1. Introduction

Various methods of spatiotemporal structuring can be used to generate a so-called “flying-focus” pulse with an intensity peak that propagates at a focal velocity that is independent of the native group velocity of the medium [1–4]. This capability may fundamentally impact laser-based secondary source experiments by enabling velocity matching for several physical processes at the core of applications spanning nonlinear optics and plasma physics. For example, the realization of a dephasingless laser wakefield accelerator (DLWFA) produced by driving a plasma wake near the vacuum speed of light may produce TeV-class electron beams in a single stage [5,6]. Two-color photoionization with a flying-focus has also been predicted to enhance the conversion efficiency and focused intensity of THz radiation by matching the THz phase velocity to the focal velocity [7]. Moreover, theoretical studies on photon acceleration [8], plasma parametric amplification [9], and nonlinear Thomson scattering with spatiotemporal control [10] have further illustrated the potential impact of flying-focus pulses on a range of laser-based secondary source experiments.

Experimental demonstrations utilizing a flying focus have lagged the burgeoning theoretical work since most applications require the challenging experimental realization of an ultrafast (<100-fs) flying-focus that propagates over an extended focal region with a controllable focal velocity. Although experimental research related to this topic began over 20 years ago, the first experiments involved the production and characterization of the superluminal and uncontrolled focal trajectory produced using Bessel-like beams [11,12]. More-recent experimental research has sought to devise schemes for controlling the focal trajectory for use in applications. For instance, a diffractive lens and a chirped pulse was used to produce a high-power flying focus with a controllable velocity and extended focal region [2,3,13,14]. Similarly, the longitudinal chromatism from refractive lenses combined with a chirped pulse was applied to create a tunable flying-focus pulse, albeit over approximately one Rayleigh range [4]. Finally, controllable spatio-temporal structuring using spatial light modulators in combination with 4 f pulse shapers has been used to synthesize optical space-time wave packets in the form of a 2-D “light sheet” that can propagate at widely tunable group velocities [15–17]. These chromatic methods can produce a superluminal, subluminal, and even backward-propagating focal intensities and have been applied to produce plasma channels [13,14] and to match the velocities of the pump and seed in a soft x-ray lasing experiment [18].

Despite the utility afforded by these highly chromatic methods of producing a flying focus, these methods spread the bandwidth of the laser across the focal region, thereby limiting the minimum laser pulse duration to 10–100 ps. This precludes the use of these methods in applications such as DLWFA or THz production via two-color photoionization that require <100-fs pulse durations. Various achromatic concepts have been proposed to circumvent this limitation such as the use of all reflective systems including axiparabolic [19,20] or parabolic mirrors used in combination with echelons [5] or active optical elements such as spatial light modulators and deformable mirrors [21,22]. Finally, the production of a femtosecond-class flying focus pulse obtained through interference between two chromatic flying foci has also been proposed and studied theoretically [23]. These methods have yet to be demonstrated and present a trade-off between flexibility in tuning the focal trajectory and potential power scaling. The use of refractive elements provides a promising method to rapidly tune the trajectory of the peak intensity but these elements are limited by issues related to B-integral, damage threshold, and dispersion. The use of a static, all-reflective echelon, on the other hand, offers scalability to extremely high peak powers.

In this article, we report the first experimental demonstration of an ultrabroadband (∼30-fs), flying-focus pulse with a designed and constant focal velocity. The measured ultrabroadband frequency response marks an ∼ 100 × increase in the maximum spectral bandwidth as compared to previous flying-focus demonstrations [2–4,13,14] and an ∼ 2 × increase in the maximum bandwidth as produced in a 2-D light sheet [15]. Ultrabroadband flying foci were produced using an axiparabola [19,20] and novel radial echelons. Spherical aberration produced by the axiparabola focused different annuli of the laser pulse to different locations in the extended focal region; the novel radially stepped echelons provided the requisite group delay to control the timing of the foci, which produced the designed focal trajectory. Spectral interferometry was applied to measure the focal trajectory and effective duration of the moving focus for an axiparabola alone and an axiparabola paired with the two echelons that were fabricated for this study. Measurements of the spectral phase and spatial beam profile indicate that a nearly transform- and diffraction-limited intensity peak can be created in the focal region.

