Mach-Zehnder interferometer

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Mach-Zehnder interferometer.

The Mach-Zehnder interferometer (named after physicists Ernst Mach and Ludwig Zehnder) is used to determine the phase shift caused by a small sample which is to be placed into one of the two collimated beams (thus having plane wavefronts) (then called sample beam (SB) as opposed to the reference beam (RB)) from a coherent light source.

There are - in contrast to the Michelson interferometer - two detectors: 1 and 2.

[edit] Function

A coherent beam is split up by a half-silvered mirror and each one is reflected by a mirror. The two beams pass a second half-silvered mirror and enter detector 1 and 2, respectively.

There are some simple rules for phase shifts due to material (i.e. non-vacuum, which has a refractive index of exactly n = 1):

reflection or refraction at a surface behind which is a medium with lower n causes NO phase shift
reflection at a surface behind which is a medium with higher n causes a phase shift of half a wavelength, or $π$ radians
the speed of light is slower in media with n > 1, specifically, its speed is: $v = \frac{c}{n}$ , where c is the speed of light in vacuum. This causes a phase shift proportional to n * length_traveled.

Given the above rules, mirrors, including half-silvered mirrors, have the following properties:

a ½ wavelength phase shift occurs upon reflection from the front of a mirror, since the medium behind the mirror (glass) has a higher refractive index than the medium the light is traveling in (air).
let k be the constant phase shift incurred by passing through a standard glass plate on which a mirror resides
a total of 2k phase shift occurs when reflecting off the rear of a mirror, since light traveling toward the rear of a mirror will enter the glass plate, incurring k phase shift, and then reflect off the mirror with no additional phase shift since only air is now behind the mirror, and travel again back through the glass plate incurring an additional k phase shift.

This effect of a sample can be measured with this setup as every slab of material will change the initial situation. Without a sample there is no phase difference for the two beams in detector 1, thus yielding constructive interference: both have incurred wavelength + k phase shift due to two front-side reflections and one transmission through a glass plate. On the other hand, at detector 2 there is complete destructive interference: the lower route beam has experienced ½ wavelength + 2k phase shift for its single front-side reflection and two transmissions through a glass plate, whereas the upper route beam has incurred 1 wavelength + 2k phase shift for its two front-side reflections and one rear-side reflection, thus yielding a phase difference of exactly half a wavelength, implying that the crest and troughs of the two waves cancel. Therefore, when there is no sample, only detector 1 receives light. If a sample is now placed into beam SB, there will be a variation in the intensities for 1 and 2, which allows the calculation of the phase shift.