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ULTRA-FAST VCSEL CAVITY SIMULATION USING PARAXIAL MODE EXPANSION Spilios Riyopoulos SAIC McLean, VA 22102

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Talk Outline Case for paraxial mode expansion for VCSEL cavity modes Simulation study of generic VCSEL behavior Comparison with experiments

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I. PARAXIAL MODE EXPANSION MOTIVATION: SPEED UP simulations Eliminate space grid / finite differencing Retain (axisymmetric) 2-D effects Retain multimode / spatial hole burning Expand radiation profile into cavity modes Ultra fast computation –Mode finder (PREVEU): 100 ms for 25 modes –Dynamic simulation (FLASH): 1 sec / 2000 ps ( 3 modes )

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Challenge: find these modes Buried heterostructureEtched mesaOxide aperture - Multilayered structure - No Obvious lateral confinement: Radial Boundary Conditions ?

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Lateral Losses: Diffraction / Scattering

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Lateral (radial) power losses Diffraction losses not addressed by guided-mode theory Index-guided modes Metal boundary waveguide modes –Zero Radial Power Flux need to include radiating mode continuum - cumbersome Paraxial modes inherently include diffraction losses Need k / k << 1 to keep losses small –Cavity eigenmodes = paraxial mode superposition Scattering Losses not addressed by paraxial modes –Scattered Radiation is a total loss

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Observations supporting lateral mode losses Large increase in threshold current density at small apertures –J th should remain constant without lateral losses Thermal index guiding ? Small n/ T~ 10 -4 /K –Opposite than expected modal stability trends Wide aperture VCSELs mode-switch near lasing threshold, despite lower T (insufficient V-parameter for higher modes). Narrow apertures have much better mode stability Mode structure exists below threshold, or in pulsed operation Spontaneous emission modes / LED operation

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Paraxial (GL) Mode Expansion: Paraxial theory to its fullest extend No a-priori index guiding (thermal or aperture) Small k / k, necessary for confinement Expand cavity modes in paraxial (GL) modes Evolve paraxial propagator in real space –Maps GL modes into GL of re-scaled spot size/curvature –Easier to treat finite diameter/scattering effects Round-trip matrix diagonalization (Fox-Li) –Obtain eigenmodes/eigenvalues algebraically –No numerical iteration involved

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G-L intensity profiles (p=0, m=0) (p=0, m=1) (p=0, m=2) (p=1, m=0) + = + = x-polarizedy-polarized

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Effective Cavity Model Replace DBRs with flat mirrors at effective cavity length(s) Axial phase advance/standing wave : Phase penetration L Wavefront curvature evolution : Use diffraction length L

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Included Features Reflection matrix R –Wing-clipping from finite mirror radii Gain matrix G –Selective on-axis gain Scattering matrix –Edge scattering losses (aperture/mesa) –Aperture phase-shift Diffraction / Self-interference matrix P –Diffracted, curved wavefront projected onto original HOW MANY BASIS MODES NEEDED FOR ACCURATE REPRESENTATION ?

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Expansion Coefficients-Eigenvalues vs mode waist w Coefficients sensitive to choice of waist Eigenvalue independent of choice of waist 7 x 3 mode basis

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Optimum Waist: Minimize Round-Trip Losses optimum waist: Cavity Eigenmode G-L Mode non-optimum: Cavity Eigenmode G-L Mode a 0 /w 0 Per-Pass Loss Gain overlap Loss Mirror Spill-over Loss Aperture Scattering Loss Increasing Diffraction Loss minimum-loss optimum 0.0 1.0 2.0 3.0 4.0 6.0 5.0 0.0 1.0 2.0 3.0 4.0 6.0 5.0 1 2 3 4 1 2 3 4

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Cavity mode representation by Pure Gauss-Laguerre modes Geometry parameterized by Two other parameters: –bulk gain g –DBR reflectivity r All matrices non-diagonal –Yet, off-diagonal terms small: Clipping losses Interference losses Optimize w: make round-trip matrix as diagonal as possible Optimum w Round-Trip Matrix Steady-state

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Etched Mesa VCSEL Results Current = 4.3 mA w hor = 1.25 m w vert = 1.28 m Current = 2.0 mA w hor = 1.33 m w vert = 1.24 m w exp = 1.31 m 0.052 w theory = 1.33 m

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Near Field Data Fit by Theoretical Mode Profile Admixtures of fundamental and first cavity mode Optimized waist prescribed by the model Proton implanted, wide aperture 850 nm VCSEL a = 7.5 m ARL, Oxide Confined 980 nm VCSEL a = 3.5 m Theory = solid line

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Cavity Eigenmodes Represented by pure, optimized waist, GL modes Analytic formula w(a; g, r, N) in laterally open cavities Waist-aperture relation determines: -Blue shifting of cavity modes / mode separation -Increase in round-trip losses / threshold current density -Differentiation among modal losses / cavity stability -J-Threshold vs. aperture location in standing wave

