![Interference of two waves](https://upload.wikimedia.org/wikipedia/commons/0/0f/Interference_of_two_waves.svg)
In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, interference is a phenomenon in which two
wave
In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...
s combine by adding their displacement together at every single point in space and time, to form a resultant wave of greater, lower, or the same
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
. Constructive and destructive interference result from the interaction of waves that are correlated or
coherent
Coherence, coherency, or coherent may refer to the following:
Physics
* Coherence (physics), an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference
* Coherence (units of measurement), a deri ...
with each other, either because they come from the same source or because they have the same or nearly the same
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
. Interference effects can be observed with all types of waves, for example,
light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 tera ...
,
radio
Radio is the technology of signaling and communicating using radio waves. Radio waves are electromagnetic waves of frequency between 30 hertz (Hz) and 300 gigahertz (GHz). They are generated by an electronic device called a transmit ...
,
acoustic,
surface water waves,
gravity waves, or
matter wave
Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality. All matter exhibits wave-like behavior. For example, a beam of electrons can be diffracted just like a beam of light or a water wav ...
s.
Etymology
The word ''interference'' is derived from the
Latin
Latin (, or , ) is a classical language belonging to the Italic branch of the Indo-European languages. Latin was originally a dialect spoken in the lower Tiber area (then known as Latium) around present-day Rome, but through the power of the ...
words ''inter'' which means "between" and ''fere'' which means "hit or strike", and was coined by
Thomas Young in 1801.
Mechanisms
![Two sources interference](https://upload.wikimedia.org/wikipedia/commons/2/2c/Two_sources_interference.gif)
The
principle of superposition of waves states that when two or more propagating waves of the same type are incident on the same point, the resultant
amplitude
The amplitude of a periodic variable is a measure of its change in a single period (such as time or spatial period). The amplitude of a non-periodic signal is its magnitude compared with a reference value. There are various definitions of amplit ...
at that point is equal to the
vector sum
In mathematics, physics, and engineering, a Euclidean vector or simply a vector (sometimes called a geometric vector or spatial vector) is a geometric object that has magnitude (or length) and direction. Vectors can be added to other vectors a ...
of the amplitudes of the individual waves. If a
crest
Crest or CREST may refer to:
Buildings
*The Crest (Huntington, New York), a historic house in Suffolk County, New York
*"The Crest", an alternate name for 63 Wall Street, in Manhattan, New York
*Crest Castle (Château Du Crest), Jussy, Switzerla ...
of a wave meets a crest of another wave of the same frequency at the same point, then the amplitude is the sum of the individual amplitudes—this is constructive interference. If a crest of one wave meets a trough of another wave, then the amplitude is equal to the difference in the individual amplitudes—this is known as destructive interference. In ideal mediums (water, air are almost ideal) energy is always conserved, at points of destructive interference energy is stored in the elasticity of the medium. For example when we drop 2 pebbles in a pond we see a pattern but eventually waves continue and only when they reach the shore is energy absorbed away from the medium.
![Interference colours in soap film 1](https://upload.wikimedia.org/wikipedia/commons/2/20/Interference_colours_in_soap_film_1.jpg)
Constructive interference occurs when the
phase
Phase or phases may refer to:
Science
*State of matter, or phase, one of the distinct forms in which matter can exist
*Phase (matter), a region of space throughout which all physical properties are essentially uniform
* Phase space, a mathematic ...
difference between the waves is an even multiple of (180°), whereas destructive interference occurs when the difference is an odd multiple of . If the difference between the phases is intermediate between these two extremes, then the magnitude of the displacement of the summed waves lies between the minimum and maximum values.
Consider, for example, what happens when two identical stones are dropped into a still pool of water at different locations. Each stone generates a circular wave propagating outwards from the point where the stone was dropped. When the two waves overlap, the net displacement at a particular point is the sum of the displacements of the individual waves. At some points, these will be in phase, and will produce a maximum displacement. In other places, the waves will be in anti-phase, and there will be no net displacement at these points. Thus, parts of the surface will be stationary—these are seen in the figure above and to the right as stationary blue-green lines radiating from the centre.
