Ultrashort pulse
Encyclopedia
In optics
, an ultrashort pulse of light is an electromagnetic pulse
whose time duration is of the order of a femtosecond (
second). Such pulses have a broadband optical spectrum, and can be created by mode-locked
oscillators. They are commonly referred to as ultrafast events.
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) to picosecond (ps) range, although such pulses no longer hold the record for the shortest pulses artificially generated. Indeed, pulse durations on the attosecond time scale have been reported.
The 1999 Nobel Prize in Chemistry
was awarded to Ahmed H. Zewail for using ultrashort pulses to observe chemical reaction
s on the timescales they occur on, opening up the field of femtochemistry
.
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 an intensity function I(t) and a phase function ψ(t):
The expression of the complex electric field in the frequency domain is obtained from the Fourier transform
of E(t):
Because of the presence of the
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.
Just as in the time domain, an intensity and a phase function can be defined in the frequency domain:
The quantity S(ω) is the spectral density (or simply, the spectrum) of the pulse, and φ(ω) is the spectral phase. Example of spectral 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 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
. It results in a temporal broadening of the pulse.
The intensity functions I(t) and S(ω) determine the time duration and spectral 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). High values of the time-bandwidth product, on the other hand, indicate a more complex pulse.
, a device that can be used to control the spectral phase of ultrashort pulses. It is composed of a sequence of prisms, or gratings. When properly adjusted it can alter the spectral phase φ(ω) of the input pulse so that the output pulse is a bandwidth-limited pulse
with the shortest possible duration. A pulse shaper
can be used to make more complicated alterations on both the phase and the amplitude of ultrashort pulses.
To accurately control the pulse, a full characterization of the pulse spectral phase is a must in order to get certain pulse spectral phase (such as Transform-Limited
). Then, a Spatial light modulator
can be used in the 4f plane to control the pulse. Multiphoton Intrapulse Interference Phase Scan (MIIPS) is a technique based on this concept. Through the phase scan of the spatial light modulator
, MIIPS can not only characterize but also manipulate the ultrashort pulse to get the needed pulse shape at target spot (such as transform-limited pulse
for optimized peak power, and other specific pulse shapes). This technique features with full calibration and control of the ultrashort pulse, with no moving parts, and simple optical setup.
Methods of characterizing and controlling the ultrashort optical pulses:
(SVEA) of the electric field of a wave with central wave vector
and central frequency 
of the pulse, is given by:
We consider the propagation for the SVEA of the electric field in a homogeneous dispersive nonistropic medium. Assuming the pulse is propagating in the direction of the z-axis, it can be shown that the envelope
for one of the most general of cases, namely a biaxial crystal, is governed by the PDE
:
Optics
Optics is the branch of physics which involves the behavior and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behavior of visible, ultraviolet, and infrared light...
, an ultrashort pulse of light is an electromagnetic pulse
Electromagnetic pulse
An electromagnetic pulse is a burst of electromagnetic radiation. The abrupt pulse of electromagnetic radiation usually results from certain types of high energy explosions, especially a nuclear explosion, or from a suddenly fluctuating magnetic field...
whose time duration is of the order of a femtosecond (
second). Such pulses have a broadband optical spectrum, and can be created by mode-lockedModelocking
Mode-locking is a technique in optics by which a laser can be made to produce pulses of light of extremely short duration, on the order of picoseconds or femtoseconds ....
oscillators. They are commonly referred to as ultrafast events.
They are characterized by a high peak intensity
Irradiance
Irradiance is the power of electromagnetic radiation per unit area incident on a surface. Radiant emittance or radiant exitance is the power per unit area radiated by a surface. The SI units for all of these quantities are watts per square meter , while the cgs units are ergs per square centimeter...
(or more correctly, irradiance
Irradiance
Irradiance is the power of electromagnetic radiation per unit area incident on a surface. Radiant emittance or radiant exitance is the power per unit area radiated by a surface. The SI units for all of these quantities are watts per square meter , while the cgs units are ergs per square centimeter...
) that usually leads to nonlinear interactions in various materials, including air. These processes are studied in the field of nonlinear optics
Nonlinear optics
Nonlinear optics is the branch of optics that describes the behavior of light in nonlinear media, that is, media in which the dielectric polarization P responds nonlinearly to the electric field E of the light...
.
