J. Eur. Opt. Society-Rapid Publ. 21, 26( 2025) 257
meridian. With this factorisation, the field expression for the TE( transverse electric) modes becomes
H ¼ i
‘ ð‘ þ 1Þ xl 0 r
E ¼ FðkrÞ X‘; m
Fkr ð ÞZ‘; m þ 1 r d rF kr dr ð ð Þ
ÞY‘; m
ð2Þ
ð3Þ
Figure 12. Panels( a) and( b) Two sketches of the microbubble modeling presented in Section 3. Different colors are used to mark the three dielectric sectors: blue for core, green for walls, red for external medium. Panel( c) shows the orientation of the spherical coordinate system used in this theory with respect to a real microbubble.
are marked with arrows and the spherical coordinate system is shown. In addition, Figure 12c shows the orientation of the coordinate system with respect to a real microbubble, highlighting the correspondence of the z-axis with the capillary stem direction and of the xy-plane with the equatorial plane of the microbubble. The modeling here presented assumes uniform wall thickness along the meridians. When considering the wall tapering of the microbubble( cf. Fig. 12c), this may appear a strong approximation. However, since WGMs have a limited extension outside the equatorial plane, only the wall thickness at the equator is relevant for their properties and therefore this hypothesis does not limit the results of the model. As a consequence, only the wall thickness at the equator is relevant for the computation and it is used as the W value. A real approximation, instead, is made by neglecting the absorption of the various dielectric media, therefore assuming the refractive indexes to be real-valued parameters. This approximation produces real-valued characteristic equations and allows to implement basic numerical methods for their solution. Finally, focusing on the advantages of this modeling, we highlight its versatility in terms of refractive indexes and radii. In fact, since there are no constraints on these parameters, several experimental configurations can be simulated: both in terms of different materials by changing n c, n w, andn e, as well as in terms of different sizes by changing R c and W( or equivalently R e).
By solving the scalar Helmholtz equation in spherical coordinates and then using the Hansen method to obtain the vectorial solution to the Helmholtz equation, it is possible to write the analytical expressions for both the electric field and the magnetic fields in each sector of the MBR. In particular, the scalar solution for the Helmholtz equation( an analogous of the time independent 1D Schrödinger equation) for a dielectric sector having index n( and therefore wavenumber k = k 0 n = 2pn / k 0) is written as:
wðr
; h; / Þ ¼ FðkrÞ Y‘; m ðh; / Þ ð1Þ
where the function F( kr) accounts for the radial dependence and the spherical harmonic Y‘, m( h, /) accounts for the angular dependence. The function F( kr) is a key quantity for the computation of the modal volume, while Y‘, m( h, /) fixes the WGM distribution along the while the expression for TM( transverse magnetic) modes is
E ¼‘ ð‘ þ 1Þ kr
Fkr ð ÞZ‘; m þ 1 kr
H ¼ i k xl 0
FðkrÞ X‘; m
d rF kr dr ð ð ÞÞY‘; m ð4Þ
ð5Þ
with the auxiliary vectors X‘, m, Y‘, m and Z‘, m defining the spatial direction of the fields. These vectors are defined in equations( 6)–( 8) and form an orthogonal base, with Z‘, m being parallel to ^r and with X‘, m and Y‘, m being a combination of ^h and ^ /. Indeed, this leads to the usual classification of TE and TM modes, where the electric field E is totally tangential( i. e. orthogonal to the radial direction) for TE modes and the magnetic field H is totally tangential for TM modes.
X‘; m ¼rY‘; m ^ r ¼ 1 @ sin h @/ Y ^h‘; m � @ @ h Y ^‘; m /
Y‘; m ¼ r rY‘; m ¼ @ @ h Y ^h‘; m þ 1 @ sin h @/ Y ^‘; m /
Z‘; m ¼ Y‘; m ^r: ð6Þ
ð7Þ
ð8Þ
The solutions of the scalar Helmholtz equation are the the Bessel functions and this allows to write w( r, h, /) as
wðr; h; / Þ ¼ Fkr ð ÞY‘; m ðh; / Þ
J s ðkrÞ Y s ðkrÞ ¼ A 1 pffiffiffiffiffi þ A 2 pffiffiffiffiffi Y‘; m ðh; / Þ kr kr ð9Þ
with s =‘ + 1 / 2. This form shows explicitly the r and the k dependence of the radial function and is useful when imposing the boundary conditions on electric and magnetic fields. It is in fact from these conditions that the characteristic equations for the TE modes and the TM modes arise, limiting the possible values for the wavelength k and therefore defining the spectrum of the microbubble Whispering Gallery modes. For brevity, here we only discuss the procedure for TE modes: the procedure for the TM modes is identical and we just report the main result.
The first step of the procedure consists in reviewing the number of combination coefficients( i. e. the A 1’ sandthe A 2’ sinEq.( 9)) thatarenecessarytodefine the function F( kr). In general, since the microbubble is made of three dielectric sectors, one has a piecewise definition of F( kr)