Abstract
Threedimensional (3D) light solitons in space–time, referred to as light bullets, have many novel properties and wide applications. Here we theoretically show how the combination of diffractionfree beam and ultrashort pulse spatiotemporalcoupling enables the creation of a straightline propagation light bullet with freely tunable velocity and acceleration. This light bullet could propagate with a constant superluminal or subluminal velocity, and it could also counterpropagate with a very fast superluminal velocity (e.g., − 35.6c). Apart from uniform motion, an acceleration or deceleration straightline propagation light bullet with a tunable instantaneous acceleration could also be produced. The high controllability of the velocity and the acceleration of a straightline propagation light bullet would enable very specific applications, such as velocity and/or acceleration matched micromanipulation, microscopy, particle acceleration, radiation generation, and so on.
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Introduction
The combination of diffractionfree beam and dispersionfree pulse permits the chance to produce 3D selfsimilar spatiotemporal optical wave packets (light bullets), which could propagate over long distances and, importantly, maintain invariant intensity profiles in space and time^{1,2,3,4}. In nonlinear optics, the balance between nonlinearity and dispersion/diffraction could easily result in temporal/spatial solitons^{5,6,7,8,9,10,11,12,13}, however some applications require to generate light bullets in linear/free space. In linear optics, Bessel beam is one of the most important diffractionfree beams whose central core is propagation invariant^{14,15}, and up to now many novel methods have already been proposed^{16,17,18,19,20,21}, such as circular slit with lens^{16}, axicon^{17}, spatial light modulator^{18}, and so on. Airy beam is another interesting diffractionfree beam, and, compared with Bessel beam, its main intensity maxima and lobes could tend to accelerate during propagation along a parabolic trajectory^{22,23,24,25,26,27,28,29}. For a diffractionfree beam, when the monochromatic wave is replaced by a dispersionfree pulse, a light bullet might be produced. The simplest form is the GaussBessel/Airy light bullet in free space (nondispersion environment) or the Bessel/AiryBessel/Airy light bullet in materials (dispersion environment) ^{1,22,30,31}. To control the velocity and even the acceleration of a light bullet is an interesting but challenging work. Previously, during a short propagation distance, the group velocity given by υ_{g} = c/n_{g} could be controlled by crafting the wavelengthdependent refractive index^{32}, however some very special material or photonic systems are necessary^{33,34,35,36,37,38,39,40,41}, such as ultracold atoms^{33}, hot atomic vapors^{35}, stimulated Brillouin scattering^{36}, active gain resonances^{37}, tunneling junctions^{38}, metamaterials^{39}, photonic crystals^{40}, and so on. The problem is that, in transparent materials at wavelengths far from resonance or even in free space, the controllability of n_{g} and accordingly the group velocity υ_{g} would be very limited. Light bullet with an airy beam could selfaccelerate during propagation in free space, however which travels along a bended (parabolic) propagation trajectory^{22,23,24}. Recently, several spatiotemporal coupling methods have been proposed to control the group velocity of optical wave packets in free space^{42,43,44,45,46}. For example by controlling temporal and spatial dispersion (temporal chirp and chromatic aberration)^{42,44}, a flying focus with a tunable group velocity could be achieved, however the propagation distance is limited within the focusing region. In this article, we combined the firstorder spatiotemporal coupling (pulsefront predeformation) with the diffractionfree (GaussBessel) pulsed beam and theoretically produced a straightline propagation light bullet in free space, whose velocity and acceleration could be freely controlled, including all cases of superluminal, subluminal, acceleration, deceleration and backwardstraveling group velocities. We have mainly discussed the case of a GaussBessel pulsed beam in free space, and the method is also suitable to other types of pulsed beams, for example the AiryBessel pulsed beam in free space or materials. This velocity and acceleration tunable straightline propagation light bullet has lots of specific applications from basic sciences to industry applications.
