And the previous lecture, I have already shown you one solution for an electromagnetic wave where the
electric field, the magnetic field and the way Vector K are all perpendicular to each other.
And the example that I've shown you was a linearly polarized electromagnetic wave and yeah, I want
to show you that this is just one special solution and is the solution for one particular orientation
or one particular shape for the constant e0 and p0.
So in general, the constant e0 is a vector that consists out of a real part and an imaginary part.
So here I've split up these two contributions and I've written it down as Eaarth Plus I Times II.
And now here for the linear polarization.
We considered a special case where this vector that corresponds to the real part is perpendicular I'm
sorry is parallel to the vector that characterizes the imaginary part.
So this capital R is the same as here, actually.
So in this case, the real part of our electric field, which is the physical irrelevant quantity is
given by this this this vector that characterizes the real part time, this cosine function here.
And then we have also here, of course, if we if we introduce or if we express it in terms of cosine
plus eight times a sign of this argument, here we have another term that is eight times sine Katara
minus McHattie.
And so for the real Pardes, the only thing that contributes is now three times I e I, times I, times
I, times sine, times this argument.
So we have I square, which gives us this negative sign and we have this etai vector here.
So since these two are parallel, we can expressed is Victor II in terms of Vector E R Times II divided
by E r.
So this is just when we take this one and this term here, this is just a vector e i times sign of K
times R minus Omega T, and this all comes from the expansion of this exponential function in terms
of cosine of the argument.
Plus I times sign of the argument.
Now, we have here is some of a Cosini and a sign with a prefect here and maybe you notice for mathematics,
if we add to cosine and sine independent of their amplitudes, we can always express this in terms of
a cosine function so we can just ride our times, cosine cave times, our minds make a T and then we
get an additional face that is determined by the ratio of these two coefficients.
But it's not really that important.
The face doesn't really play that much of a role.
The only thing that matters is that in this case we express the real part of the electric field in terms
of a vector of a real vector times a cosine function.
And so this is what is shown here.
So as you can see, once again, we have already discussed this in the previous lecture.
We have an orientation for E zero.
And similarly, we have an orientation for P zero.
And both of them are perpendicular to K.
And you can see that the wave linearly fluctuates or better to say, oscillates in these planes that
are determined by these vectors.
E0 be zero and K.
Now, another very important special case is the case of the of polarization, and once again, this
is a general statement.
The vector is zero can be expressed as the real part, plus high times the imaginary part.
And here in a special case that gives rise to the political polarization is when E r is perpendicular
to Etai.
So let us consider one special example.
Just to make it a bit easier.
We say the character is pointing along the Z direction.
The real part vector of E is pointing along X and the imaginary part vector is pointing along Y so that
we can write that the X component of the electric field is determined by R times cosine.
And then we have here the Y component is characterized by times sine.
So these are actually here already the real parts of these fields that I've taken.
So we get here a cosine from this expansion here as a cosine.
And we get a sign here because we have AI times, e AI times, AI times sine, which is a real number
because we have AI Square, which is equal to minus one.
So we have here this cosine and this sine and now we use something very special.
We say that are we we just re expressed these two equations, first of all.
So we divide them by E, R and by so we get X divided by E R is equal to cosine and we get e y divided
by is equal to sign and then we square these equations.
So we got here on the right hand side, cosine square and we get sine square.
And in both cases the arguments are the same K times are minus Omega T and you know that for every element,
for every argument, doesn't matter what it looks like when as long as, as long as it's the same argument,
then we have cosine square of a plus sine square off.
The argument is always equal to one.
So we know that this term plus this term is equal to one.
So we know that this term plus this term must also be equal to one.
And if you know about geometry and about mathematics, you will realize that this is an equation.
So it tells us that our electric field and our magnetic field basically changes over time and over space
on an ellipse.
So here I have introduced a plane that cuts our way.
And as you can see, when the time changes, the orientation of E and B is always changing on an ellipse.
And this is why it's called elliptical polarisation.
Still, at every point of time and at every position vector, you will find that the electric field
is still perpendicular to the magnetic field and both of them are perpendicular to the way vector.
Now, a special case for this political polarization is, of course, the circular polarization, so
this is when in our Elipse equation here, the quantity of E, R and the I have the same magnitude.
So this is when E R is equal to plus or minus Etai.
And the sign here, either it's plus or either it's minus determines if it's a right handed circular
polarization or if it is a left handed circular polarization.
So you can see here these two vectors, they rotate counterclockwise and here they rotate clockwise.
So there's the right handed or the left handed polarization.
So this is something that is used in optics quite a lot, you can polarise light, for example, you
can take completely polarized light, you then take some polarizer and sort of some polarizer and then
polarize these different waves.
For example, you can take it out one particular linearly polarized wave, or you can also take out
one particular circularly polarised light.
And all of these different waves can have different properties and lead to different effects.
But this is something that we will not cover here, and of course, this is more a matter of optics
here.
What was really important is that we can drive the wave equation, come up with a solution, and we'll
see that different special cases for the choices of zero be zero lead to different polarizations of
light.