Fig. 1. Envelope of a focused laser pulse at different points in time along its path. The laser pulse enters the focusing geometry from the top right, traveling towards the focusing mirror below. The input pulse is under the influence of angular dispersion and, thus, has a small pulse-front tilt before focusing. Due to , spatial dispersion develops during propagation by distance to the focusing off-axis parabola (OAP). At the OAP, the pulse is deflected by and then propagates the parabola’s effective focal distance down to the focus. Details of the pulse properties depicted further downstream assume and omit pulse-front curvature. During propagation into the focus, pulse-front tilt grows and reaches a maximum some distance ahead of the focus. Then it reduces and again equals its initial value in the focus. After the focus, this pulse-front rotation continues such that the tilt becomes zero shortly behind the focus and in the following becomes opposite in direction compared to the tilt before focusing. Also during focusing, the transverse offset of frequencies from the propagation axis grows in relation to the pulse’s width during propagation from the OAP to the focus. However, the effect of propagation with angular dispersion on the value of spatial dispersion is negligible. It remains almost constant at the focal value throughout propagation. After the focus, pulse-front rotation continues until the tilt reaches a maximum, before it falls off again.

Fig. 2. Frequency–space domain visualization of the paths of two specific frequencies belonging to the spectrum of a Gaussian pulse that is under the influence of angular dispersion and spatial dispersion. These frequencies are transversally Gaussian distributed, and the rays represent the path of the respective distribution center. The pulse’s propagation direction is defined by the propagation direction of the central frequency . The propagation direction of frequency encloses the angle with the central frequency’s propagation direction in the focal plane. This expresses immanent angular dispersion of the focusing Gaussian pulse, which can originate from both angular dispersion and spatial dispersion before the focusing off-axis parabola. In the focal plane , the spatial offset between the centers of beams and along the transverse direction expresses immanent spatial dispersion of the Gaussian pulse, which originates from angular dispersion before the off-axis parabola.

Fig. 3. Propagation of rays of different frequency during focusing of a laser pulse at an OAP. The central frequency’s incident ray (orange) propagates parallel to the axis of the OAP. The incidence plane is perpendicular to the ray and located at the point of incidence of the ray on the OAP surface. The ray encloses with the OAP’s surface normal the angle , which determines the angle of deflection . During subsequent propagation into the focus, the central frequency ray covers the effective focal distance . The focal plane is perpendicular to the central frequency ray and located in the OAP’s focus. A second ray belonging to frequency (green) encloses the angle with the central frequency ray and has a transverse spatial offset of at the incidence plane. The propagation angle is negative in this setup. Compared to the central frequency ray, the second ray propagates an additional distance until it is incident on the mirror surface. Its deflection angle , effective focal distance , propagation angle and propagation distance until the focal plane differ from the central frequency ray. The point where the second ray pierces the focal plane defines its transverse spatial offset .

Fig. 4. Pulse-front tilt and pulse duration in the course of propagation of a μm, fs, mm laser pulse through the focus of the short focal range setup without spatial dispersion before the focusing mirror. The colors of the lines represent angular dispersion values before focusing μrad/nm. Originating from , there is angular dispersion, and hence pulse-front tilt, in the focus . Correspondingly, the position of zero pulse-front tilt along the beamline is outside the focus, as shown in the inset. Since absolute values of pulse-front tilt in the focus are below for all values of , this offset is negligible in practice for this particular example.

Fig. 5. Pulse-front tilt and pulse duration in the course of propagation through the focus of the long focal range setup without spatial dispersion before focusing. Parameters are equal to the short focal range setup (see Figure 4).

Fig. 6. Distribution of the time–space domain intensity envelope along the transverse direction and time at different distances from the focus. Pulse parameters are equal to Figure 4 with μrad/nm. All distributions are normalized to the respective expected maximum value in the focus , cf. Equation (6). Colored lines mark pulse-front contours as expected from analytic and numeric determination of the pulse-front tilt angle, Equations (18) and (27), respectively. In addition, the duration of the field envelope is provided, which is obtained from the least square fit of a Gaussian curve to the 1D intensity distribution along .