Frequency-resolved optical gating

In optics, frequency-resolved optical gating (FROG) is a derivative of autocorrelation, but is far superior in its ability to measure ultrafast optical pulse shapes. In the most common configuration, FROG is simply a background-free autocorrelator followed by a spectrometer. It is the two-dimensional nature of the FROG trace that allows the extraction of the actual pulse shape from the data.

The basics

Mathematically the FROG trace is simply a spectrogram but with an unknown gate function:

$I_{sig}(\omega ,\tau )=\left|\int _{-\infty }^{\infty }P(t)G(t-\tau )e^{-i\omega t}dt\right|^{2}$

where $P(t)$ is the "probe" pulse and $G(t)$ is the "gate" pulse. The prope and gate pulses are determined by the nonlinear interaction used, and it is the form of the probe and gate that distinguishes the different types of FROG from each other. Some of the most common are:

$P^{SHG}(t)=E(t)$	$G^{SHG}(t)=E(t)$	: Second-harmonic generation FROG
$P^{PG}(t)=E(t)$	$G^{PG}(t)=|E(t)|^{2}$	: Polarization gating FROG
$P^{SD}(t)=E(t)^{2}$	$G^{SD}(t)=E(t)$	: Self-diffraction FROG
$P^{THG}(t)=E(t)$	$G^{THG}(t)=E(t)^{2}$	: Third-harmonic generation FROG

A schematic of a typical, experimental SHG FROG setup.

For example second-harmonic generation FROG (SHG FROG) would be:

$I_{sig}^{SHG}(\omega ,\tau )=\left|\int _{-\infty }^{\infty }E(t)E(t-\tau )e^{-i\omega t}dt\right|^{2}$

and PG FROG would be:

$I_{sig}^{PG}(\omega ,\tau )=\left|\int _{-\infty }^{\infty }E(t)|E(t-\tau )|^{2}e^{-i\omega t}dt\right|^{2}$

Traditional inversion algorithms for spectrograms requires perfect knowledge of the gate function ($G(t)$), however, FROG does not have this luxury. Instead an interative alogrithm is used. The algorithm uses both the data (FROG trace) and form of the nonlinearity to achieve a best match between the real FROG trace and the "retrieved" FROG trace. The retrieved FROG trace is created synthetically from the best guess for $E(t)$.

FROG algorithm

The FROG algorithm "life cycle."

The FROG algorithm is all about phase retrieval. The FROG trace measured in the lab is the exact intensity of $I_{sig}(\omega ,\tau )$; however, it is missing the phase information. To start off with define a signal field:

$E_{sig}(t,\tau )=P(t)G(t-\tau )$

and further defining:

${\tilde {E}}_{sig}(\omega ,\tau )=\int _{-\infty }^{\infty }E_{sig}(t,\tau )e^{-i\omega t}dt$

given this the FROG trace becomes:

$I_{sig}(\omega ,\tau )=\left|{\tilde {E}}_{sig}(\omega ,\tau )\right|^{2}$

inverting this:

$\left|{\tilde {E}}_{sig}(\omega ,\tau )\right|={\sqrt {I_{sig}(\omega ,\tau )}}$

so the amplitude of ${\tilde {E}}_{sig}(\omega ,\tau )$ is known, but not its phase.

Competing techniques

• Spectral interferometry for direct electric field reconstruction

Further reading

• . ISBN 1-4020-7066-7. {{cite book}}: Missing or empty |title= (help); Unknown parameter |Author= ignored (|author= suggested) (help); Unknown parameter |Publisher= ignored (|publisher= suggested) (help); Unknown parameter |Title= ignored (|title= suggested) (help); Unknown parameter |Year= ignored (|year= suggested) (help)

External links

• FROG Page by Rick Trebino (the inventor of FROG)