Abstract
Terahertz pulse timedomain holography is the ultimate technique allowing the evaluating a propagation of pulse broadband terahertz wavefronts and analyze their spatial, temporal and spectral evolution. We have numerically analyzed pulsed broadband terahertz GaussBessel beam’s both spatiotemporal and spatiospectral evolution in the nonparaxial approach. We have characterized twodimensional spatiotemporal beam behavior and demonstrated all stages of pulse reshaping during the propagation, including Xshape pulse forming. The reshaping is also illustrated by the energy transfer dynamics, where the pulse energy flows from leading edge to trailing edge. This behavior illustrates strong spatiotemporal coupling effect when spatiotemporal distribution of Bessel beam’s wavefront depends on propagation distance. The spatiotemporal and spatiospectral profiles for different spectral components clearly illustrate the model where the Bessel beam’s wavefront at the exit from the axicon can be divided into radial segments for which the wave vectors intersect. Phase velocity via propagation distance is estimated and compared with existing experimantal results. Results of the phase velocity calculation depend strongly on distance increment value, thus demonstrating superluminal or subluminal behavior.
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Introduction
Bessel beam is a theoretical model of nondiffracting light beam with an infinite number of rings that can cover an infinite distance and require an infinite amount of power. Although true nondiffractive Bessel beam can not be created in practice, several approximations can be made. The beams obtained in such approximations are called quasiBessel beams and exhibit low or no diffraction over a limited propagation distance. One of the most efficient ways for quasiBessel beams generation is Gaussian beam focusing by an axicon or conical lens. GaussBessel beam formation by the axicon occurs due to a linear phasedelay in a transverse coordinate for an incident light field. A GaussBessel light beam is generated at the axicon exit with an amplitude distribution in the radial crosssection described by the square of the zerothorder Bessel function. Incident radiation can be both continuous wave or pulsed wave.
At present, quasiBessel light beams attract wide attention from scientists due to a number of unique properties^{1}: high intensity in the nearaxis region over a limited propagation distance, small diffraction divergence of the central maximum in comparison with traditional Gaussian beams, and reconstruction properties. QuasiBessel beams are also actual in case of nonlinear lightmatter iteractions: these beams enhance ionization and laserinduced plasma effects^{2}. QuasiBessel beams are widely used in optical coherence tomography^{3}, control of micro and nanoparticles^{4}, highprecision microprocessing^{5}, detection of rotating objects^{6} and threedimensional imaging^{7}.
The concept of nondiffractive beams can be theoretically implemented for any range of electromagnetic radiation. Recently, studies of terahertz (THz) Bessel beams properties began, and Bessel beams of narrow and broadband THz radiation have already found their application in several fields. For example, usage of Bessellike beams in THz imaging results in significantly improved depth of focus^{8}. Another actively developing application based on THz radiation is THz communications, where the Bessel beam’s potential for wireless communications was studied^{9}. THz communications are a promising technology to satisfy the increasing requirements on a capacity and speed of data transmission in wireless systems^{10,11,12}. THz communication systems were developed on the basis of continuous and narrowband THz signal sources. However, the possibilities of using broadband THz signals, maximizing the available bandwidth of THz frequencies, are already being discussed^{13}. Broadband THz systems applicability for communications is restricted by the attenuation and diffraction during the propagation in the atmosphere. This problem is especially important for communications. One possible solution which may be addressed to this problem is the application of broadband THz Bessel beams for data signal transfer. Spreading immunity of quasiBessel beams seems even more useful for the case of pulsed radiation, when we should control both spatial and temporal light behavior. However, to apply nondispersive or, which is more exact, limited dispesive properties of pulsed Bessel beams for practical purposes, it is necessary to investigate the formation and propagation features of pulsed Bessel beams of broadband THz radiation.
First of all, pulsed broadband Bessel beams (also called “Xshaped waves”) are different from monochromatic Bessel beams as they contain multiple frequencies. Therefore, such beams are localized (i.e. maintain their shape) at a single frequency, but become dispersive for multiple frequencies, because the phase velocity of each frequency component is different. Thus, Xshaped waves are nondispersive only in isotropichomogeneous media. Secondly, propagation behavior of Xshaped waves is much more complex, since superluminal effects occur in their speed evolution. The term Xshaped waves appeared because they represent X shaped intensity distribution over time and radius and result from the interference of fewcycle wavepackets from ultrashort pulse light sources. Some papers mentioned that due to the “scissors effect”, the point of intersection of two or more crossing wave fronts should not obey the restrictions imposed by the principle of relativity^{14,15}. As a result, the interference maximum at the apex of the axicon can propagate at a velocity exceeding the speed of light in vacuum. The observation of this effect^{16} led to series of discussions^{14,15,17,18}. Although it is obvious that the energy velocity is always subluminal, one can observe an effect that beam propagating along the optical axis has superluminal group and phase velocity value. In general, these superluminal effects can be explained both by a specific geometry of the wave front during the passage of THz radiation through the axicon, and by the large anomalous dispersion of the propagation media^{19}. However, axicon generated Bessel beam’s fasterthanlight propagation is not related to the absorption or dispersion anomaly but arises due to the interference between the plane wave components of the beam in free space^{20}.
