Interference refers to the phenomenon where two coherent waves, meeting under certain conditions, superimpose. In the superposition region, the vibration at some points is always reinforced, while at other points it is always weakened, meaning there is a stable spatial distribution of vibration intensity within the interference region.

See also: Superposition principle and Coherence

When two waves propagate in the same medium and overlap, the particles of the medium within the overlapping region are simultaneously subjected to the actions of both waves. If the amplitude of the waves is not too large, the vibrational displacement of the particles in the overlapping region equals the vector sum of the displacements caused by each individual wave; this is known as the superposition principle of waves. If the crests (or troughs) of both waves arrive at the same point simultaneously, the waves are said to be in phase at that point, and the interfering waves produce the maximum amplitude, known as constructive interference. If the crest of one wave and the trough of the other arrive at the same point simultaneously, the waves are said to be out of phase at that point, and the interfering waves produce the minimum amplitude, known as destructive interference.

Theoretically, the superposition of two infinite monochromatic waves can always produce interference. However, in practical physical models, wave trains cannot be infinite in length. Furthermore, from the perspective of the microscopic mechanism of wave generation, both the amplitude and phase of waves exhibit random fluctuations. Therefore, strictly monochromatic waves do not exist in reality. For example, the light waves emitted by the Sun originate from interactions between electrons and hydrogen atom in the photosphere, with each interaction lasting on the order of 10 seconds. Consequently, wave trains generated from interactions that are temporally distant cannot interfere with each other. For this reason, the Sun can be considered an extended light source composed of many mutually incoherent point sources. Thus, sunlight has a very broad frequency spectrum, with its amplitude and phase undergoing rapid random fluctuations. Ordinary physical instruments cannot track such rapid fluctuations, and therefore we cannot observe the interference of light waves using sunlight. Similarly, for two light waves from different sources, if the fluctuations in their amplitudes and phases are uncorrelated with each other, we say these two waves are not coherent. Conversely, if two light waves originate from the same point source, their fluctuations are generally correlated with each other, and in this case, the two waves are perfectly coherent.

To generate two coherent waves from a single incoherent source, two different methods can be employed: one is called wavefront splitting. For a wave source with sufficiently small geometric dimensions, the wave train it produces is passed through slits placed side by side. According to the Huygens-Fresnel principle , the secondary waves generated at the wavefront are mutually coherent. The other method is called amplitude splitting, where a half-silvered mirror, which is semi-transmissive and semi-reflective, is used to split a light wave into two, creating a transmitted wave and a reflected wave. The reflected and transmitted waves produced in this way originate from the same wave source and possess high coherence. This method is also applicable to extended wave sources.

When two beams of light interfere, the intensity distribution of the interference fringes is related to the optical path difference/phase difference between the two beams: the intensity is maximum when the phase difference is an integer multiple of the period; the intensity is minimum when the phase difference is an odd multiple of half the period. From the sum and difference of the maximum and minimum intensities, the interference visibility can be defined as a measure of the clarity of the interference fringes.

Light, as an electromagnetic wave, has its intensity defined as the time average of the energy per unit area perpendicular to the direction of propagation per unit time, which is the time average of the Poynting vector:

Thus, the light intensity can be characterized by this quantity. For a monochromatic light wave field, the electric vector can be written as

Here, is the complex amplitude vector, which can be written in component form in the Cartesian coordinate system.

Here, is the (real) amplitude on the three components; for a plane wave, is constant in all directions. is the phase on the three components, and , , and are constants characterizing the polarization.

To calculate the intensity of this plane wave, first compute the square of the Electric field strength:

For a time interval much longer than one period, the average values of the first two terms in the above equation are both zero, so the intensity is

For two monochromatic plane waves of the same frequency, if they overlap at a point in space, then according to the superposition principle, the electric field strength at that point is the vector sum of the two:

Then the intensity at that point is

Where \(I_1\) and \(I_2\) are the light intensities of the two waves considered independently, and \(I_{12}\) is the interference term. Using \(\mathbf{E}_1\) and \(\mathbf{E}_2\) to represent the complex amplitudes of the two waves, the interference term \(I_{12}\) can be written as

The first two terms average to zero over time, thus the contribution of the interference term to the light intensity is

According to the definition of complex amplitude given earlier, \(\mathbf{E}_1\) and \(\mathbf{E}_2\) can be decomposed in the Cartesian coordinates as

and

Substituting the component forms into the intensity of the interference term above yields

If the phase difference between the two is the same and constant in all directions, that is,

where \(\lambda\) is the wavelength of the monochromatic light, and \(\delta\) is the optical path difference between the two waves arriving at the same point in space.

At this point, the contribution of the interference term to the light intensity is

Light waves are transverse waves with the electric vector perpendicular to the direction of propagation. Considering a simple yet general case: linearly polarized light, with the electric vector along the x-axis and the propagation direction along the z-axis, then the amplitudes of the two waves in other directions are zero:

Substituting into the total intensity formula:

Therefore, the intensity after interference is a function of the phase difference \(\Delta\varphi\). It reaches a maximum when \(\Delta\varphi = 0, 2\pi, 4\pi, \ldots\); and a minimum when \(\Delta\varphi = \pi, 3\pi, 5\pi, \ldots\).

Specifically, when the intensities of the two waves are equal, that is, when , the above formula can be simplified to

Clearly, for different interference conditions, the differences between the resulting maximum and minimum values are different. Therefore, the visibility of the fringes can be defined as a measure of fringe clarity:

Although the above discussion is based on the assumption that both waves are linearly polarized light , it also holds true for unpolarized light, because natural light can be considered as the superposition of two mutually perpendicular linearly polarized light waves.

Young's Double Slits

Main article: Double-slit experiment

Young's double-slit experiment was the first proposed demonstration experiment of Interference of Light ( Thomas Young , 1801). The significant importance of this experiment lies in its strong support for the wave theory of light. Since the observed interference fringes were a phenomenon that could not be explained by the corpuscular theory of light represented by Newton, the double-slit experiment led most physicists to gradually accept the wave theory of light. The setup of Young's double-slit experiment involves a monochromatic wave from a point source propagating to a screen with two slits. The distances from the point source to the two slits are equal, and the distance between the two slits is very small. Because the distances from the point source to the two slits are equal, these two slits become secondary monochromatic point sources in phase. The coherent light emitted from them interferes, producing interference fringes on a distant screen.

If the distance between the two slits is , and the perpendicular distance from the slits to the observation screen is , then according to geometric relationships, on the observation screen with the symmetric center point as the origin and coordinate , the optical path lengths of the two coherent beams are respectively

When the vertical distance from the slits to the observation screen is much greater, the difference in the lengths of these two optical paths can be approximately expressed as: in the Right triangle formed by drawing a perpendicular from slit 1 to optical path 2, the side opposite angle θ. According to geometric approximation, this difference is d sin θ.

If the experiment is conducted in a vacuum or air, the Refractive index of the medium is considered to be 1. Thus, the optical path difference is d sin θ, and the phase difference is (2π/λ) d sin θ.

Based on the previous conclusion, when the phase difference equals 2mπ (where m is an integer), the light intensity has a maximum value. Therefore, when d sin θ = mλ, there is a maximum. When the phase difference equals (2m+1)π, the light intensity has a minimum value. Therefore, when d sin θ = (m+1/2)λ, there is a minimum. Thus, Young's double-slit interference produces equally spaced alternating bright and dark fringes, with a spacing of Δy = (λD)/d.

