This lecture is about the polarization of light. I'll start with these summary slides
displaying everything we'll be going over. This is most of the information,
and we'll also be talking about a few graphs of polarized light.
To understand polarization you need to understand the difference between polarized
and unpolarized light. In unpolarized light, the electric field oscillates in every 2d plane
along the direction of the light's motion. As an example, the red arrow I've drawn on the left
shows light moving to the right. If i place a two-dimensional surface along that line,
if the light is unpolarized i know that no matter what direction that 2d plane exists in,
there's definitely an oscillation of an electric field along that plane. That was a vertical plane.
And if i make a horizontal plane I also know that the electric field is oscillating along that plane
because it oscillates along every possible 2-dimensional plane along the line of the
light's motion. If you look at the bottom, I have a view of the light from the front. So you
can see right now I have the electric field oscillating up and down and left and right,
but in unpolarized light that oscillation actually happens along every possible 2-dimensional plane.
So any line that I choose would have an oscillation on it, so that's the definition
of unpolarized light. In unpolarized light, the electric field oscillates along every 2D plane
along its axis of motion. In polarized light, the electric field oscillates along just one
2-dimensional plane along the direction of the light's motion. So this is what polarized light
would look like in comparison to unpolarized light. If you look at the view from the front,
you can see that it's only oscillating along one plane. In this case it's only oscillating
along that up-down line, which we can see is a 2-dimensional plane if we look at the side view,
but it's only oscillating in that one up-down way. It's not oscillating along any other plane.
That's the difference between unpolarized and polarized light. This is the notation that we
use for unpolarized and polarized light. You can see that with unpolarized light
it oscillates along every direction whereas with polarized light it only oscillates in one plane,
so we use those arrows to show which direction the light is oscillating as it moves forward.
Just to put that in writing: the unpolarized light notation symbolizes light oscillating
along every possible plane. The view is as if the light is moving straight towards us,
and the polarized light notation symbolizes light oscillating along only one plane. The angle of the
line shows the angle of the plane. The view is as if the light is moving straight toward us.
You're now in a position where you can understand polarization, which is the process of changing
unpolarized light into polarized light: changing from light with an oscillating electric field
along every 2D plane to only one 2D plane. There are two ways we can polarize light. The
first way is to pass it through a polarizer, which is a filter that only opens along one
axis and polarizes unpolarized light that passes through it along that one axis. So if
this unpolarized light encounters that polarizer, the polarizer polarizes the light along that axis.
So that would be what the polarized light would look like. It would specifically only oscillate
along the plane opened up by the polarizer. The other way to polarize light is to reflect light
off a surface. When unpolarized light is reflected off a surface, it leaves the surface as polarized
light only oscillating along the 2D plane parallel to the surface itself. So if this unpolarized
light encounters a horizontal surface, the only part of the light that will be reflected is the
horizontal polarized component of the light. So it would leave the surface as horizontally polarized
light. When light is polarized its intensity is divided by 2. One way to think of this is that all
planes of the unpolarized light can be thought of as made up of 2 perpendicular components,
like an x-axis and a y-axis, and if you take away one of the components you lose 1/2 of the energy
that the wave is carrying, so the oscillations of unpolarized light can ultimately be thought of as
having components in 2 perpendicular directions whereas polarized light only has components in
one direction, so you basically took away half the light, so you're left with half of the intensity.
This is probably the most common thing students forget when doing polarization problems.
