Fundamentals of Plasma Physics 2025, 29 - Homogeneous Plasma

Introducing non-uniformity along one coordinate direction, in a high-frequency EM plasma. Ignore ion motion,

The last statement is true to a cutoff, which might happen with only mild inhomogeneities. To arrive at the functional dependence for inhomogeneity in all three dimensions, begin with the dispersion relation as D(k, x) = 0, and introduce the perturbative δk, δx before applying Taylor expansion. The result gives a coupled equations, which make the following Hamiltonian dispersion relation for particular mode possible: D(k, x) = ∑_ijα_ijk_ik_j + g(ω, n(x), B(x)) = 0. For an EM mode in unmagnetized plasma with nonuniform density, ω²D(k, x) = c²k² - ω² + ²_pe(x) = 0. This can be modeled analogously to a particle in a potential well, with a kinetic term and potential energy (the ω-terms).

For abrupt transitions from plasma to vacuum (introducing an edge to the plasma volume), begin with Maxwell equations

For azimuthally symmetric TM modes propagating in a uniform cylindrical plasma with radius a in a vacuum, drop the ∂_z terms from the field descriptions. By using symmetry, for the radial wave numbers: κ²_p = k²-ω²/c²P, and κ²_e = k²-ω²/c². The electric field in this situation should be finite, and E_z is finite at r = 0. Only the I₀(κ_pr) solution is allowed in the plasma region. The K₀(κ_vr) solution is allowed in the vacuum region. The parallel electric field is also continuous across the surface. Essentially, E_z^vacL - E_z^plasmaL = 0. By integration, P(P - k²c²/ω²)^-1∂E_z/∂r.