Reciprocating propagation of laser pulse intensity in free space

The constant-speed straight-line propagation in free space is a basic characteristic of light. Recently, several novel spatiotemporal coupling methods, for example flying focus (or named sliding focus), are developed to control light propagation including velocity and direction. In the method of flying focus, where temporal chirp and longitudinal chromatism are combined to increase the degree of freedom for coherent control, tunable-velocities and even backward-propagation have been demonstrated. Herein, we studied the transverse and longitudinal effects of the flying focus in space and time, respectively, and found in a specific physics interval existing an unusual reciprocating propagation that was quite different from the previous result. By significantly increasing the Rayleigh length in space and the temporal chirp in time, the newly created flying focus can propagate along a longitudinal axis firstly forward, secondly backward, and lastly forward again, and the longitudinal spatial resolution for a clear reciprocation flying focus improves with increasing the temporal chirp. When this new type of light is applied in the radiation pressure experiment, a reciprocating radiation-force can be produced in space-time accordingly. This finding further extends the control of light and might enable important potential applications.


Introduction
Optical pulse propagation, including velocity and direction, is a very basic characteristic for applications like optical information/communication, laser-matter interaction, and so on [1][2][3][4][5][6]. In linear physics, an optical pulse propagates along a straight-line trajectory at the velocity of c/n, where c is the speed of light in the vacuum and n is the refractive index of the medium. In this case, the propagation velocity and/or direction can be controlled by crafting the spectrumand space-dependent refractive index [7][8][9][10][11][12]. However, this kind of method cannot be directly applied in free space, where the refractive index is constant n = 1. Another approach is by shaping beam in space or pulse in time to change the propagation velocity (e.g., superluminal or accelerating velocity) and direction (e.g., straight-line or bended trajectory) in free space [13][14][15][16][17][18][19][20][21][22][23]. For example, a Bessel beam can propagate along a straight-line trajectory at a superluminal velocity governed by c/cos in free space, where  is the half conical angle of the conical superposition [13,14]; an Airy beam can propagate along a parabolic trajectory at an accelerating superluminal group velocity in free space [15][16][17][18][19][20][21][22]; an Airy-Bessel pulsed beam (called light bullet) can propagate along a straight-line trajectory at an accelerating superluminal group velocity in a dispersion medium [23].
In the above methods, we can find that the created optical pulse/wave-packet can only propagate towards one certain direction (i.e., forward or backward), although the propagation velocity (i.e., superluminal or subluminal, and accelerating or decelerating) and trajectory (i.e., straight-line or bend) can be well controlled. In this article, we report a new phenomenon of a reciprocation FLFO, whose motion form includes three steps: forward-propagation firstly, backward-propagation secondly, and forward-propagation again lastly. For the FLFO created by longitudinal chromatism and temporal chirp [24,26], whose group velocity within the extended Rayleigh length has been well studied under the geometry optics, however the influence of the Rayleigh length that determined by the F-number is not well considered. Here, we significantly increase the Rayleigh length with respect to the focus-separation induced by longitudinal chromatism and then compare the propagation dynamics of the FLFO with the previous case. The result shows, the FLFO experiences a forward-backward-forward reciprocating propagation from the appearance to the disappearance. This unusual motion of the reciprocation FLFO may bring new potentials in applications like, particle manipulation, laser acceleration, radiation generation, and so on. In this article, we took the nanoparticle ultrafast trapping or manipulation as an example and analyzed the ST-variable radiation-force induced by this reciprocation FLFO, which may bring new opportunities to this field.

