Carrier-envelope Phase
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The carrier-envelope phase (CEP) or carrier-envelope offset (CEO) phase is an important feature of an ultrashort
laser A laser is a device that emits light through a process of optical amplification based on the stimulated emission of electromagnetic radiation. The word ''laser'' originated as an acronym for light amplification by stimulated emission of radi ...
pulse and gains significance with decreasing
pulse duration In signal processing and telecommunications, 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 amplitu ...
, in a regime where the pulse consists of a few
wavelength In physics and mathematics, wavelength or spatial period of a wave or periodic function is the distance over which the wave's shape repeats. In other words, it is the distance between consecutive corresponding points of the same ''phase (waves ...
s. Physical effects depending on the carrier-envelope phase fall into the category of highly
nonlinear optics Nonlinear optics (NLO) is the branch of optics that describes the behaviour of light in Nonlinearity, nonlinear media, that is, media in which the polarization density P responds non-linearly to the electric field E of the light. The non-linearity ...
.


CEP in the

time domain In mathematics and signal processing, the time domain is a representation of how a signal, function, or data set varies with time. It is used for the analysis of mathematical functions, physical signals or time series of economic or environmental ...

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 mathematica ...
between the
carrier wave In telecommunications, a carrier wave, carrier signal, or just carrier, is a periodic waveform (usually sinusoidal) that conveys information through a process called ''modulation''. One or more of the wave's properties, such as amplitude or freq ...
and the position of the intensity envelope of the pulse (cf. figure in the time domain). In a
train A train (from Old French , from Latin">-4; we might wonder whether there's a point at which it's appropriate to talk of the beginnings of French, that is, when it wa ... , from Latin , "to pull, to draw") is a series of connected vehicles th ...
of multiple pulses it is usually varying due to the difference between
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 mathematica ...
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 (waves), envelope'' of the wave—propagates through space. For example, if a stone is thro ...
. 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. A femtosecond is to a second, as a second is to approximately 31.6 ...
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 In mathematics, physics, electronics, control systems engineering, and statistics, the frequency domain refers to the analysis of mathematical functions or signals with respect to frequency (and possibly phase), rather than time, as in time ser ...
, a pulse train is represented by a
frequency comb A frequency comb or spectral comb is a spectrum made of discrete and regularly spaced spectral lines. In optics, a frequency comb can be generated by certain laser sources. A number of mechanisms exist for obtaining an optical frequency comb, i ...
. 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 Bandwidth commonly refers to: * Bandwidth (signal processing) or ''analog bandwidth'', ''frequency bandwidth'', or ''radio bandwidth'', a measure of the width of a frequency range * Bandwidth (computing), the rate of data transfer, bit rate or thr ...
of at least one octave. A long-wavelength part of the pulse is
frequency doubled Second-harmonic generation (SHG), also known as frequency doubling, is the lowest-order wave-wave nonlinear interaction that occurs in various systems, including optical, radio, atmospheric, and magnetohydrodynamic systems. As a prototype behav ...
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 phase-locked loop or phase lock loop (PLL) is a control system that generates an output signal whose phase is fixed relative to the phase of an input signal. Keeping the input and output phase in lockstep also implies keeping the input and ou ...
, 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