SUPPLEMENTARY INFORMATION

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1 SUPPLEMENTARY INFORMATION DOI: /NPHYS2397 Strong-field physics with singular light beams M. Zürch, C. Kern, P. Hansinger, A. Dreischuh, and Ch. Spielmann Supplementary Information S.1 Spectrometric evaluation The XUV beam carrying a vortex was focused onto the entrance slit of the spectrometer in the spectrometer beamline (see Fig. S2b). Technical details are depicted in section S.2. The entrance slit of the spectrometer was opened, allowing imaging of the HHG source along with spectral separation of single harmonics. Spectral measurement of the XUV vortex beam reveals that the double lobe structure as it was seen in direct imaging experiments (Fig. 1b) is detected for every harmonic order. It was expected that the TC is increasing by two from one observed harmonic order to the next 16. This should result in an increased diameter of the doughnut mode or, in the case presented here, an increased distance between the lobes. The recorded images as displayed in Fig. S1, show the same lobe separation for all observed harmonics. It can be concluded from Fig. S1, that the TC is not increasing with the order of the harmonics. If however the TC is increased in the fundamental beam, an increased distance of the lobes is measured in the XUV for all detectable harmonics (red curve in Fig. S1). This suggests that the TC is directly transferred to the XUV radiation, and that the TC is the same for different harmonic orders. Figure S1, Spectrometric measurement of the double lobe structure: The distance between the two lobes is a measure for the TC. For TC=1 of the fundamental beam (black squares) the XUV lobe distance decreases slightly with decreasing wavelength or increasing harmonic order, which can be explained by the lower diffraction for shorter wavelengths. For TC =2 in the fundamental (red circles) a significantly larger distance of the lobes is measured. NATURE PHYSICS 1

2 The indicated wavelength information originates from a calibration measurement with closed slit and Gaussian mode XUV beam. The inset shows the original spectra for both cases measured with an imaging spectrometer. S.2 Experimental setup The femtosecond laser system delivers pulses with 30fs pulse duration at 800nm wavelength and energies up to 1mJ at a repetition rate of 1kHz. The phase modulation is imprinted with a reflection type spatial light modulator (SLM). The reflection type method is favored due to negligible dispersion introduced into the beam. The SLM (type Hamamatsu LCOS-SLM X10468) enables local phase shifts of the wave front in the range from zero to 2π at a spatial array of 800 by 600 pixels, which are distributed over an area of 16 by 12 mm (Ref. S1). A useful feature of this setup is the ability to transform it into a simple mirror by employing a flat phase to the SLM (blank screen). This is used as a reference in the presented measurements. Typical phase masks (see Fig S3 b) as they are applied to produce a beam with a TC of one consist of a screw-like transition from black to white, which corresponds to a relative phase shift of 2π. Beams with TC > 1 can be produced by phase wrapping at 2π (see Fig S3c/d). These shaped laser pulses are then focused (f=300mm, f/#=15) into the interaction region to a diameter of roughly 40µm, resulting in peak intensities of approximately Wcm 2 (Ref. 29). The interaction region consists of a nickel tube (2mm diameter) sealed at the end and squeezed (d~1mm) to a flat surface facing the laser beam. The visible light is blocked with two thin aluminium filters (0.2µm and 0.3µm) downstream of the source. The transmitted XUV radiation is analysed with two different beamlines. The first beam line (Fig. S2a) contains a large area back-illuminated x-ray CCD detector (Andor ikon-l, 2048x2048 pixels) placed ~60cm downstream from the source. The second beam line (Fig. S2b) for spectroscopic analyses comprises an additional vacuum chamber wherein a gold coated toroidal mirror (f=328mm at 4,4 grazing incidence) is placed in a 2f-2f-setup imaging the source onto the entrance slit of an XUV-spectrometer (McPherson 248/310G with imaging option, 300 lines/mm grazing incidence grating employed) S2. Typical HHG spectra obtained by this setup range from the 11 th to the 33 rd harmonic order, for backing gas pressures ranging from 50mbar to 200mbar argon. The strongest emission is found around the 23 rd harmonic due to phase matching S3. 2

3 Figure S2, Experimental setup of the two beamlines used in the experiment: a, The beamline for spatial HHG imaging with the SLM is shown. The vortex phase profile is added to an input laser beam, which is then focused into an argon jet. Aluminium filters block the fundamental light, and an x-ray CCD is used for spatial detection of the generated radiation. b, In the spectrometry beamline a toroidal mirror images the XUV source onto the entrance slit of an XUV spectrometer. The large distance microscope for observing the laser focus is not shown in this figure. 3

4 Figure S3, Typical intensity profiles and phase masks used for the generation OVs: a, intensity distribution of the fundamental (800nm) laser beam carrying an OV with a TC = 1, b, phase profile applied to the SLM to produce an OV with TC =1 c & d, same as a and b but for a TC = 2. S.3 Evaluation on spatial phase matching A known effect in HHG is off-axis phase matching due to non-uniform intensity profiles S4. However, it can safely be excluded in the presented experiment that the observed XUV profile is subject to simple off-axis phase matching effects. For this, phase matching conditions were varied, while the XUV yield was monitored. One accessible parameter that influences phase matching is the gas backing pressure, which changes linear and plasma dispersion. If off-axis phase matching were an issue, the distance of the lobes would change on the detector. To verify this, the harmonic at 36nm wavelength was observed for different gas backing pressures with the spectrometer setup (Fig. S2 b). The distance of the lobes on the detector was determined using a lineout. The result is shown in figure S4. Despite changing the gas backing pressure significantly, the lobe structure was unaffected. Similar behaviour was observed for other wavelengths. 4

