- Modal Analysis for Bi-directional Optical Propagation FIMMPROP-3D - Modal Analysis for Bi-directional Optical Propagation Dominic F.G. Gallagher Dominic 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 in a waveguide, any solution to Maxwell’s equations may be expanded: forward backward mth mode profile

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 Nèinfinity Sj

Wide Angle Propagation Photonic crystals have light travelling at wide angles Here we have no paraxial approximation Just add more modes 45°

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

take advantage of repetition: B C 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 Traditional Tool: 5 mins 5 mins 5 mins 5 mins 5 mins 5 mins FIMMPROP-3D: 5 mins 3 mins

More Design Curves offset alter offset at joins

Memory: Speed: 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 Analysis effect of high Dn Dn = n2-n1 n1 area: A1 Consider the simulation cross-section: Dn = n2-n1 number of modes with neff between n1 and n2 is approximately: n1*(A1-A2) + n2*A2 In FMM method, time taken to compute each mode is approx. proportional to (Dn)3 n1 area: A1 n2 A2

FIMMWAVE the 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

Cylindrical Solver A holey fibre High delta-n vectorial

The Mode Matching Method 1D modes propagate propagate layers slices

(b2D)2 = (b1D,m)2 + (kx,m)2 1D mode axis y z x beta(2D) defines propagation direction of 1D mode (b2D)2 = (b1D,m)2 + (kx,m)2

Algorithm M(beta).u=0 This is a highly non-linear eigensystem: 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 A Si/SiO2 (SOI) waveguide air Si SiO2

weakly coupled waveguides - no problem

Near cut-off modes - no problem