^{1}

^{1}

^{2}

^{1}

^{1}

^{2}

In this paper, we present a theoretical study of a Surface Plasmon Resonance Sensor in the Surface Plasmon Coupled Emission (SPCE) configuration. A periodic planar array of core-shell gold nanoparticles (AuNps), chemically functionalized to aggregate fluorescent molecules, is coupled to the sensor structure. These nanoparticles, characterized as target particles, are modeled as equivalent nanodipoles. The electromagnetic modeling of the device was performed using the spectral representation of the magnetic potential by Periodic Green’s Function (PGF). Parametric results of spatial electric and magnetic fields are presented at wavelength 632.8nm. We also present a spectral analysis of the magnetic potential, where we verify the appearance of the surface plasmon polariton (SPP) waves. To validate the analytical method, we compared the limit case of small concentration of nanoparticles with published works. We also present a convergence analysis of the solution as a function of the concentration of nanoparticles in the periodic array. The results show that the theoretical method of PFG can be efficiently used as a tool for design of this sensing device.

In the last two decades, the development of new optical devices based on metallic structures has been of great interest. This is due to the interesting optical and electromagnetic properties that the metals present in the regime of high frequency [

Recently, research has shown that, on particular conditions, SPP waves can be efficiently used to control the near field intensification in metal nanostructures [

In [

The electromagnetic response of a multilayer structure excited by a planar array of nanosources can be described by the method of discrete spectral representation by Periodic Green’s Function (PGF). However, when the concentration of AuNps becomes small, the discrete spectrum converges to a continuous spectrum, which is similar to that analyzed in [

The objective of this work is to present a electromagnetic model of a surface plasmon resonance sensor in the SPCE configuration, coupled with a planar periodic array of Core-Shell AuNps. For this sensor, we obtained a spectral representation of the magnetic vector potential by the Periodic Green’s Function. To validate the method, we compared the limit case of low concentration of nanoparticles with the similar case analyzed in [

The principle of operation of the SPCE sensor is based on the interaction of the field re-irradiated by the target particle on the sensor structure, exciting plasmon modes that creates polarized propagating waves at the sensor output [

Functional illustration of the SPCE sensor coupled to a microfluidic channel.

The physical structure of the sensor is formed by a gold layer deposited above a dielectric prism and below a microfluidic channel where multicompounds, fluorophores, and Core-Shell AuNps are present in suspension. On the gold layer there is a binding substance with gold/target particle affinity that immobilizes the AuNps aggregated with fluorophores on the surface of the sensor. This substance, considered to be electrically inert, is used as a chemical spacer and acts to separate the immobilized particles and the gold sheet.

The excitation, realized by a monochromatic optical laser with wavelength

Although the radiation scattered by the target particle is not polarized, the field that mates in the region of the prism is highly polarized in TM. This characteristic can be understood by the opposite process, which occurs in the excitation of SPP waves by plane waves, exclusively polarized in TM, as in the Kretschmann configuration. For this reason, the SPCE sensor acts as a natural filter for TM polarization.

Depending on the electromagnetic characteristics of the sample formed by the multicompounds, in other words, the dielectric characteristics of the environment formed by the medium and multicompounds, in the microfluidic channel, the intensity, geometry, and angle

The electromagnetic interaction between the laser source and the Core-Shell AuNps can be described by the quasi-static Rayleigh scattering, since the wavelength of the source is much larger than the particle dimensions [

Due to the dimensions of the Core-Shell AuNps and the fluorophores, we can characterize the target particle excited by an effective dipole moment:

Equivalent between excited target particle and hertzian dipole.

In the microfluidic channel, the target particles are distributed randomly in space. Due to the fact that these are immobilized near the surface of the sensor, and the low concentration, we can approximate the sample as a planar array of uniformly distributed nanodipoles, equally distant from the gold sheet.

Since the sample was modeled as a periodic planar array, we can perform the field analysis by defining a particular cell with width

Definition of the analysis cell. (Left) 3D view of the three layers (microfluidic region

The analysis region is delimited by three volumes

Analysis volume.

The electric and magnetic fields were determined by the magnetic potential method, defined by the solution of the Helmholtz equation in the three media [

In order for the electromagnetic field to obey the periodic conditions, the potential field on surfaces

At the interfaces

From the Partial Differential Equation (PDE) defined in (

The Periodic Green’s Function is written in terms of the discrete spectral expansion in

Applying the Green identity in each medium, using the boundary conditions defined in (

The elements of the magnetic potential tensor (

The Fresnel transmission and reflection coefficients of the TE and TM modes follow the definition [

The generalized reflection and transmission coefficients TE and TM are defined in (

The

Given the magnetic potential, the magnetic and electric fields can be easily obtained through differential operations [

The discrete spectral representations in the solutions of the PGF in (

Note that, by changing the exponential base functions to cosines base functions, the domain of the spectral representation is reduced to

In [

In order to verify the validity of the PGF method, we have chosen to test the limiting case in which the nanoparticles are distant from each other in the planar array, comparing the analysis cell with the case studied in [

Component

According to [

We have also compared the results obtained from the numerical integration technique of the spectral representation used in [

As can be seen in Figures

Results of the PGF and [

From the graphical results of Figure

Based on the model presented in Section

Particularization of the equivalent electromagnetic model for the SPCE sensor.

