Rays

It gets to be pretty cumbersome and visually complicated to make wave animations for everything. It turns that we have a rather powerful alternative to these animations and we have already used it many times. The wave vector Links to an external site. for a wave has a magnitude and a direction and points in the direction perpendicular to surfaces of constant phase. We can, define the k-vector as the gradient of the phase of a wave inasmuch as

$\overset{\rightharpoonup }{k}\left(\overset{\rightharpoonup }{r}\right)\equiv \overset{\rightharpoonup }{\nabla }\varphi \left(\overset{\rightharpoonup }{r},t\right)$

where we have already [Phase and phase velocity] seen that the phase of a wave is the argument of the cosine or in complex notation for a plane wave (this wave is just one particular example, what we show here will be quite general)

$\overset{\rightharpoonup }{E}\left(\overset{\rightharpoonup }{r},t\right)=\hat{x}E_0e^{i\left(\overset{\rightharpoonup }{k}\cdot \overset{\rightharpoonup }{r}-\omega t\right)}$

here, the electric field of the plane wave is 'polarized' in the x-direction and the phase is given by

$\varphi \left(\overset{\rightharpoonup }{r},t\right)=\overset{\rightharpoonup }{k}\cdot \overset{\rightharpoonup }{r}-\omega t$

then, the k-vector is

$\overset{\rightharpoonup }{k}\equiv \overset{\rightharpoonup }{\nabla }\varphi =\overset{\rightharpoonup }{\nabla }\left(\overset{\rightharpoonup }{k}\cdot \overset{\rightharpoonup }{r}-\omega t\right)=\partial _x\left(\overset{\rightharpoonup }{k}\cdot \overset{\rightharpoonup }{r}-\omega t\right)\hat{x}+\partial _y\left(\overset{\rightharpoonup }{k}\cdot \overset{\rightharpoonup }{r}-\omega t\right)\hat{y}+\partial _z\left(\overset{\rightharpoonup }{k}\cdot \overset{\rightharpoonup }{r}-\omega t\right)\hat{z}$

writing this out, we have

$\overset{\rightharpoonup }{k}\equiv \overset{\rightharpoonup }{\nabla }\varphi =\partial _x\left(k_xx+k_yy+k_zz-\omega t\right)\hat{x}+\partial _y\left(k_xx+k_yy+k_zz-\omega t\right)\hat{y}+\partial _z\left(k_xx+k_yy+k_zz-\omega t\right)\hat{z}$

and carrying out the derivatives

$\overset{\rightharpoonup }{k}\equiv \overset{\rightharpoonup }{\nabla }\varphi =k_x\hat{x}+k_y\hat{y}+k_z\hat{z}$

(In this particular example, we must have k_x=0 because the electric field is polarized in the x-direction and we have already seen that in an isotropic medium $\overset{\rightharpoonup }{k}\cdot \overset{\rightharpoonup }{E}_0=0$ [Light is a Transverse Wave])

It is worth noting that it is not always possible to define a wave vector. In the case where the amplitude of the wave is zero (when there is no light in a particular area) the phase is undefined

$\varphi _x\left(\overset{\rightharpoonup }{r},t\right)\equiv \tan ^{-1}\left(\frac{\mathcal{I}\mathit{m}\left(E_x\right)}{\mathcal{R}\mathit{e}\left(E_x\right)}\right)$

so that we get indeterminate in the argument of the arctangent. It is also worth noting that even though we usually associate the k-vector with the direction of wave propagation, this is not always the case. The true direction of wave propagation is the direction of energy flow, which is given by the Poynting vector.

In any case, we return to our discussion of reflection, now using a single ray, or k-vector for each plane wave

In the figure, $\overset{\rightharpoonup }{k}_p$ is the primary wave's k-vector and is shown perpendicular to the wave fronts. On the right panel, $\overset{\rightharpoonup }{k}_s$ is the secondary (or scattered) wave's k-vector and is perpendicular to those wave fronts. We can now develop a new picture for the law of reflection in terms of the k-vectors as

and even more abstracted, removing all representations of the actual wave, which allows us to draw this in three-dimensions a bit easier.

 

This figure also defines the plane-of-incidence, which is the plane that contains the incident or primary wave vector and the surface normal vector.