## 3 Correlation Measurement  
Correlation is a technique to estimate the pulse intensity \(I(t)\) in time-domain without obtaining the phase information, so it does not implement a complete measurement. However, correlation is irreplaceable for its simple, robust, and ultrahigh signal-to-noise ratio (SNR). The simplest correlator, called field-autocorrelator (or linear autocorrelator), is composed of a Michelson interferometer (MI) and an integrating detector. It operates in the time domain, and the acquiring signal is \(\Gamma^{(2)}(\tau)=\int\!E(t)E^{\prime}(t-\tau)\mathrm{d}t\) when neglecting the constant term, where \(\tau\) is the delay between two pulses from two arms in MI. The Fourier transform of \(\Gamma^{(2)}\) is just the spectrum \(S(\omega)\) of the test pulse, so the field-autocorrelator can also work as a Fourier-transform spectrometer [17,18] which cannot give us the time information of the pulse. An effective method to give us the time information of the femtosecond pulses is intensity-autocorrelator, for which we only need to make a minor alter to the field-autocorrelator, by introducing a nonlinear crystal before the detector for second-harmonic generation (SHG). For convenience, when we mention autocorrelator later in the article, it means intensity-autocorrelator. The measured signal is expressed as  
\[\mathrm{A}^{(2)}(\tau)=\int\!I(t)I(t-\tau)\mathrm{d}t, \tag{4}\]  
and its intensity is related to the temporal overlap of the pulse replicas, so autocorrelation can reflect the pulse length. It is easy to see \(\mathrm{A}^{(2)}(\tau)\) is an even function and symmetric with \(\tau=0\). However, because the probe and the test pulse are divided from the same pulses, \(\mathrm{A}^{(2)}(\tau)\) is not an accurate method to describe, because it is affected by the pulse shapes. The correction factors between pulse-widths to \(\mathrm{A}^{(2)}(\tau)\)-widths, called the deconvolution factor, are given in Ref. [19]. Autocorrelator can realize measurement in single-shot based on space-time correlation of tilt incidence pulses, i.e., time-to-space encoding. If we use a collinear design for Autocorrelator, the sum-frequency (SF) term superposes with the self-frequency-doubling terms, the output signal is  
\[\mathrm{A}_{\mathrm{int}}{}^{(2)}(\tau)=\int\!|[E(t)+E(t-\tau)]^{2}|^{2} \mathrm{d}t, \tag{5}\]  
which gives an autocorrelation signal with interferometric fringes [20,21], called interferometric autocorrelation. From the structure of fringes, the chirp of the pulse can be estimated. Nowadays, autocorrelation is a proven technique, and have been applied widely due to its simplicity and cost-of-effectivity. It has been realized with many commercial versions [22, 23, 24], based on single-shot or scanning, with a wavelength from 200 nm to 12 \(\upmu\)m, pulse widths from sub-10 fs to 500 ps, using a photodiode detector, photomultiplier, or two-photon detector as the detector. For a laser with peak power higher than terawatt, its contrast, which defines as the ratio of the pulse peak to the peaks of the pre- or post-subpulses, is one of the primary parameters, because the laser is so intense at its focus spot even the pre-subpulses can ionize the target and break the original construction before the main peak comes. The contrast of intense pulse is usually required to be higher than \(10^{6}\) in terawatt laser, even \(10^{10}\) in petawatt systems [25]. The dynamic range of Spectrogram or Spectral Interferometry methods, based on frequency-resolved measurement, cannot be higher than \(\sim\)\(10^{6}\)[25]. For such high contrast, correlation is the irreplaceable method for its simple and robust structure, and more importantly, the phase-matching requirement of SHG which highly improves the measuring SNR. However, the symmetry of \(\mathrm{A}^{(2)}(\tau)\) limits its applications because it cannot distinguish the pre-subpulses from the post-subpulses.