1. 1 Dominic F.G. Gallagher

2. 2 Outline Requirements for a PIC simulator Dividing the problem Modelling passive components using EME The circuit simulator Examples

3. 3 TFF SOA / EAM Bragg reflector Feedback loop passive elements Fibre I/O PIC Elements

4. 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. 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. 6 Quantum Well Gain Model for active elements Maxwell Solver for passive element analysis TDTW Algorithm (PICWave) Post-processing – spectral analysis etc FIR Filter Generator Gain Fitting Modelling Strategy Active PIC

7. 7 section Z-element external injection distributed feedback Interface losses dz Segmentation of a Device lateral segmentation into “cells” dz=v g.dt TDTW: Travelling Wave Time Domain Method

8. 8 TDTW: Advection Equations A B spontaneous emission detuninggaingrating feedback Remove fast term exp(j   t +/- j   z), giving: Consider forward and backward fields.

9. 9 A TDTW path network representing a PIC Propagate just mode amplitudes scattering matrix defines coupling at junctions

10. 10 TE00-mode TM00-mode Cross-coupling between waveguides TE00-mode Straight waveguide transmitting TE00 and TM00 modes Y-junction coupling two TE00 modes – one from each arm, into a TE00 and TE01 mode modes Two distinct types of section...

11. 11 mode1 Multi-mode Model mode2 mode3 mode4 Mode5 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. 12 TE00-mode TM00-mode TE00-mode TM00-mode Multi-mode Model Polarisation-dependent directional coupler model implemented Independent phase index, group index and mode loss for each polarisation Coupling defined as dA tm /dz = kappa.A te - constant along length Coupling between polarisations ignored in this version TDTW Model of coupler Directional Coupler supporting both TE00 and TM00

13. 13 Example - Polarisation- dependent MZI TE in TM in 150um length 100um length

14. 14 re-write advection equations in matrix form: grating feedback gain/loss term detuning from Bragg frequency noise sources (spontaneous emission)

15. 15 Matrix coefficients: Index, gain and loss grating effects determined by relationship between K AB and K BA.:

16. 16 Spontaneous emission Random number with inverse normal distribution Spontaneous coupling factor (geometric only - i.e. due to N.A of waveguide) carrier density spontaneous recombination lifetime i n - uncorrolated in time -> white noise source i n - uncorrolated in space - assume sampling interval dz is much longer than diffusion length.

17. 17 IIR Gain Filter A(t) B(t) Lorentzian wavelength response: IIR Filter Pseudo-Lorentzian response:

18. 18 Lorenzian approximation of actual gain spectrum

19. 19 increase Ne Harold solve heterostructure problem Curve Fitting PICWave g pk (N) g 2 (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. 20 Multi-Lorentzian Model

21. 21 Multi-Lorentzian Model – Original vs Fitted Spectra original spectra fitted spectra free spectral range increasing Ne

22. 22 Carrier Rate Equation photon number for z-element For one z-element we have: carrier volume photon generation rate (measure this from inspection of gain filter output) carrier density current density noise term assume quantum conservation  N=-  P

23. 23 Extension to 3D 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. 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. 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. 26 Importing EME Results into Circuit Model EME is a frequency domain method TDTW is time domain - must convert Use FIR filter (finite impulse response) a 1 (t) b(t) FIR Filter a 2 (t) FIR Filter +

27. 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. 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. 29 FIR Filter Response - Bragg Reflector original response FIR response Force s(  f) to be periodic between -1/2dt to +1/2dt

30. 30 Modelling a 60um diameter ring resonator

31. 31 Resonator - response

32. 32 Ring Resonator FDTD time: 14 hrs on a 3GHz P4 - 2D only! (Using Q-calculator) Circuit simulator: modelling the coupler (EME): few mins running circuit model (TDTW): few secs

33. 33 Optical 2R Regenerator Both passive and active elements - highly non-linear

34. 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. 35 The Sampled Grating DBR Laser

36. 36 4 Section SG-DBR - vary current in Grating A & B together

37. 37 4 Section SG-DBR - vary Grating A & B current and tuning current

38. 38 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. 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