Ultrashort Pulse - Wave Packet Propagation in Nonisotropic Media | Wave Packet Propagation Nonisotropic Media

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 and central frequency 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 for one of the most general of cases, namely a biaxial crystal, is governed by the PDE:

$\frac{\partial \textbf{A} }{\partial z } = ~-~ \beta_1 \frac{\partial \textbf{A} }{\partial t} ~-~ \frac{i}{2} \beta_2 \frac{\partial^2 \textbf{A} }{\partial t^2} ~+~ \frac{1}{6} \beta_3 \frac{\partial^3 \textbf{A} }{\partial t^3} ~+~ \gamma_x \frac{\partial \textbf{A} }{\partial x} ~+~ \gamma_y \frac{\partial \textbf{A} }{\partial y}$

$~~~~~~~~~~~ ~+~ i \gamma_{tx} \frac{\partial^2 \textbf{A} }{\partial t \partial x} ~+~ i \gamma_{ty} \frac{\partial^2 \textbf{A} }{\partial t \partial y} ~-~ \frac{i}{2} \gamma_{xx} \frac{\partial^2 \textbf{A} }{ \partial x^2} ~-~ \frac{i}{2} \gamma_{yy} \frac{\partial^2 \textbf{A} }{ \partial y^2} ~+~ i \gamma_{xy} \frac{\partial^2 \textbf{A} }{ \partial x \partial y} + \cdots$

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 dispersion (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:

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

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.

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