We investigate the effects of beam ellipticity on the dynamics of multiple filamentation. We find that increasing the ellipticity of the initial beam decreases the power required for multiple filamentation. At lower input ellipticities, the beam breaks into filaments along its widest dimension, whereas for higher ellipticities the pulse breaks into bands and then into filaments as the power is increased. The breakup patterns of the beam along the wider dimension are consistent with the modulational instability, and these patterns are independent of polarization and noise. Numerical simulations are in qualitative agreement with these features of multiple filamentation breakup.
©2005 Optical Society of America
Self-focusing dynamics of ultrashort pulses in bulk media with cubic nonlinearity is an active area of research in nonlinear optics, and applications include LIDAR and remote sensing . Critical to these applications is an understanding of multiple filamentation (MF), which typically occurs when a pulse with a peak power several times larger than the critical power Pcr for self-focusing propagates through a nonlinear medium. The pulse self-focuses until the intensity is sufficiently high that higher-order nonlinear effects such as plasma generation halt collapse. For powers slightly above Pcr , the beam is observed to take on a circular self-similar shape known as the Townes profile . At higher powers the interplay between nonlinear self-focusing, defocusing and linear diffraction result in the pulse breaking into several filaments in the transverse dimensions. Exactly what controls the number and placement of the filaments is not well understood. Traditionally, noise was thought to determine the formation of the filaments . However, recent theoretical work predicted that an elliptical input beam can produce deterministic MF patterns independent of noise [5, 6, 7]. These predictions have been experimentally verified in water  and air . An analysis of these experiments also indicate that the behavior should be general to all media with a Kerr nonlinearity.
In this paper we investigate how the filamentation process depends on the ellipticity of the input beam in a bulk glass sample. We find that elliptically-shaped beams undergo MF at much lower powers than those that are circularly shaped. Filamentation of an elliptically shaped beam was first studied in water for a single value of the ellipticity . In our experiments, we investigate for a range of beam ellipticities, we find that increasing the ellipticity of the input beam leads to a corresponding reduction in the threshold for multiple filamentation and that highly elliptical beams produce previously unobserved patterns of MF.
The experimental setup is shown in Fig. 1. A 42-fs, 780-nm pulse from a multipass Ti:sapphire amplifier is sent through a spatial filter and is passed through a pair of anamorphic prisms that shape the beam to a desired ellipticity e over the range from 2 to 6. The pulse then travels through a 30-cm sample of BK7 glass, and the beam profile at the output face is imaged onto a CCD camera (Cohu 4910 Monochrome). The pulse energy is controlled by a half-wave plate and a polarizing beam cube. As a result of group-velocity dispersion from the beam-shaping optics, the pulse broadens to 180 fs just before entering the glass. The energy per pulse is measured after the pulse travels through the glass. For each ellipticity the glass is removed and the input pulse profile is recorded with the camera. Figure 1 shows several of the input pulse profiles. These pictures are used to measure the size of the beam and to determine the diffraction length Ldf for each ellipticity, which we define as
for a beam of the shape E(x,y)=F(x 2/a 2+y 2/b 2), where k=2πn/λ, n is the refractive index, and λ is the wavelength. Since each profile has a different value of Ldf , the more highly elliptical beams will undergo greater amounts of diffraction in propagating through the glass sample.
The experiment consists of first removing the prisms and recording profiles at incremental energies for a circular beam. The prisms are replaced, and the same procedure is repeated for the various elliptical beams. Multiple filaments are not observed for circular input beams even at the highest energies studied here, although supercontinuum generation is observed for powers >350 MW. For elliptical beams the energy is increased until supercontinuum is generated in the filaments. The resulting behavior is illustrated for three separate ellipticities in Fig 2. For e=2.1(e=b/a) the beam breaks into filaments along the wide dimension at high energies. The center filament contains most of the energy with lower energy filaments on either side. For e=4.1 the pulse breaks apart in a manner similar to that for e=2.1 but in place of the filaments wide bands occur. At higher energies, these bands subsequently break apart into individual filaments, and at input energies E just below our experimental threshold (E=38 µJ) for generating supercontinuum the center filament breaks into two side filaments leaving a gap in the center of the beam. In the final set of pictures, a value of e=5.75 yields bands at E=26 µJ, and at higher energies the bands break into individual filaments. Just before the point at which supercontinuum is generated, the center filament no longer is present which leaves a gap in the center of the beam with a band above and beneath with filaments on either side.
