We study diffraction of Bessel vortex beams with topological charges of ±1 and ±2 and a wavelength of 130 µm on two-dimensional amplitude periodic gratings. Results of simulations and experiments at the Novosibirsk Free Electron Laser facility show that there appear periodic patterns in the planes corresponding to the classical main and fractional Talbot planes, but instead of self-images of the holes, there are observed periodic lattices of annular vortex microbeams with topological charges corresponding to the charge of the beam illuminating the grating. The ring diameters are the same for beams with different topological charges, but they are proportional to the grating period and inversely proportional to the diameter of the beam illuminating the grating.
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The classical Talbot effect [1–3] is exact self-reproduction of the wavefront behind a periodical grating illuminated by a plane wave in a sequence of so called “main Talbot planes” situated at distances4,5] located at distances described by following expression: 6], in which the self-imaging effect was applied to ablation of substrates with the use of pulsed Nd:YAG laser and a microlens array. To protect the lenses and to increase the spatial frequency of the ablation spots, the substrates were placed in one of the fractional Talbot planes behind the intermediate hole array illuminated by the microlenses. Almost all studies of the Talbot effect have been performed so far in the visible wavelength range. The classical Talbot effect in the terahertz range was first demonstrated in , where it was applied to the measurement of the gap between a germanium window and an array of sensors inside a 2D microbolometer camera , as well as to the measurement of the wavelength of free electron laser radiation.
Another area of optics still of interest is beams with an orbital angular momentum, or vortex beams [9–11]. However to date, only a few papers have been published in which the Talbot geometry was applied either to study of vortex beams or to formation of vortex-beam lattices. In  the two-channel moiré deflectometry was used to measure both the wavefront and the transverse component of the Poynting vector of an optical vortex beam. In  a vortex beam with a topological charge of 1 to 3 illuminated a linear amplitude grating, behind which diffraction patterns were observed in the distance z′ = 5LT/2. The only purpose of the experiment was to demonstrate that distortion of grating slit images reflects the magnitude and sign of the topological charge of the beam. Although this task was achieved, it could be achieved in a more straightforward way, e.g. by diffraction of a vortex beam on a double slit, as it was demonstrated in [14,15]. In recent paper , diffraction of vortex beams from amplitude radial grating was investigated.
A greater number of studies have been devoted to the formation of lattices of vortex beams by embedding elements to form vortex beams directly into the openings of the gratings. In work , a periodic structure of fork-like holograms was created using a spatial light modulator (SLM). A plane wave passing through the holograms formed square or hexagonal arrays of vortex beams. Similar interferometric techniques using lateral shearing of interfering plane or spherical waves were applied in  for generation of a regular network of fork fringes. A lattice of microscopic spiral phase plates fabricated by photopolymerization was used for creation of vortex beam arrays in . In , the incident plane wave illuminating a two-dimensional lattice of holographic phase elements, each forming an individual vortex beam. It was shown that in this case a lattice of vortex microbeams appears in the main and fractional Talbot planes. The advantage of this scheme is the possibility of using an SLM to form in each cell a beam with any given topological charge. In  two artificially formed quasi-Bessel vortex beams with different transverse wave numbers and individual topological charges interfered in free space, demonstrating that the positions of the Talbot planes do not depend on the magnitude of the topological charges. A similar geometry with annular arrangement of elements to generate vortex beams was used in . For completeness, it is worth mentioning a number of publications [23–28] directly or indirectly related to the problem we are considering.
In this paper we first study the diffraction pattern that appears behind two-dimensional amplitude periodic gratings of round holes illuminated by a wide vortex beam. Since a vortex beam has more independent parameters than a plane wave does, we intended to find out how the diffraction pattern depends on the beam and grating parameters and to what extent the results agree with the classical Talbot effect. We did not expect obtaining exact self-imaging of the grating, which is a distinctive feature of the Talbot effect, but assumed that we would see some similarities with the properties of the classical Talbot effect. Almost all the experiments were performed using the high power radiation of the Novosibirsk free-electron laser (NovoFEL)  at a wavelength of 130 µm, and one experiment was carried out at a wavelength of 49 µm. A large wavelength makes it easy to manufacture the amplitude gratings used in the experiments. An additional advantage in the use of terahertz radiation is the opportunity of reaching maximum possible spatial resolution because the size of the sensitive elements of the microbolometer array [30,31] is equal to 51 µm.
