Maria Bayard Dühring - Solid Mechanics

Design of photonic-bandgap fibers by topology optimization
M. B. Dühring*, O. Sigmund* and T. Feurer**
* Department of Mechanical Engineering, Solid Mechanics, Technical University of Denmark, 2800
Kgs. Lyngby, Denmark.
** Institute of Applied Physics, University of Bern, 3012 Bern, Switzerland.
Abstract
A method based on topology optimization is presented to design the cross section of hollow core
photonic-bandgap fibers for minimizing the energy loss by material absorption. The optical problem
is modeled by the time-harmonic wave equation and solved with the finite element program Comsol
Multiphysics. The optimization is based on continuous material interpolation functions between the
refractive indices and is carried out by the Method of Moving Asymptotes. An example illustrates
the performance of the method where air and silica are redistributed around the core such that
the overlap between the magnetic field distribution and the lossy silica material is reduced and the
energy flow is increased 375% in the core. Simplified designs inspired from the optimized geometry
are presented, which will be easier to fabricate. The energy flow is increased up to almost 300% for
these cases.
Keywords: finite element analysis, wave equation, photonic-crystal fiber, morphology filter, Comsol
Multiphysics, optimized design
1 Introduction
Photonic crystals were first described in the two papers [1, 2] from 1987. They consist of periodically
structured dielectric materials in one, two or three dimensions that can prohibit the propagation of
electromagnetic waves at certain frequencies such that band gaps are created. Point and line defects
can be introduced in the structures to localize and guide optical waves. The first 2D photonic crystal
was fabricated for optical wavelengths in 1996, see [3], and applications are found in filters, splitters
or resonant cavities, see [4]. Another application of the photonic crystal is the photonic-crystal fiber
developed in the 1990s [5, 6]. Conventionally, optical fibers are made as step-index fibers where an
index difference between the core and the cladding confines the optical wave to the core region [7].
These types of fibers are extensively used in telecommunication. In contrast to the conventional
optical fibers, the optical wave in photonic-crystal fibers is guided in a core region surrounded by
a 1D or 2D periodic structured material, see [8, 4] for an overview. Depending on the periodic
structure the wave is confined either by index guiding or the band-gap effect. Because of the bandgap
effect it is possible to guide the optical wave in an air core such that losses and unwanted
dispersion and nonlinear effects from the bulk materials can be reduced. The first photonic-crystal
fibers were produced for commercial purposes in 2000 and are fabricated by a drawing process.
In this work holey fibers are considered, which denote photonic-crystal fibers with air cores
surrounded by periodic arrays of air holes. The cladding material is typically silica as it is suitable
for fabrication with the drawing process. However, silica has higher loss for most optical wavelengths
away from 1.55 µm, which is used in telecommunication. For optical wavelengths in general the
photonic-crystal fibers are therefore not convenient for long fiber links, but rather for short distance
applications. An example of a short distance application is laser surgery where wavelengths between
2-10 µm are used for various purposes in medical application as drilling holes in teeth and tissue
removal. Lasers directed by mirrors are normally used for these purposes, but by employing fibers
it will furthermore be possible to do surgery inside the body without opening it. A first example of