Abstract
During the process of Bessel beam generation in free space, spatiotemporal optical wavepackets with tunable group velocities and accelerations can be created by deforming pulsefronts of injected pulsed beams. So far, only one determined motion form (superluminal or luminal or subluminal for the case of group velocity; and accelerating or uniformmotion or decelerating for the case of acceleration) could be achieved in a single propagation path. Here we show that deformed pulsefronts with welldesigned axisymmetric distributions (unlike conical and spherical pulsefronts used in previous studies) allow us to obtain nearlyprogrammable group velocities with several different motion forms in a single propagation path. Our simulation shows that this unusual optical wavepacket can propagate at alternating superluminal and subluminal group velocities along a straightline trajectory with corresponding instantaneous accelerations that vary periodically between positive (acceleration) and negative (deceleration) values, almost encompassing all motion forms of the group velocity in a single propagation path. Such unusual optical wavepackets with nearlyprogrammable group velocities may offer new opportunities for optical and physical applications.
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Introduction
Recent studies of nondiffraction beams include the Bessel, X, Y, Airy, and parabolic waves in linear mediums^{1,2,3,4,5,6}, the selftrapping, selffocusing, and nonlinear X wave in nonlinear mediums^{7,8,9,10,11}, and multidimensional solitons in complex systems^{12,13,14,15}. Based on these studies, threedimensional (3D) localized spatiotemporal optical wavepackets with long propagation distances and nearlyinvariant intensity profiles have been widely demonstrated in both linear^{16,17,18} and nonlinear^{19,20,21,22} mediums. Applications of such wavepackets range from particlemanipulation to bioimaging and plasmaphysics. Beside the wellknown properties of longdistance selfsimilarity and/or selfhealing of these localized spatiotemporal optical wavepackets, tunable group velocity is another degree of freedom relevant to some novel applications. In nonlinear optics, the group velocity υ_{g} = c/n_{g} (n_{g} is the group refractive index and c is the speed of light) of an optical wavepacket can be well controlled by crafting the wavelengthdependent refractive index n_{λ} ^{23}. However, some very special materials or systems are required^{24,25,26,27,28,29,30}, challenging the application of this principle to linear systems where the controllability of n_{g} is very limited.
The Bessel beam is a famous family of diffractionfree beams resulting from conical superposition, which usually propagates with a constant superluminal group velocity in free space^{31,32,33,34,35,36,37,38,39}. By phasemodulating the incident crosssection, it is possible to produce selfaccelerating Bessellike beams having arbitrary curved trajectories^{40,41,42,43}. In this case, the combination of a Bessel beam and a pulse can produce a superluminal and/or selfaccelerating optical wavepacket^{18,44,45}. For example, directly combining a Bessel beam with an Airy pulse can create a selfaccelerating optical wavepacket in transmission materials^{18}; however, the accelerating value is limited within the short pulse duration range. The X wave, demonstrated in both linear and nonlinear mediums, is another important limited diffraction beam^{46,47,48}. In linear optics, broadband superposition of slightly distorted Bessel beams can create BesselX spatiotemporal optical wavepackets^{49,50}, and in nonlinear optics pulsed beams with nonlinear material responses can generate Xshaped light bullets^{51}. However, the controllability of group velocities of these Xshaped optical wavepackets is not high. The Airy beam also is a wellstudied diffractionfree beam in which main intensity maxima and lobes tend to propagate^{52,53,54,55,56,57}. By combing an Airy beam with a pulse, the resulting selfaccelerating optical wavepackets can propagate at superluminal group velocities along parabolic trajectories in free space. In the above methods, group velocityvariable optical wavepackets in free space usually correspond to curved or bended propagation trajectories, allowing for novel applications that include particles guiding/trapping along curved paths and selfbending plasma channels generation^{58,59,60}. In some other applications instead, straightline propagation optical wavepackets with tunable group velocities have irreplaceable advantages.
