Consider a wave propagating in space. Coherence is a measure of the correlation between its phases, measured at various points. The coherence of a wave depends on the characteristics of its source.
Two types of coherence
Let's look at a simple example. Imagine two floats rising and falling on the surface of the water. Suppose that the source of the waves is a single stick, which is harmoniously immersed and removed from the water, violating the calm surface of the water surface. In this case, there is an ideal correlation between the movements of the two floats. They may not rise and fall exactly in phase when one goes up and the other down, but the phase difference between the positions of the two floats is constant in time. A harmonically oscillating point source produces an absolutely coherent wave.
When the coherence of light waves is described, its two types are distinguished - temporal and spatial.
Coherence refers to the ability of light to produce an interference pattern. If two light waves are brought together and they do not create areas of increased and decreased brightness, they are called incoherent. If they produce an “ideal” interference picture (in the sense of the existence of regions of complete destructive interference), then they are completely coherent. If two waves create a "less perfect" picture, then they are considered to be partially coherent.
Michelson Interferometer
Coherence is a phenomenon that is best explained by experiment.
In the Michelson interferometer, the light from the source S (which can be any: the sun, a laser, or stars) is directed to a translucent mirror M 0 , which reflects 50% of the light in the direction of the mirror M 1 and passes 50% in the direction of the mirror M 2 . The beam is reflected from each of the mirrors, returns to M 0 , and equal parts of the light reflected from M 1 and M 2 are combined and projected onto screen B. The device can be adjusted by changing the distance from the mirror M 1 to the beam splitter.
The Michelson interferometer essentially mixes the beam with its own version delayed in time. The light that passes along the path to the mirror M 1 must travel a distance 2d more than the beam that moves to the mirror M 2 .
Coherence Length and Time
What is observed on the screen? At d = 0, many very distinct interference fringes are visible. When d increases, the stripes become less pronounced: dark areas become brighter, and light areas become dimmer. Finally, at very large d exceeding a certain critical value of D, the light and dark rings disappear completely, leaving only a blurry spot.
Obviously, a light field cannot interfere with a time-delayed version of itself if the time delay is large enough. 2D distance is the length of coherence: interference effects are noticeable only when the path difference is less than this distance. This value can be converted at time t by dividing it by the speed of light with: t c = 2D / s.
The Michelson experiment measures the temporal coherence of a light wave: its ability to interfere with a delayed version of itself. For a well-stabilized laser, t c = 10 -4 s, l c = 30 km; for filtered thermal light, t c = 10 -8 s, l c = 3 m.
Coherence and Time
Temporal coherence is a measure of the correlation between the phases of a light wave at different points along the propagation direction.
Suppose the source emits waves of length λ and λ ± Δλ, which at some point in space will interfere at a distance l c = λ 2 / (2πΔλ). Here l c is the coherence length.
The phase of the wave propagating in the x direction is specified as φ = kx - ωt. If we consider the pattern of waves in space at time t at a distance l c , the phase difference between two waves with vectors k 1 and k 2 that are in phase at x = 0 is Δφ = l c (k 1 - k 2 ). When Δφ = 1, or Δφ ~ 60 °, the light is no longer coherent. Interference and diffraction have a significant effect on contrast.
In this way:
- 1 = l c (k 1 - k 2 ) = l c (2π / λ - 2π / (λ + Δλ));
- l c (λ + Δλ - λ) / (λ (λ + Δλ)) ~ l c Δλ / λ 2 = 1 / 2π;
- l c = λ 2 / (2πΔλ).
A wave travels through space at a speed of c.
Coherence time t c = l c / s. Since λf = c, it follows that Δf / f = Δω / ω = Δλ / λ. We can write
- l c = λ 2 / (2πΔλ) = λf / (2πΔf) = s / Δω;
- t c = 1 / Δω.
