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Nanostructures Research Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH
DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited.




                                                                               http://www.iqol.uwaterloo.ca




                                                                                  http://oxfordplasma.de
Nanostructures Research Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH
DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited.

                                                                                                     Si




•  Crystal – periodic arrangement of
             atoms/molecules
    •  Propagating electrons “see” periodic potential
     due to atoms/molecules.
    •  Conduction properties dictated by crystal
     geometry.
    •  Crystal lattice introduces energy bandgap (Eg)
                                                                 freq [c/a]


•  Optical analogy is photonic crystal
    •  Periodic potential due to lattice of dielectric
     material.
    •  Propagation of photons controlled by dielectric
     contrast and (r/a) ratio.
                                                                                                               Band Gap
    •  Can engineer a photonic bandgap



      Nanostructures Research Group
     CENTER FOR SOLID STATE ELECTRONICS RESEARCH
DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited.




•  Photonic crystals/PBM shown great deal of promise for
 true integrated optics.
•  Waveguides with small bends possible making compact
 integrated photonic circuits (IPCs) achievable.
•  Design/fabrication is challenging
•  Efficient simulation tools needed to realize very low
 loss IPCs
                                                                              rods
                      http://www/photonics.tfp.uni-karlsruhe.de/research.html
                      http://pages.ief.u-psud.fr
         bend                           splitter              resonator                    cavity




  Nanostructures Research Group
  CENTER FOR SOLID STATE ELECTRONICS RESEARCH
DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited.

                                                              y

                                                                    x
                                                                          2D slice         εr = 12.0
                                                              z
  •  Specific PC geometry requires many grid
   cells (~107 cells) to resolve even a limited
   number of periods.
  •  Memory intensive computations.
  •  Modeling of 3D PCB structures dictates
   vital need for parallel HPC architectures
   with optimized domain decomposition.




                                                                          εr = 1.0          a


                                                                         Simple example:
                                                                         370 x 520 x 50 (~107 grid points)
                                                                                4.5Gb RAM needed
(Target SPAWAR 3D structure)
                                                                         Realistic grids > 109 points
        Nanostructures Research Group
       CENTER FOR SOLID STATE ELECTRONICS RESEARCH
DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited.




     In isotropic medium, Maxwell’s curl
      equations are:
                         
                      ∂H
            ∇ × E = −µ
                        ∂t
                       
                     ∂E 
            ∇× H = ε     +J
                      ∂t

                                                                                                 K.S. Yee, IEEE Trans. Antennas
                                                                                                 Propagat., 14(302) 1966
     ∂H x ∂Ez ∂E y                   ∂Ex ∂H z ∂H y
−µ       =    −                  ε       =    −    − Jx,
      ∂t   ∂y   ∂z                    ∂t   ∂y   ∂z
                                                                        
Direct explicit solution of Maxwell’s equations
     ∂H y        ∂Ex ∂Ez             ∂E y       ∂H x ∂H z                 (i.e. no matrix inversion required).
−µ           =       −          ε           =       −     − Jy,
      ∂t          ∂z   ∂x             ∂t         ∂z   ∂x                 2nd order accurate.
     ∂H z ∂Ex ∂E y                   ∂Ez ∂H y ∂H x                       Complete “full-wave” method without
−µ       =    −                  ε       =    −    − Jz.
      ∂t   ∂y   ∂x                    ∂t   ∂x   ∂y                        approximation (i.e. no pre-selection of output
                                                                          modes or solution form necessary.)
                                                                         Easy to parallelize.
            Nanostructures Research Group
            CENTER FOR SOLID STATE ELECTRONICS RESEARCH
DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited.




   • 
Introduces “artificial” anisotropic
    electric/magnetic* conductivities
    within domain boundaries allowing for
    absorption/attenuation waves.
   •  Employs a numerical “split-field”
    approach allowing perfect
    (theoretical) transmission into
    absorbing layer (regardless of
    frequency, polarization, or angle of
    incidence).
   •  Perfect electric conductor (PEC)
    surrounds PML ABC
   •  Technique “simulates” effect of                    J. P. Bérenger, IEEE Trans. Antennas Propagat., 44(110) 1996.
    outward propagation of EM waves
    to infinity.
                      Ex = Exy + Exz
                                                                          ∂Exy                     ∂ ( H zx + H zy )
  ∂Ex ∂H z ∂H y       H y = H yz + H yx                               ε          + σ y Exy =
ε     =    −    − Jx, H = H + H                                          ∂t                     ∂y
   ∂t   ∂y   ∂z        z     zx     zy
                                                                        ∂Exz               ∂ ( H yz + H yx )
                                                                      ε      + σ z Exz = −
         Nanostructures Research Group                                   ∂t                       ∂z
        CENTER FOR SOLID STATE ELECTRONICS RESEARCH
DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited.




