THz generation using a reflective stair-step echelon

We present a novel method for THz generation in lithium niobate using a reflective stair-step echelon structure. The echelon produces a discretely tilted pulse front with less angular dispersion compared to a high groove-density grating. The THz output was characterized using both a 1-lens and 3-lens imaging system to set the tilt angle at room and cryogenic temperatures. Using broadband 800 nm pulses with a pulse energy of 0.95 mJ and a pulse duration of 70 fs (24 nm FWHM bandwidth, 39 fs transform limited width), we produced THz pulses with field strengths as high as 500 kV/cm and pulse energies as high as 3.1 $\mu$J. The highest conversion efficiency we obtained was 0.33%. In addition, we find that the echelon is easily implemented into an experimental setup for quick alignment and optimization.


I. INTRODUCTION
The generation of high power THz radiation has led to the development of THz nonlinear optics and spectroscopy [1][2][3] . Nonlinear THz-driven responses in the 0.1-5 THz range have been observed in semiconductors [4][5][6][7] , graphene 3 , quantum confined systems 8, 9 , liquids 10 , gases 11,12 , molecular crystals 13 , and correlated electron materials 14,15 . This advancement has been primarily enabled by the ability to use lithium niobate (LN) for THz generation by optical rectification. Prior to this, the most common sources of high-power THz radiation were either free-electron lasers 16 which have limited accessibility, QCLs 17 which have low peak fields, and gas lasers which have long pulse durations. Compared to other widely used nonlinear materials like zinc telluride (ZnTe) or gallium phosphide (GaP), LN has a larger nonlinear coefficient (d ZnT e ≈ 67 pm/V, d GaP ≈ 44 pm/V, d LN ≈ 198 pm/V 18 ) for optical rectification, which leads to greater THz generation efficiency. The major drawback of LN, however, is the large index mismatch between the ultrafast infrared (IR) or near infrared (NIR) pump pulse that is rectified and the THz field that is produced (n gr LN ∼ 2.2, n T Hz LN ∼ 5). This causes the pump pulse to outrun the generated wave, and the THz field is consequently emitted as a cone of Cherenkov radiation 19 . While this prevents collinear phase matching in LN, it is possible to efficiently generate THz fields using noncollinear phase matching with tilted optical pulse fronts 20,21 . Typically, this is accomplished by using a grating to introduce angular dispersion into the beam, resulting in a continuously tilted pulse front 22 . This tilted pulse front is them imaged 23 into the LN prism to produce a pulse with the correct tilt angle so the optical intensity front moves laterally in the LN at a speed that allows matching of the THz phase velocity.
To date, there have been considerable experimental [24][25][26][27][28][29][30][31] and theoretical [32][33][34] efforts aimed at improving the generation efficiency of the tilted pulse front technique. Numerical studies suggest 32,33 that the three major factors limiting the generation length in the LN crystal come from (1) the degradation of the pulse duration away from the image plane due to the angular dispersion introduced 35 and (2) the imperfect imaging of the tilted pulse front from the grating. In addition to these, recent work 27,34,36 has proposed that (3) the inherent coupling between the generated THz radiation and the pump pulse leads to detrimental effects which significantly reduce the interaction length of the pump pulse and necessarily limit the generation efficiency.
While the coupling effects are difficult to manage practically, several suggestions have been proposed to mitigate the other issues. The first problem can be compensated for by using pulses with smaller bandwidths and hence longer transform limited pulse durations, but this reduces the generated THz bandwidth since there are fewer frequency components to use for optical rectification. Moreover, many high-field THz generation systems are based on amplified Ti:sapphire lasers whose pulse durations are tens of femtoseconds. Such short pulses are often desirable in a nonlinear THz spectroscopy measurement or for electro-optic sampling. For these systems, increasing the transform limited duration by filtering would come at the cost of both time resolution and reduced optical pulse energy. It has also been proposed 37 that using a direct-contact grating would mitigate the problem of imperfect imaging. However, this approach reduces the effective generation length compared to that reached by imaging of the beam from a typical grating. This is because when imaging the grating, temporal focusing of the pump pulse and associated optimization of THz generation efficiency occurs on both sides of the image plane over a total distance 2L d where L d is the dispersion length 38 . However, with the contact grating method, the length over which the optical pulse duration is short is reduced by a factor of 2. Furthermore, this still requires the use of a highly dispersive grating that leads to limited generation lengths, especially when using pulses with considerable bandwidth.
An alternate scheme which ideally avoids the introduction of large amounts of angular dispersion is a discretely tilted pulse front from a reflective echelon 39 . The reflective echelon consists of many small steps that act as mirrors and split a single input beam into many smaller time-delayed "beamlets", which together create a discretely tilted pulse front as shown in figure 1(a)-(b). The beamlets are then imaged into the LN prism and act as nearline sources to produce Cherenkov sheets of THz radiation. As the spacing of the imaged steps is small compared to THz wavelengths, the discretely tilted pulse front is effectively continuous over a portion of the THz bandwidth. Here, each beamlet is assumed to generate a THz field independently, provided that the duration of each beamlet is short relative to the time delay between them 40 . Figure 1(c) shows results of a simple 2D calculation of the THz sheet generated from a single infinitely tall beamlet with a Gaussian spatiotemporal profile 41 . In this case, the frequency content of the resulting THz field is set by the beamlet's lateral size. When the tilt angle is set appropriately, the THz fields resulting from the many responses superpose constructively on one side, leading to coherent field addition and efficient THz generation.
In conventional THz generation with a highly dispersive grating, a 1-lens imaging system is used to image the pump beam into the LN prism, and beam divergence is largely ignored because angular dispersion and pulse front distortions in the image plane are more far detrimental 32 . For the echelon, the beamlet spot size w 0 in the image plane is large relative to the pump pulse wavelength λ and the Rayleigh range z R = πw 2 0 /λ is independent of the pulse duration. As a result, the beamlets can be of shorter pulse duration and can propagate for a long distance 2z R without appreciable divergence. Pump beam divergence and THz absorption losses in the LN prism then become limiting factors. This suggests additional benefits by using multiple lenses for imaging to produce collimated beamlets at the image plane that allow exploitation of the long Rayleigh range, and by cryogenic cooling of the LN prism 33,42 to reduce THz absorption which is due primarily to phonon damping.
In this work, we demonstrate THz generation using a discretely tilted pulse front produced by a reflective echelon as described above. Experiments were performed using a 1-lens imaging system and a 3-element varifocal zoom-lens capable of continuously tunable demagnification from M = 0.16 to M = 0.29 43 at both room temperature (RT) and cryogenic temperatures (CT, T ∼ 100 K). Using discrete optical pump beams to deliver spatially and temporally shifted pulses for THz generation in LN or lithium tantalate has been reported based on different approaches including beamsplitting to produce just two beams 44 , spatiotemporal pulse shaping 45,46 , and transmission through a homebuilt stairstep echelon 47 .
In these cases the THz wave remained inside the generation crystal and the conversion efficiency was not measured. Our measurements of the output THz power and field strength show that by using the echelon, it is possible to achieve conversion efficiencies better than those obtained using a grating at room temperature using broadband pulses with sub 100 fs duration from a commercial Ti:sapphire laser. It may also be possible to use pulses with even shorter durations (<50 fs) since the discretely tilted pulse front should not suffer from the adverse effects introduced by angular dispersion.