Interference of light is a unique phenomenon in that we can never observe superposition of the EM field directly as we can for example in water. Superposition in the EM field is an assumed and necessary requirement, fundamentally 2 light beam pass through each other and continue on their respective paths. Light can be explained classically by the superposition of waves, however a deeper understanding of light interference requires knowledge of
wave-particle duality of light which is due to
quantum mechanics
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, ...
. Prime examples of light interference are the famous
double-slit experiment
In modern physics, the double-slit experiment is a demonstration that light and matter can display characteristics of both classically defined waves and particles; moreover, it displays the fundamentally probabilistic nature of quantum mechanics ...
,
laser speckle
Speckle, speckle pattern, or speckle noise is a granular noise texture degrading the quality as a consequence of interference among wavefronts in coherent imaging
Imaging is the representation or reproduction of an object's form; especially ...
,
anti-reflective coatings and
interferometers. Traditionally the classical wave model is taught as a basis for understanding optical interference, based on the
Huygens–Fresnel principle however an explanation based on the Feynman path integral exists which takes into account quantum mechanical considerations.
Derivation
The above can be demonstrated in one dimension by deriving the formula for the sum of two waves. The equation for the amplitude of a
sinusoidal wave traveling to the right along the x-axis is
where
is the peak amplitude,
is the
wavenumber
In the physical sciences, the wavenumber (also wave number or repetency) is the '' spatial frequency'' of a wave, measured in cycles per unit distance (ordinary wavenumber) or radians per unit distance (angular wavenumber). It is analogous to te ...
and
is the
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
of the wave. Suppose a second wave of the same frequency and amplitude but with a different phase is also traveling to the right
where
is the phase difference between the waves in
radian
The radian, denoted by the symbol rad, is the unit of angle in the International System of Units (SI) and is the standard unit of angular measure used in many areas of mathematics. The unit was formerly an SI supplementary unit (before tha ...
s. The two waves will
superpose and add: the sum of the two waves is
Using the
trigonometric identity
In trigonometry, trigonometric identities are Equality (mathematics), equalities that involve trigonometric functions and are true for every value of the occurring Variable (mathematics), variables for which both sides of the equality are defined. ...
for the sum of two cosines:
this can be written
This represents a wave at the original frequency, traveling to the right like its components, whose amplitude is proportional to the cosine of
.
* ''Constructive interference'': If the phase difference is an even multiple of :
then
, so the sum of the two waves is a wave with twice the amplitude
* ''Destructive interference'': If the phase difference is an odd multiple of :
then
, so the sum of the two waves is zero
Between two plane waves
![Interferences plane waves](https://upload.wikimedia.org/wikipedia/commons/d/d8/Interferences_plane_waves.jpg)
A simple form of interference pattern is obtained if two
plane wave
In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space.
For any position \vec x in space and any time t, ...
s of the same frequency intersect at an angle.
Interference is essentially an energy redistribution process. The energy which is lost at the destructive interference is regained at the constructive interference.
One wave is travelling horizontally, and the other is travelling downwards at an angle θ to the first wave. Assuming that the two waves are in phase at the point B, then the relative phase changes along the ''x''-axis. The phase difference at the point A is given by
It can be seen that the two waves are in phase when
and are half a cycle out of phase when
Constructive interference occurs when the waves are in phase, and destructive interference when they are half a cycle out of phase. Thus, an interference fringe pattern is produced, where the separation of the maxima is
and is known as the fringe spacing. The fringe spacing increases with increase in
wavelength
In physics, the wavelength is the spatial period of a periodic wave—the distance over which the wave's shape repeats.
It is the distance between consecutive corresponding points of the same phase on the wave, such as two adjacent crests, tro ...
, and with decreasing angle .
