We investigate 3D spatio-temporal focusing of elliptically-shaped beams in a bulk medium with Kerr nonlinearity and anomalous group-velocity dispersion (GVD). Strong space-time localization of the mode is observed through multi-filamentation with temporal compression by a factor of 3. This behavior is in contrast to the near-zero GVD regime in which minimal pulse temporal compression is observed. Our theoretical simulations qualitatively reproduce the experimental results showing the highly localized spatio-temporal profile in the anomalous-GVD regime, which contrasts to the weakly localized pulse in the normal-GVD regime.
© 2011 OSA
High-power, ultrashort laser propagation in bulk media has drawn significant interest for more than a decade due to its potential applications in remote sensing, lightning guiding, and pulse compression [1–6]. When the laser power P is higher than the critical power P cr, the pulse undergoes self-focusing and as it collapses, other nonlinearities such as multi-photon absorption (MPA) and plasma formation can counterbalance self-focusing, resulting in long-distance confinement of the beam, which is known as filamentation. When the power is much higher than Pcr, the beam breaks up into smaller multiple beams [i.e., multiple-filamentation (MF)], which can be due to modulational instability  or as Fibich et al.  have shown, beams with moderate powers (P < 15P cr) collapse and form a single filament first and then MF occurs after experiencing multiple focusing-defocusing cycles due to nonlinear effects such as plasma de-focusing. These noise-generated MF patterns can be random and unpredictable. However, recent theoretical and experimental studies [9–20] show that MF patterns can be controlled using elliptically-shaped input beams and vortex beams .
Liu et al.  used elliptical beams to generate (1 + 1 + 1)-D (one transverse dimension + time + propagation direction) solitons in quadratic nonlinear media. Furthermore, they investigated  whether multi-filaments from the elliptical input beam may form a 3D solitary wave (i.e. light bullets) . However, angular dispersion, which was used for controlling (1 + 1 + 1)-D soliton generation, prevented the formation of the light bullet. More recently, Eisenberg et al.  studied elliptical beams in a planar waveguide in the anomalous group-velocity dispersion (GVD) regime and achieved spatio-temporal focusing with Kerr nonlinearity. Their calculations showed nearly stable propagation of the focused beam for several diffraction/dispersion lengths as evidence for a (1 + 1 + 1)-D soliton. However they did not investigate the possibility of a full 3-dimensional spatio-temporal localization.
Here we investigate the collapse dynamics and 3D spatio-temporal focusing of a highly elliptical femtosecond laser pulse propagating in a Kerr medium (BK7 glass) in the anomalous-GVD regime. The high-power elliptical beam breaks up into a controllable MF pattern, and each filament undergoes strong spatial localization, showing a symmetric Townes-like beam profile [26,27]. We also observe simultaneous temporal focusing in which the pulse compresses by a factor of 3. For comparison, similar elliptical beams in the zero-GVD regime exhibit minimal pulse compression even though similar MF is observed. According to our simulations, after strong spatio-temporal collapse due to the Kerr nonlinearity and anomalous GVD, the beam can be localized in space and time mainly due to the balance between self-focusing and diffraction in space and between anomalous GVD and self-phase modulation (SPM) in time. Pulse compression and spatio-temporal localization in our case is different from that in the normal-GVD regime using elliptical beams by Majus et al.  since pulse compression and the formation of weakly-localized stationary waves, called the X-waves in the normal-GVD regime involves more complicated phenomena such as pulse splitting and plasma defocusing [4,5] than spectral broadening by SPM.
In our experiments, pulses from an amplified Ti: Sapphire laser system operating at a 1-kHz repetition rate (805 nm, 100 fs FWHM) are passed through an optical parametric amplifier (OPA) to produce pulses in the anomalous- (1600 nm, 300 fs FWHM, 450 fs autocorrelation FWHM) and zero- (1320 nm, 110 fs FWHM, 160 fs autocorrelation FWHM) GVD regions of BK7 glass. To generate an elliptical beam, we first expand and collimate the beam horizontally using a cylindrical lens telescope and then focus the beam vertically using another cylindrical lens onto the front face of a 30-cm-long BK7 sample. The spot size is 2.5 mm by 0.1 mm in the horizontal and vertical directions, respectively. The idler signal of the OPA is blocked by a linear polarizer, and the laser energy is controlled by a combination of a half-wave plate and another linear polarizer. We record the mode profiles after 30-cm propagation at the output face of the sample by imaging onto an IR CCD cameras with a f/2 achromat lens. The pulse duration of the apertured central part of the output beam (about 1-mm diameter) is measured using a Si photodiode two-photon autocorrelator .
