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**Eigenmode Expansion Methods for **

Simulation of Silicon Photonics - Pros and Cons Dominic F.G. Gallagher Thomas P. Felici

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**EME = “Eigenmode Expansion”**

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

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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”.

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**Basic Theory propagation constant electric field magnetic field**

forward amplitude backward amplitude mode profiles exact solution of Maxwell’s Equations bi-directional

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**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

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**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

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

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**Example: S-matrix decomposition of an MMI coupler**

B C D E

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**Evaluating S-matrix of device**

B C D E AB C DE ABC DE ABCDE

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**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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**0 Order (Staircase Approximation)**

hn hn Set of local modes computed at discrete positions along taper Simple to implement theoretically accurate as Nsliceà ¥ errors grow as Nsliceà ¥ so practical limit on Nslice can get spurious resonances between modes for long structures and small Nslice

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**1st Order (Linear Approximation)**

hn analytic integration hn 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

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**Showing zero order versus first order result**

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**Periodic 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

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**Bends transmission = (Sj)N Sj periodic structure**

bend is just periodic repeat of straight waveguide sections!

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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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**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!

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**Effect of PML on guided and radiating modes**

guided mode unbound mode PML core

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**PML’s with segmented waveguide**

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**Why Use EME? EME Advantages**

1. rigorous solution of Maxwell's Equations - rigorous as Nmode ® infinity - large delta-n

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**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

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**Input 1 ® calculate ® Result 1 Input 2 ® calculate ® Result 2 **

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

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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: ….

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**Design Curve Generation**

Traditional Tool: 5 mins 5 mins 5 mins 5 mins 5 mins 5 mins EME/FIMMPROP: 5 mins 3 mins

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**EME Advantages 5. Wide-angle capability**

- wider angle - just add more modes - adapts to problem

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**EME 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 dimensions no problem for EME/FIMMPROP algorithm does not need to discretise the structure

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**Right: plasmon between silver plates**

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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**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.

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**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

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

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Applications We present a variety of examples illustrating EME features

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SOI Waveguide Modes Ex Field Ey field

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**The MMI MMI ideal for EME - inherently a modal phenomenon 8 modes**

Illustrate design scan simulations for price of two!

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

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**50 simulations in scan 6.5s per simulation (in 3D!) 1 0.9 0.8 0.7 0.6**

taper efficiency 0.5 0.4 0.3 0.2 0.1 2000 4000 6000 8000 10000 taper length (mm) 50 simulations in scan 6.5s per simulation (in 3D!)

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**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 10 20 30 40 50 60 70 80 90 100 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

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**Co-directional Coupler**

remember logarithmic with N periods very thin layer - no problem

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**propagation at sub-wavelength scales, including metal features**

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**Ring Resonator Nmodes = 60 (for one ring)**

Nmodes much higher here - wide angle propagation. BPM gives nonsense

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**Photonic Crystal Design**

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

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**VCSEL Modelling Rtop Rbot Resonance Condition top DBR mirror**

active layer lower DBR mirror Rtop Rbot

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

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

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**BPM – Capability Scores**

Aspect Performance Score/10 Speed FD-BPM scales linearly with area and can take fairly long steps in propagation direction - Memory Usage scales linearly with c/s area NA Best 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-n Best with low delta-n simulations. 5 Polarisation Semi-vectorial versions work best. Still problems modelling mixed or rotating polarisation structures accurately Lossy materials Can model modest losses efficiently. Most versions cannot deal well with metals 7 Reflections Some success in implementing reflecting/bi-directional BPM but generally avoided due to slow speed and stability problems 3 Non-linearity FD-BPM can model non-linearity. 9 Dispersive Being a frequency-domain algorithm this is easy 10 Geometries The BPM grid allows diffuse structures to be modelled easily. Problems modelling non-rectangular structures accurately on the rectangular grid ABCs PMLs available and work well

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**FDTD – Capability Scores**

Aspect Performance Score/10 Speed Scales as V (device volume) but grid size is small so not as good as BPM or EME for long devices. - Memory NA Omni-directional algorithm is agnostic to direction of light – great when light is travelling in all directions 10 Delta-n Rigorous Maxwell solver, happy with high delta-n, but slows down somewhat with high index. 9 Polarisation Rigorous Maxwell Solver is polarisation agnostic Lossy materials Can model even metals accurately with a fine enough grid and small modifications to the algorithm. Reflections Yes – easy and stable even when there are many reflecting interfaces. Non-linearity Yes – non-linear algorithm relatively easy to do Dispersive Have 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 Geometries Fine rectangular grid can do arbitrary geometries easily, though there are problems approximating diagonal metal surfaces 8 ABCs Yes – very effective and easy to use

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**EME – Capability Scores**

Aspect Performance Score/10 Speed EME 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. - Memory Memory increases at rate between A2 and A3 (A is c/s area), but very efficient for long or periodic devices. NA Can model wide-angle beams by increasing the number of modes in the basis set at expense of speed and memory. 7 Delta-n Rigorous Maxwell Solver can accurately model high delta-n 8 Polarisation Rigorous Maxwell Solver is polarisation agnostic 10 Lossy materials Depends on mode solver used. Reflections Yes – easy and stable even when there are many reflecting interfaces. Non-linearity Difficult – have to iterate, and then only modest non-linearity levels will converge 3 Dispersive Being a frequency-domain algorithm this is easy Geometries Depends on the mode solver used. Can use different structure discretisations in different cross-sections, so solver can better adapt to the geometry. ABCs Depends 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.

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**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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