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Optics Express

Optics Express

  • Editor: C. Martijn de Sterke
  • Vol. 20, Iss. 11 — May. 21, 2012
  • pp: 12563–12578
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Ultrashort highly localized wavepackets

M. Bock, S. K. Das, and R. Grunwald  »View Author Affiliations


Optics Express, Vol. 20, Issue 11, pp. 12563-12578 (2012)
http://dx.doi.org/10.1364/OE.20.012563


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Abstract

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

1. Introduction

The confinement of light in space and time is a fundamental challenge in physics [1

1. P. Saari, “How small a packet of photons can be made?” Laser Phys. 16(4), 556–561 (2006). [CrossRef]

3

3. P. Saari, M. Menert, and H. Valtna, “Photon localization barrier can be overcome,” Opt. Commun. 246(4-6), 445–450 (2005). [CrossRef]

] 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 [4

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] [PubMed]

]. 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) [5

5. S. Trillo and W. Torruellas Eds, Spatial Dolitons (Springer, Berlin, 2001), pp. 73–74.

]. Self-induced spectral reshaping via conical refraction [6

6. Y. S, Kivshar and G. P. Agrawal, Optical solitons - From fibers to photonic crystals (Academic Press, Elsevier Science, Amsterdam, 2003), pp. 226–228.

8

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] [PubMed]

] 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 [9

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] [PubMed]

].

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

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] [PubMed]

,40

40. R. Grunwald, Thin-film microoptics - new frontiers of spatio-temporal beam shaping (Elsevier, Amsterdam, 2007).

]. 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

13. H. E. Hernández-Figueroa, M. Zamboni-Rached, and E. Recami, eds., Localized Waves, Theory and Experiments (Wiley & Sons, New York, 2008).

,40

40. R. Grunwald, Thin-film microoptics - new frontiers of spatio-temporal beam shaping (Elsevier, Amsterdam, 2007).

]. 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

40. R. Grunwald, Thin-film microoptics - new frontiers of spatio-temporal beam shaping (Elsevier, Amsterdam, 2007).

,41

41. M. Bock, S. K. Das, and R. Grunwald, “Programmable ultrashort-pulsed flying images,” Opt. Express 17(9), 7465–7478 (2009). [CrossRef] [PubMed]

]. 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 [39

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] [PubMed]

]. 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 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

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] [PubMed]

,40

40. R. Grunwald, Thin-film microoptics - new frontiers of spatio-temporal beam shaping (Elsevier, Amsterdam, 2007).

] is of particular relevance for applications (e.g. generation of “flying images” [41

41. M. Bock, S. K. Das, and R. Grunwald, “Programmable ultrashort-pulsed flying images,” Opt. Express 17(9), 7465–7478 (2009). [CrossRef] [PubMed]

]). 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):
D(z)=f(z)λ2nsinθ
(3)
(n = refractive index of air). For polychromatic beams with sufficiently narrow or symmetric spectral profiles, λ can be replaced by the center-of-gravity wavelength λ0. The scaling factor f mainly depends on the divergence of the illuminating beam. Further modifications arise from the diffraction at the edges of the programmed axicons. At small conical angles, the vanishing “contact angle” at the rim leads to a phase apodization which reduces the diffraction. The decay of intensity in the SLM plane along the radius r results from the coherent superposition (spectral interference) of all conical contributions. In the center of the zone of constructive interference at the distance z0, it is proportional to the square of the first-kind, zero-order Bessel function J02(r) within the limits of the first zero (first dark ring):
INB(r)f(z0)J02for r<r1
(4a)
INB(r)=0for r>r1
(4b)
(r1 = radius of the first minimum of J02). The axial extension of the nondiffracting propagation zone (“confocal parameter”) can be defined by two characteristic distances z1 and z2 where, in analogy to the Gaussian beam description, the beam area is doubled and the intensity at the axis is reduced by a factor of 2 (FWHM extension). A more comprehensive theoretical description of the beam quality of localized beams (e.g. by means of the Wigner function) would have to take into account that inside the zone of constructive interference, each point in space is the origin of a bundle of rays (corresponding to a local wavefront ambiguity).

