Operational Characteristics of

1
Department of Electronic and Electrical
Engineering
Operational Characteristics of Vertical Cavity
Surface Emitting Lasers (VCSELs) with laterally
non-uniform parameters T.C. Woo, J. Sarma, F.
Causa
Acknowledgements ORS and University of Bath
Scholarship
2
Department of Electronic and Electrical
Engineering
Operational Characteristics of Vertical Cavity
Surface Emitting Lasers (VCSELs) with laterally
non-uniform parameters
T.C. Woo, J. Sarma, F. Causa
Acknowledgements ORS and University of Bath
Scholarship
3
Present-Day VCSELs Oxide Apertured VCSELs
Schematic of Oxide Apertured VCSEL

• Largely homogeneous medium with no built-in
  lateral index guiding.
• Traditional modal field analysis not readily
  applicable.
• Optical field profile best described by
  diffraction-type analysis.
• Oxide aperture provides both current and optical
  confinement.
• Non-uniform refractive index profile ?(r,?,z) due
  to joule heating (cavity) and carrier (active
  layer).

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Optical Field Profile

• Scalar wave equation Field distribution in the
  laser cavity obtained by solving the following
  scalar wave equation

(1)
Total field expansion
(2)
(3)

• where exp(-jpz) Fast varying
  longitudinal phase term
• Laguerre-Gauss (LG) Functions ?km(r)

(4)

• Slowly varying F(r,?,z) expanded in terms of LG
  functions where Lkm(r) is the generalised
  Laguerre polynomial and Cmk the corresponding
  normalisation constant.
• LG functions are complete and discrete set.
• Suitable in describing
• weakly diffracting fields in homogeneous medium
• waveguide-like fields in laterally inhomogeneous
  medium.

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Field Propagation in Inhomogeneous Medium

• Representing axial(z) dependence of refractive
  index as piecewise constant segments
  ?(r,?,z)?q(r,?) where zq lt z lt zq1
• Field propagation in each segment described by
  paraxial wave equation

(5)
where (5) obtained using eqns (1)-(3), and
satisfies the paraxial approximation
(6)
Initial Value Problem

• A set of coupled ordinary differential equations
  for field propagation in each segment obtained
  using (2)-(5) and LG orthogonal condition

(7)
where ? r/w and ?i(?) contains ?q(r,?)
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LG Collocation Method

• Discretises radial (r) axis, rt taken as zeros of
  Nth order Laguerre polynomial

(8)

• Transforms integrations in (7) into matrix
  multiplications using Gaussian Quadrature Formula

(9)
Eigenmode Solutions for (Arbitrary) Index Guided
Structure ? Separated Variable Analysis

• For index guided structure, F(r,?,z) F(r,?) in
  (3).
• Hence (7) reduces to the following eigenvalue
  problem

(10)
where, now, p corresponds to the eigenvalue,
i.e. propagation constant of
corresponding eigenmode.
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Eigenmode Solution for Lateral Index Guiding
Structure

• Step Index Profile

?1 3.5, ?2 3.496, a 5um

• Exponential Index Profile

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Initial Value Problem Analysis Diffraction in
Homogeneous Medium

• The solution of paraxial wave equation (5) in
  homogeneous medium is Laguerre-Gauss Beam.
• Field profile obtained using LG function
  expansions compared with LG Beam at different
  propagation length L.

Matching of the diffracted field profile (IVP)
with the LG beam (LGB) in a homogeneous medium at
propagation distances of 10?m, 20?m and 30?m.
Propagation in Index Guided Medium

• Single mode step index fibre with ?? 0.01 and
  radius 0.5?m.

Comparison of computed field intensity profile at
different propagation distances with LP01 mode
(squares) of a step index fibre. Field matches
and stabilises after 20?m length.
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Resonant Cavity Finite Mirrors with Unity
Reflectivity

• Using Fox and Li approach to find the
  lowest-order resonator transverse mode.

Resonant cavity length L 1?m, end mirror radius
r 2?m and reflectivities R1 R2 1
Field intensity plot for initial(z0) and final
field(zL) for the last 10 round-trips(391-400)

• After 400 round-trips, field converges in form to
  the lowest order eigenmode of the resonator.
• Truncation of field at finite end mirrors shapes
  the resonant field profile.
• Larger mirror radius ? no. of round-trips ?

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Self-Consistent VCSEL Model

• Carrier Distribution N(r) in active layer
• Carrier Diffusion Equation
• LG function expansion for carrier density
• Boundary condition
• Optical Intensity P ? F(r,z) 2
• Hole-burning effect modifies carrier distribution

(11)
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Carrier Distribution in Active Layer

• Self-consistent carrier distribution solved in
  iterative manner using LG Collocation Method.
• Uniform current injection (J) with disc contact
  radius(A).

Plot of carrier distribution with varying contact
radius (A2-5um), diffusion coefficient
Dn5cm2s-1.
Plot of carrier distribution with varying
diffusion coefficient (Dn5-10 cm2s-1), disc
contact radius A5um
Constants J 10kA/cm2, Br 10-10 cm3s-1, N0
1x1016cm-3, d 0.5um
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Oxide Apertured VCSEL Characteristics
Schematic of oxide apertured VCSEL L 1.46um, d
0.25um, A 2um, R1 R2 0.9995, J 2kA/cm2
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Index Guided VCSEL Characteristics
Schematic of index guided VCSEL L 1.46um, d
0.25um, A 2um, R1 R2 0.9995, J 2kA/cm2 ,
?1 3.4, ?2 3.39
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Conclusions

• A versatile model for VCSEL was developed using
  Laguerre-Gauss Function Expansion.
• LG Collocation Method applied in eigenvalue
  problem for index guided structure gives very
  accurate results.
• Flexibility of LG functions in describing field
  diffraction/propagation in both homogeneous and
  inhomogeneous medium has been demonstrated.
• A self-consistent model for active VCSEL
  structure with or without lateral index guiding
  has been developed.