Superluminal effects for evanescent waves have been demonstrated in tunneling experiments^{21}. This effect can be revealed only over short distances due to the evanescent field properties. Then there were attempts performed to extend this effect over larger distance: Mugnai et al.^{16}. demonstrated such a possibility in the propagation of localized microwaves over the distance of some tens of wavelengths. Experimental evidence of localized light waves in a centimeter range was given in^{22}, demonstrating a practical way of obtaining Xshaped waves. These waves were theoretically predicted as Bessel beams in the 1980’s^{23}, and then they were investigated in connection with their superluminal behavior^{24}. Recently, measuring of a spatiotemporal field structure of Bessel Xshaped beams has been performed for ultrashort pulses in paper^{25}.
In the THz frequency range Bessel beams evolution deserves additional attention because of ultimate case of a pulsed radiation consisting of a few electric field oscillations. Since the appearance of the coherent detection technique, which allows direct measurement of THz amplitude field in temporal domain, there have been several attempts to estimate experimentally superluminal effect of pulsed THz Bessel beam propagation. For example, in the work of Lloyd et al.^{26} fasterthanlight behavior was observed, but only in the optical beam axis. Pulse reshaping was also determined for paraxial case, but it does not illustrate full spatiotemporal pulse behavior during propagation, especiaclly in presence of spatiotemporal coupling (STC), analyzed for ultrashort pulsed THz radiation^{27,28}. Moreover, the sampling distance step in experimental measurements of superluminal effect in^{26} for THz waveforms exceeded almost an order of value of the size occupied by the beam in space.
Since the key issue is the experimental realization of Xwaves and the monitoring of their evolution, there is ongoing effort in developing methods and tools for such studies. THz imaging is a possible solution allowing mathematical modeling and simulation of THz field passed through the object^{29,30,31}. However, most of the imaging techniques do not provide a full description of spatiotemporal evolution of THz field, and can only form images of simple binary amplitude objects. Holographic approach outperforms imaging based on timedomain spectroscopy, which is more powerfull for the chemical sciences^{32}, or tomographic^{33} techniques in acquisition time. THz holography^{34,35} provides also better spatial resolution, and needs less computational powers for numerical reconstruction than tomography. THz pulse timedomain holography (THz PTDH)^{36} as a specific case of the holography method applied for pulsed THz radiation provides information about temporal and complex spectral characteristics of the wave field. Field propagation through an arbitrary amplitudephase object allows its relief and optical characteristics reconstruction. THz PTDH developed for measuring the amplitudephase characteristics of a field passed through an object provides wide possibilities for analyzing a dynamics of complex wavefront^{37}. This technique is a powerful tool that allows extrapolation of the THz field behavior in spatial regions located at different distances from the plane where the measurements are provided.
Research of pulsed THz GaussBessel beam’s full spatiotemporal and spatiospectral evolution via propagation distance is actual and has not performed yet by our knowledge. The purpose of the research is to study the propagation dynamics of the THz pulse twodimensional profile during the passage through the phase axicon in the spatiotemporal and spatiospectral representations, as well as the estimation of the phase velocity behavior for THz GaussBessel beam in nonparaxial approach and comparison with existing experimental results.
The paper is organized as follows. Section “Results” considers the spatiotemporal and the spatiospectral evolution of pulsed broadband THz GaussBessel beam. Spatio temporal coupling and pulse reshaping are demonstrated in timedomain. In the spectral domain the broadband evolution is demonstrated as well as dynamics for the individual frequency components, thus illustrating that the wavefront after the axicon consists of radially symmetric segments which propagate at certain angles to the optical axis. Broadband longitudinal spectral evolution is additionally presented. The section “Phase velocity estimation” provides the description of pecularities of superluminal effect calculation, estimating the contribution of distance increment along zaxis.