Results and discussion
Generation mechanism. Figure 1a shows the schematic of the FLFO that frequently appears in an ultrafast optical experiment. Before the iris diaphragm [not shown in Fig. 1a], the pulse duration is stretched by a dispersion material (e.g., bulk material or fiber) with positive chirp in time, and the multi-spectral wave-fronts are deformed by a chromatic-aberration telescope (e.g., confocal transmission telescope only for the center frequency) with normal dispersion in space. The result is the low-frequency component is flying at the temporal leading edge and is focused onto the location far away from the focusing lens, vice versa. The propagation of this FLFO has already been well studied, and the velocity equation has been derived by Quéré, et al. [24,25] and Froula, et al. [26,50] independently. If based on geometrical optics and linear approximation, the velocity equation can be simplified as where, z c and z f are temporal chirp and longitudinal chromatism induced longitudinal pulse-separation and focusseparation, respectively, which corresponds to the pulse bandwidth , and c is the light speed in free space. When z c /z f > 1 (i.e., z c > z f > 0 for the case of positive temporal chirp and normal spatial dispersion that is the case in this article, or z c < z f < 0 for the case of negative temporal chirp and abnormal spatial dispersion), the result of Eq. (1) is negative, showing a backward-propagation FLFO [see Fig. 1d]. The mechanism of the velocity control is that the longitudinal chromatism spatially separates the frequency-dependent focuses on the propagation axis and the temporal chirp temporally controls the time of every frequency arriving at its focus, resulting in a moving pulse-intensity-peak (i.e., FLFO). Figure 1b schematically shows: at time t 1 , both low-and high-frequencies are before their geometrical focuses, and no FLFO exists; at time t 2 , the low-frequency arrives at its geometrical focus, while the high-frequency doesn't (due to z c > z f > 0), and the FLFO appears at the ST position of the low-frequency; at time t 3 , the low-frequency has passed through its geometrical focus, while the high-frequency is now at its geometrical focus (due to z c > z f > 0), and the FLFO flies backwards to the ST position of the high-frequency; finally at time t 4 , both low-and highfrequencies have passed through their geometrical focuses, and the FLFO disappears. During this process, all frequencies move forwards at the light speed in free space c, but the FLFO flies backwards at the velocity governed by Eq. (1), which can be superluminal or subluminal. Figure 1d shows the dependence of the velocity of this FLFO with the temporal chirp induced pulse-separation z c that is normalized by the longitudinal chromatism induced focusseparation z f . temporal chirp controls moments of frequencies arriving at their focuses in time, resulting in a moving pulse-intensity-peak, i.e., a velocity tunable FLFO. b Typical backward-propagation FLFO with short Rayleigh lengths, at time t 1 all frequencies are before their focuses; at time t 2 lowfrequency arrives at its focus, while high-frequency doesn't; at time t 3 high-frequency arrives at its focus, while low-frequency leaves its focus; at time t 4 all frequencies leave their focuses. FLFO moves backwards from the focus of low-frequency to that of high-frequency from time t 2 to t 3 . z c and z f are temporal chirp and longitudinal chromatism induced on-axis pulse-separation and focus-separation corresponding to the pulse bandwidth, respectively. Here z c > z f > 0 with positive temporal chirp and normal spatial dispersion, i.e., low-frequency locates at the temporal leading edge and has a long focal length. c Reciprocation FLFO with enlarged Rayleigh lengths, at time t 1 low-frequency enters its Rayleigh length, while highfrequency doesn't; at time t 2 low-frequency arrives at its geometrical focus, while high-frequency doesn't; at time t 3 high-frequency arrives at its geometrical focus, while low-frequency leaves its geometrical focus; at time t 4 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 t 1 to t 2 , backwards from the geometrical focus of low-frequency to that of high-frequency from time t 2 to t 3 , and finally forwards again within the high-frequency's Rayleigh length from time t 3 to t 4 . d Group velocity of FLFO  FLFO as a function of z c that is normalized by z f . e On-axis location of reciprocation FLFO during ① forward-, ② backward-, and finally ③ forward-propagations with corresponding velocities  FLFO . The distance/position is normalized by z f .
