Editing Ultrashort pulse (section)

==Definition==
[[Image:Ultrashort pulse.svg|thumb|upright=1.5|A positively chirped ultrashort pulse of light in the time domain.]]
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 [[Refractive index|index of refraction]] of the medium through which it travels, whereas "Ultrashort" refers to the temporal width of the pulse [[Wave packet|wavepacket]].<ref>{{cite web|url=https://www.rp-photonics.com/ultrashort_pulses.html|title=Encyclopedia of Laser Physics and Technology - ultrashort pulses, femtosecond, laser|first= Rüdiger|last=Paschotta|website=www.rp-photonics.com}}</ref>

A common example is a chirped Gaussian pulse, a [[wave]] whose [[Absolute value|field amplitude]] follows a [[Gaussian function|Gaussian]] [[envelope (waves)|envelope]] and whose [[instantaneous phase]] has a [[chirp|frequency sweep]].

===Background===
The real electric field corresponding to an ultrashort pulse is oscillating at an angular frequency ''ω''<sub>0</sub> 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 ''ω''<sub>0</sub> 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''):

: <math>E(t) = \sqrt{I(t)}e^{i\omega_0t}e^{i\psi(t)}</math>

The expression of the complex electric field in the frequency domain is obtained from the [[Fourier transform]] of ''E''(''t''):

: <math>E(\omega) = \mathcal{F}(E(t))</math>

Because of the presence of the <math>e^{i\omega_0t}</math> term, ''E''(''ω'') is centered around ''ω''<sub>0</sub>, and it is a common practice to refer to ''E''(''ω''-''ω''<sub>0</sub>) by writing just ''E''(''ω''), which we will do in the rest of this article.

Just as in the time domain, an intensity and a phase function can be defined in the frequency domain:

: <math>E(\omega) = \sqrt{S(\omega)}e^{i\phi(\omega)}</math>

{{anchor|Spectral phase}}The quantity <math>S(\omega)</math> is the ''[[power spectral density]]'' (or simply, the ''spectrum'') of the pulse, and <math>\phi(\omega) </math> is the ''[[phase spectral density]]'' (or simply ''spectral phase''). Example of spectral phase functions include the case where  <math>\phi(\omega) </math> is a constant, in which case the pulse is called a [[bandwidth-limited pulse]], or where  <math>\phi(\omega) </math> is a quadratic function, in which case the pulse is called a [[chirp]]ed 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 (optics)|dispersion]]. It results in a temporal broadening of the pulse.

The intensity functions—temporal <math> I(t) </math> and spectral <math>S(\omega)</math> —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 <math>\phi(\omega) </math>. High values of the time-bandwidth product, on the other hand, indicate a more complex pulse.