INTRODUCTION

The Mach-Zehnder interferometer is an important diagnostic tool. It is most frequently used in the fields of plasma physics, aerodynamics, and heat transfer to measure density, pressure, and temperature changes in gases. Because of its relatively large and freely accessible working space and flexibility of location of the fringes, it is the most suitable interferometer to study the airflow around models of aerodynamic structures.

This module will familiarize the student with the characteristics of the Mach-Zehnder interferometer and some of its applications. It will also give practical experience in building, aligning, and using this versatile optical device.

MODULE PREREQUISITES

The student should have completed Module 1-1 through 1-6 of Course 1, "Introduction to Lasers"; Modules 6-1 through 6-8 of Course 6, "Laser and Electro-Optic Components"; and have a basic knowledge of laser safety, including eye hazards and electrical safety, as well as knowledge of algebra and interference of optical waves. The student should also be able to operate a HeNe laser and adjust a beam-expanding telescope.

Upon completion of this module, the student should be able to:

1. Demonstrate knowledge of Mach-Zehnder interferometers by drawing and labeling a diagram as shown in the text. Explain the basic design of the instrument and how it operates, including the following items:

a. Interference between emerging beams to form a fringe pattern on the screen.

b. Effect on fringe pattern when a parallel glass plate is inserted into one of the beams.

c. Reason for fringe localization.

d. Use of fringe localization in the examination of airflow over the model of an aircraft wing.

2. Assemble a Mach-Zehnder interferometer and align the interferometer using a HeNe laser and beam-expanding telescope according to procedures outlined in the text..

3. With the Mach-Zehnder interferometer as previously assembled, perform experiments according to given procedures to accomplish the following tasks:

a. Observe the fringe pattern on a screen.

b. Measure the index of refraction of a gas.

c. Observe and measure the optical fringe inaccuracies in a flat plate of glass.

The Mach-Zehnder interferometer is basically a very simple device. A schematic diagram of the instrument and associated optical equipment is shown in Figure 1. The light beam from a source, say a HeNe laser, is first expanded by a telescope. The expanded laser beam is divided at the 50% reflecting/50% transmitting surface of a plane parallel glass plate P₁ into two beams of equal intensity. After reflection at the 100% reflecting plan mirrors M₁ and M₂, the two beams are recombined at the 50% reflecting/50% transmitting surface of plane parallel plate P₂. The recombined beams emerge from P₂ and are intercepted by a well-corrected positive lens L₂, which brings both beams to the focus at S¢ . This will always be true if the four reflecting surfaces, whose centers are at the corners of a parallelogram, are perfectly parallel. (Figure 1 shows a rectangle which is a special case of the parallelogram.)

When all reflecting surfaces are perfectly parallel, the recombined beams do not produce interference fringes. If, on the other hand, the four reflecting surfaces are only approximately parallel, the recombined beams will interfere and produce interference fringes. The exact location of the fringes depends on the size of the light source S and the relative alignment of the four reflecting surfaces.

Let us briefly consider the mechanism of interference as it applies to the Mach-Zehnder interferometer. The two beams of light of equal amplitude and wavelength originating from the same source are brought to a focus at S¢ and S² if the reflecting surfaces are nearly parallel to each other. As indicated in Figure 2, the light from the images S¢ and S² proceeds on, and the beams interfere in much the same way as in Young’s famous double slit experiment. In a plane (the screen) at a right angle to the direction of propagation of the combined beams, an interference fringe pattern will be observed. This means that bright lines will be seen where the peaks of the waves reinforce and dark lines where the troughs of the waves from one beam coincide with the peaks of the waves from the other beam. The end result is a series of nearly parallel light and dark lines varying sinusoidally in intensity. If the intensity were measured as a function of position across the beams, the pattern shown in Figure 2b would be observed.

If the reflecting surfaces are adjusted so that the images S¢ and S² lie on a horizontal line (into the paper), the fringes will be vertical; if S¢ and S² lie on a vertical line (in the plane of the paper), the fringes will be horizontal. Fringe spacing—that is, the distance between light lines (or dark lines)—depends on the distance between S¢ and S² . For instance, fringe spacing can be increased (more nearly equal path lengths) by adjusting one mirror in such a way that the distance between S¢ and S² is decreased. If lens L₂ is removed, fringes will still be observed, but the coherent sources S¢ and S² are now virtual and located at infinity.

What happens to the fringes if the length of the optical path of one of the beams is changed (say the test beam P₁M₂P₃ in Figure 1) by inserting a plane parallel glass plate into the path of the beam? By causing one beam to propagate through the plane parallel glass plate, a new phase shift is introduced between S¢ and S² which causes a shift of the fringes. The extent of the shift is a sensitive measure of the change of the optical path length over the two dimensional fields of view.

For example, by inserting a glass slide only 0.0002 millimeter thick into one of the beams (which is assumed to originate from a HeNe laser at 632.8 nm), the fringe pattern would reverse itself; that is, the dark and light lines would exchange positions.

