eigenmode analysis of vertical-cavity lasershiptmair/org/macsinet/slides/...• semiconductor stack...

$: Eigenmode Analysis of Vertical-Cavity Lasershiptmair/org/macsinet/slides/...• Semiconductor Stack Low Refractive Index Contrast Many Layers (30..40) • How to Find Fundamental and$
ZürichIntegrated Systems LaboratoryOptoelectronics Modeling Group Zürich

Eigenmode Analysis ofVertical-Cavity Lasers

Andreas Witzig, Matthias Streiff, Wolfgang Fichtner

Integrated Systems LaboratorySwiss Federal Institute of Technology,

ETH Zurich Switzerland


Overview

• Motivation

• Solution of the Optical Modesin Laser Cavities➜ Discretization of Vectorial Helmholtz Equation

by linear and quadratic Nédélec Elements

➜ Solution of the Algebraic Eigenvalue Problemby Jacobi-Davidson QZ Algorithm

• Spectral Portrait➜ Identification of Optical Eigenmodes

➜ Target Values

➜ Tracking of Optical Modes

• Computational Effort➜ Linear vs. Quadratic Edge-Elements

• Conclusion / Outlook


d / 2

λ-cavity

15 Bragg pairs

20 Bragg pairs

GaAs Substrate

metal contact

Si3N4 isolation

ApertureMetal Contact

Si3N4 isolation25 Layer Pairs

λ-Cavity, Active Region

35 Layer Pairs

Vertical-Cavity Surface-Emitting Lasers (VCSELs)

• Application➜ Promising Light Source for Optical

Communication Systems

➜ Semiconductor Laser (PiN-Diode)

➜ Light Generation by RadiativeRecombination of Electrons and Holes

➜ Optical Resonator (Layered Medium)

• Challenge➜ Calculate Optical Eigenmodes

➜ Calculate Resonance Frequency

➜ Simulation of Opto-Electro-ThermalDevice Characteristics

10µm

Ligh

t Out

put

Standing Wave

Metal Contact

GaAs Substrate


Problem Formulation:

Calculation of Optical Eigenmodes

➜ Frequency-Domain Solution ofHomogeneous Maxwell ’s Equations(=Helmholtz Equations)

➜ Radiation Boundary Condition

➜ Non-Hermitian ComplexGeneralized Eigenproblem

➜ Real Part of Eigenvalue= Resonance Frequency(Free-Space Resonance Wavelength λ)

➜ Imaginary Part of Eigenvalue= Field Decay Constant(related to Photon Lifetime τph)

Optical Field

20µm

+ Radiation Boundary Condition


Temperature

Optical Field

20µm

Peak Change in

by ≈ 2%Refractive Index

Application:

Coupled Opto-Electro-ThermalVCSEL Simulation

Refractive Index Depends on Temperatureand Carrier Density

➜ Need Volume Discretization (Finite-Ele-ment Type, Full-Vectorial)

Solution of Optical Cavity Problem is Iteratedwith Electronic Equations

➜ Need a Fast Solver (in Compensation,allow to use a lot of memory)

Current Density


Discretization

• Discretization by N édélec Elements➜ Unstructured Grid (typ. order 500 ’000)

➜ Full-Vectorial Complex Optical Field

➜ "curl"-Conforming Discretization

➜ Linear and Quadratic Elements

• Solution in 2D or "2.5D"➜ Body-of-Revolution

Fourier Series Expansion in φ-Direction

ρ

edgesnodes

ϕρ

z

z


r�

�[�r]�1

� r ��

�� k20["r]� = 0:

["r] =2

4� 0 0

0 � 0

0 0 �

�1

35[�r] =

24

� 0 0

0 � 0

0 0 ��1

35

Solution ofThe Open Cavity Problem

• Radiation to In finity➜ Essential Result of the Simulation

(Directly proportional to Photon Life Time,Laser Threshold, Filter Linewidth, etc)

• Perfectly Matched Layer (PML) Concept➜ Arti ficial Material for Absorption

➜ Outside of PML, use Dirichlet BoundaryConditions.

