The two-plasmon instability in warm inhomogeneous plasma for a normally incident pump is considered. The complex eigenfrequencies of the absolute instability are obtained by reducing the linearized fluid equations to a Schrödinger equation in wavenumber space. These eigenvalues are obtained in several ways. One is by combining a perturbation expansion in powers of the reciprocal scale length with WKB theory. The resulting algebraic equations are solved by three analytical approximations and by direct numerical solution. A second way is by analysis of the Schrödinger equation using an interactive WKB computer code. A third way is by the use of a shooting code. These methods are all used and compared for threshold curves and growth rates above threshold. Some eigenfunction forms are also obtained. The threshold is near (v0/ve)2k0 L =3, and varies weakly with β≂v4e/v20c2, rising from near 2 to about 4 over six decades of variation of β. The corresponding critical value of (ky/k0)2 is near 0.2/β over this range. Above threshold, there is a smooth variation of the growth rate with (ky/k0)2, peaking at some intermediate value. The perturbation method is in good agreement there with more exact calculations. Experimental implications of these results are discussed.
The Doppler shift of light reflected from a plane stratified expanding plasma is analyzed. Nonlinear effects are not considered and oblique incidence is restricted to the case of s polarization. The frequency shift is shown to consist of two components, one due to the motion of the reflecting surface, and another due to plasma flow through that surface. The shifts have a different dependence on the angle of incidence. Typically, the two contributions are comparable in laser fusion applications, being of order dw/w∼cs/c∼10−3, where cs = (Z Te/M)1/2 is the ion sound speed in the underdense plasma. In general, the Doppler shift has a time variation which induces a bandwidth in the reflected wave.
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