Frequency-resolved optical gating

Introduction

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 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, we do not have this luxury with FROG. 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.

External Links

Rick Trebino's FROG Page (the inventor of FROG)

Further Reading

Frequency-Resolved Optical Gating: The Measurement of Ultrashort Laser Pulses: ISBN 1-4020-7066-7