Very good Results, Problem with Layered Media PML


Solution ofAlgebraic Eigenproblem

• Jacobi-Davidson QZ Algorithm➜ Low-Dim. Search Subspace (e.g. order 10)

➜ High-Order Correction Equation (BiCGstab)

➜ Preconditioner: Matrix Inversion,using Direct Solver PARDISO

• In Practice...➜ Find a few inner Eigenmodes (1..5)

➜ Need Good Target Value (know λ within 0.1%)

➜ Need a Lot of Memory for Computation (6GB)

➜ Quick Method to Find Eigenvalue (20min)


101

102

103

104

105

106

10−10

10−8

10−6

10−4

10−2

rela

tive

err

or

matrix order

linear

quadratic

Reference I:Metal Box Resonator

• Analytical Solution Known

• Comparison of Linear and QuadraticElements

• Symmetry similar to VCSEL(Body-of-Revolution, Fourier Expansion of 3DFields)


Reference II:Dielectric Pill

Resonator

• Cylinder with n = 5.9in Vacuum

• Includes Radiation➜ Good Test for Absorbing

Boundaries

➜ Non-Hermitian Matrices, Com-plex Eigenvalue

• Symmetry similar to VCSEL➜ Body-of-Revolution, Fourier

Expansion

Real Part of the Electric Field

m = 1 m = 0


2676 2680 2684 2688 2692

54

55

56

57

linear

quadratic

1

2

31

23Reference [1]

Resonance Wavelength [nm]

Ph

oto

n L

ifet

ime

[ps]

Reference II:Dielectric Pill

Resonator

• Accuracy Dependent onMesh Refinement:

• 1: 10 Cells/λ (8x8 in Pill)linear: order 9’000quad.: order 36’000



[1] Tsuji et. al. "On the Complex Resonant Frequency of Open Dielectric Resonators",IEEE Trans. on Microwave Theory and Techniques, Vol. 31, No. 5 May 1983


Example I:Airpost VCSEL

• Semiconductor Stack➜ Low Refractive Index Contrast

➜ Many Layers (30..40)

• How to Find Fundamentaland Higher Order Modes?➜ Spectral Portrait

Fundamental Mode


730 740 750Resonance Wavelength [nm]

0

4

8

12

16

20

Pho

ton

Life

Tim

e [p

s]

m=0m=1m=2m=3m=4m=5


• Spectral Portrait➜ Mode Selection??

➜ Mode with Smallest ImaginaryPart and Large Overlap withActive Region...

• Note: Different ExpansionNumber m

Spectral Portrait



• Spectral Portrait➜ Mode Selection??

• Mode with SmallestImaginary Part andLarge Overlap withActive Region...

Spectral Portrait


730 740 750Resonance Wavelength [nm]

0

4

8

12

16

20P

hoto

n Li

fe T

ime

[ps]

Spectral Portrait

Inve

rse

Ph

oto

n L

ife

Tim

e [1

/ps]



7.56 7.57 7.58 7.59x 10

−7

0

500

1000

1500

2000

2500

lambda

A

B C

D E

alph

a

8

9

11

11

11 9

9

6

6

6

7

4

4

4

9

4

3

3

3

7

4

2

1

5

2

3

4

4

2

2

5 3

3 3

4

2

4

4

5

4

5

5 4

3

3

4 3

4

3

3

4

3

2

2

3

6

5

5

4

4

4

4

6

4

4

3

6

3

7

7

4

4

4

4

3

2

2

3

A

BC

D E

2.5

2.0

1.5

1.0

0.5

0

744.6 744.8 745.0 745.2


How to Find Complete Spectrum?

• "Brute Force": Scan Target Value➜ Faster Convergence for Better Target Value

(Number of Jacobi Correction Equations Solved)

➜ New Preconditioner for Each Eigenvalue?