The fringes are observed wherever the two waves overlap and the fringe spacing is uniform throughout.
Between two spherical waves
![wavepanel](https://upload.wikimedia.org/wikipedia/commons/f/f1/Wavepanel.png)
A
point source
A point source is a single identifiable ''localised'' source of something. A point source has negligible extent, distinguishing it from other source geometries. Sources are called point sources because in mathematical modeling, these sources can ...
produces a spherical wave. If the light from two point sources overlaps, the interference pattern maps out the way in which the phase difference between the two waves varies in space. This depends on the wavelength and on the separation of the point sources. The figure to the right shows interference between two spherical waves. The wavelength increases from top to bottom, and the distance between the sources increases from left to right.
When the plane of observation is far enough away, the fringe pattern will be a series of almost straight lines, since the waves will then be almost planar.
Multiple beams
Interference occurs when several waves are added together provided that the phase differences between them remain constant over the observation time.
It is sometimes desirable for several waves of the same frequency and amplitude to sum to zero (that is, interfere destructively, cancel). This is the principle behind, for example,
3-phase power and the
diffraction grating
In optics, a diffraction grating is an optical component with a periodic structure that diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form of structur ...
. In both of these cases, the result is achieved by uniform spacing of the phases.
It is easy to see that a set of waves will cancel if they have the same amplitude and their phases are spaced equally in angle. Using
phasors, each wave can be represented as
for
waves from
to
, where
To show that
one merely assumes the converse, then multiplies both sides by
The
Fabry–Pérot interferometer uses interference between multiple reflections.
A
diffraction grating
In optics, a diffraction grating is an optical component with a periodic structure that diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form of structur ...
can be considered to be a multiple-beam interferometer; since the peaks which it produces are generated by interference between the light transmitted by each of the elements in the grating; see
interference vs. diffraction for further discussion.
Optical interference
![Optical flat interference](https://upload.wikimedia.org/wikipedia/commons/4/42/Optical_flat_interference.svg)
Because the frequency of light waves (~10
14 Hz) is too high for currently available detectors to detect the variation of the electric field of the light, it is possible to observe only the
intensity of an optical interference pattern. The intensity of the light at a given point is proportional to the square of the average amplitude of the wave. This can be expressed mathematically as follows. The displacement of the two waves at a point is:
where represents the magnitude of the displacement, represents the phase and represents the
angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
.
The displacement of the summed waves is
The intensity of the light at is given by
This can be expressed in terms of the intensities of the individual waves as
Thus, the interference pattern maps out the difference in phase between the two waves, with maxima occurring when the phase difference is a multiple of 2. If the two beams are of equal intensity, the maxima are four times as bright as the individual beams, and the minima have zero intensity.
Classically the two waves must have the same
polarization to give rise to interference fringes since it is not possible for waves of different polarizations to cancel one another out or add together. Instead, when waves of different polarization are added together, they give rise to a wave of a different
polarization state.
Quantum mechanically the theories of Paul Dirac and Richard Feynman offer a more modern approach. Dirac showed that every quanta or photon of light acts on its own which he famously stated as "every photon interferes with itself". Richard Feynman showed that by evaluating a path integral where all possible paths are considered, that a number of higher probability paths will emerge. In thin films for example, film thickness which is not a multiple of light wavelength will not allow the quanta to traverse, only reflection is possible.
Light source requirements
The discussion above assumes that the waves which interfere with one another are monochromatic, i.e. have a single frequency—this requires that they are infinite in time. This is not, however, either practical or necessary. Two identical waves of finite duration whose frequency is fixed over that period will give rise to an interference pattern while they overlap. Two identical waves which consist of a narrow spectrum of frequency waves of finite duration (but shorter than their coherence time), will give a series of fringe patterns of slightly differing spacings, and provided the spread of spacings is significantly less than the average fringe spacing, a fringe pattern will again be observed during the time when the two waves overlap.
