The recently introduced concept of radially non-oscillating, temporally stable ultrashort-pulsed Bessel-like beams we referred to as needle beams is generalized to a particular class of highly localized wavepackets (HLWs). Spatio-temporally quasi-nondiffracting pulses propagating along extended zones are shaped from Ti:sapphire oscillator radiation with a spatial light modulator and characterized with spatially resolved second order autocorrelation. Few-cycle wavepackets tailored to resemble circular disks, rings and bars of light represent the closest approximation of linear-optical light bullets known so far. By combining multiple HLWs, complex pulsed nondiffracting patterns are obtained.
© 2012 OSA
The confinement of light in space and time is a fundamental challenge in physics [1–3] with various practical implications, e.g. for high-resolution microscopy, manipulation and acceleration of particles or cells, precision metrology, or nanostructuring. In recent experiments, for example, the spatial compression of femtosecond laser pulses down to the diffraction limit was demonstrated . This limit can be broken with methods which typically take advantage of evanescent fields, plasmons, nonlinear processes or singular optics (superresolution). Another objective, however, is the localization of free propagating optical fields in the angular domain (supercollimation). Perfect supercollimation would mean that a wavepacket of finite diameter would propagate without any spread in transversal direction so that it behaves quasi particle-like. In nonlinear optics, this is approximated in media by steady-state self-trapping (needle solitons) . Self-induced spectral reshaping via conical refraction [6–8] leads to the formation of light bullets of characteristic X-shape in space and time coordinates (nonlinear X-pulses). Furthermore, supercollimation was achieved in nano-engineered materials with anomalous photonic properties .
Theoretical approaches for linear localized wave phenomena (also referred to as “undistorted progressive waves” , “nondiffracting beams” , “diffraction-free beams” , etc.) are known for many decades as special solutions of wave equations for light or matter waves (for an overview, compare  and references cited there). Ideal Bessel beams were described as propagation-invariant solutions of the Helmholtz equation for conical waves [11,12] with infinite extension, radial field profiles governed by Bessel functions, non-zero axial field vectors, self-reconstruction properties , accumulating Gouy phase  and (depending on the parameters) subluminal, luminal or even superluminal group velocities (not related to any violation of the relativistic limitations of velocity). Because of the infinite generating fields, theoretical Bessel beams carry infinite total energy and are physically irrelevant. An alternative theoretical approach on the basis of superimposed Hankel functions  delivers Bessel beams of finite transversal extension. Physical approximations corresponding to finite energies can be obtained experimentally with optical systems transforming the input field into a wave with a conical angular distribution [17,18], e.g. by diffractive, refractive or reflective axicons [19,20] or holographic components . Axicons  enable to generate finite, but (in comparison to focused Gaussian beams) significantly more extended focal zones . The field profile of the perfect theoretical Bessel beams oscillates in all spatial directions including a vector component parallel to the propagation axis. In the simplest case, this spatio-temporal field can be described by the expression23–25]. An important example are Bessel-Gaussian beams which result from Gaussian beam illumination [17,26]. Higher order Bessel beams with central intensity minima are obtained by transforming Laguerre-Gaussian beams with axicons . The constructive interference of spectral field components of broadband, ultrashort-pulsed Bessel-like beams leads to linear X-pulses [28–30] which were known in acoustics from optimizing the operation of ultrasound transducers [31,32]. Superluminal as well as luminal localized pulses (“light bullets”) as solutions of the wave equations were previousy predicted by theoretical works [33–36]. It was shown that “frozen waves” with spatial localization can be generated by a continuous superposition of waves and that arbitrary shapes like spikes, donuts or cylindrical surfaces can be obtained. By breaking the radial symmetry, other types of localized waves can be generated. Airy beams with asymmetric fringe distributions and distinctly curved propagation paths were formed linearly and nonlinearly. This type of wavepackets was also referred to as “light bullets” [37,38]. The degree of localization in Bessel, Bessel-Gauss, and Airy beams, however, is limited to a certain extent because of the energy contained in the wings of radially oscillating profile functions. The more the propagation path is curved, the higher is the number of sideband oscillation peaks.
