1. S. Shiri November, 2005Optics,TNT 3D Vector Representation of EM Field in Terrestrial Planet Finder Coronagraph Shahram (Ron) Shiri Code 551, Optics Branch Goddard Space Flight Center

2. S. Shiri November, 2005Optics,TNT Content Investigated vector nature of light in modeling of Terrestrial Planet Finder Coronagraph (TPFC) masks Formulated a mathematical representation of vector EM theory Developed and employed an edge-based finite element method (FEM) to solve vector electric field Verified the FEM Model in waveguides Simulated the EM field propagation through a circular metallic mask

3. S. Shiri November, 2005Optics,TNT Visible Light in TPF Visible light coronagraph in TPF requires detection of planet which is 10 order magnitude dimmer than the central stellar source –Ratio of planetary flux to stellar diffracted/scattered flux should exceed unity Optical structures require same accuracy in intensity Light is an electromagnetic wave and naturally polarized Light has both electric and magnetic field in 3D vector fields

4. S. Shiri November, 2005Optics,TNT The 3D Vector Wave Equation for Electric Field which has the vector field solution: This is just 3 independent wave equations, one for each x-, y-, and z -components of E. Derived from Maxwell’s Equations

5. S. Shiri November, 2005Optics,TNT Vector Helmholtz Equation Helmholtz Equation in free space derived from wave equation The complex electric field has six numbers that must to be specified to completely determine its value x -component y -component z -component

6. S. Shiri November, 2005Optics,TNT Scalar Diffraction Theory Solves the scalar form of Helmholtz equation –Assumes the boundaries are perfect conductors –Valid for apertures and objects >> –Not valid for very small apertures, fibre optics, planar waveguides, ignore polarization Incident Field Over Boundary Object Transmission Function

7. S. Shiri November, 2005Optics,TNT Vector Solutions Scalar Approximations Full Wave Solutions Rayleigh- Sommerfeld & Fresnel- Kirchoff Fresnel (Near Field) Fraunhofer (Far field) Z >> λZ >> z Micro-Systems Wave front - Spherical Wave front - Parabolic Wave front - Planar Validity of Scalar Field

8. S. Shiri November, 2005Optics,TNT Finite Element Formulation Second order linear elliptic partial differential equation in 1D Subject to Boundary Conditions –Dirichlet B.C.where The domain Ω of problem is discretized into a non-overlapping set of elements. Piecewise linear finite element approximation, where are piecewise linear basis functions for i = 1,..,N

9. S. Shiri November, 2005Optics,TNT FEM Global Matrix The set of equations may be written in matrix notation as Where, General solution Procedure : Calculate the stiffness coefficients of all the elements Assemble the global stiffness matrix A Solve the system of equations using an iterative algorithm –Biconjugate gradient method suitable for Helmholtz equation

10. S. Shiri November, 2005Optics,TNT Vector Finite Element Method Vector Finite Element Method is very similar to Traditional (Scalar) Finite Element Method except the basis functions are vector based instead of scalar Vector Finite Element (edge-based) Scalar Finite Element (node-based) Unknown are components of the field at the nodes of each element Unknown are components of the field along the edges of each element

11. S. Shiri November, 2005Optics,TNT 1 2 3 4 5 6 7 8 1 5 14 18 6 8 11 13 15 17 2 4 9 7 12 10 16 3 Tetrahedral 1 Tetrahedral 2 4 8 7 6 Tetrahedral 3 4 6 2 1 1 4 7 3 4 1 6 7 5 Tetrahedral 4 1 6 7 Tetrahedral 5 Organization of Tetrahedrons in Each Hexahedron

12. S. Shiri November, 2005Optics,TNT Advantages of Vector Finite Element Vector Finite Element is based on tangential edge-based elements which overcomes the spurious modes present in node based finite element It could be used for inhomogeneous medium with irregular shapes The Maxwell’s boundary condition (continuity of tangential component) are preserved using edge-based elements along the interfaces between different materials The divergence across each element is zero It solves the open boundary problems by using factitious absorbing boundary conditions such as Perfectly Match Layer (PML)

13. S. Shiri November, 2005Optics,TNT Sparsity of FEM Global Matrix Percent of Zeros in Global Matrix

14. S. Shiri November, 2005Optics,TNT Computational Assessment of Helmholtz Solver Using Krylov Subspace

15. S. Shiri November, 2005Optics,TNT Transverse Electric Field (TE) Propagating from Front Panel in Rectangular Waveguide is Characterized by, and and only mode of operation is In Dominant Mode TE 10, can be derived from Subject to Boundary Conditions, and Then, and y x z a Where, Solution Derived from “Foundations of Optical Waveguides” by G. Owyang Vector Model Verification Using Analytic Waveguide Propagation in Rectangular Slab

16. S. Shiri November, 2005Optics,TNT Waveguide Validation: Electric Field Propagation in Lossless Hollow Rectangular Waveguide Geometry: Hollow rectangular box 8 x 4 x 48 cm Incident Beam: Y-Polarized incident from left, Wavelength = 15 cm Permittivity = 3.0 and Conductivity = 0.02

