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# Ultrashort pulse

In optics, an ultrashort pulse of light is an electromagnetic pulse whose time duration is on the order of the femtosecond (10 − 15 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 reactions on the timescales they occur on, opening up the field of femtochemistry.

## Definition

An ultrashort pulse of light in the time domain. In this figure, the amplitude and intensity are Gaussian functions. The phase function is quadratic, resulting in an instantaneous frequency sweep sometimes called a chirp, in analogy to the sound of some birds.

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):

$E(t) = \sqrt{I(t)}e^{i\omega_0t}e^{i\psi(t)}$

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

$E(\omega) = \mathcal{F}(E(t))$

Because of the presence of the $e^{i\omega_0t}$ 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.

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

$E(\omega) = \sqrt{S(\omega)}e^{i\phi(\omega)}$

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 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 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.

## 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 compressor[1], 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.

## Measurement techniques

Several techniques are available to measure ultrashort optical pulses:

• intensity autocorrelation: 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 reconstruction (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 gating (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 approximation (SVEA) of the electric field of a wave with central wave vector $\textbf{K}_0$ and central frequency ω0 of the pulse, is given by:

$\textbf{E} ( \textbf{x} , t) = \textbf{ A } ( \textbf{x} , t) \exp ( i \textbf{K}_0 \textbf{x} - i \omega_0 t )$

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 $\textbf{A}$ for one of the most general of cases, namely a biaxial crystal, is governed by the PDE[2]:

$\frac{\partial \textbf{A} }{\partial z } = ~-~ \beta_1 \frac{\partial \textbf{A} }{\partial t} ~-~ \frac{i}{2} \beta_2 \frac{\partial^2 \textbf{A} }{\partial t^2} ~+~ \frac{1}{6} \beta_3 \frac{\partial^3 \textbf{A} }{\partial t^3} ~+~ \gamma_x \frac{\partial \textbf{A} }{\partial x} ~+~ \gamma_y \frac{\partial \textbf{A} }{\partial y}$
$~~~~~~~~~~~ ~+~ i \gamma_{tx} \frac{\partial^2 \textbf{A} }{\partial t \partial x} ~+~ i \gamma_{ty} \frac{\partial^2 \textbf{A} }{\partial t \partial y} ~-~ \frac{i}{2} \gamma_{xx} \frac{\partial^2 \textbf{A} }{ \partial x^2} ~-~ \frac{i}{2} \gamma_{yy} \frac{\partial^2 \textbf{A} }{ \partial y^2} ~+~ i \gamma_{xy} \frac{\partial^2 \textbf{A} }{ \partial x \partial y} + \ldots$

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. β1 is the inverse of the group velocity projection. The term in β2 is the group velocity dispersion (GVD) or second-order dispersion; it increases the pulse duration and chirps the pulse as it propagates through the medium. The term in β3 is a third-order dispersion term that can further increase the pulse duration, even if β2 vanishes. The terms in γx and γy describe the walk-off of the pulse; the coefficient $\gamma_x ~ (\gamma_y )$ is the ratio of the component of the group velocity $x ~ (y)$ and the unit vector in the direction of propagation of the pulse (z-axis). The terms in γxx and γyy describe diffraction of the optical wave packet in the directions perpendicular to the axis of propagation. The terms in γtx and γty containing mixed derivatives in time and space rotate the wave packet about the y and x axes, respectively, increase the temporal width of the wave packet (in addition to the increase due to the GVD), increase the dispersion in the x and y directions, respectively, and increase the chirp (in addition to that due to β2) when the latter and/or γxx and γyy are nonvanishing. The term γxy rotates the wave packet in the xy 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.[3] To third order, the RHS of the above equation is found to have these additional terms for the uniaxial crystal case[4]:

$\ldots ~+~ \frac{1}{3} \gamma_{t x x } \frac{\partial^3 \textbf{A} }{ \partial x^2 \partial t} ~+~ \frac{1}{3} \gamma_{t y y } \frac{\partial^3 \textbf{A} }{ \partial y^2 \partial t} ~+~ \frac{1}{3} \gamma_{t t x } \frac{\partial^3 \textbf{A} }{ \partial t^2 \partial x} + \ldots$

The first and second terms are responsible for the curvature of the propagating front of the pulse. These terms, including the term in β3 are present in an isotropic medium and account for the spherical surface of a propagating front originating from a point source. The term γtxx can be expressed in terms of the index of refraction, the frequency ω and derivatives thereof and the term γttx also distorts the pulse but in a fashion that reverses the roles of t and x (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 γnl | A | 2A have shown that such terms have a profound effect on wave packet, including amongst other things, a self-steepening of the wave packet.[5] 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,[6] 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.[7]

## Notes

1. ^ J. C. Diels, Femtosecond dye lasers, in Dye Laser Principles, F. J. Duarte and L. W. Hillman (Eds.) (Academic, New York, 1990) Chapter 3.
2. ^ M. Trippenbach, and Y.B. Band, "Optical Wave-Packet Propagation in Nonisotropic Media", Phys. Rev. Lett. 76 (1457) 1996. [1]
3. ^ C. Radzewicz, J. S. Krasinski, M. J. laGrone, M. Trippenbach, and Y. B. Band, “Interferometric measurement of femtosecond wave-packet tilting in rutile crystal", J. Opt. Soc. Am. B 14 (420-424) 1997. [2]
4. ^ M. Trippenbach, T.C. Scott, and Y.B. Band, "Near-and Far Field Propagation of Beams and Pulses in Dispersive Media", Opt. Lett. 22 (579) 1997. [3] [4]
5. ^ M. Trippenbach and Y.B. Band, "Dynamics of short-pulse splitting in dispersive nonlinear media", Phys. Rev. A, 56 (4242) 1997. [5]
6. ^ Kinsler, P. (2010). "Optical pulse propagation with minimal approximations". Phys. Rev. A 81: 013819. arXiv:0810.5689. Bibcode 2010PhRvA..81a3819K. doi:10.1103/PhysRevA.81.013819.
7. ^ Genty, G.; Kinsler, P.; Kibler, B.; Dudley, J. M. (2007). "Nonlinear envelope equation modeling of sub-cycle dynamics and carrier shocks and harmonic generation in highly nonlinear waveguides". Opt. Express 15: 5382–5387. Bibcode 2007OExpr..15.5382G. doi:10.1364/OE.15.005382.

## References

• Hirlimann, C. (2004). "Pulsed Optics". In Rullière, Claude. Femtosecond Laser Pulses: Principles and Experiments (2nd ed. ed.). New York: Springer. ISBN 0-387-01769-0.

Andrew M. Weiner (2009). Ultrafast Optics. Hoboken, NJ: Wiley. ISBN 978-0-471-41539-8.

J. C. Diels and W. Rudolph (2006). Ultrashort Laser Pulse phenomena. New York, Academic. ISBN 978-0-12-215493-5.

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