## Abstract

We report the generation of programmable two-dimensional arrangements of ultrashort-pulsed fringe-less Bessel-like beams of extended depth of focus (referred to as needle beams) without truncating apertures. A sub-20-fs Ti:sapphire laser and a liquid-crystal-on-silicon spatial light modulator (LCoS-SLM) of high-fidelity temporal transfer in phase-only operation mode were used in the experiments. Axicon profiles with ultra-small conical angles were approximated by adapted gray scale distributions. It was demonstrated that digitized image information encoded in amplitude-phase maps of the needle beams is propagated over considerably large distances at minimal cross talk without the need for additional relay optics. This experiment represents a physical realization of Saari’s proposal of spatio-temporally nondiffracting “flying images” on a few-femtosecond time scale.

©2009 Optical Society of America

## 1. Introduction

Adaptive microoptical systems for encoding and robust transfer of complex, ultrashort-pulsed image information are important building blocks for applications in ultrafast optical processing, free-space communication, coherent control, nonlinear spectroscopy, metrology and microstructuring. Femtosecond laser pattern shaping with reflective mode spatial light modulators (SLMs) in conjunction with imaging components was reported [1]. Photorefractive volume reflection holograms were utilized for pulsed-image generation at pulse durations about 200 fs [2]. This method, however, is limited with respect to larger bandwidths because of the diffractive functionality and requires an imaging cross-correlator setup with moving parts as a detecting system. The approximation of propagation invariant transverse intensity distributions by diffractive elements and transforming optics was shown in theoretical and experimental studies with random spots and spot arrays [3-5] but is essentially limited by diffraction and losses at narrow slits of Fourier filters as well. An alternative, more promising approach is based on the combination of wavefront division multiplexing of ultrashort pulses with the generation of Bessel-type pseudo-nondiffracting beams each carrying the information of one pixel. This concept called “flying images” was proposed in 1996 by Peeter Saari [6] and first demonstrated by the authors in collaboration with other co-workers [7,8]. In these experimental studies, an amplitude mask was programmed into an SLM and the encoded beam was subsequently transformed into multiple Bessel-like sub-beams by a separate microaxicon array.

Here we report on an advanced concept based on a more compact, flexible and robust arrangement for generating flying images where the basic functionalities of microoptical beam shaping and spatial addressing are integrated in a single reflective SLM. By replacing Bessel beams by fringe-free sub-beams (“needle beams”) with very extended focal zones and of high-fidelity pulse transfer in few-femtosecond range, the free-space projection of high-power laser ultrafast pixelated flying images without the need of any additional imaging optics is well enabled. We demonstrate that a high-accuracy phase control at ultrasmall conical angles and a defined background texturing can be exploited to minimize the cross-talk of multiple needle beams.

