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Cold-plasma left-handed wave 2026-10-24

With the same conventions as the cold-plasma right-handed wave, this transverse electromagnetic wave has

E_{y} = - i E_{x}

and refractive index

n^{2} = S + g

. With immobile ions,

L = 1 - ω_{p e}^{2} / [ω (ω + Ω_{e})]

.

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Past exam of the collisionless plasma physics course of the University of Oxford / 2025 / 1 / b / Solution 2026-10-25

The Faraday law and Ampere-Maxwell law give

δ B = (c / ω) k \times δ E

and

k \times (k \times δ E) + \frac{ω ^{2}}{c ^{2}} ϵ \cdot δ E = 0.

With refractive index

n = k c / ω

, the linear system is therefore

S - n^{2} cos^{2} θ - i g n^{2} sin θ cos θ i g S - n^{2} 0 n^{2} sin θ cos θ 0 P - n^{2} sin^{2} θ δ E = 0.

R = S - g, L = S + g, R L = S^{2} - g^{2} .

Here

R

labels the cold-plasma right-handed wave and

L

the cold-plasma left-handed wave, with their polarizations fixed explicitly in d. In the electron-only response,

R = 1 - \frac{ω _{p e}^{2}}{ω ( ω - Ω _{e} )}, L = 1 - \frac{ω _{p e}^{2}}{ω ( ω + Ω _{e} )} .

N = n^{2}

, expansion of the determinant yields the dispersion relation

A N^{2} - BN + C = 0, A = S sin^{2} θ + P cos^{2} θ, B = R L sin^{2} θ + PS (1 + cos^{2} θ), C = PR L .

Equivalently,

P (N - R) (N - L) cos^{2} θ + (SN - R L) (N - P) sin^{2} θ = 0.

tan^{2} θ = - \frac{P ( n ^{2} - R ) ( n ^{2} - L )}{( S n ^{2} - R L ) ( n ^{2} - P )} .

The undivided determinant equation remains valid when this rational form has simultaneous zero factors.

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Past exam of the collisionless plasma physics course of the University of Oxford / 2025 / 1 / c / Solution 2026-10-25

A plasma wave resonance has

n^{2} \to \infty

. At finite coefficients of the magnetized cold-plasma dielectric tensor, its generic condition is the vanishing of the coefficient of

n^{4}

:

S sin^{2} θ + P cos^{2} θ = 0, tan^{2} θ = - P / S .

A plasma wave cutoff has

n^{2} = 0

, so the constant term of the dispersion relation vanishes:

PR L = 0, that is P = 0, R = 0, or L = 0.

Both statements are branch conditions: check the undivided dispersion relation at special angles and simultaneous zero factors. In particular, the parallel plasma oscillation at

P = 0

is a longitudinal branch with arbitrary

k

in the cold limit, while the transverse parallel branches have plasma wave cutoffs at

R = 0

or

L = 0

. Poles of the magnetized cold-plasma dielectric tensor can give additional limiting plasma wave resonances, such as the parallel cold-plasma right-handed wave at

ω = Ω_{e}

; these are not obtained by assuming finite

S, P, g

.

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Past exam of the collisionless plasma physics course of the University of Oxford / 2025 / 1 / d / Solution 2026-10-25

For parallel propagation, use the undivided dispersion relation:

P (n^{2} - R) (n^{2} - L) = 0.

The five requested cases are:

Plasma oscillations: $P = 0$ and $δ E ∥ B_{0} ∥ k$ . With negligible ion response, $ω^{2} = ω_{p e}^{2}$ , independently of $k$ in this cold approximation.
Cold-plasma right-handed waves: $n^{2} = R$ , $E_{y} = i E_{x}$ , $E_{z} = 0$ , for propagation along $+ B_{0}$ . At a fixed point, the real electric field rotates from $+ \hat{x}$ towards $+ \hat{y}$ , in the sense of electron gyration.
Cold-plasma left-handed waves: $n^{2} = L$ , $E_{y} = - i E_{x}$ , $E_{z} = 0$ . The real electric field rotates in the opposite sense.
Ordinary plasma waves: for perpendicular propagation, $θ = π /2$ , $n^{2} = P$ , and $δ E ∥ B_{0}$ . Their dispersion relation is $ω^{2} = ω_{p e}^{2} + c^{2} k^{2}$ .
Extraordinary plasma waves: for perpendicular propagation, $n^{2} = R L / S$ , $E_{z} = 0$ , and $E_{y} = i g E_{x} / (S - n^{2})$ . In general the electric field has both longitudinal and transverse components.

For the perpendicular cases the undivided dispersion relation factors as

(n^{2} - P) (S n^{2} - R L) = 0

, up to an overall nonzero factor. This also identifies the extraordinary plasma wave plasma wave resonance at

S = 0

.

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