 •  Stability limit, called the CFL                                                             1
                                                          Δt FDTD ≤
                                                                                           2             2           2
  criterion limits maximum timestep                                               ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞
   for solution of PDEs on a finite                                      υ max    ⎜    ⎟ +⎜
                                                                                          ⎜ Δy ⎟ + ⎜ Δz ⎟
                                                                                               ⎟
                                                                                  ⎝ Δx ⎠ ⎝     ⎠ ⎝ ⎠
  grid.
                                                              R. Courant, et al. , IBM Journal , 215(1967).

 •  For example, a uniform grid of
  1nm in Si (εr=12) results in:                               Alternate-Direction Implicit Approach
                                                              •  Timestep split into (2) sub-iterations
                                12                            •  E-fields updated implicitly along
Δt FDTD ≤
                                                   2             specific directions.
                           ⎛     1    ⎞
            (3 × 108 m ) 3 ⎜          ⎟                       •  H-fields updated explicitly throughout.
                      s            −9
                           ⎝ 1× 10 m ⎠
                                                                        y
                                                          z
                                                                                               RELAXES
Δt FDTD ≤ 6.7 × 10−17 s ⇒ 0.067 fs                                                             CFL LIMIT

Long simulation times !                                                                    T. Namiki, IEEE MTT 47(10), 2003
                                                                                            (1999).
                                                                                           F. Zheng, et. al, Microwave Guided
      Nanostructures Research Group                                                   x     Wave Lett., 9(11), 441 (1999).
     CENTER FOR SOLID STATE ELECTRONICS RESEARCH
DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited.




       z




                                      PML
                                                                                                                    y
 Air
PC Slab

 Air
                                      PML                                                 370 x 520 x 50 grid
                                                                                   x
                                                                                          ~107 grid points
                                                                                          Si slab (εr =12.0)
            (PML completely surrounds simulated structure)
                                                                                           cylinders (εr =6.0)
                                                                                          50 Si cylinders
          Nanostructures Research Group
          CENTER FOR SOLID STATE ELECTRONICS RESEARCH
370 x 520 x 50 grid
                                                         Source plane
DISTRIBUTION STATEMENT A: Cleared for public releases;   Gaussian pulse   ~107 grid points
                                                         tw = 15 ps       Si slab (εr =12.0)
distribution is unlimited.
                                                                          50 cylinders (εr =6.0)
           Nanostructures Research Group
          CENTER FOR SOLID STATE ELECTRONICS RESEARCH
Source plane    370 x 520 x 50 grid
                                                         Bipolar pulse   ~107 grid points
DISTRIBUTION STATEMENT A: Cleared for public releases;
distribution is unlimited.                                               Si slab (εr =12.0)
                                                                         44 cylinders (εr =6.0)
           Nanostructures Research Group
          CENTER FOR SOLID STATE ELECTRONICS RESEARCH
DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited.




•  When cylinders are built in domain, each
 grid cell is divided into 9 subcells
•  The dielectric contribution of each 1/9 of a
 grid cell is computed for those subcells
 completely within the cylinder radius.
•  Results in smoothing around the stair
-cased edges of cylinders




                                                               y


                                                                         x
                                                              z

       Nanostructures Research Group
       CENTER FOR SOLID STATE ELECTRONICS RESEARCH
DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited.




Task 0
        Setup
                                                                                Parallel Region

 Initial Scatter

                                     Task 1                          Task N
        BC's                                  BC's                            BC's

   calc H field                           calc H field                  calc H field

   calc E field                          calc E field                   calc E field

                         Communication – plane exchange

                                      Output & Finish
Nanostructures Research Group
CENTER FOR SOLID STATE ELECTRONICS RESEARCH
DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited.