In the specialized literature, "ultrashort" refers to the femtosecond (fs) to picosecond (ps) range, although such pulses no longer hold the record for the shortest pulses artificially generated. Indeed, pulse durations on the attosecond time scale have been reported.
The 1999 Nobel Prize in Chemistry
Nobel Prize in Chemistry
The Nobel Prize in Chemistry is awarded annually by the Royal Swedish Academy of Sciences to scientists in the various fields of chemistry. It is one of the five Nobel Prizes established by the will of Alfred Nobel in 1895, awarded for outstanding contributions in chemistry, physics, literature,...
was awarded to Ahmed H. Zewail for using ultrashort pulses to observe chemical reaction
Chemical reaction
A chemical reaction is a process that leads to the transformation of one set of chemical substances to another. Chemical reactions can be either spontaneous, requiring no input of energy, or non-spontaneous, typically following the input of some type of energy, such as heat, light or electricity...
s on the timescales they occur on, opening up the field of femtochemistry
Femtochemistry
Femtochemistry is the science that studies chemical reactions on extremely short timescales, approximately 10–15 seconds .-Introduction:...
.
Definition
Analytic signal
In mathematics and signal processing, the analytic representation of a real-valued function or signal facilitates many mathematical manipulations of the signal. The basic idea is that the negative frequency components of the Fourier transform of a real-valued function are superfluous, due to the...
corresponding to the real field.
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):
The expression of the complex electric field in the frequency domain is obtained from the Fourier transform
Fourier transform
In mathematics, Fourier analysis is a subject area which grew from the study of Fourier series. The subject began with the study of the way general functions may be represented by sums of simpler trigonometric functions...
of E(t):
Because of the presence of the
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.Just as in the time domain, an intensity and a phase function can be defined in the frequency domain:
The quantity S(ω) is the spectral density (or simply, the spectrum) of the pulse, and φ(ω) is the spectral phase. Example of spectral phase functions include the case where φ(ω) is a constant, in which case the pulse is called a bandwidth-limited pulse
Bandwidth-limited pulse
A bandwidth-limited pulse is a pulse of a wave that has the minimum possible duration for a given spectral bandwidth. Optical pulses of this type can be generated by mode-locked lasers...
, or where φ(ω) is a quadratic function, in which case the pulse is called a chirp
Chirp
A chirp is a signal in which the frequency increases or decreases with time. In some sources, the term chirp is used interchangeably with sweep signal. It is commonly used in sonar and radar, but has other applications, such as in spread spectrum communications...
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
Dispersion
Dispersion may refer to:In physics:*The dependence of wave velocity on frequency or wavelength:**Dispersion , for light waves**Dispersion **Acoustic dispersion, for sound waves...
. It results in a temporal broadening of the pulse.
The intensity functions I(t) and S(ω) determine the time duration and spectral bandwidth of the pulse. As stated by the uncertainty principle
Uncertainty principle
In quantum mechanics, the Heisenberg uncertainty principle states a fundamental limit on the accuracy with which certain pairs of physical properties of a particle, such as position and momentum, can be simultaneously known...
, 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). High values of the time-bandwidth product, on the other hand, indicate a more complex pulse.
Pulse shape control
Although optical devices also used for continuous light, like beam expanders and spatial filters, may be used for ultrashort pulses, several optical devices have been specifically designed for ultrashort pulses. One of them is the pulse compressorPrism compressor
A prism compressor is an optical device used to shorten the duration of a positively chirped ultrashort laser pulse by giving different wavelength components a different time delay. It typically consists of two prisms and a mirror. Figure 1 shows the construction of such a compressor...
, a device that can be used to control the spectral phase of ultrashort pulses. It is composed of a sequence of prisms, or gratings. When properly adjusted it can alter the spectral phase φ(ω) of the input pulse so that the output pulse is a bandwidth-limited pulse
Bandwidth-limited pulse
A bandwidth-limited pulse is a pulse of a wave that has the minimum possible duration for a given spectral bandwidth. Optical pulses of this type can be generated by mode-locked lasers...
with the shortest possible duration. A pulse shaper
Femtosecond pulse shaping
In optics, femtosecond pulse shaping is a technique that modifies the temporal profile of an ultrashort pulse from a laser. Pulse shaping can be used to shorten/elongate the duration of optical pulse, or to generate more complex pulses.-Introduction:...
can be used to make more complicated alterations on both the phase and the amplitude of ultrashort pulses.
