Question 8.16: A uniform plane wave of a wavevector ki = ax + √3 ay + 2√3 a......
A uniform plane wave of a wavevector \pmb{k}_{i} = \pmb{a}_{x} + \sqrt{3} \pmb{a}_{y} + 2\sqrt{3} \pmb{a}_{z} propagates in free space in the region z < 0 , and impinges on the surface of a dielectric of \epsilon _{r} = 1.69 and \mu _{r} = 1, occupying the region defined by z ≥ 0 . Find (a) \pmb{k}_{r}, and (b) \pmb{k}_{t}.
(a) The law of reflection comprises of two parts:
(I) \theta _{r} = \theta _{i} \\ (II) \pmb{k}_{i} and \pmb{k}_{r} are in the plane of incidence.
Thus
\pmb{k}_{r} = \pmb{a}_{x} + \sqrt{3} \pmb{a}_{y} – 2\sqrt{3} \pmb{a}_{z}(b) The law of refraction comprises of two parts:
(I) n_{i} \sin \theta _{i} = n_{t} \sin \theta _{t} \\ (II) \pmb{k}_{i} and \pmb{k}_{t} are in the plane of incidence
From Snell’s law, the tangential component of \pmb{k}_{t} is the same as that of \pmb{k}_{i}, that is,
\pmb{k}_{t} = \pmb{a}_{x} + \sqrt{3} \pmb{a}_{y} + k_{tz} \pmb{a}_{z}where k_{tz} is an unknown.
Using \pmb{k}_{t} = n_{t}k_{i} = 1.3 \times 4 in the above equation, we write
(5.2)^{2} = 1 + (\sqrt{3} )^{2} + (k_{tz})^{2}Thus,
\pmb{k}_{t} = \pmb{a}_{x} + \sqrt{3} \pmb{a}_{y} + 4.8 \pmb{a}_{z}