Chapter 1: Introduction
1.2 Thesis Outline
In chapter 2, we introduce the fundamentals of the finite-difference time-domain (FDTD) [28-33] method and Drude model [34-35]. In 1966, Yee proposed the finite-difference time-domain technique to solve Maxwell’s equations. Yee’s method can be used to solve a lot of scattering problems on dielectrics and electromagnetic absorption at microwave frequencies [36]. There was a little attention for the finite-difference time-domain technique in the early years because of lacking sufficient computing resources [37]. As the computer gets more powerful and faster the FDTD, the finite-difference time-domain technique has become a well-known method to solve electromagnetic problems in periodic structure.
In chapter 3, we have successfully proposed a novel all-optical plasmonic logic gate based on nano-disk cavity filled with optical nonlinear Kerr material in metal-insulator-metal waveguides. At first, we investigated the radius of the disk resonators. We find that a linear relationship exists between the transmitted-trough wavelength and the disk radius of the cavity. Reveals that the transmitted-peak wavelength has a inversely relationship with the coupled-aperture width of the nano-disk resonators. The active tuning of the pumping light also makes the wavelength shift. According to this result, we using the eight disk resonator can simply filter out about 70 nm band-stop at the desired wavelength by properly adjusting the radius and coupled-aperture width of the cavity. By changing the control power, we can implement logic gate. According to the numerical results, the normalized transmission of the high level logic state is larger than 90% and the low level logic state is 0.1%. In this chapter
we found that the proposed all-optical logic gate all exhibits an excellent effect. It would be a potential key component in the application of the ultra-high-speed data processing system.
In chapter 4, we have successfully proposed a novel triplexer filter structure based on nonlinear triangle resonators in metal-insulator-metal waveguides. At first, we investigated the height of the triangle resonators. We find that the longer height we set, the longer wavelength trough wavelength shifts. According to this result, we can simply filter out the desired wavelength by adjusting the height of the triangle. We also discuss the coupling length, we find that the coupling length increases, the quality factor is also increases. By simulation result, the quality factor of the filter can be controlled by changing the coupling length. We add a reflector and find out the best reflective length to improve transmission efficiency. By applying above methods, our proposed triplexer filter is designed. The triplexer that we proposed can accurately wavelength at 1310nm, 1490nm and 1550nm with transmission efficiency about 90%. The triplexer filter shows good promising for the FTTH applications. The total size of the proposed optical triplexer is only 3.2μm×1.5μm.
In chapter 5, the thesis and suggestions for the future researches in the paper will be discussed.
Chapter 2
The Basic Theory and Method
2.1 Introduction
The electromagnetic wave propagates through a structure containing a periodic modulation of material characteristics on the scale of its wavelength is extremely different from that of the homogeneous case. This is famous known in solid-state physics, where the periodicity is responsible for the formation of electronic states in metals, insulators, and semiconductors. An electromagnetic wave propagates through the periodic medium that will show a lot of interesting and potentially helpful phenomena, such as the diffraction of light from the periodic strain accompanying a sound wave. Some characteristics of the electromagnetic propagation in periodic media are analogous to the quantum theory of electrons in atomic crystals.
There are a lot of methods to calculate the characteristics of the electromagnetic field such as: the finite difference time domain (FDTD) method, finite element method (FEM), transfer matrix method (TMM). One of these calculating methods which FDTD is familiar calculated the characteristics of the electromagnetic field. FDTD is calculated the movement behavior of electromagnetic wave on time domain and solved the transient effect problem or complex boundary. In this chapter, we will introduce the famous numerical methods, FDTD.
2.2 Finite-Difference Time-Domain Method
In 1966, Yee introduced the finite-difference time-domain (FDTD) method [37]. There are several research departments working in the areas motivated large-scale solutions of Maxwell’s equations during the 1970s and 1980s. The entire field of computation electrodynamics is shifting quickly in high-speed communications and computing.
During the 1990s, engineers in the ordinary electromagnetic community became aware of the modeling capabilities afforded by FDTD and related techniques, and it has expanded well beyond defense technology in this area. The main reason to introduce FDTD method to solve electromagnetic field is that when the structure is too complex, it is hard to solve Maxwell’s equation in frequency domain. FDTD provide a robust method to solve it in time domain. On the other hand, there are many advantages in FDTD method. First, FDTD is exact and robust. The sources of error are well known.
Second, FDTD avoids the difficulties with linear algebra that limit the size of frequency-domain integral-equation when starts a fully detailed computation. When the differential forms of Maxwell's equations are inspected, they can be seen that the time derivative of the E field is related to the curl of the H field (∇×H). This can be simplified to state that the rate of the change in the E field depends on the change in the H field across space. The results in the basic FDTD equations are that the new value of the E field is related to its previous one and the difference of old values of the H fields on either side of the E field point in space.