To accurately control the pulse, a full characterization of the pulse spectral phase is a must in order to get certain pulse spectral phase (such as Transform-Limited
Bandwidth-limited pulse
A bandwidth-limited pulse is a pulse of a wave that has the minimum possible duration for a given spectral bandwidth. Optical pulses of this type can be generated by mode-locked lasers...
). Then, a Spatial light modulator
Spatial light modulator
A spatial light modulator is an object that imposes some form of spatially-varying modulation on a beam of light. A simple example is an overhead projector transparency. Usually when the phrase SLM is used, it means that the transparency can be controlled by a computer. In the 1980s, large SLMs...
can be used in the 4f plane to control the pulse. Multiphoton Intrapulse Interference Phase Scan (MIIPS) is a technique based on this concept. Through the phase scan of the spatial light modulator
Spatial light modulator
A spatial light modulator is an object that imposes some form of spatially-varying modulation on a beam of light. A simple example is an overhead projector transparency. Usually when the phrase SLM is used, it means that the transparency can be controlled by a computer. In the 1980s, large SLMs...
, MIIPS can not only characterize but also manipulate the ultrashort pulse to get the needed pulse shape at target spot (such as transform-limited pulse
Bandwidth-limited pulse
A bandwidth-limited pulse is a pulse of a wave that has the minimum possible duration for a given spectral bandwidth. Optical pulses of this type can be generated by mode-locked lasers...
for optimized peak power, and other specific pulse shapes). This technique features with full calibration and control of the ultrashort pulse, with no moving parts, and simple optical setup.
Measurement techniques
Several techniques are available to measure ultrashort optical pulses:- intensity autocorrelationOptical autocorrelationIn optics, various autocorrelation functions can be experimentally realized. The field autocorrelation may be used to calculate the spectrum of a source of light, while the intensity autocorrelation and the interferometric autocorrelation are commonly used to estimate the duration of ultrashort...
: gives the pulse width when a particular pulse shape is assumed.
- spectral interferometry (SI): a linear technique that can be used when a pre-characterized reference pulse is available. Gives the intensity and phase. The algorithm that extracts the intensity and phase from the SI signal is direct.
- Spectral phase interferometry for direct electric-field reconstructionSpectral phase interferometry for direct electric-field reconstructionIn ultrafast optics, spectral phase interferometry for direct electric-field reconstruction is an ultrashort pulse measurement technique.-The basics:...
(SPIDER): a nonlinear self-referencing technique based on spectral shearing interferometry. The method is similar to SI, except that the reference pulse is a spectrally shifted replica of itself, allowing one to obtain the spectral intensity and phase of the probe pulse via a direct FFT filtering routine similar to SI, but which requires integration of the phase extracted from the interferogram to obtain the probe pulse phase.
- Frequency-resolved optical gatingFrequency-resolved optical gatingIn optics, frequency-resolved optical gating is a derivative of autocorrelation, but is far superior in its ability to measure ultrafast optical pulse shapes...
(FROG): a nonlinear technique that yields the intensity and phase of a pulse. It's just a spectrally resolved autocorrelation. The algorithm that extracts the intensity and phase from a FROG trace is iterative. - Grating-eliminated no-nonsense observation of ultrafast incident laser light e-fields (GRENOUILLE), a simplified version of FROG.
Methods of characterizing and controlling the ultrashort optical pulses:
- MIIPS Multiphoton Intrapulse Interference Phase Scan, a method to characterize and manipulate the ultrashort pulse.
Wave packet propagation in nonisotropic media
To partially reiterate the discussion above, the slowly varying envelope approximationSlowly varying envelope approximation
In physics, the slowly varying envelope approximation is the assumption that the envelope of a forward-travelling wave pulse varies slowly in time and space compared to a period or wavelength...
(SVEA) of the electric field of a wave with central wave vector
and central frequency of the pulse, is given by:We consider the propagation for the SVEA of the electric field in a homogeneous dispersive nonistropic medium. Assuming the pulse is propagating in the direction of the z-axis, it can be shown that the envelope
for one of the most general of cases, namely a biaxial crystal, is governed by the PDEPartial differential equation
In mathematics, partial differential equations are a type of differential equation, i.e., a relation involving an unknown function of several independent variables and their partial derivatives with respect to those variables...
