Editing Optical autocorrelation (section)

==Intensity autocorrelation==
To a complex electric field <math>E(t)</math> corresponds an intensity <math>I(t) = |E(t)|^2</math> and an intensity autocorrelation function defined by

: <math>A(\tau) = \int_{-\infty}^{+\infty}I(t)I(t-\tau)dt</math>

The optical implementation of the intensity autocorrelation is not as straightforward as for the field autocorrelation. Similarly to the previous setup, two parallel beams with a variable delay are generated, then focused into a second-harmonic-generation crystal (see [[nonlinear optics]]) to obtain a signal proportional to <math>(E(t)+E(t-\tau))^2</math>. Only the beam propagating on the optical axis, proportional to the cross-product <math>E(t)E(t-\tau)</math>, is retained. This signal is then recorded by a slow detector, which measures

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

<math>I_M(\tau)</math> is exactly the intensity autocorrelation <math>A(\tau)</math>.

[[Image:optical-intensity-autocorrelation.png|thumb|upright=1.75|left|Two [[ultrashort pulse]]s (a) and (b) with their respective intensity autocorrelation (c) and (d). Because the intensity autocorrelation ignores the temporal phase of pulse (b) that is due to the instantaneous frequency sweep ([[chirp]]), both pulses yield the same intensity autocorrelation.  Here, identical Gaussian temporal profiles have been used, resulting in an intensity autocorrelation width 2<sup>1/2</sup> longer than the original intensities. Note that an intensity autocorrelation has a background that is ideally half as big as the actual signal. The zero in this figure has been shifted to omit this background.]]

The generation of the second harmonic in crystals is a nonlinear process that requires high peak [[power (physics)|power]], unlike the previous setup. However, such high peak power can be obtained from a limited amount of [[energy]] by [[ultrashort pulse]]s, and as a result their intensity autocorrelation is often measured experimentally. Another difficulty with this setup is that both beams must be focused at the same point inside the crystal ''as the delay is scanned'' in order for the second harmonic to be generated.

It can be shown that the intensity autocorrelation width of a pulse is related to the intensity width.  For a [[Gaussian function|Gaussian]] time profile, the autocorrelation width is <math>\sqrt{2}</math> longer than the width of the intensity, and it is 1.54 longer in the case of a [[hyperbolic functions|hyperbolic secant]] squared (sech<sup>2</sup>) pulse.  This numerical factor, which depends on the shape of the pulse, is sometimes called the ''deconvolution factor''.  If this factor is known, or assumed, the time duration (intensity width) of a pulse can be measured using an intensity autocorrelation. However, the phase cannot be measured.
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