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Spatio-spectral and spatio-temporal transfer and intensity propagation of truncated ultrashort-pulsed Bessel-Gauss beams were investigated. Extended needle-shaped focal zones were generated using a compact setup with a reflective small-angle axicon and self-apodized truncation by an adapted aperture. Spectral maps of Bessel-Gauss beams were analyzed on the basis of higher order statistical moments. Compared to focused pulsed Gaussian beams with their spectrally dependent propagation, an ultrabroadband spatio-spectral transfer function was detected over Rayleigh ranges exceeding 10 cm. These results indicate favorable pseudo-nondiffracting characteristics not only from the point of view of spatial propagation but also with respect to the spectral and temporal stability.
©2008 Optical Society of America
1. Introduction
With the availability of ultrashort-pulse lasers that cover one octave of bandwidth or even more, application of these pulses often mandates or benefits from control of the propagation over extended interaction zones and minimization of spectral and temporal distortions. As it is well-known from Gaussian beam optics, the transformation laws for focus parameters are governed by diffraction and are therefore spectrally dependent. For example, the Rayleigh range z0 of a perfect monochromatic Gaussian beam is inversely proportional to its wavelength λ. Consequently, in the polychromatic case of focused ultrashort pulsed Gaussian beams, the spectral transfer functions are influenced by the initial spatio-spectral distributions [1–5,10]. Even for a relatively moderate pulse duration and bandwidth, diffraction gives rise to a spectrally dependent beam waist [1,4]. For few-cycle pulses, the propagation is even more severely influenced by the bandwidth [6–7] due to spatio-temporal coupling [8–12]. Additionally, the presence of spatial or temporal chirps may further increase the complexity of the focus [8,9]. Therefore, application of ultrashort pulses did often not exploit their full potential and remained limited to rather modest focusing, or a significant reduction of aperture or pulse duration had to be tolerated. Here we report on an alternative method to generate extended single-maximum focal zones based on the self-apodizing truncation of pulsed Bessel-Gauss beams (BGBs). The stability of the generated needle-shaped beams is proven by their spatio-spectral and temporal propagation characteristics. Results of numerical simulations and experiments for BGBs are compared with their polychromatic Gaussian counterparts. To the best of our knowledge, the pseudo-nondiffracting nature of truncated Bessel-like few-cycle wavepackets is investigated by highly resolved spatio-spectral mapping for the first time. A quantitative analysis of the spectral maps is performed by computing higher order statistical moments of local spectra. Further evidence for the propagation stability of truncated BGB over distances in the range of the Rayleigh range is given by nonlinear autocorrelation.
2. Broadband ultrashort-pulsed Bessel-Gauss beams
Our method for the generation of pulsed single-maximum or “needle beams” [13] of large axial extension exploits (i) the existence of propagation invariant solutions of the Helmholtz equation that are referred to as progressive undistorted waves [14], focus wave modes [15], diffraction-free beams [16] or “nondiffracting” beams. Additionally, we benefit from (ii) high-contrast transversal fringe structures containing distinct minima and maxima. The radial field distribution U(r) of a mathematically perfect monochromatic Bessel beam is described by the zero order Bessel function J0 of the first kind
i.e., the corresponding local intensity is proportional to J 2 0. Bessel-like beams [17] of slightly modified characteristics can experimentally be approximated by Fourier imaging of slits, axicon lenses, or mirrors [18]. Replacing the plane waves assumed in (1) by Gaussian beams or applying Gaussian window functions, BGBs are generated [19]. In the absence of nonlinear optical effects, spectral interference within polychromatic Bessel beams or BGBs increases the uniformity of the axial profiles [20], and, at sufficiently broad bandwidth, it leads to characteristically X-shaped spatio-temporal energy distributions. In analogy to conical wave phenomena in acoustics [21], these distributions are called “X-waves”. At ultrashort pulse durations corresponding to about 4 cycles of the optical field or less (sub-10-fs Ti:sapphire laser at a center wavelength of 800 nm), the bandwidth is sufficiently broad to enable the creation of X-wave features with detectable contrast. For practical applications of white-light nondiffracting beams, e.g., micromanipulation of atoms and mesoscopic particles, fringe contrast and envelope functions have to be optimised. Such functionality strongly depends on spatial and temporal coherence [22].
