Numerical Simulation of the Field Confinement in a Quasiperiodic Multilayered Microsphere as an Application of the Software Engineering

We numerically study control of the frequency spectrum of nanoemitters placed in a recently constructed microsphere with a quasiperiodic spherical stack. The spectral evolution of transmittance at the change of thickness of two-layer blocks, constructed following the Fibonacci sequence, is investigated. We found essentially nonlinear behaviour in such a system: when the number of layers (Fibonacci order) increases the structure of transmittance spectrum acquires a fractal form. Our calculations show the confinement and gigantic field enhancement when the ratio of layers widths in the stack (control parameter) is close to the golden mean value. The necessity of object-oriented structure of the program code for numerical study of such a nonlinear system is also discussed.

parameter is discussed.We fund the enhanced field peaks if sunch a parameter is cleso to the golden mean value.In the last Section, we summarize our results.

Basic equations.
The In this case, a nanosource corresponds to a nanorod or quantum dot that recently was employed in experiments with microspheres, see [10], [7] and references therein.The spatial scale of the nanoemitter objects (1÷100)nm is at least one order of magnitude smaller than the spatial scale of the microspheres (10 3 ÷10 4 )nm.Therefore, in the coated microsphere we can represent the nanoemitter structure as a point source placed at r´ and having a dipole moment d 0 .
It is well known that the solution of the wave equation for the radiated electromagnetic field E(r) is due to a general source J(r), [8], [9] where G(r, r´, ) is the dyadic Green function (DGF), which contains all the physical information necessary for describing the multilayered structure.
Following the approach [5], we write down DGF of such a system as follows:  2) is given by where G cm l  f,e  r, r ,   is the particular Green tensor, m is the spherical, l is the azimuth quantum numbers, and   Fibonacci is F 9 ).We observe from Figure 1 that such a spectrum consists of peaks with various amplitudes; however, the most intensive peak with W=87 is located at 436.1THz (details of this peak are shown in the inset).Thus, even though the periodicity of the stack is broken, well defined intensive peak of the field is clearly seen.
We have also calculated the evolution of a spectrum for different values  close to = 1/Γ 0 for a fixed number of layers in the spherical stack (Fibonacci order F n ).The results are displayed in Figure 3 for the range   4 for the most intensive resonance at f 0 =436.098THz (see Figure 5), when the quasiperiodicity parameter =0.618.
We observe from Figure 6 that Im(G (r, a,   ) has a very sharp peak in the place of the nanosource location.Such a spatial field structure may be treated as a confinement of the electromagnetic energy Im(G (r, r) [6] inside the coated microsphere.The leakage of photons through such a structure into the outer space obviously is small.We observe from Figure 6 that the field structure inside of the quasiperiodic stack is anisotropic and quite intricate, but the field distribution beyond the coated microsphere has a periodic character.

Fractal structure.
As mentioned above, with a further increase of the Fibonacci order F j , the structure of T becomes more indented, see However, the latter is not necessary in a case where only the transmittance properties of a quasiperiodic spherical stack are of interest.
The use of microcavities and microspheres in advanced optoelectronics has provided a new view of various effects and interactions in highly integrated, functional photonic devices.A fundamental question in this area is how to drastically increase the spectral optical field strength, using artificially produced alternating layers on the surface of a microsphere.Nowadays, the basic regime of the operation for bare (uncoated) dielectric microspheres is the whispering gallery mode (WGM) [1].But since fabricating the coated dielectric spheres of the submicron sizes [2], the problem arises to study the optical oscillations in microspheres beyond the WGM regime for harmonics with small spherical numbers.These possibilities allow expansion of the essential operational properties of microspheres with attractive artificial light sources for advanced optical technologies.The important optical property of a periodic alternating spherical stack is the possibility of confining the optical radiation (see ([3] and references therein).However, is periodicity necessary for such resonant optical effects?In order to answer this question, we have studied the optical radiation of a nanosource (nanometer-sized light source) placed into a microsphere coated by a quasiperiodic multilayered structure (stack) constructed following the Fibonacci sequence.Such structures are called quasiperiodic and, lie outside the constraints of periodicity.One of the main properties of such a stack is re-reflections of the electromagnetic waves from the interfaces of the layers that result in the collective wave contributions.The collective optical effects in a quasiperiodic spherical stack are appreciated only if the number of layers in the stack is large enough.In this case, various approximations based on the decomposition of field states in the partial spherical modes have insufficient accuracy, so a deeper insight requires more advanced approaches.Our approach is based on the dyadic Green's function technique [5] that provides an advanced approximation for a multilayered microsphere with nanoemitters [6].We have applied this approach to a quasiperiodic spherical stack and found substantially enhanced optical resonances (Green function strength), when the ratio of the layers width in two-layer blocks in the stack (quasiperiodicity parameter) is close to the golden mean value.As far as the author is aware, the optical fields of nanoemitters placed in a microsphere with a quasiperiodic spherical stack still have been poorly considered, though it is a logical extension of previous work in this area.This Report is organized as follows.In Section 2, we formulate our approach and basic equations for optical fields in a dielectric multilayered microsphere.We outline the numerical scheme of applying the dyadic Green function (DGF) technique to evaluate the spectrum of a nanoemitter placed in such a quasiperiodic system.We found the enhanced field peaks of such a parameter is close to the golden mean value.Numerical Simulation of the Field Confinement in a Quasiperiodic Multilayered Microsphere 15 Section 3, outlying the structure of our objectoriented program code and the details of uur program realization is discussed.In Section 4, we present our numerical results on the structure of the cavity field states and resonances.In the Section 5 the fractal structures of the resonances in a microsphere with quasiperiodic spherical stackmdependently of the quasiperiodecity
r´, ) represents the contribution of the direct waves from the radiation sources in the unbounded medium, and G (fs) (r, r´, ) describes the contribution of the multiple reflection and transmission waves due to the layer interfaces.The dyadic Green's tensor G V (r, r´, ) in Eq. ( is a rather difficult problem in the use of computer resources, and also on the complexity of the program code organization.Our realization includes two main blocks.The first one contains the code with a description of the structure of a layered microsphere, while the second block is responsible for the dynamic calculation of a frequency spectrum and the radial distribution of an electromagnetic field in the microsphere.Other blocks of the program realize the graphic support, the data exchange, and also the mathematical library of the special functions used for evaluation of the Green function.

Fig. 2 .
Fig. 2. The hierarchy of main classes in the program code.

Fig. 3 .
Fig. 3. Graphical user interface and example calculation of the reflection and transmittance coefficients.

Fig. 6 .
Fig. 6.Spatial structure W(r,)=Im(G  (r,a, )) in a cross-section 0<r<21µm and 0<<2 of the microsphere with quasiperiodic stack for eigenfrequency f=436.09THz.A nanoemitter is placed at point a=900nm.Other parameters are as in Figure 2. One can see the confinement of field in the stack.Outer cycle indicates the external boundary R ext =13.8µm of the quasiperiodic spherical stack.

L is replaced by LS, S is replaced by L. As a result, we obtain a new sequence: LSL. Iteratively applying
field, we apply the Green function technique.