For a complex electric field ${\displaystyle E(t)}$ , the field autocorrelation function is defined by

${\displaystyle A(\tau )=\int _{-\infty }^{+\infty }E(t)E^{*}(t-\tau )dt}$ 

The Wiener-Khinchin theorem states that the Fourier transform of the field autocorrelation is the spectrum of ${\displaystyle E(t)}$ , i.e., the square of the magnitude of the Fourier transform of ${\displaystyle E(t)}$ . As a result, the field autocorrelation is not sensitive to the spectral phase.

The field autocorrelation is readily measured experimentally by placing a slow detector at the output of a Michelson interferometer.^[4] The detector is illuminated by the input electric field ${\displaystyle E(t)}$  coming from one arm, and by the delayed replica ${\displaystyle E(t-\tau )}$  from the other arm. If the time response of the detector is much larger than the time duration of the signal ${\displaystyle E(t)}$ , or if the recorded signal is integrated, the detector measures the intensity ${\displaystyle I_{M}}$  as the delay ${\displaystyle \tau }$  is scanned:

${\displaystyle I_{M}(\tau )=\int _{-\infty }^{+\infty }|E(t)+E(t-\tau )|^{2}dt}$ 

Expanding ${\displaystyle I_{M}(\tau )}$  reveals that one of the terms is ${\displaystyle A(\tau )}$ , proving that a Michelson interferometer can be used to measure the field autocorrelation, or the spectrum of ${\displaystyle E(t)}$  (and only the spectrum). This principle is the basis for Fourier transform spectroscopy.

To a complex electric field ${\displaystyle E(t)}$  corresponds an intensity ${\displaystyle I(t)=|E(t)|^{2}}$  and an intensity autocorrelation function defined by

${\displaystyle A(\tau )=\int _{-\infty }^{+\infty }I(t)I(t-\tau )dt}$ 

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 ${\displaystyle (E(t)+E(t-\tau ))^{2}}$ . Only the beam propagating on the optical axis, proportional to the cross-product ${\displaystyle E(t)E(t-\tau )}$ , is retained. This signal is then recorded by a slow detector, which measures

${\displaystyle I_{M}(\tau )=\int _{-\infty }^{+\infty }|E(t)E(t-\tau )|^{2}dt=\int _{-\infty }^{+\infty }I(t)I(t-\tau )dt}$ 

${\displaystyle I_{M}(\tau )}$  is exactly the intensity autocorrelation ${\displaystyle A(\tau )}$ .

The generation of the second harmonic in crystals is a nonlinear process that requires high peak power, unlike the previous setup. However, such high peak power can be obtained from a limited amount of energy by ultrashort pulses, 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 time profile, the autocorrelation width is ${\displaystyle {\sqrt {2}}}$  longer than the width of the intensity, and it is 1.54 longer in the case of a hyperbolic secant squared (sech²) 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.

As a combination of both previous cases, a nonlinear crystal can be used to generate the second harmonic at the output of a Michelson interferometer, in a collinear geometry. In this case, the signal recorded by a slow detector is

${\displaystyle I_{M}(\tau )=\int _{-\infty }^{+\infty }|(E(t)+E(t-\tau ))^{2}|^{2}dt}$ 

${\displaystyle I_{M}(\tau )}$  is called the interferometric autocorrelation. It contains some information about the phase of the pulse: the fringes in the autocorrelation trace wash out as the spectral phase becomes more complex.^[5]

The optical transfer function T(w) of an optical system is given by the autocorrelation of its pupil function f(x,y):

${\displaystyle T(w)={\frac {\int _{w/2}^{1}\int _{0}^{\sqrt {1-x^{2}}}f(x,y)f^{*}(x-w,y)dydx}{\int _{0}^{1}\int _{0}^{\sqrt {1-x^{2}}}f(x,y)^{2}dydx}}}$ 

• Autocorrelator
• Convolution
• Degree of coherence
• Frequency-resolved optical gating
• Multiphoton intrapulse interference phase scan
• Spectral phase interferometry for direct electric-field reconstruction