In linear optics, it has recently been shown that spatiotemporal coupling permits high controllability of the group velocity and/or the acceleration of spatiotemporal optical wavepackets. Saari et al. and Abouraddy et al. invented a new type 2D spatiotemporal optical wavepacket by manipulating the spatial and temporal degrees of freedom jointly, where both diffractive spreading and pulse broadening are eliminated^{61,62}. This optical wavepacket can be described by a spectral trajectory resulting from the intersection of the lightcone (k_{x}, k_{z}, ω/c) with a tilt plane (k_{z}, ω/c) in spectralspace, where k_{x} and k_{z} are the transverse and longitudinal wavenumbers, respectively, x and z are the transverse and longitudinal coordinates, respectively, and ω is the angular frequency. By adjusting the tilt angle of the plane (k_{z}, ω/c), the group velocity of the optical wavepacket in free space or in transmission materials can be controlled, including all motion forms, i.e. superluminal, subluminal, accelerating, decelerating, and backwardpropagation^{63,64,65,66,67,68}. Another spatiotemporal coupling method is to control the group velocity of the intensity peak of a focused ultrashort pulse within the extended Rayleigh length (named as sliding focus or flying focus) by combining temporal chirp and longitudinal chromatism. This method was independently demonstrated by Quéré et al. in theory^{69,70} and Froula et al. in experiments^{71}. In this method, the longitudinal chromatism separates wavelengthdependent focuses along the propagation axis and the temporal chirp controls the appearance times of these focuses, so that the sliding/flying focus possesses a tunable effective group velocity, also achieving all motion forms (superluminal, subluminal, accelerating, decelerating, and backwardpropagation). More recently, we theoretically demonstrated a third spatiotemporal coupling method to generate group velocity and acceleration tunable 3D optical wavepackets in free space^{72}. The pulsed beam used for the Bessel beam generation is deformed to have an axisymmetric pulsefront which deviates from its plane phasefront. In the generation of the Gauss−Bessel optical wavepacket, the plane phasefront determines the Bessel beam generation in space, while the deformed pulsefront determines the optical wavepacket propagation in time. Consequently, the group velocity and acceleration of the optical wavepacket can be adjusted by changing the pulsefront deformation, also including all motion forms of superluminal, subluminal, accelerating, decelerating, and backwardpropagation group velocities. In all these spatiotemporal coupling methods however, by controlling one degree of freedom it is possible to achieve only one determined motion form (superluminal or luminal or subluminal for the case of group velocity; and accelerating or uniformmotion or decelerating for the case of acceleration) in a single propagation path.
In this article, by combining welldesigned complex axisymmetric pulsefront deformations with our recently reported method^{72}, we achieve a nearly programmable group velocity with several different motion forms in a single propagation path. The created optical wavepacket can fly with superluminal and subluminal group velocities alternately, and the corresponding instantaneous acceleration varies between positive (accelerating) and negative (decelerating) values. For example, when periodically distributed axisymmetric pulsefront deformations are introduced, the group velocity and acceleration of optical wavepackets display periodical variations during propagation, showing an alternate appearance of superluminal−subliminal group velocities and accelerating−decelerating accelerations. This unusual optical wavepacket presenting different group velocity motion forms in a single propagation path may provide new opportunities for applications.
Results
Setup of the method
The schematic diagram of the method is shown in Fig. 1a. A collimated pulsed beam is reflected by a deformable mirror (DM) (or a freesurface mirror) to shape its pulse and phasefronts from a flat surface to a required axisymmetric surface. A transmission spatial light modulator (SLM) is positioned just behind DM to correct the phasefront back to the original flat surface for longdistance propagation while keeping the shaped axisymmetric pulsefront unchanged^{73}. A beam splitter (BS) samples the shaped pulsed beam into a parabola telescope, which is used for three purposes: first, to image the shaped pulsed beam into the Bessel beam generation region formed by an axicon for suppressing propagation diffraction; second, to enhance the spatial resolution of the phasefront correction that is limited by the pixel size of SLM by beam reduction (ten times in this article); and third, to increase the instantaneous pulsefront variation across the beam aperture also by beam reduction. Finally, an ideal thin axicon is used to generate a Bessel beam in the conical superposition region.
