Deflection of ultra slow light under gravity

Recent experiments on ultra slow light in strongly dispersive media by several research groups reporting slowing down of the optical pulses down to speeds of a few metres per second encourage us to examine the intriguing possibility of detecting a deflection or fall of the ultra slow light under Earth’s gravity, i.e., on the laboratory length scale. In the absence of a usable general relativistic theory of light waves propagating in such a strongly dispersive optical medium in the presence of a gravitational field, we present a geometrical optics based derivation that combines {\it the effective gravitational refractive index} additively with the usual optical dispersion. It gives a deflection, or the vertical fall $\mathrm{\Delta }$ for a horizontal traversal $L$ as $$\mathrm{\Delta }=\frac{L^2}{2}\left(\frac{R_{\oplus G}}{R_{\oplus }^2}\right)n_g\left(\frac{1}{1+n_g\frac{R_{\oplus G}}{R_{\oplus }}}\right),$$ where $R_{\oplus G}/R_\oplus$istheratioofthegravitationalEarthradius($R_{\oplus G}$) to its geometrical radius $R_{\oplus }$, and $n_g$ is the group refractive index of the strongly dispersive optical medium. The expression is essentailly that for the Newtonian fall of an object projected horizontally with the group speed $v_g=c\mathrm{/}{n}_g$, and is tunable refractively through the index $n_g$. For $L \sim 1 m$and$n_g = c/v_g \sim 10^8$(correspondingtotheultra-slowpulsespeed$\sim few \times 1 ms^{-1}$),weobtainafall$\Delta \sim 1 \mu m$, that should be measurable $-$ in particular through its sensitive dependence on the frequency that tunes $n_g$.