CHAPTER 2 COMPUTATIONAL METHOD
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 (αn<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 (εSiO2=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 in such slab waveguides. One of them is “slow light” [20].
Slow light, as implied by the name, is that the speed of light wave is slowed down to the value much smaller than that in vacuum. It is expected that a great variety of applications, such as optical delay lines, buffer devices, and memory to store optical signals could be carried out if the light speed can be controlled at will. Since the photonic crystals are a category of band structure engineering, the slope of the corresponding dispersion curves, dω/dk, would determine the energy propagation velocity of light waves in the crystal patterns. Accordingly, it makes possible to retard the propagation speed of light waves if the photonic crystal patterns are designed properly. In the band diagram of photonic crystal slab waveguides, the defect bands are formed within the photonic band gap because of the introduction of line defects.
Waveguide modes corresponding to the defect bands would propagate along the slab waveguide at different velocities if the shapes of the defect bands are not linear lines.
Therefore, it is practicable to observe propagation light waves at extremely slow velocity in photonic crystal waveguides.
In the following, we start with slow light by taking 2D photonic crystal slab waveguides with single line defect (W1 type). First, the band diagrams of the waveguides were calculated by using plane wave expansion (PWE) method with effective index approximation. Next, we demonstrated the performance of the fabricated W1 type slab waveguides in experiments. After that, the deduced group velocities as a function of wavelength were obtained by means of frequency-domain
measurement (Fabry-Perot method) and time-domain measurement (phase-delay approach), respectively. Finally, the advantages and disadvantages of the two measurement methods will be showed and an improved way to measure the ultra slow light in photonic crystal waveguides will be proposed.
3.2 Band Diagram
Figure 3.1(a) is a pattern of 2D photonic crystal slab waveguide used in our calculations. A row of crystals is removed from the otherwise hexagonal lattices to form a so-called W1 type photonic crystal waveguide. The radius and the refractive index of crystals are assumed r0=0.255a and nair=1.0, respectively. Si substrate was used as the material of the slab in simulation so as to meet the practical samples. The effect of the thickness of the slab was approximated by the approach of effective index of guided modes in the dielectric slab [48]. In this way, calculation dimension can be reduced from 3D to 2D without altering other simulated results, such as the field distribution of eigenmodes. A supercell was used as a primitive unit cell to construct the waveguide in spatial domain, as shown in Fig 3.1(b). And then, by using the PWE method, this supercell was applied to Maxwell equations (described in Chapter 2) to solve eigenmodes of 2D photonic crystal slab waveguides.
Figure 3.1(c) is the calculated band diagram for TE-like modes of a W1 type photonic crystal waveguide. The dash line, marked as light line, results from the air-cladding (nair=1.0) and indicates a demarcation between the leaky mode region and photonic crystal slab mode region. Waveguide modes above this line are leaky.
While the gray region shows the photonic crystal slab modes which can propagate inside the slab, the white area represents PBG where light transmission is forbidden.
Fig. 3.1 (a) Schematic of a W1 type photonic crystal waveguide. (b) A selected 1x10 supercell. (c) Band diagram for TE-like modes of W1 type waveguides. (d) Y-components of magnetic field in the defect waveguide. (Left) Odd modes (Right) Even modes.
Of interest are the two unusual defect bands within the PBG. These two bands allow a certain frequency range of waveguide modes to propagate along photonic crystal line defects without leakage because of in-plane PBG confinement and TIR in the vertical direction. Close inspection of the field distributions of the guided modes in the two bands exhibits highly symmetry along the central axis of the defect waveguide, named odd and even modes (see Fig. 3.1(d)).
Furthermore, it clearly appears that the gradients of the two bands under the light line become flat, especially at the zone edge of the bands. This suggests that group velocities of the guided modes within the narrow bandwidth region would be very slow. In order to further understand, we will examine the properties of even bands for the fabricated 2D photonic crystal slab waveguides by using frequency-domain and time-domain measurements in the following section.