## 2. Pseudo-nondiffracting needle beams

Idealized Bessel-beams consist of concentric fringes with a Bessel function as their radial field profile and represent propagation invariant solutions of the Helmholtz wave equation. They result from the constructive interference of conical partial waves. Because of propagating unchanged over infinite distances, these kinds of theoretical Bessel beams are often considered “diffraction-free” [9,10] or “nondiffracting”. By Fourier-transforming waves transmitted through (lossy) annular slits or by deflecting light with conical, refractive or reflective axicons [11], nondiffracting beams can physically be approximated in finite but (e.g. in comparison to Gaussian beams) extended focal zones [12-14]. The axial oscillations of such zones can be modified or even depleted by certain amplitude and/or phase functions of the illuminating beams [15,16]. Spatial frequencies and contrast of the radial oscillations (Bessel fringes) depend on spectrum, angular distribution and temporal coherence [17]. A further advantage of Bessel-like beams is their robustness against distortions (self-reconstruction) [18]. If broadband ultrashort laser pulses are applied to generate pulsed Bessel-like beams or Bessel pulses [19-21], the interplay of spectral bandwidth and angular spectrum in the linear regime (low intensities, free-space) with conical beam-shapers [22,23] as well as in the nonlinear regime (high intensities, nonlinear media) with spontaneous self-focusing [24] leads to more complex, in particular x-shaped spatio-temporal interference patterns. Such “X-pulses” also display self-reconstruction properties in spatial and temporal domain [7]. The time-integrated spectral analysis of the fringes of ultrashort pulsed Bessel beams shows that the spectral bandwidth of outer fringes is significantly reduced whereas the central maximum remains nearly unchanged [25]. This spectrally nondiffracting characteristics of the central lobe corresponds to a temporally nondiffracting characteristics (i.e. constant pulse duration). In this context it seems to be justified to use the term “supercollimation” (which is mostly applied to particle beams of narrow angular spectra) to emphasize the unique propagation properties of these particular types of optical wavepackets. There are two different options to produce arrays of such ultrashort-pulsed fringe-less beams (in the following referred to as “needle beams” [7,8]) of excellent pulse transfer which are closely connected with specific advantages and disadvantages: (I) spatial filtering, (II) small angles. In case (I), the concentric fringe patterns of Bessel-like beams have to be spatially filtered by truncating apertures. To minimize the diffraction at the rims of the apertures, the first minimum has to coincide with the radius of the aperture (self-apodization) [25]. In this way, a higher spatial resolution can be obtained but the energy of outer fringes is waisted. Recently it was shown by experiments that needle beams of extremely high aspect ratio (i.e. the ratio of Rayleigh range to the initial radius) can be shaped while the temporal pulse characteristics remains unchanged over focal zones > 10 cm for 10-fs pulses of a Ti:sapphire laser oscillator. Aspect ratios of a few 10^{2} (maximum achievable about 700:1) were realized. In this sense, the propagation behavior can be regarded as being “pseudo-nondiffracting”. Foci of comparable depth and simultaneously comparable spectral and temporal stability can not be generated with Gaussian beams because of the inverse wavelength dependence of the beam parameters. The main disadvantage of type-I-needle beam shaping is that large arrays of high-quality needle beams can hardly be generated because of the critical adjustment of many apertures (in particular with non-planar wavefronts).

The alternative solution (II) is to work at sufficiently small angles so that only those partial beams interfere which create the central lobe. This setup works without external truncation (in “self-truncation”) and thus can be made more compact and less complicated to adjust. Another advantage of type-II-needle beams is that larger Rayleigh ranges can be obtained for the price of larger waist diameters. By writing ultraflat axicon profiles with conical angles far below 1° into the phase of the SLM corresponding to option (b), arrays of needle beams can easily be obtained as we will show later on. In addition to this, small-angle operation is advantageous for reflective setups because of the extraordinarily high angular tolerance of ultraflat axicons so that an off-axis design is not necessary [26]. It has further to be mentioned that both types of needle beams were found to behave self-reconstructing like Bessel beams [7,8,25]. This property can be utilized for stabilizing the image transfer under the influence of distortions.

## 3. Spatial light modulators for shaping of ultrashort pulses in spatial domain

For encoding amplitude and/or phase maps in reconfigurable arrays of ultrashort wavepackets, SLMs have to fulfill specific requirements: (a) high spatial resolution, (b) excellent pulse transfer, and (c) high damage resistance. Acousto-optic modulators (AOMs) are proven tools for controlling amplitude and phase in temporal shaping but neither reach the necessary spatial resolution nor enable for a spatial wavefront manipulation. With pixelated liquid crystal displays (LCDs), high-definition wavefront control can be realized by changing the phase via birefringence [27]. Such devices are operated as temporal pulse shapers where the spectral phase is manipulated in the Fourier-plane of a zero-dispersion setup [28]. Because of the parallel processing of spectral channels, temporal transfer and damage are uncritical and a linear array of phase elements is sufficient. A first step toward a spatio-temporal programming was realized by individual temporal shaping along one spatial coordinate [1,29]. Non-adaptive spatial shaping was achieved with volume holograms [2,30]. Polarization properties of LCDs were also exploited [31]. Flexible mirrors [32] and micro-electro-mechanical systems (MEMS) are attractive for femtosecond applications because of low losses and minimal dispersion. Under realistic conditions, however, limitations by slow heat transfer, dynamic distortions and spatial resolution have still to be overcome.