The plan for this article is as follows. In Section 2 we describe the experimental setup. Results of simulations are presented in Section 3. In Section 4 we compare the results of numerical simulations with the experimental results. Finally, all the results and their possible application are discussed in Section 5.
2. Experimental configuration
The experimental schematic is shown in Fig. 1. The Gaussian beam of the NovoFEL with an average power of about 100 W was transformed by a silicon phase binary axicon with a diameter of 30 mm and a helical structure of the zones into a vortex Bessel beam with a topological charge of ±1 or ±232]. Application of these beams for formation of surface plasmon polaritons is described in . The diameters of the maxima of the first rings of the beams were 1.9 and 3.0 mm. The beams were expanded by a telescopic system consisting of a silicon lens and a parabolic mirror, the magnification of which was M = f2/f1 (in the experiments M = 5 or 3.3). The distance from the mirror to the grating was 600 mm. The power of the Bessel beam incident on the grating was approximately 10 W due to large Fresnel losses in the silicon optical elements and radiation loss in the positive diffraction orders on the axicon. The gratings were copper or aluminum plates with a thickness of 200 to 800 µm, in which round holes of diameter D = 1 or 2 mm with a period P = 2 to 6 mm were drilled. The diameters of the gratings were much larger than the diameter of the axicon and the parabolic mirror. Further on, the gratings will be denoted by the abbreviation DdPp, where d and p are the numerical values of the diameter of the holes and the grating period in mm.
The diffraction pattern was recorded by a 320 × 240 microbolometer terahertz camera [30,31], which was moved by a linear translator stage along the optical axis, with a frame rate of 30 frames per second. The length of recorded video films was 750 frames. The physical size of the images was 16.32 × 12.24 mm2. The simulations were performed for an area of 30 × 30 mm2, but the calculated patterns are presented in the figures in the same size as the experimental data. On the one hand, such a representation makes it easy to compare numerical calculations with experimental results. On the another hand, due to geometric limitations, a number of periods on a 30-mm grating in the terahertz range cannot be more than 5 – 15. As is known (see, for example, ), in this case the Talbot images are not distorted only in the central part of the pattern.
3. Results of simulations
The calculations of the electromagnetic field behind the grating were performed by Matlab R2016b using the Rayleigh-Sommerfeld integral, the applicability of which for our geometry follows from Fig. 5 in . The diffraction patterns are presented as they are seen when an observer looks along the z axis. In this case, when the beam has a positive topological charge, its phase increases clockwise. Figures 2 and 3 show the calculated diffraction patterns of a plane wave and a vortex wave with a topological charge l = +1 behind the gratings with openings of 1 mm in diameter and a period increasing from 2 to 6 mm for the Talbot planes with Z = z/LT = 1/6 to Z = 1.
The diffraction patterns for a plane wave (Fig. 2) completely correspond to the expectations and indicate that our simulation methods are correct. First, in the Z = 1 and 1/2 planes, the images of the grating are precisely reproduced (the blurring of the circle boundaries at large P is explained by the decrease in the number of interfering waves). Second, in the Z = 1/4 and 1/6 planes, the images of the double and tripple spatial frequencies are clearly distinguishable (they disappear when the spatial orders overlap). Third, the calculated phase distributions correspond to the expected distributions. For example, in the groups of four points in the 1/4 and 3/4 planes the phase difference, in exact agreement with the theory [4,5], changes clockwise as follows: 0, π/2, π, and π/2. For a given wavelength, the diffraction patterns in the case of the classical Talbot effect depend only on the grating period, and the effect of self-imaging is perfectly demonstrated.
