Real-valued wave functions

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

$${\displaystyle W_{1}(x,t)=A\cos(kx-\omega t)}$$

where ${\displaystyle A}$ is the peak amplitude, ${\displaystyle k=2\pi /\lambda }$ is the wavenumber and ${\displaystyle \omega =2\pi f}$ is the angular frequency of the wave. Suppose a second wave of the same frequency and amplitude but with a different phase is also traveling to the right

$${\displaystyle W_{2}(x,t)=A\cos(kx-\omega t+\varphi )}$$

where ${\displaystyle \varphi }$ is the phase difference between the waves in radians. The two waves will superpose and add: the sum of the two waves is

$${\displaystyle W_{1}+W_{2}=A[\cos(kx-\omega t)+\cos(kx-\omega t+\varphi )].}$$

Using the trigonometric identity for the sum of two cosines: ${\textstyle \cos a+\cos b=2\cos \left({a-b \over 2}\right)\cos \left({a+b \over 2}\right),}$ this can be written

$${\displaystyle W_{1}+W_{2}=2A\cos \left({\varphi \over 2}\right)\cos \left(kx-\omega t+{\varphi \over 2}\right).}$$

This represents a wave at the original frequency, traveling to the right like its components, whose amplitude is proportional to the cosine of ${\displaystyle \varphi /2}$.

• Constructive interference: If the phase difference is an even multiple of π: ${\displaystyle \varphi =\ldots ,-4\pi ,-2\pi ,0,2\pi ,4\pi ,\ldots }$ then ${\displaystyle \left|\cos(\varphi /2)\right|=1}$, so the sum of the two waves is a wave with twice the amplitude
  $${\displaystyle W_{1}+W_{2}=2A\cos(kx-\omega t)}$$

  
• Destructive interference: If the phase difference is an odd multiple of π: ${\displaystyle \varphi =\ldots ,-3\pi ,\,-\pi ,\,\pi ,\,3\pi ,\,5\pi ,\ldots }$ then ${\displaystyle \cos(\varphi /2)=0\,}$, so the sum of the two waves is zero
  $${\displaystyle W_{1}+W_{2}=0}$$

  

Between two plane waves

Geometrical arrangement for two plane wave interference

Interference fringes in overlapping plane waves

A simple form of interference pattern is obtained if two plane waves 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

$${\displaystyle \Delta \varphi ={\frac {2\pi d}{\lambda }}={\frac {2\pi x\sin \theta }{\lambda }}.}$$

It can be seen that the two waves are in phase when

$${\displaystyle {\frac {x\sin \theta }{\lambda }}=0,\pm 1,\pm 2,\ldots ,}$$

and are half a cycle out of phase when

$${\displaystyle {\frac {x\sin \theta }{\lambda }}=\pm {\frac {1}{2}},\pm {\frac {3}{2}},\ldots }$$

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

$${\displaystyle d_{f}={\frac {\lambda }{\sin \theta }}}$$

and d_f is known as the fringe spacing. The fringe spacing increases with increase in wavelength, and with decreasing angle θ.

The fringes are observed wherever the two waves overlap and the fringe spacing is uniform throughout.

Between two spherical waves

Optical interference between two point sources that have different wavelengths and separations of sources.

A point source 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 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 ${\displaystyle Ae^{i\varphi _{n}}}$ for ${\displaystyle N}$ waves from ${\displaystyle n=0}$ to ${\displaystyle n=N-1}$, where

$${\displaystyle \varphi _{n}-\varphi _{n-1}={\frac {2\pi }{N}}.}$$

To show that

$${\displaystyle \sum _{n=0}^{N-1}Ae^{i\varphi _{n}}=0}$$

one merely assumes the converse, then multiplies both sides by ${\displaystyle e^{i{\frac {2\pi }{N}}}.}$

The Fabry–Pérot interferometer uses interference between multiple reflections.

A diffraction grating 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 wave interference

Creation of interference fringes by an optical flat on a reflective surface. Light rays from a monochromatic source pass through the glass and reflect off both the bottom surface of the flat and the supporting surface. The tiny gap between the surfaces means the two reflected rays have different path lengths. In addition the ray reflected from the bottom plate undergoes a 180° phase reversal. As a result, at locations (a) where the path difference is an odd multiple of λ/2, the waves reinforce. At locations (b) where the path difference is an even multiple of λ/2 the waves cancel. Since the gap between the surfaces varies slightly in width at different points, a series of alternating bright and dark bands, interference fringes, are seen.

Because the frequency of light waves (~10¹⁴ 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 r is:

$${\displaystyle U_{1}(\mathbf {r} ,t)=A_{1}(\mathbf {r} )e^{i[\varphi _{1}(\mathbf {r} )-\omega t]}}$$

$${\displaystyle U_{2}(\mathbf {r} ,t)=A_{2}(\mathbf {r} )e^{i[\varphi _{2}(\mathbf {r} )-\omega t]}}$$

where A represents the magnitude of the displacement, φ represents the phase and ω represents the angular frequency.