Results
Spatiotemporal evolution
The electric field of any ultrashort laser pulse often fails to be separated purely into a temporal and spatial factors. These effect known as spatiotemporal coupling (STC) is observed for instance in review^{28}. STC is important in many studies connected with wavefront propagation. THz pulse consisting of only several oscillations of electric field is a special case of pulsed electromagnetic radiation which should be also investigated according to this STC effect. Moreover, GaussBessel beam’s evolution represents a nonparaxial case of wavefront propagation due to the fact that radially symmetric segments of the wavefront propagate along optical zaxis at some angle after the axicon. In this case temporal (or spectral) and spatial (or angular) properties of THz ultrashort pulse are interdependent.
The results are obtained with the use of THz PTDH technique described in detail in section “Methods”. The results shown in Fig. 1 illustrate the dynamics of GaussBessel beam propagation in twodimensional representation. Since the GaussBessel beam is circularly symmetric, the spatiotemporal distribution in (x,t) plane provides comprehensive information about overall spatiotemporal beam properties at each coordinate of propagation axis. Temporal field evolution is depicted in coordinate system with time delay τ = t − z/c. The coordinate system moves at a constant group velocity taken equal to speed of light in vacuum c along the optical axis. This coordinate system minimizes temporal pulse shift and thus allows clear observation of pulse wavefront reshaping since for short pulses observations their shape changing is more important than its uniform propagation at a group velocity^{38}. Figure 1 demonstrates the distribution of the THz field E(t, x) in the central crosssection of GaussBessel beam via propagation distance in the range from 0 to 25 mm. In order to demonstrate the beam evolution, these temporal forms are normalized to global electric field amplitude maximum. Temporal wavefront evolution during its propagation along zaxis over the distance range from axicon plane to z = 25 mm (where the Bessel beam exists) are presented with small step increment Δz = 100 μm (see movie 1).
2D field patterns in the Fig. 1 show the wavefront inversion as the propagation distance increases. Moreover, during the propagation strong pulse reshaping and Xshape structure formation occurs (Fig. 1c). This Xshape spatiotemporal distribution was investigated previously with the usage of optical 2f2f imaging system, and the results of the study showed strong STC in the propagation of fewcycle pulses^{27}. STC in this case was explained by the cutoff effect introduced by the focusing lens. Moreover, the cutoff effect appears strongly for fewcycle pulses in contrast to the long pulse, where its spatial distribution is time independent. Pulse reshaping in our research could be visually illustrated by energy transfer dependency via propagation, when energy flows from leading edge to trailing edge (see Fig. 2).
Indeed, THz pulse is displaced at the temporal scale for ~5 ps (here we use running temporal axis). This is explained by the nonparaxial propagation of lateral wavefronts and their interference. More clearly it is shown in 1D and 3D representation in Fig. 3, which depicts THz pulse evolution along the optical zaxis.
Spatiospectral evolution
Due to the broadband nature of pulsed THz GaussBessel beam it is useful to observe its spatiospectral evolution via propagation distance. Figure 4 illustrations provide 2D dependencies of spectral amplitude in central crosssection of the GaussBessel beam. The sequence of images illustrates changes in the spectral amplitude distribution for the corresponding distances from 0–25 mm with increment Δz equal to 5 mm. In detail these 2D spectral amplitude profiles U(x, ν) consist of a bright central spot surrounded by weaker interference rings. This spatiospectral pattern forms after axicon and changes during propagation. At distance z = 25 mm the central peak of spectral amplitude becomes equal to the amplitude distribution at the edge of the beam, thus illustrating that the beam is already disintegrated. Moreover, each pattern in Fig. 4 corresponds to the firstorder Bessel functions which depend on THz frequencies. The spatiospectral lateral decreasing behavior is attributed to the propagation of wavefronts at some angle to the optical axis and their mutual interference. This behavior could be refered to the frequency factor in wavefront propagation equations (see section Methods, eqs. (5), (7)). Analyzing the spectral evolution with small increment Δz = 100 μm (see movie 2) in region from axicon plane to z = 25 mm it becomes obvious that the GaussBessel beam exists in this distance range.
The longitudinal evolution is demonstrated in Fig. 5 for several monochromatic components as well as for summarized spectral range from 0,05 to 2 THz. Figure 5f provides detailed consideration showing crosssections for fixed distance z = 9,5 mm (the crosssection is indicated by dot line in Fig. 5a). One can see that different frequency components provide different patterns of maximum and minimum of spectral amplitude due to the mutual interference of radially symmetric wave planes formed after axicon and then propagating at some angle to the optical axis. The crosssection corresponding to the broadband frequency range 0,05–2 THz illustrates spreading of lateral interference pattern for the fixed distance z. Thus, the frequency summarized structure in Fig. 5a shows the narrow GaussBessel nondiffractive beam in the integral form without interference rings how it could be detected in the experiment.
