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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

Dispersion Relation Wave Packet Group Velocity Analytic Property Integration Path 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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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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