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II. SIMULATIONS of GENERIC VCSEL BEHAVIOR G-L expansion algorithm used in mode finder PREVEU (Paraxial Radiation Expansion for VCSEL Emulations) Simulate generic behavior vs. aperture size : Versatility: most VCSEL types DIFFRACTION & SCATTERING LOSSES DOMINATE AT SMALL APERTURERS MUST BE INCLUDED FOR CORRECT CAVITY BEHAVIOR

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Mode waist vs. Aperture Proton implantEtched Mesa

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Mode waist vs. Aperture Etched MesaOxide Aperture

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Cavity Blue Shifting Decreasing aperture / decreasing w Increasing k = 1/w = c p k z ( 1 + k 2 / 2k z 2 ) Blue Shift: / ~ 2 / (k z w) 2 Etched Mesa Theory: 2.28 n m Observed: 2 00 0.30 n m

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Threshold Gain Proton Implant Etched Mesa Nominal g o = (1-r)/2, r = DBR reflectivity

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Round Trip Losses Etched MesaOxide aperture Diffraction and Scattering losses dominate at small apertures Much higher than DBR reflectivity losses

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Modal Stability Small aperture stable / large unstable, relative to mode switching Differentiaton among various mode losses determines cavity stability Fundamental mode stability factor: S 01 = (R 01 -R 00 )/ R 00 Etched mesaOxide Aperture

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Gain Lensing Increasing gain causes waist to shrink Shrinking waist increases k perp = 1/w for given k z Lensing causes blue shifting Opposite to red shift from cavity thermal expansion Proton Implant

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Comparison : GL vs. waveguide modes Fundamental LP 01 almost identical to GL 00 for However: two approaches give different results for w

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Adaptation: Thermal index Guiding Parabolic Index Profile: G-L modes (again!) NO-DIFFRACTION fixed waist size Evaluate edge-clipping, scattering as before

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III. COMPARISON WITH EXPERIMENTS MODE STRUCTURE using PREVEU ( Paraxial Radiation Expansion for VCSEL Emulations ) DYNAMIC SIMULATION using FLASH (Fast Laser Algorithm for Semiconductor Heterostructures)

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Higher Mode Losses Proton ImplantEtched mesa 1-R 2 Different Optimum waist for each cavity mode

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NF data fit - Wide aperture, proton implant I/I th = 1.05 w x = 3.42 umw y = 3.32 um w = 3.42 um (0,0) + (0,1) w=1/e 2 w th = 3.90 um

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NF data fit - 980nm oxide aperture 1/e 2 = 45.5 1/e 2 = 47.6 1/e 2 = 124 w/a experiment: 0.36 - 0.38 theory : 0.47 (for a = 3.5um) I = 1.00 mA I = 1.10 mA I = 0.67 mA

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UCSB 1.55 m etched mesa VCSEL D. I. Babic, PhD Thesis "Double-fused long wavelength VCSELs", Un. California Santa Barbara, 1995. g = 846 ln(J/J tr )cm -1, J tr = 76.6A/cm 2

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Comparison with oxide aperture VCSEL* Circular aperture of equal power flux with square * K. Choquette et al, APL 70, 823 (1997); Hegarty et al, JOSA B 16, 2060 (1999)

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USC 980 nm oxide VCSEL A. E. Bond, P. D. Dapkus and J. D. O'Brien, "Design of low-loss single-mode VCSELs", IEEE Selected Topics in Quant. Electronics 5, 574 (1999).

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Comparison With Other Codes Sensitive Quantities –Threshold vs. aperture –Wavelength separation Diffraction & Scattering losses dominate at small apertures – PREVEU yields –Higher losses –Smaller waist (higher w P. Bienstman, R. G. Baets et al "Comparison of optical VCSEL models of the simulation of position dependent effects of thin oxide apertures” http://www.ele.kth.se/COST268/WG1/WGExcercise1.html

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Aperture location in standing wave LOWER THRESHOLD FOR NULL (NODE) PREVUE agrees with experiment Scattering & diffraction losses dominate guiding benefits Underestimating diffraction/scattering yields OPPOSITE trends nodeanti-node

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Dynamic multimode simulation Non-axisymmetric modes Axisymmetric density –if two polarizations are equally excited No grid, coupled ODEs at discrete radii Parametric coefficient dependence on T Green’s function for temperature Carrier diffusion: Hankel transform

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L-I curves: Motorola 780 nm etched mesa VCSEL I (mA) P tot (mW) No metalization Ring With metalization ring 0 0.5 1 1.5 2 2.5 0 0.5 1 1.5 2 2.5 1.522.533.544.555.5 3 mA 4.8 mA 7.9 mA 5.8 mA 10.0 mA 3.0 mA

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ULTRA-FAST SIMULATIONS USING PARAXIAL MODE EXPANSION Lateral diffraction, wide-angle scattering included –On-axis gain compensates spreading (steady-state) Mode waist determined from current aperture –Optimization between opposing trends : diffraction vs confinement Unified explanation of aperture size dependence –Blue shifting, threshold current, mode switching –Etched mesa: Higher threshold than oxide aperture Correct dependence on aperture location –Lower threshold for placement at node Agreement with experiments - more testing

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