If polarized light is passed through a second filter, its intensity is reduced by a specific
amount relative to the angle between the first and second filter. We call the second filter the light
passes through an analyzer. The only part of the polarized light that can pass through the analyzer
is the component parallel to the analyzer's opening. This means the light leaving the
analyzer will have a smaller amplitude than the light entering it, so we'll follow through this
full process: unpolarized light encounters a polarizer and is then polarized along that one
axis, and now exists as polarized light when it encounters the analyzer. The only component
of it that can pass through the analyzer is the component parallel to the opening of the analyzer,
which is smaller than that original polarized light. It has a reduced amplitude, so I remember
the amplitude of a wave is the distance from the center of a wave to the top of the wave,
and I can see that the distance on this final polarized light from the center to the top is less
than that distance in the original unpolarized light. So after the light leaves the analyzer,
if there's any angular difference between the analyzer and the polarizer,
the light will have a reduced amplitude. Notice here that passing unpolarized light through a
polarizer cannot reduce the amplitude of the light. No matter the angle, the final amplitude
is the same as the original unpolarized light, so the polarizer does not reduce the amplitude. It's
the analyzer that does that later on in the second part of polarization. We call the specific angled
lines of the polarizer or analyzer the "axis of transmission" or the "transmission axis."
As an example, the difference in angle between the axes of transmission of this analyzer and
polarizer is 30 degrees, because one is angled at 120 and the other is angled at 90. We would say
that the difference between their axes of transmission is 30 degrees. We can determine the
intensity of the light after it passes through the analyzer using geometry. Remember from a previous
lecture that light intensity is proportional to the light's amplitude squared. This means that
the intensity is equal to a constant times the amplitude squared. We can use this to prove a new
equation. So i'm going to imagine that we have polarized light and it encounters the analyzer,
and I can see that the amplitude of the wave leaving the analyzer is smaller than the wave
going in, but I want to understand exactly how those 2 amplitudes compare to each other,
so I'm going to break this down with geometry. Just as a note,
whenever I talk about theta in these problems, I mean the angle between the polarizer and analyzer,
so if I just kind of erase this analyzer I can see that this is the amplitude of the original light.
This is the new amplitude of the light after it goes through the analyzer. Remember,
amplitude is just the distance from the center of the wave to the highest point of the wave,
and this is the angular difference between the axis of transmission of the polarizer and the
axis of transmission of the analyzer. It's just the angle difference in how the two things open.
I can see that the original amplitude and the new amplitude are connected by cosine. The new
amplitude is equal to the original amplitude times cosine of theta. I want an equation for
the light's intensity, so I'm going to get this in terms of amplitude squared by squaring both sides
of the equation. I'll then pretend that I multiply both sides of that equation by that constant C,
and when I do that, because I have the new amplitude on the left and the original amplitude
on the right, those two things are equal to the new intensity on the left and the original
intensity on the right. This equation is more commonly written like this: intensity is equal
to original intensity times cosine squared. This equation is usually called Malus's Law
and it governs how the intensity of polarized light changes when it passes through an analyzer.
Just to put that in writing, the intensity of polarized light passed through an analyzer
is equal to the intensity of the polarized light before it passes through the analyzer
multiplied by the cosine of the angle between the analyzer and the polarizer, so this is the angle
that I'm referring to if I redraw the opening of the analyzer over the polarizer. This is not
the end of the lecture, but i'm going to pause and do some example problems with what we've
covered so far. In Example 1: unpolarized light has an intensity of 50.0 watts per meter squared.
It passes through a vertical polarizer and then an analyzer at a 30 degree angle from each other.
What is the intensity of the light leaving the analyzer? So we'll just start to draw this out.
Before going through the polarizer, the intensity is 50.0 watts per meter squared.
After passing through the polarizer, I remember that unpolarized light being polarized
is always multiplied by 0.5 (by one half) so the new intensity I is equal to 25.0
watts per meter squared. The polarizer always multiplies the intensity by 0.5,
and that's probably the most common thing that students forget when they're doing these problems.