2.2.1. Maxwell’s Equations in Three-Dimension
Maxwell’s equations in an isotropic medium are [37]:
ρ
Maxwell’s equations describe a situation in which the temporal change in the E r
field is dependent upon the spatial variation of the H
r
field. The FDTD method solves Maxwell’s equations differences in time and space and then numerically solving these equations. Yee [36] defines the grid coordinates (i, j, k) as
(
i j k, ,) (
= ∆ ∆ ∆i x j y k z, ,)
(2.11) Any function of space and time is written as(
, , ,)
i j kn, ,U i x j y k z n t∆ ∆ ∆ ∆ =U (2.12) where ∆x, ∆y, and ∆z are the lattice space increments in the x, y, and z coordinate directions, and i, j, k, and n are integers. ∆t is the time increment. The spatial and temporal derivatives of U written using central finite difference approximations as [37]
(
, , ,)
Uin1/ 2, ,j k Uin1/ 2, ,j kAnd expression for the first time partial derivative of U, evaluated at the fixed space point (i, j, k), follows by analogy:
The familiar method to solve Maxwell’s curl equations is based on Yee’s mesh and computes the E
v
and H v
field components at points on a grid with grid points spaced
∆x, ∆y, ∆z apart. The method results in six equations that can be utilized to compute the
In order to explain the method, we consider a scattering problem in two dimensions.
We assume that the fields components do not depend on the z coordinate of a point.
Besides, we take ε and µ to be constants and J≡0. The only source of our problem is then an “incident” wave. This incident wave will be “scattered” after it encounters the obstacle. The obstacle will be of a few “wavelengths” in its linear dimension. We can decompose any electromagnetic field into “transverse electric” and “transverse
magnetic” fields if ε and µ are constants. The two modes of electromagnetic waves are small compared to the wavelength. We can write the finite difference equations for the TE and TM waves.
(1, 1/ 2) ( 1/ 2, ) (1/ 21/ 2, 1/ 2) (1/ 21/ 2, 1/ 2) approximation, laying a foundation for explaining the optical characteristics of metals and other conducting materials in 1900. Since one hundred years ago the electronic theory had been grown [34]. However, the purely classical Drude model still utilized extensively to explain a lot of novel problems. Since the beginning of the generally research of the physical characteristics of semiconductors, the Drude model has been became an important implement, it offers the main way for studying their low-frequency optical absorption due to the current carriers. The complex interaction of the set of flowing electrons with the atomic cores can be considered simply by introducing an effective mass (m*) and the sign of the carrier charge. It was noted that in his theory [35], Drude also presented the existence of two types of carriers with
additive contributions to the absorption. He predicted the discovery of semiconductors and importance of his model for researching them. The next peak in the requirement for the Drude model came with the discovery of high-temperature superconductors (HTSC), for some form of the Drude model is utilized in the majority of papers to explain the results of optical researches in the infrared. Because of the low concentration of carriers, the infrared spectra of HTSC's can exhibit phonon peaks, and the visible spectra have strong bands of transitions. A sufficient description of the observed spectra can be accomplished by supplementing the Drude model with Lorentzian oscillators [38-39].
Although the Drude model of modifications with an increased number of parameters are utilized in some papers [40–42], the Drude model for a normal metal is generally related [43] to the Mattis–Bardeen model [44]. The expressions obtained in the frame of the Drude model for the real part of the dynamic dielectric functionε1
( )
ω and the dynamic conductivityσ( )
ω are known as the Drude–Zener formulae [34], the Drude–Lorentz formulae, and the Kramers–Kronig relations [45]:( )
Where N and e are the concentration and charge of the current carriers
σ
0 is the staticconductivity, and the plasma frequency is given by the relation
ω
2p=4π
Ne2/m*ε
∞. Weabbreviate the detailed indication of the frequency dependence of the optical functions in those cases where it is obvious. The electron energy loss function is related to the dynamic dielectric function of the system. It can be measured by pure optical modes, by
exciting plasma oscillations of the electrons for the polarization of an electromagnetic wave in the case of oblique incidence on the reflecting surface of a thin metal film [35].
The expressions for n and k obtained from (2.35) are considerably amenable to analytical study after substantial simplifications. It usually involves the approximation of low (or high) frequencies in comparison with the values of ωP.
Chapter 3
All-Optical Logic Gate Based on Nano-Disk Resonators with Kerr Nonlinear medium in Metal-Insulator-Metal
Waveguides
3.1 Introduction
Recently, the development of optical components is growing prosperity. A promising solution to this problem is to use optical logic gates for all optical signal processing.