Using the Simulation model and parameters given in the “Methods” section, the optical fields at different locations are calculated. Because the optical fields are always axisymmetric about the propagation axis, only the 2D distributions in the lateral plane of the r−z plane, where r and z are the transverse and longitudinal coordinates, are given. Figure 1b shows the input pulsed beam, and the carrier frequency is multiplied by 0.1 to avoid too fast oscillations for observation. Figure 1c shows the optical field, when only DM (or a freesurface mirror) is considered, both pulse and phasefronts are deformed, and Fig. 1f gives the corresponding surface of DM. Figure 1d shows the optical field, when SLM is considered, which has a deformed pulsefront and an unchanged (plane) phasefront, and Fig. 1g gives the corresponding phasefront correction of SLM. Figure 1e shows the optical field after 5m free propagation, and some diffraction distortions can be found, showing the necessity of the image relay telescope. In Fig. 1c–e, g, a small range is enlarged for observation. The continuous surface of DM won’t change the smooth pulse and phasefronts in Fig. 1c, while the segmented phasefrontcorrection of SLM (limited by the SLM pixel size shown in Fig. 1g) generates fine structure modulations in Fig. 1d. The pixel−pixel gap of SLM introduces netlike spatial amplitude modulation, and the pixel size of SLM keeps a very small phasefront tilt within each segment spatial range. In real cases, these two high spatialfrequency modulations can be reduced by suitable propagation diffraction in Fig. 1e. All in all, SLM introduces optimized (−π, π) phasecorrections across the beam aperture (like a Fresnel lens) and then collimates propagation directions (or wave vectors) at different transverse positions. In this case, after the ideal thin axicon, a Gauss−Bessel optical wavepacket can be created in the superposition region. Due to the pulsefront deformation, the created optical wavepacket does not possess a constant superluminal group velocity governed by υ_{b} = c/cos α anymore^{39}, and the detail is going to be introduced in the following section.
Variablegroup velocity and acceleration
Figure 2 shows the simulated intensity distributions (illustrated by red−yellow distributions) in 2D x−z/t and 3D x–y−z/t spatiotemporal domains. Because of the axisymmetric profiles about the propagation axis, here we only discuss the results in the 2D lateral plane x−z/t containing the propagation axis. The axicon spatially divides the input pulsed beam into two and changes their traveling directions (illustrated by green dashline arrows) with (half) conical angles of α = ±0.5°. The plane phasefronts (illustrated by white solid lines) of two pulsed beams are always perpendicular to the traveling directions. In the superposition region, a Gauss−Bessel spatiotemporal optical wavepacket is generated at the core position resulting from conical superposition. We keep the same definitions as that in our previous article^{72}: in space, the propagation axis of the optical wavepacket (or Bessel beam) is defined as the zaxis and the geometrical center of the superposition region is defined as the spatial origin of (x = y = z = 0) (see Fig. 1a); in time, the moment when the intersection of two phasefronts (the intersection of two white solid lines) arriving at z = 0 is defined as the temporal origin of t = 0.
