1-1 Photonic crystal
Over the recent twenties years, photonic crystals (PCs) and the related photonic bandgap devices have attracted a great deal of attentions in fabricating all optical integrated circuits.
The concept of the PC was first proposed by Eli Yablonovitch [1] and Sajeev John [2] in 1987.
Yablonovitch presented that a structure can offer a photonic bandgap (PBG), hence electron-hole pairs of a semiconductor cannot recombine within a range of frequencies and therefore no spontaneous emission occurs. Sajeev John also proposed photons can be localized by the PC defect, similar as the physical property of electrons. Afterward two professors used the PC and the PBG to denominate related researches of these new fields.
A PC is usually composed of periodically arranging dielectric materials with large refractive index difference. Such structure possesses a complete PBG [3]. Within the PBG of the PC, a certain range of frequencies of electromagnetic (EM) waves is disallowing to propagate and therefore suppress a band of frequencies from existing. Because there are no corresponding propagation modes in the bandgap, the incident EM wave is completely reflected. We can well design and construct PCs with PBGs to prevent light from propagating in certain directions with specified frequencies.
There are no extended states in the PBG of a perfect PC. As we introduce a defect to break the translational symmetry, the defect may permit localized modes to exist with frequencies within the bandgap. Introducing defects to PCs means to have the lattice points or locations dissimilar from the perfect arranged structure. Defects can change the band structure of a perfect PC and allow guided modes to exist inside the bandgap. For a point defect, a mode can be localized whenever its frequency is in the PBG. By using line defects,
one can guide light from one location to another. We can alter the mechanisms of PCs by introducing various defects. The characteristic of the PBG provides many novel applications, such as using a PC block to form photonic reflector [4], introducing line defects in the PC to construct a waveguide, or creating a point defect (or several defects) to form a PC resonant cavity [5].
1-2 Photonic crystal waveguide
The photonic crystal waveguide (PCW) can be created by modifying a linear sequence of unit cells. As we introducing a line defect into a perfect PC, a defect-guided mode is formed within the bandgap. A light wave will be restricted to transmit in this channel. In traditional total internal reflection waveguides, light is only restricted to propagate in high refractive index materials. Besides, the bending angle for changing the light propagation direction of traditional waveguides cannot exceed 1 degree, otherwise the energy loss is significant; this disadvantage make the scale of optical devices increases. Conversely, PCWs allow light to propagate in a low refractive index medium, like air. Light that propagates in the PCW with a frequency within the bandgap of the PC is confined to the defects and can directly transfer along the defects. Consequently, PCWs provide low energy loss and high light confinement even through a sharply bend [6]. The PCWs are the most promising elements of PCs for designing photonic integrated circuits.
1-3 Directional coupler
A directional coupler (DC) which can be used in light switches [7,8], beam splitters [9,10], and modulators [11] is formed by arranging a pair of parallel PCWs separated by one or several rows of partition rods or holes. A symmetric DC possesses two dispersion curves with an even mode and an odd mode, respectively. As two waveguides are quite close, their fields will overlap. An EM wave with a given frequency is incident into one PCW of the
coupler will transfer entirely to the other waveguide as transmitted a certain distance, which is called the coupling length. The coupling length is defined as π Δk, where Δk is the wavevector mismatch of the two dispersion curves at the operation frequency. As designing the wavelength-selective devices such as the demultiplexer [12], the coupling length of a DC is considerable. The transmission efficiency of the PC coupler can be improved by properly tuning the formation of defects. Different claddings of the coupler also vary vertical confinements of light, which propagates in the PCWs of the DC. With various applications, DCs are also major devices for the optical communication.
1-4 Numerical methods
To investigate the propagation of EM waves through a PC structure, there are several efficient and accurate algorithms such as plane wave expansion method (PWEM) [13] and finite-difference time-domain (FDTD) method [14]. The PWEM is well at calculating the bandgap for a specific polarization, dispersive properties, and eigenmodes of infinite periodic structures. The FDTD method is widely used to estimate transmission and reflection spectra for computational EM problems.
1-4-1 Plane wave expansion method
From the Maxwell’s equations, we know
( , )
density, ρ( , )r tK is the free charge density, and rK
is the spatial coordinate, respectively.
Assume the dielectric materials are lossless, linear, isotropic, non-dispersive, and non-magnetic, these equations can be rewritten as
0 permittivity of free space. In form of harmonic fields:
( , ) ( ) i t and substituting Eq. (1.9) and Eq. (1.10) into Eq. (1.5) and Eq. (1.6), one obtains
( , ) 0 ( , ) Calculating the curl of Eq. (1.12):
0 The PC is a periodic structure, hence the dielectric function can be written as
( )r (r i) ε K =ε K+ aK
, i=1,2,3, (1.17) where {aKi
} are the primitive lattice vectors of the PC. We can also define the reciprocal lattice vector G as
} are the reciprocal lattice basic vectors and
i⋅ j =2πδij and applying Bloch’s theorem to the fields, we can derive the following eigenfunctions:
( ) ( ) exp[ ( ) ]
indicates the wavevector and n denotes the band index. Substituting Eq. (1.21) and Eq. (1.22) into Eq. (1.15) and Eq. (1.16), we get eigenvalue problem of these two equations, we obtain the band diagram of a PC structure.
1-4-2 Finite-different time-domain method
The FDTD numerical technique is proposed by Yee in 1966 [15] to solve the Maxwell’s curl equations. The method can be used to deal a real PC with finite boundary, which is hard to be done by the PWEM. The FDTD method provides a straight-forward way to directly derive Maxwell’s equations in the time-domain on a space grid, avoiding mathematical
difficulties of solving frequency-domain problems. Since the FDTD is a time-domain method, it is usually used to study characteristics of the EM wave propagating in a PC structure at different time.