In the above typical FLFO, the Rayleigh length Z R () comparing with the longitudinal chromatism induced focusseparation z f is very short and negligible [see Fig. 1b]. When dramatically increasing the Rayleigh length Z R () by reducing the beam diameter (or numerical aperture, NA) at the focusing lens, the typical backward-propagation FLFO is changed into a reciprocation FLFO. Figure 1c schematically shows: at time t 1 , the low-frequency is before its geometrical focus but has entered its Rayleigh length, while the high-frequency doesn't, showing a forwardpropagation FLFO; from time t 2 to t 3 , it becomes the typical backward-propagation FLFO; at time t 4 , the highfrequency has passed through its geometrical focus but is still within its Rayleigh length, while the low-frequency doesn't, showing a forward-propagation FLFO again. The result is the FLFO has a reciprocating motion along the propagation axis. When the origin of the propagation z-axis is defined at the midpoint of z f (i.e., the focus of the center frequency  0 for linear longitudinal chromatism), Fig. 1e shows the propagation of the reciprocation FLFO is divided into three steps: firstly, forward-propagation The two forward-propagations have a constant velocity c in free space, but the backward-propagation has a tunable velocity governed by Eq. (1) and shown by Fig. 1d. Here, because a limited bandwidth  is considered, the frequency-dependence of the Rayleigh lengths Numerical demonstration. The numerical demonstration is carried out by using the model given in the "Methods" section and the following parameters: the pulse has an 800 nm center wavelength and a 20 nm flat-top bandwidth; the focusing lens has a 400 mm (for 800 nm) focal length; the BK7 confocal telescope f 1 − f 2 for longitudinal chromatism introduces z f = 5 mm focus-separation (for 790  810 nm), when f 1 = f 2 = 50 mm (for 800 nm). The temporal chirp (i.e., z c ) is modulated for FLFOs with different velocities and longitudinal spatial resolutions, and the beam diameter is adjusted by the iris diaphragm for different Rayleigh lengths Z R . When the temporal chirp induced pulse-separation z c is 10 mm (2z f ) and the Rayleigh length Z R is 50 m (0.01z f ), Fig. 2a shows the time-integrated extended Rayleigh length within a 0.1 mm  12 mm lateral region, and Fig. 2b gives the dynamics at different propagation times t, showing a typical backward-propagation FLFO. The propagation time t in this article is defined as t = z/c, and the moment of the center frequency  0 (or the FLFO for linear temporal chirp and linear longitudinal chromatism) arriving at the space origin z = 0 (or the focus of the center frequency  0 for linear longitudinal chromatism) is the propagation time origin t = 0. Remaining the temporal chirp unchanged, when the Rayleigh length is enlarged to 0.75 mm (0.15z f ) by reducing the beam diameter from 57.2 mm to 14.8 mm, Fig. 2c shows the time-integrated extended Rayleigh length is stretched and thickened. Figure 2d shows the corresponding dynamics at different propagation times t, and the reciprocating motion of the FLFO can be found. Next, keeping the enlarged Rayleigh length (0.15z f ) unchanged, when the temporal chirp induced pulse-separation z c is increased from 10 mm (2z f ) to 50 mm (10z f ) by increasing the temporal chirp, the time-integrated extended Rayleigh length remains unchanged [see Fig. 2e], however the longitudinal spatial resolution of the FLFO is enhanced, showing a clear reciprocation FLFO [see Fig. 2f]. Figure 2 indicates two results: first, increasing the Rayleigh length (by reducing NA) is necessary for the reciprocation FLFO [see Fig. 2d]; second, dramatically increasing the temporal chirp can enhance the longitudinal spatial resolution of the reciprocation FLFO [see Fig. 2f]. The previous results show a large temporal chirp would reduce the velocity of the backward-propagation. The blue and gray spots in Fig. 1d show the backward-propagation velocities in Figs. 2b/d and 2f are −c and −0.11c, respectively, which is why the time interval during the backward-propagation in Fig. 2f is increased.
In summary, to generate a reciprocation FLFO: firstly, the temporal chirp induced pulse-separation should be longer than the longitudinal chromatism induced focus-separation z c /z f > 1 (i.e., z c > z f > 0 for the case of positive temporal chirp and normal spatial dispersion that is the case in this article, or z c < z f < 0 for the case of negative temporal chirp and abnormal spatial dispersion), generating a typical backward-propagation FLFO; secondly, the Rayleigh length Z R should be obviously enlarged Z R >> 0, showing obvious forward-propagation FLFOs before and after the backward-propagation; thirdly, the Rayleigh length should be shorter than the longitudinal chromatism induced focus-separation z f  > Z R > 0, improving the longitudinal spatial resolution of the FLFO; finally, the temporal chirp should be significantly enhanced z c /z f >> 1, further improving the longitudinal spatial resolution for a clear reciprocation FLFO. The requirement then becomes z c  >> z f  > Z R >> 0 with z c /z f > 0.  