Wave interference

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
  $${\displaystyle W_{1}+W_{2}=0}$$

  

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.

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.

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.

The discussion above assumes that the waves which interfere with one another are monochromatic, i.e. have a single frequency—this requires that they are infinite in time. This is not, however, either practical or necessary. Two identical waves of finite duration whose frequency is fixed over that period will give rise to an interference pattern while they overlap. Two identical waves which consist of a narrow spectrum of frequency waves of finite duration (but shorter than their coherence time), will give a series of fringe patterns of slightly differing spacings, and provided the spread of spacings is significantly less than the average fringe spacing, a fringe pattern will again be observed during the time when the two waves overlap.

Conventional light sources emit waves of differing frequencies and at different times from different points in the source. If the light is split into two waves and then re-combined, each individual light wave may generate an interference pattern with its other half, but the individual fringe patterns generated will have different phases and spacings, and normally no overall fringe pattern will be observable. However, single-element light sources, such as sodium- or mercury-vapor lamps have emission lines with quite narrow frequency spectra. When these are spatially and colour filtered, and then split into two waves, they can be superimposed to generate interference fringes.[5] All interferometry prior to the invention of the laser was done using such sources and had a wide range of successful applications.

A laser beam generally approximates much more closely to a monochromatic source, and thus it is much more straightforward to generate interference fringes using a laser. The ease with which interference fringes can be observed with a laser beam can sometimes cause problems in that stray reflections may give spurious interference fringes which can result in errors.

Normally, a single laser beam is used in interferometry, though interference has been observed using two independent lasers whose frequencies were sufficiently matched to satisfy the phase requirements.[6] This has also been observed for widefield interference between two incoherent laser sources.[7]

White light interference in a soap bubble. The iridescence is due to thin-film interference.

It is also possible to observe interference fringes using white light. A white light fringe pattern can be considered to be made up of a 'spectrum' of fringe patterns each of slightly different spacing. If all the fringe patterns are in phase in the centre, then the fringes will increase in size as the wavelength decreases and the summed intensity will show three to four fringes of varying colour. Young describes this very elegantly in his discussion of two slit interference. Since white light fringes are obtained only when the two waves have travelled equal distances from the light source, they can be very useful in interferometry, as they allow the zero path difference fringe to be identified.[8]

To generate interference fringes, light from the source has to be divided into two waves which have then to be re-combined. Traditionally, interferometers have been classified as either amplitude-division or wavefront-division systems.

In an amplitude-division system, a beam splitter is used to divide the light into two beams travelling in different directions, which are then superimposed to produce the interference pattern. The Michelson interferometer and the Mach–Zehnder interferometer are examples of amplitude-division systems.

In wavefront-division systems, the wave is divided in space—examples are Young's double slit interferometer and Lloyd's mirror.

Interference can also be seen in everyday phenomena such as iridescence and structural coloration. For example, the colours seen in a soap bubble arise from interference of light reflecting off the front and back surfaces of the thin soap film. Depending on the thickness of the film, different colours interfere constructively and destructively.

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,[9] 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.

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.[10]

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.[11]

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.

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.