# Presentation: 183788

## 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
- 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

## 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

## 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

## 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

## 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

## Scalar Approximations

**Vector**

**Solutions**

**Scalar Approximations**

**Full Wave**** ****Solutions**

**Rayleigh-Sommerfeld**** ****& Fresnel-Kirchoff**

**Fresnel (Near Field)**

**Fraunhofer (Far field)**

*Z >> λ*

*Z >>*

*Z >>*

**z**

**Micro-Systems**

**Wave front -Spherical**

**Wave front -Parabolic**

**Wave front -Planar**

- Validity of Scalar Field

## 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

## 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

## 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

## Organization of Tetrahedrons in Each Hexahedron 

- 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

## 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)

## Sparsity of FEM Global Matrix

- Percent of Zeros in Global Matrix

## Computational Assessment of Helmholtz Solver Using Krylov Subspace 

## 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

- In Dominant Mode TE10 ,               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

## 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

## 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

## 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.

- 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

## 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

## 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)

## 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)

## FEM Verification: Electric Field Propagation around Gold Metallic Mask

-  = 0.6 m, 6 m of Gold thick
- on 5 m of glass, 18 x 18 um box

- Profile of Magnitude of Electric Field

- Thickness (microns)

## 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.

## Columbia Project Vector Finite Element Simulation

- After Mask

- Before Mask

- X- axis Profile

- X- axis Profile

**Electric Field Intensity Log Profile**

**Electric 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 microns			**Boundary**: PML absorbing boundaries

## 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.

## 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

## 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