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Shape Calculus in Nano-Optics

This seminar presents a study on the effect of geometry perturbation and the use of shape calculus in nano-optics, specifically in the field of electromagnetic wave scattering problems. Numerical results and conclusion are also discussed.

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Shape Calculus in Nano-Optics

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  1. Shape calculus in nano-optics SaharSargheini, Alberto Paganini, Ralf Hiptmair, Christian Hafner

  2. Outline • Introduction • PDE Constraint Shape calculus • Electromagnetic wave scattering problem • Numerical results • Conclusion SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

  3. Introduction What is shape calculus? Study the effect of geometry perturbation Nano-Optics SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

  4. Introduction • Nanoantenna • Production based variation: • sensitivity analysis by deriving shape gradient • SNOM • Reconstructing shape and electric properties is inverse problem. • reformulate inverse problem into a PDE constraint optimization problem (using descent approach along shape gradients). SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

  5. Outline • Introduction • PDE Constraint Shape Calculus • Electromagnetic wave scattering problem • Numerical results • Conclusion SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

  6. PDE Constraint Shape optimization Problem: find the optimal admissible geometry • Solution procedure options: • Use parametric model (a few design variables) • Consider boundaries as manifolds (infinite-dimensional minimization problems) SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

  7. PDE Constraint Shape optimization Tsis the flow of a sufficiently smooth (parameter dependent) vectorfield V Vector field: Eulerian derivative of J in the direction V at t=0 is defined by Material derivative: Shape derivative in the direction of V is: SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

  8. Outline • Introduction • PDE Constraint Shape optimization • Electromagnetic wave scattering problem • Numerical results • Conclusion SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

  9. Electromagnetic wave scattering problem Γ0 D Γ1 on Γ Ω Objective function: SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

  10. Electromagnetic wave scattering problem Shape derivative: Adjoint Equation: SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

  11. Electromagnetic wave scattering problem • Steps to be done in each iteration • Solve state problem • Solve adjoint problem • Compute shape gradient • Move boundary nodes • Smooth the mesh FEM Method LehrFEM Library based on Matlab SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

  12. Electromagnetic wave scattering problem • Compute shape gradient But we would like: • Smoothing the mesh Laplace Smoothing SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

  13. Outline • Introduction • PDE Constraint Shape optimization • Electromagnetic wave scattering problem • Numerical results • Conclusion SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

  14. Numerical results (Example1) Meshing of first Iteration structure Meshing of reference structure SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

  15. Numerical results(Example1) SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

  16. Numerical results(Example1) Final iteration structure solution Reference structure solution SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

  17. Numerical results(Example2) Meshing of first Iteration structure Meshing of reference structure SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

  18. Numerical results(Example2) SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

  19. Numerical results(Example2) Final iteration structure solution Reference structure solution SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

  20. Conclusion • Shape gradients can be used to find the optimal shape. • Method of mapping provides an analytical representation of the shape gradients. • Using FEM as a solver we don’t have access to shape gradients on boundary nodes directly and some approximations are necessary. • Despite this drawback, we experienced good convergence in our simulations. Work in Progress: improve gradient recovery, use of second order information SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

  21. Thank you for your attention Any Question? SAM-Seminar of Applied Mathematics / IFH - Electromagnetic Field Theory Group

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