Mach-Zehnder Interferometer

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Mach-Zehnder interferometerThe Mach-Zehnder interferometer, invented over one hundred years ago, is still used for many optical measurements. The "Mach" is the same man who proposed Mach's Principle and for whom a unit for the measurement of the speed of sound is named. Here we describe the details of how a simple version of the interferometer works; the discussion is largely non-mathematical but somewhat lengthy. A figure of the interferometer appears to the right.

The "legend" for the figure is:

Light source: Light source Mirror: Mirror
Detector: {short description of image} Half-silvered mirror: Half-silvered mirror

It turns out that, despite the figure, all of the light from the source ends up at detector 1; no light gets to detector 2. We will prove that this is so.

First, we will need to know the following facts from optics

Note we have labelled the two detectors 1 and 2, and have labelled the upper path of the light U and the down path of the light D. We consider the two paths for light arriving at detector 1:

Path "U":
  1. Reflected by the front of the first beam splitter, giving a phase change of one-half a wavelength.
  2. Reflected by the upper-left mirror, giving a further phase change of one-half a wavelength.
  3. Transmitted through the upper-right beam splitter, giving some constant phase change.
Path "D":
  1. Transmitted through the lower-left beam splitter, giving some constant phase change.
  2. Reflected by the front of the lower-right mirror, giving a phase change of one-half a wavelength.
  3. Reflected by the front of the second beam splitter, giving a phase change of one-half a wavelength.

Adding up all the contributions for the two paths, we see that they are the same. Thus light entering detector 1 via the two paths is in phase. Thus we get constructive interference for the light entering detector 1.

Now we consider light entering detector 2:

Path "U":
  1. Reflected by the front of the first beam splitter, giving a phase change of one-half a wavelength.
  2. Reflected by the upper-left mirror, giving a further phase change of one-half a wavelength.
  3. Transmitted through the second beam splitter, giving some constant phase change.
  4. Reflected by the inner surface of the second beam splitter, giving no phase change.
  5. Transmitted through the beam splitter a second time, giving an additional constant phase change.
Path "D":
  1. Transmitted through the lower-left beam splitter, giving some constant phase change.
  2. Reflected by the front of the lower-right mirror, giving a phase change of one-half a wavelength.
  3. Transmitted through the second beam splitter, giving some constant phase change.

Adding up all these, we see that the total difference between the two paths is that the U path has gone through one additional phase change of one-half a wavelength. Therefore, there will be complete destructive interference, and no light will reach detector 2.

Thus we have proved that, regardless of the wavelength of the light, it all goes to detector 1.

The interferometer is used to measure the phase shift of a thin sample of, say, glass. The sample is placed in either the U or D beam. The phase shift of the sample alters the phase relationships between the two beams that we have just described, and there is no longer complete destructive interference at detector 2. Measuring the relative amount of light entering detector 1 and detector 2 allows a calculation of the phase shift produced by the sample.


This document is Copyright 1999 © David M. Harrison. This is version 1.2, date (m/d/y) 08/17/05.

Creative Commons License This work is licensed under a Creative Commons License.