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 [54

54. G. Rousseau, N. McCarthy, and M. Pichãé, “Description of pulse propagation in a dispersive medium by use of a pulse quality factor,” Opt. Lett. 27(18), 1649–1651 (2002). [CrossRef] [PubMed]

]:
P2=4πσυσt
(5)
where σν and σt are the variances in spectral and temporal domain, respectively. Thus, a dimensionless spatio-temporal localization parameter L2 of an ultrashort-pulsed wavepacket can be defined by the root of the product of M2 and P2 [55

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.

]:
L2=M2P2=4πσ0σfσυσt
(6)
The quality parameters are spatially (M2) and temporally integrated (P2) so that this approximation is valid only in the paraxial case (small angles, negligible travel time effects). For non-paraxial propagation (large angles), L2 cannot be applied because spatial and temporal features are not fully separable. In the most experiments reported here, the paraxial case was well approximated by working at extremely small conical angles and can serve as a figure of merit for the localization of wavepackets in space and time. It has to be noted, however, that the localization of HLW cannot completely be described by the used approach. Improved models require to include temporal changes of spatial and angular parameters (e.g. in frame of the Wigner function). For pulses with pulse durations beyond the area of validity of the slowly varying envelope approximation, however, the determination of the temporal and spectral statistical parameters requires a more sophisticated analysis.

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
Fig. 1 Principle of generating highly localized wavepackets (HLWs). Self-apodized (diffraction-free) spatial filtering of a pulsed Bessel beam (BB) is used to generate a radially symmetric needle beam (NB). By linear and circular transform algorithms (circular and double arrow), non-radially symmetric profiles of stretched (linear beam, LB) and tubular structure (tubular beam, TB) can be obtained. The propagation remains quasi-nondiffracting in the spatial and temporal domain.
.

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
Fig. 2 Setup for the generation and characterization of programmable HLWs (schematically). The pulses emitted by a Ti:sapphire oscillator (Venteon, minimum pulse duration 6 fs, center wavelength 800 nm, FWHM spectral bandwidth 300 nm, pulse energy 7 nJ, repetition frequency 80 MHz) are shaped with an LCoS-SLM. The time-integrated intensity distribution is detected with a CCD or EMCCD camera (EMCCD). For a 2D spatially resolved analysis of the temporal pulse properties, second order autocorrelation is performed with a balanced interferometer (M1-M4 = mirrors, BS = beam splitter, BBO = beta barium borate crystal for SHG) by tuning the length one interferometer arm. The resulting time delay Δτ is doubled in a round trip and leads to a final delay of 2Δτ between the pulse replicas. Spectral maps are detected with a position-controlled fiber spectrometer (Ocean optics).
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.

Beside approximately cone-shaped phase profiles (Fig. 3(a)
Fig. 3 Selected basic types of axicons for the generation of highly localized wavepackets: (a) conus axicon shaping needle beams, (b) torus axicon shaping tubular beams, and (c) stretched conus axicon with central bi-prismatic zone shaping linear beams (effective height profiles, schematically). Possible apodization (e.g. by flattened contact angles) is not shown here.
), Fresnel-axicons (“fraxicons” [62

62. I. Golub, “Fresnel axicon,” Opt. Lett. 31(12), 1890–1892 (2006). [CrossRef] [PubMed]

]) consisting of concentric rings similar to zone lenses, torus-shaped axicons (Fig. 3(b)) [52

52. M. Bock, S. K. Das, and R. Grunwald, “Adaptive shaping of complex pulsed nondiffracting light fields,” Proc. SPIE 7716, 7950–7958 (2011).

], 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 [63

63. R. Grunwald and M. Bock, “Programmable microoptics for ultrashort pulses,” Proc. SPIE 7716, 77160P, 77160P-8 (2010). [CrossRef]

]. 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 [59

59. M. Bock, S. K. Das, C. Fischer, M. Diehl, P. Börner, and R. Grunwald, “Reconfigurable wavefront sensor for ultrashort pulses,” Opt. Lett. 37(7), 1154–1156 (2012). [CrossRef] [PubMed]

]. If the self-apodizing condition [39

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] [PubMed]

41

41. M. Bock, S. K. Das, and R. Grunwald, “Programmable ultrashort-pulsed flying images,” Opt. Express 17(9), 7465–7478 (2009). [CrossRef] [PubMed]

] 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
Fig. 4 Comparison of the propagation characteristics of pulsed diffractive and nondiffractive twin beams, left: phase distributions of pairs of (a) hard circular apertures and (d) flat axicons programmed into the SLM as gray value maps. The maximum brightness in (a) indicates the largest phase value for this device (π). Pictures (b),(e) and (c),(f) show intensity profiles measured at distances of z1 = 18.5 mm and z2 = 138.5 mm, respectively. Double beams were chosen to study cross-talk and interference effects of neighboring beams (radii: 380 µm; average conical angle of axicon: 0.13°, incident angle about 20°). The field of view was 2.3 x 2.3 mm2 in all cases.
, 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.

