Homogenization and the Neumann Poincar´ e operator Eric Bonnetier, Charles Dapogny, Faouzi Triki Outline: 1. Motivation : resonant frequencies in metallic nanoparticles 2. The NP operator/the Poincar´ e variational problem for a periodic collection of inclusions 3. The limiting spectra 4. Consequences concerning the homogenization of inclusions with non-positive conductivities 5. High contrast 6. Conclusion
1. Resonant frequencies of metallic nanoparticles [Gang Bi et al, Optics Comm., 285 (2012) 2472] Very small metallic particles exhibit interesting diffractive phenomena, related to resonances : localization and extremely large enhancement of the electromagentic fields in their vicinity Many potential applications : nanophotonics, nanolithography, near field microscopy, biosensors, cancer therapy 2 main ingredients : - The wavelength of the incident excitation should be larger than the particle diameter - the real part of the electric permittivity ε ( ω ) inside the particle is negative
Typical model problem D ⊂ R d , bounded C 2 domain with | D | = 1 The nanoparticle is centered at a fixed z ∈ R d and occupies D δ = z + δD δ small ω ∈ C is a resonant frequency of the nanoparticle D δ if there exists a non-trivial solution U to the PDE (TE polarization): R d \ D δ ∪ D δ ∆ U + ω 2 ε ( x, ω ) µ 0 U 8 = 0 in > ⌊ εU ⌋ = 0 on ∂D δ > < j k ∂U = 0 on ∂D δ ∂ν > > : radiation condition where for the electric permittivity ε , we consider the Drude model for x ∈ R d \ D δ 8 ε 0 < ε ( x, ω ) = „ « ω 2 ε 0 ˆ ǫ ( ω ) = ε 0 ε ∞ − P for x ∈ D δ ω 2 + iω Γ :
The change of variable x = z + x/δ ˜ transforms the original PDE into in R 2 \ D ∪ D 8 ∆ ˜ U + δ 2 ω 2 ε ( x, ω ) µ 0 ˜ U = 0 > > j k < ε ˜ U = 0 on ∂D j ∂ ˜ k U > = 0 on ∂D > : ∂ν ˜ and one expects that ε ˜ where U ( x ) = U (˜ x ) U converges to a solution of the quasistatic problem in R d div(1 /ε ( ω ) ∇ u ) = 0 u → 0 as | x | → ∞ [Mayergoyz-Fredkin-Z Zhang Phys. Rev. B 2005, Grieser Rev. Math. Phys. 14, Hai Zhang]
Seek u in the form u ( x ) = S D ϕ ( x ) where S D is the single layer potential on ∂ Ω Z x ∈ R d S D ψ ( x ) = G ( x, y ) ψ ( y ) dσ ( y ) , ∂D 1 8 2 π ln | x − y | if d = 2 > < G ( x, y ) = | x − y | d − 2 if d ≥ 3 > : (2 − d ) ω d For ψ ∈ L 2 ( ∂D ) , the function S D ψ is harmonic in D and in R d \ D , continuous across ∂D and satisfies the Pelmelj jump relations ± 1 / 2 ψ + K ∗ ∂ ν S D ψ | ± = D ψ The operator K ∗ D (or its adjoint) is the Neumann-Poincar´ e operator Z ν ( x ) · ( x − y ) K ∗ D ψ ( x ) = ψ ( y ) dσ ( y ) | x − y | 2 ∂D
The layer potential ϕ yields a solution to the PDE provided ( λ ( ω ) I − K ∗ D ) ϕ = 0 1 / ˆ ǫ ( ω ) + 1 is thus an eigenvalue of K ∗ where λ ( ω ) = D 2(1 / ˆ ǫ ( ω ) − 1) - When D is smooth ( C 1 ,α ), K ∗ D is compact consisting of a countable sequence of eigenvalues accumulating at 0 - When D is Lipschitz, K ∗ D may have continous spectrum - σ ( K ∗ D ) ⊂ [ − 1 / 2 , 1 / 2] - Goal in applications: tune the shape of D to trigger resonant frequencies at desired values of ω (cancer therapy, single molecule imaging, optoelectronics,...) - The Neumann-Poincar´ e operator naturally appears also in other situations: cloaking, pointwise estimates on gradients of solutions to elliptic PDE’s in composite media [Ammari-Ciraolo-Kang-Lim, Perfekt-Putinar, Ola, Kang-Lim-Yu, EB-Triki]
2. The Neumann-Poincar´ e operator/ Poincar´ e variationnal problem for a periodic collection of inclusions Y Ω ω Consider Ω ⊂ R 2 , smooth bounded domain, that contains a periodic collection of smooth inclusions D = ω ε = ∪ i ∈ N ε ( ω ε,i ) ω ε,i = z ε,i + εω, i ∈ N ε,i Model PDE : given f ∈ L 2 (Ω) , seek u ∈ H 1 0 (Ω) such that k in ω ε − div( A ( x ) ∇ u ) = f in Ω , A ( x ) = 1 otherwise What are the resonant frequencies of such a system ? Are there collective effects ? What happens as ε → 0 ?
