# Design of Photonic Molecule-Based Multiway Beam Splitter/Coupler with Variable Division Ratio

## Abstract

**:**

## 1. Introduction

## 2. PM-BS Structural Model and Working Principle

_{0}. By using the coupled mode theory [35], the evolution in time (t) of two eigenmodes a

_{1}and a

_{2}existing in a cavity pair is described by the following equations:

_{i}is field amplitude decrement due to intrinsic losses in the cavity (absorption, radiative escape, etc.), μ

_{w}and μ

_{c}stay for normalized coupling parameters of resonators with a waveguide and with each other, respectively. Obviously, the inter-cavity coupling μ

_{c}depends on the value of the interatom gap (g). In the following, for simplicity, we take A = 1, and then the solution to Equation (1) for any of the normal modes, e.g., a

_{1}, can be written as follows:

_{t}= (μ

_{I}+ μ

_{w}) and j

^{2}= −1.

_{0}possess two collective resonances—the supermodes—with the eigenfrequencies, ω

_{1,2}= ω

_{0}± μ

_{c}, equidistantly shifted from the resonant frequency of the isolated cavity by the value of the coupling coefficient μ

_{c}and having the power ratio ${a}_{1}^{2}/{a}_{2}^{2}\approx {\left({\mathsf{\mu}}_{c}-{\mathsf{\omega}}_{0}\right)}^{2}/{\left({\mathsf{\mu}}_{c}+{\mathsf{\omega}}_{0}\right)}^{2}$. At the same time, the magnitude of the phase shift Δφ between the excitation channels of the photonic duet controls the amplitudes of these supermodes. Particularly, in two extreme cases, Δφ = 2πm and Δφ = (2m + 1)π, where m is an integer, the anti-bounding or bounding modes are completely suppressed, respectively. Importantly, the dips in the PM waveguide transmittances in Figure 1b indicate that almost all incoming optical power is trapped in PM supermodes maintaining the energy recirculation along the resonators’ surface. If yet another optical waveguide (drop port) is placed in the gap between the atoms, it can capture part of the SM energy and redirect out of the photonic molecule, thus forming a drop channel. Qualitatively, this is the way a PM-based splitter/coupler works.

_{2}) substrate and is structurally characterized by the interatomic gap, g = 300 nm. Crystalline silicon (Si) with refractive index n = 3.5 and almost zero optical absorption (κ ~ 10

^{−11}) in the telecommunication spectral band centered on the wavelength λ = 1330 nm [34] is chosen as the material for photonic atoms. To enhance functionality, the considered photonic splitter is composed of two PMs coupled through a central waveguide (Figure 3a,c) acting as an input port. On the lateral sides of the splitter structure, two more waveguides are mounted at a certain distance (s) from the outermost atoms, s = 250 nm, serving for the collection of the divided optical beams. All waveguide feeders are considered to have the same width (1000 nm) and height (600 nm) and can be fabricated from the core of a single-mode optical fiber (SiO

_{2}), or through the etching of a dielectric photoresist (e.g., PMMA). The refractive index of the waveguides during the simulation was chosen as n

_{1}= 1.5.

_{7,1}in a silicon cylinder is chosen as the fundamental resonance (Figure 3d) having quality factor Q = 8500 at the wavelength λ = 1338.2 nm and the linewidth of about 0.2 nm. Note that, according to the adopted notations for the electromagnetic eigenmodes of a sphere, the subscripts “7,1” denote the semi-number of optical standing wave antinodes along the azimuthal direction (along the cylinder rim) and the number of maxima along the radial coordinate, respectively.

**E**is a vector with the components along each of the coordinate axes. Worthwhile noting, in the case of a spherical or cylindrical particle, the analytical solution to Equation (3) is widely known in the form of infinite series on the vector spherical harmonics, called the “Mie series” in the literature [38]. For two or more touching particles, although such partial wave expansion can be obtained and used (T-matrix method [39], multisphere scattering formulation [40], discrete dipole approximation [41]), it results in too sophisticated a formulation and is time/memory consuming in the calculations. Therefore, in the following, we use the direct numerical integration of the wave Equation (3) to derive the electromagnetic field distribution.

^{®}5.1 software, which exploits the finite element method (FEM). Without limiting the generality, we use the 2D formulation of the problem as shown in Figure 3c,d. In this case, the whole photonic structure is surrounded by a rectangular region of perfectly absorbing layers (PMLs) to minimize wave reflection from the boundaries. The optical radiation is input through the corresponding COMSOL digital ports. For the numerical discretization of all simulation domains, a mesh with triangular elements and a maximum edge size of λ/40 is used. The wavelength sweep is performed by creating a parametric study in the range of 1333 nm ≤ λ ≤ 1343 nm, which includes all supermodes of the considered PM topologies based on the TE

_{7,1}resonance of single atom.