Superluminal and subluminal light bullets
An axicon can transform a plane wave into a conical wave and generate a Bessel beam in the overlap region due to interference^{17}. In vacuum, the propagation velocity (group velocity) of this Bessel beam is υ_{b} = c/cosα, where c is the light speed in vacuum and α is the (half) conical angle (relevant to both the refractive index and the wedge angle of the axicon)^{47}. It can be found that this velocity is faster than the light speed in vacuum c, which also increases with increasing the conical angle α. Figure 1a shows the key idea of the method proposed in this article that the input pulsed beam possesses a distortionfree (plane) phasefront but a predeformed (conical) pulsefront, which is quite different from the previous case with both distortionfree (plane) phase and pulsefronts. It is necessary to introduce that the phasefront is the surface perpendicular to the propagation direction while the pulsefront is the surface coinciding with the peak of a pulse, which are respectively determined by the phase and groupvelocities^{48}. The result of the method is that the velocity of the generated light bullet could be freely controlled. As an example we begin the simulation with an initial input pulsed beam with a 30 fs (FWHM) pulse in time, a 2 mm (1/e^{2}) beam in space and a concave conicalpulsefront (CCCPF) in space–time, and the generation and the propagation of the light bullet in the 2D propagation section of the x–z plane is shown in Fig. 1b. The axicon spatially divides the input pulsed beam into two GaussGauss ones and changes their travelling directions (green arrows) with symmetrical angles of α = ± 0.5°. The phasefronts of two pulsed beams are illustrated by white lines, and the pulsefronts are shown by red distributions. It can be found that in the overlap region a light bullet (in free space) is generated due to interference, which in time is a Gauss pulse and in space is a Bessel beam. Because the pulsefront is predeformed to deviate from the phasefront with a tilt angle of β = 6.6°, the generated light bullet is not located at the intersection of the phasefronts anymore, and we can say the light bullet is temporally or spatially delayed along the longitudinal axis. Here, we make two definitions: in space, the propagation (longitudinal) axis of the Bessel beam is the z–axis and its geometrical center in the overlap region (formed by two pulsed beams after the axicon) is the origin of z = 0; and in time, the moment when the intersection of two phasefronts (the original location of the light bullet if without any pulsefront predeformation) arriving at z = 0 is the zero time of t = 0. Figure 1b gives the detailed distributions of the pulsefronts, the phasefronts and the light bullet at different times of t = − 120, − 60, 0, 60 and 120 ps during propagation. According to the previous result^{41}, the intersection of two phasefronts travels at a velocity of 1.00004c governed by υ_{b} = c/cosα. However, due to the pulsefront predeformation, the light bullet is behind the intersection of phasefronts and the longitudinal gap Δz (distance between the light bullet and the intersection of phasefronts) decreases with time during propagation. This phenomenon indicates the travelling velocity of the light bullet is faster than that of the intersection of phasefronts, i.e., a superluminal light bullet is produced. Red curve in Fig. 1d shows the variation of the longitudinal gap Δz with time t is linear, and accordingly the light bullet should possess a constant velocity. The simulated value is 1.001c which is faster than the velocity of the intersection of phasefronts of 1.00004c. In geometrical optics, within the Rayleigh range, the velocity of this light bullet satisfies
which is governed by both the conical angle α (determined by the axicon) and the pulsefront tilt angle β (determined by the pulsefront predeformation). Thus, the pulsefront tilt angle β is another degree of freedom to control the velocity υ_{b} of the light bullet.
In another case, when a convex conicalpulsefront (CVCPF) with β = − 6.6° is predeformed and all the other parameters remain unchanged, Fig. 1c shows although the light bullet is still behind the intersection of the phasefronts, the longitudinal gap Δz is gradually increasing (instead of decreasing in Fig. 1b) during propagation. Therefore, the velocity of this light bullet is slower than that of the intersection of phasefronts. The detailed variation of the longitudinal gap Δz with time t is shown by blue curve in Fig. 1d, which is linear, too (constant velocity). The simulated value is 0.999c and slower than the velocity of the intersection of phasefronts of 1.00004c, and accordingly a subluminal light bullet is produced.
