In recent years, slow light in photonic-crystal waveguides (PCWs) has attracted much attention, since the effect remains excellent at room temperature and corresponding devices can be fabricated using traditional semiconductor technologies, providing many potential advantages in signal processing, optical storage [1], tunable delay [2], modulator and filter [3-5]. Generally a PCW is obtained by introducing a line defect into a perfect PC structure, which results in waveguide modes within the photonic band gap of the host PC and excites slow light near the Brillouin zone’s edges. The slow-light performance parameters of the PCWs to which attention is usually paid are the group index n_g and the bandwidth Δω. However, there is interaction between the group index and bandwidth, increasing the former results in decrease of the latter [6]. The method of solving this conflict involves the idea of obtaining flat bands on dispersion curves by locally modifying the effective index in the surroundings of the PCWs, which can be achieved by varying the radius or position of the PC air holes adjacent to the waveguide’s center [7-10], or by selective optofluidic filling of the structure [11-14].

An annular photonic crystal (APC) [15] is an unusual type of PC structure composed of a two-dimensional (2D) lattice of annular dielectric rods in air, or annular air voids in a dielectric background. This structure has been proposed to achieve a large absolute band gap [16], polarization-independent slow light [17], and polarization beam splitting [18]. Because it is a composite unit, an APC presents more possibilities than a normal PC in dispersion engineering. Therefore, in this research we chose an APC as the host of the PCWs, to explore the possibility of obtaining flat-band slow light. Combined with structural-parameter modification and selective optofluidic injection, optimized APCWs can support high-performance slow light with a constant group index, broad bandwidth, and low group-velocity dispersion.

Inspired by reference 16, we designed an APC slab based on a triangular lattice above a SiO₂ layer with a refractive index of 1.45. The remaining portion within the slab was silicon (Si) with a refractive index n_Si = 3.4, and the slab thickness was 220 nm. Figure 1 shows the two-dimensional x-z spatial distribution of the proposed APC structure. The inner and outer radii of the annuli in the figure are denoted by r and R respectively. According to [16], it has been proven that for low air-volume filling factors around 15%, APC slabs can exhibit a significantly larger photonic bandgap (PBG) than conventional PC slabs. A large PBG is unquestionably very important for obtaining broadband slow light. Therefore, we set R = 0.32a and r = 0.21a, and the corresponding air-volume filling factor was 20%.

Figure 1. A schematic of the line-defect waveguide with APC structure: r denotes the inner radius of all annular air holes; R₁ denotes the outer radius of the second row of the annular air holes adjacent to the line defect; and R indicates the outer radius of the other annular air holes.

To save calculation time, it was necessary to calculate the dispersion properties of the APC slabs using a 2D simulation instead of a 3D simulation. As 2D approximation methods, the effective-index method [19, 20] and effective-period method [12, 21] have been successfully applied to the design of PC devices. Especially in Ref. [21], the latter method was shown to provide more accurate estimation of device behavior than the former. Therefore, we utilized 2D simulations by means of the effective-period method. In this method, the 2D effective period for our structure was calculated to be , where n_eff is the fundamental-mode effective index of the ideal multilayer slab waveguide, for a wavelength of 1550 nm. We took the lattice constant to be a = 480 nm, and the effective index to be n_eff = 2.835. Then the effective period was calculated to be a_eff = 400 nm. Using the 2D plane-wave-expansion (PWE) method, a broad PBG for the transverse electric (TE) polarization lay between the normalized frequencies 0.254 (2πc/a) and 0.314 (2πc/a) for R = 0.32a and r = 0.21a. It is worth mentioning that the PBG here matched very well the PBG obtained by the 3D PWE method in Ref. [16] for an APC with the same structural parameters. This proved that the 2D method can accurately model 3D APC slabs with low AVFs.

A line-defect APC waveguide with as formed by removing a row of annuli in the x-direction. The corresponding band structure for the APCW in the x-direction for TE polarization is shown in Fig. 2(a). In the figure, the light line for SiO₂ is depicted with a dotted blue line, and the odd and even modes are represented by dotted black and solid green curves respectively. In a realistic structure based on a slab waveguide, out-of-plane losses can be considerable, but we focused only on the region of interest in reciprocal space where the APCW mode remained under the light line, thus mitigating such losses. Moreover, since the even modes couple dominantly to the incident external optical field in typical applications, all the following discussions will focus on the even modes only. The group index (defined as n_g = c/v_g , where c is the speed of light in vacuum) corresponding to the even mode is shown in Fig. 2(b). The group velocity v_g is the first derivative of the APCW dispersion, i.e. v_g = |∂ω/∂k|. Evidently the group index decreased very rapidly with increasing frequency, which means that such a waveguide mode could not provide a constant group index with a wide bandwidth. The way to eliminate this problem was to obtain flat bands from the dispersion curves.

Figure 2. (a) Dispersion curve for the original APCW with an effective lattice constant of a_eff = 400 nm. (b) The group-index curve for the even mode in (a).

Based on previous work, some parameters are known to influence the dispersion, to obtain polarization-independent waveguiding and slow light in 2D APCWs [17, 22]. Such parameters include the radii and positions of the first two rows of the annuli, the width of the waveguide, and so on. To achieve a flat band, we altered the outer radius R₁ in the second row of annuli adjacent to the line defect based on the W1.1 APCW (i.e. ), while the inner radii were maintained at r = 0.21a. Figure 3(a) shows the calculated dispersion for various R₁ values. When R₁ increased from 0.32a to 0.39a, the dispersion curves moved toward higher frequencies, and we realized that when R₁ was near 0.37a or beyond, the flat-band region emerged in the dispersion diagram. Figure 3(b) shows the curve of the group index corresponding to the waveguide modes in Fig. 3(a). Bands with constant group indices appeared in three curves for R₁ = 0.37a, 0.38a, and 0.39a. Inspecting Fig. 3(a) again, we found that with increasing R₁ the dispersion curve had a faster blue shift in the region around k = 0.4 (2π/a) than the other region of the same dispersion curve. Therefore, the regions with constant group indices in Fig. 3(b) slowly changed from chairlike to U-shaped, which resulted in a smaller group index and a narrower bandwidth for slow light.