In recent experiments with reflective axicons we demonstrated that fringe-free, high-aspect-ratio Bessel-like distributions (“needle beams”) can be formed. Non-oscillating profiles require a self-apodized truncation with apertures matched to a central maximum [39,40]. In this case, the energy of outer wings is lost (or one has to recycle it by coherent addition in a resonator structure). It was demonstrated that femtosecond needle beams also show a self-reconstruction phenomenon like pulsed Bessel beams [13,40]. An alternative, aperture-less method to realize the concept of needle beams is to reduce the axicon angle until the radius of the nondiffracting zone exactly matches the central lobe of the resulting Bessel distribution (supposed that magnetic field effects can be neglected). Both arrangements can be coupled with lenses or telescopes to further reduce or enhance the beam diameter. Reconfigurable axicon arrays were programmed in the phase map of a spatial light modulator (SLM) [40,41]. For sub-20-fs pulses, the spectro-temporal profiles of needle beams were found to propagate nearly unchanged whereas they are corrupted in outer fringes of Bessel beams and extended focal zones of Gaussian beams . This better temporal stability compared to the foci of polychromatic Gaussian beams has to be emphasized as a specific advantage of needle beams and might be of importance for future ultrafast optical applications.
As we will show by experimental results with spatially programmable few-cycle pulses, such supercollimated, radially non-oscillating and temporally localized polychromatic beams are not restricted to needle beams. In this paper we propose to generalize these type of beams to a particular class of pulsed, “highly localized wavepackets” (HLW) which closely approximate perfect linear-optical light bullets. Spatio-temporal and angular free-space propagation properties for different types of HLW are demonstrated. Differences to other approaches of linear light bullets and potential applications will be discussed.
2. Spatial localization
The concept of nondiffracting beams was controversially discussed in comparison to Gaussian beams [42,43]. For a clear classification and a defined control of SLM-based systems for shaping localized beams, the spatial and temporal beam quality have to be quantitatively expressed by appropriate figures of merit. It is well known that the resolution of focusing systems is basically limited by an optical uncertainty relation between the widths in the spatial and spatial-frequency domain (positional and directional intensity [44,45]). Supercollimation can be approximated by subsequently generating focal spots at different positions along the optical axis. This can be achieved by the transversal-to-axial transformation by axicons. The interference of the superimposed coherent, conical partial waves results in the formation of Bessel-like waves or (at sufficiently large bandwidth) X-waves traveling along extended zones . In general, the angular and spatial confinement of 3D-beams can roughly be described by the beam propagation factor M2  which is related to the radial and angular variances σ0 and σf in the near- and far-field, respectively:48,49]. The application of these approximations, however, to the more general situation with adaptive compensation of input beam divergence, radially variable conical angles and axially variable beam quality is not simply possible and a comparable beam propagation factor can hardly be defined. Instead of that, radial and axial decay parameters can be used to define “beam waists” and “confocal parameters” in analogy to the Gaussian beam. For nonlinear applications or adaptive reshaping of beam profiles (e.g. off-axis arrangements or imaging through distorting media) where small variations of the intensity distribution are of particular importance, higher order statistical moments like kurtosis  and skewness [51,52] can be used as additional, compact measures for peakedness and symmetry. In the literature it was shown that dark hollow beams (DHBs) with central intensity minima can also be characterized by M2 and kurtosis .
The special case of needle beam arrays that are generated by dividing the wavefront of an extended Gaussian beam with ultraflat phase axicons programmed into an SLM [39,40] is of particular relevance for applications (e.g. generation of “flying images” ). Self-apodization without truncation is obtained, if the geometrical parameters are chosen to generate exactly the central lobe of a Bessel distribution and no outer fringes. In this configuration, the parts of the wave illuminating the segments can be regarded, in good approximation, to be plane waves. The foot-to-foot-diameter D of a monochromatic needle beam at an axial position z depends on the wavelength λ and the conical angle θ (half angle against the optical axis):
3. Spatio-temporal localization
In analogy to the spatial beam quality, the temporal shape of ultrashort pulses can formally be described by a pulse propagation factor P2 :55]:
4. The class of highly localized wavepackets (HLWs)
Pulsed needle beams (needle pulses) are localized both in space and in time. As mentioned, the main difference to other linear light bullets like pulsed Bessel beams is to appear without any radial oscillations of the intensity distribution. This makes them, finally, to the true candidates for real linear light bullets (in particular for the case of single cycle pulses where both the spatial and the temporal profiles exhibit only single maxima of the optical field).
