1. “propagating ideas”

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

3. “propagating ideas” Fields governed by the source free Maxwell equations The field equations

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

5. “propagating ideas” Mode Matching Method x yz S input output Section modes Propagation constants Continuity at interfaces  elimination of intermediate coefficients

6. “propagating ideas” MMI tapers Mode converters Photonic crystals Possible applications

7. “propagating ideas” Band Gap + line defect Choose working =1.34  m Vary wavelength... Tot. power: Tot. power

8. “propagating ideas” Field plots in line defect (arbitrary input) =1.34  m Only 1 mode excited =1.43  m 2 modes excited

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

10. “propagating ideas” Optimising the y-junction The initial structure... Wavelength response 50% transmission

11. “propagating ideas” Setting up the optimisation D1 D2 L

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

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

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

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

16. “propagating ideas” Optimisation results A A B B

17. “propagating ideas” D1= 0.38  m, D2 = 0.31  m, L= - 0.17  m Optimal point A: transmission=99.8%! Wavelength response VERY BAD! Resonant transmission

18. “propagating ideas” Optimal point B: transmission=99.5%! D1= 0.12  m, D2 = 0.47  m, L = 0.15  m Wavelength response MUCH better steering transmission

19. “propagating ideas” Bend optimisation D D L L

20. “propagating ideas” Optimisation results Best point

21. “propagating ideas” Best shape : transmission=97%! L= 0.24  m, D = 0.47  m Wavelength response Resonant transmission FAIRLY good: variation = 8%

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

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

24. “propagating ideas” Optimisation results 2% variation 0.5% variation

25. “propagating ideas” The complete crystal 98% transmission, 1% variation!!!

26. “propagating ideas” Optimal taper design

27. “propagating ideas” Large losses …Argh.. Not very good! 56% transmission

28. “propagating ideas” Could make it longer... Reduced losses 40  m Too long! 95% transmission

29. “propagating ideas” Keep length fixed... Maximise power output Deform shape...

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

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

32. “propagating ideas” The first example: length 14um... Rix = 2.5 Rix = 1.0 P = 84% Vary ends |C 1 + | 2

33. “propagating ideas” Much better... P = 91% |C 1 + | 2

34. “propagating ideas” Design of optimal taper injector Replace with artificial input …and width Vary taper length Excite fundamental mode of input waveguide Optimise offset 5m5m

35. “propagating ideas” Choose 9  m Optimal results for length range

36. “propagating ideas” Field plot at length=9  m 99%

37. “propagating ideas” The complete result!!! 97% transmission, variation 5%!!!

38. “propagating ideas” IMPROVE TAPER FURTHER? x1x1 x2x2 x3x3 xNxN Optimization problem: find (x 1, x 2,..., x N ) that maximise P Could also parametrize shape...

39. “propagating ideas” Here was the original... P = 56%

40. “propagating ideas” Here is the optimal design... 15 nodes P = 88%

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

42. “propagating ideas” Increasing the number of nodes... Optimisation problem becomes ill posed! P P+  P For “thin enough”  :  p  0 E,F are bounded, so

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

44. “propagating ideas” 3D simulations Air holes membrane with refractive index 2.5 Vary height

45. “propagating ideas” FDTD 3D Probes just inside crystal Input waveguide

46. “propagating ideas”

47. RAM requirements: > 1 Gb - you need at least 2Gb of RAM for better performance; updated version: only 765 Mb !! 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

48. “propagating ideas”