Organization of Dissertation

The dissertation is organized as follows.

Chapter 2 provides the theoretical fundamentals of photonic crystal structures.

Propagation light wave within the periodic dielectric materials is discussed by the use of Maxwell equations. Plane wave expansion (PWE) method to solve the Maxwell equations is elucidated in the chapter. The concept of photonic band diagram and photonic band gap are then described with the help of eigenfunctions obtained from the method. The supercell technique employing on PWE method is introduced for the

analyses of photonic crystal defect structures. The widely-used finite difference time domain (FDTD) method to simulate dynamic field transition in photonic crystal is detailed as well.

In Chapter 3, we present the band-edge slow light modes in the single line defect photonic crystal waveguides. The propagation loss of the fabricated samples is evaluated by means of the cut-back method. The group index dispersion curve is obtained from the measured spectra by applying Fabry-Perot method. The increasing group delay time at the cutoff frequency is also observed in the phase-detection measurements and related with the band-edge defect modes of photonic crystal waveguides.

Following these results, 2D slab photonic crystal coupled waveguides with a unique flat band allowing inflection point slow light modes propagation are systematically studied in Chapter 4. The theoretical band structure is first designed to obtain S-shaped coupled band. Then, the existence of the coupled modes is examined by transmission spectra and Mach-Zehnder interference curves. Finally the time-resolved measurements on temporal optical pulses through the coupled waveguides are conducted.

In Chapter 5, we demonstrate an integrated photonic crystal nanocavity coupled laser structure. A practical fabrication process of photonic crystal nanocavity integrated with electrically driven quantum well laser diodes is successfully developed. Light emission from the nanocavity is observed and compared with that from the cleaved facet of the integrated laser diode. The temperature stability of the nanocaviy modes and the capability of multiple wavelength emission from side by side slightly different nanocavities are also investigated, respectively.

The conclusion and suggestion for future work are found in Chapter 6.

Chapter 2

Computational Method

This chapter provides the theoretical fundamentals revelant to the study of photonic crystal structure. In the beginning, Maxwell equations will be intoduced to discuss the propagation of light wave within a linear, isotropic, and homogenious medium. Then, light behaviors in the periodic dieletric materials will be analyzed with the help of plane wave expansion (PWE) method. To determine field modes for a 2D photonic crystal, the photonic band diagram and the photonic band gap are elucidated. Besides, the supercell technique employing on the PWE method to solve defect crystal structures will also be described. Finally, the finite difference time domain (FDTD) method will be detailed to calculate the space and time changes in the electromagnetic fields of light.

2.1 Full Maxwell’s Equations

According to the electromagnetic model, all the light waves are governed by four well-known Maxwell equations (SI units):

(2.1a)

Here B, E, H, and D are the magnetic flux density, electric field, magnetic field and electrical displacement field, respectively. J is the free current density and ρ is the free

charges. These Maxwell's equations are generally applied to macroscopic averages of the fields, which vary wildly on a microscopic scale in the vicinity of individual atoms [21]. It is only in this averaged sense that one can handily treat light behaviors in various kinds of dielectric media. Detailed description of these equations from their microscopic correspodent can be found in [21-22].

Here let us consider a simple case of a uniform and transparent medium for propagation light. In this medium, no external light or current source is assumed. That is, J and ρ are all 0. We also restrict ourselves to the linear regime of the medium so that higher oder terms of dielectric constant related to nonlinearity can be ignored here. To simplify the system, frequency dependence of dielectric constant is neglected.

In addition, we treat this medium as a non-dispersive, isotropic, and lossless dielectric material. Therefore, D and B fields related to E and H can be given by

).

where ε(r) and µ(r) are the electrical permittivity (dielectric constant) and the magnetic permeability of the material, respectively. ε0 and µ0 denote the proportionality constants in vacuum and hold the relation of ε0

µ

0 =1/c², where c is the speed of light in vacuum. Since µ(r) is very close to unity for most dielecric materials of interest, we set B(r)= µ0

H(r) in the following derivations.