When the deformed pulsefront has a cosinefunctionlike axisymmetric distribution, Fig. 2a shows at the very beginning the generated optical wavepacket (the core of red−yellow distributions) has the same location with the intersection of the phasefronts (the intersection of two white solid lines); however, during the propagation, the optical wavepacket will be temporally delayed along the zaxis. From left to right in Fig. 2a, the detailed distributions of the pulsefronts (red−yellow distributions), the phasefronts (white solid lines), and the optical wavepacket (the core of red−yellow distributions) at different propagating times of t = −150, −100, −50, 0, 50, 100 and 150 ps are illustrated. The intersection of the phasefronts travels at a constant velocity of 1.00004c governed by υ_{g} = c/cos α, while the optical wavepacket has a variablegroup velocity. In the first half propagation from t = −150 ps to t = 0, the longitudinal gap Δz between the intersection of the phasefronts and the optical wavepacket increases from zero to the maximum with the propagating time t, while in the secondhalf propagation from t = 0 to t = 150 ps, it gradually decreases from the maximum to zero again. Figure 3a shows the variation of the longitudinal gap Δz with the propagating time t (or position z), which has a cosinefunctionlike distribution about the propagating time t (or position z). The blue curve in Fig. 3b shows the instantaneously variable subluminal group velocities in the first half propagation, reaching the minimum at around t = −100 ps, and the red curve illustrates the instantaneously variable superluminal group velocities in the secondhalf propagation, reaching the maximum at around t = 100 ps. Figure 3c shows the variation of the instantaneous acceleration during the entire propagation, and the optical wavepacket experiences decelerating, then accelerating and finally decelerating motions in three temporal periods of from the appearance time to around t = −100 ps, from around t = −100 ps to around t = 100 ps, and from around t = 100 ps to the disappearance time, respectively. Figure 2b illustrates three isosurface plots at 10% and 80% of the maximum intensity showing the dynamics at three different propagating times of t = −100 ps, −50 ps, and 0. It is clear that at different propagating times the relative temporal (or zaxis) locations of the optical wavepacket in the entire pulsed beam are different. At the moment of the perfect overlap (i.e. t = 0), a Bessel beam with a central core and a series of concentric rings appears; however, while comparing with the traditional Bessel beam, the central core and the concentric rings have different temporal (or zaxis) locations, which is dominated by the pulsefront deformation.
Next, when the deformed pulsefront has an opposite cosinefunctionlike axisymmetric distribution (or with a π phaseshift), Fig. 2c shows the simulated distributions of the pulsefronts, the phasefronts, and the optical wavepacket at different propagating times of t = −150, −100, −50, 0, 50, 100 and 150 ps. At the very beginning, the optical wavepacket has the maximum temporal delay along the zaxis relative to the intersection of the phasefronts, i.e., the maximum longitudinal gap Δz. In the first half propagation from t = −150 ps to t = 0, the longitudinal gap Δz decreases to zero with the propagating time t, and in the secondhalf propagation from t = 0 to t = 150 ps, it increases back to the maximum again. Figure 3d shows the variation of the longitudinal gap Δz with the propagating time t (or position z). The red curve in Fig. 3e shows the instantaneously variable superluminal group velocities in the first half propagation, reaching the maximum at around t = −100 ps, and the blue curve shows the instantaneously variable subluminal group velocities in the secondhalf propagation, reaching the minimum at around t = 100 ps. Figure 3f shows the variation of the instantaneous acceleration, and the optical wavepacket experiences accelerating, then decelerating and finally accelerating motions in three temporal periods of from the appearance time to around t = −100 ps, from around t = −100 ps to around t = 100 ps, and from around t = 100 ps to the disappearance time, respectively.
By comparing Figs. 2a with 2c or 3a–c with 3d–f, the motions of the optical wavepacket in two cases are opposite, which is determined by the two opposite pulsefront deformations. Figure 3a, d shows the variation value of the longitudinal gap Δz (between the intersection of the phasefronts and the optical wavepacket) is dominated by that of the pulsefront deformation. Because the intersection of the phasefronts has a constant velocity of c/cos α (for an ideal thin axicon), the instantaneous group velocity of the optical wavepacket can be calculated by the derivative operation of υ_{b} = c/cos α − d(Δz)/dt, and the instantaneous acceleration of the optical wavepacket can be obtained by a = dυ_{b}/dt. In this case, by controlling the value of the pulsefront deformation, the instantaneous longitudinal gap Δz, and accordingly the instantaneous group velocity and acceleration of the optical wavepacket, in theory, can be well controlled. Furthermore, by introducing some unusual shapes of the pulsefront deformations, optical wavepackets with unusual motion forms (e.g., periodically variablegroup velocity and—acceleration here) can also be created.
Moreover, in Fig. 2a, c, the best image relay position by the parabola telescope is at z = 0 (or t = 0), and when the pulsed beam propagates at z = ±45 mm (or t = ±150 ps) the propagationdiffractioninduced slight distortions can be found. In this case, when reducing the conical angle α to increase the propagation distance, the propagation diffraction should be considered, and the details are going to be discussed in the “Discussion” section.
