Editing Mach–Zehnder interferometer (section)

== Operation ==
[[file:Mach-zender-interferometer.png|right|thumb|350px|Figure 3. Effect of a sample on the phase of the output beams in a Mach–Zehnder interferometer]]

The collimated beam is split by a [[half-silvered mirror]].  The two resulting beams (the "sample beam" and the "reference beam") are each reflected by a [[mirror]].  The two beams then pass a second half-silvered mirror and enter two detectors.

The [[Fresnel equations]] for reflection and transmission of a wave at a dielectric imply that there is a phase change for a reflection, when a wave propagating in a lower-[[refractive index]] medium reflects from a higher-refractive index medium, but not in the opposite case. A 180° 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). No phase shift accompanies a rear-surface reflection, since the medium behind the mirror (air) has a lower refractive index than the medium the light is traveling in (glass).

The speed of light is lower in media with an index of refraction greater than that of a vacuum, which is&nbsp;1.  Specifically, its speed is: ''v''&nbsp;=&nbsp;''c''/''n'', where ''c'' is [[the speed of light in vacuum]], and ''n'' is the index of refraction.  This causes a phase shift increase proportional to (''n''&nbsp;−&nbsp;1)&nbsp;×&nbsp;''length traveled''. If ''k'' is the constant phase shift incurred by passing through a glass plate on which a mirror resides, a total of 2''k'' phase shift occurs when reflecting from the rear of a mirror. This is because light traveling toward the rear of a mirror will enter the glass plate, incurring ''k'' phase shift, and then reflect from 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.

The rule about phase shifts applies to [[beamsplitter]]s constructed with a [[dielectric]] coating and must be modified if a metallic coating is used or when different [[Polarization (waves)|polarizations]] are taken into account. Also, in real interferometers, the thicknesses of the beamsplitters may differ, and the path lengths are not necessarily equal. Regardless, in the absence of absorption, conservation of energy guarantees that the two paths must differ by a half-wavelength phase shift. Also beamsplitters that are not 50/50 are frequently employed to improve the interferometer's performance in certain types of measurement.<ref name=Zetie/>

In Fig.&nbsp;3, in the absence of a sample, both the sample beam (SB) and the reference beam (RB) will arrive in phase at detector&nbsp;1, yielding constructive [[Interference (wave propagation)|interference]]. Both SB and RB will have undergone a phase shift of (1 &times; wavelength&nbsp;+&nbsp;''k'') due to two front-surface reflections and one transmission through a glass plate. At detector&nbsp;2, in the absence of a sample, the sample beam and reference beam will arrive with a phase difference of half a wavelength, yielding complete destructive interference. The RB arriving at detector&nbsp;2 will have undergone a phase shift of (0.5 &times; wavelength&nbsp;+&nbsp;2''k'') due to one front-surface reflection and two transmissions.  The SB arriving at detector&nbsp;2 will have undergone a (1 &times; wavelength&nbsp;+&nbsp;2''k'') phase shift due to two front-surface reflections, one rear-surface reflection.  Therefore, when there is no sample, only detector&nbsp;1 receives light. If a sample is placed in the path of the sample beam, the intensities of the beams entering the two detectors will change, allowing the calculation of the phase shift caused by the sample.