Abstract
Constantspeed straightline propagation in free space is a basic characteristic of light, and spatiotemporal couplings recently were used to control light propagation. In the method of flying focus, where temporal chirp and longitudinal chromatism were combined, tunablevelocities and even backwardpropagation were demonstrated. We studied the transverse and longitudinal effects of the flying focus in spacetime and found in a specific physics interval existing an unusual reciprocating propagation that was quite different from the previous result. By increasing the Rayleigh length in space and the temporal chirp in time, the created flying focus can propagate along a longitudinal axis firstly forward, secondly backward, and lastly forward again, and the longitudinal spatial resolution improves with increasing the temporal chirp. When this light is applied in a radiation pressure simulation, a reciprocating radiationforce can be produced accordingly. This finding extends the control of light and might enable important potential applications.
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Introduction
Optical pulse propagation, including velocity and direction, is a very basic characteristic for applications like optical information/communication, lasermatter interaction, and so on^{1,2,3,4,5,6}. In linear physics, an optical pulse propagates along a straightline trajectory at the group velocity of c/n_{g}, where c is the speed of light in the vacuum and n_{g} is the group refractive index in the medium. In this case, the propagating velocity and/or direction can be controlled by crafting the spectrum and spacedependent refractive index n_{λ}(x, y, z)^{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 propagating velocity (e.g., superluminal or accelerating velocity) and direction (e.g., straightline 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 straightline trajectory at a superluminal velocity governed by c/cosα in free space, where α 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 AiryBessel pulsed beam (called light bullet) can propagate along a straightline trajectory at an accelerating superluminal group velocity in a dispersion medium^{23}.
Currently, spatiotemporal (ST) couplings are frequently used to modulate the propagation or structure of a pulsed beam, which permits both velocity control (i.e., superluminal or subluminal, and accelerating or decelerating) and direction control (i.e., forward or backward)^{24,25,26,27,28,29,30,31,32,33,34,35,36,37,38}. The first example is the threedimensional (3D) flying focus (FLFO) within the extended Rayleigh length independently demonstrated by Quéré, et al.^{24,25} (originally named “sliding focus”) and Froula et al.^{26,27} (originally named “flying focus”), respectively, which can propagate at an arbitrary group velocity in free space including all motion forms of superluminal or subluminal, accelerating or decelerating, and forward or backward propagations. The second example is the twodimensional (2D) optical ST wavepacket demonstrated by Abouraddy et al.^{28,29,30,31,32,33,34}, which can also propagate at an arbitrary group velocity in free space, including all above motion forms. The third example is the 3D ST GaussBessel pulsedbeam (or 3D optical wavepacket) created by predeforming the pulsefront of the input Gauss pulsedbeam that is for the generation of a GaussBessel pulsedbeam^{35,36}, similarly whose group velocity is tunable, too. Apart from the above motion forms (i.e., superluminal or subluminal, accelerating or decelerating, and forward or backward propagations), a compound motion with several different motion forms in a single propagation path is also possible^{36}.
In the above methods, we can find that the created optical pulse/wavepacket can only propagate towards one certain direction (i.e., forward or backward), although the propagating velocity (i.e., superluminal or subluminal, and accelerating or decelerating) and trajectory (i.e., straightline or bend) can be well controlled. In this article, we report a phenomenon of a reciprocating FLFO, whose motion form includes three steps: forwardpropagation firstly, backwardpropagation secondly, and forwardpropagation 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 geometrical optics, however, the influence of the Rayleigh length as determined by the Fnumber is not well considered.