Controllability of variablegroup velocity
The above result shows that the shape and the value of the pulsefront deformation dominates the motion form and the group velocity value of the optical wavepacket, respectively. In our previous article^{72} with the simplest form of pulsefront tilt (PFT), i.e., a linearly tilt pulsefront, the optical wavepacket has a constant group velocity governed by
where α is the (half) conical angle formed by the ideal thin axicon, and β is the tilt angle between pulse and phasefronts.
For a complex pulsefront deformation, for example the cosinefunctionlike profile here, Fig. 4a shows the instantaneous PFT can be obtained by calculating the tangent angle β of the tangent line at the intersection of two pulsefronts. Figure 4b shows the dependence of the optical wavepacket group velocity υ_{b} on PFT for three different (half) conical angles α. The tunable (or variation) range of the group velocity υ_{b} increases with increasing PFT, and the capability can be further enhanced by choosing a larger (half) conical angle α. If PFT is enlarged, the difficulty of generating the required phasecorrection by SLM will increase. Figure 4c shows when the pulsefront deformations have the same cosinefunctionlike shape but different peakvalley (PV) values, the required phasecorrections by SLM are quite different. When the PV value is enlarged, the instantaneous PFT changes drastically along the transverse axis, accordingly the variable range of the group velocity υ_{b} increases; however, a high spatialfrequency phasecorrection is also required, challenging the SLM resolution. Figure 4d shows the number of the phasecorrection periods (−π, π) within 1 mm increases linearly with increasing PFT. If one modulation period (−π, π) contains at least five pixels, the required SLM pixel size has a negative exponential function distribution with respect to PFT. For example, with reference to an available 4 μm pixel size, when the absolute value of PFT is larger than 280 fs mm^{−1}, beam reduction by the image relay telescope becomes necessary. This is another reason why a parabola telescope with ten times beam reduction is used in Fig. 1a. And the third reason is to reduce the beam size to increase the instantaneous PFT across the beam aperture and, accordingly, increase the variation range of the group velocity υ_{b}.
The variation range of the (propagating time/position dependent) group velocity υ_{b} is dominated by that of PFT across the deformed pulsefront. For the case of a cosinefunctionlike pulsefront deformation used in this article, the pulsefront is z = L/2·cos(r/D·2π), where L and D are the longitudinal PV value and the transverse period, respectively. The tangent angle can be obtained β = −L/D·π·sin(r/D·2π), and the extremum is β_{max/min} = ±L/D·π. Based on an available commercial DM with a 100 μm PV value and a 20 mm diameter^{74}, i.e., L = 100 μm and D = 10 mm for an axisymmetric cosinefunctionlike pulsefront deformation, the extremum is β_{max/min} ≈ ±31.41 mrad (or PFT ≈ ±104.7 fs mm^{−1}). When the ten times beam reduction by the parabola telescope is considered, the extremum is increased to β_{max/min} ≈ ±314.1 mrad (or PFT ≈ ±1047 fs mm^{−1}). By the substitution of Eq. (1) with the calculated extremum, for different (half) conical angles of α = 0.5°, 5°, and 10°, the variation range of the group velocity υ_{b} is (0.997c, 1.003c), (0.976c, 1.033c), and (0.96c, 1.077c), respectively. We can find that the overall variation range of the group velocity is limited, and if required, which can be slightly increased by replacing DM with a freesurface mirror and/or increasing the magnification of the beam reduction telescope.
