Two-Dimensional Pulse Propagation without Anomalous Dispersion

Anomalous dispersion is a surprising phenomenon associated with wave propagation in an even number of space dimensions. In particular, wave pulses propagating in two-dimensional space change shape and develop a tail even in the absence of a dispersive medium. We show mathematically that this dispersion can be eliminated by considering a modified wave equation with two geometric spatial dimensions and, unconventionally, two timelike dimensions. Experimentally, such a wave equation describes pulse propagation in an optical or acoustic medium with hyperbolic dispersion, leading to a fundamental understanding and new approaches to ultrashort pulse shaping in nanostructured metamaterials.

Figure 1

Numerical solution of the wave equation in three cases: (a) conventional 3-D wave equation $u_{tt}=c^2(u_{xx}+u_{yy}+u_{zz})$; (b) conventional 2-D wave equation $u_{tt}=c^2(u_{xx}+u_{yy})$ showing anomalous dispersion; (c) modified two-dimensional wave equation $u_{tt}-(1/t)u_t+(1/t^2)u=c^2(u_{xx}+u_{yy})$ with no anomalous dispersion. The insets show the solution at a fixed time. The initial conditions are zero initial value $u(r,t=0)=0$ and a radially symmetric initial time derivative Gaussian distribution $u_t(r,t=0)=\exp [-r^2/(2{\sigma }_r^2)]$ with ${\sigma }_r=0.01$. The radial variable is $r=\sqrt{x^2+y^2+z^2}$ for (a) and $r=\sqrt{x^2+y^2}$ for (b) and (c).