1-1 Photonic crystal and Photonic crystal slab
1-1.1 Photonic crystal (PC)
In the last few decades, there have been many works on developing integrated photonic circuits, which hold potential to confine and control electromagnetic (EM) waves on the scale comparable to modern electronic devices and also can be created with integrated electronic circuits to improve the device performance. Photonic crystal (PC) is composed of multi -dimensional periodically arranged dielectric materials with large refraction index difference, which can provide a periodic dielectric function for light traveling. This kind of structures was first proposed by Eli Yablonovitch [1] and Sajeev John [2] in 1987, and have attracted a great deal of attentions because of low loss and high confinement of light, strong dispersive properties [3,4] and capability of fabricating in silicon substrates.
In the PCs, EM wave propagation is inhibited in certain ranges of frequencies, which is called photonic bandgap (PBG), and the photons with these frequencies can be trapped and guided by creating defects in the crystal lattice. These features are much analogous to the electronic band structures of semiconductors. In the PBG, the incident waves satisfy the Bragg condition and will be completely reflected owing to the vanish of corresponding eigenmodes, so the PC can be designed to prevent light propagating in certain direction with specific frequencies. Furthermore, if the interaction between light and the lattice is sufficiently strong, the PBG can extended to cover all possible directions, becoming a complete PBG [5], which can be used to form omni-directional reflectors [6] and can act as an efficient optical insulator. When we introduce some defects to break the translational
symmetry in the perfect PCs, which means to change the locations or sizes of the lattice points, that can provide extended defect modes with frequencies inside the PBG. Therefore, the light with certain frequencies or directions can easily be controlled in the structures.
These characteristics of the PBG offers many novel applications of optical mode confinement. For point defects, the defect modes can be strongly localized as in the optical resonant cavities with high quality factor [7]. On the other hand, by using line defects, the photonic crystal waveguides (PCWs) [8] can be created to restrict and to guide light from one position to another. Different from the index-guiding in the traditional total internal reflection (TIR) waveguides, whose disadvantages such as high energy loss and small bending angle will increase the scale of optical devices, the PCWs allow light to propagate in a medium with relatively low refraction index by the Bragg reflection, and can provide low energy loss even through a sharply bending [9] for wide range of frequencies.
1-1.2 Photonic crystal slab (PCS)
Generally, researches and applications are mainly about two-dimensional (2D) PC structures because the design and fabrication of which are relatively easy. However, for investigating the dispersion properties of practical PC devices with finite thickness, the structures called photonic crystal slab (PCS) should be considered, which promises easier fabrication using existing techniques. The PCS is a three-dimensional (3D) PC structure with 2D periodicity on a plane whose height is comparable to the lattice constant [10], and can use index guiding to confine light in the finite dimension, says the vertical direction.
This structure can be simply classified into two types [11] : the dielectric rods in low index materials and the air holes in planar dielectric substrates. In general, the air-hole PCS is relatively easy to design and to avoid fabrication errors, so that is widely used to form the practical devices. To obtain the larger PBG in designing, the dielectric-rod and air-hole
structures are usually arranged with square lattice and triangular lattice individually, as shown in Fig. 1.1(a) and Fig. 1.1(b), where a is the lattice constant. Furthermore, to prevent from the multi-mode propagation in devices under operation frequencies [12], the air defects made by reducing the radius of dielectric rods or enlarging the radius of air holes (reducing the effective refraction index) are the better choices.
(a) (b)
Fig. 1.1 Photonic crystal slabs made of (a) dielectric rods in air with a square lattice and (b) air holes in dielectric slab with a triangular lattice. The marked parameters are used in the simulations in Chapter 3.
In the PCS, owing to the lack of the translational symmetry in the vertical direction, there are no pure transverse-electric (TE) modes and transverse-magnetic (TM) modes, but rather the TE-like (even) modes and the TM-like (odd) modes, which are determined by that the electric fields are mainly parallel or vertical to the slab plane. Additionally, because of the finite thickness in the vertical direction, there exists a light line in the photonic band diagram. The propagation modes under this line can localize around the defects and be guided in waveguides. On the other hand, the radiation modes in the region above the light line will leak their energy outward the slab. Therefore, for the photonic devices, we should mainly focus on the propagation modes.
1-2 Coupled resonant optical waveguide (CROW)
The coupled resonant optical waveguide (CROW) is composed of linear periodic arrays of identical point-defect cavities in the PC. This kind of structures was first proposed and analyzed by Yariv et al. [13,14] in 1999. In the CROWs, electric fields are strongly localized in the defect cavities with high quality factor, and the propagation of EM waves can be accomplished by the evanescent-field coupling or photon hopping between the adjacent defects. Because of large group delay caused by the weak tunneling of waves among cavities, the group velocity of light in the CROWs can be several orders of magnitude smaller than that in bulk materials with the same refraction index, and the dispersion curve of defect modes is nearly flat. Such slow-light propagation has great significance for several devices of optical communication systems, such as group velocity dispersion (GVD) compensators [15,16], optical buffers [17] and delay lines [18,19]. Furthermore, the high-Q cavities lead to large field amplitude of defect modes, so the nonlinear interaction between photons and materials will be enhanced. This property is useful for many applications such as optical pulse propagation [20-22], soliton optics [23-25], holographic recording [26] and efficient second-harmonic generation (SHG) process [27,28].