Discussion
The pulsefront deformation would shorten the propagation distance of a pulsed beam due to propagation diffraction, although the phasefront is corrected. When keeping all simulation parameters given in the “Methods” section and the setup shown in Fig. 1a unchanged, we remove the axicon and simulate the propagation of the pulsed beam around the image relay position. Figure 5a shows the optical fields at different positions of z = 0, 0.005Z_{R}, 0.01Z_{R}, and 0.015Z_{R} (where z = 0 is the image relay position and Z_{R} = 4 m is the Rayleigh length of the corresponding monochromatic Gaussian beam) and no serious diffraction distortion is found. When the Gaussian pulse bandwidth (FWHM) is increased from 10 to 20 nm, Fig. 5b shows after z = 0.005Z_{R} the propagation diffraction seriously distorts the pulsed beam. Keeping the 10 nm bandwidth unchanged, when the PV value of the pulsefront deformation is enlarged from 300 to 600 fs, Fig. 5c shows that also after z = 0.005Z_{R} the propagation diffraction seriously distorts the pulsed beam. In this case, for a broadband pulsed beam or a largevalue pulsefront deformation, the propagation distance is limited by propagation diffraction. In this article, the length of the superposition region is around 115 mm (i.e., the beam waist 1 mm divided by the (half) conical angle 0.5°). If the pulsefront deformed pulsed beam is imaged into the center of the superposition region of z = 0, as shown in Fig. 5a, within the propagation range of (−0.015Z_{R}, 0.015Z_{R}) the spatiotemporal distribution has no serious distortion, which can cover the whole superposition range. However, if the pulse bandwidth or the pulsefront deformation is enhanced, the flying length of the optical wavepacket is reduced. If the (half) conical angle α is enlarged to increase the variation range of the group velocity υ_{g} (see Fig. 4b), the length of the superposition region is dramatically reduced, which relaxes the limitation induced by the propagation diffraction. We should also emphasize that in theory, the ideal Bessel beam can propagate over an infinite distance without any spread. However, in a real experiment, due to a finite beam aperture, the propagation invariant length is restricted. In this article, the pulsefront deformation is added onto the input pulsed beam, which enhances the propagation diffraction and accordingly shortens the propagation invariant length. In this case, the optical wavepacket slightly diverges during propagation, although both the quantitative analysis in this paragraph and the simulations in Figs. 2a, c and 5a show that the diffraction distortion within the conical superposition region is small.
Using the spatiotemporal coupling to control the group velocity and acceleration of an optical wavepacket or a focused intensity peak recently is an interesting and valuable technology. The spatiotemporal spectrum method invented by Abouraddy et al.^{65} can change the group velocity of the optical wavepacket in a very large range by changing the tilt angle of the plane (k_{z}, ω/c) with respect to the lightcone (k_{x}, k_{z}, ω/c) in spectralspace. Group velocities varying from −4c (in the backward direction) to 30c (in the forward direction) were measured, and in theory, arbitrary group velocities can be generated. The spatiotemporal dispersion method simultaneously demonstrated by Quéré et al. and Froula et al. can also adjust the group velocity of the focused intensity peak within a large range by changing the longitudinal chromatism and the temporal chirp^{69,70,71}, and −0.09c to 39c flying focuses were measured in experiments. This work of the pulsefront deformed Bessel beam generation provides a third spatiotemporal coupling method to control the group velocity and acceleration^{72}. Compared with the previous two methods, it provides an opportunity to precisely control the variations of both the group velocity and the acceleration of the optical wavepacket, although the tunable range of the group velocity is limited by the amount of the pulsefront deformation. Because the diversity of the pulsefront deformation makes the diversity of the group velocity (also the acceleration) possible, we can create some optical wavepackets with unusual motion forms, for example, the optical wavepacket with a nearly programmable group velocity (or acceleration) theoretically demonstrated in this article.
In conclusion, we have theoretically demonstrated an optical wavepacket having a nearly programmable group velocity by introducing a complex axisymmetric pulsefront deformation into the traditional Bessel beam generation. Different from the previous results of optical wavepackets displaying only a single motion form (superluminal or luminal or subluminal for the case of group velocity; and accelerating or uniformmotion or decelerating for the case of acceleration), the optical wavepackets here can propagate with nearly programmable motion forms during a single propagation path (e.g., superluminal followed by subluminal for the case of group velocity; and accelerating followed by decelerating for the case of acceleration). In this article, due to a periodically distributed pulsefront deformation along the transverse axis, the optical wavepacket propagates with superluminal and subluminal group velocities periodically along the longitudinal axis, and the corresponding instantaneous acceleration also varies between negative and positive values periodically. In this case, the propagating timedependent motions can be well controlled by carefully optimizing the shape of the pulsefront deformation, creating optical wavepackets with unusual motion forms. As regards applications, we believe it can be used in some propagating velocity matched experiments, such as bioimaging, particlemanipulation, particle acceleration, and radiation generation^{75,76,77}, and the high spatiotemporal controllability could also offer new opportunities for fundamental studies in optics and physics.