Since the eccentricity e of the beam is 25, self-focusing and beam collapse initially occur along the narrower dimension. In this case, as demonstrated by several groups [12–16,18,19], the beam breaks up into a deterministic MF pattern along its wider dimension when the laser power is several times the critical power Pcr. We can control the number and the position of filaments using elliptical beams in contrast to the random, noise-generated filaments for a circular beam . According to calculations [29,30], the critical power for elliptical beams is larger than that for circular beams (Pcr = αλ 0 2/4πn 0 n 2 ~12 MW at 1600 nm, where α = 1.8962 is a constant associated with the initial Gaussian beam , λ 0 is the central wavelength, n 0 is the refractive index of BK7, and n 2 is the nonlinear index coefficient.) by a factor (0.4e/2 + 0.6) for large values of e. Thus the critical power for the elliptical beam for our experiment is 70 MW and the beam with P = 67 MW does not collapse as is shown in Fig. 1 . For P = 117 MW, we observe the bright central portion due to self-focusing. For P = 143 MW, two filaments with Townes-like beam profiles [26,27] form near the central region where the intensity is highest. At higher powers, the number of filaments increases and filaments in the central region become less intense and larger than those in the outer region. This can be explained as follows: as the intense parts of the beam collapse and form filaments, those filaments diffract due to nonlinear losses such as multi-photon absorption (MPA) by the time they reach the output face of the sample. For the case of the zero-GVD regime, similar MF patterns also occur along the wider dimension (Fig. 2 ). Since the critical power is proportional to λ 0 2, beam collapse and filamentation for 1320 nm occurs at smaller powers.
For the anomalous-GVD regime (1600 nm), autocorrelation measurements show that there is simultaneous pulse compression during beam collapse and MF [Figs. 3(a) and 3(b)]. When the power is low (P < 100 MW) such that there is no collapse, pulse compression is negligible since the sample length (30 cm) is much shorter than the dispersion length (290 cm). There is slight pulse compression near 100 MW due to spectral broadening by SPM in the anomalous-GVD regime . However as the power increases and SPM becomes stronger, temporal focusing is observed at the output face with a minimum duration of 150 fs FWHM autocorrelation, which corresponds to approximately a 100 fs-pulse duration and represents compression of the input pulse by a factor of 3 [Fig. 3(b)]. For comparison, the autocorrelation trace in the zero-GVD regime with P = 136 MW increases to more than 250 fs FWHM due to SPM as compared to 160 fs at P = 45 MW, which approximately corresponds to an autocorrelation of the 110-fs input pulse. However, there is little change in the pulse duration after the initial broadening even though the power increases further [Fig. 3(c)] . The aperture in front of the autocorrelator transmits 3 or 4 filaments near the center of the beams.
Although filamentation and pulse compression in the anomalous-GVD regime with circular beams were experimentally and theoretically investigated before [32–37], the filamentation patterns using circular beams are random and thus cannot be easily controlled since they are generated by random noise compared with the elliptical beams. The pulse compression observed here is also different from that with elliptical beams in the normal-GVD regime since drastic pulse reshaping such as (multiple) pulse-splitting occurs in the normal-GVD regime .