•  Both 1D and 3D decompositions
 have been implemented within the
 MPI framework of the simulator

•  In order to reduce the computation
   steps, redundant calculations at
  boundary regions were employed
[Hanawa et al., IEEE Trans. on Mag, 43(4),
1545 (2007)]




                                                             Speedup of SPAWAR vs. ASU code

                                                          •  Initial 1D decomposition resulted in good
                                                           scaling for long crystal geometry
       Interprocessor boundary
    Nanostructures Research Group
    CENTER FOR SOLID STATE ELECTRONICS RESEARCH
DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited.




•  Speedup for increasingly larger
 domains.




      Interprocessor boundary
    Nanostructures Research Group
   CENTER FOR SOLID STATE ELECTRONICS RESEARCH
DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited.




•    3D decomposition worked best
  for more general geometries and
   particularly for large problem
 domains

•  This is the default decomposition
 in the code delivered to DoD user
 community.




     Nanostructures Research Group
     CENTER FOR SOLID STATE ELECTRONICS RESEARCH
DISTRIBUTION STATEMENT A: Cleared for public releases;
                                                    distribution is unlimited.



    Benchmarking of parallel ADI-FDTD code. (reduced simulation times)

    Demonstration of 3-layer 3D PCG structures and circuits.




                                                     J. S. Rodgers, “Quasi-3D photonic crystals for nanophotonics,”
                                                      Proceedings of SPIE, vol. 5732, Quantum Sensing and Nanophotonic
                                                      Devices II, Manijeh Razeghi, Gail J. Brown, Editors, March 2005, pp.
      Nanostructures Research Group                   511-519.
      CENTER FOR SOLID STATE ELECTRONICS RESEARCH

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Parallel 3D FDTD Simulator for Photonic Crystals