Methods
Simulation model
The ideal thin axicon transfers a plane wave into a conical wave and generates the Bessel beam in the superposition region. Due to the axisymmetric distribution, the description is carried out in the 2D lateral plane containing the propagation axis for simplification, for example the x−z or y−z plane shown in Fig. 1a, and the divided half beams have individual traveling directions symmetrically bout the propagation axis. Both the input beam and the generated Bessel beam are described in the coordinate system of r−z, where r is the radial axis and z is the propagation axis. The divided half beam after the axicon is described in its own propagation (rotated) 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 the same location at the geometrical center of the superposition region. When the clockwise rotation of the (half) conical angle α induced by the axicon is defined as the positive, and two coordinate systems of r−z and r_{α}−z_{α} satisfy the rotation relationship
Under the paraxial approximation and the plane wave approximation, at the center of the superposition region (or the coordinate origin), the spectral optical field of a divided half beam is described in its rotated coordinate system of r_{α}−z_{α} and given by
where A(r_{α}) and A(ω) are the spatial and spectral profiles of the amplitude, respectively, A_{SLM}(r_{α}) is the spatial amplitude modulation by SLM, ϕ_{DM}(r_{α}) and ϕ_{SLM}(r_{α}) are the phasemodulations by DM and SLM, respectively. In this article, because of the image relay by the parabola telescope, we assume the shaped pulsed beam appears at the image relay position of z_{α} = 0. The spatial amplitude modulation A_{SLM}(r_{α}) by SLM is due to the netlike pixel−pixel gaps, which is described as
where N is integer, rect() is the rectangularfunction, d is the SLM pixel−pixel gap, p is the overall size of the SLM pixel (including the gap d), and then p−d is the effective size of the SLM pixel. Because DM has a continuous surface, it has no spatial amplitude modulation and only introduces a continuous phase modulation across the beam aperture. In this article, two axisymmetric cosinefunctionlike phasemodulations ϕ_{DM}(r_{α}) by DM are used, respectively, and given by
where k is the wavenumber, and L and D are the longitudinal PV value and the transverse period of the modulation, respectively. SLM in theory needs to introduce a conjugated [0, 2π) phasecorrection for restoring a plane phasefront. However, the influence of the spatial resolution limited by the pixel size should be considered, and then the phasefront correction ϕ_{SLM}(r_{α}) by SLM is described as
where ⌊⌋ is the floorfunction that gives as output the greatest integer less than or equal to the input, mod() is the modulofunction to return the positive remainder of a division, and k_{0} is the wavenumber for the center wavelength (corresponding to SLM). The floorfunction describes the spatially discrete phase induced by the SLM pixel size p, the modulofunction describes the [0, 2π) phasevariation by SLM, and finally the conjugated phasecorrection is produced and moved to the region of (−π, π].
The angular spectrum method is used for modeling the propagation diffraction, and the optical field after z_{α} propagation is given by
where, at the initial position of z_{α} = 0, the planewave angular spectrum and the corresponding optical field satisfy the Fouriertransform relationship
After propagation, by using Eq. (2), the spectral optical fields of two divided half beams are described in the nonrotated coordinate system of r−z, and the coherent superposition (or interference) is given by
The temporal optical field is obtained by the Fouriertransform in spectrum and given by
Finally, because the pulsed beam is axisymmetric about the propagation axis (zaxis), the 3D distribution is achieved by rotating the 2D result about the zaxis with r^{2} = x^{2} + y^{2}.
