1

Dominic F.G. Gallagher

2

Outline

• Requirements for a PIC simulator• Dividing the problem• Modelling passive components using EME• The circuit simulator• Examples

3

TFF

SOA / EAM

Bragg reflectorFeedback loop

passive elements

Fibre I/O

PIC Elements

4

Fabry Perot laser

DFB Laser

Tuneable DFB

External Cavity laser with FBG

Sampled Grating Tuneable Laser

Ring cavity laser

Branched Tuneable Laser

Laser Geometries

5

Requirements for a PIC Simulator

• Must be able to model passive elements correctly - tapers, y-junctions, MMIs, AWGs

• Capable of modelling active elements correctly - SOAs, modulators, laser diodes

• Hybrids• Capable of modelling reflections - bidirectional• Capable of retaining any physical processes that

interact - e.g. effect of diffusion on dynamics• Capable of computing time response• Capable of multi-wavelength modelling• All of this must be able to scale to large circuits!

6

Quantum Well GainModel for active

elements

Maxwell Solver forpassive element

analysis

TDTW Algorithm(PICWave)

Post-processing –spectral analysis etc

FIR FilterGenerator

GainFitting

Modelling Strategy Active PIC

7

sectionsection

Z-element

external injection

distributed feedback

Interface losses

dz

Segmentation of a Device

lateral segmentation into “cells”

dz=vg.dt

TDTW: Travelling Wave Time Domain Method

8

TDTW: Advection Equations

)().(..1

)().(..1

eBg

eAg

NFBjgAjz

B

t

B

v

NFAjgBjz

A

t

A

v

AB

spontaneous emission

detuninggaingrating feedback

Remove fast term exp(jt +/- jz) , giving:

Consider forward and backward fields.

9

A TDTW path network representing a PIC

Propagate just mode amplitudes

scattering matrix defines coupling at junctions

10

TE00-modeTM00-mode

Cross-couplingbetweenwaveguides

TE00-mode

Straight waveguide transmitting TE00and TM00 modes

Y-junction coupling two TE00 modes –one from each arm, into a TE00 andTE01 mode modes

Two distinct types of section...

11

mode1

Multi-mode Model

mode2mode3mode4Mode5

• The TDTW engine can now propagate multiple modes, eg of different polarisation.

• Independent phase index and mode loss for each mode

• For now, group index is same for each mode - changing group index requires different segmentation since vg = dz / dt

12

TE00-modeTM00-mode

TE00-modeTM00-mode

Multi-mode Model

• Polarisation-dependent directional coupler model implemented

• Independent phase index, group index and mode loss for each polarisation

• Coupling defined as dAtm/dz = kappa.Ate - constant along length

• Coupling between polarisations ignored in this version

TDTW Model of coupler

Directional Couplersupporting both TE00 and TM00

13

Example - Polarisation-dependent MZI

TE in

TM in

150um length

100um length

14

bP

aP

F

F

tzzB

tzA

mm

mmzjg

ttzB

ttzzA

,

,

2221

1211

),(

),(..exp

),(

),(

re-write advection equations in matrix form:

grating feedback

gain/loss term

detuning from Bragg frequency

noise sources

(spontaneous emission)

15

*212122

*121211

21

12

1

1

mmm

mmm

zm

zm

AB

BA

Matrix coefficients:

glrBA

glrAB

**

*

Index, gain and loss grating effects determined by relationship between KAB and KBA.:

16

),(2

ztitN

hF nNA

r

eP

Spontaneous emission

Random number with inverse normal distribution

Spontaneous coupling factor (geometric only - i.e. due to N.A of waveguide)

carrier densityspontaneous recombination lifetime

• in - uncorrolated in time -> white noise source

• in - uncorrolated in space - assume sampling interval dz is much longer than diffusion length.

17

IIR Gain Filter

A(t)B(t)

02

0 1/)()( ggLorentzian wavelength response:

IIR Filter

MM

M

Kgg

21

sin21

cos

21

cos)()(

222

2

0Pseudo-Lorentzian response:

18

Material gain

Lorenzian gainfilter response

photon energy

Lorenzian approximation of actual gain spectrum

19

increase Ne

Harold

solve heterostructure problem

Curve Fitting

PICWave

gpk(N)

g2 (N)

pk (N)

spon

...

Harold/PICWave Interaction

• solve heterostructure just a few times at start of simulation.

