# The complex refractive index vs. permittivity

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1. Nov 4, 2014

### Plant_Boy

I have a process of thought and would like to run past some other minds to point out if I am incorrect in my thinking.

I am looking into conductivity in high frequencies and a lot of papers I am looking up list a complex refractive index. They list something as in nAg = 0.1453 + j11.3587. (Excuse the imaginary symbol, j, I come from an electrical engineering background.)

Various sources inform that $n = \sqrt {ε_r}$. [Link]

Also that $ε = ε' - jε'' = ε_1 - j \frac {σ}{ω}$ [Electromagnetics for Engineers; Fawwaz T Ulaby]

We can get from [Wikipedia.org] that:
$ε = ε_1 + jε_2 = (n + j κ)^2 = n^2 + j 2nκ - κ^2$
$ε_1 = n - κ^2; ε_2 = 2nκ$
*Possible contradiction in Wikipedia vs. Ulaby*
Ulaby states - $ε = ε' - jε''$
Wikipedia states - $ε = ε_1 + jε_2$
So, does:
$2nκ = \frac {σ}{ω}$
Where:
n - real part refractive index
κ - Complex part refractive index
σ - conductivity
ω - angular frequency
I am kind of running this by so that someone can say "Yup" but also, I think, writing it down helps me to understand a little better. Also, this is the first time of me using LaTeX and wanted to keep trying it out.

Last edited: Nov 4, 2014
2. Nov 4, 2014

### DrDu

I don't know why Ulaby uses a minus sign in the definition of epsilon''. Must be an engineering convention.

3. Nov 4, 2014

### dlgoff

Isn't that just an approximation?
If you mean this equation, I believe it's a dash; not a negative sign.

4. Nov 5, 2014

### Plant_Boy

It was a proof from Maxwell's Equations
$\nabla \times H(t)= J(t) + jωεE(t)$
$J(t) = σE(t)$

Substitute
$\nabla \times H(t) = (σ + jωε) E(t)$
$\nabla \times H(t) = jω (ε + \frac {σ}{jω}) E(t)$
$\nabla \times H(t) = jω (ε - j \frac {σ}{ω}) E(t)$
$\nabla \times H(t) = jω (ε' - j ε'') E(t)$
Where:
H(t) - Time Varying Magnetic Field with direction (Still getting used to the $coding) E(t) - Time Varying Electric Field with direction J(t) - Current Density σ - Conductance ω - Angular Frequency (2π f) ε - Permittivity​ Though I could well be wrong in my assumptions as this provides a relationship between Time Varying (TV) Magnetic fields and TV Electric fields... 5. Nov 6, 2014 ### Plant_Boy I suppose the question would then be, is the permittivity gained through complex refractive index similar to that of the permittivity relating Magnetic Fields and Electric fields? 6. Nov 6, 2014 ### DrDu It certainly is, but usually, we use it in other frequency regions. There are also different conventions. E.g. in optics it is usual to set B=H. All potential magnetic effects are included in a dependence of the dielectric function on the wavevector, i.e. $\epsilon(\mathbf{k},\omega)$. 7. Nov 13, 2014 ### Plant_Boy Previously I had given the proof: [itex]\varepsilon - j \frac{\sigma}{\omega} = n^2 - K^2 + j2nK$
Should it be:
$\varepsilon_r = n^2 - K^2 + j2nK$
And so:
$\varepsilon_0 \varepsilon_r = \varepsilon_0 \varepsilon_r - j \frac{\sigma}{\omega}$
$\varepsilon_r = \varepsilon_r - j\frac{\sigma}{\omega \varepsilon_0}$
Therefore:
$\varepsilon_r - j \frac{\sigma}{\omega \varepsilon_0} = n^2 - K^2 + j2nK$
Does this sound correct?

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