Maria Bayard Dühring - Solid Mechanics

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28 Chapter 5 Design of photonic-crystal fibers by topology optimization [P2] Figure 5.1 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. radius of the air core in the center is R = Λ. The boundary condition along all the borders is the perfect electric conductor. The chosen optical wavelength in free space is λ0 = 2 µm and the refractive indexes for this wavelength are na = 1 for air and ns = 1.43791 + 0.0001i for silica. The imaginary part of ns indicates the absorbing coefficient αs. It is assumed that the propagating optical modes have harmonic solutions on the form Hp,ν(x1, x2, x3) = Hp,ν(x1, x2)e −iβνx3 , (5.1) where Hp,ν is the magnetic field of the optical wave and βν is the propagation constant for a given optical mode ν. This is entered into the time-harmonic wave equation ∂ ∂Hp,ν eijk bklblmemnp − k ∂xj ∂xn 2 0Hi,ν = 0, (5.2) where k0 = 2π/λ0 is the free space propagation constant, eijk is the alternating symbol and biknkj = δij. For the problem considered in this chapter the materials are assumed to be isotropic and therefore bklblm reduces to n −2 . For the given value of k0 the propagation constant βν for the possible modes are found by solving the wave equation as an eigenvalue problem. In order to define the problem for the topology optimization, the propagation constant for the guided mode β1 is calculated. The wave equation is then solved in a way where both k0 and β1 are fixed, and the guided mode is excited by applying a magnetic source term in the center of the fiber, which is directed in the horizontal direction. The design variable ξ takes the value 0 for air and 1 for silica and the refractive index is interpolated linearly between the two material phases. The objective function Φ is an energy measure and is defined as
5.2 Results 29 the sum of the squared magnetic field amplitudes in the x1- and x2-direction in the core area, Ωc. Hence, the optimization problem is given as max ξ log(Φ) = log Ωc j=1 2 |Hj(r, ξ(r))| 2 dr, objective function (5.3) subject to 0 ≤ ξ(r) ≤ 1 ∀ r ∈ Ωd, design variable bounds (5.4) The morphology-based filter described in section 3.5 is applied to avoid meshdependent solutions and to enforce 0-1 designs. To prevent convergence to a local optima a continuation method is applied where αs has a big value from the beginning of the optimization such that the problem is modified to a smoother problem. During the optimization αs is gradually reduced to its more realistic value. Vector elements are used for H1 and H2 in order to avoid spurious modes. Lagrange elements of second order are used for H3 and of zero order for ξ. 5.2 Results An example of the optimization is shown in figure 5.2(a) and 5.2(b) where the initial and the optimized designs are plotted, respectively. The distribution of the energy measure from the objective function normalized with its maximum value for the initial design, Φmax, is indicated with the contour lines. The objective function is increased 3.75 times for the optimized design. All parts are connected in the optimized geometry and it is close to be a 0-1 design. The symmetry around the vertical axis is kept in the optimized design, but the 60 degree symmetry from the initial design has vanished. The objective function in the optimized design has been redistributed such that the modes shape is extending more in the vertical direction Figure 5.2 The center region of the holey fiber where black indicates silica and white is air. The contour lines show the distribution of the energy measure from the normalized objective function Φ/Φmax. (a): Initial design. (b): Optimized design.
Page 1: Optimization of acoustic, optical a
Page 4 and 5: Title of the thesis: Optimization o
Page 6 and 7: Resumé (in Danish) Forskningsomr˚
Page 8 and 9: Publications The following publicat
Page 10 and 11: vi Contents 6 Design of acousto-opt
Page 12 and 13: 2 Chapter 1 Introduction waves. The
Page 14 and 15: 4 Chapter 2 Time-harmonic propagati
Page 20 and 21: 10 Chapter 2 Time-harmonic propagat
Page 22 and 23: 12 Chapter 3 Topology optimization
Page 32 and 33: 22 Chapter 4 Design of sound barrie
Page 40 and 41: 30 Chapter 5 Design of photonic-cry
Page 42 and 43: 32 Chapter 5 Design of photonic-cry
Page 44 and 45: 34 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,
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566 ARTICLE IN PRESS Fig. 6. The di
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568 ~kðxÞ 1 ¼ k1 ARTICLE IN PRES
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570 ARTICLE IN PRESS M.B. Dühring
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572 ARTICLE IN PRESS M.B. Dühring
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574 frequency or the frequency inte
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Publication [P2] Design of photonic
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photonic-crystal fibers used for me
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2.2 Design variables and material i
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two air holes, is Λ = 3.1 µm, the
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Figure 4: The geometry of the cente
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performance will decrease and some
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[23] J. Riishede and O. Sigmund, In
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Surface acoustic wave driven light
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IDT GaAs AlGaAs 0.10 0.05 0.00 Ampl
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Publication [P4] Improving the acou
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083529-2 M. B. Dühring and O. Sigm
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Publication [P5] Design of acousto-
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Figure 1: The geometry used in the
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where ˜pijkl are the rotated strai
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where ∂Φ ∂w R 2,n − i ∂Φ
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Figure 3: The distribution of the o
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[12] J.S. Jensen and O. Sigmund, To
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Energy storage and dispersion of su
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093504-3 Dühring, Laude, and Kheli
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093504-5 Dühring, Laude, and Kheli
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Publication [P7] Improving surface
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FIG. 1: The geometry of the acousto
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finer mesh. For the coupled model i
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FIG. 4: Results for the finite mode
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FIG. 6: Results for the acousto-opt
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Suzuki, A. Onoe, T. Adachi and K. F
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Maria Bayard Dühring - Solid Mechanics

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