Instituto Tecnol´ogico y de Estudios Superiores de Monterrey

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Campus Monterrey

School of Engineering and Sciences

Numerical Modelling of a Nanoplasmonic Biosensor based on a Mach-Zehnder Interferometer

A thesis presented by

Ulises F´elix Rend´on

Submitted to the

School of Engineering and Sciences

in partial fulfillment of the requirements for the degree of Master of Science

in

Nanotechnology

Monterrey, Nuevo Le´on, June 3^rd, 2020

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To my parents.

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I would like to express my deep gratitude to my research supervisor Dr. Israel de León Arizpe for the continuous support and guidance during these past years. Besides my advisor, I would like to thank my committee members, Dr. Sergio Omar Mart´ınez Chapa and Dr. Raúl Ignacio Hernández Aranda for their patience, willingness and helpful comments towards this work.

Finally, I would like to thank the Tecnol´ogico de Monterrey for this opportunity and its support on tuition, and CONACYT for the financial support provided for living.

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on a Mach-Zehnder Interferometer by

Ulises F´elix Rend´on Abstract

In the last few decades, optical biosensors based on surface plasmon resonance (SPR) have attracted increasing attention as a label-free alternative for the detection of small traces of biological and chemical markers, for application ranging from drug discovery and medical diagnosis to food quality and national defense. These approaches exploit the high sensitivity of surface plasmons polaritons (SPPs) to variations in the refractive index of the medium surrounding a thin metal film, which is caused by adsorption of the analyte molecules in the metal-dielectric interface. However, nowadays the plasmonic biosensor platforms with best performance require of complex optical configurations and bulky instrumentation, which difficult its miniaturization capability and portability, limiting its integration with other bioanalytical tools.

In this work, we propose a novel design based on a Mach-Zehnder interferometer (MZI), consisting on a gold layer with a subwavelength aperture surrounded by grooves, and a detection system based on intensity interrogation. Our proposed architecture contemplates independent control of the reference and sensing arms in a planar disposition, which allows the biosensor to operate in the region of maximum sensitivity for low-analyte concentration and avoid the requirement of using complex multilayer fabrication techniques. Through numerical simulations using the FDTD-method, we found that our platform performed satisfactorily compared to previously reported designs. Moreover, its miniaturization potential, small footprint, and simple illumination scheme make it an ideal candidate for use in integrated sensing systems, which can be further enhanced by multiplexing.

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1.1 Ilustrations of different schemes for plasmonic biosensing: (a) Kretschmann configuration, (b) nanohole array lattice, (c) groove-slit plasmonic interferometer, (d) bullseye plasmonic interferometer, (e) vertical Mach-Zehnder plasmonic interferometer, and (f) metal-insulator-metal Mach-Zehnder plasmonic interferometer . . . 3 2.1 (a)Real and (b) imaginary parts of the dielectric function for gold according to

Drude’s model in Eq. (2.19), with ωp = 1.36 × 10¹⁵Hz and Γ = 9.99 × 10¹³ Hz. . . 11 2.2 (a) Real and (b) imaginary parts of the contribution of bounded electrons to

the dielectric function of gold, according to Eq.(2.22). . . 11 2.3 (a) Real and (b) imaginary parts of the dielectric function for gold accord-

ing to Drude’s model incorporating the effect of a single interband transition.

Solid line is the analytic expression from Eq. (2.23), and circles represent the experimental data. . . 12 2.4 (a) Real and (b) imaginary parts of the dielectric function for gold using the

Drude-Lorentz model for 5 interband transitions. Solid lines represents the analytical results, expressed in Eq. (2.24) and circles represent the experimental data. . . 13 2.5 Schematic illustration of a (a) TE and (b) TM polarized wave in a system with

a x-z plane of incidence and an interface at z = 0. Medium 1, at z > 0, is characterized by relatives permittivity and permeability ₁ and µ₁, respectively; while medium 2, at z < 0,is described by 2and µ2. . . 15 2.6 Propagation length vs wavelength for an air-gold interface . . . 21 2.7 (a) Schematic representation of the penetration length in each medium of a

SPP mode at a dielectric/gold interface. (b) Penetration length vs wavelength in the air (δ1, blue solid line) and gold (δ2, orange dashed line) mediums. . . . 22

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plasmons, presented as red solid lines. The blue shaded area represent the light cone of the incident light. . . 23 2.9 Attenuated total reflection (ATR) method for the coupling of SPPs. Schematic

illustration of the (a) Otto and (b) Kretschmann configurations. (c) Dispersion curves of surface and bulk plasmons including the light cone produced by ATR methods. . . 24 2.10 Grating method for coupling of surface plasmon waves. The parameter a is

the lattice constant of the grating . . . 25 2.11 Schematic illustration of the Yee cell for the implementation of the FDTD

method. Blue arrows indicate the positions in the cell where the components of the electric field E are estimated, while red arrows point the spots where the magnetic field strength H is calculated. . . 28 2.12 Schematic illustration of the subgriding mesh scheme. Red solid lines indicate

the coarse mesh, and dashed orange lines show the region with finer mesh.

Circles point the positions where the magnetic field strength is calculated, while crosses indicate the electric field. . . 30 2.13 Schematic illustration of the PBC in the y-direction for normal incident illu-

mination . . . 32 2.14 Schematic illustration of the PBC in the y-direction for oblique incident il-

lumination. In the left panel, the wave launched has a cos(ωt) dependence, while in the right one the incident wave has sin(ωt) dependence. . . 32 2.15 Schematic illustration of PML boudary conditions. Γj indicates the reflection

coefficient in each interface, for j = 1, 2, 3. . . 34 3.1 Illustration of the biosensor architecture: a gold layer is deposited on a sub-

strate, and a central hole is milled in the metallic film, surrounded by circular grooves. Here, the system is illuminated by a radially polarized beam, with the electric field (shown as black arrows) pointing in the radial direction. As a result, SPPs are excited and propagate toward the nanohole. . . 37 3.2 Schematic illustration of the proposed approach to simulate the plasmonic

biosensor architecture. The simulation is splitted in two stages: (a) coupling of the radially polarized beam to SPPs; and (b), propagation and interference of the focusing SPPs. The two-stage simulation of the problem is necessary to reduce the computational-power demands so that it can be handled by a standard personal computer. . . 38