3.3 Device Fabrication and Characterization
Based on the theoretical results given in Section 3.1, a 4-inch SOI wafer with a 2µm buried oxide and a 205nm thin silicon guiding layer were used as a slab dielectric material. W1 type photonic crystal waveguides with triangle lattice constant of a=432 nm and hole radius of r=110 nm were patterned by E-beam lithography and transferred upon the Si guiding layer by ion couple plasma (ICP) dry etching. Then the underlying SiO2 cladding layer was removed by selective wet etching using HF solution to form an Air/Si/Air membrane structure, as shown in Fig. 3.2. Si tapered ridge waveguides were connected to both sides of photonic crystal waveguides. W1 type slab waveguides with different lengths (L=50~3024µm) were prepared for the following propagation loss measurements. The total lengths of all samples, including
Fig. 3.2 SEM micrographs of the fabricated W1 type photonic crystal waveguides. The lattice constant and hole radius are 432nm and 110nm, respectively. The thickness of the slab is 205nm.
Fig. 3.3 Schematic illustration of the measurement system [49]. The inset shows the taper structure between Si and photonic crystal waveguides.
Si taper waveguides and photonic crystal waveguides, were fixed at 6mm.
Figure 3.3 shows the schematic of the measurement setup. A long-wavelength tunable laser with wide bandwidth of λ=1490~1620 nm was used as light source.
Linearly TE-polarized light (E // PhC plane) was coupled into the samples by using polaizers, single-mode optical fibers and objective lens. Light passing through photonic crystal waveguides was then analyzed by optical spectrum analyzer (OSA).
A typical transmission spectrum of the fabricated W1 type photonic crystal waveguide with 54µm long is shown in Fig. 3.4. In this case, the structural parameters are
Fig. 3.4 Experimental transmission spectrum of a W1 type photonic crystal waveguide with a 54μm length. The inset is the magnification of the cutoff region. Δλmeans the Fabry-Perot oscillation period.
designed to be the same as that in Fig. 3.1. The span from 1510nm to 1593nm corresponds to the bandwidth of the even modes calculated in Fig. 3.1. Similarly, other samples with different PhC waveguide lengths but the same structure parameters were also measured and shown in Fig. 3.5. Random dips observed in the
Fig. 3.5 Measured spectra of the samples with different waveguide lengths.
Fig. 3.6 Transmittance against photonic crystal waveguide lengths at different launch wavelengths.
spectra of longer waveguides are due to fabrication disorder, i.e. broken crystals.
Different position of the cutoff in the curve results from the propagation loss of the waveguides, which is in proportion to waveguide length.
To estimate the propagation loss of the fabricated W1 type photonic crystal waveguides, the so-called “cut-back” method was used. The details of the method are described as follows. First, the original measured transmission curves were rearranged as a function of PhC waveguide length. Then, straight lines were applied to each modified curve by using the approach of least squares, shown in Fig. 3.6. Finally, the slopes of each fitting lines, presenting propagation loss, were plotted against the wavelength. Figure 3.7 shows the results of propagation loss measurements of the fabricated W1 type photonic crystal waveguides. It is clearly seen that the propagation loss in the range of 1530nm~1580nm is around 2~3dB/mm. In other words, if the device size is shrunk to 50µm, only 0.1~0.15 dB loss is shown. Such loss is too small to affect the normal operation of practical devices, hence our fabricated photonic crystal waveguides are considered ready to be applied to photonic integrated circuits.
This value, to our knowledge, is the smallest one among those ever reported studies on photonic crystal waveguides.
Fig. 3.7 Propagation loss spectrum for the fabricated W1 type photonic crystal waveguide. It is clearly that propagation loss is only 2~3dB/mm in the transmission region.
3.4 Group Velocity Measurement
3.4.1 Frequency-Domain Method
Since Si ridge waveguides were designed to connect with both sides of the photonic crystal waveguides (see the inset in Fig. 3.3), light reflection from the interface of Air and Si waveguide or Si and PhC waveguides resulted in the obvious Fabry-Parot (FP) interference patterns in spectra, as shown in Fig. 3.4. These FP patterns, however, exhibited abnormal oscillation periods (∆λ), which is getting smaller as wavelength increases. Based on FP formula, given by
2 ,
2
Lng
λ = λ
∆
with λ being the FP resonant peak, L being the length of PhC waveguides, and ng
being the group index, a reduction of ∆λ implies the increase of ng. Well-known in solid state physics, group index is defined as the ratio of light speed in vacuum to that in material, expressed as ng
= c/v
gwhere c is the speed of light in vacuum and v
g is the group velocity. Therefore, a large value of ng means that light wave propagates through materials at a small group velocity. Fig. 3.8 shows the deduced group indexes from Fig. 3.4 as a function of wavelength λ. The group index rapidly increases from 20 to 200 at around wavelength λ=1592-1596nm and reaches its highest value ng=300 at wavelength λ=1597nm near the cutoff. This indicates a swift change of the energy propagation velocity of light in photonic crystal waveguides larger than that in air by a factor of 20-300.Fig. 3.8 Group index dispersion curve. The red dots show experimental ng deduced from Fig. 3.4 by applying the Fabry-Perot formula. The dash line represents theoretical ng, which is derived from the even mode band in Fig 3.1(c).