They'll often just assume the light doesn't lose any intensity in the polarizer and record the
intensity as the same before and after, but it's actually always multiplied by 0.5 when
it goes from unpolarized light to polarized light. After passing through the analyzer,
the intensity before the analyzer is multiplied by cosine of 30 degrees squared because of Malus's
Law. So multiplying that out, the new intensity I is equal to 18.8 watts per meter squared. Again,
Malus's Law is telling us that the intensity of light after it passes through an analyzer
is equal to the intensity of the polarized light before it passes through the analyzer
multiplied by cosine of the angle of difference between the polarizer and analyzer squared. When I
plug in 25.0 by cosine of 30 degrees squared I get a final answer of 18.8 watts per meter squared,
so that's the intensity of the light leaving the analyzer. I can also work backwards and say
light leaving an analyzer has an intensity of 30 watts per meter squared. If the analyzer makes a
60 degree angle with the polarizer, what was the intensity of the unpolarized light entering the
polarizer? So we have this light coming out, an intensity of 30.0 watts per meter squared,
and because the analyzer multiplies the intensity by cosine of 60 degrees squared,
the light before the analyzer has an intensity equal to the intensity after
divided by cosine of 60 degrees squared. So just working backwards, if light going through
the analyzer is multiplied by cosine of theta squared, tracing our steps backwards would require
us to divide by cosine of theta squared to get an intensity before of 120 watts per meter squared,
and the polarizer multiplies the intensity by 0.5, so the light before the polarizer's intensity
equals the intensity after divided by 0.5. So we're just working backwards here. Therefore,
before going through the polarizer the light had an intensity of 240 watts per meter squared.
In Example Problem 3: polarized light leaves a polarizer with an angle of transmission of
90 degrees. The polarized light has an intensity of 10 watts per meter squared.
It impacts an analyzer and leaves with an intensity of 6 watts per meter squared.
What is the angle of transmission of the analyzer? So we know that we have 10 watts per meter squared
going in and 6 watts per meter squared going out, and I know that Malus's Law applies here,
so I can plug in the numbers that I have for the original and the new intensity
and then isolate theta by itself, and I'm left with 39.6 degrees and that's theta,
so that's equal to the difference between the two angles of transmission. I remember that theta in
Malus's Law is equal to the difference between the two angles of transmission
so that means that 90 minus the angle of the analyzer is equal to that difference,
and that difference could be plus or minus. It's just the difference between them,
so it could be either positive or negative 39.6 degrees. It has the same result in Malus's Law,
so either one is an option. Isolating the missing variable gets me that the
angle of transmission of the analyzer is either 129.6 degrees or 50.4 degrees,
so that's an important note: the angle in Malus's Law is the difference between the two angles of
transmission. It's not either one of the angles of transmission, it's the difference between them.
Moving back into the concepts, sunglasses are a common real-world example of polarization.
Sunglasses are meant to reduce glare, so the goal is to reduce the intensity of light
passing through as much as possible, and because light reflecting off of horizontal surfaces is
polarized horizontally, vertically polarized glasses will reduce the intensity the most
because cosine of 90 degrees is zero. Because most surfaces that we look at are horizontal,
it makes sense to polarize the glasses vertically because when light from the sun is polarized on
the surface it's polarized horizontally, so when that horizontal light impacts the
vertically polarized glasses the angle between the polarized light and the opening of the glasses
is 90 degrees, and we can use Malus's Law to show that when you have cosine of 90 degrees there the
final intensity passing through the sunglasses will just be zero watts per meter squared.