As the definition of the Neumann-Poincar´ e depends on the number of inclusions, we rather work with the Poincar´ e operator T ε : H 1 0 (Ω) → H 1 0 (Ω) Z Z ∀ v ∈ H 1 0 (Ω) , ∇ T D u · ∇ v = ∇ u · ∇ v Ω ω ε 1. T ε has norm 1 and is self-adjoint, σ ( T ε ) ⊂ [0 , 1] 2. Ker ( T ε ) = { u = c i,ε on each connected component of ω ε } 3. Ker ( T ε − I ) = { u = 0 in Ω \ ω ε } ≡ H 1 0 ( ω ε ) 4. H 1 0 (Ω) = Ker ( T ε ) ⊕ Ker ( T ε − I ) ⊕ H where H is the set of u ∈ H 1 0 (Ω) such that ∆ u = 0 in ω ε ∪ (Ω \ ω ε ) ∂ω ε,i ∂ ν u | + R = 0 i ∈ N ε
for some u ∈ H 1 0 (Ω) and β ∈ R , then for any v ∈ H 1 If T ε u = βu 0 (Ω) Z Z Z ∇ T ε u · ∇ v = β ∇ u · ∇ v = ∇ u · ∇ v Ω Ω ω ε so that Z Z ∇ u · ∇ v + (1 − 1 /β ) ∇ u · ∇ v = 0 Ω \ ω ε ω ε It follows that ( λI − K ∗ u = S ω ε ϕ with ω ε ) ϕ = 0 , λ = 1 / 2 − β σ ( T ε ) = 1 / 2 − σ ( K ∗ We conclude that ω ε ) when the former is considered as an operator H ⊕ Ker ( T ω ε ) 0 − → H ⊕ Ker ( T ω ε ) 0 Our goal is to analyze ε → 0 σ ( K ∗ lim ω ε ) := { λ ∈ [0 , 1] , ∃ ε n → 0 , λ ε n ∈ σ ( T ε n ) , lim n → 0 λ ε n = λ }
- It is more convenient to work with T ε (domains of definition easier to handle) - Our work is largely inspired by the analysis of [Allaire-Conca] who studied the high frequency limit of spectra of diffusion equations using Bloch wave homogenization - As ε → 0 , the operators T ε converge to a limiting operator T ∞ defined on H 1 0 (Ω) by Z Z ∀ v ∈ H 1 0 (Ω) ∇ T ∞ u · ∇ v = | ω | ∇ u · ∇ v Ω Ω However, this convergence is weak only, and thus does not yield any information on lim ε → 0 σ ( T ε ) - To take into account the microscopic effects in the limit, we define a 2-scale version ˜ T ε of T ε on the larger space L 2 (Ω , H 1 ( ω ) / R ) , which has the same spectrum - We show that the operators ˜ T ε converge strongly to a limiting operator ˜ T 0 , and lim ε → 0 σ ( T ε ) ⊃ σ ( ˜ thus T 0 )
