10.1: Intensity Autocorrelation

Pulse duration measurements using second-harmonic intensity autocorrelation is a standard method for pulse characterisation. Figure 10.1 shows the setup for a background free intensity autocorrelation. The input pulse is split in two, and one of the pulses is delayed by \(\tau\). The two pulses are focussed into a nonliner optical crystal in a non-colinear fashion. The nonlinear optical crystal is designed for efficient second harmonic generation over the full bandwidth of the pulse, i.e. it has a large second order nonlinear optical suszeptibility and is phase matched for the specific wavelength range. We do not consider the \(z\)—dependence of the electric field and phase—matching effects. To simplify notation, we omit normalization factors. The induced nonlinear polarization is expressed as a convolution of two interfering electric—fields \(E_1(t)\), \(E_2(t)\) with the nonlinear response function of the medium, the second order nonlinear susceptibility \(\chi^{(2)}\).

\[P^{(2)} (t) \propto \int \int_{-\infty}^{\infty} \chi^{(2)} (t - t_1, t - t_2) \cdot E_1 (t_1) \cdot E_2 (t_2) dt_1 dt_2\nonumber \]

Image removed due to copyright restrictions.

Please see:
Keller, U., Ultrafast Laser Physics, Institute of Quantum Electronics, Swiss Federal Institute of Technology, ETH Hönggerberg—HPT, CH-8093 Zurich, Switzerland.

Figure 10.1: Setup for a background free intensity autocorrelation. To avoid dispersion and pulse distortions in the autocorrelator reflective optics can be and a thin crystal has to be used for measureing very short, typically sub-100 fs pulses.

We assume the material response is instantaneous and replace \(\chi^{(2)} (t - t_1, t - t_2)\) by a Dirac delta-function \(\chi^{(2)} \cdot \delta (t - t_1) \cdot \delta (t - t_2)\) which leads to

\[P^{(2)} (t) \propto E_1 (t) \cdot E_2 (t) \nonumber \]

Due to momentum conservation, see Figure 10.1, we mayseparate the product \(E(t) \cdot E(t - \tau)\) geometrically and supress a possible background coming from simple SHG of the individual pulses alone. The signal is zero if the pulses don’t overlap.

\[P^{(2)} (t) \propto E(t) \cdot E(t - \tau). \nonumber \]

Image removed due to copyright restrictions.

Please see:

Keller, U., Ultrafast Laser Physics, Institute of Quantum Electronics, Swiss Federal Institute of Technology, ETH Hönggerberg—HPT, CH-8093 Zurich, Switzerland.

Table 10.1: Pulse shapes and its deconvolution factors relating FWHM, \(\tau_p\), of the pulse to FWHM, \(\tau_A\), of the intensity autocorrelationfunction.

The electric field of the second harmonic radiation is directly proportional to the polarization, assuming a nondepleted fundamental radiation and the use of thin crystals. Due to momentum conservation, see Figure 10.1, we find

\[I_{AC} (\tau) \propto \int_{-\infty}^{\infty} |A(t) A(t - \tau)|^2 dt.\label{eq10.1.3} \]

\[\propto \int_{-\infty}^{\infty} I(t) I(t - \tau) dt, \nonumber \]

with the complex envelope \(A(t)\) and intensity \(I(t) = |A(t)|^2\) of the input pulse. The photo detector integrates because its response is usually much slower than the pulsewidth. Note, that the intenisty autocorrelation is symmetric by construction

\[I_{AC} (\tau) = I_{AC} (- \tau). \nonumber \]

It is obvious from Eq.(\(\ref{eq10.1.3}\)) that the intensity autocorrelation does not contain full information about the electric field of the pulse, since the phase of the pulse in the time domain is completely lost. However, if the pulse shape is known the pulse width can be extracted by deconvolution of the correlation function. Table 10.1 gives the deconvolution factors for some often used pulse shapes.