Considering medium 1, where the nanoparticles are immobilized, as the reference medium and electrically inert (i.e., any variation in the refractive index will be compared with the reference medium), we can approximate their relative permittivity by

First, the magnetic potential field (

For better visualization and analysis, the graphical results in Figure

Magnetic Potential Field

From the differential relations (

Electric and Magnetic Field.

In order to verify the influence of the dipole orientation in the array, electric and magnetic field results of the

Differently from the results of magnetic potential (Figure

In summary, we can see that the HED mode, which predominantly excites TE modes, does not efficiently excite plasmons. In contrast, the VED mode actively excites SPP waves, thus, coupling highly polarized TM waves in the prism region, with the latter being the mode that presents the best feature in relation to the SPCE sensor. From this point, we will adopt the VED polarization as effective and consider only the case

Finally, it is worth showing that the periodic conditions are met, for both the electric and magnetic field. Figure

Component

From this result, the interaction (interference) between the fields re-irradiated by two near dipoles is verified, where, at the boundary of the cells of analysis, the periodic boundary conditions are obeyed.

Originally, in the inverse double transform of the Fourier series, which defines the magnetic potential at (

The analysis of the spectral terms

Spectral distribution in the mn plane:

For the case of Figures

A fundamental result is the characteristic of cylindrical symmetry that the spectral terms present. This is because the eigenvalues

Setting

Normalized spectral distribution of

Spectral Coefficient

Figure

It is also seen that, as the period of the cell

The results in Figure

As the cell period increases, we approach the limiting case, where the radiation comes from a single isolated particle. This case was studied in [

In order to verify the effect of the SPP1 and SPP2 poles for different values of term

By the graphical analysis of the spectral distribution in Figures

Electric and Magnetic Fields in the

For better visualization the graphs in Figure

Note in Figure

By the relation (

Generalized transmission coefficient in the gold/prism interface as a function of the coupling angle

At first, we can verify that, with the increase of refractivity index in medium 1, we have a displacement of the plasmonic angle in the 90° direction and decay of the transmitted signal. For this reason, we can assume that these will be the changes in the optical response of the sensor.

In this work a theoretical electromagnetic model was presented for a SPR sensor in the SPCE configuration, coupled with a periodic planar array of equivalent nanodipoles. The Periodic Green’s Function method was applied to the magnetic potential, where the spectral representation method of the Fourier Double Complex Series was used.

From the magnetic potential field results, we find that

In the spectral analysis, we verified the emergence of the SPP poles in the spectral domain

As the period of the analysis cell increases, the results approach the limiting case, where there are no interactions between the nanoparticles in the array. The convergence of the method depends strongly on the cell period, being faster for relatively smaller cells and becoming slow as the cell period increased. Thus, we see the need for a greater number of terms, in the convergence of the method, for relatively larger cell periods. In this situation, we have the transition from the discrete spectrum to the continuous spectrum, when the summation in the spectral representation becomes an integral. However, the transition from discrete to continuous spectrum still encounters some difficulties, since in the discrete spectrum the propagation constant

The field results showed consistency with the observation of polarized TM waves, with high directivity in the prism region, which in the far field should form the characteristic of the sensor output, the light cone. In general, the results obtained showed good agreement, both mathematical and physical, demonstrating that the PGF method is efficient and can be used as a tool in the design and optimization of the sensor in the SPCE configuration.

For future work, we propose to verify the possibility of far field calculation from the obtained electromagnetic model, thus verifying the directivity and intensity of far field in function of the refractive characteristics of the sample. In addition, we will verify the temporal decay characteristics of the radiation emitted by the fluorophores and the total response of the sensor excited by a plane wave.

Surface Plasmon Resonance

Localized Surface Plasmon Resonance

Surface Plasmon Polaritons

Periodic Green’s Function

Surface Plasmon Coupled Emission

Gold Nanoparticles

Discrete Complex Images Method

Surface Wave.

The data used to support the findings of this study are available from the corresponding author upon request.

The authors declare no conflicts of interest.

Thanks are due to the members of the Nanophotonics and Nanoelectronics Laboratory from UFPA, Nanotribo. National Council for Scientific and Technological Development - CNPq (grant numbers: 423614/2018-5).