In an isotropic and lossless medium, Eqs. (1.5) and (1.6) are equivalent to the following scalar equations in the rectangular coordinate system:
0 To denote a grid point of the space as
( , , )i j k = Δ(i x j y k z, Δ , Δ (1.31) ) and for any function of space and time, we set
( , , , ) n( , , )
F i x j y k z n tΔ Δ Δ Δ =F i j k , (1.32) where Δx, Δ , and zy Δ are spatial discretizations, and Δt is the time step, i, j, k, and n are integers. Applying the central-difference approximations for both the spatial and temporal differential equations gives
Substitute Eqs. (1.34) – (1.36) into Eq. (1.25), we can get the following finite-difference time-domain expression for the x component of magnetic field:
1 1 Through the same procedures, we also obtain
1 1
1 1
Equations (1.37) – (1.39) are finite difference equations for the transverse magnetic (TM) wave, and Eq. (1.40) – (1.42) are equations for the transverse electric (TE) wave. A TE wave is defined as the EM wave for which the magnetic field is polarized vertical to the plane of a waveguide. Figure 1.1 is the Yee’s cell to describe the various field components.
Assuming the E-components are in the middle of the edges and the H-components are in the center of the faces to satisfy the curl relations of Maxwell’s equations.
Fig. 1.1 Field components of a three-dimensional Yee cell.
In using the FDTD method, we have to set absorbing boundary layers to truncate the calculation domain without reflection. As simulating a wave transferring in the free space, these ideal layers can absorb the EM wave, which propagates outward. The perfectly matched layers absorbing boundary conditions proposed by Berenger [16] are the most efficient and widely used mechanisms. Besides, in order to ensure the values will not diverge, the time step Δt should satisfy the restriction
2 2 2
For the purpose of increasing the simulate accuracy, a smaller grid size is expected.
However, the time step also reduces with the grid size, which makes time-consuming in the
computation. In applying the FDTD method, electric and magnetic fields are derived by iterating each other in time and spatial domains. Therefore, we can obtain the propagation characteristics of EM waves in the PC.
1-5 Motivation
Because the design and the fabrication of two-dimensional (2-D) PCs are easy, researches and applications are mainly about 2-D PC structure. The DCs are frequently assumed with infinite height; however, a simply 2-D structure cannot strictly confine the radiation energy loss of vertical direction. Besides, a 2-D PC device with infinite height is not suitable for the realistic usage. As a result, we investigate the properties of the slab with 2-D PCs in this thesis. The PC slab is a three-dimensional structure with finite thickness to confine light in the vertical direction. Generally, the PC slab can be classified into two forms: One is constructed of dielectric rods fabricated in air, as shown in Fig. 1.2(a); and the other is composed of a thin planar dielectric substrate with air holes, surrounded by dielectric claddings as Fig. 1.2(b).
Fig. 1.2 Two types of PC slabs: (a) dielectric rod type and (b) air-hole type.
The coupler made of two identical PCWs supports two dispersion curves, with an even mode and an odd mode, respectively. It is called a symmetric DC. Two dispersion curves of a symmetric DC with triangular lattice should cross. The crossing point is called the decoupling point with infinite coupling length [17]. When an EM wave propagates at the
frequency of the decoupling point, energy is never couple into the other waveguide [18].
While the DC is designed as a demultiplexer or a switch, to decouple two waveguides might be necessary sometimes. Therefore, as designing a DC, tuning the crossing point and the dispersion relation to get the proper coupling length at the wanted range of frequencies is an important issue [19].
Although the PCWs of a DC are usually made by removing dielectric rods or air holes due to less scattering loss caused by the structure disorder [8] and easier to be fabricated, the PCW made by removing rods or holes usually guides multimodes and the coupling length are quite long [9,10]. In order to prevent from multimode propagation at the operation frequency or to reduce the coupling length, a DC made by reducing the radius of the dielectric rod or enlarging the radius of the air hole seems to be a better choice if the structure disorder can be minimized.
Besides, dispersion curves of the symmetric DC usually cross in the triangular lattices, but rarely cross in the square lattices. This phenomenon causes restrictions as designing the DC. In order to solve such problems, we adjust the DC by shifting defects of the PCWs to tune the dispersion relation and find a design rule for designing the coupler.
Applying the PWEM can derive the band diagram of the DC. And from the FDTD method, we can get characteristic of EM waves propagating within the PC. These two methods are regularly utilized to simulate a periodical PC structure. However, such numerical methods cannot provide a useful analytical result for realizing the coupling properties of the coupler or PCWs. The tight-binding theory (TBT) [20,21], which provides analytic equations, can be applied to illustrate the coupling effects between near defects of PCWs and to infer the shift trend of dispersion curves as shifting defects of the coupled PCWs.
We anticipate using the TBT to derive an analytic description to explain physical properties of the PC coupler.
1-6 Organization of the thesis
In this thesis, we first introduce the extended TBT to describe the dispersion relation of a single PCW and the coupled PCWs made of slab in Chapter 2. Second, by transversely and longitudinally shifting all defects in DCs, we survey variations of coupling coefficients and their influences on the dispersion curves using the TBT. From the analytical derivation, we acquire the designing rules to tune the eigenfrequency of the decoupling point and the coupling length by suitably shifting line defects (PCWs) in Chapter 3. Finally, numerical simulations based on the PWEM and the FDTD verify the validity of the analytical results derived from the TBT for the PC couplers. The conclusion and the perspectives will be presented in Chapter 4.