Maria Bayard Dühring - Solid Mechanics
Maria Bayard Dühring - Solid Mechanics 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
Figure 1: Geometry of the photonic-crystal fiber used in the optimization. Black indicates silica and white is air. The design domain is Ωd and Ωc is the core region where the objective function is optimized. is zero at the outer boundary of the fiber and the perfect electric conductor boundary condition is applied. The guided mode for the geometry used here is twofold symmetric and therefore only half the geometry is needed. At the symmetry border the perfect electric conductor is applied as well, such that the electric field is polarized in the vertical direction. For the given value of k0 the propagation constant β for the guided mode is found by solving the wave equation as an eigenvalue problem. In order to define the problem for the topology optimization β is first computed. Then the wave equation is solved in a time-harmonic way where both k0 and β are fixed, and the guided mode is excited by introducing a small half-circle in the middle of the fiber where a magnetic forcing term in the horizontal direction is applied. To get a realistic problem for the topology optimization, material damping is introduced in the silica cladding by adding an imaginary part to the refractive index. This means that the magnetic field components are complex valued. The mathematical model of the optical problem is solved by finite element analysis. The complex fields Hj are discretized using sets of finite element basis functions {φj,n(r)} Hj(r) = N Hj,nφj,n(r). (3) n=1 The degrees of freedom are assembled in the vectors Hj = {Hj,1, Hj,2, ...Hj,N} T . A triangular element mesh is employed and vector elements are used for H1 and H2 in order to avoid spurious modes. Second order Lagrange elements are used for H3. The optical model is solved by the commercial finite element program Comsol Multiphysics with Matlab [24] with the module “Perpendicular waves, hybrid-mode waves” for the three magnetic field components. This results in the discretized equations for the time-harmonic problem used in the topology optimization 3 SkjHj = fk, (4) j=1 where Skj is the complex system matrix and fk is the load vector. The problem in (4) is solved as one system in Comsol Multiphysics. 3
- Page 52 and 53: 42 Chapter 6 Design of acousto-opti
- Page 54 and 55: 44 Chapter 6 Design of acousto-opti
- Page 56 and 57: 46 Chapter 6 Design of acousto-opti
- Page 58 and 59: 48 Chapter 6 Design of acousto-opti
- Page 60 and 61: 50 Chapter 6 Design of acousto-opti
- Page 62 and 63: 52 Chapter 6 Design of acousto-opti
- Page 64 and 65: 54 Chapter 6 Design of acousto-opti
- Page 66 and 67: 56 Chapter 7 Concluding remarks des
- Page 68 and 69: 58 Chapter 7 Concluding remarks
- Page 70 and 71: 60 References [14] E. Yablonovitch,
- Page 72 and 73: 62 References [42] A. A. Larsen, B.
- Page 74 and 75: 64 References [67] K. Svanberg, “
- Page 77: Publication [P1] Acoustic design by
- Page 80 and 81: 558 In Ref. [3] it is studied how t
- Page 82 and 83: 560 2.2. Design variables and mater
- Page 84 and 85: 562 When the mesh size is decreased
- Page 86 and 87: 564 objective function, Φ [dB] 130
- Page 88 and 89: 566 ARTICLE IN PRESS Fig. 6. The di
- Page 90 and 91: 568 ~kðxÞ 1 ¼ k1 ARTICLE IN PRES
- Page 92 and 93: 570 ARTICLE IN PRESS M.B. Dühring
- Page 94 and 95: 572 ARTICLE IN PRESS M.B. Dühring
- Page 96 and 97: 574 frequency or the frequency inte
- Page 99: Publication [P2] Design of photonic
- Page 104 and 105: 2.2 Design variables and material i
- Page 106 and 107: two air holes, is Λ = 3.1 µm, the
- Page 108 and 109: Figure 4: The geometry of the cente
- Page 110 and 111: performance will decrease and some
- Page 112 and 113: [23] J. Riishede and O. Sigmund, In
- Page 115 and 116: Surface acoustic wave driven light
- Page 117 and 118: IDT GaAs AlGaAs 0.10 0.05 0.00 Ampl
- Page 119: Publication [P4] Improving the acou
- Page 122 and 123: 083529-2 M. B. Dühring and O. Sigm
- Page 124 and 125: 083529-4 M. B. Dühring and O. Sigm
- Page 126 and 127: 083529-6 M. B. Dühring and O. Sigm
- Page 128 and 129: 083529-8 M. B. Dühring and O. Sigm
- Page 131: Publication [P5] Design of acousto-
- Page 134 and 135: Figure 1: The geometry used in the
- Page 136 and 137: where ˜pijkl are the rotated strai
- Page 138 and 139: where ∂Φ ∂w R 2,n − i ∂Φ
- Page 140 and 141: Figure 3: The distribution of the o
- Page 142 and 143: [12] J.S. Jensen and O. Sigmund, To
- Page 145 and 146: Energy storage and dispersion of su
- Page 147 and 148: 093504-3 Dühring, Laude, and Kheli
- Page 149 and 150: 093504-5 Dühring, Laude, and Kheli
- Page 151: Publication [P7] Improving surface
Figure 1: Geometry of the photonic-crystal fiber used in the optimization. Black indicates silica<br />
and white is air. The design domain is Ωd and Ωc is the core region where the objective function is<br />
optimized.<br />
is zero at the outer boundary of the fiber and the perfect electric conductor boundary condition is<br />
applied. The guided mode for the geometry used here is twofold symmetric and therefore only half<br />
the geometry is needed. At the symmetry border the perfect electric conductor is applied as well,<br />
such that the electric field is polarized in the vertical direction.<br />
For the given value of k0 the propagation constant β for the guided mode is found by solving the<br />
wave equation as an eigenvalue problem. In order to define the problem for the topology optimization<br />
β is first computed. Then the wave equation is solved in a time-harmonic way where both k0 and<br />
β are fixed, and the guided mode is excited by introducing a small half-circle in the middle of the<br />
fiber where a magnetic forcing term in the horizontal direction is applied. To get a realistic problem<br />
for the topology optimization, material damping is introduced in the silica cladding by adding an<br />
imaginary part to the refractive index. This means that the magnetic field components are complex<br />
valued.<br />
The mathematical model of the optical problem is solved by finite element analysis. The complex<br />
fields Hj are discretized using sets of finite element basis functions {φj,n(r)}<br />
Hj(r) =<br />
N<br />
Hj,nφj,n(r). (3)<br />
n=1<br />
The degrees of freedom are assembled in the vectors Hj = {Hj,1, Hj,2, ...Hj,N} T . A triangular element<br />
mesh is employed and vector elements are used for H1 and H2 in order to avoid spurious<br />
modes. Second order Lagrange elements are used for H3. The optical model is solved by the commercial<br />
finite element program Comsol Multiphysics with Matlab [24] with the module “Perpendicular<br />
waves, hybrid-mode waves” for the three magnetic field components. This results in the discretized<br />
equations for the time-harmonic problem used in the topology optimization<br />
3<br />
SkjHj = fk, (4)<br />
j=1<br />
where Skj is the complex system matrix and fk is the load vector. The problem in (4) is solved as<br />
one system in Comsol Multiphysics.<br />
3
Change language