1-1 Background
During the past decade the use of photonic crystals (PhCs) has been studied and risen from an indistinct technology to a prominent field of research [1,2]. This is mainly because of their potential ability to well control the propagation of light. Eli Yablonovitch [3] and Sajeev John [4] initially predicted the idea that a periodic structure consisting of materials with different dielectric constants possesses bandgaps for certain ranges of the frequency, in much the same way as an electronic bandgap exists in semiconductor materials. Photonic crystal with defects can be found much more applications. Defects in photonic crystals means the points or places different from perfectly arrayed structures. Defects just like missing a point, line or dislocations can create defect modes within the photonic band gap.
Using this property, photonic crystals can modify the spontaneous emission efficiency and the propagation of light, leading to novel applications in splitter, waveguides (Fig. 1-1), defect-mode light-emitters, electro-optical switch [5], Mach-Zehnder interferometer [6], and micro-cavity lasers (Fig. 1-2) [7–10], etc. This is why many scholars believe that the PhCs bring us a possible solution and unlimited vision of creating large-scale photonic integrated circuits (PICs) in the future and have done more and more studies on photonic crystals.
Numbers of reports focusing on the design of PhC’s devices in PICs have been published in the last few years [11].
(a) (b)
Fig. 1-1 (a) splitter (b) waveguide
Fig. 1-2 Photonic-crystal micro-cavity laser
Two-dimension photonic crystals are regarded as the hottest topic nowadays, because they offer the possibility of fabricating high-Q cavities [12-13] and waveguide devices [14] on the scale of the wavelength in the semiconductor-based structures (i.e. GaAs/AlGaAs or SOI).
Photonic integrated circuits of similar integration density so far only known as electronic VLSI (Very Large Scale Integrated Circuits) can be imagined. Photonic crystal waveguide (PCW) is an important basic element in PICs [15, 16] as important as the electric wire in the electric circuits. It is the key component of interconnect between optical circuits. Optical waveguiding in two-dimension photonic crystals is achieved by introducing line defects in the structure that is otherwise periodic in two dimensions.
When we take photonic crystal as basic structure of waveguide, another important characteristic of photonic crystal is its unusual dispersion property. Group velocity
dispersion of line defect in photonic crystal slabs is experimentally proved to be extremely large, and can be tuned via adjusting the widths of defects [17]. In conventional total internal reflection (TIR) waveguide, the bending angle for changing light propagation direction cannot be over 1o, otherwise the loss will be quite big. Different from the conventional waveguides, photonic crystal band gap (PBG) and large group velocity of PCWs can still keep well guiding the signal even if they form sharp-bend, as shown in Fig. 1-3.
Fig. 1-3 Distribution of the real part of electric field in a 90o bend of the dielectric rods PCs. The red color shows positive
amplitude of electric field and the blue for negative amplitude.
Two closely parallel waveguides can be used as a directional waveguide coupler [18-22].
A directional waveguide coupler is also one of key components for optical communication.
They can be used as wavelength-selective power dividers, switches, modulators, etc. [23, 24]
Besides, it might be desirable to decouple the two waveguides to minimize cross talk between them, for example, when envisioning closely packed photonic wires in integrated optical circuits [25].
Other phenomena of two-dimension PhCs had also been widely discussed, including coupling/decoupling, energy flow [26], and extremely low group velocity [27-28]. All of those researches make us getting closer and closer to entirely grasp this new technologies.
1-2 Motivation
In order to prepare for arrival of the next-generation optical communication, many scholars try to develop new optical devices which possess tiny scale, high efficiency, integrabilility and easy fabrication. Fortunately, people found some kinds of man-made materials called photonic crystals that make all our imagination realizable. By introducing different defects into perfect photonic crystals, many abilities such as wave-guiding, light-trapping, filtering, slowing light and light coupling could all be generated at will. With integrating such devices in a single chip, large photonic integrated circuits provide a wide view of future information technology. People even predict the coming of the photonic computer in the next ten years.
For optical communication systems used now, the size of the wavelength dependent power splitter is about hundreds of micrometer. If one can reduce the size of photonic crystal directional coupler devices to ten of micrometers, it should provide a great advantage for wavelength division multiplexing (WDM) systems. This provides the motivation to develop an effective numerical method for analyzing coupling between channel waveguides in a two-dimension photonic crystal. In the previous research, a photonic crystal waveguide is formed by a chain of point defects, so the waveguide can be regarded as a coupled-cavity waveguide (CCW), in which the energy can hop from a cavity to the neighbor one. The propagation of wave through a CCW is exactly the classical wave analog of the tight-binding (TB) method in solid state physics. It also indicates that there exists a large potential in designing various compact photonic devices by using the large dispersion of coupled mode splitting. According to this idea, we can do the design of PhC devices applying in optical communication with micrometer scale. In the following chapter, we will present two topics focusing on physical insight in PhC waveguides with tight-binding theory and optical devices
such as WDM based on photonic crystal with silicon rod array. In order to design WDM, we need to know the coupling length at each frequency. According to the coupling length formula L=π/Δk, we must know the value of Δk in order to calculate the coupling length.
Although we can obtain a band structure through the plane wave expansion (PWE) method, it needs to extensive calculation to generate good resolution of dispersion curve, especially for the decoupling point of two identical photonic crystal waveguides (PCWs). By using the dispersion function derived from the tight-binding theory, we can well fit the calculated dispersion curves of the derived dispersion function. In turn, we can easily calculate the coupling length at corresponding frequency using the dispersion relation function. Therefore, few data of dispersion relation calculated from the PWE are enough to determine the dispersion function and the decoupling point.
1-3 Organization of the thesis
We divided this thesis into four chapters. We have narrated a brief statement to the background and history of the photonic crystal and also our research motivations in chapter 1.
The main theory and numerical analysis methods we depended will put in chapter 2. After that, in the chapter 3 we will describe our approaches to the coupling problem between PCWs and our PhC optical device design and the simulation results. In the end, the final conclusion will be presented in chapter 4.