The Bloch Transform and Scattering from Perturbed Periodic - PowerPoint PPT Presentation

Zentrum f¨ ur Scattering from Perturbed Periodic Structures Technomathematik The Bloch Transform and Scattering from Perturbed Periodic Structures Ruming Zhang, joint with Armin Lechleiter Center for Industrial Mathematics, University of Bremen Linz, October 2016 Scattering problem Bloch Transform Numerical Analysis Ruming Zhang 1 / 19

Zentrum f¨ ur Scattering from Perturbed Periodic Structures Technomathematik Non-destructive Testing of Nano-grass Applications: Optics, sensors, self-cleaning materials, . . . Scattering problem Bloch Transform Numerical Analysis Ruming Zhang 2 / 19

Zentrum f¨ ur Scattering from Perturbed Periodic Structures Technomathematik Non-destructive Testing of Nano-grass Simple model: Scattering from perturbed periodic structure Maybe possible (but more complicated): Scattering from random structure (?) Scattering problem Bloch Transform Numerical Analysis Ruming Zhang 3 / 19

Zentrum f¨ ur Scattering from Perturbed Periodic Structures Technomathematik Direct Scattering Problems Given: Periodic structure with local perturbation Incident wave field (non-periodic!) Searched-for: Scattered wave field (analytically/numerically) Setting is less important – possible to look at . . . acoustics and Dirichlet/impedance boundary condition or inhomogeneous medium . . . inhomogeneous electromagnetic medium . . . Scattering problem Bloch Transform Numerical Analysis Ruming Zhang 4 / 19

Zentrum f¨ ur Scattering from Perturbed Periodic Structures Technomathematik Periodic Scattering Problems Almost classical if everything is (quasi-)periodic but difficult to treat scattering problems in case of modes propagating along structure Fliss, Joly, Li: Scattering in closed periodic waveguides (’05, ’09, ’16) Coatl´ even: Scattering in periodic medium + defect (’12) Hoang, Radosz: Limiting absorption principle for periodic waveguides (’11,’14) Hohage, Soussi: Riesz bases for the translation operator in periodic waveguides (’13) Scattering problem Bloch Transform Numerical Analysis Ruming Zhang 5 / 19

Zentrum f¨ ur Scattering from Perturbed Periodic Structures Technomathematik 1 Dirichlet Scattering Problem 2 Bloch Transform 3 Numerical Analysis Scattering problem Bloch Transform Numerical Analysis Ruming Zhang 6 / 19

Zentrum f¨ ur Scattering from Perturbed Periodic Structures Technomathematik Scattering from Rough Dirichlet Surfaces Domain Ω = { x ∈ R 2 : x 2 > ζ ( x 1 ) } with ∂ Ω = { ( x 1 , ζ ( x 1 )) : x 1 ∈ R } for function ζ : R → R Incident field u i solves ∆ u i + k 2 u i = 0 in Ω Seek total field u such that ∆ u + k 2 u = 0 in Ω , u | ∂ Ω = 0 and u s = u − u i radiates: For x 2 > H , � ξx 1 + √ � � − i k 2 −| ξ | 2 ( x 2 − H ) u s ( x ) = � u s ( ξ, H ) d ξ e Γ H Scattering problem Bloch Transform Numerical Analysis Ruming Zhang 7 / 19

Zentrum f¨ ur Scattering from Perturbed Periodic Structures Technomathematik Existence and Uniqueness Ω H = { x ∈ R 2 : ζ ( x 1 ) < x 2 < H } ⊂ R 2 for ζ : R → R Weighted Sobolev spaces H s r (Ω H ) with norm � � � (1 + x 2 � 1 ) r/ 2 u u �→ H s (Ω H ) , s, r ∈ R Incident field u i solves ∆ u i + k 2 u i = 0 in Ω and belongs to H 1 r (Ω H ) for | r | < 1 r (Ω H ) � H 1 Seek total field u ∈ H 1 loc (Ω) such that ∆ u + k 2 u = 0 in Ω , u | ∂ Ω = 0 and u s = u − u i radiates Chandler-Wilde & Elschner ’10: Scattering problem r (Ω H ) � H 1 solvable in � H 1 loc (Ω) if ζ Lipschitz continuous Scattering problem Bloch Transform Numerical Analysis Ruming Zhang 8 / 19

