# Ultrashort Vortex Pulses with Controlled Spectral Gouy Rotation
## 1 Introduction  
Orbital angular momentum (OAM) beams or vortex beams [1; 2; 3; 4] propagate with twisted wavefronts and belong to \"core concepts\" in optical technologies [5]. Their topological charges are a measure of OAM density. The specific topological degrees of freedom can be exploited for optical tweezing [6; 7; 8], momentum transfer to atoms and molecules [9; 10] microstructuring of materials [11; 12; 13; 14], microscopy [15], optical processing [16], communication [17; 18; 19; 20], and other advanced applications [21; 22]. For information transfer, tuning and switching of topological charges is an obvious option. In addition to the azimuthal degrees of freedom, the axial propagation characteristics can be structured as well. It was found that an axial shaping of local topological charges can be obtained by the interference of multiple Bessel-like modes, while the OAM is globally conserved [23]. The influence of amplitude and phase gradients as well as intensity-induced nonlinearities on the vortex propagation was investigated [24; 25]. The combination of twisted wavefront and ultrashort pulse duration opens specific channels for high-data-rate optical communication, nonlinear excitation, or optical processing. Ultrafast singularoptics is per se polychromatic and enables for multidimensional structuring of light in spectral, temporal, and spatial domains. At extremely short pulse durations, spatio-temporal coupling effects caused by pulse travel-time and angular geometry can lead to highly complex propagation effects and related limitations of the topological charge spectrum [26]. At moderate pulse durations and corresponding spectral bandwidths in 10-nm range, however, space-time coupling can typically be neglected. Instead, spatio-spectral maps in the vicinity of phase singularities reveal kinds of spectral anomalies with peculiar properties, so-called \"spectral eyes\". Spectral eyes have been well-known in singular optics for a relatively long time [27; 28; 29; 30; 31; 32; 33; 34] and represent regions with blue- and red-shifted spectral centers of gravity (COG) of polychromatic OAM light fields. The spectral redistribution basically results from wavelength-dependent diffraction at a phase discontinuity, e.g., the phase step of a spiral phase plate (SPP), the transition between regions of different spatial frequencies in a forked grating or the asymmetric phase structure of a spiral phase grating (SPG). In all cases, the vortex is generated by symmetry-breaking geometrical configurations. Recently, we found that, for appropriately chosen parameters, spectral eyes of Bessel-Gauss beams with OAM undergo a characteristic rotation during pulse propagation [35]. This behavior was explained by a wavelength-dependent Gouy phase shift of a polychromatic vortex and therefore referred to as \"spectral Gouy rotation\" [36; 37]. Until that point, the influence of the Gouy phase on the propagation dynamics was mostly discussed with respect to rotating intensity patterns [38]. From experiments with distorted OAM beams, it was also known that an initial intensity distortion leads to rotation in intensity maps as a function of the axial distance of the detection plane [39]. This type of rotation directly reflects the wavefront twist and can be applied to metrology applications. For precise translation measurements, e.g., revolutions of multi-zeros can be counted [40], and interferometry with conjugate twisted beams is another option [41]. Unlike OAM vortices, vector vortices are based on the spin angular momentum and can be tuned by polarization control. With pairs of chirped vortex pulses and polarization-dependent vortex conversion, ultrafast rotating ring-shaped intensity lattices were obtained and applied to the coherent excitation enhancement of a quasi-particle [42]. Variable intra-pulse rotation (\"self-torque\") was obtained by superposition of chirped vortex pulses [43]. In contrast to the rotating intensity patterns, spectral Gouy rotation is a more subtle effect.