1. Introduction

Bessel beams (BBs) are one of the four known types of exact solutions of the Helmholtz wave equation describing non-diffracting beams in circular cylindrical coordinates [1,2]. Non-diffracting means that the central maxima of these beams are remarkably resistant to diffractive spreading [3,4]. Precisely speaking, an ideal BB has an infinite number of rings carrying infinite power/energy, and hence, cannot be generated in the exact sense. After the pioneering publication of Durnin on BBs [1] it was shown that their reasonably good approximations can be created by axicons [4], by deformable mirrors reproducing the phase structures of reflective axicons [5], and by imprinting axicon phases on the beams when suitably programmed spatial light modulators (SLMs) are used [6]. The use of an SLM allows reproduction of computer-generated holograms of axicons [7] as well. Since the angular spectrum of the Bessel function is a ring, one can generate BBs by placing an annular slit in the back focal plane of a lens [4,8,9] or by using a cylindrical lens that is morphed to a closed ring-like form [10].

Due to the high aspect ratio (length/diameter) of BBs, they are of interest for numerous applications including laser welding with large focal position-tolerant zone [11], multifocal optical coherence tomography [12], Bessel-Bessel light bullets [13], optical trapping and tweezing [14,15], secure sharing of cryptographic keys [16], real-world free-space optical communications [17], and light detection and ranging (lidar) [18]. Another class of applications calls for intense, ultrashort pulses. Examples are high-order harmonic generation [19], laser particle acceleration [20,21], filament formation [22], and nanochannel machining [23]. It is obvious that this class of applications requires ultrashort pulses, i.e. ultrafast BBs are of outstanding interest. Particularly obvious is the example of femtosecond material processing, which is well-known for its superior precision and quality since the processed material does not heat up appreciably. With Gaussian beams, it is not possible to drill deep holes of constant diameter. BB offer a solution. However, for any new approach to generate them, one needs to prove that the pulse duration remains ultrashort.

An intuitively well motivated approach to create an achromatic broad-band BB would be to use a SLM as a diffractive optical element, in combination with a dispersive element (e.g. prism) [24]. The approach described in this work is different. It is entirely based on linear singular optics and was recently demonstrated with continuous wave beams to reach meters of propagation lengths [25,26]. Here we report unambiguous data that zeroth- and first-order long-range Gauss-Bessel beams (GBBs) can be generated in the fields of few-cycle femtosecond laser pulses by initially nesting and subsequently annihilating multiple-charged optical vortices (OVs) in the laser beam, finally Fourier-transforming the resulting ring-shaped beam with a large ring radius-to-width ratio ($>$10) by a thin lens (focusing mirror). The approach relies in several respects on interference, which implies that it is not obvious how well the above idea works for broadband pulses, let alone few-cycle pulses. Rather, it is necessary to demonstrate that the pulse duration is preserved for broadband BBs.

One key concept of the present work is the topological charge of the OVs. Due to their characteristic screw phase profiles, the OVs are the only known truly two-dimensional phase singularities [27]. The total phase change around the OV beam axis in azimuthal direction is $2\pi m$, where the integer number $m$ is the topological charge (TC) of the OV. Its sign (positive or negative) determines the azimuthal phase gradient. Because of the on-axis singularity, the intensity profiles of the OVs are ring-like, often denoted as doughnut-shaped. Important for our approach are four well established facts. (i) The ratio of ring diameter to ring width increases for increasing $m$ [26]. (ii) Multiply charged OVs with a TC $|m|>1$ split into $|m|$ singly charged OVs [28] in the presence of even small coherent perturbations (e.g. by the weak diffraction originating from the pixelated structure of SLM’s active area and/or by the spatio-temporal overlap of the incoming unmodulated and the reflected modulated wave from the SLM). (iii) OVs with the same signs of their TCs rotate around the beam axis and repel each other [28,29]. (iv) Closely neighboring OVs with opposite TCs, when nested on a common background beam, attract each other and annihilate [28,29].