II. EXPERIMENTAL SETUP
A schematic illustration of the echelon is shown in figure 1. The echelon was fabricated by Sodick F.T. (Japan) by cutting a stair-step pattern into a block of steel followed by set so that the width ("horizontal") direction of the steps is at normal incidence to the incident beam. Each step produces a single reflected "beamlet" whose pulse is time delayed relative to an adjacent beamlet pulse due to the additional 2H path length. (c) Calculated Cherenkov wavefronts produced by a single beamlet pulse propagating through a LN crystal. The beamlet is polarized along the extraordinary (z) axis, which is parallel to the LN crystallographic c-axis. The spatially and temporally shifted beamlets all generate THz wavefronts that superpose constructively on one side.
gold plating for high reflectivity. The step width and height were W = 150.0 ± 0.2 µm and H = 69.00 ± 0.02 µm, respectively. As shown in figure 1(a), the height determines the time delay between adjacent pulses which is given by t = 2H/c = 460.31 ± 0.07 fs, and the width determines the initial width of the beamlet (whose size in the orthogonal direction is ∼ 9 mm) that is reflected by each step surface. The echelon was rotated such that the direction along the 150-µm widths of the steps was normal to the incident beam path. Upon reflection, the input beam was split into many beamlets, resulting in a discrete tilted pulse