We will show that the characteristic properties of ultrashort-pulsed needle beams addressed here can also be realized in further types of quasi-nondiffracting wavepackets. By particularly breaking the symmetry of radially symmetric needle beams but preserving their key propagation properties, the case of needle beams can be generalized to a class of radially non-oscillating wavepackets with spatially and temporally undistorted propagation over extended ranges we refer to as “highly localized wavepackets” (HLWs). The basic idea of combining diffraction-free spatial filtering with certain geometrical transforms is illustrated in Fig. 1 .
The intensity profiles of HLWs are related to extremum properties. Similar to transform-limited pulses in temporal Fourier transform, the spatial frequency spectrum of HLWs is coupled with a minimum diffraction because of apodization. The experimental realization of such HLWs is enabled by the capability of spatial light modulators to flexibly tailor arbitrary angular distributions of phase elements in extremely narrow angular intervals. The HLWs of most simple structure (“elementary HLWs”) can be used as building blocks for composing complex patterns . Contrary to alternative approaches based on enhancing the depth of focus of complete, monochromatic, coherent images by optimizing the diffraction with adapted amplitude-phase-masks , the image information is encoded in an finite number of channels which keep separated over an extended (nondiffracting) zone. Because of their high aspect ratios, self-reconstructing behavior  and tilt tolerance , needle beams were recently applied to spatially resolved autocorrelation , reconfigurable wavefront sensing  and high-speed optical data transfer  with Ti:sapphire laser pulses in 10-20-fs range.
In the following sections, the experimental realization of HLWs with programmable generalized axicons at pulse durations down to the sub-3-cycle range (6 fs) is reported. Spatial and temporal localization and pulse transfer properties of HLWs with needle-shaped, tubular and linear intensity maps are discussed. The idea to synthesize quasi-continuous complex nondiffracting images by fusing such dissimilarly shaped “elementary” HLWs is considered.
5. Experimental realization of HLWs
5.1. Experimental techniques
Figure 2 shows the experimental setup consisting of a pulsed light source (Ti:sapphire laser oscillator VENTEON PULSE: ONE PE, minimum pulse duration 6 fs, center wavelength 800 nm, FWHM spectral bandwidth 300 nm, pulse energy up to 7 nJ, repetition frequency 80 MHz), an interferometer, an active beam shaper and (depending on the measuring task) adapted detection systems.
Initial pulse duration and spectral phase were performed with a second-harmonic autocorrelator (Mini, APE) and an FC-SPIDER (few-cycle spectral phase interferometry for direct electric field reconstruction) (APE) (not drawn in the schematic representation). For a flexible shaping of HLWs, different types of low-dispersion LCoS-SLMs were used depending on the requirements (maximum phase step, resolution, fill factor). All devices worked in reflection (liquid crystal layer on mirror) with either a parallel or vertical alignment and a layer thickness between 3 and 18 µm. Maximum phase differences up to > 4 π at a wavelength of 800 nm could be obtained. To approximate HLWs with quasi-nondiffracting propagation characteristics, single and multiple conical phase elements (axicons) and related structures (generalized axicons) were programmed into the phase map of the LCoS-SLM under test. This was possible on the basis of known calibration curves for the spectral phase depending on the gray values of the graphic processor software of the SLM . In these previous experiments it was found that Gires-Tournois interference is responsible for slight periodical spectral-dependent phase distortions can be minimized by a proper choice of the SLM parameters. For example, the corresponding pulse lengthening without re-compression was in the range of 2 fs for input pulses of 15 fs (4 µm LC layer).
Beside approximately cone-shaped phase profiles (Fig. 3(a) ), Fresnel-axicons (“fraxicons” ) consisting of concentric rings similar to zone lenses, torus-shaped axicons (Fig. 3(b)) , elliptical axicons, linearly-conical axicons (Fig. 3(c)), bi-microprisms and combinations of such elements were used to generate different types of HLWs. The time-integrated spatial intensity profile was detected with a CCD (not drawn) or a cooled EMCCD camera (drawn). The highly sensitive EMCCD detection was necessary in case of very weak signals, e.g. from 2D second harmonic generation (SHG) excited by the few-nJ pulses. For a spatially resolved temporal pulse analysis, collinear autocorrelation was measured by tuning the arm length of a Michelson interferometer (formed by the of mirrors M1, M2 and M3 and a symmetric broadband beam splitter BS). Additionally, 2D spectral maps were detected with a fiber-based spectrometer (Ocean optics) on a three-axis translation stage. The second order autocorrelation function was obtained by analyzing the SHG signal generated in a thin beta-barium borate (BBO) crystal. In few-cycle experiments with pulse durations < 10 fs at the input of the SLM, the dispersion of the optical components was pre-compensated by a pair of chirped mirrors.
