Presentation on theme: "Eigenmode Expansion Methods for"— Presentation transcript:
1Eigenmode Expansion Methods for Simulation of Silicon Photonics - Pros and ConsDominic F.G. GallagherThomas P. Felici
2EME = “Eigenmode Expansion” OutlineEME = “Eigenmode Expansion”Introduction to the EME method- basic theory- stepped structures- smoothly varying- periodicWhy use EME?Comparison to BPM and FDTDExamples
3ABmodes:“The fields at AB of any solution of Maxwell’s Equations may be written as a superposition of the modes of cross-section AB”.
4Basic Theory propagation constant electric field magnetic field forward amplitudebackward amplitudemode profilesexact solution of Maxwell’s Equationsbi-directional
5so 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
6Applying continuity relationships, eg: LHSforwardbackwardforwardbackwardRHSWith orthogonality relationships and a little maths we get an expression of the form:SJ is the scattering matrix of the joint
7Straight WaveguideABtrivial - the scattering matrix is diagonal:
8Example: S-matrix decomposition of an MMI coupler BCDE
9Evaluating S-matrix of device BCDEABCDEABCDEABCDE
10Smoothly Varying Elements Problem: modes are changing continuously along elementThus each cross-section requires a large computational effort to locate the set of modesThis was the major hurdle that has in the past restricted application of EMEFIMMPROP (our implementation of EME) has tackled these problems, enabling EME to be used realistically for the first time even for 3D tapering structures.
110 Order (Staircase Approximation) hnhnSet of local modes computed at discrete positions along taperSimple to implementtheoretically accurate as Nsliceà ¥errors grow as Nsliceà ¥ so practical limit on Nslicecan get spurious resonances between modes for long structures and small Nslice
121st Order (Linear Approximation) hnanalytic integrationhnSet of local modes computed atdiscrete positions along taperMore complex to implementgood accuracy for modest Nsliceerrors à 0 for modest Nsliceneed only small number of modes
14Periodic Structures A B A B A B A B A B P1 P1 P1 P1 P1 P2 P2 P3 S compute time µ log(Nperiod)i.e. almost as quick as a straight waveguide!S
15Bends transmission = (Sj)N Sj periodic structure bend is just periodic repeat of straight waveguide sections!
16Boundary ConditionsPEC & PMC (perfect electric/magnetic conductors) - useful for exploiting symmetriestransparent boundary conditionsPML’s - perfect matched layers (with PEC or PMC)Transparent BC’s are naturally formed at input and output of EME computationfinding 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.
17The Perfect Matched Layer (PML) d1 (real)d2=a+jbPECPECwaveguide core/claddingPMLImaginary thickness of PML absorbs light propagating towards boundaryas much absorption as we wish with no reflection at cladding/PML interface!guided modes not absorbed at all - nice!
18Effect of PML on guided and radiating modes guided modeunbound modePMLcore
20Why Use EME? EME Advantages 1. rigorous solution of Maxwell's Equations- rigorous as Nmode ® infinity- large delta-n
21EME 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
22Input 1 ® calculate ® Result 1 Input 2 ® calculate ® Result 2 EME Advantages3. 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 1Input 2 ® calculate ® Result 2Input 3 ® calculate ® Result 3EME/FIMMPROP:Calculate S matrixInput 1 ® Result 1Input 2 ® Result 2Input 3 ® Result 3
23EME Advantages4. 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:schange period - time: ……change offset - time: ….
24Design Curve Generation Traditional Tool:5 mins5 mins5 mins5 mins5 mins5 minsEME/FIMMPROP:5 mins3 mins
25EME Advantages 5. Wide-angle capability - wider angle - just add more modes - adapts to problem
26EME Advantages very thin layers - wide range of dimensions 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 dimensionsno problem for EME/FIMMPROP algorithm does not need to discretise the structure
27Right: plasmon between silver plates PlasmonicsRight: plasmon between silver platesEME is a rigorous Maxwell solution and can model many plasmonic devices (provided basis set is not too large).
28Why 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.
29Computation Time Compare computation time with BPM and FDTD Restrict our discussion to 2D - i.e. z and one lateral dimensionBoth BPM and FDTD require a finite difference grid sampling the structure, this same grid used for optical fieldEME does not need a structure grid (FMM Solver)Equivalent of grid in EME is the number of modesFor straight wg, EME particularly efficientPeriodic section - logarithmic timeArbitrary z-variations - all 3 methods have similar dependence with z-complexity/resolutionIn lateral dimension EME gets high spatial resolution almost for free. (c.f. BPM, non-uniform grid problems…)But lateral optical resolution - compute time µ N3the limiting factor in EME
34The MMI MMI ideal for EME - inherently a modal phenomenon 8 modes Illustrate design scan simulations for price of two!
