Bessel-Gaussian beam’s paraxial diffraction

The investigation scheme of Bessel-Gaussian beams’s paraxial diffraction is demon-strated in Fig. 3.1. The phase axicon is illuminated by the wide-aperture THz beam.

Amplitude and phase function of the phase axicon could be described by general eq. (2.41), which gives the complex wavefront in the axicon plane. Then we simu-late this wavefront propagation in the air.

Fig. 3.2 demonstrates the Bessel beam evolution in time-domain. Gauss-Bessel beam is circularly symmetric, thus its spatio-temporal structureE(x,t)gives a sufficient information about general space-time beam’s character. Time-domain dynamics is shown in coordinate system with running time, whereτ=t−z/c. It means that the coordinates move with a speed corresponding to the beam’s group velocity. Such system minimizes the temporal beam shift and helps to observe the wavefront reshaping[63]. The inset in Fig. 3.1 illustrates the THz field structure E(x,t)in the center of the Gauss-Bessel beam versus distance of propagation in the interval 0-25 mm. For clear observation the wavefront is normalized to the global maximum of amplitude.

THz Gauss-Bessel beam has a broadband spectrum, thus we can also analyze its spatio-spectral dynamics versus distance of propagation. Fig. 3.3 demonstrates distri-butions of spectral amplitude in the center of the Gauss-Bessel beam for distances in the interval 0–25 mm with the step∆z=5 mm. In the middle distances (5-20 mm) it is observed that the spectral amplitudeG(x,ν)contains the strong central disloca-tion encircled by the weaker interference rings. Atz=25 mm the central dislocation becomes comparable to the distribution at the edge of the beam, thus proving that it is already degraded. Such spatio-spectral dynamics could be explained by the mutual interference of wavefronts during their propagation at some angle to the optical axis.

Figure 3.1 The setup for pulsed THz Gauss-Bessel beam propagation. The phase axicon is irradiated by collimated THz beam. The fs wide-aperture probe beam propagates the same way and goes to the detection crystal. Fs probe beam changes its polarization under the impact of THz beam according to the electrooptical effect in ZnTe crystal. Hence, the polarization of fs probe field is relative to the amplitude of the THz field. This is registered on the CCD matrix in terms of intensity. The polarization state is controlled by two Glane prisms (G) in a crossed position at 90 degrees. The time-domain forms of THz signal is provided by the scanning using the optical delay line which changes the intersection time of the probe and THz beams in the ZnTe crystal. Placing the axicon in different positions we can get wavefronts corresponding to various propagation distances of THz Gauss-Bessel beam. n the numerical simulation the following parameters were used: THz pulse durationτ=0.38ps, number of points in time-domainN =2048, time window size˜t₀=100ps, central frequencyν₀=0.66THz. The size of axicon 20 mm and the maximum height of axiconH=9.33mm. It gives the base angle 43 degrees. This parameters correspond to the experimental work described in [48], in order to directly compare the simulation results with experimental. The refractive index of the axiconn_{ob j}=1.46^.

Figure 3.2 Spatio-temporal structure of THz Gauss-Bessel beam for propagation distances 0 mm (a), 5 mm (b), 10 mm (c), 15 mm (d), 20 mm (e), 25 mm (f).

Figure 3.3 Spatio-spectral structure of the THz Gauss-Bessel beam for propagation distances 0 mm (a), 5 mm (b), 10 mm (c), 15 mm (d), 20 mm (e), 25 mm (f).

THz holography provides a possibility to analyze the behavior of spectral data for each individual frequencies. Fig. 3.4 demonstrates the longitudinal dynamics for different frequency components and for summarized spectral interval 0.05-2 THz.

The comprehensive analysis for fixed distance z =9.5 mm is presented in Fig. 3.4.

Various spectral components give various distributions of maximum and minimum of amplitude because of the interference of radially symmetric wavefronts generated after axicon and then propagated at some angle to the optical axis. Fig. 3.4-a corre-sponds to frequency interval 0.05-2 THz which demonstrates the spreading of lateral interference structure versusz. We observe the narrow non-diffractive Gauss-Bessel beam in the integral form without any interference rings.

Figure 3.4 Longitudinal structure of Gauss-Bessel beam versus propagation length z for frequencies 0.05-2 THz (a), 0.5 THz (b), 1 THz (c), 1.5 THz (d), 2 THz (e). (f) shows the cross-sections for lengthz= 9.5 mm. The cut line is shown by dotted line in section (a).

On of the advantage of THz hyperspectral holography method is its possibility to observe wavefields at different propagation distances z. Thus, we can estimate the changes of wavefield parameters such as amplitude or phase versusz. Here the phase velocity is estimated by equationV_{p h} = 2πνz/φ. Besides, the calculation of derivative in discrete form may be significant, because the derivative definition supposes the calculation in ultra-small intervals. But there are some works, where, differentiation intervals were sufficiently large, for example, in deterministic phase

retrieval[49]. In the paper[48], the derivative estimation over large interval could be performed because of the experimental scheme peculiarity. Thus, the solution for theV_{p h}versus grid pitch decreasing should be additionally estimated. Thus, for THz pulse durationτ =2 ps (or 600µm in space), step∆z =5 mm which corresponds to≈10 longitudinal beam sizes could be huge for pure observation of superlumi-nal effect. Here we show that THz hyperspectral holography allows to investigate the dynamics of the broadband THz wavefront for different distances with different step size. Fig. 3.5 (a–d) illustrates the spatio-spectral structure of the phase velocity V_{p h}(x,ν) using step∆z =5 mm beginning from z = 4.5 mm as measured in the work[48]. For phase velocity calculation with the small∆zwe provided the wave-front propagation dynamics using length increment 10µm(Fig. 3.5 (e–h)).

Figure 3.5 Phase velocity versus THz spectrum for various length increment∆zequal to 5mm(a-d) and 5µm(e-h). (i) correspond to the central cross-section shown in (a), (j) corresponds to the central cross-section shown in (e).

Paper[48]demonstrates results averaged over several pairs of measurements in

interval 4.5-24.5 mm with length increment 5 mm. Fig. 3.5-(i,j) shows the aver-aged phase velocity for the central cross-section pointed by dotted line in Fig. 3.5-(a,e). Fig. 3.5-(i) illustrates the frequency interval similar to the results in Fig.4 in work[48], demonstrating the approximate accordance of the phase velocity dynam-ics. Note, that phase velocity oscillations in Fig.4 in[48]could be caused by experi-mental measurement features, because calculated graphs do not suppose reasons for oscillations. However, estimation with small propagation length increment ∆z = 10µm illustrates thatV_{p h} behavior does not show any superluminal effect. Phase velocity dynamics is subluminal with asymptotic drift to speed of light in vacuum.

In conclusion note, that investigation of different pulsed THz beams evolution needs solid method. Currently existing THz imaging could be a solution allowing experimental and numerical analysis of THz beams[2]. But, existing imaging tech-niques could not provide a full spatio-temporal and spatio-spectral dynamics of THz wavefront. Hyperspectral holographic, presented in this work, is type of hologra-phy realized for pulsed THz radiation and it allows full spatial, temporal and spectral investigation of THz wavefront. Moreover, THz hyperspectral holography allows the numerical wavefront propagation to the spatial areas at various distances from the plane where measurements were performed.