Here, we significantly increase the Rayleigh length with respect to the focusseparation induced by longitudinal chromatism and then compare the propagating dynamics of the FLFO with the previous case. The result shows, the FLFO experiences a forwardbackwardforward reciprocating propagation from the appearance to the disappearance. This motion form of the reciprocating FLFO may bring 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 STvariable radiationforce induced by this reciprocating FLFO, which may bring 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 focusing optics, the pulse duration is stretched by a dispersion element (e.g., grating pair, bulk material or fiber, etc.) with temporal chirp in time, and the spectrumdependent wavefronts are deformed by a chromatic element [e.g., confocal (only for the center frequency) transmission telescope] with longitudinal chromatism in space (two elements are not shown in Fig. 1a). As illustrated in Fig. 1a, the lowfrequency component is flying at the temporal leading edge and is focused onto the location far away from the focusing optics, 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.^{26,39} independently. If based on geometrical optics and linear approximation, the velocity equation can be simplified as
with
where, L_{f} and L_{c} are total focusseparation and total pulseseparation (corresponding to the pulse bandwidth Δω) on the propagation axis induced by longitudinal chromatism and temporal chirp, respectively, f is the focal length of the focusing optics, α is a parameter describing the pulsefront curvature (PFC) or named radial group delay (RGD) of GD(r) = αr^{2} in the nearfield induced by longitudinal chromatism^{24,25}, c is the lightspeed in free space, ω_{0} is the center frequency, and ϕ_{2} is the groupvelocity dispersion (GVD). It is important to note that L_{f} and L_{c} have directions, which are defined from the lowestfrequency (“red” light) to the highest frequency (“blue” light) on the propagation axis. The derivations of Eqs. (1), (2), and (3) are given in Supplementary Information Notes 3, 2, and 1, respectively. When L_{c}/L_{f} > 1 (i.e., L_{c} < L_{f} < 0 for the case of positive temporal chirp ϕ_{2} > 0 and concave PFC α < 0 that is the case in this article, or L_{c} > L_{f} > 0 for the case of negative temporal chirp ϕ_{2} < 0 and convex PFC α > 0), the result of Eq. (1) is negative, showing a backwardpropagating FLFO [see Fig. 1d]. The mechanism of the FLFO velocity control is that the longitudinal chromatism spatially separates the frequencydependent foci on the propagation axis and the temporal chirp temporally controls the time of every frequency arriving at its focus, resulting in a moving pulseintensitypeak (i.e., FLFO). Figure 1b schematically shows: at time t_{1}, both low and highfrequencies are before their geometrical foci, and no FLFO exists; at time t_{2}, the lowfrequency arrives at its geometrical focus, while the highfrequency does not (due to L_{c} < L_{f} < 0), and the FLFO appears at the ST position of the lowfrequency; at time t_{3}, the lowfrequency has passed through its geometrical focus, while the highfrequency is now at its geometrical focus (due to L_{c} < L_{f} < 0), and the FLFO flies backwards to the ST position of the highfrequency; finally at time t_{4}, both low and highfrequencies have passed through their geometrical foci, and the FLFO disappears. During this process, all frequencies move forwards at the lightspeed 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 pulseseparation L_{c} that is normalized by the longitudinal chromatism induced focusseparation L_{f}.
In the above typical FLFO, the Rayleigh length Z_{R}(ω) comparing with the longitudinal chromatism induced focusseparation L_{f} is very short and negligible [see Fig. 1b]. When dramatically increasing the Rayleigh length Z_{R}(ω) by reducing the beam diameter (or numerical aperture, NA) at the focusing optics, the typical backwardpropagating FLFO is changed into a reciprocating FLFO. Figure 1c schematically shows: at time t_{1}, the lowfrequency is before its geometrical focus but has entered its Rayleigh length, while the highfrequency does not, showing a forwardpropagating FLFO; from time t_{2} to t_{3}, it becomes the typical backwardpropagating FLFO; at time t_{4}, the highfrequency has passed through its geometrical focus but is still within its Rayleigh length, while the lowfrequency does not, showing a forwardpropagating FLFO again. The result is the FLFO has a reciprocating motion on the propagation axis. When the origin of the propagation axis (zaxis) is defined at the midpoint of L_{f} (i.e., the geometrical focus of the center frequency ω_{0} for linear longitudinal chromatism), Fig. 1e shows the propagation of the reciprocating FLFO is divided into three steps: firstly, forwardpropagation from z = L_{f}  /2−Z_{R} to z = L_{f}  /2; secondly, backwardpropagation from z = L_{f}  /2 to z = −L_{f}  /2; thirdly, forwardpropagation from z = −L_{f}  /2 to z = −L_{f}  /2+Z_{R}. The two forwardpropagations have a constant velocity c in free space, but the backwardpropagation has a tunable velocity governed by Eq. (1) and shown by Fig. 1d. Here, because a limited bandwidth Δω is considered, the frequencydependence of the Rayleigh lengths Z_{R}(ω) is neglected, and then Z_{R}(ω) is represented by Z_{R}.
