Record transmissions achieved in photonic crystal waveguide components through

1. Record transmissions achieved in photonic crystal waveguide components through novel automatic optimisation techniques. Thomas Felici Dominic Gallagher Andrei Lavrinenko Tom Davies Thomas Felici Dominic Gallagher Andrei Lavrinenko Tom Davies

2. D=0.35 L=0.5 Rix=2.5 Rix=1 Our Aim

3. The field equations Fields governed by the source free Maxwell equations

4. The Field solver Robust: Must be capable of dealing with any taper shape Accurate: correctly model high contrast structures Use mode matching method

5. x y z Propagation constants Section modes Continuity at interfaces  elimination of intermediate coefficients S input output Mode Matching Method

6. Possible applications MMI tapers Mode converters Photonic crystals

7. Tot. power: Tot. power Choose working l=1.34mm Band Gap + line defect Vary wavelength...

8. l=1.43mm 2 modes excited l =1.34mm Only 1 mode excited Field plots in line defect (arbitrary input)

9. Design an “artificial” waveguide s.t. its fundamental mode has 100% transmission W W Choose w=0.351 Exciting the PC mode

10. Wavelength response 50% transmission Optimising the y-junction The initial structure...

11. D1 D2 L Setting up the optimisation

12. P L,D, or W Many local minima Problem! Many local minima holes can overlap and vanish different topological configurations

13. Evolution algorithms (statistical in nature) • Not guaranteed to find global optimum • Loose a lot of information on the way! Global optimisation Search whole function space in intelligent way

14. Deterministic global optimisation These are algorithms that systematically search the parameter space. • Splitting algorithms: • successively subdivide regions in systematic way. • Divide more quickly where optima are “more likely” to exist. Etc...

15. Monitoring interface define your own objective! Specify your independent variables... Connect them to any structure parameter

16. B B A A Optimisation results

17. Wavelength response Resonant transmission Optimal point A: transmission=99.8%! VERY BAD! D1= 0.38mm , D2 = 0.31mm , L= - 0.17mm

18. Wavelength response steering transmission Optimal point B: transmission=99.5%! MUCH better D1= 0.12mm , D2 = 0.47mm , L = 0.15mm

19. D L D L Bend optimisation

20. Best point Optimisation results

21. Wavelength response Resonant transmission Best shape : transmission=97%! FAIRLY good: variation = 8% L= 0.24mm , D = 0.47mm

22. Input from here Wavelength scan Bend + y junction transmission=97%! Pretty good!

23. D OFF OFF L D L Bend optimisation II Idea: try to find optimal steering transmissions

24. Optimisation results 2% variation 0.5% variation

25. The complete crystal 98% transmission, 1% variation!!!

26. Optimal taper design

27. Large losses …Argh .. Not very good! 56% transmission

28. Reduced losses 40 mm Could make it longer ... 95% transmission Too long!

29. Keep length fixed ... Maximise power output Deform shape ...

30. second order convergence, but requires derivatives per iteration. Could approximate these using finite differences: The local optimisation algorithm Use an iterative technique (the quasi-Newton method). …but this requires N field calculations per iteration!

31. We can derive analytic expressions for Taper region Change in permettivity due to shape deformation Electric field (solution of wave equations) Adjoint electric field (solution of adjoint wave equations) GOOD NEWS!  Only 2 field calculations per iteration!

32. Rix = 2.5 Rix = 1.0 Vary ends |C1 +|2 The first example: length 14um... P = 84%

33. P = 91% |C1 +|2 Much better ...

34. Vary taper length Replace with artificial input …and width Optimise offset Excite fundamental mode of input waveguide Design of optimal taper injector 5mm

35. Choose 9mm Optimal results for length range

36. 99% Field plot at length=9mm

37. The complete result!!! 97% transmission, variation 5%!!!

38. x1 x2 x3 xN IMPROVE TAPER FURTHER? Could also parametrize shape ... Optimization problem: find (x1 , x2 , ... , xN) that maximise P

39. Here was the original ... P = 56%

40. 15 nodes Here is the optimal design ... P = 88%

41. 39 nodes “Resonant” region Fwd/bwd power Using lots of nodes P = 97%

42. P P+dP Increasing the number of nodes... Optimisation problem becomes ill posed! E,F are bounded, so For “thin enough” de: dp  0

43. homing on optimum becomes more difficult: Power transmission becomes less sensitive to variation of any individual node Numerical instabilities - inverse problems Use regularisation techniques. Consequences Can improve transmission, but ... there could be more minima, On Shape Optimisation of Optical Waveguides Using Inverse Problem Techniques Thomas Felici and Heinz W. Engl, Industrial Mathematics Institute, Johannes Kepler Universität Linz

44. Vary height 3D simulations Air holes membrane with refractive index 2.5

45. FDTD 3D Probes just inside crystal Input waveguide

47. Computational performance Numerical space consists of 290x92x452 grid points ( 12 million points) we use 8 thousand time steps Hence we have 96 billion floating point operations per simulation!! CPU time: weeks??? - impossible due to the lack of memory (HP station at COM); days??? Feasible but very slow due to usage of hard disk memory(Pentium 4 PC); updated version:only3 hours and 55 minutes!! Speed is even less than in Example1: 142.7 ns per grid point RAM requirements: > 1 Gb - you need at least 2Gb of RAM for better performance; updated version:only 765 Mb !!

48. The end