Several important features are determined from these pictures. First, the increasing ellipticity decreases the threshold for MF, which is similar to the behavior observed in water. Second, the patterns produced are not the result of noise since the patterns do not change from shot to shot. Third, and most interesting, is the previously unobserved phenomenom that the more highly elliptical beams form bands and the bands split into filaments. These bands start to form when the ellipticity is e=3 and are in the same vertical location as the circular filaments for e=2.
It has been shown that for tightly focused beams the polarization orientation of a linearly polarized beam could determine MF patterns [11, 12]. We checked this issue and observed that the direction of the polarization had no affect on the output pattern. This is as expected since in order for polarization effects to initiate the MF, the beam size needs to be on the order of the wavelength. For most materials, a beam with a power several times the Pcr and a diameter of the wavelength would be far above the damage threshold.
The placement of the filaments indicates a regular modulation along the wider dimension of the profiles. As the input ellipticity is increased and the beam more closely approximates a one-dimensional plane wave, modulational instability could initiate the beam breakup in the long dimension . The plots in Fig. 3 show this modulation clearly. While the standard modulational instability can explain the breakup along the longer axis of the beam, it is not sufficient to explain the disappearance of the center filament which requires a more complete non-perturbative treatment of the dynamics.
We performed numerical simulations in an attempt to understand the full dynamics. Using a (2+1)D model of continuous-wave beams in a medium with a saturable nonlinearity. Our model includes diffraction, Kerr nonlinearity, and a nonlinear saturation term. A complete theoretical description of these pulses should include a full 3+1 simulation including dispersion and other temporal effects. However, such a model would be computationally expensive, and we find that most of the qualitative features are captured by the 2+1 model. The diffraction length is as defined in Eq. 1, and the nonlinear length is defined as
which yields the following equation
where is the transverse Laplacian, A 0 is the peak input amplitude, and Nsat is the nonlinear saturation parameter. This last term on the right-hand side of Eq. (3) prevents the beam from undergoing complete collapse. This model does not include dispersion which is non-negligible in the experiment since the pulse does broaden in time as it propagates through the glass. As a result, the peak power decreases as it propagates, which is an issue when making quantitative comparison between theory and experiment.
Figure 3 shows our theoretical results for a series of profiles for beams with e=2 and e=6. The e=2 beam first forms a central filament with smaller peaks on either side along the longer axis of the beam. These profiles show similar shapes to those in Fig. 2(a). At higher powers the central filament widens into a band. The band could not be observed in the experimental profiles for e=2.1 since supercontinuum generation occurred before the input power was sufficiently high. The series of profiles for e=6 show the development of the bands in which the power flows to the side bands and eventually the central filament disappears. The formation of the bands and eventual disappearance of the central filament is also evident in the experimental profiles for both the e=4.1 and e=5.75.
A more detailed comparison of the MF patterns from the numerical simulations with the experimental data is not justified without a more detailed model including all temporal effects. This is because the true mechanisms for arresting collapse and forming the bands are more complicated than the simple saturation term. However, the close qualitative agreement of these features suggests that this model captures reasonably well the underlying dynamics governing multiple filamentation.
In conclusion, we demonstrate that as the ellipticity of a beam increases the threshold power for multiple filamentation decreases. At high ellipticities bands are generated before filamentation occurs, and these patterns are independent of noise and polarization. We find that in the initial stages of multiple filamentation, the oscillatory structure is consistent with one-dimensional modulational instability. At higher powers the generated profiles can only be explained qualitatively via numerical simulations using the nonlinear Schrödinger equation model.
This work is supported by the National Science Foundation under grant PHY-0244995.
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