• maintain speed of PicWave

• out-of-bound detectors ensure simulation stays within fit range.

gain spectra

20

IIR-1

IIR-2

IIR-3

+

z-element

Multi-Lorentzian Model

-400

-200

0

200

400

600

800

1000

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Frequency offset/FSR

gain (a.u)

Lorentzian-2

Lorentzian-1

Lorentzian-3

Lorentzian-4

Lorentzian-1

-800

-600

-400

-200

0

200

400

600

800

1000

-0.5 -0.4 -0.3 -0.2 -0.1 0 0.1 0.2 0.3 0.4 0.5

Frequency offset/FSR

fitted

measured

measured

fitted

gain (a.u.)

21

Multi-Lorentzian Model – Original vs Fitted Spectra

original spectra

fitted spectra

free spectral range

increasing Ne

22

Carrier Rate Equation

Nees

e

stimN

e Fdq

J

N

N

dt

dP

Vdt

dN

.)(

1 #

photon number for z-elementFor one z-element we have:

carrier volume

photon generation rate (measure this from inspection of gain filter output)

carrier densitycurrent density

noise term

assume quantum conservation N=-P

23

Extension to 3D

lateral diffusion

Active layers

contacts

computation cells

current flows

In TDTW method, extension to include lateral carrier profile Ne(x,y) is simple. Instead of 1 carrier density in each z-element we have nx.ny discrete densities.

24

Integration with Frequency Domain Models

Two main choices:•BPM - beam propagation method•EME - eigenmode expansion

For circuit modelling EME is better:• Bidirectional - takes account of all reflections• Scattering matrix - integrates well with circuit model

TDTW cannot predict e.g. the scattering loss of a y-junction - this must be computed with a more rigorous EM solver.

25

FIMMPROP

Compute lambda-dependent scattering matrix using rigorous Maxwell solver

FIR filter generation

PICWave

EME (FIMMPROP)/PICWave Interaction

• Rigorous analysis of waveguide components - tapers, y-junction, MMI etc done in FIMMPROP.

• PICWave generates an FIR (time domain) filter corresponding to the s-parameter spectra.

S-parameter spectra

26

Importing EME Results into Circuit Model

EME is a frequency domain methodTDTW is time domain- must convert

Use FIR filter (finite impulse response)

a1(t)

b(t)FIR Filter

ip

N

kipjpipkjp kitacitb

1,, ][][

a2(t)FIR Filter

+

27

1. Input s() from EME

2. Compute FIR filter coefficients

3. Launch impulse into filter

4. Measure impulse response function - FFT -> spectrum

FIR Filter Response - Bragg Reflector

28

FIR Filter Response - Bragg Reflector

original response

FIR response

Simple FIR filter works poorly - s(f) is not periodic in FSR of TDTW

29

FIR Filter Response - Bragg Reflector

original response

FIR response

Force s(f) to be periodic between -1/2dt to +1/2dt

30

Transmission

Drop

C oup lersm odelled byEM so lver.

Modelling a 60um diameter ring resonator

31

Resonator - response

32

Ring Resonator

FDTD time:14 hrs on a 3GHz P4 - 2D only! (Using Q-calculator)

Circuit simulator:modelling the coupler (EME): few minsrunning circuit model (TDTW): few secs

33

Optical 2R Regenerator

SOA

SOA

data 2

control 1 data out 2

A

B

C

D

Both passive and active elements - highly non-linear

34

Optical 2R Regenerator

2GB/s NRZ bit pattern - optical input

Input: 5:1 on/off

But: noise

Output: 25:1 on/off

Gain: 25x

35

The Sampled Grating DBR Laser

Grating A Grating BGainSection

PhaseSection

36

4 Section SG-DBR - vary current in Grating A & B together

37

4 Section SG-DBR - vary Grating A & B current and tuning current

38

0

0.5

1

1.5

2

2.5

3

3.5

0 2 4 6 8 10

lateral position (um)

carr

ier

den

sity

(x1

e18/

cm3)

Optical 2R Regenerator

Transverse Carrier Density

Start of SOA 3900 A/cm2

End of SOA 4900 A/cm2

=> Can take account of lots of physics if designed carefully

39

Conclusions

• presented strategies for modelling large circuits including both active and passive elements

• TDTW can be easily coupled with Maxwell Solvers using FIR filters

• Can create very high speed algorithm while maintaining a lot of physics if system is designed carefully

• Have developed a product PICWave to implement this circuit simulator

• EME ideal method for integration with circuit model