A bandwidth-limited pulse (also known as Fourier-transform-limited pulse, or more commonly, transform-limited pulse) is a

pulse In medicine, a pulse represents the tactile arterial palpation of the cardiac cycle (heartbeat) by trained fingertips. The pulse may be palpated in any place that allows an artery to be compressed near the surface of the body, such as at the nec ...

of a

wave In physics, mathematics, and related fields, a wave is a propagating dynamic disturbance (change from equilibrium) of one or more quantities. Waves can be periodic, in which case those quantities oscillate repeatedly about an equilibrium (res ...

that has the minimum possible duration for a given

spectral ''Spectral'' is a 2016 3D military science fiction, supernatural horror fantasy and action-adventure thriller war film directed by Nic Mathieu. Written by himself, Ian Fried, and George Nolfi from a story by Fried and Mathieu. The film stars ...

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 ...

. Bandwidth-limited pulses have a constant

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 ...

across all frequencies making up the pulse.

Optical Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...

pulses of this type can be generated by mode-locked

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" is an acronym for "light amplification by stimulated emission of radiation". The fir ...

s. Any waveform can be disassembled into its

spectral component In telecommunications, spectral component is any of the waves that range outside the interval of frequencies assigned to a signal. Any waveform can be disassembled into its spectral components by Fourier analysis or Fourier transformation. The ...

s by

Fourier analysis In mathematics, Fourier analysis () is the study of the way general functions may be represented or approximated by sums of simpler trigonometric functions. Fourier analysis grew from the study of Fourier series, and is named after Josep ...

Fourier transform A Fourier transform (FT) is a mathematical transform that decomposes functions into frequency components, which are represented by the output of the transform as a function of frequency. Most commonly functions of time or space are transformed, ...

ation. The length of a pulse thereby is determined by its complex spectral components, which include not just their relative intensities, but also the relative positions (

spectral phase The power spectrum S_(f) of a time series x(t) describes the distribution of Power (physics), power into frequency components composing that signal. According to Fourier analysis, any physical signal can be decomposed into a number of discre ...

) of these spectral components. For different pulse shapes, the minimum duration-bandwidth product is different. The duration-bandwidth product is minimal for zero phase-modulation. For example,

\mathrm

pulses have a minimum duration-bandwidth product of 0.315 while gaussian pulses have a minimum value of 0.441. A bandwidth-limited pulse can only be kept together if the

dispersion Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns *Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variatio ...

of the medium the wave is travelling through is zero; otherwise

dispersion management Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns * Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variati ...

is needed to revert the effects of unwanted spectral phase changes. For example, when an ultrashort pulse passes through a block of glass, the glass medium broadens the pulse due to

group velocity dispersion In optics, group velocity dispersion (GVD) is a characteristic of a dispersive medium, used most often to determine how the medium will affect the duration of an optical pulse traveling through it. Formally, GVD is defined as the derivative of the ...

. Keeping pulses bandwidth-limited is necessary to compress information in time or to achieve high field densities, as with

ultrashort pulse 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 ...

s in modelocked lasers.

Further reading