Reflection and Transmission of Electromagnetic Waves: Part 3
Recap
It has been a while since I was last able to focus on publishing, a fact that can be attributed mostly to an intensive study schedule, and in part to my own laziness. Either way, let me link the first two parts of this series here:
If you recall our previous discussion, we were focusing on calculating the amplitudes of reflected and transmitted electromagnetic waves after being incident on a dielectric boundary. In shorter terms, we were trying to derive Fresnel’s equations.
Among our endeavors, part 1 was the relatively simple case of normal incidence, while part 2 discussed oblique incidence with magnetic field vectors perpendicular to the incident plane. In this final part, we will turn our attention to the case when electric field vectors are perpendicular to the incident plane, making magnetic field vectors parallel to the incident plane. Here is a diagram.
As before, I apologize for my artistic incompetence. Let me explain the diagram.
- The electric field vectors always point out of your screen, which is along the y-axis.
- The incident magnetic wave vector is perpendicular to the direction of propagation and, as expected by the right-hand rule, lies in the plane of the screen. It points downwards.
- The reflected wave has changed direction. The electric wave vector points outwards. Using the right-hand rule, the direction of the magnetic wave vector is now reversed. It now points upwards and perpendicular to the direction of propagation.
To make this concept even clearer, I will write down the components of all the wave vectors here. I strongly recommend that you try to figure out the directions of the wave vectors via pen, paper, and the right-hand rule. Finding the correct directions is crucial before continuing to apply the boundary conditions.
Notice how the incident and reflected magnetic wave vectors both make use of the incident angle. This is because we have already derived the laws of reflection and refraction in our previous article. We can directly make use of the fact that the incident and reflected waves lie at the same angle with respect to the surface normal.
The Boundary Conditions
The boundary conditions are a sore topic in electrodynamics. Truth be told, despite their rigorous application in this field, I am yet to arrive at a mnemonic for remembering them at a moment’s will. If you believe you can list them with accuracy without help, well done. For those of you who can’t, here they are:
The Electric Wave Vectors
There are no components of electric field perpendicular to the dielectric interface and thus, equation (i) is trivial. We turn our attention to equation (iii). The sum of the amplitudes of electric fields on either side of the interface must be equal. That is,
The Magnetic Wave Vectors
The discussion of the magnetic wave vectors won’t be so straightforward. As always, we would first need to resolve their components along and normal to the boundary. This, I have already done above in Equation Set 1, but if it is not clear, you should try and draw a diagram like the one we drew in the previous article.
Another small detail you must remember is the relation between the amplitudes of electric and magnetic fields. The final equations we derive will be fully in terms of the electric field amplitudes, thanks to the following relation:
While applying the boundary conditions for the magnetic vectors, we will use the above substitution to simplify our problem. Now, if you apply equation (ii) and then use Snell’s law (which we derived in the previous part), you will be directed back to what equation (iii) suggests. The only remaining useful equation for us is equation (iv), and here is what it entails:
Simplifying Our Equations
Our task is pretty much done at this point and all that remains is to solve equations 5 and 6. To that end, I define the two quantities alpha and beta as before. Take a look.
As you can see, there’s nothing fancy about these two quantities and they just serve as a way to make our relations neater.
Uncoupling the incident, transmitted, and reflected wave amplitudes from the above equation set, we arrive at Fresnel’s Relations for the case of oblique incidence of electromagnetic wave with electric wave vectors perpendicular to the plane of incidence.
And that finishes our task. Once again, we have a relation in terms of alpha and beta, the former of which depends on the angle of incidence of the wave. Thus, the intensities of the reflected and transmitted waves also depend on this angle. As for the reflection and transmission coefficients, they are pretty easy to calculate and left as an exercise to the reader.
This concludes my series on Reflection and Transmission of Electromagnetic Waves.
