[UPSC IFoS 2011] Polarisation and Modern Optics (Q401, 10M)

Consider a Gaussian pulse propagating in pure silica, at \lambda_0 = 0\cdot 85\\ \mu\text{m}, along z-direction. As the pulse propagates, it gets broadened according to the formula \tau^2(z) = \tau_0^2 \left( 1 + \frac{4 \alpha^2 z^2}{\tau_0^4} \right), \quad \alpha = \left. \frac{d^2 k}{d \omega^2} \right|_{\omega=\omega_0} where \tau(z) is the pulse width after propagation through a distance z and \tau_0 is the pulse width at z = 0. Given that \alpha = 3 \times 10^{-26}\\ \frac{\text{S}^2}{\text{m}} at \lambda_0 = 0\cdot 85\\ \mu\text{m}. Calculate the distance at which \tau(z) = \sqrt{2} \cdot \tau_0.