Optical autocorrelation

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.

Other techniques based on two-photon absorption may also be used in autocorrelation measurements,[1] 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.

, the field autocorrelation function is defined by The Wiener-Khinchin theorem states that the Fourier transform of the field autocorrelation is the spectrum of

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

, or if the recorded signal is integrated, the detector measures the intensity

, proving that a Michelson interferometer can be used to measure the field autocorrelation, or the spectrum of

This principle is the basis for Fourier transform spectroscopy.

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

Only the beam propagating on the optical axis, proportional to the cross-product

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.

For a Gaussian time profile, the autocorrelation width is

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

Classification of the different kinds of optical autocorrelation.
Setup for a field autocorrelator, based on a Michelson interferometer . L : modelocked laser , BS : beam splitter , M1 : moveable mirror providing a variable delay line , M2 : fixed mirror, D : energy detector.
Two ultrashort pulses (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 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).
Two ultrashort pulses (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 1/2 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.
Setup for an interferometric autocorrelator, similar to the field autocorrelator above, with the following optics added: L : converging lens , SHG : second-harmonic generation crystal , F : spectral filter to block the fundamental wavelength.
Two ultrashort pulses (a) and (b) with their respective interferometric autocorrelation (c) and (d). Because of the phase present in pulse (b) due to an instantaneous frequency sweep ( chirp ), the fringes of the autocorrelation trace (d) wash out in the wings. Note the ratio 8:1 (peak to the wings), characteristic of interferometric autocorrelation traces.