Bright and dark alternating fringes from double-slit interference under different slit separation conditions. The corresponding slit separation a = 0.250mm, and the corresponding slit separation a = 0.500mm. The central bright fringe seen in the photograph is brighter than the bright fringes on either side, which is due to the diffraction effect of the slits.

If the linear width of the slits along the line connecting the two slits is increased in double-slit interference to the extent that the slits can no longer be considered point sources, the resulting extended source can be regarded as a collection of multiple continuously distributed point sources. Due to their different positions, these point sources will cause different phase differences at the same point on the screen, potentially causing the maxima and minima of interference from each point source to coincide, leading to a decrease in fringe visibility.

Fresnel Double Mirror [edit]

Geometric schematic of Fresnel double mirror interference

The Fresnel double mirror is an instrument that can directly produce two coherent light sources. The Fresnel double mirror consists of two plane mirrors, M1 and M2, of equal length, arranged at a very small angle α relative to each other. When a light wave from a point source S is incident on the two mirrors and reflects from each, it forms two Virtual image , S1 and S2. Since they are virtual images of the same source, they are coherent sources. The blue shaded area represents the region of interference between the two beams.

The geometric relationship of interference with the Fresnel double mirror is the same as that of Young's double-slit experiment. Therefore, once the distance d between the two virtual images is determined, the positions of the interference fringes can be deduced. If the distance from the light source S to the intersection point A of the two plane mirrors is b, then based on mirror symmetry, the distance from the two coherent sources to the mirror intersection point is also equal to b, that is, SA = b.

The angles between the virtual optical paths S1A, S2A, and the bisector (the horizontal dotted line) are all equal to the inclination angle α of the plane mirrors, thus we have d = 2b sin α.

This distance is equivalent to the slit separation in Young's double-slit experiment. Substituting it into the formula mentioned earlier yields the positions of the interference fringes. When the light wave is incident on the two mirrors, each reflection undergoes a phase change of π, which does not affect the final phase difference between the two beams. Therefore, the shape of the interference fringes in the Fresnel double mirror experiment is exactly the same as that in Young's double-slit experiment, consisting of equally spaced bright and dark fringes, with the central fringe being the zeroth-order bright fringe.

Fresnel Double Prism [Edit]

Geometric schematic of interference with the Fresnel double prism

The Fresnel double prism (Fresnel double prism) is an instrument similar to the Fresnel double mirror for creating coherent light sources. It consists of two identical thin triangular prism placed base-to-base, with a very small refracting angle α, and their refracting edges are parallel to each other. When a point source S located on the axis emits light, the incident light is partially refracted upward and partially downward by the two prisms, thus forming two symmetrical virtual images. These two virtual images serve as the two coherent sources.

If the apex angle of the prism is α and its refractive index is n, then for small α, the deflection angle of a light ray due to refraction is β ≈ (n - 1)α.

If the distance from the point source S to the prism is a, then based on geometric relationships, the distance between the two coherent sources is d = 2aβ ≈ 2a(n - 1)α.

The following calculation for fringe spacing is identical to that for Young's double-slit experiment.

Lloyd Mirror [Edit]

The Lloyd mirror is a simpler wavefront-splitting interferometer, essentially consisting of a flat mirror M placed horizontally. A point source S is located relatively far from and quite close to the plane of the mirror M, resulting in a very small angle of incidence for the incoming light. The point source S and its virtual image S' formed by the mirror constitute a pair of coherent sources. Based on geometric relations, if the distance from the point source S to the mirror plane is d, then the distance between the two coherent sources is 2d. Since one of the two coherent light paths undergoes reflection from the mirror surface, only one coherent beam experiences a reflection-induced phase change of π. For this reason, the center of the interference fringe pattern is a zero-order dark fringe.

Michelson Stellar Interferometer[edit]

Main article: Michelson stellar interferometer

Basic optical path diagram of the Michelson stellar interferometer

The Michelson stellar interferometer is an interferometer used to measure the angular diameter of stars by utilizing the principle that the visibility of interference fringes decreases as the linear size of the extended light source increases (see the section on spatial coherence below). The basic optical path of the instrument was first conceptualized by American physicist Albert A. Michelson and French physicist Armand Fizeau in 1890. The first measurement of a star's angular diameter using this interferometer was conducted by Michelson and American astronomer Francis G. Pease in 1920 at the Mount Wilson Observatory. The Michelson stellar interferometer was approximately 6 meters in length and was mounted on the 2.5-meter aperture Hooker Telescope. The maximum separation between the two flat mirrors M1 and M2 was 6.1 meters and was adjustable; the positions of flat mirrors M3 and M4 were fixed at a separation of 1.14 meters. When starlight enters the interferometer, the optical paths formed by the two sets of mirrors are of equal length, producing equally spaced straight interference fringes with a fringe spacing of

The Michelson stellar interferometer mounted on the Hooker Telescope, now preserved at the American Museum of Natural History

Here, f is the focal length of the telescope, and b is the distance between the flat mirrors M3 and M4. The distance between the flat mirrors M1 and M2 corresponds to the linear size of the extended source. When M1 and M2 are very close together, the fringe visibility is close to 1; as their separation increases, the visibility gradually decreases to zero. If the star is considered to be a circular light source with a Uniform Distribution intensity distribution and an angular diameter of 2α, its visibility is given by the following formula

where is the Bessel Function . As the distance between the plane mirrors M1 and M2 is gradually increased, the visibility first drops to zero when the following relationship is satisfied:

The first star successfully measured by the Michelson stellar interferometer was Betelgeuse , with an angular diameter of 0.047 arcseconds. Based on its distance from the Sun (approximately 600 light-years), its diameter was calculated to be about 4.1×108 km, which is 300 times the diameter of the Sun. In fact, this particular Michelson stellar interferometer could only measure giant stars with diameters hundreds of times that of the Sun, because measuring smaller stars required a greater distance between M1 and M2, and constructing such a large interferometer was quite difficult with the technology of the time.

Equal Inclination Interference

A monochromatic point source S emits electromagnetic waves that are incident on a transparent parallel-plane plate. Reflection and refraction occur at the upper surface of the plate. The refracted light is then reflected by the lower surface, and this reflected light is refracted again at the upper surface back into the original medium. This refracted light will necessarily coincide in space with another beam of light that was directly reflected from the upper surface. Since they are both parts of the electromagnetic wave emitted from the same source, they are coherent light, and non-localized interference fringes will form. If the source is an extended source, the visibility of the interference fringes generally decreases. However, if we consider the case where the two reflected beams are parallel, meaning the point of coincidence is at infinity, localized equal inclination interference fringes will form. According to geometric relationships, the optical path difference between the two beams can be expressed as

where is the refractive index of the parallel-plane plate, and is the refractive index of the surrounding medium. The specific lengths can be expressed as

where is the thickness of the parallel-plane plate, is the angle of incidence, and is the angle of refraction, with the two satisfying the law of refraction.

The resulting optical path difference is , and the corresponding phase difference is . Additionally, considering the phase change upon reflection at either the upper or lower surface, the phase difference should be

According to the conditions for constructive and destructive interference, bright fringes occur when , where m is an integer, and dark fringes occur when m is a half-integer.

