1. - Modal Analysis for Bi-directional Optical Propagation
FIMMPROP-3D
- Modal Analysis for Bi-directional Optical Propagation
Dominic F.G. Gallagher
Dominic F.G. Gallagher

2. 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

3. Local Mode Approximation
in a waveguide, any solution to Maxwell’s equations may be expanded:
forward
backward
mth mode profile

4. Instant Propagation Traditional tool: many steps
FIMMPROP-3D: one step per section

5. Scattering Matrix Approach
Solves for all inputs
Component framework
Port=mode (usually)
Alter parts quickly

6. Bi-directional Capability
Unconditionally stable
Takes any number of reflections into account
NOT iterative
Even resonant cavities
Mirror coatings, multi-layer

7. Fully Vectorial
glass
air
Ey Field
Ex Field

8. Periodic Structures
Very efficient - repeat period: S=(Sp)N
A mode converter
TE00
TE01

9. Bends
Transmission: T= (Sj)N
exact answer as Nèinfinity Sj

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

11. Rigorous Diffraction
Metal plate

12. propagation at sub-wavelength scales, including metal features

13. Photonic Crystals! Can take advantage of the periodicity
In fact can take advantage of any repetition

14. take advantage of repetition:B C
Here we need just 3 cross-sections

15. A hard propagation problem
very thin layers - wide range of dimensions
no problem for FIMMPROP-3D - algorithm does not need to discretise the structure

16. Design Curve Generation
Traditional Tool:
5 mins
5 mins
5 mins
5 mins
5 mins
5 mins
FIMMPROP-3D:
5 mins
3 mins

17. More Design Curves
offset
alter offset at joins

18. 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.

19. 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
n A2

20. 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.

21. Rectangular geometry
Cylindrical geometry
General geometry

22. Cylindrical Solver
A holey fibre
High delta-n
vectorial

23. The Mode Matching Method
1D modes
propagate
propagate
layers
slices

24. (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

25. 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

26. Devices with very thin layers - no problem

27. A Si/SiO2 (SOI) waveguide
High delta-n waveguides - no problem
A Si/SiO2 (SOI) waveguide
air Si
SiO2

28. weakly coupled waveguides - no problem

29. Near cut-off modes - no problem