WAVES AND VIBRATIONS IN INHOMOGENEOUS STRUCTURES ...

6 th World Congresses of Structural and Multidisciplinary Optimization
Rio de Janeiro, 30 May - 03 June 2005, Brazil
Topology Optimization of Building Blocks for Photonic Integrated Circuits
Jakob Søndergaard Jensen and Ole Sigmund
Department of Mechanical Engineering, Solid Mechanics
Nils Koppels Allé, Building 404, Technical University of Denmark
DK-2880 Kgs. Lyngby, Denmark
jsj@mek.dtu.dk
1. Abstract
Photonic integrated circuits are likely candidates as high speed replacements for the standard electrical
integrated circuits of today. However, in order to obtain a satisfactorily performance many design problems
that up until now have resulted in too high losses must be resolved. In this work we demonstrate
how the method of topology optimization can be used to design a variety of high performance building
blocks for the future circuits.
2. Keywords: photonic crystals and wires, integrated circuits, topology optimization.
3. Introduction
A key component in photonic integrated circuits (PICs) [1] is the photonic crystal (PhC). A PhC is an
optical material that has a periodic modulation of the refractive index, e.g. obtained by a dielectric base
material such as silicon perforated with circular air holes in a triangular pattern. Such materials may
display large band gaps, i.e. frequency ranges for which light cannot propagate. Functional components
can be created from the PhCs, e.g. by removing a single hole to form a resonating cavity, removing a line of
holes to create a waveguide, and by combining cavities and waveguides in order to design more advanced
functionality such as frequency selective components, dispersion compensators, etc. Basic PhC waveguide
components such as various bends and splitters have recently been designed using topology optimization
(e.g. [2]), which led to a significant (up to orders of magnitude) increase in component performance.
Another candidate for basic signal transmission is photonic wires (PhWs), created by making simple
waveguide strips of high refractive material surrounded by air. These PhWs have been reported to have
superior loss characteristics for straight waveguides compared to PhC straight waveguides [3].
In this paper we demonstrate new topology optimized splitters based on PhWs that display a good
performance compared to the optimized bend and splitters in PhC waveguides. Additionally we demonstrate
a new optimized PhC-based component with advanced functionality that includes bends, splitters
and crossings. The optimization is based on a topology optimization algorithm [4] using a SIMP-like
material model to distribute two material phases (dielectric and air), using analytical sensitivity analysis
and the mathematical programming software MMA [5]. As the objective we wish to distribute the
two materials in a designated design domain at various trouble spots in the component, such that the
energy flow through the component is maximized. In order to create broadband components we can
maximize for several wave frequencies simultaneously using a strategy in which we repeatedly update
the target frequencies to be the most critical ones. The update is based on using Padé approximants
to facilitate fast frequency sweeps with high resolution. In order to avoid gray elements in the final
design we introduce a penalization method (PAMPING) in which elements with densities between 0 and
1 artificially cause dissipation and thus cause undesired loss of energy. Although developed for photonics
[6] the methods directly apply to elastic vibration and wave propagation problems as well.
4. Photonic Crystal Waveguides: Bends and Splitters
Since the primary publications ([7], [8]), PhCs have received large attention primarily due to their
potential application in photonic integrated circuits.
PhCs are optical materials that have a periodic modulation of their refractive indices [9]. The
periodicity gives rise to gaps in the band structure that characterizes the wave propagation in the
material, i.e. the so-called band gaps. Within the band gap frequency range, light cannot propagate
regardless of direction so that the material acts as a perfect omnidirectional mirror.
Photonic crystals that consist of dielectric columns placed in a rectangular or triangular pattern
display large band gaps and have been subjected to numerous theoretical investigations [9]. However,
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