# Ultrashort Vortex Pulses with Controlled Spectral Gouy Rotation
## 3 Experimental Techniques and Mathematical Tools  
### Shaping of Femtosecond Vortex Pulses with Stationary and Adaptive Components  
The experimental setup for rotational pulse control in spectral domain with pulsed polychromatic Bessel-Gauss beams is schematically drawn in Figure 2. The studies were performed with a linearly polarized Ti: sapphire laser oscillator as the light source (Tsunami, Spectra Physics, formerly Mountain View, CA, USA) maximum spectral FWHM width 12 nm, minimum pulse duration 80 fs, central wavelength 795 nm, repetition rate 80.2 MHz, pulse energy \(<\)8 nJ) [44]. The capability of adaptive and passive optical components to generate OAM beams with variable spatio-spectral parameters, in particular to control the spectral Gouy rotation of pairs of blue-and red-shifted spectral eyes, was studied. Bessel-Gauss beams were either shaped by static lithographically structured, transmissive spiral phase gratings (SPG) [51; 52; 53] and continuous-relief spiral phase plates (SPPs) [54], or spiral and forked gratings flexibly programmed into the phase map of a liquid-crystal-on-silicon (LCoS) spatial light modulator (SLM) (GAEA 2, HOLOEYE Photonics AG, Berlin, Germany) [44]. The SLM enables one to individually address \(4160\times 2464\) pixels with 3.74 um pitch at a fill factor of 90% with a frame rate of 58 Hz and 8 Bit dynamic range. The reflective operation mode requires working at an oblique angle of incidence to avoid the insertion of dispersive beam splitters. The resulting geometrical distortions in one spatial direction can easily be corrected by an adapted linear transformation (stretching or shrinking) with the programmable display [44]. The average laser power density on the SLM was always chosen to ensure a linear operation.  
With the scanning fiber-coupled spectrometer, full local spectra were detected for each distance \(z\) and further processed with advanced statistical methods [55].  
Areas in the proximity of the phase singularities with diameters at few-micrometer scale were sampled with a truncated fiber (detecting aperture 10 um) in combination with a 100\(\times\) achromatic microscope ([PERSON]). To reconstruct the spectral propagation characteristics, two-dimensional  
Figure 1: Numerical simulation of rotating anomalies in the spatio-spectral propagation of femtosecond vortex pulses in air (FOV = 12 × 12 μm2). The centers of gravity (COG) of local spectral moments at distances \(z\) between 4 and 11 mm show the circulation of spectral eyes around the singularity during a full period of spectral Gouy rotation. To induce the orbital angular momentum, a binary spiral phase grating with a period of 8 μm and a phase stroke of 2\(\pi\) at a central wavelength of 800 nm was illuminated by a Gaussian beam of a full-width-at-half-maximum (FWHM) spectral bandwidth of 12 nm with a waist radius of w\({}_{0}\) = 100 μm. The color code reaches from a lower limit (blue tail) of \(\lambda_{\text{min}}\) = 793 nm to an upper limit (red tail) of \(\lambda_{\text{max}}\) = 808 nm (\(\pm\)0.9 nm).  
maps of complete local spectra for variable axial distances \(z\) were analyzed in a first step by a grating spectrometer (AvaSpec-2048-USB2, Avantes, 600 lines/mm grating, \(<\)0.7 nm resolution, 2048 pixel CCD ILX554, Sony, Avantes, Apeldoorn, The Netherlands). In a second step, the spectra were post-processed. The positioning accuracy of the three-axis translation stages (AG-LS 25-27, Newport) was limited to +/-100 nm. The spectral resolution was found to be closely sufficient to clearly indicate the relevant spectral structures. In addition to sequential spectral scanning, intensity patterns were monitored by a high-quantum-efficiency EMCCD camera (iXon, Andor Technology, Belfast, UK).