Maria Bayard Dühring - Solid Mechanics

photonic-crystal fibers used for medical purposes is found in [9] where a 1D, rotational symmetric,
photonic-crystal fiber is employed for laryngeal and airway surgery. However, as the silica is lossy
at these optical wavelengths it is important to design the cross sections such that the losses in the
cladding material are as small as possible.
Low loss photonic-bandgap fibers have been designed by parameter studies in various papers
[10, 11, 12, 13, 14], where geometry parameters as the core size, the thickness and shape of the
core boundary, the air filling fraction and the size of the fingers pointing towards the core have
been varied. In the present work, we suggest to use topology optimization in order to design low
loss holey fibers with cladding material that itself is lossy. Topology optimization is a computer
based method and was originally developed in 1988 to maximize the stiffness of structures for
a limited amount of material [15]. The method has since then been developed and applied to
other engineering fields as mechanism design, heat transfer and fluid flow problems, see [16] for an
introduction to the method and the different applications. By topology optimization air and solid
material can be distributed freely in a design domain and the size and the number of the holes are
determined. The method therefore offers more freedom in the design than obtained by size and shape
optimization alone. The method was extended to electromagnetic wave propagation problems in
1999 where photonic-bandgap materials were designed in [17] for plane waves. Structures as bends
and splitters based on 2D band-gap materials were designed in [18, 19] for single frequencies or
frequency intervals. Planar photonic-crystal waveguide components optimized for low loss and high
bandwidth and transmission have been fabricated and characterized, see [20, 21]. The tailoring of
dispersion properties by topology optimization is considered for photonic-crystal waveguides in [22]
and for step-index optical fibers in [23].
In this work we extend the method of topology optimization to holey fibers. The aim is to
design the cross section of the fiber for a fixed wavelength relevant for medical purposes in order
to optimize the energy flow through the core region. The optical model and optimization problem
are described in section 2. An example of the performance of the method is given in section 3 and
simplified designs inspired from the optimized geometry are studied.
2 Description of the optimization problem
2.1 The optical model
The optimization problem is based on a photonic-bandgap fiber with the geometry indicated in
figure 1. It is a holey fiber where the optical wave is guided in a hollow core region, which is
surrounded by a two-dimensional periodic cladding of air holes and silica at IR wavelengths. Silica
is a highly lossy material and the goal of the optimization is to improve the energy transport in
the core region, or in other words, to modify the mode profile such that its overlap with the silica
becomes as small as possible. This is done by redistributing air and silica in the design domain Ωd
such that the objective function Φ, which is an expression related to the power flow, is optimized in
the inner part of the core region denoted Ωc.
It is assumed that the propagating optical modes of order ν have harmonic solutions on the form
Hp,ν(x1, x2, x3) = Hp,ν(x1, x2)e −iβνx3 , (1)
where Hp,ν is the magnetic field of the optical wave and βν is the propagation constant for a given
optical mode ν. In the following only the guided mode is considered and therefore the notation with
ν is omitted. The magnetic field is entered into the time-harmonic wave equation
eijk
∂
∂xj
n −2 eknp
∂Hp
∂xn
− k 2 0Hi = 0, (2)
where k0 is the free space propagation constant and eijk is the alternating symbol. As the energy
of guided optical modes is concentrated in the core region, it can be assumed that the electric field
2