3. Theoretical simulations
Modeling the spatio-temporal behaviors of elliptical beams by performing a full 3-D spatio-temporal simulation is highly numerically intensive. As a result, we perform numerical simulations ignoring diffraction in the minor (shorter) axis since the beam profile does not change much in the minor axis (see Figs. 1 and 2), and MF only occurs along the major axis. This approach can yield qualitative spatio-temporal behavior that is very similar to that observed in our experiments. The nonlinear propagation equation is given by,
where u is the electric field amplitude normalized by the initial electric field amplitude E 0, ξ = z/Ldf is the propagation distance normalized by the diffraction length Ldf = n 0 πw 0 2 /λ 0, w 0 = is the effective spot size for the elliptical beam, = 2.5 mm is the spot size in the horizontal direction, = 0.1 mm is the spot size in the vertical direction, n 0 = 1.5 is the refractive index of BK7, λ 0 is the central wavelength, µ = x/w 0 is the transverse coordinate along the major axis, normalized by the spot size w 0, τ is the retarded time normalized by the 1/e 2 pulse duration τp, L 2 = τp 2 /β 2 is the dispersion length, L 3 = τp 3/β 3 is the third-order dispersion length, β 2 = −3.09 x 10−28 s2/cm (1.906 x 10−30 s2/cm) is the GVD parameter at 1600 nm (1320 nm), and β 3 = 1.77 x 10−42 s3/cm (0.88 x 10−43 s3/cm) is the third-order dispersion at 1600 nm (1320 nm). Lnl = c/(ωn 2 I 0) is the nonlinear length, n 2 = 2.2 x 10−16 cm2/W (2.5 x 10−16 cm2/W) is the nonlinear index coefficient at 1600 nm (1320 nm) , I 0 = cn 0 |E 0 |2/2π is the peak input intensity of the laser, Lmp = 1/[β(K)I 0 (K- 1 )] is the K-photon absorption length, β(K) = 1.4 x 10−75 cm11/W6 (1.8 x 10−50 cm7/W4) is the 7 (5)-photon absorption coefficient at 1600 nm (1320 nm) , Lpl = 2/(σρ 0 ωτ c) is the plasma length, σ is the inverse bremsstrahlung cross section, ρ 0 = β(K)I 0 K τp/(Kℏω) is the total electron density that is produced by the input laser pulse via multi-photon ionization, τc ~2.33 x 10−14 s is the electron-ion collision time [40,41], and η = ρ/ρ 0 is the normalized electron density. Each term on the right side of Eq. (1) represents diffraction in the major axis, dispersion, third-order dispersion, Kerr-nonlinearity, MPA, and plasma defocusing and collisional absorption, respectively. Considering collisional ionization, multiphoton ionization, and recombination for the plasma, the normalized electron density satisfies the equation,
Since we ignore diffraction in the minor axis, we expect that there are discrepancies between experiments and simulations, particularly in the distance to collapse. Therefore, we choose a propagation distance z in the simulation to approximately match the measured MF patterns. We include spatial noise of 2% in the initial intensity profiles, which results in slight asymmetry similar to that observed in our experiments (see Figs. 1 and 2). Large noise (>10%) results significant asymmetry in the filament patterns.
Figure 4 shows examples of the calculated spatio-temporal profiles at z = 13 cm in the anomalous-GVD [Figs. 4(a)-4(d)] and the zero-GVD [Figs. 4(e)-4(h)] regimes. In each figure, time-integrated spatial profiles which represent beam profiles along the major axis are shown on the top. The calculations qualitatively reproduce the experiments showing MF patterns and diffraction of central filaments due to MPA. The most notable feature about the filament in the anomalous-GVD regime is formation of the highly-localized, spatio-temporal profile [Figs. 4(c) and 4(d)] which is not observed in the zero-GVD regime. This is also in contrast to the case of the normal-GVD regime in which multiple peaks in the temporal domain take place mainly due to pulse splitting . Small oscillations of the trail edges in the temporal domain in Figs. 4(d) and 4(h) are due to third-order dispersion .
The calculated autocorrelation traces and pulse durations also agree qualitatively with those observed in the experiments (Fig. 5 ). For 1320-nm pulses with P >200 MW, the pulse duration increases. However, the calculation does not show the saturation of pulse broadening at higher powers which is observed in the experiment [see Fig. 3(c)].
For comparison, if we ignore diffraction in the major axis, the calculated autocorrelation traces and pulse durations (Fig. 6 ) at z = 30 cm agree better with those observed in the experiments. However fully 3-dimensional simulations are required for more quantitative comparisons.
In summary, we observe spatio-temporal focusing of an elliptical laser beam in a bulk medium with Kerr nonlinearity and anomalous GVD in a controllable manner. The beam breaks into a deterministic MF pattern, with each filament showing a Townes-like symmetric profile [26,27]. Due to the combined effect of spectral broadening via SPM and anomalous GVD, pulse compression by a factor of 3 simultaneously occurs. The simulation results suggest that the highly localized spatio-temporal wave can form via multifilamentation, which is in contrast to the formation of multiple peaks via pulse splitting in the normal-GVD regime . In the future, we will investigate whether each filament generated from the break-up of the elliptical beam can propagate as a 3-D optical bullet under appropriate conditions.
This work was supported by NSF under Grant No. PHY-0703870 and the Army Research Office under Grant No. 186695-PH. The authors gratefully acknowledge useful discussions with M. Foster and A. Chong.
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