Conical emission from short-pulse propagation in normally dispersive media

A uniform asymptotic approximation of Maxwell's equations for propagation of a short pulse in Kerr media yields the well-known nonlinear Schrodinger equation 2ik (∂A÷∂z + k′ ∂A÷∂t) + ▽_⊥² - kk″ ∂²A÷∂t² + 2k²n₂÷n |A|²A = 0, (1) which describes the dynamical evolution of the light pulse subject to diffraction, normal dispersion, and cubic nonlinearity. This equation reproduces many of the features associated with short-pulse propagation that are observed experimentally. In this paper we show that a form of conical emission, due to a four-wave interaction, is captured by Eq. (1). In Fig. 1 the evolution of the intensity and spectrum of a pulse, calculated by integrating Eq. (1), is shown before and after pulse splitting. As the pulse propagates, its bandwidth spreads both in transverse wave number, k_⊥, and in frequency, ω-0$/ + Ω. Note that the energy of the pulse is not redistributed uniformly. Instead, energy is preferentially coupled into the band of wave numbers and frequencies indicated by the cross in Fig. 1, on which Ω² ∝ k_⊥². In Fig. 2 the farfield time-integrated image of the pulse is shown both with and without a filter. Whereas the unfiltered image has no noticeable structure, the filtered image clearly shows off-axis emission at the frequency of the filter. Figure 1 shows that conical emission predicted by Eq. (1) occurs at frequencies of the pulse. The emission angle increases with the magnitude of the shift. As the pulse power is increased, a threshold is reached where the conical signal grows explosively. Here self-focusing plays an important role by reducing the scale length for growth. The same instability that is responsible for conical emission causes an enhancement in the broadening of spectrum of the pulse. An explosive broadening of the spectrum also occurs at this threshold.