1. Introduction

High-energy single-cycle terahertz pulses (“THz”), available from table-top setups, are in high demand to advance fields such as THz strong field physics [1–3] or novel THz based particle accelerators and manipulators [4–7]. Due to both its versatility and scalability, intra-pulse difference frequency generation (DFG) of intense ultrashort laser pulses in nonlinear materials developed into the most applied method for the generation of high-energy single-cycle THz radiation. Among a wide range of nonlinear materials, lithium niobate (LN) has become a popular choice due to its high effective nonlinear coefficient, the availability of cm-sized crystals and high optical damage threshold as well as low absorption [8]. However, low frequency phonon resonances cause a significant difference between the refractive index at THz and optical frequencies, effectively preventing the applicability of broadband collinear phase-matching techniques. A solution to this problem was pioneered by Hebling and coworkers via utilizing a pump pulse with continuously tilted intensity front such that the projected group velocity of the laser pulse matched the THz wave velocity [9]. This solution allows the THz pulse to build up coherently as it slides along the pump laser intensity front and significantly improved conversion efficiencies of broadband table-top THz sources. The generated THz levels were pushed to new limits by reducing the free-carrier generation through use of longer pump wavelengths [10–15], reducing the THz absorption via cryogenically cooling the crystal [16], chirping the pump pulses [17,18] as well as optimization of the imaging configuration [15,19–22] used to image the grating into the nonlinear crystal. Current reported conversion efficiencies reach beyond the so called Manley-Rowe limit by a process known as cascaded DFG where redshifted pump photons are effectively reused multiple times for generating THz photons [23–27]. A consequence of the cascaded nonlinear interactions is a spectral broadening of the pump pulse which, together with the angular dispersion required for phase-matching, causes a temporal break-up of the pump pulse which ultimately limits the conversion efficiency [28]. While modified setup schemes are in development to overcome these limitations [21,29–32], the conventional tilted-pulse-front scheme remains the method of choice for high energy single-cycle THz generation with pulse energies recently reported beyond the mJ-level [33].

Despite the current ubiquity of the tilted-pulse-front technique, the complexities associated with the geometry make the practical implementation and optimization difficult and non-transparent to the non-expert. In particular, fine-tuning of the performance and tailoring of the THz beam properties requires a detailed understanding of the dependences on the setup parameters. While extensive parameter studies have been carried out via simulations [34–39], there has been a deficit of experimental work that corroborates the predictions and verifies the dependence of the performance and THz-pulse properties on the key parameters.

In this work, we report on the results of multi-dimensional parameter scans to systematically map out the performance of our THz source on the primary interaction parameters. These include the pump-pulse fluence, the pulse-front tilt γ, the distance of the pump beam from the crystal apex d_a and the size of the pump beam σ_x^f. We compared our experimental results to the simulations in [35,37] and confirm several predictions for the first time, including the effect of pump fluence on the optimum interaction length. In addition, we find an optimum for the pump-beam size and characterize the THz spatial mode behaviour as a function of various experimental parameters. Finally, we outline a step-by-step optimization procedure to robustly determine the global optimum of performance for tilted-pulse-front THz setups independently of the pump wavelength and pulse duration. This procedure provides a methodology to disentangle the complexities of the tilted-pulse-front geometry and enables consistent comparison of THz setups which can help to experimentally answer remaining questions such as the optimum pump pulse duration.

2. Experimental setup

The tilted-pulse-front setup shown in Fig. 1(a) was pumped by a 2-stage cryogenic Yb:YLF amplifier system delivering high energy pulses centered at 1020 nm with a pulse duration of ≈ 1.35 ps at 10 Hz repetition rate [40]. Two half-wave plates (λ/2) adjust the pump polarization for optimum grating diffraction efficiency and maximal effective nonlinear coefficient of the nonlinear conversion process. The grating parameters and lenses for the imaging system were selected such that the image of the grating was approximately aligned with the tilted pulse-front of the pump pulse inside the crystal. The holographic gold-coated diffraction grating (1500 l/mm), mounted on a precision rotation stage, introduces angular dispersion to the pump beam resulting in a tilted-pulse-front. This tilted-pulse-front is imaged into the nonlinear crystal (congruent 5% MgO doped lithium niobate, cut angle of 62.4°) via a cylindrical Keplerian telescope consisting of two cylindrical lenses (L₁ and L₂) with a constant demagnification factor of M = f₂/f₁= 0.6721 ± 0.0007 (see Fig. 1(b)). The pump energy was controlled via a half-wave plate mounted in a motorized rotation stage before a thin-film-polarizer. The transmission through this assembly was calibrated vs the waveplate angle for a given incident energy measured by energy meter (Coherent Inc.). The size of the pump beam was adjusted by insertion of cylindrical Galilean telescopes with fixed magnification factors upstream of the tilted-pulse-front setup (not shown in Fig. 1(a)). For determination of the pump fluence, the pump beam size was characterized via CCD beam profilers (DataRay Inc. and Ophir). The crystal was positioned on top of a motorized translation stage for movement along the z-axis. The transverse crystal position along the x-axis was adjustable via a manual translation stage. This way, the length L of material the pump pulse propagates through was controlled via setting the distance between crystal apex and pump d_a (see Fig. 1(c)).

 

Fig. 1. (a) Schematic of the experimental setup with the relevant setup parameters and coordinate frames. (b) Determination of the magnification M of the imaging telescope (L₁ and L₂) by the square root of the derivative of the image distance d₂ with respect to the object distance d₁. Here, M was determined as 0.6721 ± 0.0007. (c) Schematic illustrating the interaction lengths L and L’ along the path of propagation for pump and the THz beam respectively. Due to the non-collinear interaction geometry in tilted-pulse-front THz setups, both these lengths must be considered for an effective interaction. L and L’ are controlled via the setup parameters d_a and σ_x^f.

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The generated usable THz pulse-energy was measured using two large-area pyro-electric THz detectors: An uncalibrated but sensitive detector (SDX-1152, MT-coating) from Gentec-EO, and a calibrated but less sensitive THz detector from SLT Sensor- und Lasertechnik GmbH (THz 20, calibration certificate 73426 PTB 18) with a flat frequency response across the 0.1–3 THz range. A cross-calibration of both detectors with broadband THz pulses centered around ≈ 200 GHz yielded a calibration factor of (74 ± 3) mV/µJ for the signal provided by the Gentec detector, from which the signal was acquired using a digital storage oscilloscope. The beam profile of the THz pulses was recorded using a sensitive, low-noise THz detector with a small 2 × 2 mm active element (WiredSense GmbH) that was raster-scanned through the THz beam using a motorized x’-y’-z’-translation stage assembly. The detector signal was integrated, and background corrected using a fast-readout digital oscilloscope. The same detectors were used to record the THz spectrum via Fourier transforming the recorded interferogram from a Michelson type THz interferometer utilizing a pellicle beam splitter. To avoid distortions of the measurement by residual infrared or second harmonic light, all THz detectors were covered by a sheet of black polyethylene (1.9 mm thickness) with a measured average calibrated transmission of ≈ 54%. The setup performance was studied at both room-temperature (≈ 293 K) and with a cryogenically cooled crystal (≈ 82 K). For experiments at cryogenic temperatures, the crystal was mounted inside a vacuum chamber with AR-coated windows for the pump beam and a 5 mm thick TPX window for the THz beam to exit. The crystal was cooled via a copper cold-finger indium bonded to a reservoir of liquid nitrogen and its temperature was monitored with a cryogenic-temperature sensor. If not explicitly marked otherwise, the THz energy values reported in this work refer to the amount of usable THz energy measured outside of the lithium niobate crystal (room-temperature) or outside the vacuum chamber (cryogenic temperature). For the evaluation of the degree of optimization as well as convenient comparison to simulated results in the literature, efficiencies inside the material are also estimated based on the calibrated transmission of windows and the crystal refractive indices in both the infrared and THz frequency range.

