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And the previous lecture, I have already shown you one solution for an electromagnetic wave where the



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electric field, the magnetic field and the way Vector K are all perpendicular to each other.



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And the example that I've shown you was a linearly polarized electromagnetic wave and yeah, I want



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to show you that this is just one special solution and is the solution for one particular orientation



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or one particular shape for the constant e0 and p0.



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So in general, the constant e0 is a vector that consists out of a real part and an imaginary part.



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So here I've split up these two contributions and I've written it down as Eaarth Plus I Times II.



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And now here for the linear polarization.



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We considered a special case where this vector that corresponds to the real part is perpendicular I'm



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sorry is parallel to the vector that characterizes the imaginary part.



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So this capital R is the same as here, actually.



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So in this case, the real part of our electric field, which is the physical irrelevant quantity is



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given by this this this vector that characterizes the real part time, this cosine function here.



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And then we have also here, of course, if we if we introduce or if we express it in terms of cosine



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plus eight times a sign of this argument, here we have another term that is eight times sine Katara



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minus McHattie.



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And so for the real Pardes, the only thing that contributes is now three times I e I, times I, times



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I, times sine, times this argument.



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So we have I square, which gives us this negative sign and we have this etai vector here.



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So since these two are parallel, we can expressed is Victor II in terms of Vector E R Times II divided



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by E r.



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So this is just when we take this one and this term here, this is just a vector e i times sign of K



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times R minus Omega T, and this all comes from the expansion of this exponential function in terms



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of cosine of the argument.



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Plus I times sign of the argument.



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Now, we have here is some of a Cosini and a sign with a prefect here and maybe you notice for mathematics,



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if we add to cosine and sine independent of their amplitudes, we can always express this in terms of



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a cosine function so we can just ride our times, cosine cave times, our minds make a T and then we



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get an additional face that is determined by the ratio of these two coefficients.



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But it's not really that important.



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The face doesn't really play that much of a role.



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The only thing that matters is that in this case we express the real part of the electric field in terms



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of a vector of a real vector times a cosine function.



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And so this is what is shown here.



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So as you can see, once again, we have already discussed this in the previous lecture.



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We have an orientation for E zero.



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And similarly, we have an orientation for P zero.



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And both of them are perpendicular to K.



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And you can see that the wave linearly fluctuates or better to say, oscillates in these planes that



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are determined by these vectors.



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E0 be zero and K.



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Now, another very important special case is the case of the of polarization, and once again, this



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is a general statement.



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The vector is zero can be expressed as the real part, plus high times the imaginary part.



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And here in a special case that gives rise to the political polarization is when E r is perpendicular



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to Etai.



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So let us consider one special example.



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Just to make it a bit easier.



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We say the character is pointing along the Z direction.



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The real part vector of E is pointing along X and the imaginary part vector is pointing along Y so that



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we can write that the X component of the electric field is determined by R times cosine.



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And then we have here the Y component is characterized by times sine.



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So these are actually here already the real parts of these fields that I've taken.



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So we get here a cosine from this expansion here as a cosine.



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And we get a sign here because we have AI times, e AI times, AI times sine, which is a real number



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because we have AI Square, which is equal to minus one.



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So we have here this cosine and this sine and now we use something very special.



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We say that are we we just re expressed these two equations, first of all.



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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



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by is equal to sign and then we square these equations.



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So we got here on the right hand side, cosine square and we get sine square.



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And in both cases the arguments are the same K times are minus Omega T and you know that for every element,



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for every argument, doesn't matter what it looks like when as long as, as long as it's the same argument,



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then we have cosine square of a plus sine square off.



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The argument is always equal to one.



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So we know that this term plus this term is equal to one.



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So we know that this term plus this term must also be equal to one.



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And if you know about geometry and about mathematics, you will realize that this is an equation.



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So it tells us that our electric field and our magnetic field basically changes over time and over space



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on an ellipse.



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So here I have introduced a plane that cuts our way.



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And as you can see, when the time changes, the orientation of E and B is always changing on an ellipse.



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And this is why it's called elliptical polarisation.



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Still, at every point of time and at every position vector, you will find that the electric field



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is still perpendicular to the magnetic field and both of them are perpendicular to the way vector.



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Now, a special case for this political polarization is, of course, the circular polarization, so



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this is when in our Elipse equation here, the quantity of E, R and the I have the same magnitude.



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So this is when E R is equal to plus or minus Etai.



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And the sign here, either it's plus or either it's minus determines if it's a right handed circular



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polarization or if it is a left handed circular polarization.



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So you can see here these two vectors, they rotate counterclockwise and here they rotate clockwise.



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So there's the right handed or the left handed polarization.



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So this is something that is used in optics quite a lot, you can polarise light, for example, you



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can take completely polarized light, you then take some polarizer and sort of some polarizer and then



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polarize these different waves.



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For example, you can take it out one particular linearly polarized wave, or you can also take out



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one particular circularly polarised light.



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And all of these different waves can have different properties and lead to different effects.



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But this is something that we will not cover here, and of course, this is more a matter of optics



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here.



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What was really important is that we can drive the wave equation, come up with a solution, and we'll



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see that different special cases for the choices of zero be zero lead to different polarizations of



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light.