17. S. Shiri November, 2005Optics,TNT Waveguide Validation: Electric Field Propagation in Lossy Hollow Rectangular Waveguide Geometry: Hollow rectangular box 8 x 4 x 48 cm Incident Beam: Y-Polarized incident from left, Wavelength = 15 cm Permittivity = 3.0 and Conductivity = 0.02

18. S. Shiri November, 2005Optics,TNT Dielectric Slab: |E| Electric Field Propagation in a box (8x4x48 Microns) with conductor walls on the sides. The box consist of two mediums of lossy medium at the front and lossless medium at the back. Beam incident (wavelength=15 micron) on the front wall, Back wall is open-ended. Dielectric Slab Cross Section: Electric Field (Real Part) propagation in a box with conductor walls on the sides. The material on the front side of the box is lossy medium (permittivity=4.0 and conductivity=0.02). The material on the end of the box is lossless medium (permittivity=2.0). Beam incident on the front wall, Back wall is open-ended. Lossy medium Nonlossy medium Waveguide Validation: Electric Field Propagation in Hollow Rectangular Waveguide with Mixed Permittivity

19. S. Shiri November, 2005Optics,TNT Waveguide Validation: Electric Field Propagation in Hollow Rectangular Waveguide with Mixed Permittivity Geometry: Hollow rectangular box 8 x 4 x 48 cm Incident Beam: Y-Polarized incident from left, Wavelength = 15 cm Permittivity = 3.0 and Conductivity = 0.02

20. S. Shiri November, 2005Optics,TNT Vector FEM Verification: Rectangular Waveguide Case – Lossy Medium VFEM Verification: Geometry - Hollow rectangular waveguide 8 x 4 x 32 cm, Incident Beam – Planar 15 cm wavelength. Boundaries - Perfect conductor at the walls. Filling Medium – Non magnetic dielectric with permittivity = 3.0 + 0.04i Z-axis (m)

21. S. Shiri November, 2005Optics,TNT Vector FEM Verification: Rectangular Waveguide Case – Lossless Medium VOM Verification: Geometry - Hollow rectangular waveguide 8 x 4 x 32 cm, Incident Beam – Planar 15 cm wavelength. Boundaries - Perfect conductor at the walls. Filling Medium – Non magnetic dielectric with permittivity = 2.0 Z-axis (m)

22. S. Shiri November, 2005Optics,TNT Glass Gold Vacumm Plane Wave = 0.6 mm 18  m 10 1 10 0 10 -2 10 -3 10 -4 10 -5 10 -1 = 0.6  m, 6  m of Gold thick on 5  m of glass, 18 x 18 um box FEM Verification: Electric Field Propagation around Gold Metallic Mask Profile of Magnitude of Electric Field Thickness (microns)

23. S. Shiri November, 2005Optics,TNT Columbia Project Vector Finite Element Simulation Simulation of vector diffraction model: Geometry: 20x20x10 microns Filling: Air Mask: Silver, 5x5x1 microns Incident Beam: Planar, y-polarized, wavelength of 5 microns Boundary: PML absorbing boundaries X-cross section, Red line: Intensity profile before mask Y-cross section, Black line: Intensity profile after mask Z-cross section, Red line: Intensity profile before mask Black line: Intensity profile after mask Numerical Complexity: Edges: 6 millions Nodes: 646400 Memory: 350 Gig. Bytes CPUs: 100 Convergence Time: 7 Hrs.

24. S. Shiri November, 2005Optics,TNT Columbia Project Vector Finite Element Simulation After Mask Before Mask X- axis Profile Electric Field Intensity Log ProfileElectric Field Intensity Linear Profile Columbia Project vector finite element simulation of planar incident beam into geometry air filled centered with circular silver mask. Beam Properties: Planar, Wavelength of 5 microns, y-polarized Geometry: 20 x 20 x 10 micronsBoundary: PML absorbing boundaries

25. S. Shiri November, 2005Optics,TNT Electric Field Before and After Circular Mask Profile of electric field before and after a circular mask in a vacuum. The mask is silver metallic disk. The incident beam is planar and polarized in y direction with wavelength of 5 microns.

26. S. Shiri November, 2005Optics,TNT Limitations of Vector FEM Computationally expensive to achieve 10 8 accuracy or higher A sample realistic simulation requires –Large amount of memory (Gigabytes) –Matrix Solver based on MPI requires cluster to reach convergence in a reasonable time frame –More than 72 samples per wavelength Selecting appropriate Absorbing Boundary Condition (ABC) far away from the object

27. S. Shiri November, 2005Optics,TNT Summary Formulated a vector model for the electric field around the mask in TPF coronagraph Incorporated and configured the Vector Finite Element Method (VFEM) for this problem Verified the accuracy of VFEM for TPF using analytical solutions in rectangular waveguide to 10 –VFEM shows promise of being used for further highly accurate models around or near the mask