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 flattop bandwidth; the focusing optics has a 400 mm (for the center wavelength 800 nm) focal length; the longitudinal chromatism is generated by a BK7 confocal (only for the center wavelength 800 nm) telescope f_{1} − f_{2} with a L_{f} = −5 mm focusseparation (from the longest wavelength 810 nm to the shortest wavelength 790 nm), when f_{1} = f_{2} = 50 mm (for the center wavelength 800 nm). The temporal chirp (i.e., L_{c}) is modulated for FLFOs with different velocities and longitudinal spatial resolutions, and the beam diameter is adjusted for different Rayleigh lengths Z_{R}. When the temporal chirp induced pulseseparation is L_{c} = −10 mm (2L_{f}) (from the longest wavelength 810 nm to the shortest wavelength 790 nm) and the Rayleigh length is Z_{R} = 50 μm (0.01L_{f}  ), Fig. 2a shows the timeintegrated extended Rayleigh length within a 0.1 × 12 mm lateral region, and Fig. 2b gives the dynamics at different propagating times t, showing a typical backwardpropagating FLFO. The propagating time t in this article is defined as t = z/c, and the time of the center frequency ω_{0} (i.e., the FLFO for linear temporal chirp and linear longitudinal chromatism) arriving at the space origin z = 0 (i.e., the geometrical focus of the center frequency ω_{0} for linear longitudinal chromatism) is defined as the propagating time origin t = 0. Keeping the temporal chirp unchanged, when the Rayleigh length is enlarged from 50 μm (0.01L_{f}  ) to 0.75 mm (0.15L_{f}  ) by reducing the beam diameter from 57.2 to 14.8 mm, Fig. 2c shows the timeintegrated extended Rayleigh length is stretched and thickened. Figure 2d shows the corresponding dynamics at different propagating times t, and the reciprocating motion of the FLFO can be found. Next, keeping the enlarged Rayleigh length (0.15L_{f}  ) unchanged, when the temporal chirp induced pulseseparation L_{c} is enlarged from −10 mm (2L_{f}) to −50 mm (10L_{f}) by increasing the temporal chirp, the timeintegrated extended Rayleigh length remains unchanged [see Fig. 2e], however, the longitudinal spatial resolution of the FLFO is enhanced, showing a clear reciprocating FLFO [see Fig. 2f]. Figure 2 indicates two results: first, increasing the Rayleigh length (by reducing NA) is necessary for the reciprocating FLFO [see Fig. 2d]; second, dramatically increasing the temporal chirp can enhance the longitudinal spatial resolution of the reciprocating FLFO [see Fig. 2f]. The previous results show a large temporal chirp would reduce the velocity of the backwardpropagation. The blue and gray spots in Fig. 1d show the backwardpropagating velocities in Fig. 2b/d and 2f are −c and −0.11c, respectively, which is why the time interval during the backwardpropagation in Fig. 2f is increased.
In summary, to generate a reciprocating FLFO: firstly, the temporal chirp induced pulseseparation should be longer than the longitudinal chromatism induced focusseparation L_{c}/L_{f} > 1 (i.e., L_{c} < L_{f} < 0 for the case of positive temporal chirp ϕ_{2} > 0 and concave PFC α < 0 that is the case in this article, or L_{c} > L_{f} > 0 for the case of negative temporal chirp ϕ_{2} < 0 and convex PFC α > 0), generating a typical backwardpropagating FLFO; secondly, the Rayleigh length Z_{R} should be obviously enlarged Z_{R} ≫ 0, showing obvious forwardpropagating FLFOs before and after the backwardpropagation; thirdly, the Rayleigh length should be shorter than the longitudinal chromatism induced focusseparation L_{f} > Z_{R} > 0, improving the longitudinal spatial resolution of the FLFO; finally, the temporal chirp should be significantly enhanced L_{c}/L_{f} ≫ 1, further improving the longitudinal spatial resolution for a clear reciprocating FLFO. The requirement then becomes L_{c} ≫ L_{f} > Z_{R} ≫ 0 with L_{c}/L_{f} > 0.
