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

10where j l (h (1)l) are spherical Bessel functions (Hankel functions of the rst kind) oforder l and ! X lm = [l(l +1)]1=2 ! L Y lm and Y lm are vector spherical harmonic functionswith ! L = ! r ¢( ir). These functions (X lm and Y lm ) describe the sinusoidal natureof the eld corresponding to reections of the ray o the sphere surface.The three mode numbers, n, l, and m, are used to characterise the radial, theequatorial, and the polar components of the electromagnetic eld.Mode numbern (n = 1; 2; 3; : : :) gives the number of maxima in the radial electromagnetic elddistribution. The fundamental (n = 1) TM radial mode is depicted in Fig. 1.2(a)and two radial modes are depicted in Fig. 1.2(b). The second mode number, m(m = l; l + 1; : : : ; 0; : : : ; l 1; l), describes the number of maxima in the sinusoidalvariation in eld intensity in the equatorial or latitudinal direction.Finally, themode number l describes the eld intensity distribution in the polar or longitudinaldirection, and can take values l = 0; 1; 2; : : : with the number of maxima present inthe polar eld distribution given by l jmj + 1.For a perfect sphere (zero ellipticity), modes with the same l but dierent m valueshave a 2l + 1 degeneracy, and, therefore, the mode numbers n and l are sucient todescribe the WGM. In reality, some ellipticity exists in the sphere and this removesthe degeneracy. For minimal mode volumes one wants n to be small and m l,as shown in Fig. 1.3. The spheres used in the present studies have a diameter inthe range 50 - 200 m with < 5% ellipticity.Figure 1.3 shows a cross-section ofthe eld for dierent values of m and l. We see that the mode volume is lowest forthe fundamental mode and increases for more complicated mode structures. In theray optics picture, the fundamental mode represents a ray with the smallest reectionangle, the lowest diraction losses, the lowest mode volume, and the highest Q factor.