Figure 3 shows the temporal chirp z c /z f (normalized by z f ) determines both the backward-propagation velocity and the longitudinal spatial resolution of the FLFO. When the normalized temporal chirp z c /z f is larger than 10 [z c /z f = 10 is the case in Fig. 2f], the parameter M approaches infinitely to its minimum value 1, generating a clear reciprocation FLFO with almost the shortest longitudinal width or with almost the highest longitudinal spatial resolution. However, the backwardpropagation velocity of the FLFO is limited less than 0.11c in free space, which would further decrease with increasing the temporal chirp z c /z f . Noting that, the forward-propagation velocity of the FLFO will not be affected by adjusting the temporal chirp, which always is constant c in free space. Finally, the result is that a clear reciprocation FLFO exists in the interval of z c /z f  [10, ), where it has a [-0.11c, 0) subluminal backward-propagation velocity and a light-speed c forward-propagation velocity in free space. This phenomenon can be found in the simulated dynamics as shown in Fig. 2f. Another parameter affecting the longitudinal spatial resolution of the reciprocation FLFO is the Rayleigh length Z R , which generally cannot be too small (reducing the forward-propagation distance) or too large (depredating the longitudinal spatial resolution), and the simulation here indicates it is better to keep it within around 0.1z f  0.2z f .  [39][40][41][42][43], the dynamic radiation-force induced by the reciprocation FLFO may bring new possibilities. In the Rayleigh scattering regime where the particle is sufficiently smaller than the laser wavelength, Fig. 4a shows the radiation force exerted on a small dielectric sphere (treated as an induced, simple point dipole) can be divided into two components: a scattering force F scat and a gradient force F grad [42,44]. The scattering force F scat is proportional to the product a 6 I and points in the direction of the beam, and the gradient force F grad is proportional to the product a 3 I and points in the direction of the intensity gradient, where a is the radius of the sphere and I is the laser intensity. Recently, pulsed Gaussian beams also are used as optical tweezers which can produce much stronger radiation forces than those by continuous-wave Gaussian beams [45,46]. A study on the instantaneous radiation forces produced by the pulsed Gaussian beams shows that, for long pulses (typically longer than 1 ps), stable optical trapping/manipulation like by the continuous-wave Gaussian beams can be obtained [47]. Figure 2f shows, because both the temporal chirp and the Rayleigh length are enlarged for a clear reciprocation FLFO, the temporal length of the reciprocation FLFO is around 5 ps (much longer than 1 ps), and consequently the induced radiation forces F scat and F grad can be described well by the steady-state model given in the "Methods" section. Figure 4b schematically illustrates the FLFO experiences a reciprocating propagation trajectory within the extended Rayleigh length, in theory, which can produce a reciprocating radiation force in space-time. Based on the clear reciprocation FLFO shown in Fig. 2f, the induced radiation force is simulated. Figure 4c shows the spectrum of the input pulsed beam, and Fig. 4d shows the on-axis intensity distribution of the FLFO at different propagation times from its appearance (at around t = −83 ps) to its disappearance (at around t = 83 ps). Because of the subluminal velocity of the backward-propagation, the scale of the time axis (vertical axis) of Fig. 4d from t = −75 ps to t = 75 ps is reduced. When the total input energy is 1 mJ and the relative refractive index is m = 1.511/1, for a dielectric sphere radius a = 2 nm, Fig. 4e shows the on-axis gradient forces F grad , scattering force F scat , and net force F net at the propagation time t = 0. The gradient force is significantly stronger than the scattering force, and the sphere can be trapped at the position z = 0 by the net force. Figure 4f shows the dynamics of the on-axis net force F net at different propagation times from the appearance (at around t = −83 ps) to the disappearance (at around t = 83 ps), showing an ultrafast reciprocating trapping force. When the radius of the dielectric sphere is increased to a = 10 nm, Figs. 4g and 4h show the on-axis radiation forces at the propagation time t = 0 and the dynamics of the onaxis net force from t = −83 ps to t = 83 ps, respectively. The ultrafast reciprocating trapping force is now changed into an ultrafast reciprocating pushing force, because the scattering force is much larger than the gradient force that dominates the net force. The two examples given in Figs. 4f and 4h are based on small particles (dipole) and show the induced reciprocating radiation-force that may bring new possibilities to advance ultrafast optical trapping or manipulation or acceleration. Besides that, for a large particle (multipoles), because backward scattering forces exist [48], once it is combined with the reciprocation FLFO, an ultrafast reciprocating trapping force for multipoles may also be achieved. Longitudinal axial (at z-axis) intensity of FLFO at different propagation times t from appearance to disappearance, showing reciprocating propagation. Because the forward-propagation velocity c is much faster than the backward-propagation velocity -0.11c, the propagation time axis (vertical axis) is zoomed for the forward-propagation. For a very small Rayleigh dielectric sphere (2 nm radius) in the air, e induced longitudinal axial (at z-axis) radiation forces at the propagation time t = 0, and f induced ultrafast reciprocating trapping forces at different propagation times t from appearance to disappearance. The net force is F net = F grad + F scat . For a relatively big Rayleigh dielectric sphere (10 nm radius) in the air, g induced longitudinal axial (at z-axis) radiation forces at the propagation time t = 0, and h induced ultrafast reciprocating pushing forces at different propagation times t from appearance to disappearance.

