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Hybrid singularity for the oblique incidence response of a cold plasma

In this paper, we study the electromagnetic response of a cold plasma with dissipation ν > 0 in a general slab geometry (plasma density n0(x) and background magnetic field B0(x)~e3). We express the behavior of the electromagnetic field ( ~ E, ~ H) (at a given frequency ω) in the neighborhood of the hybrid resonance xh solution of ω = ωh(x). This generalizes to any incident wave vector (k1, k2, k3) the result obtained by Despres, Imbert-Gerard and Lafitte in the normal incidence case (the incoming electromagnetic field has its wave vector belonging to the orthogonal plane to ~e3, id est k3 = 0), hence treating a fully coupled system of ODEs of order 4. This is done by deducing from the complete system of ODEs an integro-differential system which differential part contains a regular singular point at ν → 0+. This new system is then solved in the neighborhood of this regular singular point using the variation of parameters method. In the oblique incidence case at the limit ν → 0+, we observe that (E2, E3, H2, H3) are integrable in the neighborhood of xh uniformly in ν → 0+, while the most singular term of the limit of ε0E1 when ν → 0+ is proportional to [P.V.( 1 x− xh )− iπsign(∂xωh(xh))δxh + a ln(x− xh) ].