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)

Figure 2 - Formation of circular fringes for a divergent beam.

If the mirror M₁ and the virtual image M₂' are not exactly parallel, fringes will still be seen with monochromatic light for small path differences, but the pattern loses its circular shape (ref. 2,6). The fringes appear as parallel lines that can be adjusted, e.g., to be vertical in the field of view.

Thus, one of the mirrors in the M.I. - usually the stationary one - has tilt controls so that one can tune the instrument to produce either a circular, or a parallel pattern of fringes of equal intensity.

You will note that Fig. 1 has an additional parallel plate of glass, B. The light reflected from M₁ passes through A three times before reaching the eye. The light reflected from M₂ passes through A once and through B twice. Thus, the purpose of the compensating plate B is to make the two optical paths, ray (1) and ray (2), traverse equal amounts of glass. This is not essential for producing fringes in monochromatic light, but it is indispensable when white light is used.

Experimental Procedure

NOTE:The interferometer is a precise optical system that can be easily damaged. DO NOT put your fingers on any reflecting surface.

1. Initial adjustment of the Michelson Interferometer

   Verify that the components are in the correct position for the Michelson interferometer. The angular tilt of mirror M₁ is fixed, but it can be displaced using a linkage (lever) to a micrometer. Mirror M₂ can be tilted but not displaced. Place the mercury lamp before the diffusing screen and affix a pointer needle vertically to the screen.

   CAUTION: The mercury lamp emits UV radiation. You have to wear safety glasses to protect your eyes. Never operate the mercury lamp without a glass plate attached.

   Turn the mercury lamp on and align the eye with the viewing axis about ten inches (25 cm) from the instrument. Two pairs of reflected images of the pointer will be seen, one coming from reflection at the front surface of A, and the other from reflection at its back surface. When the tilting screws on M₂ are now turned until one pair of images falls exactly on the other, the interference fringes should appear. Acquaint yourself with the instrument, and try to obtain a circular, as well as a parallel pattern. Try to produce a circular pattern with a small number of fringes?

   As mirror M₁ is moved a distance d, the pattern of bright (dark) fringes shifts according to eq 5 (eq. 6). Thus, if one changes d and counts how many bright (dark) fringes pass a reference point (e.g. the image of the pointer), the wavelength λ of the monochromatic light can be determined. On the other hand, one can use this method with light of a known wavelength in order to calibrate the mechanism for the mirror movement, i.e., to determine the distance d that corresponds to a given reading on the micrometer. In the following steps you will first perform a calibration of the linkage, and then determine λ for a source of unknown wavelength.

   Calibration of the mirror movement

   The mercury lamp produces a spectrum of lines of discrete wavelengths (consult a spectral chart to verify, cf.. Table I in Exp. 9) .The green filter used in front of the Hg lamp isolates the 5461Ǻ (=546.1 nm) line. Measure the distance on the micrometer screw that produces a shift of 50 fringes. Repeat the measurement for ten sets of 50 fringes. Carefully deduce the mechanical ratio of the micrometer-mirror linkage with its error.

2. Determination of the wavelength of the Na doublet lines

   It is possible to use the Michelson interferometer to measure the wavelength difference between two lines closely spaced in wavelength and of similar intensities, without finding the wavelength of either line to high precision. In this part we determine the separation and average wavelength of the sodium doublet, known for its characteristic yellow lines.

   The two lines with wavelengths λ₁ and λ₂ are closely spaced for the following reason. At the level of nonrelativistic quantum mechanics a single line is predicted for the transition corresponding to a de-excitation of the Na(3p) to the Na(3s) ground-state level. Note that in contrast to the hydrogen spectrum (a pure -Ze²/r potential) the Na energy spectrum does have a dependence on the principal quantum number n as well as on the magnitude of angular momentum l , i.e., 3s and 3p are not degenerate. In a more refined treatment the orbital angular momentum (l = 1) state Na(3p) can couple with the electron spin (s = 1/2) in two ways: a state with total angular momentum projection j_z = 3/2, and another state with projection j_z = 1/2 along the z-axis. The energies for these states are slightly different. The separation of the two wavelengths (that you determine in this experiment) is a measure of the effect of the spin-orbit coupling: a tiny energy offset that arises as a result of the interaction between the magnetic moments associated with the orbital and spin angular momenta of the electron (cf.. ref. 3).

   The presence of a superposition of two monochromatic waves in the M.I. complicates the analysis somewhat. One obtains a superposition of two slightly different fringe patterns due to the difference between the wavelengths λ₁ and λ₂. For some choices of d the fringes for the two wavelengths are aligned, while for other values the maximum of intensity for one wavelength lies between the maxima of the other wavelength, leading to uniform illumination (as opposed to a fringe pattern). The difference in the mirror displacements for which a uniform field is observed can be used to measure the wavelength difference as shown below.

   When a certain point in the visual field is a maximum for both wavelengths (aligned fringes) then the phase differences introduced by the path difference between the mirrors must be integer multiples of 2π. Let δ₁ be the phase difference for wavelength λ₁ between the beams arriving at P from the two mirrors (i.e., δ₁= φ₂ - φ_i for λ₁). Then it follows that δ₁ = (2π/λ)2 d = m₁(2π), where m₁ is an integer, and correspondingly for the other wavelength δ₂ = m₂(2π), where m₂ is another integer.