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of the electromagnetic field covered by said source. . . 39 3.4 Magnitude of the electric and magnetic fields components of the SPP wave

propagating in the x-direction. As shown in the inset, the system is composed by a air/gold interface at y = 0 and a SiO2substrate for y < 0.2µm. . . 41 3.5 Magnitude of the electric and magnetic field components of the customized

source located on the right side of the simulation region, transversal to the x-direction, as depicted in Fig.3.3. . . 42 3.6 Magnitude of the electric and magnetic field components obtained with the

customized source implementation for a cylindrical SPP wave focusing at the origin . . . 43 3.7 k-space representation of the electric field components shown in the upper

row of Fig. 3.6. . . 44 4.1 Schematic illustrations of the proposed architectures for plasmonic biosensor

based on Mach-Zehnder interferometry. In (a), a central slit is flanked by parallel grooves that couples the incident x-polarized light to SPPs propagating towards the slit; while in (b) a nanohole is surrounded by circular grooves, which couple light from a radially-polarized beam to SPPs converging towards the nanohole. The blue and purple areas corresponds to the sensing and reference microfluidic channels, respectively. The grating and aperture areas are covered by CYTOP. (c) shows a transverse plane representation of the physical mechanisms in the plasmonic biosensor. (d) shows a typical transmission and bulk sensitivity curves, which can be shifted by changing the refractive index nrof the fluid in the reference arm. . . 47 4.2 Wavelength-dependent H term, expressed in Eq. (4.22) for a water/gold in-

terface. Given that S_b ∝ H, we can expect that the bulk sensitivity follows the same trend as the one presented here, with a maximum at λ = 1200 nm. . 53 4.3 Schematic representation of the parameters that will be optimized. Here, t is

the thickness of the gold layer, d the width of the central aperture, L the length of the dimension occupied by the microfluidic channel in each arm, and λ the wavelength of the incident light . . . 54 4.4 Transmission (left column) and sensitivity (right column) curves as a function

of the arm length for (a)-(b) λ = 700 nm, and (c)-(d) λ = 1200 nm . . . 55

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wavelength. The blue solid line indicates the numerically results, while the red dashed line shows the propagation length L_SPP for each case, which according to Eq.(4.15) is the optimal length. . . 56 4.6 Maximum bulk sensitivity for the biosensor with the slit architecture, as ob-

tained from Eq. (4.1) The data is plotted in terms of the slit width vs wavelength. The marked red dashed line shows one of the regions where the sensitivity approaches to zero. . . 57 4.7 Sensitivity curves (left column) and plots of the transmission curves (right

column) of the wavelengths marked with dashed lines. The results in (a) were obtained using a slit width of of 400 nm, while those (b) shows the results for a slit width of 600 nm. . . 58 4.8 Maximum bulk sensitivity for the biosensor with the bull’s eye structure. The

data is plotted in terms of the hole diameter vs wavelength . . . 59 4.9 Sensitivity curves (left column) and plots of the transmittance curves (right

column) of the wavelengths marked with dashed lines for the bull’s eye structure. (a) corresponds to a nanohole with a diameter of 600 nm, while (b) shows the results for a hole diameter of 800 nm . . . 60 4.10 The effect of the shape of the aperture in the sensitivity results for the bull’s

eye structure under bulk sensitivity. In (a), an schematic illustration of an elliptical hole is presented, with different axis length along each coordinate.

The x-axis was fixed to (b) 500 nm, (c) 700 nm, and (d) 900 nm, and the z-axis was varied from 500 to 2000 nm. The figures show the maximum bulk sensitivity for each case . . . 61 4.11 Maximum bulk sensitivity as a function of the gold layer thickness t . . . 62 4.12 Schematic representation of the groove structure where the incident light is

coupled to SPPs. h refers to the height of the groove, w is the width, s the distance from the end of one groove to the beggining of the next one, and p = w + s is the period of the grating . . . 63 4.13 Optimization of the coupling efficiency η for incident light with = 1200nm.

The s and w parameters are shown in the schematic illustration in Fig. 4.12.

Three groove heights are considered: (a) h = 20 nm, (b) h = 80 nm, and (c) h = 140 nm. Each fringe of maximum sensitivity corresponds a constant optimal p = w + s, marked as a dotted line. The value of p for each fringe is shown in the upper part of each colormap. . . 64

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4.15 (a) Transmission and (b) bulk sensitivity curves for the optimized slit structure, for λ = 1200nm and a slit width d = 200nm. The maximum bulk sensitivity achieved is 4.4 RIU⁻¹ . . . 65 4.16 (a) Transmission and (b) bulk sensitivity curves for the optimized bull’s eye

design, for λ = 1200nm and a hole diameter d = 900nm. The maximum bulk sensitivity achieved is 4.01 RIU⁻¹ . . . 66 5.1 Schematic illustrations of the proposed architectures for plasmonic biosensor

based on Mach-Zehnder interferometry in a surface sensitivity scheme. First, in (a), a straight slit is milled in a gold layer deposited on a SiO₂ substrate, and parallel grooves couples the incident x-polarized light to SPPs that propagates to the center. In (b) a central hole is surrounded by concentric circular grooves, and a radially-polarized beam excites the SPP mode, creating a focusing plasmon wave. The blue and purple areas corresponds to the sensing and reference microfluidic channels, respectively. (c) shows a transverse plane representation of the physical mechanisms in the plasmonic biosensor. The bi- olayer in the sensing arm with thickness a grows when the analyte molecules are adsorbed to the metal surface previously functionalized. The change ∆a in the thickness modify the phase-difference between SPPs in different arms, resulting in changes in the transmitted light. (d) Shows a typical transmission and surface sensitivity curves, which can be shifted by changing the refractive index nrof the fluid in the reference arm . . . 70 5.2 Wavelength-dependent G parameter, expressed in Eq. (5.9) for a water/gold

interface. Given that Ss ∝ G, we can expect that the surface sensitivity follows the same trend as the one presented here. As we can see, the value of G remains close to the maximum one for a wavelength range from λ = 800 nm to λ = 1100 nm, which could lead to maximum surface sensitivity values in a broad range. . . 73 5.3 Maximum surface sensitivity for the biosensor with the slit architecture. The

data is plotted in terms of the slit width vs wavelength . . . 74 5.4 Sensitivity curves (left column) and transmission curves (right column) of the

wavelengths marked with dashed lines. (a) corresponds to a slit width d = 400 nm, while (b) shows the results for a slit width d = 600 nm . . . 75 5.5 Maximum surface sensitivity for the biosensor with the bull’s eye structure.

