In optics, various autocorrelation functions can be experimentally realized. The field autocorrelation may be used to calculate the spectrum of a source of light, while the intensity autocorrelation and the interferometric autocorrelation are commonly used to estimate the duration of ultrashort pulses produced by modelocked lasers. The laser pulse duration cannot be easily measured by optoelectronic methods, since the response time of photodiodes and oscilloscopes are at best of the order of 200 femtoseconds, yet laser pulses can be made as short as a few femtoseconds. In the following examples, the autocorrelation signal is generated by the nonlinear process of second-harmonic generation (SHG). Other techniques based on two-photon absorption may also be used in autocorrelation measurements, as well as higher-order nonlinear optical processes such as third-harmonic generation, in which case the mathematical expressions of the signal will be slightly modified, but the basic interpretation of an autocorrelation trace remains the same. A detailed discussion on interferometric autocorrelation is given in several well-known textbooks.
Field autocorrelation
For a complex electric field E ( t ) {\displaystyle E(t)} , the field autocorrelation function is defined by
A ( τ ) = ∫ − ∞ + ∞ E ( t ) E ∗ ( t − τ ) d t {\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 E ( t ) {\displaystyle E(t)} , i.e., the square of the magnitude of the Fourier transform of E ( t ) {\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. The detector is illuminated by the input electric field E ( t ) {\displaystyle E(t)} coming from one arm, and by the delayed replica E ( t − τ ) {\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 E ( t ) {\displaystyle E(t)} , or if the recorded signal is integrated, the detector measures the intensity I M {\displaystyle I_{M}} as the delay τ {\displaystyle \tau } is scanned:
I M ( τ ) = ∫ − ∞ + ∞ | E ( t ) + E ( t − τ ) | 2 d t {\displaystyle I_{M}(\tau )=\int _{-\infty }^{+\infty }|E(t)+E(t-\tau )|^{2}dt}
Expanding I M ( τ ) {\displaystyle I_{M}(\tau )} reveals that one of the terms is A ( τ ) {\displaystyle A(\tau )} , proving that a Michelson interferometer can be used to measure the field autocorrelation, or the spectrum of E ( t ) {\displaystyle E(t)} (and only the spectrum). This principle is the basis for Fourier transform spectroscopy.
Intensity autocorrelation To a complex electric field E ( t ) {\displaystyle E(t)} corresponds an intensity I ( t ) = | E ( t ) | 2 {\displaystyle I(t)=|E(t)|^{2}} and an intensity autocorrelation function defined by
A ( τ ) = ∫ − ∞ + ∞ I ( t ) I ( t − τ ) d t {\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 ( E ( t ) + E ( t − τ ) ) 2 {\displaystyle (E(t)+E(t-\tau ))^{2}} . Only the beam propagating on the optical axis, proportional to the cross-product E ( t ) E ( t − τ ) {\displaystyle E(t)E(t-\tau )} , is retained. This signal is then recorded by a slow detector, which measures
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