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Abstract
Spatiotemporal pulse shaping provides control over the trajectory and range of an intensity peak. While this control can enhance laser-based applications, the optical configurations required for shaping the pulse can constrain the transverse or temporal profile, duration, or orbital angular momentum (OAM). Here we present a novel technique for spatiotemporal control that mitigates these constraints by using a "stencil" pulse to spatiotemporally structure a second, primary pulse through cross-phase modulation (XPM) in a Kerr lens. The temporally shaped stencil pulse induces a time-dependent focusing phase within the primary pulse. This technique, the "flying focus X," allows the primary pulse to have any profile or OAM, expanding the flexibility of spatiotemporal pulse shaping for laser-based applications. As an example, simulations show that the flying focus X can deliver an arbitrary-velocity, variable-duration intensity peak with OAM over distances much longer than a Rayleigh range.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Spatiotemporal pulse shaping offers a new paradigm for controlling laser intensity. By exploiting space-time correlations in the amplitude or phase of a laser pulse, a number of recent techniques have created arbitrary-velocity intensity peaks that remain nearly propagation invariant over distances much longer than a Rayleigh range [1–8]. These features promise to revolutionize a wide range of laser-based applications, including high-power amplifiers [9], radiation sources [10–12], and compact accelerators [6,7,13–15]. Nevertheless, each of the existing techniques requires an optical configuration that constrains properties of the intensity peak, such as its transverse or temporal profile, duration, or orbital angular momentum (OAM). As an example, the original, “chromatic" flying focus uses a chirp and a chromatic lens to control the time and location at which each temporal slice of a laser pulse comes to focus, respectively [2,3]. While this technique offers some flexibility to shape the transverse profile of the far-field, the chromatic aberration and chirp can place a lower bound on the duration of the intensity peak that is much larger than the transform-limited duration [2,16,17]. An alternative technique, the “achromatic" flying focus, employs the spherical aberration of an axiparabola [18,19] to focus different annuli in the near-field to different axial locations in the far-field and an echelon to adjust their relative timing [6]. Here, the intensity peak can have a near transform-limited duration, but the flattop transverse profile required in the near-field and the spherical aberration of the axiparabola fully determine the far-field profile.
Both the chromatic and achromatic flying focus techniques use linear optical elements to structure a pulse with space-time correlations, but nonlinear processes, such as self-phase modulation, can also produce space-time correlations [20–24]. Self-phase modulation (SPM) refers to the phase accrued by a laser pulse as it traverses a nonlinear medium and modifies the refractive index. In a Kerr nonlinear medium, the change in the refractive index ($\delta n$) is proportional to the intensity ($I$): $\delta n = n_2I$, where $n_2$ is the nonlinear refractive index and linear polarization is assumed throughout. When $n_2>0$, the phase velocity is reduced in regions of high intensity, creating a wavefront curvature that can counteract diffraction, i.e., the pulse self-focuses [20,21,25,26]. At sufficiently high power, this curvature can even overcome diffraction, resulting in the transverse collapse of the pulse until the intensity reaches a threshold for activating an arrest mechanism [23,27–29]. The recently described “self-flying focus” combines this effect with temporal pulse shaping to control the trajectory of an intensity peak in a Kerr medium [24]. This approach can deliver an arbitrary velocity intensity peak over distances comparable to the focal length, but nonlinear evolution of the transverse and temporal profiles during self-focusing collapse and arrest significantly complicate any attempt at far-field shaping. Further, the arrest mechanism, e.g., ionization, can irreversibly damage the medium.
Cross-phase modulation (XPM) refers to the phase that one laser pulse induces onto another as they copropagate through a nonlinear medium [30,31]. For two pulses ($P$ and $S$ for specificity) overlapped in a Kerr medium, the change in refractive index experienced by pulse $P$ ($\delta n_P$) will depend on both its intensity ($I_P$) and that of pulse $S$ ($I_S$): $\delta n_P = n_2(I_P + \alpha I_S)$, where $\alpha$ depends on their relative polarizations. As a consequence, the phase velocity of pulse $P$ can be lower in regions where either pulse has a high intensity. If $I_S\gg I_P$, the intensity profile of pulse $S$ will dictate the local phase velocity and resulting wavefront curvature acquired by pulse $P$. That is, a high-intensity pulse can induce the focusing of a colocated low-intensity pulse [32–34].
Here we describe a novel technique for spatiotemporal control: the “flying focus X,” which combines temporal pulse shaping with XPM to produce an ultrashort, arbitrary velocity intensity peak, with or without OAM, over distances far greater than a Rayleigh range. Specifically, a temporally shaped, high-intensity “stencil" pulse induces the time-dependent focusing of a second, primary pulse through XPM in a Kerr lens—a lens-shaped optic in which the refractive index changes dynamically via the Kerr effect (Fig. 1). The minimum and maximum intensity of the stencil pulse ($S$) set the focal range of the primary pulse ($P$), while its duration sets the velocity of the resulting intensity peak. The use of a stencil pulse requires more laser energy than other techniques, but mitigates the constraints on the primary pulse, allowing for spatiotemporal control independent of properties such as the far-field duration, transverse profile, or OAM. In effect, these constraints are offloaded onto the stencil pulse. Specifically, the flying focus X improves upon the "chromatic" flying focus by removing the lower bound on the duration imposed by chromatic aberration, the "achromatic" flying focus by allowing for any transverse profile, and the self-flying focus by avoiding nonlinear modifications to the profile of the pulse in the far field. As a result, the flying focus X provides unprecedented flexibility for structuring the far-field properties of the intensity peak, promising to further enable or enhance a wide range of laser-based applications.