With all of these assumptions done, the Maxwell’s equations [(2.1a) ~(2.1d)]

are reduced to

(2.3a) Here E(r, t) is the electric field and H(r, t) is the magnetic flux density at the point r and time t. Since these equations are linear, we can separate out the time dependence by expanding the fields into a set of harmonic modes:

.

Equation (2.3) then becomes

(2.4a)

Now take a look at Eq. (2.4a) and (2.4b). They mean that the divergence of a magnetic field or elcectrical displacement field is always zero. If considser the propagation of a uniform plane wave along the arbitrary direction within the medium, we will find that a constant vector (amplitude) of the magnetic field (or electric field ) is always transverse to the direction of propagation. Take a plane wave of a magnetic filed H(r)=H₀e^ik⋅ras an example [23]. Here H0 is a constant vector, and k is a

By replacing H(r), the divergence equation of ∇⋅H(r)=0 becomes assuming a uniform plane wave of . Thus this proves that both E and H are transverse to the direction of propagation.

r

eik

E r

E( )= ₀ ^⋅

Based on reference [22], E and H are also shown perpendicular to each other in a homogenous, source-free, and lossless medium. Generally, we name such particular field configuration transverse electromagnetic (TEM) wave.

Next, we pay our attention to the last two curl equations, Eq. (2.4c) and (2.4d).

These two equations can be combined together by taking the curl on both sides of the equations. Then the equations become entirely in E(r) or H(r):

). can be solved if a dielectric function ε(r) is given.

2.2 Plane Wave Expansion Method

Since photonic crystal is a kind of periodically varied dielectric materials, its nature of highly symmetries in space, such as translational symmetry, rotational symmetry, mirror symmetry and so on [23-24], is helpful to shorten the calculation time as we determine field modes by Eq. (2.5a) or (2.5b). These special characteristices are very

similar to those of electron systems so that some of the theorems and terminologies of solid state physics can be inherited by photonic crystal in slightly modified forms. In the well-known Schrödinger equation for band calculations of electrons, for example, periodic Coulomb potential in crystals is expressed by Fourier expansion and an electron wave function is supposed in Bloch form [25-26]. When substituted into the Schrödinger equation, the allowed eigenstate of electron energy as a function of its kinetic momentum is thus solved. This approach is named plane wave expansion

(PWE) mehtod because an electron wave function expressed as a superposition of

plane waves is used [3, 27-29].

The same derivation can be applied to the propagation of lightwaves in photonic crystals. The Coulomb potential of an electron is now replaced by the perodic dielectric constant ε(r) and given by

∑

^⋅

where G is the reciprocal lattice vector to characterize the periodicity of photonic crystal and Vc is the volume of the unit cell. The wavefuntion of an electric field E(r) or magnetic field H(r) can also be expressed in the form of the Bloch type, given by

. equation [(2.5a) or (2.5b)], and their corresponding unit vectors are perpendicular to

(k+G) because of the transverse nature of H and D (

∇⋅H(r)=0= ). k is an arbitrary wavevector within the first Brillouin zone (because any k located outside the

) (r

⋅D

∇

Brillouin zone can be replaced by (k-G) without the results changed, as introduced in solid state physics [25-26]). Substituting Eq. (2.6) and (2.7) into Eq. (2.5), we obtain the following matrix [28-29]:

). with wavevector (G’-G). Both equations are the k-dependent linear matrixes and can be solved by using standard matrix-diagonalization method [29].

Futhermore, Eq. (2.8) indicates that the dielectric materials enter the calculation only through the position-dependent dielectric function ε(r), which is evaluated on the fine grid in the real space of unit cell and Fourier transformed into the reciprocal space [3]. Therefore, any periodic arrangement of dielectric crystals with any shapes and filling ratios can be exactly treated in the calculation of field eigenmodes. This was first proposed by Ho et al. in 1990 and is noted for its efficiency and accurancy in calculation

[3]. They used the calculated 750 plane waves for a 3D fcc dielectric

structure and achieved eigenfrequency convergence within 1%. Today, of course, the improvement of the numerical approaches and the rapid development of computers have reduced a lot of time in calculating 3D crystal structures, which used to require thousands of plane waves for accuracy.