Zentrum f¨ ur Scattering from Perturbed Periodic Structures Technomathematik Existence and Uniqueness Ω H = { x ∈ R 2 : ζ ( x 1 ) < x 2 < H } ⊂ R 2 for ζ : R → R Weighted Sobolev spaces H s r (Ω H ) with norm � � � (1 + x 2 � 1 ) r/ 2 u u �→ H s (Ω H ) , s, r ∈ R Incident field u i solves ∆ u i + k 2 u i = 0 in Ω and belongs to H 1 r (Ω H ) for | r | < 1 r (Ω H ) � H 1 Seek total field u ∈ H 1 loc (Ω) such that ∆ u + k 2 u = 0 in Ω , u | ∂ Ω = 0 and u s = u − u i radiates Chandler-Wilde & Elschner ’10: Scattering problem r (Ω H ) � H 1 solvable in � H 1 loc (Ω) if ζ Lipschitz continuous Scattering problem Bloch Transform Numerical Analysis Ruming Zhang 8 / 19

Zentrum f¨ ur Scattering from Perturbed Periodic Structures Technomathematik Existence and Uniqueness Ω H = { x ∈ R 2 : ζ ( x 1 ) < x 2 < H } ⊂ R 2 for ζ : R → R Weighted Sobolev spaces H s r (Ω H ) with norm � � � (1 + x 2 � 1 ) r/ 2 u u �→ H s (Ω H ) , s, r ∈ R Incident field u i solves ∆ u i + k 2 u i = 0 in Ω and belongs to H 1 r (Ω H ) for | r | < 1 r (Ω H ) � H 1 Seek total field u ∈ � H 1 loc (Ω) such that ∆ u + k 2 u = 0 in Ω , u | ∂ Ω = 0 and u s = u − u i radiates Chandler-Wilde & Elschner ’10: Scattering problem r (Ω H ) � H 1 solvable in � H 1 loc (Ω) if ζ Lipschitz continuous Scattering problem Bloch Transform Numerical Analysis Ruming Zhang 8 / 19

Zentrum f¨ ur Scattering from Perturbed Periodic Structures Technomathematik Main Tool: Floquet–Bloch Transform Bloch transform � “periodic” Fourier transform Period Λ > 0 ⇒ α -quasiperiodicity: f ( x + Λ) = e − i α Λ f ( x ) 1D-Bloch transform J defined for f ∈ C ∞ 0 ( R ) by � Λ � 1 / 2 � f ( x + Λ j ) e i α Λ j J f ( α, x ) = 2 π j ∈ Z � � J f ( · , x ) is periodic with period one ⇒ set W ∗ = − 1 2 , 1 2 J f ( α, · ) is α -qp with period 2 π ⇒ set W = ( − π, π ] J maps H s ( R ) into L 2 � � ; H s α () Scattering problem Bloch Transform Numerical Analysis Ruming Zhang 9 / 19

Zentrum f¨ ur Scattering from Perturbed Periodic Structures Technomathematik Main Tool: Floquet–Bloch Transform Bloch transform � “periodic” Fourier transform Period 2 π ⇒ α -quasiperiodicity: f ( x + 2 π ) = e − 2 π i α f ( x ) 1D-Bloch transform J defined for f ∈ C ∞ 0 ( R ) by � f ( x + 2 πj ) e 2 π i αj J f ( α, x ) = j ∈ Z � � J f ( · , x ) is periodic with period one ⇒ set W ∗ = − 1 2 , 1 2 J f ( α, · ) is α -qp with period 2 π ⇒ set W = ( − π, π ] J maps H s ( R ) into L 2 � � ( − 1 / 2 , 1 / 2); H s α ( − π, π ) Scattering problem Bloch Transform Numerical Analysis Ruming Zhang 9 / 19