Fig. 1. Schematic of the flying focus X. A high-intensity stencil pulse with a flattop transverse and shaped temporal profile co-propagates with a low-intensity primary pulse. The stencil induces a time-dependent focusing phase on the primary in a parabolic Kerr lens. A preliminary parabolic mirror (not shown) compensates the linear focusing of the Kerr lens. A filter removes the stencil, while an ideal lens provides additional focusing for the primary. The resulting focus travels over a distance $L_f$ at a velocity $v_f$, both of which can be tuned by shaping the temporal profile of the stencil. A stencil pulse with an intensity that ramps up in time can produce either negative (left) or positive superluminal (right) focal velocities, depending on its duration ($\tau _S$). On the left: despite having a longer focal length, the leading temporal slice of the primary pulse is sufficiently advanced with respect to the trailing slice that it comes to focus earlier in time. On the right: despite being delayed with respect to the initial time slice, the trailing temporal slice has a shorter focal length and comes to focus earlier in time.
2. Flying focus X
Figure 1 illustrates how temporal pulse shaping and XPM can be used to control the trajectory and range of an intensity peak. A stencil pulse and a primary pulse, overlapped in space and time, co-propagate along the z axis and enter a convex, parabolic Kerr lens. The two pulses have distinct wavelengths or polarization vectors and $I_S \gg I_P$. The stencil pulse has a flattop transverse profile, which, in combination with the parabolic shape of the Kerr lens, ensures that the primary pulse acquires an aberration-free wavefront curvature. The instantaneous intensity of the stencil dictates the time dependence of that curvature. Specifically, the primary pulse exits the Kerr lens with the phase
where $\tau = t - z/c$ denotes time within the primary pulse, $k_0$ its vacuum wavenumber, and $R$ the Kerr lens radius of curvature (see Appendix I). Here the linear focusing phase of the Kerr lens has been compensated by a preliminary parabolic mirror. The phase ($\phi _K$) is equivalent to that applied by an ideal lens, but with a time-dependent focal length: $f_K(\tau )=R/\alpha n_2I_s(\tau )$. After the Kerr lens, a wavelength or polarization filter removes the stencil pulse. The primary pulse continues through a second, linear lens such that its total phase ($\phi _P$) becomes $\phi _P = \phi _K + \phi _l$, where $\phi _l = -k_0r^{2}/2f_l$ and $f_l$ is the linear lens focal length.Upon exiting the linear lens, each time slice within the primary pulse will have a focal length determined by its total wavefront curvature:
A time slice that was overlapped with a low stencil intensity in the Kerr lens will focus farther from the optical assembly and closer to the linear lens focal point; a time slice that was overlapped with a high stencil intensity will focus closer to the optical assembly and farther from the linear lens focal point. Over the duration of the primary pulse, the focal point will move within a range ($L_f$) defined by the minimum and maximum intensity of the stencil pulse:By appropriately shaping the temporal profile of the stencil pulse, the focal point can be made to move through the focal region with a desired trajectory. Each time slice within the primary pulse will focus at a time $t_f = \tau + f(\tau )/c$. A general focal trajectory, $f(t_f)$, can be found by iterating to solve for $f(\tau )$. For a focal point moving at a constant velocity ($v_f$), $f(\tau ) = f_0 + u\tau$, where $f_0$ is the focal point of the initial time slice and $u = cv_f/(c-v_f)$ is the reduced velocity. Using this expression for $f(\tau )$, one can invert Eq. (2) to find the required temporal profile of the stencil pulse:
For either a superluminal ($v_f>c$) or backwards ($v_f < 0$) focal velocity, $u<0$ and the stencil intensity must ramp up in time [Fig. 2(a)]. For a subluminal focal velocity ($0<v_f<c$), $u>0$ and the stencil intensity must ramp down in time. The total duration of the stencil and primary pulses ($\tau _S$) depends only on the focal range and the reduced velocity, i.e., $\tau _S = L_f/|u|$. Because the focal range ($L_f$) is typically much smaller than the linear lens focal length ($f_l$), the stencil profile can often be approximated as a simple linear ramp: $I_S(\tau ) \approx (R/\alpha f_0n_2)(1 - f_0/f_l - u\tau /f_0)$.Fig. 2. Design space for a flying focus X pulse. (a) The intensity profile for a ramp-up ($u<0$) stencil pulse as a function of time. Starting at $\tau =0$ represents a pulse with $f_0 = f_l$; starting at a time $\tau =\Delta \tau$ represents a pulse with $f_0 = f_l - L_f\Delta \tau /\tau _S$. The intensity profile for a ramp-down pulse ($u>0$) corresponds to flipping the horizontal axis and appropriately shifting in time for a particular $f_0$. (b) The focal velocity as a function of the stencil pulse duration and focal range.
From the stencil pulse duration, the focal velocity can be expressed in terms of physical parameters of the flying focus X: $v_f/c = (1\mp c\tau _S/L_f)^{-1}$, where the minus and plus signs correspond to ramp-up ($u<0$) and ramp-down ($u>0$) pulses, respectively [Fig. 2(b)]. This illustrates two points. First, for a desired focal range, the focal velocity can be tuned by adjusting the stencil pulse duration. Second, the transition from a superluminal to backwards travelling focus occurs when the pulse duration of a ramp-up pulse ($u<0$) becomes longer than the focal range (i.e., $\tau _S>L_f/c$).
The primary pulse can have any initial transverse profile. Because the combined Kerr and linear lenses create an ideal parabolic wavefront, a natural choice is a Laguerre-Gaussian (LG) mode, which, in addition to allowing for OAM, has well-known expressions for its spot size, radius of curvature, and Gouy phase. With this choice, the spot size for each time slice within the primary pulse is given by $w_P(\tau )=w_0[(1-z/f(\tau ))^{2} + (z/Z_0)^{2} ]^{1/2}$, where $w_0$ is the spot size in the near-field ($z=0$) and $Z_0 = k_0w_0^{2}/2$. From the spot size and conservation of power in each time slice of the primary pulse, one can derive the spatiotemporal profile of the intensity peak
The duration of the intensity peak ($\tau _f$) can be interpreted as the rate at which adjacent time slices within the primary pulse come in and out of focus, i.e., $Z_R/|u|$. The duration can be tuned by adjusting the focal geometry: stronger (weaker) focusing shortens (lengthens) the Rayleigh range, which decreases (increases) the duration. In general, the intensity peak can have an ultrashort duration that is much smaller than that of the primary pulse. The time-dependent refractive index experienced by the primary pulse in the Kerr lens provides the bandwidth necessary to support the ultrashort duration of the intensity peak in the far-field. The total bandwidth acquired by the primary pulse can be calculated using the linear ramp approximation for the stencil profile and the Kerr lens phase (Eq. (1)): $\Delta \omega = 2\partial _{\tau } \phi _K |_{r=w_0}$, or $\Delta \omega = 2 |u|/Z_R \propto \tau _f^{-1}$, consistent with Eq. (6). Note that the required bandwidth increases rapidly as $v_f$ approaches $c$. As an example, with $v_f = 1.01\,c$ and $Z_R = 300\,\mathrm {\mu m}$, $\tau _f = 10 \,\mathrm {fs}$.
