Concurrent Terahertz Generation via Quantum Interference in a Quadratic Media

The strive for efficiency in the generation of terahertz (THz) waves motivates intense research on novel field–matter interactions. Presently, THz generation via quadratic crystals remains the benchmark thanks to its simple and practical deployment. An interesting problem is whether new mechanisms can be exploited to elicit novel generation approaches and forms of control on the THz output in existing systems. THz generation via quantum interference (QI) leverages a third‐order nonlinear response under resonant absorption, and it has been recently explored to access surface generation in centrosymmetric systems. Its deployment in standard THz quadratic sources can potentially create a physical setting with the concurrence of two different mechanisms. Here, THz generation via QI in noncentrosymmetric crystals concurrent with phase‐matched quadratic generation in a bulk‐transmission setting is demonstrated. Beyond investigating a new physical setting, it is demonstrated that conversion efficiencies much larger than those typically associated with the medium become accessible for a typically adopted crystal, ZnTe. An inherent control on the relative amplitude and sign of the two generated THz components is also achieved. This approach provides disruptive boost and management of the optical‐to‐THz conversion performance of a well‐established technology, with significant ramifications in emerging spectroscopy and imaging applications.


Introduction
The nonlinear generation of terahertz (THz) waves from ultrafast optical excitation is a relatively established approach to generating intense broadband pulses. [1][2][3][4] The strive for better efficiency and larger THz fields motivated the creation of a broad spectrum of nonlinear media with more significant nonlinearities [5,6] or more complex interaction geometries. [7][8][9] From a fundamental point of view, the opticalto-THz conversion efficiency is limited by the need to maintain the phase-matching condition between fields across the entire generated bandwidth. [10] Hence, the need for large bandwidth conversion results in very thin crystals. In most practical scenarios, however, the nonlinear net conversion becomes vanishingly low for thicknesses below the optical pump wavelength. Nevertheless, the performance of many emerging applications, i.e., near-field imaging and microscopy, critically depends on minimizing the distance between the target and source planes. [11][12][13][14][15][16] New generation mechanisms like spintronics emission and nonlinear surface generation represent viable options [17][18][19] because of their exceptional nonlinear response per unit of length. However, noncentrosymmetric bulk media are still the platform of choice due to their well-understood physics, relative efficiency, and mature compatibility with laser sources. Notably, from the mid-90s, the broad compatibility with ultrafast Ti:Sa lasers contributed to the diffusion of the zinc telluride (ZnTe) crystal as a benchmark THz source to such a level that it is still considered a recognized reference in many works on nonlinear emitters. [18,[20][21][22][23] A more exotic and much less explored scenario exploits additional nonlinear mechanisms in commonly deployed THz bulk sources. The underlying idea is to elicit multiple concurrent physical processes contributing to the THz generation to gain performance and capabilities. In this context, we recently proposed exploiting surface THz generation, mediated by a cubic nonlinear response, via a two-color excitation scheme in media already supporting single-color-driven surface THz generation mechanisms. [24] In this example, the extreme absorption exhibited by the substrate to all pumps limited the generation mechanism to an almost 2D surface region. In fact, the application of two-color nonlinear mechanisms for THz generation in thick condensed matter bulks is generally challenged by the medium dispersion, hence the inability to achieve velocity matching between all the different waves. Conversely, the literature exploiting two-color generation in air plasma is very rich [25][26][27] and has led to some of the best-performing sources in the domain. [28][29][30] It is worth noting that other examples of multiple concurrent mechanisms to THz emission already exist in nature, as surface generation in low-bandgap semiconductors can be notoriously attributed to several different physical processes. [19,31,32] In this framework, the two-color quantum interference (QI) [33][34][35][36][37] is a microscopic mechanism that expresses a thirdorder nonlinear phenomenon in condensed matter elicited by two-color excitation and capable of generating ultrafast current transients. The QI originates from the interference between the wavefunctions of photocarriers promoted via coherent single-photon and two-photon absorption processes. It is typically achieved by exciting a semiconductor with a coherent superposition of an optical pump at angular frequency and its second-harmonic such that the medium energy band gap gap lies between the energy of the pump photons ℏ and the second harmonic ones 2ℏ , i.e., ℏ < gap < 2ℏ . Figure 1 shows a phenomenological interpretation of the typical quadratic optical rectification process (OR) and the QI. In particular, Figure 1b presents a typical phenomenological description of the origin of the QI-local currents as the unbalance of population for a specific absolute waveform momentum k (forward vs backward) for the promoted photoelectrons. The unbalance manifests in a kinetic momentum for electrons with a sign determined the interference of two phase-shifted coherent absorption pathways. In the case of ultrafast optical excitation, the broad spectrum of wavenumbers covering the population is expressed by an ultrafast current transient, which is the source of a broadband THz field pulse. [38] Relevant to this work, this also means that the fundamental wave does not experience absorption. Hence, we can explore the unexplored physical setting where QI is concurrent to, and potentially interacts with, separate nonlinear generation mechanisms supported by the nonlinear crystal.
In this work, we investigate the THz generation mechanism in a noncentrosymmetric crystal, originating from a superposition of the bulk optical rectification (OR- Figure 1a) and QI. Our experimental results demonstrate that the rapid absorption of the second harmonic excitation prevents the decoherence of the QI THz emission as it propagates, an effect that is normally elicited by the velocity mismatch of the generating pumps (Figure 1c). Although the QI emission is supported by a thin volume at the input surface, the exhibited conversion per unit of length orders of magnitude higher than the corresponding one for bulk OR, placing the two fields generated within the same order of magnitude for commonly adopted ZnTe media.

