Electromagnetic Fields and Waves: Notes, Solved Problems and Interactive Lab

Can the reflection vanish at some angle? For S polarisation Eq. (6.4.13) gives $Γ_{⊥} = 0$ when

η_{2} cos θ_{i} = η_{1} cos θ_{t} . (6.4.22)

The angle of incidence at which this happens is called the Brewster angle, $θ_{B ⊥}$ . To find it in terms of the medium parameters, eliminate $θ_{t}$ with Snell's law:

cos θ_{B ⊥} = \frac{η _{1}}{η _{2}} cos θ_{t} = \frac{η _{1}}{η _{2}} 1 - sin^{2} θ_{t} = \frac{η _{1}}{η _{2}} 1 - \frac{β _{1}^{2}}{β _{2}^{2}} sin^{2} θ_{B ⊥} .

Squaring and collecting the terms in $sin^{2} θ_{B ⊥}$ ,

1 - sin^{2} θ_{B ⊥} = \frac{η _{1}^{2}}{η _{2}^{2}} (1 - \frac{β _{1}^{2}}{β _{2}^{2}} sin^{2} θ_{B ⊥}) ⟹ sin^{2} θ_{B ⊥} (1 - \frac{η _{1}^{2} β _{1}^{2}}{η _{2}^{2} β _{2}^{2}}) = 1 - \frac{η _{1}^{2}}{η _{2}^{2}} .

Finally $η_{1}^{2} / η_{2}^{2} = μ_{1} ε_{2} / (μ_{2} ε_{1})$ and, since $η β = μ / ε ω μ ε = ω μ$ , $η_{1} β_{1} / (η_{2} β_{2}) = μ_{1} / μ_{2}$ .

Brewster angle, S polarisation

sin^{2} θ_{B ⊥} = \frac{1 - μ _{1} ε _{2} / ( μ _{2} ε _{1} )}{1 - μ _{1}^{2} / μ _{2}^{2}} . (6.4.23)

For media with the same permeability, in particular for all nonmagnetic media, the denominator vanishes while the numerator does not: there is no Brewster angle for S polarisation. For media with the same permittivity but different permeabilities, $sin θ_{B ⊥} = 1/ 1 + μ_{1} / μ_{2}$ .

In plain words

Between two ordinary dielectrics an S wave is reflected at every angle. A reflection-free S wave needs a magnetic contrast between the media, a situation that is very rare in practice.

Figure 6.4.9 shows such a rare case: air on a medium with $μ_{r 2} = 2$ and $ε_{r 2} = 1$ . Then $sin^{2} θ_{B ⊥} = (1 - \frac{1}{2}) / (1 - \frac{1}{4}) = \frac{2}{3}$ and $θ_{B ⊥} = 54. 7^{\circ}$ . Here the S wave is fully transmitted at $θ_{B ⊥}$ while the P wave is reflected ( $Γ_{∥} = 0.333$ ), the mirror image of the behaviour of ordinary dielectrics that we meet in Subsection 6.4.9.

A magnetic medium 2 (\mu_{r2}=2, \varepsilon_{r2}=1) seen from air. Both coefficients start at (\sqrt2-1)/(\sqrt2+1)=0.172. The S coefficient vanishes at \theta_{B\perp}=54.7^\circ, the P coefficient never does. — **Figure 6.4.9.** A magnetic medium 2 ( $μ_{r 2} = 2$ , $ε_{r 2} = 1$ ) seen from air. Both coefficients start at $(2 - 1) / (2 + 1) = 0.172$ . The S coefficient vanishes at $θ_{B ⊥} = 54. 7^{\circ}$ , the P coefficient never does.

6.4.7 The Brewster angle in S polarisation