Fig. 1: Generation mechanism and characteristic of reciprocating FLFO.
From: Reciprocating propagation of laser pulse intensity in free space
a Flying focus (FLFO) is formed by coupling of longitudinal chromatism and temporal chirp within an extended Rayleigh length. Longitudinal chromatism separates frequency-dependent foci in space, and temporal chirp controls moments of frequencies arriving at their foci in time, resulting in a moving pulse-intensity-peak, i.e., a velocity tunable FLFO. b Typical backward-propagating FLFO with short Rayleigh lengths, at time t1 all frequencies are before their foci; at time t2 low-frequency arrives at its focus, while high-frequency does not; at time t3 high-frequency arrives at its focus, while low-frequency leaves its focus; at time t4 all frequencies leave their foci. FLFO moves backwards from the focus of low-frequency to that of high-frequency from time t2 to t3. Lc and Lf are temporal chirp and longitudinal chromatism induced on-axis pulse-separation and focus-separation corresponding to the pulse bandwidth, respectively. Here Lc < Lf < 0 (along the negative direction of the z-axis) with positive temporal chirp and concave pulse-front curvature, i.e., low-frequency locates at the temporal leading edge and has a long focal length. c Reciprocating FLFO with enlarged Rayleigh lengths, at time t1 low-frequency enters its Rayleigh length, while high-frequency does not; at time t2 low-frequency arrives at its geometrical focus, while high-frequency does not; at time t3 high-frequency arrives at its geometrical focus, while low-frequency leaves its geometrical focus; at time t4 high-frequency is still in its Rayleigh length, while low-frequency leaves its Rayleigh length. FLFO moves forwards within the low-frequency’s Rayleigh length from time t1 to t2, backwards from the geometrical focus of low-frequency to that of high-frequency from time t2 to t3, and finally forwards again within the high-frequency’s Rayleigh length from time t3 to t4. d Group velocity of FLFO υFLFO as a function of Lc/Lf. e On-axis location of reciprocating FLFO during ①forward-, ② backward-, and finally ③ forward-propagations with corresponding velocities υFLFO. ZR is the Rayleigh length, and lengths and positions are normalized by |Lf | .
