Controlling the Angular Divergence of Terahertz Radiation Generated by Single-Color Filaments Using Phase Optical Elements

We have experimentally obtained two-dimensional distributions of terahertz radiation generated by one or four filaments formed by phase optical elements in air. It has been demonstrated that the use of the phase mask reduces the propagation angles of terahertz beam by approximately one and a half times, which is due to the interference of terahertz radiation from four sources. The use of the Dammann grating slightly enlarges these angles.


INTRODUCTION
Over the past two decades, the problems of generating and detecting radiation in the terahertz spectral range ( -10 THz) have remained one of the most pressing topics in laser physics and nonlinear optics [1][2][3].One of the terahertz radiation sources proposed over the years is plasma formed during filamentation of femtosecond laser pulses in air [4,5].In case of single-color filamentation, the radiation pattern of this source is a hollow cone with the opening angle determined as , where is terahertz radiation wavelength and is filament plasma channel length [5][6][7].In some cases, the distribution of terahertz radiation in a plane perpendicular to laser pulse propagation direction can have a strong modulation [8] or represent two maxima located on an axis perpendicular to laser polarization plane [9][10][11], while on the propagation axis, a minimum of the terahertz signal is still observed.It was demonstrated that the efficiency of terahertz radiation generation in a singlecolor filament grows significantly with beam numerical aperture increase, while the cone opening angle widens, reaching more than 60°, and even at the angle of 90° from axis appreciable terahertz signals are observed [12].Obviously, it is very difficult to diagnose and apply such a divergent terahertz beam.Therefore, in a number of works authors tried to reduce the divergence of a terahertz beam.For example, in the paper [13] it was shown in numerical simulations that the interference of signals from several filaments formed with a time shift can give one narrow maximum directed at an angle to the axis.However, introducing the necessary time shifts for a large array of filaments seems to be a rather difficult experimental task.Later, in [14], also using computer simulations, it was shown that the terahertz radiation cone can be narrowed many times by creating a regular array of filaments with a period equal to the terahertz radiation wavelength, thereby increasing the brightness of the terahertz source.
The aim of our work is to experimentally demonstrate the possibility to change the pattern of terahertz radiation generated during single-color filamentation by creating an ordered array of filaments using diffractive optical elements.

METHODS AND EXPERIMENTS
In the experiments we use radiation from a Ti:Sa laser system generating pulses of 100 fs duration with a central wavelength of 750 nm.The beam diameter is 8 mm at the 1/e level.The laser beam is polarized horizontally.After the laser pulse compressor, the phase mask divides the laser beam into four equal parts with the phase shift of adjacent parts of .Such a mask forms the Hermite-Gaussian TEM 11 mode in the far field [15].With a power at each maximum several times higher than the critical self-focusing power, the phase mask creates an array of four filaments that do not merge with each other even near the focus if additional focusing is used [16].In the experiments we also use a Dammann grating, which, like the phase mask, forms four equally intense maxima at the vertices of a square in the far field.As demonstrated in [17], this phase element also creates a 4-filament structure that is stable over a wide range of laser pulse power.In addition, we carry out experiments with a Gaussian beam, i.e., without phase elements, which forms a single maximum.The laser pulse energy in the Gaussian beam is 1.5 mJ, and in the presence of phase elements 6 mJ.Thus, each maximum contains 1.5 mJ, which corresponds to approximately 5-fold excess over the critical power for self-focusing.Having passed through the phase elements (or without them), the laser beam is focused by a spherical mirror with focal length of 1 or 0.5 m.At the focal length of 0.5 m, the use of the phase mask leads to the appearance of four filaments with a distance between them in the focal plane of about 50 μm.For a Dammann grating the distance is about 250 μm.The use of the mirror with a focal length of 1 m leads to a twofold increase in the distance between the filaments.The distance is determined from the plasma channels side image obtained by a lens on a CCD camera.The plasma formed as a result of self-focusing and ionization of air is a source of terahertz radiation.We register terahertz radiation by a superconducting NbN hot electron bolometer (Scontel).A narrow-band filter with a maximum transmission of about 85% at a frequency of 1 THz with a spectral width of 0.17 THz is placed in front of the bolometer input window.The use of the filter is necessary since different terahertz frequencies propagate at different angles [18], and in the case of detecting a broadband signal, changes in the propagation angle can be "blurred" [7].The scheme in our experiments is similar to [11], which makes it possible to obtain a two-dimensional distribution of terahertz radiation.The bolometer is located at a distance of about 40 cm from the focal point of the spherical mirror.The horizontal distribution of terahertz radiation is measured by rotating the bolometer in the horizontal plane around the focal point.Then we rotate the laser pulse propagation axis by moving and adjusting the spherical mirror in the vertical plane.The position of the focal point is fixed.For each angle of the optical axis, we measure the angular distribution of terahertz radiation in the horizontal plane.As a result, a twodimensional (raster) distribution of terahertz radiation is obtained step by step.