Phase velocity estimation
Lloyd et al.^{26}. estimated the phase velocity of THz GaussBessel beam experimentally. However, these values were measured only for paraxial case, i.e. on the optical axis in a single point. Moreover, for waves of a more complicated form in contradistinction to monochromatic plane waves the phase velocity differs in general from c/n and varies from point to point even in a homogenius medium^{39}. Anyway, according to the theory of relativity, signals can never exceed c. This implies that the phase velocity cannot correspond to a signal propagation velocity.
Experimental measurements^{26} of the phase velocity was provided in case of THz waveforms with distance increment Δz equal to 5 mm along zaxis. The phase velocity is calculated according to V_{ ph } = 2πν ⋅ Δz/Δφ. Thus, calculation of the derivative value of a discrete function may play a substantial role. From a mathematical point of view, the derivative definition implies the calculation of the function change on ultrasmall intervals. However, there are examples where, from certain physical considerations, the intervals of the argument over which the differentiation is performed are chosen to be sufficiently large, for instance, in the case of deterministic phase retrieval^{40}. In work^{26}, apparently, the calculation of the derivative over such a large interval was due to the peculiarity of the experimental scheme or the difficulty of the operational measurements. Thus, the convergence to the exact value of V_{ ph } via grid pitch decreasing deserves additional approval. Due to the THz pulse duration of 2 ps (this is equal to 600 μm in space), step size of 5 mm corresponding to ~10 longitudinal beam sizes may be too big for clear superluminal effect observation. In present research we show that with THz PTDH technique it is possible to monitor the evolution of the broadband THz field both at big and small distances. Here we simulated full spatialtemporal evolution of pulsed THz GaussBessel beam with different distance increment values. Figure 6(a–d) demonstrates spatiofrequency distributions of the phase velocity V_{ ph }(x, ν) with distance increment 5 mm starting from z = 4,5 mm similar with paper^{26}. For the estimation of the phase velocity behavior for the case of the small increment we simulated the beam propagation evolution using Δz equal to 10 μm (Fig. 6(e–h)).
The experimental results in paper^{26} represented averages over several pairs of measurement in distance range from 4,5 mm to 24,5 mm with increment 5 mm. In order to compare these values Fig. 7 depicts averaged V_{ ph } for central xplane crosssection marked by dot line in Fig. 6a,e. It is clearly seen that for increment 5 mm V_{ ph } exceeds c. The inset in the Fig. 7a demonstrates the spectral range similar to the results shown in Fig. 4 in paper^{26}, showing the approximate agreement of the frequency dependent phase velocity value. In this case, the oscillations in the phase velocity graph in Fig. 4 in^{26} could be attributed to the experimental measurements peculiarities, since there are no objective reasons for their appearance, as follows from the calculated graphs. On the other hand, calculations with small distance increment Δz = 10 μm demonstrates that the phase velocity behavior does not display superluminal effect. Moreover, V_{ ph } dynamic is subluminal and approximates asymptotically to c via frequency increasing. Noisy behavior of the phase velocity at the edges of xplane is associated with pulse size and corresponding THz amplitude decreasing due to the GaussBessel beam reshaping during the propagation.
Discussion
This paper investiagates the evolution of the pulsed THz GaussBessel beam in nonparaxial case in the spatiotemporal and spatiospectral representations, thus demonstrating strong spatiotemporal coupling. We have demonstrated THz pulsed GaussBessel beam’s wavefront propagation, including formation of the Xshape structure and subsequent wavefront changes to the opposite orientation relative to the original position behind the axicon at z = 0 mm. The evolution of the phase velocity in the twodimensional representation is discussed, comparing with the experimental measurements in previous work^{26}, where THz waveforms were acquired at distance increment equal to 5 mm along zaxis. This can be crucial due to the fact that THz pulse with 2 ps duration occupies 600 μm in space. Thus, the phase velocity estimation requires spectral phase information of two waveforms measured at two different locations on zaxis according to V_{ ph } = 2πν ⋅ Δz/Δφ. Therefore, step size of 5 mm corresponding to ~10 longitudinal beam sizes may be incorrect from the mathematical point of view, since the phase velocity is calculated as the discrete derivative Δz/Δφ with the assumption that the Δz tends to zero. This can be important especially for GaussBessel beam, where plane wave components after axicon propagate at some angle to the optical axis, but waveforms measurements are provided only along zaxis. As derivative of a discrete function Δz/Δφ is more accurate when increment is smaller, the question whether it is possible to estimate the phase velocity using Δz step about 10 pulse sizes needs an additional approval. Thus, convergence to an exact value as the grid step is reduced, is a significant aspect for superluminal observation. In our research we have shown that results of the phase velocity calculation depend strongly on propagation increment value, thus demonstrating superluminal (Fig. 7a) or subluminal behavior (Fig. 7b).