3. Results and discussion

The conceptual approach for optimizing the tilted-front-setup is known and has been described in detail by Fülöp et al. in [14,15] and Ravi et al. in [37]. The primary parameters affecting the interaction are the pump-pulse intensity, the pulse-front tilt γ, the distance of the pump beam from the crystal apex d_a and the size of the pump beam σ_x^f. These “interaction” parameters govern the THz generation efficiency via the phase-matching, the effective length L the pump propagates inside the crystal and the length L’ over which the THz interacts with the pump beam before it spatially walks off, respectively. In practice, however, these interaction parameters cannot all be independently controlled. Instead, a set of experimental controls are used which result in changes to more than one of the interaction parameters, introducing ambiguities to the optimization procedure. A key factor in our approach is therefore to perform multidimensional scans to decouple the effects of each parameter and thus allow accurate characterization of the performance dependence on each. The experimental controls used are the pulse energy, the optical beam size, the grating diffraction angle, the magnification of the imaging system and the position of the LiNbO₃ crystal. The results and detailed methodology behind characterizing the THz source performance on the interaction parameters are described in three sections focusing on (3.1) the pulse front tilt, which is controlled by the grating diffraction angle and telescope magnification; (3.2) the interaction length L, which is controlled by the insertion of the crystal into the beam; and (3.3) the THz-optical overlap distance L’, which is controlled by the size of the optical beam at the crystal. The results of these investigations are then compiled into an optimization procedure which is described in the last section (3.4).

3.1 Optimizing phase-matching via pulse-front tilt

Efficient conversion from pump beam to THz requires the two beams to be phase matched. The tilted-pulse-front technique achieves phase-matched interaction in a non-collinear geometry such that the THz beam coherently builds up as it propagates along the tilted intensity front of the pump. This requires the pulse-front tilt angle γ to meet the condition ${\mathrm{\gamma }_{\textrm{opt}}} = \; {\cos ^{ - 1}}\left( {\frac{{{\textrm{n}_\textrm{g}}}}{{{\textrm{n}_{\textrm{THz}}}}}} \right)$ [9]. Mistuning of the pulse-front tilt angle results in a shorter coherence length over which the THz pulse can build up effectively, thus yielding reduced net efficiency. To avoid temporal and spatial distortions in the THz beam, the diffraction grating and imaging system should be chosen such to ensure the image of the grating inside the crystal is aligned with the tilted-pulse-front [15,19,31,41]. Once the optics are set up, finding the optimum pulse-front-tilt requires either the grating angle θ_d or the magnification M of the imaging system to be scanned. Single-lens imaging configurations are frequently used to image the tilted-pulse-front into the crystal [10,13,31,42,43] as they are simple and flexible. By adjusting the position of the lens relative to grating and crystal, the magnification M and thus the pulse-front tilt angle can be tuned. The single-lens scheme, however, comes with several disadvantages including wavefront curvature in the imaged beam [44] and poorer imaging quality, which translate directly into the THz beam properties [21,22]. Tuning the lens position also results in ambiguities in the optimization procedure because the position and angle of the grating image, the beam size and the pulse-front-tilt all change simultaneously. In principle, however, once the grating groove density is chosen, a single optimum of the magnification and grating diffraction angle exists that also ensures that the grating image is parallel to the pulse front tilt set to phase-match the IR to THz conversion, as required by theory [15]. We therefore chose a cylindrical Keplerian telescope with a fixed magnification M as close as possible to the optimum value predicted for the selected grating and tune the pulse front tilt via rotation of the grating. The grating angles scanned across can be converted to γ, which requires these angles to be calibrated and the magnification M to be measured precisely. To ensure high precision in determining the angles of incidence (θ_i) and diffraction (θ_d), the grating angular position was calibrated by confirming zero-order retroreflection of the optical beam at zero incidence angle. In addition, M was characterized carefully by imaging a resolution test target (Thorlabs) and determining the shift of its image location Δd₂ as a function of a shift of the target position Δd₁ as the target was scanned along z. The magnification was then obtained from a linear fit to the data using M² = Δd₂/Δd₁ (see Fig. 1(b)). M was measured to be 0.6721 ± 0.0007, which is very close to the value of 0.66 predicted optimum by the design calculations. The precise knowledge of the magnification and grating diffraction angle allows precise determination of γ at the image location inside the crystal via $\mathrm{\gamma } = {\tan ^{ - 1}}\left( {\frac{{\mathrm{m\lambda \rho }}}{{\textrm{M}{\textrm{n}_\textrm{g}}\cos {\mathrm{\theta }_\textrm{d}}}}} \right)$, where m is the diffraction order, λ the pump wavelength, ρ the grating groove density and n_g the group refractive index at the pump wavelength [15,19].

As the underlying nonlinear conversion process depends on intensity, and thus the pump fluence, one major challenge is to ensure consistency among the results if a setup parameter that alters the fluence is scanned. Here, efficiency vs fluence is chosen as primary metric to compare results. Thus, if any other parameter scanned affects the fluence, this requires the pump energy to be scanned along simultaneously. One such parameter is the grating angle, because a grating both (de)magnifies the pump beam along x by M_g = cos(θ_d)/cos(θ_i) and can have a diffraction efficiency that varies with angle. Therefore, simultaneous scans of the pump energy were carried out during the grating angle scans and a holographic grating was selected that provides constant diffraction efficiency over the range of angles scanned. Last, the pulse-front tilt changes upon propagation of the pump [45], while the pump pulse duration reaches a minimum at the image of the grating [15]. Since the minimum pulse duration corresponds to the highest peak intensity, the grating image position represents the center of where the nonlinear interaction can be driven the strongest. This is only true under the assumption of a fully compressed pump pulse to be incident onto the grating. For chirped pump pulses the plane of optimum conversion is shifted along z to where the GDD associated with angular chirp added by the diffraction grating compensates for the initial pump pulse GDD. As a consequence, the optimum z-position will change as the grating angle is tuned. Therefore, the z position of the crystal needs to be scanned in addition to pump pulse energy and grating angle to optimize the position the plane with optimal pulse duration relative to the end of the crystal where the THz is extracted. Due to the interdependence of these parameters, 3D-parameter scans are therefore required to systematically measure the impact of pulse-front tilt on efficiency and compare the results for optimized z-position in each case. The procedure was performed independently for room-temperature and cryogenic operation of the setup since the precise phase-matching conditions vary with temperature. This robust optimization procedure ensures that both efficiency and THz spectra could be accurately compared for the two modes of operation.