Simulation parameters
Throughout this article, the parameters of the setup shown in Fig. 1a are as follows: the input Gaussian pulse has a 800 nm center wavelength and a 10 nm (FWHM) bandwidth; the beam diameter before and after the parabola telescope is 20 and 2 mm, respectively; DM introduces a cosinefunctionlike axisymmetric deformation with a 10 mm period and a 90 μm (300 fs) PV value; SLM has [−π, π) (corresponding to the center wavelength) phasecorrection capability and a 40 μm pixel size including a 5 μm pixel−pixel gap; the parabola telescope introduces perfect ten times beam reduction; and the ideal thin axicon introduces a α = ±0.5° (half) conical angle. DM and SLM are positioned much closed to each other, and the beam divergence after DM is neglected. The parabola telescope images the pulsefront deformed pulsed beam into the geometrical center of the superposition region formed by the ideal thin axicon, where each divided half pulsed beam is assumed to have a sixorder superGaussian beam profile in the lateral plane. All simulations are accomplished in the time domain with 1.2 ps window size and 1 fs accuracy (and the corresponding spectrum domain satisfies the Fourier relationship) and in the space domain with 100 mm window size and 2 μm accuracy (and the corresponding angular spectrum domain satisfies the Fourier relationship).
Group velocity and acceleration calculation
Under the approximation with a distortionfree pulsefront in a finite propagation length, the equations for the instantaneous group velocity and acceleration of the optical wavepacket are derived. Figure 6 shows, from the propagating time t_{0} to t, the propagation distance along the z_{α}axis in the rotated coordinate system of r_{α}−z_{α} is given by
where t_{0} is the initial time when the phasefronts arrive at the backsurface of the ideal thin axicon, and the propagation distance of the intersection of the phasefronts (or the intersection of the phasefront and the zaxis) along the zaxis in the nonrotated coordinate system of z−r is given by
The propagation distance of the optical wavepacket along the zaxis in the coordinate system of r−z is given by
where Δz(t) is the instantaneous longitudinal gap between the intersection of the phasefronts and the optical wavepacket, satisfying
The function z = f_{t0}(r) is the distribution of the deformed pulsefront in the coordinate system of r−z at the initial propagating time t_{0}. In this article, the initial pulsefront distribution in the rotated coordinate system of r_{α}−z_{α} can be obtained by dividing Eq. (5) with the wavenumber k, and then, using Eq. (2), which in the nonrotated coordinate system of r−z can be obtained conveniently.
The instantaneous group velocity of the optical wavepacket is given by
and by the substitution of Eq. (15) with Eq. (13), the instantaneous group velocity can also be described as
Equation (16) shows the instantaneous group velocity is relevant to two terms: the first term is the constant velocity of the intersection of the phasefronts; the second term is the variable velocity related to the change of the longitudinal gap Δz (between the intersection of the phasefronts and the optical wavepacket), which is eventually dominated by the pulsefront deformation. This indicates why the pulsefront deformation can change the group velocity of the optical wavepacket and conforms to the phenomenon shown in Fig. 2.
By the substitution of Eq. (16) with Eq. (14), the instantaneous group velocity can be redescribed as
where f_{t0}^{′}(r) is the firstorder derivative of the pulsefront function f_{t0}(r). The instantaneous acceleration of the optical wavepacket satisfies
and by the substitution of Eq. (18) with Eq. (17), it is described as
where f_{t0}″(r) is the secondorder derivative of the pulsefront function f_{t0}(r). Equations (17) and (19) show, for a certain pulsefront deformation of z = f_{t0}(r), the instantaneous group velocity and acceleration of the optical wavepacket produced by this method can be directly calculated and, more importantly, well designed.
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, carried out the simulation, and derived the equations. Z.L. and J.K. prepared the manuscript. All authors discussed the results and commented on the manuscript.
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Li, Z., Kawanaka, J. Optical wavepacket with nearlyprogrammable group velocities. Commun Phys 3, 211 (2020). https://doi.org/10.1038/s42005020004814
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DOI: https://doi.org/10.1038/s42005020004814
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