Mach–Zehnder interferometer
The Mach–Zehnder interferometer is a device used to determine the relative phase shift variations between two collimated beams derived by splitting light from a single source.
The interferometer has been used, among other things, to measure phase shifts between the two beams caused by a sample or a change in length of one of the paths.
[3] The versatility of the Mach–Zehnder configuration has led to its being used in a range of research topics efforts especially in fundamental quantum mechanics.
In contrast to the well-known Michelson interferometer, each of the well-separated light paths is traversed only once.
If the source has a low coherence length then great care must be taken to equalize the two optical paths.
In this orientation, the test and reference beams each experience two front-surface reflections, resulting in the same number of phase inversions.
The result is that light travels through an equal optical path length in both the test and reference beams leading to constructive interference.
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).
The speed of light is lower in media with an index of refraction greater than that of a vacuum, which is 1.
This causes a phase shift increase proportional to (n − 1) × length traveled.
If k is the constant phase shift incurred by passing through a glass plate on which a mirror resides, a total of 2k 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.
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.
Both SB and RB will have undergone a phase shift of (1 × wavelength + k) due to two front-surface reflections and one transmission through a glass plate.
The RB arriving at detector 2 will have undergone a phase shift of (0.5 × wavelength + 2k) due to one front-surface reflection and two transmissions.
We can model a photon going through the interferometer by assigning a probability amplitude to each of the two possible paths: the "lower" path which starts from the left, goes straight through both beam splitters, and ends at the top, and the "upper" path which starts from the bottom, goes straight through both beam splitters, and ends at the right.
, which means that when a photon meets the beam splitter it will either stay on the same path with a probability amplitude of
The phase shifter on the upper arm is modelled as the unitary matrix
, which means that if the photon is on the "upper" path it will gain a relative phase of
A photon that enters the interferometer from the left will then end up described by the state and the probabilities that it will be detected at the right or at the top are given respectively by One can therefore use the Mach–Zehnder interferometer to estimate the phase shift by estimating these probabilities.
It is interesting to consider what would happen if the photon were definitely in either the "lower" or "upper" paths between the beam splitters.
This can be accomplished by blocking one of the paths, or equivalently by removing the first beam splitter (and feeding the photon from the left or the bottom, as desired).
[7] The Mach–Zehnder interferometer's relatively large and freely accessible working space, and its flexibility in locating the fringes has made it the interferometer of choice for visualizing flow in wind tunnels[8][9] and for flow visualization studies in general.
It is frequently used in the fields of aerodynamics, plasma physics and heat transfer to measure pressure, density, and temperature changes in gases.
[6]: 18, 93–95 Mach–Zehnder interferometers are used in electro-optic modulators, electronic devices used in various fiber-optic communication applications.
Mach–Zehnder modulators are incorporated in monolithic integrated circuits and offer well-behaved, high-bandwidth electro-optic amplitude and phase responses over a multiple-gigahertz frequency range.
In particular, optical heterodyne detection with an off-axis, frequency-shifted reference beam ensures good experimental conditions for shot-noise limited holography with video-rate cameras,[12] vibrometry,[13] and laser Doppler imaging of blood flow.
