Simulation of Optical Modes in Microcavities

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1 Simulation of Optical Modes in Microcavities Bernd Witzigmann, Matthias Streiff Computational Optoelectronics Group Integrated Systems Laboratory, ETH Zurich

2 Overview Introduction Physics of Microcavities Finite Element Method Benchmark Example Some Results Outlook & Conclusion

3 Introduction: Spontaneous Emission Engineering Fermi s Golden Rule: Spont. Emission Rate: ρ ΔE Cav Free Space Free space: Cavity modes: ρ(e) = 8πE 2 h 3 c 3 1 ρ(e) = ΔE cav V Optical Density of States E Purcell Factor: Presence of a Cavity modifies Spontaneous Emission Properties Spectral Control: Enhancement/Suppression Spatial Control: Redirection into Mode of Choice Electronic and Photonic Quantization Ref.: A. Fiore, EPF Lausanne

4 Example: VCSEL Cavity Decay Rate (1/s) F p radial F p vertical Simulation of Standard Single Mode VCSEL Cavity Extension several microns, Oxide Aperture Solution of vectorial Helmholtz Equation Radial Spont. Emission Enhancement of 2 for selected wavelengths (modes) Wavelength [nm]

, APL 1996 Goal: - Semiconductor-based high-q Microcavities with small Mode Volume - Electrical Pumping of active

5 Microcavity: Implementations Micropillar: Strong optical confinement, Optical pumping (Q 2700) VCSEL: Weak optical confinement, Electrical pumping Gerard et al., APL 1996 Goal: - Semiconductor-based high-q Microcavities with small Mode Volume - Electrical Pumping of active region Applications: - High Efficiency Nano-LEDs (tayloring output direction) - Single Photon Source (single Quantum Dot as Active Region) TCAD-based Design of Electromagnetic Properties of Microcavity

6 Optical Mode Simulation in Microcavities Requirements: - Solve Maxwell for Optical Modes and Quality Factor - Include Radiation Losses through Simulation Boundaries - Full Vectorial Solution due to Large Refractive Index Steps - Realistic Representation of Cavity (Geometry, Material) Approach: - Finite Element Solution of Complex Vectorial Helmholtz Equation - Sparse Matrix Eigenproblem - Optical Modes are Inner Eigenpairs of generalized Complex Symmetric Eigenproblem - Complex Eigenvalue: Resonance Frequency and Decay Rate:

Simulation: Finite Element Method Semi-classical Approach: Classical Maxwell and QM Polarization Expansion in Eigenfrequencies, Slowly Varying Amplitude

7 Simulation: Finite Element Method Semi-classical Approach: Classical Maxwell and QM Polarization Expansion in Eigenfrequencies, Slowly Varying Amplitude Approximation, and Adiabatic Dielectrics Yields: Complex Helmholtz Equation (3D!): Photon Rate Equation: Body of Revolution transforms 3D Helmholtz to set of 2D Solutions:

8 Absorbing Boundary Conditions Electromagnetic Radiation Effects pertinent for mode control: Perfectly Matched Layers (PML) as Free-Space BCs Impedance Matching using anisotropic magnetic Permeability: Helmholtz Equation: Output Power and Energy Transfer Rate through Top Aperture:

Accuracy: Benchmark Example Free dielectric sphere (radiation boundary), with eigenvalue: safe for Microcavity simulation Memory requirement approximately given by matrix order (1

9 Accuracy: Benchmark Example Free dielectric sphere (radiation boundary), with eigenvalue: safe for Microcavity simulation Memory requirement approximately given by matrix order (1 st and 2 nd order FE). Use higher order basis functions to improve memory/accuracy balance. Sufficient levels of accuracy attainable with tolerable memory for Microcavity structures.

Fiore, EPFL - Bottom DBR: Oxide - Top DBR: AlGaAs/GaAs - Oxide Aperture - Active

10 Simulation of Microcavity 260 nm aperture top DBR Simulation HE11 Type oxid. aperture oxid. DBR SEM: A. Fiore, EPFL - Bottom DBR: Oxide - Top DBR: AlGaAs/GaAs - Oxide Aperture - Active Region: QW or QD - Narrow Aperture => Mode Quenching - Surface/Substrate Leakage - 1-Dimensional or Effective Index Methods not applicable

11 Comparison Experiment - Simulation Experiment Dots: Experiment Line: Simulation Smaller Volume - Experiment: Electroluminescence for Different Aperture Diameters - Good Agreement Measurement - Simulation - Design Goal: Increase Q-Factor and Decrease Cavity Volume (Maximum F p )

12 Mode Analysis: A Closer Look r ox = um r ox = um Lateral optical leakage increases losses for certain oxide radii For r ox < 1 µm microcavity can no longer be considered as scaled VCSEL cavity (see Lalanne et al. APL 84, 23, 4726 )

$Diffractive Losses - Demonstrated High-Q =>$

13 Alternative Designs Micro-Pillar Micro-Lens - Benchmark Structure - Idea: Compensate Diffractive Losses - Demonstrated High-Q => Microlens - Optically Pumped - Can be pumped electrically

14 Alternative Designs: Results Pillar Lens

15 Conclusion & Outlook Finite-Element Method for 2.5D, Complex, Vectorial Helmholtz Equation Perfectly Matched Layers as Absorbing Boundary Conditions Experimental Microcavity Results as Benchmark Tool for Analysis and Design of Spontaneous Emission in Realistic Cavities Outlook: Improve Memory Consumption Investigate 3-Dimensional Effects Find better Design for Electrically Pumped Microcavity

16 Acknowledgements A. Fiore & Quantum Devices Group EPF Lausanne COE Group: - A. Baecker, V. Laino, M. Loeser, M. Luisier, S. Odermatt, L. Schneider Synopsys Switzerland Ltd.: - W. Fichtner, A. Witzig - A. Bregy, M. Pfeiffer Institute of Computational Science (ETHZ): - Peter Arbenz, Oskar Chinellato Seminar for Applied Mathematics (ETHZ): - R. Hiptmair, P. Ledger

17 Finite-Element Implementation Variational Functional of modified Helmholtz Equation: Rayleigh-Ritz: insert Edge Node Finite Element Basis: Build Variational: Sparse Matrix Eigenproblem: Optical modes: Inner Eigenpairs of large sparse generalised complex symmetric Eigenproblem. Complex Frequency Re: Resonance Freq. Im: Field Decay Rate

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

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