In applications, the above superluminal and subluminal velocities for the cases of CCCPF and CVCPF could be calculated by using Eq. (1) with a positive and a negative pulsefront tilt angle β, respectively, and thereby Eq. (1) can be directly used to describe the light bullet created by using this method.
Next, we continue to the controllability of the light bullet velocity. Figure 2a shows the variation of the light bullet velocity υ_{b} with the pulsefront tilt angle β for different conical angles α. When the conical angle α is small (for example 0.5°, 1° and 5° in Fig. 2a upper), the light bullet velocity υ_{b} and the pulsefront tilt angle β satisfy a linear relationship, and a superluminal or a subluminal light bullet could be produced by choosing a positive (CCCPF) or a negative (CVCPF) pulsefront tilt angle β. The sensitivity of the light bullet velocity υ_{b} to the pulsefront tilt angle β could be increased by increasing the conical angle α. Once the conical angle α is dramatically increased (for example 80°, 85° and 88° in Fig. 2a lower), the linear relationship disappears, and the light bullet velocity υ_{b} becomes very sensitive to the pulsefront tilt angle β, especially when α + β is close to 90°. And even a negative velocity υ_{b} (backwards travelling) could be generated, when α + β is large than 90°. We simulate this case, when the conical angle α is increased to 85° and all the other parameters used in Fig. 1b remain unchanged (the spot marked in Fig. 2a lower). Figure 2b shows the generation and the propagation of the light bullet in the x–z plane. At t = 0 two phasefronts intersect at the position of z = 0, while two pulsefronts still separate in space. From t = 175 fs to 195 fs and then to 215 fs, two pulsefronts begin to intersect with each other at the leading edge and then the intersection quickly moves towards the trailing edge, showing a backwards travelling light bullet in space–time. The propagation velocity of this light bullet is very high of − 35.6c. This phenomenon could be explained by Eq. (1). Because the sum of the conical angle and the pulsefront tilt angle α + β is slightly larger than 90° (91.6°) and the pulsefront tilt angle β is small (6.6°), so the light bullet velocity υ_{b} is negative and its absolute value is dramatically increased. The problem of this backwards travelling light bullet is that, once a large conical angle α is used, the size of the center lobe of the Bessel beam (the light bullet) would be greatly reduced and most energies would transfer from the center to side lobes (concentric rings around the light bullet in Fig. 2b)^{15}. In some applications, such as laser drilling, optical tweezing, etc., the energy loss at the center lobe might be unacceptable, however in other applications, such as particle manipulation, secondaryradiation generation, etc., a superluminal backwards travelling light bullet, as well as a series of concentric rings, may possess unique performance.
Acceleration and deceleration light bullets
When the initial input pulsed beam possesses a sphericalpulsefront and a planephasefront (Fig. 3a), which could be frequently found in a transmission telescope and usually called as “pulsefrontcurvature”^{48,49} or “radialgroupdelay”^{50}, the velocity of the generated light bullet would not be a constant during propagation. The simulation is still based on the parameters used in the above section. First we simulate the case of concave sphericalpulsefront (CCSPF), and the curvature R is − 4.2 mm. Figure 3b shows the distributions of the pulsefronts, the phasefronts and the generated light bullet at different times t in the x–z plane. The light bullet is still behind the intersection of the phasefronts, the longitudinal gap Δz also decreases during propagation (superluminal light bullet), however the decreasing amount in the first half propagation (from t = − 120 ps to 0) is very small while in the second half propagation (from t = 0 to 120 ps) becomes much obvious. Because the intersection of the phasefronts has a constant velocity of 1.00004c, then the light bullet experiences an acceleration propagation. Red curve in Fig. 3d gives the detailed variation of the longitudinal gap Δz with time t, which is no longer a perfect linear relationship anymore. The longitudinal gap Δz decreases faster and faster with time t, and consequently an acceleration light bullet is produced. Second we simulate another case of convex sphericalpulsefront (CVSPF) with a curvature R of 4.2 mm. Figure 3c shows the generation and the propagation of the corresponding light bullet, and the longitudinal gap Δz is increasing during propagation, which also becomes more and more obvious with the propagation position z or time t. Blue curve in Fig. 3d shows the variation of the longitudinal gap Δz with time t and illustrates that the longitudinal gap Δz increases faster and faster with time t, and therefore a deceleration light bullet is produced.
