ULTRA-FAST VCSEL CAVITY SIMULATION USING PARAXIAL MODE EXPANSION

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ULTRA-FAST VCSEL CAVITY SIMULATION USINGPARAXIAL
MODE EXPANSION

• Spilios Riyopoulos
• SAIC
• McLean, VA 22102

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Talk Outline

• Case for paraxial mode expansionfor VCSEL cavity
  modes
• Simulation study ofgeneric 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 heterostructure
Etched mesa
Oxide 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 ltlt 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
• 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

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Paraxial (GL) Mode ExpansionParaxial 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
(p0, m0)
(p0, m1)
(p0, m1)


x-polarized
y-polarized
(p1, m0)
(p0, m2)
(p0, m2)


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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
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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 ?

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Expansion Coefficients-Eigenvaluesvs 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
Gain overlap Loss Mirror Spill-over
Loss Aperture Scattering Loss
Increasing Diffraction Loss
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3
0.0
Cavity Eigenmode
1.0
2
2.0
3.0
4.0
1
5.0
G-L Mode
Per-Pass Loss
6.0
non-optimum
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a0/w0
3
0.0
1.0
minimum-loss optimum
Cavity Eigenmode
2
2.0
3.0
4.0
1
G-L Mode
5.0
6.0
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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
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Etched Mesa VCSEL Results
Current 4.3 mA whor 1.25 mm wvert 1.28 mm
Current 2.0 mA whor 1.33 mm wvert 1.24 mm
wexp 1.31 mm ? 0.052 wtheory 1.33 mm
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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
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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
  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
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Mode waist vs. Aperture
Proton implant
Etched Mesa
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Mode waist vs. Aperture
Etched Mesa
Oxide Aperture
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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
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Threshold Gain
Proton Implant
Etched Mesa
Nominal go (1-r)/2, r DBR reflectivity
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Round Trip Losses
Etched Mesa
Oxide aperture

• Diffraction and Scattering losses dominate at
  small apertures
• Much higher than DBR reflectivity losses

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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

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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
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Comparison GL vs. waveguide modes

• Fundamental LP01almost identical to GL00for
• Howevertwo approaches givedifferent results
  for w

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Adaptation Thermal index Guiding

• Parabolic Index ProfileG-L modes (again!)
• NO-DIFFRACTIONfixed 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 SIMULATIONusing FLASH(Fast Laser
  Algorithm for Semiconductor Heterostructures)

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Higher Mode Losses
Proton Implant
Etched mesa
1-R2
1-R2
Different Optimum waist for each cavity mode
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NF data fit - Wide aperture, proton implant
w 3.42 um (0,0) (0,1)
I/Ith 1.05
w1/e2
wth 3.90 um
wx 3.42 um
wy 3.32 um
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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
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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.
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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 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/WGExc
ercise1.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

node
anti-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
• Greens function for temperature
• Carrier diffusion Hankel transform

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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
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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