Conventional light sources emit waves of differing frequencies and at different times from different points in the source. If the light is split into two waves and then re-combined, each individual light wave may generate an interference pattern with its other half, but the individual fringe patterns generated will have different phases and spacings, and normally no overall fringe pattern will be observable. However, single-element light sources, such as
sodium- or
mercury-vapor lamp
A mercury-vapor lamp is a gas-discharge lamp that uses an electric arc through vaporized mercury to produce light. The arc discharge is generally confined to a small fused quartz arc tube mounted within a larger soda lime or borosilicate gl ...
s have emission lines with quite narrow frequency spectra. When these are spatially and colour filtered, and then split into two waves, they can be superimposed to generate interference fringes. All interferometry prior to the invention of the laser was done using such sources and had a wide range of successful applications.
A
laser beam
A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word "laser" is an acronym for "light amplification by stimulated emission of radiation". The fir ...
generally approximates much more closely to a monochromatic source, and thus it is much more straightforward to generate interference fringes using a laser. The ease with which interference fringes can be observed with a laser beam can sometimes cause problems in that stray reflections may give spurious interference fringes which can result in errors.
Normally, a single laser beam is used in interferometry, though interference has been observed using two independent lasers whose frequencies were sufficiently matched to satisfy the phase requirements.
This has also been observed for widefield interference between two incoherent laser sources.
![Soap bubble sky](https://upload.wikimedia.org/wikipedia/commons/1/18/Soap_bubble_sky.jpg)
It is also possible to observe interference fringes using white light. A white light fringe pattern can be considered to be made up of a 'spectrum' of fringe patterns each of slightly different spacing. If all the fringe patterns are in phase in the centre, then the fringes will increase in size as the wavelength decreases and the summed intensity will show three to four fringes of varying colour. Young describes this very elegantly in his discussion of two slit interference. Since white light fringes are obtained only when the two waves have travelled equal distances from the light source, they can be very useful in interferometry, as they allow the zero path difference fringe to be identified.
Optical arrangements
To generate interference fringes, light from the source has to be divided into two waves which have then to be re-combined. Traditionally, interferometers have been classified as either amplitude-division or wavefront-division systems.
In an amplitude-division system, a
beam splitter is used to divide the light into two beams travelling in different directions, which are then superimposed to produce the interference pattern. The
Michelson interferometer
The Michelson interferometer is a common configuration for optical interferometry and was invented by the 19/20th-century American physicist Albert Abraham Michelson. Using a beam splitter, a light source is split into two arms. Each of those ...
and the
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 p ...
are examples of amplitude-division systems.
In wavefront-division systems, the wave is divided in space—examples are
Young's double slit interferometer and
Lloyd's mirror Lloyd's mirror is an optics experiment that was first described in 1834 by Humphrey Lloyd in the ''Transactions of the Royal Irish Academy''. Its original goal was to provide further evidence for the wave nature of light, beyond those provided by ...
.
Interference can also be seen in everyday phenomena such as
iridescence and
structural coloration
Structural coloration in animals, and a few plants, is the production of colour by microscopically structured surfaces fine enough to interfere with visible light instead of pigments, although some structural coloration occurs in combination wi ...
. For example, the colours seen in a soap bubble arise from interference of light reflecting off the front and back surfaces of the thin soap film. Depending on the thickness of the film, different colours interfere constructively and destructively.
Applications
Beat
In
acoustics
Acoustics is a branch of physics that deals with the study of mechanical waves in gases, liquids, and solids including topics such as vibration, sound, ultrasound and infrasound. A scientist who works in the field of acoustics is an acoustician ...
, a beat is an
interference
Interference is the act of interfering, invading, or poaching. Interference may also refer to:
Communications
* Interference (communication), anything which alters, modifies, or disrupts a message
* Adjacent-channel interference, caused by extr ...
pattern between two
sound
In physics, sound is a vibration that propagates as an acoustic wave, through a transmission medium such as a gas, liquid or solid.