Spectral effect
The above numerical demonstration is based on a flattop spectrum. As the dynamics of the FLFO and the spectral profile of the pulse have a strong correlation, it is necessary to discuss the spectral effect. Keeping the parameters used in Fig. 2e, f unchanged, when the spectral profile is changed from flattop into supperGaussian [see Fig. 3a], Fig. 3b, c show the timeintegrated extended Rayleigh length and the dynamics of the FLFO at different propagating times t. The intensity and the ST position of the forwardpropagating FLFOs before and after the backwardpropagation are obviously reduced and slightly changed, respectively, which actually is caused by the “gentle” rising and falling edges of the spectrum. For observing the weak forwardpropagating FLFOs, the color scale of Fig. 3 is adjusted. Next, when the spectral profile is changed into saddleshape [see Fig. 3d], Fig. 3e shows the timeintegrated extended Rayleigh length is insensitive to this change, however, Fig. 3f shows the dynamics of the FLFO at different propagating times t obviously change because the instantaneous FLFO corresponds to different frequencies at different propagating times/positions. When comparing Fig. 3c, f with Fig. 2f, “steep” rising and falling edges of the spectrum are also required for a clear reciprocating FLFO. Moreover, the propagating dynamics of the FLFO can be freely controlled by modulating the spectral profile, which has already been introduced in the previous work^{26}.
Existence of clear reciprocating FLFO
The analytical formulas given in the “Methods” section [see Eq. (18)] show the length (the full width at half maximum, FWHM) ΔL of the reciprocating FLFO is proportional to a product M·Z_{R}, i.e., ΔL ∝ M·Z_{R}, where M = L_{c}/(L_{c} − L_{f}). When the longitudinal spatial resolution of the FLFO is defined as the ratio between the FLFO length and the longitudinal chromatism induced focusseparation R_{FLFO} = ΔL/L_{f}, the longitudinal spatial resolution R_{FLFO} ∝ M·Z_{R}/L_{f} improves with increasing the temporal chirp L_{c} (i.e., with reducing the parameter M), and this agrees well with the simulation results in Fig. 2d, f. Figure 4 shows the temporal chirp L_{c}/L_{f} (i.e., normalized by L_{f}) determines both the backwardpropagating velocity υ_{FLFO} and the longitudinal spatial resolution R_{FLFO} of the FLFO. When the normalized temporal chirp L_{c}/L_{f} is larger than 10 (L_{c}/L_{f} = 10 is the case in Fig. 2f), the parameter M approaches infinitely to its minimum value 1, where a clear reciprocating FLFO with the shortest length ΔL and the best longitudinal spatial resolution R_{FLFO} can be observed. However, the backwardpropagating velocity of the FLFO is limited less than 0.11c in free space, which would further decrease with increasing the temporal chirp L_{c}/L_{f}. Noting that, the forwardpropagating 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 reciprocating FLFO exists in the interval of L_{c}/L_{f} ∈ [10, ∞), where it has a [−0.11c, 0) subluminal backwardpropagating velocity and a lightspeed c forwardpropagating 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 R_{FLFO} of the reciprocating FLFO is the Rayleigh length Z_{R}, which generally cannot be too small (reducing the forwardpropagation distance) or too large (degrading the longitudinal spatial resolution), and the simulation here indicates it is better to keep it within around 0.1L_{f}  ∼ 0.2L_{f}  .
