Consider the setup shown. At which angles $\theta$ must the laser hit the ceiling of material 1 for the
laser light to be transmitted to material 2 and for it not to be transmitted to material 3?
Adjustable
laser
$\theta$
Air
Material 1
Material 2
Material 3
(a) Identify and illustrate the ways in which light would propagate (e.g., reflection, refraction,
etc) i) as it hits the ceiling of material 1, ii) as it enters material 2, and iii) as it reaches the
interface between materials 2 and 3. Label all important quantities in your illustration.
(b) Consider first the case where total internal reflection doesn't occur within material 2 (i.e.,
when the refracted beam is along the interface. Set up the relevant laws (e.g., law of
reflection, law of refraction) governing propagation of the laser light. All the variables in your
equations must be illustrated in your answer for part (a). Let $n_{air}$, $n_1$, $n_2$, and $n_3$ represent
the indices of refraction of air and materials 1, 2, and 3, respectively. Take $n_{air}$ to be 1.
(c) From your equations in part (b), solve for $\theta$. Express $\theta$ in terms of the given refractive indices.
(d) Solve for the possible values of $\theta$ which would allow total internal reflection to occur within
material 2. From your answers in parts (c)-(d), you may now express your final answer in
Problem 1 as an inequality (e.g., $\theta \ge \pi/2$).