Accordingly, the desired ring-shaped beam with a large radius-to-width ratio in front of the focusing lens is formed as a result of the decay of highly-charged OV imprinted on the beam after the first reflection at the SLM, followed by mutual repulsion [30] of the OVs, and, finally, by the annihilation of all OVs (for zeroth-order Gauss-Bessel beam creation) or by annihilation of all but one OV (for first-order GBB creation) at the second reflection at the SLM. As it will be shown later, the higher the TC initially imprinted and subsequently annihilated in the femtosecond beam, the larger the ring radius-to-width ratio of the ring-like beam in front of the Fourier-transforming lens, and the narrower the central peak/ring of the generated zeroth-/first-order GBB. The data in this work demonstrate quasi-non-diffracting propagation of GBBs over meters without any noticeable broadening even of few-cycle laser pulses.

The creation of optical vortices (OVs) with high topological charges (>10) and their subsequent successful annihilation is not obvious when few-cycle laser pulses with broad spectra are used. The main reason is that the used SLM providing more than 3.0$\pi$ maximal phase shift at 780 nm provides less than 2.5$\pi$ above 850 nm and more than 3.5$\pi$ below 650 nm. This means that a 2$\pi$ helical phase ramp of an "ideal" singly-charged OV at 780 nm will appear as a fractional OV at much longer and at much shorter wavelengths [31]. The erasure of such quasi-2-D dislocations in polychromatic optical fields is neither obvious, nor trivial. Nevertheless the presented results proved that it was successfully realized in the current setup even in the few-cycle regime.

2. Experimental setup

The used experimental setup is shown in the upper panel in Fig. 1. The beam from a high-power few-cycle femtosecond laser system is first controllably attenuated by a neutral density filter (NDF) in order not to damage the SLM. Then the beam passes through a pair of fused silica wedges for a precise adjustment of the minimum pulse duration achievable with this system. The pulses are characterized by using a commercial device (APE FC SPIDER) performing spectral phase interferometry for direct electric-field reconstruction [32,33]. One such typical pulse (pulse duration $\sim 7.5$ fs) is shown in Fig. 1(a). The laser source is a commercial Ti:Sapphire laser system (Femtolasers Femtopower compact PRO CEP) with a subsequent pulsed spectral broadening in a Neon-filled hollow-core fiber and final pulse recompression using broadband chirped mirrors to reach the few-cycle regime. The pulse energy is $1$ mJ. The pulse repetition rate is $4$ kHz, corresponding to a mean power of $4$ W. To protect the SLM from damaging, we limited the input power to 150 mW using a neutral density filter.

 

Fig. 1. Experimental setup: Laser source – few-cycles femtosecond laser system (see text for details); NDF – neutral density filter; W – glass wedges; SLM – computer-controlled spatial light modulator; M1 – flat silver mirror; M2 – removable flat silver mirror; SPIDER – commercial device performing spectral phase interferometry for direct electric-field reconstruction; L – biconvex uncoated focusing lens with focal length $f=100$ cm; CCD camera placed on a rail with option to follow the beam up to 160 cm after the focus of the lens. Angle of incidence of the input beam on the SLM – $2^\circ$. Distance between SLM and M1 – 15cm. SLM-to-L distance $\approx 30$ cm. Graph in panel (a) – measured time profile of the input femtosecond pulses (pulse duration $\sim 7.5 (\pm 0.5)$ fs) entering the setup. Panel (b) – phase (modulo 2$\pi$) of an OV with a TC=30. Panel (c) – typical ring-shaped beam (TC=30-30) prior to Fourier transformation.