In the case of vortex beams (Fig. 3), the patterns obey more complex regularities. The first conclusion that we can make is that the illumination of a grating by a vortex beam produces some ordered pictures with Talbot spatial periods. Remarkably, arrays of ring-like vortex beamlets can be formed in the “classical” Talbot distances (Fig. 3). This can be a reason to call this phenomenon “the quasi-Talbot effect”. Despite the variety of observed diffraction patterns, even the first glance at Fig. 3 suggests a discernible regularity in their appearance. This is clearly seen if we compare the columns of figures for the Talbot distances, equal to 1/4, 1/2, and 1, in which arrays of annular vortex beams appear (hereinafter, we normalize the distances by the Talbot length Z = z/LT ). Summarizing the results presented in Fig. 3, as well as similar compilations (which we do not show here) of the diffraction of vortex beams with different parameters, we find that five parameters should be considered in order to describe this regularity: diameter of grating openings and distance from the grating to the observation plane, grating period, size of the ring-like illuminating beam, and its topological charge.
The diameter of the holes determines the angle of diffraction of radiation behind the grating. At D = 1 mm and λ = 130 µm, the diffraction angle on a single hole is equal roughly to 0.13. For a beam of ∼ 13 mm in diameter, waves diverging from the left and right edges of the grating intersect in a distance of approximately z0 = 100 mm. Thus, the first two parameters determine the “zone of limited diffraction” (see Fig. 4) within which waves with opposite phases do not interfere. This distance corresponds to the frames located in the upper left corner of Fig. 3. In these frames, the “local” classic Talbot effect is clearly visible. This is irrefutably confirmed, for example, by the phase distribution in the above-mentioned groups of four images in the Z = 1/4 plane for gratings D1P2 and D1P3. The helicoidal nature of the illuminating beam, however, manifests itself in these frames as a general phase dependence exp (+ilθ). In distances greater than 100 mm, a periodic lattice of perfect vortex microbeams with an annular intensity distribution resembling the Fourier transform of the Bessel beam arises in the Z = 1/4, Z = 1/2, and Z = 1 planes. The topological charge of each microbeam corresponds to the topological charge of the illuminating beam, but the phase distribution Φ = exp [i(θ + ∆θsm)] is shifted depending on the microbeam position (xs, ym). The diameter of the rings in the corresponding Talbot planes increases with the grating period. For the same grating, the diameter grows up with the number of the plane. When the rings begin to overlap, complex distributions of intensity and phase arise in place of the system of rings. This corresponds to the upper region in Fig. 4.
Dependences on the topological charge, the diameter of the holes, and the diameter of the illuminating beam are clearly visible from the collections of diffraction patterns shown in Fig. 5. The diameter of the rings [Fig. 5(a)] remains constant when the sign and the magnitude of the topological charge change, which may be useful for applications. Increase in the diameter of the holes with the period unchanged does not alter the diameter of the rings [Fig. 5(b)], but the pattern is distorted when the rings begin to overlap. Another dependence [Fig. 5(c)], which should be specially noted, is the decrease in the diameter of the rings with increase in the illuminating beam diameter (in the experiment this is increase in the telescope magnification M). The dependencies shown in Fig. 5 can be represented graphically. The grating period is uniquely related to the Talbot length via relation (1), so we can present the dependence of the diameter of the rings as a function of the Talbot length [see Fig. 6(a)]. With good precision, this dependence turned out to be a linear one. On the contrary, the dependence of the diameter of the rings of the beamlets on the diameter of the beam illuminating the grating [Fig. 6(b)] is inversely proportional to the size of the latter.
In addition, we give some more numerical results. We investigated the diffraction of vortex Laguerre-Gaussian beams on the same gratings. In this case, lattices of annular microbeams are also observed in the Talbot planes. When gratings with asymmetrical holes are illuminated by Bessel beams, oval-shaped rings are observed in the Talbot planes.
4. Comparison with experiments
A set of frames extracted from terahertz videos recorded by the microbolometer array is presented in Fig. 7 for distances expressed in the Talbot length. Comparison of the frames shows that the patterns recorded in the experiments are in good agreement with the results of the numerical calculations. This is clearly seen from comparison of rows e–f and g–h. Comparing series a–b and c–d, we see that the experimental frames corresponding to the calculated ones are situated a little bit farther from the grating. This can be easily explained if we assume that in the experiments with M = 5 a slightly divergent wave was formed after the telescope, as a result of which the position of the Talbot planes shifted.
