Mach–Zehnder interferometer

Last updated
Figure 1. The Mach-Zehnder interferometer is frequently used in the fields of aerodynamics, plasma physics and heat transfer to measure pressure, density, and temperature changes in gases. In this figure, we imagine analyzing a candle flame. Either output image may be monitored. Mach Zehnder interferometer alternate candle images.svg
Figure 1. The MachZehnder interferometer is frequently used in the fields of aerodynamics, plasma physics and heat transfer to measure pressure, density, and temperature changes in gases. In this figure, we imagine analyzing a candle flame. Either output image may be monitored.

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 [1] was refined by Mach in an 1892 article. [2] Demonstrations of Mach-Zehnder interferometry with particles other than photons (particles of light) had been demonstrated as well in multiple experiments. [3]



The MachZehnder check 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 the source has a low coherence length then great care must be taken to equalize the two optical paths. White light in particular requires the optical paths to be simultaneously equalized 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 travels through an equal optical path length in both the test and reference beams leading to constructive interference. [4] [5]

Figure 2. Localized fringes result when an extended source is used in a Mach-Zehnder interferometer. By appropriately adjusting the mirrors and beam splitters, the fringes can be localized in any desired plane. Mach-Zender interferometer fringe localization.svg
Figure 2. Localized fringes result when an extended source is used in a Mach–Zehnder interferometer. By appropriately adjusting the mirrors and beam splitters, the fringes can be localized in any desired plane.

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. [6] :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 MachZehnder 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 [7] [8] 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] :18,9395

MachZehnder 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.

MachZehnder interferometers are also used to study one of the most counterintuitive predictions of quantum mechanics, the phenomenon known as quantum entanglement. [9] [10]

The possibility to easily control the features of the light in the reference channel without disturbing the light in the object channel popularized the MachZehnder configuration in holographic interferometry. 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, [11] vibrometry, [12] and laser Doppler imaging of blood flow. [13]

Method of operation


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 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).

Figure 3. Effect of a sample on the phase of the output beams in a Mach-Zehnder interferometer Mach-zender-interferometer.png
Figure 3. Effect of a sample on the phase of the output beams in a Mach–Zehnder interferometer

The speed of light is lower 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 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 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. [4]

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, one rear-surface reflection and two transmissions. 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.

Quantum treatment

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. The quantum state describing the photon is therefore a vector that is a superposition of the "lower" path and the "upper" path , that is, for complex such that .

Both beam splitters are modelled as the unitary matrix , which means that when a photon meets the beam splitter it will either stay on the same path with a probability amplitude of , or be reflected to the other 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 , and it will stay unchanged if it is on the lower path.

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). In both cases there will no longer be interference between the paths, and the probabilities are given by , independently of the phase . From this we can conclude that the photon does not take one path or another after the first beam splitter, but rather that it must be described by a genuine quantum superposition of the two paths. [14]


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, Elitzur–Vaidman bomb tester, the quantum eraser experiment, the quantum Zeno effect, and neutron diffraction. In optical telecommunications it is used as an electro-optic modulator for phase and amplitude modulation of light.

See also

Other flow visualisation techniques

Related Research Articles

Wave interference Phenomenon

In physics, interference is a phenomenon in which two waves superpose to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive interference result from the interaction of waves that are correlated or coherent with each other, either because they come from the same source or because they have the same or nearly the same frequency. Interference effects can be observed with all types of waves, for example, light, radio, acoustic, surface water waves, gravity waves, or matter waves. The resulting images or graphs are called interferograms.

In optics, polarized light can be described using the Jones calculus, discovered by R. C. Jones in 1941. Polarized light is represented by a Jones vector, and linear optical elements are represented by Jones matrices. When light crosses an optical element the resulting polarization of the emerging light is found by taking the product of the Jones matrix of the optical element and the Jones vector of the incident light. Note that Jones calculus is only applicable to light that is already fully polarized. Light which is randomly polarized, partially polarized, or incoherent must be treated using Mueller calculus.

Quantum mechanics Branch of physics describing nature on an atomic scale

Quantum mechanics is a fundamental theory in physics that provides a description of the physical properties of nature at the scale of atoms and subatomic particles. It is the foundation of all quantum physics including quantum chemistry, quantum field theory, quantum technology, and quantum information science.

Interferometry Measurement method using interference of waves

Interferometry is a technique in which waves, usually electromagnetic waves, are superimposed, causing the phenomenon of interference, which is used to extract information. Interferometry is an important investigative technique in the fields of astronomy, fiber optics, engineering metrology, optical metrology, oceanography, seismology, spectroscopy, quantum mechanics, nuclear and particle physics, plasma physics, remote sensing, biomolecular interactions, surface profiling, microfluidics, mechanical stress/strain measurement, velocimetry, optometry, and making holograms.