Possible applications
The propagation form of the reciprocating FLFO provides a tool for laser pulse intensity control in spacetime. For applications like optical trapping/manipulating/accelerating small particles with the laser radiation pressure developed by Ashkin et al., especially in biology, medicine, and nanoscience^{40,41,42,43,44}, the dynamic radiationforce induced by the reciprocating FLFO may bring some possibilities. In the Rayleighscattering regime where the particle is sufficiently smaller than the laser wavelength, Fig. 5a shows the radiationforce 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}^{43,45}. 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 radiationforces than those by continuouswave Gaussian beams^{46,47}. A study on the instantaneous radiationforces produced by the pulsed Gaussian beams shows that, for long pulses (typically longer than 1 ps), stable optical trapping/manipulation like by the continuouswave Gaussian beams can be obtained^{48}. Figure 2f shows, because both the temporal chirp and the Rayleigh length are enlarged for a clear reciprocating FLFO, the temporal length of the reciprocating FLFO is around 5 ps (much longer than 1 ps), and consequently the induced radiationforces F_{scat} and F_{grad} can be described well by the steadystate model given in the “Methods” section. Figure 5b schematically illustrates the FLFO experiences a reciprocating propagation trajectory within the extended Rayleigh length, in theory, which can produce a reciprocating radiationforce in spacetime. Based on the clear reciprocating FLFO shown in Fig. 2f, the induced radiationforce is simulated. Figure 5c shows the flattop spectrum of the pulse, and Fig. 5d shows the onaxis intensity distribution of the FLFO at different propagating times from its appearance (at around t = −83 ps) to its disappearance (at around t = 83 ps). Because of the subluminal velocity of the backwardpropagation, the scale of the time axis (vertical axis) of Fig. 5d 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. 5e shows the onaxis gradient force F_{grad}, scattering force F_{scat}, and net force F_{net} (F_{grad} + F_{scat}) at the propagating 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 5f shows the dynamics of the onaxis net force F_{net} at different propagating times from the appearance (at around t = −83 ps) to the disappearance (at around t = 83 ps), showing an ultrafastreciprocating trapping force. When the radius of the dielectric sphere is increased from a = 2 nm to a = 10 nm, Fig. 5g, h show the onaxis radiationforces at the propagating time t = 0 and the dynamics of the onaxis net force from t = −83 ps to t = 83 ps, respectively. The ultrafastreciprocating trapping force is now changed into an ultrafastreciprocating pushing force, because the scattering force is much stronger than the gradient force that dominates the net force. The two examples given in Fig. 5f, h are based on small particles (dipole) and show the induced reciprocating radiationforce that may bring some possibilities to advance ultrafast optical trapping or manipulation or acceleration. For large particles (multipoles), because backwardscattering forces exist^{49}, once it is combined with the reciprocating FLFO, an ultrafastreciprocating trapping force for multipoles might be possible, too. Besides that, the above simulation is based on a flattop spectrum, as illustrated in Fig. 3, the spectral profile can influence the dynamics of the FLFO as well as the induced instantaneous radiationforce, of course, which can also provide opportunities for active control.
In addition to optical radiation pressure experiments, the ultraintense traditional FLFO with superluminal or subluminal, accelerating or decelerating, and forward or backwardpropagating velocities recently have been used in laserplasma physics applications like dephasingless/phaselocked laserwakefield accelerator, arbitraryvelocity ionization wave generation, photon accelerator, laserplasma amplifier, and so on^{3,4,5,6,26,27,39,50}. The reciprocating FLFO introduced in this article, because of its characteristics, can further enhance the controllability of the laser pulse intensity in spacetime and may also provide some opportunities in this field.
Conclusion
In this work, we have introduced a type of laser pulse intensity propagation or control, i.e., the reciprocating FLFO. Based on the recently reported FLFO (or named sliding focus) that is 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 a different motion form: flying forward–backward–forward along a straightline in free space, showing a longitudinalreciprocating trajectory. The existence condition of a clear reciprocating FLFO with a high longitudinal spatial resolution is also analyzed and introduced. In free space, a clear reciprocating FLFO has a lightspeed forwardpropagating velocity c and a subluminal backwardpropagating velocity (typically −0.11c < υ_{FLFO} < 0), respectively. We have shown when this unique light is applied in radiation pressure simulations in the Rayleighscattering regime, for different particle radiuses an ultrafastreciprocating trapping or pushing force can be induced in spacetime. Moreover, in laserplasma physics, the ultraintensereciprocating FLFO can further extend the performance of the traditional FLFO. All in all, this light may provide useful applications from nanooptics to highfield optics.
