We're going to be talking about the reflection and The refraction of waves when they hit an interface otherwise known as Fresnel equations if you're following along in my book What we're covering is in chapter 8 and since we're going to cover a whole chapter I'm just going to give you the highlights and not give you the derivation Although the derivations of Fresnel's equations is a really good way to understand if you you really know waves or not So if you remember in our picture We essentially had an incoming electric field and that electric field would turn on and off and it would essentially Jiggle the clouds of the atom the electron clouds of the atoms up and down and we saw that this can lead both to change in the phase constant with distance the index of refraction or the relative permittivity of the material and also some absorption or some loss and If you have a material where you have some absorption and I'm going to say it's three here and a phase constant Which is three and this is the equation that describes our plane wave going into the material and I'll note that the plane wave is Here at normal incidence that essentially in the general case You're going to get a change of the wavelength of the material that arises from a change in the velocity phase velocity and you're also going to get some kind of exponential type of attenuation as the wave goes down into the material and You can calculate the absorption coefficient alpha Which is the attenuation of the wave with distance as well as the spatial phase change of the wave Using some expressions as we saw before and these are the most complicated expressions.

You can simplify them in particular cases So that's where we were.

Where are we going?

What we want to do today is talk about essentially a more general case where essentially you have the incoming Wave with an incident K vector.

We'll call it case of I for instance for incident coming in some angle And as you will see we're going to measure the angles normal to the surface because that's the only way you can accurately measure things Related to a surface is to the normal and that this incident wave is going to give rise to both the reflected wave That's going back out in that way and a transmitted wave that's going to go into the medium this way So we're we're really expanding our picture here To think about how these oscillations of the atom inside the material Aren't just caused by the incident wave but affect the transmitted wave as well as give rise to a wave reflected from the surface So rather than go into a full derivation.

Let me just cover really the basics before we get into some cases So we're interested in waves incident on surfaces and essentially what we're going to do is we're going to assume that there's some incoming wave With a K vector case of I that's the direction It's coming in and the incident angle is defined relative to the surface normal and that incident angle is theta sub I right here And so that angle is right here the angle between the dashed line Which is normal to the surface and the direction of the incoming K vector?

There's also something called the law of reflection which says the reflected angle Theta sub R is also equal To theta sub I so here we have theta sub R the angle of reflection and this angle theta sub R And this angle theta sub I are equal to one another in other words for most flat surfaces The angle of reflection doesn't go off in a different direction than the angle of incidence Another thing that's going to make our lives a lot easier is to characterize the material by the index of refraction in so essentially we Have some incident index of refraction That's the index of refraction of the medium the incident wave is going through over on this side And we also we are also going to talk about the transmitted index of refraction that's the index of refraction of this gray medium over here that the transmitted wave goes into and The index of refraction essentially in a general case is given by the square root of epsilon R and mu sub R of course most Materials are non-magnetic so mu sub R is equal to 1 so in most cases you can simply say the index of refraction of the square root of the relative permittivity and Of course since the relative permittivity in free space is 1 the index of refraction of free space is 1 as well However, we've also seen that in lossy materials you have complex permittivity and in the general case we can essentially represent the index of refraction this way is a square root of The complex epsilon divided by epsilon naught and essentially then we can basically tie that back to an alpha and J beta term and To be perfectly correct.

