Aage, Niels

Gersborg, Allan Roulund

Sigmund, Ole

This paper is concerned with topology optimization of a coupled optical and mechanical wave propagation problem in photonic crystals. It is motivated by the potential gain in functionality of optical devices where mechanical Rayleigh waves (travelling in the surface of the material) play a leading role. The practical applications cover novel optical modulators, switches and frequency shifters. The work uses COMSOL Multiphysics which is a modern and finite element based modelling language

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Downloaded from orbit.dtu.dk on: Dec 17, 2017TOPOLOGY OPTIMIZATION OF OPTICAL BAND GAP EFFECTS IN SLAB STRUCTURESMODULATED BY PERIODIC RAYLEIGHWAVESGersborg, Allan Roulund; Sigmund, Ole; Aage, NielsPublication date:2008Document VersionEarly version, also known as pre-printLink back to DTU OrbitCitation (APA):Gersborg, A. R., Sigmund, O., & Aage, N. (2008). TOPOLOGY OPTIMIZATION OF OPTICAL BAND GAPEFFECTS IN SLAB STRUCTURES MODULATED BY PERIODIC RAYLEIGHWAVES [Sound/Visual production(digital)]. 22nd International Congress of Theoretical and Applied Mechanics, Adelaide, Australia, 24/08/2008XXII ICTAM, 25–29 August 2008, Adelaide, AustraliaTOPOLOGY OPTIMIZATION OF OPTICAL BAND GAP EFFECTS IN SLAB STRUCTURESMODULATED BY PERIODIC RAYLEIGH WAVESNiels Aage, Allan Roulund Gersborg and Ole SigmundDepartment of Mechanical Engineering (Solid Mechanics)Technical University of Denmark, DK-2800 Kgs. Lyngby, DenmarkSummary This paper is concerned with topology optimization of a coupled optical and mechanical wave propagation problem inphotonic crystals. It is motivated by the potential gain in functionality of optical devices where mechanical Rayleigh waves (travellingin the surface of the material) play a leading role. The practical applications cover novel optical modulators, switches and frequencyshifters. The work uses COMSOL Multiphysics which is a modern and finite element based modelling language.INTRODUCTIONTopology optimization is a computational method that can be used to optimize the response of continuum structuressubject to given loads and constraints [1]. It has proven a very successful design tool in mechanical problems in general(see, e.g., [1]) and also in photonic crystal problems (see, e.g., [2]). This work contributes with knowledge on how toapply topology optimization to the coupled problem of a photonic crystal subjected to mechanical surface waves.The unique feature of a crystal is that a band gap can form – meaning that eigenstates within a limited band of frequenciesdo not exist, see, e.g., [3] for examples from optics. A requirement for the existence is that a material contrast is presentin the structure, e.g., for optical as well as mechanical waves, one can create a band gap in 1D by a Bragg grating andin 2D via an arrangement of circular inclusions. Computationally a unit cell analysis is often employed, motivated bythe periodicity of the crystal. Typically the relevant material parameter (e.g. the dielectric permittivity in optics andthe Youngs modulus in solid mechanics) is taken to be cell periodic whereas the state variables are represented using atime-harmonic Floquet-Bloch solution, see, e.g., [4].MODELLINGOptical modelIn this contribution we focus exclusively on the special case of band gaps between guided optical modes in a photoniccrystal slab being a structure which is in-plane periodic with a finite out-of-plane height, see figure 1(a). A guidedoptical mode is simply a localized eigenstate in the slab analogous to total internal reflection of light in ray optics. Themodelling of the guided optical modes in the slab relies on two subtle arguments as explained in [5]. When computingthe band diagram, the guided modes appear under the light line l being defined as l = |k w|/2π where |kw| is the lengthof the wave vector, see figure 1(d). Secondly, to avoid the computation with a large volume of surrounding air and non-reflecting boundary conditions, we use the trick of periodicity in the z-direction. This is acceptable since our focus isthe guided modes which have a rapidly decaying amplitude outside the slab where the z-periodicity changes the responsesignificantly. Furthermore we assume linear constitutive relations and non-magnetic materials which allow us to computethe optical state by a 3D eigenvalue problem formulation of Maxwell’s equations implemented using Nédélec elements.By evaluation of the energy in the x−, y− and z−direction a procedure is developed which enables a classification of a3D mode as being either transversely polarized (TM-like or TE-like) or as a mixed mode without a simple polarization.This is useful for comparing topology designs with