rffiffiffiffiffi A w 2 ¼ n w J s ðk w R c Þ P s ðk c R c Þ�J s ðk c R c Þ P s ðk w R c Þ n c J s ðk w R c Þ V s ðk w R c Þ�Y s ðk w R c Þ P s ðk w R c Þ Ac: ð19Þ
J. Eur. Opt. Society-Rapid Publ. 21, 26( 2025) 259
Then, using the fourth row from equation( 11), one obtains A e: rffiffiffiffiffi A e n e J s ðk w R e Þ A w 1
¼ þ Y sðk w R e Þ A w 2 ð20Þ
Y s ðk e R e Þ n w since A w 1 and Aw 2 are given by equations( 18) and( 19). The procedure to deduce the A’ s amplitudes for TM modesisthesameandthefinal results are: rffiffiffiffiffi
A w 1 ¼ n c J s ðk c R c Þ V s ðk w R c Þ�q 2 Y s ðk w R c Þ P s ðk c R c Þ J s ðk w R c Þ V s ðk w R c Þ�Y s ðk w R c Þ P s ðk w R c Þ Ac n w
n w ð21Þ rffiffiffiffiffi A w 2 ¼ n c q 2 J s ðk w R c Þ P s ðk c R c Þ�J s ðk c R c Þ P s ðk w R c Þ
J s ðk w R c Þ V s ðk w R c Þ�Y s ðk w R c Þ P s ðk w R c Þ Ac
rffiffiffiffiffi A e n w J s ðk w R e Þ A w 1
¼ þ Y sðk w R e Þ A w 2 n e Y s ðk e R e Þ ð22Þ
ð23Þ
where q = n w / n c. Comparing the expressions for the TE and TM cases, one notices that the refractive index ratio q = n w / n c plays a more important role in the TM case, due to the q 2 factors in the numerators.
Figure 13 shows a sketch of the MBR highlighting the three dielectric sectors and the trend of the radial function F( kr) for the fundamental WGM. In Figure 13, we assume a silica MBR( n w = 1.45) filled with water( n c = 1.33) in an air environment( n e = 1). Due to the higher index of the microbubble walls, the mode is confined in the walls sector and only evanescent tails are present in both the core and the external sectors. Even if these evanescent tails represent a tiny fraction of the field distribution, both have important roles for the microbubble applicability. The external tail, in fact, can couple the WGM to a waveguide mode, allowing light injection and extraction. The core tail, instead, allows to sense particles in proximity or in contact with the internal wall, enabling the implementation of the microbubble as an optical sensor.
Sumetsky [ 43 ] reported on the first solution for the eigenfrequencies of MBRs using the Wentzel – Kramers – Brillouin( WKB) approximation. One year later, Rauschenbeutel et al. [ 60 ] published a full scale wave equation calculation using the adiabatic approximation for a solid microbottle. In that modelling, the resonator has a parabolic shape along the z-axis with a small variation of the resonator radius. This small variation allows to separate the radial and the axial coordinates, obtaining a solution that can be written as E( r, q, z)= W( q, z) exp( imu). The maximum angular momentum modes are located close to the surface and therefore, the radial component of the wave vector is negligible compared to the axial and azimuthal one. The total wave vector is:
Figure 13. Sketch of the different dielectric zones of the MBR( left) and F( kr) function for the fundamental WGM( right). Colors are used to highlight the sector over which the function extends: blue for the core, green for the walls and red for the external medium. k ¼ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 z þ k2 u
¼ 2pn k ð24Þ
where k is the vacuum wavelength and n the refractive index. The symmetry of the system is cylindrical and there is no propagation at the caustics, thus the azimuthal and axial wave vectors can be written as:
k u ðzÞ ¼ kR c RðzÞ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k z ðzÞ ¼k 1 � R 2
c
RðzÞ ð25Þ
ð26Þ
where R c is the radius at the caustic, and R c k = m. The function W( q, z) can be separated as the product of the two functions U( q, z) and Z( z)( W( q, z) = U( q, z) Z( z)) with U( q, z) being the solution of the Bessel equation and Z( z) being a function which depends on the profile of the resonator. For a parabolic profile, Z( z)’ s equation is very similar to the one of an harmonic oscillator:
d 2 Z dz 2 þ
k2 � m2
R 2 0
� m2 k 2 z 2
R 2 0
Z ¼ 0
ð27Þ where the term E ¼ k 2 � m 2 = R 2
0 is equivalent to the total energy of the oscillator and the term V ¼ m 2 z 2 k 2 = R 2
0 ¼ ð E mz = 2Þ 2 is equivalent to the potential energy of the oscillator. The function Z is square integrable, and it gives a discrete set of energy levels and the following eigenvalues
|
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi |
|
k mq ¼ |
m 2
�ðq þ 1 = 2Þ E
R 2 m
:
0
|
ð28Þ |
The corresponding solution of equation( 27) is the Hermite polynomial of degree q:
r ffiffiffiffiffiffiffiffiffiffiffiffiffiffi!
E m Z mq ðzÞ ¼C mq H q z exp � E
m z 2 ð29Þ 2
4