ECE3300 Lecture 30-1 Plane Wave Reflection and Transmission, Oblique Incidence

Today's lecture is going to be how to analyze plane waves that
come on to a boundary between two media in an oblique fashion. So
here's a separation between two different materials: Material
number one and material number two. This is the normal to that
boundary, the perpendicular to the boundary, and here's a plane
wave that's coming in at a direction that is not perpendicular to
the boundary. We call this oblique incidence, just because it is
not perpendicular. This represents the frontal plane waves as
they are moving through space. There are two aspects of this
plane wave that are going to be particularly important to us. One
is the direction of propagation. We're going to call that KI,
with a unit vector, because that's representing the direction the
wave is propagating. Here's the angle theta I relative to the
normal that tells us the direction of that plane wave. There's
also going to be a reflected wave that comes off like this, the
angle of the reflected wave, theta R, is equal to the angle of the
incident wave, regardless of the two materials. We're going to
call this vector KR with a hat over it. That's a unit vector in
the reflected direction. And then finally there's going to be a
transmitted vector. We're going to call this KT, for
transmission. And there's going to be an angle here, theta T. We
use Snell's law to be able to find all of the angles. The first
thing that we indicated was that theta I and theta R are equal.
The second thing is depending on these two materials and theta I,
we'll be able to find theta T. This is how it works. We can say
that the index of refraction N, this is not eta, this is N. N1
times the sine of theta 1 is N2 sine of theta 2. This is material
1 and this is material number 2. This is N1 sine of theta
incident equals N2 sine of theta 2. And for a lossless material
is the square root of epsilon R. Square root of epsilon R1,
square root of epsilon R2. So this tells us that the angle of
incidents affects the angle of transmission in a lossless -- this
is lossless-- nonmagnetic material. Nonmagnetic means that UR is
equal to 1. This helps us analyze the different directions of
propagation, and there's one other factor that's going to be very
important. And that is the polarization of the plane wave. The
polarization is defined by the electric field vector. So I'm
going to draw two different choices for electric field vector
here. One of them would be if the electric field vector came out
of the board or the electric field vector went into the board. In
this case the electric field would be perpendicular to the board.
We call the board the plane dissonance because it includes the
direction of the propagation and it includes the normal boundary
between the two materials. So if you have an electric field in
either of these directions, this is a perpendicular polarized
plane wave. On the other hand, you might have an electric field
vector that is either up or down. And this electric field vector
can be drawn on the plane of incidence, the blackboard, and this
is therefore parallel to the plane of incidence. So we're going
to do two different cases. One of them is going to be for a plane
wave that is propagating in the perpendicular polarization and the
other one is the plane wave propagating in the parallel
polarization. And now let's also look at where the magnetic field
vector is. If I had a direction of propagation such as my
incident field here, and I had a perpendicular polarized electric
field vector, let's figure out where the magnetic field vector is.
The magnetic field vector can be found using the right-hand rule.
We put our thumb in the direction of propagation. We put our
first finger in the direction of E, and then our third finger
shows us where H is. So H in this case is down like so. Let's
look at this case. We put our thumb in the direction of
propagation. We put our finger in the direction of E. Our third
finger will show us where H is. So for the electric field vector
going up like so, H would be in this direction. So the two things
we've looked at so far are how to define the angles, how to define
the direction of the propagation, how to determine if we are
perpendicular or parallel polarize d and how to find the magnetic
field vector.