Thus, each fringe corresponds to a specific refraction/incidence angle, hence it is referred to as equal inclination interference. If the observation direction is perpendicular to the parallel-plane plate, a set of concentric circular interference fringes can be observed. Furthermore, the two parallel beams transmitted through the lower surface of the parallel-plane plate also form equal inclination interference. However, due to the absence of a reflection phase change, the phase difference does not require the addition of the π term, resulting in the bright and dark positions of the interference fringes for the transmitted light being exactly opposite to those for the reflected light.

Equal Thickness Interference

If the two surfaces of the parallel-plane plate in equal inclination interference are not strictly parallel, then for the light emitted from a monochromatic point source S, the reflected light from its upper and lower surfaces will always interfere at a certain point P in space, and the interference fringes are non-localized. At this point, the optical path difference between these two beams can be written as

Similarly, n₁ is the refractive index of the surrounding medium, and n₂ is the refractive index of the parallel-plane plate. Generally, this calculation is quite complex. However, in the case where the parallel-plane plate is sufficiently thin and the angle between its two surfaces is sufficiently small (e.g., a thin film), the optical path difference can be approximated as

where d is the thickness of the film at the reflection point C, and θ₂ is the reflection angle at that point. Consequently, the corresponding phase difference is δ = (2π/λ) * Δ.

If the light source is an extended source, it will broaden the range of phase differences for the interfering light at point P, leading to a decrease in fringe visibility. An exception occurs when point P is located on the surface of the film: for the interfering light emitted from various points of the extended source, the thickness d is the same. When the variation range of θ₂ is very small, the interference condition can be written as

Interference maxima occur when m is an integer, and minima occur when m is a half-integer. Here, θ₂ is the average value obtained by averaging over all points of the extended source, and the presence of the ± π/2 term accounts for the reflection phase change. If θ₂ is constant, the fringes are lines connecting points of constant thickness within the film, which are referred to as equal thickness fringes. Equal thickness interference is often used to detect whether the thickness of an optical surface is uniform. For the case of normal incidence, θ₂ = 0, the condition for an interference minimum is

An example of equal thickness interference is wedge interference, where light is incident normally onto a wedge-shaped film. If the refractive index of the wedge is n, then according to the previous conclusion, the interference condition is

where bright fringes correspond to m being an integer, and dark fringes correspond to m being a half-integer. The fringes are a set of Parallel Lines aligned with the wedge edge, and the edge itself corresponds to the zeroth-order dark fringe. The thickness difference corresponding to adjacent bright fringes is therefore λ/(2n).

Furthermore, the fringe spacing can be derived, where is the wedge angle, meaning the fringes in wedge interference are equally spaced.

Another famous example of equal thickness interference is Newton's rings. They are interference fringes formed by placing the convex surface of a Convex lens with a very large radius of curvature on a flat glass plate and illuminating it with normally incident parallel light. In this case, the gap between the convex lens and the glass plate forms a wedge with air (refractive index approximately 1) as the medium. Thus, the interference condition is

Let the radius of curvature of the lens be , then the relationship between the fringe radius and the wedge thickness is given by

From this, the radius of the interference fringes can be obtained as , where dark fringes occur when m is an integer, and bright fringes occur when m is a half-integer. This shows that the spacing of Newton's rings becomes progressively denser from the center outward.

Michelson interferometer

Main article: Michelson interferometer

The Michelson interferometer is a classic example of an amplitude-splitting interferometer. It works by splitting a beam of incident light into two beams. These two coherent beams are then reflected back by corresponding plane mirrors, resulting in amplitude-splitting interference. The optical path difference between the two interfering beams can be adjusted by changing the length of the interferometer arms or by altering the refractive index of the medium, allowing for the formation of different interference patterns. A famous application of the Michelson interferometer was the Michelson–Morley experiment conducted in 1887 by American physicists Albert A. Michelson and Edward W. Morley, which yielded a null result in the measurement of the "aether wind." Additionally, Michelson used it to conduct the first systematic study of the fine structure of spectral lines.

Basic construction of the Michelson interferometer: There are two optical paths between the light source and the photodetector: one beam is reflected by a beam splitter (e.g., a semi-transparent mirror) onto an upper plane mirror, reflects back to the beam splitter, and then transmits through the beam splitter to be received by the photodetector; the other beam transmits through the beam splitter, strikes a right-side plane mirror, reflects back to the beam splitter, and is then reflected again towards the photodetector. By adjusting the forward/backward position of the plane mirrors, the optical path difference between the two beams can be controlled. It is noteworthy that the beam reflected by the beam splitter passes through the beam splitter three times in total, while the transmitted beam passes through it only once. For monochromatic light, adjusting the position of the plane mirror is sufficient to eliminate this path difference; however, for polychromatic light, dispersion of different wavelengths occurs within the beam splitter medium. Therefore, a glass plate of identical material and thickness to the beam splitter, called a compensating plate, must be placed in the path of the transmitted beam to eliminate this effect.

When the two plane mirrors are strictly perpendicular, a monochromatic light source forms concentric circular fringes of equal inclination interference, localized at infinity. If one of the plane mirrors is adjusted to gradually reduce the optical path difference between the two beams, the fringes contract towards the central bright fringe until the path difference becomes zero and the interference fringes disappear. If the two plane mirrors are not strictly perpendicular and the optical path difference is small, the light source forms localized fringes of equal thickness interference, which are equidistant straight fringes equivalent to those produced by wedge interference.

Between 1905 and 1930, the Michelson–Morley experiment was repeated multiple times using the Michelson interferometer, with results not exceeding 10% of the fringe shift predicted for the existence of an aether wind. In 1979, the most precise Michelson–Morley experiment to date was conducted using a laser. The frequency of the helium–neon laser used was locked to an adiabatically stabilized Fabry–Pérot interferometer. The results showed that any possible shift in the laser frequency due to an aether wind would not exceed 5×10 of its predicted value.

Mach–Zehnder interferometer

In a Michelson interferometer, a beam splitter is also used to recombine two coherent light beams to produce interference. If an independent half-silvered mirror is employed to recombine the two beams, a Mach–Zehnder interferometer can be constructed. It was designed by the German physicists Ludwig Mach (son of Ernst Mach) and Ludwig Zehnder at the end of the 19th century. Its basic optical path is as follows: the light source is located at the focal plane of a lens. The collimated light emerging from the lens is incident on the first half-silvered mirror and splits into two beams. Each beam is reflected by a plane mirror and then recombines at a second, identical half-silvered mirror. Interference can then occur at the photodetectors in both output directions. Typically, the four reflecting surfaces within the interferometer need to be set as strictly parallel as possible, and the four reflection points should form a Parallelogram to ensure collimation. Consequently, the path length difference between the two interfering arms critically affects the interference signals received by the photodetectors in the two directions. Any minute change in the optical path difference will lead to a redistribution of the incident light energy. When the optical paths of the two interfering arms are exactly equal, and considering the multiple half-wavelength losses caused by reflections at the half-silvered mirrors and plane mirrors, it can be deduced that at this point, the two coherent beams undergo constructive interference along the path to photodetector 1, and all the incident light energy will enter photodetector 1. Conversely, along the path to photodetector 2, destructive interference occurs, and no incident light energy enters photodetector 2.

In practical operation, if one of the half-silvered mirrors and a plane mirror are slightly tilted relative to each other, wedge interference similar to that in a Michelson interferometer is formed, resulting in localized, parallel, equidistant straight fringes.