In order for the intensity peak to have a constant maximum throughout the focal region, the primary pulse must have a flattop intensity profile in time with a duration equal to that of the stencil pulse ($\tau _S$). By shaping the temporal intensity profile of the primary pulse ($I_{P0}$), the intensity peak can also exhibit more exotic patterns. For instance, modulating the temporal profile of the primary pulse will create an oscillating intensity distribution along the optical axis, similar to Ref. [16].
Designing a flying focus X pulse for an experiment or application follows a straightforward procedure. Several laser-based applications require (1) sustaining a high intensity over an extended distance and (2) that the velocity of the intensity peak conform to some process [35–48]. The first step is then to specify the focal range ($L_f$) and the focal velocity ($v_f$). Next, one would set the focal geometry ($f_l$) based on either a desired far-field spot size ($w_f$) or intensity peak duration ($\tau _f$). As demonstrated by Eq. (6), these two properties are interdependent. However, for a fixed $\tau _f$, a larger root-mean-square spot size can be achieved by using higher-order Laguerre-Gaussian modes. Finally, one would select a Kerr lens material with a large $n_2$, low absorption and weak nonlinear dispersion at the primary and stencil wavelengths, and the smallest possible radius of curvature. At optical frequencies, for instance, ZnS is an excellent candidate [49]. With all other parameters specified, the duration and profile of the stencil pulse can be calculated from $\tau _S = (L_f/c)|c/v_f-1|$ [Fig 2(b)] and Eq. (2), respectively. Note that while the previous discussion considered a positive Kerr lens, the flying focus X can also be designed with a negative Kerr lens (see Appendix II).
3. Simulation results
To demonstrate the versatility of the flying focus X, simulations (see Appendix III for details) were conducted for two focal velocities, each with relevance to laser-based applications (Table 1). The initial transverse structure of the primary pulse in each case was an LG$_{\ell,p}$ mode, where $\ell$ and $p$ denote the OAM and radial quantum numbers, respectively. The first example features a superluminal ($v_f = 1.01 c$) intensity peak with $p=\ell =0$, i.e., a standard transverse Gaussian profile, ideal for compact accelerators (Fig. 3) [6,15,44]. The second example features a backwards ($v_f=-c$) travelling intensity peak with $p=0$ and $\ell = 1$, i.e., a “donut" mode, ideal for exotic high-power amplification and photon acceleration schemes (Fig. 4) [11,50].
Table 1. Near-field parameters for the two simulated flying focus X designs. Note that only the properties of the stencil pulse change, illustrating the versatility of a single optical configuration to accommodate different focal velocities and transverse profiles. For both examples, $\lambda _0 = 2\pi /k_0 = 1.054 \,\mu \text {m}$, $L_f = 1\,\text {cm} = 33 \,Z_R$, and $w_f = 10 \,\mu \text {m}$. ZnS was used for the Kerr lens and the stencil and primary pulses had parallel polarizations, such that $\alpha = 2$ [30,49]. For a near-field primary pulse intensity of $I_{P,\mathrm {max}} = 4\times 10^{7}\,\mathrm {W/cm^{2}}$, the intensity at focus is $I_f = 1.6\times 10^{14}\,\mathrm {W/cm^{2}}$.
Fig. 3. Simulation results for an LG$_{0,0}$ flying focus with $v_f = 1.01 c$, demonstrating velocity control, an ultrashort duration, and near propagation invariance through the focal region. (a) The temporal profile of the stencil and primary pulses at the entrance of the Kerr lens. (b) The transverse electric field of the primary pulse at the entrance of the Kerr lens. (c) The on-axis intensity of the primary pulse in the far-field and a lineout of its temporal profile. Note that the abscissa is $\tau = t-z/c$ and that a lab frame trajectory $z = v_ft$ is equivalent to $z = cv_f\tau /(c-v_f)$, which, for $v_f > c$, has a negative slope. (d) The fluence of the primary pulse in the far-field. All values are normalized to their respective maxima.
Figure 3 illustrates that the flying focus X can create an ultrashort-duration intensity peak traveling near the vacuum speed of light over many Rayleigh ranges. As described by Eq. (4), the intensity of the stencil pulse ramps up linearly in time. The primary pulse has a super-Gaussian (SG20) temporal profile to ensure a constant intensity in the far-field. Throughout the focal region, spanning $z=59\,\text {cm}$ to $z=60\,\text {cm}$, the intensity peak remains ultrashort, with a duration comparable to the value predicted by Eq. (6), i.e., $\tau _f = 10\,\text {fs}$. In terms of the moving frame coordinate ($\tau = t - z/c$), the trajectory of the intensity peak ($z = v_ft$) is given by $z=cv_f\tau /(c-v_f)$, such that $\frac {dz}{d\tau }<0$ in Fig. 3(c). The intensity peak maintains its transverse Gaussian profile with a diffraction-limited spot over the entire focal range.