An acceptable compromise are liquid crystal on silicon spatial light modulators (LCoS-SLM) which combine high resolution, large fill factor and moderate damage threshold with phase shifts of up to 2π in near infrared. Such reflective microdisplays contain a thin LC-layer and have a refractive index close to that of the alignment/conducting layer. They show an excellent spectral transfer because of being designed for RGB-projection [33]. With well adapted 2D gray level maps, arbitrary phase functions (aspheric, toroidal, convex, concave etc.) can be approximated even for white light sources. For a proper spatial shaping of ultrashort pulses, in particular at few or single optical cycles and up to octave-spanning spectral bandwidths, however, specific problems arise from spatio-temporal coupling and spectrally dependent propagation effects (chromatic aberration, dispersion, diffraction).

If the SLMs are applied to video projection, gray value maps are transferred to amplitude maps by filtering of a linearly polarized input beam with an external analyzer. To approximate the phase-only operation mode, however, SLMs have to be driven without additional polarizing element. In case of an optimized input polarization, the beam shaping can be realized by phase profiles only at unchanged intensity. Although the total efficiency is typically in the range of 60%, the majority of losses are non-absorptive and the damage resistance is high (peak intensities of few GW/cm^{2}). Most of the loss energy is transformed in parasitic diffraction orders which do not affect the zero order beam. To overcome the limitations given by the maximum phase, a Fresnel zone lens approach can be applied to create flat axicons which are referred to as “Fresnel axicons” or “Fraxicons” [34]. At ultrashort pulse durations, the large spectral bandwidth and spatio-temporal coupling significantly complicate the Fraxicon design. Therefore, our investigations were focused on continuous profiles which can be controlled with acceptable phase accuracy in selected types of vertical aligned nematic (VAN) and parallel aligned nematic (PAN) SLMs [33].

## 4. Experimental techniques

Beam shaping experiments were performed with a high-definition LCoS-SLM (HoloEye) which was carefully calibrated with respect to its phase characteristics (phase as a function of gray value and wavelength). The setup is shown schematically in Fig. 1.

The SLM contains a 3 μm thick VAN-type LC layer of negative dielectric anisotropy and reaches a maximum phase shift of about π at a wavelength of 800 nm. The spatial resolution is given by the matrix of 1920 × 1200 square-shaped pixels (8.1 μm pitch, fill factor > 0.9). The phase resolution (minimum phase step at a given wavelength) resulting from the dynamic range is π/255 (corresponding to a wavelength of 800 nm). The LCoS microdisplay was illuminated with sub-20 fs pulses of a Ti-sapphire laser oscillator (Femtosource, 4 nJ pulse energy, 75.3 MHz repetition rate, center wavelength 800 nm). Time-integrated spatial patterns shaped with the LCoS-SLM were detected with a 4 million pixel CCD-camera (Vosskuehler) after magnifying with a zoom lens and a microscope objective (10x magnification). For generating focal zones in close proximity to the SLM, an incident angle of 20 degree had to be chosen. Advantageously, working with off-axis illumination at such a relatively large angle was supported by the aforementioned tilt tolerance of flat conical phase profiles [26]. To write axicon phase profiles in the LCoS-SLM, a 2D matrix of gray values was encoded by using a graphics processor unit (GPU) with generic image processing software. Gray level images were transferred from the GPU via digital visual interface (DVI) to the controller unit which further translates the information into a voltage signal changing the local alignment of the LCs, thus modifying their effective refractive index pattern and hence the entire phase map. To study the propagation conditions including spatio-spectral and spatio-temporal contrast ratio and cross-talk, *three basic geometrical arrangements* of gray value addressed phase axicons were realized for the following situations: (i) programming single needle beams (to totally exclude any cross talk effects), (ii) programming complex ray patterns consisting of reconfigurable arrays of needle beams, and (iii) arrays like above but with an additional background management. Whereas the background was chosen to be unstructured (i.e. with homogeneous phase) in cases (i) and (ii), a checkered phase pattern acting as a diffractive cross grating was programmed in the spaces between the programmed binary phase axicons (to study the influence of surrounding high-spatial-frequency textures and, in particular, to separate unshaped light from the main propagation direction of the shaped sub-beams).

