Interference of light waves 3

In physics, two wave sources are perfectly coherent if they have a constant phase difference and the same frequency. It is an ideal property of waves that enables stationary (i.e. temporally and spatially constant) interference. It contains several distinct concepts, which are limiting cases that never quite occur in reality but allow an understanding of the physics of waves, and has become a very important concept in quantum physics. More generally, coherence describes all properties of the correlation between physical quantities of a single wave, or between several waves or wave packets.

Interference is nothing more than the addition, in the mathematical sense, of wave functions. A single wave can interfere with itself, but this is still an addition of two waves (see Young s slits experiment). Constructive or destructive interferences are limit cases, and two waves always interfere, even if the result of the addition is complicated or not remarkable.

When interfering, two waves can add together to create a wave of greater amplitude than either one (constructive interference) or subtract from each other to create a wave of lesser amplitude than either one (destructive interference), depending on their relative phase. Two waves are said to be coherent if they have a constant relative phase. The amount of coherence can readily be measured by the interference visibility, which looks at the size of the interference fringes relative to the input waves (as the phase offset is varied); a precise mathematical definition of the degree of coherence is given by means of correlation functions.

Spatial coherence describes the correlation (or predictable relationship) between waves at different points in space, either lateral or longitudinal.[1] Temporal coherence describes the correlation between waves observed at different moments in time. Both are observed in the Michelson–Morley experiment and Young s interference experiment. Once the fringes are obtained in the Michelson interferometer, when one of the mirrors is moved away gradually, the time for the beam to travel increases and the fringes become dull and finally are lost, showing temporal coherence. Similarly, if in a double-slit experiment, the space between the two slits is increased, the coherence dies gradually and finally the fringes disappear, showing spatial coherence. In both cases, the fringe amplitude slowly disappears, as the path difference increases past the coherence length.

Contents
1 Introduction
2 Mathematical definition
3 Coherence and correlation
4 Examples of wave-like states
5 Temporal coherence
5.1 The relationship between coherence time and bandwidth
5.2 Examples of temporal coherence
5.3 Measurement of temporal coherence
6 Spatial coherence
6.1 Examples of spatial coherence
7 Spectral coherence
7.1 Measurement of spectral coherence
8 Polarization and coherence
9 Applications
9.1 Holography
9.2 Non-optical wave fields
10 Quantum coherence
11 See also
12 References
13 External links
Introduction[edit]
Coherence was originally conceived in connection with Thomas Young s double-slit experiment in optics but is now used in any field that involves waves, such as acoustics, electrical engineering, neuroscience, and quantum mechanics. The property of coherence is the basis for commercial applications such as holography, the Sagnac gyroscope, radio antenna arrays, optical coherence tomography and telescope interferometers (astronomical optical interferometers and radio telescopes).

Mathematical definition[edit]
A precise definition is given at degree of coherence.

The coherence function between two signals {\displaystyle x(t)} x(t) and {\displaystyle y(t)} y(t) is defined as[2]

{\displaystyle \gamma _{xy}^{2}(f)={\frac {|S_{xy}(f)|^{2}}{S_{xx}(f)S_{yy}(f)}}} {\displaystyle \gamma _{xy}^{2}(f)={\frac {|S_{xy}(f)|^{2}}{S_{xx}(f)S_{yy}(f)}}}
where {\displaystyle S_{xy}(f)} {\displaystyle S_{xy}(f)} is the cross-spectral density of the signal and {\displaystyle S_{xx}(f)} {\displaystyle S_{xx}(f)} and {\displaystyle S_{yy}(f)} {\displaystyle S_{yy}(f)} are the power spectral density functions of {\displaystyle x(t)} x(t) and {\displaystyle y(t)} y(t) , respectively. The cross-spectral density and the power spectral density are defined as the Fourier transforms of the cross-correlation and the autocorrelation signals, respectively. For instance, if the signals are functions of time, the cross-correlation is a measure of the similarity of the two signals as a function of the time lag relative to each other and the autocorrelation is a measure of the similarity of each signal with itself in different instants of time. In this case the coherence is a function of frequency. Analogously, if {\displaystyle x(t)} x(t) and {\displaystyle y(t)} y(t) are functions of space, the cross-correlation measures the similarity of two signals in different points in space and the autocorrelations the similarity of the signal relative to itself for a certain separation distance. In that case, coherence is a function of wavenumber (spatial frequency).

The coherence varies in the interval {\displaystyle 0\leqslant \gamma _{xy}^{2}(f)\leqslant 1.} {\displaystyle 0\leqslant \gamma _{xy}^{2}(f)\leqslant 1.}. If {\displaystyle \gamma _{xy}^{2}(f)=1} {\displaystyle \gamma _{xy}^{2}(f)=1} it means that the signals are perfectly correlated or linearly related and if {\displaystyle \gamma _{xy}^{2}(f)=0} {\displaystyle \gamma _{xy}^{2}(f)=0} they are totally uncorrelated. If a linear system is being measured, {\displaystyle x(t)} x(t) being the input and {\displaystyle y(t)} y(t) the output, the coherence function will be unitary all over the spectrum. However, if non-linearities are present in the system the coherence will vary in the limit given above.

Coherence and correlation[edit]
The coherence of two waves expresses how well correlated the waves are as quantified by the cross-correlation function.[3][4][5][6][7] The cross-correlation quantifies the ability to predict the phase of the second wave by knowing the phase of the first. As an example, consider two waves perfectly correlated for all times. At any time, phase difference will be constant.[clarification needed] If, when combined, they exhibit perfect constructive interference, perfect destructive interference, or something in-between but with constant phase difference, then it follows that they are perfectly coherent. As will be discussed below, the second wave need not be a separate entity. It could be the first wave at a different time or position. In this case, the measure of correlation is the autocorrelation function (sometimes called self-coherence). Degree of correlation involves correlation functions.[8]:545-550

Examples of wave-like states[edit]