It has to be mentioned that the spectral dispersion can be a limiting factor for angular shaping at large spectral bandwidth. Because of the very small applied angles, however, in particular in the generation of needle beams and other highly localized wavepackets, the overlap zone of the spectral compartments remains much larger compared to the distorted zone and the effect can be neglected in good approximation in our case.
5.2. Few-cycle pulsed needle beams
Uniform and non-uniform arrays of ultrashort-pulsed needle beams with spatial periods between p = 80 µm and p = 500 µm and different geometrical arrangements (hexagonal, rectangular) were programmed in different types of LCoS-SLMs (HoloEye) with maximum phase steps between π and 2π at the center wavelength of 800 nm . The pixel size was in all cases about 8 µm. At short pulse durations in few-cycle range, the corresponding axial extension of the individual wavepackets (few micrometers) was typically three orders of magnitude smaller than the radial dimensions of the arrays (few millimeters). In fact, ultraflat wavepackets with a low number of field oscillations (similar to a stack of pancakes) propagate through a needle-shaped zone of low distortion (nondiffracting zone). To minimize unwanted dispersion, the LCoS-SLMs were operated without beam splitters in oblique incidence at incident angles of up to 50°. The resulting aberrations (ellipticity) were adaptively corrected as it was recently demonstrated in the frame of reconfigurable Shack-Hartmann wavefront applications and two-dimensional temporal mapping with pulsed needle beams . If the self-apodizing condition [39–41] is fulfilled, the angular diameter of a needle beam is determined by the conical angle after passing the axicon, the center wavelength and the divergence of the illuminating beam. In Fig. 4 , the propagation of pulsed diffractive and nondiffractive twin beams generated from the Ti:sapphire oscillator beam are compared to each other. In the starting plane, a hard circular aperture (Fig. 4(a), left) and an ultraflat axicon (Fig. 4(a), right, concial angle 0.13°) were programmed into the SLM phase map via gray values. Both elements had the same radius of 380 µm. The resulting time-integrated intensity profiles were detected at distances of z1 = 18.5 mm and z2 = 138.5 mm (Figs. 4(b) and 4(c), respectively). The double beam structure enables to quantitatively evaluate specific multi-beam effects like cross-talk and interference.
Figures 5(a) and 5(b) show the radii (defined by the transversal decay down to 1/e2 of the maximum intensity) and the intensity as a function of the propagation distance in a simulation for a monochromatic Gaussian beam (red, circles) in comparison to experimental data for a polychromatic 6-fs pulsed needle beam (blue, squares). The green dashed line in Fig. 5(b) is related to the input intensity at the center of the cross section of the illuminating Gaussian beam. Between z = 60 and z = 100 mm, a focus zone with > 4-times enhanced intensity appears. If an equivalent parameter zN corresponding to a Rayleigh length is calculated, it is not surprising that one obtains inconsistent values on the basis of expanding radius and decaying intensity (110 and 150 mm, respectively). A Gaussian beam of comparable initial geometry behaves significantly different (zG = 85 mm) so that we find a strong indication for a quasi-nondiffracting propagation of the adaptively generated wavepackets.
These results indicate an excellent propagation behavior of the needle beams even at few-femtosecond pulse durations and correspondingly broad spectral bandwidths. For the quantitative characterization, the temporal aspect has to be carefully analyzed as well. In the case of arrays of beams, however, most of the established methods for pulse diagnostics (e.g. SPIDER or FROG) can hardly be applied. Therefore we used the nonlinear two-dimensional autocorrelation (combining an interferometer with an EMCCD detector) to diagnose the pulses with spatial resolution and high conversion efficiency . Figure 6 shows the spatially resolved temporal structure of multiple needle beams reconstructed from two-dimensional second-order autocorrelation.
