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FIMMPROP-3D - 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

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FIMMPROP-3D - Modal Analysis for Bi-directional Optical Propagation

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  1. 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: backward forward 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. air glass Fully Vectorial 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. B A A A A A A B C C A A A A take advantage of repetition: 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: 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 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

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

  25. 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’=vfor any guessv • i.e. must invert M, an N3 operation, per iteration

  26. Devices with very thin layers - no problem

  27. 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

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