Editing Ultrashort pulse (section)

==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 <math> \textbf{K}_0 </math> and central frequency <math> \omega_0 </math> of the pulse, is given by:
:<math>
\textbf{E} ( \textbf{x} , t) = \textbf{ A } ( \textbf{x} , t) \exp ( i \textbf{K}_0 \textbf{x} - i \omega_0 t )
</math>
We consider the propagation for the SVEA of the electric field in a homogeneous dispersive nonisotropic medium. Assuming the pulse is propagating in the direction of the z-axis, it can be shown that the envelope <math> \textbf{A} </math> for one of the most general of cases, namely a biaxial crystal, is governed by the [[partial differential equation|PDE]]:<ref>{{Cite journal | doi=10.1103/PhysRevLett.76.1457| pmid=10061728| bibcode=1996PhRvL..76.1457B| title=Optical Wave-Packet Propagation in Nonisotropic Media| year=1996| last1=Band| first1=Y. B.| last2=Trippenbach| first2=Marek| journal=Physical Review Letters| volume=76| issue=9| pages=1457–1460}}</ref>
:<math>
\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}
</math>
::<math>
~~~~~~~~~~~
~+~ 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} + \cdots
</math>
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-isotropic media, valid in the near-field and far-field.
<math> \beta_1 </math> is the inverse of the [[group velocity]] projection. The term in <math> \beta_2 </math> is the group velocity [[dispersion (optics)|dispersion]] (GVD) or second-order dispersion; it increases the pulse duration and chirps the pulse as it propagates through the medium. The term in <math> \beta_3 </math> is a third-order dispersion term that can further increase the pulse duration, even if <math> \beta_2 </math> vanishes. The terms in <math> \gamma_x </math> and <math> \gamma_y </math> describe the walk-off of the pulse; the coefficient <math> \gamma_x ~ (\gamma_y ) </math> is the ratio of the component of the group velocity <math> x ~ (y) </math> and the [[unit vector]] in the direction of propagation of the pulse (z-axis). The terms in <math>\gamma_{xx}</math> and <math> \gamma_{yy} </math> describe diffraction of the optical wave packet in the directions perpendicular to the axis of propagation. The terms in <math> \gamma_{tx} </math> and <math> \gamma_{ty} </math> containing mixed derivatives in time and space rotate the wave packet about the <math>y</math> and <math>x</math> axes, respectively, increase the temporal width of the wave packet (in addition to the increase due to the GVD), increase the dispersion in the <math>x</math> and <math>y</math> directions, respectively, and increase the chirp (in addition to that due to <math> \beta_2 </math>) when the latter and/or <math> \gamma_{xx} </math> and <math> \gamma_{yy} </math> are nonvanishing. The term <math> \gamma_{xy} </math> rotates the wave packet in the <math> x-y </math> 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.<ref>{{cite journal |doi=10.1364/JOSAB.14.000420|bibcode=1997JOSAB..14..420R|title=Interferometric measurement of femtosecond wave-packet tilting in rutile crystal|year=1997|last1=Radzewicz|first1=C.|last2=Krasinski|first2=J. S.|last3=La Grone|first3=M. J.|last4=Trippenbach|first4=M.|last5=Band|first5=Y. B.|journal=Journal of the Optical Society of America B|volume=14|issue=2|pages=420}}</ref> To third order, the RHS of the above equation is found to have these additional terms for the uniaxial crystal case:<ref>{{cite journal |doi = 10.1364/OL.22.000579|pmid = 18185596|bibcode = 1997OptL...22..579T|title = Near-field and far-field propagation of beams and pulses in dispersive media|year = 1997|last1 = Trippenbach|first1 = Marek|last2 = Scott|first2 = T. C.|last3 = Band|first3 = Y. B.|journal = Optics Letters|volume = 22|issue = 9|pages = 579–81 |url=http://www.bgu.ac.il/%7Eband/Tripp.OptLet22.579.97.pdf}}</ref>
::<math>
\cdots
~+~ \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} + \cdots
</math>
The first and second terms are responsible for the curvature of the propagating front of the pulse. These terms, including the term in <math>\beta_3</math> are present in an isotropic medium and account for the spherical surface of a propagating front originating from a point source. The term <math> \gamma_{txx} </math> can be expressed in terms of the index of refraction, the frequency <math> \omega </math> and derivatives thereof and the term <math> \gamma_{ttx} </math> also distorts the pulse but in a fashion that reverses the roles of <math> t </math> and <math> x </math> (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 <math> \gamma_{nl} |A|^2 A </math> have shown that such terms have a profound effect on wave packet, including amongst other things, a ''self-steepening'' of the wave packet.<ref>{{Cite journal | doi=10.1103/PhysRevA.56.4242| bibcode=1997PhRvA..56.4242T| title=Dynamics of short-pulse splitting in dispersive nonlinear media| year=1997| last1=Trippenbach| first1=Marek| last2=Band| first2=Y. B.| journal=Physical Review A| volume=56| issue=5| pages=4242–4253}}</ref> The non-linear aspects eventually lead to [[soliton (optics)|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,<ref name=kinsler2010>{{cite journal|last1=Kinsler|first1=Paul|title=Optical pulse propagation with minimal approximations|journal=Physical Review A|volume=81|issue=1|pages=013819|year=2010|issn=1050-2947|doi=10.1103/PhysRevA.81.013819|arxiv=0810.5689|bibcode=2010PhRvA..81a3819K}}</ref> 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.<ref name=genty2007>{{cite journal|last1=Genty|first1=G.|last2=Kinsler|first2=P.|last3=Kibler|first3=B.|last4=Dudley|first4=J. M.|title=Nonlinear envelope equation modeling of sub-cycle dynamics and harmonic generation in nonlinear waveguides|journal=Optics Express|volume=15|issue=9|year=2007|pages=5382–7|issn=1094-4087|doi=10.1364/OE.15.005382|pmid=19532792|bibcode=2007OExpr..15.5382G|doi-access=free}}</ref>