01.03.2017 | Ausgabe 3/2017 Open Access

# Simulation methods for multiperiodic and aperiodic nanostructured dielectric waveguides

- Zeitschrift:
- Optical and Quantum Electronics > Ausgabe 3/2017

## 1 Introduction

_{2}waveguide with aperiodic deterministic nanostructure based on a Thue–Morse binary sequence. Deterministic aperiodic nanostructures are engineered ordered nanostructures without periodicity (Dal Negro 2012a, b; Maciá 2012). Compound multiperiodic gratings and deterministic aperiodic nanostructures offer the opportunity to tailor the spectral properties and have recently been suggested for refractive index biosensing (Boriskina et al. 2008a, b; Kluge et al. 2014; Neustock et al. 2016). The grating structure allows incident light to couple to guided modes by scattering. The guided light can again couple out due to the nanostructure and thus the modes are called quasi-guided modes (QGM) or leaky modes. We investigate the case that normally incident light is coupled into and out of QGM in the waveguide structure as depicted in Fig. 1b. Reemission of the QGM in the reflection direction leads to characteristic guided-mode resonances (GMR) in the transmission spectrum, which depend on the angle of incidence, polarization of the light, refractive index of the material and the geometric properties, such as the nanostructure sequence, duty cycle and structure depth (Fan and Joannopoulos 2002). In this work we employ and compare three different simulation methods—finite element method (FEM, COMSOL Multiphysics

^{®}Wave Optics Module by COMSOL Inc.), finite difference time domain (FDTD, FDTD Solutions by Lumerical Solutions, Inc.) and rigorous coupled wave analysis (RCWA, in-house implementation)—for simulating the transmission properties of compound multiperiodic and deterministic aperiodic nanostructures. Five different structures are examined—two multiperiodic structures (one with a two-compound and one with a three-compound grating) and three binary deterministic aperiodic sequences with different degrees of disorder (Thue–Morse, Fibonacci and Rudin–Shapiro).

## 2 Structures under investigation

Name | Type | Description | Supercell length, number of recursions |
---|---|---|---|

2-compound | Compound multiperiodic, two periods | Λ _{1} = 250 nm, Λ_{2} = 300 nm duty cycles, t_{1} = 0.3, t_{2} = 0.4 | L = 1500 nm |

3-compound | Compound multiperiodic, three periods | Λ _{1} = 250, Λ_{2} = 300 nm, Λ_{3} = 350 nm, duty cycles: t_{1} = 0.3, t_{2} = 0.3, t_{3} = 0.3 | L = 10,500 nm |

Rudin–Shapiro | Deterministic aperiodic, continuous spectrum | Substitution: AA → AAAB, AB → AABA, BA → BBAB, BB → BBBA | L = 12,800 nm, N = 7 |

Thue–Morse | Deterministic aperiodic, singular continuous spectrum | Substitution: A → AB, B → BA | L = 12,800 nm, N = 9 |

Fibonacci | Deterministic aperiodic, pure-point spectrum | Substitution: A → AB, B → A | L = 11,600 nm, N = 13 |

_{2}is used as depicted in Fig. 3 (Devore 1951). The cladding refractive index is set to air (n

_{1}= 1) and the substrate is assumed to be AMONIL photo resist (n

_{3}= 1.52).

## 3 Methods

### 3.1 FDTD

### 3.2 RCWA

^{®}and based on the algorithms presented in Moharam et al. (1995a, b), suitable for an RCWA implementation for a geometry with different layers. To model the geometries shown in Fig. 1, they are divided in three layers. An upper layer modelling the variation between high index and cladding material, a middle layer consisting only of high-index material and a lower layer describing a variation of refractive index between high-index and substrate material. For each layer, the corresponding Fourier coefficients of the multiperiodic and aperiodic gratings are calculated analytically with a base frequency of the inverse length of the supercell. The number of Fourier coefficients for each grating is determined by a convergence analysis. For the aperiodic gratings and the multiperiodic grating with 3 superimposed periods, 512 Fourier coefficients are sufficient, whilst for the multiperiodic grating consisting of 2 periods only 64 Fourier coefficients are necessary. Generally, the number of required coefficients increases with smaller feature sizes with respect to supercell length. After implementing the geometry, the spectrum is calculated for each wavelength individually.

