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 

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Copyright information

© Springer Science+Business Media New York 1998

Authors and Affiliations

  • Walter Greiner
    • 1
  1. 1.Institut für Theoretische PhysikJohann Wolfgang Goethe-UniversitätFrankfurt am MainGermany

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