A. THz Characterization Procedure
To begin characterizing the THz output of the system, we measured the THz timedependent waveform by EO sampling and computed the corresponding Fourier transform, yielding the results shown in figure 3(a)-(b). The data shown were obtained using maximum pump power for the 1-lens and zoom-lens at RT and CT, and the results are summarized in table I below.
The highest peak THz electric field (E-field) measured was 500 kV/cm when using 1-lens at CT. In all cases, the peak frequency was 0.63 THz and the mean frequencies ranged from 0.9 THz to 1.1 THz. Our peak frequency is slightly lower than the peak frequencies typically obtained using a continuous tilted pulse front, which is likely a result of the finite size of the beamlets in the LN prism. Assuming independent THz generation by each of the beamlets, an estimate of the peak THz frequency, ν p , can be made (neglecting frequency-dependent THz absorption in the LN crystal) according to 41 : where n T Hz LN = 4.96 is the THz refractive index of LN and w 0 is the spot size of the beamlet. Using w 0 = 30 µm, we find ν p = 0.64 THz in almost exact agreement with our experiment.
The full-width-half maximum of the THz spectrum was ∆ν ≈ 0.8 THz in all cases except for the 1-lens at CT, which had ∆ν ≈ 1.3 THz. These values are all comparable to values obtained using 800 nm pump light with a grating to produce a continuous tilted pulse front 49 . For the echelon, the bandwidth of the resulting THz pulses depends on the available wave vector components of the beamlets, with smaller beamlets having more content. Any additional divergence in the full beam through the image plane will also lead to variations in the tilt angle and thus to changes the THz bandwidth 50 . To further characterize the THz output, we measured the spatiotemporal evolution of the THz pulse at the focus of the second parabolic reflector 27 . Instead of focusing the EO sampling beam, it was left collimated and allowed to over-fill the GaP crystal which was imaged using a 1-lens imaging system through a quarter waveplate and Wollaston prism onto a camera resulting in a pair of images. A time series of these images was collected with the THz field present and the THz field absent and then balanced to produce a movie (see Multimedia 1). using the zoom-lens at CT. The extracted signal is proportional to E(x, y) and was squared to obtain an intensity distribution I(x, y) ∝ E 2 T Hz (x, y) . The traces along the top and the side of the 2D image correspond to vertical and horizontal cross sections through the center of the spot. These were fit to Gaussian functions and values of 0.57 mm and 0.55 mm were obtained for the 1/e 2 radii along the vertical and horizontal dimensions, respectively.

B. THz pulse energy and conversion efficiency
While the peak E-field and bandwidth of the THz pulse are important for driving nonlinear effects in materials, a more generalized metric for comparison to other THz generation schemes is the pump-to-THz conversion efficiency. In order to accurately determine this, we calculated the THz pulse energy using parameters obtained by EO sampling measurements and the THz intensity profile assuming Gaussian spatial and temporal profiles. The free space pulse energy, U, is given by 51 where g = 0.94 is a correction factor assuming a Gaussian temporal shape, T = 0.72 is the power transmission coefficient into the GaP crystal, E 0 is the peak field strength in the EO crystal, τ 0 is the intensity full width-half maximum pulse duration, w x and w y are the 1/e 2 radii along x-and y-directions, and n T Hz GaP = 3.24 is the THz refractive index of GaP. By measuring the peak THz field at various pump pulse energies using the 1-lens and zoom-lens imaging systems at both RT and CT, we calculated the THz pulse energy, plotted in figure 4(a), and the corresponding conversion efficiency, plotted in figure 4(b). In all cases, below 500 µJ pump pulse energy (14.4 mJ/cm 2 fluence) the THz pulse energy increased quadratically with pump pulse energy, corresponding to a linear rise in the conversion efficiency. For the 1-lens system, above 700 µJ pump energy (20.2 mJ/cm 2 fluence) the THz pulse energy grew linearly with pump pulse energy at both RT and CT, which is also indicated by a saturation of the conversion efficiency. Previous studies using 800 nm pulses observed saturation in the conversion efficiency at lower pumping fluence 27,31 , which was attributed to unwanted effects from self-phase modulation 34 and three-photon absorption 31 .
In contrast, the pulse energy continues super-linear growth when imaged using ZL system, which is confirmed by the continued increase in the conversion efficiency.
The data are summarized in table II. From these values, we observed an enhancement in the conversion efficiency from RT to CT by a factor of 1.4× for the 1-lens imaging system and 1.6× for the zoom-lens. We obtained a peak conversion efficiency of 0.33% using 0.95 mJ energy (27.5 mJ/cm 2 fluence) pulses using the zoom-lens at CT, and our highest THz pulse energy exceeded 3.1 µJ . This maximum almost exceeds the record 0.35% using 800-nm center wavelength pulses 42 (based on pyroelectric detector measurement). In that case, the tilted pulse front was produced by a grating, and a much longer transform-limited pulse duration (> 200 fs) was used. With the echelon, we achieved a comparable conversion efficiency using a much shorter transform-limited pulse duration where the grating performance is expected to be limited 32 . The energy of the THz pulses resulting from echelon-based generation was also measured using a TK power meter and a Microtech pyroelectric detector. The values obtained from these other methods both exceeded those calculated using the parameters assuming a Gaussian beam by roughly a factor of 2. The numbers reported here, therefore, may represent a lower bound on the THz pulse energy and conversion efficiency of the echelon  Finally, we characterized the spectrum of the NIR pump pulse used for THz generation.
This is plotted in figure 4(c), and shows significant spectral broadening as well as a redshift in the center wavelength of the modulated spectrum. This red-shift is characteristic of the THz generation process 24,26,53,54 , which occurs through difference-frequency mixing among the optical spectral components 34,53,55 and which can be cascaded through multiple difference-frequency steps, leading greater than 100% photon conversion efficiencies.