In a first experiment corresponding to case (i), the diameter of a solitary axicon was adjusted to 500 μm (pitch of the SLM raster multiplied with the relevant number of pixels) with a maximum effective phase upstroke of 400 nm (π) at the center wavelength. It should be mentioned that pixellated SLMs approximate continuous phase profiles only by staircase functions with a best possible resolution of one pixel size. Thus, the specifications of programmed slope or axicon angles are related to smoothed curves (envelope functions). In the example, the phase profile was chosen to exhibit a soft decrease at the rim (no sharp edges) to prevent unwanted diffraction effects (phase apodization). In addition, a Gaussian shaped offset was applied to correct for slight distortions caused by deviations of the gray value dependent phase from linearity and to extend the depth of the focal zone in axial direction. Simultaneously, this fading leads to an axial shift of the distance where neighboring sub-beams start to overlap (reduction of cross-talk). For axicons with spatially variable phase tilt (e.g. Gaussian phase profiles), an averaged conical angle <α(r)> can be introduced as a simple means to roughly describe the focusing properties (Fig. 2(a)).

In the example, the radius of *r _{max}* = 170 μm corresponds to an average conical angle <α(r)> of 2.1 mrad. To demonstrate the propagation of nondiffracting ultrashort-pulsed images, we arranged 15 identical axicons to form a letter “E”, as presented in Fig. 2(b) as a special case of type-(ii)-shaping. To directly generate the pattern without a magnifying telescope and thus to avoid additional dispersion in lenses or aberrations from off-axis illumination, we had to accept certain restrictions. The useable area was limited to 2.0 mm

^{2}(input beam diameter 2w

_{0}of 4 mm). Consequently, it was necessary to shrink the interval between adjacent elements and to reduce the number of lines to a minimum. A special overlapping algorithm was introduced to preserve the gray level distribution of the individual axicons. The outer parts of each axicon gray value map were corrected depending on distance and diameter of the surrounding axicons. Horizontal and vertical periods were adjusted to a value of 324 μm (40 pixels). To reduce the diffraction at hard edges, profile functions with slight deviations from a linear cone were programmed. For this experiment, the above described conical axicon with a radius of

*r*= 170 μm (blue line in Fig. 2(a)) was replaced by an adapted distribution (red symbols in Fig. 2(a)) which can be described part by part by fitting Gaussian functions (black line: fit of the tailing part with a radius of 55 μm in terms of the standard deviation). The geometrical arrangement (iii) of diffractive checkered gray value patterns is shown in the inset in Fig. 2(b). The chosen minimum period was 2 pixels (1 white, 1 black) corresponding to a grating period 16.2 μm.

_{max}The following section concerns with the results of simulations as well as experiments on the propagation of spatially structured femtosecond pulses (flying images), adaptively shaped with above addressed techniques in all three modes of operation.

## 5. Results and discussion

#### 5.1. Propagation of ultrashort-pulsed needle beams

Under the assumption of a concial phase profile, a Gaussian-shaped spectrum, and a homogeneous illumination over the axicon area, the evolution of pulsed Bessel beams (PBB) can be simulated analytically in a similar way as described in Refs. [35,36]. The signal measured by a time-integrating photodetector (e.g. camera pixel) can be expressed as

where

(*c* = light velocity in vacuum, *λ*
_{0} = center wavelength, *α*(*r*) = local axicon angle at the radius *r*, *r*′ = z-dependent reconstruction parameter, *q* = integration variable, *F* = scaling factor, *τ*
_{0} = pulse duration of the electrical field related to the standard deviation, *τ _{FWHM}* = pulse duration related to the full-width-of-half-maximum value of the measured pulse intensity)

The particular mathematical terms reflect the underlying physical relationships as follows. The Gaussian term in Eq.1 describes the pulse evolution in the temporal domain (time coordinate *t*) corresponding to a Gaussian shape in spectral domain and contains a 0^{th} -order Bessel function *J*
_{0} as the solution of the wave equation (Eq. 6). The complex quantity Z (Eq. 2) is related to the overlap between the conical wave parts in time-domain. The temporal delay is necessary to take the temporal coherence into account which determines the interference contrast of the transversal fringes as well as the axial behavior. The terms responsible for axial and transversal contrast are *t _{d}* (which depends on the local axicon angle at a radius

*r*at the axial position

*z*) and

*τ*(Eq. 3 and Eq. 4). The calculation is performed by stepwise shifting a time window (integration time constants

_{a}*t*and

_{1}*t*). The constant term

_{2}*t*determines the initial delay of the time window and has to be well chosen because of its correlation to the pulse duration and geometrical conditions. The local beam divergence angle is involved as an additional angular contribution to the propagation angles

