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Eigenmode Expansion Methods for Simulation of Silicon Photonics - Pros and Cons

Eigenmode Expansion Methods for Simulation of Silicon Photonics - Pros and Cons. Dominic F.G. Gallagher Thomas P. Felici. Outline. EME = “ E igen m ode E xpansion”. Introduction to the EME method - basic theory - stepped structures - smoothly varying - periodic Why use EME?

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Eigenmode Expansion Methods for Simulation of Silicon Photonics - Pros and Cons

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  1. Eigenmode Expansion Methods for Simulation of Silicon Photonics - Pros and Cons Dominic F.G. Gallagher Thomas P. Felici

  2. Outline EME = “Eigenmode Expansion” • Introduction to the EME method • - basic theory • - stepped structures • - smoothly varying • - periodic • Why use EME? • Comparison to BPM and FDTD • Examples

  3. A B modes: “The fields at AB of any solution of Maxwell’s Equations may be written as a superposition of the modes of cross-section AB”.

  4. Basic Theory propagation constant electric field magnetic field forward amplitude backward amplitude mode profiles • exact solution of Maxwell’s Equations • bi-directional

  5. so far only z-invariant • what about a step change? am(+) bm(+) bm(-) am(-) Maxwell's Equations gives us continuity conditions for the fields, e.g. the tangential electric fields must be equal on each side of the interface

  6. Applying continuity relationships, eg: LHS forward backward forward backward RHS With orthogonality relationships and a little maths we get an expression of the form: SJ is the scattering matrix of the joint

  7. Straight Waveguide A B trivial - the scattering matrix is diagonal:

  8. Example: S-matrix decomposition of an MMI coupler A B C D E

  9. Evaluating S-matrix of device A B C D E C AB DE DE ABC ABCDE

  10. Smoothly Varying Elements • Problem: modes are changing continuously along element • Thus each cross-section requires a large computational effort to locate the set of modes • This was the major hurdle that has in the past restricted application of EME • FIMMPROP (our implementation of EME) has tackled these problems, enabling EME to be used realistically for the first time even for 3D tapering structures.

  11. 0 Order (Staircase Approximation) hn hn Set of local modes computed at discrete positions along taper • Simple to implement • theoretically accurate as Nsliceॠ• errors grow as Nsliceॠso practical limit on Nslice • can get spurious resonances between modes for long structures and small Nslice

  12. 1st Order (Linear Approximation) hn hn analytic integration Set of local modes computed at discrete positions along taper • More complex to implement • good accuracy for modest Nslice • errors à 0 for modest Nslice • need only small number of modes

  13. Showing zero order versus first order result

  14. Periodic Structures A B A B A B A B A B P1 P1 P1 P1 P1 P2 P2 P3 • compute timeµ log(Nperiod) • i.e. almost as quick as a straight waveguide! S

  15. Bends transmission = (Sj)N Sj periodic structure • bend is just periodic repeat of straight waveguide sections!

  16. Boundary Conditions • PEC & PMC (perfect electric/magnetic conductors) - useful for exploiting symmetries • transparent boundary conditions • PML’s - perfect matched layers (with PEC or PMC) • Transparent BC’s are naturally formed at input and output of EME computation • finding eigenmodes with true transparent boundary conditions leads to leaky modes - leaky modes cause problems with completeness of basis set. • PML much better suited for finding modes for EME than leaky modes - completeness better achieved.

  17. The Perfect Matched Layer (PML) d1 (real) d2=a+jb PEC PEC waveguide core/cladding PML • Imaginary thickness of PML absorbs light propagating towards boundary • as much absorption as we wish with no reflection at cladding/PML interface! • guided modes not absorbed at all - nice!

  18. guided mode unbound mode PML PML core Effect of PML on guided and radiating modes

  19. PML’s with segmented waveguide

  20. Why Use EME? EME Advantages 1. rigorous solution of Maxwell's Equations - rigorous as Nmode ®infinity - large delta-n

  21. EME Advantages EME Advantages 2. inherently bi-directional. - unconditionally stable since always express (outputs) = S.(inputs) - takes any number of reflections into account - not iterative - even highly resonant cavities - mirror coatings, multi-layer

  22. EME Advantages 3. The S-matrix technique provides the solution for all inputs! - component-like framework where the S-matrix of one component may be re-used in many different contexts. Other methods: • Input 1 ® calculate ® Result 1 • Input 2 ® calculate ® Result 2 • Input 3 ® calculate ® Result 3 EME/FIMMPROP: • Calculate S matrix • Input 1 ® Result 1 • Input 2 ® Result 2 • Input 3 ® Result 3

  23. EME Advantages 4. Hierarchical algorithm permits re-use, accelerating computation of sets of similar structures. When one part of a device is altered only the S-matrix of that part needs to be re-computed. • Initial evaluation time: ~ 2:54 m:s • change period - time: …… • change offset - time: ….

