Dependence of index of refraction on frequency

[timetraveller123]

ok so recently i was doing dispersion of light and all of sudden something that i always took for granted came across the fact that blue light refracts more than red light then i recalled snells law and realized that it had no terms with the quantity of frequency in it (n sin θ = c) then i decided to consult my physics teacher and he told me the refractive index of material varies with the frequency of the light and that came as a huge shock to me i decided to google more about it
but many sites just state n = $\frac{\lambda_o}{\lambda}$ but that just seems to be double reasoning so i googled more and found out something about electrons vibration and transimission of light energy
so this is my understanding thus far
in vacuum speed of light is just $\frac{1}{\sqrt{\mu_o \varepsilon_o}}$
so in any other medium the permittivity and permeability would have have to be replaced accordingly
why would those constants be affected by frequencies?
another reasoning was to think of light as constantly exciting electrons
every electron in the medium has its own resonance frequency
and the light that comes along has its own frequency
so essentially this is just a driven harmonic oscillator (was truly amazed there )
it is also my understanding that the delay in absorption and emission of the light is what is "slowing down the light"
so now my question is
- when the light is not of resonant frequency what exactly happens to the light i know it gets absorbed and emitted but what exactly is happening in the middle and what happens as the frequency changes
- when the light is of resonant frequency then what happens
any conceptual understanding will be useful and math would also be appreciated

 

[blue_leaf77]

when the light is of resonant frequency then what happens
any conceptual understanding will be useful and math would also be appreciated

When the light is resonant with the energy levels of the material, some electrons will be excited to a higher level. From here on, a couple of different process can occur depending on the characteristic of the material. One possibility is for the excited electrons to transfer its energy to nearby atoms or molecules until it goes back to the ground level. In this case a little to no light is being reemitted since most of the energy is converted to atomic or molecular vibration (it heats up the medium). Another possibility is to go down back to the ground state by emitting spontaneous photon which is incoherent by nature. A type of material which is rare in nature except those fabricated by human is the so-called amplifying material used in lasers. In this material, the excited electrons can also undergo relaxation by emitting a coherent light.

when the light is not of resonant frequency what exactly happens to the light i know it gets absorbed and emitted but what exactly is happening in the middle and what happens as the frequency changes

It will simply pass through the material with minimal loss.

i decided to consult my physics teacher and he told me the refractive index of material varies with the frequency of the light and that came as a huge shock to me i decided to google more about it

In the classical theory of light, the passage of light through matters is described by the so-called polarization of the medium. It describes how matter response to the passing light and it's also determined by the frequency of the light itself. Check this link https://refractiveindex.info/ .

it is also my understanding that the delay in absorption and emission of the light is what is "slowing down the light"

Absorption and dispersion are indeed mutually dependent. See this link about Kramers-Kronig relation which relates refractive index and the absorption coefficient.

 

[Gigaz]

The picture of the driven harmonic oscillator is not too bad. Essentially, light causes a time-dependent, sinusoidal polarization in a material. The amplitude and phase lag is ultimately what gives you the refractive index.

In the range of visible light, the important resonances are the plasmon resonances in the near-UV. Plasmons are very similar to sound waves, but it's not the air density that changes, but the electron density in the material.
Not all resonances come from Plasmons. In the microwave region, you can excite molecular oscillations. In the X-ray region, there are excitations of tightly bound electrons close to the nucleus, just to name a few examples.

So, what happens in a driven harmonic oscillator? Think of shaking a traffic sign or something. When you shake it with the right frequency (its resonance) it can shake quite a lot, it may even be damaged. It absorbs a lot of the energy that you put into it.
And the same is true for light - resonances are peaks in the absorption (imaginary part of n). The real part is slightly more complicated, but generally it looks similar to the derivative of the imaginary part (its actually the Kramers Kronig transformation). Anyways, the height and width of the absorption peaks depends mostly on the half-life of excited intermediate states, and that's unfortunately the point where you get better answers from a book about advanced Quantum mechanics and not from the internet.

 

[timetraveller123]

It will simply pass through the material with minimal loss.

i want to know what you mean by simply pass through i understood the rest you are talking about

phase lag is ultimately what gives you the refractive index.

i was just reading up the lorentz dispersion equation is that the phase lag you are talking about

my understanding of resonance is when maximum frequency is transferred
and what from what you are saying i gather that the loss energy goes into heating up the medium
and the closer you get to the resonant frequency the more the energy transferred hence the more the refraction
is my understanding correct please do correct me if i am wrong because i tend to jump to conclusions thanks for your help i am just starting to get a hang of this
didn't so much physics was behind a simple refraction

 

[blue_leaf77]

i want to know what you mean by simply pass through i understood the rest you are talking about

It's like when you shine a laser through a glass. Nothing perceptible occurred except that it's frequency and intensity are very slightly changed due to the nonzero absorption.