The key ingredient is the notion of 2-scale convergence [Allaire, Nguetseng] and the associated compactness properties Let u ε be a bounded sequence in L 2 (Ω) Theorem : 1. Then there exists u 0 ∈ L 2 (Ω × L 2 # ( Y )) such that u ε 2-scale converges weakly to u 0 , i.e. Z Z ∀ φ ∈ L 2 (Ω , C # ( Y )) , u ε ( x ) φ ( x, x/ε ) dx → u 0 ( x, y ) φ ( x, y ) dxdy Ω Ω × Y 2. Assume further that || u ε || L 2 (Ω) → || u 0 || L 2 (Ω × Y ) Then u ε 2-scale converges strongly, i.e., for any sequence v ε that 2-scale converges weakly to v 0 ∈ L 2 (Ω × L 2 # ( Y )) Z Z ∀ φ ∈ C (Ω , C # ( Y )) , u ε ( x ) v ε ( x ) φ ( x, x/ε ) dx → u 0 ( x, y ) v 0 ( x, y ) φ ( x, y ) dxdy Ω Ω × Y
3. Assume that a sequence ( u ε ) converges weakly in L 2 to some u 0 ∈ H 1 (Ω) . Then u ∈ L 2 (Ω , H 1 there exists ˆ # ( Y ) / R ) such that, up to a subsequence - u ε 2-scale converges to u - ∇ u ε 2-scale converges to ∇ u 0 ( x ) + ∇ y ˆ u ( x, y )
3. The limiting spectra : Bloch wave homogenization Following [Allaire-Conca] (see also [Cioranescu-Damlamian-Griso]) we define - an extension operator E ε : L 2 (Ω) − → L 2 (Ω × Y ) 8 u ( ε [ x/ε ] + εy ) if x ∈ ω ε,i ⊂ Ω < E ε u ( x, y ) = 0 otherwise : - a projection operator P ε : L 2 (Ω × Y ) − → L 2 (Ω) Z 8 φ ( ε [ x/ε ] + εz, { x/e } ) dz if x ∈ ω ε,i ⊂ Ω > < P ε φ ( x ) = Y > : 0 otherwise
Denoting Ω ε the union of all the cells ω ε,i that are fully contained in Ω , we have P1. Z Z Z ∀ φ ∈ L 1 (Ω) , φ dx = E ε φ dxdy + φ dx Ω Ω × Y Ω \ Ω ε P2. E ε and P ε are bounded operators with norm 1 P3. P ε is the L 2 -adjoint of E ε P4. P ε is almost a left inverse to E ε : for u ∈ L 2 (Ω) u ( x ) if x ∈ Ω ε P ε E ε u ( x ) = 0 otherwise P5. If u ∈ L 2 (Ω) , strongly in L 2 (Ω × Y ) E ε u → u P6. If ψ ∈ D (Ω , L 2 ( Y )) and if u ε ( x ) := ψ ( x, x/ε ) , then E ε u → ψ strongly in L 2 (Ω × Y ) P7. If φ ∈ L 2 (Ω × Y ) , then strongly in L 2 (Ω × Y ) E ε P ε φ → φ
In our setting, we should be cautious as the definition of T ε involves derivatives, whereas the operators E ε , P ε may not define functions in H 1 ˜ T ε := E ε T ◦ We set ε P ε with T ε : H 1 H 1 0 (Ω) − → 0 (Ω) ↓ ↑ T ◦ : H ε := H 1 (Ω) /C ( ω ε ) − → H ε ε E ε ↓ ↑ P ε T ε : L 2 (Ω , H 1 ( ω ) / R ) ˜ L 2 (Ω , H 1 ( ω ) / R ) − → where C ( ω ε ) = { u ∈ H 1 0 (Ω) , u = (const) i on ω ε,i } and the inner product on L 2 (Ω , H 1 ( ω ) / R ) is Z < φ, ψ > = ∇ y φ ( x, y ) · ∇ y ψ ( x, y ) dxdy Ω × Y
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