Question #43241

A thin transparent film of thickness $t$ and refractive index $n_2 = \sqrt{2}$ is coated on a thick glass substrate of refractive index $n_3 = 2$. The film is in contact with air ($n_1 = 1$). A plane light wave of wavelength $\lambda$ (in vacuum) is incident on the film from air at an angle of incidence $\theta_i = 60^\circ$. The incident light is p-polarized (electric field in the plane of incidence).

The amplitude reflection coefficient for p-polarized light moving from medium $n_a$ to $n_b$ is given by $r_p = \frac{n_b \cos \theta_a - n_a \cos \theta_b}{n_b \cos \theta_a + n_a \cos \theta_b}$, where $\theta_a$ and $\theta_b$ are the angles of incidence and refraction, respectively. A positive $r_p$ corresponds to a phase shift of $\pi$, while a negative $r_p$ corresponds to a phase shift of $0$.

Find the minimum non-zero thickness $t$ of the film for which the reflected light undergoes perfectly destructive interference.