title: Carrier-envelope phase
text: # Carrier-envelope phase
  
  The carrier-envelope phase (CEP) or carrier-envelope offset (CEO) phase is an important feature of an ultrashort laser pulse and gains significance with decreasing pulse duration, in a regime where the pulse consists of a few wavelengths. Physical effects depending on the carrier-envelope phase fall into the category of highly nonlinear optics.
  
  ## CEP in the time domain
  
  The CEP $\phi _{0}$ is the phase between the carrier wave and the position of the intensity envelope of the pulse (cf. figure in the time domain). In a train of multiple pulses it is usually varying due to the difference between phase and group velocity. The time, after which the phase increases resp. decreases by $2\pi$ is called $T_{\mathrm {CEO} }$. Ideally, it is an integer multiple of the duration $T_{\mathrm {rep} }$ between two pulses and the pulses are picked at the corresponding rate to obtain a constant phase over all picked pulses. Besides this linear evolution, fluctuations which are common in conventional femtosecond laser systems usually cause a nonlinear shot-to-shot fluctuation of the CEP. This is why measuring and controlling it is very important for many applications.
  
  ## CEP in the frequency domain and measurement
  
  In the frequency domain, a pulse train is represented by a frequency comb. Here, the carrier-envelope frequency $f_{\mathrm {CEO} }={\frac {1}{T_{\mathrm {CEO} }}}={\frac {\mathrm {d} \phi _{0}}{\mathrm {d} t}}=$ is exactly the offset frequency of the pulse train, cf. figure. This makes it possible to perform a multi-shot measurement of the CEP, for example by using an f-2f interferometer. Here, the pulses to be measured are broadened to a bandwidth of at least one octave. A long-wavelength part of the pulse is frequency doubled and the beat note between it and the short-wavelength part of the fundamental pulse is measured. This is better known as the offset phase.
  
  With a phase-locked loop, a property of the laser oscillator such as the optical path length can be adjusted correspondingly to the obtained offset frequency and thus the phase can be stabilized.
  
  