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.
- 1 Introduction
- 2 How it works
- 3 Quantum paradoxes
- 4 See also
- 5 References
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.
The Mach–Zehnder interferometer has been used for demonstrating various highly counterintuitive predictions of quantum mechanics. We will begin by describing an experiment that at first seems to represent but a modest take-off on Young's experiment, but which leads directly into deeper results that have been at the core of the so-called "second quantum revolution".
Following the analysis given above, we see that in Fig. 4a and b, 100% of the output impinges upon detector B. 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: it appears as if the wave function of each individual photon travels both paths and engages in interference at the last beam splitter, so that only the wave to B is constructive. In Fig. 4a, although the photon is illustrated as having taken the "northern" branch of the interferometer, it interferes with itself so that only detector B detects the photon. The same holds for Fig. 4b, although 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.
Fig. 4d illustrates the fundamental paradox raised by this demonstration. An individual photon arriving at detector A must have traversed the "northern" path and could not have interacted with mirror M. The arrival of a photon at detector A constitutes proof that an obstacle exists on the "southern" path, but there is no exchange of energy between the photon and obstacle M. How could the photon possibly have acquired information about M without an exchange of energy? This is an example of counterfactual measurement in quantum physics.
Elitzur–Vaidman bomb tester
The above analysis, with only trivial changes, is directly applicable to understanding the Elitzur-Vaidman bomb tester, which is a thought experiment in quantum mechanics, first proposed by Avshalom Elitzur and Lev Vaidman in 1993.
An actual experiment demonstrating the solution was constructed and successfully tested in 1995.
Consider a collection of bombs, of which some, but not all, are duds. Suppose that these bombs possess certain properties: usable (non-dud) bombs have a photon-triggered fuse, which will absorb an incident photon and detonate the bomb. Dud bombs have a malfunctioning fuse, which will not interact with the photons in any way. Thus, the dud bomb will not detect the photon and will not detonate. The problem is how to separate at least some of the usable bombs from the duds. A bomb sorter could accumulate dud bombs by attempting to detonate each one. Unfortunately, this naive process destroys all the usable bombs.
As before, the light source is attenuated so that single photons enter the apparatus. The photon sensor of the bomb being tested is set in the sample beam as shown in Fig. 5.
A photon arriving at detector B tells us nothing about whether a particular bomb is usable or a dud. As shown in Fig. 5a, the photon could have arrived at B via the northern route, bypassing the bomb completely, or, as shown in Fig. 5b, it could have arrived at B via the southern route, with the bomb being a dud, its sensor nonfunctional.
An explosion, as shown in Fig. 5c, tells us that the photon had traveled the southern route and that the bomb was usable, but of course was destroyed by the explosion.
As shown in Fig. 5d, Detection of a photon at detector A can only occur if a photon travels the northern route with a usable bomb blocking the southern route. Receipt of this photon provides us information about the bomb without any interaction having occurred. Using this protocol, up to 25% of the usable bombs can be identified on the first pass (i.e. a photon will have been received at A), 50% will be detonated, and 25% will remain unknown (i.e. a photon will have been received at B). 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%.
Other uses of the Mach–Zehnder interferometer in testing quantum mechanics
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.
- List of types of interferometers
Related forms of interferometer
Other flow visualisation techniques
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- 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.
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- 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.
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