Editing Optical autocorrelation (section)

==Field autocorrelation==
[[Image:optical-field-autocorrelation-setup.svg|thumb|upright=1.2|Setup for a field autocorrelator, based on a [[Michelson interferometer]]. '''L''': [[modelocking|modelocked]] [[laser]], '''BS''': [[beam splitter]], '''M1''': moveable [[mirror]] providing a variable [[Propagation delay|delay line]], '''M2''': fixed mirror, '''D''': [[energy]] detector.]]

For a complex electric field <math>E(t)</math>, the field autocorrelation function is defined by

: <math>A(\tau) = \int_{-\infty}^{+\infty}E(t)E^*(t-\tau)dt</math>
The [[Wiener-Khinchin theorem]] states that the [[Fourier transform]] of the field autocorrelation is the spectrum of <math>E(t)</math>, i.e., the square of the ''magnitude'' of the Fourier transform of <math>E(t)</math>. As a result, the field autocorrelation is not sensitive to the spectral ''phase''.
[[Image:optical-field-autocorrelation.png|thumb|upright=1.75|left|Two [[ultrashort pulse]]s (a) and (b) with their respective field autocorrelation (c) and (d). Note that the autocorrelations are symmetric and peak at zero delay. Unlike pulse (a), pulse (b) exhibits an instantaneous frequency sweep, called ''[[chirp]]'', and therefore contains more [[Bandwidth (signal processing)|bandwidth]] than pulse (a). Therefore, the field autocorrelation (d) is shorter than (c), because the spectrum is the Fourier transform of the field autocorrelation (Wiener-Khinchin theorem).]]

The field autocorrelation is readily measured experimentally by placing a slow detector at the output of a [[Michelson interferometer]].<ref>{{Cite journal | doi=10.1364/OE.409185| title=Fully symmetric dispersionless stable transmission-grating Michelson interferometer| year=2020| last1=Kolesnichenko| first1=Pavel| last2=Wittenbecher| first2=Lukas| last3=Zigmantas| first3=Donatas| journal=Optics Express| volume=28| issue=25| pages=37752–37757| doi-access=free| pmid=33379604| bibcode=2020OExpr..2837752K}}</ref>  The detector is illuminated by the input electric field <math>E(t)</math> coming from one arm, and by the delayed replica <math>E(t-\tau)</math> from the other arm.  If the time response of the detector is much larger than the time duration of the signal <math>E(t)</math>, or if the recorded signal is integrated, the detector measures the intensity <math>I_M</math> as the delay <math>\tau</math> is scanned:

: <math>I_M(\tau) = \int_{-\infty}^{+\infty}|E(t)+E(t-\tau)|^2dt</math>

Expanding <math>I_M(\tau)</math> reveals that one of the terms is <math>A(\tau)</math>, proving that a Michelson interferometer can be used to measure the field autocorrelation, or the spectrum of <math>E(t)</math> (and only the spectrum). This principle is the basis for [[Fourier transform spectroscopy]].
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