Fresnel Coefficients help in solving the reflection and transmission when a light wave goes from one medium to another. More importantly they form the building blocks of Transfer Matrix method which is used to solve the reflection and transmission behaviour or multilayer stacks.

Normal Incidence

Assume light is incident along the interface normal. Medium 1 is n1 and Medium 2 is n2. Interface is at z=0. This means kvector is along z. Since is incidence angle is zero, we need not consider polarization of the light. The Electric field and Magnetic can be along the two in-plane directions perpendicular to the k vector of the incident wave. There are two waves in the incident medium - the incident wave and the reflected wave. There is one wave in the second medium - the transmitted wave.

Now we apply the continuity equations.

1. Continuity of in-plane electric field boundary condition

\[ E_i + E_r = E_t \]

2. Continuity of in-plane magnetic field boundary condition

\[ H_i - H_r = H_t \]

Notice the negative sign in front of the reflected wave magnetic field.

Using: \[ H = \frac{n}{\mu_0 c} E \] in the above equations, we end up with two variables, two equations.

Reflectance and Transmittance for normal incidence

\[ E_r = \frac{n_1-n_2}{n_1+n_2} E_i \]

\[ E_t = \frac{2n_1}{n_1+n_2} E_i \]

This gives us the fresnel coeffcients that relate the electric fields :

\[ r = \frac{E_r}{E_i} = \frac{n_1-n_2}{n_1+n_2} \]

\[ t = \frac{E_t}{E_i} = \frac{2n_1}{n_1+n_2} \]

The reflectance (R) and transmittance (T) thus becomes:

\[ R = |r|^2 \]

\[ T = \frac{n_2}{n_1}|t|^2 \]

Now we move onto the oblique incidence case.

Non-Normal / Oblique Incidence

In this case, the incident light incidents at an angle theta with the interface normal. This is our new k vector. We know that the electric field and magnetic field are perpendicular to each other and to the k vector. So now we have two situations: Either the electric field is normal to the plane of incidence (formed by the k vector and the interface normal), this will make the magnetic field in the plane of incidence but forming an angle 90-theta to the interface normal. The other situation can be when magnetic field is normal to the plane of incidence (and thus lying fully in the interface plane) and the electric field in this case is in the plane of incidence perperndicular to the k vector. The former case is called as s-polarized while the latter is called as p-polarized.

s-polarized

We will apply the same field continuity equations but this time the magnetic field has only a component in the plane.

1. Continuity of in-plane electric field boundary condition

\[ E_i + E_r = E_t \]

2. Continuity of in-plane magnetic field boundary condition

\[ H_i\cos\theta_i - H_r\cos\theta_i = H_t\cos\theta_t \]

Reflectance and Transmittance for s-polarized light

Using the same relation between magnetic field and electric field, we will end up with two variables, 2 equations.

\[ r_s = \frac{E_r}{E_i} = \frac{n_1\cos\theta_i-n_2\cos\theta_t}{n_1\cos\theta_i+n_2\cos\theta_t} \]

\[ t_s = \frac{E_t}{E_i} = \frac{2n_1\cos\theta_i}{n_1\cos\theta_i+n_2\cos\theta_t} \]

The Reflectance and Tranmission would be :

\[ R_s = |r_s|^2 \]

\[ T_s = \frac{n_2\cos\theta_t}{n_1\cos\theta_i}|t_s|^2 \]

p-polarized

We will apply the same field continuity equations but this time the electric field has only a component in the plane.

1. Continuity of in-plane electric field boundary condition

\[ E_i\cos\theta_i + E_r\cos\theta_i = E_t\cos\theta_t \]

2. Continuity of in-plane magnetic field boundary condition

\[ H_i - H_r = H_t \]

Reflectance and Transmittance for p-polarized light

Using the same relation between magnetic field and electric field, we will end up with two variables, 2 equations.

\[ r_p = \frac{E_r}{E_i} = \frac{n_1\cos\theta_t-n_2\cos\theta_i}{n_1\cos\theta_t+n_2\cos\theta_i} \]

\[ t_p = \frac{E_t}{E_i} = \frac{2n_1\cos\theta_i}{n_1\cos\theta_t+n_2\cos\theta_i} \]

The Reflectance and Transmittance is :

\[ R_p=|r_p|^2 \]

\[ T_p = \frac{n_2\cos\theta_t}{n_1\cos\theta_i}|t_p|^2 \]

We now have the reflection and tranmission coefficients for any interface for light incident at any angle.