Fig. 3.9 (a) Magnified dispersion curve for the guided modes around the band-edge region. (b) Illustration of the Fabry-Perot resonant modes, which correspond to the guided modes in W1 type photonic crystal waveguides.
Since these group indexes were derived from the measured spectra by means of FP method, longer PhC waveguides would offer better opportunities to explore extremely large group indexes of light waves than short ones. This is because FP modes spacing is inversely in proportion to the waveguide length. The narrower the FP mode spacing, the more likely we are to approach the band edge points of guided modes. Figure 3.9(a) is a magnified dispersion curve around the band edge of even modes in W1 type slab waveguides. Figure 3.9(b) is a sketch of corresponding FP resonant modes in spectrum. Here, we assume that the dispersion curve is in parabolic shape and wave vector at the band edge of the first Brillouin zone is k0, which corresponds to a normalized frequency ω0. According to a parabolic approximation, we could express other normalized frequency ω around band edge as
2
where α is a numerical parameter related to hole radius, refractive index, and pattern structure, and ∆k is the wave vector difference in dispersion curve, which is given by
2 ... the dispersion curve, group velocity is given by
( )
As the waveguide length (L) increases, ∆k would become smaller. So, it should be possible to achieve vg ≈0 at
ω
≈ω
0 in theory. In practical experiments, however, a long waveguide would cause a large propagation loss. Therefore, we believe if the optimum waveguide length is found, an approximately ultimate band edge point of the guided mode, which corresponds to the extremely large group index, could beobserved.
3.4.2 Time-Domain Method
The other approach to determine velocity dispersion of photonic crystal waveguides is to monitor the phase delay of launched signal. Figure 3.10 is the schematic view of our time-domain measurement system. A 3G Hz(ω) modulated signal was steered into two branches: one is conducted to the fabricated photonic crystal waveguides and the other is directed to the phase combiner. Due to the different optical paths, the phase of the modulated signal passing through photonic crystal waveguides would differ from the original signal by an amount of ∆Φ. Since delay time is given by τ=∆Φ/ω where ω being the modulated frequency, group index of the waveguides can be determined via
n
g = c* τ /L,where L is the length of the photonic crystal waveguides.
Fig. 3.10 Illustration of the phase-delay measurement setup.
(a)
(b)
Fig. 3.11(a) Measured group delay time for the samples with different photonic crystal waveguide lengths. (b) Group index dispersion curve for the 1.3mm long W1 type photonic crystal waveguide.
The black line in (a) and (b) represents the signal transmits over a Si waveguide.
Figure 3.11(a) shows the measured relative group delay time for W1 type photonic crystal waveguides with different lengths. Here, the structural parameters of all samples are designed to be the same as those in Fig. 3.1. The resolution of
time-domain scale is 0.1psec and the fluctuation shown in the plot comes from the internal FP noise of the waveguides. It clearly shows that the delay time rapidly increases at the cutoff region. However, due to the limited function of the modulator in our measurement system, the maximum deduced group index from the longest sample (L=1.3mm) is only 40 at around λ=1590 nm, as shown in Fig. 3.11(b). The information behind this point can not be analyzed. Compare this result with what we have obtained in frequency-domain measurement, the obtained group indexes are almost the same from ng=15 atλ =1580 nm to ng=40 at λ =1590 nm. This again confirms that light waves at launched frequency close to the band-edge guided modes are rapidly retarded when passing through photonic crystal waveguides due to the unique characteristic of dispersion.
3.5 Discussion
While these two measurement methods provide convenient ways to examine the slow
While these two measurement methods provide convenient ways to examine the slow