Methods
The investigation scheme is depicted in Fig. 8. The wideaperture THz beam propagates through the phase axicon, which is formed by the corresponding amplitudephase transmission in the approximation of an infinitely thin phase object. According to works^{41,42} original THz pulse is a singlecycle pulse, whose electric field amplitude E(t) can be represented by the following function:
where E_{0} is the amplitude of electric field cycles, and τ sets the pulse duration value. The THz pulse, which is an ideal oneperiod structure with FWHM of the envelope of the modulus square of E(t) about 1 ps, decomposes through the Fourier spectrum into frequencies ν:
These spectra are complex functions \(u(x^{\prime} ,y^{\prime} ,\nu )=u(x^{\prime} ,y^{\prime} ,\nu )\exp (i\phi (x^{\prime} ,y^{\prime} ,\nu ))\) that contain both amplitude u(x′, y′, ν) and phase φ(x′, y′, ν) components in each (x′, y′) point at zplane. Then for each spectral component the field u(x′, y′, ν) is layerwise multiplied on the amplitude transparency mask T(x′, y′, ν), thus forming initial 3D data array at z = 0 mm. Here coordinates (x′, y′) indicate at the initial plane where the axicon is placed:
Here transparency mask T(x′, y′, ν) corresponds to 2D Gaussian distribution (Fig. 8, inset), which characterizes amplitude of wide collimated THz beam. If the beam is ideally collimated, the dependence of field amplitude via frequency can be neglected. However, in practice, it may not occur if to use, for instance, chromatic focusing elements. In present calculation we use the approximation where the THz beam is collimated and is not affected by diffraction divergence during small propagation distances from the THz source to the axicon plane. The phase mask corresponding to the axicon relief H(x′, y′) is transformed into a phase distribution according to the equation:
Here H(x′, y′) is axicon relief distribution, n is refractive index of the axicon’s material. Thus, a passage of the THz field through the axicon with relief H(x′, y′) is equivalent to the 3D data array of phase delay φ(x′, y′, ν) for each spectral component ν. It is also possible that the material from which the diffractive element is made has a dispersion. A detailed study on this issue is published in the paper^{44}, but now this case is beyond the scope of our consideration. Diffraction by the axicon also can be described using Maxwell theory with the finiteelement method^{45}.
Here we use a scalar theory of diffraction realized by THz PTDH^{36}. The resulting twodimensional complex wave field \(u(x^{\prime} ,\,y^{\prime} ,\,\mathrm{0,}\,\nu )=u(x^{\prime} ,\,y^{\prime} ,\,\mathrm{0,}\,\nu )\exp (i\phi (x^{\prime} ,\,y^{\prime} ,\,\mathrm{0,}\,\nu ))\) after the axicon propagates along zaxis. Wavefront propagation from object plane u(x′, y′, 0, ν) to arbitrary plane U(x, y, z, ν) is realized by the angular spectrum (AS) and RayleighSommerfeld convolution (RSC) methods^{46}. AS method for wavefront numerical propagation is defined as follows:
where
is the angular spectrum represented through the spatial frequencies (f_{ x }, f_{ y }).
Similarly, the field U(x, y, z, ν) can be calculated using the RayleighSommerfeld convolution (RSC) method:
where h is the pulse response function
and r is the distance between the object and registration planes:
In case of broadband THz radiation, the choice of the calculation method is determined by the critical frequency ν_{0}, depending on the distance z, the grid size N, the grid pitch Δx which should be equal to Δy (see the paper^{47}) and the refractive index dispersion n(ν):
Thus, AS method is used when \(\nu \ge {\nu }_{0}\), and RSC is used otherwise for ν < ν_{0}. There are some experimental techniques providing temporal forms of THz electric field. In particular the method of electrooptical detection is illustrated in Fig. 8, where crosspolarization scheme is used^{48,49}. In this configuration the registration plane represents the plane where THz detection crystal ZnTe is located, and the plane of the object is the axicon itself (see Fig. 8). Therefore, it is possible to vary the propagation distance Δz by axial displacement of the phase axicon from the detection crystal according to the assumpion that the incident THz beam with Gauss profile do not change significantly its form via axicon displacement. The main feature of the THz holography scheme is the possibility of measurement of the wide aperture collimated THz beam and the recording of THz electric field in timedomain. This method^{36} allows numerical investigation of the dynamics of wave field in temporal E(x, y, z, t) and spectral U(x, y, z, ν) representation in arbitrary plane. Thus, one can reconstruct the complex amplitudephase characteristics of radiation in an arbitrary located plane by solutions of numerical wavefront propagation equations (5) and (7). This experimental possibility allows the simulation in approximation of wide collimated pulsed THz beam.