By measuring the change in light intensity received by the photodetectors caused by variations in the optical path difference, the Mach–Zehnder interferometer is frequently used to measure changes in the refractive index within compressible gas flows. Specifically, one of the two coherent light paths serves as the reference path, while the other is placed within the gas flow under test as the measurement path. This allows for the measurement of the change in the refractive index of the gas flow, and subsequently, the change in density of the gas flow under test can be determined.

Main article: Coherence

In amplitude-splitting interference devices such as the Michelson interferometer or the Mach–Zehnder interferometer, although the two beams of light originate from the same source, experiments reveal that if the optical path difference between the two beams is increased indiscriminately, it leads to a decrease in the visibility of the interference fringes until they vanish. Similarly, in Young's double-slit interference, if the linear dimension of the two slits along the line connecting them is gradually extended, it also results in a decrease and eventual disappearance of fringe visibility. This phenomenon of the final disappearance of interference fringes is due to coherence. The former case is because actual light waves are not strictly infinite, monochromatic wave trains; they possess a finite coherence length (temporal coherence). The latter case is because an extended source reduces the mutual coherence between different points in space (spatial coherence). For example, in a Michelson interferometer, an incident wave train of finite length enters the interferometer and is split into two wave trains of equal length. If the optical path difference between the two arms of the interferometer is greater than the length of these two wave trains, then for this particular incident wave train, the two resulting partial waves cannot interfere; that is, the two wave trains are not coherent. Consequently, at any given moment, all wave trains arriving at a specific point in space are superpositions from different incident wave trains. These incident wave trains themselves have random phase and amplitude fluctuations, and their superposition over an observable time interval does not produce interference.

Temporal Coherence

As time changes, within a time interval , the amplitude of a wave with significant phase drift (red), and its amplitude delayed by time (green). At any set time , the red wave will interfere with the delayed green copy. However, since for half the time the red wave is in phase with the green wave, and for the other half, the two waves are out of phase, the time-averaged interference for this delay equals zero.

Temporal coherence is a reflection of the monochromaticity of a light wave. The better the monochromaticity of a light wave, the better its temporal coherence. That is, for a light wave train, if it is delayed by a period of time and then made to interfere with its delayed version, and if interference still occurs even when this delay time is very large, then this wave train or the corresponding source is said to have good temporal coherence. For a strictly infinite, monochromatic wave, it can still interfere with itself no matter how long the delay. For an actual finite-length wave train, however, interference becomes impossible beyond a specific time interval; this time is called the coherence time, which is essentially the duration of this light wave train. By definition, the method for describing temporal coherence is the autocorrelation function.

Consider a finite-length wave train with a duration of , meaning when . Performing a Fourier Transform on this wave train yields its frequency spectrum as

The result of this integration is a normalized Sinc function, and the squared modulus of the spectrum (the power spectrum) corresponds to the light intensity. From the function, it can be seen that the first zero of the light intensity corresponds to.

Thus, the frequency range of this finite-length wave train is obtained, i.e., the frequency range of the wave train is approximately the reciprocal of the wave train's duration. In fact, actual light waves satisfy the relation. From this, it can be understood that the laser linewidth is also a reflection of temporal coherence; the narrower the linewidth of a laser, the higher its temporal coherence.

From the coherence time, the coherence length can be further defined, which is the range of wavelengths. When the optical path difference between two light waves approaches or exceeds their coherence length, interference effects become difficult to observe.

Spatial Coherence

Spatial coherence reflects the degree of correlation between the electric fields at two points in space during the propagation of an electromagnetic wave; it is a type of cross-correlation function. If the phases at different points on the same wavefront of a propagating electromagnetic wave are highly correlated with each other, the wave is considered to have strong spatial coherence. For example, across the cross-section of a laser beam, the electric fields oscillating in different directions are highly consistent in their phase changes, even if the laser has a broad linewidth and thus poor temporal coherence. Spatial coherence is a key factor enabling lasers to maintain high directionality.

According to Fourier optics, the Fourier transform of the light intensity distribution of a source on a two-dimensional plane is the visibility function of the interference fringes. Thus, for an extended light source of linear dimension, its visibility is a Sinc function. Therefore, on a wavefront at a distance of, the range over which spatial coherence exists can be approximately expressed as

This distance is called the coherence interval, from which the coherent aperture angle can be defined. This means that within the optical field within this angular range, any two points on the wavefront possess spatial coherence.

Since the visibility of fringes in Young's Double-slit experiment is closely related to the linear extent of the slits along the line connecting them, this method can be used to measure the angular size of small light sources. This is precisely the principle behind the Michelson stellar interferometer.

In the case of equal inclination interference produced by amplitude division when incident light illuminates a parallel-plane plate, since the light reflected from the lower surface can be reflected again by the upper surface, a third transmitted beam will emerge from the upper surface and interfere with the first two beams. By extension, if the loss of electromagnetic waves by the parallel-plane plate is negligible (the medium does not absorb or scatter electromagnetic waves), theoretically an infinite number of beams will emerge from the upper surface, and all these beams are mutually coherent.

Multiple beam interference from a parallel-plane plate.

Let the Refractive index of the parallel-plane plate be n, its thickness be d, the angle of incidence of the monochromatic light be θ_i, and the angle of refraction be θ_t. According to previous conclusions, the optical path difference between adjacent reflected or transmitted beams is Δ, and the corresponding phase difference is δ.

To calculate the Interference of Light of multiple reflected or transmitted beams, it is also necessary to calculate the vector sum (or the algebraic sum if represented by complex amplitudes) of the Electric field strength of these light fields. For both the upper and lower surfaces of the parallel-plane plate, there are specific reflectance (the ratio of reflected wave amplitude to incident wave amplitude) and transmittance (the ratio of transmitted wave amplitude to incident wave amplitude). Here, let the reflectance and transmittance of the upper surface (from the surrounding medium into the plate) be r_1 and t_1 respectively, and the reflectance and transmittance of the lower surface (from the plate into the surrounding medium) be r_2 and t_2 respectively. If the complex amplitude of the incident wave at the incidence point A1 is E_0, then the complex amplitudes of the beams reflected from the upper surface are successively:

Ignoring the phase shift produced by the propagation of the first transmitted wave within the parallel-plane plate (as it is a constant present in all transmitted waves), the complex amplitudes of the beams transmitted from the lower surface are successively:

Summing the complex amplitudes of all reflected light, this is an infinite Geometric Progression , and the result is (lossless, R = |r|^2):

If we define the reflectances of the upper and lower surfaces of the parallel-plane plate as R_1 and R_2 respectively, and assume R_1 = R_2 = R, and the transmittance T = |t|^2. Reflectance and transmittance are the ratios of the energy of the reflected and transmitted waves to the energy of the incident wave, so under the condition of negligible loss, the energy conservation condition must be satisfied.

Thus, the amplitude of the reflected light can be expressed as:

The intensity of the reflected light is the squared modulus of the complex amplitude, and its expression is:

The intensity of the transmitted light can be directly obtained by subtracting the intensity of the reflected light from the intensity of the incident light, or by summing an infinite geometric series:

The expressions for reflected light intensity and transmitted light intensity are also known as the Airy formulae.

According to the expression for transmitted light intensity, the interference condition is:

A maximum in transmitted light intensity occurs when m is an integer, and a minimum occurs when m is a half-integer. Since the intensity distribution depends on the angle of inclination, the resulting fringes are equal inclination fringes.

When discussing reflected and transmitted light intensity, a parameter is typically introduced, yielding the reflectance function and transmittance function of the parallel plane plate:

The relationship between the transmittance function and the finesse. A transmittance function with higher finesse (red curve) compared to one with lower finesse (blue curve) has sharper peaks and lower transmission minima. The free spectral range of the parallel plane plate is the full width at half maximum of the transmission peak.

