Metallic nanostructures provide strong electric fields which can be exploited for surface enhanced spectroscopies [1-5]. This has been a major focus of plasmonics research over the past decade. Additionally, plasmonic nanoparticles can also induce remarkably large electromagnetic field gradients near their surfaces. Sizeable field gradients can excite dipole-forbidden transitions in nearby atoms or molecules and provide unique spectroscopic fingerprinting for chemical and bimolecular sensing [6, 7]. The enhancement of higher order molecular transitions is important when their dipole moment is vanishing. In fact, the interaction between a molecule and an electromagnetic field can be decomposed in different terms where higher order multipole moments of the molecule combine with increasing spatial derivative orders of the field. The strength of the interaction decreases with the expansion order and usually only the first term of the interaction is considered, i.e. dipole-allowed transitions. However many molecules (such as H2, N2, CO2 etc.) have vanishing dipole moments (μ=0) but finite higher order moments (e.g. Q≠0). Therefore the detection of such molecules relies in the capability of obtaining sizeable field gradients. At this purpose irradiated metallic nanoparticles can be employed since they exhibit large gradients in proximity of their surface with respect to plane waves. Whereas, on the molecular scale (a~0.1-1nm), plane waves (PW) show very low gradients, metallic nanoparticles (NP) may exhibit high electric field gradients due to the induced charge densities which accumulates at the particles surface. For example a 10 nm metallic nanoparticle could enhance a quadrupolar transition rate in the mm wavelengths range up to 1010 as shown in .
Here we present the Local Angular Momentum (LAM) as a practical and physically meaningful figure of merit to predict nanostructures capability to produce large gradients . In fact the gradient of the electric field is a 9 components tensor which can be difficult to optimize when considering complex plasmonic structures. Therefore we introduce the Local Angular Momentum (LAM) as a figure of merit which expresses the effectiveness of a plasmonic structure to induce higher order transitions . The LAM is calculated as the integration of the electromagnetic angular momentum density over a small sphere in proximity of metallic NPs. The LAM can be utilized to study complex geometries and can be associated also to the rotational content of a system. In fact, high order transitions involve a transfer of angular momentum between field and molecules. An equivalent torque can be introduced and which include the LAM in its expression. As consequence, the torque transmitted to a nearby molecule can be estimated as well .
The interaction between a molecule and an electromagnetic field can be decomposed into different terms where higher order multipole moments of the molecule combine with increasing spatial derivative orders of the field (Figure 1).
The strength of the interaction decreases with expansion order. Usually only the first term of the interaction is considered, i.e. dipole-allowed transitions. However many molecules (such as H2, N2, CO2 etc.) have vanishing dipole moments (μ=0) but finite higher order moments (e.g. Q≠0). Therefore the detection of such molecules relies in the capability of obtaining sizeable field gradients. It is found that metallic nanoparticles exhibit large gradients in proximity of their surface with respect to plane waves. Whereas, on the molecular scale (a~0.1-1nm), plane waves (PW) show very low gradients, metallic nanoparticles (NP) may exhibit high electric field gradients due to the induced charge densities which accumulate at the particles surface.
Whereas, on the molecular scale (a~0.1-1nm), plane waves (PW) show very low gradients, metallic nanoparticles (NP) may exhibit high electric field gradients due to the induced charge densities which accumulates at the particles surface. With higher order transition probabilities (Γ) being proportional to the square of the gradients, the enhancement with respect to the PW case scale as (λ/d)2n  (see Figure 2).
Metallic nanoparticles possessing sharp edges where charges can accumulate, lead to electric field gradient enhancements larger than spherical NPs. In many cases, the calculations of near-field gradients cannot be performed analytically, so numerical approaches based on spatial discretization (e.g., finite-difference time domain (FDTD) or finite element method (FEM)) need to be employed. However, spatial discretization leads to some difficulties when calculating the field gradients since they can show remarkable discontinuities in the grid, making it very difficult to analyze and map the spatial distribution of the field. Therefore, the rational design of nanostructures for enhanced multipolar transition rates would greatly benefit from a figure of merit whose values would indicate the presence of large gradients.
Thus we introduce the Local Angular Momentum (LAM) as an intuitive figure of merit to predict nanostructures capability to produce large gradients that we define as :
where V is the volume of a small sphere of radius RS centered around a point r outside the NP, and ε is the permittivity of the medium. Its dimensional unit [N·m·s/m3] is equivalent to an angular momentum (AM) density since it is derived from the definition of electromagnetic field angular momentum [9-11]. In this context the LAM is based on the electromagnetic angular momentum density  and expresses the effectiveness of a plasmonic structure to induce higher order transitions. The connection between the LAM and the field gradients can be made explicit by performing a Taylor series expansion of the fields around r  :
From equation 2 it can be observed how 7 of the 9 component of the gradients can be fully recovered by the LAM since Eij = Eji and Exx+Eyy+Ezz=0. Moreover it is found that when the missing components Ezx= Exz are large, a diagonal component is large as well. In this sense all the 9 components are virtually recovered.
RESULTS AND DISCUSSION
We consider now a spherical metallic nanoparticle and we analyze how LAM depends on the wavelength if the incident electromagnetic field and on the distance from the surface of the sphere. In Figure 3, both the wavelength and the distance dependence are presented for three different gold sphere sizes: 7, 20 and 70 nm.
In Figure 3 a and b two important results are reported. The first one is that, as expected, LAM and electric field gradients follow the same trend for all nanoparticle sizes. The second one is represented by the fact that LAM and gradient enhancements, whereas being larger at the particles plasmonic resonances, saturate at longer wavelengths. This means that plasmonic nanoparticles are promising candidates to enhance ro-vibrational dipole-forbidden transitions which may lie in the infrared spectral region. In Figure 3 c and d we observe how the LAM follows the trend of the 7 shared components of the gradient identified by equation 2 at different distances from the particle surface. However, even if the total 9-components gradient cannot be reproduced at every distance, in Figure 3a we notice how the 7-components gradient start to depart from the 9-components one only at uninteresting distances where the nanoparticle does not provide any enhancement and thus |∇ENP|2 ≅ |∇Epw|2. Therefore LAM is confirmed as a good indicator to be used to evaluate quadrupole transitions .
In order to provide a clearer view of the correspondence between gradient and LAM, in Figure 4 it is explicitly shown how 7 components of the gradient tensor can be related to the three LAM components. Dashed lines of the same color underline the redundancy of information contained in some of the gradient components as discussed before.
RESULTS AND DISCUSSION
We have presented a novel parameter, the Local Angular Momentum (LAM), which can be used as a simple figure of merit to quantify the effectiveness of nanostructures in inducing quadrupole transitions which depend on the spatial gradient of the electric field. We have shown that the LAM includes all the useful information contained in the tensor gradient to calculate higher order transitions but it simpler to calculate and it can be more easily interpreted since its definition is derived from the expression of the angular momentum of light.
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