COUPLING COEFFICIENTS OF THE SPHERICAL DIELECTRIC MICRORESONATORS WITH WHISPERING GALLERY MODES

Introduction Today the Spherical dielectric microresonators with whispering gallery (WG) modes are actively investigated for purpose of their application in different optical devices [1 10]. For calculation a scattering of electromagnetic waves on the microresonators in various structures it may be necessary a knowledge of mutual coupling coefficients between its. Mutual coupling coefficients of the Spherical dielectric microresonators are not studied in full detail. The goal of the present work is the calculation and analysis of the coupling coefficients of the Spherical microresonators with WG modes in the free space. Calculation of the eigenoscillation fields of the disk microresonator For coupling coefficients calculation it's required information about fields of isolated microresonators. The eigenmode’s electromagnetic field of the Spherical microresonator is well known [11]. In local spherical coordinate system, which is associated with microresonator center, the field of the magnetic type is described by equality to zero of the radial component of the electric field. Usually it is denoted as nml H


Introduction
Today the Spherical dielectric microresonators with whispering gallery (WG) modes are actively investigated for purpose of their application in different optical devices [1 -10]. For calculation a scattering of electromagnetic waves on the microresonators in various structures it may be necessary a knowledge of mutual coupling coefficients between its. Mutual coupling coefficients of the Spherical dielectric microresonators are not studied in full detail.
The goal of the present work is the calculation and analysis of the coupling coefficients of the Spherical microresonators with WG modes in the free space.

Calculation of the eigenoscillation fields of the disk microresonator
For coupling coefficients calculation it's required information about fields of isolated microresonators. The eigenmode's electromagnetic field of the Spherical microresonator is well known [11].
In local spherical coordinate system, which is associated with microresonator center, the field of the magnetic type is described by equality to zero of the radial component of the electric field. Usually it is denoted as nml H 0 r e  ; (  Here spherical (2)  . We shall interest to the high-Q WG modes of the Spherical DR.

Coupling coefficient calculating
The coupling coefficients of the Spherical microresonators can be obtained from already known expressions. We used analytical expressions for the coupling coefficients of the Spherical DRs, situated in the metal Rectangular waveguide [13]. At rush the waveguide walls to the infinity, the sums on the waveguide waves numbers transformed to the integrals on nondimensional parameters.
As a result of integration, after simplifications, has been obtained next relationships for two equal Spherical microresonators with magnetic modes nml H : fig. 2, a).
In the special cases of the 101 H ; 111 H modes, the obtained expression (4) coincides with known early [14]. Top relationship (4) in the square brackets corresponds to odd -odd (higher), or even-even (lower) modes that radial magnetic field proportional to the sin( ) m or cos( ) m in the local spherical coordinate system (1). The bottom relationship in the square brackets corresponds to the odd -even, or even -odd modes. It's intended, that the direction of the axis at all local coordinate systems, associated with resonator centers, are coincides.
The relationship (4) is superfluous. Most often, the coupling oscillation field in such structure forms in the view of even, or odd spatial distributions relatively symmetry plane, passing to the DR centers. It's examines the case, when the DR centers situated in the plane, that parallel to the xz : 1 2 y y  . In this case 0   and the relationship (4) can be simplified: If all resonator centers are disposed on the z -axis 0   ( fig. 2, a), then (5) takes more simple form: The integrals (4 -6) can be calculated, if used relationships, following from the non tabular integral [14]: where b, c -is the real constants.
In the particular case of 0 n m   the (7) takes more simple form, known as Sommerfeld's integral: Differentiates (8) of m times on the parameter b, obtain: Since, using the expansion: Convergence of the (7) -(10), in the area One put in the (11) 0 c  , obtain: Expressions (9); (11) do not comfortable for calculating if they contain derivatives of higher degree. In the cases of 1 m  , the derivatives can be calculated with help of the decompositions: and         The relation (13) was obtained as a result of the spherical Hankel function presentation via elementary functions as well as using the Leibniz's theorem to the product's of differencing. The decomposition (14) follows from the summation theorem for the spherical Hankel functions.
Obtained integral relations (9) -(14) allow to calculate from (4) - (6) In the common case, the integral (4) can be calculated, if it is considered, that presented items, contained derivatives as well as the Legendre polynomials, itself are the polynomials and can be expanded into the series: Substituting (17) in the (4) and using integral (7), obtains finally for equal parity each resonator oscillations: / 1 E n r n n n n n j p p q p n n j p nj p pj p q h q where also 1 0 -is the characteristic parameters of the Spherical microresonator for the electrical modes (3).

Coupling coefficient analysis
Obtained above relationships were used for the calculating of the coupling coefficients for WG modes Spherical microresonators. As can be seen from the calculation results, presented in fig. 2, 3, the coupling coefficients of the WG modes obtain sufficiently large values only in the near-field region in which the microresonators are touching by surfaces to each other. These regions have a largest field concentration on the corresponding modes (see fig. 1). Increasing distance between resonator centers accompanies by significant coupling decreasing. At that, the relative motion in the tangent directions leads to a complex interference of their mutual influence, determining by significant eigenmode field variation nearby their surfaces ( fig. 2, d -f; fig. 3, c, f).
Different orientation of fields of the microresonators relatively it movement in the radial direction also leads to a complicated correlation of the radial coupling coefficients ( fig. 2, e, g, h).
In the majority cases the imaginary part values of the coupling coefficients at least one tenth as many as it real parts. Degree of the imaginary part of the coupling approximately is equal to 1 Q  .

Conclusions
An analytical relationships for the coupling coefficients of the Spherical microresonator in the Open space has been obtained and investigated.
It's showed that the WG mode coupling coefficient describes by more complicated dependencies on the structure parameters.
The real and imaginary parts of the coupling coefficients of the WG modes can be differed more than one degree.
Obtained results showed a possibility of building different types of varied bands filters with symmetrical characteristics.