Nghiên cứu thiết kế và chế tạo kênh dẫn sóng Plasmon bề mặt Study and fabrication of surface plasmon polariton waveguides

Nghiên cứu thiết kế và chế tạo kênh dẫn sóng Plasmon bề mặt Study and fabrication of surface plasmon polariton waveguides Nghiên cứu thiết kế và chế tạo kênh dẫn sóng Plasmon bề mặt Study and fabrication of surface plasmon polariton waveguides luận văn tốt nghiệp thạc sĩ

MINISTRY OF EDUCATION AND TRAINING

HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY

INTERNATIONAL TRAINING INSTITUTE FOR MATERIALS SCIENCE

-

NGUYEN VAN CHINH

STUDY AND FABRICATION OF SURFACE PLASMON

POLARITON WAVEGUIDES

MASTER THESIS OF MATERIALS SCIENCE

ITIMS BATCHS 2014

Hanoi - 2016

MINISTRY OF EDUCATION AND TRAINING

HANOI UNIVERSITY OF SCIENCE AND TECHNOLOGY

INTERNATIONAL TRAINING INSTITUTE FOR MATERIALS SCIENCE

-

NGUYEN VAN CHINH

STUDY AND FABRICATION OF SURFACE PLASMON

POLARITON WAVEGUIDES

Specialized: Science and Engineering of Electronic Materials

MASTER THESIS OF MATERIALS SCIENCE

ITIMS BATCHS 2014

SUPERVISOR: Dr CHU MANH HOANG

Hanoi - 2016

Finally, thanks should also be given to my family and friends, who always supported me in my study

LIST OF PUBLICATIONS

1 Nguyen Van Chinh, Nguyen Thanh Huong, and Chu Manh Hoang (2015),

―Design and simulation of triangular wedge surface plasmon polariton waveguide‖,

9 th Vietnam National Conference of Solid Physics and Materials Science , pp

3 Nguyen Van Chinh, Nguyen Thanh Huong, Vu Ngoc Hung, Chu Manh

Hoang (2016), ―Characteristics of Trapezoidal-Shaped Plasmonic Waveguide‖, The

3 rd International Conference on Advanced Materials and Nanotechnology, pp

111-114, 2016, ISBN: 978-604-95-0010-7

STATEMENT OF ORIGINAL AUTHORSHIP

I hereby declare that the results presented in the thesis are performed by the author The research contained in this thesis has not been previously submitted to meet requirements for an award at this or any higher education institution

Date: 30/09/2016

Signature:

CHAPTER 1 FUNDAMENTALS OF PLASMONICS 1

1.1 History of development 1

1.2 Fundamentals of Surface Plasmon Polaritons 3

1.3 Wedge surface plasmon polariton waveguides 6

1.3.1 Conventional wedge waveguides 7

1.3.2 Hybrid wedge waveguides 9

1.3.3 Fabrication of wedge waveguides 11

1.3.4 Exciting surface plasmon polariton mode in the wedge waveguide 13

1.3.5 Applications of wedge waveguides 15

Purpose of this thesis 19

CHAPTER 2 DESIGN AND SIMULATION OF WAVEGUIDE 20

2.1 Basic theory for the simulation of waveguides 20

2.2 Triangular Waveguide 21

2.2.1 Structure of triangular waveguide 21

2.2.2 Meshing 22

2.2.3 The thickness of metal layer 23

2.2.4 The tip angle of triangular waveguide 28

2.2.5 Height of triangular waveguide 30

2.2.6 The refractive index of cladding medium 32

2.3 Trapezoidal waveguide 34

2.3.1 Model of the waveguides 34

2.3.2 Results and discussion 35

CHAPTER 3 FABRICATION OF WAVEGUIDE 40

3.1 Process of fabrication 40

3.1.1 Oxidation of SOI wafer 42

3.1.2 Photolithography 43

3.1.3 Isotropic wet–etching in buffered Hydrofluoric acid solution 45

3.1.4 Anisotropic etching in Potassium Hydroxide 46

3.1.5 Sputtering metal layer 47

3.2 Results of the fabrication of waveguide 48

CONCLUSIONS 51

SUGGESTED FURTURE WORKS 51

REFERENCES 52

LIST OF FIGURES

Figure 2.10: Propagation characteristic at different heights 32

GLOSSARY OF TERMS AND ABBREVIATIONS

CHAPTER 1 FUNDAMENTALS OF PLASMONICS

1.1 History of development

So long before scientists study about Plasmonic, ancient artisans used their

properties to generate vibrant colors in glass artifacts One of the most famous

examples is the Roman glass work dating from the Byzantine Empire in the 4th

century AD - the Lycurgus Cup (Fig.1) Under normal lighting, the cup appears

green, and when illuminated from within it becomes red color Here, gold and silver

nanoparticles of different sizes and shapes were embedded in glass to create

beautiful color

Figure 1.1 Lycurgus cup illuminated under normal external lighting (left)

and from within (right)

In the early 20th century, Robert Wood observed a pattern of extraordinary dark

and light bands in the reflected light, when he illuminated polarized light on a

mirror with a diffraction grating on its surface This is considered as the first

observation of plasmon

About fifty years later, in 1956, David Pines theorized to explain the

characteristic energy losses experienced by fast electrons traveling through metals is

due to the collective oscillations of free electrons in the metal These oscillations are

similar to the plasma oscillations in gas discharges, he call them ―plasmon‖ In

1957, Rufus Ritchie published a study on electron energy losses in thin film, in which it is shown that plasmon modes can exist near the surface of metals This study presented the first theoretical description of surface plasmons One year later, John Joseph Hopfield introduced the term ‗polariton‘ for the coupled oscillation of bound electrons and light inside transparent media

In 1968, Andreas Otto and Erich Kretschmann presented two methods for exciting the surface plasmon on metal film, making experiments on surface plasmons easily accessible to many scientists From here, the major advance in study of surface plasmon was made

Figure 1.2 Number of papers containing “surface plasmon” in the title [1]

Fig.1.2 shows the growth of the plasmonic field since 1960 to 2008 In middle 1990‘s, the number of papers on Plasmonic increased rapidly due to the achievements of nanofabrication techniques, physical analysis techniques and simulation codes

Although this is a promising field, it has not been extensively researched in Vietnam Only a few scientists interested in this field as Prof Van Hieu Nguyen, Bich Ha Nguyen, Van Hop Nguyen The study of these scientists mainly is on theoretical calculations of bulk plasmon and localized plasmon

1.2 Fundamentals of Surface Plasmon Polaritons

Surface plasmon polaritons (SPPs) are resulting from strong coupling of the electromagnetic wave with the collective oscillations of electrons at metal – dielectric interface As a surface electromagnetic wave, we can use the Maxwell theory for describing this phenomenon

Firstly, we detail in SPPs that propagate on single interface of a metal and a dielectric medium For solving this problem, we consider the haft space z < 0 is metal medium with the complex dielectric function ε1(ω) and the other haft space z

> 0 is dielectric medium with real dielectric permittivity ε2 (Figure 1.3) By choosing the x – axis in the direction of wave propagation, we get the dispersion of field at z = 0 plane [25]

with + for z > 0 and – for z < 0 Here, kx and kz is the tangential and normal component of wave vector, respectively Solving the Maxwell equations with the continuous conditions of field at the interface, we obtain the tangential component

Figure 1.3 Geometry for SPPs propagation at a single interface between a

metal and a dielectric

Typically, ( ) and | | , we get the complex

of tangential wave vector characterizes the attenuation of SPP The propagation length where the intensity reduce to 1/e is defined by

In the case of metal with low losses and | | , propagation length is given approximately by

We see that if we want the propagation length to be as large as possible, we need

a metal with a large negative real part and very low value of losses In the case of sliver, the propagation length varies in the 50 - 300 μm range for radiation wavelength in the 0.5 – 1.5 μm

We now turn to calculate the components of the wave number in the z-direction,

in order to determine the penetration depths in the dielectric and metal, respectively Using (1.2) and (1.6) we obtain

Here, d is the thickness of metal layer These equations are so complex In fact,

we only study the simple cases such as the symmetric structure In this condition, the coupling divides SPP into symmetric and anti-symmetric mode The

Figure 1.4 Geometry of the three layers system

symmetric mode reduces the wavenumber while anti-symmetric mode increases the wavenumber of SPP [15]

In the case SPP propagates in a metal wedge, we can approximate to the propagation of SPP in a thin metal film with the thickness d continuously changing

to zero

1.3 Wedge surface plasmon polariton waveguides

To achieve photonic circuits with high integration density, it is necessary to develop miniature optics components for high-speed signal generation, propagation, detection and processing A recent research direction has been devoted for developing nano-scale optical waveguides which are capacity of wave propagation with strong field confinement permitting denser waveguide packaging without crosstalk and lower waveguide bending loss Three kinds of typical nanophotonic waveguides have been developed, which includes nanophotonic wires [13], [22], photonic-crystal waveguides [23] and nanoplasmonic waveguides [2] The former two nanophotonic waveguides, which utilize nano-structures with ultra-high index contrast, are limited due to the classical optical diffraction phenomena In contrast, a nanoplasmonic waveguide can break the diffraction limit and enable deep sub-wavelength confinement and wave-guiding of light, which makes it become a very

attractive candidate for ultra-high integration density A nanoplasmonic waveguide also offers a way to merge electronics and photonics so that it is potential to realize ultra-small optoelectronic integrated circuits for low power—consumption and high speed signal generation, processing as well as detection Numerous of nanoplasmonic waveguides have been proposed and demonstrated in past year, including metal nano-slot waveguides [7], strip waveguides [10], metal V-groove waveguides [6], wedge waveguides [1] and hybrid waveguides [12][3] In this study, we concentrate on considering structures and characteristics of wedge waveguides as well as their applications

1.3.1 Conventional wedge waveguides

The wedge waveguide structure is proposed to fulfill the goal of achieving modal sizes smaller than the wavelength while keeping a relatively high propagation length Figure 1.5 shows wedge waveguide structures developing from original researches The first wedge waveguide is shown in Fig 1.5 (a), which consists of a

V wedge on the metal substrate The material around the waveguide is dielectric, which is usually to be air The second structure type of the wedge waveguide is shown in Fig 1.5 (b), which is formed from the waveguide (a) by covering the metal wedge by a dielectric layer The third typical wedge waveguide is composed

of a dielectric wedge waveguide covered by a metal layer as shown in Fig 1.5 (c)

As considering the dependence of the modal characteristics of WPPs as a function

of the relevant geometric parameters, it shows that when the height of the wedge decreases, the modal effective index neff (i.e., modal wave vector divided by wave vector in vacuum) tends to the effective index of a SPP on flat surface (for h < hc, neff reaches the effective index of a SPP and the mode is no longer guided) Notice that a low effective index is equivalent to a more extended field The mode size and propagation length decrease when the wedge height increases The behavior of the WPP () modal characteristics as the angle  increases is reminiscent to what occurs when the height h decreases There is, however, a major difference: there is

no critical angle above which the mode is no longer guided As  is increased

towards 180o, propagation length, neff, and modal size tend to those of a SPP on a flat surface Modal size rapidly increases as the angle grows, but numerical simulations show wave guiding no matter how large the angle is (whenever <

180o)

Figure 1.5 Scheme of conventional wedge waveguides; (a) a V wedge on the

metal substrate; the material around the waveguide is dielectric, which is usually to be air; (b) the waveguide is composed of a dielectric wedge waveguide covered by a metal layer; ( c) the structure similar to (a) but covered by a dielectric layer above the metal wedge, which is also called to be

dielectric-loaded wedge waveguide

Comparing the CPP and WPP modes supported by the V groove and the wedge which have exactly the same dimension, one can find that while the propagation lengths are comparable to each other, the WPP mode has a significantly smaller mode size than the CPP mode This is mainly due to the fact that the CPP mode is hybridized with wedge modes supported by the edges at both sides of the groove Moreover, it is worth to notice that the cutoff groove height of the CPP mode is larger than that of the WPP mode The cutoff groove height is understood that when the groove height is smaller this value, the mode is no longer guided So in terms of mode confinement and propagation loss, the WPP mode seems to be superior to the CPP mode

1.3.2 Hybrid wedge waveguides

The wedge waveguide structure is proposed to fulfill the goal of achieving modal sizes smaller than the wavelength [22] However, it is due to Ohmic losses in metal, which lead to relatively short propagation length in the conventional wedge waveguide In order to improve this limit, a number of hybrid surface plasmon waveguides have been developed in past years, which exploit capability of tight light confinement of metal wedge while achieving relatively long propagation length [3], [12], [17]–[19], [27], [28] Figure 1.6 shows dimensional (2D) cross-

sectional schematic of recently developed hybrid wedge waveguides

The waveguide in Fig 1.6 (a) is developed from the waveguide shown in Fig 1.5 (c), in which a high-index dielectric layer is covered on the low-index dielectric layer Owing to the superior guiding properties of wedge plasmons in conjunction with high refractive index contrast near wedge tips, the modal sizes can be squeezed into significantly smaller spaces that those of their conventional wedge counterparts Hybrid waveguides shown in Figs 1.6 (b) – (c) are composed of a low-index dielectric layer separating a high-index dielectric waveguide and metal layer Electromagnetic field is tightly confined in the low-index dielectric gap Based on the reported results, the electromagnetic field mode can be confined well

in the gap at nanoscale The propagation length of these hybrid waveguides is also confirmed to be higher than that of the conventional wedge waveguides shown in Fig 1.5 However, one shortcoming of these hybrid waveguides is that the confinement of the electromagnetic field within low-index dielectric region is poor

in the lateral direction In order to exploit the low loss propagation of the waveguides shown in Figs 1.6 (a) – (c), one uses the property of strong field confinement at the tip of wedge structures to minimize further the size of guided mode Figures 1.6 (d) – (f) show basic structures of hybrid waveguides employing wedge waveguide structures Figures 1.6 (g) – (h) show double hybrid waveguides developed from the single waveguides in Figs 1.6 (e) – (f), respectively With strong coupling between the dielectric nanowire mode and long-range surface

Figure 1.6 Two-dimensional (2D) cross -sectional schematic of t he typical

hybrid wedge waveguides: (a) hybrid dielectric -loaded wedge waveguide; (b) the structure comprises a silica buffer layer sandwiched between a silver cladding and a triangular silicon wedge; (c) consisting of a triangle semiconductor waveguide surrounding by a metal layer through an isolating gap layer; (d) consisting of a triangle semiconductor waveguide placed on a silver substrate with a gap; (e) a circular semiconductor nanowire on a metal wedge waveguide; (f) the structure composed of a wedge dielectric waveguide placed on a wedge metal waveguide; (g) and (h) are single -coupling and

double-coupling wedge waveguides respectively

plasmon polariton mode, both deep subwavelength mode confinement and low propagation loss are also achieved to the waveguide in Fig 1.6 (g) Furthermore, when compared to the waveguide in Fig 1.6 (e), this waveguide can provide an order of magnitude longer propagation length similar level of mode confinement The optimized research results show that the waveguide in Fig 1.6 (h) demonstrates

a 9-fold enhancement in mode confinement for the same propagation length or a 2.4-fold extended propagation length for the same mode confinement compared to the waveguide in Fig 1.6 (g)

1.3.3 Fabrication of wedge waveguides

Figure 1.7 Steps for fabricating the wedge waveguide s: (1) a silicon wafer

is covered with a layer of silicon oxide and photoresist, (2) resist is exposed and developed, and the pattern is transferred into the oxide, (3) V -grooves are etches in silicon, (4) gold is deposited after oxide removal, (5) nickel is deposited, (6) silicon substrate is dissolved leaving gold 70.5° -wedges

Various models of conventional and hybrid wedge waveguides have been introduced and investigated theoretically in past years However, there are few experimental demonstrations about wedge waveguides Especially, the wedge waveguides with sharp apex are not easy to fabricate In addition, the wedge waveguides are not usually fabricated directly by conventional dry-etching, but using focus ion beam (FIB) mining Using FIB, a 40o silver triangular nano-wedge was realized [23] There are also not so many reports on wedge waveguides directly fabricated from wet etching, which is well known as wet anisotropic etching of single crystal silicon in hydroxide solution This might be due to undercut etching rate of (111) plane much lower than other planes, which leads to difficulty in

achieving wedge structure However, one can obtain wedge waveguide structure by hybrid fabrication process, which combines wet etching and flip-technique to form metal wedges [5] Using this hybrid fabrication process, wedges with a larger dimension and higher angles around 700 are obtained In the ref [5], a process composed of chemically etching the V groove into a Si substrate and then transferring the pattern into the wedge structures by depositing metals into the groove and etching away the Si substrate was proposed to realize wedge angles around 700 Figure 1.7 shows the fabrication process proposed by [5] In order to achieve sharper wedge angles, trench modifying via oxidization process prior to metal layer deposition needs to be applied [8]

Figure 1.8 Schematic setup for the fabrication of SPP structures (on the left) and plasmon propagation in a dielectric waveguide fabricated by 2PP (on

the right) SPP is excited on the right hand side

Another novel method based on the application of two-photon polymerization (2PP) technique was proposed for fabricating plasmonic components [25] Here, dielectric structures are fabricated on glass or metal substrates in terms of 2PP of the inorganic-organic hybrid polymer ORMOCER provided by Microresit Technology This polymer can be polymerized by using a radical photoinitiator In order to fabricate dielectric structures by 2PP, a femto second oscillator, Spectra Physics Model Tsunami, is used This system delivers laser pulses at a wavelength of 780

nm with pulse duration of 80 fs, and repetition rate of 80MHz In this experiment,

an average laser power of 40 mW is applied The schematic setup used for the sample preparation is illustrated in Fig.1.8 Femto second laser pulses are focused

by an immersion-oil objective (Nikonwitha100×magnification and a numerical aperture of 1.3) A liquid polymer droplet is sandwiched between the substrate and a cover glass with a thickness of 150µm Their separation is fixed by a plastic frame with the size of 18×18mm2 and thickness of 100µm For the fabrication of surface structures, the laser beam is focused through the cover glass and the ORMOCER layer on the substrate surface During the structuring, the laser beam is scanned along the sample surface by a galvo-scanner system After completion of the 2PP and development of the surface polymer structures, the samples are washed in isobutyl-methylketone (4-methyl-2-pentanone) to remove liquid non-irradiated polymer At the final fabrication stage, 50nm-thin gold films can be deposited on the dried samples with surface dielectric structures by electron sputtering Figure 1.8 on the right side shows plasmon propagation in a dielectric waveguide fabricated by 2PP SPP is excited on the right hand side

1.3.4 Exciting surface plasmon polariton mode in the wedge waveguide

Figure 1.9 Prism coupling to SPPs in the Otto (a) and Kretschmann (b)

configuration

The wave number of SPP at interface of two medium is always larger than in each medium Therefore, we can‘t excite SPP directly by light There are a few methods able to match or couple the incoming electromagnetic moment with that of the SPP One of the most common methods is using the Attenuated Total Reflectance (ATR) structures Figure 1.9 show the schematic of prism coupling, one

of ATR structures The light beam is illuminated to the bottom face of the prism with the incident angle θ in the limit of the total internal reflection The incident angle θ is adjusted so that the component wave vector of photon in prism parallel to the interface equal the wave vector of SPPs

Figure 1.10 Schematic of waveguide coupling

Additionally, there is another method used ATR film to coupling SPPs, the waveguice coupling method (Fig.1.10) Excitation light with wave vector k0 was coupled into the dielectric waveguide εg> ε2 (ε2 = 1) Electromagnetic wave inside

the waveguide has wave vector kg> k0, some photon tunneling through the metal film would be excited SPPs on the interface between metal/air

Another way to obtain the wave vector conservation for SPPs excitation is to use diffraction effects A diffraction grating is created on a part of smooth metal film, component of the diffraction light which has the wave vector equal the SPPs wave vector can be coupled to surface plasmon polariton The grating with period a embedded in a dielectric with permittivity ε2 (Fig.1.11), the matching condition is given by

√ (1.16)

Figure 1.11 Scheme of grating coupling

Where θ is the incident angle, i = 1,2,3,… is an integer By choosing an incident angle suitable, we can excite SPPs wave at the interface between the metal - dielectric interface

1.3.5 Applications of wedge waveguides

Focusing and propagation of electromagnetic fields

Strong light concentration in the cross-section perpendicular to the propagation direction, SPP-based waveguides could transport the same huge bandwidth of information as in conventional (dielectric-based) photonics and yet not be limited

by diffraction to sub-micrometer cross sections Many different types of SPP-based waveguides have been developed as presented above Although, SPP waveguides suffer from ohmmic losses caused by EM penetrating into metal Furthermore, many waveguide structures have been introduced, which achieve long propagation length, while keeping deep-subwavelength light confinement [21] In addition to applications such as waveguide for guiding plasmon wave, other optical devices, which exploit the tight confinement of wedge structure to build microresonator and nanolases [5] Besides application research, the wedge surface plasmon waveguide can also be used for fundamental research such Quantum Plasmonics [14] In the following, we present several typical applications of the wedge surface plasmon

metal wedge ring has a thin tapered edge, which guides the plasmon modes along the φ direction Here, (r, φ, z) is denoted as a conventional cylindrical coordinate system The edge could be tapered by, for instance, chemical sharping or mechanical polishing [9] and rounded with radius of curvature rc It is shown that a sharper wedge is preferred to achieve higher Q factor and stronger confinement of electromagnetic field In addition, the hybrid plasmon modes still possess relatively high Q factor even if there a gap between the metallic wedge ring and the silica substrate The WPPs exhibit unique properties, especially in that they concentrate the electromagnetic field close to the nanoscale tip of the metallic wedge, which enables the proposed hybrid WPP microresonator to serve as a low-threshold plasmonicmicrolaser The wedge ring also presents a new type of microcavity for coherent light-matter coupling experiments in cavity quantum electrodynamics (cQED)

Nanolaser

Wedge waveguides can also be applied for making nanolasers This application

is based on the capacity of achieving deep-subwavelength confinement and relatively low loss propagation of wedge mode of the waveguide The nanolaser can

be made by composing a dielectric nanowire placed above a wedge waveguide via a thin dielectric layer The dielectric nanowire serves as a laser gain environment The pumping threshold needed to lasing actions is controlled by modifying the overlap mode field between the wedge mode and dielectric mode This is carried out by verifying the tip angle of triangular wedge waveguide Compared with laser structures based on interactions between the traditionally flat metal - dielectric surface with the cylindrical or square gain nanostructures, improved optical confinement in the wedge waveguide can lead to new capacities when leveraged as lasing sources Figure 1.13(a) shows a model of nanolaser based on the wedge surface plasmon waveguide The wedge waveguide is triangular-shaped Ag substrate The gap dielectric layer is MgF2 The dielectric nanowire served as a gain

environment is CdS Figure 1.13 (b) is 2D and cross-sectional electric field

distributions supported by the metal groove - based hybrid

Figure 1.13 (a) Geometries of the plasmon nanolaser based on a triangular

metal wedge substrate and (b) 2D and cross -sectional (along the dashed -lines

in the 2D plot) electric field distributions (|E(x, y)|) of the plasmonic modes supported by the metal groove - based hybrid structure (=60 0 , t=5 nm) The positions of the dashed-lines are determined by the points of tangency between the C d S nanowire and the MgF 2 layer along the x - direction, and through the

center of the nanowire for their directions [4]

structures The field well concentrated within the ultra-thin MgF2 gap layer near the wedge tip due to the strong hybridization between the wedge plasmonic mode and the dielectric nanowire mode, which indicates an ultra-small mode area achievable and meanwhile exhibiting reasonable modal overlap with the gain medium to enable laser action Such a deep-subwavelength-scale mode size offers unique advantages

to allow electronic transitions directly coupled to strongly confined optical modes, thus overcoming the challenge of delivering light from microscopic external sources into a deep-subwavelength scale Ultra-deep-subwavelength mode area is achievable when the tip angle is relatively small, which could be leveraged to enable the operation of a plasmon laser with nanoscale optical mode size The

reduced mode size also indicates a larger Purcell factor might be achieved using the

metal-wedge-based-plasmon nanolaser when compared to the based structure When the tip angle increases, the lasing pumping threshold is

flat-metal-substrate-reduced

Purpose of this thesis

Invention of plasmon effect has important impact in nano-science and technology This is a phenomena relating to radiation, confinement, and propagation

of electromagnetic wave energy in metal nanostructures, opening researches and applications without limited by optical diffraction Recent research has paid attention to waveguides for guiding surface plasmon wave In this thesis, we will therefore propose wedge plasmon waveguide structures using wet etching technology with low cost and high efficiency These plasmon waveguide structures are designed and simulated using finite element method Design models as well as fabrication process are based on deeply investigating the wet anisotropic etching property of single crystal silicon in KOH solution

CHAPTER 2 DESIGN AND SIMULATION OF

WAVEGUIDE

2.1 Basic theory for the simulation of waveguides

Maxwell theory describes the expression of SPPs wave on a metal–dielectric interface by [24]

Using parameters in Table 1, we obtain the effective refractive index and mode size of SPPs propagating along the silver–air interface being 1.0038 and 2.86 µm, respectively

On a metal tip, WPP is the coupling mode formed by two SPP waves propagating toward the tip on the two opposite faces of the wedge [11] These SPPs stop at the tip of wedge with both their phase and group velocities The group velocity tends to zero and the wave vector tends to infinite at the tip This leads to infinitesimal mode size (nanofocusing) and the wave can be propagated on very large distance

However, this model is applied only to very small wedge angle (7o) For the large wedge angle (180o – α), we must use numerical simulation methods to solve

By finite element method, we can find the propagation constant of WPP mode, that satisfies eigenvalue equation

(∇ - ik1) × ((∇ - ik1) × E1) – ko2εrE1 = 0 [2.5] (∇ - ik2) × ((∇ - ik2) × E2) – ko2εrE2 = 0 [2.6] The modal is three – dimension with the calculation size is 4x4x10 µm3 Two boundary ports condition are applied on back and front face to analyze mode that can propagate in the structure The perfectly matched layer is larger than calculation domain five times and the minimum mesh size is about 1 nanometer (at near the wedge of waveguide)

2.2 Triangular Waveguide

2.2.1 Structure of triangular waveguide

Figure 2.1 Scheme of triangular shaped plasmonic waveguide used for

simulation (left side) and the cross – sectional view (right side)

Structure of triangular waveguide is shown in figure 2.1 This structure consists

of a triangular silicon waveguide, which is fabricated on a Silicon–on–Insulator (SOI) wafer A noble metal layer is deposited onto the surface of waveguide to form metal–dielectric interface The silicon waveguide forms a mold to deposit a metal layer in inverted V-like shape The interface between the V-shaped metal layer and air medium is used for guiding surface plasmon wave The Wedge Plasmon

Polariton (WPP) mode propagates on the wedge of structure We will evaluate the dependence of transmission characteristics on the change of structural parameters such as metal layer, shape and refractive index of core medium (silicon waveguide) and cladding medium (air medium)

ns and number of elements per wavelength in cross-section nm