Figure 4 illustrates that the flying focus X can create an intensity peak with OAM that counterpropagates with respect to its phase fronts over many Rayleigh ranges. The stencil and primary pulses have the same temporal shape as in Fig. 3, but have much longer durations to accommodate the negative focal velocity [Fig. 2(b)]. As predicted by Eq. (6), the negative focal velocity results in the intensity peak having a longer effective duration, $\tau _{f}\approx 2\,\text {ps}$. The intensity peak maintains its OAM profile with a diffraction-limited spot over the entire focal range—the spot radius and transverse profile are that of a focused LG$_{1,0}$ mode, and no intensity modulation due to Gouy phase interference between additional transverse modes is observed.
Fig. 4. Simulation results for an LG$_{1,0}$ flying focus with $v_f = -c$, demonstrating velocity control, a short duration, and near propagation invariance through the focal region. (a) The temporal profile of the stencil and primary pulses at the entrance of the Kerr lens. (b) The transverse electric field of the primary pulse at the entrance of the Kerr lens. (c) The intensity of the primary pulse in the far-field at the radial location of its maximum intensity ($r = w_f/\sqrt {2}$) and a lineout of its temporal profile. Note that the abscissa is $\tau = t-z/c$ and that a lab frame trajectory $z = v_ft$ is equivalent to $z = cv_f\tau /(c-v_f)$. (d) The fluence of the primary pulse in the far-field. All values are normalized to their respective maxima.
4. Discussion
While the flying focus X offers additional flexibility to shape the far-field profile of the intensity peak, it does present some trade-offs. Foremost, the stencil pulse is filtered out and discarded after the Kerr lens, making the design inherently inefficient. This is exacerbated by the fact that the stencil pulse must be much more intense than the primary pulse to ensure that the induced focusing is not disrupted by self-phase modulation (i.e., $I_S \gg I_P/2$).
Self-phase modulation of the primary pulse can modify the time-dependent focal point and introduce spherical or higher order aberrations that broaden the confocal region of each time slice. For a flattop temporal profile, as considered here, SPM uniformly shifts the focal point for each time slice. This can be corrected by adjusting the focal length of the linear lens ($f_l$). More importantly, the spherical aberration from SPM causes each annulus of the primary pulse in the near field to focus to a different location in the far field. As a result, each annulus also focuses at the wrong time. To avoid this, one can estimate the maximum intensity of the primary pulse by equating the spread in focal lengths due to spherical aberration to the pulse length: $I_P/I_S < cf_K\tau _S/5f_l^{2}$, where a factor of 5 has been introduced based on simulations including SPM (not presented).
The inefficiency of the flying focus X can be mitigated by setting $I_{S,\text {min}}=0$, which minimizes the total energy of the stencil pulse, but fixes the maximum focal location to $f=f_l$. The constraint on the relative intensities can be eliminated if both the stencil and the primary pulses have a flattop, transverse intensity profile. From the resulting Kerr lens phase, i.e., $\phi _K(r,\tau )=-k_0n_2[\alpha I_S(\tau ) + I_P]r^{2}/2R$, the temporal profile of the stencil can be shaped to accommodate SPM for any primary intensity, while still providing control over the far-field trajectory. Specifically, $I_S(\tau )=\frac {R}{\alpha f_ln_2}(\frac {f_l}{f_0 + u\tau } -1)-\frac {1}{\alpha }I_P$. Note, however, that this approach limits the far-field transverse profile to an Airy pattern and precludes a primary pulse with OAM. To eliminate the constraint on the relative intensities for a primary pulse with OAM, the shape of the Kerr lens can be changed to compensate the transverse spatial profile of the OAM mode.
In addition to its inefficiency, the flying focus X requires stencil and primary pulses with low levels of phase and intensity noise. As with conventional optics, the primary pulse must have a clean wavefront as it enters the optical assembly to ensure near diffraction-limited focusing to the far-field. While XPM is insensitive to phase noise, the induced focusing depends sensitively on the transverse intensity profile of the stencil. A noisy stencil intensity can imprint phase aberrations onto the primary pulse that inhibit its focusing and modify the far-field properties. Further, for a large stencil intensity or in a thick Kerr lens, the intensity noise can seed filamentation [51]. To avoid unwanted, nonlinear propagation of noisy pulses, the phase acquired from XPM in the Kerr lens should be no more than a few $\pi$, i.e., $\phi _{X} = \alpha k_0n_2I_S\Delta z \lesssim 3\pi$, where $\Delta z$ is the maximum thickness of the lens [31,52]. In practice, this amounts to limiting the focal range to a few Rayleigh ranges. The use of low-noise pulses significantly relaxes this condition: $w_0^{-2}n_2I_S(\Delta z)^{2} \ll 1$, i.e., the self-focusing collapse distance must be much larger than the lens thickness [20]. Together, the conditions to avoid self-phase modulation and filamentation place an upper bound on far-field intensity of the flying focus X ($I_{f,\text {max}}$): for clean beams, $I_{f,\text {max}} \ll c\tau _Sf_KR^{2}/f_l^{2}n_2w_f^{2}$ or $I_{f,\text {max}} \ll 1\times 10^{18} \,\mathrm {W/cm^{2}}$ for the parameters in Table 1. Thus, the requirements of a low stencil pulse intensity in the near-field do not preclude a high intensity in the far-field.
Finally, the stencil pulse must have a flattop transverse profile and a ramped temporal profile. A flattop transverse profile can be achieved using a $\pi$Shaper [53], while a ramped temporal profile can be achieved using either an acousto-optic programmable dispersion filter [54] or a grating pair with a spatial light modulator [55].