Figures 2(a) and 2(b) show the sensitivity of the efficiency on the pulse-front tilt for different pump fluence levels at both cryogenic and room-temperature. Within the resolution of the scan, no change in the optimum pulse-front tilt angle with pump fluence was observed. This may differ for short pump pulses with high peak intensities because the intensity dependent refractive indices may affect the phase-matching. In our setup, the optical-to-THz conversion efficiency reaches a clear optimum for a pulse-front tilt of γ_opt = 65.64° (293 K) and γ_opt = 64.33° (82 K), indicating optimal phase-matching at this angle. The difference between room-temperature and cryogenic operation is expected since both the THz and IR refractive index of lithium niobate change with crystal temperature [16,46,47]. Therefore, the pulse-front-tilt requires a re-optimization if the setup operation temperature is changed. Here, the grating had to be rotated by 2.4° to re-optimize phase-matching upon switching from 293 K to cryogenic conditions. This is also shown in Fig. 2(c), which illustrates the sensitivity of the THz conversion efficiency to detuning of the pulse-front tilt angle at constant pump fluence level of F = 80 mJ/cm². To stay within the range of 90% of the maximum conversion efficiency, the grating angle θ_d needs to be set with an accuracy of ± 0.44° at 293 K or ± 0.32° at 82 K respectively. The THz refractive index of the crystal can be estimated via the optimum pulse-front tilt angles where phase-matching and thus the condition ${\mathrm{\gamma }_{\textrm{opt}}} = \; {\cos ^{ - 1}}\left( {\frac{{{\textrm{n}_\textrm{g}}}}{{{\textrm{n}_{\textrm{THz}}}}}} \right)$ is fulfilled [9]. Based on the measured setup parameters and the Sellmeier equations reported in [47] for the IR refractive index, we obtain n_THz ≈ 5.35 (293 K) and n_THz ≈ 5.0 (82 K). These estimated values are slightly higher than reported values of around 4.9–5.1 (300 K) in [46,48,49] and 4.7–4.9 (100 K) in [46,49] for congruent lithium niobate with slightly higher MgO-doping.

 

Fig. 2. Measurement of the conversion efficiency as a function of both pulse-front-tilt angle γ and pump fluence F at crystal temperatures of 293 K (a) and 82 K (b). Contour lines (dashed) were added as guide-to-the-eye. The optimal pulse-front tilt angle is found to not depend on the pump fluence for the long pump pulses (≈ 1.35 ps) at use and within the resolution of the measurement. (c) Direct comparison of the normalized efficiency vs pulse-front tilt at a fluence of F = 80 mJ/cm² for both 293 K and 82 K. (d) The THz spectrum in cryogenic operation broadens by ≈ 50% and shifts towards higher THz frequencies compared to the spectrum obtained at room-temperature. (e) THz energy and efficiency measured in dependence of the pump fluence at optimal pulse-front tilt angle γ_opt found for room-temperature operation, while (f) shows the results for a cryogenically cooled crystal.

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The systematic γ-F-z-scans allow for a consistent comparison of the performance between a setup operated at room-temperature and one with a cryogenically cooled crystal. The THz spectra for both configurations are shown in Fig. 2(d). The center frequency ν₀ of the cryogenically cooled setup is blue-shifted by ≈ 80 GHz and the bandwidth (FWHM) Δν increased by approximately 46% (ν₀ = 290 GHz ± 10 GHz, Δν = 190 GHz) compared to the room-temperature spectrum (ν₀ = 210 GHz ± 10 GHz, Δν = 130 GHz). This blue-shift is attributed to a reduced bandwidth of the transverse optical phonon resonance at ≈ 7.6 THz for low crystal temperatures [50]. Figures 2(e) and 2(f) show the generated usable THz energy vs the pump fluence and the corresponding efficiency values. At room temperature operation, the external efficiency reached as high as 0.12% at a fluence level of 160 mJ/cm². At the same fluence-level the conversion efficiency increases by a factor of ≈ 2.7 to 0.31% if the lithium niobate crystal is cryogenically cooled. In cryogenic operation a maximum efficiency of 0.36% was reached when pumping the conversion process in saturation with > 230 mJ/cm². Converted to internal efficiency values (accounting for THz losses at windows and interfaces) 0.22% (293 K) and 0.78% (82 K) conversion efficiency were achieved. Considering the sub-optimal long pulse-duration of the driving laser system, these values are surprisingly high. It should be noted, that during the measurement shown in Fig. 3(b), the THz output surface showed discoloration for a fluence ≈ 250 mJ/cm² which may be a preliminary indication of damage.

 

Fig. 3. a) – d) Sensitivity of the conversion efficiency on crystal position (d_a, z) and beam size σ_x^f for fixed pump fluence (F = 90 mJ/cm²). (a) THz energy measured vs the coordinates d_a and Δz for a pump beam size σ_x^f = 1.2 mm and (b) for σ_x^f = 2.0 mm. Contour lines mark where the efficiency dropped to the labeled percentage of the maximum THz energy. c) Sensitivity of the efficiency on d_a compared for different values of σ_x^f. d) Optimum values for d_a found for each beam size σ_x^f. The range within which the efficiency remains > 90% of its optimum is indicated by the grey bars. e) shows the influence of the pump fluence on the d_a -coordinate sensitivity for σ_x^f = 2.5 mm. The optimum d_a-coordinate shifts towards smaller values with increasing fluence (f), while the range within which 90% of the maximum efficiency were still achieved (grey bars) is unaffected.

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Based upon this robust determination of the optimally phase-matched configuration, it is possible to consistently study the impact of further setup parameters on the nonlinear interaction process.

