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Physics of the Phenomenon and Theoretical Background of Surface Plasmon Resonance MethodTable of Contents:
IntroductionThe study of surface plasmons is of interest both from fundamental and applications points of view [14]. By altering the structure of a metal surface, the properties of surface plasmons can be tailored, which offers the potential for developing new types of photonic devices. Surface plasmons are studied for their use in subwavelength optics [5,6], data storage, light generation [7,8], microscopy and biophotonics [911]. Therefore, surface plasmon excitation on surfaces covered by mesoparticles, bio and organic molecules, ultrathin organic and biopolymer films, and its applications are currently a highly investigated field [1215]. The surface plasmon resonance (SPR) phenomenon is most widely used in modern sensors as a sensitive method for the study of physical properties of molecular layers or coverings at the surface of noble metal thin films and nanoparticles [1619]. This chapter presents the theoretical background of SPR method from the point of view of its application primarily for sensing and related fields and results of several relevant theoretical studies. Specifically, Section 2.2 describes principal aspects of the SPR Molecular Plasmonics: Theory and Applications Volodymyr I. Chegel and Andrii M. Lopatynskyi Copyright © 2021 Jenny Stanford Publishing Pte. Ltd. ISBN 9789814800655 (Hardcover), 9780429295119 (eBook) www.jennystanford.com phenomenon in thin noble metal films and its applications. These include studies on 3D quantification of molecular covers using SPR and influence of the shape of the particles covering the metal surface on the dispersion relations of surface plasmons. Section 2.3 includes theoretical background of the localized surface plasmon resonance (LSPR) phenomenon and presents results of studies related to the LSPR sensor sensitivity optimization. These include investigations of LSPR response depending on the geometrical parameters of the sensitive element, comparative analysis of different LSPR response measurement modes, and LSPR response analysis for lightabsorbing molecular coatings. SPR Phenomenon and Theoretical Background for Its Application in SensingGeneral Interpretation of SPR Phenomenon and Most Common The oretical Research MethodsSurface plasmon polariton (SPP) oscillations on the metal surface are a density charge wave that propagates along the metal dielectric interface. It is known that SPP can be excited only when the dielectric permittivities c_{d} and e_{m} of the adjacent environments have opposite signs [20]. As a result, a surface plasmon wave cannot interact with the electromagnetic field of light that falls on a metal film, the carrier of plasmon oscillations, and the excitation of SPP can be achieved in the case of total internal reflection with the use of a prism or diffraction grating. At the same time, only ppolarized light participates in the excitation of the SPP. The dispersion equations connect the frequency of the wave with its wave vector and indicate conditions when such waves can be excited. Using Eq. (2.1), it can be noticed that the wave vector of the SPP is defined as where (y_{spp} is the SPP frequency and c is the speed of light. In Fig. 2.1, dependence ft)_{spp}(k_{spp}) is shown. Of course, with the change in surface conditions, excitation conditions ofSPPs are changing too. So the dispersion equations depend on the surface conditions where the SPP is excited. Figure 2.1 Surface plasmon polariton dispersion relation. Reprinted from Ref. [21], Copyright 2008, with permission from Elsevier. Surface plasmons cannot be excited by light falling directly from a less optically dense medium, since the value of photon wave vectors k = —JT^ is not enough. One of the simplest ways for the coordination of the mentioned vectors (maintaining the law of conservation of wave vector component parallel to the metaldielectric interface) is using the attenuated total reflection (ATR) method. For example, in the Kretschmann configuration [22] (Fig. 2.2), an optical prism with a high refractive index ^je~ > or a glass substrate coupled to the prism by an immersion liquid for optical contact is coated with a metal film. In the ATR method, light reflects with angles higher than the critical angle of the total internal reflection (#_{c}). In this case, the total internal reflection is observed (R = 1). However, part of the light passes through the glass and excites SPP in the metal when the film is thin enough to let energy flow reach the metaldielectric interface (Fig. 2.3). Therefore, the SPR phenomena appears in case k_{x} = k_{zpp} (Fig. 2.2), which can be achieved with incidence angle changing in range 9_{C} < в < 90° and can be observed as a minimum in the angular dependence of reflected ppolarized light intensity R[9) (further, SPR curve or spectrum). Rapid decrease in reflection curve, which can be observed upon angular sweep, represents absorption of light energy and appearance