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ULTRA-FAST VCSEL CAVITY SIMULATION USING PARAXIAL MODE EXPANSION

ULTRA-FAST VCSEL CAVITY SIMULATION USING PARAXIAL MODE EXPANSION. Spilios Riyopoulos SAIC McLean, VA 22102. Talk Outline. Case for paraxial mode expansion for VCSEL cavity modes Simulation study of generic VCSEL behavior Comparison with experiments. I. PARAXIAL MODE EXPANSION.

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ULTRA-FAST VCSEL CAVITY SIMULATION USING PARAXIAL MODE EXPANSION

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  1. ULTRA-FAST VCSEL CAVITY SIMULATION USINGPARAXIAL MODE EXPANSION Spilios Riyopoulos SAIC McLean, VA 22102

  2. Talk Outline • Case for paraxial mode expansionfor VCSEL cavity modes • Simulation study ofgeneric VCSEL behavior • Comparison with experiments

  3. 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 )

  4. Challenge: find these modes Buried heterostructure Etched mesa Oxide aperture - Multilayered structure - No Obvious lateral confinement: Radial Boundary Conditions ?

  5. Lateral Losses: Diffraction / Scattering

  6. 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

  7. Observationssupporting lateral mode losses • Large increase in threshold current density at small apertures • Jth should remain constant without lateral losses • Thermal index guiding ?Small Dn/DT~ 10-4 /K • Opposite than expected modal stability trends • Wide aperture VCSELs mode-switch near lasing threshold, despite lower DT (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

  8. 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 inreal 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

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

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

  11. Included Features • Reflection matrix R • Wing-clipping from finite mirror radii • Gain matrix G • Selective on-axis gain • Scattering matrix S • 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 ?

  12. Expansion Coefficients-Eigenvaluesvs mode waist w Coefficients sensitive to choice of waist Eigenvalue independent of choice of waist 7 x 3 mode basis

  13. Optimum Waist:Minimize Round-Trip Losses optimum waist: Gain overlap Loss Mirror Spill-over Loss Aperture Scattering Loss Increasing Diffraction Loss 4 3 0.0 Cavity Eigenmode 2 1.0 2.0 3.0 4.0 1 5.0 Per-Pass Loss G-L Mode 6.0 non-optimum: 4 a0/w0 3 0.0 1.0 minimum-loss optimum Cavity Eigenmode 2 2.0 3.0 4.0 1 5.0 G-L Mode 6.0

  14. Cavity mode representationby Pure Gauss-Laguerre modes Round-Trip Matrix • 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 Steady-state

  15. Etched Mesa VCSEL Results Current = 2.0 mA whor = 1.33 mm wvert = 1.24 mm Current = 4.3 mA whor = 1.25 mm wvert = 1.28 mm wexp = 1.31 mm 0.052 wtheory = 1.33 mm

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

  17. Cavity Eigenmodes • Represented by pure, optimizedwaist, 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

  18. II. SIMULATIONS of GENERIC VCSEL BEHAVIOR • G-L expansion algorithm used in mode finderPREVEU(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

  19. Mode waist vs. Aperture Proton implant Etched Mesa

  20. Mode waist vs. Aperture Etched Mesa Oxide Aperture

  21. Cavity Blue Shifting • Decreasing aperture /decreasing w • Increasing k= 1/w •  = cpkz ( 1 + k2 / 2kz2) • Blue Shift:l /l ~ 2 / (kzw)2 Etched Mesa Theory: 2.28 nm Observed: 2 000.30 nm

  22. Threshold Gain Proton Implant Etched Mesa Nominal go = (1-r)/2, r = DBR reflectivity

  23. Round Trip Losses Etched Mesa Oxide aperture • Diffraction and Scattering losses dominate at small apertures • Much higher than DBR reflectivity losses

  24. Modal Stability Differentiaton among various mode losses determines cavity stability Fundamental mode stability factor: S01= (R01-R00 )/ R00 Etched mesa Oxide Aperture • Small aperture stable / large unstable, relative to mode switching

  25. Gain Lensing • Increasing gain causeswaist to shrink • Shrinking waist increases kperp = 1/w for given kz • Lensing causes blue shifting • Opposite to red shift from cavity thermal expansion Proton Implant

  26. Comparison : GL vs. waveguide modes • Fundamental LP01almost identical to GL00for • However:two approaches givedifferent results for w

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

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

  29. Higher Mode Losses Proton Implant Etched mesa 1-R2 1-R2 Different Optimum waist for each cavity mode

  30. NF data fit - Wide aperture, proton implant w = 3.42 um (0,0) + (0,1) I/Ith = 1.05 w=1/e2 wth = 3.90 um wx = 3.42 um wy = 3.32 um

  31. NF data fit - 980nm oxide aperture I = 1.00 mA I = 0.67 mA 1/e2 = 45.5 1/e2 = 124 I = 1.10 mA w/a experiment: 0.36 - 0.38 theory : 0.47 (for a = 3.5um) 1/e2 = 47.6

  32. UCSB 1.55 mm etched mesa VCSEL  g = 846 ln(J/Jtr)cm-1, Jtr = 76.6A/cm2  D. I. Babic, PhD Thesis "Double-fused long wavelength VCSELs", Un. California Santa Barbara, 1995.

  33. 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)

  34. 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).

  35. 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 Dl~ l2/w2)  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

  36. 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 node anti-node

  37. 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

  38. With metalization ring 2.5 2.5 No metalization Ring 2 2 (mW) 1.5 1.5 tot 1 1 P 0.5 0.5 0 0 1.5 2 2.5 3 3.5 4 4.5 5 5.5 I (mA) L-I curves: Motorola 780 nm etched mesa VCSEL 3 mA 4.8 mA 7.9 mA 3.0 mA 5.8 mA 10.0 mA

  39. ULTRA-FAST SIMULATIONSUSING 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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