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Numerical evolutions of whispering gallery modes of both isotropic and anisotropic spheroidal resonators are presented and are used to build analytical approximations of these modes. Such approximations are carried out mainly to have the possibility to have manageable analytic formulas for the eigenmodes and eigenfrequencies of anisotropic resonators. A qualitative analysis of ellipsoidal anisotropic modes in terms of superposition of spherical modes is also presented.

Thanks to their extremely high quality-factor (up to

In this paper we study whispering gallery modes (WGMs) in isotropic and anisotropic dielectric ellipsoidal resonators. Whereas nonspherical WGMs have been already studied in the past, and their analytical expression was given in terms of spherical-like functions [

This paper is organized as follows: in Section

Let us consider an ellipsoid of revolution filled by a dielectric medium, rotationally invariant around the _{3}), an anisotropic crystal characterized by the following dielectric tensor:

(Color online) Sketch of the 3D geometry of the resonator (a) and its cross section in the

In order to make a suitable approximation, we compare the results obtained from this resonator with the ones obtained from a toroidal resonator with circular section like the one depicted in Figure

(Color online) Sketch of the 3D geometry of the toroidal (a) resonator that approximates the ellipsoidal resonator and its cross section in the

(Color online) Lateral (a) and top (b) view of the ellipsoidal resonator (blue solid line) and the approximating torus of circular cross section (red shaded circle). The radius of the torus cross section

Even if an analytical solution for the open ellipsoid (with the fields continuous at the resonator surface) does not exist, it is possible to transform the open resonator into a closed one (with ideal metallic boundaries, i.e., the electric field is zero at the resonator surface) by simply considering that the field of a WGM outside the resonator is evanescent (i.e., exponentially decaying from the resonator surface) and replacing the real spheroid with semiaxes

Before entering deep in the subject of this paper, we briefly intend to recall to the mind of the reader the weak formulation of Maxwell’s equations with Galerkin’s method of the weighted residuals, that we use to simulate with COMSOL whispering gallery modes in a general axisymmetric dielectric medium with permittivity tensor

We choose to implement such a model rather than using a standard mode solver because the latter cannot be easily configured to fully exploit the axial symmetry of the problem and they experience some problems when dealing with WGMs. Then, in order to obtain from them an accurate solution, a fully 3D model must be implemented, making the calculations very time-consuming and complicated.

Galerkin’s method, on the other hand, has the advantage that can easily take into account the symmetry of the problem, giving in this case the possibility to reduce the dimension of the problem from 3D to 2D, by solving the problem only in the

This method, compared with mode solvers, allow us to save a lot of computational time and memory and gives us the possibility to calculate all the fields in the post processing phase of the simulation and only if we are interested in those quantities.

This method can be applied either for solving directly the electric field components [

Our results are based on the discussion presented in [

The electromagnetic field inside the resonator obeys Maxwell’s equations in continuous macroscopic media [

In the general case, however, the medium is not isotropic, and the dielectric constant became a tensor. For the sake of clarity, we will approach the problem from a general point of view, leaving the dielectric constant as a tensor in order to obtain a general expression for the magnetic field inside such a resonator.

After obtaining the general expression, we will specialize it to the case of either isotropic (Section

By eliminating the electric field in favor of the magnetic field in the Maxwell’s equations, after some simple algebra the problem reduces to solve the following equation:

In order to have the complete solution of this equation, suitable boundary conditions must be applied: in our case we will assume that the resonator is closed and so we will use the so-called electric wall boundary conditions, namely,

From (

In order to reduce (

Then we integrate by parts over the spatial coordinates and, after the disposal of the surface integrals with the help of boundary conditions, we finally arrive to the desired weak form of (

The axial symmetry of the problem suggests to describe the resonator in cylindrical coordinates

The next step is to transform (

In Figures

(Color online) Contour plot of the fundamental isotropic WGM (

(Color online) Contour plot of the sixth isotropic WGM (

As can be seen from Figures

(Color online) Comparison of the

(Color online) Same as Figure

(Color online) Comparison of the

(Color online) Same as Figure

In order to make the approximation complete, we calculated the eigenvalues corresponding to the resonator’s eigenmodes. Figure

(Color online) Comparison between isotropic ellipsoidal (red dots) and toroidal (blue dots) eigenvalues corresponding to the first seven WGM. The inset shows the expression of the relative error used in this work to estimate the validity of our approximation.