In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the ...
s of slightly different
frequencies, ''perceived'' as a periodic variation in
volume
Volume is a measure of occupied three-dimensional space. It is often quantified numerically using SI derived units (such as the cubic metre and litre) or by various imperial or US customary units (such as the gallon, quart, cubic inch). Th ...
whose rate is the
difference
Difference, The Difference, Differences or Differently may refer to:
Music
* ''Difference'' (album), by Dreamtale, 2005
* ''Differently'' (album), by Cassie Davis, 2009
** "Differently" (song), by Cassie Davis, 2009
* ''The Difference'' (al ...
of the two frequencies.
With
tuning
Tuning can refer to:
Common uses
* Tuning, the process of tuning a tuned amplifier or other electronic component
* Musical tuning, musical systems of tuning, and the act of tuning an instrument or voice
** Guitar tunings
** Piano tuning, adjusti ...
instruments that can produce sustained tones, beats can be readily recognized. Tuning two tones to a
unison
In music, unison is two or more musical parts that sound either the same pitch or pitches separated by intervals of one or more octaves, usually at the same time. ''Rhythmic unison'' is another term for homorhythm.
Definition
Unison or per ...
will present a peculiar effect: when the two tones are close in pitch but not identical, the difference in frequency generates the beating. The volume varies like in a
tremolo
In music, ''tremolo'' (), or ''tremolando'' (), is a trembling effect. There are two types of tremolo.
The first is a rapid reiteration:
* Of a single Musical note, note, particularly used on String instrument#Bowing, bowed string instrument ...
as the sounds alternately interfere constructively and destructively. As the two tones gradually approach unison, the beating slows down and may become so slow as to be imperceptible. As the two tones get further apart, their beat frequency starts to approach the range of human pitch perception, the beating starts to sound like a note, and a
is produced. This combination tone can also be referred to as a
missing fundamental
A harmonic sound is said to have a missing fundamental, suppressed fundamental, or phantom fundamental when its overtones suggest a fundamental frequency but the sound lacks a component at the fundamental frequency itself.
The brain perceives the ...
, as the beat frequency of any two tones is equivalent to the frequency of their implied fundamental frequency.
Optical interferometry
Interferometry has played an important role in the advancement of physics, and also has a wide range of applications in physical and engineering measurement.
Thomas Young's double slit interferometer in 1803 demonstrated interference fringes when two small holes were illuminated by light from another small hole which was illuminated by sunlight. Young was able to estimate the wavelength of different colours in the spectrum from the spacing of the fringes. The experiment played a major role in the general acceptance of the wave theory of light.
In quantum mechanics, this experiment is considered to demonstrate the inseparability of the wave and particle natures of light and other quantum particles (
wave–particle duality
Wave–particle duality is the concept in quantum mechanics that every particle or quantum entity may be described as either a particle or a wave. It expresses the inability of the classical concepts "particle" or "wave" to fully describe the b ...
).
Richard Feynman
Richard Phillips Feynman (; May 11, 1918 – February 15, 1988) was an American theoretical physicist, known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, the physics of the superfl ...
was fond of saying that all of quantum mechanics can be gleaned from carefully thinking through the implications of this single experiment.
The results of the
Michelson–Morley experiment
The Michelson–Morley experiment was an attempt to detect the existence of the luminiferous aether, a supposed medium permeating space that was thought to be the carrier of light waves. The experiment was performed between April and July 1887 ...
are generally considered to be the first strong evidence against the theory of a
luminiferous aether
Luminiferous aether or ether ("luminiferous", meaning "light-bearing") was the postulated medium for the propagation of light. It was invoked to explain the ability of the apparently wave-based light to propagate through empty space (a vacuum), so ...
and in favor of
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws o ...
.
