The total electromagnetic field (near-field and far-field) properties of a metallic nanoobject depends intrinsically on the geometry and on the optical properties of the involved materials. Nanoshells are composed of a core with radius

*r*_{1} and of a metallic coating or shell of thickness

*e* (see ). The core could be made of silica or just vacuum (the commonly used name of such nanoshells is hollow nanospheres with

*n*_{1} = 1.0) whereas the shell is made of gold. The permittivity of the core and the permittivity of the coating are denoted

*ε*_{1} and

*ε*_{2}, respectively. The metallic material is characterized by a complex permittivity

*ε*_{2} =

*ε*_{r} +

*jε*_{i}, (where

*j*^{2}=−1). The nanoparticles are embedded in a non-absorbing medium with permittivity

*ε*_{m}, corresponding to biological surrounding. From these parameters (size and permittivities), the electromagnetic field can be computed accurately by a variety of methods, such as analytic Mie scattering theory for spherical geometries [

22–

26] and numerical methods such as Finite Element Methods (FEM), especially for nanoscale objects of more complex geometries [

27–

30]. In the following, we compute the absorption efficiency

*Q*_{abs} (see

Eq. (3)), which must be maximum to get the more efficient thermal effect. This absorption efficiency can be deduced from

*W*_{abs} which is the rate at which energy is absorbed by the sphere [

22]:

where

**E**_{i},

**H**_{i},

**E**_{s} and

**H**_{s} are the incident and scattered electric and magnetic components, respectively. The integration is achieved on the solid angle

*d*Ω =

*ρ*^{2} sin

*θdθdϕ*. It follows that the absorption cross section is defined by:

where

*I*_{i} is the incident intensity,

being the wave vector in the surrounding medium, and

*a*_{n} and

*b*_{n} are the scattering coefficients. The absorption efficiency

*Q*_{abs} is therefore the absorption cross section

*C*_{abs} per unit of area

*S* =

*π*(

*r*_{1} +

*e*)

^{2}:

In the limit of small particles (

*r*_{2}*/λ* << 1) [

22], the computation of

*Q*_{abs} is often achieved by neglecting the order

*n* > 1 in the expansion of the series (

Eq. (3)) [

22]. Therefore, in the limit of small particle approximation, the dipolar approximation consists in considering only the

*a*_{1} and

*b*_{1} (called dipolar electric and magnetic) terms of the series in

Eq. (3). In the present study, the question is: is this dipolar approximation valid for the considered nanoshells (i.e. limiting the

*Q*_{abs} computation with n=1)?

We show in that even for small particles as nanoshells, this dipolar approximation is not valid. We summary in the acceptable intervals of parameters for inner radius

*r*_{1}, thickness

*e*, illuminating wavelength

*λ* and the relative real and imaginary parts of the bulk permittivity of gold

*ε*_{2}(

*λ*) [

31]. show the relative error between the absorption efficiency

*Q*_{abs} computed from the dipolar approximation and the series (

Eq. (3), with 60 terms, ensuring a convergence of the series, better than 10

^{−12}). The ratios of the radius to the wavelength are

*r*_{2}*/λ* = 0.092 and

*r*_{1}*/λ* = 0.089 (), and

*r*_{2}*/λ* = 0.022 and

*r*_{1}*/λ* = 0.021 (). Even if the computation is based on the systematic variation of the permittivity of the gold shell within the domain of wavelengthes of [800;1000] nm, the external radius of the nanoshell is between 800 × 0.092 = 73.6 nm and 1000 × 0.092 = 92 nm and the relative error is greater than 25% (see ). The relative error cannot be neglected in both cases. This result shows that the computation of

*Q*_{abs} with dipolar approximation is not appropriate even for small particles with thin gold coating (see ). In the case of the present study, the series of the efficiency cannot be limited to the first order and the dipolar approximation is not relevant. The dipolar approximation cannot be used for the prediction of the location of plasmon resonance and therefore for optimization in the biological window [

32].

| **Table 1**Summary of Acceptable Intervals of Parameters *r*_{1}, *e*, *λ*, *ε*_{r}(*λ*) and *ε*_{r}(*λ*) |

Therefore, for sensitivity design, the multipolar computation of

*Q*_{abs} must be achieved (i.e. more than one term must be computed in the series). In the following, the computations of the

*Q*_{abs} are achieved by using the full Mie’s theory (i.e. the series of

Eq. (3) are computed with 60 terms, ensuring a convergence of the series, better than 10

^{−12}). In the next section, we introduce the computational Monte-Carlo method used to compute the sensitivity of the absorption efficiency

*Q*_{abs} to the various parameters of the problems. The tolerances for each parameter will be deduced, so that

*Q*_{abs} remains above a given threshold.