The influence of a dielectric shell on metallic spherical nanoparticles [core-shell nanoparticles (CSNps)] in the resonant modal response of a surface plasmon resonance (SPR)-type sensor is presented. The planar multilayer sensor structure, based on the Kretschmann and surface plasmon coupled emission (SPCE) configurations, is coupled to a periodic array of these nanoparticles. In the first configuration, the CSNps are considered as a homogeneous layer with effective permittivity given by the Clausius-Mossotti mixing formula and polarizability of a core shell for a quasi-static scattering regime. In the second configuration, it performed an evaluation via the discrete complex image method (DCIM). Electromagnetic wave propagation is evaluated by the generalized reflection coefficient for multilayer structures. The analytical results are validated by numerical simulations performed via finite element method and also by experimental data. We observed that the dielectric shell thickness affects considerably the sensibility of the sensor when analyzing the change in other parameters of the CSNps array.
- SPR sensor
- wave propagation
- modal analysis
- core-shell metallic nanoparticle
- Kretschmann sensor
- SPCE configuration
Surface plasmon resonance (SPR) sensor is a photonic device capable to detect sensitive variations in the effective electromagnetic refraction index near its multi-layered structure, which can be related to intermolecular interactions or the detection of immobilized analytes, from the interaction between the analyzed samples and the evanescent field generated by surface plasmon polaritons (SPPs) wave, which propagate in the metal-dielectric interface .
Despite the first observations of the SPP that have been referenced at the beginning of the last century [2, 3], only at the beginning of the 1980s, SPP-based devices began to be applied to optical sensors with applications in gas detection and biosensors [4, 5], characterizing and quantifying biomolecular interactions , medical diagnostics, and viral monitoring , among others. The researches in SPR sensors have been increased mainly due to the development of modern nanofabrication techniques, such as the colloidal lithography, focused ion beam (FIB), and electron beam lithography (EBL) .
We evaluate an SPR sensor based on Kretschmann configuration (KR)  and surface plasmon coupled emission (SPCE)  coupled to the periodic array of (CSNps), which can represent the surface immobilization of metal nanopollutants generated, for example, from the nanocomposites manufacturing process . The former has a structure (Figure 1) comprising a multilayer formed by a prism (dielectric), a thin metal film (gold), a dielectric spacer (silicon dioxide), the periodic array of CSNps and air. The second one has a similar structure (Figure 10) and differs from the first one by the direct incidence of the optical excitation over the immobilized nanoparticles and by the suppression of a layer.
2. Kretschmann configuration
2.1. Functioning description
A functional illustration of the SPR sensor based on the Kretschmann configuration is shown in Figure 1, where the structure is coupled to a microfluidic channel with a sample flowing at a controlled rate, while a ligand substance immobilizes only the target nanoparticles (analytes) in the functionalized sensor surface. The optical excitation, coupled through the prism, is linearly polarized on transversal magnetic (TM) or transversal electric (TE) and configured in angular modulation, this is with fixed wavelength λ = 632.8 nm and variable incidence angle
For TM polarization, the SPP is excited in the gold-SiO2 interface (Figure 1) when the phase condition matches only for
The alterations in Γ(
The extra dielectric layer allows the excitation of multiple resonant wave modes, like guides modes, even in TE polarization . Using both TE and TM Γ(
2.2. Theoretical modeling
The SPR sensor in Figure 1 is modeled by the multilayer planar structure depicted in Figure 2. The incident beam, reflected beam, and incidence angle
The applied relative permittivity was prism (SF4)
In Eq. (1): is the CSNps volume fraction in the planar array; and the parameter
The propagation of the electromagnetic wave in the sensor planar structure (Figure 1(b)) is performed, in the frequency domain with time dependence of exp(−
For the recursive expressions, Eqs. (3)–(5): = 1 is the incident field amplitude in prism layer; is the propagation constant in the
2.3. Model validation and modal analysis
Herein, we compare the approximate analytical model with the results obtained by numerical simulations and experimental data to achieve the theoretical consistency between the models and study the parametric interval for validation. The numerical results were obtained through the 3D simulation environment COMSOL Multiphysics, based on the finite element method . We obtained the experimental data from the SPR spectrometer described in , which uses a He-Ne laser as the excitation source and a rotary base to control the incident angle. The sensor’s structure is fabricated by e-beam vacuum deposition process.
In Figure 3, we compare the analytical (An.), numerical (Num.), and experimental (Exp.) Γ(
The deviations An.-Exp. and Num.-Exp. in Figure 3 are 2.39 and 2.12% for the TM curves, and 1.97 and 2.05% for the TE curves, showing high accuracy for the numerical simulation and the analytical model. The differences may be due to measurement errors and roughness in the fabricated multilayer structure [15, 24].