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 [59

59. M. Bock, S. K. Das, C. Fischer, M. Diehl, P. Börner, and R. Grunwald, “Reconfigurable wavefront sensor for ultrashort pulses,” Opt. Lett. 37(7), 1154–1156 (2012). [CrossRef] [PubMed]

]. Figure 6
Fig. 6 Nonlinear characterization of multiple few-cycle wavepackets at a distance of 100 mm generated by an array of conical profiles (axicons) programmed in the phase map of an LCoS-SLM: (a) intensity distribution at zero time delay, (b) autocorrelation trace for a selected position in comparison to a theoretical bandwidth-limited pulse (flat spectral phase), (c) Visualization of the spatio-temporal structure by post-processed data of spatio-temporal 2nd order autocorrelation (fixed DC value added, size and brightness encoded according to the absolute value, perspective drawing with vanishing point). Conditions: axicon period 720 µm, effective height 400 nm, conical angle 0.13°, incident angle 43°; source: Ti:sapphire oscillator, nonlinear converter: 10 µm thick BBO crystal, pulse duration 6.5 fs, detector: EMCCD).
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 [60

60. R. Grunwald and M. Bock, “Spatially encoded localized wavepackets for ultrafast optical data transfer,” JEOS:RP (submitted to).

], the value for the beam propagation factor M2 was determined to be 1.8.

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 [63

63. R. Grunwald and M. Bock, “Programmable microoptics for ultrashort pulses,” Proc. SPIE 7716, 77160P, 77160P-8 (2010). [CrossRef]

]. The phase profiles of two selected elements as gray value maps and cut curves with effective heights can be found in Fig. 7
Fig. 7 Programmable torus axicons: comparison of two discretized structures of different symmetry. (a) and (b): 2D gray value maps (1 pixel), (c) linear cuts with effective height profiles. The corresponding symmetry factors were SF = 0.50 in the picture (a) and the curve with red dots in (c), and SF = 1.00 in (b) and the curve with blue squares in (c).
. To classify toroidal axicons of different symmetry of their the ring profile, we define a symmetry factor SF as the ratio
SF=2riR
(7)
where ri and R are the inner radius (center-to-peak) and outer radius (center to rim) (see Fig. 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
Fig. 8 Geometry of hollow beams as a function of the axicon symmetry: (a) intensity profile at a distance of z = 8 mm for SF = 0.5 (blue squares), 0.7 (green circles) and 0.9 (red triangles). At this distance, the best overall contrast was obtained. Inner and outer contrast Ci = 84% and Ca = 55% keep nearly constant over the considered range of SF; (b) dependence of the hollow beam diameter (peak-to-peak distance for central cut) on SF. One recognizes that the diameter scales linearly with this parameter in fairly good approximation.
demonstrates the propagation properties of the individual ring-shaped HLWs which were sub-beams of a hexagonal array (period about 430 µm, Fig. 9
Fig. 9 Propagation of a hollow beam array (a) 3D-reconstruction from measured x-, y- and z- intensity data (b) Intensity map of a hexagonal array of hollow beams generated with an LCoS-SLM (period about 430 µm, distance z = 8 mm, ellipticity corrected by a linear transform factor SF = 0.9 in one direction, field of view: 1.2 x 0.8 mm2).
). 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 [59

59. M. Bock, S. K. Das, C. Fischer, M. Diehl, P. Börner, and R. Grunwald, “Reconfigurable wavefront sensor for ultrashort pulses,” Opt. Lett. 37(7), 1154–1156 (2012). [CrossRef] [PubMed]

].