1-3 Directional coupler (DC)
The directional coupler (DC), which can be created by placing a pair of parallel waveguides closely in the PC [29] as in Fig. 1.2, is a kind of optical components for mixing or separating the guided EM waves. For symmetric DCs, there are two dispersion curves corresponding to one odd parity mode and one even parity mode, with respect to the symmetry plane between the two waveguides. After an operation frequency is selected, the guided modes can be expressed as the linear combination of these eigenmodes, and the
coupling length can be defined as π/∆k, where ∆k is the wavevector difference between the two dispersion curves. The light guided in one waveguide of the DC will be completely coupled into another one after travels a coupling length as in Fig. 1.3 (a). In some cases, the dispersion relations may have a crossing point due to the degeneracy of eigenmodes [11]. At this so-called decoupling point, the coupling length becomes infinite, so the power transfer between the two waveguides is eliminated as in Fig. 1.3 (b). These properties of optical signal coupling are especially important for integrated photonic circuits so the DC has been widely used in many devices, such as beam splitters [30,31], optical switches [32-34], add/ drop filters [35], and wavelength multiplexers/demultiplexers [36,37]. Therefore, turning the decoupling point to get the proper coupling length at a certain range of frequencies is an important issue.
Fig. 1.2 An optical switch made of a directional coupler in the photonic crystal slab. [34]
(a) (b)
Fig. 1.3 Numerical simulation results of EM waves propagating in the DC (a) with a finite coupling length or (b) operated under the decoupling point (with an infinite coupling length).
1-4 Numerical methods
To investigate the EM wave propagation in such photonic devices, several numerical efficient algorithms such as the plane-wave expansion method (PWEM) [38,39] and the finite-difference time-domain (FDTD) method [40,41] are generally used to simulate the dispersion characteristics of the PC structures.
1-4.1 Plane-wave expansion method (PWEM)
The PWEM is good at analyzing the photonic band structures and waveguide modes for a specific polarization of infinite periodic structures. It is operated by formulating and
where E(r , t) is the electric field intensity, H(r , t) is the magnetic field intensity, D(r , t) is the electric flux density, B(r , t) is the magnetic flux density, ρ(r , t) is the electric charge density, J(r , t) is the electric current density, and r is the spatial coordinate, respectively.
Assume the dielectric materials are source-free, transparent, linear, isotropic, non-dispersive, and non-magnetic, these equations can be written as
where ε0 is the permittivity of the free space, εr(r) is the relative permittivity, and µ0 is the permeability of the free space. In form of time-harmonic electric fields and magnetic fields
E(r , t) = E(r) exp(iωt) H(r , t) = H(r) exp(iωt) ,
we substitute Eq. (1.9) and Eq. (1.10) into Eq. (1.6) and Eq. (1.8), and obtains
▽×E(r) + iωµ0 H(r) = 0 where ΘH is the hermitian operator. Similarly, we can obtain from Eq. (1.11)
ΘE E(r) = [εr(r)]-1 ▽×[▽×E(r)] = (ω / c0)2 E(r) .
Because the PC is a periodic structure, the dielectric function ε(r) can be written as εr(r) = εr(r + T)
T = u1 a1 + u2 a2 + u3 a3 ,
where T is the lattice translation vector, {ai} are primitive translation vectors, u1, u2 and u3 are integers, respectively. Therefore, we can also define the reciprocal lattice vector as
G = m1 b1 + m2 b2 + m3 b3 ,
where {bi} are primitive translation vectors, m1, m2 and m3 are integers, and ai•bj = 2πδij
with the Kronecker’s delta function δij. By expanding the [ε(r)]-1 into the Fourier series as [εr(r)]-1 = ∑ κ(G) exp(iG•r)
where VC is the volume of an unit cell. Applying the Bloch’s theorem to the fields, we derive the following eigenfunctions for Eq. (1.15) and Eq. (1.16)
Ekn(r) = ∑ Ekn(G) exp[i(k + G)•r]
Hkn(r) = ∑ Hkn(G) exp[i(k + G)•r] ,
where k indicates the wavevector and n denotes the index of photonic bands. By substituting Eq. (1.23) and Eq. (1.24) into Eq. (1.15) and Eq. (1.16), we can obtain the eigenvalue equations as
∑ κ(G − G’) (k + G’) × [(k + G’) × Ekn(G’)] = −(ωkn / c0)2 Ekn(G)
∑ κ(G − G’) (k + G) × [(k + G’) × Hkn(G’)] = −(ωkn / c0)2 Hkn(G) ,
where ωkn is the eigenfrequency for the specific fields Ekn(r) and Hkn(r). By solving these two equations numerically, we can obtain the photonic band diagram of the PC structure.
