Main Page/PHYS 3220/Interferometer

The Michelson Interferometer

In this experiment a Michelson interferometer is calibrated using a known mercury line, and then used to make a wavelength determination for the Na doublet. In addition, the index of refraction of a piece of plastic, and that of air at room pressure is determined.

Introduction

The Michelson interferometer is well known for the role it played in finding that the speed of light is the same in every reference frame, and thus to begin the development of the special theory of relativity. It is a simple set-up by means of a half-silvered and two ordinary mirrors to split a beam of light into two, allowing one beam to travel a different path length than the other in order to superimpose them. Small variations in the path length difference d allow us to observe the wave nature of light, i.e., to observe how the two beams can add to produce an interference pattern. An observation of the interference pattern as d is changed allows us to draw conclusions about the wavelength of coherent monochromatic light.

An interferometer is one of several instruments to demonstrate the wave nature of light (ref. 1). Other important two-beam experiments are, e.g., the Fresnel mirror, Young’s double-slit experiment, a thin film, or a grating (cf. the spectrometer used in Expt 9). A more sophisticated version of the interferometer used extensively in research is the Fabry-Perot interferometer. Use the references to understand how these interferometers work and what is meant by spatial and temporal coherence of light coming from an extended source.

Figure 1 - The Michelson Interferometer.

In Fig. 1 the schematic diagram is shown for the M.I. with the semi-transparent mirror positioned at 45 degrees splitting the source beam into two halves that then travel toward the mirrors M₁ and M₂. The electric field associated with a monochromatic planar wave with circular frequency ω=2πc/λ, and wavenumber k=2π/λ is given by

(1)

The waves that are reflected at M1 and M2 are again split by the half-silvered mirror. We are interested in the signal arriving at P (the observer’s eye). Let M₂' be the virtual image of M₂ as seen by P (along the line of sight of M₁), and d be the separation between M₁ and M₂'.The mirrors are carefully adjusted such that M₁ and M₂' are parallel. The difference in path length of the light is thus given by 2d. The amplitudes of the two waves reaching the point P simultaneously are equal, and independent of the reflectivity of the half-silvered mirror (!), whereas their respective phases φ_i are going to be different depending on the traversed path lengths.

(2)

From (1) it is obvious that the phase difference depends on the path length difference 2d and on the wavelength of the monochromatic light:

(3)

The time-averaged intensity of the light I_T observed at P is given by

(4)

Consequently the M.I. acts as a wavelength-dependent mirror, and depending on the phase difference Δφ , I_T varies between 0 and the time-averaged incident intensity I₀.

Equations (3,4) yield the conditions for constructive and destructive interference respectively: complete passage of the incident light occurs for the wavelengths determined by Δφ = m2π, i.e.,

(5)

while complete reflection occurs for Δφ = (m+1/2)2π, i.e.,

(6)

For wavelengths in between these values the light is partially reflected and partially transmitted. Note that according to eq. (4) the intensity is independent of the order m. This is not exactly realized in practice due to imperfections in the glass.

The mathematical description of the two waves for the M.I. still omits one point. Light reflected at a boundary surface from an optically rare to a dense dielectric medium experiences a phase change of π (ref. 4,5). Considering the M.I. for d=0, we find from eq. 5 that m=0 corresponds to the condition for maximum intensity. If we treat the half-silvered mirror as a thick glass plate, then if we follow the path of the two beams we observe that one beam suffers an additional phase change of π while reflecting at the back surface of the mirror. Thus, destructive, and not constructive, interference should occur. However, the treatment is non-trivial as the metallic coating (of a few atomic layers) can introduce phase changes in both the transmitted and reflected waves. This can introduce an overall phase difference Δφ between 0 and π depending on the nature of the reflecting surface. The effect of this phenomenon when using white light will be considered in a later section.

Usually the light entering the interferometer is not parallel, but somewhat divergent. The partial beams have an angle of inclination θ as shown in Fig. 2. Owing to the several reflections in the real interferometer we may now think of the extended source as being at L, behind the observer, and as forming two virtual images L1 and L2 in M1 and M2'. The path length difference 2d now depends on θ, and is due to axial symmetry with respect to the central axis owing to this. A system of circular fringes is produced in the field of view. The difference in path length for beams with inclination θ becomes 2d cos(θ), and thus the condition of constructive interference becomes 2d cos(θ) = mλ. The phase difference for the partial waves arriving at P is given by

(7)