Electromagnetic eigenmodes of spherical thin films

Eigenmodes of the spherical and ellipsoidal dielectric thin films have been analyzed. The results of calculations of frequencies of natural electromagnetic oscillations of spherical thin films depending on their sizes and dielectric permeability of external environment are presented.


Introduction
One of the objectives of physical electronics is the investigation of physical processes in thin films, particularly, the influence of electromagnetic wave on them. Accordingly, it is useful to know the natural frequency of electromagnetic oscillations of thin films. Thin films find wide application in microelectronics. There are many types of it, and there are many methods of obtaining it. One of thin films is a bilayer -a double molecular layer formed by polar lipids in the aqueous medium. [1]. In lipid bilayer, molecules are oriented in such a way that their polar fragments are directed towards the aqueous phase and form two hydrophilic surfaces, and nonpolar "tails" form a hydrophobic region inside the bilayer. The bilayer is a thermodynamically advantageous form of polar lipids association in the aqueous medium. It forms a closed surface that is geometrically an ellipsoid of revolution.
The surface of the phospholipid bilayer is a complex formation. The polar heads of phospholipid molecules bordering the electrolyte form a surface layer (0.6 -1 nm thick) filled with electric charges and dipoles. Part of these charges and dipoles belong to the heads, the other part is made up of water molecules and electrolyte ions. Therefore, the terms «surface charges» and «surface dipoles» are largely academic. Charge and dipoles of real phospholipid surfaces are distributed in the near-surface layer. As a rule, it is considered that the thickness of this layer is negligibly small and the charges are at the boundary of the section [2].
Thus, the phospholipid bilayer can be represented as a resonator, which is a two-layer ellipsoid. To find the eigenfrequencies it is necessary to solve the Maxwell equations in spheroidal coordinates -in this case the boundary conditions will depend on only one coordinate [3].

The Model
Spheroidal coordinates are rotation-symmetric coordinate system that can be obtained by rotating around the symmetry axes of the flat elliptical coordinate system consisting of mutually orthogonal focal ellipses and hyperbols. There are 2 types of spheroidal coordinates: rotation about the focal axis of the ellipse (the symmetry axis on which the foci are located) produces prolate spheroidal coordinates, rotation about the other axis produces oblate spheroidal coordinates. As it can be seen, in our case it is more convenient to use prolate spheroidal coordinates. It can be designated the focal length distance in the elliptical coordinate system d and the axis z can be considered the axis of rotation.
Prolate spheroidal coordinates are related with Cartesian (rectangular) and spherical coordinates by the following equations: by the planes y x0 and z y0 .
In a general orthogonal coordinate system, the curl is given by the expression k is the wave number, can be written in the form: The variables in these equations can be separated if the components E and H do not depend on the angular coordinate  . This condition is fulfilled for the lower electrical and magnetic oscillations. With that in mind, the previous system of equations will accordingly take the form: It is possible to separate variables using scalar functions    , P and    , Q , known as Abraham's potentials: . , From the second, fourth and fifth equations of the system (2) it is possible to obtain: Or using the scale factors (1): . Similar equations can be obtained for the potential P . Separating the variables in the last equation by presenting the potential Q in the form of , the system of equations is obtained: where  is the separation constant, . To find the eigenfrequencies of electromagnetic oscillations of the ellipsoidal dielectric electromagnetic resonator, it is necessary to know the dependence of the fields on the coordinate, i.e. the solution of the first equation of the system (3). Substituting U in an explicit form in this equation the following expression can be obtained: The solution of this differential equation is the prolate radial spheroidal wave functions of the first and second order  . Figure 3 shows the dependence of the 101 H and 101 E oscillation frequencies on the form of the resonator (b = 10 µm). Similar results were obtained in [9]. As it can be seen, the frequency changes proportionally to the ratio of half-axes and at a slight difference of the form of the resonator from the sphere ( b a  ) the frequencies will also slightly differ from the eigenmods of the spherical resonator. The presented calculations are made for prolate spheroidal coordinates, similar results are obtained for oblate spheroidal coordinates. Since small deviations from sphericity do not significantly change the eigenfrequencies, it is possible to apply the spherical functions of Bessel and Neumann (   From the condition of continuity of fields on the boundary of the media interface, it is possible to get the equation for the calculation of wave numbers [10]: R -inner and outer radii of the thin layer. Figure 4 shows the dependence of the lower frequencies of magnetic and electrical oscillations for dielectric and closed resonators on its size. Interesting results can be obtained by finding the dependence of natural frequencies on dielectric permittivity of the external medium. It is possible because thin lipid bilayer is a border between inner and outer media. When water is diluted with another liquid, the dielectric permittivity changes, and there is no change in the area limited by the bilayer. To calculate the frequencies (see figure 5), the following parameters were taken: the radius of the resonator 10 1  R µm, the thickness of the thin film 10 1 2   R R nm, the relative permittivity of the inner space, limited by the film 5 1   . It can be seen that the natural frequency of oscillations is highly dependent on the dielectric permittivity of the areas separated by a thin film and will be maximum when they are equal.
Thus, it can be concluded that the natural frequencies of electromagnetic oscillations are mainly in the terahertz range, very promising at the present stage of electronics development. The strong correlation between natural frequencies and the dielectric permittivity of the medium can be used to control and measure the concentration of various substances dissolved in water.