# Plane wave at dielectric interface

Welcome, in the last lecture we investigated the wave propagation in some arbitrary direction with respect to the coordinate axis. Our main objective is to find out the wave propagation in a bound medium and it has been mentioned earlier that when you are having a bound medium the freedom of choosing the coordinate axis is not really there because choosing the coordinate axis in a particular direction may simplify the problem at least algebraically. So normally when we are having the medium boundaries we choose the coordinate axis so that it aligns along the boundaries and then the wave propagates in arbitrary direction with respect to the coordinate axis because the wave is now incident on the boundary or some arbitrary angle In today’s lecture we will investigate the propagation of a plane wave at media interface What you are now saying is instead of having an infinite medium if I divide the medium into two semi infinite media so I have a media interface, on the left side of this line we are having a medium which is infinite, on the right side we have a medium which is again infinite and the medium properties abruptly change at this location called a media interface So let us say we have Medium 1 on the left side of this and we have Medium 2 on the right side and let us assume that the conductivity for both the media is still zero that means the media is still lossless but the permeability and permittivity are different for these two media. So let us say the permeability for this Medium 1 is given by μ1, permittivity is given by ε1 and for Medium 2 the permeability is given by μ2 and the permittivity is given by ε2 Let us say now we orient the coordinate system such that the media interface is in the xy-plane So along this direction we have coordinate axis x perpendicular to this axis z and perpendicular to the plane of the paper that is the arrow coming outwards normal to the plane of the paper that is the y direction. We can again verify that we must have the right handed coordinate system so if my fingers go from x to y where y is coming outwards from the paper then my thumb must point in the direction of z so this is the correct right handed coordinate system Now let us say we have a wave which is incident on this dielectric interface at some arbitrary angle with respect to the coordinate axis So I have a wave which is incident at some arbitrary angle like that, this is the direction of the wave so essentially this is the wave vector for the wave which is incident on the dielectric Now specifically we are asking questions like when this uniform plane wave is incident on this dielectric interface then what will happen to this wave. Intuitively it appears that part of the energy will get transferred to the second medium so that will again constitute some kind of wave propagation but also what is not obvious at this moment is that part of the energy will get from the interface, this is what essentially we will argue that if the wave is incident on this interface we essentially require two kinds of fields, one which is in the second medium and also the fields in the first medium all have to

be modified to satisfy the boundary conditions So essentially when the wave is incident on the dielectric interface, part of the energy will be transferred to the second medium but also the part of the energy will come back to the first medium and that is what essentially we will investigate When the energy comes back into the first medium or goes to the second medium what happens to plane wave nature, which direction the energy will be going are questions essentially we will have to ask, also we have to ask for what is the magnitude of the field which goes to second medium, how much power is going to transferred to second medium So the analysis of the plane wave at interface essentially includes finding out the direction in which the waves will be moving in the two media, how much power get transferred from one medium to another medium, how much power comes back from the interface to the first medium itself, what happens to the direction of the electric field that means what happens to the polarization of the electromagnetic wave and so on So in this lecture and in the following lectures we will essentially discuss these issues related to the propagation of uniform plane wave across a media interface. And since you have taken conductivity zero for both these media we can at moment we call this as Dielectric Media so this is the Dielectric Medium 1, this is the Dielectric Medium 2. So essentially at the moment we are investigating propagation of the uniform plane wave at a dielectric media interface, on both sides we are having media which have dielectrics Now let us say this is making an angle with respect to this direction that this angle is given by some θi Now you find a certain quantities here that is if I say that xy-plane is the media interface that means in that plane suddenly the medium properties change the medium is uniform in the xy direction only suddenly it changes this point along this z direction and without losing generality we can say this quantity z = 0. So let us say the origin of this coordinate axis at the interface and the wave vector is making an angle θi with respect to this direction. Now this z direction is perpendicular to the media interface so we call this as the normal to the media interface. So this line the z axis is the normal to the dielectric media interface We are now having a wave traveling at an angle θi with respect to the normal to the media interface that is the problem essentially we are having and now you want to ask when this wave is incident at this angle, what will happen to this wave. So first of all let us represent this wave in the form that is the phase function which has amplitude, let us say without losing generality without specifically saying we are talking about electric field or magnetic field you are having some field vector which is associated with this wave but it has a definite phase function because it is traveling at an angle θi with respect to coordinate axis So in this case if I write down as we discussed in the last lecture the wave vector makes three angles øx, øy and øz with three coordinate axis so these three angles we can write down in that terminology so øx is the angle which the wave vector makes with the x axis if I extend this such that this angle will be θi so this angle which the wave vector makes with the x axis will be π/2 – θi, the angle which the wave vector makes with the y axis which is perpendicular to the plane of paper will be ninety degrees so in this case øy is π/2 and the angle which this makes with the z axis is øz = θi Once I know these three angles which the wave vector makes with the three coordinate axes then I can write down the direction cosines and I can write down the phase function. Since the medium properties for this Medium 1 are μ1 and ε1 we have the phase constant of

this first medium is β1 that is equal to ω square root μ1 ε1 and in the second medium we have phase constant β2 that is equal to ω square root μ2 ε2 So I know in both media the phase constant for a plane wave and we know the angle which the wave vector makes with the three coordinate axes. So let us say I have some field represented by this wave which is having a vector and the phase function will be given as we saw last time as follows Let us say I have some field which is incident and let me can call that as some Fi bar where F could represent the electric field or magnetic field which is having a magnitude term so let us say F0i bar which is a vector and then you are having a phase function which is e to the power -j since the wave is incident in Medium 1 so this phase constant is β1 so this is β1 into (cos øx x + cos øy y + cos øz z) Now since øx is π/2 – θi so cos øx is sin θi, cos øy which is cos π/2 is zero and cos øz is cos θi. So from here we get cos øx is cos(π/2 – θi) which is again equal to sin θi, cos øy = cos π/2 that is zero and cos øz = cos θi that will be cos θi If I substitute this in the expression for the fields in general the incident field we call this wave as the incident wave this is incident on the dielectric media and this is what the suffix denotes this i essentially gives you the incident field so let us call this as incident wave so we have the field for the incident wave which is this magnitude amplitude term e to the power -jβ1 (x sin θi + z cos θi) where this quantity is zero because of øy is zero So the wave which is traveling at an angle θi with respect to the normal to the interface can be represented by field which is like this so if I look at some instant of time if I look what is the variation of the field amplitude and the function of x and z and also y we can obtain the field variation by taking the real part of this quantity so if I take the field amplitude at some instant of time the amplitude per magnitude will be the real part of this quantity so real part of Fi bar will be F0i bar cosine of this angle which is β1 times this quantity Let us say I am interested in finding out what is the variation of the phase in the media interface so I am not worried about how the phase is varying in z direction just