This is not an equal sign this essentially says the in the complex index of refraction Essentially is related to alpha and beta, but is not equal to alpha and beta So let me just clarify that since that's a mistake in the slide Now if the material is very lossy or if it's something like a metal then then most of the incident radiation is going to be Reflected, but if it's a dielectric material then the transmitted wave which essentially has a k vector K sub T here is coming in a different angle So theta sub T is not necessarily equal to theta sub I or theta sub R the transmitted wave goes in a different direction than the incident wave and the reflected wave and We can calculate the angle of that transmitted wave using something called Snell's law, which is given right here that the the Index of refraction in I of the wave coming in times the sine of the incident angle incident angle is equal to the index of refraction for the material the waves getting transmitted into times the angle of the Transmitted wave so if we know in sub I in sub T and the incident angle theta we can do a simple algebraic manipulation to calculate the angle of transmission And so those really are the basics of waves and incident on surfaces if you understand these basics the rest of it Although there are some long equations follows pretty straightforwardly now the first complication we run across is that the Reflection and transmission that we see depend on the direction of the electric field And so the first direction of electric field we're going to talk about is called s or perpendicular Polarization and the way we we define this is we think as we've talked about before of a plane of incidence and this plane of incidence is the plane that contains both the incident and the In the plane of incidence the normal to the surface lies in the plane of incidence and if the electric field is sticking up out Of the plane of incidence so there's essentially a 90-degree angle Between the electric field and the plane of incidence then we call it s or perpendicular polarization where perpendicular means It's perpendicular to the plane of incidence Now if the electric field is perpendicular to the plane of incidence We can calculate the magnitude of the reflected field and the transmitted field given the magnitude of the incident field so essentially this incident field is going to come in like this some of its going to bounce off and there will also be an electric field pointing in the same direction that we're going to call e Sub r or the reflected field and the relationship between the incident field e sub i Right here and the reflected field e sub r right here is The reflected field and notice we have this little perpendicular sign that says it's perpendicular to the plane of incidence is equal to the coefficient r perpendicular times the the incident field and Essentially without deriving it the reflection coefficient for perpendicularly or s polarized radiation is given by this it depends both on the incident index of refraction the index of refraction of the material that the waves reflecting from are getting transmitted into and the cosine of both the incident angle and The angle the waves being transmitted at and remember Snell's law says in sub i sine of theta sub i is Equal to the index of refraction of the material that the waves getting transmitted into Times the sine of that angle and so we calculate theta sub t from this equation right here So overall this gives the reflected field Once we know all of these things similarly, there's going to be an electric field pointing in the same direction That's transmitted into the material We can call it e sub t here and the relationship between the transmitted field and the incident field is given by a very similar expression, but we use the term t perpendicular for the We find it depends on the incident and transmitted indices of refraction the incident and transmitted angles And if we plug it into this equation, we'll know what fraction of the electric field gets transmitted in other words the ratio Between the incident field the magnitude of the incident field on the transmitted field and because they're vectors the vectors are going to point in the same direction Now I'm using a lot of Lines here, but we have to remember that these are actually waves coming in as shown in the bottom figure down here So they oscillate up and down and they're plane waves So we assume that everywhere on these planes that are represented by these little green squares The electric field is a vector field the vectors all point in the same direction But that direction can vary with time and space as we saw in previous talks And again, the angles are just given by the K vectors K incident K transmitted K reflected we have exactly the same situation or very close to the same situation if the Notice that instead of the electric field pointing up and down like before Perpendicular to the plane of incidence now the electric field lies in the plane of incidence and we call this P or parallel polarization Because the electric field is in the plane of incidence.

It's parallel to the plane of incidence Again exactly the same thing happens.

We have a reflected electric field also lying in the plane of incidence e sub R We have a transmitted electric field We're going to call e sub T and the relationship between the incident field and the reflected field is given by those sets of equations The relationship between the incident field and the transmitted field is given by those sets of equations again The fields point in pretty much the same direction The electric fields in both cases are parallel to the plane of incidence Except the magnitude of the field the amount that gets through varies depending on the reflection coefficient And the transmission coefficient again very very similar the equations are slightly different but you do the calculation once you know the the Index of refraction on either side of the material the incident angle and from Snell's law you calculate the transmitted angle And again, let's stress is shown in the bottom that this is a wave coming in It's not just a line although we can represent it as a line and that the wave essentially maps an electric field to every point in space Given by planes that are perpendicular to the propagation direction of the wave So what you'll see in a lot of books or figures that look like this for s polarization We have the electric field sticking up out of the screen right at you for perpendicular Or excuse me for parallel or p polarization.