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
Fig. 10 Stable propagation zone of a pulsed hollow beam generated with an asymmetric toroidal axicon (SF = 0.8): (a) intensity profiles measured at distances between 0 and 15 mm; (b) peak intensity (red dots) and normalized diameter (related to a reference diameter of 394 µm) as a function of distance (please notice the different scales for CCD-signal and diameter).
. 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
Fig. 11 Temporal properties of a solitary few-cycle ring-shaped HLW generated by programming a toroidal axicon (outer diameter 3.2 mm, conical beam angle 0.029°) into an LCoS-SLM: (a) 2nd order autocorrelation function (ACF) measured at a distance of 400 mm (FWHM 9.8 fs, green line: derived intensity autocorrelation), (b) propagation-dependent FWHM of the ACF (black squares) and peak-to-peak ring diameter (blue circles). The averaged pulse duration along the zone was 6.7 ± 0.2 fs.
). 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
Fig. 12 Reconstructed time-dependent E-field of the HLW. To indicate the structure of a ring-shaped light bullet, the iso-electric-field surface of 6.8-fs pulses was retrieved from spatially resolved second-order autocorrelation. For visualization purposes, a DC-field (corresponding to the maximum field amplitude) was added [64] (red circles: field maxima, d = maximum-to-maximum ring diameter, Δτcycle = field oscillation period in time).
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
Fig. 13 Stack of pulsed nondiffracting linear HLWs (light blades) generated from the Ti:sapphire oscillator beam: (a) 3D plot with a nonlinear dependence of the color coded structure on the intensity (initial pulse duration 6.5 fs, perspective drawing with vanishing point, see Media 1); (b) axially dependent thickness of the light blades. Within the first 15 mm of propagation it was found to be about 60 µm.
. 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.

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.

6. Conclusions

For quasi-nondiffracting light fields at pulse durations where approximations like slowly varying envelope approximation and narrow bandwidth models break down [2

2. P. Saari, “Photon Localization Revisited,” in: Quantum Optics and Laser Experiments Sergiy Lyagushin Ed. (InTech - Open Access Publisher, Croatia, 2012), 49–66.

] 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.

Acknowledgments

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] [PubMed]

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] [PubMed]

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] [PubMed]

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] [PubMed]

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]

12.

J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987). [CrossRef] [PubMed]

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]

15.

P. Martelli, M. Tacca, A. Gatto, G. Moneta, and M. Martinelli, “Gouy phase shift in nondiffracting Bessel beams,” Opt. Express 18(7), 7108–7120 (2010). [CrossRef] [PubMed]

16.

S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).

17.

K. Reivelt and P. Saari, “Bessel-Gauss pulse as an appropriate mathematical model for optically realizable localized waves,” Opt. Lett. 29(11), 1176–1178 (2004). [CrossRef] [PubMed]

18.

K. Reivelt and P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys. 66(5), 056611 (2002). [CrossRef] [PubMed]

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]

21.

J. Turunen, A. Vasara, and A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt. 27(19), 3959–3962 (1988). [CrossRef] [PubMed]

22.

J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett. 13(2), 79–80 (1988). [CrossRef] [PubMed]

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]

24.

R. M. Herman and T. A. Wiggins, “Bessel-like beams modulated by arbitrary radial functions,” J. Opt. Soc. Am. A 17(6), 1021–1032 (2000). [CrossRef] [PubMed]

25.

R. M. Herman and T. A. Wiggins, “Apodization of diffractionless beams,” Appl. Opt. 31(28), 5913–5915 (1992). [CrossRef] [PubMed]

26.

F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun. 64(6), 491–495 (1987). [CrossRef]

27.

J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett. 25(4), 191–193 (2000). [CrossRef] [PubMed]

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] [PubMed]

32.

J. Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control 39(3), 441–446 (1992). [CrossRef] [PubMed]

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] [PubMed]

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] [PubMed]

40.

R. Grunwald, Thin-film microoptics - new frontiers of spatio-temporal beam shaping (Elsevier, Amsterdam, 2007).

41.

M. Bock, S. K. Das, and R. Grunwald, “Programmable ultrashort-pulsed flying images,” Opt. Express 17(9), 7465–7478 (2009). [CrossRef] [PubMed]

42.

P. Sprangle and B. Hafizi, “Comment on nondiffracting beams,” Phys. Rev. Lett. 66(6), 837 (1991). [CrossRef] [PubMed]

43.

J. Durnin, J. J. Miceli Jr, and J. H. Eberly, “Durnin, Miceli, and Eberly Reply,” Phys. Rev. Lett. 66(6), 838 (1991). [CrossRef] [PubMed]

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]

48.