THz PTDH technique provides a possibility to estimate the evolution of electric field of beams by estimating and measuring their propagation dynamics in the representation of 5Ddata: threedimensional spatial coordinates, temporal and spectral scales, including amplitude and phase information in the complex spectrum data. This method is a powerful tool of obtaining an amplitudephase THz fields with high resolution, which allows to display the spectroscopic information of the object under study. It also allows the accounting of optical parameters, such as a refractive index dispersion of the object material and propagation media. The method demonstrates the ability to reconstruct smooth and stepped relief objects or an object that is transparent in the THz region, as well as to estimate the complex spatialtemporal evolution of the complex THz field. Theoretical plausibility and repeatability of THz PTDH is confirmed in a variety of experimental and simulation works^{36,37,44,46,50,51,52}.
Conclusion
THz PTDH method could be applied for arbitrary beams which can be formalized by initial amplitudephase spatial distribution. In this case THz PTDH approach can perform simulation of wide aperture beams, thus expanding numerical solutions of wavefront propagation to nonparaxial case. This is especially actual for the beams which propagates at some angle to the optical axis, for example GaussBessel beams, formed by axicon with high base angle. Reliability of THz PTDH was proved including by comparison of simulated wavefront with the wave field experimentally obtained after its passing through an arbitrary phase object (see Fig. 5 in paper^{36}). In present research we observed the evolution of THz pused broadband GaussBessel beam in spatiotemporal and spatiospectral representation, demonstrating all stages of pulse reshaping during propagation, including Xshape pulse forming and analyzing the energy transfer dynamics, where pulse energy flows from leading edge to trailing edge. THz PTDH also provided the illustration of STC effect of Bessel beam’s wavefront propagation. The beam’s profile structure in the temporal and spectral domain clearly illustrated the model when the wave front of the GaussBessel beam at the exit from the axicon can be divided into radial segments for which the wave vectors intersect during propagation. The phase velocity via propagation distance was estimated and compared with existing experimantal results^{26}. The superluminal or subluminal behavior depends strongly on distance increment value in compariosn with the size occupied by the THz GaussBessel pulse in space: for the high value of Δz like in^{26} superluminal effect is observed and for the small Δz subluminal effect is observed.
References
McGloin, D. & Dholakia, K. Bessel beams: diffraction in a new light. Contemporary Physics 46, 15–28 (2005).
Faccio, D. et al. Nonlinear lightmatter interaction with femtosecond highangle bessel beams. Physical Review A 85, 033829 (2012).
Lee, K.S. & Rolland, J. P. Bessel beam spectraldomain highresolution optical coherence tomography with microoptic axicon providing extended focusing range. Optics letters 33, 1696–1698 (2008).
McGloin, D., GarcésChávez, V. & Dholakia, K. Interfering bessel beams for optical micromanipulation. Optics letters 28, 657–659 (2003).
Alexeev, I., Leitz, K.H., Otto, A. & Schmidt, M. Application of bessel beams for ultrafast laser volume structuring of non transparent media. Physics Procedia 5, 533–540 (2010).
Fu, S., Wang, T., Zhang, Z., Zhai, Y. & Gao, C. Nondiffractive besselgauss beams for the detection of rotating object free of obstructions. Optics Express 25, 20098–20108 (2017).
Planchon, T. A. et al. Rapid threedimensional isotropic imaging of living cells using bessel beam plane illumination. Nature methods 8, 417–423 (2011).
Bitman, A., Moshe, I. & Zalevsky, Z. Improving depthof field in broadband thz beams using nondiffractive bessel beams. Optics letters 37, 4164–4166 (2012).
Kollarova, V. et al. Application of nondiffracting beams to wireless optical communications. In Proc. SPIE, vol.6736, 67361C (2007).
