Fig. 1. (a) Conceptual sketch of the optical system (FST pulse shaper) required for the generation of flying structured ultrashort laser pulses, e.g., in the form of spring and multi-ring ones. In this example, a flying spring pulse in the form of a helix of circular geometry is created near the focal plane. It propagates towards the focal plane of the focusing lens, which focuses the input collimated laser pulse modulated by the SLM. The intensity distribution of the spring pulse is shown in the right panel, corresponding to the numerical propagation of the wavefield E˜0(r0) given by Eq. (1). (b) Intensity and phase distributions of two wavefield spectral-ring sets whose interference yields a flying spring pulse; its intensity peak is located at α=0 for the time t=0. In this example, the prescribed phase is Ψ1(α)=7α and Ψ2(α)=6α. The generated flying spring pulse has a phase distribution Ψ(r,α,t) described by Eq. (10), which is displayed in (b) for t=0. (c) Spatio-spectral coupling configuration: each spectral-ring set (of radii R1 and R2, with R2>R1) has a frequency-scaled radial intensity profile resulting in the sketched spectral superposition.

Fig. 2. Families of flying structured pulses with circular geometry: (a) spring, (b) closed multi-ring, and (c) open multi-ring. The expected trajectory h(α) [see Eq. (17)] of the peak intensity is represented for linear and different non-linear prescribed phase functions ΔΨ(α). Specifically, the trajectory h(α) displayed in (a) corresponds to the case of a single-helix spring pulse (N=Q2−Q1=−1) for a linear phase ΔΨL1(α)=−α and two different non-linear phase functions ΔΨNL1(α)=−α4/(2π)3 and ΔΨNL2(α)=−α−4 cos(2α). For the closed multi-ring (b) the linear phase is ΔΨL2(α)=0 and the two different non-linear phase functions are ΔΨNL3(α)=−4(α−π)2/2π and ΔΨNL4(α)=−4 cos(2α). In the case of the open multi-ring (c) the linear phase is ΔΨL2(α)=−0.5α and the non-linear phase is ΔΨNL5(α)=−0.5α−cos(2α+π/2). Note that in each case the master curve segment comprising h(α), and corresponding to a constant value n(α), is also indicated using different colors.

Fig. 3. First family of flying structured pulse: single-helix spring (Ravg=545  μm) with linear prescribed phase ΔΨL1(α)=−α. (a) Intensity and phase distributions observed at the focal plane for times t=0  fs and t=δt/2=22  fs. In the second and third rows, the displayed phase distributions correspond to a flying spring pulse with phases Ψavg(α)=19.5α and Ψavg(α)=3.5α, respectively. These numerical and analytical results are also provided in Visualization 1 showing the complete temporal evolution. (b) Volumetric xyz–plot for the intensity of this single-helix flying spring pulse at t=0. (c) Time lapse volumetric xyt–plot of the rotatory motion of the pulse peak intensity, spinning at frequency δt−1∼23  THz, at the focal plane. Since ΔΨL1(α)=−α holds for each case (Qavg=19.5 and Qavg=3.5), the rotatory motion of the intensity peak is the same, as seen in (c).

Fig. 4. First family of flying structured pulse, single-helix spring (Ravg=545  μm) with non-linear prescribed phase: (a) ΔΨNL1(α)=−α4/(2π)3 and (b) ΔΨNL2(α)=−α−4 cos(2α). The corresponding intensity and phase distributions (numerical simulation) are shown at the focal plane, for times t=0  fs and t=δ/2=22  fs (see also Visualization 2 and Visualization 3). The time lapse volumetric xyt−plot is also shown for each case. (c) Intensity peak dynamics α(t) displayed for the case of non-linear phase function ΔΨNL1(α) (orange plot line) as well as for the linear one ΔΨL1(α) (blue plot line) displayed here for direct comparison. In one period, e.g., interval [0,δt], several master curve segments [n(α)] can be involved in the intensity peak dynamics α(t) of the flying spring pulse, as indicated in (c) and (d) for ΔΨNL1(α) (orange plot lines) and ΔΨNL2(α) (violet plot lines), respectively. The angular positions of the observed intensity peak sections are indicated by colored points in (a) and (b), as well as their corresponding dynamic plots α(t) in (c) and (d). The plot for the normalized angular speed Ω(α)/Ω0 is displayed in (e) for each case.

Fig. 5. Second family of flying structured pulse: closed multi-ring set (Ravg=545  μm) with the following prescribed phase: (a) ΔΨL2(α)=0, (b) ΔΨNL3(α)=−4(α−π)2/2π, and (c) ΔΨNL4(α)=−4 cos(2α). The corresponding intensity and phase distributions observed at the focal plane (numerical simulation) are shown for times t=0  fs and t=δ/2=22  fs (see Visualization 4, Visualization 5, and Visualization 6 for the complete temporal evolution sequence in each case). The peak dynamics α(t) is displayed in the second row for each case. The corresponding time lapse volumetric xyt–plot of the rotatory motion of the pulse’s peak intensity is displayed in the third row.

Fig. 6. Third family of flying structured pulse: open multi-ring set (Ravg=545  μm) with the following prescribed phase: ΔΨL3(α)=−0.5α (a), ΔΨNL5(α)=−0.5α−cos(2α+π/2) (b). The corresponding intensity and phase distributions observed at the focal plane (numerical simulation) are shown for times t=0  fs and t=δ/2=22  fs (see Visualization 7 and Visualization 8 for the complete temporal evolution sequence). The plots of the peak dynamics α(t) and normalized angular speed Ω(α)/Ω0 are displayed for each case in (c) and (d), respectively.

Fig. 7. (a) Intensity and phase (Qavg=14.5) distributions of a flying triangular-spring pulse (a Reuleaux triangle). The peak intensity propagates along the triangle tracing the triangular helix observed in the displayed time lapse volumetric xyt–plot. See Visualization 9. In the 3D volumetric phase structure, the color represents the pulse phase evaluated at time t=0, ranging from 0 (red) to 2π (pink). Furthermore, the brightness of each color is directly proportional to the pulse intensity.

Fig. 8. Volumetric xyz–plots for the intensity and phase of some of the developed flying structured pulses, provided here to help their comparison. In the 3D volumetric phase structure, the color represents the pulse phase evaluated at time t=0, ranging from 0 (red) to 2π (pink). Furthermore, the brightness of each color is directly proportional to the pulse intensity.