_{c}*θ*′ in Eqs. 4,6 and 7. The factor

*F*is used for the numerical procedure to adapt the transversal scale in the plane of interest to the beam deformation caused by the divergence. The relation in Eq. 5 contains the well known transform factor between first and second order Gaussian functions. To analyze the intensity propagation of a needle beam with an initial pulse duration of

*τ*= 20 fs and a center wavelength of 800 nm, the shape of the experimentally generated propagation zones was reconstructed by determining the beam radii at each z-position. Noise and fluctuations by speckles were eliminated by an averaging procedure based on radial cuts in 1000 rotational steps of 0.36°. This rotation averaging was applied to solitary needle beams as well as to selected needle beams from a complex pattern, with black as well as with checkered background. For the scaling factor, a value of

_{FWHM}*F*= 7.69 was chosen.

The measured and simulated propagation of a solitary needle beam generated with a uniform phase background and a smoothed transition between minimum and maximum phase is plotted in Figs. 3a,b. The corresponding experimentally determined and Gaussian fit spectra can be found in Fig. 3(c),(d) respectively.

The measured and the calculated propagation zones agree relatively well in their qualitative features. A detailed analysis, however, reveals certain deviations which are caused by unavoidable parasitic diffraction at sharp sub-structures (SLM pixels), non-Gaussian fine structure of the real spectrum and chirp effects. The measured needle beam shows a notably extended region of approximately constant intensity caused by the broader angular spectrum compared to the narrow one of a conical axicon. In the important central region of the needle beam, the simulated transversal beam diameter was found to be in excellent agreement with the experimental results (a quantitative analysis is given later on). Although the interference pattern should disappear at a distance of approximately 40 mm (*z _{max}* =

*r*/<2

_{max}*θ*>), the observed depth of the focal zone was significantly larger. This effect is explained by the smoothed Gaussian-like edges of the axicon phase function which induce flatter conical angles and consequently lead not only to an apodization but also to an axially asymmetric, more extended interference zone.

A central aspect of the propagation analysis is the evolution of the radial shape function. Its mathematical description should be compact and selective but keeping track of significant physical information. Therefore, well-known tools of higher order statistical moments were applied. In particular, the fourth order statistical moment *K* (kurtosis) of central cuts through measured profiles in transversal direction was calculated to extract specific information about peakedness (positive value) or flatness (negative value). This enables us to evaluate also non-uniform shapes with high sensitivity. The propagation dependent kurtosis of the transversal shape functions in Fig. 4 is nearly identical for single (blue curve) and arrayed needle beams (green curve) if no additional grating is programmed in the dead space. However, the presence of such a phase structure (black curve) leads to significant deviations over most parts but a convergence between *z* = 38 and 50 mm. The abrupt changes of the kurtosis from positive to negative values can be attributed to the peculiarities of the formation of realistic broadband Bessel-like beams of finite energy in contrast to idealized Bessel beams.

The deviations from the kurtosis value of −0.53 of the central maximum of an ideal squared mathematical Bessel function *J*
_{0}
^{2} are a measure of the non-conicity of the generating wave. On the basis of this criterion, we found the solitary needle beam and the sub-beams of needle beam arrays with homogeneous background in the experiment to well approximate *J*
_{0}
^{2} between *z* = 19 mm and *z* = 50 mm. In the case of a checkered phase pattern in the dead space between the axicons, *J*
_{0}
^{2} is approximated from larger distances (about *z* = 38 mm).

For a further quantitative comparison of the propagation of different types of needle beams, we introduce a set of characteristic beam parameters in analogy to the description of Gaussian beams. With the Rayleigh length *z _{0}* (distance at which the cross sectional beam area has doubled) and the beam waist radius

*w*(1/e

_{0}^{2}of the transversal intensity distribution) we define an aspect ratio of

*z*/

_{0}*w*as a measure for the depth of the focal zone of the needle beams. The aspect ratio of solitary needle beams and sub-beams of needle beam arrays was found in all cases to exceed 300:1 (see Tab. 1).

