Light Waves

• Walter Greiner
Part of the Theoretical Physics book series (CLASSTHEOR)

Abstract

For light waves
$$\omega (k) = \frac{{ck}}{{n(k)}}{\text{ }}or{\text{ }}k(\omega ) = n(\omega )\frac{\omega }{c}$$
(19.1a)
where c is the speed of light in vacuum and n(k) is the index of refraction of the corresponding medium. According to equation (18.3) we obtain the phase velocity
$${\upsilon _p} = \frac{{\omega (k)}}{k} = \frac{c}{{n(k)}}{\text{ }}or{\text{ }}{\upsilon _p} = \frac{\omega }{{k(\omega )}} = \frac{c}{{n(\omega )}}$$
To calculate the group velocity, we write formally
$$1 = \frac{{d\omega }}{{d\omega }}{\text{ }}or{\text{ }}1 = \frac{{dk}}{{dk}}$$
and with equation (19.1 a) we obtain
$$1 = \frac{c}{{{n^2}}}\left( {\left. {n\frac{{dk}}{{d\omega }} - k\frac{{dn}}{{d\omega }}} \right)} \right.{\text{ }}or{\text{ }}1 = \frac{n}{c}\frac{{d\omega }}{{dk}} + \frac{\omega }{c}\frac{{dn}}{{d\omega }}\frac{{d\omega }}{{dk}}$$
(19.1b)

if the functions w(k) and n(w) are differentiable so that dn/dk = (dn/dw)(dw/dk).

Keywords

Microwave Convolution Refraction