In physics, 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. The apparatus is named after the physicists Ludwig Mach (the son of Ernst Mach) and Ludwig Zehnder: Zehnder's proposal in an 1891 article was refined by Mach in an 1892 article.
The Mach–Zehnder interferometer is a highly configurable instrument. In contrast to the well-known Michelson interferometer, each of the well separated light paths is traversed only once.
If it is decided to produce fringes in white light, then, since white light has a limited coherence length, on the order of micrometers, great care must be taken to simultaneously equalize the optical paths over all wavelengths or no fringes will be visible. As seen in Fig. 1, a compensating cell made of the same type of glass as the test cell (so as to have equal optical dispersion) would be placed in the path of the reference beam to match the test cell. Note also the precise orientation of the beam splitters. The reflecting surfaces of the beam splitters would be oriented so that the test and reference beams pass through an equal amount of glass. 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 traveling an equal optical path length in the test and reference beams produces a white light fringe of constructive interference.
Collimated sources result in a nonlocalized fringe pattern. Localized fringes result when an extended source is used. In Fig. 2, we see that the fringes can be adjusted so that they are localized in any desired plane.:18 In most cases, the fringes would be adjusted to lie in the same plane as the test object, so that fringes and test object can be photographed together.
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 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.:18,93–95
Mach–Zehnder interferometers are used in electro-optic modulators, electronic devices used in various fibre-optic communications 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 GHz frequency range.
How it works
A 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 fully silvered and half-silvered surfaces of all mirrors, except the last, face the inbound beam, and the half-silvered surface of the last mirror faces the outbound beam exiting in the same orientation as the original collimated beam. That is, if the original beam is horizontal, the half-silvered surface of the last mirror should face the horizontally outbound beam.
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 reflects off a change from low to high refractive index but not when it reflects off a change from high to low.
In other words:
- A 180 degree 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).
We also note that:
- The speed of light is slower in media with an index of refraction greater than that of a vacuum, which is 1. Specifically, its speed is: v = 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 − 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 off 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 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.
Caveat: The rule about phase shifts applies to beamsplitters constructed with a dielectric coating, and must be modified if a metallic coating is used, or when different 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 note that beamsplitters that are not 50/50 are frequently employed to improve the interferometer's performance in certain types of measurement.
Observing the effect of a sample
In Fig. 3, in the absence of a sample, both the sample beam SB and the reference beam RB will arrive in phase at detector 1, yielding constructive interference. 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.
At detector 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 2 will have undergone a phase shift of 0.5×(wavelength) + 2k due to one front-surface reflection and two transmissions. The SB arriving at detector 2 will have undergone a (1×wavelength + 2k) phase shift due to two front-surface reflections and one rear-surface reflection. Therefore, when there is no sample, only detector 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.
Use of the Mach–Zehnder interferometer
The versatility of the Mach–Zehnder configuration has led to its being used in a wide range of fundamental research topics in quantum mechanics, including studies on counterfactual definiteness, quantum entanglement, quantum computation, quantum cryptography, quantum logic, the quantum eraser experiment, the quantum Zeno effect, and neutron diffraction. See their respective articles for further information on these topics. Some other examples are:
Following the analysis given above, we see that in Fig. 4a and b, 0% of the output impinges upon detector A. This is true even if the light intensity is reduced so that only one photon at a time travels through the apparatus. The explanation for this is the same as for the double-slit experiment: the wave of each individual photon travels both paths and has destructive interference in detector A. In Fig. 4a, the photon is illustrated as having taken the "northern" branch of the interferometer and both waves interferes so that only detector B detects the photon. The same holds for Fig. 4b, where the photon is considered to have taken the "southern" branch of the interferometer.
Fig. 4c illustrates the situation where an obstacle has been introduced on the "southern" branch of the interferometer. 50% of the photons are deflected by mirror M, and the remaining photons are split 25% to detector A, and 25% to detector B.