Given these trade-offs, a flying focus X with a far-field intensity $I_{f}<1\times 10^{17}\,\mathrm {W/cm^{2}}$ can enable or improve a wide-range of laser-based applications, including the formation of long plasma channels for advanced accelerators or directed energy [16], photon accelerators for creating ultrashort pulses of extreme ultraviolet radiation [11], Raman amplification with quasi-stationary plasma conditions [9], and laser wakefield accelerators with minimal phase-slippage between relativistic electrons and the plasma wave [6,7]. For creating long plasma channels, a backwards travelling intensity peak, as in Fig. 4, mitigates plasma refraction, allowing for the creation of long, contiguous plasma columns. Temporal slices that occur later in the pulse focus closer to the optics assembly and ionize the medium before they propagate through the downstream plasma created by earlier time slices. For photon acceleration, a superluminal flying focus X, composed of orthogonally polarized LG$_{0,0}$ and LG$_{1,0}$ modes, can ionize a background medium, creating a plasma density profile that guides the primary pulse as it frequency upshifts in the moving refractive index gradient. With the accompanying increase in group velocity, the upshifted photons catch up to the unshifted, downstream photons that are coming into focus. This forms an optical shock that enhances the ionization and the frequency upshifting. For Raman amplification, the flying focus X can provide a pump pulse with a backwards travelling intensity peak, which, by ensuring that the seed pulse experiences near-constant plasma conditions throughout the interaction, increases the transfer of energy from the pump to the seed. Finally, for laser wakefield acceleration, a superluminal flying focus X can compensate the subluminal group velocity in a plasma and drive a plasma wave with a phase velocity equal to or greater than the vacuum speed of light. This can prevent relativistic electrons from outrunning the accelerating phase of the wave, greatly extending the acceleration length.
5. Summary and conclusions
A new technique for spatiotemporal control, the flying focus X, can create an ultrashort, arbitrary trajectory intensity peak, with any transverse profile or OAM, over distances much greater than a Rayleigh range. The technique combines cross-phase modulation in a Kerr lens with temporal pulse shaping. In a Kerr lens, a shaped auxiliary, or “stencil," pulse imprints a second, primary pulse with a time-dependent focusing phase. In the far-field, the intensity peak formed by the time-dependent focusing of the primary pulse follows a trajectory dictated by the shape of the stencil. Depending on the shape, the peak can move forward or backwards and at any velocity. The use of a stencil pulse and XPM requires more laser energy, but overcomes constraints inherent to the optical configurations employed in previous techniques for spatiotemporal control. Specifically, the flying focus X provides independent control over the shape of the transverse profile, OAM, and duration of the intensity peak. Ultimately, this additional flexibility could further enhance the utility of spatiotemporally shaped pulses for laser-based applications.
Appendix I: Kerr lens model
To determine the profile of the primary pulse after the Kerr lens, consider the wave equation for the transverse electric field ($\text {E}_P$) in a time-dependent, Kerr-nonlinear medium:
where $n(\textbf {x},t) = n_0(\textbf {x})+\alpha n_2(\textbf {x})I_S(\tau )$ is the total refractive index, $n_0$ is the linear refractive index, $\tau = t - \int {n_0dz}/c$, and $I_S \gg I_P$ has been assumed. The spatial dependence of $n_0(\textbf {x})$ and $n_2(\textbf {x})$ indicates that the refractive index changes across the Kerr lens-vacuum interface: outside the lens $n_0=1$ and $n_2=0$. Upon switching to the coordinates $z$ and $\tau$, using $n_0\gg \alpha n_2I_S$, and noting that the lens is thin enough to neglect diffraction, Eq. (7) becomes This equation admits analytic solutions. For the special case of $I_S(\tau ) = (R/\alpha f_0n_2)(1 - f_0/f_l - u\tau /f_0)$, valid when $L_f \ll f_l$, the solution is $\text {E}_P = e^{\int \hspace {-.2em}\kappa dz}\text {E}_0[\textbf {r},\psi (\textbf {x},\tau )]$, where $\text {E}_0[\textbf {r},\tau ]$ is the electric field of the primary pulse at the entrance to the Kerr lens, $\kappa = Ru/cf_0^{2}$, $\psi (\textbf {x},\tau ) = e^{\int \hspace {-.2em}\kappa dz}(\tau -\gamma ) + \gamma$, and $\gamma = (1-f_0/f_l)(f_0/u)$. The exponential factor captures the temporal broadening (compression) and the corresponding decrease (increase) in amplitude resulting from the time-dependent refractive index induced by the ramp-up (down) stencil.The boundaries of the Kerr lens are defined by its entrance plane at $z=z_0<0$ and its exit surface: $s(r)=z_1-r^{2}/2R$, where $z_0<z_1<0$. The thickness and maximum radius are then $d = z_1-z_0$ and $r_{\text {max}} = (2Rd)^{1/2}$, respectively. For the Kerr lenses of interest, $\kappa d \ll 1$. As a result, the exponential factor in $\psi$ can be Taylor expanded. At the exit of the Kerr lens, i.e., $z=z_1$, $\int \hspace {-.2em}\kappa dz = \kappa (d-r^{2}/2R)$ and
The electric field of the primary pulse at the entrance to the Kerr lens can be expressed as a rapidly oscillating carrier modulating a slowly varying envelope: $\text {E}_0 = \frac {1}{2}e^{-i\omega _0\tau }E_0(\textbf {r},\tau ) + \text {c.c.}$, where $\omega _0=ck_0$ is the central frequency of the pulse. Performing the substitution $\tau \rightarrow \psi (r,\tau )$, in accordance with the initial condition, illustrates that the lens phases acquired in the optical assembly appear in the exponential prefactor, while the radial group delay and dispersion appear in $E_0$. Inclusion of the radial group delay and dispersion are critical for correctly modeling the duration of the intensity peak.