results from the literature.Mechanical modelThe Rayleigh wave is represented as a time-harmonic Floquet-Bloch solution and we consider the (typical) case where theRayleigh wave is unaffected by the optical wave such that there is a one-way coupling in the problem. The wave lengthof the Rayleigh wave is approximately one order of magnitude larger than the size of the periodic cell which justifies theassumption that the geometric features of the perforated layer do not change the Rayleigh wave. Thus this model does notcover the full geometric coupling.The opto-mechanical interaction is modelled geometrically based on the amplitude field of the Rayleigh wave. In thisway the optical state is computed using the eigenvalue formulation but now with a geometrically distorted unit cell.Optimization problemWe apply the gradient driven topology optimization procedure described in [6] and use the dielectric permittivity as thedesign variable. Rather than the double bound formulation known from the literature [1] this work uses an active setstrategy to select a small set of critical eigenvalues on which the mathematical optimization problem is based.(a) View of the unit cell with a topology optimizeddesign. The design is restricted to have rectangularsymmetry and the design domain is −0.5 ≤ z ≤0.5. Outside the design domain the slab is suspendedin air. The Colorbar: the dielectric permittivity.(b) Band diagram for figure 1(a). Grey box relativebandgap beteween TM-like modes of 11.5%. The fatline is the light line. Mode symbols: ’•’ TM-like, ’×’TE-like and ’’ mixed.(c) View of the unit cell with a topology optimizeddesign (which is hollow). The design is restricted tohave rectangular symmetry and the design domain is−0.5 ≤ z ≤ 0.5. Outside the design domain theslab is suspended in air. The Colorbar: the dielectricpermittivity.0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.800.10.20.30.40.50.60.7k−Bloch/2πω(d) Band diagram for figure 1(c). Grey box relativebandgap between guided modes below the light line of6.9%. The fat line is the light line. Mode symbols: ’•’TM-like, ’×’ TE-like and ’’ mixed.Figure 1. Topology design and bandgaps for the optical problem. The relative bandgap measure used is 2(ωi+1 − ωi)/(ωi+1 + ωi).PRELIMINARY RESULTSTopology optimization of photonic crystal slabs is introduced by comparing the band gaps of topology optimized slabsto the literature [5]. As seen from figure 1(a) this allows for an increase of known band gaps between TM-like modes.Furthermore, figure 1(c) shows a slab structure which displays a complete bandgap. The results of using topology opti-mization to design crystal slabs with a 3D model are novel, unpublished and form a necessary prerequisite for solving thecoupled problem.It is natural to extend this study with the slight variation where the photonic slab is not suspended in air, but mechanicallysupported on an optical insulator. Having established a well-posed mechanical problem, the final goal of this work isto apply topology optimization to the coupled problem where a periodic model of a Rayleigh wave is introduced in theoptical slab model as described above.A small part of this talk has its focus on the implementation in COMSOL Multiphysics which is a finite element basedmodelling software which is sufficiently flexible to provide analytical sensitivities. Moreover, computational performanceresults such as efficiency of the 3D eigenvalue solver and parallel scalability are discussed. These issues are criticallyimportant since they limit the resolution in the topology design.References[1] Bendsøe M. P., Sigmund O.: Topology Optimization theory, methods and applications. Springer-Verlag Berlin Heidelberg, 2003.[2] Borel P.I., Harpøth A. et al.: Topology optimization and fabrication of photonic crystal structures. Optics Express, 12(9): 1996–2001, 2004.[3] Busch K., Freymann G. von, Linden S., Mingaleev S.F. et al.: Periodic nanostructures for photonics. Physics Reports 444: 101-202, 2007.[4] Joannopoulos J.D., Meade R.D.,Winn J.N.: Photonic Crystals: Molding the Flow of Light. Princeton University Press, 1995.[5] Johnson S.G., Fan S., Villeneuve P.R. et al.: Guided modes in photonic crystal slabs. Physical Review B, 60(8): 5751-5758, 1999.[6] Jensen J.S., Sigmund O.: Systematic Design of Photonic Crystal Structures using Topology Optimization: Low-loss Waveguide Bends. AppliedPhysics Letters 84(12): 2022–2024, 2004.

TOPOLOGY OPTIMIZATION OF OPTICAL BAND GAP EFFECTS IN SLAB STRUCTURES MODULATED BY PERIODIC RAYLEIGHWAVES

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TOPOLOGY OPTIMIZATION OF OPTICAL BAND GAP EFFECTS IN SLAB STRUCTURES MODULATED BY PERIODIC RAYLEIGHWAVES

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