Both reflectance and transmittance are functions of wavelength. The wavelength interval between two adjacent transmission peaks on the transmittance function is called the free spectral range (FSR), given by:

where is the center wavelength of the nearest peak.

Dividing the free spectral range by the full width at half maximum (FWHM) of the transmission peak at half its maximum height yields a value called the finesse:

For high reflectance ratios, the finesse can often be approximated by:

From this formula, it can be seen that a higher reflectance ratio results in higher finesse, corresponding to a sharper transmission peak shape.

Fabry–Pérot interferometer

Main article: Fabry–Pérot interferometer

The Fabry–Pérot interferometer is a multi-beam interferometer consisting of two parallel glass plates. Its fundamental principle is identical to the interference principle of the parallel-plate discussed in the previous section. The inner surfaces of the two glass plates are coated to have very high reflectivity, ensuring that interference fringes with sufficiently high finesse can be obtained. Since a parallel plate exhibits transmission maxima only for specific wavelengths of light, the Fabry–Pérot interferometer can transmit or reflect light whose frequency satisfies its Resonance condition, achieving very high transmission and reflection rates. For this reason, it is also called a Fabry–Pérot resonator or a Fabry–Pérot etalon. The Fabry–Pérot interferometer is widely used in fields such as telecommunications, lasers, and spectroscopy, primarily for the precise measurement and control of light frequency and wavelength. For example, optical wavelength meters employ a combination of several Fabry–Pérot interferometers whose resonance frequencies differ from each other by a factor of ten. After the incident light to be measured undergoes interference within these interferometers, its wavelength can be determined by measuring the spacing of the bright fringes produced by each. Furthermore, in laser technology, Fabry–Pérot interferometers are used to suppress spectral line broadening to obtain single-mode lasers. In Gravitational wave detection, Fabry–Pérot interferometers are combined with Michelson interferometers. By causing Photon to oscillate repeatedly within the resonator, the effective arm length of the Michelson interferometer is increased.

To observe the equal inclination interference fringes of a Fabry–Pérot interferometer, a lens must be placed perpendicular to the propagation direction of the transmitted light. When the optical axis of the lens is perpendicular to the screen, the equal inclination interference fringes appear as a set of concentric circles, with the center corresponding to the focal point of the normally incident transmitted light. In this case, due to normal incidence, it reaches its maximum value in the interference condition:

Generally, is not an integer. If its integer part is denoted as and its fractional part as , such that , then the angular radius of the -th bright fringe from the central bright fringe is

Thus, the diameter of the circular fringes satisfies

where is the focal length of the lens.

The three important characteristic parameters of a Fabry–Pérot interferometer are the finesse (the ratio of the free spectral range to the full width at half maximum of the transmission peak), the peak transmission (the maximum ratio of transmitted light intensity to incident light intensity), and the contrast factor (the ratio of the maximum to the minimum of the transmitted-to-incident light intensity ratio). However, because a higher reflectivity is required to achieve higher finesse, it is not possible for both the transmission and the finesse/contrast factor to be very high simultaneously.

See also: Double-slit experiment #Results in Quantum mechanics and Dynamics of photons in the double-slit experiment

The Double-slit experiment conducted by emitting single electrons one at a time yields results analogous to those obtained with photons. It describes the distribution of electrons arriving at the screen as they accumulate over time.

Between 1905 and 1917, Albert Einstein, building upon Max Planck 's hypothesis of energy quantization and his explanation of the Photoelectric Effect , proposed in papers such as "On a Heuristic Viewpoint Concerning the Production and Transformation of Light," "On the Development of Our Views Concerning the Nature and Constitution of Radiation," and "On the Quantum Theory of Radiation" that the energy of Electromagnetic wave is composed of discrete Quantum of energy. These quanta were called light quanta (Photon), and Electromagneticradiation must possess both wave-like and particle-like natural properties simultaneously, which is known as Wave–particle duality . Since Robert Millikan completed a series of experiments on the photoelectric effect in 1916, and Arthur Compton observed the scattering of X-ray by free electrons in 1923 and measured the momentum of photons in 1926, the physics community gradually accepted the fact that electromagnetic waves also possess particle nature. However, if we attempt to understand the phenomenon of interference from the perspective of photons, the following problem arises: when two corresponding photons from two coherent beams interfere with each other, constructive interference would require the generation of four photons from two, and destructive interference would require the two photons to cancel each other out, which violates the Law of conservation of energy .

To resolve this problem, the Copenhagen interpretation of Quantum mechanics posits that the interference of photons is the superposition of the probability amplitudes of the wave function of a single photon. The wave function is a probability wave, and the square of the modulus of its complex amplitude (probability amplitude) is proportional to the probability of the corresponding state (eigenstate) occurring. Taking double-slit interference as an example, for each photon, its state is a superposition of the quantum states of passing through each of the two slits:

Here, and correspond to the quantum states of passing through slit 1 and slit 2, respectively. The probability amplitudes and correspond to the respective probabilities of this photon emerging from slit 1 and slit 2; each is itself a complex number.

The probability of a light detector detecting this photon, which statistically corresponds to the light intensity detected, is the square of the modulus after the superposition of the probability amplitudes:

This expression bears a strong resemblance to the vector superposition of classical electromagnetic waves—indeed, if the quantum states above are substituted with specific electromagnetic wave forms, that is, using the electromagnetic field to represent the photon's wave function, one can formally arrive at the same conclusion as classical interference. However, this equivalence is fundamentally erroneous because the electromagnetic field is an observable quantity, whereas the wave function in the Copenhagen interpretation is a non-observable quantity; the Double-slit experiment viewed from the perspective of photons is the interference of the probability wave of a single photon itself, and the probability is the likelihood of a single photon appearing in a specific quantum state, not the number of photons in that specific quantum state. On this point, Paul Dirac provided clarification in *The Principles of Quantum Mechanics*:

"The association of waves with particles was not made until quite recently. It was made by Einstein in connection with his theory of the photoelectric effect. He showed that the energy of a photon is proportional to the frequency of the associated wave. Later, de Broglie suggested that the same proportionality holds for material particles, the constant of proportionality being Planck's constant. This suggestion was confirmed by experiment. The development of the theory led to the present quantum mechanics, in which waves and particles are on the same footing, the waves being the probability waves associated with the particles. The theory is essentially statistical in character. It does not answer the question: 'What is the state of the system?' but only: 'What is the probability of finding the system in a given state?' The waves give us information about these probabilities. They do not tell us how many photons are present, but only the probability of finding a photon in a particular place. The importance of this distinction can be seen in the following way. Suppose we have a beam of light split into two beams of equal intensity. According to the idea that the intensity is connected with the number of photons, we should have half the total number of photons going into each beam. Now, if the two beams are allowed to interfere, we require that a photon in one beam can interfere with a photon in the other beam. In some cases, the two photons would cancel each other, and in other cases, they would produce four photons. This would contradict the conservation of energy. The new theory, which connects the wave function with the probability for one photon, overcomes this difficulty, because it makes each photon go partly into each of the two beams. Thus, each photon interferes only with itself. Interference between two different photons never occurs." — Paul Dirac, The Principles of Quantum Mechanics, Fourth Edition, Chapter 1, Section 3 photon "The association of waves with particles was not made until quite recently. It was made by Einstein in connection with his theory of the photoelectric effect. He showed that the energy of a photon is proportional to the frequency of the associated wave. Later, de Broglie suggested that the same proportionality holds for material particles, the constant of proportionality being Planck's constant. This suggestion was confirmed by experiment. The development of the theory led to the present quantum mechanics, in which waves and particles are on the same footing, the waves being the probability waves associated with the particles. The theory is essentially statistical in character. It does not answer the question: 'What is the state of the system?' but only: 'What is the probability of finding the system in a given state?' The waves give us information about these probabilities. They do not tell us how many photons are present, but only the probability of finding a photon in a particular place. The importance of this distinction can be seen in the following way. Suppose we have a beam of light split into two beams of equal intensity. According to the idea that the intensity is connected with the number of photons, we should have half the total number of photons going into each beam. Now, if the two beams are allowed to interfere, we require that a photon in one beam can interfere with a photon in the other beam. In some cases, the two photons would cancel each other, and in other cases, they would produce four photons. This would contradict the conservation of energy. The new theory, which connects the wave function with the probability for one photon, overcomes this difficulty, because it makes each photon go partly into each of the two beams. Thus, each photon interferes only with itself. Interference between two different photons never occurs." — Paul Dirac, The Principles of Quantum Mechanics, Fourth Edition, Chapter 1, Section 3 quantum mechanics "The association of waves with particles was not made until quite recently. It was made by Einstein in connection with his theory of the photoelectric effect. He showed that the energy of a photon is proportional to the frequency of the associated wave. Later, de Broglie suggested that the same proportionality holds for material particles, the constant of proportionality being Planck's constant. This suggestion was confirmed by experiment. The development of the theory led to the present quantum mechanics, in which waves and particles are on the same footing, the waves being the probability waves associated with the particles. The theory is essentially statistical in character. It does not answer the question: 'What is the state of the system?' but only: 'What is the probability of finding the system in a given state?' The waves give us information about these probabilities. They do not tell us how many photons are present, but only the probability of finding a photon in a particular place. The importance of this distinction can be seen in the following way. Suppose we have a beam of light split into two beams of equal intensity. According to the idea that the intensity is connected with the number of photons, we should have half the total number of photons going into each beam. Now, if the two beams are allowed to interfere, we require that a photon in one beam can interfere with a photon in the other beam. In some cases, the two photons would cancel each other, and in other cases, they would produce four photons. This would contradict the conservation of energy. The new theory, which connects the wave function with the probability for one photon, overcomes this difficulty, because it makes each photon go partly into each of the two beams. Thus, each photon interferes only with itself. Interference between two different photons never occurs." — Paul Dirac, The Principles of Quantum Mechanics, Fourth Edition, Chapter 1, Section 3

Although it is theoretically possible to emit only one photon at a time from a coherent light source in a double-slit interference experiment, and according to the statistical interpretation of the wave function, classical interference fringes will be obtained on the screen after a long accumulation time; however, with current technology, obtaining a single-photon state remains very difficult—even when using a single-mode laser as the coherent source, multiple photons can still enter the photodetector very close to each other, which is a quantum effect of photons as Boson . A relatively feasible method in practical operation is to generate photon pairs, which can serve as an approximation to a single-photon state; in such a pair, the frequency and propagation direction of the second photon are correlated with those of the first, allowing it to be regarded as a Fock state of a single photon. One common method for generating photon pairs is atomic cascade. In experiments, calcium atoms are excited to the 6S0 state; they return to the ground state via a second-order radiation process, emitting a photon pair with wavelengths of 551.3 nanometers and 422.7 nanometers, respectively. Another more common method utilizes parametric down-conversion in nonlinear optics, where a single ultraviolet photon is used as the pump light in a Crystal . Through nonlinear effects, it generates a signal photon and an idler photon, both with wavelengths approximately twice that of the pump photon and polarization directions perpendicular to that of the pump photon. By using a birefringent crystal, phase matching between the pump light and the down-converted light can be achieved, maximizing the output light intensity. Both generated down-converted photons carry the phase information of the pump photon, placing them in an entangled state. Any measurement on the signal photon affects the quantum state of the idler photon, and vice versa.

Main article: Interferometry

Interferometry is a general term for techniques based on the theory of wave interference, which involves detecting interference patterns, frequencies, amplitudes, phases, and other properties of coherent Electromagnetic wave and applying them to various related measurements. Instruments used to implement interferometry are called interferometers. Interferometry plays a significant role in numerous scientific fields today, including astronomy, fiber optics, and engineering metrology. Generally, interferometry is divided into two basic types: homodyne detection and heterodyne detection.

Homodyne Detection

In interferometry, the interference of two waves as described earlier can be considered homodyne detection, meaning the two interfering electromagnetic waves have the same (carrier) frequency or wavelength. In homodyne detection, the measured electromagnetic wave is mixed with a known reference signal (often called a local oscillator), and the carrier frequencies of the measured signal and the reference signal are identical. The resulting interference field can eliminate the effects of frequency noise inherent to the electromagnetic wave itself. A typical optical homodyne detection setup is the Mach–Zehnder interferometer, where the measured signal and the reference signal originate from the same wave source.

Heterodyne Detection

An example of heterodyne interference: monochromatic waves with frequencies of 1 kHz, 1.4 kHz, 1.8 kHz, and 2.2 kHz undergo heterodyne interference, resulting in a beat frequency of 400 Hz.

Heterodyne detection involves the interference of two coherent electromagnetic waves with different but close frequencies. It was first mentioned in the research of the American- Canada inventor Reginald Fessenden. It achieves frequency modulation of the measured electromagnetic wave by mixing it with a reference signal. This method is now widely used in signal detection and analysis in fields such as telecommunications and astronomy, most commonly with Radio wave , Infrared , and visible light. The frequencies of the measured signal and the reference signal are similar but not identical. In heterodyne detection, both waves are incident on a mixing device—typically a (photo) Diode —where they undergo heterodyne interference.

If the electric field of the measured signal is denoted as, and the electric field of the reference signal is denoted as, then the light intensity received by the mixing device after heterodyne interference is

The final result shows that the interference light intensity originates from three distinct contributions: the direct current term (constant term), the high-frequency term, and the beat frequency term (low-frequency term). In heterodyne interference, the first two are typically Filter ed out by a Filter, retaining only the lower-frequency beat signal. In 1962, it was observed that the interference of two laser beams with very close frequencies on a photodetector would produce a beat frequency. Since then, heterodyne detection technology has undergone rapid development. The measurement of the beat frequency or phase can achieve very high precision, which has had a profound impact on length measurement by interferometry.

Optical Interferometry

Interferometry with Visible Light was the first category of interferometry to be developed and remains the most widely applied. Early practical applications included measurements of stellar angular diameters using the Michelson stellar interferometer. However, the challenge of obtaining stable coherent light sources was a major factor limiting the development of optical measurements for a long time. It was not until the 1960s that optical interferometry technology saw rapid advancement. This progress is attributed to the invention of the laser as a high-intensity coherent light source, the greatly enhanced capability of digital integrated circuits like computers to acquire and process data from interferometers, and the application of single-mode optical fibers, which increased the effective optical path length in experiments while maintaining very low noise. The development of electronics allowed researchers to move beyond merely observing the interference fringes produced by interferometers and instead directly measure the phase difference of coherent light. Important applications of optical interferometry in various areas are listed below.