In physics, two wave sources are perfectly coherent if their frequency and waveform are identical and their phase difference is constant. Coherence is an ideal property of waves that enables stationary interference. It contains several distinct concepts, which are limiting cases that never quite occur in reality but allow an understanding of the physics of waves, and has become a very important concept in quantum physics. More generally, coherence describes all properties of the correlation between physical quantities of a single wave, or between several waves or wave packets.

Michelson interferometer

The Michelson interferometer is a common configuration for optical interferometry and was invented by Albert Abraham Michelson. Using a beam splitter, a light source is split into two arms. Each of those light beams is reflected back toward the beamsplitter which then combines their amplitudes using the superposition principle. The resulting interference pattern that is not directed back toward the source is typically directed to some type of photoelectric detector or camera. For different applications of the interferometer, the two light paths can be with different lengths or incorporate optical elements or even materials under test.

Beam splitter

A beam splitter is an optical device that splits a beam of light in two. It is a crucial part of many optical experimental and measurement systems, such as interferometers, also finding widespread application in fibre optic telecommunications.

In physics, the Hanbury Brown and Twiss (HBT) effect is any of a variety of correlation and anti-correlation effects in the intensities received by two detectors from a beam of particles. HBT effects can generally be attributed to the wave–particle duality of the beam, and the results of a given experiment depend on whether the beam is composed of fermions or bosons. Devices which use the effect are commonly called intensity interferometers and were originally used in astronomy, although they are also heavily used in the field of quantum optics.

Sagnac effect

The Sagnac effect, also called Sagnac interference, named after French physicist Georges Sagnac, is a phenomenon encountered in interferometry that is elicited by rotation. The Sagnac effect manifests itself in a setup called a ring interferometer. A beam of light is split and the two beams are made to follow the same path but in opposite directions. On return to the point of entry the two light beams are allowed to exit the ring and undergo interference. The relative phases of the two exiting beams, and thus the position of the interference fringes, are shifted according to the angular velocity of the apparatus. In other words, when the interferometer is at rest with respect to a nonrotating frame, the light takes the same amount of time to traverse the ring in either direction. However, when the interferometer system is spun, one beam of light has a longer path to travel than the other in order to complete one circuit of the mechanical frame, and so takes longer, resulting in a phase difference between the two beams. This arrangement is also called a Sagnac interferometer. Georges Sagnac set up this experiment to prove the existence of the aether that Einstein's theory of special relativity had discarded.

Time-bin encoding is a technique used in quantum information science to encode a qubit of information on a photon. Quantum information science makes use of qubits as a basic resource similar to bits in classical computing. Qubits are any two-level quantum mechanical system; there are many different physical implementations of qubits, one of which is time-bin encoding.

The theoretical and experimental justification for the Schrödinger equation motivates the discovery of the Schrödinger equation, the equation that describes the dynamics of nonrelativistic particles. The motivation uses photons, which are relativistic particles with dynamics described by Maxwell's equations, as an analogue for all types of particles.

In quantum mechanics, a weak value is a quantity related to a shift of a measuring device's pointer when usually there is pre- and postselection. It should not be confused with a weak measurement, which is often defined in conjunction. The weak value was first defined by Yakir Aharonov, David Albert and Lev Vaidman, published in Physical Review Letters 1988, and is related to the two-state vector formalism. There is also a way to obtain weak values without postselection.

Quantum mechanics was first applied to optics, and interference in particular, by Paul Dirac. Richard Feynman, in his Lectures on Physics, uses Dirac's notation to describe thought experiments on double-slit interference of electrons. Feynman's approach was extended to N-slit interferometers for either single-photon illumination, or narrow-linewidth laser illumination, that is, illumination by indistinguishable photons, by Frank Duarte. The N-slit interferometer was first applied in the generation and measurement of complex interference patterns.

The closure phase is an observable quantity in imaging astronomical interferometry, which allowed the use of interferometry with very long baselines. It forms the basis of the self-calibration approach to interferometric imaging. The observable which is usually used in most "closure phase" observations is actually the complex quantity called the triple product. The closure phase is the phase of this complex quantity, but the phrase "closure phase" is still more commonly used than the more accurate phrase "triple product".

A common-path interferometer is a class of interferometers in which the reference beam and sample beams travel along the same path. Examples include the Sagnac interferometer, Zernike phase-contrast interferometer, and the point diffraction interferometer. A common-path interferometer is generally more robust to environmental vibrations than a "double-path interferometer" such as the Michelson interferometer or the Mach–Zehnder interferometer. Although travelling along the same path, the reference and sample beams may travel along opposite directions, or they may travel along the same direction but with the same or different polarization.

