Fresnel coefficient

In wave optics, Fresnel coefficients are defined as the ratio of the reflected (or transmitted) $\vec{E}$ and the incident $\vec{E}$. We say that $r$ is the ratio of the reflected to incident wave. And $t$ is the ratio of the transmitted to incident wave.

We first define two types of polarisation:

• TE polarisation (transverse electric) is when $\vec{E}$ is polarised normal to the plane of incidence.
• TM polarisation (transverse magnetic) is when $\vec{E}$ is polarised tangential to the plane of incidence.

At the interface of two media, there are a few things to observe depending on the angle of incidence ${\theta }_i$:

• ${\theta }_p$ is a crossover point called the Brewster angle. This has some physical significance: $r_{TM}=0$, so TM-polarised light is completely transmitted with no reflection.
• A negative $r$ means one field is pointing in the other direction. This is to do with the sign convention, and has no real significance.
• The critical angle ${\theta }_c$ defines the boundary between total internal reflection. For ${\theta }_i<{\theta }_c$, there is still some light transmitted

We also define the reflectance as the ratio between the reflected power $P_r$ and the incident power $P_i$.

$$R=r^2$$

The transmittance is defined as:

$$T=\frac{P_t}{P_i}=\frac{n_t\cos {\theta }_t}{n_i\cos {\theta }_i}{t}^2$$

By the conservation of energy:

$$R+T=1$$