The matrix of generating axicons was programmed into the phase-map of an SLM (period 720 µm, phase modulation depth 400 nm, conical angle 0.13°) which was illuminated by a 6.5-fs Ti:sapphire oscillator pulse (incident angle 43°). A 10 µm thick BBO crystal was used as frequency converter. For a better visualization, the time-dependent electrical field amplitude was modified by adding a fixed DC value and encoding size and brightness of the maxima according to the absolute value of the field. The radial beam waist radius w0 at the chosen distance of z = 100 mm was found to be 150 µm. The distances of neighboring maxima indicate the cycles of the electric field (2.7 fs at 800 nm central wavelength).
The analysis of individual sub-beams in space and time enables to describe the localization of the wavepackets in a first approximation (i.e. within the frame of a simplified model as described in section 3). According to the formalism reported in , the value for the beam propagation factor M2 was determined to be 1.8.
From the autocorrelation measurement, a pulse propagation factor of P2 = 1.75 can be assumed. Both factors yield a spatio-temporal localization parameter according to Eq. (6) of L2 = 1.77 related to the conditions of our experiment. To compare, a double-transform limited HLW of Gaussian shape in space and time would have a theoretical localization factor of 1. It has to be noted that the application of M2 (as defined for Gaussian beams) to nondiffracting HLWs delivers a quantitative description but has to be treated with caution because of the ambiguity of the wavefront in the superposition zone. Modified localization parameters could consider a propagation dependence and differ between near and far field.
As already mentioned, needle beams can also be shaped by zone axicons (“fraxicons”). In the case of a monochromatic cw-source, the discrete profile can be optimized by implementing steps of exactly 2π at the transition between neighboring zones. For a polychromatic ultrashort-pulsed source, however, the axicons can only be perfectly adapted to a narrow wavelength interval. Moreover, the pulse travel time experiences a delay between adjacent zones. Therefore, the fidelity of pulse transfer with fraxicons depends stronger on the pulse duration and axicons with continuous phase profiles are more suitable.
Because of diffraction and scattering caused by substructures of the shapers (pixels, phase steps) and overlap of multiple beams, a non-negligible background signal is generated. This reduces the energy transfer efficiency significantly (depending on the system configuration). Further improvements of the contrast should be possible by spatial filtering.
5.3. Few-cycle nondiffracting light rings
Ultrashort-pulsed dark hollow beams were shaped by programming toroidal axicons in LCoS-SLMs . The phase profiles of two selected elements as gray value maps and cut curves with effective heights can be found in Fig. 7 . To classify toroidal axicons of different symmetry of their the ring profile, we define a symmetry factor SF as the ratioFig. 7(c)). The symmetry factors were SF = 0.50 and 1.00 in Figs. 7(a), 7(b) and in the curves with red dots and blue squares in Fig. 7(c), respectively.
The steps in Fig. 7(c) result from discretized phase levels. Figure 8 demonstrates the propagation properties of the individual ring-shaped HLWs which were sub-beams of a hexagonal array (period about 430 µm, Fig. 9 ). The profiles with blue squares, green circles and red triangles in Fig. 8(a) show central cuts through the intensity maps shaped at SF values of 0.5, 0.7 and 0.9, respectively. The best overall contrast was obtained at a distance of z = 8 mm whereas inner and outer contrast Ci = 84% and Ca = 55% were nearly constant for all values of SF. The hollow beam diameter (peak-to-peak distance for central cuts) depends, in good approximation, linearly on the SF (Fig. 8(b)). An intensity map of a part of the hexagonal beam array detected at a distance of 8 mm is shown in Fig. 9(b). A 3D-plot of the hollow beam array (see Fig. 9(a)) was reconstructed using all measured depth information. The period was about 430 µm after slightly correcting for elliptical distortion introduced by the deviation from normal angle of incidence (20°). The procedure was similar to the correction of aberrations described in a recent paper on adaptive wavefront sensing of ultrashort pulses with needle beams in a reflective setup .