Length Measurement

Length measurement is one of the most common applications of optical interferometry. To measure the absolute length of a sample, one of the simplest methods is to count the interference fringes generated by interference. In cases where the fringe count is non-integer, the wavelength of the coherent light can be successively doubled to produce narrower interference fringes until the desired measurement accuracy is achieved. Common methods also include the Hewlett-Packard interferometer developed by Hewlett-Packard. It operates a helium-neon Laser at two closely spaced frequencies by applying an axial magnetic field, causing the laser to emit two beams with a frequency difference of 2 Hertz . A polarizing beam splitter is then used to make these two beams produce heterodyne interference. The resulting difference frequency signal is recorded by a photodetector, and the change in optical path difference caused by the sample under test can be expressed by a counter as an integer multiple of the light wavelength. The Hewlett-Packard interferometer can measure lengths up to about 60 meters. With additional optical components, it can also be used for measuring angles, thickness, flatness, and other parameters. Furthermore, the difference frequency signal can be obtained through modulation methods, which can achieve higher difference frequencies, allowing for higher counting resolution from the difference frequency signal.

Another category of length measurement involves measuring changes in length. A common method utilizes heterodyne interference generated by modulation, where the phase difference carried by the beat frequency signal is recorded by a photodetector to obtain the length change. For measuring the coefficient of thermal expansion of materials with very low coefficients, such as fused quartz, a more precise method is often employed: two partially transmitting, partially reflecting glass plate are placed at both ends of the sample under test, forming a Fabry–Pérot interferometer. Two laser beams undergoing heterodyne interference are used, and through feedback, the frequency of one laser is locked to one of the transmission peak frequencies of the Fabry–Pérot interferometer. Consequently, when thermal expansion of the sample changes the length of the Fabry–Pérot interferometer, the shift in the transmission peak frequency causes a corresponding change in the locked laser frequency. This change is also reflected in the heterodyne signal and thus detected.

Optical Testing

Optical testing encompasses the inspection and evaluation of optical components and systems. Applications in this category include using equal thickness interference fringes to measure the thickness at various points on a glass plate and measuring the modulation transfer function (MTF) of camera lenses, among others. The most common method for testing the flatness of a sample surface using equal thickness interference is the Fizeau interferometer. It utilizes collimated parallel light that reflects off the sample surface and interferes with the incident light to produce equal thickness fringes. Alternatively, the Twyman–Green interferometer, an improvement upon the Michelson interferometer, can be used. The Twyman–Green interferometer also uses a collimated parallel light source and, being derived from the Michelson design, allows the optical paths of the two coherent beams to be very close. This reduces the requirement for the source's coherence length compared to the Fizeau interferometer.

Another type of interferometer widely used for testing optical surface figures, optical system aberrations, and measuring optical transfer functions is the shearing interferometer. It splits the wavefront exiting the sample under test into two, displacing them relative to each other by a certain distance (called the shear). The overlapping portions of the two wavefronts then produce an interference pattern. Shearing interferometers are categorized into tangential shearing, normal shearing, and rotational shearing types. A tangential shearing interferometer typically consists of a parallel plate or a slightly angled wedge; collimated light incident on the parallel plate produces two displaced coherent beams. Normal shearing interferometers are similar in principle to Fizeau and Twyman–Green interferometers. The advantage of shearing interferometers is that they eliminate the need for a separate reference optical surface, resulting in a simpler structure and nearly equal optical paths for the two coherent beams. Their disadvantage lies in the more complex numerical analysis required for the interference patterns.

Interference Spectroscopy

The solar Solar Corona observed using the LASCO C1 camera aboard the SOHO satellite. Multiple wavelengths of the Fe XIV 5308 Å spectral line were precisely measured using a Fabry–Pérot interferometer. These wavelengths experience Doppler shifts due to the relative motion between the Plasma in the corona and the observing satellite. Different colors are used to represent different degrees of Doppler shift in the images, thus different colors also indicate different relative velocities.

The ratio of the central wavelength of two spectral lines that a spectrometer can resolve to the minimum resolvable wavelength difference is called the chromatic resolving power of the spectrometer. For prism spectrometers utilizing the Dispersion effect and grating spectrometers utilizing the diffraction effect, their chromatic resolving power typically does not exceed an order of magnitude of 10. However, by employing a Fabry–Pérot interferometer, since the half-width of the transmission peak equals the free spectral range of the interferometer divided by its finesse:

And by substituting the interference condition, we obtain

Thus, the chromatic resolving power of a Fabry–Pérot interferometer is . Generally, the interference order , and the finesse is at least , so the chromatic resolving power of an interference spectrometer is on the order of 10^5 to 10^6 or higher.

Another important application of interferometers is in the construction of wavelength meters, which are further divided into dynamic wavelength meters and static wavelength meters. The former contains movable components to adjust the optical path difference, while the latter is composed of multiple Michelson interferometers with optical path differences that are multiples of each other or multiple Fabry–Pérot interferometers with free spectral ranges that are multiples of each other. Furthermore, by utilizing heterodyne interference with lasers, combined with a Fabry–Pérot interferometer, the frequency of a laser or the frequency difference between two lasers can be measured more precisely. Additionally, by employing modulation and fiber delay, the linewidth of a laser can also be measured.

Astrometry

Before the invention of the Michelson stellar interferometer, measuring the angular diameter of stars remained a significant challenge in astronomy, as even the largest known stars had angular diameters of only about 10 Arcsecond . However, even the Michelson stellar interferometer had a limited resolution, capable of measuring only the angular diameters of certain giant stars and was ineffective for stars of somewhat smaller mass. It was the invention of laser and heterodyne interference technology that sparked a revolution in the field of stellar interferometry starting in the 1970s. In these improved interferometers, starlight collected by telescopes undergoes heterodyne interference with a local laser source. The frequencies of the two light sources are very close, generating a beat frequency signal in the radio frequency domain. Furthermore, because the intensity of this beat signal originates from the product of the intensities of the starlight and the laser light, this interference method achieves higher resolution. Additionally, most of these experiments utilized carbon dioxide lasers with a wavelength of 10.6 micrometers, as longer wavelengths enhance the resolution of heterodyne interference. In 1974, Johnson, Betz, and Townes constructed a heterodyne interferometer with a baseline length of 5.5 meters. It employed a frequency-stabilized carbon dioxide laser with a power of 1 watt, operating at a wavelength of 10.6 micrometers. They used this interferometer to observe a series of infrared sources, including M-type supergiants and Mira variables, obtaining information such as the temperature and mass distribution of circumstellar dust shells. Today, with advancements in technology and manufacturing processes, the baseline length of such interferometers can be extended to hundreds of meters, thereby overcoming the difficulties initially encountered by the Michelson stellar interferometer.

Another issue in astrometry concerns the measurement of the positions and motions of celestial bodies. By precisely determining the positions of stars, the observed positions of radio sources can be compared with their corresponding observed optical positions. This allows for the direct measurement of their parallax and the establishment of the cosmic distance scale. Furthermore, such measurements can help determine the size and shape of Binary system orbits. Interferometers of this type include the Naval Prototype Optical Interferometer (NPOI) located in Arizona . It consists of four elements arranged in a Y-shape, with interferometer arm lengths of 20 meters between them. The NPOI can achieve astrometric positioning of celestial bodies at the milliarcsecond level. Another example is the Astronomical Interferometer for Extrasolar Planets (ASEPS-0), which studies extrasolar planets by monitoring the reflex motion of a star caused by planets orbiting it.

