FIMMPROP-3D

- Modal Analysis for Bi-directional Optical Propagation

Dominic F.G. GallagherDominic F.G. Gallagher

What is FIMMPROP-3D?

• a tool for optical propagation• rigorous solutions of Maxwell’s Equations • compare ray-tracing and BPM – the latter solve approximate

equations• sub-wavelength effects, diffraction/interference, good for

small cross-sections, not for telescope lenses!• 3D• full vectorial• uses modal analysis• much faster than previous techniques for many applications• much more accurate in very many cases too

Local Mode Approximation

m

zim

zimm ececyxzyx )..(),(),,(

mth mode profileforward backward

in a waveguide, any solution to Maxwell’s equations may be expanded:

Instant Propagation

Traditional tool: many steps

FIMMPROP-3D: one step per section

Scattering Matrix Approach

•Solves for all inputs

•Component framework

•Port=mode (usually)

•Alter parts quickly

Bi-directional Capability

• Unconditionally stable

• Takes any number of reflections into account

• NOT iterative

• Even resonant cavities

• Mirror coatings, multi-layer

Fully Vectorial

glass

air

Ey Field

Ex Field

Periodic Structures

Very efficient - repeat period: S=(Sp)N

A mode converter

TE00 TE01

Bends

Transmission: T= (Sj)N

exact answer as Ninfinity

Sj

Wide Angle Propagation

•Photonic crystals have light travelling at wide angles

•Here we have no paraxial approximation

•Just add more modes45

Rigorous Diffraction

Metal plate

propagation at sub-wavelength scales, including metal features

Photonic Crystals!

• Can take advantage of the periodicity• In fact can take advantage of any repetition

A A A A A A A A A AB C C B

take advantage of repetition:

Here we need just 3 cross-sections

A hard propagation problem

• very thin layers - wide range of dimensions

• no problem for FIMMPROP-3D - algorithm does not need to discretise the structure

Design Curve Generation

5 mins

5 mins

5 mins 5 mins

5 mins

5 mins

5 mins

3 mins

Traditional Tool:

FIMMPROP-3D:

More Design Curves

alter offset at joins

offset

Memory:increase area by factor of 2

- need 2x number of modes

- each mode needs 2x number of grid points

therefore memory proportional to A2

Speed:increase area by factor of 2

- need 2x number of modes

- each mode needs time An, (1<n<3, depending on method)

therefore time to build modes proportional to A.An

overlaps: # of points x # of modes

therefore time to calculate overlaps proportional to A3

- overlap integrals will eventually limit modal analysis for very large calculations.

Modal Analysiseffect of high n

n1 area: A1

n2 A2

n = n2-n1

number of modes with neff between n1 andn2 is approximately:

n1*(A1-A2) + n2*A2

In FMM method, time taken to compute each mode is approx. proportional to (n)3

Consider the simulation cross-section:

FIMMWAVEthe mode solver

• We need a very reliable, fast mode solver to do propagation using modal analysis.

• Photon Design has many years experience in finding waveguide modes - FIMMWAVE is probably the most robust and efficient mode solver available.

Rectangular geometry

Cylindrical geometry

General geometry

A holey fibre

•High delta-n

•vectorial

Cylindrical Solver

The Mode Matching Method

1D

mo

des

propagate propagate

slices

layers

y z

x

1D mode axis

beta(2D) defines propagationdirection of 1D mode

2D)2 = 1D,m)2 + (kx,m)2

Algorithm• Find all (N) TE and TM 1D modes for each slice

• Build overlap matrices between 1D modes at each slice interface

• Guess start beta

• From given beta and LHS bc, propagate to middle, ditto from RHS

• Generate error function at middle boundary

• Loop until error is small

• Done

This is a highly non-linear eigensystem:

M(beta).u=0

• solve using: M(beta).u’=v for any guess v • i.e. must invert M, an N3 operation, per iteration

Devices with very thin layers - no problem

A Si/SiO2 (SOI) waveguide

High delta-n waveguides - no problem

SiO2

Si

air

weakly coupled waveguides - no problem

Near cut-off modes - no problem