We have the electric field in the plane of the screen We have essentially a distance between peaks We're going to call the wavelength when we go into a material the wavelength changes because the phase changes when we go into the material and Essentially through Snell's law we can relate the incident angle and the transmitted angle We know the incident angle is equal to the reflected angle and we can calculate the magnitudes of e sub i well If we're given e sub i and we can calculate the magnitudes of e sub t and e sub r through the reflection and transmission Coefficients for either the perpendicular polarization case or the parallel polarization case quick summary And these are the types of figures that you're going to see in most of your textbooks So let's consider the gray plane here in this figure being the plane of incidence What happens if we have an electric field that's not in the plane of incidence in other words this green line that we're going to Doesn't stick in either the s-plane perpendicular to the plane of incidence or the p-plane parallel to the plane of incidence This is pretty straightforward.

This is just like we worked with the parallelization essentially We're going to break it into two components We're going to say there is one component that is s polarized or perpendicular to the plane of incidence We're going to add a second component.

That's parallel polarized parallel to the plane of incidence or the electric field points and Simply by doing the summation of these two components.

We can apply the Fresnel equations for the reflection or transmission for each component separately and use superposition to sum up and get the overall electric field Quick message here if you are given a problem where you have an electric field that's not polarized Perpendicular or parallel to the plane of incidence use superposition to break it up Essentially if you know this angle then the parallel component is going to be the cosine of this angle.

Let's call it phi the Perpendicular component is going to be proportional to the sine of phi through simple sine of phi through simple geometry and Essentially we just do our calculation twice one for parallel one for perpendicular Sum it up at the end to get the overall field superposition works here as well So I've taken you on kind of a whirlwind tour of Fresnel's equations Let's actually stop for a minute use some real numbers put in some Representative values and try to figure out what we're doing here What I'm going to assume is that I've got essentially a block of material The index of refraction on this side is 1 so we assume the incident wave is going through free space the index of refraction e sub in sub t is 2 in the material and I just chose those numbers I could have chosen any set of numbers But these are the ones I chose so our incident wave is going to come in here at some angle theta sub I to the normal and We want to know how strong or what the magnitude of the reflected and transmitted fields are In essence, what I've done is I went into the computer program I like to use for plotting called MATLAB and I've essentially plotted the equations for the reflection coefficient and the transmission coefficient as a function of incidence angle theta I and so theta I is given in degrees right here and Let's first look at the electric field that's reflected So if the electric field is P polarized, so our electric field vector points in that direction right there Then the green line essentially represents the strength of the electric field that's reflected So you can see if you come in at normal incidence in a material Whose indices of refraction are given by one on the incident wave and two on the transmitted side about 0.35 of the electric field or 35 percent is going to be reflected if the incidence angle is zero or the beams coming in straight On in that direction, or if you've come out to about a 30 degree angle that hasn't dropped very much But by the time you get to about 50 degrees Only about maybe 0.16 of the incident field 16 percent is reflected And as you drop down to 90 degrees, you can see that pretty much all the light So I've taken you kind of on a whirlwind tour of Fresnel's equation So let's stop for a minute and essentially do the calculations let's actually calculate the reflection and transmission coefficients and do it for a Set of materials where essentially the incident wave is coming in with an index of refraction 1 So we're in free space over on this side and the index of refraction of the transmitted wave for the index of refraction the material the waves going into is 2 and Essentially if we do that we can use Fresnel's equations given the incident electric field to calculate Relatively how strong the reflected and transmitted electric fields are so let's first look at the reflected electric field If we do this calculation and essentially plug in theta sub I and I've represented the angle of incidence here in degrees So this is a 30 degree angle of incidence a 70 degree angle of incidence a zero degree angle of incidence down here is Essentially going to be when the wave is coming in at normal incidence because the angle between the incident k vector and the surface normal is Zero, there's no difference between them for normal incidence.

You can see that for the electric field.

That's polarized S or perpendicular to the plane of the electric field in other words We're talking about electric fields that are pointing up in this direction in this case What you're going to see is that for normal incidence?

