Eigenmode Expansion Methods for Simulation of Silicon Photonics - Pros and Cons Dominic F.G. Gallagher Thomas P. Felici
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
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”.
Basic Theory propagation constant electric field magnetic field forward amplitude backward amplitude mode profiles • exact solution of Maxwell’s Equations • bi-directional
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
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
Straight Waveguide A B trivial - the scattering matrix is diagonal:
Evaluating S-matrix of device A B C D E C AB DE DE ABC ABCDE
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.
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
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
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
Bends transmission = (Sj)N Sj periodic structure • bend is just periodic repeat of straight waveguide sections!
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.
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!
guided mode unbound mode PML PML core Effect of PML on guided and radiating modes
Why Use EME? EME Advantages 1. rigorous solution of Maxwell's Equations - rigorous as Nmode ®infinity - large delta-n
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
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
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: ….
Design Curve Generation Traditional Tool: 5 mins 5 mins 5 mins 5 mins 5 mins 5 mins EME/FIMMPROP: 5 mins 3 mins
EME Advantages 5. Wide-angle capability - wider angle - just add more modes - adapts to problem
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
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).
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.
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
Memory Requirements • Very efficient as a function of z-resolution - N0 scaling for straight or periodic
Applications • We present a variety of examples illustrating EME features
SOI Waveguide Modes Ey field Ex Field
The MMI • MMI ideal for EME - inherently a modal phenomenon • 8 modes • Illustrate design scan - 100 simulations for price of two!
Tapered Fibre • 6 modes • “how long for 98% efficiency?” - ideal question for EME
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!)
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
Co-directional Coupler • remember logarithmic with N periods • very thin layer - no problem
Ring Resonator • Nmodes = 60 (for one ring) • Nmodes much higher here - wide angle propagation. • BPM gives nonsense
Photonic Crystal Design • Nmodes = 60 (for one ring) • Nmodes much higher here - wide angle propagation. • BPM gives nonsense
Rtop Rbot VCSEL Modelling • Resonance Condition top DBR mirror active layer lower DBR mirror
Showing the domains of applicability of FDTD, BPM and EME to varying delta-n and device length.
Showing the domains of applicability of FDTD, BPM and EME to varying numerical apperture and cross-section size.
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.