Microwave Optics - Thin Film Interference

Aim

In this experiment, we aim to: (a) determine the refractive index n of a polymer slab from its constructive and destructive thin-film microwave interference; and (b) determine the lattice constant a of an array of thin copper rods from its Bragg diffraction of the incident microwave.

Introduction

Thin Film Interference

When plane electromagnetic wave of wavelength λ is incident on a dielectric surface, part of the electromagnetic wave is reflected, and part of it is transmitted. If the dielectric is thin, i.e. its thickness t is comparable to λ, the plane electromagnetic wave will be reflected again at the second dielectric surface, as shown in Figure 1.

Figure 1. Electromagnetic waves incident at angle θ1 reflected from the top and bottom surfaces of a thin dielectric film of thickness t, and refractive index n.

After emerging from the top surface of the dielectric, the ray B which propagated through the thin film would incur an optical path difference of

(1)

compared to ray A. With the help of the law of refraction

(2)

the path difference between rays A and B can be simplified to

(3)

Because ray A suffers a 180° phase shift when it reflects off the top dielectric surface, the two rays interference destructively if their path difference is an integer multiple of λ,

(4)

and destructively if their path difference is a half-integer multiple of λ,

(5)

Suppose the wavelength λ of the electromagnetic wave, and the thickness t of the dielectric are known. Suppose we further determine experimentally the angles and of successive reflection minimum and maximum, we can then solve Equation (4) and Equation (5) simultaneously for the interference order m and the refractive index n.

Bragg's Law

Figure 2. Plane electromagnetic waves scattering off parallel planes of atoms in a crystal lattice. In this figure, the rays make an angle θ with the atomic planes, which are spaced d apart.

When plane electromagnetic waves of wavelength λ scatter off parallel planes of atoms in a crystal lattice (see Figure 2), the scattered waves interfere destructively for nearly all incident