Computational Optoelectronics Group, Integrated Systems Laboratory, ETH Zurich 2. Institute of Photonics and Quantum Electronics, EPF Lausanne

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1 Modelling the Purcell Effect in Photonic Crystal Cavities with a 3D Finite Element Maxwell Solver Friedhard Römer 1, Andrea Fiore 2, Laurent Balet 2, and Bernd Witzigmann 1 1 Computational Optoelectronics Group, Integrated Systems Laboratory, ETH Zurich 2 Institute of Photonics and Quantum Electronics, EPF Lausanne

2 Overview Photonic Crystal Cavities The 3D FEM Maxwell Solver Simulation & Characterization Results Conclusion & Outlook

3 Photonic Crystal Cavities Design Goal cavities for efficient generation of incoherent light or single photons for sensing applications Photonic Crystal Cavities Requirements High spontaneous emission enhancement (Purcell factor) High extraction efficiency Mode stability Design and Optimization requires efficient simulation tool for computing the eigenmodes and the Purcell factor

4 The Purcell Effect Spontaneous Emission Enhancement by an optical cavity: Free space SE rate Cavity modifies SE Cavity SE at resonance Γ is proportional to the local density of states (LDOS) and to the Green s function G 11 (r,r,ω) Approximate solution: A high Purcell factor F P =Γ max /γ f demands: high quality Q small cavity volume V atomic dipole parallel to mode field pê ( QD issue)

5 Solver prerequisites Universality PCC geometry thin semiconductor slab periodic arrangement of holes high aspect ratio restricts applicability of 2D methods 3D FEM Maxwell applicable to any cavity type no analytic Green s functions for special geometry features required restricts applicability of mode expansion Result computation Local density of optical modes (spontaneous emission enhancement) Farfield, nearfield and cavity efficiency target of postprocessing Coupled simulation Fast operation for coupled electroopto-thermal device simulation Compatibility with Newton scheme restricts applicability of FDTD

6 Solving the Maxwell Equations with FEM Wave equation Domain discretization Base function expansion variational of ψ: Variation Functional [1] Solution vector, eigenpair Impose boundary conditions Dirichlet: E t = 0, von Neumann: E n = 0, Perfectly Matched Layer Minimize variational [1] J. Jin, The Finite Element Method in Electromagnetics, John Wiley & Sons,

7 Computing LDOS and Green s functions LDOS definition Extracting the Green s functions from the FEM scheme requires computing the response of the cavity to a test dipole p located at the position r. Define dipole source and probe Definition of Green s function: Solve wave equation with dipole

8 Benchmarks 3D FEM Maxwell solver eigenmode results compared to different solution approaches Dielectric sphere benchmark: FEM on one eighth of the sphere Comparison with analytic solution Relative target error: Photonic crystal benchmark: FEM on one eighth of photonic crystal cavity Comparision with 3D FDTD solution [1] Relative target difference: for x-polarized mode for y-polarized mode [1] Jelena Vuckovic, et al., Design of photonic crystal microcavities for cavity QED, Phys. Rev. E, vol. 65, , 2001

9 Photonic crystal cavity eigenmodes H1: single defect H2: hexagonal defect L3: linear defect 1e5 1e4 1e Intensity (a.u.) k 0 = m -1 Q = k 0 = m -1 Q = 1729 k 0 = m -1 Q = nm GaAs slab, 340nm hole distance, filling factor: 35%

10 Photonic Crystal Cavity Eigenmodes H2 cavity: cut through the cavity center normal to the z-axis, intensity pattern k 0 = m -1 Q = 1729 k 0 = m -1 Q = 1014

11 Photonic crystal cavity results Eigenmode: k 0 = m -1 Q = Single defect cavity 320nm GaAs slab 340nm hole distance filling factor: 35% Source sweep and PL characterization Problem size: DOFs CPU Memory: 12GB, CPU Time: 4500s/eigenmode, 420s/source step

12 Photonic crystal cavity results Eigenmode: k 0 = m -1 Q = 4686 Linear defect cavity 320nm GaAs slab 340nm hole distance filling factor: 35% Source sweep and PL characterization Problem size: DOFs CPU Memory: 12GB, CPU Time: 4500s/eigenmode, 420s/source step

13 Conclusion Flexible Full 3D Maxwell solver for cavity problems convenient and efficient eigenmode computation using source/probe computation benchmark examples show excellent accuracy provides potential for Density of Modes computation enables the self consistent electro-opto-thermal simulation of photonic crystal light emitters Acknowledgements Part of the solver implementation has been provided by Oscar Chinellato and Peter Arbenz from the Institute of Computational Science, ETH Zürich. Funding by the ESF under the COST P11 project C05.70

14 Outlook Under development: Farfield computation: coding completed testing is ongoing LDOS and Green s function computation: design and coding is ongoing More benchmarks (together with COST P11 WG2) Under investigation Perturbation and parametric derivatives with FEM for simulating devices with strong coupling of optics and electronics and for geometry optimization