3.2 Dependencies on the transverse crystal position d_a

Using the optimization of the phase matching from the previous section, the effects of the interaction lengths L and L’ on the performance could then be studied. L is the effective length over which the pump beam propagates through the crystal, while L’ is the length over which the THz beam interacts with the IR beam before it spatially walks off. These two characteristic lengths determine the amount of crystal volume within which THz is generated, the amount of material within which the THz beam experiences absorption and the length over which the pump beam can effectively generate THz. Both L and L’ are accessible via tuning the distance between pump beam and the crystal apex d_a and the size of the pump beam σ_x^f (1/e² intensity radius) as shown in Fig. 1(c). Scanning d_a requires a simultaneous scan of the crystal z-position as otherwise the relative distance between THz output surface and plane of shortest pump duration changes. This way, for a given fluence, the effective length L can be probed via scanning d_a and z. Characterizing the effects caused by changes in L’ is more difficult as it requires a change in the pump size σ_x^f. Beam reshaping upstream of the grating and simultaneous energy scans were required to preserve constant fluence as metric. Changes in pump fluence, however, require re-optimization of d_a, which in turn requires readjustment of the z-position. The reason is that cascading, and therefore the break-up of the pump pulse, increases with fluence. Due to this effect, L is expected to decrease with increasing fluence [37]. Consequently, for each σ_x^f, a complete d_a-z-F-scan needs to be repeated. In total, systematic measurement of the dependence of performance on L and L’ requires a 4-D parameter scan of (σ_x^f, d_a, z and F). First, the focus is set on mapping out the dependence on d_a and a systematic comparison between optima for different pump sizes and fluence levels. In addition, the impact onto the shape of the THz profile is measured. These studies were performed at room temperature with the grating set to γ_opt = 65.64°.

3.3 Impact of the effective interaction length L on the conversion efficiency

The length L over which the pump beam propagates through the crystal can be directly tuned by adjusting the transverse crystal position relative to the beam d_a. As d_a increases, so does L ≈ d_a tan α, where α is the crystal cut angle. On the one hand, longer L means an extended evolution of the nonlinear interaction and a larger portion of material to contribute to the THz generation. On the other hand, the pump pulse broadens spectrally upon further propagation caused by cascaded THz generation as well as self-phase-modulation. In combination with material dispersion and angular dispersion, this spectral broadening degrades the pump pulse temporally and sets a limit to the maximum effective value for L. Last, long values of L mean that THz generated close to the entrance surface for the pump must travel a longer distance to the output surface and thus is absorbed more significantly. To map the dependence of the efficiency on L, however, it is not sufficient to solely scan d_a because this causes a relative shift of the THz output surface with respect to the image of the grating. Such shift Δz matters, because several pump parameters such as beam size, pulse duration and pulse-front tilt change with distance Δz from the image of the grating: the pump beam size increases with distance Δz from the grating image due to the angular spread introduced by the grating. Similarly, the pulse duration increases with Δz due to the group-delay dispersion (GDD) attributed to the angular dispersion [51]. This effect becomes more pronounced for larger bandwidths of the pump pulse, leading to an increased sensitivity on Δz. Simultaneously, the GDD acting on the spatially chirped pump pulse leads to a reduction of the pulse front tilt with increasing distance from the angular disperser or its image respectively [45]. Therefore, the longitudinal crystal position must be adjusted simultaneous to d_a to decouple the effects. To simultaneously study effects of the beam size σ_x^f, the scanned parameter space was extended to a 4D-scan spanning (σ_x^f, F, d_a, z).

In the experiment, the transverse crystal position x was calibrated to d_a by moving the crystal into the beam until half the pump beam was clipped by the crystal apex, where d_a = 0 mm. The longitudinal coordinate Δz refers to the displacement from the z-position found to yield the maximum efficiency. For reference, the fluence values quoted refer to the fluence at the image location of the grating in air. A deviation Δz from the grating image, however, causes the beam to expand due to the angular spread Δθ. Here, Δθ ≈ 10.6 mrad in air (≈ 4.8 mrad in the crystal estimated via Snell’s law) leads to an expansion of σ_x^f by ≈ 50 µm (≈ 24 µm inside the crystal) over a scanning range Δz = ± 10 mm and thus was neglected. For extended scans of the z-position that are long compared to the change in beam size the resulting change in fluence would need to be considered. σ_x^f was adjusted via tuning the size of the beam incident on the diffraction grating σ_xⁱ (1/e² intensity radius). This could be done via a single lens, such as performed in [52]. However, this has been shown to curve the pump wavefront, which affects both the optimum value for d_a and to enhance the sensitivity of the efficiency vs Δz. This enhanced sensitivity is caused by a rapidly changing pulse-front tilt with Δz which translates into less favorable phase-matched THz generation for non-optimal z-positions [44]. Therefore, in this work 4 different cylindrical Galilean telescopes with fixed magnification factors were set up upstream of the tilted-pulse-front setup (not shown in Fig. 1(a)) to each alter the size of the pump beam. In addition to the scenario, where no such telescope was inserted, this resulted in 5 pump beam sizes within the range of 0.5 mm – 3.0 mm that were tested.

Figures 3(a) and 3(b) show two cross-sections through such datasets for a constant fluence level of F = 90 mJ/cm² for 2 different pump beam sizes. As discussed, the two coordinates Δz and d_a cannot be optimized independently from each other. The longer the initial pump pulse durations, however, the less pronounced this effect becomes, which is why the sensitivity on d_a is observed to be higher than on Δz. The datasets presented in Figs. 3(a) and 3(b) qualitatively match simulated results in [44], which however concentrated on much shorter pump pulses (< 500 fs), which is why the optimum efficiency peaks at different coordinates (d_a, z).

Figure 3(c) shows how the efficiency scales with d_a for increasing pump size σ_x^f at fixed pump fluence F = 90 mJ/cm², which was the highest fluence common to all the parameter scans performed for different σ_x^f. At first, the THz yield increases monotonically with growing d_a. This is because for small d_a, the interaction length L is small and only a small section of the crystal is illuminated by the pump. As d_a is increased, so does L and with it the overall amount of material that is pumped. This causes the THz yield to increase. Once a significant portion of the pump beam is set onto the crystal, effects such as THz absorption and optimum interaction length become relevant. That is why the efficiency peaks at an optimum (around d_a ≈ 3.5 mm ± 0.5 mm), after which the efficiency starts to decrease for larger d_a. This decrease in efficiency is based on two effects: First, larger distance from the crystal apex means portions of the generated THz beam need to propagate through an increasing amount of unpumped crystal volume on their way to the crystal output surface, thus increasingly suffering absorption. Second, an increased length L over which the pump beam propagates in the crystal is accompanied by increased cascading-induced spectral broadening, which in combination with angular- and material-dispersion leads to temporal degradation of the pump pulse [37]. Thus, after an optimum value of d_a that balances the effects mentioned above, the efficiency drops with increasing distance of the pump beam from the crystal apex. Since the amount of cascading that occurs depends on the peak intensity, this behavior may differ for broadband pulses because they are only temporally compressed within a short distance Δz from the grating image. Therefore, they can be expected to not be affected as much by cascading effects while they propagate through material before the image plane. On the other hand, more broadband pulses suffer more severely from material dispersion, which limits the amount of material the pulse should be sent through. The slope by which the efficiency drops for d_a > d_a^opt depends on σ_x^f. The larger σ_x^f the less the efficiency drops once the center of the pump beam is moved further from the optimum. This can be understood because for a large beam the relative change in peak intensity with the transverse coordinate is smaller than for a small pump beam. In addition, larger beams average over a larger range of d_a, which means a given movement therefore makes a smaller difference to the average.

