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

11The fundamental mode in Fig. 1.3(a) is located on the equatorial plane of thesphere. Although modes with l jmj > 0 travel with greater inclinations with respectto the equator, they still have the same resonant wavelength as the fundamental mode,since the curvature of the sphere precisely compensates the greater incline. The pathlength for ljmj > 0 modes in Fig. 1.3(a) to Fig. 1.3(d) are all equal, i.e. theyare degenerate.Therefore, the equatorial mode number, m, is superuous whendescribing the resonance wavelength.1.4 Resonance PositionsThe continuity of the tangential components of the electric and magnetic elds at thesphere surface (see Eqn. 1.2) must satisfy a characteristic equation,ns2b [n s kaj l (n s ka)] 0j l (n s ka)=hn s kah (1)l(ka)h (1)l(ka)i 0; (1.3)where b is the polarisation (equal to 1 for TM modes and 0 for TE modes). Theprime denotes dierentiation with respect to the argument. The position and widthof the resonances are obtained by numerically solving this characteristic equation.The solution yields discrete values of frequency at which resonances are possible. Inpractice, the locations of these resonances are found by scanning a tuneable diodelaser over the free spectral range of the sphere.The solution to Eqn. 1.3 requires a signicant amount of computational time asthe resonances have sharp Lorentzian line-shapes. Schiller [27] has used an asymptoticformula to yield an approximation for the resonance frequency in terms of the size