You can see this in this video that this actually does happen. When you put on polarized sunglasses,
the intensity of the sunlight is very drastically reduced because it's bouncing off the horizontal
water, and the sunglasses are vertically polarized so they cancel out the maximum amount of sunlight,
and you're left with much less glare and can see a lot more under the surface of the water as a
result. This is the last part of the lecture where we'll talk about intensity versus angle
graphs for polarized light. We can draw a graph of the intensity of the transmitted light versus the
angle of the polarizer or analyzer. You can see I have the intensity of the transmitted light on the
y-axis and the change in the angle of the analyzer on the x-axis, and this is going to be a graph of
light passing through this specific polarizer and analyzer on the left and I'm going to be graphing
the value of I with respect to I-0. Just an important note: the angle being graphed is not the
angle in Malus's Law I =I0 times cosine of theta squared. The angle in Malus's Law is the angle
difference between the angles of transmission of the polarizer and analyzer. The angle being
graphed is just the change in angle of the analyzer from an arbitrary starting point. That
was a long pretty complicated definition, so i'm going to also write down what I mean here. Malus's
law says that the intensity will be equal to the original intensity times cosine of theta squared,
and I'm going to rewrite that as the original intensity times cosine of this specific angle
formula squared. I can see that the polarizer and analyzer are starting with no angular difference,
and then the additional angle between them will just be any change I make in
the angle of the analyzer, so purple theta there represents the change in angle of the analyzer
and the analyzer is starting at 0 degrees and the original angle between the analyzer and polarizer
is also 0 degrees. So this is starting at the value of I0 times cosine of zero squared and
I know that cosine of 0 is 1, so this graph will physically start at the position I0 and then take
the shape of cosine of theta squared starting from an angle of 0. So you can see I'm turning
the analyzer and this is the resulting intensity of the light coming out of the analyzer, so this
graph shows the intensity of the transmitted light on the y-axis and the change in the angle
of the analyzer on the x-axis. I could start the analyzer at a 90 degree angle to the polarizer,
and in this case the difference in angle at the beginning is 90 degrees, and I'm starting
at a point where the analyzer hasn't been turned more yet, so theta is going to start off at 0,
and because the purple theta is starting off at 0 I'm dealing with the original intensity times
cosine of 90 degrees squared, and I know that cosine of 90 degrees is 0, so this whole thing
will be 0 at the beginning and so now as I turn the analyzer and increase the angle of cosine,
this is going to take the form of a cosine squared graph. Just starting at what would be 90 degrees
instead of 0 degrees, so this is what that graph of intensity versus the change in the angle of
the analyzer would look like. In example number 3 the original light is now being passed through the
polarizer, so it's starting off as unpolarized light and emerging as polarized light and then
going through the analyzer, and we're graphing the light that comes out of the analyzer,
so in this case the intensity is actually going to be multiplied by 0.5 as it goes through the
polarizer and then be multiplied by cosine of theta squared as it goes through the analyzer,
so the full equation for the intensity will be 1/2 of the original intensity times cosine of
theta squared, and here we're starting at an angular difference of 0 degrees between the
polarizer and analyzer. They're starting at the same angle in comparison to each other,
so if purple theta is also 0, cosine of 0 is 1, so the value at 0 degrees
is going to be 1 times 0.5 times the original intensity, so that will be where it starts.
This is then going to take the form of a cosine squared graph that starts at 0 degrees so as the
analyzer turns this is what that graph is going to look like it's never going to get higher than 0.5
times the original intensity, because that's the intensity of the light going into the analyzer,
so it's never going higher than this line here. In example number 4 I'm going to turn the polarizer
instead of the analyzer. Everything is starting the same, so this is going to start at the same
position. I'm just doing this to show you that when I turn the polarizer instead of the analyzer,
that actually doesn't make a difference because what's affecting this graph is the difference
between the two angles, so whether I'm turning the polarizer or analyzer doesn't actually result in
a different graph at the end because what matters is the difference in angle between the polarizer
and analyzer if i'm using both and observing the light that comes out of the analyzer.
There's a common trick question that students are given where you have unpolarized light passing
through a polarizer, but there's no analyzer involved. You're just asked: what would the
graph look like as the polarizer turns where the unpolarized light is the original intensity and
the intensity that you're graphing is the light leaving the polarizer? So remember that no matter
what angle the polarizer is at, it's always going to reduce the original intensity by 0.5.
There's no cosine squared involved because we're not using the analyzer on the polarized light,
so no matter what the polarizer is just going to reduce the original intensity by 0.5,
so it's going to start here, and no matter how we turn the polarizer it's always going to just
polarize the light and reduce its intensity by one half, so that's kind of a trick question that
shows up every now and then. So a graph of just a polarizer turning where unpolarized light enters
and polarized light leaves is just 0.5 times the original intensity. So this is the summary,
this is everything that we talked about. The definition of unpolarized and polarized light,
polarization, Malus's Law, and an everyday example of polarization with sunglasses,
and the shapes of a few common graphs that appear in intensity versus angle graphs.
So that's what you need to know about the polarization of light.