# Ultrashort Vortex Pulses with Controlled Spectral Gouy Rotation
## 3 Experimental Techniques and Mathematical Tools
### Rotation Control and Relevant Parameters  
To control the spectral Gouy rotation, key influencing parameters have to be identified. The dependence of the Gouy phase shift on the wavelength and the conical beam angle enables one to adjust the spectral rotation by (a) shaping the spectral distribution function and (b) shaping the angular profile. The spectral parameters can be influenced by tuning the center wavelength of the source by mode management, or spectral filtering. The angular profile can be adjusted by grating periods of spiral shapers and by the divergence or convergence of the illuminating beam. For microoptical vortex generators, the diffraction at the edges essentially contributes to the angular profile so that the Fresnel number of the aperture also has to be taken into account as an additional relevant parameter.  
Furthermore, radial gradients of spectral and/or angular parameters can be exploited to axially tune the rotation velocity. In time domain, a slow tuning by spatial light modulation with piezo-actuated or liquid-crystal spiral phase shapers or a fast tuning by intrapulse frequency chirp is possible. The experiments to be presented here were focused on the variation of bandwidth, center wavelength, grating period, and aperture diameter.  
The identification of COG positions of spectral eyes as a function of the propagation distance \(z\) provides a simple description of the helical spectral characteristics of interest. From the axial rotation cycles, which will be referred to as \"periods\", an axial angular velocity \(\Omega_{z}\) can be defined as the axial derivative of the rotation angle \(\varphi\) of the connecting line between these COG coordinates:  
\[\Omega_{z}\ =\ \frac{d\varphi}{dz} \tag{4}\]  
The related temporal angular velocity \(\Omega_{t}\) follows as  
\[\Omega_{t}\ =\ \frac{d\varphi}{dt}\ =\ c\cdot\Omega_{z} \tag{5}\]  
where \(c=z/t\) is the velocity of light in vacuum. For a constant axial angular velocity (comparable to the \"thread pitch\" of a screw), the number of axial and temporal periods \(P_{0}\) and \(T_{0}=P_{0}\)/c within a given propagation distance \(z\) is directly proportional to \(\Omega_{z}\) and \(\Omega_{t}\), respectively. The rotation angle as a function of the distance can be written as  
\[\varphi(z)\ =\ 2\pi\frac{z}{P_{0}} \tag{6}\]  
In general, however, the axial angular velocity can be a function of propagation distance \(z\) and propagation time \(t\), and variable axial and temporal periods \(P(z)\) and \(T(t)=P(z)\)/c can be determined. In this case, equation (6) cannot be applied and has to be replaced by a differential approach:  
\[\frac{d\varphi(z)}{dz}\ =\ \frac{2\pi}{P(z)} \tag{7}\]  
Because of the relationship between spatial and temporal rotation, the distance-dependent spectral Gouy rotation of light pulses in flight can be interpreted as an ultrafast temporal phenomenon correlated to dynamic spectral maps. In both cases, the actual information can be encoded in first- or higher-order derivatives of the rotation angle of the spectral anomalies. For example, radial periodicity gradients of spiral gratings can be used to modulate the axial angular velocity. To further extend the range of free parameters, rotation velocity and rotational acceleration can also be combined with targeted modifications of the statistical moments of the spectral eyes.