optical characterisation of rare-earth doped fluoride and phosphate ...

107detuning can be written as a function of the input pump intensity, x x0 2= + KI inc jc j 2 ; (4.9)¡xwhere K is a constant dening the amount of spatial overlap between the perturbationand the energy distribution of the mode. This equation produces the black line inthe gure which scans across jc j 2 with increasing values of incident pump I inc . Theintersection points between the line and jc j 2 are solutions to the cubic equation inEqn. 4.7. The cubic equation in Eqn. 4.7 produces a response similar to that shownin Fig. 4.11(b). In reality, these intersections describe the positioning of the cavityresonance relative to the pump frequency (c.f. the movie linked in Fig. 4.11). In caseswhere the line intersects at three points for increasing I inc , the most stable positionis that with minimum detuning. Alternatively, for decreasing I inc , the most stableposition is that with maximum detuning. There are no stable solutions where the lineintersects two points. The internal microsphere intensity amplitude is jc j 2 E 2 o whereI inc / E 2 o. A theoretical bistable plot of the internal sphere eld is plotted in Fig.4.11(c) and clearly mimics the results in Fig. 4.6.Comparison of the bistability plots for IR lasing in Fig. 4.5 and uorescence in Fig.4.6 indicates somewhat unusual behaviour around the switch-low position. The IRlasing suddenly peaks at 10 mW launched power before switching to the lower state.All spheres have demonstrated this behaviour to a greater or lesser degree. Despitethis, the uorescence never shows a sudden jump before switching to the lower state.In particular, a sudden increase in absorption (as determined from the ratio of thetwo green emissions discussed in Section 4.3.3) in the sphere at this point, is neverobserved. There seems to be no similar eect observed in Fabry-Perot cavities andother resonators, which might have shed some insight onto the problem. A possible