Zentrum f¨ ur Scattering from Perturbed Periodic Structures Technomathematik Main Tool: Floquet–Bloch Transform Bloch transform � “periodic” Fourier transform Period 2 π ⇒ α -quasiperiodicity: f ( x + 2 π ) = e − 2 π i α f ( x ) 1D-Bloch transform J defined for f ∈ C ∞ 0 ( R ) by � f ( x + 2 πj ) e 2 π i αj J f ( α, x ) = j ∈ Z � � J f ( · , x ) is periodic with period one ⇒ set W ∗ = − 1 2 , 1 2 J f ( α, · ) is α -qp with period 2 π ⇒ set W = ( − π, π ] J maps H s ( R ) into L 2 � � W ∗ ; H s α ( W ) for all s ∈ R Scattering problem Bloch Transform Numerical Analysis Ruming Zhang 9 / 19

Zentrum f¨ ur Scattering from Perturbed Periodic Structures Technomathematik Bloch Transform on Domains Periodic domain Ω Ω H = { x ∈ R 2 : ζ ( x 1 ) < x 2 < H } (Partial) Bloch transform in 2D: � u ( x 1 +2 πj ) e 2 π i αj Ω H J u ( α, x ) = x 2 j ∈ Z J : H s (Ω H ) → L 2 (( − 1 / 2 , 1 / 2); H s α (Ω 2 π H )) Inverse transform J − 1 : Ω 2 π H � W ∗ w ( α, x ) e 2 π i αj d α, J − 1 w ( x 1 +2 πj x ∈ Ω 2 π ) = H , j ∈ Z x 2 Scattering problem Bloch Transform Numerical Analysis Ruming Zhang 10 / 19

Zentrum f¨ ur Scattering from Perturbed Periodic Structures Technomathematik Bloch Transform on Domains Periodic domain Ω Ω H = { x ∈ R 2 : ζ ( x 1 ) < x 2 < H } (Partial) Bloch transform in 2D: � u ( x 1 +2 πj ) e 2 π i αj Ω H J u ( α, x ) = x 2 j ∈ Z J : H s (Ω H ) → L 2 ( W ∗ ; H s α (Ω 2 π H )) Inverse transform J − 1 : Ω 2 π H � W ∗ w ( α, x ) e 2 π i αj d α, J − 1 w ( x 1 +2 πj x ∈ Ω 2 π ) = H , j ∈ Z x 2 Scattering problem Bloch Transform Numerical Analysis Ruming Zhang 10 / 19

Zentrum f¨ ur Scattering from Perturbed Periodic Structures Technomathematik Bloch Transform on Domains Periodic domain Ω Ω H = { x ∈ R 2 : ζ ( x 1 ) < x 2 < H } (Partial) Bloch transform in 2D: � u ( x 1 +2 πj ) e 2 π i αj Ω H J u ( α, x ) = x 2 j ∈ Z J : H s r (Ω H ) → H r p ( W ∗ ; H s α (Ω 2 π H )) for s, r ∈ R Inverse transform J − 1 : Ω 2 π H � W ∗ w ( α, x ) e 2 π i αj d α, J − 1 w ( x 1 +2 πj x ∈ Ω 2 π ) = H , j ∈ Z x 2 Scattering problem Bloch Transform Numerical Analysis Ruming Zhang 10 / 19

Zentrum f¨ ur Scattering from Perturbed Periodic Structures Technomathematik Bloch Transform on Domains Periodic domain Ω Ω H = { x ∈ R 2 : ζ ( x 1 ) < x 2 < H } (Partial) Bloch transform in 2D: � ) e 2 π i αj u ( x 1 +2 πj Ω H J u ( α, x ) = x 2 j ∈ Z J : � p ( W ∗ ; � H s r (Ω H ) → H r H s α (Ω 2 π H )) for s, r ∈ R Inverse transform J − 1 : Ω 2 π H � W ∗ w ( α, x ) e 2 π i αj d α, J − 1 w ( x 1 +2 πj x ∈ Ω 2 π ) = H , j ∈ Z x 2 Scattering problem Bloch Transform Numerical Analysis Ruming Zhang 10 / 19