Figure 3(d) shows this sensitivity, where the range for d_a within which the efficiency is > 90% of the optimum is indicated by grey bars. This range is observed to double upon increasing σ_x^f from 1.2 mm to 2.5 mm. The optimum apex-distance d_a^opt does not significantly vary with increasing σ_x^f within the accuracy of the measurement. This qualitatively matches theoretical predictions in [37], where d_a^opt shifted on the order of ± 0.1 mm when the pump beam radius was scanned across the same range.

As the nonlinear interaction sensitively depends on fluence, we also investigated the effect of a change in fluence on the optimum d_a. We found the optimum efficiency occurs at smaller values of d_a if the fluence is increased. This is because higher pump fluence leads to a reduction in the effective interaction length L due to stronger cascading. This shift towards smaller d_a is observed in Fig. 3(e), where the THz yield was measured vs d_a for fluences within the range of 30 mJ/cm² – 130 mJ/cm² at fixed σ_x^f = 2.5 mm. The optimum d_a shifts with fluence by approximately -0.8 mm ± 0.2 mm per 100 mJ/cm² increase in fluence, which was obtained via a linear fit to the data shown in Fig. 3(f). This can be converted to how much the optimum effective length L_opt reduces with increasing fluence, namely, -1.5 mm ± 0.4 mm per 100 mJ/cm² increase in fluence. This observed fluence dependence of L_opt will be interesting to benchmark to simulations. The range of d_a for which the efficiency stays > 90% of the optimum (indicated by the grey bars) was observed to be insensitive to changes in fluence. On average, this range spans 3.3 mm, which makes optimization of d_a in setups pumped by long pump pulses such as ours convenient. However, as we will show next, d_a also affects the spatial profile of the THz beam and thus, it is still important to be set carefully.

3.4 Impact of the effective interaction length L on the shape of the THz beam

Besides the efficiency, another important metric for the performance of a tilted-pulse-front THz setup is the shape of the generated THz beam. As the effective propagation length of the pump beam impacts the conversion efficiency, the same is expected for the spatial shape of the THz beam. Therefore, here we map the cross-section along x’ of the THz beam, recorded as close as possible to the output surface of the lithium niobate prism. The impact of the apex distance d_a on the shape of the THz beam profile was mapped out via raster-scanning the center cross-section of the THz beam (y’ = 0, z’ = 5 mm) vs d_a for both the smallest (0.5 mm) and the largest pump size σ_x^f (3.0 mm) used in the experiments. These measurements were carried out at low THz energy (≈ 1.8 µJ) via reduced pump energy to avoid distortions in the response of the sensitive THz detector used to sample the beam while maintaining a high signal-to-noise ratio.

Figure 4(a) shows the recorded profiles normalized to their maximum for all d_a-values scanned across at σ_x^f = 0.5 mm. Three observations can be made: First, the size of the THz beam increases with d_a. This is matching expectations since the width of the pumped crystal section projected onto the THz output surface increases with d_a. Second, the position of the THz beam tracks the position of the laser beam as expected for a prism-shaped crystal: The peak in THz intensity shifts further from the crystal apex (towards larger x’) as d_a increases. For values d_a > 3.5 mm, however, the position of the THz peak hardly is affected by any change in d_a. In addition, for such large d_a the beam profile is observed to asymmetrically broaden towards larger x’, which is the third observation. This behavior differs for σ_x^f = 3.0 mm (Fig. 4(b)), where this trend seems to be smoothed over by the larger pump beam size. Figures 4(c) and 4(d) show profiles of the THz beam along x’ for selected equidistant steps along the d_a-scan, normalized to the global maximum amplitude of each d_a-x’ scan. The further the pump beam is moved from the apex, the more the THz beam profile is observed to distort from a Gaussian shape. Similarly to the observations made for the conversion efficiency earlier, portions of the pump beam that are located at d_a > d_a^opt ≈ 3.5 mm contribute less to the net generated THz because the respective portion of the pump beam suffers from dispersive effects and the generated THz experiences higher losses due to absorption as it propagates to the output crystal surface. This leads to an increasing distortion of the THz beam profile with larger values of d_a and towards larger x’-coordinates. Our observation of spatially distorted THz beams for large d_a match predictions by simulations [35,37].

 

Fig. 4. Cross-sections through the THz beam along the x’-axis (crossing through the peak of the beam) while the distance between pump beam center and crystal apex d_a was scanned. Larger x’ coordinates are further from the crystal apex. a) Cross-sections normalized to their maximum vs d_a for a small pump beam size σ_x^f = 0.5 mm. and for a large pump beam with σ_x^f = 3.0 mm (b). (c) Selected beam profiles at equidistant d_a normalized to the global maximum amplitude of the scan for σ_x^f = 0.5 mm (c) and for σ_x^f = 3.0 mm (d). Dashed lines show the beam profile at d_a = 4 mm set for optimum efficiency (Gaussian) as well as for d_a = 8 mm where the THz profile takes on a symmetric super-Gaussian like shape.

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This deformation observed for the THz beam profile is specifically relevant for applications, where the THz beam of a tilted-pulse-front source is imaged onto a sample and the distribution of THz intensity matters. On the other hand, this indicates, that the THz beam shape can to some extend be controlled via setting the apex distance to a different value at the cost of some overall net conversion efficiency. This could be helpful for THz applications that require a more super-Gaussian shaped THz beam instead.

3.5 Impact of the pump size σ_x^f on conversion efficiency and THz beam profile

For efficient intra-pulse difference-frequency generation (DFG) one important macroscopic parameter is the interaction length L’ that specifies the length over which pump beam and generated THz spatially and temporally overlap. The interaction length L’ can be controlled via setting σ_x^f and is limited by absorption, degradation of the pump pulse and other effects which we will discuss here.

Impact of the effective interaction length L’ on the conversion efficiency

In a collinear-interaction scheme, the interaction length would be determined by the length of the nonlinear crystal and the temporal walk-off. For the non-collinear geometry in tilted-pulse-front THz setups, one must rather think in terms of an effective interaction volume: Both the length over which the pump pulse propagates through the crystal as well as its transverse size define the region of interaction. In practice, these characteristic lengths are controlled via setting d_a and the pump beam size σ_x^f. These two parameters need to be set such that they balance effects that increase the THz yield such as pumped crystal volume that contributes to the THz generation and effects that degrade the conversion efficiency such as break-up of the optical pump, THz absorption and spatial walk-off [37]. Here, we focus on the interaction length L’ oriented along the z’-axis.

