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

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

Introduction

Plan of the Presentation Introduction

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

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

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

Modellisation

ModellisationIntroduction

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

CMT applied to 3x3 homogeneous array

Near fields Amplitude Far fields Intensities

In-phase mode

Out-of-phase mode

Limited far fieldpattern

ModellisationIntroduction

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

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

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

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

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

Optical Properties

Optical PropertiesModellisation

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

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

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

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

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

Questions

Thank you for your attention

QuestionsConclusion