Methods
Numerical simulation of reciprocating FLFO
The propagation of the FLFO in a paraxial cylindrical coordinate system rφz can be simulated by the Collins diffraction integral, and the optical field at the output of the ABCD system (i.e., spatial focusing and free propagation) is given by
with
where, f(ω) is the frequencydependent focal length of the focusing optics (or a constant f for a focusing mirror), and z_{1} and z_{2} is the input and output onaxis position, respectively. The optical field at the input of the ABCD system is given by
where, A(ω) is the spectral amplitude, ϕ_{2} is GVD denoting the temporal chirp, w_{in} is the input beam waist, and Δz(r_{1},ω) is the frequencydependent wavefront denoting the longitudinal chromatism. For example, when a transmission telescope is used for generating the longitudinal chromatism, Δz(r_{1},ω) is a quadratic function with respect to the transverse coordinate Δz(r_{1},ω) = a(ω)r_{1}^{2} + b, where a(ω) is a frequencydependent coefficient and b is a constantcoefficient^{51,52,53,54}, and accordingly PFC or named RGD of GD(r_{1}) = αr_{1}^{2} will be produced in the nearfield^{25}. After the Fouriertransform, the output temporal optical field is given by
where τ is the local time. By choosing different propagating positions z_{2}, the dynamics of the FLFO at different propagating 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}/  L_{f}  ) and the temporal chirp (i.e., normalized as L_{c}/L_{f}) directly. In this case, an approximately analytical description is also required.
Analytical description of reciprocating FLFO
In the theory of linear pulse propagation^{55}, the temporal profile of a deeply chirped pulse can be approximately described by its spectral profile, i.e., I(t) ∝ I(ω). In this case, the dynamics of a deeply chirped FLFO in spacetime can be approximately described by that in spacespectrum^{56}.
Figure 6 shows in the lateral plane of the coordinate system rz, the space origin z = 0 is at the geometrical focus of the beam for the center frequency ω_{0}, and the time origin t = 0 is the time of the center frequency ω_{0} arriving at the space origin z = 0. Then, the instantaneous position of the center frequency is at z = tc. In space, the longitudinal chromatism moves the geometrical focus of the beam for an arbitrary frequency ω = ω_{0} + δω to
which is similar to Eq. (2) and given by SainteMarie et al.^{24} and Jolly et al.^{25}. In spacetime, the temporal chirp moves the instantaneous position of an arbitrary frequency ω = ω_{0} + δω to
where, the first term on the right side is the relative shift with respect to the center frequency [similar to Eq. (3)], and the second term on the right side is the instantaneous position of the center frequency. z_{f} is a function of frequency ω, while z_{c} is a function of both frequency ω and propagating time t. When z_{c} = z_{f}, different frequencies ω_{f} = ω_{0} + δω_{f} arrive at their geometrical foci at times
which shows a linear relationship between the appearance time of the backwardpropagating FLFO with the frequency.
We assume the beam for every frequency has a Gaussian profile. Then, an arbitrary frequency ω at an arbitrary propagating time t is at an instantaneous position z_{c} and has an instantaneous beam radius w_{z}, which is given by
with
where, r is the transverse coordinate, I_{0}(ω) is the spectral intensity, w_{0} is the beam waist, z_{f} is the geometrical focus (waist) position, Z_{R} is the Rayleigh length, k is the wavenumber, f is the focal length of the focusing optics, and w_{in} is the input beam radius at the focusing optics. w_{0} and Z_{R} are functions of frequency ω, while w_{z} and I(r, w_{z}) are functions of both frequency ω and propagating time t.