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After passing the wedges, the beam reaches the first half of the SLM programmed with the phase distribution of a highly charged optical vortex. In panel (b) of Fig. 1, we show a phase profile (modulo 2$\pi$) corresponding to a TC=30 and the generated ring-shaped beam (panel (c)) in front of the focusing lens. The spatial light modulator used in this experiment is model PLUTO (HoloEye Photonics) with an active area of 15.36 mm $\times$ 8.64 mm (1920 pix. $\times$ 1080 pix., pixel pitch – 8 $\mu$m). The diameter of the beam at the first reflection at the SLM is measured to be 2.7 mm FWHW. The phase and the amplitude of the beam become modulated and the beam is redirected to a flat silver mirror M1. This mirror is aligned to perform a second reflection of the beam at the other half of the appropriately programmed SLM. After this second reflection, the TC of the highly charged OV generated from the first reflection is reset to zero or one. The typical ring-shaped beam in the plane of the lens L is shown in Fig. 1(c). Please note the large radius-to-width ratio. In this particular case OVs with TCs $30$ and $-30$ are created and annihilated. The higher the used TCs, the larger radius-to-ring width ratio at the second reflection from the SLM. As a result, the central peak/ring of the GBBs is narrower after the Fourier-transformation to the artificial far field and the number of the well formed surrounding rings is larger. The SLM-mirror-SLM distance is, however, geometrically limited by the size of the active area of the used SLM (15.36 mm $\times$ 8.64 mm). More precisely, the ring diameter at the second beam reflection form the SLM is limited to $\sim$7 mm, i.e. to somewhat less than the half of the length of the SLM. For the particular SLM-to-mirror M1 distance of 15 cm we could increase the topological charge up to 50. The lens L (uncoated, biconvex, focal length $f=100$ cm) is performing a spatial Fourier transformation of the ring-shaped beam thus resulting in a Gauss-Bessel beam in its focal plane.

Due to the two reflections from the SLM (single reflection coefficient $R_{SLM}$ $\sim$67%), one reflection from the flat silver mirror ($R_{M1}$ $\sim$97%), and the use of an uncoated focusing lens (transmission coefficient $T_L$ $\sim$92%), the overall transmission efficiency of the used setup exceeds 40% slightly. The evolution of this GBBs is followed up to 160 cm behind the focus of the lens. The flat silver mirror M2 was placed on a flip mirror mount so that it can steer the beam carrying the vortex ring towards the SPIDER device for measuring the duration of the pulses. In this device there is a mirror focusing the input beams/pulses into a non-linear crystal. Instead of the lens L, this mirror was used to conveniently Fourier-transform the beam into a Gauss-Bessel beam in the course of the measurement of the pulse duration. The distance between the SLM and the lens is approximately 30 cm.

As seen in Fig. 1(c), the decisive final step in generating GBBs is the Fourier-transformation of the ring-shaped beam. It is apparent that the radius-to-width ratio is much larger than one, as required by the theoretical model in [26]. For generating zeroth-order GBB, the phase profile of this ring has to be flat, while for first-order GBB the beam should carry an on-axial optical vortex with unit topological charge. The approach we used in generating the desired ring-shaped beams with the described phase profiles in the fields of few-cycle laser pulses is similar to the one developed for cw laser beams [25,26] with two SLMs. In summary, we:

3. Results and discussion

In Fig. 2(a), we show the radial cross-section of the generated zeroth-order Gauss-Bessel beam. The square data points are obtained from the energy density profile shown as an inset to the graph. The energy density profiles are captured 90 cm behind the focal plane of the lens. The solid red curve in Fig. 2(a) is a numerically generated radial cross-section of a perfect Bessel beam. The experimental modulation depth (down to zero) of the rings is evident from the data and is a result of the $\pi$ phase difference between adjacent rings surrounding the central peak, as known from the literature ([35], see also Fig. 3(c) in [26]). In Fig. 2(b), following the same style of presentation as in Fig. 2(a), we show the corresponding results for the first-order GBBs. Again, the modulation depth of the central ring (carrying an optical vortex with unit topological charge) and of the satellite rings (due to the mentioned radial $\pi$ phase jumps) is remarkable. One possible reason for the slight displacement between the positions of the experimental and the theoretically predicted curve after the second bright ring in Fig. 2(a) could be the following. Due to the limited half-size of the used SLM, a ring-shaped beam with a finite radius-to-ring width ratio is formed in front of the focusing lens (mirror). As a result, we are able to generate Gauss-Bessel beams with finite number of surrounding rings, not an ideal Bessel beam with infinite number of rings. The better radial overlap of the experimentally recorded and the numerically simulated peaks in the case of first-order GBB (Fig. 2(b)) we attribute to the presence of an on-axis singly-charged OV. Its presence results in a toroidal central bright structure. This torus is 1.5 times wider (measured as a diametrical peak-to-peak distance) than the central peak (FWHM) of the zeroth-order GBB. Because of the $\pi$-phase jumps between the neighboring rings of the (Gauss)-Bessel beams (see Fig. 7(c) in [26]), the mutual repulsion between the rings is stronger in the case of the first-order GBB. Because of this, the rings stretch stronger radially and match better to the theoretical curve. Potential imperfect alignment of the setup cannot be excluded, although we have done our best to minimize it.

