The Brewster angle for parallel-polarized electromagnetic waves incident on a relativistically moving isotropic plasma half-space is investigated. There exist two Brewster angles for certain ranges of the medium velocity irrespective of whether the medium moves parallel or normal to the interface. In the case of parallel motion, it is found that there are no Brewster angles for wave frequencies less than the plasma frequency.

Yeh [1] investigated the Brewster angle for a dielectric medium moving at a relativistic speed. Lee and Lo [2] studied, among other things, the conditions for total transmission for electromagnetic waves incident at an interface between a dielectric medium and an uniaxial plasma moving parallel to the interface. The plane of incidence, however, was confined to be parallel to the direction of motion. Later, Kong and Cheng [3] and Pyati [4] removed this restriction and considered an arbitrary orientation of the plane of incidence. In a recent paper, Chuang and Ko [5] obtained the generalized Brewster law for a dielectric medium moving parallel to its surface.

This communication reports the Brewster-angle phenomenon for a moving plasma halfspace. The results are expected to be useful in maximizing the strength of a transmitted signal through a relativistically moving bounded plasma. The plasma medium being dispersive in nature, the Brewster angle will, in general, be a function of the incident wave frequency as well as the medium velocity. The problem in its full generality too would be involved and as such the present treatment is restricted to the case in which the plane of incidence is parallel to the direction of medium motion. The motions of the medium both parallel and perpendicular to the interface are considered.

Consider a parallel-polarized plane electromagnetic wave obliquely incident (xz-plane of incidence) on an isotropic plasma half-space $({\mathrm{\mu }}_0,{ϵ}^{\prime })$ moving relativistically through the free space $({\mathrm{\mu }}_0,{ϵ}_0)$. The power reflection coefficient (ρ_x for the medium moving parallel to the interface, i.e., $\mathrm{\upsilon }={\mathrm{\upsilon }}_0\widehat{x}$, and ρ (for normal motion, i.e., $\mathrm{\upsilon }={\mathrm{\upsilon }}_0\widehat{z}$) can be easily obtained by the conventional theory of reflection and transmission:

In the above, q′, the root of the dispersion equation, is given by

and

where C = cos θ and S = sin θ (θ is the angle of incidence).

The symbols and expressions used are as in References 6,7.

The power reflection coefficient is zero when $q^{\prime }={ϵ}^{\prime }C$. This condition really means that the ratio of the tangential electric and magnetic fields in plasma $\left(E_x^{\prime }{}^{\text{p}}\text{}\text{/}\text{}{H}_y^{\prime }{}^{\text{p}}={\mathrm{\eta }}_0{q}^{\prime }\text{}\text{/}\text{}{ϵ}^{\prime }\right)$ is the same as that in the free space $\left(E_x^{\prime }\text{/}{H}_y^{\prime }={\mathrm{\eta }}_0{C}^{\prime }\right)$, thus giving a perfect match of the plasma-free space system. Substituting for q′, the above conditions may be recast in the form

For a moving dielectric medium, Equation 14D.2 will lead to a quadratic equation in S(C) for parallel (normal) motion which can be easily solved to give the Brewster angle [1]. However, when the moving medium is a plasma, the relative permittivity ${ϵ}^{\prime }$ is a function of the angle of incidence, the medium velocity and the incident wave frequency leading now to a quartic equation in S or C:

where

and

where

and

The variation of the Brewster angle is now examined as a function of the medium velocity with incident wave frequency as the parameter.

Case 1: Plasma half-space moving parallel to the interface $\left(\mathrm{\upsilon }={\mathrm{\upsilon }}_0\widehat{x}\right)$.

The Brewster angle for this case is sin⁻¹ S, S being given by Equation 14D.3. This is an eighth-degree curve in S – β and after tracing this complicated curve it is found that, for Ω < 1, no portion of this curve exists within the physically meaningful limits of S and β (0 < S < 1 and −1 < β < 1) relevant to the problem. Thus, there is no Brewster angle for Ω < 1. This is because, in this range, plasma supports [6] evanescent waves only.

Figure 14D.1 shows the sine of the Brewster angle as a function of the medium velocity with the parameter Ω in the range Ω > 1. It is seen that the Brewster angle depends on the magnitude as well as the direction of the velocity of motion; for parallel motion (β positive) the angle increases from its stationary value of sin⁻¹{[(Ω² − 1)(2Ω² − 1)⁻¹]^1/2} with velocity, while for the antiparallel motion (β negative) the angle decreases with velocity. Another Brewster angle appears at β = β₀ = [(1 – Ω²) + Ω² (Ω² − 1)^1/2]^1/2 and extends over a narrow range of medium velocity beyond β₀. For a given velocity, the Brewster angle increases with the incident wave frequency.

Case 2: Plasma half-space moving normal to the interface $\mathrm{\upsilon }={\mathrm{\upsilon }}_0\widehat{z}$.

The Brewster angle for this case is cos⁻¹ C, C being obtained by the solution of Equation 14D.4. The C versus β curve was traced over the complete range of Ω and, as in parallel motion, some branches of the curve are outside the physically meaningful limits of C and β.

Figure 14D.2 incorporates the allowed branches and shows the cosine of the Brewster angle as a function of the medium velocity. The Brewster angle for the stationary case is given by cos⁻¹(2 – l/Ω²)^−1/2. When the medium moves away from the incident wave (β positive), there is an additional angle of incidence for which the reflected wave as seen from the observer’s frame grazes the interface, thus giving zero-reflected power [7]. This is given by β = (1 – S)/C, that is, C₀ = 2β/(1 + β²) and is included as a dashed curve in Figure 14D.2. The existence of C₀ is independent of the nature of the moving medium and as such this aspect checks with Yeh’s conclusions for a dielectric medium [1].

There is no Brewster angle between β = (1 – S)/C and β = C (this is the range where the reflected wave travels toward the moving medium [7]) for $\mathrm{\Omega }<2\sqrt{2}$. The reason is that for this range of β and Ω, plasma supports only evanescent waves. For $\mathrm{\Omega }<2\sqrt{2}$, the relevant Brewster angle is given by the portion of the curve boxed by the curves C₀ and β = C. There exist two Brewster angles for large Ω, and one of them, due to the branch to the right of β = C line, corresponds to the case of the medium impinging on the wave. It should be further noted that β = (Ω² – 1)/(Ω² + 1) corresponds to C = 1 on the C versus β curve which means that for these medium velocities the Brewster angle is 0°.

Finally, for the medium moving toward the incident wave (β negative), the Brewster angle is seen to increase with velocity, becoming 90° at $-{\mathrm{\beta }}_0=-{\left\{\frac{1}{2}\left[1-2{\mathrm{\Omega }}^2+{\left(4{\mathrm{\Omega }}^4+1\right)}^{1/2}\right]\right\}}^{1/2}$. For –β₀ > β > −1, there is no Brewster angle.