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What we'd like to do now is,

look at the simplest possible model for

the constitutive relationship between

electric field and the displacement field of the material,

or really the polarizability of the material.

And this is important to understand in the context of optical design,

because what we find is that the material has an impulse response,

where we use here a damped harmonic oscillator version,

that impulse response the material going buoying to an electric field.

When we fully transform that to understand its frequency response,

results in the index of refraction depending on frequency, and therefore, wavelength.

And that has really important implications, for example,

how lenses focus light that isn't monochromatic,

so the visible spectrum I would say.

So, we're going to use the simplest possible model,

developed by a guy named Lorentz,

fully classical and he said let's imagine that a material is made up of atoms.

Those atoms have the massive positively charged nucleus

and a lighter negatively charged electron cloud around them.

We're going to put an electric field that goes in some direction on the material E,

and we're going to see the electron cloud moves in the opposite direction,

of course because there's a negative charge,

and we'll assume that the electron cloud is so light in comparison to the nucleus,

that we can really just think about the electron cloud itself moving.

What we want to find is how this leads to polarizability of the material,

for which eventually is going to give us our refractive index.

So, the simple classical mechanical model of that,

is that the electron cloud is represented by some mass,

the electrostatic attraction between the electrons and

the nucleus is a spring that's linear in its force relative to the displacement.

And because we expect that the electron cloud is

not going to oscillate forever if it's hit by an impulse,

we'll include a viscous damping term.

This is called a dashpot, in the language of mechanics.

It simply means that plunger in some viscous liquid that dissipates energy.

That being a fully classical system,

we can write down a differential equation for

the displacement or the distance that the electron cloud is displaced.

And that's just Newton's law.

On the right hand side, we have the force,

that's the electric field times the negatively charge electrons.

On the right hand side, we have the acceleration m a second derivative respect

with time of ohm and we have two more force terms that we put over here,

the first is the spring which is linear in its force with respect to distance.

And we've written the spring constant k,

in terms of the resonant frequency of the system and that's just

because we know where we're going with this but

this is essentially the k term right here.

And then there's a viscous damping term that depends on the time rate of change

or the velocity of a system with some constant gamma that expresses the viscous damping.

So that's a pretty simple differential equation.

We could go solve that.

And then we'd get an impulse response.

Then we fully transform that impulse response into the temporal frequency domain

to understand how our response with respect to the frequency of electric field.

A much smarter thing to do however,

is to reverse those operations to fourier transform

this equation because remember in the temporal fourier transform,

we know the time dependence of all of the variables like R and E,

it's E to the J omega T. That means we can take the time derivatives,

each time derivative become simply a multiplication by J omega.

And so this will become minus omega squared.

So what that does is it turns the differential equation into an algebraic equation for R,

and we can simply divide by all terms over

here and that so we get right here that's our solution.

So we see Q over the mass of the electron

and then the differential equation is spread out here in the denominator.

Let's plot that, that seems important.

Notice that this is a complex equation,

we have J in it here and it's got something magic

happening when the temporal frequency gets near

this resonance frequency that was why we went ahead and wrote the spring

constant in terms of that resonance frequency and this shows what happens.

The imaginary part of this,

that's the response of the electron cloud that's out of phase,

90 degrees with the forcing function,

is essentially zero far away from resonance,

but near resonance goes through a maximum.

And the real part is also essentially zero, far from resonance.

That's typical, this is a damped harmonic oscillator.

If we pump the harmonic oscillator far from resonance, it doesn't respond.

If you've ever pushed a kid on a swing,

you know that you need to match the resonant frequency of kid on

swing in order to get significant energy into that system.

But the real part is a very different character.

It's antisymmetric as opposed to being symmetric like the imaginary part.

And that has some really important implications for how optical materials respond,

and so let's look at those next.

So, what we just found was the dipole response of a single atom,

and the dipole moment is defined by the separation that we get

between the positively charged nucleus in

the electron cloud times the charge of the cloud.

And let's assume we have a single electron now and

this is the fundamental electron charge.

So that single dipole moment is related to

the polarizability of the material, the polarization field.

Basically, you multiply by the number density of the labels.

Let's pull out here the electric field because of

a perfectly linear system to find how the material responds to electric field.

And we had electric field in our expression just a minute ago so that's no problem.

And so, this which is formerly called the susceptibility,

lots of words here that are terribly important,

has that Lorentzian that we just saw in it.

And so this is the key thing is that this microscopic field describes how the material

responds to an invisible electric field

has this Lorentzian character that we just derived.

And the key thing, and I'm skipping all the steps here,

because we just want to get to the fundamentals for this course.

The key thing is that the index of refraction is related to the susceptibility,

let me make that index refraction squared.

So let's look at, now,

let's take those plots that we just saw and

see what that means for refractive index which remember,

this is a complex quantities so we're going to have

real and imaginary parts to this expression.

So, let's look at the character of this Lorentzian

and its impact on the real and imaginary part of the refractive index.

I've subscripted in here with real to remind us this is the real part,

but normally we would simply call the real part of this expression, the refractive index.

And we see something kind of important,

at high temporal frequency which is therefore small wavelength can,

remember they're related as inverses,

so is the blue end of the spectrum and this the red.

We see that as we go towards long wavelengths,

the slope of the refractive index is decreasing.

We have a lower refractive index as we go towards the red.

Until we get near the resonance, where D increases,

and then once we get away from the resonance, once again,

we see that as we go towards the red,

the index is decreasing.

So to say that again, index is larger in

the blue and smaller in the red everywhere except,

very close to the resonance.

And it turns out we almost never use

materials in this region where we see this anomalous,

it's called, this irregular increase of

refractive index as we go towards the red and this is why over here.

The imaginary part of the refractive index tells us about

loss and what this says is we have low loss that is,

we have transparent materials away from this resonance.

But when you get close to the resonance all of a sudden,

the material becomes very absorptive,

and we typically don't like to make lenses out of absorptive material.

So, we don't operate in this region close to the resonance.

So, this region away from the resonance where the refractive index

is increasing as we go towards longer wavelength here and here,

is called the regular dispersion regime and that's

the reason in prisms blue is bent most,

the refractive index is highest in the blue.

And you very rarely see anomalous dispersion where you

have the opposite slope of lower refractive index in the blue.

And the reason is, for normal materials,

the material would be very absorptive there.

That's the important character to get out of this argument we just went through with

the Lorentzian damped harmonic oscillator for an electron cloud is that,

near the resonance, you're transferring energy to that dashboard,

you're effectively heating up the material with

your light and you tend not to make optics and lenses there.

So, that also means that there is a consistent slope on this refractive index and it's

always that refractive index is

decreasing as we go towards longer wavelengths or the red,

and that's the reason the prism does what it does.

So, giving you the expressions here for how

the real and imaginary part of

the refractive index relates to the quantities you care about,

and it's really that we pick up phase based on how the real part of

their refractive index impacts the phase velocity

through the material and we lose energy,

this is the Beer-Lambert law, right here,

related to the imaginary part of the refractive index.

So that's why these two things are important,

one loses the energy or gains the energy if we have a negative quantity here.

And one relates to how much we speed up or slow

down light as we go through the material due to its phase velocity.