Considering fixed pump fluence, two effects lead to an increase in the efficiency with increasing σ_x^f (and L’ respectively): Coherent build-up of THz over a longer distance z’ and a favorable ratio between pumped crystal volume to unpumped volume in which the THz is absorbed. The larger the pump beam, the later the THz will walk off from it transversely. Thus, the distance L’ over which the THz can build up coherently increases with σ_x^f. Moreover, a larger pump beam means a larger portion of the crystal contributes to the THz generation process. Simultaneously, the region through which the THz that has walked off transversely from the beam propagates through decreases. In this unpumped region, the THz suffers from absorption. Thus, the ratio between net THz generation and absorption scales favorably as the beam size increases. These effects are counteracted by the degradation of the pump beam portions located further from the apex that experience a larger than optimum L. For this portion of the pump beam, dispersive effects mainly due to cascading in the THz generation process grow until they degrade portions of the pump beam to an extent that prevents further efficient THz generation, thus reducing the overall efficiency.

To the best of our knowledge only one experiment in which the pump beam size was tuned systematically to maximize the setup performance was performed so far: Meyer and coworkers tested 3 pump beam radii σ_x^f within ≈ 0.2 mm – 0.5 mm in the limit of high MHz repetition-rate at low pump energy (≈ 10 µJ) while keeping other setup parameters such as d_a constant [52]. In this work, we use the 4-dimensional parameter scans described in the beginning of the previous section to study the efficiency scaling with σ_x^f at constant pump fluence levels and at re-optimized crystal position (d_a, z) for each pump size under study. Focus is set on the regime of high pump fluences and high THz pulse energies critical for THz strong-field applications. To robustly identify the optimum pump size for maximum conversion efficiency we emphasize the importance of using fluence instead of pump energy as a metric for such measurement.

Figure 5 shows how the THz energy (Fig. 5(a)) and the conversion efficiency (Fig. 5(b)) depend on the pump radius σ_x^f, recorded for fixed fluence levels of 50 mJ/cm², 70 mJ/cm² and 90 mJ/cm². Each data-point represents the optimum efficiency found during the scans discussed in Sec. 4.2. The efficiency improves monotonically with increasing pump size up to an optimum which is observed for a pump radius σ_x^f ≈ 2.5 mm. For larger beam size σ_x^f, the efficiency is observed to roll over and slightly decrease. To approximately stay within 90% of the optimum performance level, σ_x^f should not deviate from the optimum by more than ± 0.4 mm. These observations qualitatively match the predictions made in simulations by Ravi et al. [37] where, in contrast to undepleted pump models, the decrease in efficiency for larger pump beams is mainly attributed to the dispersive broadening of the pump initiated by the cascaded nonlinear interaction. As our data indicates an optimum for σ_x^f ≈ 2.5 mm, it supports the findings by Ravi et al. that for the optimization of the pumped crystal volume break-up of the pump pulse due to cascading needs to be considered. The optimum pump size found in the experiment can be used to estimate the optimum interaction length L’: Since the underlying process is a χ⁽²⁾-process, the effective diameter of the pump contributing to THz generation approximately is 2σ_x^f/$\sqrt 2 $. L’ can then be approximated via projecting this width onto the z’-axis of the THz beam as L’ ≈ $\sqrt 2 $σ_x^f/sin(γ). We thus obtain L’_opt ≈ 4 mm for σ_x^opt = 2.5 mm for fluence levels within 50–90 mJ/cm². The magnitude of L’_opt is expected to decrease with increasing pump fluence due to faster break-up of the pump beam. Consequently, the optimum beam size σ_x^f should decrease with increasing pump fluence. However, within the resolution along σ_x^f in our scans such shift could not be resolved within the examined fluence range (50–90 mJ/cm²).

 

Fig. 5. Scaling of the THz energy (a) and conversion efficiency (b) with the pump radius σ_x^f (1/e²) for fluence levels of 50–90 mJ/cm². Blue lines act as guide-to-the-eye. Error-bars (grey) are only shown for F = 90 m/cm² in (b) for improved visibility.

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The results show that setting the interaction length L’ via the transverse beam size is very important to achieve efficient conversion from IR to THz radiation. Furthermore, the results proof that tilted-pulse-front setups should rather be scaled towards higher pump energies via enlarging σ_y^f (after σ_x^f has been used to optimize L’), as recently demonstrated in [33].

Size and propagation of the THz beam for different σ_x^f

For almost any experiment that utilizes tilted-pulse-front setups as source, the THz pulses need to be transported to the point-of-interest. For a proper design of such optical transport line, it is crucial to know the characteristics of the THz beam upon propagation out of the source. Here, we have investigated the impact of the pump size σ_x^f on the output THz beam size and its propagation.

The THz caustics were mapped out after the crystal position (d_a, z) had been optimized for maximum efficiency for each σ_x^f under test. A small and sensitive THz detector mounted on a motorized linear translation stage assembly was used to 3D-raster-scan the THz beam in (x’, y’, z’). Similarly to the measurements of the THz beam profiles vs d_a, these 3D-raster scans were carried out at low THz energy (≈ 1.8 µJ). The 1/e²-radii σ_x’^THz and σ_y’^THz of the THz beam were obtained via fitting a Gaussian through the centroid both horizontally and vertically. The obtained widths were deconvolved by the point-spread-function of the small THz detector element to obtain the actual size of the THz beam.

Figures 6(a) and 6(b) show the recorded widths σ^THz (1/e² radius of the intensity distribution) of the THz beam vs distance z’ from the THz output surface of the crystal for the investigated pump beam sizes σ_x^f. The results in Fig. 6(a) show that σ_y’^THz, or the THz beam divergence respectively, are rather insensitive to changes in σ_x^f, as postulated for example in [35]. As the phase-matching process in tilted-pulse-front THz setups is set up in the x-z-plane, the THz generation is translation symmetric along the y-direction. Like in other collinear 2^nd-order nonlinear interactions σ_y’^THz is therefore expected to approximately be $\mathrm{\sigma }_\textrm{y}^\textrm{f}/\sqrt 2 $, which we roughly find fulfilled ($\mathrm{\sigma }_\textrm{y}^\textrm{f}/\sqrt 2 $ ≈ 2.9 ± 0.2 mm, σ_y0’^THz ≈ 3.4 ± 0.1 mm). This translational symmetry is violated within the x-z plane due to the non-collinear interaction geometry connected to the tilted-pulse-front based phase-matching. Paired with the cascaded nonlinear conversion processes acting back on the pump beam, no such scaling law for σ_x’^THz can be expected. Indeed, the caustics for σ_x’^THz show some peculiar features (see Fig. 6(b)). First, the THz beam waists are within the range of ≈ 3 - 4 mm, even for the smallest pump size used (σ_x^f = 0.5 mm). This can be understood considering the geometrical aspects of the interaction: If we imagine a pump beam focused to a thin line and say all the THz generated along its propagation length L can be observed as THz beam, then the diameter of the THz beam can be approximated by the projection of L onto the x’ axis. For d_a ≈ 3.75 mm ± 0.25 mm this would yield an estimated THz beam diameter of 2σ_x’^THz ≈ 7 mm ± 1 mm, which is on the order of the beam size observed here. Even if the underlying processes are significantly more complex, within this simple picture it can be understood why THz beams with considerably larger beam size than the input pump are observed (see Fig. 1(c)). Second, while for small σ_x^f, the THz beam diverges Gaussian-like as it propagates further from the crystal, a focusing THz beam was observed for the two largest values of σ_x^f, that hardly is affected by the increase in 20% of pump size when increasing σ_x^f from 2.5 mm to 3.0 mm. While explanation of the observed focus is beyond the scope of this work, the insensitivity of the THz caustic to further increased pump size is consistent with the results found for the efficiency-scaling with σ_x^f: As an optimum efficiency was found for σ_x^f = 2.5 mm where the interaction length between THz and pump pulse is set to optimize efficient THz build-up, THz absorption and pump pulse break-up, larger σ_x^f values lead to portions of the beam not contributing efficiently to the THz beam anymore. Thus, one can expect the THz beam coupled out of the crystal to also not be affected by further increases of the pump size. This also matches predictions made in simulations qualitatively [35].

