Sub-wavelength terahertz beam profiling of a THz source via an all-optical knife-edge technique

We propose an all-optical Knife Edge characterization technique and we demonstrate its working principle by characterizing the sub-{\lambda} features of a spatially modulated Terahertz source directly on the nonlinear crystal employed for the Terahertz generation.

THz radiation makes it of extreme interest for the imaging of biological samples, since it does not carry the hazard issues typically related to ionizing fields 8 .
In 1995 9 , Hu and Nuss pioneered THz imaging via the use of ultra-short THz pulses. Their work stimulated the exploration of novel imaging concepts relying on the time-domain discrimination of the THz electric field 10,11 . Such THz imaging systems are able to collect the three-dimensional distribution of the field, with a working principle recalling that of ultrasound medical machines. In order to provide an imaging system sensitive to sub-millimetre details, as required by several applications such as biological imaging, resolutions largely exceeding the limitations imposed by standard far-fieldbased imaging approaches (e.g. 300 μm at 1 THz) are commonly required. For this reason, several near-field techniques have been implemented to obtain deeply sub-wavelength (sub-) resolutions (even below /100) [12][13][14][15][16] . A typical approach relies on a mechanical sub- probe, such as a tip or an aperture, able to probe the near-field of the THz field impinging on the sample. A possible alternative for the characterization of sub- THz features relies on the implementation of a knife-edge (KE) measurement of the field radiating from the surface of the sample under investigation 17,18 . The KE technique involves the partial clipping of a propagating field induced by a thin shield with a flat boundary, i.e. a blade. In the standard setting, a power meter collects the transmitted power. The blade is then translated and the correlation between the blade position and the recorded power enables the reconstruction of the field profile. The KE technique has been successfully implemented to characterize the profiles of a variety of THz sources. In particular, we recently highlighted that the reconstruction of sub- sources using KE techniques is in general characterized by severe image aberrations due to the non-separable space-time nature of the radiated non-paraxial field. Yet, exploiting the inherent field sensitivity of time-domain THz detection techniques, we demonstrated that the exact sub- reconstruction of the field profiles is indeed feasible 18 . For a proper sub-λ characterization via KE, a physical blade must be located and translated in close proximity (at a sub-λ) distance from the sample, a quite unpractical constraint in many real-world scenarios.
In this work, we propose an all-optical KE characterization technique and we demonstrate its working principle by characterizing the sub-λ features of a spatially modulated THz source directly on the nonlinear crystal employed for the THz generation. In our experiment an ultraviolet (UV) optical beam is projected on the output facet of a generation crystal (featuring sub-λ thickness) where it induces a thin layer (< 100 nm) of photo-excited carriers. Such conductive mask blocks THz radiation, thus acting as the blade in a KE measurement, directly on the THz source plane. We then use this approach to implement a KE measurement enabled by the optically induced virtual blade.
This optical KE (OKE) technique eliminates the need of a moving physical blade close to the sample. It is also worth noticing that the OKE enables us to characterize the emission geometry inside the generation crystal, overcoming the significant refractive index mismatch between the crystal and the air, well known for filtering out the high frequency components of the field spatial spectrum. We employ this approach to demonstrate the imaging of a THz source with sub- features (< 30 m) directly in its emission plane.

Results
The experimental setup used for the demonstration of the OKE approach is shown in Fig. 1a. Terahertz pulses are generated via optical rectification in a 20 m-thick, free standing, <110>-cut Zinc Telluride (ZnTe) crystal. The THz pulse is detected by electro-optics sampling in a 3mm-thick, <110>-cut ZnTe crystal at the Fourier plane using a parabolic mirror-based system. For the sub- characterization by the KE technique it is indeed essential to detect the central part of the spatial Fourier transform of the investigated field pattern 18 (note that this configuration differs from the standard setup in THz spectroscopic systems). The virtual blade on the output facet of the generation crystal is formed by UV femtosecond pulses ( = 400 nm, having a photon energy of 3.1 eV) that photo-excite the carriers into the conduction band of the ZnTe semiconductor in a sharp blade-shaped area projected by a telescope system (lenses L1 and L2 in Fig. 2) to form the desired sharpness and beam waist (the beam waist is determined to be = 3mm through a conventional KE measurement, i.e. performed by translating a physical blade to block the UV beam -in the absence of the ZnTe generation crystal -at the THz generation plane). OKE measurement is implemented by translating a physical metallic blade before the telescope in the x0-direction, which controls the boundary position of the optical-induced conductive layer, the virtual knife-edge, on the crystal surface. generation. The red curve shows the THz peak field against the delay between the UV pump and the THz pulse, tpump,UV. Approximating the photo-excited region as a steep-wall metal layer, the dark grey plot shows the estimated exponential decay d = ln (AR/AS), i.e. the product between the effective layer thickness d and its attenuation coefficient , where AR is the reference THz field and As is the signal detected after propagation through the layer 19 . The maximum attenuation is reached for delays exceeding 6 ps, with approximately 85% of the total THz energy reflected by the free carriers. In the inset the peak field, plotted up to a maximum delay of 140 ps, highlights a carrier recombination time within the scale of 100 ps, consistent with the excitation of a very high photo-carrier population in the conduction band.  Fig. 3b where the THz power obtained by the integral of 8 the squared field over time is plotted versus the blade position. Noteworthy, as the THz beam waist is significantly super-, the OKE does not introduce any significant changes in the field phase 18 (i.e. the waveform just scales as the beam is clipped by the edge). This is also highlighted in Fig. 3c, which shows the OKE measurements resolved around the frequency 1 THz ( = 300 m), extracted as the square-root of the THz power spectral density at 1 THz: the curve matches closely the one obtained from the full spectrum, at any tested delay. The OKE measurement for a UV pump delay of 6 ps compared with the beam waist of the optical pump. The inset shows the reconstructed spatio-temporal map.
The spatio-temporal reconstruction of the THz field, ETHz (x0, y0, t), is thus obtained at a pump delay tpump,uv = 6 ps, under the common hypothesis of a completely separable x and y dependence of the field profile (i.e. ETHz (x0, y0, y0, t)). The temporal THz waveform collected by the TDS system for each blade delay provides a spatio-temporal map, EM (x0, t). The Given the fact that we are mapping the THz source inside the generation crystals -one of the main targets in this paper-, it is important to highlight that among several crystals that are mid-to-large band-gap nonlinear semiconductors, ZnTe is by far the most popular and widely deployed for THz generation due to the very favourable phase matching condition (occurring around the Titanium:Sapphire laser central frequency, i.e. at 800 nm). Thus, this approach has a significantly large applicability in the field (this may include also semiconductor-based THz generating devices such as quantum cascade lasers (QCLs)).

