Reflection and Transmission of Electromagnetic Waves: Part 1
I previously authored an article discussing the evolution of Maxwell’s Equations and Displacement Current. I now quote it once more to stress the importance of these equations and the role they play in electrodynamics:
In fact, such is the power embedded within the four equations given above that together with the Lorentz Force Law, they suffice in explaining the entirety of electromagnetism on their own!
Not only that, but Maxwell’s equations are further used when we look at electromagnetic radiation in the context of wave theory. As we will come to see, solutions to Maxwell’s equations satisfy the wave equation as well, much like all other waves. However, Maxwell’s equations impose further restrictions on solutions to the wave equation. Solutions to the former are necessarily solutions to the latter, while the converse is not true.
The Boundary Conditions
Using the integral form of Maxwell’s equations, and using a Gaussian pill box, the following boundary conditions can be derived for a medium that is free of sources and currents.
I am not going into too much detail about this here since the derivation is quite trivial and there is enough material on the topic out there. Instead, let us introduce electromagnetic waves and jump straight into reflection and transmission at boundaries.
Looking at Electromagnetic Waves
Maxwell’s equations are four coupled equations in electric field and magnetic flux density vectors, E and B. In the case of a linear medium that is devoid of charges and sources, ρ vanishes, as does the current density, J. Thus, we are left with the following equations.
The next step is de-coupling these equations, which is a trivial process if you take the curl of the equations on the right, something that mathematicians refer to as the vector triple product. The identity for this expression states that ∇ × (∇ × A) = ∇ (∇.A) — ∇²A. The divergence of E and B will vanish in this scenario, and you will arrive at the classical wave equation in three dimensions. Further, monochromatic plane electromagnetic waves of frequency ω can be written in complex notation in the following form:
E (r, t) = E₀ exp (i(k.r — ωt)), and B (r, t) = B₀ exp (i(k.r — ωt))
For the moment, the phase factor isn’t important, and either way, it can be included in the amplitude portion by taking out the exp (iδ). Now, we look at the result of such an electromagnetic wave falling on an interface between two linear, homogenous, dielectric media. Specifically, we would like to see what would happen to the amplitudes of the incident, the reflected, and the transmitted waves, and derive a relation between the three quantities. Our work will be divided into three categories:
- Normal incidence at the boundary.
- Oblique incidence with electric wave vector polarized perpendicular to the plane of incidence.
- Oblique incidence with magnetic wave vector polarized perpendicular to the plane of incidence.
Due to the extensive nature of these topic, I will handle each of these cases in separate articles. Let us start with the first one.
Normal Incidence
In the image attached here, the X-Y plane is considered as the interface between two media of refractive indices n₁ and n₂. The z-axis is the direction of propagation.
Recall the right-hand rule. Considering the electric vectors to be polarized along the x-axis, the magnetic field will be along the y-axis, which, in this case, is the axis coming out of your screen. For the reflected wave, B will change direction by 180⁰ and point into your screen (I urge you to comment why that happens). For the transmitted wave, the directions won’t change. Here is a diagram:
It is easy to see that there is no component of B or E that is perpendicular to the boundary and thus, two of the boundary conditions, viz, (i) and (ii) are out of the picture. The remaining two will be satisfied by the combined values of Eᵢ and Eᵣ on the left, along with Eₜ on the right. The same is true for the magnetic fields as well. Let us first represent all the waves in complex notation.
As previously stated, there are no perpendicular components and thus, two boundary conditions are trivial. We turn our attention to the component of electric field parallel to the interface. As per what we know, these components must be continuous. Thus, we have:
Further, the perpendicular components of magnetic field must also be continuous due to the absence of free charges and currents. These are written in terms of the electric field themselves and we have:
Let β = μ₁v₁/μ₂v₂ = μ₁n₂/μ₂n₁, where n is the refractive index. Thus, we can solve the above equations for the reflected and transmitted electric field amplitudes and arrive at an expression for those in terms of the incident electric field amplitude. Naturally, since the magnetic field amplitude is proportional to its electric counterpart, half the job done in this case, is the whole job done. The expressions will take the following form:
Finishing Touches
Notice the dependence of the sign of the reflected intensity on the velocities of the waves in the two media. If v₂ is smaller than v₁, or in other words, the refractive index of the second medium is larger than the first one, then the reflected wave is out of phase with the incident wave. This is a familiar result we have seen before in optics, which says that rays travelling from rarer to denser medium suffer a phase change of π upon reflection.
All that remains now is to arrive at the expressions for reflection and transmission coefficients. These two quantities are immensely useful since they connect the incident, reflected, and transmitted intensities instead of the field amplitudes. With I = ½ εvE₀², we define the reflection and transmission coefficients, R and T as the ratios of reflected and transmitted intensities to the incident intensity, respectively. Naturally, since there is no loss of energy, R + T = 1, and their values are:
Conclusion
There we have it. We were able to find expressions for reflection and transmission coefficients, as well as the field intensities for all the scenarios. I will reiterate, though, that this is not the end. Indeed, two situations still need to be tackled, which require quite a bit of algebra. I will upload another article for the same quite soon. Till then, happy studying!
Update 28th February 2022: The second part of this article is now up. Read about oblique incidence at the following link.