_{0}The comparison of relevant beam parameters in Tab. 1 indicates no significant dependence of the Rayleigh range on the background phase structure. This was proved by switching from a solitary needle beam to an array of needle beams with non-textured background. The position of the beam waists, however, was found to be slightly shifted towards the CCD detector if the checkered background phase pattern was applied. Axially-dependent beam radii extracted from measured and simulated data are shown in Fig. 5. Each waist radius was determined by sectional fitting the progressing curve by a 3^{rd} order polynomial. In the case of a non-uniform background, the beam radius increased significantly in the proximity of the SLM where a transient superposition of needle beam and the radiation diffracted at the checkered phase area is expected. For z > 25 mm, the axial dependence of the radius within the Rayleigh range was found to be nearly identical again for all curves including the simulation. That implies that the cross-talk between adjacent optical channels remains low over large propagation distances.

#### 5.2. Ultrashort-pulsed flying images

In a proof-of-principle study on ultrashort-pulsed flying images we tested the performance of our approach by programming well-recognizable patterns (letter “E”) into the gray value map of an LCoS-SLM (for the principle, see Fig. 2(b)). The minimum period was chosen to be 324 μm. The detected intensity maps show the evolution of the pattern in axial direction with background structuring (animation (Media1) in Fig. 6(a)). To better visualize the beam propagation in 3D, the data were further processed by means of a tomographic-based software (animation (Media 2) in Fig. 6(b)).

The pseudo-nondiffracting signature of the partial beams is clearly indicated by the nearly depth-invariant distribution. The propagating image appears to be stable with respect to angular relationships and pattern essentials (image information) but is slightly magnified by the divergence. One should draw the attention to the difference in the transversal and axial scales in Fig. 6(b) and the related movies. Compared to the transversal plane, the axial coordinate was shrinked by a factor of ~1:46 for reasons of 3D visualization. The key problem for any image reconstruction is the contrast in a relevant parameter map (here: intensity). The intensity along a cut through the 2^{nd} column in y-direction of a nondiffracting “E” at an axial distance of *z* = 50 mm was detected for both a homogeneous (HB) and a checkered (CB) phase background (Fig. 7).

The contrast *C* = (*I*
_{max}-*I*
_{min})/(*I*
_{max}+*I*
_{min}) appeared to be clearly higher if the SLM was operated with structured background (*C* = 0.66) whereas it was found to be lower in the other case (*C* = 0.59). Because of the spatial envelope of the beam, the contrast undergoes local variations. Nevertheless, the qualitative behavior could be verified at all positions in a similar way. In the presence of a checkered phase background, the contrast increases towards larger axial distance whereas it is depleted in the near zione. The spatial fine structure of a selected sub-beam propagation zone can be recognized in the intensity map of radially averaged cuts determined along the propagation path for a checkered phase background (Fig. 8). The Rayleigh range was *z _{2}*-

*z*= 27.2 mm in this case (compared to a value of 23 mm without structured background). The reduction factor for the axial scale in Fig. 8 is still about 1:17 but the needle shape can well be imagined.

_{1}#### 5.3. Spectral analysis of flying images with statistical moments

The spectral properties of flying images (letter “E”) were analyzed with high resolution. To realize a 2D spectral mapping by acquiring complete spectra at each position, we replaced the camera by a fiber-based spectrometer [25,26]. The spatial resolution was limited by an aperture of 15 μm diameter in front of the fiber. The pattern was scanned in transversal steps of 30 μm at distances of 50 mm and 60 mm, respectively. By compressing the data to a set of four statistical moments and peak signal, an unambiguous reconstructing essential spectral features and a quantitative comparison were enabled. Exemplary, the spatial and axial dependences of the standard deviation of a selected sub-beam (central peak in Fig. 7) are shown in Fig. 9 for two distances and allow to recognize the transversal structure even with a noisy background. Other statistical moments like skewness *S*, kurtosis *K* and center of gravity *C _{o}G* were determined as well. Spatially averaged values are listed in Tab. 2.

Within the area of interest, the spectral bandwidth was obviously transferred over a distance of 10 mm (> 10 array periods) with low changes. A comparison of the averaged statistical moments in Tab. 2 confirms the broadband spectral transfer of needle beams generated by this LCoS-SLM. The averaged skewness (*S*) and kurtosis (*K*) values indicate the symmetry and a certain flattening of the spectrum of the sub-beam. It has to be mentioned that the bandwidth can differ from sub-beam to sub-beam because of the laser properties. The quantitative interpretation has to take into account that the spectral FWHM can directly be derived from the standard deviation (by applying the well-known factor of 2.35) only if the deviation from a Gaussian distribution (where *S* = 0, *K* = 0) is negligible. The spectral data were confirmed by autocorrelation measurements for a selected sub-beam (central peak in Fig. 7) of a flying image. At a distance of *z* = 50 mm from the SLM (distance from oscillator: 1.5 m), a pulse duration of 23 fs was determined.

