Introduction to Electrodynamics/Chapter 8/2/5

8.2.5 Reflection and Transmission at Oblique Incidence

What happens when an EM plane wave travels at an oblique angle towards a boundary between two different linear media?

Hi, I'm Jonathan Gardner. Last time in section 8.2.4, we considered what happens for EM plane waves striking perpendicular. This time, in section 8.2.5 of Griffith's Introduction to Electrodynamics, 2nd Edition, we will consider what happens at an angle.

Setup

The boundary is the y-z plane at x=0. On the left, we have \epsilon_1 and \mu_1, and EM waves travel at v1. The index of refraction is n_1. On the right, the same things but for 2 - \epsilon_2, \mu_2, v_1, n_1.

We have the incident wave coming in from the left side, forming some angle \theta_I with the x axis. It will transmit and reflect, those waves forming an angle \theta_R and \theta_T. Each of these waves may have different wave numbers \kappa_I, \kappa_R, \kappa_T. Thankfully, they all share the same \omega. The reasoning is simple: We have some boundary conditions to meet at the boundary, and that requires that the waves wave at the same frequency.

Fields

As before, we have E and B fields for each of the waves. However, this time we can't use the simple kappa-x. We need to use the kappa vector we discussed earlier in 8.2.1. The r vector is the vector pointing from the origin to some arbitrary point in space. The Kappa vectors have a direction matching the direction of the wave, and a magnitude reflecting the wave number rather than the speed. The speed, of course, depends on the material and not the wave.

Note that our E_0's are vectors. The direction is the polarization of the wave.

<math> \begin{align}

\tilde{\vec{E_I}}(\vec{r}, t) &= \tilde{\vec{E_{0_I}}}e^{i(\vec{\kappa_I} \cdot \vec{r} - \omega t)} & \qquad \tilde{\vec{B_I}}(\vec{r}, t) &= {1 \over v_1} (\hat{\kappa_I} \times \tilde{\vec{E_I}}) \\

\tilde{\vec{E_R}}(\vec{r}, t) &= \tilde{\vec{E_{0_R}}}e^{i(\vec{\kappa_R} \cdot \vec{r} - \omega t)} & \tilde{\vec{B_R}}(\vec{r}, t) &= {1 \over v_1} (\hat{\kappa_I} \times \tilde{\vec{E_R}}) \\

\tilde{\vec{E_T}}(\vec{r}, t) &= \tilde{\vec{E_{0_T}}}e^{i(\vec{\kappa_T} \cdot \vec{r} - \omega t)} & \tilde{\vec{B_T}}(\vec{r}, t) &= {1 \over v_1} (\hat{\kappa_I} \times \tilde{\vec{E_T}}) \\

\end{align} </math>

Of course, the total E and B fields are given by the sums of the incident and reflected waves on the left side, and the transmitted wave on the right.

Remember that these are plane waves. Perpendicular to the direction, the E and B fields are the same. (Draw dotted perpendicular lines)