The data is plotted in terms of the hole diameter vs wavelength . . . 76 x

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5.7 The effect of the shape of the aperture in the sensitivity results for the bull’s eye structure for surface sensitivity. (a) Schematic illustration of an elliptical hole, with different axis length along each coordinate. The x-axis was fixed to (b) 500 nm, (c) 700 nm, and (d) 900 nm, and the z-axis was varied from 500 to 2000 nm. The figures show the maximum surface sensitivity for each case . . . 78 5.8 Maximum surface sensitivity as a function of the gold layer thickness t . . . . 79 5.9 Optimization of the coupling efficiency η for two wavelengths of incident

light: = 950 nm, shown in the upper row, and = 900 nm, in the lower row.

The s and w parameters are shown in the schematic illustration in Fig. 4.12.

Three heights are considered for each wavelength: (a) h = 20 nm, (b) h = 80 nm, and (c) h = 140 nm for λ = 950 nm; and (d) h = 20 nm, (e) h = 60 nm and (f) h = 100 nm for λ = 900 nm. Each fringe of maximum sensitivity corresponds a constant optimal p = w + s, marked as a dotted line. The value of p for each fringe is shown in the upper part of each colormap. . . 80 5.10 Maximum surface sensitivity as a function of the gold layer thickness t, for

an incident light wavelength of (a) λ = 950 nm, and (b) λ = 900 nm . . . 81 5.11 (a)Transmission and (b) surface sensitivity curves for the optimized slit struc-

ture, for λ = 950nm and a slit width d = 550 nm. The maximum surface sensitivity achieved is 2.06 × 10⁻³nm⁻¹ . . . 81 5.12 (a) Transmission and (b) surface sensitivity curves for the optimized bull’s eye

structure, for λ = 900 nm and a hole diameter of d = 700 nm. The maximum surface sensitivity achieved is 2.02 × 10⁻³nm⁻¹ . . . 82 6.1 Schematic illustration of two SPP waves with different wavelength under bulk

(upper row) and surface (lower row) sensing. The dots for the bulk sensing illustrates the homogeneous refractive index change in the whole medium, while the red layer in the surface sensing depicts a refractive index change lo- calized at the interface. The blue line represents the magnitude of the electric field as a function of the y-coordinate . . . 85

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through a linear polarizer (shown in green) and the resultant linearly-polarized light incides on an array of q-plates (shown in blue), producing radially- polarized beams centered in each of the individual biosensors. Transmitted light is collected and coupled to a CCD camera. The inlets and outlets of the microfluidic channels for the sensing (light blue) and reference (purple) arms are shown. For clarity,the microfluidic channel arrangement inside the chip is not included. (b) 3 x 3 array of bull’s eye biosensors for multiplexed sensing. The microfluidic channels for the sensing and reference arms are shown in blue and purple, respectively. (c) Representation of the CCD image of the light after transmission. The individual intensities recorded by the camera are integrated, improving the overall system’s sensitivity. . . 89

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A.1 Performance results of plasmonic biosensors reported in literature . . . 90 A.2 Performance results of plasmonic biosensors reported in literature (continuation) 91

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Abstract v

List of Figures xii

List of Tables xiii

1 Introduction 1

1.0.1 Thesis outline . . . 7

2 Theoretical framework 8 2.1 Optical properties of metals . . . 8

2.1.1 Maxwell equations . . . 8

2.1.2 Drude’s model . . . 9

2.2 Plasmons . . . 14

2.2.1 SPPs in a planar metal-dielectric interface . . . 14

2.2.2 Dispersion relation and propagation length . . . 19

2.2.3 Penetration depth . . . 21

2.3 Coupling of surface plasmons . . . 22

2.3.1 ATR methods . . . 23

2.3.2 Grating method . . . 25

2.4 FDTD method . . . 26

2.4.1 Principles of FDTD algorithm . . . 26

2.4.2 Time and space steps . . . 29

2.4.3 Boundary conditions . . . 31

3 Implementation of a convergent cylindrical SPP source 36 3.1 Problem statement . . . 36

3.2 Field components of the customized source . . . 39

3.3 Numerical implementation . . . 41

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4 Design and analysis of plasmonic architecture for bulk sensing 46

4.1 Theoretical analysis of the plasmonic MZI . . . 48

4.2 Numerical optimization . . . 53

4.2.1 Microfluidic channel length . . . 54

4.2.2 Aperture dimensions and wavelength . . . 56

4.2.3 Thickness of the gold film . . . 62

4.2.4 Grating coupler . . . 63

4.3 Results for the optimized design . . . 65

5 Design and analysis of plasmonic architecture for surface sensing 69 5.1 Theoretical analysis of the plasmonic MZI . . . 71

5.2 Numerical optimization . . . 73

5.2.1 Aperture dimension and wavelength . . . 74

5.2.2 Thickness of the gold film . . . 78

5.2.3 Grating coupler . . . 79

5.3 Results for the optimized design . . . 80

6 Discussion and Conclusions 84 6.1 Analysis of the results . . . 84

6.2 Proposal for experimental implementation of the plasmonic biosensor . . . . 86

6.3 Conclusions and future work . . . 88

Appendix A 90

Appendix B 92

Bibliography 105

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Introduction

A biosensor is an integrated device capable of providing analytic information about a biological sample, translating from the concentration of a specific analyte to a useful signal [1]. Since the proposal of the fist glucose biosensor based on an enzymatic electrode, by Leland Clark and Champs Lyon in 1962 [2], interdisciplinary research and development around biosensors has sparked and been constantly growing in the last few decades, leading to the conception of biosensors based on different physical mechanisms and the creation of over 500 companies [3]. Nowadays, biosensors are an indispensable tool in multiple areas, ranging from drug discovery [4] and medical diagnosis [5] to food quality [6] and national defense [7].

In its broadest definition, a biosensor is formed by a bioreceptor, which is the molecule or biological recognition element that recognizes the analyte of interest, and a transducer, which is the component that converts the bio-recognition event into a measurable signal [8].

Additionally, it requires an electronic part, which allows the signal to be processed and dis- played.

The performance of a biosensor can be evaluated attending to certain statistical and dy- namic parameters, which are sought to be optimized when improving a biosensor. These include the selectivity or specificity, which is the proficiency on detecting an specific desired analyte in a medium with contaminants; the sensitivity, which refers to the minimum measurable change in the amount or concentration of the analyte; the reproducibility, which refers to the ability to obtain identical results in a series of measurements; and the stability, referring to the susceptibility of the biosensor to disturbance in the medium [8].

