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We report the effect of scattering of electromagnetic plane waves by two cylinders on whispering gallery mode (WGM) formation in a cylinder. WGM can occur because of the presence of additional cylinder scatterers at specific location, while WGMs can only form in a single cylinder for specific cylinder radius and/or wavelength values, the matching accuracy required would be much greater than that required in our model for the additional cylinders locations. Analysis of the general solution to the problem showed that the effect can be explained by the interference of waves scattered by additional cylinders and incident on the main cylinder.

The term “whispering gallery modes” (WGMs) was introduced by Lord Rayleigh to explain the effects of sound propagation a circular gallery [

Our model consists of two cylinders. One of these (marked as A on

was used as the incident wave. The mesh grid size in the space was equal to 0.04 μm (0.075 of wavelength). The electric permittivity ε = 1.59 (quartz glass), and cylinder’s A radius R_{A} = 4λ. For cylinder B (where the latter is introduced below), the radii R_{B} = 0.25 R_{A} were used. All distances below are measured in μm.

The calculated distribution of the absolute field intensity value is shown on _{x} = −2.7, L_{y} = 0.5).

To determine how the positions of cylinder B affect the maximal absolute field value inside the cylinder A, we varied its positions by moving the centers within the ring defined in polar coordinates as interval of radii [R_{A} + R_{B}, R_{A} + 1.6*R_{B}]. Each step in the ring was 0.06 R_{B}/10 along the radius, with an angle step of (p/2)/75. The resulting picture is shown in

We associate the near-surface area of high field intensity with WGM propagation, because the specific feature of WGMs is that the high intensity field in these modes is concentrated near the cavity walls. The absence of high field intensity areas inside a single scattering cylinder A indicates that WGM is caused by the presence of the additional cylinder B.

In general, the WGM are characterized by the specific value of the following relationship: cylinder radius/wavelength. As an example we consider simple expression 2pRn/l = T_{ml} [_{ml} is the lth root of the mth order Bessel function. This means that if the wavelength is known, then to determine the propagation of (ml) mode we must choose the cylinder radius based on the expression above. Additionally, when this mode makes a larger contribution to the field intensity, then the radius must be defined more accurately [^{−4} l [

In the case of two or more cylinders, the equation for the derivation the WGM will contain a contribution caused by the presence of satellite cylinder. First of all, this means that now WGM are derived by many parameters (radii of the satellite cylinders, distance between cylinders, their mutual orientation, dielectric permittivity), but not only by a relation R/l, as it was for single cylinder (it is confirmed by comparison of

and waves scattered by additional cylinder. The latter also follows from the general formulas describing multiple cylinders scattering. Now we analyze the general solution to the scattering problem on multiple cylinders. Our task in this analysis is reduced to an assessment of the contributions to the field inside cylinder A of the components that are associated with cylinder B. Let’s consider the expression that corresponds to the solutions to Maxwell’s equations for our model, which were given in [

where k_{m} = 2p/l, a_{l} = R_{A}, J_{n} ? Bessel function of first kind, R_{lp} ? distance between point P inside lth cylinder and its center, and

The expansion coefficients (a_{jn}, b_{jn}) are related to the single cylinder scattering coefficients (

where

Equation (3) allows us to conclude that in the presence of an additional cylinder the amplitude of a certain mode of the first cylinder contains contributions from all modes of the second cylinder. These contributions decrease as the distance between the cylinders increases. Therefore, the resonance is observed only when the location of the second cylinder is close to the edge of the first cylinder.

We now consider why the intensity is maximal in a narrow strip near the edge, but does not decrease gradually if the additional cylinder moves away from the main cylinder. To find a solution, we consider Equation (1) assuming asym- ptotic expressions for the Bessel and Hankel functions for large values of their arguments. Indeed, the arguments are much more than 1 for the parameters of our model (for example: k_{m}a_{l} = 2p/l R_{A} = 2p/l*4l = 8p; k_{l}R_{lj} < 2p/l*(R_{A} + R_{B}) > 8p). Additionally, in Equation (1), we use only the second term of Equation (2) for A_{ln}, and only the first term in Equation (3) for a_{1n}. From the above, the contribution to the field of the 2-nd cylinder is described as follows:

Here we have deal with sum of waves with different phases and amplitudes. A well-known gain condition leads to the following equality

where p represents integer numbers. Analysis of this equation for different signs leads to the relationship R_{lj} − (R_{A} + R_{B}) < λ. This means that the maximum field intensity value occurs if the distance between the centers of the cylinders does not exceed the minimal possible value more than wavelength.

We have simulated the scattering of a plane wave by the pair of cylinders. It was found that WGMs can be formed inside the basic cylinder in the presence of an additional cylinder, without any special requirement for the wavelength or for the radii of the cylinders. However, additional cylinder must be located in specific positions near the edge of main cylinder. The accuracy required for these cylinder locations is much lower than the accuracy required for the setting of the resonant radius of the cylinder at which the WGMs can be observed. Analysis of the general solution to our model has shown that these effects can be explained by the interference of the waves that are scattered by the extra cylinder. Therefore, our work describes a new method for the formation of WGMs in a cylinder.

The work was supported by State Key Laboratory of Meta-RF Electromagnetic Modulation Technology (2011DQ782011), Guangdong Key Laboratory of Meta-RF Microwave Radio Frequency (2011A060901010), Composite intelligent materials engineering laboratory, Shenzhen Key Laboratory of Ultrahigh Refractive Structural Material (CXB201105100093A), Shenzhen Key Laboratory of Data Science and Modeling (CXB201109210103A), Shenzhen Science and Technology Plan(JSGG20150917174852555, JCYJ20151015165322766 and JCYJ2015 1015165557141), the introduction of innovative R&D team program of Guangdong Province (NO. 2011D024).

Abramov, A., Yue, Y.T., Ji, C.L., Liu, R.P., Li, X., Zhou, J.H. and Kostikov, A. (2017) Whispering Gallery Modes Formed by Scattering of an Electromagnetic Plane Wave by Two Cylinders. Journal of Applied Mathematics and Physics, 5, 785-791. https://doi.org/10.4236/jamp.2017.54067