Interferometry has been used in defining and calibrating
length standards. When the metre was defined as the distance between two marks on a platinum-iridium bar,
Michelson and Benoît used interferometry to measure the wavelength of the red
cadmium
Cadmium is a chemical element with the symbol Cd and atomic number 48. This soft, silvery-white metal is chemically similar to the two other stable metals in group 12, zinc and mercury. Like zinc, it demonstrates oxidation state +2 in most of ...
line in the new standard, and also showed that it could be used as a length standard. Sixty years later, in 1960, the metre in the new
SI system was defined to be equal to 1,650,763.73 wavelengths of the orange-red emission line in the electromagnetic spectrum of the krypton-86 atom in a vacuum. This definition was replaced in 1983 by defining the metre as the distance travelled by light in vacuum during a specific time interval. Interferometry is still fundamental in establishing the
calibration
In measurement technology and metrology, calibration is the comparison of measurement values delivered by a device under test with those of a calibration standard of known accuracy. Such a standard could be another measurement device of know ...
chain in length measurement.
Interferometry is used in the calibration of
slip gauges (called gauge blocks in the US) and in
coordinate-measuring machine
A coordinate measuring machine (CMM) is a device that measures the geometry of physical objects by sensing discrete points on the surface of the object with a probe. Various types of probes are used in CMMs, the most common being mechanical and l ...
s. It is also used in the testing of optical components.
[RS Longhurst, ''Geometrical and Physical Optics'', 1968, Longmans, London.]
Radio interferometry
![USA](https://upload.wikimedia.org/wikipedia/commons/6/63/USA.NM.VeryLargeArray.02.jpg)
In 1946, a technique called
astronomical interferometry
An astronomical interferometer or telescope array is a set of separate telescopes, mirror segments, or radio telescope antennas that work together as a single telescope to provide higher resolution images of astronomical objects such as stars, ne ...
was developed. Astronomical radio interferometers usually consist either of arrays of parabolic dishes or two-dimensional arrays of omni-directional antennas. All of the telescopes in the array are widely separated and are usually connected together using
coaxial cable
Coaxial cable, or coax (pronounced ) is a type of electrical cable consisting of an inner conductor surrounded by a concentric conducting shield, with the two separated by a dielectric ( insulating material); many coaxial cables also have a p ...
,
waveguide
A waveguide is a structure that guides waves, such as electromagnetic waves or sound, with minimal loss of energy by restricting the transmission of energy to one direction. Without the physical constraint of a waveguide, wave intensities de ...
,
optical fiber
An optical fiber, or optical fibre in Commonwealth English, is a flexible, transparent fiber made by drawing glass ( silica) or plastic to a diameter slightly thicker than that of a human hair. Optical fibers are used most often as a mea ...
, or other type of
transmission line
In electrical engineering, a transmission line is a specialized cable or other structure designed to conduct electromagnetic waves in a contained manner. The term applies when the conductors are long enough that the wave nature of the transmi ...
. Interferometry increases the total signal collected, but its primary purpose is to vastly increase the resolution through a process called
Aperture synthesis
Aperture synthesis or synthesis imaging is a type of interferometry that mixes signals from a collection of telescopes to produce images having the same angular resolution as an instrument the size of the entire collection. At each separation and ...
. This technique works by superposing (interfering) the signal waves from the different telescopes on the principle that waves that coincide with the same phase will add to each other while two waves that have opposite phases will cancel each other out. This creates a combined telescope that is equivalent in resolution (though not in sensitivity) to a single antenna whose diameter is equal to the spacing of the antennas farthest apart in the array.
Acoustic interferometry
An
acoustic interferometer An acoustic interferometer is an instrument, using interferometry, for measuring the physical characteristics of sound waves in a gas or liquid. It may be used to measure velocity, wavelength, absorption, or impedance. A vibrating crystal
...
is an instrument for measuring the physical characteristics of sound waves in a
gas
Gas is one of the four fundamental states of matter (the others being solid, liquid, and plasma).
A pure gas may be made up of individual atoms (e.g. a noble gas like neon), elemental molecules made from one type of atom (e.g. oxygen), or ...
or liquid, such
velocity
Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...