  24. Design Curve Generation Traditional Tool: 5 mins 5 mins 5 mins 5 mins 5 mins 5 mins EME/FIMMPROP: 5 mins 3 mins

  25. EME Advantages 5. Wide-angle capability - wider angle - just add more modes - adapts to problem

  26. EME Advantages 6. The optical resolution and the structure resolution may be different. - c.f. BPM (stability problems with non-uniform grid) • very thin layers - wide range of dimensions • no problem for EME/FIMMPROP algorithm does not need to discretise the structure

  27. Plasmonics Right: plasmon between silver plates EME is a rigorous Maxwell solution and can model many plasmonic devices (provided basis set is not too large).

  28. Why Use EME? Disadvantages • Structures with very large cross-section are less suitable for the method since computational time typically scales in a cubic fashion with e.g. cross-section width. • The algorithms are much more complex to write - for example it is very difficult to ensure that a mode has not been missed from the basis set. • EME is not a "black box" technique - the operator must make some effort to understand the method to use it to his best advantage.

  29. Computation Time • Compare computation time with BPM and FDTD • Restrict our discussion to 2D - i.e. z and one lateral dimension • Both BPM and FDTD require a finite difference grid sampling the structure, this same grid used for optical field • EME does not need a structure grid (FMM Solver) • Equivalent of grid in EME is the number of modes • For straight wg, EME particularly efficient • Periodic section - logarithmic time • Arbitrary z-variations - all 3 methods have similar dependence with z-complexity/resolution • In lateral dimension EME gets high spatial resolution almost for free. (c.f. BPM, non-uniform grid problems…) • But lateral optical resolution - compute time µ N3 the limiting factor in EME

  30. Memory Requirements • Very efficient as a function of z-resolution - N0 scaling for straight or periodic

  31. Applications • We present a variety of examples illustrating EME features

  32. SOI Waveguide Modes Ey field Ex Field

  33. The MMI • MMI ideal for EME - inherently a modal phenomenon • 8 modes • Illustrate design scan - 100 simulations for price of two!

  34. Tapered Fibre • 6 modes • “how long for 98% efficiency?” - ideal question for EME

  35. 1 0.9 0.8 0.7 0.6 taper efficiency 0.5 0.4 0.3 0.2 0.1 0 0 2000 4000 6000 8000 10000 taper length (mm) • 50 simulations in scan • 6.5s per simulation (in 3D!)

  36. 0 50 60 70 80 90 100 10 20 30 40 Effective indices of first 5 modes 1.472 1.470 1.468 1.466 1.464 Mode eff. index 1.462 1.460 1.458 1.456 z-position (um) • Strong coupling occurs when the effective indices are close - telling us where the trouble spots of the device are. • Powerful diagnostic - tells designer where to improve the design

  37. Co-directional Coupler • remember logarithmic with N periods • very thin layer - no problem

  38. propagation at sub-wavelength scales, including metal features

  39. Ring Resonator • Nmodes = 60 (for one ring) • Nmodes much higher here - wide angle propagation. • BPM gives nonsense

  40. Photonic Crystal Design • Nmodes = 60 (for one ring) • Nmodes much higher here - wide angle propagation. • BPM gives nonsense

  41. Rtop Rbot VCSEL Modelling • Resonance Condition top DBR mirror active layer lower DBR mirror

  42. Showing the domains of applicability of FDTD, BPM and EME to varying delta-n and device length.

  43. Showing the domains of applicability of FDTD, BPM and EME to varying numerical apperture and cross-section size.

  44. BPM – Capability Scores

  45. FDTD – Capability Scores

  46. EME – Capability Scores

  47. Conclusions • Powerful compliment to BPM and FDTD • Exceedingly efficient/fast for wide range of examples • Rigorous Maxwell solver, bi-directional, wide angle • mode data provides important insight into workings of device.

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