Ultrashort pulse

Definition

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.

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The central angular frequency ω0 is usually explicitly written in the complex field, which may be separated as an intensity function I(t) and a phase function ψ(t):

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: E(t) = sqrt{I(t)}e^{iomega_0t}e^{ipsi(t)}

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The expression of the complex electric field in the frequency domain is obtained from the Fourier transform of E(t):

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: E(omega) = F{E(t)}

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Because of the presence of the e^{iomega_0t} term, E(ω) is centered around ω0, and it is a common practice to refer to E(ω-ω0) by writing just E(ω), which we will do in the rest of this article.

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

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: E(omega) = sqrt{S(omega)}e^{iphi(omega)}

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Example of phase functions include the case where φ(ω) is a constant, in which case the pulse is called a bandwidth limited pulse, or where φ(ω) 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.

Related Topics:
Bandwidth limited pulse - Chirp - Dispersion

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The width intensity functions I(t) and S(ω) determine the time duration and spectral bandwidth of the pulse. As stated by the incertainty principle, their product (sometimes called the time-bandwidth product) has a lower bound. This minimum value depends on the definition used and on the shape of the pulse. For a given spectrum, the minimum time-bandwidth product is obtained by a transform-limited pulse, i.e., for a constant spectral phase φ(ω). High values of the time-bandwidth product, on the other hand, reveal the pulse complexity.

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