Modeling and Methodology
We consider the generation of a 0.5mm-thick 〈110〉 ZnTe crystal. We define E( ) the field of the optical pump at angular frequency and E(2 ) its second harmonic. The central wavelength of the pump photons is set to fulfil the phase-matching condition for THz generation via bulk OR, i.e., = 800 nm for ZnTe crystals. A full description of the experimental setup is included in the Experimental Section and a schematic is available in Figure 2d. A summary of the light-matter interactions in our experimental conditions is illustrated in Figure 1. Bulk OR is a difference frequency generation between the broadband pump frequency components, such that the generated THz field E OR is proportional to the intensity of the optical pump field via the second-order susceptibility of the material 2 , e.g., E OR ∝ 2 E( )E*( − ). For traditional bulk sources, the bandwidth of the nonlinear conversion is essentially limited by the phase mismatch of the propagating fields accumulated in propagation and by frequency-dependent phonon-driven absorption within the medium. [2] The QI generation process, conversely, is triggered by introducing an additional synchronous second harmonic excitation ( 2 = 400 nm) (i.e., simultaneous one-and two-photon absorption pathways), which exhibits a penetration depth of L 2 ≈ 70 nm [39] (the ZnTe energy bandgap corresponds to a photon wavelength gap ≈ 514 nm). Within this propagation length, phasematch-driven decoherence does not play any significant role. Following canonical third-order nonlinear modeling, we can express the THz field generated via QI as Microscopically, the interference between two-and singlephoton absorption promoted photoelectrons corresponds to the local injection of a current J i , where the subscript i indicates a spatial coordinate, acting as the source of the third-order nonlinear process. [40] The current evolution is expressed as follows [40,41] where is a fourth-rank tensor element related to a divergent, imaginary part of a third-order optical susceptibility 3 (0; , , −2 ) (i.e., a nonlinear absorption coefficient) for two-color optical rectification, [38,40] and , 2 are the phases of the fundamental and second harmonic field components. R represents the characteristic carrier relaxation/recombination time. The phase term in Equation (1) highlights that both the amplitude and direction of the injected current are controlled by the phase difference between the two fields. In the case of ultrafast optical excitation, introducing a controllable delay between the two pulses leads to a fast periodic beating along the delay coordinate superimposed to a wide slope originating from the crosscorrelation of the two pulse envelopes. [24,42] Figure 2 shows a typical experimental result highlighting the cooperation/competition between the two generation mechanisms. Specifically, we present the time-domain spectroscopy (THz-TDS) of the generated THz wave for different delays between the two pumps. By changing the delay between the fundamental and second-harmonic excitation pulses (Figure 2b), the generated field is characterized by constructive and destructive interference between the OR-and QI-generated contributions. As illustrated in Figure 2e, the interference pattern along the delay coordinate at the TDS peak exhibits a bell-shaped envelope of the overall interference pattern, resulting from the change of overlap between the two pump pulses (i.e., the change of their cross-correlation product).

