What is Paraxial equation in laser plasma interaction?

Answer 1

The paraxial equation refers to the reduction of Maxwell's hyperbolic wave equation to a parabolic equation, appropriate if the radiation has slow spatio-temporal variation with respect to its central frequency and wavevector. Specifically, we write Maxwell's equation for the vector potential A (sorry for the poor looking equations, and I use Gaussian units)

[∂^2/∂t^2 - c^2(∂^2/∂x^2 + ∂^2/∂y^2 + ∂^2/∂z^2)]A = 4*π*c*J,
where J is the current in the plasma. We express A(x,y,z,t) = a(x,y,z,t)*exp(-iwt + ikz), with |∂a/∂t| << w*|a|, |∂a/∂z| << k*|a|. Dropping the second order time derivatives and using the vacuum dispersion relation w^2 = c^2k^2, the equation above becomes
[∂/∂t + c∂/∂z - (i*c^2/2w)(∂^2/∂x^2 + ∂^2/∂y^2)a = i*4*c*π*J/(2w)*exp(iwt - ikz).
This is the paraxial wave equation. In laser-plasma studies the current J is related to the plasma density n and velocity v via J = n*v. Finally, if the transverse (i.e., in the x-y plane) variation is slow, the transverse canonical momentum g*m*v - e*A/c (with m the particle mass, e its charge, and g = 1/sqrt(1-v^2/c^2) the relativistic factor) is approximately conserved. Writing the current J = -e*n*v = -e(n_0 + dn)*v = -e(n_0 + dn)*p/(g*m) = -e^2(n_0 + dn)*A/(g*m*c) (where n_0 is the equilibrium plasma density and dn is the change in density), we can use the plasma EM dispersion relation w^2 = w_p^2 + c^2k^2, with w_p^2 = <4*π*e^2/(g*m)> to simplify the equation, with <> an appropriate averaging. This becomes nice and familiar in the non-relativistic limit for which w_p^2 -> 4*π*e^2/m and the paraxial equation becomes
[∂/∂t + (c*k/w)∂/∂z - (i*c^2/2w)(∂^2/∂x^2 + ∂^2/∂y^2)a = -i*4*c*π*e^2*dn*A/(2*m*w),
with w^2 = c^2k^2 + w_p^2. Hope this helps.