Appendix II: Defocusing Kerr lens
A Kerr lens with a concave, quadratic shape acts as a defocusing lens. In this case, the primary pulse exits the Kerr lens with the phase
such that each time slice has a focal length This exhibits a somewhat counterintuitive effect: cross-phase modulation in a self-focusing medium (i.e., one with a positive $n_2$) can induce defocusing. Here, a time slice that was overlapped with a low stencil intensity in the Kerr lens will focus closer to the optical assembly and the linear lens focal point, while a time slice that was overlapped with a high stencil intensity will focus farther from the optical assembly and the linear lens focal point.For the same maximum stencil intensity, a defocusing Kerr lens can produce a longer focal range than a focusing Kerr lens. The focal point of a defocusing Kerr lens is bounded by $f=f_l$ and $f = \infty$, while that of a focusing Kerr lens is bounded by $f=0$ and $f = f_l$. Specifically, the focal range for a defocusing Kerr lens is given by
To create an intensity peak that travels through the focal region at a constant velocity ($v_f$), the intensity profile of the stencil pulse must have the form
where $u = cv_f/(c-v_f)$. In contrast to a focusing Kerr lens, superluminal or backwards focal velocities ($u<0$) require a stencil intensity that ramps down in time, while subluminal focal velocities ($u>0$) require a stencil intensity that ramps up in time. For focal ranges much smaller than the linear lens focal length ($L_f \ll f_l$), the stencil profile can be approximated as a simple linear ramp: $I_S(\tau ) \approx (R/\alpha f_0n_2)(f_0/f_l - 1 + u\tau /f_0)$.Appendix III: Simulation details
The simulations solved for the evolution of the primary pulse in three steps. The first step used the analytic model presented in Appendix I to determine the profile of the primary pulse after the optical assembly. The second step used a frequency domain Fresnel integral, as described in the Appendix of Ref. [16], to propagate the envelope of the primary pulse to the far-field. The final step used the modified paraxial wave equation to propagate the primary pulse through the far-field. The results shown in Figs. 3 and 4 were obtained from this last step.
To solve for the evolution of the primary pulse after the optical assembly, $\text {E}_P$ is written as a carrier modulating an envelope: $\text {E}_P = \frac {1}{2}e^{-i\omega _0\tau }E_P(\textbf {r},z,\tau ) + \text {c.c.}$, with the initial condition $E_P(\textbf {r},0,\tau ) = e^{\kappa (d-r^{2}/2R) -i\omega _0(\psi -\tau ) }E_0[\textbf {r},\psi (r,\tau )]$. Propagation from the optical assembly to the far-field is calculated using the frequency-domain Fresnel integral:
The model described above does not account for the dispersion of $n_2$ or four-wave mixing within the Kerr lens. Simulations of a 20 fs pulse using the focal geometry described in Table 1 predict that the dispersion of $n_2$ in ZnS increases the duration by $\approx 10\,\text {fs}$. Based on the relative intensities of the stencil and primary pulses and dispersion in ZnS, the strongest four-wave mixing process would be generation of light at a frequency $\omega _M = 2\omega _S - \omega _P$, where $\omega _S$ and $\omega _P$ are the frequencies of the stencil and primary pulses, respectively. Following a standard nonlinear optics calculation [31], one can find the maximum intensity of the mixed light: $I_{M,\mathrm {max}} = 4(n_P/n_M)\omega _M^{2}[2\omega _S(n_S-n_M)-\omega _P(n_P-n_M)]^{-2}(n_2I_S)^{2}I_P$, where $n_P$, $n_M$, and $n_S$ are the refractive indices evaluated at the respective frequencies. For a 10% difference in $\omega _S$ and $\omega _P$, dephasing limits the intensity of the mixed light ($I_M$) to about 2% of the primary pulse intensity, i.e., $I_M \approx.02I_P$.
Funding
U.S. Department of Energy (DE-NA0003856); Fusion Energy Sciences (DE-SC0016253).
Acknowledgments
The authors would like to thank C. Dorrer, J.L. Shaw, and K.L. Nguyen for fruitful discussions. The work published here was supported by the U.S. Department of Energy Office of Fusion Energy Sciences under contract no. DE-SC0016253, the Department of Energy under cooperative agreement no. DE-NA0003856, the University of Rochester, and the New York State Energy Research and Development Authority.
This report was prepared as an account of work sponsored by an agency of the U.S. Government. Neither the U.S. Government nor any agency thereof, nor any of their employees, makes any warranty, express or implied, or assumes any legal liability or responsibility for the accuracy, completeness, or usefulness of any information, apparatus, product, or process disclosed, or represents that its use would not infringe privately owned rights. Reference herein to any specific commercial product, process, or service by trade name, trademark, manufacturer, or otherwise does not necessarily constitute or imply its endorsement, recommendation, or favoring by the U.S. Government or any agency thereof.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
1. H. E. Kondakci and A. F. Abouraddy, “Diffraction-free space–time light sheets,” Nat. Photonics 11(11), 733–740 (2017). [CrossRef]
2. A. Sainte-Marie, O. Gobert, and F. Quere, “Controlling the velocity of ultrashort light pulses in vacuum through spatio-temporal couplings,” Optica 4(10), 1298–1304 (2017). [CrossRef]
3. D. H. Froula, D. Turnbull, A. S. Davies, T. J. Kessler, D. Haberberger, J. P. Palastro, S.-W. Bahk, I. A. Begishev, R. Boni, S. Bucht, J. Katz, and J. L. Shaw, “Spatiotemporal control of laser intensity,” Nat. Photonics 12(5), 262–265 (2018). [CrossRef]
4. H. E. Kondakci and A. F. Abouraddy, “Optical space-time wave packets having arbitrary group velocities in free space,” Nat. Commun. 10(1), 929 (2019). [CrossRef]
5. Z. Li and J. Kawanaka, “Velocity and acceleration freely tunable straight-line propagation light bullet,” Sci. Rep. 10, 1–9 (2020). [CrossRef]
6. J. P. Palastro, J. L. Shaw, P. Franke, D. Ramsey, T. T. Simpson, and D. H. Froula, “Dephasingless laser wakefield acceleration,” Phys. Rev. Lett. 124(13), 134802 (2020). [CrossRef]
7. C. Caizergues, S. Smartsev, V. Malka, and C. Thaury, “Phase-locked laser-wakefield electron acceleration,” Nat. Photonics 14(8), 475–479 (2020). [CrossRef]
8. S. W. Jolly, O. Gobert, A. Jeandet, and F. Quéré, “Controlling the velocity of a femtosecond laser pulse using refractive lenses,” Opt. Express 28(4), 4888–4897 (2020). [CrossRef]
9. D. Turnbull, S. Bucht, A. Davies, D. Haberberger, T. Kessler, J. Shaw, and D. Froula, “Raman amplification with a flying focus,” Phys. Rev. Lett. 120(2), 024801 (2018). [CrossRef]
10. A. Howard, D. Turnbull, A. Davies, P. Franke, D. Froula, and J. Palastro, “Photon acceleration in a flying focus,” Phys. Rev. Lett. 123(12), 124801 (2019). [CrossRef]
11. P. Franke, D. Ramsey, T. T. Simpson, D. Turnbull, D. H. Froula, and J. P. Palastro, “Optical shock-enhanced self-photon acceleration,” Phys. Rev. A 104(4), 043520 (2021). [CrossRef]
12. D. Ramsey, B. Malaca, A. Di Piazza, P. Franke, M. Formanek, D. H. Froula, M. Pardal, T. T. Simpson, J. Vieira, K. Weichman, and J. P. Palastro, “Nonlinear thomson scattering with ponderomotive control,” arXiv preprint arXiv:2108.04044 (2021).