Although the PWE method has advantages of accuracy and efficiency, there are still two limitations. One is for dispersive material and the other is for nonperiodic

structure. The former contributes complex part to the dielectric function. This

contradicts the assumption that the refractive index distribution must be identical at all frequencies of the modes being found. The alternative solution is to use FDTD method, which will be introduced in Section 2.5. The latter usually occurs when we treat defect structures but can be solved by employing supercell technique, which will be described in Section 2.4.

2.3 Photonic Band Structure

Since Eq. (2.8) is also a kind of Hermitian eigenvalue problems, all of the eigenvalues (ω/c)² can be obtained for a given value of k [23, 29]. If we plot the eigen-frequency (ω) as functions of wavevectors (k) in the first Brillouin zone, the so-called photonic band structure would be obtained. Take a 2D photonic crystal sttructure as an example.

Assume that the air crystal patterns are embedded in a dielectric background ε(r) and arranged in hexagonal array. The radius of the crystal is r and the pitch (lattice constant) is a, as shown in Fig. 2.1(a). Because it is a two dimensional structure, there are only two primitive lattice vectors, ā1 and ā2, in real space [26]. In this case, they which corresponds to the reciprocal lattice vectors

3 ,

, as shown in Fig. 2.1(b). The realation between

a

₁ a , ₂ and

b

¹ b , ² is defined by Fourier coefficients and given by

) .

φ

, represents Fourier coefficients of each elctric field component and needs to be determined later.

Fig. 2.1 Schematic of a 2D photonic crystal structure with triangular lattices. (a) Real lattice vectors, a1

and a2, and a dielectric backgroundε(r). (b) Reciprocal lattice vectors, b1 and b2, and a wave vector k.

Next, substituting Eq. (2.11) and (2.12) back into Eq. (2.5a) and performing a series of algebraic simplification, we can obtain a modified eigenvalue equation:

,

where O is a matrix in which each entry is a function of wavevector k. (The step- by-step derivation of Eq. (2.14) can be found in [29].) Finally, the eigenfrequcny ω as functions of k can be determined with the help of linear algebra and diagonalization.

Figure 2.2 shows the calculated dispersion curve ω-k (or named as photonic band diagram) of a 2D photonic crystal structure. In this case, the dielectric constant of air crystal and dielectric background are 1 and 13, respectively. The hole radius r is 0.33a. Because the light wave now propagates in the xy plane, the filed modes are seperated into two distinct polarizations, TE-like (E in the xy plane) and TM-like (H in the xy plane) modes. Here, only TE-like modes are concerned.

From Fig. 2.2, we found a special mode gap in a certain frequency region. That is, the density of states, the number of possible modes per unit frequency, is zero within the region. So this gap is called the photonic band gap. All light emission and propagation are prohibited in it. Many applications of this property to nanocavity lasers and waveguides are rapidly developed. A good example will be detailed in Chapter 5. Also, Figure 2.2 reveals that the group velocity of light waves, defined by

dω/dk, is zero at some symmetric points, like Γ

and K. This is very interesting because as light speed is retarded, the photon-material interaction time will be extended; thus, the properties, such as optical gain and nonlinearity effect, would be enhanced. Such topic is categorized as slow light phenomenon and also the focus of this dissertation. More details can be seen in Chapter 3 and 4.

Fig. 2.2 Band diagram for the TE-like modes of a 2D triangular lattice structure. The gray area shows the photonic band gap where light propagation is forbidden. Γ, M, and K are symmetric points in the first Brillouin zone.

