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“propagating ideas” Dominic F.G. Gallagher Thomas P. Felici

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

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“propagating ideas” 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”. A B

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“propagating ideas” magnetic field electric field Basic Theory mode profiles forward amplitude backward amplitude propagation constant exact solution of Maxwell’s Equations bi-directional

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“propagating ideas” a m (+) a m (-) b m (+) b m (-) so far only z-invariant what about a step change? 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

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“propagating ideas” forwardbackward forward LHS RHS Applying continuity relationships, eg: With orthogonality relationships and a little maths we get an expression of the form: S J is the scattering matrix of the joint

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“propagating ideas” Straight Waveguide trivial - the scattering matrix is diagonal: A B

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“propagating ideas” ABCDE Example: S-matrix decomposition of an MMI coupler

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“propagating ideas” ABCDE ABDE ABC DE ABCDE C Evaluating S-matrix of device

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“propagating ideas” 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.

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“propagating ideas” Set of local modes computed at discrete positions along taper h 0 Order (Staircase Approximation) h Simple to implement theoretically accurate as Nslice errors grow as Nslice so practical limit on Nslice can get spurious resonances between modes for long structures and small Nslice

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“propagating ideas” analytic integration Set of local modes computed at discrete positions along taper 1st Order (Linear Approximation) h h More complex to implement good accuracy for modest Nslice errors 0 for modest Nslice need only small number of modes

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“propagating ideas” Showing zero order versus first order result

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“propagating ideas” Periodic Structures BABABABABA P1 P2 P3 S compute time log(Nperiod) i.e. almost as quick as a straight waveguide!

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“propagating ideas” Sj transmission = (Sj) N periodic structure Bends bend is just periodic repeat of straight waveguide sections!

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“propagating ideas” 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.

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“propagating ideas” PEC d1 (real) d2=a+jb PML waveguide core/cladding The Perfect Matched Layer (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!

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“propagating ideas” guided mode unbound mode PML core Effect of PML on guided and radiating modes

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“propagating ideas” PML’s with segmented waveguide

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“propagating ideas” Why Use EME? 1. rigorous solution of Maxwell's Equations - rigorous as Nmode infinity - large delta-n EME Advantages

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“propagating ideas” 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 EME Advantages

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“propagating ideas” 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. EME Advantages Other methods: Input 1 calculate Result 1 Input 2 calculate Result 2 Input 3 calculate Result 3 Calculate S matrix Input 1 Result 1 Input 2 Result 2 Input 3 Result 3 EME/FIMMPROP:

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“propagating ideas” 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. EME Advantages Initial evaluation time: ~ 2:54 m:s change period - time: …… change offset - time: ….

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“propagating ideas” 5 mins 3 mins Traditional Tool: EME/FIMMPROP: Design Curve Generation

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“propagating ideas” 5. Wide-angle capability - wider angle - just add more modes - adapts to problem EME Advantages

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

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“propagating ideas” 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).

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“propagating ideas” Why Use EME? 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. Disadvantages

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“propagating ideas” 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 N 3 Computation Time Compare computation time with BPM and FDTD Restrict our discussion to 2D - i.e. z and one lateral dimension the limiting factor in EME

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“propagating ideas”

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Memory Requirements Very efficient as a function of z-resolution - N 0 scaling for straight or periodic

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“propagating ideas” Applications We present a variety of examples illustrating EME features

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“propagating ideas” SOI Waveguide Modes Ex Field Ey field

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“propagating ideas” The MMI MMI ideal for EME - inherently a modal phenomenon 8 modes Illustrate design scan simulations for price of two!

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“propagating ideas” Tapered Fibre 6 modes “how long for 98% efficiency?” - ideal question for EME

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“propagating ideas” taper length ( m) 50 simulations in scan 6.5s per simulation (in 3D!)

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“propagating ideas” Effective indices of first 5 modes 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

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“propagating ideas” Co-directional Coupler remember logarithmic with N periods very thin layer - no problem

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“propagating ideas” propagation at sub- wavelength scales, including metal features

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“propagating ideas” Ring Resonator Nmodes = 60 (for one ring) Nmodes much higher here - wide angle propagation. BPM gives nonsense

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“propagating ideas” Photonic Crystal Design Nmodes = 60 (for one ring) Nmodes much higher here - wide angle propagation. BPM gives nonsense

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“propagating ideas” VCSEL Modelling Resonance Condition active layer top DBR mirror lower DBR mirror R top R bot

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“propagating ideas” Showing the domains of applicability of FDTD, BPM and EME to varying delta-n and device length.

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“propagating ideas” Showing the domains of applicability of FDTD, BPM and EME to varying numerical apperture and cross-section size.

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“propagating ideas” BPM – Capability Scores AspectPerformanceScore/10 SpeedFD-BPM scales linearly with area and can take fairly long steps in propagation direction - MemoryUsage scales linearly with c/s area- NABest 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. 4 Delta-nBest with low delta-n simulations.5 PolarisationSemi-vectorial versions work best. Still problems modelling mixed or rotating polarisation structures accurately 5 Lossy materialsCan model modest losses efficiently. Most versions cannot deal well with metals 7 ReflectionsSome success in implementing reflecting/bi-directional BPM but generally avoided due to slow speed and stability problems 3 Non-linearityFD-BPM can model non-linearity.9 DispersiveBeing a frequency-domain algorithm this is easy10 GeometriesThe BPM grid allows diffuse structures to be modelled easily. Problems modelling non-rectangular structures accurately on the rectangular grid 7 ABCsPMLs available and work well9

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“propagating ideas” AspectPerformanceScore/10 SpeedScales as V (device volume) but grid size is small so not as good as BPM or EME for long devices. - MemoryScales as V (device volume) but grid size is small so not as good as BPM or EME for long devices. - NAOmni-directional algorithm is agnostic to direction of light – great when light is travelling in all directions 10 Delta-nRigorous Maxwell solver, happy with high delta-n, but slows down somewhat with high index. 9 PolarisationRigorous Maxwell Solver is polarisation agnostic10 Lossy materials Can 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.10 Non-linearityYes – non-linear algorithm relatively easy to do9 DispersiveHave 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. 7 GeometriesFine rectangular grid can do arbitrary geometries easily, though there are problems approximating diagonal metal surfaces 8 ABCsYes – very effective and easy to use9 FDTD – Capability Scores

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“propagating ideas” EME – Capability Scores AspectPerformanceScore/10 SpeedEME scales poorly with cross-section area – as A 3 (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 A 2 and A 3 (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. 7 Delta-nRigorous Maxwell Solver can accurately model high delta-n8 PolarisationRigorous Maxwell Solver is polarisation agnostic10 Lossy materials Depends on mode solver used.7 ReflectionsYes – easy and stable even when there are many reflecting interfaces.10 Non-linearityDifficult – have to iterate, and then only modest non-linearity levels will converge 3 DispersiveBeing a frequency-domain algorithm this is easy10 GeometriesDepends on the mode solver used. Can use different structure discretisations in different cross-sections, so solver can better adapt to the geometry. 7 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. 7

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“propagating ideas” 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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