Experimental Results
Interestingly, as visible in Figure 2c,e, the QI contribution leads to a peak field enhancement exceeding 16% (≈35% in power) at the peak field delay when compared to the OR generation alone. Considering that the quantum interference generation occurs within one penetration depth (70 nm), while the OR scales with the full crystal thickness (0.5 mm) therefore we infer conversion efficiencies per unit length on the scale of 2000 times greater than OR. Furthermore, the QI generation is effectively independent of the overall thickness of the generation medium. Hence, for commercially available crystals of 80 μm thickness [2] the QI could effectively double the absolute conversion efficiency. For thinner crystals, it would likely represent the dominant process.

QI+OR Symmetry and Crystal Orientation
An interesting feature of this combined generation setting is that the second and third-order nonlinear interactions inherently manifest different orientational symmetry properties due to the different structure and symmetry of the two nonlinear tensors. [43][44][45] We explored this aspect by analyzing the THz emission for different rotations of the nonlinear crystal about the 〈110〉 face. Our experimental results are shown in Figure 3a,b, where we recorded the dependence of the THz peak field on the ZnTe rotation angle at different delays CP . In Figure 3c, we isolated the relative OR and QI contribution by subtracting the minimum signal from the maximum signal, dividing by two. We observe a faint, yet visible threefold symmetry. This suggests that the medium's quadratic response couples the coherent fundamental and second-harmonic pumps, which is the same of considering a faint second harmonic contribution converted from the pump in the ZnTe, interfering with second harmonic pump. Thus, we attribute the specific behavior to a perturbative change in the second harmonic contribution as the crystal is rotated. Indeed, this suggests an extremely weak inherent dependence of the QI process on the crystal rotation. On the one side, this is coherent with the observation that the optical absorption does not exhibit significant change. However, this can also stem from the way the different contribution of the projections of the imaginary nonlinear tensor composes in the rotation (similar to THz generation in the cubic surface optical rectification from 〈100〉 InAs surfaces). [46] Our experimental results show that we can gain control over the nonlinear processes. By coupling the crystal orientation with the rotation of the CP, we are able to tailor the emissions of both optical rectification and quantum interference.

Conclusion
We demonstrated QI-mediated THz generation in a quadratic crystal hosting a concurrent phase-matched optical rectification generation process. By employing ZnTe, an established reference emitter in the field, we demonstrate that the combination of OR and QI easily surpasses the current nonlinear conversion limits. Since the QI generation mechanism is drastically localized on the crystal surface region, it is unaffected by phase mismatch. Yet, its contribution to the generated THz field can match, and in terms of emission per unit length, even surpass the one from standard bulk emission, significantly raising the efficiency bar for a quite deployed emission configuration. Furthermore, the coherent in-terference with the quadratic optical rectification realizes a quite exotic physical setting and enables new forms of control over the generated field.

Experimental Section
The experimental setup comprised a fundamental beam (FH-= 800 nm, 80 fs, 1 kHz ≈1.0 mJ, beam diameter 3.5 mm) copropagating with its second-harmonic (SH-= 400 nm, ≈35.4 μJ) generated by a type-I process in a thin (0.1 mm) BBO crystal. The generated SH was cross-polarized with respect to the fundamental beam. A birefringent CP (a standard group velocity compensation plate from EKSMA Optics) introduced a tuneable phase delay CP (via rotations perpendicular to the extraordinary plane), between the two cross-polarized fields as a result of the change in the optical path length between the fundamental and second harmonic. A second HWP centered at 800 nm rotated the polarization of the fundamental field resulting in the two pumps being co-polarized when they impinged on a 0.5 mm thick ZnTe 〈110〉 which served as generation crystal. The THz signal was collected through standard TDS detection, implemented via electro-optical sampling exploiting a THz field-induced birefringence in a second 〈110〉 ZnTe crystal of thickness 3.5 mm.
The thickness of the detection crystal limited the detection bandwidth to ≈1.2 THz at −20 dB to the spectral peak.
It was worth noting that the measured THz bandwidth was hard-limited by the thickness of the detection crystal, which was much larger than the emission crystal thickness. A complete schematic of the optical setup, including the waveplates and calcite configurations, is available in ref. [24].