# Ultrashort Vortex Pulses with Controlled Spectral Gouy Rotation
## 4 Experimental Results and Discussion
### Variation of Grating Parameters  
A variation of the axial angular velocity \(\Omega_{z}\) of the spectral Gouy rotation was obtained by (i) subsequently inserting static SPGs with different structural periods, and (ii) by adaptively modifying the periods of spiral phase gratings programmed into grey value maps of an LCoS-SLM. The diameter of the illuminating beam was about 4 mm, and the generated OAM beams were regarded to be Bessel-Gauss beams. For the smallest elements, this diameter was by a factor of 20 larger compared to the element diameter, so a plane wave illumination was assumed. The set of propagation-dependent first spectral moment (COG) maps in Figure 6a clearly indicates the spectral Gouy rotation for a particular configuration. Here, the SPG acts as an axicon and generates a Bessel-Gauss beam with OAM accumulating the spectral Gouy phase shift. In the second configuration in Figure 6b, the phase is only shifted axially by a staircase-like phase profile of a programmed SPP. In this case, no significant spectral Gouy rotation appears because the phase shift is not accumulated (there is no Bessel-like beam). It should be mentioned that a simple control of the rotation is even possible in the second case if the SPP phase map on the SLM is rotated. This trivial approach enables one to easily change the direction of rotation and to realize arbitrary angles at each distance. By varying the geometrical parameters, the angular velocity of spectral Gouy rotation can be controlled over an extended range. This is demonstrated in Figure 7 for programmable (SLM) and fixed (DOE) configurations by comparing the rotation cycles for equal propagation paths \(\Delta z\). Grating structure periods \(p\) and overall diameters \(D\) are found in the Figure caption. Figure 5: Spectral transfer into vortex beams as a function of input spectral FWHM bandwidth for three wavelength ranges around (1) 798 nm, (2) 800 nm, and (3) 802 nm (spectral profiles in Figure 4a). Filled symbols: upper limits, hollow symbols: lower limits of spectral content in the spectral eyes. The related parameter space is presented in Figure 8. The colored plane shows how the rotation speed can be tuned by adjusting the initial parameters (a) beam convergence angle and (b) aperture diameter both for DOEs and SLM. Figure 6: Propagation-dependent maps of 1 st spectral moment (COG) of experimentally detected maps of local spectra for two different vortex shaper configurations programmed in an LCoS-SLM (aperture \(D\) = 2 mm): (**a**) generation of a Bessel–Gauss beam with a spiral phase grating (SPG) with spectral Gouy rotation (spiral grating period \(p\) = 16 \(\upmu\)m) (see also video S1 in the supplementary material); (**b**) vortex generation with a spiral phase plate (SPP), without spectral Gouy rotation. The little grey scale pictures at the left side symbolize the type of element (SPG, SPP). Figure 7: Controlling the angular velocity by varying geometrical parameters. The modulo of the rotation angle is plotted as a function of propagation distance for equal paths \(\Delta z\). Different grating structure periods \(p\) and overall diameters D for programmable (SLM) and fixed (DOE) configurations are compared: (**a**) SLM, \(p\) = 32 \(\upmu\)m, \(D\) = 2 mm; (**b**) SLM, \(p\) = 16 \(\upmu\)m, \(D\) = 2 mm; (**c**) DOE, \(p\) = 32 \(\upmu\)m, \(D\) = 0.4 mm; (**d**) DOE, \(p\) = 16 \(\upmu\)m, \(D\) = 2 mm; (**e**) DOE, \(p\) = 16 \(\upmu\)m, \(D\) = 0.2 mm; and (**f**) DOE, \(p\) = 16 \(\upmu\)m, \(D\) = 0.4 mm. The axial period of rotation grows with increasing diameter \(D\) and decreasing conical angle \(\theta\). The SPG periods of 16 \(\upmu\)m and 32 \(\upmu\)m correspond to conical half angles of the first diffraction orders of 2.87\({}^{\circ}\) and 1.43\({}^{\circ}\), respectively. If the bandwidth is small enough, the components act like axicons and Gouy phase is continuously accumulated with increasing propagation distance.