 

Fig. 2. (a): Radial cross-section (black squares) of the experimentally obtained zeroth-order Gauss-Bessel beam and theoretically generated radial cross-section of a BB (solid red curve). Inset frame – experimentally recorded energy density distribution of a zeroth-order GBB obtained by the creation and subsequent annihilation of an OV with |TC|=30, captured 90 cm after the focal plane of the lens. (b): The same as in graph (a) but for the first-order GBB generated by annihilating OV with TC $=31$ and $-30$. (c) and (d): Longitudinal evolution of the transverse cross-sections of zeroth- (c) and first-order (d) GBBs for propagation distances from 20 cm to 140 cm behind of the focus of the lens L (see Fig. 1). (e): Typical reconstructed time profile of the generated (zeroth- and first-order) GBBs, retrieved from the SPIDER device. Pulse duration $\sim 7.5(\pm 0.5)$ fs.

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In panels (c) and (d) of Fig. 2 we show the longitudinal (axial) profiles of the normalized intensities of zeroth- (c) and first-order GBBs (d) retrieved from sets of experimental data. For better visualization, at each step the peak intensity of each intensity profile is set to unity. This allows recognizing the relative fast reshaping of the GBBs up to some 30 cm after the lens’ focus. Some weak disbalance between the intensities of the rings on both sides of the beam axis for the first-order GBB can be seen in Fig. 2(d) above 120 cm. Nevertheless, it is well pronounced that the central maxima of the GBBs are remarkably resistant to diffractive spreading, while the radii of the surrounding rings slowly increase vs. propagation distance. One can also recognize that the number of the well-formed outer-lying rings increases with propagation distances. This visualization allows for a better comparison between the peak intensities of the central peak/ring and the surrounding rings, however, there is no equivalent of an "energy conservation" between the individual slices. Nevertheless, the slow increase of the width of the central peak and of the radii of the surrounding rings of the zeroth-order GBB vs. distance shown in the graph in Fig. 3(b) are clear indication that the peak intensity of the GBBs slowly decreases with propagation distance due to the weak diffraction (diffraction half-angle less than 40 $\mu$rad). All data are recorded with sub-8-fs pulses, as measured with SPIDER-device. The typical shape of the reconstructed $\sim$ 7.5-fs-pulse is shown in Fig. 2(e).

 

Fig. 3. Zeroth-order GBBs. (a): Width of the central peak for different (subsequently annihilated) TCs vs. propagation distance after the focus of the lens. (b): Width of the central peak (p) and diameters of the first four satellite rings of the GBB generated by annihilating |TC|=47 vs. distance. For comparison, the width of the unmodulated Gaussian beam is shown as well. Middle frames: experimental data for zeroth-order GBBs generated by annihilating |TC|=27 and 47 at 140 cm after the lens’ focus.

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Figure 3 is devoted to the longitudinal evolution of a zeroth-order femtosecond GBB. In Fig. 3(a) we show the widths (FWHM) of the central peaks of such GBBs generated using OVs with |TC|=17, 27, 37, and 47 vs. propagation distance. It is clearly seen that the higher the initial (and subsequently annihilated) TC, the narrower the central peak of the generated GBB beam in and after the focal plane of the lens. This is due to the observed tendency that the increase of the ring radius in front of the lens, while keeping the ring width much smaller, leads to a narrowing of the central peak of the GBB [25,26]. Recalling the similarity theorem of the Fourier transformation, this behavior is expected. Another illustration of this fact is given by the two density plots sandwiched between Fig. 3(a) and (b) when annihilating TCs of 27 and 47. Both frames are recorded 140 cm behind the focal plane of the lens L and display a remarkably good Gauss-Bessel form of the beam, even at such large propagation distances. In Fig. 3(b) we present data for the longitudinal evolution of the width of the central peak (p) and of the diameters of the four satellite rings (1..4) located closest to the beam axis (peak-to-peak distances measured in radial cross-sections). For comparison, the width of the pure Gaussian beam is shown as well. The beams are followed up to 160 cm behind the focus of the lens. The estimated divergence half-angle of the central peak of the zeroth-order GBB generated by OVs with |TC|=47 is 36($\pm$4) $\mu$rad. In view of their negligible spatial dynamics we can consider the generated GBBs as quasi-non-diffracting.

