# Speed of light of different wavelengths in medium

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1. Sep 1, 2015

### LmdL

Hello,
As we all know, speed of light of different wavelengths (let's say red and blue) in vacuum is same. However, in medium (like glass) it's different and depends on a wavelength and a refractive index.
Let's say we send a short pulse of white light incident into a piece of glass (of length L). Red light will come out of it first and a blue last. But how can I calculate the time difference (or a displacement) of red and blue light at the exit from the glass?
I know the velocity in medium is v=c/n, where c is constant (speed of light in vacuum) and n is a refractive index of the medium. But how can I get a relation between v (speed of light in medium) and its wavelength? I guess I need to connect between refractive index n and a wavelength of the light first. Thought to use a Cauchy's equation (or a Sellmeier equation), but maybe there is a easier way?
Thanks!

2. Sep 1, 2015

### nasu

I am afraid not. The variation of the index of refraction with wavelength is determined experimentally. Either you look at a dispersion curve, use a table or an empirical formula.

3. Feb 9, 2016

### akaaty

wave number k is 2pi/wavelength ...also k1/k2= lamda2/lamda1= velocity2/velocity1 ....where 1 and 2 represent medium 1 and 2. These are simply the results of applying the boundary conditions to a wave passing from one medium to another. Since red and blue light cover the same distance through slab, L ....knowing a relationship between their velocities should tell you the difference between their time of arrival just outside the slab.
the relation ship between their velocities inside glass can be found out from their respective wavelengths...since frequency stays constant when light passes through any medium, frequency of red= frequency of blue , or velocity of red/lamda of red = velocity of blue/lamda of blue.

4. Feb 10, 2016

### LmdL

You are right, but I still need their velocity to know the difference between exit times.
Red light velocity:
$$\frac{v_{R}}{\lambda_{R}}=\frac{v_{B}}{\lambda_{B}}\rightarrow v_{R}=\frac{\lambda_{R}}{\lambda_{B}}v_{B}$$
Time taken to red light to pass:
$$t_{R}=\frac{L}{v_{R}}=\frac{\lambda_{B}}{\lambda_{R}}\frac{L}{v_{B}}$$
Time taken to blut light to pass:
$$t_{B}=\frac{L}{v_{B}}$$
Exit time difference:
$$\Delta t=t_{B}-t_{R}=\left (\frac{L}{v_{B}}-\frac{\lambda_{B}}{\lambda_{R}}\frac{L}{v_{B}} \right )=\left (1-\frac{\lambda_{B}}{\lambda_{R}} \right )\frac{L}{v_{B}}$$

So, I still need velocity of either blue or red light.

5. Feb 10, 2016

### blue_leaf77

Well, you indeed need to know the dependency between refractive index and wavelength to get $v(\lambda)$ for different colors. However, there is no such a thing as a general equation describing refractive index as a function of wavelength. The function $n(\lambda)$ is material dependent, the most common way is to use Sellmeier equation. There is no easier than using this equation.
Where does that equation come from? You cannot simply assume that the velocity is linear with respect to wavelength.

6. Feb 10, 2016

### LmdL

That's what akaaty suggested, and I showed that I still need a velocity of either red or blue light (from Cauchy/Sellmeyer).

7. Feb 10, 2016

### blue_leaf77

In akaaty's formula $v_1$ and $v_2$ are defined to be the velocities at two different medium, they do not correspond to velocities for waves of different wavelengths. Moreover about the formula itself, I don't think that represents a correct relationship between the velocities of light in different media for this relationship is given by
$$\frac{v_1}{v_2} = \frac{n(\lambda_2)}{n(\lambda_1)}$$
I guess he/she was taking an analogy with a mechanical waves in a boundary between two media, and that the velocities he was using are the oscillation velocities.

8. Feb 10, 2016

### nasu

No, this is wrong. The frequencies of the red and blue light are not the same. And this is true in any medium. So the relationship between wavelengths and velocities is not valid.
It is true that the frequency does not change when light passes through a medium but this does not help you to find the speed in the medium.
The dependence of velocity on frequency is a property of the medium, you cannot find it from these manipulations.

9. Feb 10, 2016

### Staff: Mentor

Correct, you have to look for data related to the specific medium that you're interested in. Try Googling for phrases similar to "optical dispersion in xxx".