 

[timetraveller123]

so it is still a process of absorption and emission is it ?
and why would frequency be changed ?
and why would intensity be changed ?

 

[Gigaz]

i was just reading up the lorentz dispersion equation is that the phase lag you are talking about

my understanding of resonance is when maximum frequency is transferred
and what from what you are saying i gather that the loss energy goes into heating up the medium
and the closer you get to the resonant frequency the more the energy transferred hence the more the refraction
is my understanding correct please do correct me if i am wrong because i tend to jump to conclusions thanks for your help i am just starting to get a hang of this
didn't so much physics was behind a simple refraction

You see, the index of refraction n that is taught in school is generally a simplification.

All materials have a refractive index, but that's not obvious because most are not transparent.

n is actually composed of a real and an imaginary part n=a+ib (Ask wikipedia about complex numbers)
The imaginary part gives you the absorption. Most materials absorb visible light quite quickly.

The energy is usually going into heat, but the material may of corse also be a solar cell that generates electricity from light. That would not be very different.

However - you must carefully distinguish absorption and refraction. Refraction means: Photon goes in, Photon goes out. Absorption means: Photon goes in. Two very different processes. But they are of corse related, so when the absorption gets stronger, so does the refraction (at least in the majority of cases).

 

[timetraveller123]

ok at this point it would be good if i define refraction and absorption

refraction is the bending due to the slowing down of light which occurs dues to absorption
but the particle could not just keep absorbing it what they should emit it at some point
it we be good if you could help me to define these

 

[blue_leaf77]

so it is still a process of absorption and emission is it ?
and why would frequency be changed ?
and why would intensity be changed ?

It's actually more precise to talk in terms of spectrum rather than frequency. When a light passes through a medium, despite not being resonant, it still experiences some small amount of absorption. This absorption modifies the spectrum very slightly if the medium's absorption is very low within the range of the light's spectrum. The change (decrease) in intensity due to absorption goes without saying.

 

[timetraveller123]

but how can the medium keep absorbing it has to emit it at some point to return to ground state

 

[blue_leaf77]

but how can the medium keep absorbing it has to emit it at some point to return to ground state

Read again the 1st part in post #2. Apart from reemitting photons, the excited atoms can also transfer its energy to any nearby atoms increasing their overall vibrational motion. Physically you feel this vibration as a rise in temperature of the medium. As the temperature of the medium reaches the equilibrium with the ambient one, the atoms go back to the ground state. This is called non-radiative relaxation because the medium does not reemit light upon going back to the ground level.

 

[timetraveller123]

so is the light energy converted to heat energy

 

[blue_leaf77]

so is the light energy converted to heat energy

Yes (btw it's just "heat", no need to attach energy).

 

[timetraveller123]

sorry for the late reply have been having busy academic
let me tell you about my understanding of this phenomenon so far correct me if i am wrong

so when there is light incident on the medium
electrons absorb it then two things can happen

if the frequency of light is close to its resonant frequency most of the light is absorbed which is converted to heat via its oscillations
so little light is transmitted is this why some objects are not transparent ? as it was mentioned absorption happens to be a part of complex refractive index

but if the light is far away from the resonant frequency light is still absorbed by the electrons but is emitted back after some time causing a phase lag and hence apparent change in the group velocity of the light known as refraction
is this correct?

 

[blue_leaf77]

if the frequency of light is close to its resonant frequency most of the light is absorbed which is converted to heat via its oscillations
so little light is transmitted is this why some objects are not transparent ?

Yes.

but if the light is far away from the resonant frequency light is still absorbed by the electrons but is emitted back after some time causing a phase lag and hence apparent change in the group velocity of the light known as refraction
is this correct?

Yes in some sense. In general, when the light is not absorbed by the medium it will just proceed through the medium without loss in intensity (of course this is an idealization since the absorption can never be exactly zero). Refraction is just a special case of this phenomena where the incoming light is in oblique incidence with respect to the medium's surface.

 

[timetraveller123]

so when it is not at resonant frequency the problem is just a matter of collision is it mostly?