 

Fig. 6. Evolution of the THz beam radius σ^THz (1/e²) with increasing distance z’ from the crystal output surface (z’ = 0) for each investigated pump size σ_x^f. (a) The vertical size of the THz beam σ_y’^THz is hardly affected by σ_x^f and expands as expected for a Gaussian beam (blue line). (b) In contrast, σ_x’^THz is affected by the choice of σ_x^f. For improved visibility of each dataset, lines were added as guide-to-the-eye.

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To conclude, the results show that while σ_y’^THz is somewhat well-behaved, σ_x’^THz needs to be characterized carefully if the THz beam is to be transported to a point-of-interest in an experiment.

3.6 Optimization procedure based on measured sensitivities

Since the performance of tilted-pulse-front THz setups depends on pump characteristics such as wavelength, pulse duration and spectral content, the global optimum performance for each tilted-pulse-front setup is very specific for the pump laser system at use. Capturing all the complex dynamics involved in the non-collinear interaction in simulations is computationally costly which is why extensive parameter scans require similar effort as in real experiments [44]. Thus, an experimental procedure to find robust optima and to systematically measure the relevant setup characteristics is important. In this section, we discuss aspects geared towards the systematic optimization of tilted-pulse-front setups, adding to the setup design suggestions and general approach regarding parameter tuning laid out in the first paragraphs of the previous subsections. Such procedure may help to design and robustly optimize tilted-pulse-front THz setups for applications that require reliable high-field single-cycle THz sources.

Pump polarization

Prior to optimization of the phase-matching condition, the pump polarization should be set to maximize the effective nonlinear coefficient d_eff by tuning the polarization angle ϑ of the pump beam relative to the crystal c-axis. As shown in Fig. 7, the measured total THz energy vs ϑ is the superposition (S_o + S_e) of the THz generated by the projected portion of the full pump energy onto the principal crystal axes (ordinary axis: S_o, extraordinary axis: S_e). Maximum conversion is achieved for an e-polarized pump beam corresponding to a maximized d_eff. The individual contributions S_o and S_e approximately follow a cos⁴-dependence on ϑ.

 

Fig. 7. Impact of the pump polarization orientation on the THz energy. ϑ is pump polarization angle with respect to the optical axis of the lithium niobate crystal. The measured signal is the sum of THz generated by o-polarized (S_o) and e-polarized pump light (S_e).

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3.3 Phase-matching

The pulse-front tilt is one of the key parameters to achieve high conversion efficiency. The FWHM-sensitivity on γ was found to be ≈ 3° (293 K) and ≈ 2° (82 K), which matches simulated values in [34,44]. To achieve high conversion efficiency within a 10%-range of the optimum value, it is necessary to set the pulse-front-tilt angle with a precision of 0.2–0.3° (see. Figure 2(c)). Otherwise, phase-mismatch prevents efficient build-up of the generated THz beam. In more practical terms, this means the grating angle θ_d needs to be set with an accuracy of ± 0.4° at 293 K or ± 0.3° at 82 K to not suffer from a loss in efficiency by more than 10% compared to the optimum configuration. In our work, we relied on using efficiency vs fluence as the primary metric to evaluate the setup performance. However, using a constant-energy metric during the pulse-front tilt optimization would still have resulted in a setup performance within 10% of the optimum performance. This is very convenient, as such optimization with fixed pump energy is easier to perform experimentally. The optimization is easier to perform at low to moderate fluence levels because accidental damage of the crystal is prevented. For long pump pulses, such as used in this work, the optimum tilt-angle γ_opt was found to be robust vs changes in fluence such that no separate optimization at high fluence levels would be required. This may not hold for shorter pump pulses, where higher peak intensities are reached such that the nonlinear refractive indices could play a role in the phase-matching mechanism. Note, that at lower fluence levels the THz energy scales more sensitively with fluence and thus is affected stronger by changes due to the grating magnification. This means the error of a constant-energy metric during the grating angle optimization is larger at lower pump energy or fluence levels. As a rule of thumb, the pump beam size σ_x^f during the grating angle optimization should not be too small, as the interaction length between THz and pump beam is required to be larger than the coherence length of the phase-mismatched conversion process to effectively detect the improvement achieved via better phase-matching. During the optimization of our THz setup, we simultaneously scanned the crystal along z, to avoid optimizing the pulse front tilt for a sub-optimal longitudinal location.

Effective interaction lengths L and L’ and longitudinal crystal positioning Δz

Once the phase-matching condition is met via an optimized pulse-front-tilt, the THz signal can only build up optimally, if the interaction lengths L and L’ between THz radiation and the pump beam are set correctly. The optima for these interaction lengths change and require re-adjustment of the setup whenever fluence, spectrum or pulse duration of the pump pulse change [37]. That is why, the fluence level during the optimization should be chosen close to the level at which the setup is planned to be operated at. This optimization involves scanning σ_x^f which is preferably performed using telescopes set up upstream of the THz setup, such that curvature in the pump beam wavefront is minimized as this can affect the optimum to be found for d_a. [44]. To select a global optimum for σ_x^f, it is crucial to use efficiency vs fluence as a metric as an energy-based metric may not reveal the global optimum. This causes the optimization of the pump beam size to be experimentally challenging, as an adjustment of pump energy is required to keep the pump fluence constant while σ_x^f is altered.

During the optimization of the effective interaction length L, a couple of iterated scans to tune d_a and the crystal z-position are sufficient if long pump pulses on the order of 1 ps (such as in this work) are used. Shorter pump pulses require a more careful optimization of the crystal position as the prism-shaped crystal geometry couples changes in d_a with the effects caused by a shift Δz of the crystal output surface relative to the grating image. Furthermore, our measurements of the THz beam profile show that beam distortions can occur, if the apex-distance d_a was set larger than optimum, which must be considered.

