2D- Photonic Crystals based on Vertical Cavity Surface Emitting Lasers (VCSELs) arrays
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2D-Photonic Crystals based on Vertical Cavity Surface Emitting Lasers (VCSELs) arrays
Presentation for the Photonic Crystal Course, June 2009
Elodie LamotheIng. Microtechn. Dipl. EPFPhD. Student in Photonic SchoolLPN EPF Lausanne
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Plan of the PresentationIntroduction
Vertical Cavity Surface Emitting Laser (VCSEL)Photonic crystal based on VCSEL
Modellisation of VCSELs-arrayFormalism of coupled mode theoryFabry-Perot cavity modelEquivalent 3D photonic crystal model
Optical PropertiesHomogeneous structuresHeterostructure and mode confinement Coupling between two confine modes
Conclusion
Plan of the Presentation
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Introduction
Plan of the Presentation Introduction
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VCSEL Description
p-DBR (AlGaAs/GaAs)
active region(InGaAs)
p-contact
n-DBR(AlGaAs/GaAs)
n-contact
1) Two Distributited Bragg Reflectors (DBR) define the cavity2) Light is amplified by stimulated emission in the active region3) Emission of the ligth through the lower DBR (n-DBR)
hole
electron
Plan of the Presentation Introduction
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Photonic Crystals based on VCSELPhotonic crystals are obtained by modulating the reflectivity of the top DBR reflector: => RAu>RCr
Optical coupling between adjacent microcavities via diffraction of the optical field at the edges of the pixels
• Such structures incoporate gain and losses• Optical Bloch waves are stimulated at each
lattice site
Active Photonic crystal
CrAup-contact
n-contact
Optical coupling
2D-Photonic crystal
Au
Cr
Plan of the Presentation Introduction
H. Pier and al., Nature (London), 407,880-883, 2000
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Condition on the wavelength
)sin(
)sin(
kk p
Bragg condition
Usual photonic crystals Photonic crystal based on VCSELs
kaa 2
nmm
96065
Photonic crystal based on VCSEL have lattice constants significantly exceeding the optical wavelength.
Condition
ÞOnly the transversal component of the wavevector undergoes Bragg conditionÞ |kp| << |kz|
Plan of the Presentation Introduction
k
pk
zk
z
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Modellisation
ModellisationIntroduction
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Couple Mode Theory (CMT)1) Consider an isolated waveguide (WG)
=> slight perturbation of the fields at the WG
3) Each solitary WG is placed in a periodic lattice
=> weak coupling between adjacent WGs
2) Electric field distribution is obtained by solving Helmolz equation for each WG=> Set of orthogonal eigenmodes
propagation constant
WGn
4) Total field : SUPERMODE = superposition of the separated orthogonal WG modes
ModellisationIntroduction
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CMT applied to 3x3 homogeneous array
Near fields Amplitude Far fields Intensities
In-phase mode
Out-of-phase mode
Limited far fieldpattern
ModellisationIntroduction
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Fabry-Perot cavity approach1) 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
propagator : ),K(functionty reflectivi :)(
field optical:)(
12
1
2
rrrrV
Rayleight-Sommerfeld integral is solved iteratively by numerical computation.
A
rdrVrrKrrV 111212 )(),()()(
A. E. Siegman, Lasers, University Science, Mil Valley, CA, 1986
ModellisationIntroduction
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Equivalent 3D-Photonic Crystal (1)
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, 2003D. L. Boiko and al., Opt. Express,12, 2597-2602, 2004
ModellisationIntroduction
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Equivalent 3D-Photonic Crystal (2)
)()(ˆpkmkmpkm rvrvH
|||| zp kk
1R
paraxial approximation
small reflectivity modulation
2D-Hamiltonien eigenvalue problem in transversal plan
Model of the VCSEL-based photonic crystal
Brillouin zone of theequivalent 3D photonic crystal
Master Equation
G. Guerrero, PhD Thesis, Thèse N°2837, EPFL, Lausanne, Switzerland, 2003D. L. Boiko and al., Opt. Express,12, 2597-2602, 2004
Z
T
ModellisationIntroduction
k
pk
zk
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Band Diagram
Lowest loss mode T5
pRkipkmppkm ervRrv
)()(
),,(5 zT kk
ipTkmppTkmervRrv )()(
55
694.0
5.4
2
2
aFF
m
plane-in xy vector lattice pR
984.0 , 991.0960
CrAu RRnm
Phase difference between complex reflection coeffecient Au and Cr
L.D.A. Lundeberg and al., IEEE J. Top. Quant. Elec., 13,5, 2007
Parameters Photon energy Mode Losses
=> No Bandgap for photon energy
Real part of the eigenvalue
rad41016.9
Imaginary part of the eigenvalue
=> Bandgap in terms of losses
out-of-phase relationship between adjacent lattice site
Bloch theorem
ModellisationIntroduction
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Lowest Loss Mode:Simulations of the Optical Field
Near Field
Far FieldFrauhenoferdiffraction
Amplitude
Phase
Geometrical ModelNumerical Solution of Master Equation
out-of-phase coupling between VCSELs
pi phase shift between adjacent VCSELs
L.D.A. Lundeberg, Thèse N°3911, EPFL, Lausanne, Switzerland
ModellisationIntroduction
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Optical Properties
Optical PropertiesModellisation
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Homogeneous Structures
Near Field Patterns Far Field Patterns
m50 10
mm
5constant lattice44 size pixel
pixels square 1010latticer Rectangula
2
pixels hexagonal 1111lattice Hexagonal
4 lobes
out-of-phase lasing mode
SpontaneousEmission
StimulatedEmission
StimulatedEmission
H. Pier and al., Nature (London), 407,880-883, 2000
Optical PropertiesModellisation
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Modes Confinement
Numerical Calculation Measurement
• Mode confinement can be achieved by creating photonic crystal heterostructure• Domain with lower fill factor FF presents higher loss
out-of-phase relationship between adjacent VCSEL elements is maintain
Confinement Structure
Þ Rectangular shape PhC island with higher FF in a sea of lower FF material confines supermodes
L.D.A. Lundeberg and al., App. Phys. Lett.,87, 241120, 2005L.D.A. Lundeberg and al., IEEE J. Top. Quant. Elec., 13,5, 2007
Optical PropertiesModellisation
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Coupled IslandsNumerical Analysis
Coupling between two islands
Þ Bonding state |B>Þ Anti-bonding state |A>
|B>
|A>
Structure
Near Field Far Field
L.D.A. Lundeberg and al., App. Phys. Lett.,87, 241120, 2005
Far field intensity distribution of one principal lobe along θx
FFisland = 0.694FFsea = 0.25λ=960nmΛ=6μm
Optical PropertiesModellisation
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Coupled IslandsMeasurementModal loss considerations
|B> : This phase relationship is maintained => lowest loss
Bloch part of the wave function gives an out-of-phase relationship between adjacent pixels:
|A>: This phase relationship is altered => higher loss
Bright fringe in the centre of the lobes=> Bonding state |B> is lasing
L.D.A. Lundeberg and al., App. Phys. Lett.,87, 241120, 2005
Optical PropertiesModellisation
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Conclusion 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 supermodeA coupling between two confined supermodes can be achieved=> This coupling results in a bonding state.
ConclusionOptical Properties
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Questions
Thank you for your attention
QuestionsConclusion