RESULTS AND DISCUSSION
At first, we experimentally obtain two-dimensional distributions of terahertz radiation generated in plasma during single-color filamentation of a Gaussian beam without modulation, a Gaussian beam using the phase mask and the Dammann grating (Figs.1a-1c, respectively).The beam is focused by the mirror with a focal length of 1 m.Similar to the earlier works [9][10][11], the pattern of the terahertz emission at 1 THz, generated during filamentation of the Gaussian laser beam, consists of two pronounced maxima located on the axis perpendicular to the laser pulse polarization plane (Fig. 1a).
The use of the phase elements (phase mask and Dammann grating) does not change the structure of the terahertz radiation distribution: two maxima are also observed on the vertical axis.However, the use of the phase mask significantly (by about one and a half times) reduces the terahertz radiation propagation angles (Fig. 1b).Apparently, this is due to the interference of terahertz radiation, which has been shown numerically in the paper [14].In the case of using the Dammann grating, the propagation angles of terahertz radiation, on the contrary, slightly increase (Fig. 1c).Indeed, after the grating, the beam is split not into four, but into many sub-beams, which form four local maxima only in the far field.Thus, the effective numerical aperture corresponds to the original Gaussian beam (the sub-beams converge from different regions of the original beam).However, when focusing and propagating a Gaussian beam, a phase shift occurs due to Kerr nonlinearity and self-focusing.Whereas each of the many sub-beams has a relatively small power, and Kerr nonlinearity begins to act only when the sub-beams have already largely merged in space, i.e., near the beam waist [17].Due to this effect, selffocusing does not have a significant impact on the beam propagation after the Dammann grating, which results in a considerable reduction in the plasma channels length and, as a consequence, an increase in the terahertz emission propagation angles [4,7,12].It should be noted that with phase elements the amplitude of the terahertz signal is slightly higher, which can be explained by a four times higher energy of the initial laser pulse.
Experiments carried out with a spherical mirror with a shorter focal length (0.5 m) show similar results (Fig. 2).The use of the phase mask due to the creation of four sources and their interference makes it possible to significantly reduce the terahertz radiation propagation angles (Fig. 2b) compared to a single filament (Fig. 2a).In addition, Fig. 2b shows weak (not much higher than noise) additional local maxima predicted in [14].Figure 2c shows vertical sections of the terahertz patterns from Figs. 2a, 2b.The phase mask narrows down the terahertz radiation cone by almost one and a half times.

CONCLUSIONS
Two-dimensional distributions of terahertz radiation at a frequency of 1 THz generated by one and four filaments are experimentally obtained.The terahertz beam has the form of two maxima located on the axis perpendicular to the polarization plane of the laser beam.Four filaments are created using the phase mask and the Dammann grating.When terahertz radiation is generated by four plasma channels created by the phase mask, the propagation angles of the terahertz beam are reduced by approximately one and a half times due to the interference of terahertz radiation from several sources.The use of the Dammann grating, on the contrary, slightly increases these angles, which is apparently due to a significant filaments shortening in this mode compared to the regime with the phase mask and to the Gaussian beam without additional phase modulation.

Fig. 1 .
Fig. 1. (Color online) Normalized two-dimensional angular distributions of terahertz radiation at frequency of 1 THz for (a) the Gaussian beam and (b, c) the beam modulated by (b) the phase mask or (c) the Dammann grating focused by the mirror with a focal length of 1 m.