Edu

Let's consider the optical properties of a thin dielectric film on a substrate. The system is in air, the refractive index $n_1$ of the film is real and the refractive index $n_2=n'_2-i n''_2$ of the substrate has the imaginary part.

Let $d$ be the thickness of the film, so the complex amplitude of the electric field $\tilde{E}$ could be written as
$\tilde{E} = E_0 \begin{cases} e^{ikx} + r e^{-ikx}, &\text{for } x < 0\\ A e^{in_1kx} + B e^{-in_1kx}, &\text{for } 0 < x < d\\ t e^{in_2k(x-d)}, &\text{for } x > d\\ \end{cases}$

A1 Using the boundary conditions for the electric field $\vec{E}$ and the magnetic field $\vec{B}$, find the amplitude reflection coefficient $r$ in such a form:
$r = \frac{a + b e^{i\phi}}{ab + e^{i\phi}}.$

The material of the substrate is silicon with $n_2 = 4.32 - i \cdot 0.073$ and the material of the film is sapphire $\rm Al_2O_3$ with $n_1 \approx 1.7$ (the refractive indices are given for the wavelength $\lambda = 500~\text{nm}$).

With thin_film.py you can simulate the experiment and plot the dependence of $R = |r|^2$ as a function of the thickness $d$ of the thin film on the silicon. The refractive index $n_1$ of the film is defined in the 23rd line.

The data shown below corresponds to the cyclic chemical deposition of $\rm Al_2O_3$ on the silicon substrate. At the beginning of the experiment, the thickness $d$ of the sapphire is zero. The intensity $I$ of the reflected light is given as a function of the number $N$ of deposition cycles.

A2 What is the refractive index $n_1$ of the sapphire thin film?

A3 Find the thickness $d_0$ of the sapphire film grown on the substrate in one deposition cycle. Consider that this value does not depend on the thickness of the already grown $\rm Al_2O_3$.