Spectral interferometry

Spectral interferometry (SI) or frequency-domain interferometry is a linear technique used to measure optical pulses, with the condition that a reference pulse that was previously characterized is available. This technique provides information about the intensity and phase of the pulses. SI was first proposed by Claude Froehly and coworkers in the 1970s.^[1][2]

A known (acting as the reference) and an unknown pulse arrive at a spectrometer, with a time delay ${\displaystyle \tau }$ between them, in order to create spectral fringes. A spectrum is produced by the sum of these two pulses and, by measuring said fringes, one can retrieve the unknown pulse. If ${\displaystyle E_{un}(\omega )}$ and ${\displaystyle E_{ref}(\omega )}$ are the electric fields of the unknown and reference pulse respectively, the time delay can be expressed as a phase factor ${\displaystyle e^{-i\omega \tau }}$ for the unknown pulses.^[3] Then, the combined field is:

${\displaystyle E_{SI}=E_{ref}(\omega )+E_{un}(\omega )e^{-i\omega \tau }}$

The average spacing between fringes is inversely proportional to the time delay ${\displaystyle \tau }$.^[4] Thus, the SI signal is given by:

${\displaystyle S_{SI}=S_{ref}(\omega )+S_{un}(\omega )+2{\sqrt {S_{ref}(\omega )}}{\sqrt {S_{un}(\omega )}}cos[\phi _{SI}]}$

where ${\displaystyle \phi _{SI}=\phi _{un}(\omega )-\phi _{ref}(\omega )+\omega \tau }$ is the oscillation phase.

Furthermore, the spectral fringes width can provide information on the spectral phase difference between the two pulses ${\displaystyle \Delta \phi =\phi _{un}(\omega )-\phi _{ref}(\omega )}$; narrowly spaced fringes indicate rapid phase changes with frequency.^[5]