Sub- beam profiling.
We chiefly investigated the OKE as a mean for the characterization of sub- THz features, and specifically at the THz source plane. A pump illumination pattern consisting of narrow bright fringes is generated by a Fresnel biprism made of BK7, featured by a refractive index of 1.51, a refractive angle of 1.5 and an apex angle of 177. In order to generate the fringes, the biprism is mounted in the THz pump beam path, before the ZnTe generation crystal. The fringe spacing, dp, originating from the field interference is simply determined by geometrical optics as dp = /[2(n-1)r] = 30 m, where  is the 800 nm pump wavelength, n is refractive index of the biprism and r is the refractive angle, as shown in Fig. 4a. Figure 4b-c illustrate the OKE measurement applied on the sub- modulated THz source and its reconstructed spatio-temporal profile with blade movement steps of 5 m. Figure 4c shows a better resolved acquisition performed using 1 m blade position steps ranging from 0.5 to 0.7 mm.

Discussion
The reconstructed THz pulse in Fig. 5a shows a very strong contribution of the sub- fringes, indicating that the data are collected in proximity of the generation plane. This is particularly evident when the reconstructed field in the transformed space E (R) (k, ) is observed as in Fig. 5c, along with the calculated E (i) (k, ) in Fig. 5d where c is the speed of light. Such function cuts the super luminal components corresponding to the region bounded by the straight lines / THz spectral region. Specifically, our setting is directly applicable to THz field-profiling inside the generation crystal. We believe that this approach could readily lead, as a next step, to the realization of reconfigurable (active) metamaterials or other structures directly imprinted on the generating crystal.
We foresee the usage of a spatial light modulator (SLM) to dynamically create and move the optically induced blade.

Grating-patterned (rectangular-shaped) sub- THz source. A thin biprism
with refractive index n, refraction angle r and maximum thickness d, as shown in Fig. 4a, forms fringes when illuminated by coherent radiation. In the small angle approximation the intensity pattern after the biprism (at distance z) reads as 23 : where k =2 is the propagation constant. From simple algebra the fringes spacing results: where d is the deflection angle. In our experiment, a Fresnel biprism is mounted prior to the ZnTe crystal, and non-diffractive fringes are projected onto the ZnTe to generate a sharply sub- modulated THz source.
Numerical calculation of the THz source distribution. The optical pump is modelled as a one dimensional source distribution along x in the plane z = 0, generating THz radiation by optical rectification. In the hypothesis of nodepletion, the optical source intensity Io (x, z,  acts as an equivalent current source in the Helmholtz equation for the THz electric field. In our experiment the optical field is polarized along y. By computing the nonlinear second order tensor we calculate that the equivalent current source is polarized along y, together with the generated electric field, obtained from the equation: The electric field is calculated by solving equation (7) with the two dimensional green integral 24 .
KE+TDS system transfer function. The THz electric field is defined in time, t, and in frequency, ω, by the Fourier relation: Considering z the propagation coordinate, a spatial Fourier transform for the transverse coordinates x, y is defined as: For simplicity we consider an incident field distribution ( )Ê ( , ) i x y  that is invariant and polarized along y, i.e. parallel to the blade edge. The blade is in the plane z = 0. At a coordinate z = 0 + , the field transmitted by the blade can be calculated via the Sommerfeld integral 24 : For each position of the blade moving along x, the TDS system implements the integral of the transmitted field along x and y for a specific polarization, i.e. for a field polarized along y we collect the quantity: Deriving along x0 and transforming back in the temporal domain we get the reconstructed field: That leads to the transfer function:

Acknowledgments
The work has been supported by the MERST (Ministère de l'Enseignement supérieur, de la Recherche et de la Science) and by the NSERC (National