Filament propagation

A laser beam traversing a medium can modulate the refractive index of medium as[5]

${\displaystyle n={n_{0}+{\bar {n}}_{2}I}}$

where ${\displaystyle n_{0}}$, ${\displaystyle {\bar {n}}_{2}}$ and ${\displaystyle I}$ are linear refractive index, second order refractive index and intensity of propagating laser field respectively. Self-focusing occurs when the phase shift due to Kerr effect compensates for the phase shift because of Gaussian beam divergence. Phase change due to diffraction for a Gaussian beam after traversing a length of ${\displaystyle \Delta z}$ is

${\displaystyle \phi _{diffraction}={k\Delta z \over 2\rho _{0}^{2}}r^{2}}$

and phase change because of Kerr effect is

${\displaystyle \phi _{Kerr}={2\pi {\bar {n}}_{2}I_{0}\Delta z \over \lambda }exp({-2r^{2} \over w_{0}^{2}})\approx {2\pi {\bar {n}}_{2}I_{0}\Delta z \over \lambda }(1-{2r^{2} \over w_{0}^{2}})}$.

where ${\displaystyle k={2\pi n_{0} \over \lambda }}$, ${\displaystyle \rho _{0}={\pi w_{0}^{2}n_{0} \over \lambda }}$(Rayleigh range) and ${\displaystyle w_{0}}$ is the waist of Gaussian beam. For self-focusing to happen the one have to satisfy the condition of ${\displaystyle r^{2}}$ terms be equal in magnitude for both Kerr and diffraction phases. Hence

${\displaystyle I_{0}={w_{0}^{2} \over 4\rho _{0}^{2}{\bar {n}}_{2}}}$.

On the other hand, we know that area of a Gaussian beam at its waist is ${\displaystyle \pi w_{0}^{2} \over 2}$. Therefore[6]

${\displaystyle P_{c}={\lambda ^{2} \over 8\pi n_{0}{\bar {n}}_{2}}}$.

Note

${\displaystyle {\bar {n}}_{2}\left({cm^{2} \over W}\right)=n_{2}n_{0}\epsilon _{0}c}$

Self-focusing needs a laser peak power higher than the critical power ${\displaystyle P_{c}}$ (order of gigawatts in air[7]), however, for infrared (IR) nanosecond pulses with peak powers higher than the critical power self-focusing is not possible. Multiphoton ionization, inverse Bremsstrahlung and electron avalanche ionization are three major results of gas and laser interaction. The later two processes are collisional-type interactions and take time to accomplish (picosecond to nanosecond). A nanosecond pulse is long enough to develop the air breakdown before the power reaches the GW order required for self-focusing. Breakdown of gas produces plasma that has absorbing and reflecting effect so self-focusing is prohibited.[7]