The acceleration or the deceleration propagation of a light bullet could be explained by Eq. (1). The sphericalpulsefront can be considered as the superposition of a series of conicalpulsefronts. Figures 3b,c show that, during propagation, different parts of two pulsefronts contribute to the creation of the light bullet, which corresponds to different pulsefront tilt angles β at different propagation times t or positions z, and consequently, the velocity υ_{b} varies with time t or position z during propagation, showing acceleration or deceleration.
The propagation position z (or time t) dependent velocity υ_{b} and acceleration a of a light bullet for different geometries are simulated based on the parameters used in above, and the result is shown in Fig. 4. When the conical angle α is 0.5° and the sphericalpulsefront curvature R is ± 4.2 mm (negative and positive for CCSPF and CVSPF, respectively), red curves in Fig. 4a,b illustrate the variation of the velocity υ_{b} and the acceleration a of CCSPF (acceleration) and CVSPF (deceleration) with the propagation position z. To analyze the influence of the conical angle α and the curvature R, we modify the two parameters individually. When the conical angle α is enlarged by four times from 0.5° to 2°, green curves show that: first the tunable range of the velocity υ_{b} is increased, and second the absolute value of the acceleration a is enlarged (significant acceleration or deceleration). However, the tunable range of the acceleration a during propagation is not increased, but unfortunately the propagation distance of the light bullet (Bessel beam) is reduced a lot which is determined by the conical angle α. When the absolute value of the sphericalpulsefront curvature R is reduced by four times from 4.2 mm to 1.05 mm, blue curves illustrate that: first the tunable ranges of both the velocity υ_{b} and the acceleration a are dramatically increased, and more importantly the long propagation distance of the light bullet (Bessel beam) remains unchanged. In this case, adjusting the sphericalpulsefront curvature R is an ideal approach to control the propagation position (or time) dependent velocity υ_{b} and acceleration a of an acceleration or deceleration light bullet, which would be very attractive in some special applications, such as particle acceleration^{51,52}, plasma channel generation^{53}, and so on.
Discussion and conclusion
In this article, we introduced the method based on a GaussBessel pulsed beam, which in free space can be considered as diffractionfree and dispersionfree (light bullet). According to previous publications^{1}, if the thirdorder dispersion (cubic spectral phase) is introduced into the initial pulse, the GaussBessel pulsed beam would be conveniently transferred into an AiryBessel pulsed beam, which is a light bullet in dispersion materials. This process won’t influence the method and the result proposed here, and consequently a velocity and acceleration tunable straightline propagation light bullet in dispersion materials could also be produced. In principle, this method is applicable to any other type of pulsed beams, and we won’t repeat the detail again.
In conclusion, we have theoretically proposed a method that by combining the traditional diffractionfree beam (Bessel beam in this article) with the firstorder spatiotemporal coupling ultrashot pulse (pulsefront predeformation), a straightline propagation light bullet with freely tunable velocity (superluminal and subluminal) and acceleration (acceleration and deceleration) could be produced, and a backwards traveling superluminal light bullet could also be created. This highly tunable light bullet has a broad range of applications, for example a superluminal or subluminal light bullet is quite useful to timedependent pumpprobe measurement/microscopy, and an acceleration or deceleration light bullet can be used to match a flying particle in particle acceleration experiment.
