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

2. Talk Outline Case for paraxial mode expansion for VCSEL cavity modes Simulation study of generic 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 heterostructureEtched mesaOxide 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. 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

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 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

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

10. 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

11. 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 ?

12. Expansion Coefficients-Eigenvalues vs 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: 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

14. 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

15. 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

16. 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

17. 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

18. 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

19. Mode waist vs. Aperture Proton implantEtched Mesa

20. Mode waist vs. Aperture Etched MesaOxide Aperture

21. 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

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

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

24. 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

25. 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

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

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

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

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

30. 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

31. 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

32. 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

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    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

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 nodeanti-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. 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

39. 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