Bessel-Gauss pulses carry finite energy, are non-spreading over certain distances and physically realizable [23,24]. Further advantageous properties include self-reconstruction [25] and superluminal group velocities [26]. This renders pulsed BGBs an attractive alternative to pulsed Gaussian beams, enabling robust and ultrafast laser-matter interaction schemes [27]. We used refractive and reflective thin-film axicons to generate pseudo-nondiffractive beams. Our method avoids the small energy transfer and parasitic diffraction of beam generators based on the Fourier imaging of slits [28]. Also, as a linear method by the use of axicons is typically superior to spontaneous shaping of X-pulses in nonlinear media [29], which may give rise to a nonlinear amplification of inhomogeneities. Using our method, arrayed, sub-mm-diameter Bessel-like X-pulses with Rayleigh ranges in the mm range and pulsed BGBs with Rayleigh lengths in the decimeter range were already demonstrated at pulse durations of a few optical cycles (8–10 fs), including a direct detection of their X-pulse characteristics by fully spatio-temporally resolved nonlinear autocorrelation [30,31].
3. Spatial filtering of Bessel beams by self-apodized truncation
Generally, all beams with Bessel-like features arising from Eq. (1) exhibit radial intensity distributions with concentric fringes, which have to be truncated for transformation into single-maximum beams using suitable apertures. The beam propagation of the transmitted central lobe is essentially determined by the aperture function [32]. For optimum spatio-spectral transfer, minimum diffraction is ensured by matching the inner perimeter of the aperture to the first zero of the intensity profile (Fig. 1).
Fig. 1. Principle of the self-apodized truncation of Bessel-like beams (schematically): (a) Transmitted Bessel-like intensity fringe pattern for an aperture diameter d large compared to the diameter of the first minimum 2r0, (b) truncation of the intensity profile (radial cut) by an optimized aperture of d=2r0, and (c) transmitted central lobe shaping a single-maximum beam (“needle beam”) without diffraction at the aperture rim.
This spatial filtering method is referred to as self-apodization and is well-known from the generation of high harmonics in hollow fibers where truncated Bessel-like beams are intrinsically generated from propagation modes in rotationally-symmetric waveguides [33]. In recent experiments with adapted circular apertures, we demonstrated that self-apodization can be applied to unguided beams as well [13]. Self-apodizing spatial filtering can also be induced by virtually vanishing rims of ultraflat structures or by exploiting self-masking by absorption gradients resulting from thickness profiles. In these two cases, however, a higher background due to forward propagating light or the spectral dependence of absorber materials have to be tolerated, respectively. In the self-apodized truncation setup the influence of these effects is widely eliminated. The only trade-off of using a diaphragm is a reduction of the total energy. The maximum intensity, however, is expected to remain constant. In the following paragraphs, the results of numerical simulations and experimental investigations of ultrashort-pulsed truncated BGBs will be presented.
4. Simulation of the beam propagation
The propagation of broadband paraxial BGBs can be described analytically by a superposition of monochromatic Bessel-Gauss beams of different frequencies with spectrally independent Rayleigh ranges and angles. For simply structured spectra, e.g., Gaussians, closed-form analytical expressions exist [24], however, for truncated beams with strongly structured spectra as used in our experiments the propagation cannot be described by a simple analytical model. Therefore, we numerically simulated the propagation of Bessel-Gauss pulses in comparison to Gaussian pulses. The applied formalism for the solution of the diffraction integral is based on the propagation of an angular spectrum of monochromatic plane waves [34,35]. It is assumed that the field of a polychromatic BGB at the spatio-temporal coordinate (r,z,t) can be written as the integral
where (x,y) is the transversal position, z is the propagation distance, ω is the angular frequency, r is the radius, S(ω) the frequency-dependent amplitude, and, finally, kρ and kz are the radial and axial components of the wave vector. To numerically simulate the propagation, the diffraction integral has first to be solved for the complete frequency spectrum. The temporal field can then be calculated by an inverse Fourier transform at each point in propagation space. The propagation of a 10-fs BGB without truncation [Figs. 2(a) and 2(b)] and with truncation in self-apodizing setup [Figs. 3(a) and 3(b)] was simulated on the basis of experimental parameters, i.e., conical angle of the reflective axicon structure 0.027°, initial Gaussian beam waist radius on the axicon 2.4 mm, wavefront radius of curvature of the illuminating beam -4 m, and central wavelength 800 nm.