2.4 Supercell Techniques

Based on the PWE method (Eq. 2.8), infinite periodicity of dielctric materials in all directions is a requirement for band diagram and field mode calculation. However, if a defect is introduced into the otherwise periodic structures, the original periodicity of system would be altered and a defect mode would arise in the photonic band structure.

To avoid this, the technique of “supercell” is often employed in PWE method. The approach is to replace the original unit cell of the structure by a complicated unit cell, where a defect is surrounded by a set of crystals. The new defined cell, called

supercell, retains its periodicity in every direction so that PWE method can be applied

to solve defect modes [28-29].

Figure 2.3 is an example of cavity defect modes calculated by a 5x5 supercell technique. The supercell size depends on design structures and user’s demand, but it shoud be large enough to avoid interaction with adjacent supercell. Generally, the larger the supercell, the more accurate the defect states are and the longer the calculation time takes. In this case, only a single defect is concerned so that a small size of 5x5 supercell is used. If defect studctures become complicated, such as several missing holes around crystals, larger cells are necessary for studying confined modes with higher accuracy. The same technique can applied to photonic crystal slabs. These structures are periodic in two dimension (i.e. xy plane), but the dielectric funtion ε(r) in the third dimension (i.e. z direction ) still varies because of the finite thickness of slab. If we are able to define an appropriate cell plane, the PWE methood can be used.

Fig. 2.3 Concept of the supercell technique. (a) A 2D photonic crystal defect structure. (b) A selected 5x5 supercell, including the defect. (c) Repeated supercell in all directions for the PWE method. (d) Calculated defect modes in each identical supercell. Clearly it is seen that no interaction occurs between two adjacent supercells.

2.5 Finite-Difference Time-Domain Method

Another widely used numerical technique for computating the propagation of light waves in the medium is the finite-difference time-domain (FDTD) method [30-32].

The FDTD method is a rigid solution to the Maxwell’s equations. Different from the PWE method which merely provides stationary states (eigenmodes), the FDTD method can directly explore the dynamic motion of light waves. Besides, there is no theroretical restriction and approximations in the FDTD method. Any finite space including objects can be straightforward analyzed by the FDTD technique.

The main algorithm of FDTD method builds on solving two Maxwell’s curl equations, as depicted in Eq. (2.3c)and (2.3d). If we rewrite the equations in Cartesian coordinates, we would obtain six scalar equations [33]:

1 .

All of these equations indicate that any spatial variations of one field would cause the other field to change dynamically. In other words, the electricl field and the magnetic field are interlaced in temporal and spacial domain. The FDTD techniques to solve the equations are based on Yee’s approach published in 1966 [31]. Yee dividied the analytical domain into small regutangular cells with edges ∆x= ∆y= ∆z, and assumed that each E field vector component is located midway between a pair of H field vector components inside the cells (see Fig. 2.4), and vice versa. Then, he discreted the time step so that E and H field components can be computed at every short time interval ∆t.

Fig. 2.4 Illustration of the Yee’s cell used for FDTD method [31].

For example ,the field components, Ez an Hz, in Eq. (2.15) at time t= n∆t (n is an integer) can be solved by

),

if a cell, denoted by integers (i, j, k), is given [33]. This scheme, now called Yee’s cell or Yee’s mesh, proves to be essential to many current FDTD softwares.

There is an important issue to be addressed while using FDTD method. It is

boundary condition. In the real world, the space is continuous and infinite, but in the

FDTD method, the computational domain is finite and limited by the computer memeory. Therfore, as the light waves reach the boundary surface of the computational domain, relfection would occur and disturb the excited fields. To solve this problem, many simulations introduce an absorbing boundary condition (ABC),

which eliminates any outward-going waves that impinge on the domain boundries, into their models [33]. Among a number of available highly effective absorbing boundary conditions, Mur’s condition and Bereger’s perfectly matched layer (PML) condition are the two most commonly used techniques [30-31]. The former calculates fields so as to cancel the reflection on the boundary, and the latter places absorbing materials (conductive medium) close to the boudary to suppress the reflection. In general, PML can provide orders-of-magnitude lower reflection than Mur’s condition so that it is widely adoptedly by FDTD users [28]. In the following chapters, our FDTD simulation results are all based on the PML boundary conditon.

