Dynamics of fast and slow pulse propagation through a microsphere–optical-fiber system

We discuss the mechanism of fast and slow light generation when optical pulses propagate through a fiber taper coupled to an ultrahigh-$Q$ microsphere system in the time domain. Fast and slow light are explained as interference effects between ballistic light through the fiber and circulated light within the sphere. One of the striking effects predicted by this model is dynamic pulse splitting in the transmitted pulse at the critical coupling condition, where the coupling strength between the sphere and fiber equals the round-trip loss in the sphere. By realizing this critical coupling condition experimentally, we observed this dynamic pulse splitting effect.

Figure 1

Schematic illustration of the microsphere-fiber-taper system.

Figure 2

(a) Illustration of the fast light mechanism in the time domain in the undercoupling condition. The ballistic light is drawn upward, while the circulated light is shown downward, as the phase of the circulated light at the resonance frequency is shifted by $\pi$ with respect to the ballistic light. These curves are drawn in intensity, but the output pulse is calculated by adding the amplitudes of the two waves. (b) Temporal profile of the output pulse for different parameters of the coupling constant. From top to bottom, $y=1$, 0.99997, 0.99995, and 0.999903, respectively. The magnitude of the curve for $y=1$ is scaled by a factor of $1∕2$.

Figure 3

(a) Illustration of the slow light mechanism in the overcoupling condition. These curves are drawn in intensity, but the output pulse is calculated by adding the amplitudes of the two waves. (b) Temporal profile of the output pulse for different parameters of the coupling parameter $y$. From bottom to top, $y=0.999903$, 0.9998, 0.9997, and 0.9995, respectively.

Figure 4

(Color) (a) The time integrated transmission intensity as a function of the detuning frequency. (b) Two-dimensional contour plot of the transmitted pulse intensity as a function of time and frequency detuning of the laser at the critical coupling condition. The intensity is plotted on a logarithmic scale. The time origin is set at the peak arrival time for a pulse at a far detuning frequency of $100\mathrm{MHz}$. (c) The transmitted pulse profile at the resonance frequency along the dashed line in Fig. 4b. The solid thin curve is the experimental result. The dotted curve is the theoretical pulse profile derived from Eqs. (1, 2) in the text. The solid thick curve is the calculated profile when 2.9% of the circulated light is assumed to leak into higher modes in the fiber.