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Mathematical modelling of plasmonic strain sensors

  • Habib Ammari EMAIL logo , Pierre Millien and Alice L. Vanel
Published/Copyright: November 7, 2020

Abstract

We provide a mathematical analysis for a metasurface constructed of plasmonic nanoparticles mounted periodically on the surface of a microcapsule. We derive an effective transmission condition, which exhibits resonances depending on the inter-particle distance. When the microcapsule is deformed, the resonances are shifted. We fully characterize the dependence of these resonances on the deformation of the microcapsule, enabling the detection of strains at the microscale level. We present numerical simulations to validate our results.

MSC 2010: 35R30; 35C20

Award Identifier / Grant number: 200021–172483

Funding statement: This work was supported in part by the Swiss National Science Foundation grant number 200021–172483.

A Periodic Green’s function

Definition A.1.

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

(A.1) Δ G ( ξ ) = n δ 0 ( ξ + ( n d δ , 0 ) ) .

Lemma A.2.

Let ξ = ( ξ 1 , ξ 2 ) . Then

G ( ξ ) = 1 4 π ln [ sinh 2 ( π δ d ξ 2 ) + sin 2 ( π δ d ξ 1 ) ]

satisfies (A.1).

Proof.

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

Let us set G ( ξ , ζ ) := G ( ξ - ζ ) .

Lemma A.3.

The following expansions hold for G at infinity:

G ( ξ ) = δ ( ξ 2 - ζ 2 ) 2 d - ln 2 2 π + 𝒪 ( exp ( - ξ 2 ) ) as  ξ 2 + ,
G ( ξ ) = - δ ( ξ 2 - ζ 2 ) 2 d - ln 2 2 π + 𝒪 ( exp ( ξ 2 ) ) as  ξ 2 - .

Proof.

As ξ 2 + , we have

G ( ξ , ζ ) = 1 4 π ln [ sinh 2 ( π δ d ( ξ 2 - ζ 2 ) ) + sin 2 ( π δ d ( ξ 1 - ζ 1 ) ) ] ,
= 1 2 π ln [ sinh ( π δ d | ξ 2 - ζ 2 | ) ] + 𝒪 ( 1 + 1 sinh 2 ( ξ 2 ) ) ,
= 1 2 π ln [ exp ( π δ d | ξ 2 - ζ 2 | ) - exp ( - π δ d | ξ 2 - ζ 2 | ) ] - ln 2 2 π + 𝒪 ( ln ( 1 + exp ( - 2 ξ 2 ) ) ) ,
= 1 2 π ln [ exp ( π δ d | ξ 2 - ζ 2 | ) ] - ln 2 2 π + 𝒪 ( exp ( - ξ 2 ) ) ,
= δ ( ξ 2 - ζ 2 ) 2 d - ln 2 2 π + 𝒪 ( exp ( - ξ 2 ) ) .

The proof is similar for ξ 2 - . ∎

B Periodic boundary integral operators

In what follows, let H s ( B ) be the usual Sobolev space of order s on B and let H 0 denote the zero-mean subspace of H.

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 [ × of class 𝒞 1 , α for some 0 < α < 1 :

𝒮 B , : H - 1 2 ( B ) H loc 1 ( 2 ) , H 1 2 ( B ) ,
ϕ 𝒮 B , [ ϕ ] ( x ) = B G ( x , y ) ϕ ( y ) d σ ( y ) , x 2 , x B ,
𝒟 B , : H 1 2 ( B ) H loc 1 ( 2 ) , H 1 2 ( B ) ,
ϕ 𝒟 B , [ ϕ ] ( x ) = B G ( x , y ) ν ( y ) ϕ ( y ) d σ ( y ) , x 2 B , x B ,
𝒦 B , * : H - 1 2 ( B ) H - 1 2 ( B ) ,
ϕ 𝒦 B , * [ ϕ ] ( x ) = B G ( x , y ) ν ( x ) ϕ ( y ) d σ ( y ) , x B .

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 [ × 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 0 * ( B ) , which is H 0 - 1 / 2 ( B ) equipped with the following inner product:

    u , v 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 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 , be the eigenvalues and normalized eigenfunctions of 𝒦 B , * in * ( 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 0 * ( B ) ϕ j for all  ϕ 0 * ( B ) .

The following result on the shape derivative of the eigenvalues of 𝒦 B , * follows from [3].

Lemma B.3.

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

(B.1) λ j ( B η ) = λ j ( B ) - η ( λ j - 1 2 ) ( λ j + 1 2 ) B | ϕ j | 2 d σ + η B | 𝒮 D , T [ ϕ j ] | 2 d σ + 𝒪 ( η 2 ) .

Proof.

Following [3, p. 54], we have

𝒦 B η , * = 𝒦 B , * + η [ 𝒟 B , ν - 2 𝒮 B , T 2 ] + 𝒪 ( η 2 ) ,

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

λ j ( B η ) = λ j ( B ) + η 𝒟 B , [ ϕ j ] ν - 2 𝒮 B , [ ϕ j ] T 2 , ϕ j 0 * ( B ) + 𝒪 ( η 2 )
= λ j ( B ) - η B [ 𝒟 B , [ ϕ j ] ν - 2 𝒮 B , [ ϕ j ] T 2 ] 𝒮 B , [ ϕ j ] d σ + 𝒪 ( η 2 )
= λ j ( B ) + η B 𝒟 B , [ ϕ j ] 𝒮 B , [ ϕ j ] ν d σ - η B | 𝒮 B , T [ ϕ j ] | 2 d σ + 𝒪 ( η 2 )

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

λ j ( B η ) = λ j ( B ) + η ( λ j - 1 2 ) ( λ j + 1 2 ) B | ϕ j | 2 d σ - η B | 𝒮 D , T [ ϕ j ] | 2 d σ + 𝒪 ( η 2 ) .

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Received: 2020-03-11
Accepted: 2020-10-04
Published Online: 2020-11-07
Published in Print: 2022-02-01

© 2020 Walter de Gruyter GmbH, Berlin/Boston

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