FEM simulations of nanocavities for plasmon lasers S.Burger, L.Zschiedrich, J.Pomplun, F.Schmidt Zuse Institute Berlin JCMwave GmbH 6th Workshop on Numerical Methods for Optical Nano Structures ETH Zürich, July 07, 2010
Outline Introduction (FEM) Plasmonic nanocavities: Propagating modes in plasmonic waveguides Simulate transmission of waveguide modes (light scattering simulations) Transmission through plasmonic waveguide cavities Direct simulation of cavity resonances (eigenvalue problem) Conclusion 2
Finite Element Method Choose computational domain with appropriate boundary conditions Sub-divide the geometry into patches Expand the electric/magnetic fields with local ansat functions which are defined on the triangles and plug into weak formulation of Maxwell s equations Solve sparse matrix equation with fast numerics 3
Features of our FEM code FEM package JCMsuite (Version 2.2.7, 07/2010): solvers for time-harmonic Maxwell problems high-order edge elements p = 1 9 adaptive mesh refinement (goal oriented error estimation) domain decomposition algorithms heat conduction, elastomechanics automatic differentiation 4
Convergence ~6s 11s Comparison to plane wave method : Ca. 1 day for result with low accuracy! 19s 40s ~100s [R. Hollöhner, et al. Efficient optimiation of hollow-core photonic crystal fiber design using the finite element method. JEOS 1, 06011 (2006).] 5
Application Examples VCSEL resonances Photonic crystal fibers Nanodisc resonators Photolithography masks Metamaterials Circular Grating Resonator Plasmonic antennas 6 [Image: MPIP Main]
Plasmonic lasers Plasmon lasers: New physics and applications, e.g. biosensing on a nanoscale Plasmonic lasers demonstrated using different designs (Spherical, Bow-Ties, Rectangular nanopillars, nanowires on top of a silver substrate, etc.) [Bergman, Stockman, Phys. Rev. Lett. 90, 027402 (2003). Anker, et al, Nature Materials 7, 442 (2008). Noginov, et al, Nature 460, 1110 (2009). Hill, et al, Opt. Express 17(13), 11107 (2009). Oulton, et al, Nature 461, 629 (2009). Chang, et al, Opt. Express 16(14), 10580 (2008).] In some of the designs, plasmonic waveguides play a crucial role. Plasmonic waveguides allow to concentrate light at deep subwavelength scale with relatively low material loss. 7
Waveguide Setup Design of Zhang group (Berkeley) [Oulton, et al, Nature 461, 629 (2009).] CdS nanowire, D=50nm (n=2.5) Thin MgF2 layer, h=5nm (n=1.38) Silver layer (optically thick) (n=0.05+3i) Guided mode / emitted laser radiation Mesh 8
Waveguide Mode 2D Geometry (invariant in -dir) Triangular mesh 9 Triangular mesh, after some automatic refinement steps
r r E = E Waveguide Mode Guided mode E field at wavelength 489 nm (pseudo-colors, log-scale) pm ( x, y)exp( ik ) E y E r 50 nm Guided mode is sub-wavelength by two orders of magnitude [cf. Oulton, et al, New Journal of Physics 10, 105018 (2008).] 10
Convergence How accurate and at what computational cost can we compute these eigenmodes?: n = k λ / 2π eff 0 Convergence Real / Imaginary parts of effective refractive index (eigenvalue) Corresp. computation time 11
3D simulations Simulate light transmission through finite section of infinite plasmonic waveguide. 3D geometry Incident field Record transmitted flux Transmission = transmitted flux / incident flux Non-standard computational task due to multiple scales of the 3D problem: deep-sub-nanometer meshing required at corners and edges, but micrometer scales also present. 12
x A 3D simulations 3D light scattering simulation results: Images of Re(E y ) in different cross sections through the 3D computational domain y A x B y B L = 1500nm 13
3D simulations / Convergence Effective refractive index of 2D waveguide mode gives quasi-exact result to compare with 3D simulation results: T = exp( 4πLI( n eff ) / λ0) Accuracy better 1% reached at moderate computational effort. 14
Plasmonic cavity Waveguide of finite length L with mirrors formed by air-gaps: Plasmonic cavity y x Gap Cavity Gap Waveguide for Coupling (Input) Waveguide for Coupling (Output) 15
Plasmonic cavity Input waveguide mode at 489nm vary cavity length L different gap widths W Observe Fabry-Perot fringes. 16
Plasmonic cavity 370nm 1360nm y B x Field distributions, log scale, pseudocolors Standing-wave patterns. A x A y B L = 370nm L = 1360nm 17
Direct simulation of resonances For determining main resonance properties (Q-factor and resonance wavelength): Solve eigenvalue problem, i.e., search electromagnetic field and complex eigenfrequency which solve Maxwell s equations without sources. Find ( ω, E) ε 1 µ 1 such that E 2 = ω E. Energy stored Q = 2π Energy dissipated per cycle R( ω) Q = 2I( ω) Advantages w.r.t. scattering simulations: No problems with scattered field for low-q resonances; wavelength-scans not required ( faster). 18
Direct simulation of resonances Mode solver B y x Visualiation of computed resonance mode. (Ey component, pseudo-colors, log scale). A L = 374nm x A y B 19
Direct simulation of resonances Convergence of complex eigenvalue (corresp. resonance wavelength and resonance quality). High numerical accuracy reached. 20
Cavity design How does a change of cavity length influence resonance wavelength and Q-factor? Cavity for 489nm resonance frequency @L=374nm with Q-factor of ~2. Excellent agreement with results from light scattering simulations. 21
Summary Numerical investigation of resonances in finite sie plasmonic waveguides using time-harmonic FEM Propagation mode solver (waveguide mode) Light scattering solver (transmission analysis) Resonance mode solver (cavity resonances) Demonstrated convergence Influence of cavity sie variations on nanometer scale 22
High-Q resonators, large computational domains Thank you for your attention! Workshop: Computational Nanooptics, International Conference on Numerical Algebra and Applied Mathematics (ICNAAM), 19.- 26.9.2010, Rhodes, Greece Info: frank.schmidt@ib.de 23