In Fig. 4, following the same style of presentation as in Fig. 3, we present details regarding the behavior of few-cycle first-order GBBs. The significant difference in comparison to the preceding case is that a first-order GBB is carrying a residual topological charge equal to $\pm$1. As a result, instead of a central peak, one can observe a central bright OV ring with |TC|=1 nested in it. As seen in Fig. 4(a), the higher the TCs (as an absolute value) of the encoded and annihilated OVs, the smaller the diameter of the first ring at a fixed distance (see also the grayscale images in the middle of Fig. 4, recorded at a distance of 140 cm) and the lower its longitudinal dynamics. The data for this first central bright ring and for the three neighboring rings vs. propagation distance are shown in Fig. 4(b). The estimated divergence half-angle of the first central ring of the first-order GBB generated by OVs with residual |TC|=47-46 is 35($\pm$4) $\mu$rad. This value is, once again, a clear manifestation for the negligible spatial dynamics of the generated few-cycle first-order GBBs and justifies to characterize them as quasi-non-diffracting.

 

Fig. 4. First-order GBBs. (a): Diameter of the central ring for different initial TCs (subsequently annihilated, except one TC) vs. distance after the focus of the lens. (b): Diameters of the central four rings of the first-order GBB generated by annihilating TC $=47$ and $-46$ vs. propagation distance. For comparison, the width of the unmodulated Gaussian beam is shown as well. Middle frames: Experimental data for the first-order GBBs generated by annihilating TC $=27$ and $-26$ (upper panel) and TC $=47$ and $-46$ (lower panel) 140 cm after the lens’ focus.

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The broad spectrum of the sub-8-fs laser pulses as measured by the SPIDER device is shown in Fig. 5(c). Its width exceeds the interval spanning from 650 to 950 nm. When working with such a broad bandwidth, the problem of the diffraction of the different spectral components deserves special attention. In Fig. 5 we show graphs in which we follow the transverse positions (ring diameters) of the five bright rings supporting the central peak of the zeroth-order GBB (see Fig. 5(a)) and the central six bright rings of the generated first-order GBB (see Fig. 5(b)) as a function of wavelength. To this end, a set of interference filters of different central transmission wavelength $\lambda _c$ and equal bandwidth (50 nm) was used. It can be seen that the "coloring" of the satellite rings increases with $\lambda _c$ – the diameter of the rings depends on wavelength. For the first, i.e. the innermost ring, the variation is almost negligible, but gets the more pronounced, the higher the ring number. As a consequence, the outer rings develop increasingly pronounced color fringes. Generally, under comparable conditions, the higher the wavelength, the stronger pronounced the diffraction.

 

Fig. 5. Coloring of the peak/rings. Change of the positions of the satellite rings relative to the central peak of the zeroth-order GBB (a) and relative to the central bright ring for the first-order GBB (b) for different 50 nm wide spectral intervals around central wavelength $\lambda _c$ of the transmission of the used filter. The inserted slices from experimental frames correspond to $\lambda _c = 900$ nm. The presented data are recorded at a distance of 160 cm behind the focus of the lens L, referring to TC $=41$ and $-41$ (graph (a)) and to TC $=41$ and $-40$ (graph (b)). Typical spectrum of the used sub-8-fs laser pulses as measured by the SPIDER device (c).

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The almost parallel vertical lines for the position of the central rings vs. wavelength (Fig. 5(a), the ring surrounding the central peak of the zeroth-order GBBs) indicate that the spatial chirp is negligible for the central portion of the beam. The spatial dispersion is more pronounced for the further outside lying bright rings. Nevertheless the quasi-non-diffracting properties and the main part of the carried energy are associated with the central peak (or with the central ring), where the influence of the spatial dispersion is negligible. The measurements of the pulse duration of the ultrashort GBBs with the SPIDER device showed the same sub-8-fs pulse durations as measured when the SLM was switched off (thus acting as a mirror). In view of this, one can conclude that the influence of the spatial dispersion on the outside-lying low-energy rings and on the duration of the pulse carrying such GBBs remains negligible even in the few-cycle regime.