C. Discussion of Comparison
In comparing the measured conversion efficiencies, the 1-lens and zoom-lens imaging systems show markedly different behavior with respect to pump pulse energy. At both RT and CT, the 1-lens system conversion efficiency appears to saturate, while the zoom-lens system conversion efficiency increases due to continued superlinear growth. In addition, the 1-lens system shows a noticeable increase in the THz bandwidth at CT as a result of the dramatic reduction in THz absorption at higher frequencies 33 , while the bandwidth obtained from the zoom-lens system does not noticeably change. This behavior could be due to the larger divergence and wavevector spread of the beamlets with the 1-lens imaging system, which would allow phase matching over a wider range of frequencies as described in in Sec. 3.1. In contrast, the divergence of the beamlets is diffraction limited with the zoom-lens system, and hence there is negligible change in the generated bandwidth. The beam divergence in the 1-lens system also leads to a range of tilt angles that expands more rapidly in front and back of the image plane. While this may diminish the overall conversion efficiency, it may at the same time increase conversion into higher-frequency THz spectral components.
Our THz generation system has some similarities to a phased antenna array 56 . Treating each beamlet as an element of such an array, the superposition of THz waves along the Cherenkov angle is reduced when λ T Hz ≤ 30 µm or f T Hz ≥ 2 THz, as energy is diffracted into higher orders by the imposed periodic structure. A smaller echelon step width may therefore improve the conversion efficiency by reducing the amount of diffraction of high frequency waves. This would require a proportional reduction in the step height. The reductions would ultimately be limited by the pulse duration since for pulses with durations τ > 2H/c, it is possible that the echelon would act as an echelle grating that produces a beam with low angular dispersion because of its significantly reduced groove density. In that case, the larger demagnification used here (M ≈ 5) would increase the angular dispersion to be the same as that from a conventional grating. At the image plane the pump pulse from the echelon would therefore suffer the same degradation, limiting the generation efficiency.
Our assumptions that the discrete tilted pulse front properties are insensitive to the pump pulse duration and that each beamlet acts as an independent line source in the LN crystal, with a Rayleigh range given by its spot size, would no longer be valid in this limit.
Careful theoretical treatment of THz generation using a continuous tilted pulse front from a grating has shown that the dramatic modification of the pump spectrum is significantly influenced by the nonlinear coupling between the THz and optical fields which candramatically reduce the effective THz generation length 34,36 . Similar calculations are under way for the echelon generation that we have demonstrated here, and the results may help understand the limitations that are faced in this approach.

IV. CONCLUSIONS
We have demonstrated efficient THz generation using an easily implemented 800-nm discrete tilted pulse front produced by a reflective stair-step echelon structure. We produced THz pulses with field strengths as high as 500 kV/cm and pulse energies exceeding 3 µJ. The peak conversion efficiency at cryogenic temperature was 0.33%, which is an improvement over values routinely reported in the literature for sub-100 fs pulse durations. Further improvements to the echelon could be made to increase the THz output, such as reducing the step width to facilitate the generation of higher frequencies.
It is worth noting that higher conversion efficiencies have been observed using even longer transform-limited pulses at 1030 nm center wavelength [28][29][30] . When generating using 1030 nm, saturation of the conversion efficiency occurs at higher intensity because free carriers are generated by four-photon absorption, rather than three-photon absorption at 800 nm 57 .
Furthermore, the high conversion efficiencies for these longer transform-limited pulses were attributed mainly to the reduced pump pulse bandwidth, which increases the coherence length and decreases the effects from stimulated Raman scattering [32][33][34]52 . It is likely that the use of an echelon would allow shorter pump pulse durations to be used at similar pump wavelengths, resulting in high conversion efficiencies with greater THz bandwidths.