• Repeated Solution of Perturbed Problem➜ Track Eigenvalue

Inve

rse

Ph

oto

n L

ife

Tim

e [1

/ps]


Example II:Tunable Filter

• Air-Gap Mirrors➜ High Refractive Index Contrast

➜ Need Only 3..4 Layers

• Fabrication:➜ Selectively Ethching Sacrifice Layers

➜ Air Bridges Hold the Layers

• Concept Allows Tuning

• Filter: Incident Light from Top➜ Resonances in the Cavity Cause Sharp

Spectral Dip in the Reflected Beam


Example II:Tunable Filter

• Charge Center Layers to ControlAir-Gap Distance

• Critical Design Issue: Bending of theLayers

• Sensitivity Analysis by Simulation

Tuning

Bend

< 20nm


1..20nm


1580 1585 1590 1595 1600 1605

1500

1550

1600

1650

1700

1750al

ph

a [1

/m]

lambda [nm]

20 nm

0 nm

Convex Mirrors

➜ Trajectory for 0nm..20nm bend

➜ Mode Relations are Preserved

➜ Better Confinement (Smaller Decay Const.)

➜ Shift to Smaller Resonance Wavelengths


1580 1590 1600 1610 16201000

1500

2000

2500

3000

3500

alp

ha

[1/m

]

lambda [nm]

20 nm

0 nm

previous slide

Concave Mirrors

➜ Trajectory for 0nm..20nm bend

➜ Resonances

➜ Shift to Higher Resonance Wavelengths


Example III:VCSEL Simulation

• Separation between Longitudinal andTransverse Direction Impossible➜ Diffraction at Oxide Confinement or Mesa

Structure

➜ Remedy for Rotationally Symmetric Struc-tures: Expand by Fourier Series (Body-of-Revolution)

• Solution of Large Eigenvalue Problem➜ Finite-Element Discretization

➜ Need to Resolve Bragg Mirrors

➜ Jacobi-Davidson Method

20µm

ZürichIntegrated Systems LaboratoryOptoelectronics Modeling Group ZürichZürichIntegrated Systems LaboratoryOptoelectronics Modeling Group Zürich

1.

2.3.

4.

Zoom-In:

• Standing Waves➜ High Field Intensities

in the Centre

• Traveling Waves1.Laser Output

2.Bragg Losses

3.Radiating Waves

4.Guided Surface Waves

• Diffraction at theOxide Confinement➜ Important to Calculate Pho-

ton Lifetime and ThresholdCurrent


Typical Simulation Tasks

• Device Optimization➜ Mode Discrimination➜ High Output Power➜ Good Fiber Coupling

• Design of Top Contact➜ Aperture Diameter➜ Material

• However...➜ Results Questionable for

Optics-Only Simulation➜ Need To Simulate Fully-Coupled

Opto-Electro-Thermal Physics

2µm

3µm

4µm

5µm

Aperture

0.6 1.81.2 2.4

Distance from Centre [µm]

Op

tica

l Po

wer

Den

sity

[a.

u.]

3.0 3.6 4.20.0


1 2 3 4 5 6 70

0.02

0.04

0.06

0.08

0.1

0.12

0.14

dielectric aperture radius [um]

(QH

E11

−QH

E21

)/Q

HE

11

Simulation Results (Optical): Mode DiscriminationFurukawa Electric 850 nm VCSEL with oxide aperture:Optimal single-mode performance (cold cavity)

➜ Adjust dielectric aperture to obtain maximum separation betweenfundamental HE11 and next higher order HE21 optical mode.

HE11 (m=1)

HE21 (m=2)

1 2 3 4 5 6 70

2

4

6

8

10

12

dielectric aperture radius [um]

mo

dal

loss

[1/

cm]

total loss HE11top radiation HE11total loss HE21top radiation HE21


Electronics Optics

n, p: Electron/Hole DensityV: Electrostatic PotentialT: Lattice TemperatureG, R: Optical Gain/Recombinationf: Lasing FrequencyA: Optical Wave AmplitudeS: PhotonNumberK: Quantum Mechanical Wavefunction

Poisson EquationV, n, p, T, K

Quantum Carrier TreatmentK,n,p,V

Continuity Equationn, p, V, G, R, S, A, T

Temperature EquationT, V, n, p

Complex Wave EquationA, n, p, f, G

Photon Rate EquationS, G, R, f, n, p

Gain ModelG, n, p, f, T

Self-Consistent Solution


Example III:Opto-Electro-Thermal

VCSEL Simulation

• Separate meshes and interpolateelectro-thermal: ~ 10’000 FEoptical: ~ 200’000 FE

• Newton scheme to solve coupled non-linear set of electro-thermal equations.Parallelised direct solver (PARDISO) tosolve linear equations in Newtonscheme