One cannot neglect the fact that in the SPIDER device a sum-frequency mixing process is used. In this second-order nonlinear process the central peak/ring of the GBB should be expected to dominate over the satellite rings. However, the quasi-non-diffracting nature of the GBBs is attributed, to the spatial behavior of their central peak/ring. The result that the durations of the central peaks of the GBBs remain unchanged even in the few-cycle laser fields is a valuable result.

Fig. 5(c) shows that the wavelength range between 700 nm and 850 nm determines the mean spectral power of the used pulses. The detailed inspection of the data reveals that the relative power of the spectral components decreases by a factor of about 5 around 650 nm and 900 nm, while beyond 950 nm and below 600 nm this decrease is 3 orders of magnitude or more. This means that these spectral components have minor (if any) relative contribution to the spectral content of the generated GBBs. In obtaining the data used for Fig. 5(a) and (b) we intentionally enhanced these short- and long-wavelength signals in order to present this tendency.

The "coloring" effect could be further minimized by replacing the used uncoated biconvex lens by a focusing mirror with the same focal length. The influence of the pixelated structure of the SLMs and the resulting weak spatial dispersion could be removed by using reflective etched vortex phase plates, which can also survive much higher pulse energies/powers. However this will be at the cost of losing the reconfiguration flexibility provided by the SLMs.

A characteristic feature of BBs is the ability to reconstruct their initial profile in free-space propagation after being disturbed by an obstacle [36]. In view of potential long-range applications, it is interesting to learn how the GBBs carried by few-cycle laser pulses behave in this respect. In Fig. 6 we present experimental proof for the self-healing of zeroth-order GBB (frames (a1) and (a2)) and of first-order GBB (frames (b1) and (b2)). Strong perturbations are introduced in the form of a needle tip (diameter 130 $\mu$m) at 20 cm after the focus of the lens L, as shown in frames (a1, b1). Frames (a2) and (b2) are recorded after an additional propagation distance of 100 cm. One can qualitatively state that at this distance the GBBs are already self-healed. In this example, the zeroth-order GBB is generated by annihilating OVs with TC$=37$ and $-37$, while the first-order GBB is obtained by annihilating TC$=37$ and $-36$. All frames in Fig. 6 are post-processed for better visibility of the outer-lying rings. The length of the scale bars is 300 $\mu$m.

 

Fig. 6. Perturbation for the zeroth- and first-order GBBs (a1) and (b1) 20 cm after the focus of the lens. Self-healing of the zeroth-order GBB (a2) and of first-order GBB (b2) 100 cm after the plane of the perturbation. Scale bars – 300 $\mu$m. See text for details.

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4. Conclusion

In this work we experimentally demonstrated that zeroth- and first-order long-range Gauss-Bessel beams can be generated in few-cycle femtosecond laser fields by initially nesting and subsequently annihilating multiply charged optical vortices and then Fourier-transforming the resulting ring-shaped beam by a thin lens. The estimated diffraction half-angles for the zeroth- and first-order GBBs of less than 40 $\mu$rad justify classifying them as quasi-non-diffracting. Even for sub-8-fs pulses and a spectrum spanning more than 300 nm no noticeable influence on the measured pulse duration was observed. The self-healing nature of the BBs is confirmed for the few-cycle GBBs studied in this work.

Noteworthy is the plainness of the used experimental scheme: It does not require special optical elements except a single reflective spatial light modulator allowing reconfiguration of the order of the GBBs. It is shown that at large propagation distances, the quality of the generated GBBs significantly surpasses this of GBBs created by low angle axicons [25]. The approach is easy to apply for generating high-order GBBs.

Additional measurements (not shown here) were carried out with 30-fs pulses from another laser oscillator switched between mode-locking and continuous-wave regime without any realignment of the setup. These experiments showed a remarkable good agreement between the beam profiles. Regrettably, SLMs do not survive high intensities/pulse energies. When no reconfiguration of the GBBs is needed, i.e. when a fixed TC is sufficient, reflective etched vortex phase plates can be used. In order to keep the option for switching between different orders of GBBs in such fields, one option is the use of digital micromirror devices. An obviously reasonable choice when working with few-cycle pulses is to use a focusing mirror instead of a focusing lens.