Mandehgar, M., Yang, Y. & Grischkowsky, D. Experimental confirmation and physical understanding of ultrahigh bit rate impulse radio in the thz digital communication channels of the atmosphere. Journal of Optics 16, 094004 (2014).
Grachev, Y. V. et al. Wireless data transmission method using pulsed thz sliced spectral supercontinuum. IEEE Photonics Technology Letters in print, 103–106 (2017).
Liu, C., Wang, C. & Cao, J. Multipath propagation channel modeling and capacity analysis for terahertz indoor communications. Journal of Optical Technology 84, 53–61 (2017).
Akyildiz, I., Jornet, J. & Han, C. Teranets: Ultrabroadband communication networks in the terahertz band. IEEE Wireless Communications 21, 130–135 (2014).
Rodrigues, W., Thober, D. S. & Xavier, A. Causal explanation for observed superluminal behavior of microwave propagation in free space. Physics Letters A 284, 217–224 (2001).
Sauter, T. & Paschke, F. Can bessel beams carry superluminal signals? Physics Letters A 285, 1–6 (2001).
Mugnai, D., Ranfagni, A. & Ruggeri, R. Observation of superluminal behaviors in wave propagation. Physical review letters 84, 4830 (2000).
Lunardi, J. T. Remarks on bessel beams, signals and superluminality. Physics Letters A 291, 66–72 (2001).
Ringermacher, H. & Mead, L. R. Comment on “observation of superluminal behaviors in wave propagation”. Physical review letters 87, 059402 (2001).
Boyd, R. W. & Gauthier, D. J. Slow and fast light. Tech. Rep., ROCHESTER UNIV NY INST OF OPTICS (2001).
Kuntz, K. et al. Spatial and temporal characterization of a bessel beam produced using a conical mirror. Physical Review A 79, 043802 (2009).
Steinberg, A. M., Kwiat, P. G. & Chiao, R. Y. Measurement of the singlephoton tunneling time. Physical Review Letters 71, 708 (1993).
Saari, P. & Reivelt, K. Evidence of xshaped propagationinvariant localized light waves. Physical Review Letters 79, 4135 (1997).
Durnin, J., Miceli, J. Jr. & Eberly, J. Diffractionfree beams. Physical review letters 58, 1499 (1987).
Recami, E. On localized “xshaped” superluminal solutions to maxwell equations. Physica A: Statistical Mechanics and its Applications 252, 586–610 (1998).
Bowlan, P. et al. Measuring the spatiotemporal field of ultrashort besselx pulses. Optics letters 34, 2276–2278 (2009).
Lloyd, J., Wang, K., Barkan, A. & Mittleman, D. M. Characterization of apparent superluminal effects in the focus of an axicon lens using terahertz timedomain spectroscopy. Optics communications 219, 289–294 (2003).
Jiang, Z. & Zhang, X.C. 2d measurement and spatiotemporal coupling of fewcycle thz pulses. Optics express 5, 243–248 (1999).
Akturk, S., Gu, X., Bowlan, P. & Trebino, R. Spatiotemporal couplings in ultrashort laser pulses. Journal of Optics 12, 093001 (2010).
Ahi, K. Mathematical modeling of thz point spread function and simulation of thz imaging systems. IEEE Transactions on Terahertz Science and Technology 7, 747–754 (2017).
Ahi, K. & Anwar, M. Developing terahertz imaging equation and enhancement of the resolution of terahertz images using deconvolution. In Proc. SPIE, vol.9856, 98560N (2016).
Ahi, K. & Anwar, M. Modeling of terahertz images based on xray images: a novel approach for verification of terahertz images and identification of objects with fine details beyond terahertz resolution. In Proc. SPIE, vol.9856, 985610 (2016).
McIntosh, A. I., Yang, B., Goldup, S. M., Watkinson, M. & Donnan, R. S. Terahertz spectroscopy: a powerful new tool for the chemical sciences? Chemical Society Reviews 41, 2072–2082 (2012).
Li, B. et al. Continuouswave terahertz digital holographic tomography with a pyroelectric array detector. Optical Engineering 55, 053106–053106 (2016).
Rong, L. et al. Dynamic dehydration observation based on terahertz inline digital holography. In Digital Holography and ThreeDimensional Imaging, DTh1A–2 (Optical Society of America, 2015).
Huang, H. et al. Continuouswave terahertz multiplane inline digital holography. Optics and Lasers in Engineering 94, 76–81 (2017).