2. Method

Measurements of the focal trajectory and effective temporal pulse duration of the ultrabroadband flying-focus produced using an axiparabola alone and axiparabola-echelon pairs were performed using spectral interferometry [24–30]. Detailed measurements were enabled by using an ultrabroadband superluminescent diode (SLED) as a surrogate for a femtosecond laser pulse. Spectral interferometry was used to determine the spatially and spectrally dependent transfer function that can completely describe a linear optical system. The SLED was coupled through a single-mode fiber to produce a spatially coherent but spectrally incoherent source that was applied to measure the spatiotemporal couplings (STC) induced by the axiparabola and echelon pair. This source provided a convenient and functionally equivalent method for measuring STC via spectral interference as that performed using a femtosecond source [4]. Techniques based on test + reference interferometry can fully characterize the transfer function of an optical system, and in particular the induced spatio-temporal coupling, with high resolution and without uncertainties arising from phase-matching conditions in diagnostics based on nonlinear optics [24,30,31].

Figure 1(a) illustrates the axiparabola-echelon configuration. The echelon design [Fig. 1(b)] and interferometric measurements [Figs. 1(c) and 1(d)] of a radially stepped echelon are shown in Fig. 1. These 100-mm-diameter echelons were produced using an evaporative coating technique and designed to have half-wavelength (∼500-nm) steps to produce radial group delay without distorting the phase front. The echelon shapes were designed based on an analytic model that is described in detail elsewhere [22] and provides a relationship between the radial group delay imparted by the echelon , desired focal velocity ${v_f}$, and the radially dependent focal length of the axiparabola $f(r )$:

 

Fig. 1. (a) A schematic depicting the formation of an ultrafast flying-focus laser pulse by subsequent reflection from a radially stepped echelon and an axiparabola [19,20]. (b) Designed and [(c)-(e)] measured step profiles for a concave echelon that was used in this experiment. (d) Zoomed version of (c) in the vicinity of a single half-wavelength step, and (e) the profile indicated by the white line on (d).

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The radially dependent focal length

An axiparabola [19,20] with a f₀ = 50-cm nominal focal length and nominal focal region of L = 1 cm was used in combination with two radially stepped echelons designed and fabricated for this experiment: (1) a convex echelon [ Fig. 2(c)] designed to produce a uniform focal velocity near the vacuum speed of light and (2) a concave echelon that is the inverse of the former and designed to produce a superluminal and accelerating focal velocity [Fig. 1(b)]. The axiparabola and echelons were designed with a 100-mm diameter to accommodate high-power experiments. A continuous-wave, ultrabroadband [∼50-nm full-width at half-maximum (FWHM)] SLED with a central wavelength near 1 µm illuminated the test and reference arms of the spectral interferometer that is depicted schematically in Fig. 2(a). The test arm was transmitted through an achromatic telescope to increase the Gaussian beam diameter to the 100-mm diameter of the echelons or a flat mirror used to study the moving focus produced using an axiparabola alone. The beam was reflected back through the system and recollimated through an identical achromatic telescope before being focused by the f/5 axiparabola through a 2-cm diameter hole in a transport mirror. The focused beam was coupled into a single-mode, polarization-maintaining fiber that is mounted on a translational stage used to scan through the focal region. The reference arm was coupled into another single-mode, polarization-maintaining fiber that is the same length as the test arm. The test and reference arms of the spectrometer were combined in the fiber with a nominal delay controlled by a fiber-based variable delay box to produce spectral interference when analyzed with a spectrometer. Figure 2(b) displays a typical spectral interferogram illustrating interference across ∼100 nm of bandwidth.

 

Fig. 2. (a) A schematic depicting the spectral interferometer used to characterize the ultrafast flying focus, (b) a typical spectral interferogram, and (c) the step profile of the convex echelon. (d) The spatial beam profile measured at the best focus of the axiparabola.

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Measurements of the focal trajectory (v_f = Δz/Δt) were realized by stepping the test arm fiber launcher through the focal region and observing a change in the group delay (Δt) as determined from typical Fourier analysis [24] of the spectral interference pattern. The displacement of the stage (Δz) was measured with a precision of ∼100 nm using a separate, HeNe-laser–based Michelson interferometer (not shown in Fig. 2) with one arm formed via reflection from a mirror mounted on the translational stage. The ratio of the spatial displacement, measured using the Michelson interferometer, to the change in group delay, measured using the spectral interferometer, allowed for an accurate determination of the focal velocity (v_f = Δz/Δt).

The minimum effective pulse duration of the flying focus was determined by analyzing the measured spectral phase as extracted from the spectral interferograms and the spatial beam profile was measured using an achromatic microscope objective with 20 × magnification. The best focus of the axiparabola is presented in Fig. 2(d) and indicates that a near-8 µm, FWHM spot was created. All data acquisition was automated to accommodate scans of a 1-cm-long focal region with ∼20-µm spatial resolution. Note that a similar scanning spectral interferometric technique was previously applied to measure spatiotemporal couplings arising from longitudinal chromatism [27,28] and noncollinear optical parametric amplification [29].

