Topology optimization of manufacturable photonic crystals with complete bandgaps

Periodic structures consisting of dielectric material, i.e. photonic crystals, are capable of prohibiting the transmission of electromagnetic waves within frequency ranges referred to as bandgaps. This principle was first demonstrated with alternating slabs of material and later demonstrated with three-dimensional (3D) structures capable of reflecting waves from any incident angle. A number of potential applications for photonic bandgap structures exist, including waveguides, integrated circuits, fiber optics, and photonic cavities. Additionally, many other optical devices could benefit from the perfect-mirror behavior of photonic crystals. The design of photonic crystals for complete bandgap has challenged researchers for the past three decades. Bandgap structures are often quite complicated and therefore difficult to design heuristically. Thus, the application of automated design tools, such as topology or shape optimization, is very attractive. Unfortunately, bandgap analysis is very computationally intensive, and it is difficult to employ effective low-dimensional design parameterizations capable of generating bandgap structures. Until recently, computational power was insufficient to design 3D structures with complete bandgaps. The development of computing clusters has reduced this burden significantly, although computational cost remains a challenge. A major obstacle when numerically designing for bandgap, or any other design metric derived from eigenvalues, is the presence of degenerate eigenmodes. Optimal bandgap structures often possess many planes of symmetry; this is helpful to reduce the overall cost of the required dispersion analysis, but it often leads to wave frequencies that have multiple propagation directions, the physical result of degenerate eigenmodes. Herein lies the challenge; we would like to use gradient-based optimization algorithms to design bandgap structures, but the presence of degenerate eigenmodes renders our design metric non-smooth. Solving this conundrum by leveraging symmetric polynomials was a major contribution of this work. Further, an efficient sensitivity analysis and a successive mesh refinement strategy were developed to augment the design framework. Finally, it was observed that bandgap structures often exhibit poor stiffness properties, sometimes even making the structures unable to be physically realized. A series of physics-based, nonlinear constraints were developed to ensure the algorithmically-generated structures are manufacturable. These constraints were demonstrated by designing a series of photonic crystals that were fully self-supporting without the presence of enclosed void space. Additionally, the trade-off between bandgap and bulk stiffness was investigated. The proposed design framework is the first of its kind; a technique able to leverage traditional, gradient-based nonlinear programming solvers to generate 3D bandgap structures with manufacturing constraints.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mechanical Engineering

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2022

Author dc:creator

• Swartz, Kenneth E

Contributors dc:contributor

• James, Kai A
• Tortorelli, Daniel A
• Jin, Jianming
• Matlack, Kathryn H

• Mechanical science

Statement dc:rights

• Copyright 2021 Kenneth Swartz

Language dc:language

en, eng

Handle dc:identifier

http://hdl.handle.net/2142/113848

OAI record oai:identifier

https://www.ideals.illinois.edu/oai-pmh?verb=GetRecord&metadataPrefix=etdms&identifier=oai%3Awww.ideals.illinois.edu%3A2142%2F113848