Maximizing the efficiency using the given pump energy

Finally, after phase-matching and proper build-up of the THz along the interaction length has been ensured, the setup can be optimized such that it is operated at optimal fluence levels when the full energy of the laser is used. The findings presented in Sec. 4.3 show, that σ_x^f is not a feasible parameter to scale the setup to lower or higher pump pulse energies as it affects the nonlinear-conversion process significantly. The pump beam size σ_y^f, however, is perpendicular on the phase-matching plane and thus allows adjustment such, that optimal conversion efficiency is reached at exactly the maximum pump laser energy that is available - given a sufficiently tall crystal. This can be achieved via appropriately setting σ_yⁱ with cylindrical optics placed upstream of the THz setup.

Optimization of multiple setups for use with the same laser source

Such optimization and characterization of a tilted-pulse-front THz setup requires considerable effort, which is why it is advantageous if the knowledge of optimal parameters from one setup can be transferred to a potential second or third setup to be powered by the same laser. Based on the parameters M and θ_d at which γ was found optimum in the previous setup, the optimal diffraction angle θ_d^new in a new setup can be computed via $\mathrm{\theta }_\textrm{d}^{\textrm{new}} = \; {\cos ^{ - 1}}\left( {\frac{\textrm{M}}{{{\textrm{M}_{\textrm{new}}}}}\cos {\mathrm{\theta }_\textrm{d}}} \right)$, where M_new is the magnification of the telescope in the new setup. For this to work, the pulse-front tilt angle found optimum and the magnification factor of the imaging setup in the tilted-pulse-front setup need to be determined precisely. Therefore, the method for the telescope characterization shown in Fig. 1(c) was developed. Via these simple measures, the effort for future setup design, construction and optimization can be significantly reduced.

4. Summary and outlook

To conclude, we have studied the impact of crucial setup parameters such as pulse-front-tilt, pump size and crystal location on the conversion efficiency and THz beam profile. Our study was carried out with a setup that decoupled relevant parameters to a large extent. Via the use of extensive multi-dimensional parameter scans, the sensitivity on the individual setup parameters was mapped out systematically for the first time. Therefore, these findings are important to further advance tilted-pulse-front setups as sources for efficient table-top single-cycle THz pulses.

First, the sensitive dependence of the setup performance on the pulse-front-tilt angle, which optimizes the non-collinear phase-matching, was mapped out. Even a small deviation of the grating angle θ_d by ± 0.4° (293 K) or ± 0.3° (82 K) causes the conversion efficiency to drop by 10%. The optimum angle was found to stay unaffected by the pump fluence level within the resolution and range of our measurements, suggesting this parameter can be safely optimized at low fluence levels. Based on the multi-dimensional parameter scans carried out for the optimization, a robust comparison between performance at room-temperature and cryogenically cooled setup was enabled: To switch from phase-matched operation at room-temperature to operation at 82 K, the pulse-front-tilt angle had to be adjusted by Δγ = 1.3° from γ_opt^293K = 65.64° to γ_opt^82K =64.33° via a rotation of the grating by 2.4°. Cryogenic cooling resulted in an enhanced conversion efficiency for usable THz by a factor of 2.7.

Second, the effect of the interaction lengths L and L’ for both pump and THz beam on both conversion efficiency and spatial THz profile were systematically measured for the first time. As L and L’ are controlled by d_a and σ_x^f, but also are affected by the choice of pump fluence and crystal z-position, simultaneous scans of the crystal along z, d_a, the pump energy and the beam size σ_x^f allowed evaluating the conversion efficiency for constant fluence levels. The results verified trends predicted in [37], which were not yet experimentally verified. We were able to observe an increase in efficiency with growing σ_x^f that decreased for beams larger than σ_x^f = 2.5 mm. The optimum values for σ_x^f and d_a balance the amount of crystal volume illuminated by the pump and effects that degrade the efficiency such as THz absorption, spectral broadening in combination with dispersion acting on the pump beam as well as spatial walk-off. Furthermore, d_a^opt was measured to decrease with increasing fluence, which means that the optimum effective propagation length L decreases with increasing fluence. The rate of this change can be used to benchmark future simulations and ensure better quantitative agreement between simulations and experiments. Moreover, the efficiency hardly varies (< 10%) if d_a is detuned within a range of ≈ 3.3 mm around the optimum. However, such detuning towards larger than optimum d_a was found to lead to distortions in the spatial profile of the THz beam. Measurements of the THz beam profile as a function of the pump size showed that σ_y’^THz is insensitive to changes in σ_x^f and an approximate waist σ_y’^THz of $\mathrm{\sigma }_\textrm{y}^\textrm{f}/\sqrt 2 $ was observed. In contrast, σ_x’^THz was found to not follow any simple scaling law related to σ_x^f. The two largest pump beams resulted in a THz beam that hardly changed in shape when enlarging the input beam size from σ_x^f = 2.5 mm to σ_x^f = 3.0 mm. This insensitivity of the THz shape to large σ_x^f can be explained by the same mechanism that leads to an optimum for the efficiency: Larger than optimum pump beams contain portions that do not significantly contribute to the THz output anymore, as the pump beam is distorted by the interplay of cascading induced spectral broadening and dispersion. This effect causes the decrease in efficiency with larger σ_x^f and the reduced sensitivity of the THz caustic for large beams. Surprisingly, the beams were found to focus ≈ 3 cm outside of the crystal which would be interesting to be topic of a dedicated future investigation.

Lastly, based on the results, important aspects for a robust optimization of tilted-pulse-front THz setups were discussed. Considering the optimum configuration for a tilted-pulse-front setup sensitively depends on the pump pulse characteristics such as pulse duration and wavelength such a systematic optimization procedure should help construct high-field THz sources. In our case, the systematic multi-dimensional optimization procedure resulted in ≈ 96 µJ of usable THz energy from a setup pumped by 27 mJ pulses with a relatively long pulse duration of 1.35 ps. This corresponds to a conversion efficiency of ≈ 0.36% (≈ 0.78% internal conversion efficiency) and was sufficiently strong to power a novel compact THz-powered photo-gun [53].

One major unsolved problem in the optimization of tilted-pulse-front THz setups remains the solid experimental verification of the optimum pump pulse duration suggested by simulations. Since our study was carried out for one single pump pulse duration (≈ 1.35 ps), follow-up investigations on the dependencies shown in this work for other laser sources would finally pave the way towards such verification. This would be a major step in advancing single-cycle THz sources based on the tilted-pulse-front scheme. The results presented in this work thus are relevant for understanding and improving the performance of table-top tilted-pulse-front THz setups which are important to advance fields in which strong single-cycle THz pulses are required. Applications include, but are not limited to, electron beam acceleration and manipulation as well as strong-field THz experiments such as in THz plasmonics.