Maria Bayard Dühring - Solid Mechanics

More documents

Recommendations

Info

6 Chapter 2 Time-harmonic propagating waves medium can change the phase velocity of the optical wave because of higher order effects. Another example is the photoelastic effect where the refractive index can be changed by applying an external stress field. The change in refractive index is related to the stress or the strain through the stress-optical or strain-optical relation, respectively, where the stress-optical and strain-optical coefficients are collected in second-rank tensors. By utilizing this effect it is possible to change an isotropic material to an optically anisotropic material by applying a stress field. This stressinduced birefringence was first studied in 1816 by Brewster. If an elastic wave is propagating in a medium, the refractive index will change periodically because of the photoelastic effect. The phenomenon is known as the acousto-optical effect. The acousto-optical interaction was first investigated in 1922 by Brillouin in order to diffract an optical beam [19]. Experiments that confirmed the effect were performed in 1932 [20]. Further investigation of the effect showed that elastic waves can be used to change properties as intensity, frequency and direction of optical waves. The other way around, optical waves can be used to measure characteristics of elastic waves as attenuation and radiation patterns. This research led to the development of spectrum analyzers, tunable optical filters and variable delay lines, where it is used that the velocity of the elastic wave is typically 10 5 times smaller than for electromechanical waves. One of the most popular devices is the acousto-optical modulator that is used for instance in laser printers. As SAWs are confined to a material surface they have a potential in integrated optics where they can interact efficiently with optical waves in waveguides close to the surface. This concept is currently being invested for fast and compact devices where optical waves are modulated by SAWs for signal generation in semiconductor structures. An introduction is given in [21], where the concepts for SAW generation and the mechanisms for the acousto-optical interaction in different structures are reviewed. In [22] experimental results for a compact and monolithic modulator consisting of a SAW driven Mach-Zehnder interferometer (MZI) are presented, and the device is modified for more efficient modulation in [23]. The concept of acoustooptical multiple interference devices are suggested in [24] where several MZIs are combined in parallel or series in order to design ON/OFF switching, pulse shapers and frequency converters. In [25] the working principle of a SAW driven optical frequency shifter based on a multi-branch waveguide structure is presented. These devices are expected to be very compact and compatible with integrated optics based on planar technology for different material systems. 2.2 Modeling of wave problems In this work, three different types of wave propagating problems with increasing complexity are investigated. The first is about propagation of acoustic waves in air and the next treats optical waves propagating in a photonic-crystal fiber. In the last problem the propagation of SAWs in piezoelectric materials and their interaction with optical waves in waveguides are studied. All the different types of waves vary
2.2 Modeling of wave problems 7 in space and time and can be described in a similar way by second order differential equations. In the case of elastic waves in a solid material, the displacement ui of a point with the coordinates xk varies with time t, such that ui = ui(xk, t). By combining Newton’s second law with Hooke’s law for a solid inhomogeneous material the wave equation is obtained [3] ∂ ∂ul cijkl − ρ ∂xj ∂xk ∂2ui = 0, (2.1) ∂t2 where cijkl are the stiffness constants and ρ is the mass density, which all depend on the position xk. The expression in (2.1) is a system of three second order differential equations for an anisotropic, inhomogeneous material in 3D. In the case of a fluid the wave equation has the scalar form ∂ ∂u ∂ α1 − α2 ∂xi ∂xi 2u = 0, (2.2) ∂t2 where u is the pressure, α1 is the inverse of the mass density and α2 is the inverse of the bulk modulus. If u describes the out of plane displacements in a solid material, α1 is the shear modulus and α2 denotes the mass density, equation (2.2) models the behavior of elastic shear waves. In a similar way, the electromagnetic waves can be described by a system of three second order differential equations in the general case, which can be reduced to the scalar equation (2.2) for the special cases of in-plane modes. In the case of transverse electric modes u describes the electric field, α1 is the inverse of the dielectric constant and α2 is the product of the vacuum permittivity and the vacuum permeability. In many situations it can be assumed that the forces, and thus the waves, are varying harmonically in time with the angular frequency ω. It is therefore convenient to eliminate the time-dependency in the equations. In the case of for instance a fluid, the field variable is described on the complex exponential form u(xk, t) = û(xk)eiωt , where û describes the complex amplitude and the phase indirectly. The exponential represents the time variation. This means that the operator ∂2 ∂t2 can be written as −ω2 and when ûeiωt is introduced in the wave equation (2.3) and the hat is omitted it reduces to the simpler form ∂ ∂u α1 + ω ∂xi ∂xi 2 α2u = 0. (2.3) This equation is known as the Helmholtz equation and is valid for the field oscillating at the single frequency ω. The real physical value of u is found by taking the real part of the complex solution to the problem. The conversion from a time dependent to a time-harmonic problem can be done in an equivalent way for the other wave problems and only these types of problems are considered in this work. A solution to a wave problem can be found by applying a set of boundary conditions that give information about free and clamped surfaces and interfaces between
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 18 and 19: 8 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 38 and 39: 28 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,
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 102 and 103:
photonic-crystal fibers used for me
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:
Page 126 and 127:
Page 128 and 129:
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
Page 154 and 155:
FIG. 1: The geometry of the acousto
Page 156 and 157:
finer mesh. For the coupled model i
Page 158 and 159:
FIG. 4: Results for the finite mode
Page 160 and 161:
FIG. 6: Results for the acousto-opt
Page 162:
Suzuki, A. Onoe, T. Adachi and K. F
show all

Maria Bayard Dühring - Solid Mechanics

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?