How Is the Sellmeier Equation Derived from Complex Dielectric Constants?

[lillemy]

Homework Statement

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

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

Homework Equations

[itex]\omega = \frac{2\pi c}{\lambda_{vac}}[/itex]

[itex]\omega^{2}_{p}= \frac{Nq^{2}_{e}}{\epsilon_{0}m_{e}}[/itex]

The Attempt at a Solution

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

[itex]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}}[/itex]

i.e. everything is ok expect that i have on extra of [itex]\lambda^{2}_{0,vac}[/itex] 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]

[itex]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}}[/itex]

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

Nothing is wrong. That "extra" [itex]\lambda^{2}_{0,vac}[/itex] 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!

 

[HalfLostAndSearching]

The Sellmeier equation is derived from the classical electromagnetic wave equation, which relates the refractive index (n) to the complex dielectric constant (\epsilon). The equation is given as:

(n^{2}-\kappa^{2}) = 1 + \frac{\omega^{2}_{p}}{\omega^{2}-\omega^{2}_{0}+i\omega\gamma}

Where \omega_{p} is the plasma frequency, \omega_{0} is the natural frequency of the material, and \gamma is the damping constant.

To derive the Sellmeier equation, we start with the complex dielectric constant (\epsilon) given by:

\epsilon = \epsilon_{0}(1+\frac{\omega^{2}_{p}}{\omega^{2}-i\omega\gamma})

The refractive index (n) is related to \epsilon by the equation n = \sqrt{\epsilon_{r}} where \epsilon_{r} is the relative permittivity. Therefore, we can rewrite the above equation as:

n^{2}-1 = \frac{\epsilon_{0}}{\epsilon_{r}}\frac{\omega^{2}_{p}}{\omega^{2}-i\omega\gamma}

Next, we use the definition of refractive index \omega = \frac{2\pi c}{\lambda} to substitute for \omega. This gives:

n^{2}-1 = \frac{\epsilon_{0}}{\epsilon_{r}}\frac{4\pi^{2}c^{2}}{\lambda^{2}}\frac{Nq^{2}_{e}}{m_{e}}\frac{1}{\omega^{2}-i\omega\gamma}

where N is the number density of electrons, q_{e} is the charge of an electron, and m_{e} is the mass of an electron.

We can then simplify this equation by defining A = \frac{\epsilon_{0}}{\epsilon_{r}}\frac{4\pi^{2}c^{2}}{\lambda^{2}}\frac{Nq^{2}_{e}}{m_{e}}, which is a constant for a given material. This gives us the final form of the Sellmeier equation:

n^{2} = 1 + \frac{A}{\omega^{2}-i\omega\gamma}

To convert this equation to a form with wavelength (\lambda) instead of frequency, we