### 3.3 FEM model

### 3.4 Fabrication and measurement setup

## 4 Results

### 4.1 Near field simulations

### 4.2 Multiperiodic nanostructures

_{2}layer, having a lower refractive index than bulk TiO

_{2}as found in the literature (Devore 1951). No experimental dispersion relation of the sputtered TiO

_{2}is available at the moment. Another reason for the spectral mismatch might be a lower height of the high index layer as a result of the sputtering process, which showed to have an accuracy of a few nanometers. The transmission of the measurement is lower in general, compared to the simulation, which we attribute to material absorption and additional scattering at the imperfect material boundaries. Even though the measurement and simulation are not perfectly matched, the results show, that the general behavior—the resonance position and shape—of multi-periodic nanostructures can be predicted by all three methods. Knowledge of the exact material parameters seems to be crucial in absolute prediction of the spectral position and transmission values.

### 4.3 Aperiodic nanostructures

### 4.4 Comparison of simulation methods

^{®}Xeon

^{®}CPU E5-2637 v3 @ 3.50 GHz, 512 GB of RAM). The first scenario is the computation of the transmission spectrum over the visible range as shown above (430–750 nm, 641 spectral points). The second scenario is the electrical field calculation as shown in Fig. 5 for a single wavelength and the 2-compound structure detailed in Table 1. The third scenario is the computation of both transmission and near-field data for the entire spectrum. The FDTD simulations and the FEM simulations are natively parallelized. We have parallelized the in-house RCWA code with Matlab

^{®}’s parallel processing toolbox. For the present comparison, we restricted all concurrent simulations to 8 workers (CPU cores).

^{®}offers a broad range of tools for subsequent analysis. The broadband nature of FDTD leads to the shortest simulation time if one is interested in full spectra alongside field plots for all involved wavelengths. On the other hand, this nature also leads to significant time drawbacks, when exercising the first two scenarios, as compared to the frequency domain methods. For extensive parameter sweeps, in RCWA, less Fourier components may be used to have an even faster calculation with less details. Concurrently, an approach with lower spectral and meshing resolution in the COMSOL

^{®}’s Wave Optics Module and FDTD Solutions is thinkable to reach shorter simulation times. The 2-compound structure under investigation is the smallest structure we investigated in this study. Simulation time will scale with the number of Fourier coefficients in the RCWA case and with the size of the supercell (number of mesh elements) in the FEM and FDTD case. Please note that our in-house implementation has still potential for further runtime optimization (Hench and Strakoš 2008).

Method | Scenario 1: transmission spectrum | Scenario 2: single- wavelength field | Scenario 3: full Spectrum and field | Pros | Cons |
---|---|---|---|---|---|

Finite Element Method (FEM, COMSOL Multiphysics ^{®} Wave Optics Module) | Slowest, all fields have to be calculated, 13 min, 11 s | Fastest, 6 s | 13 min, 11 s | Fast single field profile calculation; extensive toolbox | Proprietary |

Finite Difference Time Domain (FDTD, FDTD Solutions by Lumerical Solutions, Inc.) | 1 min 7 s | 59 s | Fastest, 5 min, 4 s | Fast full-spectrum full-field calculation | Proprietary |

Rigorous Coupled Wave Analysis (RCWA, in-house implementation) | Fastest, 18 s | 12 s | 7 min, 10 s | Fast spectrum calculation | Non-intuitive implementation, no user interface |

## 5 Conclusion

## Acknowledgements

## Appendix: comparison of different models

_{2}) ridges embedded in the substrate.

_{1}= 1, n

_{2}= 2.44, n

_{3}= 1.52. Figure 9 shows FDTD simulations of the transmission characteristics of the two models as well as the measured transmission of the Thue–Morse nanostructured waveguide. The refractive index of the high-index layer differs from the values used for Fig. 7, which includes material dispersion. This accounts for the differences in resonance position and shape with regard to Fig. 9.