Comparing the results of the experiments and calculations, we see that in both cases bright spots are observed in the center of the rings formed in the Talbot L = 1/2 plane, and in the experimentally recorded frames they are somewhat brighter. The emergence of this spots are easily explained by the fact that, in addition to the main Bessel beam, a weaker beam with a plane wave front is formed in our experiments behind the binary silicon axicon. The appearance of the spots is caused by two reasons. The binary axicon used by us was calculated for a wavelength of 141 µm, whereas the experiments were carried out at a wavelength of 130 µm, and therefore a small part of the radiation propagated as a zero-order wave of diffraction. This portion of the radiation was also taken into account in the calculations. However, one more source of the plane wave was not taken into account in the calculations: the Fresnel reflections inside the axicon. In our case, the interference of these waves turned out to be constructive (for details, see Sec. III(b) in ), which additionally increased the intensity of the undifferentiated wave in the experiment. Thus, these waves are responsible for the appearance of the classical Talbot pattern in the Z = 1/2 plane.
Fig. 7(b,d) shows that in the experiment the rings are transformed in spirals. It is known that such patterns appear when the vortex beam interferes with a spherical wave. Our numerical calculations of Talbot carpets, which we do not present here, have shown that, because of the finite dimensions of the incident beam, the electromagnetic wave behind the grating is divergent. The interference of the vortex waves with the spherical wavefront of the non-diffracted portion of the beam forms helical (a) and double-helical (b) patterns. In the figure they are not seen well enough due to poor spatial resolution, but a similar diffraction pattern with an excellent resolution is shown in Fig. 26 in a review .
In addition to the experiments described above, we carried out an experiment on the diffraction of vortex beams on gratings at the wavelength of the NovoFEL equal to 49 µm. At this wavelength, we also observed the formation of annular beams.
The Talbot effect is revisited for the case of vortex Bessel beams illuminating two-dimensional gratings. Regular patterns with spatial frequencies corresponding to the main and fractional Talbot planes are observed in respective distances. The diffraction patterns are more complex than in the case of the classical Talbot effect and depend on a larger number of parameters. Instead of images of the holes, there are ring-shaped microbeam arrays formed in the Talbot planes within a certain distance interval. For each of these rings, as it was shown in the calculations, the topological charge reproduces that of the illuminating beam. The greater is the grating period, the larger are the ring diameters. Expansion of the illuminating beam leads to decrease in the ring diameters. Keeping the diameter of the illuminating beam unchanged and increasing the lattice period, we find that the diameter of the rings increases linearly. “Images” of holes in the Talbot planes are ideal single rings. Recall that the spatial Fourier spectrum of Bessel beams is a ring and that the diameter of this ring for beams created by a binary axicon does not depend on the topological charge, and if the beam is expanded by a telescopic system, its angular spectrum decreases linearly. Recall also that the Talbot effect can be described by means of a fractional Fourier transform . Taking into account what was said above, one can conclude from the dependences obtained in this paper that the properties of the “ideal” vortex beams observed in the Talbot planes (the beams whose diameter does not depend on the topological charge) are determined by the Fourier-transform properties of the Bessel beams.” More detailed analytical calculations will be carried out later.
The phenomenon described in the paper can be used in various applications, for example, to create an array of optical traps, in micromachining, and in microfluidic devices. This method has a useful property of the production in the planes with Z = 1/4 of arrays of beamlets with a double spatial frequency, which densely fill the space. The formation of a vortex beam array with a simple metal grating allows using high power laser beams. This is especially important in the terahertz optics, where metals reflect radiation practically completely, whereas dielectric optical elements have higher absorption as compared with elements in visible range, and it is difficult to create, for example, terahertz light modulators commonly used for transformation of wavefronts.
RFBR grant (15-02-06444); Russian Science Foundation (14-50-00080); The Ministry of Education and Science of the Russian Federation (RFMEFI62117X0012, 16.7894.2017/6/7).
The authors are grateful to G. N. Kulipanov, V. G. Serbo and V. A. Soifer for stimulating discussions, B. O. Volodkin for technical support, and Ya. V.Getmanov, V. V. Kubarev, T. V.Salikova, M. A. Scheglov, O. A.Shevchenko, D. A. Skorokhod, and other members of the NovoFEL team for the invaluable support of the experiments.
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