White light interferometry

As described here, white light interferometry is a non-contact optical method for surface height measurement on 3-D structures with surface profiles varying between tens of nanometers and a few centimeters. It is often used as an alternative name for coherence scanning interferometry in the context of areal surface topography instrumentation that relies on spectrally-broadband, visible-wavelength light.

Linear Optical Quantum Computing or Linear Optics Quantum Computation (LOQC) is a paradigm of quantum computation, allowing universal quantum computation. LOQC uses photons as information carriers, mainly uses linear optical elements, or optical instruments to process quantum information, and uses photon detectors and quantum memories to detect and store quantum information.

The KLM scheme or KLM protocol is an implementation of linear optical quantum computing (LOQC), developed in 2000 by Knill, Laflamme and Milburn. This protocol makes it possible to create universal quantum computers solely with linear optical tools. The KLM protocol uses linear optical elements, single photon sources and photon detectors as resources to construct a quantum computation scheme involving only ancilla resources, quantum teleportations and error corrections.

Self-mixing or back-injection laser interferometry is an interferometric technique in which a part of the light reflected by a vibrating target is reflected into the laser cavity, causing a modulation both in amplitude and in frequency of the emitted optical beam. In this way, the laser becomes sensitive to the distance traveled by the reflected beam thus becoming a distance, speed or vibration sensor. The advantage compared to a traditional measurement system is a lower cost thanks to the absence of collimation optics and external photodiodes.


  1. Zehnder, Ludwig (1891). "Ein neuer Interferenzrefraktor". Zeitschrift für Instrumentenkunde. 11: 275–285.
  2. Mach, Ludwig (1892). "Ueber einen Interferenzrefraktor". Zeitschrift für Instrumentenkunde. 12: 89–93.
  3. Ji, Yang; Chung, Yunchul; Sprinzak, D.; Heiblum, M.; Mahalu, D.; Shtrikman, Hadas (March 2003). "An electronic Mach–Zehnder interferometer". Nature . 422 (6930): 415–418. arXiv: cond-mat/0303553 . doi:10.1038/nature01503. ISSN   0028-0836.
  4. 1 2 Zetie, K. P.; Adams, S. F.; Tocknell, R. M. "How does a Mach–Zehnder interferometer work?" (PDF). Physics Department, Westminster School, London. Retrieved 8 April 2012.
  5. 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.
  6. 1 2 Hariharan, P. (2007). Basics of Interferometry. Elsevier Inc. ISBN   978-0-12-373589-8.
  7. 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.
  8. Ristić, Slavica. "Flow visualization techniques in wind tunnels – optical methods (Part II)" (PDF). Military Technical Institute, Serbia. Retrieved 6 April 2012.
  9. Paris, M. G. A. (1999). "Entanglement and visibility at the output of a Mach–Zehnder interferometer" (PDF). Physical Review A. 59 (2): 1615–1621. arXiv: quant-ph/9811078 . Bibcode:1999PhRvA..59.1615P. doi:10.1103/PhysRevA.59.1615. Archived from the original (PDF) on 10 September 2016. Retrieved 2 April 2012.
  10. Haack, G. R.; Förster, H.; Büttiker, M. (2010). "Parity detection and entanglement with a Mach-Zehnder interferometer". Physical Review B. 82 (15): 155303. arXiv: 1005.3976 . Bibcode:2010PhRvB..82o5303H. doi:10.1103/PhysRevB.82.155303.
  11. Michel Gross; Michael Atlan (2007). "Digital holography with ultimate sensitivity". Optics Letters. 32 (8): 909–911. arXiv: 0803.3076 . Bibcode:2007OptL...32..909G. doi:10.1364/OL.32.000909.
  12. Francois Bruno; Jérôme Laurent; Daniel Royer; Michael Atlan (2014). "Holographic imaging of surface acoustic waves". Applied Physics Letters. 104 (1): 083504. arXiv: 1401.5344 . Bibcode:2014ApPhL.104a3504Y. doi:10.1063/1.4861116.
  13. Caroline Magnain; Amandine Castel; Tanguy Boucneau; Manuel Simonutti; Isabelle Ferezou; Armelle Rancillac; Tania Vitalis; José-Alain Sahel; Michel Paques; Michael Atlan (2014). "Holographic imaging of surface acoustic waves". Journal of the Optical Society of America A. 31 (12): 2723–2735. arXiv: 1412.0580 . Bibcode:2014JOSAA..31.2723M. doi:10.1364/JOSAA.31.002723. PMID   25606762.
  14. Vedral, Vlatko (2006). Introduction to Quantum Information Science. Oxford University Press. ISBN   9780199215706. OCLC   442351498.