Petrov, N. V., Kulya, M. S., Tsypkin, A. N., Bespalov, V. G. & Gorodetsky, A. Application of terahertz pulse timedomain holography for phase imaging. IEEE Transactions on Terahertz Science and Technology 6, 464–472 (2016).
Petrov, N. V., Gorodetsky, A. A. & Bespalov, V. G. Holography and phase retrieval in terahertz imaging. In Terahertz Emitters, Receivers, and Applications IV, vol. 8846 (2013).
Shpolyanskiy, Y. A. Envelope, phase, and frequency of ultrabroadband signal in a transparent medium. Journal of Experimental and Theoretical Physics 111, 557–566 (2010).
Born, M. & Wolf, E. Principles of optics: electromagnetic theory of propagation, interference and diffraction of light (Elsevier, 2013).
MartnezCarranza, J., Falaggis, K. & Kozacki, T. Deterministic phase retrieval techniques employing spherical illumination. In Proc. of SPIE Vol, vol.9525, 952503–1 (2015).
Koulouklidis, A., Fedorov, V. Y. & Tzortzakis, S. Spectral bandwidth scaling laws and reconstruction of thz wave packets generated from twocolor laser plasma filaments. Physical Review A 93, 033844 (2016).
Kapoyko, Y. A., Drozdov, A. A., Kozlov, S. A. & Zhang, X.C. Evolution of fewcycle pulses in nonlinear dispersive media: Velocity of the center of mass and rootmeansquare duration. Physical Review A 94, 033803 (2016).
Lee, K. & Ahn, J. Singlepixel coherent diffraction imaging. Applied Physics Letters 97, 241101 (2010).
Kulya, M. et al. Computational terahertz imaging with dispersive objects. Journal of Modern Optics 64, 1283–1288 (2017).
Khonina, S. N. & Degtyarev, S. A. Analysis of the formation of a longitudinally polarized optical needle by a lens and axicon under tightly focused conditions. J. Opt. Technol. 83, 197–205, https://doi.org/10.1364/JOT.83.000197, http://jot.osa.org/abstract.cfm?URI=jot834197 (2016).
Balbekin, N. S., Kulya, M. S., Rogov, P. Y. & Petrov, N. V. The Modeling Peculiarities of Diffractive Propagation of the Broadband Terahertz Twodimensional Field. In Physics Procedia, vol. 73, 49–53 (2015).
Petrov, N. V., Bespalov, V. G. & Volkov, M. V. Phase retrieval of thz radiation using set of 2d spatial intensity measurements with different wavelengths. In Proc. SPIE, vol. 8281, 82810J (2012).
Jiang, Z. & Zhang, X.C. Terahertz imaging via electrooptic effect. IEEE Transactions on microwave theory and techniques 47, 2644–2650 (1999).
Wu, Q., Hewitt, T. & Zhang, X.C. Twodimensional electrooptic imaging of thz beams. Applied Physics Letters 69, 1026–1028 (1996).
Semenova, V. et al. Amplitudephase imaging of pulsed broadband terahertz vortex beams generated by spiral phase plate. In Infrared, Millimeter, and Terahertz waves (IRMMWTHz), 2016 41st International Conference on, 1–2 (IEEE, 2016).
Gredyukhina, I. et al. Influence of the degree of neutralization of acrylic acid and crosslinking agent on optical properties and swelling of sodium polyacrylate. Optics and Spectroscopy 122, 877–879 (2017).
Balbekin, N. S., Kulya, M. S. & Petrov, N. V. Terahertz pulse timedomain holography in dispersive media. Computer Optics 41, 348–355 (2017).
Acknowledgements
This work was supported by the Ministry of Education and Science of the Russian Federation, Project No. 3.1893.2017/4.6. The Authors also expresses gratitude to Arkadiy A. Drozdov for useful discussions about theoretical aspects.
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M.S.K. developed the software, performed numerical simulation of THz GaussBessel beam’s wavefront propagation and prepared the core of the manuscript. N.V.P. conceived the idea of application of THz PTDH for wavefront sensing of arbitrary broadband beams and layed the concept of the numerical investigations of their spatiotemporal dynamics. V.A.S. prepared the review in the introduction about the current state of research in the THz Bessel beams. V.G.B. is the establisher of THz PTDH concept. M.S.K. and N.V.P. contributed to the manuscript preparation.
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Kulya, M.S., Semenova, V.A., Bespalov, V.G. et al. On terahertz pulsed broadband GaussBessel beam freespace propagation. Sci Rep 8, 1390 (2018). https://doi.org/10.1038/s4159801819830z
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DOI: https://doi.org/10.1038/s4159801819830z
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