Mathematical modelling of plasmonic strain sensors

Definition A.1.

Let us define the one-dimensional periodic Green’s function in ℝ 2 as the function G ♯ : ℝ 2 → ℂ satisfying

Lemma A.2.

Let ξ = ( ξ 1 , ξ 2 ) . Then

satisfies (A.1).

Proof.

The proof can be found in [2] in the special case d / δ = 1 . Adding the multiplicative factor is straightforward. ∎

Lemma A.3.

The following expansions hold for G ♯ at infinity:

Proof.

As ξ 2 → + ∞ , we have

The proof is similar for ξ 2 → - ∞ . ∎

Definition B.1.

We define the one-dimensional periodic single- and double-layer potentials and the one-dimensional periodic Neumann–Poincaré operator, respectively, for B ⋐ ] - d 2 ⁢ δ , d 2 ⁢ δ [ × ℝ of class 𝒞 1 , α for some 0 < α < 1 :

Lemma B.2.

We recall the following classical results [2]:

1. For any ϕ ∈ H - 1 2 ⁢ ( ∂ ⁡ B ) , 𝒮 B , ♯ is harmonic in B and in ] - d 2 ⁢ δ , d 2 ⁢ δ [ × ℝ ∖ B ¯ .

2. The following Plemelj’s symmetrization principle identity (also known as Calderón’s identity) holds:

   𝒦 B , ♯ ⁢ 𝒮 B , ♯ = 𝒮 B , ♯ ⁢ 𝒦 B , ♯ *   on  ⁢ H - 1 2 ⁢ ( ∂ ⁡ B ) ,

   where 𝒦 B , ♯ is the L 2 -adjoint of 𝒦 B , ♯ * .

3. The operator

   𝒦 B , ♯ * : H 0 - 1 / 2 ⁢ ( ∂ ⁡ B ) → H 0 - 1 / 2 ⁢ ( ∂ ⁡ B )

   is self-adjoint in the Hilbert space ℋ 0 * ⁢ ( ∂ ⁡ B ) , which is H 0 - 1 / 2 ⁢ ( ∂ ⁡ B ) equipped with the following inner product:

   〈 u , v 〉 ℋ 0 * ⁢ ( ∂ ⁡ B ) = - 〈 u , 𝒮 B , ♯ ⁢ [ v ] 〉 - 1 2 , 1 2 ,

   with - 〈 ⋅ , ⋅ 〉 - 1 / 2 , 1 / 2 being the duality pairing between H 0 - 1 / 2 ⁢ ( ∂ ⁡ B ) and H 0 1 / 2 ⁢ ( ∂ ⁡ B ) , which makes ℋ 0 * ⁢ ( ∂ ⁡ B ) equivalent to H 0 - 1 / 2 ⁢ ( ∂ ⁡ B ) .

4. If ∂ ⁡ B is of class 𝒞 1 , α for some α > 0 , then 𝒦 B , ♯ * is compact. Let ( λ j , ϕ j ) j ∈ ℕ , be the eigenvalues and normalized eigenfunctions of 𝒦 B , ♯ * in ℋ * ⁢ ( ∂ ⁡ B ) . Then λ j ∈ ] - 1 2 , 1 2 ] , λ 0 = 1 2 and λ j → 0 as j → ∞ .

5. Since 𝒦 B , ♯ ⁢ [ 1 ] = 1 2 , it holds that

   ∫ ∂ ⁡ B ϕ j ⁢ d σ = 0   for  ⁢ j ≠ 0 .

6. The following trace formulae hold for ϕ ∈ H - 1 2 ⁢ ( ∂ ⁡ B ) :

   𝒮 B , ♯ [ ϕ ] | + = 𝒮 B , ♯ [ ϕ ] | - ,

   𝒟 B , ♯ [ ϕ ] | ± = ( ∓ 1 2 I + 𝒦 B , ♯ ) [ ϕ ] ,

   ∂ ⁡ 𝒮 B , ♯ ⁢ [ ϕ ] ∂ ⁡ ν | ± = ( ± 1 2 I + 𝒦 B , ♯ * ) [ ϕ ] .

7. The following representation formula holds:

   𝒦 B , ♯ * ⁢ [ ϕ ] = ∑ l = 0 ∞ λ j ⁢ 〈 ϕ , ϕ j 〉 ℋ 0 * ⁢ ( ∂ ⁡ B ) ⁢ ϕ j   for all  ⁢ ϕ ∈ ℋ 0 * ⁢ ( ∂ ⁡ B ) .

Lemma B.3.

Let B η = { x + η ν ( x ) , x ∈ ∂ B } for | η | small enough. Suppose that λ j ⁢ ( B ) is simple. Then

Proof.

Following [3, p. 54], we have

where ∂ ⋅ / ∂ T denotes the tangential derivative. Therefore, since λ j is assumed to be simple,

by a standard perturbation argument. Hence, by using the jump relations in Lemma B.2 (vi), it follows that