:
-
where the coefficients contains diffraction and dispersion effects which have been determined analytically with computer algebra and verified numerically to within third order for both isotropic and non-istropic media, valid in the near-field and far-field.
is the inverse of the group velocity projection. The term in 
is the group velocity dispersionDispersion (optics)In optics, dispersion is the phenomenon in which the phase velocity of a wave depends on its frequency, or alternatively when the group velocity depends on the frequency.Media having such a property are termed dispersive media...
(GVD) or second-order dispersion; it increases the pulse duration and chirps the pulse as it propagates through the medium. The term in
is a third-order dispersion term that can further increase the pulse duration, even if 
vanishes. The terms in 
and 
describe the walk-off of the pulse; the coefficient 
is the ratio of the component of the group velocity 
and the unit vector in the direction of propagation of the pulse (z-axis). The terms in 
and 
describe diffraction of the optical wave packet in the directions perpendicular to the axis of propagation. The terms in 
and 
containing mixed derivatives in time and space rotate the wave packet about the 
and 
axes, respectively, increase the temporal width of the wave packet (in addition to the increase due to the GVD), increase the dispersion in the 
and 
directions, respectively, and increase the chirp (in addition to that due to 
) when the latter and/or 
and 
are nonvanishing. The term 
rotates the wave packet in the 
plane. Oddly enough, because of previously incomplete expansions, this rotation of the pulse was not realized until the late 1990s but it has been experimentally confirmed. To third order, the RHS of the above equation is found to have these additional terms for the uniaxial crystal case:
-
The first and second terms are responsible for the curvature of the propagating front of the pulse. These terms, including the term in
are present in an isotropic medium and account for the spherical surface of a propagating front originating from a point source. The term 
can be expressed in terms of the index of refraction, the frequency 
and derivatives thereof and the term 
also distorts the pulse but in a fashion that reverses the roles of 
and 
(see reference of Trippenbach, Scott and Band for details).
So far, the treatment herein is linear, but nonlinear dispersive terms are ubiquitous to nature. Studies involving an additional nonlinear term
have shown that such terms have a profound effect on wave packet, including amongst other things, a self-steepening of the wave packet. The non-linear aspects eventually lead to optical solitons.
Despite being rather common, the SVEA is not required to formulate a simple wave equation describing the propagation of optical pulses.
In fact, as shown in, even a very general form of the electromagnetic second order wave equation can be factorized into directional components, providing access to a single first order wave equation for the field itself, rather than an envelope. This requires only an assumption that the field evolution is slow on the scale of a wavelength, and does not restrict the bandwidth of the pulse at all—as demonstrated vividly by.
Applications
- Micro-machiningMachiningConventional machining is a form of subtractive manufacturing, in which a collection of material-working processes utilizing power-driven machine tools, such as saws, lathes, milling machines, and drill presses, are used with a sharp cutting tool to physical remove material to achieve a desired...
- FemtochemistryFemtochemistryFemtochemistry is the science that studies chemical reactions on extremely short timescales, approximately 10–15 seconds .-Introduction:...
- Medical imagingMedical imagingMedical imaging is the technique and process used to create images of the human body for clinical purposes or medical science...
: Ultrashort laser pulses are used in multiphoton fluorescence microscopeFluorescence microscopeA fluorescence microscope is an optical microscope used to study properties of organic or inorganic substances using the phenomena of fluorescence and phosphorescence instead of, or in addition to, reflection and absorption...
s - Terahertz (T-rays) generation and detection.
- frequency combFrequency combA frequency comb is the graphic representation of the spectrum of a mode locked laser. An octave spanning comb can be used for mapping radio frequencies into the optical frequency range or it can be used to steer a piezoelectric mirror within a carrier envelope phase correcting feedback loop...
Further reading
External links
- Multiphoton fluorescence microscopy tutorial
- Spectral interferometry (SI) http://ultrafast.physics.ox.ac.uk/spider/res.html
- The virtual femtosecond laboratory Lab2
- Lecture notes for Ultrafast Optics at MIT OpenCourseWareMIT OpenCourseWareMIT OpenCourseWare is an initiative of the Massachusetts Institute of Technology to put all of the educational materials from its undergraduate- and graduate-level courses online, partly free and openly available to anyone, anywhere. MIT OpenCourseWare is a large-scale, web-based publication of...
- Studies of Nonlinear Femtosecond Pulse Propagation in Bulk Materials with a chapter on ultrashort pulse measurement techniques, PhD thesis (by Hilary K. Eaton)
- Animation on Short Pulse propagation in random medium (YouTube)
- Micro-machining
-