M. Mlejnek, E. M. Wright, and J. V. Moloney
Arizona Center for Mathematical Sciences, and Optical Sciences Center,
University of Arizona, Tucson, Arizona 85721
Recently, there has been considerable excitement regarding experimental
demonstrations of propagation of femtosecond pulses over 
m
in air [1, 2, 3, 4]
due to potential applications in, e.g.
lightning channeling [2] and LIDAR [3].
To determine the utility of this phenomenon
for these and other applications the underlying physics needs clarifying.
The critical power for self-focusing in air is 
GW,
and catastrophic collapse is avoided by a combination of
multi-photon ionization (MPI), and
absorption and defocusing by the electron-plasma generated by
MPI. The question
we have addressed is how do these mechanisms conspire to produce long
distance propagation?
To address this issue we have performed numerical simulations
using a comprehensive air propagation model [5].
Typical results for a 780 nm pulse of duration 200 fs (FWHM) and peak
power 
GW are shown in Fig. 1. The surface plots show
the intensity versus time (in a frame moving at the group velocity)
and transverse dimension x for various propagation distances z,
and the insets show the maximum on-axis intensity.
The top plot for z=61 cm is close to
the paraxial collapse distance, but
MPI and plasma defocusing arrest the collapse yielding
a stabilized pulse. This creates the impression that long distance
propagation is due to the stabilization of collapse by MPI and plasma
defocusing [1]. However, upon
further propagation this pulse decays due to absorption, but the
fascinating result is that a new pulse grows from the trailing edge
pulse: This phenomenon is
shown in the middle plot for z=148 cm where the leading edge pulse
is decaying as the trailing edge pulse is growing. The formation of
the trailing pulse is due to the re-self-focusing of power that was
displaced into spatial rings by the plasma-defocusing imposed on the
trailing edge of the incident pulse by the collapsing front edge.
In the bottom plot for z=161 cm the trailing edge pulse has now
replaced the leading one.
This process, which we term dynamic spatial replenishment, in which the initial pulse forms, is absorbed, and is replenished by refocusing of the trailing edge, can occur several times for high-peak-power incident pulses and create the illusion of a single pulse propagating over a long distance. For experiments in which only the fluence is monitored time integration masks the dynamics and creates the impression of one stabilized light filament. However, our simulations [5] show that the situation is more dynamic, and Brodeur et. al. [4], have presented experimental evidence of this.
GW. An input spot size of 
mm was
employed so that numerical simulations could be performed on the scale
of 1 m, but the distance for larger spot sizes scales as 
.