The propagation zone of a pulsed hollow beam generated with a torus-shaped axicon of a symmetry factor SF = 0.8 is shown in Fig. 10 . Two-dimensional intensity maps were detected with a CCD camera for distances between 0 and 15 mm. The plot in Fig. 10(a) represents a central radial cut through the obtained image 3D data. Within the first few millimeters, the four separated light fields shaped by the torus profile start to overlap and to create an extended tubular zone of stable propagation. The contrast (peak-to-center) reaches values > 0.8. Corresponding curves for peak intensity (red dots) and normalized diameter D/D0 (related to a reference diameter of D0 = 394 µm) are drawn in Fig. 10(b). It is evident that the radial extension of the hollow beam experiences no significant changes over the whole propagation distance. The temporal transfer was characterized by measuring the second order autocorrelation function. At a fixed axial distance the measured pulse duration was still 6.5 fs.
The propagation behavior was also studied for single hollow beams of 8-times larger diameter (Fig. 11 ). It was found that the FWHM of the autocorrelation function varies only slightly with the increasing distance (at a correspondingly larger Rayleigh length). The iso-field representation in Fig. 12 clearly indicates a few-cycle ring-shaped wavepacket (“ring bullet”) obtained by linear-optical shaping by broadband free-space conical beams.
5.4. Few-cycle nondiffracting light blades and patterns composed of linear elements
To obtain radially stretched quasi-nondiffracting focal zones, ultraflat bi-prisms were programmed. A stack of nondiffracting pulsed light blades with a period of 200 µm perpendicular to the optical axis is displayed in Fig. 13 . Such light blades are of increasing interest for the spatially selective multiphoton fluorescence excitation of gases (e.g. for spectroscopy and turbulence analysis of flames in combustion engines) or biological matter (e.g. brain tissue). Other applications of linear nondiffracting focal zones are the high-speed depth-independent laser-machining of non-planar surfaces (e.g. 3D-texturing), transient poling of nonlinear materials, manipulation and acceleration of particles, and non-uniform plasmon excitation.
Recently we demonstrated that linear elementary HLWs can also be applied to for the spatial encoding of few-cycle pulsed “flying images” [41,63]. As an example, the nearly undistorted propagation of a pattern (logo of MBI) is shown in Fig. 14 . A selected feature of this image was analyzed more in detail. The intensity profile along a cut of 160 µm through one partial line focus (Fig. 15(a) ) was measured as a function of the propagation distance (Fig. 15(b)). The FWHM radius (Fig. 15(c)) shows a moderate increase and an aspect ratio (ratio of Rayleigh range: initial radius) of about 200:1. Image information can be extracted over a much more extended range by applying standard image processing procedures.
Quasi-nondiffracting ultrashort-pulsed patterns of even higher complexity can be composed by combining different types of elementary HLWs. The temporal transfer was not directly measured for the complex HLWs shown in this section. From measurements with needle and tubular beams with feature sizes in a comparable range (sub-100 µm field structures) we expect similar pulse durations.
The concept of pulsed needle beams as a particular approximation of spatio-temporally undistorted progressive waves can be generalized to a class of pulsed beams we refer to as “highly localized wavepackets” (HLWs). Mathematically, HLWs can be generated from needle beams by applying simple geometrical transforms (linear stretching and/or rotation). This leads to modified symmetry properties at, at the same time, persistently small spatial and temporal distortions at pulse durations down to a few femtoseconds. In previous experiments, “flying images” were composed of addressable needle beams of identical shape. In contrast to this, HLWs of circular, elliptical, linear or ring shape can serve as building blocks (“elementary” HLWs) of even more complex, segmented nondiffracting images encoded in ultrashort-pulsed light fields. The freedom of programming enables for generating arbitrary shapes. In experiments at pulse durations of down to 6 fs, needle beams, tubular beams (dark hollow beams) and linear beams (light blades) of flexible parameters were obtained by programming circular, toroidal and bi-prismatic shaped axicons in the phase map of an LCoS-SLM. Aberrations caused by working at oblique angles are corrected by an adapted off-axis design. It should be emphasized that the generated needle-type HLWs can be regarded as linear-optical “light bullets” which do not require any nonlinear effects to compensate the diffraction over significantly extended depth of focus.
For quasi-nondiffracting light fields at pulse durations where approximations like slowly varying envelope approximation and narrow bandwidth models break down  it is expected that the limits of localization have to be redefined and standardized quantitative measures like the beam propagation factor have to be replaced by more adequate parameters in future.
