Nonlinear Spatiotemporal Control of Laser Intensity

Spatiotemporal control over the intensity of a laser pulse has the potential to enable or revolutionize a wide range of laser-based applications that currently suffer from the poor flexibility offered by conventional optics. Specifically, these optics limit the region of high intensity to the Rayleigh range and provide little to no control over the trajectory of the peak intensity. Here, we introduce a nonlinear technique for spatiotemporal control, the"self-flying focus,"that produces an arbitrary trajectory intensity peak that can be sustained for distances comparable to the focal length. The technique combines temporal pulse shaping and the inherent nonlinearity of a medium to customize the time and location at which each temporal slice within the pulse comes to its focus. As an example of its utility, simulations show that the self-flying focus can form a highly uniform, meter-scale plasma suitable for advanced plasma-based accelerators.


I. INTRODUCTION
places a lower bound on the duration of the peak, and the bandwidth of the pulse limits the range of high intensity to a small fraction of the focal length. An alternative technique employs an axiparabola [27] to focus different radial locations in the near field to different axial locations in the far field and a second optic (i.e., an echelon, spatial light modulator, or deformable mirror) to adjust the relative timing of the radial locations. Along with velocity control, the features of this technique-an extended focal range and a near transform-limited intensity peak in the far field-have enabled a novel regime of laser-wakefield acceleration that eliminates dephasing and greatly decreases the accelerator length [22,23]. Nevertheless, each velocity requires a unique axiparabola-optic pair, hindering its tunability.
Each of these methods uses linear optical elements in the near field to structure a pulse with advantageous space-time correlations, but nonlinear processes, such as self-focusing, can also give rise to these correlations [16,28,29]. Self-focusing occurs when the nonlinear optical response of a medium, quantified by the nonlinear refractive index (n 2 ), reduces the phase velocity in regions of high intensity. The ratio of the instantaneous pulse power, P (t), to the critical power, P c = ηλ 2 0 /4πn 0 n 2 , parameterizes the effect, where λ 0 is the central, vacuum wavelength of the pulse, η depends on its transverse profile, and n 0 is the linear refractive index [28]. For temporal slices within a pulse with P (t) > P c , self-focusing overcomes diffraction. These slices undergo transverse collapse until their intensity reaches a threshold for activating a mechanism that can arrest the collapse. Because the distance over which a slice collapses depends on its value of P (t)/P c , the temporal profile of the power correlates time within the pulse to a collapse location [ Fig. 1(a)].
Here we describe the first nonlinear technique for spatiotemporal control: the "self-flying focus." The technique combines temporal pulse shaping with the inherent nonlinearity of a medium to control the velocity of an intensity peak over distances comparable to the focal length. Specifically, the instantaneous power determines the collapse location for each temporal slice, with the minimum and maximum powers setting the collapse range, while the pulse shape determines the time at which the intensity peak moves through these locations.
A self-focusing arrest mechanism with an intensity threshold, such as ionization refraction, ensures that the maximum intensity of the peak remains nearly constant throughout the collapse range. Unlike linear techniques, which use specially designed optics to manipulate the space-time structure in the near field, the self-flying focus mixes temporal shaping in the near field with spatial shaping through nonlinear self-focusing in the far field. As an example of its utility, simulations demonstrate that a self-flying focus pulse with an intensity peak that counter-propagates with respect to the group velocity can create a highly uniform, meterlong plasma channel-a critical component of advanced laser and beam-driven accelerators. While the constant velocity collapse trajectory shown in Fig. 1(b) could have utility for a number of applications, the self-flying focus has much more flexibility. In fact, a power profile can be found for any desired collapse trajectory. Using a slice-by-slice picture (i.e., a moving focus model [16,29,30]) in conjunction with the source dependent expansion (SDE) method [31] provides the power profile