Example A demonstrates how the Mach-Zehnder interferometer can be used to calculate the thickness of a glass slide that will cause a reversal in the fringe pattern.

Example A: Reversal in Fringe Pattern Caused by Glass Slide Thickness
Given:	The index of refraction of the glass slide is 1.5. To produce a reversal in the fringe pattern, the following condition must be met:
	where:	phase change = p (180° ) = (k¢ – k) d k¢ = k = d = The thickness of the slide.
Find:	Assume l = 0.63 ´ 10–4 cm (HeNe laser). The thickness of a glass slide that will cause a reversal in the fringe pattern.
Solution:	d = = = d = 6.3 ´ 10–5 cm, or 6.3 ´ 10–4 mm

Let us illustrate the sensitivity of this device with one more example. The index of refraction of fluids depends on the temperature, among other things. If we fill a cell of length L = 10 cm with water and insert this cell into the test volume between M₂P₂ and an identical air-filled cell into the path M₁P₁ of Figure 1, and raise the water temperature by 1/20°C, we would observe a full shift of one fringe corresponding to an optical path length change D nL of l (l = 632.8 nm). Since 1/10 fringe shift can be accurately measured, it is possible to measure temperature changes of water by as little as 1/200°C. Thus, the interferometer can act as a very sensitive thermometer.

Before turning our attention to the more technical aspects of the Mach-Zehnder interferometer, such as its construction, alignment, and use, let us discuss one final but very important characteristic of this device. If, instead of the well-collimated beam which is essentially derived from a point source, we had to work with an extended source of light, fringes may still be observed. But now they will be sharp only in a particular plane. The reason for this is that instead of the two coherent point images S¢ and S² in Figure 2, we now have many pairs of coherent points, covering the whole extent of the images centered about S¢ and S² .

It is possible to show that for extended sources, fringes are located in the region in which the two emerging rays originating from the same incident may meet. This principle is illustrated in Figure 3a where mirror M₂ has been rotated counterclockwise by a small amount.

Fig. 3
Fringe localization with extended sources

It can be seen that only the reflected beam to P₂ will be affected. The reference beam propagating through the outer part of the interferometer is not affected and arrives at P₂ as before. The beams of light indicated by rays 1 and 2 emerging from P₂ will no longer intersect. However, by extending both rays back through beam splitter P₂, as is done in Figure 3a, it can be seen that the virtual rays intersect at a point between M₂ and P₂. The region where these virtual rays meet becomes the virtual region of fringe localization. This region of fringe localization can be moved anywhere between mirror M₂ and beam splitter P₂ by simply rotating P₂ clockwise a small amount. The effect of a small rotation of P₂ on the virtual location of the fringes is shown in Figure 3b. In this way, one can actually place the fringes at infinity or before or behind beam splitter P₂.

This flexibility in fringe localization gives the Mach-Zehnder interferometer one very important advantage over many other interferometers. For instance, when the Mach-Zehnder interferometer is used for the examination of airflow around a section of an aircraft wing, a model of the wing is enclosed in a test cell that is located in one leg of the interferometer. Once the model is in position, a high-velocity airstream is blown past it. The airstream forms a pressure pattern around the model which causes local changes in the index of refraction which, in turn, can be measured from the resultant change in the fringe pattern.

For this kind of experimental work, it is necessary to photograph both the interference fringes and test model simultaneously so that both are in focus. Therefore, it is necessary that the fringes be located (or localized) in the region where the test object is located. The plane where the object and the fringes more or less coincide can now be imaged on a screen or on a photographic plate.

In the case of a wind tunnel or shock-wave tube experiment, photographs of the fringe pattern are obtained with and without gas flow. The change of fringes, that is, the number of times the intensity at a selected point P changes from say bright to dark to bright, is counted. This number, which is designated by D m, is given by Equation 1.

Candle

Although many Mach-Zehnder interferometers use a rectangular arrangement, the parallelogram arrangement shown in Figure 4 has certain advantages. For one, the effect of imperfections in the optical flats used for the 50% reflecting mirrors decreases with decreasing angle of incidence of the light beams on these flats. Another reason is that the aperture, or useful field width to mirror width, increases as the angle of incidence is reduced. We will select a parallelogram arrangement with an acute angle of 60° , giving an angle of incidence of 30° .

A typical measurement for air at STP with a test cell of L = 5 cm, l = 632.8 nm, and D m = 21 resulted in an index of refraction of 1.000253. This procedure can be repeated at several pressures and a plot of refractive index versus pressure obtained.

The final step in this experiment consists of an examination of the fringe pattern at the specimen. Depending on the polishing imperfections, the fringe pattern may resemble a contour map. The difference in the thickness D L from one point to another point on the specimen is determined by counting the number of fringes D m between these two points and using Equation 3.