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 signal processing and telecommunication, pulse duration is the interval between the time, during the first transition, that the amplitude of the pulse reaches a specified fraction (level) of its final amplitude, and the time the pulse amplitude ...

, 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 Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform * Phase space, a mathematic ...

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

and

group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope'' of the wave—propagates through space. For example, if a stone is thrown into the middl ...

. The time, after which the phase increases resp. decreases by

2 \pi

is called

T_\mathrm

. Ideally, it is an integer multiple of the duration

T_\mathrm

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 A femtosecond is a unit of time in the International System of Units (SI) equal to 10 or of a second; that is, one quadrillionth, or one millionth of one billionth, of a second. For context, a femtosecond is to a second as a second is to about 31. ...

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=\frac=\frac =

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

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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.

Bibliography

* * {{cite journal, last1=Krausz, first1=Ferenc, last2=Ivanov, first2=Misha, title=Attosecond physics, journal=Reviews of Modern Physics, date=2 February 2009, volume=81, issue=1, pages=163–234, doi=10.1103/RevModPhys.81.163, bibcode = 2009RvMP...81..163K , url=https://nrc-publications.canada.ca/eng/view/fulltext/?id=1245a958-9c93-4116-bfdb-f447e8a53c48 Nonlinear optics