Figure 4 shows the magnitudes of the transversal fields, in the
To validate the analytical model, in Figure 5, we compare it with Num. simulations in three cases for the sensor: (i) No CSNps; (ii) CSNps with
The hypothesis that generally increases the relative deviation characterizes the An. model limitation, such as (A) scattering losses [13, 21]; (B) dipole field interaction between CSNps in the array ; and (C) the restriction as thin of the effective layer thickness [12, 19]. As (A) grows with the CSNp size (parameters
In Figure 6, we compare the An. and Num. real magnetic fields, in the
Based on Figure 6, in Figure 7, we compare the An. and Num. transversal fields for the TE curve
2.4. Sensitivity analysis
To evaluate the sensibility, we vary the CSNps array parameters
2.4.1. Sensitivity to the shell thickness b
In Figure 8, we present the curves of
Note in Figure 8(a), Δ
2.4.2. Sensitivity to the distance d
Figure 9 presents the Δ
In Figure 9(a) we note that the increasing of
3. SPCE configuration
3.1. Functional description
The functioning of an SPR sensor in SPCE configuration (Figure 10) is based on the interaction of the field radiated by the immobilized analytes with the sensor structure, composed by a thin metal film deposited on the prism . These interactions generate the SPP wave on the air-gold surface and radiating modes in the prism, that is a high directional emission in a specific SPCE angle and depending on the nanoparticle . The high directional nature of the SPCE emission also increases the efficiency of coupled emission detection . Similar to the sensor in the Kretschmann configuration, here the SPCE configuration is excited by a laser beam operating at the wavelength of 632.8 nm.
In Figure 10(a), the sensor in the SPCE configuration is illustrated. First, a solution with the suspended analytes flows in the microfluidic channel while the target nanoparticles are immobilized on the sensor surface by a specific ligand substance. Then, the solution flow is cut off and drained until only the immobilized CSNps remain to be analyzed. The CSNps can be held by the ligand substance at
An approximate model of the sensor in Figure 10(a) is presented in Figure 10(b), where the structure is a planar multilayer with three layers: air, thin gold film, and SF4 optical prism, all represented by their respective complex permittivity. The interaction of the laser beam with the analytes and their re-annealing is equivalently modeled by a dipole, which represents the immobilized CSNps and is situated at the height
Although the dipole-type optical emitter is nonpolarized, the coupled field targeting the detector in Figure 10(b) is highly polarized in the TM . This occurs because part of the CSNps emission is naturally in the TM polarization and can excite the SPP wave on the air-gold interface, which evanescent wave passes through the thin metallic layer and radiate in the prism as a propagating wave polarized in the TM polarization. Therefore, the SPR sensor in the SPCE configuration can be understood as a reverse functioning of the Kretschmann configuration.
Note the existence of different nanoparticles in the fluidic channel (Figure 10(a)); however, only the target nanoparticles are immobilized on the sensor surface. Here, the sensor is analyzed with only immobilized CSNps and the result is a radiating TM field in a specific angle of coupling in the prism that corresponds to this nanoparticle. However, when using different ligand substances, for multichannel evaluation, different coupling angles would be detected, each angle related to a different particle of interest .
3.2. Theoretical modeling
For the SPR sensor in SPCE configuration, the SPP wave is created from interactions of the sensor structure and immobilized nanoparticles, which emit radiation and evanescent field when excited by a source. In the SPCE sensor, the nanoparticles on the substrate have dimensions smaller than the excitation wavelength, so they are represented here by infinitesimal dipoles with equivalent dipole moments or by elementary currents given by Eq. (6) [31, 32]:
The elementary current of the equivalent dipole is orientated by the laser source. To determine the induced dipole moments of an array of
For the total in Eq. (6), that has four terms, is considered a linear dependence with the dipole. The first term of represents the incident field, the second one, the reflection of the field incident on the structure, the third term, the radiation of each dipole, and the fourth term, the reflections in the structure of the field radiated by each dipole.
where is a unitary dyad, is the point of observation, and is the source point. The dipoles irradiated nonpolarized spherical waves; thus, a spherical wave radiated can be expanded as an integral of conical or cylindrical waves in the direction
The identity in Eq. (8) can be obtained from the solution of the scalar wave equation, obtained first in spherical coordinates, and later in rectangular coordinates using the three-dimensional Fourier transform. For simplicity, only the
When solving problems involving integrals like Eq. (10), several recent approaches have been proposed [36, 37], all consist of the evaluation of spectral functions using the Sommerfeld Identity with variants of the discrete complex image method (DCIM) as an acceleration tool. Here, we evaluate the integral equations directly from the electric field into a versatile application for the use of DCIM and applies the DCIM directly on the integral field equations.
