|Fig. 1: Typical picture of a filament.|
From the laser's invention in the late 1950s, a steady research initiative has been the development temporally shorter pulses and higher peak power. This combination has exposed a number of nonlinear optical phenomena, pushing new ideas about light propagation and opening a variety of new applications.
As ever higher powers are accessed, some processes that were previously observed only in dense media begin to be readily observed in gaseous materials, such as air. An example of one of these processes recently unveiled in air by modern lasers is the self focusing and filamentation of high powered optical pulses. 
Shortly after the development of the ruby laser, scientists theorized about the self focusing of optical pulses under certain conditions.  Experimental evidence was provided in 1964 when a beam focused into glass produced a 1 cm track of permanent damage. The diameter of the damage track was a small fraction of the expected focus diameter and led to hypotheses that the propagating pulse, through its interaction with the medium, produced a self-induced waveguide counteracting diffraction. 
It wasn't until the mid 1990s and the advent of CPA laser systems that similar effects were first observed in air.  In a 1994 paper, researchers at the University of Michigan reported filamentation in air at a power of roughly 10 GW. Using an 800 nm beam with pulse duration of 200 fs, they varied the energy from 2 mJ up to 50 mJ. At low energies, they reported small-scale filamentation and at high energies, multiple filament formations. 
A typical picture of a filament in air is shown in Fig. 1. In short, a powerful pulse self focuses until it reaches high enough intensities to ionize the medium in its path and leave behind a narrow column of plasma. The formation of the plasma causes local defocusing that halts the collapse of the beam as interplay between diffraction, self-focusing, and plasma defocusing ensues. The energy required for continued ionization detracts from the pulse energy so that the self focusing gradually weakens and is eventually overcome by diffraction and plasma defocusing, ultimately terminating the filament. The length of the ionization channel can be on the order of a few meters, and has been demonstrated to extend hundreds of meters in outdoor propagation. 
The crucial element in filament formation is the self-focusing; without it, there would be no filaments. Sure, it is possible to focus a pulse through a lens and ionize a medium, but the extended column of ionization would not occur without self-focusing. The self-focusing effect of high powered pulses is a 3rd order nonlinear optical process known as the optical Kerr effect. This effect prescribes an intensity-dependent index of refraction :
where n0 is the linear refractive index, n2 is the nonlinear refractive index, and I(r,t) is the intensity profile of the laser pulse. Typical values of n0 and n2 for common media are listed in Table I.
|Table 1: Refractive index values for common media |
For pulses of sufficient peak power, the intensity-dependent refractive index enables the pulses to overcome the natural diffraction spreading and begin to self-focus. For pulses to do this, they must have an input power exceeding the critical power, described for Gaussian pulses by:
For an 800nm pulse in air, this amounts to:
Once the pulse begins to self focus, its intensity continues to rise as photons become more localized towards the pulse center. This inevitably leads to multiphoton ionization of the medium and the creation of an electron plasma in the central region of the pulse. In the ionizing portion of the pulse, high electron densities lower the susceptibility of the medium, thereby locally reducing the index of refraction. This central reduction in the refractive index, while the peripheral region continues to self focus leads to interesting energy dynamics within the pulse as it propagates. 
After ionization begins, the transverse beam profile is distinctly separated into two regions: the inner ionization region and the outer energy reservoir. As shown in figure (2), the inner ionization region has the highest intensity and a diameter of 50-100 μm. The outer region, coined the energy reservoir, can extend to several millimeters in diameter and carries a much lower overall intensity, but often a majority of the total pulse energy. In a healthy, unrestrained filament, photons in the central ionizing region experience a reduction in the refractive index and tend to move out from the beam center. At the same time, peripheral photons lying outside of the ionization region continue to self-focus, moving towards the beam center. It is the balance of these two processes that allow sustained filament propagation. 
|Fig. 2: Transverse profile of ionizing region of a filament.|
Experiments show that the outer energy reservoir is vital to the stability of the filament ionization core. When a tight pinhole is placed in the filament's path such that only the central ionizing portion passes, the filament quickly terminates beyond the pinhole. Not until the pinhole is expanded to the order of 2 cm does the filament propagate freely as if the pinhole were not there.  Conversely, if a stopper is placed in the central region of the filament and the outer regime allowed clearance, experiments and theory suggest regeneration of the filament beyond the stopper. [8-10]
These discoveries are easily understood in terms of self-focusing and the ionization-induced plasma defocusing. Blocking the outer reservoir removes the abundance of self-focusing photons that replenish the depleting central region. Light evacuating the central region may begin to self focus again, though the majority of the pulse energy was lost to the pinhole, and the filament quickly terminates. On the other hand, when the central portion is blocked, some of the outer reservoir photons flood the center and quickly begin the ionization process. The abundance of energy left in the reservoir allows continuation of the filament propagation.
These considerations, particularly the regeneration aspects, suggest that filamentation may be stable when propagating through adverse conditions that may be encountered in long range air propagation. 
Pulses that have undergone filamentation experience a number of spatiotemporal effects from the combination of the self focusing and ionizing processes. One of the most notable of these is wide spectral broadening; a filamenting pulse centered at 800nm will generate a white-light spectrum spanning well over the visible range. 
One application that takes advantage of filament's long range propagation ability and wide spectral broadening is Light Detection and Ranging (LIDAR), a remote sensing technique used to determine constituents of a distant target. Conventional LIDAR techniques are inhibited by the fact multiple shots are required with a tunable wavelength source. The backscattered light from each shot is captured, the data is analyzed collectively, and a general picture of the target constituents is provided. The advantage to filament-based LIDAR techniques is that the pulse's broad spectrum is able to provide all of the same information as conventional LIDAR in a single shot. Furthermore, because it is a single shot technique, time-resolved data collection gives a clear picture of the constituent concentrations as a function of distance or altitude. 
The development of high powered lasers enabled the observation of laser pulse filamentation in air. Filaments are generally described as the self-focusing of powerful pulses, which leads to ionization of the medium. The following dynamic balance of energy within the filament structure allows propagation over long distances. Certain characteristics of filamenting pulses, such as wide spectral broadening, makes filamentation an interesting topic of research for both nonlinear optical processes and general applications.
© David Nicholson. The author grants permission to copy, distribute and display this work in unaltered form, with attribution to the author, for noncommercial purposes only. All other rights, including commercial rights, are reserved to the author.
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