From Eq. (12), the onaxis intensity distribution of FLFO can be simplified as
with
where, ω_{r} and ω_{b} are the lowest and the highest frequencies within the pulse bandwidth Δω, and ω_{f} is a variable frequency arriving at its geometrical focus at the time t_{f} during the backwardpropagation and is governed by Eq. (11). Equations (16) and (17) show in three propagation stages FLFO overlaps with different frequencies, where z_{c} − z_{f}  has its minimum values and I_{FLFO} has the corresponding maximum values, i.e., the movement of FLFO. Equations (16) and (17) (and Fig. 1c) show: in the first forwardpropagation, only the lowestfrequency ω_{r} (“red” light) enters its Rayleigh length [i.e., minimum z_{c} − z_{f}  only for ω_{r}, see Eq. (17)’s first formula], and FLFO overlaps with the lowestfrequency ω_{r} and propagates forwards at the velocity c; in the backwardpropagation, different frequencies ω_{f} from the lowest one ω_{r} to the highest one ω_{b} arrive at their geometrical foci in turn [i.e., minimum z_{c} − z_{f}  = 0 for different frequencies ω_{f}, see Eq. (17)’s second formula], and FLFO overlaps with different frequencies ω_{f} (from the lowest one ω_{r} to the highest one ω_{b}) at different propagating times t_{f} governed by Eq. (11) and propagates backwards at the velocity υ_{FLFO} governed by Eq. (1); in the second forwardpropagation, only the highest frequency ω_{b} (“blue” light) is still in its Rayleigh length [i.e., minimum z_{c} − z_{f}  only for ω_{b}, see Eq. (17)’s third formula], and FLFO overlaps with the highest frequency ω_{b} and propagates forwards at the velocity c. It is important to note that, as shown in Fig. 1, Eq. (17) is based on the case of positive temporal chirp ϕ_{2} > 0 (i.e., L_{c} < 0) and concave PFC α < 0 (i.e., L_{f} < 0), if for the other case of negative temporal chirp ϕ_{2} < 0 (i.e., L_{c} > 0) and convex PFC α > 0 (i.e., L_{f} > 0), the parameters ω_{r} and ω_{b} in Eq. (17) should be interchanged.
By substituting Eq. (17) with Eq. (16), once neglecting the overall spectral profile, the first forwardpropagating FLFO has a positiondependent increasing peak intensity of [0.5, 1) from z = L_{f}  /2 − Z_{R} to L_{f}  /2; the backwardpropagating FLFO has a constant peak intensity of 1 from z = L_{f}  /2 to −L_{f}  /2; the second forwardpropagating FLFO has a positiondependent decreasing peak intensity of (1, 0.5] from z = −L_{f}  /2 to −L_{f}  /2 + Z_{R}.
From Eqs. (16) and (17), the FWHM length ΔL of FLFO is derived and given by
and Supplementary Information Note 4 gives the detailed derivations. The first and the third formulas of Eq. (18) show the forwardpropagating FLFOs have variable lengths, which decreases and increases in the first and the second forwardpropagations, respectively. While the second formula of Eq. (18) shows the backwardpropagating FLFO has a constant length. This result agrees well with the simulation shown in Fig. 2f. Equation (18) also shows the lengths ΔL of FLFOs are proportional to a fixed product of M·Z_{R} where M = L_{c}/(L_{c} − L_{f}), although the values of ΔL are slightly different in forward and backward propagations. Then, M and Z_{R} are key parameters to obtain a clear reciprocating FLFO.
Radiationforces in the Rayleighscattering regime
Under the zerothorder approximation of the continuouswave Gaussian beam or a weakly focused continuouswave Gaussian beam, the formulas of the scattering and gradient forces F_{scat} and F_{grad} exerted on a Rayleigh dielectric sphere in a steadystate are given by^{43,45}
and
where, n_{1} and n_{2} are the refractive indexes of the dielectric sphere and the environment, m = n_{1}/n_{2} is the relative refractive index, a is the radius of the dielectric sphere, I(x, y, z) is the laser intensity, and ∇I(x, y, z) is the laser intensity gradient.
Data availability
The data that support the findings of this study are available from the corresponding author upon reasonable request.
Code availability
The code that support the findings of this study are available from the corresponding author upon reasonable request.
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Acknowledgements
This work was supported by the JSTMirai Program, Japan, under contract JPMJMI17A1.
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Z.L. developed the concept, derived the equations, performed the simulations, and wrote the manuscript. Y.G. commented on the application part, and J.K. commented on the optical propagation part. All authors discussed the results and commented on the manuscript.
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Li, Z., Gu, Y. & Kawanaka, J. Reciprocating propagation of laser pulse intensity in free space. Commun Phys 4, 87 (2021). https://doi.org/10.1038/s42005021005908
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DOI: https://doi.org/10.1038/s42005021005908
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