Deduction of Electric Field Module in a Multilayer of Isotropic Materials to Detect Surface Plasmons with a Graphical User Interface

Brazilian Microwave and Optoelectronics Society-SBMO received 18 June 2020; for review 22 June 2020; accepted 6 Nov 2020 Brazilian Society of Electromagnetism-SBMag © 2021 SBMO/SBMag ISSN 2179-1074 Abstract—Electric field module for any isotropic multilayer thin film structure, is presented as analytical deduction. Analytic expressions for the electric field distribution are developed initially for a monolayer isotropic system based on Airy’s formulae and boundary conditions, with an incident monochromatic source of light. The transfer matrix method 2×2, is used to deduce the distribution of the forward and backward electric field amplitudes on the inner layers in a general multilayer thin film structure. Analytical results are simulated in Transverse-Magnetic (TM) and Transverse-Electric (TE) modes making evident (when takes place) an electric field enhancement due to surface plasmons resonance. A graphical user interface is created to make steady simulations and create new structures as desired, minimizing time and optimizing resources.


I. INTRODUCTION
Light propagation over multilayer thin film structures have been widely studied in optics [1] - [2] , applied physics [3] - [4] , bio-sensing [5] - [6] and applied electromagnetism [7] - [8] . This article presents an analytical deduction of electric field module in multilayer systems of isotropic layered thin film media, a graphical user interface (GUI) is designed to plot optical functions of reflectance, transmittance and absorptance depending on the incidence angle of a monochromatic source of light.
GUI also plot electric field module as function of transversal structure coordinate, assuming known the parameters: thicknesses of each layer, refractive index, and source wavelength. Initially, a monolayer system is analyzed, using the Airy's formulae [9] - [10] , it will helped to plot optical functions in Kretschmann & Raether geometry [11] whose results are consistent with data reported [12] . Optical functions were found from a general problem of propagation of light on isotropic multilayer systems using Yeh's matrix transfer method [13] . Optical functions are determined for Transverse-Electric (TE) and Transverse-Magnetic (TM) modes. Calculations of electric field module are made from the boundary conditions of electromagnetic fields on materials [14]. Optical  II. PROPOSED METHODOLOGY Development stages are mentioned below treating optical functions and deducing for a monolayer and multilayer structure, the electric field module from boundary conditions.

A. Optical Functions and surface plasmons
Optical functions of reflectance, transmittance and absorptance have been widely studied and simulated for multilayer thin film structures of isotropic materials. One of the most analyzed structures was created by Kretschmann & Raether which have optical crystal (known as BK7) as incident medium, gold as inner layer and air as substrate. If there is a monochromatic incident source of light toward the structure with p-polarization, then, will be an unusual absorption of light at an specific angle [11] . The angle where this phenomenon takes place is called plasmon angle due to plasmonic resonances in the interface gold-air [17]. Plasmonic excitation can be achieved in two ways, from multilayer structures [18] , or through nanoparticles [19] - [20] . On the first technique, there are Surface Plasmon Polaritons (SPP's), generated via thin film structures. The second one, nanoparticles are immerse into an external electric field generating Localized Surface Plasmons (LSP's) [21]- [22].
The treatment herein is only for the case of SPP's.
There is no way to do an experimental measure for electric (or magnetic) field, because of the inner layer thicknesses are in order of nanometers, so, the electric field enhancement due to plasmonic resonances could not be treated directly in laboratory for the TM mode. Analytical deduction and simulation of electric field distribution depending on the transversal coordinate of the structure, becomes useful to determinate how the plasmon resonance enhance the electric field amplitude.
Electric field distribution on a multilayer system has been analyzed using matrix methods [23] , and without using them [24].

B. Monolayer Structures
A monolayer structure is determined by a semi-infinite medium of incidence, an inner layer and a substrate considered as well semi-infinite. Kretschmann & Raether geometry exhibits surface plasmon resonances for an specific angle, there is another geometry created by Andreas Otto [25], consisting of optical glass (BK7) as medium of incidence, air as inner layer and gold as substrate, this geometry also evidence resonance, but, is less analyzed because of its difficult experimental fabrication. Here, 1 and 2 are the dielectric constants in this case for gold and air, respectively, is the angular frequency of the source and is the speed of light in vacuum.
In a general monolayer structure, the analysis developed by the astronomer George Bidell Airy (1.833) allows to find Fresnel coefficients [9] - [10] , for a source of light coming from the incident medium, travelling along the three media (multiple reflections are considered) giving the equations (2) and (3) as follows: Analytical deductions are performed to find explicit expressions for electric field in the inner layer, expressions for magnetic field could be obtained using Maxwell equations. For s-polarization or TE mode, the electrical field is transverse (perpendicular) to the wave's incidence plane supposing there is no waves coming from the substrate, the incident reflected and transmitted waves are given by the equation (4): are the forward and backward electric field amplitudes in each layer, refractive indexes are: 1 , 2 , 3 . It will assume, = 1 or a known amplitude defined by the source, so, = and = , are Fresnel coefficients for reflection and transmission in s-polarization. Setting the parameters: materials refraction index, inner layer thickness and wavelength of the source, the goal herein is to find and . Using boundary conditions [14], the parallel component of the electric field, must be continuous at the interfaces for = 0 and = , solving for and from the equation (4), are found explicit expressions in equations (5) and (6): In a similar way, for p-polarization, boundary condition for tangential continuous electric field is given by: Journal Performing similar algebraic steps, from equation (7), and solving for and (forward and backward) components of electric field, doing the same assumptions as s-polarization, is possible to get: , by using Snell Law's equations (8) and (9) can be written as functions only of incidence angle 1 .