The approximation is very good for the first few WGMs, where the mode is highly confined near the resonator surface both in

When higher order modes are considered (like, e.g., the one depicted in Figure

For the case of anisotropic resonators, results are reported in Figures

(Color online) Same as Figure

(Color online) Same as Figure

(Color online) Same as Figure

(Color online) Same as Figure

(Color online) Same as Figure

(Color online) Same as Figure

The geometry of the anisotropic resonators is the same as the one described before for the isotropic case. The value of the dielectric function in the

Figure

(Color online) Comparison between anisotropic ellipsoidal (red dots) and toroidal (blue dots) eigenvalues corresponding to the first seven WGMs. The inset shows the expression of the relative error used in this work to estimate the validity of our approximation.

The approximations shown in the previous sections are of great interest, since they give us the possibility to approximate with a very good level of accuracy the WGM solutions of Helmholtz equations for an ellipsoid, by means of the WGM solutions of the Helmholtz equation in a toroidal cavity with circular cross section. This is very important from the theoretical point of view, because, as the WGM eigenmodes of the torus approximate well the WGM eigenmodes of the ellipsoid, then it is possible to use simple analytic expressions (as, e.g., the solutions of the Helmholtz equation for the circle) to describe both isotropic and anisotropic ellipsoidal resonators.

The results presented here show that if we take a circumference whose radius is equal to the rim radius of the actual spheroid near its boundary, the solutions for this geometry very well approximate the

Having this goal in mind, let us first consider the radial wave equation for the circle, that has the following form:

With these expressions for the WGMs it is then possible to fit the ellipsoidal modes. For example, the fundamental WGM described in Figure

Proceeding along this way, we have then the possibility to approximate, with a large degree of precision, all the WGMs of the ellipsoidal resonator (either isotropic or anisotropic) with a

This is of great advantage because it guarantees more manageable expressions and allows one to easily understand the physics that lives behind these modes, as one can interpret them as just a linear combination of few modes from a circle.

In the same manner it is possible to analytically approximate the angular distribution of such modes, that is, the WGM structure along the

It has to be pointed out, however, that while for an uniaxial anisotropy in a sphere the anisotropy will affect (at the first order) just the radial part of the mode [

For example, for the sixth mode of the anisotropic ellipsoidal resonator as the one shown in Figure

As can be seen, the value of the effective anisotropy parameter in the argument of the angular functions is very low but still different from zero. This means that while for a spherical anisotropic resonator (if the anisotropy is small) the angular function is not affected by the anisotropy, in the case of the anisotropic ellipsoidal resonator, this is also true for the low order WGMs. As the order of the WGM grows, then an effective anisotropy (arising from the different shapes of the spherical and ellipsoidal boundaries that the mode sees) starts to play an important role even in the angular distribution of the mode.

In this work we have used a numerical finite element solver like COMSOL to find the field distribution of WGMs in a spheroidal resonator. We have proposed to approximate the spheroidal resonator near its boundaries with a toroidal resonator with circular cross section, having the radius equal to the major axis of the spheroid and the radius of the circular cross section equal to the rim radius of the spheroid neat its surface. We presented results for both isotropic and anisotropic resonators and we pointed out that, especially for anisotropic resonators, thanks to the numerical verification of our approximation, it is possible to approximate WGMs of a spheroid with a suitable superposition of modes of the circle. The advantage of our proposed analytical approximation is two-fold: from one side it gives the possibility to easily represent the ellipsoidal WGMs as superposition of few circular eigenmodes rather than infinite series of spherical harmonics. On the other side, for the case of anisotropic resonator, it gives easy manageable formulas to describe modes that otherwise will not have an analytical representation. This could be of great help when challenging complicated calculations on such resonators because at first it allows us to use analytical (i.e., simple and practical) methods to have a first educated guess about how things should go inside the considered system. Second, a posteriori, availability of simple approximate formulas permits very powerful qualitative analysis of the quantitative numerical results. Last, but not least, WGMs found, in the last decade, many different applications, ranging from sensing [

The authors declare that there is no conflict of interests regarding the publication of this paper.

_{2}whispering gallery mode resonator stabilized fiber ring laser