4.3 Filament Instability of Lasers
The soliton solution was obtained with the assumption of one-dimensional
geometry. It is natural question if the soliton propagates stationary in the threedimensional geometry. This is essentially the same question about the stability of
intense laser beam propagation in plasmas. In what follows, consider the physics for
the case where the intense laser propagates in x-direction with a beam size of radius
R in the perpendicular direction. It is assumed that R is much longer the wavelength
of the laser. If this is the case of laser propagation in the vacuum, the laser light
cannot propagate as plane wave for a long distance because of diffraction effect, and
it diverges with an angle of roughly θ ~ k 0 R, where k 0 is the wavenumber of the
laser wave.
The effect of the ponderomotive force, however, is an attractive force effectively
to keep the high-intensity region by depressing the electron density. It is clear that if
the ponderomotive effect to laser intensity change is stronger than the diffraction
effect, the laser intensity at the beam central may increase, and the initial structure
becomes unstable. If the laser intensity is not so high to result the enough nonlinear
force, on the other hand, the diffraction is dominant, and the beam intensity
decreases as a function of time or space. The precise calculation on this
filamentation and related modulation instabilities is intensively studied from the
start of the research on laser-plasma interaction physics, and their precise
mathematical treatment is given, for example, in Ref. [1].
In the present study, it is shown intuitively why the laser propagation is unstable
and the instability condition of the filamentation instability without precise
calculations. Start with the equation of laser propagation in three-dimensional
space, where the effect of the density modification by the ponderomotive force is
included with the use of the relation (4.2.10):
Fig. 4.1 The structure of an envelope soliton Ψ and the density profile to balance the
ponderomotive force
136
4 Nonlinear Physics of Laser-Plasma Interaction
The soliton solution was obtained with the assumption of one-dimensional
geometry. It is natural question if the soliton propagates stationary in the threedimensional geometry. This is essentially the same question about the stability of
intense laser beam propagation in plasmas. In what follows, consider the physics for
the case where the intense laser propagates in x-direction with a beam size of radius
R in the perpendicular direction. It is assumed that R is much longer the wavelength
of the laser. If this is the case of laser propagation in the vacuum, the laser light
cannot propagate as plane wave for a long distance because of diffraction effect, and
it diverges with an angle of roughly θ ~ k 0 R, where k 0 is the wavenumber of the
laser wave.
The effect of the ponderomotive force, however, is an attractive force effectively
to keep the high-intensity region by depressing the electron density. It is clear that if
the ponderomotive effect to laser intensity change is stronger than the diffraction
effect, the laser intensity at the beam central may increase, and the initial structure
becomes unstable. If the laser intensity is not so high to result the enough nonlinear
force, on the other hand, the diffraction is dominant, and the beam intensity
decreases as a function of time or space. The precise calculation on this
filamentation and related modulation instabilities is intensively studied from the
start of the research on laser-plasma interaction physics, and their precise
mathematical treatment is given, for example, in Ref. [1].
In the present study, it is shown intuitively why the laser propagation is unstable
and the instability condition of the filamentation instability without precise
calculations. Start with the equation of laser propagation in three-dimensional
space, where the effect of the density modification by the ponderomotive force is
included with the use of the relation (4.2.10):
Fig. 4.1 The structure of an envelope soliton Ψ and the density profile to balance the
ponderomotive force
136
4 Nonlinear Physics of Laser-Plasma Interaction