Photonic devices can provide high speed, high capacity and low loss. Due to the diffraction limit of light in the photonic devices, they are all big in size, low density, and low efficiency. Surface plasmon polaritons (SPPs) have promising application on the devices of highly integrated optical circuits because they overcome the conventional diffraction limit and can manipulate light on sub-wavelength scales [46-49]. Some devices based on SPPs have been proposed and demonstrated, such as Nano-antenna [50-51], splitters [52-53]. Utilizing optical devices in communication system has many advantages like wider communication bandwidth and higher transmission speed.
Therefore, some all-optical logic gates have been proposed such as photonic crystal [54-55], plasmonic waveguide [56-57], silicon micro-ring resonator [58] and other several different types of all-optical logic gate [59-64].By using metal-insulator-metal (MIM) waveguide, all-optical logic gates can provide many advantages such as making the device size miniaturized and high efficiency. The metal-insulator-metal (MIM)
structures consist of a dielectric waveguide and two metallic claddings, which strongly confine the incident light in the insulator region, allow the control and transmission of light at the nano-scale. The waveguides with nonlinear Kerr effect logic gate also attract a lot of attention. Several all-optical switching and logic devices using optical nonlinearity have ever been proposed and implemented [65-66]. All-optical logic gates devices will become important components for all optical signals processing, because all-optical logic gates can avoid complex of optical-to-electrical or electrical-to-optical conversions. The application of signal processing in optical integrated circuits will be very interesting and unique in the future. By using metal-insulator-metal (MIM) waveguide, all-optical logic gates provide many advantages such as making the device size miniaturized and high efficiency. The MIM structures consist of a dielectric waveguide and two metallic claddings, which strongly confine the incident light in the insulator region, allow the control and transmission of light at the nano-scale. However, useing the optical nonlinearity to design all-optical logic gate devices not common.
Therefore, we try to use the optical nonlinearity materials to design all-optical logic gate.
3.2 Analysis and Numerical Results
In this chapter, we proposed and numerically investigated all-optical plasmonic logic gates based on the nonlinear nano-disk cavity resonators. We used bus waveguides and filled with optical nonlinear materials nano-disk resonators to construct logic gates based on MIM waveguide structure.
The dielectric constant of the Kerr-type nonlinear medium in the resonator is changed
by varying the pump light and the signal transmission can thus be controlled. The logic gate performance was analyzed and simulated by finite difference time domain (FDTD) method. The performance of the proposed devices has advantages of small size, requirement of low pumping light intensity that have potential applications in ultra compact all-optical integrated photonic circuits.
Under normal circumstances, the interfaces between semi - infinite materials having negative and positive dielectric constants can effectively guide the transverse magnetic (TM) surface waves. Since the width of the MIM plasmonic waveguide is much smaller than the wavelength, just the fundamental TM waveguide modes can spread. For TM modes, the dispersion equation is given by [67]：
0
k is the free-space wave vector. The propagation constant β is represented
as effective index neff = β / k0 of the waveguide for SPP.
In the chapter, the dielectric is assumed to be air with
ε
d =1, and the metal to be silver. The dielectric constantε
mof silver can be calculated by Drude model [68]：( ) ( )
value of 3.7, ωp =1.38×1016Hz is the bulk plasma frequency, which represents the
natural frequency of the oscillation of free conduction electrons.
γ
=2.73×1013Hzis the damping frequency of the oscillation, andω is the angular frequency of the incident electromagnetic radiation. The SPPs are pleased with inputting a TM - polarized plane wave. The transmission of the structures is defined as T =Ptr/Pin [69]. Pin presentsthe total incident power, and Ptr is the transmission power. The refractive index of the Kerr-type nonlinear material can be expressed as
n = n0 + n2 I (4.3) where the value of the linear refractive index n0 is set as 1.47. The Kerr-type nonlinear material is assumed to be Au-SiO2, and its nonlinear refractive index is n2 = 2.07 × 10-9 m2/V2 and I is the pumping beam intensity [70].