13. A. Debus, R. Pausch, A. Huebl, K. Steiniger, R. Widera, T. E. Cowan, U. Schramm, and M. Bussmann, “Circumventing the dephasing and depletion limits of laser-wakefield acceleration,” Phys. Rev. X 9(3), 031044 (2019). [CrossRef]
14. D. Ramsey, P. Franke, T. T. Simpson, D. H. Froula, and J. P. Palastro, “Vacuum acceleration of electrons in a dynamic laser pulse,” Phys. Rev. E 102(4), 043207 (2020). [CrossRef]
15. J. Palastro, B. Malaca, J. Vieira, D. Ramsey, T. Simpson, P. Franke, J. Shaw, and D. Froula, “Laser-plasma acceleration beyond wave breaking,” Phys. Plasmas 28(1), 013109 (2021). [CrossRef]
16. J. P. Palastro, D. Turnbull, S.-W. Bahk, R. K. Follett, J. L. Shaw, D. Haberberger, J. Bromage, and D. H. Froula, “Ionization waves of arbitrary velocity driven by a flying focus,” Phys. Rev. A 97(3), 033835 (2018). [CrossRef]
17. D. H. Froula, J. P. Palastro, D. Turnbull, A. Davies, L. Nguyen, A. Howard, D. Ramsey, P. Franke, S.-W. Bahk, I. A. Begishev, R. Boni, J. Bromage, S. Bucht, R. K. Follett, D. Haberberger, G. W. Jenkins, J. Katz, T. J. Kessler, J. L. Shaw, and J. Vieira, “Flying focus: Spatial and temporal control of intensity for laser-based applications,” Phys. Plasmas 26(3), 032109 (2019). [CrossRef]
18. S. Smartsev, C. Caizergues, K. Oubrerie, J. Gautier, J.-P. Goddet, A. Tafzi, K. T. Phuoc, V. Malka, and C. Thaury, “Axiparabola: a long-focal-depth, high-resolution mirror for broadband high-intensity lasers,” Opt. Lett. 44(14), 3414–3417 (2019). [CrossRef]
19. K. Oubrerie, I. A. Andriyash, R. Lahaye, S. Smartsev, V. Malka, and C. Thaury, “Axiparabola: a new tool for high-intensity optics,” arXiv preprint arXiv:2106.06802 (2021).
20. J. Marburger, “Self-focusing: Theory,” Prog. Quantum Electron. 4, 35–110 (1975). [CrossRef]
21. Y. Shen, “Self-focusing: Experimental,” Prog. Quantum Electron. 4, 1–34 (1975). [CrossRef]
22. A. Brodeur, C. Chien, F. Ilkov, S. Chin, O. Kosareva, and V. Kandidov, “Moving focus in the propagation of ultrashort laser pulses in air,” Opt. Lett. 22(5), 304–306 (1997). [CrossRef]
23. A. Couairon and A. Mysyrowicz, “Femtosecond filamentation in transparent media,” Phys. Rep. 441(2-4), 47–189 (2007). [CrossRef]
24. T. T. Simpson, D. Ramsey, P. Franke, N. Vafaei-Najafabadi, D. Turnbull, D. H. Froula, and J. P. Palastro, “Nonlinear spatiotemporal control of laser intensity,” Opt. Express 28(26), 38516–38526 (2020). [CrossRef]
25. P. Kelley, “Self-focusing of optical beams,” Phys. Rev. Lett. 15(26), 1005–1008 (1965). [CrossRef]
26. G.-Z. Sun, E. Ott, Y. Lee, and P. Guzdar, “Self-focusing of short intense pulses in plasmas,” Phys. Fluids 30(2), 526–532 (1987). [CrossRef]
27. G. Fibich, “Small beam nonparaxiality arrests self-focusing of optical beams,” Phys. Rev. Lett. 76(23), 4356–4359 (1996). [CrossRef]
28. O. Bang, W. Krolikowski, J. Wyller, and J. J. Rasmussen, “Collapse arrest and soliton stabilization in nonlocal nonlinear media,” Phys. Rev. E 66(4), 046619 (2002). [CrossRef]
29. O. Kosareva, J.-F. Daigle, N. Panov, T. Wang, S. Hosseini, S. Yuan, G. Roy, V. Makarov, and S. L. Chin, “Arrest of self-focusing collapse in femtosecond air filaments: higher order kerr or plasma defocusing?” Opt. Lett. 36(7), 1035–1037 (2011). [CrossRef]