Fig. 2. Numerical simulations of time-integrated propagation of an untruncated pulsed Bessel-Gauss beam: (a) and (b) show the axial propagation in the r-z-plane for intensity and square root of amplitude, respectively. The propagation invariance can be recognized in corresponding radial cuts at different distances in (c)–(f). Dashed lines in (a) and (b) mark the positions of these planes of interest. zD, marks the position of the diaphragm in the subsequent simulation, cf. Fig. 3. Pulse duration and spectral distribution correspond to experimental parameters of a 10-fs Ti:sapphire laser oscillator. The calculations were performed on the basis of angular spectrum representation.
This is compared to the case of a focused polychromatic Gaussian beam [Figs. 3(c), 3(d)] of the same spectrum. For the measured spectral distribution see Fig. 4. The radius w0 of the Gaussian beam at the waist position zW was adapted to the radius r0 of the central lobe of the pulsed BGB by matching the second order moments of the radial profile functions. To enhance the visibility of field details, which can hardly be recognized in the intensity representation, the square root of the temporally and spectrally integrated field amplitude distributions is shown in Figs. 3(b), 3(d). It has to be mentioned, however, that the fringes in Fig. 3(b) are actually so weak that they will vanish below the noise level in real experiments.
Fig. 3. Simulated propagation of a pulsed Bessel-Gauss beam truncated by a diaphragm D with a diameter d=2rD located a few cm in front of the intensity maximum (a,b) and propagation of an idealized pulsed Gaussian beam of comparable waist radius w0 (c,d). The intensity representations in (a,c) show needle-like focal zones (notice the different scales for z and r). To enhance the visibility of details, the square root of field amplitudes is plotted in (b) and (d). Pulse duration and spectral distribution were assumed to be identical to Fig. 2. To obtain compatible spatial parameters, the second moments at the positions of maximum intensity of both types of focal zones were matched. The dashed line marks the position of the Gaussian beam waist.
The axial distance between the truncating spatial filter and the axicon (zD=110 cm) was chosen to truncate the beam a few centimeters in front of the point of highest intensity. As we found in simulations as well as in experiments, this leads to an effective lengthening of the focal zone. At the same time, it reduces the risk of destroying the aperture rims in the case of any transversal misalignment at high peak powers.
The simulated intensity focal zones of Gaussian beam and BGB seem to have similar geometrical properties at the first glance (compare Figs. 3(a), 3(c). A closer look, however, reveals a significantly different propagation characteristics. The beam radius corresponding to the standard deviation of radial intensity profiles is depicted in Fig. 5. The curves were normalized with respect to the values at the waist positions. The different propagation behavior for both types of beams is mainly caused by the small difference between the initial amplitude distribution functions in the aperture plane and the position of the diaphragm. Thus, within reasonable statistical limits, a zone with propagation invariant beam profile (“nondiffracting zone”) appears in the near-field of the truncated pulsed BGB, even when the diaphragm is directly placed at the location of maximum intensity of the untruncated beam. Within this zone the radial waist radius of the truncated BGB is smaller than the radius of the Gaussian beam. Behind this zone, the beam diameters converge again and, finally, towards the far field, the pulsed Gaussian beam propagates with a slightly smaller divergence.
An important observation is that axicons enable the generation of an extended focal zone at a much shorter distance from the beam shaping element than in the case of Gaussian beams. This is particularly relevant for focusing of ultrashort pulses, both, because of a strong reduction of air dispersion and because it allows for significantly more compact systems.
Fig. 4. Spectral distribution used for the numerical simulation of polychromatic pulsed BGB and Gaussian beams as shown in Figs. 3(a)–3(d). The FWHM bandwidth was about 120 nm.