Chapter 3

Band-Edge Slow Light Modes in Two-Dimensional Slab Photonic Crystal Waveguides

3.1 Introduction

Photonic crystal (PhC), consisting of periodically-varying dielectric materials, exhibits the ability to manipulate the propagation of light in both passive and active optical devices [23]. If properly designed in geometry, a photonic band gap (PBG), analogous to electron band gap in semiconductors, would be created [34-35]. It is believed that photons can be amenably controlled by introducing artificial defects into the ordered crystals [2, 36]. For example, point defects, where a single or a few periodic dielectric materials are removed, have been realized as cavities at nanometer scale [37-38].

The simplest type of photonic crystal consists of two alternative thin films with different refractive indexes [39]. This 1D photonic crystal, however, dose not have the flexibility for being used in a complicate photonic circuit. A full blown 3D crystal with a complete band gap, in which light at the frequency of defect states can be trapped or guided without losing any energy, would be a good candidate for the realization of omni-functional photonic chips [4, 40]. However, the complicated fabrication of 3D structures keeps them from being further applied. As for 2D crystals, they have many advantages, such as well in-plane light confinement and excellent integration with photonic and electronic devices. But the deficiency of vertical confinement would create another path for light to escape, resulting in serious energy

leakage.

Recently, 2D photonic crystal slab structures, where dielectric slab with planar photonic crystal patterns are sandwiched between two low dielectric constant materials, were proposed and studied extensively [10, 41]. Such structures not only maintain the superiority as 2D crystals, but also effectively confine light in the vertical direction by means of total internal refraction. If we could create high refractive index contrast in between the dielectric slabs and choose the adequate thickness of the sandwiched layer, a pseudo 3D PBG will be formed. One of the most accomplished structures that have been reported so far is that fabricated on silicon-on-insulator (SOI) substrate [42]. This structure consists of Si/SiO2/Si (from surface layer to substrate) and has been maturely developed as the platform for ultra large scale integrated (ULSI) circuits in current semiconductor industry. Strictly speaking, Si is not a good candidate for photonic devices because of its natural characteristic of indirect band gap, but its superior properties, such as high refractive index (nsi=3.4) and low absorption coefficient (α_n<1cm^-1, at n₀=8*10¹⁶ cm^-3 and λ=1.55µm) meet the needs in optical communication network [43]. In addition, the low dielectric constant material of SiO2 (ε^SiO2=3.9) can be easily removed with solution, buffered HF (BOE), thus turning SOI substrate into a membrane structure of air/Si/air/Si bulk. The suspended membrane hence possesses fairly large index contrast (nsi

/n

air =3.4/1.0) which can reinforce the confinement of light in a vertical direction. Moreover, the thickness of the membrane (Si layer of SOI) is always chosen to be half of the wavelength at which light transmits in the material to satisfy the lowest-order modes of standing waves. This is especially important for Si-based planar photonic crystal devices aimed at optical applications.

Examples of such most studied slab devices are 2D photonic crystal slab waveguides

[44]. By inserting line defects into the periodically-arranged crystal

lattices, a certain frequency of light waves will be guided along the defect waveguides with less intensity loss because of the formation of defect modes in above-mentioned pseudo 3D PBG. Over the last decade, various applications of these waveguides, like photonic beam splitter [45], channel-drop filter [46], all-optical switch [47] and so on, were developed. Moreover, many interesting physical phenomena have been observed

lattices, a certain frequency of light waves will be guided along the defect waveguides with less intensity loss because of the formation of defect modes in above-mentioned pseudo 3D PBG. Over the last decade, various applications of these waveguides, like photonic beam splitter [45], channel-drop filter [46], all-optical switch [47] and so on, were developed. Moreover, many interesting physical phenomena have been observed

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