, wavelength,
absorption, or
impedance. A vibrating
crystal
A crystal or crystalline solid is a solid material whose constituents (such as atoms, molecules, or ions) are arranged in a highly ordered microscopic structure, forming a crystal lattice that extends in all directions. In addition, macro ...
creates ultrasonic waves that are radiated into the medium. The waves strike a reflector placed parallel to the crystal, reflected back to the source and measured.
Quantum interference
Quantum interference is quite different from the
classical wave interference described above. Below, an enumeration of the important differences is provided. Quantum interference is, however, similar to
optical interference
In physics, interference is a phenomenon in which two waves combine by adding their displacement together at every single point in space and time, to form a resultant wave of greater, lower, or the same amplitude. Constructive and destructive ...
.
Let
be a
wavefunction
A wave function in quantum physics is a mathematical description of the quantum state of an isolated quantum system. The wave function is a complex-valued probability amplitude, and the probabilities for the possible results of measurements ...
solution of the
Schrödinger equation
The Schrödinger equation is a linear partial differential equation that governs the wave function of a quantum-mechanical system. It is a key result in quantum mechanics, and its discovery was a significant landmark in the development of th ...
for a quantum mechanical object. Then the
probability
Probability is the branch of mathematics concerning numerical descriptions of how likely an Event (probability theory), event is to occur, or how likely it is that a proposition is true. The probability of an event is a number between 0 and ...
of observing the object at position
is
where * indicates
complex conjugation. Quantum interference concerns the issue of this probability when the wavefunction is expressed as a sum or
linear superposition
The superposition principle, also known as superposition property, states that, for all linear systems, the net response caused by two or more stimuli is the sum of the responses that would have been caused by each stimulus individually. So tha ...
of two terms
:
Usually,
and
correspond to distinct situations A and B. When this is the case, the equation
indicates that the object can be in situation A or situation B. The above equation can then be interpreted as: The probability of finding the object at
is the probability of finding the object at
when it is in situation A plus the probability of finding the object at
when it is in situation B plus an extra term. This extra term, which is called the ''quantum interference term'', is
in the above equation. As in the
classical wave case above, the quantum interference term can add (constructive interference) or subtract (destructive interference) from
in the above equation depending on whether the quantum interference term is positive or negative. If this term is absent for all
, then there is no quantum mechanical interference associated with situations A and B.
The best known example of quantum interference is the
double-slit experiment
In modern physics, the double-slit experiment is a demonstration that light and matter can display characteristics of both classically defined waves and particles; moreover, it displays the fundamentally probabilistic nature of quantum mechanics ...
. In this experiment, electrons, atoms or other quantum mechanical objects approach a barrier with two slits in it. If the quantum object succeeds in passing through the slits, its position is measured with a detection screen a certain distance beyond and behind the barrier. For this system, one lets
be that part of the wavefunction that passes through one of the slits and lets
be that part of the wavefunction that passes through the other slit. When the object almost reaches the screen, the probability of where it is located is given by the above equation. In this context, the equation says that the probability of finding the object at some point just before it hits the screen is the probability that would be obtained if it went through the first slit plus the probability that would be obtained if it went through the second slit plus the quantum interference term, which has no counterpart in classical physics. The quantum interference term can significantly change the pattern observed on the detection screen.
The separation of
is particularly clear in the
path integral formulation
The path integral formulation is a description in quantum mechanics that generalizes the action principle of classical mechanics. It replaces the classical notion of a single, unique classical trajectory for a system with a sum, or functional i ...
of quantum mechanics in the
context of the double-slit experiment.
consists of the path integral contributions in which the paths pass through the first slit;
consists of the path integral contributions in which they pass through the second slit.
Here is a list of some of the differences between classical wave interference and quantum interference:
- In classical interference, two different waves interfere; In quantum interference, the wavefunction interferes with itself.
