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3D Vector Representation of EM Field in Terrestrial Planet Finder Coronagraph

3D Vector Representation of EM Field in Terrestrial Planet Finder Coronagraph. Shahram (Ron) Shiri Code 551, Optics Branch Goddard Space Flight Center. Content. Investigated vector nature of light in modeling of Terrestrial Planet Finder Coronagraph (TPFC) masks

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3D Vector Representation of EM Field in Terrestrial Planet Finder Coronagraph

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

  2. 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 Optics,TNT

  3. 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 Optics,TNT

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

  5. x-component y-component z-component 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 Optics,TNT

  6. 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 Object Transmission Function Incident Field Over Boundary Optics,TNT

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

  8. 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 Optics,TNT

  9. 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 Optics,TNT

  10. 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 along the edges of each element Unknown are components of the field at the nodes of each element Optics,TNT

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

  12. 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) Optics,TNT

  13. Sparsity of FEM Global Matrix Percent of Zeros in Global Matrix Optics,TNT

  14. Computational Assessment of Helmholtz Solver Using Krylov Subspace Optics,TNT

  15. Vector Model Verification Using Analytic Waveguide Propagation in Rectangular Slab Solution Derived from “Foundations of Optical Waveguides” by G. Owyang • Transverse Electric Field (TE) Propagating from Front Panel in Rectangular Waveguide is Characterized by, and and only mode of operation is • In Dominant Mode TE10 , can be derived from • Subject to Boundary Conditions, • and • Then, and y z x a • Where, Optics,TNT

  16. 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 Optics,TNT

  17. 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 Optics,TNT

  18. Waveguide Validation: Electric Field Propagation in Hollow Rectangular Waveguide with Mixed Permittivity 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. Nonlossy medium 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 Optics,TNT

  19. 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 Optics,TNT

  20. Vector FEM Verification: Rectangular Waveguide Case – Lossy Medium Z-axis (m) 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 Optics,TNT

  21. Vector FEM Verification: Rectangular Waveguide Case – Lossless Medium Z-axis (m) 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 Optics,TNT

  22. Plane Wave l = 0.6 mm 101 Vacumm 100 10-1 10-2 10-3 Gold 10-4 10-5 Glass 18 mm FEM Verification: Electric Field Propagation around Gold Metallic Mask Profile of Magnitude of Electric Field l = 0.6 mm, 6 mm of Gold thick on 5 mm of glass, 18 x 18 um box Thickness (microns) Optics,TNT

  23. Columbia Project Vector Finite Element Simulation Y-cross section, Black line: Intensity profile after mask Z-cross section, Red line: Intensity profile before mask Black line: Intensity profile after mask X-cross section, Red line: Intensity profile before mask Numerical Complexity: Edges: 6 millions Nodes: 646400 Memory: 350 Gig. Bytes CPUs: 100 Convergence Time: 7 Hrs. 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 Optics,TNT

  24. Columbia Project Vector Finite Element Simulation Electric Field Intensity Log Profile Electric Field Intensity Linear Profile After Mask Before Mask X- axis Profile X- axis 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 microns Boundary: PML absorbing boundaries Optics,TNT

  25. 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. Optics,TNT

  26. Limitations of Vector FEM • Computationally expensive to achieve 108 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 Optics,TNT

  27. 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 Optics,TNT

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