35Tapered Fibre6 modes“how long for 98% efficiency?” - ideal question for EME
3650 simulations in scan 6.5s per simulation (in 3D!) 1 0.9 0.8 0.7 0.6 taper efficiency0.50.40.30.20.1200040006000800010000taper length (mm)50 simulations in scan6.5s per simulation (in 3D!)
37Effective indices of first 5 modes 1.4721.4701.4681.4661.464Mode eff. index1.4621.4601.4581.456102030405060708090100z-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
38Co-directional Coupler remember logarithmic with N periodsvery thin layer - no problem
39propagation at sub-wavelength scales, including metal features
40Ring Resonator Nmodes = 60 (for one ring) Nmodes much higher here - wide angle propagation.BPM gives nonsense
41Photonic Crystal Design Nmodes = 60 (for one ring)Nmodes much higher here - wide angle propagation.BPM gives nonsense
42VCSEL Modelling Rtop Rbot Resonance Condition top DBR mirror active layerlower DBR mirrorRtopRbot
43Showing the domains of applicability of FDTD, BPM and EME to varying delta-n and device length.
44Showing the domains of applicability of FDTD, BPM and EME to varying numerical apperture and cross-section size.
45BPM – Capability Scores AspectPerformanceScore/10SpeedFD-BPM scales linearly with area and can take fairly long steps in propagation direction-MemoryUsage scales linearly with c/s areaNABest with low NA simulations. Versions using Pade approximants can model a beam travelling at a large angle but still cannot deal well with light simultaneously travelling at a wide range of angles.4Delta-nBest with low delta-n simulations.5PolarisationSemi-vectorial versions work best. Still problems modelling mixed or rotating polarisation structures accuratelyLossy materialsCan model modest losses efficiently. Most versions cannot deal well with metals7ReflectionsSome success in implementing reflecting/bi-directional BPM but generally avoided due to slow speed and stability problems3Non-linearityFD-BPM can model non-linearity.9DispersiveBeing a frequency-domain algorithm this is easy10GeometriesThe BPM grid allows diffuse structures to be modelled easily. Problems modelling non-rectangular structures accurately on the rectangular gridABCsPMLs available and work well
46FDTD – Capability Scores AspectPerformanceScore/10SpeedScales as V (device volume) but grid size is small so not as good as BPM or EME for long devices.-MemoryNAOmni-directional algorithm is agnostic to direction of light – great when light is travelling in all directions10Delta-nRigorous Maxwell solver, happy with high delta-n, but slows down somewhat with high index.9PolarisationRigorous Maxwell Solver is polarisation agnosticLossy materialsCan model even metals accurately with a fine enough grid and small modifications to the algorithm.ReflectionsYes – easy and stable even when there are many reflecting interfaces.Non-linearityYes – non-linear algorithm relatively easy to doDispersiveHave to approximate the dispersion spectrum with one or more Lorentizans but exact fit to the spectrum over a wide wavelength is difficult and the algorithm also slows down.7GeometriesFine rectangular grid can do arbitrary geometries easily, though there are problems approximating diagonal metal surfaces8ABCsYes – very effective and easy to use
47EME – Capability Scores AspectPerformanceScore/10SpeedEME scales poorly with cross-section area – as A3 (A is c/s area). However it can efficiently model very long structures especially if their cross-section changes only slowly or occasionally. Periodic structures scale as log(number of periods) – so can compute efficiently. S-matrix approach allows a set of similar simulations to be done very quickly – parts of previous calculation can be reused.-MemoryMemory increases at rate between A2 and A3 (A is c/s area), but very efficient for long or periodic devices.NACan model wide-angle beams by increasing the number of modes in the basis set at expense of speed and memory.7Delta-nRigorous Maxwell Solver can accurately model high delta-n8PolarisationRigorous Maxwell Solver is polarisation agnostic10Lossy materialsDepends on mode solver used.ReflectionsYes – easy and stable even when there are many reflecting interfaces.Non-linearityDifficult – have to iterate, and then only modest non-linearity levels will converge3DispersiveBeing a frequency-domain algorithm this is easyGeometriesDepends on the mode solver used. Can use different structure discretisations in different cross-sections, so solver can better adapt to the geometry.ABCsDepends on the mode solver used. E.g. a finite-difference solver can be readily constructed to implement PMLs. However, PML’s are more difficult to use with EME than with BPM or FDTD.
48Conclusions Powerful compliment to BPM and FDTD Exceedingly efficient/fast for wide range of examplesRigorous Maxwell solver, bi-directional, wide anglemode data provides important insight into workings of device.