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where in each sector a couple( A 1, A 2) is used, leading to a total of 6 A’ s coefficients. However, since the Bessel functions Y p( x) haveadivergenceinx = 0, they are not suitable for the core sector and therefore one must have A c 2 ¼ 0in that sector. At the same time, since the Bessel functions J p( x) feature slowly decaying oscillations in the limit of high x, they are not suitable to describe the evanescent tail of a Whispering Gallery mode in the external sector and therefore one must have A e 1
¼ 0 in that sector. In conclusion, only four coefficients are necessary to define F( kr), which can be written as
Fkr ð Þ ¼
8 A c J s ðk c rÞ pffiffiffiffiffiffiffi r < R c k c r
>< A w J s ðk w rÞ
1 pffiffiffiffiffiffiffi þ A w Y s ðk w rÞ
2 pffiffiffiffiffiffiffi R c < r < R e k w r k w r A e Y s ðk e rÞ
>: pffiffiffiffiffiffiffi r > R e: k e r ð10Þ
In this equation, k c, k w, p andk ffiffiffi e are the wavenumbers in the
different media( k ¼ k 0 r ¼ k0 n ¼ 2pn = k 0).
Combining equation( 10) with equations( 2)–( 5) to impose the continuity of the E and the H tangential component at each dielectric interface produces a series of constraints on the A’ s amplitudes. These constraints can be resumed using the matrix formalism and, for TE modes, one has
0
M TE B @
A c
A w 1
A w 2
A e
1 0 1 0
C A ¼ 0
B C @ 0 A 0
0
J s ðk c R c Þ pffiffiffiffiffi
� J sðk w R c Þ pffiffiffiffiffi n c n w P s ðk c R c Þ pffiffiffiffiffi
� P sðk w R c Þ pffiffiffiffiffi n c n w M TE ¼
0 � J sðk w R e Þ
pffiffiffiffiffi n w
B @
0 � P sðk w R e Þ
pffiffiffiffiffi n w with ð11Þ
� Y 1
sðk w R c Þ pffiffiffiffiffi
0 n w
� V sðk w R c Þ pffiffiffiffiffi
0 n w
� Y sðk w R e Þ Y s ðk e R e Þ pffiffiffiffiffi pffiffiffiffiffi n w n e
� V C
sðk w R e Þ V s ðk e R e Þ A pffiffiffiffiffi pffiffiffiffiffi n w n e ð12Þ
with representing the usual row-by-column product. For TM modes, instead, one has
0
A c 1 0 1 0
A w 1
M TM B @ A w C
A ¼ 0
B C with ð13Þ @
2
0 A A e 0
0
P s ðk c R c Þ pffiffiffiffiffiffiffiffiffiffiffi
� P sðk w R c Þ pffiffiffiffiffiffiffiffiffiffi n 2 c n w n e n 2 w n c n e
J s ðk c R c Þ pffiffiffiffiffiffiffiffiffiffiffi
� J sðk w R c Þ pffiffiffiffiffiffiffiffiffiffi n w n e n c n e M TM ¼
0 � P sðk w R e Þ pffiffiffiffiffiffiffiffiffiffi n 2 w n c n e
B @
0 � J sðk w R e Þ pffiffiffiffiffiffiffiffiffiffi n c n e
� V 1
sðk w R c Þ pffiffiffiffiffiffiffiffiffiffi
0 n 2 w n c n e � Y sðk w R c Þ pffiffiffiffiffiffiffiffiffiffi 0 n c n e � V sðk w R e Þ V s ðk e R e Þ
: pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi n 2 w n c n e n 2 e n w n c � Y C
sðk w R e Þ Y s ðk e R e Þ A pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffiffi n c n e n c n w ð14Þ
In both cases, the auxiliary functions P s( x) = xJ s�1( x) �‘ J s( x) andV s( x)= xY s�1( x) �‘ Y s( x) are introduced for more compact expressions.
To�have a solution with physical meaning, the amplitudes A c; A w 1; Aw 2; Ae have to be non-zero and this implies that M TE and M TM must have null determinant:
det ½ M TE Š ¼ 0 for TE modes ð15Þ
det ½ M TM Š ¼ 0 for TM modes: ð16Þ
In the end, equations( 15) and( 16) are the characteristic equations that define the WGM spectrum of the microbubble resonator. In fact, if the geometry and the indexes of the MBR are fixed, the remaining variables are the vacuum wavelength k 0( which appears through the wavevectors k c, k w and k e) andtheinteger‘. In particular, by fixing the‘ value( e. g.‘ =‘ 1) and then scanning k 0, onefinds a series of roots( k ð‘ 1; 1Þ 0, k ð‘ 1; 2Þ
0, k ð‘ 1; 3Þ
0,...) which are the WGMs resonance wavelengths of that specific‘ family. By repeating the root-finding procedure for different‘ values( e. g.‘ 1,‘ 2,‘ 3,...), one can reconstruct the overall WGM spectrum. ð‘; nÞ
� Once the roots k0 are found, the amplitudes A c; A w 1; Aw 2; Ae of each WGM can be found by inverting equations( 11) and( 13). We highlight that since k 0’ sand‘’ s values are now known, the matrices M TE and M TM are fully computable, without any unknown parameter. For brevity, we report the inversion procedure only for theTEmodes: theprocedurefortheTMmodesisanalogous. Since M TE has null determinant � from the characteristic equation( 15), thevector A c; A w 1; Aw 2; Ae in equation( 11) can be inverted up to a free component( i. e. three A’ s can be written as a function of the fourth). We choose A c as the free amplitude and rearrange the system in equation( 11) so that terms with A c are on the right side. Rows two and three give
0 1 0
1 J s ðk w R c Þ Y s ðk w R c Þ
J s ðk c R c Þ pffiffiffiffiffi pffiffiffiffiffi n w n w A w pffiffiffiffiffi
A c
1 n c
B
C @ P s ðk w R c Þ V s ðk w R c Þ A A w ¼ B
C @
2
P s ðk c R c Þ pffiffiffiffiffi pffiffiffiffiffi pffiffiffiffiffi
A c A n w n w n c ð17Þ which can rffiffiffiffiffi
be resolved using Cramer’ s rule: A w 1 ¼ n w J s ðk c R c Þ V s ðk w R c Þ�Y s ðk w R c Þ P s ðk c 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 ð18Þ