3. Results

An ultrabroadband flying-focus corresponding to a near transform- and diffraction-limited pulse was produced using the axiparabola alone and the axiparbola in combination with the convex and concave echelons. Figure 3 shows the peak fluence as a function of focal coordinate obtained with the axiparabola and convex echelon as measured with a camera illustrating an extended focal region ∼50× larger than the Rayleigh range of the full aperture focal spot (∼3.8-µm FWHM obtained near the end of the extended focus). Bessel-like beam profiles characteristic of axiparabola focusing [19,20] were recorded and some representative profiles are displayed as insets in Fig. 3. The FWHM of the beam was near 8 µm at the location with the highest fluence before evolving to an ∼3.8-µm, f/5 diffraction-limited spot further into the extended focal region. This spatial evolution is expected since annuli corresponding to a larger f-number form the dominant contribution to the fluence nearest the nominal focus [19,20]. Also plotted on Fig. 3 are theoretical curves obtained by performing a Fresnel integral for an equivalent bandwidth pulse initiated with phase terms accounting for the convex echelon and axiparabola shapes [5,22]. The dashed black line in Fig. 3 depicts an ideal case corresponding to a flattop beam incident on the axiparabola echelon pair while the solid red line is a calculation performed using the spatial beam profile of the SLED source. Note that the use of a 100-mm diameter (1/e²) Gaussian beam with a 20-mm hole at its center reduced the extended focal region as compared to the ideal case. The disagreement between the experimental and theoretical curves at longer focal distances is attributed to imperfections in the radially stepped echelon and will be discussed in more detail below.

 

Fig. 3. The theoretical (dashed black and solid orange lines) and experimental (blue points) peak fluence versus focal location for a laser focused by the f/5 axiparabola with a convex echelon. The dashed black line corresponds to the ideal case whereby the axiparabola is illuminated with a flat top spatial intensity profile while the solid orange line corresponds to a calculation performed using the experimental beam profile. The insets are experimental spatial beam profiles from the data points marked by arrows.

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Figure 4 shows the experimentally measured group delay data as a function of the distance along the laser’s propagation axis. The vertical axis of Fig. 4 corresponds to a time in a reference frame moving at the vacuum speed of light (ξ = t – z/c) such that a horizontal line on this plot indicates a signal moving at the vacuum speed of light (see the dashed lines to guide the eye in Fig. 3). Plotted along the experimental data points are theoretical group delay curves that were obtained using the frequency-domain Fresnel integral described above [5,22]. In all three cases, the experimental data agree well with the theory.

 

Fig. 4. The delay data plotted at a coordinate moving at the vacuum speed of light versus the distance relative to the nominal focus (f₀) and corresponding to trajectories from the (a) axiparabola alone, (b) axiparabola with a convex echelon, and (c) axiparabola with a concave echelon.

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The spectral phase introduced by the spatio-temporal shaping setup was analyzed to determine the minimum effective pulse duration of the flying-focus near the peak fluence of the axiparabola focus [24]. Figure 5 summarizes these results and depicts the input spectrum [Fig. 5(a)] and transform-limited pulse [Fig. 5(d)] that corresponds to the Fourier transform of the measured spectrum in Fig. 5(a). As can be seen in Fig. 5(d), the fastest pulse that can be produced with this bandwidth has an ∼15-fs FWHM. Figures 5(b) and 5(c) show the spectrum and measured spectral phase of the moving focus after it has been focused by an axiparabola alone [Fig. 5(b)] and focused by an axiparabola used in combination with the convex echelon [Fig. 5(c)]. Note that the large, ∼10,600-fs², second-order spectral phase that was measured and plotted in Figs. 5(b) and 5(c) resulted from the group delay dispersion (GDD) in the achromatic telescopes used in the setup [see Fig. 2(a)]. The GDD was removed from the data to calculate the effective pulse durations plotted as blue solid curves in Figs. 5(e) and 5(f) along the theoretical temporal pulse durations obtained using the frequency-domain Fresnel integral [see orange dashed lines in Fig. 5(e) and 5(c)]. It should be noted that this telescope was required to expand the ∼ 10 mm SLED beam to the 100 mm diameter required for the large diameter axiparabola and echelons designed for high-power experiments. Future high-power demonstrations of the ultrabroadband flying focus will either use a large beam or an all-reflective telescope thereby justifying the removal of the GVD imposed by the refractive optics to calculate the effective pulse durations depicted in Figs. 5(e) and 5(f). The third- and higher-order phase terms inherent in the transmissive optics were not removed from the measurement.

 

Fig. 5. (a) The initial superluminescent diode (SLED) spectrum, (b) the spectra and measured spectral phase corresponding to a flying focus produced using an axiparabola alone, and (c) that produced using an axiparabola with the convex echelon. (d) The initial transform-limited temporal pulse profile of the SLED and the [(e),(f)] group delay dispersion (GDD) corrected temporal pulse profiles and theoretical temporal pulse profiles of the flying-focus pulse created using the axiparabola alone [(e)] and the axiparabola in combination with the convex echelon [(f)].