In Fig. 4d an individual photon arriving at detector A has traversed the "northern" path and while its wave through the southern path was intercepted by mirror M. There is no destructive interference anymore, so the photon can be detected in A. This is an example of counterfactual measurement in quantum physics.
Elitzur–Vaidman bomb tester
Elitzur-Vaidman bomb tester is a version of the above experiment. It is a thought experiment in quantum mechanics proposed by Avshalom Elitzur and Lev Vaidman in 1993. An actual experiment was successfully tested in 1995.
In the above experiment mirror M is replaced by a bomb with a photon detector as detonator. In some duds this detonator is not present. Now if a photon is detected in A, the bomb did not pass the wave, so had a detonator. The presence of the detonator was tested without exploding the bomb. Of course still 50% of the good bombs will explode (by particles taking the southern path) and it gives no information if the detonator is present but defect.
By repeating the process with the unknowns, the ratio of surviving, identified, usable bombs approaches 33% of the initial population of usable bombs. Alternative approaches to performing interaction-free measurements have been devised with much higher bomb detection efficiencies. In principle, it should be possible to achieve usable bomb detection efficiencies approaching 100%.
- List of types of interferometers
Related forms of interferometer
Other flow visualisation techniques
- Zehnder, Ludwig (1891). "Ein neuer Interferenzrefraktor". Zeitschrift für Instrumentenkunde 11: 275–285.
- Mach, Ludwig (1892). "Ueber einen Interferenzrefraktor". Zeitschrift für Instrumentenkunde 12: 89–93.
- Zetie, K.P.; Adams, S.F.; Tocknell, R.M. "How does a Mach–Zehnder interferometer work?". Physics Department, Westminster School, London. Retrieved 8 April 2012.
- Ashkenas, Harry I. (1950). The design and construction of a Mach-Zehnder interferometer for use with the GALCIT Transonic Wind Tunnel. Engineer's thesis. California Institute of Technology.
- Hariharan, P. (2007). Basics of Interferometry. Elsevier Inc. ISBN 0-12-373589-0.
- Chevalerias, R.; Latron, Y.; Veret, C. (1957). "Methods of Interferometry Applied to the Visualization of Flows in Wind Tunnels". Journal of the Optical Society of America 47 (8): 703. doi:10.1364/JOSA.47.000703.
- Ristić, Slavica. "Flow visualization techniques in wind tunnels – optical methods (Part II)". Military Technical Institute, Serbia. Retrieved 6 April 2012.
- Paris, M.G.A. (1999). "Entanglement and visibility at the output of a Mach-Zehnder interferometer". Physical Review A 59 (2): 1615–1621. arXiv:quant-ph/9811078. Bibcode:1999PhRvA..59.1615P. doi:10.1103/PhysRevA.59.1615. Retrieved 2 April 2012.
- Haack, G. R.; Förster, H.; Büttiker, M. (2010). "Parity detection and entanglement with a Mach-Zehnder interferometer". Physical Review B 82 (15). arXiv:1005.3976. Bibcode:2010PhRvB..82o5303H. doi:10.1103/PhysRevB.82.155303.
- Elitzur, A. C.; Vaidman, L (1993). "Quantum-Mechanical Interaction-free Measurements". Foundations of Physics 23: 987–997. arXiv:hep-th/9305002. Bibcode:1993FoPh...23..987E. doi:10.1007/BF00736012. Retrieved 16 March 2013.
- Kwait, P.; Weinfurter, H.; Herzog, T.; Zeilinger, A.; Kasevich, M. (1995). "Experimental Realization of Interaction-free Measurements". Annals of the New York Academy of Sciences 755: 383–393. doi:10.1111/j.1749-6632.1995.tb38981.x.
- Kwiat, P. G. (1998). "Experimental and Theoretical Progress in Interaction-free Measurements". Physica Scripta: 115. doi:10.1238/Physica.Topical.076a00115.