In optics, an ultrashort pulse, also known as an ultrafast event, is an electromagnetic pulse whose time duration is of the order of a picosecond (10−12 second) or less. Such pulses have a broadband optical spectrum, and can be created by mode-locked oscillators. Amplification of ultrashort pulses almost always requires the technique of chirped pulse amplification, in order to avoid damage to the gain medium of the amplifier. They are characterized by a high peak intensity (or more correctly, irradiance) that usually leads to nonlinear interactions in various materials, including air. These processes are studied in the field of nonlinear optics. In the specialized literature, "ultrashort" refers to the femtosecond (fs) and picosecond (ps) range, although such pulses no longer hold the record for the shortest pulses artificially generated. Indeed, x-ray pulses with durations on the attosecond time scale have been reported. The 1999 Nobel Prize in Chemistry was awarded to Ahmed H. Zewail, for the use of ultrashort pulses to observe chemical reactions at the timescales on which they occur, opening up the field of femtochemistry. A further Nobel prize, the 2023 Nobel Prize in Physics, was also awarded for ultrashort pulses. This prize was awarded to Pierre Agostini, Ferenc Krausz, and Anne L'Huillier for the development of attosecond pulses and their ability to probe electron dynamics.
Definition
There is no standard definition of ultrashort pulse. Usually the attribute 'ultrashort' applies to pulses with a duration of a few tens of femtoseconds, but in a larger sense any pulse which lasts less than a few picoseconds can be considered ultrashort. The distinction between "Ultrashort" and "Ultrafast" is necessary as the speed at which the pulse propagates is a function of the index of refraction of the medium through which it travels, whereas "Ultrashort" refers to the temporal width of the pulse wavepacket. A common example is a chirped Gaussian pulse, a wave whose field amplitude follows a Gaussian envelope and whose instantaneous phase has a frequency sweep.
Background The real electric field corresponding to an ultrashort pulse is oscillating at an angular frequency ω0 corresponding to the central wavelength of the pulse. To facilitate calculations, a complex field E(t) is defined. Formally, it is defined as the analytic signal corresponding to the real field. The central angular frequency ω0 is usually explicitly written in the complex field, which may be separated as a temporal intensity function I(t) and a temporal phase function ψ(t):
E ( t ) = I ( t ) e i ω 0 t e i ψ ( t ) {\displaystyle E(t)={\sqrt {I(t)}}e^{i\omega _{0}t}e^{i\psi (t)}}
The expression of the complex electric field in the frequency domain is obtained from the Fourier transform of E(t):
E ( ω ) = F ( E ( t ) ) {\displaystyle E(\omega )={\mathcal {F}}(E(t))}
Because of the presence of the e i ω 0 t {\displaystyle e^{i\omega _{0}t}} term, E(ω) is centered around ω0, and it is a common shorthand to refer to E(ω-ω0) by writing just E(ω), which will be followed for the remainder of this article. Just as in the time domain, an intensity and a phase function can be defined in the frequency domain:
E ( ω ) = S ( ω ) e i ϕ ( ω ) {\displaystyle E(\omega )={\sqrt {S(\omega )}}e^{i\phi (\omega )}}
The quantity S ( ω ) {\displaystyle S(\omega )} is the power spectral density (or simply, the spectrum) of the pulse, and ϕ ( ω ) {\displaystyle \phi (\omega )} is the phase spectral density (or simply spectral phase). Example of spectral phase functions include the case where ϕ ( ω ) {\displaystyle \phi (\omega )} is a constant, in which case the pulse is called a bandwidth-limited pulse, or where ϕ ( ω ) {\displaystyle \phi (\omega )} is a quadratic function, in which case the pulse is called a chirped pulse because of the presence of an instantaneous frequency sweep. Such a chirp may be acquired as a pulse propagates through materials (like glass) and is due to their dispersion. It results in a temporal broadening of the pulse. The intensity functions—temporal I ( t ) {\displaystyle I(t)} and spectral S ( ω ) {\displaystyle S(\omega )} —determine the time duration and spectrum bandwidth of the pulse. As stated by the uncertainty principle, their product (sometimes called the time-bandwidth product) has a lower bound. This minimum value depends on the definition used for the duration and on the shape of the pulse. For a given spectrum, the minimum time-bandwidth product, and therefore the shortest pulse, is obtained by a transform-limited pulse, i.e., for a constant spectral phase ϕ ( ω ) {\displaystyle \phi (\omega )} . High values of the time-bandwidth product, on the other hand, indicate a more complex pulse.
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