How Is the Sellmeier Equation Derived from Complex Dielectric Constants?

[lillemy]

Homework Statement

Derive the Sellmeier equation
n^{2} = 1 + \frac{A\lambda^{2}_{vac}}{\lambda^{2}_{vac}-\lambda^{2}_{0,vac}}
from
(n+i\kappa)^{2}= 1 + \frac{\omega^{2}_{p}}{\omega^{2}_{0}-<br /> i\omega\gamma - \omega^{2}}

for a gas or glass with negligible absorption (i.e. \gamma ≈ 0, valid far
from resonance \omega_{0}, where \lambda_{0,vac}
corresponds to frequency \omega_{0} and A is a constant.

Homework Equations

\omega = \frac{2\pi c}{\lambda_{vac}}

\omega^{2}_{p}= \frac{Nq^{2}_{e}}{\epsilon_{0}m_{e}}

The Attempt at a Solution

Since the absorption is negligible, \gamma = 0 we can drop the imaginary part , and I will substitute directly for \omega and \omega_{p} from the above equations. It gives this result:

1+ \frac{\lambda^{2}_{vac}\lambda^{2}_{0,vac}\frac{Nq^{2}_{e}}{4\pi^{2}c^{2}\epsilon_{0}m_{e}}}{\lambda^{2}_{vac}-\lambda^{2}_{0,vac}}

i.e. everything is ok expect that i have on extra of \lambda^{2}_{0,vac} in the numerator. What have I done wrong? Very thankful for all help:)

 

[Brian Coffey]

I know the thread is 3 years old but any idea on this question? Have a similar problem, appreciate any help

 

[ehild]

1+ \frac{\lambda^{2}_{vac}\lambda^{2}_{0,vac}\frac{Nq^{2}_{e}}{4\pi^{2}c^{2}\epsilon_{0}m_{e}}}{\lambda^{2}_{vac}-\lambda^{2}_{0,vac}}

i.e. everything is ok expect that i have on extra of \lambda^{2}_{0,vac} in the numerator. What have I done wrong? Very thankful for all help:)

Nothing is wrong. That "extra" \lambda^{2}_{0,vac} is included into the constant A.

 

[Brian Coffey]

That's what I was thinking but wasn't sure since that term appeared elsewhere in the formula, thanks for your help!