R. Borghi and M. Santarsiero, “M2 factor of Bessel-Gauss beams,” Opt. Lett. 22(5), 262–264 (1997). [CrossRef] [PubMed]

49.

R. M. Herman and T. A. Wiggins, “Rayleigh range and the M2 factor for Bessel-Gauss beams,” Appl. Opt. 37(16), 3398–3400 (1998). [CrossRef] [PubMed]

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).

53.

Z. Mei and D. Zhao, “Controllable dark-hollow beams and their propagation characteristics,” J. Opt. Soc. Am. A 22(9), 1898–1902 (2005). [CrossRef] [PubMed]

54.

G. Rousseau, N. McCarthy, and M. Pichãé, “Description of pulse propagation in a dispersive medium by use of a pulse quality factor,” Opt. Lett. 27(18), 1649–1651 (2002). [CrossRef] [PubMed]

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.

56.

B. Salik, J. Rosen, and A. Yariv, “Nondiffracting images under coherent illumination,” Opt. Lett. 20(17), 1743–1745 (1995). [CrossRef] [PubMed]

57.

R. Grunwald, S. Huferath, M. Bock, U. Neumann, and S. Langer, “Angular tolerance of Shack-Hartmann wavefront sensors with microaxicons,” Opt. Lett. 32(11), 1533–1535 (2007). [CrossRef] [PubMed]

58.

R. Grunwald, U. Neumann, U. Griebner, K. Reimann, G. Steinmeyer, and V. Kebbel, “Ultrashort-pulse wave-front autocorrelation,” Opt. Lett. 28(23), 2399–2401 (2003). [CrossRef] [PubMed]

59.

M. Bock, S. K. Das, C. Fischer, M. Diehl, P. Börner, and R. Grunwald, “Reconfigurable wavefront sensor for ultrashort pulses,” Opt. Lett. 37(7), 1154–1156 (2012). [CrossRef] [PubMed]

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]

62.

I. Golub, “Fresnel axicon,” Opt. Lett. 31(12), 1890–1892 (2006). [CrossRef] [PubMed]

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.

OCIS Codes
(320.0320) Ultrafast optics : Ultrafast optics
(320.5540) Ultrafast optics : Pulse shaping

ToC Category:
Ultrafast Optics

History
Original Manuscript: March 19, 2012
Revised Manuscript: May 4, 2012
Manuscript Accepted: May 4, 2012
Published: May 18, 2012

Citation
M. Bock, S. K. Das, and R. Grunwald, "Ultrashort highly localized wavepackets," Opt. Express 20, 12563-12578 (2012)
http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-20-11-12563