1-4.2 Finite-difference time-domain (FDTD) method
On the other hand, the FDTD method is regularly used to estimate the transmission and reflection spectra for computational electromagnetic problems, and can deal with the structures with finite boundary, which is hard to be done by the PWEM. This method is directly derive from the Maxwell’s equations in the time-domain on a space grid to study the characteristics of EM wave propagation at different time, and the fields are obtained by
µ0 (∂Hx / ∂t) = (∂Ey / ∂z) − (∂Ez / ∂y) µ0 (∂Hy / ∂t) = (∂Ez / ∂x) − (∂Ex / ∂z) µ0 (∂Hz / ∂t) = (∂Ex / ∂y) − (∂Ey / ∂x) ,
where E(r , t) = (Ex , Ey , Ez) and H(r , t) = (Hx , Hy , Hz) . To denote a grid point of space as (i , j , k) = (i∆x , j∆y , k∆z)
for any function of space and time, we assume
F(i∆x , j∆y , k∆z , n∆t) = F n(i , j , k),
where ∆x , ∆y , and ∆z are spatial discretizations, ∆t is the time step, and i, j, k, n are integers.
By applying the central-difference approximations for both the spatial and temporal differential equations, we obtain time-domain expression for the x component of electric fields :
Ex n+1
Through the same procedures, we can obtain Ey n+1
Ez n+1
Eqs. (1.39) − (1.41) are the finite difference equations for the transverse electric (TE) waves, and Eqs. (1.42) − (1.44) are the equations for the transverse magnetic (TM) waves. In addition, Fig. 1.4 is the Yee’s cell used to describe the various field components for the FDTD method that assume the components of electric fields are in the middle of the edges and of magnetic fields are in the center of the faces to satisfy the curl relations of the Maxwell’s equations.
Fig. 1.4 Components of the electric fields and the magnetic fields in the Yee’s cell.
(1.42)
(1.43)
(1.44)
When using the FDTD method, we must set ideal absorbing boundary layers to keep the calculating region from the reflection of EM waves, and the boundary condition of perfect matched layers are the most efficient and widely used. Additionally, in order to ensure the values converge to stable solutions, the time step should satisfy the restriction :
c0 ∆t ≤ 1 / √[(∆x)−2 + (∆y)−2 + (∆z)−2] .
For increasing the simulation accuracy, a smaller grid size should be considered, but the appropriate time step is also reduced, which makes the time-consuming in the computation.
1-5 Motivation
The CROWs made by periodic arrays of point defects in the PC will lead to a large group delay for the propagation of EM waves, and the defects can act as the optical cavities with high quality factor. Therefore, the group velocity of light can be reduced by several orders of magnitude, and the nonlinear interaction between the fields and materials can be enhanced.
On the other hand, the DC consist of a pair of waveguides in the PC support two dispersion curves with one odd mode and one even mode, and can be used to mix or separate the optical signals of different frequencies by controlling the coupling length. In the symmetric structures, these two dispersion curves may cross at a so-called decoupling point, and the EM waves propagate under this point will have no energy coupling between the two waveguides.
For integrated photonic circuits, combining different types of structures is an important issue, because that can hold more abilities and improve the performance of the whole systems. In addition, 3D structures as the slab should be considered to approach actual devices. To our best of knowledge, there is no research demonstrating the DCs based on CROWs in the PCS, which holds potential to combine the applications of slow-light propagation, nonlinear optical processes and optical signal coupling. Therefore, it is worth discussing the dispersion behaviors and providing the effective models for this kind of structures.
(1.45)
To design such photonic devices, numerical methods such as the PWEM and the FDTD method are regularly used to simulate the band diagram and the characteristics of EM wave propagation. However, these methods cannot provide a direct physical insight or a good explanation for the simulation results, and they also require concentrated computing to reach the coupling behaviors within the designed models. In the CROWs, the strong localization of electric fields and the weak tunneling of EM waves have much in common with the electronic transport in crystalline solids, which possess strong periodic potential from the localized lattice atoms. Therefore, the tight-binding theory (TBT) in solid-state physics [42,43] can be used to express the eigenmodes in the CROWs, and the simple equations of the dispersion relation can be derived analytically, which have the ability to describe the coupling effects between the defect cavities [13]. This method and the derived equations have been successively used in many studies of both linear and nonlinear optical properties of the CROW structures [13-16, 19-22, 24-27], so we anticipate using it to investigate and provide a design concept on our structures. types from the relative positions of the PCS-CROWs and the composition of the PCS. Then, by modifying the size of the defects in the designed structures, we can obtain the variation of electric fields among the cavities, and realize the changes of the parameters in the derived equations that influence on the dispersion curves. In Chapter 3, we first discuss the simulation errors due to the finite size of the super cell in the 3D PWEM, and the difference
between the simulation results of 2D and 3D structures. Then, we show that the dispersion behaviors of the PCS-CROWs and the PCS-DCs analytically predicted by the TBT agree well with the numerical results simulated by the PWEM, and the design rules of these structures are given. Additionally, we also discuss the possible applications for the different types of PCS-DCs. Finally, the conclusion and perspectives are presented in Chapter 4.