# Complex refractive index

## Main Question or Discussion Point

The index of refraction is a complex number if the mediums conductivity is non zero.

n=Re(n)+Im(n)*i

We know that for real refractive index:

v=c/n

How is the corresponding relation for complex n? I would expect (and got from my calculations)

v=c/Re(n),

but since Re(n) may become arbitarilly small (by chosing the frequency of the wave suitably) I distrust the above.

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Your thinking is correct. Actually there is no corresponding relation fo complex part of n. You can simply see the meaning of complex part if you put it in the plane wawe:

$$e^{i(kx-\omega t )} ~ where ~ k=\frac{\omega n}{c} ~ and~ n=n_0+i\kappa$$

you get:

$$e^{-\omega \kappa x /c } ~ e^{i(kx-\omega t) } ,~ k~ is~ now ~ k=\frac{\omega n_0}{c}$$

So in the phase velocity there is only real part of n. Imaginary part brings you only a damping factor that decreases the amplitude exponentially.

I hope it's was clear enough :)

Thanks! That fine example was precisely what gave birth to my curiosity! But if the relation I got

$$v = \frac{c}{\Re{n}}$$

is correct, isn't it strange that, by making $$\Re{n}$$ small we can achieve virtually unlimited phase velocity? We know that practically zero real part of $$n$$ can be observed for certain frequencies of EM waves in a medium.

That alerts me, even though there shouldn't be any violation of relativistic principles, since $$v$$ is a phase velocity.

Thank you again :-)

olgranpappy
Homework Helper
Thanks! That fine example was precisely what gave birth to my curiosity! But if the relation I got

$$v = \frac{c}{\Re{n}}$$

is correct, isn't it strange that, by making $$\Re{n}$$ small we can achieve virtually unlimited phase velocity...
no, it's not strange, since the group velocity
$$\frac{d\omega}{dk}$$
is still less than c.

f95toli
Gold Member
The phase velocity is quite often much larger than c in transmission lines (if I am not misstaken this happens in e.g. cylindrical waveguides). But, as olgranpappy pointed out, there is no violation of SR since the group velocity (which is the speed you can transfer information at in this case) is always smaller than c.

Thank you both!

I was aware of the point that there is no violation of SR, but still suprised that v can exceed 3x10^8 m/s by that much. Does anyone know how big v has been observed to become?

Thanks again :-)

olgranpappy
Homework Helper
if you are asking how small n has been known to get then the answer is zero... both the real and imaginary parts of the dielectric function go to zero at the plasma frequency.

v, in this context, has no other meaning that some quantity which is inversely proportional to n...

Ah, I think I understand where my mistake is: so v cannot be measured directly?

For large enough frequencies I have, too found

$$p=\frac{1}{\omega}\sqrt{{\omega}^2-{\omega_p}^2}$$

which, as you said, vanishes at the plasma frequency.

Thank you!

This post reminds me one of mu question. Does anyone know something called linewidth enhancement factor? It is the ratio of imaginary part of refractive index to its real part. I have a feeling that, by writing a plain wave as e^-i(omega * t - k * r) or e^i(omega * t - k * r), this value will have different sign. Could anybody give any comment on it?

olgranpappy
Homework Helper
This post reminds me one of mu question. Does anyone know something called linewidth enhancement factor? It is the ratio of imaginary part of refractive index to its real part. I have a feeling that, by writing a plain wave as e^-i(omega * t - k * r) or e^i(omega * t - k * r), this value will have different sign. Could anybody give any comment on it?

the real and imaginary parts of the refractive index are properties *of the material*, not of the probe which you send into the material and certainly not of the conventions which you use for plane-waves.

That being said, if you are asking how the ratio Im(N(f))/Re(N(f)) (where N is the complex refractive index; $$N^2=\epsilon$$ where $$\epsilon$$ is the dielectric constant) changes when f --> -f (where f is frequency) I believe that you can figure it out by the fact that when f --> -f we know:

$$Im(\epsilon) \to -Im(\epsilon)$$

and

$$Re(\epsilon)\to Re(\epsilon)$$.