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The time-domain representations calculated from the measured spectral phases are depicted in Fig. 5 and indicate that the FWHM temporal pulse length is approximately 20 fs when using the axiparabola alone and 30 fs when using the convex echelon axiparabola pair. We have measured the GDD of both radially stepped echelons and found them to be ∼100 to 200 fs², of the order of the GDD of chirped mirrors. This temporal broadening was also accompanied by a slight degradation in the ∼100 fs contrast visible in Fig. 5(f). As can be inferred by Figs. 5(e) and 5(f), the measured temporal/spectral changes in the field was inconsistent with the Fresnel integration that predicted an ∼15-fs FWHM pulse. The slight chromaticity of these optics and apparent contradiction with theoretical predictions will be discussed in more detail below. Note that the spectrum and spectral phase measured when using the axiparabola-concave echelon configuration was qualitatively consistent through the focal region and to that measured using the axiparabola-convex echelon combination.

4. Discussion

The measurements presented above demonstrate that a nearly transform- and diffraction-limited flying-focus with a controlled trajectory and extended focal region can be produced with an axiparabola-echelon pair. Moreover, due to the use of a large-aperture (100-mm-diam) all-reflective optical system, this technique is scalable to the high-peak powers required for experiments at relativistic intensity such as dephasingless laser wakefield acceleration of electrons [5,6].

Evident in the spectral phase measurements was an increase of the effective temporal pulse duration from the 15-fs transform-limit to approximately 30 fs when using the axiparabola and echelon pair. This temporal broadening of the pulse was accompanied by an apparent degradation in the ∼ 100-fs contrast, which is clearly visible in Fig. 5(c). Although some chromatism may be expected and unavoidable due to the connection between radial group delay and longitudinal chromatism [30], this effect should be small for focal velocities near the vacuum speed of light and was not predicted using the frequency-domain Fresnel integral calculations [see Figs. 5(e) and 5(f)]. It is more likely that the inconsistency between the experimental and theoretical pulse durations arise due to limitations in the experimental set-up and fabricated optics.

Aside from the uncompensated higher-order dispersion in the setup discussed above, the 6-µm inner diameter of the optical fiber was large enough such that multiple annuluses of the flying focus (see Fig. 3) may couple into the spectral interferometer. These annuluses have slightly different group delays and may appear as additional spectral modulation, as observed in Figs. 5(b) and 5(c). Moreover, since polarization maintaining fibers have two polarization modes, imperfect input polarization may additionally contribute to the observed spectral interference and temporal degradation. Finally, limitations in the echelon manufacturing technique result in finite ramps as opposed to vertical steps [see Fig. 1(e)] that may impact their performance. The imperfections in the fabricated echelons are more pronounced at large radii where the steps become closely spaced, which may explain the increased disagreement between the experimental and theoretical fluence data further into the focal range (Fig. 3). Note that the deviation between the designed and actual axiparabola shape was found to be within ∼100 nm as determined by white-light interferometry suggesting that the slight disagreement between the experimental and theoretical temporal pulse profiles obtained with the axiparabola alone may be related to limitations in the spectral interferometer.

Future work will be devoted to applying this technique to produce a high-power flying focus that can be applied for proof-of-principle experiments on, e.g., the efficient production of THz via two-color photoionization [7]. Such an application is relatively insensitive to slight degradation in contrast, may be performed using a high-repetition rate laser, and will provide a means to study the how optical nonlinearities and photoionization may affect the flying-focus pulse, thereby informing future designs to apply this technique for experiments at relativistic intensities. Finally, a modification of the echelon fabrication technique to produce a compound axiparabola-echelon optic that may greatly simplify the application of this technique is being pursued.

5. Conclusion

We have experimentally demonstrated that a nearly transform- and diffraction-limited ultrabroadband flying-focus with a controllable focal velocity and extended focal region can be produced using an axiparabola-echelon pair. Spectral interferometric measurements of the focal velocity and effective pulse duration as determined from the measured spectral phase are consistent with analytic calculations and indicate that an ∼30-fs pulse moving at a constant velocity near the vacuum speed of light can be created using this system. Future work will be dedicated to the realization and characterization of a high-power (∼100-GW) ultrafast flying focus with this scheme that will be applied for proof-of-principle experiments dedicated to nonlinear optics with spatiotemporal control. Subsequent effort will focus on incrementally increasing the peak power of the system to the ∼100-TW peak powers required to drive a dephasingless laser wakefield accelerator [5,6].