C. Multilayer structures
Henceforth, Yeh's deduction [13] is followed to obtain Fresnel coefficients and optical functions in multilayer thin film structures for isotropic materials. Anisotropic treatment for multilayers presented by Yeh has an 4 × 4 matrix transfer method, nevertheless, there is a singularity reports by [27] . In Fig. 2 there is a schematic graphic showing the medium of incidence of the light source, inner layers and substrate. All materials are labeled from 0 for incident media, to + 1 in the substrate, for alayers system. In Fig. 2, are the refractive index of each material, and represent forward and backward electric field amplitudes, for = 0,1,2, … + 1 and will be thicknesses for inner layers, for = 1,2, …, . Transfer matrix method is compound by two kind of 2 × 2 matrices, dynamical matrix, and propagation matrix. Dynamical matrix is defined for s and p-polarization as shown in equations (10) and (11). Propagation matrix is defined for both polarizations as: If only two materials are involved, i.e., there is no inner layer, electric field amplitudes are obtained through dynamical matrices as: For all article purposes, 0 (incident field amplitude) is assumed known, also, there is no wave coming from the substrate, this implies: 1 ' = 0 . Equation (13) is going to generate a couple of equations with a couple of unknowns 0 and 1 ', for this case they will be Fresnel coefficients for reflection and transmission.
Electric field amplitudes become relevant from three media (one inner layer). In Fig. 3 a monolayer generic structure with incident (forward) and reflected (backward) electric field amplitudes, is shown.
In presence of four materials there are two inner layers, will be six equations with six unknown quantities, in general for + 2 materials with layers, will be 2 + 1 equations with 2 + The -Matrix relates amplitudes on incident medium with the amplitude on the substrate. This matrix is obtained with the thickness in each layer, the angle of incidence of the source of light and the wavelength of the incident beam, for both modes of propagation TE and TM, is given by: In addition, Fresnel coefficients are given by the equations (20) and (21) Doing a forward substitution, knowing the values 0 and 0 , 1 ' and 1 ' are obtained replacing in (15). Having 1 ' and 1 ' , 2 ' and 2 ' are found by the equation (16), and so on, for any number of layers. Since, previous procedure gives electric field total amplitudes in each layer, a plot as a function of the structure's transverse coordinate is capable of being programmed for all inner layers. For -th layer, equation (23) is employed to calculate the total electric field:

III. RESULTS AND DISCUSSION
Simulations from theoretical expressions deduced in the preceding section will exhibit behaviour of optical functions and electric field module. First, optical functions for a monolayer known geometry Kretschmann & Raether are calculated as so to achieve a plasmon angle in which calculate electric

B. Trilayer Structure analysis
Calculations are carried out for a Trilayer structure compound by BK7 as medium of incidence,  Kretschmann & Raether geometry is widely used on biosensing due to is possible set an organic sample in the air layer, so, reflectance curve slightly changes the resonant angle, this implies detection of urea and creatinine in solution of various concentrations [31] , or molecular interactions [32] , alternatively gold can be changed by low cost materials as or 2 3 having plasmonic resonances as well [33]. Multilayer structures simulated are the base in analysis in magneto-plasmonic structures as Au||Co||Au in which an external magnetic field (Transversal, longitudinal, and Polar) has shown an enhancing the magneto-optic Kerr effect [34].

C. Graphical User Interface
A graphical user interface involving all parameters for the multilayer thin film structure is usefulness in order to plot optical functions and electric field module [35] . The code was written on MATLAB, to run in any computer requires Runtime 8.5 or newest versions. It is possible to find similar software as the one developed by D. L. Windt [15] in which the primal approach is to find optical properties on multilayer films as reflectance or transmittance, more than to explore the presence of surface plasmons and produced enhancement of the electric field, analytical deduction for optical functions and electric field has other approach as presented here. Opti-layer software designed by A.
Tikhonravov [16], is an alternative software that focus on reflectance, transmittance and ellipsometric aspects more than electric field or surface plasmons. in each of the inner layers, as can be seen in Fig. 8. A couple of flux diagrams of back end programming for optical functions and electric field distribution are shown in figures Fig. 9 and Fig. 10.
IV. CONCLUDING REMARKS On this paper was carried out the analytical deductions for total electric field module for monolayer system, based on the Airy's equations. The general case, for propagation of light on isotropic multilayer thin film structure, 2 × 2 transfer matrix method is used, and electric field amplitudes is derived in each layer. When surface plasmons resonances (unusual absorptions) are involved in multilayer structures there is an enhancement of the electric field, present only in p-polarization impossible to measure in a laboratory because of the dimensions of the structure.
Computational tool created has made possible to calculate optical functions and electric field module, for an arbitrary number of layers, saving time, money, and resources, creating experimental structures on lab only when the structure has been optimized via software. It is essential to note, that the GUI was register in Dirección Nacional de Derechos de Autor, Bogotá, Colombia, recognized as unpublished work.
Finally, is important to exalt that calculations derived from the software were useful in the develop of thesis: Evaluación del límite de detección y sensibilidad de detectores basados en resonancias plasmónicas, Optimización de la resonancia plasmónica en multicapas | 2 | and Control de la respuesta óptica efectiva de multicapas mediante la excitación de diferentes modos plasmónicos.
Whose results were tested in lab showing consistence with theoretical and simulations framework presented.
ACKNOWLEDGEMENT This project was supported by Universidad Santo Tomás de Aquino, Bogotá, giving the time to the authors to design the GUI (Front end) and its code (Back end).