5 Figure S4, Distance of the vortex lobes for different gas backing pressures: The upper part shows the image recorded with the spectrometer for 90mbar backing pressure. For a more quantitative analysis we plotted lineouts for the range of 36nm. The lobe distance is independent of the gas backing pressure, in contradiction to off-axis phase matching, but in agreement with the existence of an OV. S.4 Detailed discussion on proposed vortex decay From the presented data in this letter the questions arises as to why OVs with TCs equal to one and two in the fundamental beam yield OVs with the same TC in the HHG signal. Our intuitive understanding is the following. At least the third- and fifth-order nonlinearities should be non-resonant and positive which means that the OV cores spread out faster than in the linear regime. The overall regime is dominated by the nonlinearity and, hence, the OV core dynamics is probably most influenced by the positive nonlinearity, not by the diffraction. We would like to support the simple explanation of (OV) decay by some references. In the case of an m-fold charged screw dislocation, the addition of a small coherent pedestal gives rise to its splitting in m dislocations of unit charge S5. A theoretical and experimental study on the propagation and decay of highly charged optical vortices in a medium with an anisotropic nonlocal nonlinearity can be found in S6. Even for an isotropic nonlinearity, the vortex dynamics of the nonlinear wave equation shows that OV beams of charge m 2 are topologically unstable S7,S8. Eq.(2.12) in Ref. S9 also indicates that the increase of the TC m leads to higher values for the critical transverse spatial frequency of the perturbation Ω crit and, therefore, to an enhanced modulational sensitivity of multiple-charged OVs. This is clearly seen in the experimental data shown in Fig. 1 in Ref. S9 and in the numerical simulations. The result in Ref. S9 is more general in accounting for the nonlinearity saturation, but this is not relevant here. According to Eq.(27) in Ref. S10 the transverse velocity of a vortex has two components arising separately from the transverse phase and intensity gradients of its background field. The first component is directed normal to the wave-front of the background, 5

6 which is in the direction of transverse energy flow in the background field, giving rise to radial motion of a vortex in a Gaussian beam. The second component is directed along the intensity contour of the background upon which the vortex is positioned. Experimental results confirming the model in Ref. S10 are published in Ref. S11. In the present manuscript we have positive nonlinearity. Comparative numerical analyses carried out in self-defocusing and self-focusing Kerr media showed S12 that the basic scenarios (attraction/repulsion, translation/rotation vs. background) in the interaction of two and three vortices with equal and alternative TCs are the same in both media. However, the vortex dynamics under selffocusing conditions is gradually influenced by the reshaping of the surrounding part of the background (see e.g. Figs. 3 and 7 in Ref. S12). S.5 On the simulation of the thin wire diffraction In Fig. 2 of the manuscript the measured diffraction pattern from a thin wire is compared to a simulation of the diffraction. Although the wire is opaque, the phase difference between the beam portions passing just by the sides of the wire have well pronounced influence on the diffraction structure observed after certain free space propagation to the detector plane, to be precise, diffraction about a wire works equivalent to division-of-wave-front interferometer S13. In Fig. S5 we present the steps made for the simulation to resolve possible misunderstanding. The expected patterns differ much for different topological charges, if Fig. S5e and f are compared. Please note that Fig. S5e is the same picture as the false colour one in Fig. 2b of the manuscript. Figure S5, Simulation of vortex diffraction at a thin wire for different TC: a&b, amplitude and phase of a vortex carrying TC of one, c&d, input intensity and phase for the simulation with the wire in place, e, shows the distribution at a detector placed two Rayleigh diffraction lengths away from the wire (no beam focusing is accounted for) for TC=1, f, shows the same as e but for a TC=3. 6

7 References S1. Kuang, Z., et al. Diffractive multi-beam surface micro-processing using 10 ps laser pulses. Appl. Surf. Sci. 255, (2009). S2. Koláčeka, K., et al. McPherson s XUV grazing incidence spectrograph for plasma diagnostics and its calibration. German-Polish Conference on Plasma Diagnostics for Fusion and Applications, Greifswald, Germany, September S3. Spitzenpfeil, R., et al. Enhancing the brilliance of high-harmonic generation. Appl. Phys. A 96, (2009). S4. Fomichev, S.V., Breger, P., Carre, B., Agostini, P. & Zaretsky, D.F., Non-collinear high-harmonic generation. Laser Phys. 12, 383 (2002). S5. Basistiy, I. V., Bazhenov, V. Yu., Soskin, M. S. & Vasnetsov, M. V. Optics of light beams with screw dislocations. Opt. Commun. 103, 422 (1993). S6. Mamaev, A. V., Saffman, M. & Zozulya, A. A. Decay of high order optical vortices in anisotropic nonlinear optical media. Phys. Rev. Lett. 78, 2108 (1997). S7. Neu, J. C. Vortices in complex scalar fields. Physica D 43, (1990). S8. Neu, J. C. Vortex dynamics of the nonlinear wave equation. Physica D 43, (1990). S9. Dreischuh, A., et al. Modulational instability of multiple-charged optical vortex solitons under saturation of the nonlinearity. Phys. Rev. E 60, 7518 (1999). S10. Kivshar, Y. S., Christou, J., Tikhonenko, V., Luther-Davies, B., Pismen, L. M. Dynamics of optical vortex solitons. Opt. Commun. 152, 198 (1998). S11. Christou, J., Tikhonenko, V., Kivshar, Y. S., Luther-Davies, B. Vortex soliton motion and steering. Opt. Lett. 21, 1649 (1996). S12. Hansinger, P., Dreischuh, A., Paulus, G. G. Optical vortices in self-focusing Kerr nonlinear media. Opt. Commun. 282, 3349 (2009). S13. Peele, A.G., et al. Phase retrieval from coherent soft X-ray optics. J. Electron. Spectrosc. Relat. Phenom. 144, 1171 (2005). 7