Gravitational wave detection

Gravitational wave are disturbances in spacetime that propagate at the speed of light, as predicted by the theory of General Relativity . Although the interaction between gravitational waves and matter is extremely weak, indirect astronomical observations, such as those of binary star systems, have provided evidence for their existence and demonstrated their significant influence on the physical properties of such celestial objects. The direct detection of gravitational waves can not only verify general relativity but, more importantly, provides a new observational method distinct from traditional astronomy based on electromagnetic wave observations. Furthermore, due to the different properties of electromagnetic waves and gravitational waves, gravitational wave astronomy will study another aspect of the universe that is inaccessible through electromagnetic wave observations. Since the 1970s, it has been gradually recognized that gravitational wave detectors based on the principle of interference are a promising design. The basic configuration of such detectors is an equal-arm Michelson interferometer. Essentially, a laser interferometer gravitational-wave detector measures changes in the lengths of the interferometer arms and analyzes the observed data, with the hope of identifying the effects caused by gravitational waves. That is, the ratio of the change in interferometer arm length caused by a gravitational wave to the arm length itself.

Here, h+ and h× are the two polarization states of the gravitational wave, F+ and F× are the detector's responses to these two polarization states respectively, and h is the strain amplitude of the gravitational wave. In practice, noise from external vibrations, molecular thermal motion, and shot noise from the photodetector readout superimposes onto the observation data. Therefore, to detect gravitational waves from typical astronomical sources, the detector's sensitivity must be better than 10^(-21) and other noise sources must be minimized as much as possible. By using longer interferometer arms, adding Fabry–Pérot resonators at both ends, and employing power recycling techniques, among other methods, noise can be effectively reduced and the interferometer's sensitivity can be improved.

The Laser Interferometer Gravitational-Wave Observatory (LIGO) in Louisiana and State of Washington , USA, is a typical ground-based gravitational wave detector based on the Michelson interferometer and Fabry–Pérot resonators. It is designed to detect gravitational wave signals in the frequency range of 20 Hz to 10 kHz. Ground-based detectors with the same architecture include VIRGO in The Republic of Italy , GEO600 in Germany, TAMA300 in Japan, and the planned LCGT. The National Aerospace (NASA) and the European Space Agency (ESA) are collaborating on the development of the Laser Interferometer Space Antenna (LISA) project, which plans to perform laser interferometry similar to a Michelson interferometer in space to detect gravitational waves in the low-frequency band (30 microhertz to 0.1 Hz). Additionally, Japan's planned Deci-hertz Interferometer Gravitational Wave Observatory (DECIGO) is also a space-based project. It is hoped that DECIGO will be able to detect gravitational waves in the decihertz range, thereby filling the gap between the operational frequency domains of LIGO and LISA.

Radio interferometry

Main article: Radio astronomy

The angular resolution of a telescope is proportional to the wavelength divided by the aperture. Since the wavelength of radio waves is much longer than that of visible light, a single radio telescope cannot achieve the resolution required to observe typical radio sources (for example, to achieve a resolution of 1 milliarcsecond using radio waves with a wavelength of 2.8 centimeters would require a telescope aperture of up to 6,000 kilometers). For this reason, British astronomer Sir Martin Ryle and others invented radio interferometry in 1946, using a radio interferometer consisting of two antennas to observe the Sun. Radio interferometry employs an array of multiple discrete radio telescopes. During observation, all telescopes are aimed at the same radio source, and the signals obtained from each are connected via Coaxial Cable , waveguides, or optical fibers to produce interference. This interference not only enhances the strength of the observed signal but also increases the effective aperture of the observation due to the long baseline distances between the telescopes. Because the telescopes are located at different positions, there is a time delay in the arrival of the same wavefront at each telescope. Therefore, appropriate delays must be applied to the signals that arrive earlier to maintain temporal coherence among the signals. Furthermore, the greater the number of telescopes involved in the interferometer, the better. This is because when observing the intensity distribution across the surface of a radio source, an interferometer composed of only two telescopes can only sample one spatial frequency of the Fourier transform (i.e., the visibility) of the intensity distribution (here, spatial frequency refers to the Fourier frequency describing how quickly the intensity changes in different directions). By using an array of multiple telescopes, the radio source can be observed at multiple spatial frequencies. The observed visibility function is then subjected to an inverse Fourier transform to obtain the intensity distribution of the radio source. This method is called aperture synthesis. For example, the Very Large Array (VLA) in New Mexico consists of 27 radio telescopes, each comprising a parabolic antenna 25 meters in diameter. Together, they form 351 independent interferometric baselines, with the longest effective baseline reaching up to 36 kilometers.

In the late 1960s, as the performance and stability of radio telescope receivers improved, it became possible to produce interference between the same radio signals observed by telescopes located very far apart worldwide (and even in Earth's orbit ). This technique is known as Very Long Baseline Interferometry (VLBI). VLBI does not require a physical connection between the observing signals. Instead, timing information calibrated by atomic clock is embedded within the signal data itself, and this data is later correlated. Since this data is observed from widely separated locations, the equivalent baseline can be extremely long. Operational VLBI instruments include the Very Long Baseline Array (VLBA), located in the contiguous United States and its overseas territories (with a baseline length of 8,611 kilometers), and the European VLBI Network (EVN), spread across the Eurasian and African continents. These interferometric arrays typically operate independently, but they can achieve simultaneous observations in specific projects, thereby forming a global Very Long Baseline Interferometry network.

When two waves propagate in the same medium and overlap while traveling in opposite directions, the particles of the medium within the overlapping region are simultaneously acted upon by both waves. If the amplitude of the waves is not too large, the displacement of the vibrating particles in the overlapping region equals the vector sum of the displacements caused by each individual wave. This is known as the principle of superposition of waves. (Interference phenomena also occur during the propagation of light waves, but in that case, there are no particles of a medium being acted upon.)

In phase: If the crests (or troughs) of two waves arrive at the same point simultaneously, the waves are said to be in phase at that point.

Out of phase (half-cycle out of phase): If the crest of one wave and the trough of the other wave arrive at the same point simultaneously, the waves are said to be out of phase at that point.

The waveform and propagation speed of the two waves after they meet will not change as a result of having overlapped.

Fundamental Principle: A beam of light emitted from a single point of a light source is divided into two beams, which are then made to meet.

Two fundamental methods:

Wavefront splitting method (e.g., Young's double-slit interference, Lloyd mirror, Fresnel's double mirror, and Fresnel's biprism) and amplitude splitting method (e.g., thin-film interference, wedge interference, Newton's ring , and Michelson interferometer).

1. Constructive interference:

When two waves overlap, if the amplitude of the resulting wave is greater than that of the component waves, it is called constructive interference.

If the two waves interfere exactly in phase, the maximum amplitude is produced, which is called fully constructive interference.

2. Destructive interference:

When two waves overlap, if the amplitude of the resulting wave is less than that of the component waves, it is called destructive interference.

If the two waves interfere exactly out of phase, the minimum amplitude is produced, which is called fully destructive interference.

When two sine waves with identical amplitude, wavelength, and period travel in opposite directions, they undergo interference and form a standing wave . For details, see: The concept of standing waves.

In genetics, the phenomenon where the occurrence of a single crossover influences the occurrence of another single crossover in its vicinity is called interference.

When several waves meet, each wave can maintain its own state and continue to propagate forward in its original direction, without affecting each other, as if it had not encountered other waves.