II. COLLAPSE DYNAMICS AND THE SELF-FLYING FOCUS
where is the desired, time-dependent collapse trajectory, v c is the collapse velocity, f is the lens focal length, w i is the spot size at the lens plane, and w f is the linear focal spot. The profile depends only on the ratio of spot sizes (w i /w f ) and the chosen collapse trajectory.
The trajectory persists over a range defined by the minimum and maximum collapse points, L c = z c (P min ) − z c (P max ), or, more explicitly, Notably, this range can be tuned through the focal geometry or power and can approach distances comparable to the focal length. Scaling P min and P max in tandem shifts the range along the propagation axis, while adjusting the difference between P min and P max alters the total distance. Alternatively, one can alter the distance through w i /w f . Decreasing w i decreases the distance it takes a temporal slice to collapse radially, while increasing w f corresponds to weaker linear focusing, which enhances the relative effect of self-focusing. 0 duration of the intensity peak formed by the collapse of adjacent time slices within the pulse can be much shorter. In media where the self-focusing arrest mechanism has an intensity threshold (I a ), e.g., due to ionization, the effective duration will scale as t e ∝ w 2 a /v c λ 0 , where w a ≈ 2P/πI a is the spot size at arrest. That is the effective duration of the intensity peak is proportional to the difference in time over which adjacent slices self-focus and refract.
While arrest always occurs before collapse, the trajectory of the resulting intensity peak will nearly match that of the collapse point as long as w i >> w a . This property-that the duration of the intensity peak can be substantially shorter than the pulse duration-opens up the possibility of using long pulses for applications that typically require short pulses. As a result, the self-flying focus could take advantage of existing long pulse, high energy laser systems with advanced pulse shaping capabilities, such as at the National Ignition Facility or the OMEGA laser. The energy expenditure for a self-flying focus pulse can be tuned to meet the requirements of a broad range of laser systems (Fig. 3). Integrating the power in Eq. (1) over time and defining the normalized energy as ε ≡ |v c |E/P c f provides where z 0 ≡ z c (0)/f and the ± accounts for positive and negative collapse velocities, respectively. For a particular lens (i.e., a focal length) and a desired L c /f , the required energy can be adjusted through w i /w f . Along an L c /f contour, the energy requirement grows with increasing spot ratio and peak power. Larger energies correlate with larger maximum powers, which, in turn, will result in larger spot sizes at arrest. As also indicated by Fig.   2, for fixed w i /w f , a larger maximum power lengthens the collapse range (L c /f ). However, this has a potential downside of greater variation in the spot size at arrest: 2P min /πI a to 2P max /πI a . The scaling E = εP c f /|v c | exhibits the expected behavior that a self-flying focus can be created with less energy in media with a stronger nonlinearity (i.e., a smaller critical power). Of particular note is that the required pulse energy decreases linearly with increasing |v c | due to the shorter pulse duration, t p = L c /|v c |. This property could be useful for applications in which precise velocity matching is unnecessary, but controlling the direction of v c is, such as the formation of plasma channels or in directed energy.
The equations presented for power, collapse range, and energy all depend on the collapse distance (z c ). No exact expression exists for this distance for an arbitrary transverse profile.
The SDE method used in Eqs. (1)-(3) assumes that the transverse profile remains Gaussian until collapse [31]; physically, a beam undergoing collapse will evolve to the characteristic Townes profile [32]. While the simplicity is convenient for developing theoretical scalings and illustrating the salient physics, more precise predictions require a collapse distance determined by simulations or measurements. In general, the self-focusing collapse distance of a focused temporal slice can be expressed as where α, β, and γ are curve-fitting parameters that depend on the transverse profile. The SDE method predicts α = 1, β = 0, and γ = 1, but a more accurate simulation analysis of a Gaussian profile conducted by Marburger found α = 0.367, β = 0.852, and γ = 0.0219 [28].
In all figures and simulations presented here, the Marburger values were used.
In arriving at Eqs. (1)- (3), it was assumed that f πw 2 f n 0 /λ 0 is the linear Rayleigh range of a focused slice. Further, the maximum power was limited to ≈ 10P c to minimize the likelihood of transverse breakup and altered scalings prior to collapse. For pulses with reasonably low levels of phase and intensity noise, this limit should be conservative and could potentially be pushed to ≈ 40P c [33]. As a final note, multiple self-focusing/refocusing cycles could extend the region of high intensity beyond L c and the lens focal length [34].