of resonance in a surface layer of electron plasma. The minimum reflectance intensity corresponds to the resonance angle 0_{spp}, which can be calculated using the equation where only a real part of the complex dielectric function e_{m}(ft)) = e^(co)+e"(co) was used. Of course, surface plasmons cannot exist without damping in metal film and but in most cases, е'_{1}(ю)»е"(ш). In Fig. 2.4 one can observe the SPR curve for gold. The dielectric coating located at the metal surface, for example, as a layer of adsorbed molecules, results in an increase in surface plasmon wave vector value: Figure 2.2 Kretschmann configuration of surface plasmon resonance excitation using the ATR method. Figure 2.3 SPP wave vector and electric field distribution at the metal dielectric interface. Figure 2.4 Dependence of reflected light intensity on the angle of incidence (SPR curve) for gold. According to Eq. (2.2), this leads to a shift in the minimum position d6 of the SPR curve. Using Fresnel equations for d9 calculation, it becomes possible to determine the dielectric layer's optical thickness. Using Maxwell's equations, it is possible to describe the propagation of plane, monochromatic, linearly polarized electromagnetic field in a multilayer thinfilm system. Experimentally obtained SPR curve depends on optical constants (refractive indices and absorption coefficients) of all phases, which interact with electromagnetic wave (material of prism, metal layer, surfaceadsorbed substance, external medium, and other phases, which can be included in the system under study depending on measurement conditions) and also on geometrical thickness of all layers, including gold film and adsorbed layer. This dependence can be represented as [23]:
where R_{p}[@) stands for the reflection coefficient of ppolarized electromagnetic wave, which is incident on the interface at angle 9, and Y_{p} is the total admittance of all reflective layers for a specific wavelength, which can be calculated with the following equation:
where m is the total number of layers for multilayer system (Fig. 2.5), excluding external medium; j is the number of considered layer; /?, is the phase thickness of the yth layer
Uj_{P} is an admittance of the ;th layer (for ppolarized electromagnetic waves)
where Nj = rij  ikj is the complex refractive index of investigated layer, Oj is the angle of incidence in the yth layer, A is the wavelength, dj is the y'th layer thickness, and 6_{0} is the external angle of incidence. Admittances u_{0p} and u_{m+lp} describe the prism and the external medium, respectively. Figure 2.5 Schematic representation of a multilayer system model. Other theoretical approach for description of multilayer structures uses the integral Fresnel reflection coefficient for ppolarization [24]: where r_{01}, r_{12}, r_{23} are Fresnel reflection coefficients for corresponding interfaces and p_{v} p_{2} ^{are} the phase thicknesses of layers. Equation (2.8) allows calculations ofthreelayer structures. In the case of more layers, it is convenient to use calculations within the framework of scattering matrix formalism [25]. The scattering matrix 5 is a 2x2 matrix connecting the complex strengths of electric fields on the first and last interfaces of the multilayer structure: where the indexes and denote two components of the total field propagating in positive and negative directions relative to the z axis. The matrix S can be represented as a product of the interface matrices 1 and the layer matrices L, describing the influence of the individual layers and interfaces in the multilayer structure:
Interface and layer matrices can be expressed as follows:
where t_{j(}j _{+} ц and r_{;U +} ц are the amplitude Fresnel transmission and reflection coefficients of ppolarized light for interface j(J + 1), Pj = 2ndjNjCosGj/к is the phase thickness of the yth layer, dj is the thickness of the y'th layer, N_{t} is the complex refractive index of the y'th layer, and 9j is the angle of incidence of light in the y'th layer. Amplitude Fresnel transmission and reflection coefficients of ppolarized light for interface j{j + 1) are defined by the following equations [24]:
By calculating the scattering matrix of a multilayer system, one can determine the intensity reflection coefficient of ppolarized light from a multilayer structure:
where the indexes denote the corresponding elements of the matrix S. From the calculated R[9_{0}) dependence for the investigated multilayer structure at 9_{0} > 6_{C}, an angular position of the minimum 0_{spp} can be obtained, which corresponds to the SPR phenomenon. Computer modeling showed that both approaches produce nearly the same results. Although Eq. (2.4) describes SPR phenomenon as a function dependent on the incidence angle в of monochromatic light (which means Я is a constant), it can be easily changed to obtain the reflectance equation as a function of A, where в is constant. In practice for the creation of SPR sensors, both angular and wavelength spectra are used. 
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