- Classical interference is obtained simply by adding the displacements from equilibrium (or amplitudes) of the two waves; In quantum interference, the effect occurs for the probability function associated with the wavefunction and therefore the modulus of the wavefunction squared.
- The interference involves different types of mathematical functions: A classical wave is a
real
Real may refer to:
Currencies
* Brazilian real (R$)
* Central American Republic real
* Mexican real
* Portuguese real
* Spanish real
* Spanish colonial real
Music Albums
* ''Real'' (L'Arc-en-Ciel album) (2000)
* ''Real'' (Bright album) (2010) ...
function representing the displacement from an equilibrium position; a quantum wavefunction is a complex
Complex commonly refers to:
* Complexity, the behaviour of a system whose components interact in multiple ways so possible interactions are difficult to describe
** Complex system, a system composed of many components which may interact with each ...
function. A classical wave at any point can be positive or negative; the quantum probability function is non-negative.
- In classical optical interference the energy conservation principle is violated as it requires quanta to cancel. In quantum interference energy conservation is not violated, the quanta merely assume paths per the path integral. All quanta for example terminate in bright areas of the pattern.
See also
*
Active noise control
Active noise control (ANC), also known as noise cancellation (NC), or active noise reduction (ANR), is a method for reducing unwanted sound by the addition of a second sound specifically designed to cancel the first. The concept was first develop ...
*
Beat (acoustics)
In acoustics, a beat is an interference pattern between two sounds of slightly different frequencies, ''perceived'' as a periodic variation in volume whose rate is the difference of the two frequencies.
With tuning instruments that can produce ...
*
Coherence (physics)
*
Diffraction
Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...
*
Haidinger fringes Haidinger fringes are interference fringes formed by the interference of monochromatic and coherent light to form visible dark and bright fringes. Fringe localization is the region of space where fringes with reasonably good contrast are observed ...
*
Interference lithography
Interference lithography (or holographic lithography) is a technique for patterning regular arrays of fine features, without the use of complex optical systems or photomasks.
Basic principle
The basic principle is the same as in interferometry o ...
*
Interference visibility The interferometric visibility (also known as interference visibility and fringe visibility, or just visibility when in context) is a measure of the contrast of ''interference'' in any system subject to wave superposition.
Examples include as opti ...
*
Interferometer
*
Lloyd's Mirror Lloyd's mirror is an optics experiment that was first described in 1834 by Humphrey Lloyd in the ''Transactions of the Royal Irish Academy''. Its original goal was to provide further evidence for the wave nature of light, beyond those provided by ...
*
Moiré pattern
In mathematics, physics, and art, moiré patterns ( , , ) or moiré fringes are large-scale interference patterns that can be produced when an opaque ruled pattern with transparent gaps is overlaid on another similar pattern. For the moiré ...
*
Multipath interference
In radio communication, multipath is the propagation phenomenon that results in radio signals reaching the receiving antenna by two or more paths. Causes of multipath include atmospheric ducting, ionospheric reflection and refraction, and ref ...
*
Newton's rings
Newton's rings is a phenomenon in which an interference pattern is created by the reflection of light between two surfaces, typically a spherical surface and an adjacent touching flat surface. It is named after Isaac Newton, who investigated the ...
*
Optical path length
*
Thin-film interference
Thin-film interference is a natural phenomenon in which light waves reflected by the upper and lower boundaries of a thin film interfere with one another, either enhancing or reducing the reflected light. When the thickness of the film is an ...
*
Rayleigh roughness criterion
*
Upfade
In wireless communications, fading is variation of the attenuation of a signal with various variables. These variables include time, geographical position, and radio frequency. Fading is often modeled as a random process. A fading channel is a ...
References
External links
Easy JavaScript Simulation Model of One Dimensional Wave Interference
{{DEFAULTSORT:Interference (Wave Propagation)
Wave mechanics
ca:Interferència (propagació d'ones)#Interferència òptica