_{1}-port transmittance. As seen, the spectrum of such a molecule contains a total of 4 collective supermodes, each of them being fourfold degenerate due to the high spatial symmetry in the PM atomic structure. Here, three groups of resonant supermodes can be distinguished [31]: (i) anti-bonded modes with λ = 1335.1 nm and 1335.3 nm, (ii) bonded modes with λ = 1340.68 nm, and (iii) a mixed-type supermode centered at λ = 1338 nm. The blue-shifted anti-bonded modes have close resonance wavelengths, so in the following, we consider the only one of them, with λ = 1335.1 nm, which exhibits more pronounced transmission dip.

**E**and

**H**are electric and magnetic field vectors, respectively, and c is the speed of light), are plotted for each of the selected supermodes. As seen, in resonance, the streamlines of vector

**S**are captured from the feeder waveguide by nearby atoms of the molecule and then directed into the hybridized field of the PM supermode formed by the coupling of atomic resonant modes. Photonic molecules on both sides of the feeder waveguide always have a counter-directed circulation of optical energy and produce regions of field singularity such as optical vortices and saddles, which is typical for standing wave resonators [42]. The most distant atoms of photonic molecules (relative to the input waveguide) drop a part of the energy flux into the lateral waveguides and the rest of mode optical energy is looped again in the molecular supermode.

## 3. Discussion

#### 3.1. Structural Types of PM-Based Beam Splitter

^{5}) resonances at the wavelengths 1337 nm and 1339.4 nm can be found in the BS spectral response function. At each of these supermodes, a different form of beam splitting is realized. Importantly, at the red-shifted bonded supermode, the input beam is divided equally among all ports of the splitter, while at the blue-shifted anti-bonded resonance, the power in the passthrough port is twice as high as that in remaining drop ports. Meanwhile, out of resonance the majority of input power passes unchanged into port 1, and the photonic splitter is in the off-line state.

#### 3.2. Design of PM-Based Optical Coupler

## 4. Conclusions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**(

**a**) Schematic of two coupled resonant cavities (atoms) excited through two optical channels with a phase mismatch Δφ. (

**b**) Spectral transmittance |A

_{1}/A|

^{2}of right waveguide channel in a silicon two-atom PM (Δφ = π/2) demonstrating anti-bonding and bonding PM supermodes (SMs). Single-atom fundamental resonance is shown for comparison.

**Figure 2.**Structural types of PM-based optical splitters with (

**a**,

**b**) serial (2s and 4s) and (

**c**) serial–parallel (2s-p) electromagnetic coupling of atoms in photonic molecules PM1 and PM2 (outlined with dashed ovals). Optical signal is injected through the central waveguide (red arrow) and collected in the through and drop ports labeled as P

_{1}and P

_{2}to P

_{5}, respectively.

**Figure 3.**Three-dimensional design of proposed photonic (

**a**) splitter and (

**b**) coupler on the SOI platform with input and drop port labeling. (

**c**) PM-BS structural scheme and (

**d**) the working principle.

**Figure 4.**(

**a**) Resonance spectrum of a 2s-p photonic molecule showing different types of supermodes. (

**b**–

**d**) Two-dimensional distributions of the normalized amplitude |E

_{z}| (color maps) and the Poynting vector

**S**(x,y) for three principal PM supermodes: anti-bonded (

**b**), mixed (

**c**), and bonded (

**d**). The numbers indicate the proportions of the optical energy splitting that are inputted through the central waveguide.

**Figure 5.**(

**a**) Splitting ratio for a pair of microrings in resonance. The inset shows the Poynting vector streamlines. (

**b**) Spectral dependence of the relative power directed in the ports of proposed PM-BS based on two diatomic 2s-PMs.

**S**streamlines are shown for the two supermode resonances.

**Figure 6.**Spectral dependence of the dividing ratio of PM-PS designed on a pair of (

**a**) 4s and (

**b**) 2s-p molecules.

**Figure 7.**Spectral dependence of relative optical power in the output ports of PM-based coupler for (

**a**) 2s и (

**b**) 4s PM structures.

**Figure 8.**(

**a**) Dependence of relative power in the optical ports of optical coupler on the left port dephasing Δϕ. (

**b**,

**c**) Poynting vector streamlines in the 2s PM coupler at (

**b**) Δφ = 0 and (

**c**) Δφ = π radians at 1339.4 nm bonded supermode resonance.

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**MDPI and ACS Style**

Geints, Y.E.
Design of Photonic Molecule-Based Multiway Beam Splitter/Coupler with Variable Division Ratio. *Photonics* **2024**, *11*, 600.
https://doi.org/10.3390/photonics11070600

**AMA Style**

Geints YE.
Design of Photonic Molecule-Based Multiway Beam Splitter/Coupler with Variable Division Ratio. *Photonics*. 2024; 11(7):600.
https://doi.org/10.3390/photonics11070600

**Chicago/Turabian Style**

Geints, Yury E.
2024. "Design of Photonic Molecule-Based Multiway Beam Splitter/Coupler with Variable Division Ratio" *Photonics* 11, no. 7: 600.
https://doi.org/10.3390/photonics11070600