Fig. 5. Beam radii (standard deviation) of a truncated pulsed Bessel-Gaussian beam and a pulsed Gaussian beam with an idealized spectrum calculated from simulated near field intensity propagation and for time-integrated detection. In proximity of the diaphragm, the truncated pulsed BGB is less divergent compared to a Gaussian beam, which is why we refer to this interval as “nondiffracting zone”. However, towards the transition to the far field, i.e., well behind the Rayleigh range, the pulsed Gaussian beam displays a somewhat lower divergence.
Compared to Gaussian beam foci, the necessary distance between beam shaper and the center of the focal zone amounts to a factor of 2–5 in our case. Of course this depends on the tolerance for matching waists and/or Rayleigh ranges. It has to be further noted that we used an idealized fully coherent pulsed Gaussian beam for the simulation shown in Fig. 5. Here all spectral contributions have been assumed to display identical spatial profiles in the waist plane. In practice, however, the propagation depends on the history of the generation, in particular on the position, shape, and dispersion of the shaping component and the angle of incidence, thus influencing the experimental results in a complex manner.
5. Experimental results
Truncation experiments were performed with a Ti:sapphire laser system consisting of a commercial oscillator (Femtosource Scientific PRO, pulse duration about 10 fs, power 300 mW, spectral FWHM 120 nm, repetition rate 87 MHz), an amplifier (pulse duration about 35 fs, power 700 mW, repetition rate 1 kHz), two argon-filled hollow fiber waveguides for self-phase modulation, and chirped mirrors for pulse compression (minimum pulse duration <4 fs, spectral FWHM >450 nm, power 20 mW). Spectral phase and pulse shape were analyzed by spectral phase interferometry for direct electric-field reconstruction (SPIDER, [36, 37]) and an autocorrelator [38]. To generate a small-angle BGBs with an about 1-m-deep zone of non-spreading propagation, a flat concave, circularly symmetric axicon of 1 cm diameter was used. The axicon consists of a gold-coated conical copper-layer on a glass substrate with a base angle of 0.027°. The variable-thickness copper layer was fabricated by vapor deposition using a rotating shadow mask [39, 40]. The surface profile of the axicon in Fig. 6(a) was inspected interferometrically. Figure 6(b) shows the time-integrated spatial intensity profile of a Bessel-Gauss pulse, generated by illuminating this axicon with 10-fs Ti:sapphire laser pulses.
Fig. 6. Generation of pulsed Bessel-Gauss beams by conical axicons: (a) measured thickness profile of a thin-film axicon (total diameter 1 cm, Au-coated Cu structure, apex angle 0.027°, (b) Intensity distribution of the pulsed BGB (central lobe and first maximum) generated by illuminating the axicon with 10-fs Ti:sapphire laser pulses (spectral FWHM 120 nm, imaged plane at an axial distance z=87.5 cm from the axicon, field of view 1×1 mm2).
The far field distribution was found to be ring-shaped as is typical for Bessel-like beams [39]. In the center of the axicon in about 1/50 of the total area, a small deviation from the perfect cone shape appears. The beam diameter was adapted by reflective beam expanders, consisting of convex and concave gold mirrors. The axicon was illuminated at angles of incidence <20°. Intensity distributions at different propagation distances were imaged onto a CCD camera (Vosskuehler CCD-4000, 2048×2048 Pixels, optical system: zoom objective 1:1.2/12.5-75 mm combined with a microscope objective of M=4× magnification). The experimental setups for spatio-spectral mapping and propagation measurements are schematically drawn in Figs 7(a), 7(b).
To control the self-apodization, matching of the diameters of the truncating apertures to the first minima of the intensity distributions was monitored. For this purpose, the camera was overexposed to enhance the visibility of the fringes. Spatially resolved spectra were measured by a fiber-coupled spectrometer (Ocean optics), with a 15-µm pinhole mounted in front of the fiber input on a 2-axis translation stage [Fig. 7(a)]. To compress the large amount of data generated by two-dimensional spectral mapping and to visualize significant spectral features, 1st–4th order statistical moments of the local spectral distributions were calculated.