The proposed of single aperture-side-coupled dielectric nano-disk resonators is shown in Fig. 3.1, which is consisted of a waveguide coupled to aperture-side-coupled dielectric nano-disk cavity filling with a Kerr-type nonlinear material. The parameters of the structure are set to be w = 50nm, w1 = 20nm, h = 100nm, and R = 290nm. We studied the influence of the radius of the nano-cavity on the resonance wavelengths by the FDTD method. The radius is set as variable while the other parameters are fixed as above. Figures 3.2 (a) and (b) show the transmission spectra with different radius R. The resonant peak-wavelengths have a red-shift with increasing of the radius. Fig. 3.2 (a) shows the different radius R=250nm to 370nm and the Fig. 3.2 (b) shows the relationship between the resonant trough-wavelength and the radius of nano-cavity is
approximately linear. And the coupled-aperture width is set as variable while the other parameters are fixed as above. Fig. 3.3 (a) and (b) show the transmission spectra with different width w1. The resonant trough-wavelengths have a red-shift with reducing of the width. Fig 3.3 (a) shows the different width w1=10nm to 50nm and Fig. 3.3 (b) shows the relationship between the resonant trough-wavelength and the coupled-aperture width w1 of the nano-disk resonators is approximately inversely. Then we use the same method to investigate the channel length h between the nano-disk resonators and the bus waveguide. In Figs. 3.4 (a)-(f), the length h is varying from 40nm to 140nm. As the numerical results shown above, the operating wavelength can be easily tuned by changing the radius and the coupled-aperture width of the nano-disk resonator.
Fig.3.5 (a) shows the proposed two nonlinear aperture-coupled dielectric nano-disk resonators. We use the same radii of the nano-disk resonators and two different coupled-aperture width of cavity to find out the influence. The parameters of the proposed structure are setting to be R = 290nm, h = 100nm, d = 240nm, w1 = 20nm, w2
= 30nm, and w = 50nm. According to the above results, the transmission spectra of the output nano-disks possess band-stop wavelengths of 1310nm and 1300nm, respectively.
As shown in Fig. 3.5 (b) it is found that the resulting band-stop about 10nm, where the transmission waves through two mirrors generate the destructive interference around the transparency wavelength. Next, we investigate the distance d between the nano-disk cavities. In Figs. 3.6 (a)-(h), the distance d is varied from 0nm to 320nm. When d
=240nm, we can get the better band-stop and band-pass results.
Finally, we set the two nano-disk resonators as a unit cell nano-disk resonator and
investigate the influence of the number of the nano-disk resonators. The number and the radii of the are set as variable while the other parameters are fixed as shown in Tab. 3.1.
Figures 3.7(a)-(f) show the transmission spectra with different number of the cavity, the number of cavity are two to twelve. Numerical results of the number of the nano-disk cavity is eight. The best band-pass and the band-stop is eight nano-disk resonators. Next, we analyze AND, OR, NOR, XNOR, NAND and XOR logic gates by using the eight nano-disk resonators filled with a Kerr-type nonlinear material, as shown in Fig. 3.8.
Before designing the logic gates, we suit investigate the parameters of the proposed structure to obtain the optimal values for designing the logic devices.
3.3. Simulation and Results
3.3.1. AND gate
We use the proposed plasmonic waveguide structure with eight-disk resonators filled with a Kerr-type nonlinear material to design an AND logic gate, as shown in Fig.
3.8. Before design the logic gate, we must first investigate the two straight pumping waveguide L1 and L2 and coupling length g the influence. First, we are the proposed plasmonic waveguide structure, as shown in Fig. 3.8, to design an all-optical AND logic gate. The parameters of the proposed structure are chosen R1 = 295 nm, R2 = 290nm, R3
= 285 nm, R4 = 280 nm, h = 100 nm, w = 50 nm, w1 = 20 nm, w2 = 30 nm, d1 = 20nm, d2 = 240nm. In the Figure 3.9 and Tab 3.2 shows the band-stop efficiency in wavelength of 1310nm for different length of pumping waveguide. Basis the simulation results, we found that the best length L1 is 400 nm. Next, we change L2 from 100nm to 500nm. In Figure.3.10 and Tab. 3.3 we find the transmission efficiency at L2 = 300nm, the trough
wavelength at 1310nm minimum transmittance is 0.002%. Next, we investigate the distance d1 between the nano-disk cavities and waveguide. In Figs. 3.11, the distance d1
is varying from 0nm to 220nm. When d1 =20nm, we can get the better band-stop and band-pass results. Then we use the same method to investigate the transmission spectra for different gap distance between the straight pumping waveguide and the bus waveguide. In Fig. 3.12 shows that when g = 50nm, we can obtain the best band-pass transmission efficiency. In this structure, the signal port is always ON with the input
is varying from 0nm to 220nm. When d1 =20nm, we can get the better band-stop and band-pass results. Then we use the same method to investigate the transmission spectra for different gap distance between the straight pumping waveguide and the bus waveguide. In Fig. 3.12 shows that when g = 50nm, we can obtain the best band-pass transmission efficiency. In this structure, the signal port is always ON with the input