Fig. 2. Numerical demonstration and comparison. When a, b Z
In addition to optical radiation pressure experiments, the ultra-intense traditional FLFO with superluminal or subluminal, accelerating or decelerating, and forward-or backward-propagation velocities recently have been used in laser-plasma physics applications like dephasingless/phase-locked laser-wakefield accelerator, arbitrary-velocity ionization wave generation, photon accelerator, laser-plasma amplifier, and so on [3-6, 26, 27, 49, 50]. The reciprocation FLFO introduced in this article, because of its new characteristics, can further enhance the controllability of the laser pulse intensity in space-time and may also provide some new opportunities in this field.

Conclusion
In this work, we have introduced a new type of laser pulse intensity propagation or a new type of optical control, i.e., the reciprocation FLFO. Based on the recently reported FLFO (or named sliding focus) that created by combining temporal chirp and longitudinal chromatism together [24][25][26][27], when both the Rayleigh length and the temporal chirp are dramatically increased, different from the previous result the produced FLFO would present an unprecedented motion form: flying forward-backward-forward along a straight-line in free space, showing a longitudinal reciprocating trajectory. The existence condition of a clear reciprocation FLFO with a high longitudinal spatial resolution is also analyzed and introduced. In free space, a clear reciprocation FLFO has a light-speed forwardpropagation velocity c and a subluminal backward-propagation velocity (typically −0.11c <  FLFO < 0), respectively.
We have shown when this unique light is applied in radiation pressure experiments in the Rayleigh scattering regime, for different particle radiuses an ultrafast reciprocating trapping or pushing force can be induced in space-time. Moreover, in laser-plasma physics, the ultra-intense reciprocation FLFO can further extend the performance of the traditional FLFO. All in all, this new type of light may provide useful and unique applications from nano-optics to high-field optics.

Methods
Numerical simulation of reciprocation FLFO. The propagation of the FLFO in a paraxial cylindrical coordinate system r--z can be simulated by the Collins diffraction integral, and the optical field after an ABCD system is given by where, f() is the frequency-dependent focal length after the focusing lens, and z 0 and z is the input and output onaxis position, respectively. The input optical field is given by where, A() is the spectral amplitude,  2 is the group-velocity dispersion (GVD) denoting the temporal chirp, w in is the beam waist, and z(r 1 ,) is the frequency-dependent wave-front denoting the longitudinal chromatism. For example, when a transmission telescope is used for generating the longitudinal chromatism, z(r 1 ,) is a quadratic function with respect to the transverse coordinate z(r 1 ,) = a()r 1 2 + b, where a() is a frequency-dependent coefficient and b is a constant-coefficient [51][52][53][54]. After the Fourier-transform, the output temporal optical field is given by where  is the local time. By choosing different propagating positions z, the dynamics of the FLFO at different propagation times t (t = z/c) can be obtained. The numerical simulation can show the dynamics of FLFOs well however cannot give the mathematical influence of the Rayleigh length (i.e., normalized as Z R /z f ) and the temporal chirp (i.e., normalized as z c /z f ) directly. In this case, an approximately analytical formula is also required.
Analytical formula of reciprocation FLFO. In the theory of linear pulse propagation [55], a deeply chirped Gaussian pulse can be described by a complex Gaussian parameter  = a−ib, where, a and b here are constant coefficients denoting the pulse duration and the temporal chirp, respectively. The instantaneous frequency of this pulse at time t is (t) =  0 +2bt, which shows the temporal profile of a deeply chirped Gaussian pulse can be approximately described by its spectral profile, i.e., I(t)  I(). In this case, the dynamics of a deeply chirped FLFO in space-time can be approximately described by that in space-spectrum [56].

Figure 1
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