The untruncated BGB was found to be slightly modulated in axial direction with an intensity maximum at about 90 cm, which is in agreement with the theoretical simulation. Inserting a diaphragm of about 450 µm inner diameter at this distance, a single maximum beam was obtained. In accordance with the theoretical prediction, the detected time-integrated intensity propagation shows an extended focal zone (transversal cuts at different distances (zD-z) in Fig. 8, radial-axial cut in Fig. 9).
The Rayleigh range defined in analogy to a focused pulsed Gaussian beam, i.e., the distance for a decrease of intensity to 1/e of the maximum, was found to be about 10 cm. It could be slightly enhanced by placing the diaphragm a few centimeters in front of the intensity maximum. The largest observed Rayleigh range exceeded 13 cm. The aspect ratio between Rayleigh range and initial waist radius in the plane of the diaphragm was found to be 440:1 at a pulse duration of 10 fs (corresponding to Fig. 9) and reached up to 520:1 at a pulse duration of <5 fs [40].
Fig. 7. Experimental setup for the self-apodized truncation of a pulsed Bessel-Gauss beam generated by a thin-film axicon. (a) Spatio-spectral mapping by scanning with a fiber-based spectrometer, utilizing a 15 µm pinhole in proximity to the fiber. (b) 3D analysis of the beam propagation by microscopic imaging onto a CCD camera (schematically).
Fig. 8. Radial cuts through the measured time integrated intensity propagation of a truncated pulsed Bessel-Gauss beam generated by a 10-fs Ti:sapphire laser pulse (diameter of diaphragm d=450 µm at a distance of 110 cm behind the axicon, FOV=960×960 µm2). The distance z-zD is the propagation distance relative to the truncating diaphragm.
Because of their particular shape, the propagation zones of BGBs that were truncated in a self-apodizing arrangement were referred to as “needle beams” in Ref. [13,40,41]. For a deeper analysis, we plot two-dimensional maps of selected spectral parameters obtained with and without truncation of the BGB in Figs. 10(e)–10(h) and Figs. 10(a)–10(d), respectively.
Fig. 9. Spatial propagation of a truncated pulsed Bessel-Gauss beam generated from a 10-fs Ti:sapphire laser pulse with a reflective axicon (axicon diameter 1 cm, conical angle 0.027°) for an axial position of the diaphragm at about z=110 cm. In agreement with the theoretical simulations, a needle-shaped propagation zone appears at a time-integrated detection with a CCD camera (Vosskuehler, 4 Megapixels). The plot shows a two-dimensional cut (r-z-plane) of a 3D propagation zone (normalized intensity represented by contour levels increasing in linear steps).
Fig. 10. Spectral mapping of pulsed BGBs represented by two-dimensionally resolved spectral moments in planes perpendicular to the optical axis for the untruncated (a–d) and truncated beam (e–h), M1=first order moments (a,e), M2=second order moments (b,f), M3=third order moments (c,g), M4=fourth order moments (d,h). The spectra were detected with a time-integrating detector. The rim of the truncating diaphragms is schematically indicated by the red dashed circle in (a).
These parameters are defined on the basis of statistical moments and are capable to pinpoint even very small changes in distribution functions and to enable an effective data reduction [42,43]. Whereas M1 and M2 are identical with center of gravity (1st moment) and standard deviation (square root of 2nd moment) of all completely measured spectra in a plane of interest, the maps of M3 and M4 (3rd moment or skewness and 4th moment or kurtosis) yield information about the spectral asymmetry and peakedness. The detailed analysis of the time-integrated fringe patterns of the untruncated beam reveals a non-uniform radially oscillating spatio-spectral structure. The oscillating centers of gravity of the spectrum (1st moments, M1) result from spatio-spectral interference inside a conical broadband beam. The second-order moment identifies the central region as the area of maximum bandwidth whereas the FWHM decreases significantly in the outer fringes. The 3rd and 4th moments can be explained by a splitting in multiple peaks of different intensity, leading to a spectral asymmetry. The spectral bandwidth in the center is obviously transferred over more than a Rayleigh length, which was found to reach 13 cm here. This finding has to be interpreted, in addition to the propagation behavior, as a strong indication for the nondiffracting nature of the Bessel-Gauss pulse by its spatio-spectral characteristics.