30. G. P. Agrawal, Nonlinear fiber optics (Springer, 2000).
31. R. W. Boyd, Nonlinear optics (Academic Press, 2019).
32. C. McKinstrie and D. Russell, “Nonlinear focusing of coupled waves,” Phys. Rev. Lett. 61(26), 2929–2932 (1988). [CrossRef]
33. G. P. Agrawal, “Induced focusing of optical beams in self-defocusing nonlinear media,” Phys. Rev. Lett. 64(21), 2487–2490 (1990). [CrossRef]
34. B. Hafizi, J. P. Palastro, J. R. Penano, D. F. Gordon, T. G. Jones, M. H. Helle, and D. Kaganovich, “Stimulated raman scattering and nonlinear focusing of high-power laser beams propagating in water,” Opt. Lett. 40(7), 1556–1558 (2015). [CrossRef]
35. J. Hebling, G. Almasi, I. Z. Kozma, and J. Kuhl, “Velocity matching by pulse front tilting for large-area thz-pulse generation,” Opt. Express 10(21), 1161–1166 (2002). [CrossRef]
36. C. D’Amico, A. Houard, M. Franco, B. Prade, A. Mysyrowicz, A. Couairon, and V. T. Tikhonchuk, “Conical forward thz emission from femtosecond-laser-beam filamentation in air,” Phys. Rev. Lett. 98(23), 235002 (2007). [CrossRef]
37. K.-Y. Kim, A. Taylor, J. Glownia, and G. Rodriguez, “Coherent control of terahertz supercontinuum generation in ultrafast laser–gas interactions,” Nat. Photonics 2(10), 605–609 (2008). [CrossRef]
38. A. Rundquist, C. G. Durfee, Z. Chang, C. Herne, S. Backus, M. M. Murnane, and H. C. Kapteyn, “Phase-matched generation of coherent soft x-rays,” Science 280(5368), 1412–1415 (1998). [CrossRef]
39. C. G. Durfee, A. R. Rundquist, S. Backus, C. Herne, M. M. Murnane, and H. C. Kapteyn, “Phase matching of high-order harmonics in hollow waveguides,” Phys. Rev. Lett. 83(11), 2187–2190 (1999). [CrossRef]
40. T. Popmintchev, M.-C. Chen, D. Popmintchev, P. Arpin, S. Brown, S. Ališauskas, G. Andriukaitis, T. Balčiunas, O. D. Mücke, A. Pugzlys, A. Baltuška, B. Shim, S. E. Schrauth, A. Gaeta, C. Hernández-García, L. Plaja, A. Becker, A. Jaron-Becker, M. M. Murnane, and H. C. Kapteyn, “Bright coherent ultrahigh harmonics in the kev x-ray regime from mid-infrared femtosecond lasers,” Science 336(6086), 1287–1291 (2012). [CrossRef]
41. C. Durfee III and H. Milchberg, “Light pipe for high intensity laser pulses,” Phys. Rev. Lett. 71(15), 2409–2412 (1993). [CrossRef]
42. H. Milchberg, T. Clark, C. Durfee III, T. Antonsen, and P. Mora, “Development and applications of a plasma waveguide for intense laser pulses,” Phys. Plasmas 3(5), 2149–2155 (1996). [CrossRef]
43. T. Tajima and J. M. Dawson, “Laser electron accelerator,” Phys. Rev. Lett. 43(4), 267–270 (1979). [CrossRef]
44. E. Esarey, C. Schroeder, and W. Leemans, “Physics of laser-driven plasma-based electron accelerators,” Rev. Mod. Phys. 81(3), 1229–1285 (2009). [CrossRef]
45. V. Malkin, G. Shvets, and N. Fisch, “Fast compression of laser beams to highly overcritical powers,” Phys. Rev. Lett. 82(22), 4448–4451 (1999). [CrossRef]
46. R. Trines, F. Fiuza, R. Bingham, R. Fonseca, L. Silva, R. Cairns, and P. Norreys, “Simulations of efficient raman amplification into the multipetawatt regime,” Nat. Phys. 7(1), 87–92 (2011). [CrossRef]
47. S. Wilks, J. Dawson, and W. Mori, “Frequency up-conversion of electromagnetic radiation with use of an overdense plasma,” Phys. Rev. Lett. 61(3), 337–340 (1988). [CrossRef]
48. J. Dias, C. Stenz, N. Lopes, X. Badiche, F. Blasco, A. Dos Santos, L. O. e Silva, A. Mysyrowicz, A. Antonetti, and J. Mendonça, “Experimental evidence of photon acceleration of ultrashort laser pulses in relativistic ionization fronts,” Phys. Rev. Lett. 78(25), 4773–4776 (1997). [CrossRef]
49. M. Sheik-Bahae, D. C. Hutchings, D. J. Hagan, and E. W. Van Stryland, “Dispersion of bound electron nonlinear refraction in solids,” IEEE J. Quantum Electron. 27(6), 1296–1309 (1991). [CrossRef]
50. J. Vieira, R. M. G. M. Trines, E. P. Alves, R. A. Fonseca, J. T. Mendonça, R. Bingham, P. Norreys, and L. O. Silva, “Amplification and generation of ultra-intense twisted laser pulses via stimulated raman scattering,” Nat. Commun. 7(1), 10371 (2016). [CrossRef]
51. E. Yablonovitch and N. Bloembergen, “Avalanche ionization and the limiting diameter of filaments induced by light pulses in transparent media,” Phys. Rev. Lett. 29(14), 907–910 (1972). [CrossRef]
52. G. Fibich, S. Eisenmann, B. Ilan, Y. Erlich, M. Fraenkel, Z. Henis, A. L. Gaeta, and A. Zigler, “Self-focusing distance of very high power laser pulses,” Opt. Express 13(15), 5897–5903 (2005). [CrossRef]
53. Flat top beam shapers, https://www.edmundoptics.com/f/flat-top-beam-shapers/15036/ (2021).
54. F. Verluise, V. Laude, Z. Cheng, C. Spielmann, and P. Tournois, “Amplitude and phase control of ultrashort pulses by use of an acousto-optic programmable dispersive filter: pulse compression and shaping,” Opt. Lett. 25(8), 575–577 (2000). [CrossRef]
55. A. M. Weiner, “Femtosecond pulse shaping using spatial light modulators,” Rev. Sci. Instrum. 71(5), 1929–1960 (2000). [CrossRef]
56. W. Zhu, J. Palastro, and T. Antonsen, “Studies of spectral modification and limitations of the modified paraxial equation in laser wakefield simulations,” Phys. Plasmas 19(3), 033105 (2012). [CrossRef]