The DCIM method expands the integral equations of (11) and (12) into a sum of complex terms, that is, it estimates values of complex integrals over an integration path in the complex domain, usually with a range of (0, ∞), by a finite number of samples of the integrand. A solution based on a two-level path is used . We use a sophisticated scheme, where the integrand is approximated by a superposition of complex exponentials, and this approximation is semi-analytical since it is not an exact but approximate solution.
3.3. Modal analysis
In this section, we analyze the SPR sensor of Figure 10(b) using the same relative permittivity presented in Section 2.2 for the excitation source wavelength of λ = 632.8 nm. It is considered that the radiation of the analyte occurs at this same wavelength. The results are presented for the near and far field.
3.3.1. Numerical example
Figure 11(a) shows the real part of the field
Figure 12(a) shows the two-dimensional radiation diagram of the SPCE sensor, where maintaining the operating wavelength at λ = 632.8 nm makes an evaluation of the intensity of the distant field at different heights:
Figure 12(b) shows the far field three-dimensional diagram of the SPCE sensor for the permissiveness values presented above and optimized height for
3.4. Reciprocity between SPR sensors in the KR and SPCE configuration
The SPCE sensor is physically the reverse structure of the KR sensor. In this topic, a previous evaluation of the electromagnetic reciprocity between the KR and SPCE configuration sensors is presented. In both configurations, the same materials are used, that is, air, gold, and prism (SF4). The sensor operates on a standard λ = 632.8 nm wavelength, with a gold layer of 50 nm thickness.
Figure 13 presents results for evaluation of reciprocity between the KR and SPCE sensors with the configurations described above, where Figure 13(a) shows the real part of the
Figure 13(d) illustrates the
So, a coupling angle was set on the SPCE sensor equal to the angle of plasma resonance occurred at the KR sensor, which was actually found
In this article was presented a theoretical analysis of an SPR sensor in Kretschmann (KR) and SPCE configurations, when a periodic array of core-shell nanoparticles (CSNps) is immobilized on the sensor sensitive surface. For the SPR sensor in KR configuration, the CSNps array has approximated by an effective homogeneous layer to treat the resultant structure as a planar multilayer, which improved the computational processing. For the SPCE configuration, the CSNps array has been treated as equivalent dipoles and the study is performed by the discrete complex image method (DCIM).
The approximate model of the KR sensor was validated for low size of the CSNp, parameters
The modal analysis of the KR configuration reveled that, besides the SPP surface wave, multiples guide wave modes can be excited, even in TE polarization. The thickness of the SiO2 layer can alter the order of these guide modes and for configured value guide wave modes of order 1 and 2 was observed. The characteristic field of the guide modes in TM polarization presents a surface wave in the gold-SiO2 interface, such as the SPP wave. We observed better validation of the approximate model for the TE curves.
There are evidences, such as the higher field intensity in the CSNps array region, that can indicate a greater sensitivity response for the wave modes which the minimum point is closest to the critical ATR angle. As the SiO2 layer thickness can regulate the minimum point position, this parameter can also improve the sensor sensibility.
The sensitivity analysis revealed that both radius core and shell thickness increase the sensor response, but the radius core presented a greater influence in this behave, showing larger sensitivity to the parameter
To develop the modal analysis of the SPCE sensor, we focus in the solution of the field equations for a resonant dipole over a multiple planar structure. The equations are optimized for direct application via DCIM method. The evaluation of the near field is presented and we observe that the waves radiated by the immobilized nanoparticles induce the surface plasmonic mode in the air-gold interface and radiating modes in the prism concentrated at specific angles.
The DCIM method was applied for a general solution of multilayer media using the generalized reflection coefficients. The far field results are presented by numerical simulations performed via finite element method and we observe that in SPCE configuration, the intensity of the lower lobes increases with the decrease in height z’. It is observed that the far field strength is greater for height
By the last analysis of the sensor in the SPCE configuration, we demonstrated the reciprocity of the SPP modes in the configurations KR and SPCE. It has been found that the coupling angle of the SPP mode in the SPCE configuration is equal to the angle of maximum coupling of TM0 mode KR configuration.
- Obtained from the Lorentz-Drude model with one term of interband and time dependence with exp(−iωt).
- The SPP wave is named TM0, this is a zero-order guide mode in TM polarization.