E_p(x) E_i^*(x)\} - i k_s(\omega_{s,0})E_s(k_x) + i \frac{k_x^2}{2k_s(\omega_{s,0})} E_s(k_x)$$

(6b)$$\begin{aligned}d_{z}E_p(k_x) = i& \frac{\omega_{p,0} d_{\textrm{eff}}}{n_p c} \mathcal{F} \{E_s(x) E_i(x)\} - i k_p(\omega_{p,0})E_p(k_x) + i \rho_{p} k_x E_{p}(k_x)\\ &+ i \frac{k_x^2}{2k_p(\omega_{p,0})} E_p(k_x) \end{aligned}$$

(6c)$$\begin{aligned}d_{z}E_i(k_x) = i& \frac{\omega_{i,0} d_{\textrm{eff}}}{n_i c}\mathcal{F} \left\{E_p(x) E_s^*(x) + E_{2i}(x) E_i^*(x)\right\} -i k_i(\omega_{i,0})E_i(k_x)\\ &+ i \frac{k_x^2}{2k_i(\omega_{i,0})} E_i(k_x) \end{aligned}$$

(6d)$$\begin{aligned} d_{z}E_{2i}(k_x) = i& \frac{\omega_{2i,0} d_{\textrm{eff}}}{2n_{2i} c} \mathcal{F} \{E_i^2(x)\} - i k_{2i}(\omega_{2i,0})E_{2i}(k_x) + i \rho_{2i} k_x E_{2i}(k_x)\\ &+ i \frac{k_x^2}{2k_{2i}(\omega_{2i,0})} E_{2i}(k_x) \end{aligned}$$

where $E_j(k_x)=A_j (k_x)e^{i k_j(\omega _{j,0})z}$, and $A_j$, $k_j$, and $n_j$ are the signal, pump, idler, and idler SH electric field amplitudes, wave vectors in the nonlinear medium, and indices of refraction, respectively, and where $k_x$ and $z$ are the transverse spatial and propagation coordinates. The effective quadratic nonlinear coefficient is given by $d_{\textrm {eff}}$ and $c$ is the speed of light. Terms quadratic in $k_x$ represent diffraction in the paraxial regime. Terms linear in $k_x$ represent Poynting vector walk-off, which in CSP is present for pump and idler SH but not signal and idler, as both OPA and SHG processes are Type-I ($o + o = e$). The walk-off angles for the pump and idler SH are given by $\rho _p$ and $\rho _{2i}$, respectively.

Figure 14 shows the results for the CSP device simulated in Section 3.1, where both diffraction and spatial walk-off are of concern. $\rho _p$ and $\rho _{2i}$ for the 1.03 $\mu$m pump and 3.25 $\mu$m idler SH in this case are 16.64 $\mu$rad and 16.82 $\mu$rad, respectively. For the 2.55 mm optimal crystal length used in the simulation, this corresponds to $\sim$40 $\mu$m of walk-off for the pump. Figure 14 shows the spatial simulation results of solving Eqs. (S1)-(S4) for three beam sizes, $1/e^2$ beam radii 1, 0.5, and 0.2 mm. For a 1 mm beam radius (the pump beam radius used in the CSP device example in the main text), we find negligible contribution from diffraction and walk-off, resulting in dynamics identical to the simulations where these effects are not included (top two panels). Decreasing to a 0.5 mm beam radius, a very slight deviation appears in the last 10% of the crystal length, with minor back-conversion setting in just as the crystal terminates. Decreasing even further to a 0.2 mm beam radius, there is no longer a region within the crystal where conversion cycles are fully damped out. These results show that ignoring spatial effects is an excellent approximation for a 1.0 mm pump beam radius and marginal at a 0.5 mm beam radius. At 0.2 mm radius, diffraction and spatial walk-off have a strong effect on the evolution by inducing mixing between fields at different spatial coordinates that disturbs the local dynamics of SHA.

 

Fig. 14. Spatial beam propagation dynamics for Type-I ($ooe$) phase-matched OPA and idler SHG in CSP at time = 0 for (a) pump beams of radius 1, 0.5, and 0.2 mm (1/$e^2$) and (b) the corresponding signal. The first panels of (a) and (b) show the 1 mm beam radius case where all spatial propagation effects are neglected from the simulation. Simulation parameters correspond to the device parameters in Section 3.1.

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This analysis illustrates that spatial effects can disturb the SHA dynamics, preventing uniform spatiotemporal amplification. However, the use of suitably large beams circumvents the problem.

Appendix 2: numerical model for full spatiotemporal propagation analysis

For the CSP and LNB devices simulated in the main text, diffraction and spatial walk-off were found to have negligible effect as a result of the large beam size. Thus full spatiotemporal evolution (two transverse spatial dimensions plus one temporal dimension) along the propagation axis could be solved without spatial derivative terms. Each coordinate of a radially symmetric transverse spatial axis ($r$) was calculated independently with a temporal grid to capture temporal propagation effects. Using a Fourier split-step method, we solved the following four coupled pulse propagation equations for OPA and idler SHG (shown in the frequency domain), accounting for the exact frequency-dependent dispersion, $k_j(\omega )$, given by published Sellmeier equations:

(7a)$$d_{z}E_s(\omega,r) = i \frac{\omega_{s,0} d_{\textrm{eff}}}{n_s c} \mathcal{F} \{E_p(t,r) E_i^*(t,r)\} - i k_s(\omega)E_s(\omega,r) $$

(7b)$$d_{z}E_p(\omega,r) = i \frac{\omega_{p,0} d_{\textrm{eff}}}{n_p c} \mathcal{F} \{E_s(t,r) E_i(t,r)\} - i k_p(\omega)E_p(\omega,r) $$

(7c)$$d_{z}E_i(\omega,r) = i \frac{\omega_{i,0} d_{\textrm{eff}}}{n_i c} \mathcal{F} \left\{E_p(t, r) E_s^*(t,r) + E_{2i}(t,r) E_i^*(t,r)\right\} - i k_i(\omega)E_i(\omega,r) $$

(7d)$$d_{z}E_{2i}(\omega,r) = i \frac{\omega_{2i,0} d_{\textrm{eff}}}{2n_{2i} c} \mathcal{F} \{E_i^2(t,r)\} - i k_{2i}(\omega)E_{2i}(\omega,r) $$

where $E_j (\omega ,r)=A_j (\omega ,r)e^{i k_j (\omega _{j,0})z}$. The power and power spectrum can then be computed by integration over space: $P(t)=4\pi c \epsilon _0 \int _0^\infty |E_j(t,r)|^2rdr$ and $S(\omega )=4\pi c \epsilon _0 \int _0^\infty |E_j(\omega ,r)|^2rdr$, respectively. Integrating either of these over time yields the total energy. This model allowed us to capture all non-negligible propagation effects consistent with a collinear geometry. Nonlinear polarization terms beyond quadratic order were not included.

Funding

NSF Directorate for Engineering (ECCS-1944653); Directorate for Mathematical and Physical Sciences (DMR-1719875).

Acknowledgments

This work was partially supported by the Cornell Center for Materials Research with funding from the NSF MRSEC program, Directorate for Mathematical and Physical Sciences (DMR-1719875).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are available in Ref. [54].

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