  • 1. DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited. Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  • 2. DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited. http://www.iqol.uwaterloo.ca http://oxfordplasma.de Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  • 3. DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited. Si •  Crystal – periodic arrangement of atoms/molecules •  Propagating electrons “see” periodic potential due to atoms/molecules. •  Conduction properties dictated by crystal geometry. •  Crystal lattice introduces energy bandgap (Eg) freq [c/a] •  Optical analogy is photonic crystal •  Periodic potential due to lattice of dielectric material. •  Propagation of photons controlled by dielectric contrast and (r/a) ratio. Band Gap •  Can engineer a photonic bandgap Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  • 4. DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited. •  Photonic crystals/PBM shown great deal of promise for true integrated optics. •  Waveguides with small bends possible making compact integrated photonic circuits (IPCs) achievable. •  Design/fabrication is challenging •  Efficient simulation tools needed to realize very low loss IPCs rods http://www/photonics.tfp.uni-karlsruhe.de/research.html http://pages.ief.u-psud.fr bend splitter resonator cavity Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  • 5. DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited. y x 2D slice εr = 12.0 z •  Specific PC geometry requires many grid cells (~107 cells) to resolve even a limited number of periods. •  Memory intensive computations. •  Modeling of 3D PCB structures dictates vital need for parallel HPC architectures with optimized domain decomposition. εr = 1.0 a Simple example: 370 x 520 x 50 (~107 grid points) 4.5Gb RAM needed (Target SPAWAR 3D structure) Realistic grids > 109 points Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  • 6. DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited. In isotropic medium, Maxwell’s curl equations are:   ∂H ∇ × E = −µ ∂t   ∂E  ∇× H = ε +J ∂t K.S. Yee, IEEE Trans. Antennas Propagat., 14(302) 1966 ∂H x ∂Ez ∂E y ∂Ex ∂H z ∂H y −µ = − ε = − − Jx, ∂t ∂y ∂z ∂t ∂y ∂z   
Direct explicit solution of Maxwell’s equations ∂H y ∂Ex ∂Ez ∂E y ∂H x ∂H z (i.e. no matrix inversion required). −µ = − ε = − − Jy, ∂t ∂z ∂x ∂t ∂z ∂x   2nd order accurate. ∂H z ∂Ex ∂E y ∂Ez ∂H y ∂H x   Complete “full-wave” method without −µ = − ε = − − Jz. ∂t ∂y ∂x ∂t ∂x ∂y approximation (i.e. no pre-selection of output modes or solution form necessary.)   Easy to parallelize. Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  • 7. DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited. • 
Introduces “artificial” anisotropic electric/magnetic* conductivities within domain boundaries allowing for absorption/attenuation waves. •  Employs a numerical “split-field” approach allowing perfect (theoretical) transmission into absorbing layer (regardless of frequency, polarization, or angle of incidence). •  Perfect electric conductor (PEC) surrounds PML ABC •  Technique “simulates” effect of J. P. Bérenger, IEEE Trans. Antennas Propagat., 44(110) 1996. outward propagation of EM waves to infinity. Ex = Exy + Exz ∂Exy ∂ ( H zx + H zy ) ∂Ex ∂H z ∂H y H y = H yz + H yx ε + σ y Exy = ε = − − Jx, H = H + H ∂t ∂y ∂t ∂y ∂z z zx zy ∂Exz ∂ ( H yz + H yx ) ε + σ z Exz = − Nanostructures Research Group ∂t ∂z CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  • 8. DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited. •  Stability limit, called the CFL 1 Δt FDTD ≤ 2 2 2 criterion limits maximum timestep ⎛ 1 ⎞ ⎛ 1 ⎞ ⎛ 1 ⎞ for solution of PDEs on a finite υ max ⎜ ⎟ +⎜ ⎜ Δy ⎟ + ⎜ Δz ⎟ ⎟ ⎝ Δx ⎠ ⎝ ⎠ ⎝ ⎠ grid. R. Courant, et al. , IBM Journal , 215(1967). •  For example, a uniform grid of 1nm in Si (εr=12) results in: Alternate-Direction Implicit Approach •  Timestep split into (2) sub-iterations 12 •  E-fields updated implicitly along Δt FDTD ≤ 2 specific directions. ⎛ 1 ⎞ (3 × 108 m ) 3 ⎜ ⎟ •  H-fields updated explicitly throughout. s −9 ⎝ 1× 10 m ⎠ y z RELAXES Δt FDTD ≤ 6.7 × 10−17 s ⇒ 0.067 fs CFL LIMIT Long simulation times ! T. Namiki, IEEE MTT 47(10), 2003 (1999). F. Zheng, et. al, Microwave Guided Nanostructures Research Group x Wave Lett., 9(11), 441 (1999). CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  • 9. DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited. z PML y Air PC Slab Air PML 370 x 520 x 50 grid x ~107 grid points Si slab (εr =12.0) (PML completely surrounds simulated structure) cylinders (εr =6.0) 50 Si cylinders Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  • 10. 370 x 520 x 50 grid Source plane DISTRIBUTION STATEMENT A: Cleared for public releases; Gaussian pulse ~107 grid points tw = 15 ps Si slab (εr =12.0) distribution is unlimited. 50 cylinders (εr =6.0) Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  • 11. Source plane 370 x 520 x 50 grid Bipolar pulse ~107 grid points DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited. Si slab (εr =12.0) 44 cylinders (εr =6.0) Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  • 12. DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited. •  When cylinders are built in domain, each grid cell is divided into 9 subcells •  The dielectric contribution of each 1/9 of a grid cell is computed for those subcells completely within the cylinder radius. •  Results in smoothing around the stair -cased edges of cylinders y x z Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  • 13. DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited. Task 0 Setup Parallel Region Initial Scatter Task 1 Task N BC's BC's BC's calc H field calc H field calc H field calc E field calc E field calc E field Communication – plane exchange Output & Finish Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  • 14. DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited. •  Both 1D and 3D decompositions have been implemented within the MPI framework of the simulator •  In order to reduce the computation steps, redundant calculations at boundary regions were employed [Hanawa et al., IEEE Trans. on Mag, 43(4), 1545 (2007)] Speedup of SPAWAR vs. ASU code •  Initial 1D decomposition resulted in good scaling for long crystal geometry Interprocessor boundary Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  • 15. DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited. •  Speedup for increasingly larger domains. Interprocessor boundary Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  • 16. DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited. •  3D decomposition worked best for more general geometries and particularly for large problem domains •  This is the default decomposition in the code delivered to DoD user community. Nanostructures Research Group CENTER FOR SOLID STATE ELECTRONICS RESEARCH
  • 17. DISTRIBUTION STATEMENT A: Cleared for public releases; distribution is unlimited.   Benchmarking of parallel ADI-FDTD code. (reduced simulation times)   Demonstration of 3-layer 3D PCG structures and circuits. J. S. Rodgers, “Quasi-3D photonic crystals for nanophotonics,” Proceedings of SPIE, vol. 5732, Quantum Sensing and Nanophotonic Devices II, Manijeh Razeghi, Gail J. Brown, Editors, March 2005, pp. Nanostructures Research Group 511-519. CENTER FOR SOLID STATE ELECTRONICS RESEARCH