Biosensors can be classified according to their bioreceptor or their transducer. The choice between one and the other depends on the application or the nature of the analyte.

The most common kinds of bioreceptors are based on antibody-antigen interaction, nucleic acid interaction, enzymatic interaction, cellular interaction and interaction using biomimetic materials [9]. From these, the enzyme-based biosensors, which rely on physical or chemical

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changes in the receptor due to catalysis, are the most widely used, because the high turnover rates of biocatalysts and the high specificity of enzyme-analyte interactions [10]. However, the popularity of antibody-based bioreceptors has significantly increased in the last few years, because the specificity in the binding between the antibody and the tagged antigen allows the selective sensing of analytes in the presence on interfering substances, together with providing extremely low detection limits. [9, 11].

In the case of transducers, they can be classified according to the nature of the de- tectable signal in electrochemical, optical, magnetic, calorimetric or mass change [9]. The most popular kind of transducer are electrochemical, in which the presence of the analyte generates an electrical signal in the form of a current (amperometric biosensor), a potential difference (potentiometric biosensor) or a measurable change in the conductive properties of a medium (conductometric biosensor) [12]. However, there are some drawbacks associated to their use, such as ineffective electron transfer from the receptor to the electrode surface or malfunction in real-world applications as a result of electrode fouling [13]. Moreover, amperometric biosensors, as the ones used in glucose sensors, frequently suffer from poor selectivity, because all components in the sensed solution with a potential smaller than the operating potential of the sensor contribute to the faradaic current [14].

Biosensors based on optical transducers are an attractive alternative, because they pro- vide high sensitivity and selectivity, isolation to electromagnetic interference, capability of remote monitoring in hazardous or inaccessible environments, flexibility for miniaturization and potential for multiplexing [6]. Since the proposal of the first fiber-optic biosensor for measurement of pCO₂ or pO in fluids and gases in 1975 [15], the popularity of optical biosensors has been steadily rising, leading to great improvement in their sensitivity and the development of commercial platforms already in the market [16, 17].

In general, optical biosensors comprise any kind of sensor with a transducer in which the information signal is collected by the measurement of photons in the form of absorbed, reflected and transmitted emissions in the ultraviolet (UV), visible or near-infrared (NIR) spectrum of the light. These include biosensors based on optical fibers [18], surface plasmon resonance sensors [19], resonant mirrors [20], fluorescence [21], integrated interferometers [22], surface-enhanced Raman spectroscopy [23], and chemoluminiscence [24], to name a few.

Optical biosensors can be classified into two categories: label-based biosensors, where an auxiliary molecule is used to detect the analyte by a luminescent or fluorescent method; and label-free biosensors, in which the signal to be measured is directly produced as a results of the interaction between the analyte and the transducer [25]. Although the former allows extremely sensitive sensing, label-free methods are cheaper and easier to perform, since there is no need

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of a pre-labeling process that may interfere with the analyzed biomolecule. Additionally, they allow real-time measurement of molecular interactions [19].

Surface plasmon resonance (SPR) biosensors have attracted increasing attention as a label-free method for the detection of small traces of biological and chemical markers. This approach exploits the high sensitivity of surface plasmons polaritons (SPPs) to variations in the refractive index of the medium surrounding a thin metal film, which is caused by adsorption of the analyte molecules in the metal-dielectric interface [26].

There are several approaches for the implementation of plasmonic biosensors [27, 28].

Appendix A presents a literature review with some of the platforms used in recent years and their performance.

Figure 1.1: Ilustrations of different schemes for plasmonic biosensing: (a) Kretschmann configuration, (b) nanohole array lattice, (c) groove-slit plasmonic interferometer, (d) bullseye plasmonic interferometer, (e) vertical Mach-Zehnder plasmonic interferometer, and (f) metal-insulator-metal Mach-Zehnder plasmonic interferometer

The Kretschmann configuration is one of the most popular setups for plasmonic sensing.

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As shown in Fig. 1.1 (a), a metal layer is coated on top of a high refractive index prism, and the presence of analyte in the sensing medium is identified by shining light at the prism-metal interface and monitoring the change in the reflected light [29]. In this case, the coupling between the evanescent wave of the prism and the surface plasmon wave of the metal layer is realized as a dip in the reflectivity spectrum, which is shifted when the refractive index in the sensing medium changes.

Depending on the experimental arrangement, there are several interrogation schemes for the Kretschmann configuration [30]. If the angle of the incident light is fixed, one can use a broadband light source and detector and track the spectral position of the resonance dip as the refractive index in the sensing medium changes. In this scheme, known as spectral interrogation [31], an increase in the refractive index at the sensing medium produces a red- shift in the reflectivity curve. In this approach, the sensitivity of the setup is measured in terms of nm/RIU, where RIU stands for refractive index units. In this interrogation scheme the order of magnitude for the limit of detection (LOD), which is the smallest amount of analyte that can be detected by the biosensor, is around 10⁻⁶ RIU [32].

On the other hand, if the wavelength of the incident light is fixed, the reflectance is measured as a function of the incidence angle. Then, the angle at which the coupling dip is located can be related to the analyte concentration [33]. In this scheme, known as angular interrogation, the sensitivity is given in terms of degrees/RIU, with an usual LOD lower than the spectral case, around 5 × 10⁻⁷RIU [32].

Fixing both the angle and the wavelength of the incident light, one can also monitor the change in intensity, known as intensity interrogation [34], or the variation in the phase of the reflected light, called phase interrogation [35]. Although the former alternative represents the most simple scheme from the instrumentation point of view, it results in relatively poor performance, with a typical LOD on the order of 10⁻⁵ RIU [32]. Contrarily, due to a sharp phase change in the resonance dip, phase interrogation results in a lower LOD than the other alternatives, around 10⁻⁸ RIU [32], but requires complex optical configurations to retrieve the phase change, including methods such as ellipsometry, heterodyne detection and interferometry [36].

Although the Kretschmann configuration results in high sensitivities and it is used in most commercial SPR systems [37], there are some drawbacks with its implementation. It requires steep-angle illumination to ensure total internal reflection, complicating its optical design and alignment [38]. Moreover, its intrinsic size difficults its miniaturization capability and portability, limiting its integration with other bioanalytical tools [38, 39].

Since the discovery of extraordinary optical transmission (EOT), where the light transmitted through a sub-wavelength aperture is enhanced via surface plasmons on metal films

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with nanohole arrays [40], or with a single nanohole or slit surrounded by periodic grooves [41, 42], research has sparked around the mechanisms [43, 44, 45] and applications [46] of this phenomena. In particular, there has been increasing interest in its exploitation for biosensing, because it allows the building of platforms with collinear geometry, using a standard illumination setup at normal incidence, which facilitates its alignment and integration with additional optical elements [47]. Moreover, this alternative has a smaller footprint than the one required for the Kretschmann configuration, which improves its potential for miniaturization and integration into microfluidic architectures [48].

Fig. 1.1 (b) illustrates a sensor based on a nanohole array [49]. In this approach, a periodic square array of sub-wavelength holes is patterned in a metallic film deposited on a substrate, and white light is illuminated at normal incidence. When the refractive index of the medium adjacent to the metal layer increases, the resonance peak in the transmission spectra is red-shifted, resulting in a sensitivity of 400 nm/RIU [49]. Additional studies have been made changing the hole shape to rectangles [50] and double-holes [51] producing similar results.

However, the achieved sensitivity is still lower than the ones reported for plasmonic sensors with the Kretschmann configuration, in the range of 3200 to 8000 nm/RIU [52]. An alternative approach for the nanohole-array architecture is the sensing through intensity interrogation, resulting in an improved limit of detection as low as 6.4 × 10⁻⁶ RIU [53].

One of the reasons for the underperformance of biosensors with nanohole array architecture is the broad resonance linewidths in the transmission spectrum, which result in low sensitivities [54]. Taking this challenge into account, surface plasmons interferometers have emerged as a strong alternative for biosensing, since they allow the engineering of the resonance line shape by tailoring the parameters of the interfering channels, resulting in greater control of the transmission spectrum and lower sensitivity results [54].

Fig. 1.1 (c) shows one of the most widely used architectures for surface plasmon interferometers. The biosensor consists of a thin metallic film on a glass substrate with a central slit and one or more grooves on each side. When the interferometer is illuminated at normal incidence, light that reaches the central slit interferes with the surface plasmon polaritons that travelled from the lateral grooves, resulting in an interference pattern in the transmitted light.

By varying the slit-groove distance, one can determine the phase properties and amplitude of the surface plasmons, which allows to tailor the bandwidth of the interference peaks and increase the sensitivity [54]. In the spectral interrogation scheme, this setup can achieve a limit of detection around 3 × 10⁻⁷ RIU [55]. Also, this device has great potential for real time, multiplexed sensing under intensity interrogation [54].

An alternative configuration for plasmonic interferometry is illustrated in Fig. 1.1 (d).

Here, a central hole in the metal film is surrounded by concentric circular grooves. Similar

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to the previous case, a collimated beam is used to illuminate the structure, and the light that arrives to the nanohole interferes with SPPs, which are excited and focused by the grooves [56]. A detection limit of 1.55 × 10⁻⁶RIU is achieved for spectral interrogation.

In the previous cases, the interference pattern in transmission depended on the phase difference between the plasmon polaritons and the original incident light when they arrive to the central hole or slit. Alternatively, there are approaches for nanoplasmonic interferometry where the transmission is only affected by the phase difference of the SPP waves, which are directly correlated to the refractive index in the sensing areas. Fig 1.1 (e)-(f) show these proposals, inspired in a Mach-Zehnder interferometer (MZI).

In the simplest architecture, in Fig. 1.1 (e) two slits are patterned in a metal film on top of a substrate. One of the slits is illuminated, so that SPPs are excited in both the top and bottom interfaces, and they propagate and interfere at the second slit. In this way, the bottom interface constitutes the reference arm of the MZI and the top one is the sensing arm. When the refractive index in the sensing interface changes, the plasmon arrives to the second slit with a different phase, resulting in a change in the transmission spectrum [57]. The vertical integration allows a dense array packing, and its simple structure makes it easy to fabricate in large-area arrays [57].

The vertical plasmonic MZI reaches a maximum theoretical spectral sensitivity of 9.2 × 10⁴ nm/RIU, which is achieved when the refractive indices in both the reference and sensing arm closely match [57]. Experimentally, the reported minimum limit of detection of the platform is around 1.5 × 10⁻⁵RIU [58].

A drawback in this approach is that only one slit must be illuminated, which represents a challenge for the experimental arrangement if multiplexed array sensing is sought to be implemented. One alternative, presented in Fig. 1.1 (f), is to use a metal-insulator-metal (MIM) architecture, where a dielectric layer (in this case SiO2) is placed below the top metal layer, and an extra metallic layer with just one slit is added [39, 59]. In this architecture, the reference arm is in-between the metallic layers, and a MIM SPPs interferes with the single- interface SPPs at the top interface to produce the transmission signal. This approach achieves a spectral sensitivity around 2.2 × 10³nm/RIU.

In all the alternatives for nanoplasmonic interferometry reviewed above, the only parameter used to control the phase at which the biosensor is working is the distance between slit-groove, hole-slit or slit-slit. Then, the performance of the biosensors heavily relies on the accuracy during the nanofabrication of the structure. Moreover, there is no way to independently optimize the slit or hole dimensions, or the size of the sensing area without affecting the coupling and phase between surface plasmons and the incident light.

In this work, a plasmonic platform for biosensing based on a Mach-Zehnder integrated

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interferometer is proposed. Our proposal contemplates independent control of the reference and sensing arms in a planar disposition, which allows the biosensor to operate in the region of maximum sensitivity. Taking advantage of the potential of low-cost, small footprint, miniaturization and high-multiplexing of sensors based on nanostructured metallic, the purpose of this thesis is to develop a novel integrated platform for plasmonic biosensing based on intensity interrogation with potential for multiplexing, and evaluate its performance through numerical simulation,

1.0.1 Thesis outline

First, Chapter 1 reviewes the state of the art of plasmonic sensing in Chapter 1, and Chapter 2 covers the theoretical framework involving surface plasmons in metal structures, as well as the theory of the FDTD method, which was implemented for the numerical simulations.

Chapter 3 presents the implementation of a source for focusing surface plasmons with cylindrical geometry, which will be used in later chapters. Chapter 4 and 5 cover the simulation results for bulk and surface sensing, respectively, including the parameter optimization of the platform. Finally, Chapter 6 presents the discussion of results, including a realistic proposal for experimental validation of our design, and summarizes the main results.

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Theoretical framework

2.1 Optical properties of metals

2.1.1 Maxwell equations

The characteristic optical properties of metals can be modeled starting from the macroscopic Maxwell’s equations, given by:

∇ · D = ρ, (2.1)

∇ · B = 0, (2.2)

∇ × E = −∂B

∂t, (2.3)

∇ × H = J + ∂D

∂t , (2.4)

here D denotes the electric displacement, B the magnetic flux density, E the electric field, H the magnetic field strength, J the current density and ρ the charge density.

Maxwell’s equations relate the macroscopic fields with currents and charges within the medium. However, we still need relations that describe the behavior of matter under the influence of the electric and magnetic fields. Defining the polarization density P and the magnetization density M as

P = ₀χ_eE, (2.5)

M = ₀χ_mH, (2.6)

where χeand χmare respectively the electric and magnetic susceptibilities, such relations are

8

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given by

D = ₀E + P, (2.7)

H = 1

µ₀B − M, (2.8)

where 0 and µ0 are the electric permittivity and magnetic permeability, respectively. More- over, for a linear, isotropic and non-dispersive medium, equations (2.7) and (4.18) lead to the following constitutive relations:

D = 0E, (2.9)

B = µ₀µH, (2.10)

where = 1 + χ_eis the dielectric constant of the medium and µ = 1 + χ_m the permittivity.

Here, we are limiting ourselves to nonmagnetic media, so µ = 1.

2.1.2 Drude’s model

The optical properties of metals have their origin in the presence of free conduction electrons, moving around the metal in a fixed background of positive ions. When an external electric field E impinges into the medium, the equation of motion corresponding to a small perturbation r of the electron is given by:

m d²r

dt² + Γdr dt

= eE, (2.11)

where e and m are the electron’s charge and mass, respectively, and Γ is a damping constant, corresponding to Γ = 1/τ , τ being the average time between collisions. Assuming a harmonic time dependence in the external electric field as

E(t) = E₀e^−iωt, (2.12)

we look for a solution for the small perturbation of the form

r(t) = r0e^−iωt. (2.13)

Substituting the Eqs. (2.12) and (2.13) into Eq.(2.11) and solving for r(t) we obtain:

r(t) = − e/m

(ω²+ iΓω)E(t). (2.14)

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If the molecules composing the medium have a density of N molecules per unit volume, we can add the individual contribution of each molecule to the total induced polarization as P = N er. Substituting this expression into Eq.(2.21). we get

P = − N e²

m (ω²+ iΓω)E(t). (2.15)

Using this results and Eq.(2.7), we can express the electric displacement as

D(t) = ₀E(t) − N e²

m (ω²+ iΓω)E(t)

= ₀

1 − N e²

₀m (ω²+ iΓω)

E(t)

= ₀

1 − ω²_p (ω²+ iΓω)

E(t),

(2.16)

where

ω_p = s

N e²

0m (2.17)

is known as the plasma frequency of the free electron gas. Comparing this expression with the constitutive relation in Eq.(2.9), we arrive to an expression for the dielectric function:

(ω) = 1 − ω_p²

(ω²+ iΓω). (2.18)

This is known as the Drude’s model for the electric permittivity. Rewriting the expression in its real and imaginary parts leads us to:

(ω) = 1 − ω²_p

ω²+ Γ² + i Γω_p² ω(ω²+ Γ²),

= ⁰+ i⁰⁰.

(2.19)

For gold, the plasma frequency and damping constants are ω_p = 1.36 × 10¹⁵ Hz and Γ = 9.99 × 10¹³Hz, respectively [60]. Fig. 2.1 shows the plot of the real and imaginary part of the dielectric function.

Although Drude’s model is useful to illustrate some of the most important optical properties of metal, such as a permittivity with negative real part and a large imaginary component, this model only accounts for the movement of free electrons in the conduction band. Given that not all electrons in a metal are free, it is also necessary to consider the contribution of bound electrons to model the electric permittivity at short wavelengths. In this regime,

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(a) (b)

ε' ε''

(a)

Figure 2.1: (a)Real and (b) imaginary parts of the dielectric function for gold according to Drude’s model in Eq.

(2.19), with ωp= 1.36 × 10¹⁵Hz and Γ = 9.99 × 10¹³Hz.

photon-energy is high enough to promote electrons from deeper electronic levels to the conduction band, which are known as interband transitions.

Wavelength (nm)

300 400 500 600 700 800 900 1000

ib

0 1 2 3 4 5

(b)

Wavelength (nm)

300 400 500 600 700 800 900 1000 -2

-1 0 1 2 3

(a)4

ε '

ib

ε ''

ib

Figure 2.2: (a) Real and (b) imaginary parts of the contribution of bounded electrons to the dielectric function of gold, according to Eq.(2.22).

The contribution due to bounded electrons in metals is analogous to the Lorentz model- ing of electrons in a dielectric media. Being r the displacement of a bound electron due to an electric field E, the equation of motion of the system is

mb

d²r

dt² + mbγdr

dt + αr = eE, (2.20)

where m_b is the effective mass of the bound electron, γ the damping constant and α is the spring constant of the bounded electrons. Assuming a harmonic dependence in the electric field and in the displacement, just as in equations (2.12) and (2.13), and solving we get:

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r(t) = e/m_b

(ω²₀− ω²− iγω)E(t), (2.21)

which is similar to the expression in Eq.(2.21) except from an additional term w₀ = pα/m, denoting the oscillation frequency of the bound electron. Solving as in the case of Drude’s model, we arrive to the expression of the electric permittivity due to interband transitions:

ib(ω) = 1 + ω_ib²

(ω₀²− ω²− iγω), (2.22)

where w_ib is a term analogous to the plasma frequency in free electrons, related to the frequency of the interband transition.

Fig. 2.2 illustrates the contribution of the interband transition to the real and imaginary parts of the electric permittivity in gold, for ω_in = 4.49 × 10¹⁵ Hz, γ_in = 8.96 × 10¹³ Hz and ω0 = 4.18 × 10¹⁵ Hz [60]. As we can see, the permittivity display resonance behaviour at 450 nm, where the imaginary part presents a peak. At this wavelength the absorption is maximum, which explains the color of gold when is illuminated by white light, absent of the absorbed blue and green portions in the visible range. Moreover, in the vicinity of the resonance the real part of the permittivity increases along the frequency, delimited by a minimum and a maximum value at each side. The region between this extremes is called the abnormal dispersion region [61].

Wavelength (nm)

300 400 500 600 700 800 900 1000

ib

0 2 4 6

(b)8

Wavelength (nm)

300 400 500 600 700 800 900 1000

ib

-50 -40 -30 -20 -10

(a)0

ε' ε''

Figure 2.3: (a) Real and (b) imaginary parts of the dielectric function for gold according to Drude’s model incorporating the effect of a single interband transition. Solid line is the analytic expression from Eq. (2.23), and circles represent the experimental data.

Combining the Drude’s model in Eq.(2.18) with the Lorentz model for interband transitions (2.22), we arrive to an expression of the dielectric function considering both free and bounded electrons:

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(ω) = 1 − ω_p²

(ω²+ iΓω) + ω²_ib

(ω₀²− ω²− iγω). (2.23) Fig. 2.3 shows the real and imaginary componentes of the electric permittivity considering a single interband transition. Solid lines are obtained from the analytic expression in Eq. (2.23), while the circles represent the experimental data taken from [62]. As we can see, the incorporation of the effect of bounded electrons produces a valley in the real part of the dielectric function and a peak in the imaginary part, caused by the resonance at λ =450 nm.

However, at wavelengths below 500 nm there is still discrepancy between the experimental data and our model. This is caused by the complex band structure of bound electrons in noble metals, which presents numerous interband transitions in the UV regime [63].

Wavelength (nm)

300 400 500 600 700 800 900 1000

ib

1 2 3 4 5 6 7 8

Wavelength (nm)

300 400 500 600 700 800 900 1000 -50

-40 -30 -20 -10

0 (b)

(a)

ε' ε''

Figure 2.4: (a) Real and (b) imaginary parts of the dielectric function for gold using the Drude-Lorentz model for 5 interband transitions. Solid lines represents the analytical results, expressed in Eq. (2.24) and circles represent the experimental data.

Taking into account multiple interband transitions, one can approximate the dielectric function using the modified Drude-Lorentz model [61]:

(ω) = 1 − ω²_p

(ω²+ iΓω) +

K

X

i=1

fiω_ib²

(ω_i²− ω²− iγ_iω). (2.24) where ω_iand γ_irefers to the resonance frequencies and damping for each interband transition, respectively, and f_iare weightning coefficients. Based on the parameters reported in [64], we evaluate the Drude-Lorentz model with 5 interband transitions, and the results are plotted in Fig. 2.4. Although the fitting improved compared to previous expressions, it is necessary to take into account a greater number of interband transitions to obtain a better result.

As stated before, one of the most distinctive features of noble metals is their dielectric function, with a negative real part and a large imaginary component. The negative value

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implies that electrons in metals oscillate 180^◦out of phase with respect to the driving electric field. This causes the reflectivity of metals to come close to unity, since most incident photons for wavelengths away from the resonances are reflected at the interface [63]. Moreover, the particular characteristics of the dielectric function results in a short skin depth and allows the existence of surface plasmons, that will be discussed in the next section.

2.2 Plasmons

At optical frequencies, free electrons in metals behave like plasma, with collective responses when they are subject to electromagnetic fields [60]. The quantum of plasma excitation are known as plasmons. In this sense, plasmons are quasi-particles describing the excitation of the charge density oscillations in a plasma [65].

Plasma oscillations can be sustained inside the metal (volume or bulk plasmons) or in the surface, with different resonance frequencies. In this work, we will focus on surface plasmons waves, propagating at a dielectric/metal interface as a mixed mode between the charge density wave (plasmon) and the electromagnetic wave (photon) [65]. These plasmon modes, theoretically predicted by Ritchie in 1957 [66] and experimentally confirmed by Powell and Swan in 1959 [67], are known as surface plasmon polaritons (SPPs).

2.2.1 SPPs in a planar metal-dielectric interface

Surface plasmon polaritons are bounded to the interface between a dielectric and a metal. To study its properties, we start by considering a system composed of two semi-infinite media with their interface at z = 0, characterized by their dielectric functions 1 (purely real) at z > 0 and ₂ (complex) at z < 0. Both regions are non-magnetic, so that µ1 = µ₂ = 1. We assume that the wave propagates in the x-z plane, and shows no variation in the y-direction.

We start our analysis applying Maxwell equations to this system. If there are no current densities nor external charges in the medium, Eqs. (2.3) and (2.4), together with the constitutive relations in Eqs. (2.9) and (2.10) and the cross product identity ∇ × (∇ × A) =

∇(∇ · A) − ∇²A lead to the homogeneous electromagnetic wave equation

∇²E − c²

∂²E

∂t² = 0. (2.25)

Assuming a harmonic time dependence in the electric field,

E(r, t) = E(r)e^−iωt, (2.26)

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where ω is the angular frequency of the wave, and E(r) is given by

E(r) = E₀e^ik·r, (2.27)

k being the wave vector, Eq. (2.25) yields

∇²E(r) + k₀²E(r) = 0, (2.28)

where k₀ = ^ω_c is the magnitude of the wave vector in vacuum, is the electric permittivity of the medium and c is the speed of light in vacuum. This expression is known as the Helmholtz equation. An analog result is obtained for the magnetic field strength H.

Figure 2.5: Schematic illustration of a (a) TE and (b) TM polarized wave in a system with a x-z plane of incidence and an interface at z = 0. Medium 1, at z > 0, is characterized by relatives permittivity and permeability

1and µ1, respectively; while medium 2, at z < 0,is described by 2and µ2.

Additionally, from Maxwell equations in Eqs.(2.3) and (2.4), and remembering the harmonic dependence of the electric and magnetic fields _∂t^∂ = 0, and the homogeneity in the y-direction

∂

∂y = 0

, we arrive to the set of differential equations:

∂E_y

∂z = −iωµ₀H_x, (2.29)

∂E_x

∂z − ∂E_z

∂x = iωµ₀H_y, (2.30)

∂E_y

∂x = iωµ₀H_z, (2.31)

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∂Hy

∂z = iω0Ex, (2.32)

∂E_x

∂z − ∂H_z

∂x = −iω₀E_y, (2.33)

∂H_y

∂x = −iω₀E_z. (2.34)

This differential equation system allows two set of solutions [68]: transverse electric modes, also known as TE-modes or s-polarization, where the electric field of the wave is perpen- dicular to the plane of incidence (x-z plane); and transverse magnetic modes, also called TM-modes or p-polarization, where the polarization of the wave is parallel to the plane of incidence. Fig. 2.5 shows schematics for each polarization. In TE-modes, the H_x, H_z and E_y components of the electromagnetic field are nonzero, while in TM-modes H_y, E_x, and E_zare nonzero.

Before analyzing each polarization independently, we might first study the properties of the wave vector, which is the same for both TE-modes and TM-modes. We assumed that the SPPs propagate along the x-direction and it is bounded to the interface, so the field decays in the z-direction. Therefore, we get that ky = 0 in both media, and the x-component of the wave vectors is a complex number, also known as the propagation constant. Given that the wave vector components parallel to the interface must be conserved, we conclude that

k_1x= k_2x= k_x, (2.35)

where

k₁² = 1k²₀ = k_x²+ k_1z² , (2.36) k₂² = ₂k²₀ = k_x²+ k_2z² . (2.37) Regarding k_z, to get exponentially decaying solutions this component must be purely imaginary; therefore, k_jz = ik⁰⁰_jz, for j = 1, 2. Substituting the wave vector components in Eq.

(2.27) we get for media 1 at z > 0:

E₁(r) = E₀₁e^ik^x^xe^−k^1z^z, (2.38) and for media 2 at z < 0:

E2(r) = E₀₂e^ik^x^xe^k^2z^z. (2.39) Similar expression are obtained for the magnetic field strengths. It should be noted that the

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argument of the exponential functions of the z-components have opposite sign in each media, to ensure the decay of the fields away from the interface. Next, we will analyze each polarization case separately.

TE-mode

Fig. 2.5 (a) illustrate the electric and magnetic fields of the surface plasmon waves at each media. In TE-modes the electric field is oriented in the y-direction, while the magnetic field strength have components in x and z. Therefore, from Eqs. (2.38) and (2.39), the electric field is given by:

E_1y = A₁e^ik^x^xe^−k^1z⁰⁰^z, (2.40) E_2y = A₂e^ik^x^xe^k^2z⁰⁰^z, (2.41) where A₁ and A₂ are the complex amplitudes of each wave. Substituting these expressions in Eqs. (2.29) and (2.31), we get

H_1x= −iA₁

ωµ₀k_1ze^ik^x^xe^−k^1z^z, H_1z = A₁

ωµ0

k_xe^ik^x^xe^−k^1z^z,

(2.42)

for media 1, and

H_2x= iA2

ωµ₀k_2ze^ik^x^xe^k^2z^z, H_2z = A₂

ωµ₀k_xe^ik^x^xe^k^2z^z,

(2.43)

for media 2. Boundary conditions requires that the components of the fields parallel to the interface are equal in both side. Therefore, we have to ensure the equality E1y = E2y. This results in:

E_1y= E_2y,

A1e^ik^x^xe^−k^1z⁰⁰⁽⁰⁾ = A2e^ik^x^xe^k⁰⁰^2z⁽⁰⁾, A₁ = A₂.

(2.44)

Similarly, for the magnetic field strength boundary conditions at z = 0 lead to H_1x= H_2x:

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H1x= H2x

−iA₁ k⁰⁰_1z

ωµ₀e^ik^x^xe^−k⁰⁰^1z⁽⁰⁾ = iA₂ k⁰⁰_2z

ωµ₀e^ik^x^xe^k^2z⁰⁰⁽⁰⁾

−A₁k_1z⁰⁰ = A₂k⁰⁰_2z A₁(k_1z⁰⁰ + k_2z⁰⁰ ) = 0

(2.45)

Since we require that k_1z⁰⁰ , k⁰⁰_2z > 0 for the mode to be confined at the interface, we conclude that A₁ = 0. Therefore, there are no surface plasmon polaritons with TE-polarization.

TM-modes

Fig. 2.5 (b) shows the electric and magnetic field strengths for TM-polarized waves. In this case, the electric field have x and z components, while H is oriented in the y-direction.

Analogous to the TE-mode case, from Eqs. (2.38) and (2.39), we get:

H_1y = C₁e^ik^x^xe^−k⁰⁰^1z^z, (2.46) H_2y = C₂e^ik^x^xe^k⁰⁰^2z^z. (2.47) where C1and C2are the complex amplitudes of the magnetic field strengths. Substituting H1y

and H_2yin Eqs. (2.32)-(2.33) we arrive to the expressions for the electric fields components

E_1x= iC₁

ω₀₁k_1ze^ik^x^xe^−k^1z^z, E1z = − C₁

ω₀₁kxe^ik^x^xe^−k^1z^z,

(2.48)

for media 1, and

E_2x= − iC₂

ω₀₂k_2ze^ik^x^xe^k^2z^z, E2z = − C₂

ω₀₂kxe^ik^x^xe^k^2z^z,

(2.49)

for media 2. Due to boundary conditions, the magnetic field strength parallel to the interface must be continuous, therefore H1y = H_2y. Using the expressions in equations (2.46) and

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(2.47), we get:

H_1y= H_2y,

C₁e^ik^x^xe^−k^1z⁰⁰⁽⁰⁾ = C₂e^ik^x^xe^k^2z⁰⁰⁽⁰⁾, C₁ = C₂,

(2.50)

while the boundary condition E1x= E_2xresults in E_1x= E_2x, iB₁ k_1z⁰⁰

ω₀₁e^ik^x^xe^−k^1z⁰⁰⁽⁰⁾ = −iB₂ k_2z⁰⁰

ω₀₂e^ik^x^xe^k⁰⁰^2z⁽⁰⁾, B₁k_1z⁰⁰

1

= −B₂k_2z⁰⁰

2

, k_1z⁰⁰

k_2z⁰⁰ = −₁

₂.

(2.51)

Given that both k_1z⁰⁰ and k⁰⁰_2zare positive to ensure confinement at the interface, and taking into account that ₁ is real and positive, we conclude that Re(₂)< 0. As we learned in the previous section, the dielectric function of nobel metals at the visible range display negative values in its real component. Therefore, they are perfect candidates to support surface plasmon polaritons at their interface when they are adjacent to a dielectric.

2.2.2 Dispersion relation and propagation length

Now, we can derive the expression for the dispersion relation of the SPPs. Remembering that k_jz = k⁰⁰_jz, for j = 1, 2 we substitute this into the wave vector relations in Eqs. (2.52) and (2.53), one obtains

k⁰⁰²_1z = k_x²− k₀²₁, (2.52) k⁰⁰²_2z = k_x²− k₀²2. (2.53) Next, substituting k_2z⁰⁰ = ₂k⁰⁰_1z/₁ in Eq.(2.53) we get:

k_2z⁰⁰ ² = ²₂k_1z⁰⁰²

²₁ = k²_x− k₀²₂, k_1z⁰⁰ ² = ²₁

²₂ k²_x− k₀²₂ .

(2.54)