The results concerning the spectral transfer were furthermore confirmed by second order autocorrelation experiments for the central region. Typical measured noncollinear autocorrelation traces for two different distances (0 cm and 14.5 cm) are plotted in Fig. 11.
Fig. 11. Temporal transfer of the ultrashort-pulsed BGB after self-apodized truncation proved by second order autocorrelation: noncollinear autocorrelation traces (normalized, after subtracting background and averaging positive and negative envelope functions) measured at two different distances from the diaphragm (z-zD=0 cm and z-zD=14.5 cm).
It was found that the truncated beam is not only propagation invariant with respect to the time integrated spectral content but also remains nearly unchanged with respect to its pulse duration (temporal invariance). Directly in front of the diaphragm, an average pulse duration of 10.2(±0.5) fs was detected. After self-apodized truncation and passing a distance of 14.5 cm, the duration was only slightly increased to about 10.4(±0.5) fs.
Finally we compare the results of spectrally invariant truncated pulsed BGBs beams to pulses with a focused pulsed Gaussian beam using an identical laser source (Ti:sapphire laser oscillator). The focus of the pulsed Gaussian beam was generated by a concave mirror at a distance of about f=2.6 m (see contour plots in Fig. 12). Spatio-spectral maps were detected at two different distances (z1=256 cm, z2=276 cm) by vertically sampling the beam with the fiber-based spectrometer in steps of 0.02 mm. The normalized intensities are represented by the color code. It can clearly be recognized that the spectral intensity profiles along the scan direction change during the propagation even under this moderate focusing conditions rather than staying constant as in the case of the truncated BGBs.
Fig. 12. Spatio-spectral mapping of transversal cuts through the focal region of a pulsed Gaussian beam at a pulse duration of 10 fs) by a vertical scan of the spectrometer-fiber. Normalized spectra (color code) were measured at two different axial distances: (a) z=256 cm, (b) z=276 cm.
6. Conclusions
To conclude, a self-apodizing truncation of few-cycle pulsed BGBs was demonstrated, which was shaped by a reflective axicon with a conical angle of 0.027°. Self-apodization was obtained by matching the diameter of a spatial filter to the first intensity minimum of the beam at distances in the range of 1 m. A needle-shaped propagation region of the truncated beam was observed over Rayleigh ranges >10 cm. This “needle beam” did not display any diffraction fringes. In good agreement with theoretical simulations, a zone of low divergence, i.e., a “nondiffracting zone”, was observed in the near field. This zone results from constructive spectral interference. The beam radius defined by the standard deviation remains significantly smaller than that of a Gaussian focus of comparable Rayleigh range. By placing the truncating spatial filter in front of the axial intensity maximum, the nondiffracting zone can further be extended. At pulse durations of about 10 fs, spectral bandwidth and pulse duration remained invariant within the error bar. Therefore, the shaped wavepackets are nondiffracting not only with respect to the invariance of their intensity profiles but also to their spectral and temporal characteristics. Spatially induced group-velocity dispersion which is known to be relevant in the nonparaxial case [44] was obviously negligible. Thus, a particular proof of the paraxiality was given as well by the spectral and time domain experiments. In comparison to pulsed Gaussian beams, the total optical path length can be significantly reduced, such that distortions by air dispersion and turbulence are minimized. First measurements at pulse durations below 5 fs show more complex propagation characteristics that require a detailed analysis in future investigations. Further improvements of the truncation setup could be obtained by combining it with coherent addition in resonator structures. Finally, the integration of adaptive components, such as spatial light modulators, promises a higher flexibility in shaping needle beams.
Acknowledgments
The authors like to thank T. Elsaesser, W. Jüptner, H. Weber, U. Griebner, C. von Kopylow, P. Corkum, P. Saari, K. Reivelt, M. Fortin, N. McCarthy, P. Herman and J. Li for stimulating discussions. Thin-film components were fabricated by J.-F. Fortin and G. Rousseau (Laval University, Quebec) and H.-J. Kühn (Berliner Glas). The work was supported by DFG project no. GR 1782/2 and German-Canadian Collaboration project no. CAN 00/016.
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