Exercise 24

As with all of the exercises you will complete in this course, you will be expected to turn your *.nb file in to the course TA before the next lecture.  Please do so by e-mailing your completed exercise file to Lan Yu (ece329.uiuc@gmail.com), make sure you name the file using the following notation: “Exn_Last_First.nb”, where n is the Exercise number (in this case, 24), and First and Last would be your first and last name.  So my submission to Lan for this exercise would be titled “Ex24_Wasserman_Dan.nb”.  You will receive full credit for your exercise if it turned in before the next lecture (Lecture n+1).  No late submissions will be accepted.

Objective/Overview:

Today in class we signal/carrier waves and polarization of waves, in particular circularly polarized light.  Both of these are fundamentally important for a range of applications in electricity and magnetism.  Today we are going to work with polarized waves in our exercise.  Hopefully we will learn how to visualize these waves and understand what it means when we say a wave is circularly, eliptically, or linearly polarized.

Wave Polarization

In a previous lecture, we were able to animate propagating waves in 3D, using the Animate and ParametricPlot3D commands.  In todays lecture we will build on this to see if we can do the same thing for different kinds of polarization.  To simplify things, we are going to assume we are in vacuum ((Subscript[ϵ, r])=(Subscript[μ, r])=1), and less realistically (meaning totally unphysical), that  (Subscript[ϵ, o])=(Subscript[μ, o])=1, which will make c=η=β=1, which again, is not realistic but will mean that our H and E fields have the same magnitude and are thus easier to plot on the same axes.  Let’s also assume that our wave has a radial frequency of ω=1.  For today’s lecture we don’t really care about anything other than the polarization of the wave, so these are reasonable simplifications to make.  Let’s also assume our waves are, for starters, propagating in the +(OverHat[x]) direction.
We’ll start by writing an expression for the E and H fields.  We can write these in phasor or standard notation, so let’s start with standard notation.

e1[e1y_,e1z_]:={0,e1y*Cos[t-x],e1z*Sin[t-x]}

Since our wave is propagating in the +(OverHat[x]) direction, we can use the RHR on each component of the E field to find our H field.  Please note that when we scale by η=1, we will end up with an H field that has the same magnitude as our E field (unrealistic, but who cares?):

h1[e1y_,e1z_]:={0,-e1z*Sin[t-x],e1y*Cos[t-x]}

We can quickly figure out if this is RH or LH polarized by ploting the time evolution of the E field at the x=0 plane. Let’s start by assuming e1y=e1z=1.

Animate[VectorPlot[{e1[1,1][[2]]/.{x->0,t->to},e1[1,1][[3]]/.{x->0,t->to}},{y,-2,2},{z,-2,2}],{to,0,5},AnimationRepetitions->1]

We are plotting the vector field on the y,z plane, which means the positive x direction is coming out of the page.  Following the rotation of the arrows tells us that the field is RH circularly polarized.
However, plotting the whole vector field is cumbersome, especially since along any plane of constant x, the field will be the same across y and z, so let’s replace the vector field with an arrow.

Animate[Graphics[{Red,Arrowheads[0.1],Arrow[{{0,0},{e1[1,1][[2]]/.{x->0,t->to},e1[1,1][[3]]/.{x->0,t->to}}}]},PlotRange->{{-1,1},{-1,1}}],{to,0,5},AnimationRepetitions->1]

Let’s add the H field

Animate[Graphics[{Red,Arrowheads[0.1],Arrow[{{0,0},{e1[1,1][[2]]/.{x->0,t->to},e1[1,1][[3]]/.{x->0,t->to}}}],Blue,Arrowheads[0.1],Arrow[{{0,0},{h1[1,1][[2]]/.{x->0,t->to},h1[1,1][[3]]/.{x->0,t->to}}}]},PlotRange->{{-3,3},{-3,3}}],{to,0,10},AnimationRepetitions->1]

Try changing the e1y and e1z above so that they are not equal.  This is an eliptically polarized wave.  What if you make one of the amplitudes negative?
You should see the direction of rotation change!!  You have now turned your wave into a LH polarized wave!!
Let’s plot our waves in 3D, using the ParametricPlot3D command

Manipulate[Animate[ParametricPlot3D[{x,e1[e1y,e1z][[2]]/.{t->to},e1[e1y,e1z][[3]]/.{t->to}},{x,0,20}],{to,0,10},AnimationRepetitions->1],{e1y,{-4,-2,0,2,4}},{e1z,{-4,-2,0,2,4}}]

Now add the H field and plot both simultaneously:

Determine combinations of e1y and e1z that make the wave: RHC, LHC, RH Elliptical, LH Elliptical and Linearly polarized in (OverHat[y]) and linearly polarized in (OverHat[z]):

RHC:

LHC:

RHE:

LHE:

LP in (OverHat[y]):

LP in (OverHat[z]):

Now create the equations and the animated plot (like we did above), for a circularly polarized wave propagating in the +(OverHat[z]) direction: