Study of Stripe-shaped Optical Filter and Plasmonic Analogue of EIT Effect based on Hybrid Plasmonic Waveguides

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Academic year 2012-2013

Study of Stripe-shaped Optical Filter and Plasmonic Analogue of

EIT Effect based on Hybrid Plasmonic Waveguides

Xu Sun

Master thesis

Supervisor: Lech Wosinski

TRITA-ICT-EX-2013:86

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Study of Stripe-shaped Optical Filter and Plasmonic Analogue of

EIT Effect based on HP Waveguide

TRITA-ICT-EX-2013:86

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Abstract

The rapid growth of nanotechnology leads to the possibility to fabricate ultra compact optical devices on a single chip. However, miniaturization of photonic circuits and devices is restricted by the diffraction limit. A promising solution to this problem is by exploiting plasmonic systems for guiding and manipulating signals at visible or communication wavelengths. Among different kinds of plasmonic waveguide, hybrid plasmonic (HP) waveguide shows a sub-wavelength confinement as well as long propagation distance, which has the potential to be applied in next-generation integrated circuits.

This thesis is aimed to design optical devices based on HP waveguide, including optical filters and analogue of electromagnetically induced transparent (EIT) effect. The study methods are Scatter Matrix Method (SMM) and Finite Element Method (FEM) based simulation software, where SMM is used for calculate the numerical solutions of optical filter and FEM based simulation software is introduced to design, simulate and characterize the optical devices. We use the double-slot HP waveguide with metal cladding to realize the optical devices, which provide a good optical confinement. A stripe-shaped optical filter based on HP waveguide is designed and studied. The extinction ratio of the optical filter can be as low as 0.01 with a Q factor around 50. The analogue of EIT effect is also realized in this HP waveguide system, which is composed of double stripe-shaped resonators in form of resonant stubs with different lengths located on each side of the HP waveguide.

At two resonant frequencies (150THz and 222THz) resonance is obtained in respective stub, where the transmittance can be as low as 0.01. In the frequency range of 150THz and 222THz, a constructive interference is obtained between the two resonant stubs, where a maximum transmittance occurs, which can be as high as 0.7 at the frequency of 184THz (λ=1.63μm). Accompanying with the transparency ‘window’, the group velocity can be slowed down as much as 30 times in respect to the velocity of light in vacuum.

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Acknowledgements

My deepest gratitude goes first and foremost to Dr. Lech Wosinski, my supervisor, for his constant encouragement and guidance. He has led me through all the stages of the research and writing of this thesis. Without his consistent and illuminating direction, this thesis could not reach its present form.

I also want to thank to Professor Lars Thylén for being my examiner at KTH for his valuable suggestions and assistance.

Special thanks to Fei Lou, my good friend and advisor, for not only teaching me simulation methods but also sharing his valuable experiences with me.

I would like to express my heartfelt gratitude to all my friends at KTH who offered me their help and time in listening to me.

Finally, I am indebted to my parents for their continuous love and supports. Without their consideration and encouragement, I would never be able to complete this task. I also owe my sincere gratitude to my girlfriend for her unswerving love.

Xu Sun, May 15, 2013,

Stockholm, Sweden.

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Chapter 1 Introduction

1.1 Background

In 1969, Stewart E. Miller of Bell Labs has put forward the concept of integrated optics [1], which is a term similar to integrated electronics. After decades of development in nanotechnology, ultra fast optical interconnects and photonic integrated circuits in the size range of hundreds of nanometers are possible to be fabricated. Their evolution is even faster than the Moore’s Law predicts for electronics [2]. In comparison with the conventional electronic circuits, photonic circuits demonstrate several significant advantages, such as lower power consumption, higher operation speed and much higher bandwidth of the transmitted and processed optical signals. Due to their appealing characteristics, they have received a lot of attention worldwide in recent decades. Currently, integrated optics has been widely introduced into wireless and fiber communication systems.

The rapid growth of nanotechnology leads to the possibility to fabricate ultra compact optical devices on a single chip. Nevertheless, due to the diffraction limit of light, the size reduction of traditional waveguides has been limited to the order of ( is the wavelength and is the effective refractive index of the waveguide) [3]. Hence, the reduction of optical device’s size has become one of the hot spots in optical integration domain. Optical devices based on Surface Plasmon Polaritons (SPPs) [4] shed new light and opportunity for integrated optics. SPPs are generated when optical waves interact with the electrons at the metal surface. In other words, free electrons will oscillate, when they are irradiated by electromagnetic wave and their frequency is equal to the frequency of the irradiating wave. As SPP has an exponential decay in the metal media, optical wave can be exceedingly confined at the metal surface. The property of confining electromagnetic energy into nano-scale region makes SPP waveguides the most promising information carriers for nano-integrated optical systems. Different sub-wavelength waveguides have been designed and experimentally confirmed in recent years, such as metal-insulator-metal (MIM) plasmonic waveguide, dielectric loaded plasmonic waveguide [5], hybrid plasmonic (HP) waveguide [6-8], etc.

Taking advantage of SPP waveguides, various functional plasmonic structures have been designed and fabricated, such as optical couplers [9-10], T-shape [11] and Y-shape [12] splitters and combiners as well as Mach-Zehnder interferometers [13-14] and ring resonators [15]. Recently, interest in cavity resonators based on SPP waveguide to achieve wavelength filtering function has been further fueled by the theoretical predictions and experimental demonstrations [16-19]. In photonic circuits and optical interconnects, filter, one of the most important optical devices, is used for wavelength selection of electromagnetic waves. Normally, the function of wavelength selection is realized by different kinds of resonators, such as circular resonators [16], tooth-shaped resonators [20-21], Fabry-Perot

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resonators [22-23] and stripe-shaped cavity resonators [24-25]. Among different types of resonators, resonant modes of the stripe-shaped resonators, where the resonator is formed by side-coupling of one or more short waveguides (stubs) forming a resonant cavity, are mostly geometry-dependent. By adjusting the geometry parameters of the stub cavity, the resonant wavelength can be easily tuned. In addition to this filtering effect of the stripe-shaped resonators, an optical analog to a well-known phenomenon originating from atomic system, Electromagnetically Induced Transparency (EIT), has also been observed in SPP waveguide systems. The observation of EIT in the plasmonic system has broken through the rigorous conditions in the original atomic system, such as low temperature and stable gas lasers. The plasmonic analogue of EIT effect can be applied in noise decreasing, optical switches and sensitive interferometers. At present, a lot of work has focused on the phenomenon of plasmonic analogue of EIT effect in the coupled optical resonator systems [26-29].

1.2 Motivation and Objective

Since the existing research on optical filters and analogue of EIT based on plasmonic waveguide system deals mainly with MIM waveguides, little work has been done in the HP waveguide system.

Nevertheless, taking the advantages of HP waveguides, such as long propagation length and small effective mode area, it is valuable to design and evaluate such optical devices based on HP waveguide structures. Due to lower losses one can expect even more relaxed conditions than using pure plasmonic system and may be other interesting properties.

The main objectives of this thesis are described below.

The first objective is to design a stripe-shaped optical filter based on HP waveguide. This optical filter is made of a series of simple stripe-shaped resonators aside the HP waveguide, which is expected to work in the region of communication wavelengths. The optical filter can also be tuned by adjusting the parameters of the resonant cavities.

The second objective is to design a HP waveguide structure to realize the hybrid plasmonic analogue of EIT. The HP waveguide system is composed of double stripe cavities, which are located on both sides of the HP waveguide. Due to the resonant effects between two cavities, a transparent

‘window’ for selected wavelengths can be expected. The sudden change of transmittance will generate a large dispersion, ( is the propagation constant of the whole system, and is the angular frequency), at the transparent wavelength area. Accompanying with the large dispersion at this transparent ‘window’, large group index ( ) can be observed.

1.3 Research Summary

In this thesis, the methods used for studying the HP waveguide systems are scattering matrix computation method (SMM) and finite element method (FEM)-based simulation software. SMM is used to calculate the numerical solutions of the optical filter system, and the FEM based simulation software is employed to numerically design, simulate and characterize the HP waveguide system.

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1.4 Thesis Outline 3

The optical filter has a series of stripe resonant cavities aside the HP waveguide. The second order resonant mode is used to realize the destructive interference. The resonant wavelength is located at the communication wavelength range (~1550nm), and the extinction ratio is about 0.01. The optical filter can be also easily tuned by adjusting the geometry parameters, such as length and width of the resonant cavities.

The plasmonic analogue of EIT effect has been observed in a HP waveguide system with two resonant cavities [28]. These two resonant cavities generate a constructive interference at resonant wavelength. Meanwhile, on both sides of this transparent wavelength, two destructive interferences are generated at each resonant cavity respectively. In the transparent ‘window’, the maximum group index is around 30 at the communication wavelength.

1.4 Thesis Outline

After the introduction in chapter 1, basic principles of wave equations and Surface Plasmon Polaritons are discussed in Chapter 2, where the study methods (SMM and FEM) and the dispersion model are explained briefly. In Chapter 3, a comparison of different optical waveguides is made, including planar dielectric waveguide, surface plasmonic waveguide and hybrid plasmonic waveguide. The characteristics, such as propagation length and confinement factor, are studied.

Chapter 4 presents the design and simulation of a stripe-shaped filter based on HP waveguide. The transmittance curves have been illustrated by 2D and 3D simulation. Chapter 5 describes the realization of the plasmonic analogue of EIT effect in a HP waveguide system with two stripe-shaped cavities. The characteristics like group velocity and phase shift within the transparency ‘window’ are analyzed. Finally, the conclusions and future work are discussed in Chapter 6.

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Chapter 2 Basic Principles and Study Methods

Starting with the Maxwell’s equation, this chapter will introduce the basic wave equations of Surface Plasmon Polaritons (SPPs) at the metal-dielectric interface. Maxwell’s equations describe how electric and magnetic fields are generated and altered by each other. The equation system is named by Scottish physicist and mathematician James Clerk Maxwell who published the earliest form of those equations between 1861 and 1862. Maxwell’s equations are widely used to deduce and analyze characteristics of electromagnetic wave.

For many practical applications, since it needs a large amount of work to solve these equations, a number of approximation methods have been developed and studied due to the progress in computer science, such as Finite Element Method (FEM) [29], Method of Moment (MoM) [30], Finite- Difference Time-Domain (FDTD) [31], etc. These numerical methods can get accurate solutions if there are enough gridding sizes and computer capacity. Therefore in this chapter, the different numerical methods will also be presented briefly.

When the numerical methods are applied to study the characteristics of optical systems, various dispersion relations are required to calculate the optical properties of materials. Hence in the last section of this chapter, the Drude and Lorentz models of noble metals will be illustrated.

2.1 Surface Plasmon Polaritons

Surface Plasmon Polaritons (SPPs) [3] are mixed excited states of metal surface free electrons and photons, and the electromagnetic excitations are propagated at the interface between metal and dielectric. In the perpendicular direction, due to the huge loss in the metal, the electromagnetic fields can be evanescently confined at the interface. Normally, the depths of light penetration into metal and dielectric are around 10nm and 100nm respectively. Therefore the waveguides based on SPPs have a sub-wavelength optical confinement, which can be applied in ultra small photonic circuits.

Now, we will begin with the Maxwell’s equations to draw forth the SPPs wave mode.

2.1.1 The Electromagnetic Wave Equation

The electromagnetic wave equation is an important second-order linear partial differential equation, which describes the propagation of electromagnetic wave. It can be applied to study the characteristics of optical waveguides and devices.

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The electromagnetic wave equation can be deduced from the Maxwell’s equations:

(2.1a) (2.1b)

(2.1c)

. (2.1d) Here, D is the dielectric displacement, is the external charge, is the current density, E is the electric field, H is the magnetic field, and B is the magnetic induction (magnetic flux density).

From the Maxwell’s equations, in the case of absence of external charge and current density, the curl Eqs. 2.1c and 2.1d can be combined to:

, (2.2) where is the variation of the dielectric profile over the propagation distance, which is negligible on the order of one optical wavelength.

Then, Eq. 2.2 can be simplified as

. (2.3) This is the central equation of electromagnetic wave theory.

By extracting the time domain part of the electric field:

. (2.4) Eq. 2.3 can be written as

. (2.5) This is the well-known Helmholtz equation, where is the wave vector of the propagation wave in vacuum.

With different polarizations, two different modes can be supported: the transverse magnetic (TM) mode, for which the components Ex, Ez and Hy exist and the transverse electric (TE) mode, for which only the components Ey, Hx and Hz exist.

Next, we will introduce the SPP waves at the metal-dielectric interface from the Helmholtz equation.

2.1.2 Surface Plasmon Polaritons at the Metal-dielectric Interface

In order to analyze the wave propagation properties at the interface between metal and dielectric, we firstly put the metal-dielectric interface into a Cartesian coordinate system: the input direction is along the x-axis, and the spatial variation is in the direction of z-axis ( ). The plane is the interface between metal and dielectric layer, as shown in Fig. 2.1.

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2.1 Surface Plasmon Polaritons 7

Figure 2.1: Schematic of metal and dielectric interface.

At the interface between metal and dielectric, (see Fig. 2.1), the conducting half ( ) has a complex permittivity . This is because metals are lossy materials, and the imaginary part of permittivitity expresses the loss. While the non-absorption half ( ) has a positive real permittivity

, which corresponds to the dielectric materials.

The propagating wave can be described as

, (2.6) where is the propagation constant of the traveling wave, which corresponds to the component of the wave vector in the direction of propagation, i.e. x-direction as we set above.

By inserting Eq. 2.6 into the Helmholtz equation, the wave equation of electric field E is:

. (2.7) A similar equation also exists for a magnetic field H, which is

. (2.8) By solving the Helmholtz equation, the TM mode at the metal-dielectric interface can be written as:

For ,

(2.9a) (2.9b)

. (2.9c) For

(2.10a) (2.10b)

Metal Dielectric

z

y

x

Propagation direction kx

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. (2.10c) Here, are the components of the wave vectors perpendicular to the interface of the two media, i.e. z-axis direction as shown in Fig. 2.1. is the evanescent decay length of the fields perpendicular to the interface, which is related to the optical confinement of the propagation mode.

The electromagnetic fields at the metal-dielectric interface and the variation of field Ez in z-axis direction are shown in Fig. 2.2.

Figure 2.2: (a) The electromagnetic field at the metal ( )-dielectric ( ) interface, where the “+” and “-”

correspond to the positive and negative charge at the metal surface, and the arrows mean the interaction between free electrons and photons. (b) The variation of the field Ez in z-direction.

From Eqs. 2.9 and 2.10, we notice that due to the convention of the signs in these expressions, confinement of the wave at the interface requires if . In other words, the surface waves with TM mode exist only at the interface of metal and dielectric. By inserting Eq. 2.9a and Eq.

2.10a into the wave equation of magnetic field H (Eq. 2.8), we can obtain:

(2.11a) . (2.11b) The continuity of and at the interface requires , and then

. (2.12) By combining Eqs. 2.11 and Eq. 2.12, we can arrive at the dispersion relation of SPPs propagating at the interface between the two half spaces:

+ + + + + + + +

_ + _ _ _ _

_ _

_ _ _ + _ _ _ + + + _ _ _ + + _ _

z x

Metal ( ) Dielectric ( )

z

Ez

(a) (b)

Dielectric (z>0)

Metal (z<0)

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2.2 Numerical and Simulation Methods 9

. (2.13)

Similarly, we can get the expression of TE mode at the metal-dielectric interface, which is:

For ,

(2.14a) (2.14b)

. (2.14c) For ,

(2.15a) (2.15b)

. (2.15c) From the Eqs. 2.14 and Eqs. 2.15, we notice that to satisfy the convention of the signs, it requires and . The condition is only satisfied when . Thus, no surface wave exists for TE polarization.

For the metal-dielectric interface planar structure, Surface Plasmon Polaritons (SPPs) only exist for TM polarization [32].

2.2 Numerical and Simulation Methods

To study the characteristics of optical waveguides and devices, the numerical and simulation methods can be applied. In computational electromagnetic area, the scattering matrix method (SMM) is a widely used numerical method to solve the complicated optical systems. Full vectorial solver of COMSOL Multiphysics based on finite element method (FEM) is widely employed simulation software to design, optimize and characterize the optical devices. In this section, these two study methods will be introduced.

2.2.1 Scattering Matrix Method

In electromagnetic region, scattering matrix method (SMM) relates the initial and final states of an optical system undergoing a scattering process. To reveal the basic principles of SMM, we firstly consider a simple mirror reflectance system as an example.

The mirror reflectance system is shown in Fig. 2.3. We set E+1 and E-2 as the incident waves traveling toward planes (1) and (2), and E-1 and E+2 are outgoing waves away from the mirror system.

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Figure 2.3: The schematic of a mirror system calculated by SMM

The S-parameters are defined as:

: the power that reflects back from plane (1);

: the power that transmits from plane (1) to plane (2);

: the power that transmits from plane (2) to plane (1);

: the power that reflects back from plane (2).

In planes (1) and (2), the incident and reflected waves are in anti-phase, which is related by:

(2.16a) , (2.16b) here, and are the reflectance coefficients.

The scattering matrix elements for S11 and S22 can be written as:

(2.17a)

. (2.17b) If there is no loss in the system, then the S-matrix parameters should satisfy the following equations:

(2.18a) (2.18b)

. (2.18c) where S^* corresponds to the conjugate matrix of S.

Due to the recipropcal condition, S12 and S21 have the same value. Then we have:

(2.19a) . (2.19b)

(2) (1)

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2.2 Numerical and Simulation Methods 11 That is to say, even though the system is asymmetric, the matrix elements and have the same value ( ). Then we can yield

. (2.20) As equals to 1, is purely imaginary, we can set

, (2.21) here, t is the ‘transmittance’: .

The S-matrix of this mirror reflectance system can be written as:

. (2.22) Here, r is the reflectance coefficient of the mirror, and t is the transmittance coefficient of it.

. (2.24) We can make a transformation of this S-matrix, which is given by:

. (2.25) This is the S-matrix equation for a single mirror reflectance system. For a more complicated system, the analytical results can be gained by multiplying S-matrix for each component.

We can take an optical system with N components as an example. The S-matrix equation for such system can be written as:

. (2.26) The transmittance of this system is:

. (2.27) The introduction above is the basic calculation process of SMM, which can be applied to complicated optical systems with multi-components. Sometimes, we can also set the optical system which has N ports instead of two ports, and then the S-matrix of the optical system become a matrix. The calculation process in this case is similar to the two ports system.

2.2.2 Finite Element Methods (FEM)

A fast development of theoretical and experimental investigations of electromagnetic phenomena has been pushed forward after the establishing of Maxwell’s equation. In a real electromagnetic system, calculations of Maxwell’s equation are usually very complicated, and the analytical solutions do not exist. Due to the complexity of the calculation process in Maxwell’s equation, different numerical methods have been developed to model complex electromagnetic problems using computers. In this section, we will briefly introduce the finite element simulation method (FEM), since this is one of the major study methods used in this thesis.

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FEM is a numerical method for solving partial differential problems. Especially with the help of computer techniques, the FEM simulation method becomes one of the most important numerical methods for solving electromagnetic field problems. In FEM method, the major domain will be divided into numerous sub-domains, and the final results will be obtained by summarizing the calculation results in each sub-domain.

Figure 2.4: (a) Sub-domain model in a HP waveguide simulation (b) the electric field distribution in HP waveguide

Fig. 2.4(a) shows the mesh graph of a 2D HP waveguide in COMSOL, each triangle inside this graph is a sub domain used for calculating the electromagnetic characteristics. Fig. 2.4(b) is the simulation result of electric field distribution in such HP waveguide. Moreover, other characteristics, such as effective index, phase, propagation constant, etc., can also be derived from the simulation process.

FEM is a simple numerical method with a straightforward calculation process to solve the complicated mathematical problems. To introduce the principle of FEM, we can start from a 1-D function.

Assume we have a 1-D function in the domain ,

. (2.28)

Figure 2.5: Expansion approximation based on sub-domain .

x0 xZ

f(x) f(x)

x0 xZ

(a) (b)

x y

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2.3 Dispersion Model of Noble Metals 13 We can approximate this function by dividing the major domain into , , … , as shown in Fig. 2.5.

Then, we can write Eq. 2.28 in the form:

. (2.29) Here, is the base function, we adopt in Eq. 2.28; it is defined in sub- domain . The expansion coefficients are .

As it is seen from Fig. 2.5, the numerical method adopts very simple linear function, , to make an approximate expansion. If we divide the independent variable range, , into enough amount of sub-domains, then the accurately approximated results can be obtained. The disadvantage of this numerical method is the massive calculation work. However, with help of rapid progress in computer science, methods based on this approximation become increasingly matured.

Using enough fast computer, the advantages of this method can be easy confirmed.

The introduction above shows the basic concept of FEM method: by simplifying the complicated functions into numerous simple functions and summarizing the calculation results gained from each sub-domain, the infinitely accurate solutions can be obtained. In the following, we will use this numerical method to study the optical systems.

2.3 Dispersion Model of Noble Metals

The interest of using simulation methods for the study of different electromagnetic phenomena has increased constantly in the past few years. On the other hand, the lack of analytical models of materials dispersion is a limiting factor when we use FEM simulation method to different kind of materials. Typically, the analytical models of dispersion used for calculating the optical constant of noble metals are the Lorentz or Drude dispersion model. In this section, we will present detailed study of the dispersion models of Noble metals and compare them with the experimental results.

2.3.1 The Drude Model

It is well known that in near infrared region, the relative permittivity of Silver and Gold can be described by the expression of Drude model [33]:

, (2.30) where _ stands for the dielectric constant at infinite angular frequency; is the bulk plasma frequency, it represents the natural frequency of the oscillations of free conduction electrons; is the damping frequency of the oscillation; is the angular frequency of the electromagnetic wave.

Fig. 2.6 illustrates the curves of permittivity ( ) of gold and silver, it shows the real and imaginary components and calculated by Drude model fitted to the experimental results [34].

The values of parameters used for Drude Model are depicted in Table 2.1.

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Table 2.1: Values of parameters used for Drude Model [33].

(THz) (THz)

Gold 9.0685 2155.6 18.36

Silver 3.7 2196 3.772

Figure 2.6: Real and imaginary part of permittivity for gold and silver, the red dashed curves are the results calculated by Drude model, the blue circles are the experimental results [34], (a) the real part of permittivity of gold, (b) the imaginary part of permittivity of gold, (c) the real part of permittivity of silver and (d) the imaginary part of permittivity of silver.

From the permittivity curves shown in Fig. 2.6, we can see that the real part of permittivity for both gold and silver are increasing with the energy ( , h is the Plank constant, and ν is the frequency of electromagnetic wave) and tend to saturate below zero for high energies, which is in good agreement with the experimental results. In addition, the real part of permittivity of gold is a little higher than this of silver at the same energy level.

By comparing the imaginary part of silver and gold, we find that the imaginary part of silver is much smaller than this of gold. Moreover, in energy region higher than 1.5eV, the Drude model cannot match well with the experimental results of gold and silver, which is due to the interband transitions of free electrons. Another analytical dispersion model, extended Drude model (Drude- Lorentz model), can be applied to overcome this mismatch problem.

Gold

Silver

(b)

(d)

(c) (a)

Energy (eV) Energy (eV)

(b)

Energy (eV) Energy (eV)

(d)

Real part of permittivityReal part of permittivity Imaginary part of permittivityImaginary part of permittivity

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2.3 Dispersion Model of Noble Metals 15

2.3.2 Extended Drude Model

The expression for extended Drude model is formulated as:

, (2.31) where and stand for the oscillator strength and the spectral width of the Lorentzian oscillators, respectively. can be interpreted as a weighting factor.

The values of parameters used for Drude model and Drude-Lorentz model are depicted in Table 2.2.

Table 2.2: Values of parameters used for Drude Model and Drude-Lorentz Model [33, 34].

(THz) (THz) (THz) (THz)

Drude 9.0685 2155.6 18.36

Drude-Lorentz 3.7 2196 3.772 650.07 104.86 1.09 14.521

Figure 2.7: Permittivity of gold as published by Johnson and Christy, calculated with single Drude Model and Drude-Lorentz Model, (a) real part of permittivity, (b) imaginary part of permittivity.

Fig. 2.7 shows the comparison of different analytical models of dispersion of gold (Drude and Drude-Lorentz models), where the Drude and Drude-Lorentz models are expressed by red and blue line respectively. The black circles are the experimental results. From this figure, we can see that the real part of permittivities of gold calculated by analytical models agree with experimental results, while for imaginary part of permittivites, due to the interband transition of free electrons, the Drude model is mismatched at higher energy level (higher than 1.5eV). The Drude-Lorentz model is taking into consideration the problem with the interband transitions of free electrons and therefore is more acurate than Drude model.

2.3.3 Optical Constant of Gold

In the simulation process, we normally use the optical constant to identify the characteristics of metals, the optical constant for metals is composed of a real part and an imaginary part, which can be calculated by:

(a) (b)

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(2.32a) , (2.32b) where and are the real and imaginary part of permittivity, respectively. and are the real and imaginary part of refractive index.

Fig. 2.8 shows the curves of optical constant of gold, the red and blue lines are the results calculated by Drude and Drude-Lorentz model, respectively, and the black circles represent the experimental results. In the near infrared region, both of the dispersion models agreed well with the experimental results. In the region of higher energy, the Drude model cannot be employed due to the effect of interband transition.

Figure 2.8: Comparison of the refractive index of gold, (a) real part, (b) imaginary part.

Above we discussed different analytical models for dispersion for Noble metals. We have analyzed the values of permittivity of silver and gold, as these two metal materials are most widely used in SPP waveguides and devices. In Drude Model, the imaginary part of gold and silver are not matched well with the experimental results due to the interband transition. However, since the normally operated wavelengths (less than 1.5eV) are below the unmatched region, we can use the Drude model to calculate the permittivity of metal in our following work.

2.4 Summary

In this chapter, the basic wave equations and SPPs have been introduced. These wave equations are derived from Maxwell’s equations, the deriving process has been illustrated in brief. The wave equations of SPP wave show the sub-wavelength optical confinement at the metal-dielectric surface for TM mode, which is due to the very short decay length inside the metal (~10nm).

Next, we have depicted the different study methods in this thesis, which are Scattering Matrix Method (SMM) and Finite Element Method (FEM). SMM shows the possibility to calculate the propagation properties of optical systems. Beside this, the FEM based software, COMSOL

(a) (b)

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2.4 Summary 17

Multiphysic, can be used to design, optimize and characterize the optical system. In the following chapters, we will show the numerical and simulated results of these study methods.

In the last section, in order to set the values of refractive indices into the numerical and simulating processes, we introduced different dispersion models of silver and gold. Among them, the Drude model has a simplier expression and it can match well with the experimental results in infrared light region, hence we can use this dispersion model to calculate the optical constant of noble metals.

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Chapter 3 Comparison of Different Optical Waveguides

An optical waveguide is a physical dielectric structure, where guided electromagnetic waves in the optical wavelength region propagate along the optical medium with high refractive index (core), surrounded by low refractive index media (cladding). By classifying different waveguides with geometries, they can be divided into planar or slab optical waveguides, which in cross section consist of layers of high and low index materials, and non-planar waveguides, which in cross section can have different form of core surrounded by cladding material: circular, like optical fiber, and channel waveguides: ridge, rib, stripe-loaded, buried- and slot channel waveguides. By classifying them with refractive indexes, they can be divided into weakly guiding optical waveguides (the refractive index difference between core and cladding is smaller, normally less than 1%) and strongly guiding waveguides (larger refractive index difference between core and cladding materials).

Channel dielectric waveguides utilize the difference of refractive index of dielectric materials to guide optical wave in the core. There exist also other methods for guiding electromagnetic waves, such as photonic crystal waveguides, where light is confined by periodicity of the structure in one or more directions as well as surface plasmon polariton (SPP) waveguides that use short decay length in metal to gain an ultra-compact optical confinement at the metal-dielectric interface. Different geometries of plasmonic waveguides are possible. They offer a good optical confinement in micrometer or nanometer scale, which can be used for example in optical interconnect and photonic circuits to connect ultra small optical devices. Electromagnetical characteristics of channel dielectric and plasmonic waveguides are different: SPP waveguide can provide superior light confinement in ultra compact photonic devices; channel dielectric waveguide can offer a long-range propagation with low losses. This trade-off between light confinement and propagation length was one of the hot spots in the research of optical waveguides. A novel mixed waveguide, hybrid plasmonic (HP) waveguide, has demonstrated a good performance in photonic circuits. HP waveguide is an optical waveguide with a mixture of channel dielectric guiding and SP guiding, which can offer a fairly good optical confinement with considerably extended propagation length in comparison to SPP waveguides.

Due to these advantages, HP waveguides have recently attracted lot of attention and many structures of HP waveguides have been designed and reported, e.g. dielectric-cylinder HP waveguide [6], metal-GaAs-gap HP waveguide [35], the HP waveguide structure of a silicon nanowire with a metal cap [7], and double low-index slot HP waveguide [8]. The different structures of HP waveguides can be optimized to provide various optical performances, when applied in nanophotonic intergrated circuits (nPICs), such as long propagation length or excellent optical confinement.

In this chapter, we will study different characteristics of these optical waveguides, such as the electromagnetic field distributions, propagation lengths and effective refraction indexes.

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3.1 Dielectric Waveguides

The key components of photonic circuits and optical interconnects are dielectric waveguides with the structure that consists of different layers of dielectric materials. The basic structure of a dielectric waveguide is built of a longitudinally extended high-index dielectric medium, called core, which is transversely surrounded by low-index media, called cladding. The guided electromagnetic wave propagates in the waveguide along the longitudinal direction.

By classifying dielectric waveguides with dimensions of optical confinement, there are two basic types of dielectric waveguides: planar dielectric waveguide (optical wave confined in one transversal direction) and nonplanar dielectric waveguide (optical wave confined in two transversal directions). In this section, we will introduce them briefly.

3.1.1 Planar Dielectric Waveguide

Planar dielectric waveguide is also called dielectric slab waveguide, since it is made of different dielectric slabs [36]. Normally, planar dielectric waveguide is composed of guided wave layer, cover layer and substrate layer, and the electromagnetic energy is only confined in one transversal direction.

In Fig. 3.1, we put the planar dielectric waveguide into a Cartesian coordinate system. The propagation direction is along the z-axis, and no spatial variation of refractive index exists in x- direction. The relative permittivity is changed only in y-direction. The refractive indexes of planar layer materials are set as nc (cover), nco (guided layer) and nsub (substrate).

Figure 3.1: Schematic diagram of planar dielectric waveguide in Cartesian coordinate system. n_c, n_co and n_sub stand for refractive indices of the cover layer, guided wave layer and substrate, respectively.

To confine electromagnetic wave in the guided layer (core), the refractive indexes of the layers should satisfy: nc< nco> nsub. If the refractive indexes of cover and substrate are equal nc = nsub , we call this planar waveguide as symmetrical planar waveguide.

Planar dielectric waveguide can only confine electromagnetic waves in one direction, and therefore it has little practical applications. Hence in next section, the widely applied waveguides, nonplanar dielectric waveguides, will be introduced.

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3.1 Dielectric Waveguide 21

3.1.2 Nonplanar Dielectric Waveguide

In opposite to planar dielectric waveguides, nonplarnar dielectric waveguides have two-dimensional transversal optical confinement. The core is surrounded by the cladding in all transversal directions in such structures as buried channel, strip-loaded, ridge, rib and diffused waveguides. Fig. 3.2 shows corss sections of the most common geometries of nonplarnar dielectric waveguides.

Figure 3.2: Nonplanar dielectric waveguide types: (a) buried channel, (b) strip-loaded, (c) ridge, (d) rib.

1) Buried Channel Waveguide

Fig. 3.2(a) shows a buried channel waveguide, it consists of a high-index waveguiding core buried in a low-index cladding. The optical wave can be confined in two dimensions due to differences of refractive index between the core and the cladding.

2) Strip-loaded Waveguide

Fig. 3.2(b) is the geometry of a strip-loaded waveguide, which is composed of three dielectric layers: a substrate, a planar layer, and then a ridge. The planar waveguide (without the strip) already provides optical confinement in the vertical direction (y-axis), and the addition strip can offer localized optical confinement under the strip, due to the locally increase of effective refractive index.

3) Ridge Waveguide

Fig. 3.2(c) is the ridge waveguide, which is a step-index structure. The different between dielectric layers at the sides of the guide, as well as the top and bottom faces, can confine the optical wave in two dimensions.

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4) Rib Waveguide

Fig. 3.2 (d) is the corss-section of a rib waveguide. The guiding layer basically consists of the slab with a strip (or several strips) superimposed onto it, which has a similar structure with the strip-loaded waveguide, and the strip is part of the waveguiding core.

The nonplanar dielectric waveguides can provide the two-dimensional optical confinement, hence they are extensively used in integrated optics. Next, we will choose a typical ridge silicon waveguide, to make further analysis.

In Fig. 3.3, the electric field distribution of a silicon-on-insulator waveguide is shown, which is a ridge dielectric waveguide. The cover is air (n_air=1), the substrate is SiO2 (n_SiO2=1. 45) and the guided layer is Si (n_Si=3. 45). The width and height of the silicon waveguide are 450nm and 200nm, respectively. The operated wavelength is 1550nm. The simulation results are obtained from FEM based software, COMSOL Multiphysics.

Figure 3.3: Electric field distribution of TE mode in a silicon planar dielectric waveguide, the yellow and red curves express the amplitude distribution in x- and y-axis directions, respectively; the substrate material is SiO2, and the cover is air. The guided layer is made of Silicon material with the geometry parameters of:

height=200nm and width=450nm.

As for silicon waveguide, the refractive indexes of different layers are satisfying the guiding condition ( , k0 and β are the propagation constant in vacuum and waveguide, respectively), hence the electromagnetic wave can be confined inside the silicon core in both y- and x- direction. The effective refractive index of this silicon waveguide can also be extracted from the simulation results, which should be in a complex form ( ). It is worth noting that the imaginary part (κ) has a relationship with the loss of the waveguide. Normally, the loss of optical waveguide can be replaced by propagation length.

The propagation length is defined as the distance that the amplitude of the field attenuates to 1/e, which is

, (3.1)

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3.2 MIM and IMI Plasmonic Waveguide 23

where, is the operated wavelength and neff is the effective refractive index of the optical waveguide.

The propagation length of dielectric waveguide can reach at the centimeter level. Due to the advantage of long propagation length, this kind of optical waveguide has been widely used in long- range optical communication systems.

3.2 MIM and IMI Plasmonic Waveguide

According to the introduction of SPPs in chapter 2, we know that SPPs can propagate at the interface between metal and dielectric layers. Nevertheless, in the applications, the optical waveguides based on SPPs have a more complicated structure rather than a single metal-dielectric interface. Normally, the surface plasmon polariton (SPP) waveguides are composed of several dielectric and metal layers, in order to give a better optical confinement compared with single metal-dielectric interface.

Insulator-metal-insulator (IMI) and metal-insulator-metal (MIM) are two kinds of widely used SPP waveguides. The geometries of these two SPP waveguides are shown in Fig. 3.4.

Figure 3.4: Schematics of insulator-metal-insulator (IMI) and metal-insulator-metal (MIM) structures.

The first structure is the IMI waveguide, which is composed of a metal film inset into the dielectric background. The second one is the MIM waveguide, which is composed of a dielectric film inset into the metal background materials on the contrary. Since there are two metal-dielectric interfaces in both MIM and IMI waveguides, the SPPs at each interface will be coupled and propagated if the thickness of the guided layer (metal film for IMI waveguide and dielectric film for MIM waveguide) is small enough.

Among these two SPP waveguides, MIM waveguide is more widely used in PICs due to better optical confinement compared with IMI structure. This is because there is a large part of optical wave that propagates in the dielectric layer in IMI waveguide, which results in weaker optical confinement.

Next, we will make a further analysis of the widely used MIM waveguide.

To study the characteristics of MIM waveguide, we firstly set parameters of materials and geometry as follows: the operated wavelength is 1550nm; the metal is gold ( at the operated wavelength), whose refractive index is calculated by Drude model (see section 2.3); the dielectric material is SiO2 (n=1.45); the thicknesses of dielectric and metal layers are 25nm and 200nm, respectively. Then, we use simulation software, COMSOL Multipysics, to study this kind of optical waveguide.

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Fig. 3.5 shows the energy density distribution of MIM waveguide. The maximum electromagnetic energy exists in the dielectric layer, and very small amount of energy penetrates in the metal layers.

Figure 3.5: Energy density distribution in a MIM plasmonic waveguide. The thicknesses of dielectric and metal layers are 25nm and 200nm, respectively.

By adjusting the geometric parameters of MIM plasmonic waveguide, the optical characteristics will be modified, such as optical confinement, propagation length and effective refractive index. Fig. 3.6 shows the change of electric field amplitudes along x=0 (the black dashed line in inset), where the thicknesses of dielectric layer are 20nm, 60nm and 140nm, respectively.

We can notice that the amplitudes decrease with the increasing thickness of dielectric layer. This is because the electromagnetic wave traveling inside the guided layer is a coupled wave of SPPs at top and bottom metal-dielectric interfaces. When the thickness of the dielectric layer increases, the coupling effect is weakened, resulting in the reduction of the electric amplitudes in the guided layer (dielectric layer). In other words, if the thickness of dielectric layer is large enough, the SPPs at top and bottom metal-dielectric interfaces will propagate independently. The change of coupling effect has a significant influence on the optical confinement and propagation length of optical waveguide.

Besides, a small quantity of optical energy can penetrate the metal slab, which results in the non-zero electric filed outside the MIM structure.

In the following paragraphs, we will study the changes of optical properties by adjusting the geometry parameters.

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3.2 MIM and IMI Plasmonic Waveguide 25

Figure 3.6: Electric field amplitudes changes along x=0 (the black dashed line in inset). The thickness of dielectric layer is altered by 20nm, 60nm and 140nm.

Besides propagation length (Eq. 3.1), the confinement factor ( ) is another important characteristic of optical waveguide. It is defined as the ratio between optical power confined in the guided layer and the total incident power, which can be expressed as:

. (3.2) By changing the dielectric layer thickness from 40nm to 200nm, and keeping other parameters the same, we got the curves of confinement factor and propagation length, as shown in Fig. 3.7.

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d_dielectric=60nm

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Figure 3.7: Confinement factors and propagation lengths change with the dielectric layer thickness. Red and blue curves indicate the propagation length and confinement factor, respectively. The dielectric layer thicknesses are changed from 40nm to 200nm.

In this figure, the propagation length increases with the thickness of the guided layer, and the confinement is decreasing in the mean time. There is a trade-off between light confinement and propagation length which exists in all the types of SPP waveguides. In the applications, by carefully designing the waveguide structure, optimized optical properties of SP waveguide can be achieved.

3.3 Hybrid Plasmonic Waveguide

In the former sections, we have discussed the dielectric waveguides and pure surface plasmon polariton (SPP) waveguides (MIM and IMI waveguides). Compared with dielectric waveguides, SPP waveguides have a better optical confinement, which can be applied into ultra-compact photonic circuits. However, since such nano-scale optical waveguides are usually quite lossy (the propagation length is usually at the scale of several micrometers), SPP waveguides have certain restriction in real applications. A new type of waveguide utilizing the coupling between the SPP waveguide and the dielectric waveguide has been reported in 2008 [6]. This novel hybrid plasmonic waveguide is composed of a dielectric cylinder overlying on a metal surface. A small gap with a lower refractive index material is located between the cylinder and metal interface (as seen in Fig. 3.8). This HP waveguide can provide better performance compared with the conventional dielectric waveguides and SPP waveguides.

3.3.1 Cylindrical Hybrid Plasmonic Waveguide

The schematic of the Cylindrical HP waveguide is shown in Fig. 3.8(a). The optical guided region is the low-index dielectric gap between metal surface and high-refractive index dielectric cylinder. At the

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3.3 Hybrid Plasmonic Waveguide 27

interface between the metal surface and low-index dielectric gap, the SPP-like mode can be propagated. In the mean time, the interface between low-index gap and high-index dielectric cylinder will support another kind of propagating wave, which is similar as the propagating wave of the dielectric waveguides. If the low-index dielectric gap between the metal and high-index dielectric cylinder is narrow enough, the hybrid propagating wave will be obtained.

Figure 3.8: (a) Schematic of a cylinder hybrid plasmonic waveguide. (b) Energy density distribution in cylinder hybrid plasmonic waveguide. The wavelength is chosen at 1550nm, the radius of the dielectric cylinder is 100nm and the gap thickness of the guided region is 10nm. (c) The electric-field amplitude along x=0 (the blue dashed line in the inset) shows the optical confinement in the low-index dielectric gap (the blue shading area). (d) The electric-field amplitude along y=5nm (the blue dashed line in inset) shows the localized optical confinement under the high-index dielectric cylinder.

In Fig. 3.8(b), the energy density distribution of cylindrical HP waveguide with the operated wavelength of 1550nm is illustrated. The radius of the dielectric-cylinder is 100nm and the thickness of the guided gap is 10nm. The refractive indexes of silicon and silica are 3.45 and 1.45, respectively.

The metal is gold ( at the operated wavelength). From this figure, we can see that most of the optical energy is confined in the gap between dielectric cylinder and metal substrate.

Fig. 3.8(c) shows the electric field amplitude distribution along x=0 (the blue dashed line in the inset), which demonstrates the optical confinement of low-index dielectric gap (the blue shading area).

The electric field amplitude in the low-index dielectric gap decreases with the increasing thickness of x

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the gap, which means the optical confinement of the guided gap is weak when the thickness of the gap is large. Similar phenomenon can be observed in MIM plasmonic waveguide as well.

Fig. 3.8(d) shows the electric field amplitude distribution along y=5nm (the blue dashed line in inset), which shows the localized optical confinement under the high-index dielectric cylinder. The localized optical confinement acts similar with the strip-loaded dielectric waveguide in Fig. 3.2(b).

Next, we will consider the influences of the radius of high-index dielectric-cylinder. This can be discussed firstly from the electric field distributions in Fig. 3.9. We set the dielectric-cylinder radiuses as 150nm, 350nm and 500nm, and other parameters stay the same. When the radius of dielectric cylinder is 150nm, a good optical confinement can be formed in the low-index dielectric guiding gap.

When the radius increases to 500nm, the HP waveguide looks more like a dielectric-cylinder waveguide. A larger part of optical energy is confined in the dielectric-cylinder, resulting in a lower confinement factor of the low-index dielectric gap.

Figure 3.9: Energy density distribution in cylinderical HP waveguide, the thickness of low-index dielectric gap is 15nm. (a) Cylinder_radius=150nm, (b) Cylinder_radius=350nm, and (c) Cylinder_radius=500nm.

As we known, the increasing of the radius of dielectric-cylinder leads to the lower optical confinement in the low-index gap, which can be directly observed from the energy density distributions in the figure above. Simultaneously, the propagation length will also be changed when the geometric parameters are changed. In the following part, we will discuss the relation between propagation length and geometric parameters.

We change the radius of dielectric cylinder from 10nm to 250nm, and the thickness of the low- index gap is altered from 2nm to 50nm. The curves of propagation length can be obtained from the simulation software (COMSOL Multiphysics), which are shown in Fig. 3.10 (a).

For a large radius and gap width (r_cylinder>200nm and d_gap>50nm), the HP waveguide acts more like a low-loss cylinder-like waveguide, where the propagation length is large. On the contrary, for a small-radius cylinder (r_cylinder<80nm), the SPPs dominate the propagating wave, and the propagation length is small. The propagation length tends to the same value when the radius of the cylinder is smaller than 20nm, and this value equals to the propagation length of metal-dielectric interface (absence of the dielectric-cylinder).

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3.3 Hybrid Plasmonic Waveguide 29

Figure 3.10: (a) Propagation length changing with the cylinder radius. (b) Normalized mode area changed with the cylinder radius. The radiuses of the cylinder are changed from 10nm to 250nm, the gap thicknesses are 2nm, 5nm, 10nm, 25nm and 50nm.

Considering the gap thickness, the propagation length increases with the gap thickness in large radius region (r_cylinder>100nm), while for a small-radius cylinder (r_cylinder<70nm), the propagation length decreases with the gap thickness. Similarly, this phenomenon can be explained by the transformation between SPP-like and cylinder-like modes when the radius of dielectric-cylinder increases.

In addition to the propagation length, the effective mode area is another important parameter used to evaluate the performance of optical waveguide, which is defined by:

, (3.3) where and are the electromagnetic energy and energy density respectively.

The diffraction-limited area in free space is written as:

. (3.4) Normally, we use the normalized mode area, , to classify the optical confinement capacity of an optical waveguide.

Fig. 3.10 (b) illustrates the change of the normalized mode area with the variation of cylinder radius and gap thickness. The radius of cylinder is changed from 10nm to 250nm, and the gap thickness is altered by 2nm, 5nm, 10nm, 25nm and 50nm.

For a small-radius cylinder (r_cylinder<80nm), the HP waveguide acts like a metal-dielectric interface, where the optical confinement is weak, which results in a high normalized mode area. On the contrary, at the region of large radius (r_cylinder>150nm), the HP waveguide is close to a dielectric cylinder waveguide, which also has large values of normalized mode area. It is worth noting that there is an optimized mode area around the radius of 80-150nm, where the optical wave can be intensely confined into the gap between dielectric cylinder and metal surface. Besides, the normalized mode area increases with the gap thickness. For a large gap thickness and cylinder radius (d_gap=50nm, r>180nm), the normalized mode area of HP waveguide become to be equal to the purely dielectric cylinder waveguide (the black dashed curve). It is worth to notice that, at the radius of

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Study of Stripe-shaped Optical Filter and Plasmonic Analogue of EIT Effect based on Hybrid Plasmonic Waveguides

Academic year 2012-2013

Study of Stripe-shaped Optical Filter and Plasmonic Analogue of

EIT Effect based on Hybrid Plasmonic Waveguides

Xu Sun

Study of Stripe-shaped Optical Filter and Plasmonic Analogue of

EIT Effect based on HP Waveguide

TRITA-ICT-EX-2013:86

Abstract

Acknowledgements

Contents

Chapter 1

Introduction

1.1 Background

1.2 Motivation and Objective

1.3 Research Summary

1.4 Thesis Outline

Chapter 2

Basic Principles and Study Methods

2.1 Surface Plasmon Polaritons

2.1.1 The Electromagnetic Wave Equation

2.1.2 Surface Plasmon Polaritons at the Metal-dielectric Interface

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2.2 Numerical and Simulation Methods

2.2.1 Scattering Matrix Method

2.2.2 Finite Element Methods (FEM)

2.3 Dispersion Model of Noble Metals

2.3.1 The Drude Model

2.3.2 Extended Drude Model

2.3.3 Optical Constant of Gold

2.4 Summary

Chapter 3

Comparison of Different Optical Waveguides

3.1 Dielectric Waveguides

3.1.1 Planar Dielectric Waveguide

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3.1.2 Nonplanar Dielectric Waveguide

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3.3 Hybrid Plasmonic Waveguide

3.3.1 Cylindrical Hybrid Plasmonic Waveguide

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