If the wavelength or propagation frequency of the light source is known, l c and t c can be calculated. It is not possible to observe the interference pattern obtained by dividing the amplitude, such as thin-film interference, if the optical path difference significantly exceeds l c .
Temporal coherence indicates the monochrome source.
Coherence and space
Spatial coherence is a measure of the correlation between the phases of a light wave at different points transversely with respect to the direction of propagation.
At a distance L from a thermal monochromatic (linear) source, the linear dimensions of which are of the order of δ, two slots located at a distance exceeding d c = 0.16λL / δ no longer produce a recognizable interference pattern. πd c 2/4 is the source coherence area.
If at time t we look at a source of width δ located perpendicular to the distance L from the screen, then on the screen you can see two points (P1 and P2) separated by distance d. The electric field in P1 and P2 is a superposition of the electric fields of the waves emitted by all points of the source, the radiation of which is not interconnected. In order for electromagnetic waves leaving P1 and P2 to create a recognizable interference pattern, superpositions in P1 and P2 must be in phase.
Coherence condition
Light waves emitted by the two edges of the source at some point in time t have a certain phase difference right in the center between the two points. The ray going from the left edge of δ to the point P2 must go d (sinθ) / 2 further than the ray toward the center. The path of the ray going from the right edge of δ to the point P2 passes the path d (sinθ) / 2 less. The difference in the distance traveled for two rays is d · sinθ and represents the phase difference Δph '= 2πd · sinθ / λ. For the distance from P1 to P2 along the wave front, we obtain Δφ = 2Δφ '= 4πd · sinθ / λ. The waves emitted by the two edges of the source are in phase with P1 at time t and do not coincide in phase at a distance of 4πdsinθ / λ in P2. Since sinθ ~ δ / (2L), it follows that Δφ = 2πdδ / (Lλ). When Δφ = 1 or Δφ ~ 60 °, the light is no longer considered coherent.
Δφ = 1 -> d = Lλ / (2πδ) = 0.16 Lλ / δ.
Spatial coherence indicates the uniformity of the phase of the wave front.
An incandescent lamp is an example of an incoherent light source.
Coherent light can be obtained from a source of incoherent radiation by discarding most of the radiation. First of all, spatial filtering is performed to increase spatial coherence, and then spectral filtering is performed to increase temporal coherence.
Fourier Series
A sinusoidal plane wave is absolutely coherent in space and time, and its length, time and area of coherence are infinite. All real waves are wave pulses that last for a finite time interval and have a finite perpendicular to their direction of propagation. Mathematically, they are described by non-periodic functions. To find the frequencies present in the wave pulses, it is necessary to analyze non-periodic functions to determine Δω and the coherence length.
According to Fourier analysis, an arbitrary periodic wave can be considered as a superposition of sine waves. Fourier synthesis means that the superposition of many sinusoidal waves allows you to get an arbitrary periodic waveform.
Communication with statistics
The coherence theory can be considered as the connection of physics with other sciences, since it is the result of the merger of electromagnetic theory and statistics, just as statistical mechanics is the union of mechanics with statistics. The theory is used to quantify and characterize the effects of random fluctuations on the behavior of light fields.
It is usually not possible to measure the fluctuations of the wave field directly. The individual “rises and falls” of visible light cannot be detected directly or even with complex instruments: its frequency is of the order of 10 15 vibrations per second. Only averaged indicators can be measured.
Coherence Application
The connection of physics with other sciences by the example of coherence can be traced in a number of applications. Partially coherent fields are less susceptible to atmospheric turbulence, which makes them useful for laser communications. They are also used in the study of laser-induced fusion reactions: a decrease in the effect of interference leads to a “smooth” action of the beam on the fusion target. Coherence is used, in particular, to determine the size of stars and the allocation of binary stellar systems.
The coherence of light waves plays an important role in the study of quantum as well as classical fields. In 2005, Roy Glauber became one of the Nobel Prize winners in physics for his contribution to the development of the quantum theory of optical coherence.