The displacement of the summed waves is

$${\displaystyle U(\mathbf {r} ,t)=A_{1}(\mathbf {r} )e^{i[\varphi _{1}(\mathbf {r} )-\omega t]}+A_{2}(\mathbf {r} )e^{i[\varphi _{2}(\mathbf {r} )-\omega t]}.}$$

The intensity of the light at r is given by

$${\displaystyle I(\mathbf {r} )=\int U(\mathbf {r} ,t)U^{*}(\mathbf {r} ,t)\,dt\propto A_{1}^{2}(\mathbf {r} )+A_{2}^{2}(\mathbf {r} )+2A_{1}(\mathbf {r} )A_{2}(\mathbf {r} )\cos[\varphi _{1}(\mathbf {r} )-\varphi _{2}(\mathbf {r} )].}$$

This can be expressed in terms of the intensities of the individual waves as

$${\displaystyle I(\mathbf {r} )=I_{1}(\mathbf {r} )+I_{2}(\mathbf {r} )+2{\sqrt {I_{1}(\mathbf {r} )I_{2}(\mathbf {r} )}}\cos[\varphi _{1}(\mathbf {r} )-\varphi _{2}(\mathbf {r} )].}$$

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 2π. 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.

Quantum interference

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Quantum interference – the observed wave-behavior of matter^[9] – resembles optical interference. Let ${\displaystyle \Psi (x,t)}$ be a wavefunction solution of the Schrödinger equation for a quantum mechanical object. Then the probability ${\displaystyle P(x)}$ of observing the object at position ${\displaystyle x}$ is ${\displaystyle P(x)=|\Psi (x,t)|^{2}=\Psi ^{*}(x,t)\Psi (x,t)}$ where * indicates complex conjugation. Quantum interference concerns the issue of this probability when the wavefunction is expressed as a sum or linear superposition of two terms ${\displaystyle \Psi (x,t)=\Psi _{A}(x,t)+\Psi _{B}(x,t)}$:

$${\displaystyle P(x)=|\Psi (x,t)|^{2}=|\Psi _{A}(x,t)|^{2}+|\Psi _{B}(x,t)|^{2}+(\Psi _{A}^{*}(x,t)\Psi _{B}(x,t)+\Psi _{A}(x,t)\Psi _{B}^{*}(x,t))}$$

Usually, ${\displaystyle \Psi _{A}(x,t)}$ and ${\displaystyle \Psi _{B}(x,t)}$ correspond to distinct situations A and B. When this is the case, the equation ${\displaystyle \Psi (x,t)=\Psi _{A}(x,t)+\Psi _{B}(x,t)}$ 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 ${\displaystyle x}$ is the probability of finding the object at ${\displaystyle x}$ when it is in situation A plus the probability of finding the object at ${\displaystyle x}$ when it is in situation B plus an extra term. This extra term, which is called the quantum interference term, is ${\displaystyle \Psi _{A}^{*}(x,t)\Psi _{B}(x,t)+\Psi _{A}(x,t)\Psi _{B}^{*}(x,t)}$ in the above equation. As in the classical wave case above, the quantum interference term can add (constructive interference) or subtract (destructive interference) from ${\displaystyle |\Psi _{A}(x,t)|^{2}+|\Psi _{B}(x,t)|^{2}}$ in the above equation depending on whether the quantum interference term is positive or negative. If this term is absent for all ${\displaystyle x}$, 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 this experiment, matter waves from electrons, atoms or molecules approach a barrier with two slits in it. One slit becomes ${\displaystyle \Psi _{A}(x,t)}$ and the other becomes ${\displaystyle \Psi _{B}(x,t)}$. The interference pattern occurs on the far side, observed by detectors suitable to the particles originating the matter wave.^[10] The pattern matches the optical double slit pattern.

Beat

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 sustained tones, beats can be readily recognized. Tuning two tones to a unison 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 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,^[11] the beating starts to sound like a note, and a combination tone is produced. This combination tone can also be referred to as a missing fundamental, 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.^[8] 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). Richard Feynman was fond of saying that all of quantum mechanics can be gleaned from carefully thinking through the implications of this single experiment.^[12]

The results of the Michelson–Morley experiment are generally considered to be the first strong evidence against the theory of a luminiferous aether and in favor of special relativity.

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 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 chain in length measurement.

Interferometry is used in the calibration of slip gauges (called gauge blocks in the US) and in coordinate-measuring machines. It is also used in the testing of optical components.^[13]

Radio interferometry

The Very Large Array, an interferometric array formed from many smaller telescopes, like many larger radio telescopes.

In 1946, a technique called astronomical interferometry 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, waveguide, optical fiber, or other type of transmission line. Interferometry increases the total signal collected, but its primary purpose is to vastly increase the resolution through a process called Aperture synthesis. 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 is an instrument for measuring the physical characteristics of sound waves in a gas or liquid, such velocity, wavelength, absorption, or impedance. A vibrating crystal 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.