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References

  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. Express19(15), 14451–14463 (2011). [CrossRef] [PubMed]
  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] [PubMed]
  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] [PubMed]
  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] [PubMed]
  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. A4(4), 651–654 (1987). [CrossRef]
  12. J. Durnin, J. J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett.58(15), 1499–1501 (1987). [CrossRef] [PubMed]
  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]
  15. P. Martelli, M. Tacca, A. Gatto, G. Moneta, and M. Martinelli, “Gouy phase shift in nondiffracting Bessel beams,” Opt. Express18(7), 7108–7120 (2010). [CrossRef] [PubMed]
  16. S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt.46, 923–930 (1999).
  17. K. Reivelt and P. Saari, “Bessel-Gauss pulse as an appropriate mathematical model for optically realizable localized waves,” Opt. Lett.29(11), 1176–1178 (2004). [CrossRef] [PubMed]
  18. K. Reivelt and P. Saari, “Experimental demonstration of realizability of optical focus wave modes,” Phys. Rev. E Stat. Nonlin. Soft Matter Phys.66(5), 056611 (2002). [CrossRef] [PubMed]
  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]
  21. J. Turunen, A. Vasara, and A. T. Friberg, “Holographic generation of diffraction-free beams,” Appl. Opt.27(19), 3959–3962 (1988). [CrossRef] [PubMed]
  22. J. Durnin, J. J. Miceli, and J. H. Eberly, “Comparison of Bessel and Gaussian beams,” Opt. Lett.13(2), 79–80 (1988). [CrossRef] [PubMed]
  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. A8(5), 732–745 (1991). [CrossRef]
  24. R. M. Herman and T. A. Wiggins, “Bessel-like beams modulated by arbitrary radial functions,” J. Opt. Soc. Am. A17(6), 1021–1032 (2000). [CrossRef] [PubMed]
  25. R. M. Herman and T. A. Wiggins, “Apodization of diffractionless beams,” Appl. Opt.31(28), 5913–5915 (1992). [CrossRef] [PubMed]
  26. F. Gori, G. Guattari, and C. Padovani, “Bessel-Gauss beams,” Opt. Commun.64(6), 491–495 (1987). [CrossRef]
  27. J. Arlt and M. J. Padgett, “Generation of a beam with a dark focus surrounded by regions of higher intensity: the optical bottle beam,” Opt. Lett.25(4), 191–193 (2000). [CrossRef] [PubMed]
  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. A67(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. Control39(1), 19–31 (1992). [CrossRef] [PubMed]
  32. J. Y. Lu and J. F. Greenleaf, “Experimental verification of nondiffracting X waves,” IEEE Trans. Ultrason. Ferroelectr. Freq. Control39(3), 441–446 (1992). [CrossRef] [PubMed]
  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. D21(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. A22, 2465–2475 (2005).
  35. M. Z. Rached and E. Recami, “Subluminal wave bullets: Exact localized subluminal solutions to the wave equations,” Phys. Rev. A77(3), 033824 (2008). [CrossRef]
  36. M. Zamboni-Rached, “Unidirectional decomposition method for obtaining exact localized wave solutions totally free of backward components,” Phys. Rev. A79(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. Photonics4(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] [PubMed]
  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. Express16(2), 1077–1089 (2008). [CrossRef] [PubMed]
  40. R. Grunwald, Thin-film microoptics - new frontiers of spatio-temporal beam shaping (Elsevier, Amsterdam, 2007).
  41. M. Bock, S. K. Das, and R. Grunwald, “Programmable ultrashort-pulsed flying images,” Opt. Express17(9), 7465–7478 (2009). [CrossRef] [PubMed]
  42. P. Sprangle and B. Hafizi, “Comment on nondiffracting beams,” Phys. Rev. Lett.66(6), 837 (1991). [CrossRef] [PubMed]
  43. J. Durnin, J. J. Miceli, and J. H. Eberly, “Durnin, Miceli, and Eberly Reply,” Phys. Rev. Lett.66(6), 838 (1991). [CrossRef] [PubMed]
  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. A3(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. SPIE1224, 2–14 (1990). [CrossRef]
  48. R. Borghi and M. Santarsiero, “M2 factor of Bessel-Gauss beams,” Opt. Lett.22(5), 262–264 (1997). [CrossRef] [PubMed]
  49. R. M. Herman and T. A. Wiggins, “Rayleigh range and the M2 factor for Bessel-Gauss beams,” Appl. Opt.37(16), 3398–3400 (1998). [CrossRef] [PubMed]
  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. SPIE7716, 7950–7958 (2011).
  53. Z. Mei and D. Zhao, “Controllable dark-hollow beams and their propagation characteristics,” J. Opt. Soc. Am. A22(9), 1898–1902 (2005). [CrossRef] [PubMed]
  54. G. Rousseau, N. McCarthy, and M. Pichãé, “Description of pulse propagation in a dispersive medium by use of a pulse quality factor,” Opt. Lett.27(18), 1649–1651 (2002). [CrossRef] [PubMed]
  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.
  56. B. Salik, J. Rosen, and A. Yariv, “Nondiffracting images under coherent illumination,” Opt. Lett.20(17), 1743–1745 (1995). [CrossRef] [PubMed]
  57. R. Grunwald, S. Huferath, M. Bock, U. Neumann, and S. Langer, “Angular tolerance of Shack-Hartmann wavefront sensors with microaxicons,” Opt. Lett.32(11), 1533–1535 (2007). [CrossRef] [PubMed]
  58. R. Grunwald, U. Neumann, U. Griebner, K. Reimann, G. Steinmeyer, and V. Kebbel, “Ultrashort-pulse wave-front autocorrelation,” Opt. Lett.28(23), 2399–2401 (2003). [CrossRef] [PubMed]
  59. M. Bock, S. K. Das, C. Fischer, M. Diehl, P. Börner, and R. Grunwald, “Reconfigurable wavefront sensor for ultrashort pulses,” Opt. Lett.37(7), 1154–1156 (2012). [CrossRef] [PubMed]
  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]
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