The authors thank T. Elsaesser and G. Steinmeyer (MBI), A. Friberg (University Helsinki), P. Saari (University Tartu), H. Weber (Technical University Berlin) and M. Piché (Laval University Quebec) for stimulating discussions. We gratefully acknowledge the collaboration with HoloEye AG (Berlin), APE GmbH and metrolux GmbH (Göttingen). The work was supported in part by German Research Foundation (project no. GR 1782/7-1).
References and links
1. P. Saari, “How small a packet of photons can be made?” Laser Phys. 16(4), 556–561 (2006). [CrossRef]
2. P. Saari, “Photon Localization Revisited,” in: Quantum Optics and Laser Experiments Sergiy Lyagushin Ed. (InTech - Open Access Publisher, Croatia, 2012), 49–66.
3. P. Saari, M. Menert, and H. Valtna, “Photon localization barrier can be overcome,” Opt. Commun. 246(4-6), 445–450 (2005). [CrossRef]
4. B. Piglosiewicz, D. Sadiq, M. Mascheck, S. Schmidt, M. Silies, P. Vasa, and C. Lienau, “Ultrasmall bullets of light-focusing few-cycle light pulses to the diffraction limit,” Opt. Express 19(15), 14451–14463 (2011). [CrossRef]
5. S. Trillo and W. Torruellas Eds, Spatial Dolitons (Springer, Berlin, 2001), pp. 73–74.
6. Y. S, Kivshar and G. P. Agrawal, Optical solitons - From fibers to photonic crystals (Academic Press, Elsevier Science, Amsterdam, 2003), pp. 226–228.
7. H. Sõnajalg, M. Rätsep, and P. Saari, “Demonstration of the Bessel-X pulse propagating with strong lateral and longitudinal localization in a dispersive medium,” Opt. Lett. 22(5), 310–312 (1997). [CrossRef]
8. P. Di Trapani, G. Valiulis, A. Piskarskas, O. Jedrkiewicz, J. Trull, C. Conti, and S. Trillo, “Spontaneously generated X-shaped light bullets,” Phys. Rev. Lett. 91(9), 093904 (2003). [CrossRef]
9. P. T. Rakich, M. S. Dahlem, S. Tandon, M. Ibanescu, M. Soljacić, G. S. Petrich, J. D. Joannopoulos, L. A. Kolodziejski, and E. P. Ippen, “Achieving centimetre-scale supercollimation in a large-area two-dimensional photonic crystal,” Nat. Mater. 5(2), 93–96 (2006). [CrossRef]
10. J. A. Stratton, Electromagnetic Theory (McGraw Hill, New York, 1941), 356.
11. J. Durnin, “Exact solution for nondiffracting beams I - The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987). [CrossRef]
13. H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves, Theory and Experiments (Wiley & Sons, New York, 2008).
14. Z. Bouchal, J. Wagner, and M. Chlup, “Self-reconstruction of a distorted nondiffracting beam,” Opt. Commun. 151(4-6), 207–211 (1998). [CrossRef]
16. S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).
19. G. Scott, “Efficient generation of nearly diffraction-free beams using an axicon,” Opt. Eng. 31(12), 2640–2646 (1992). [CrossRef]
20. J. H. McLeod, “The axicon: A new type of optical element,” J. Opt. Soc. Am. 44(8), 592–597 (1954). [CrossRef]
23. P. L. Overfelt and C. S. Kenney, “Comparison of the propagation characteristics of Bessel, Bessel-Gauss, and Gaussian beams diffracted by a circular aperture,” J. Opt. Soc. Am. A 8(5), 732–745 (1991). [CrossRef]
26. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987). [CrossRef]
28. P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997). [CrossRef]
29. R. Grunwald, U. Griebner, U. Neumann, A. Kummrow, E. T. J. Nibbering, M. Piché, G. Rousseau, M. Fortin, and V. Kebbel, “Generation of ultrashort-pulse nondiffracting beams and X-waves with thin-film axicons,” in: M. Murnane, N. F. Scherer, and A. M. Weiner (Eds.), Ultrafast Phenomena XIII (Springer-Verlag, New York, 2002) 247–249.