• Photon Rate Equation Accounts forOptics

• Photon Life Time (Imaginary Part ofEigenvalue) as a Variable in the Equa-tion

r � (�r�) = �q�p� n +N+D �N�

A�

�r � S = H + ctot@tT

r � jn = q (R + @tn)

�r � jp = q (R + @tp)

S = ��thrT

jn = �q (�nnr��Dnrn+ �nnPnrT )

jp = �q (�ppr�+Dprp+ �ppPprT )

ddt

S�(t) = (G� � L�)S�(t) +Rsp


2.0x

1e2

2.0x

1e4

2.0x

1e3

[A/c

m2]

300

350

400

[K]

PML

PML

PM

L

contact

top DBR

bottom DBR

GaAs substrate

air

oxide confinement

Simulation Results (Coupled)

Electro-thermo-optical simulation:

➜ Distribution of current density

➜ temperature and optical intensity at 16.3 mA.


current [mA]0 5 10 15 20

op

tica

l po

wer

[m

W]

0

5

10

15

0.0

1.0

2.0

ano

de

volt

age

[V]

anode voltageoptical power

880

885

900

890

895

300

310

320

330

340

350

0 5 10 15 20

wav

elen

gth

[n

m]

aver

age

cavi

ty t

emp

erat

ure

[K

]

current [mA]

average cavity temperaturewavelength

Simulation Results (Coupled)Electro-thermo-optical simulation:

➜ I/P_opt and I/V curve

➜ Wavelength tuning of HE11 mode

➜ average cavity temperature versus laser current.


0 1 2 3 4 5radius [um]

op

tica

l in

ten

sity

(a.

u.)

0 [mA]5 [mA]

15 [mA]10 [mA]

20 [mA]

0 5 10 15radius [um]

loca

l op

tica

l mat

eria

l gai

n [

1/cm

]

2000

-1000

-500

0

500

1000

15001 [mA]3 [mA]7 [mA]

Simulation Results (Coupled)Electro-thermo-optical simulation:

➜ Normalised optical intensity of HE11 mode in active region versus current (thermal lensing)

➜ Optical material gain in active region versus current (spatial hole burning).


All benchmarks performed on Compaq AlphaServer ES45 1250 MHz (1 CPU)

Numerical Performance

Simulation Results (Optical):

• # optical vertices: 90’000

• job size: ~ 2 GBytes

• Total CPU Time Usage: ~3 m

Simulation Results (Coupled):

• # optical vertices: 90’000# electro-thermal vertices: 7’729

• job size: ~ 2.2 GBytes

• Total CPU Time Usage: ~8 h➜ 98 bias points➜ run time without self-consistent

optics: ~5 h


Summary

• Solve Helmholtz Equation➜ Open Resonator Requires Absorbing Boundaries

➜ Non-Hermitian Problem, Complex Eigenvalue

➜ Discretization: Linear vs. Quadratic

• Spectral Portrait➜ Difficult to Find Lasing Modes Automatically

➜ Track Modes for Perturbed Problem

• Applications➜ Tunable Air-Gap Filter

➜ Coupled Opto-Electro-Thermal VCSEL Simulation


Conclusions / Outlook

• Eigenmode Analysis of Microcavity Lasers➜ Perfectly Matched Layer (PML) Boundary Condi-

tion is a Good Choice for Absorbing Boundary

➜ Need Full-Wave Solution to Account for Diffrac-tion Losses and Calculate Photon Life Time Accu-rately

➜ Trade-Off Between Accuracy and Problem Size

• Improve Preconditioner➜ Possibility: First Order Elements for Precondi-

tioner and Second Order Elements for Jacobi Cor-rection Equation (Requires Hierarchical FiniteElements)


Acknowledgements

• Oscar Chinellato, Peter ArbenzInstitute of Scientific Computing, ETH Zurich

University of Kassel

• Avalon PhotonicsHardware Partner

• ISE Integrated Systems EngineeringSoftware Partner

• Swiss Funding Agency➜ Commission for Technology and Innovation (CTI)

➜ TOP NANO21 Programme.

eigenmode analysis of vertical-cavity lasershiptmair/org/macsinet/slides/...• semiconductor stack...

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