III. SIMULATION AND DEMONSTRATION
The ability to control the intensity trajectory over long distances makes the self-flying focus ideal for creating long plasma channels-a critical component in a number of applications, such as advanced laser-based accelerators and directed energy. Current techniques for creating long plasmas rely on filamentation through a dynamic balancing of self-focusing and plasma refraction [15,16], axicon focusing [35][36][37], variable wave front distortion [38], or the use of short wavelengths [39]. Axicon focusing, for example, can suffer from significant pump depletion and ionization refraction by the end of the medium due to the forward propagation of the intensity peak [36]. The self-flying focus has elements in common with filamentation, but offers velocity control and does not necessarily require a short-pulse laser.
Further, the ability to counter-propagate the intensity peak with respect to the pulse avoids ionization refraction, allowing for a wider range of focal geometries [26].
Here the self-flying focus is applied to the formation of a plasma channel necessary for the recently described "dephasingless" laser-wakefield accelerator [22,23]. The simulations solve the paraxial wave equation assuming cylindrical symmetry with source terms accounting for self-focusing, plasma refraction, and loss due to field ionization and inverse bremsstrahlung (c.f., Ref. [40]). The gas is initially neutral and the electron density evolves due to multiphoton and collisional ionization alongside radiative and threebody recombination.
The density of the lithium gas, N g = 10 19 cm −3 , was chosen to optimize the energy gain of electrons in a dephasingless laser wakefield accelerator: in the relatively compact, 1m plasma created here, electrons could be accelerated to ≈ 200 GeV. Lithium, in particular, was selected for its low ionization energy and critical power, P c ≈ 19MW at λ 0 = 1.054µm and the density of interest [41]. On account of the low critical power, the self-flying focus pulse only required ≈ 200 mJ of energy with a maximum power of ≈ 180 MW, which are well within the capability of existing laser systems. In fact, the laser and lens parameters, an f/500 lens, spot ratio w i /w f ≈ 6, and collapse range L c /f = 0.5 (f = 2 m, L c = 1 m), were chosen to ensure full ionization of the valence state while minimizing the pulse energy.
Throughout the collapse range, the maximum on-axis intensity remains nearly constant (within a factor of 2 of the maximum) despite the power varying by nearly a factor of 10.
The resulting plasma channel has a > 300µm radius consistent with the predicted spot size at arrest, w a ≈ 2P/πI a . Note that because L c /f < 1 a length of gas ahead of the collapse range is required for nonlinear focusing. As a result, not all of the gas is ionized. In principle, this can be overcome by using an additional Kerr lens in the near field to pre-focus each temporal slice before it enters the nonlinear medium. Regardless, the simulation illustrates that the self-flying focus can produce long uniform plasma channels with modest power and energy requirements.
The parameters of the self-flying focus can be readily tuned for applications that require lower-density plasmas (N e < 10 17 cm −3 ), such as electron or proton beam-driven wakefield accelerators [42][43][44]. Here the design would use a gas with a lower P c , (e.g. rubidium or cesium) and, ideally, a shorter wavelength laser (e.g., λ 0 = 351nm) to offset the higher P c associated with the lower gas density. While the high intensities associated with the collapse and arrest dynamics of the self-flying focus are suitable for these plasma applications, they may be undesirable in situations where ionization and material damage are unwanted, e.g., THz generation in a crystal.

IV. SUMMARY AND CONCLUSIONS
A novel technique for nonlinear spatiotemporal control delivers an arbitrary trajectory intensity peak over distances comparable to the linear focal length. The technique combines temporal pulse shaping in the near field with nonlinear focusing in the far field to control the time and location at which each temporal slice within a pulse undergoes self-focusing collapse/arrest. These self-flying focus pulses can accommodate a wide range of parameters facilitating their use on various laser systems and in diverse applications. Notably, the self-flying focus could take advantage of long-pulse, high-energy laser systems, such as the National Ignition Facility or the OMEGA laser, to create intensity peaks with durations comparable to short-pulse lasers. Simulations demonstrate that a self-flying focus pulse can create a meter-scale plasma channel with a leading edge that is velocity matched to relativistic electrons, promising to both enable and improve this necessary component of advanced accelerators. Further, this demonstration suggests that the self-flying focus could improve the formation of long, uniform plasma channels in other media for filamentation and directed energy-based applications.