30. R. Grunwald, V. Kebbel, U. Griebner, U. Neumann, A. Kummrow, M. Rini, E. T. J. Nibbering, M. Piché, G. Rousseau, and M. Fortin, “Generation and characterization of spatially and temporally localized few-cycle optical wavepackets,” Phys. Rev. A 67(6), 063820 (2003). [CrossRef]
31. J. Y. Lu and J. F. Greenleaf, “Nondiffracting X waves. Exact solutions to free space scalar wave equation and their finite aperture realizations,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39(1), 19–31 (1992). [CrossRef]
33. M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, “New localized Superluminal solutions to the wave equations with finite total energies and arbitrary frequencies,” Eur. Phys. J. D 21(2), 217–228 (2002). [CrossRef]
34. M. Zamboni-Rached, E. Recami, and H. E. Hernández-Figueroa, “Theory of 'frozen waves': modeling the shape of stationary wave fields,” J. Opt. Soc. Am. A 22, 2465–2475 (2005).
35. M. Z. Rached and E. Recami, “Subluminal wave bullets: Exact localized subluminal solutions to the wave equations,” Phys. Rev. A 77(3), 033824 (2008). [CrossRef]
36. M. Zamboni-Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A 79(1), 013816 (2009). [CrossRef]
37. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4(2), 103–106 (2010). [CrossRef]
38. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105(25), 253901 (2010). [CrossRef]
39. R. Grunwald, M. Bock, V. Kebbel, S. Huferath, U. Neumann, G. Steinmeyer, G. Stibenz, J.-L. Néron, and M. Piché, “Ultrashort-pulsed truncated polychromatic Bessel-Gauss beams,” Opt. Express 16(2), 1077–1089 (2008). [CrossRef]
40. R. Grunwald, Thin-film microoptics - new frontiers of spatio-temporal beam shaping (Elsevier, Amsterdam, 2007).
44. M. Mansuripur, “The uncertainty principle in classical optics,” Opt. & Photon. News , Jan. 2002, 44–48 (2002).
45. M. J. Bastiaans, “Uncertainty principle and informational entropy for partially coherent light,” J. Opt. Soc. Am. A 3(8), 1243–1246 (1986). [CrossRef]
46. P. Saari and K. Reivelt, “Evidence of X-shaped propagation-invariant localized light waves,” Phys. Rev. Lett. 79(21), 4135–4138 (1997). [CrossRef]
47. A. Siegman, “New developments in laser resonators,” Proc. SPIE 1224, 2–14 (1990). [CrossRef]
50. S.-A. Amarande, “Beam propagation factor and the kurtosis parameter of flattened Gaussian beams,” Opt. Commun. 129, 311–317 (1996).
51. R. Grunwald and M. Bock, “Spatio-spectral analysis and encoding of ultrashort pulses with higher-order statistical moments,” in Conference on Lasers and Electro-Optics/Quantum Electronics and Laser Science Conference and Photonic Applications Systems Technologies, OSA Technical Digest Series (CD) (Optical Society of America, 2007), paper CThM2.
52. M. Bock, S. K. Das, and R. Grunwald, “Adaptive shaping of complex pulsed nondiffracting light fields,” Proc. SPIE 7716, 7950–7958 (2011).
55. M. Piché and R. Grunwald, private communication. In the discussion, the product of M2 and P2 was considered to be used to describe the spatio-temporal beam properties of pulsed Bessel beams.
60. R. Grunwald and M. Bock, “Spatially encoded localized wavepackets for ultrafast optical data transfer,” JEOS:RP (submitted to).
61. M. Bock, S. K. Das, R. Grunwald, S. Osten, P. Staudt, and G. Stibenz, “Spectral and temporal response of liquid-crystal-on-silicon spatial light modulators,” Appl. Phys. Lett. 92(15), 151105 (2008). [CrossRef]
63. R. Grunwald and M. Bock, “Programmable microoptics for ultrashort pulses,” Proc. SPIE 7716, 77160P, 77160P-8 (2010). [CrossRef]
64. M. Bouafia, A. Bencheikh, L. Bouamama, and H. Weber, “M2 quality factor as a key for mastering laser beam propagation,” Proc. SPIE 5456, 130–140 (2004). [CrossRef]
65. R. Grunwald and M. Bock, “Programmable micro-optics for ultrashort pulses,” SPIE Newsroom (2010). http://spie.org/x39625.xml?highlight=x2422&ArticleID=x39625.