## 6. Conclusions

To conclude, the concept of Saari’s flying images was realized with programmable arrays of ultrashort-pulsed pseudo-nondiffracting fringe-free beams (“needle beams”). The shaping of a Ti:sapphire laser oscillator pulse in spatial domain was performed by a phase-only, reflective liquid-crystal-on-silicon spatial light modulator (LCoS-SLM) of proved high-fidelity transfer. Needle beams were obtained by self-truncation, i.e., by adjusting the foot-to-foot diameter of the radial intensity distribution to remain within the first minima of a Bessel function. The smooth rims of the intensity distribution lead, at the same time, to a self-apodization effect similar to the case of self-apodized truncation [25] (see Appendix). To study the time-integrated beam propagation, solitary needle beams as well as discrete images (letter “E”) composed of multiple needle beams were detected with high spatial resolution. Aspect ratios of > 300:1 were achieved. The effective aspect ratios are significantly influenced by the laser spectrum, divergence, wavefront errors and diffracting substructures (in particular for pixelated SLMs). A detailed study of the specific limitations will be a subject of future investigations. It was demonstrated that the contrast can be essentially improved by spatially separating the background light via controlled diffraction. For this purpose, an additional, high-spatial-frequency binary grating (checkered phase pattern) was programmed in the gaps between the axicons. Furthermore, statistical moments were applied to evaluate the spatially resolved spectrum of a propagating pulsed needle beam. By comparing spectral moments of four orders at different distances it was verified that no significant spatio-spectral changes emerge. The approach is robust against distortions and fluctuating initial conditions. Criteria for a reconstruction of relevant image information far behind the Fresnel domain will be a subject of continuing work. A more compact image shaper design based on radially asymmetric phase axicons is currently under investigation.

## Appendix: Self-truncation condition for the generation of needle beams with conical axicons

A condition for the generation of needle beams with conical axicons will be derived in a simplistic ray-optics picture (neglecting for diffraction). If a small-angle reflecting axicon of a finite diameter *D* is illuminated with a plane wave in normal incidence without an additional truncating aperture, the (in our case programmable) phase depth *h* has to be chosen properly so that no outer fringes of a Bessel beam appear. This can be regarded as a “self-truncation” where the finite extension of the illuminated area replaces the spatially filtering aperture (as applied, e.g., in the self-apodizing truncation setup described in Ref. [25]). The geometrical situation is schematically drawn in Fig. A1.

The diameter ∧ of the central lobe of a polychromatic Bessel beam with a symmetric spectrum (first zero) is approximately given by

(*λ _{0}* = center wavelength,

*n*= refractive index of air, 2

*θ*= total conical beam angle) If the maximum diameter of the overlapping zone (D/2) at an axial distance

*z*(center of the focal zone) equals the diameter ∧ of the central lobe it follows

_{c}with

(*h* = structure depth). Here, a factor 2 is due to the reflective geometry. From Eq. (A2), we obtain the self-truncation condition for the maximum acceptable concial beam angle

and the corresponding minimum length of the needle beam of

The last two relationships are useful for a rough design consideration. A geometric-optic estimation of a maximum aspect ratio can be found with the help of Eqs. (A5) and (A1):

For example, for *D* = 500 μm and *λ*
_{0} = 0.8 μm, one ontains a theoretical aspect ration A^{*} > 620:1. For a more realistic desription, however, a detailed simulation of the beam propagation including diffraction and initial beam divergence is inevitable.

## Acknowledgments

The authors thank T. Elsaesser, G. Steinmeyer, P. Saari, V. Kebbel, S. Huferath-von-Luepke, S. Langer, M. Piché, M. Fortin, E. Recami, M. Besieris, S. Trillo and C. Conti for stimulating discussions as well as HoloEye AG (Berlin) and APE GmbH for a close collaboration. We appreciate the assistance of C. Poppe and M. Tischer. The work was supported in part by DFG project no. GR 1782/7-1.

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