Methods
Pulsefront predeformation
The pulsefront of the initial input pulsed beam can be easily deformed by using a pair of matched transmission and reflection optics^{48,49,50,54,55,56}, and its phasefront would remain unchanged in this process due to the perfect imaging geometry. Figure 5a shows a pair of transmission convex axicon and concave conical reflector would generate CCCPF, and similarly CVCPF can be produced by a pair of transmission concave axicon and concave conical reflector (Fig. 5b). When the transmission convex or concave axicon and the concave conical reflector is respectively replaced by a transmission convex or concave spherical lens and a concave spherical reflector, CCSPF (Fig. 5c) or CVSPF (Fig. 5d) would be produced.
Coordinate rotation
The axicon transfers an input plane wave into an output conical wave, and, in the 2D propagation section, the upper and lower half beams have their individual travelling directions, which are symmetrical bout the Bessel beam propagation axis (Figs. 1a or 3a). The input plane wave and the generated Bessel beam are described in the coordinate system of rz, where r is the radial axis and z is the propagation axis. The output upper or lower half beam after the axicon is described in its own propagation coordinate system of r_{α}z_{α}, where r_{α} is the radial axis and z_{α} is the propagation axis. The origins of two coordinate systems have a same location, which is at the geometrical center of the overlap region after the axicon. Thus, two coordinate systems of rz and r_{α}z_{α} satisfy the rotation relationship
where α is the half conical angle which is also the rotation angle of the coordinate system of r_{α}z_{α}. In this article, positive and negative coordinate rotation angles α are defined for the upper and lower half beams after the axicon in the 2D propagation section, respectively.
Pulsed beam propagation and coherent superposition
After the axicon, the Efield of upper or lower half beam in the 2D propagation section of r_{α}z_{α} can be described as
where A is the amplitude, τ is the local time, T(r_{α}) is the spatiotemporal coupling term, t and z_{α} (t = z_{α}/c) are time and length from the z_{α}–axis origin of z_{α} = 0 to the current propagation position, Δτ is the pulse duration, k is the wave vector, q_{zα} is the complex Gaussian beam parameter at z_{α}, and ϕ is the initial phase. In this article, the beam waist locates at the coordinate origin of (r_{α} = 0, z_{α} = 0).
For the case of conical pulsefront, the spatiotemporal coupling term in Eq. (3) is given by
where w_{zα} is the beam waist, and we should emphasize that it is the beam waist of the upper or lower half beam after the axicon. For the case of CCCPF, the upper or lower half beam after the axicon has a positive and negative pulsefront tilt angle β, respectively, which corresponds to w_{zα} and + w_{zα}. While for the case of CVCPF, the situation is exactly the opposite: the pulsefront tilt angle β of the upper and lower half beam is negative and positive, respectively, and which corresponds to + w_{zα} and w_{zα}.
For the case of spherical pulsefront, the spatiotemporal coupling term in Eq. (3) satisfies
where R is the curvature of the pulsefront, which is negative and positive for CCSPF and CVSPF, respectively. And + w_{zα} and w_{zα} correspond to the upper and lower half beams after the axicon.
The complex Gaussian beam parameter q_{zα} at z_{α}, containing the information of both the radius of beam curvature R_{zα} and the beam size w_{zα}, satisfies
After propagation governed by Eq. (3) in the r_{α}z_{α} coordinate system, the coherently superimposed (interference) Efield in the rz coordinate system is given by
where E_{u} and E_{l} are Efields of two half beams after the axicon.
Data availability
The data 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 and carried out the simulations. Z. L. and J. K. wrote the paper. All authors discussed the results and commented on the manuscript.
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Li, Z., Kawanaka, J. Velocity and acceleration freely tunable straightline propagation light bullet. Sci Rep 10, 11481 (2020). https://doi.org/10.1038/s41598020684781
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DOI: https://doi.org/10.1038/s41598020684781
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