You're going to get a reflection coefficient of about minus point three five now What does that minus sign mean it simply means there's a hundred and eighty degree phase shift in the electric field so s or?

perpendicular polarization Sees a phase shift when it hits the interface However as the angle gets larger and larger as theta incident becomes further and further away from the normal the reflection coefficient Gets larger and larger or the magnitude does it actually gets more and more negative?

Until down here at 90 degrees the reflection coefficient is close to 1 which essentially is saying That all the light is reflected What happens if we take a look at the p polarized case or the when the electric field lies?

parallel to the plane of incident In this case we go ahead and erase these electric fields and put our electric fields in that direction You can see that we get about a thirty five percent reflection at normal incidence that Drops off until at some angle we get zero reflection here And then eventually the reflection coefficient becomes negative and we start to get that hundred eighty degree phase shift again and so essentially what this curve is telling us is that it's telling us what the phase shift and The strength of the reflected electric field is as a function of the incident angle Similarly if we want to know the the strength of the transmitted field We can plot the transmission coefficient as a function of the angle of incidence Which I've done here for perpendicular s and parallel or p polarized and essentially if we take a look at that Maybe 65% of the wave gets through This makes a lot of sense since it was only 35% over on the other side But as the wave comes in at steeper and steeper angles eventually we drop to zero transmission and all the wave gets reflected It makes a lot of sense.

So this is simply what happens if you plot those reflection and transmission coefficients All you do is you simply look up the value of the strength of the reflected and transmitted field And we know if the value is positive.

There's no phase shift if the value is negative There's a hundred and eighty degree phase shift and that's pretty much about it.

We've got a few special cases.

We need to talk about One of these cases is something called total internal reflection It turns out that if you have a wave coming from a material that has a high index of refraction So in this case in sub I is greater than in sub T As long as your angles of incidence are small or the the direction of propagation is Is pretty much close to the normal then things come out as you would expect you get a reflected field and transmitted field But as theta I increases and gets bigger as you're going from the material with higher index to the material with lower index essentially the direction case of T of This vector is going to move that way as the incident angle increases and at some point it's going to lie along the surface This means that all the radiation that comes in this direction is going to get reflected You're going to get a hundred percent reflection of the radiation and there's going to be no propagating radiation that goes out and this is called total internal reflection for the obvious reason that the total amount of the radiation gets Reflected from the surface going from a material of higher index into lower index The place this is most commonly used as an optical fibers because this is what keeps the light inside fibers to go very very long distances and Essentially if you want to calculate what the angle is the incident angle is where total internal reflection Starts to occur you simply use Snell's law you basically set theta T is equal to 90 degrees or greater and Essentially you can find that critical angle Let's call it theta C there is given the by the equation the sine of the critical angle is the Insub T the transmitted divided by the incident indices of refraction the second case we've also mentioned very briefly, but that's called Brewster's angle, and that's essentially the point where the Reflection coefficient of the parallel or P polarized electric field component is equal to zero right here in this case You get no reflection whatsoever And so for theta I equal to Brewster angles and Brewster angles given by that equation right there where theta B is Brewster angle so for theta sub I equal Theta sub B.

There is no reflection and all of the radiation is transmitted So if you need a very very high transmission of radiation Then making sure you are incident at Brewster's angle is an important thing to do notice that however at Brewster's angle Let's draw a line down here, which in this particular case happens to be about 64 degrees That there is a reflection of the perpendicular s polarized light And this is why polarizing sunglasses work because essentially Radiation polarized in one direction off of shiny surfaces like pools or oceans Experiences much less reflection than the other other polarized component in other words the parallel component has very little reflection the perpendicular component has a lot by blocking out one of those directions of optical radiation you can essentially reduce the glare and We use Brewster's angle a lot when we're building high-powered things like lasers that have so much power inside them that even a little bit of reflection or a little bit of absorption might cause the The device to function poorly or to heat up or cause some damage internally and so we use Brewster's angle when we have to minimize Reflection for certain cases, but that's beyond the scope of the class