Transcript Title of Presentation

Presentation for the Photonic Crystal Course, June 2009
Elodie Lamothe
Ing. Microtechn. Dipl. EPF
PhD. Student in Photonic School
LPN EPF Lausanne
Introduction
Vertical Cavity Surface Emitting Laser (VCSEL)
Photonic crystal based on VCSEL
Modellisation of VCSELs-array
Formalism of coupled mode theory
Fabry-Perot cavity model
Equivalent 3D photonic crystal model
Optical Properties
Homogeneous structures
Heterostructure and mode confinement
Coupling between two confine modes
Conclusion
Plan of the Presentation
Plan of the Presentation
Introduction
hole
p-contact
p-DBR (AlGaAs/GaAs)
active region(InGaAs)
n-DBR(AlGaAs/GaAs)
electron
n-contact
1)
2)
3)
Two Distributited Bragg Reflectors (DBR) define the cavity
Light is amplified by stimulated emission in the active region
Emission of the ligth through the lower DBR (n-DBR)
Plan of the Presentation
Introduction
Photonic crystals are obtained by modulating the reflectivity of the top DBR
reflector: => RAu>RCr
2D-Photonic crystal
p-contact
Au
Cr
Active Photonic crystal
• Such structures incoporate gain and losses
• Optical Bloch waves are stimulated at each
lattice site
Optical coupling
n-contact
Optical coupling between adjacent microcavities
via diffraction of the optical field at the edges of
the pixels
Au
Cr
H. Pier and al., Nature (London), 407,880-883, 2000
Plan of the Presentation
Introduction
Usual photonic crystals
Condition
Photonic crystal based on VCSELs
2
a a
k
z


k


kz

kp
Bragg condition
  5  6m
  
  960nm 
Photonic crystal based on VCSEL have lattice constants significantly exceeding the optical
wavelength.
Only the transversal component of the
wavevector undergoes Bragg condition
 |kp| << |kz|
Plan of the Presentation
Introduction
k p     k sin( )  


 sin( )
Introduction
Modellisation
1) Consider an isolated waveguide (WG)
nWG
 propagation constant
2) Electric field distribution is obtained by solving Helmolz equation for each WG
=> Set of orthogonal eigenmodes
3) Each solitary WG is placed in a periodic lattice
=> slight perturbation of the fields at the WG
=> weak coupling between adjacent WGs
4) Total field :
SUPERMODE = superposition of the separated orthogonal WG modes
Introduction
Modellisation
Near fields Amplitude
Far fields Intensities
Out-of-phase
mode
In-phase mode
Introduction
Modellisation
Limited far field
pattern
1) Replace the bottom DBR and the top DBR by
mirrors with modulated reflection
2) Consider the VCSELs-array as a Fabry-Perot
cavity with an effective length Leff
Cavity description by Rayleight-Sommerfeld diffraction integral


 


V (r2 )    (r1 ) K (r2 , r1 )V (r1 )dr1
A

V (r2 ) : optical field

 (r1 ) : reflectivi ty function
 
K( r2 , r1 ) : propagator
Rayleight-Sommerfeld integral is solved iteratively by numerical computation.
A. E. Siegman, Lasers, University Science, Mil Valley, CA, 1986
Introduction
Modellisation
…
1) VCSEL cavity is unfolded => an effective 2L-periodicity along z-axis is induced.
2) The reflections at the DBR are replaced by thin equivalent layers
3) The resulting 3D-PhC is analyzed using Orthogonal Plane Wave expansion method
G. Guerrero, PhD Thesis, Thèse N°2837, EPFL, Lausanne, Switzerland, 2003
D. L. Boiko and al., Opt. Express,12, 2597-2602, 2004
Introduction
Modellisation
Brillouin zone of the
equivalent 3D photonic crystal
Model of the VCSEL-based
photonic crystal
T
Z
Master Equation
paraxial approximation
| k p || k z |

k

kz

kp
small reflectivity modulation R  1
Introduction
Modellisation
2D-Hamiltonien eigenvalue
problem in transversal plan
 
 
Hˆ vmk (rp )  mk vmk (rp )
G. Guerrero, PhD Thesis, Thèse N°2837, EPFL, Lausanne, Switzerland, 2003
D. L. Boiko and al., Opt. Express,12, 2597-2602, 2004
Parameters
Photon energy
Mode Losses
Imaginary part of the eigenvalue
Real part of the eigenvalue
  4.5m
FF 
=> No Bandgap for photon
energy
2
a
 0.694
2
  960nm
RAu  0.991 , RCr  0.984
Phase difference between complex
reflection coeffecient Au and Cr
4
  9.16 10 rad
=> Bandgap in terms of losses
Lowest loss mode T5
Bloch theorem

   
   ikR p
vmk (rp  R p )  vmk (rp )e

 
kT5  ( , , k z )
 

R p lattice vector in xy - plane
   


vmk (rp  R p )  vmk (rp )ei
T5
T5
out-of-phase relationship
between adjacent lattice site
L.D.A. Lundeberg and al., IEEE J. Top. Quant. Elec., 13,5, 2007
Introduction
Modellisation
Near Field
Geometrical Model
Numerical Solution
of Master Equation
Far Field
Frauhenofer
diffraction
Amplitude
Phase
pi phase shift between adjacent VCSELs
out-of-phase coupling between VCSELs
L.D.A. Lundeberg, Thèse N°3911, EPFL, Lausanne, Switzerland
Introduction
Modellisation
Modellisation
Optical Properties
Near Field Patterns
Spontaneous
Emission
Stimulated
Emission
Far Field Patterns
Stimulated
Emission
4 lobes
Rectangular lattice
10  10 square pixels
out-of-phase
lasing mode
pixel size  4  4 m 2
lattice constant   5m
Hexagonal lattice
1111 hexagonal pixels
50 m
10
H. Pier and al., Nature (London), 407,880-883, 2000
Modellisation
Optical Properties
Confinement Structure
• Mode confinement can be achieved by creating photonic crystal
heterostructure
• Domain with lower fill factor FF presents higher loss
 Rectangular shape PhC island with higher FF in a sea of lower FF material
confines supermodes
Numerical Calculation
Measurement
out-of-phase relationship between adjacent
VCSEL elements is maintain
L.D.A. Lundeberg and al., App. Phys. Lett.,87, 241120, 2005
L.D.A. Lundeberg and al., IEEE J. Top. Quant. Elec., 13,5, 2007
Modellisation
Optical Properties
Structure
Numerical Analysis
Near Field
FFisland = 0.694
FFsea = 0.25
λ=960nm
Λ=6μm
Far field intensity distribution of
one principal lobe along θx
Far Field
|B>
|A>
Coupling between two islands
 Bonding state
|B>
 Anti-bonding state |A>
L.D.A. Lundeberg and al., App. Phys. Lett.,87, 241120, 2005
Modellisation
Optical Properties
Modal loss considerations
Bloch part of the wave function gives an out-ofphase relationship between adjacent pixels:
|B> : This phase relationship is maintained => lowest loss
Measurement
Bright fringe in the centre of the
lobes
=> Bonding state |B> is lasing
|A>: This phase relationship is altered => higher loss
L.D.A. Lundeberg and al., App. Phys. Lett.,87, 241120, 2005
Modellisation
Optical Properties
2D-Photonic Crystal can be realized using VCSEL-array
The lasing supermode predicted by simulation and
experiments presents an out-of-phase relationship
between each pixel
Well designed heterostructures can confine the
supermode
A coupling between two confined supermodes can be
achieved
=> This coupling results in a bonding state.
Optical Properties
Conclusion
Thank you for your attention
Conclusion
Questions