Figure 3. (a) Dispersion diagram and (b) group-index curve for the alteration of the outer radius (R₁) of the second row of the annular air holes adjacent to the line defect.

The normalized delay-bandwidth product (NDBP), a comprehensive index for evaluating the slow-light performance of an APCW, can be calculated by [23, 24]

where and Δω/ω₀ are the average group index and the normalized bandwidth in the slow-light region respectively. The criterion for a worthy region of n_g can be explained as

Usually the region of dispersion in relation to group index values within [5, 25] can be defined as an acceptable region for slow light applications.

Applying the above criterion, the group indices corresponding to R₁ = 0.37a, 0.38a, and 0.39a were 36, 33.5, and 27.3, and the NDBP values were 0.387, 0.307, and 0.221 respectively. These decreasing NDBP values imply that it was not a wise choice to improve the slow-light performance by increasing R₁ based on 0.39a.

Next, to improve the dispersion, optofluid infiltrations with different refractive indices n_f were applied to the second row of annuli adjacent to the line defect, for R₁ = 0.37a and 0.38a. The dispersion curves and group-index curves for two cases are shown in Figs. 4 and 5 respectively. The corresponding NDBP values are listed in Table 1.

As can be seen in Fig. 4(a), for R₁ = 0.37a, by increasing n_f, all of the APCW modes shifted to lower frequencies. However, since the rear portion of each curve approaching the edge of Brillouin zone was affected to a greater degree than the front portion, the bandwidth of the flat bands gradually widened, while the group indices gradually diminished. Fortunately, the areas with constant group indices in Fig. 4(b) maintained chairlike shapes for different n_f values. Therefore, the NDBP values stayed larger than 0.387 when n_f increased from 1 to 2.1, and the best NDBP value of 0.446 was achieved by infiltrating an optofluid with n_f = 1.65.

Then we turned to the case of R₁ = 0.38a. As shown in Fig. 5(a), the downward behavior of the curves is almost the same as that shown in Fig. 4(a). Hence, larger bandwidths and smaller group indexes occurred for bigger n_f values. However, in contrast to Fig. 4(b), the areas with constant group indices in Fig. 5(b) slowly changed from being U-shaped to chairlike with increasing n_f, and the inflection point occured at n_f = 1.95. Because the chairlike type was more advantageous for obtaining a broad bandwidth, the NDBP values for n_f = 1.95 and n_f = 2.1, seen in Table 1, increased significantly compared to the cases with smaller n_f values, and the highest NDBP value of 0.502 was obtained at n_f = 1.95. Although the areas with constant group indices in Fig. 4(b) maintained a chairlike shape, thanks to the wider bandwidth, the best NDBP value for R₁ = 0.38a was larger than the best value for R₁ = 0.37a.

Figure 4. (a) Dispersion diagram and (b) group-index curve for the different optofluid infiltrations into the second annular air hole row adjacent to the line defect with R₁ = 0.37a.

Figure 5. (a) Dispersion diagram and (b) group-index curve for the different optofluid infiltrations into the second annular air hole row adjacent to the line defect with R₁ = 0.38a.

TABLE 1.. Details of the group indices and the corresponding GVD, flat band’s center wavelength λ_c, bandwidth Δλ, and NDBP for infiltrating different optofluids into the second row of annular air holes adjacent to the line defect.

To illustrate the advantages of APCs over normal PCs in slow-light dispersion engineering, we compared the results of our current work to the previously reported results proposed by line-defect PCWs hosted in 2D air-hole or rod PCs. Our best NDBP value of 0.502 was higher than the results reported in Ref. [26] (NDBP = 0.48), Ref. [7] (NDBP = 0.465), and Ref. [12] (NDBP = 0.469).

As shown in Fig. 5(a), it can be found that in the same dispersion curve there are two regions where the slopes of the curves differ greatly. One region has a larger slope, because the corresponding group index is small, resulting in a large group velocity, called a fast-light mode. The other region with a smaller slope corresponds to a larger group index, and the group velocity is smaller; this region is called a slow-light mode. For further investigation, we selected two frequencies (ω₁ = 0.261 (2πc/a) and ω₂ = 0.253 (2πc/a)) in two regions in the dispersion curve of the APCW with R₁ = 0.38a and n_f = 1.95 for modal-field-distribution simulation, using the finite-different time-domain (FDTD) method with a boundary treatment of perfectly matched layers. The fast-light and slow-light modes were excited by narrow-pulse sources (Δω/ω₀ = 1/200) centered at 0.261 (2πc/a) and 0.253 (2πc/a) respectively. Figures 6(a) and 6(b) illustrate the screenshots of the stable field distribution with only two periods. For the longitudinal modal field distribution, the fast-light mode was more concentrated in the center of the waveguide, while the slow-light mode was extended more on the upper and lower sides. For the lateral field distribution, the slow-light mode showed more obvious mode compression and local field enhancement than the fast-light mode. It should be noted that the high-energy distribution of the oscillation period in the slow-light mode was much larger than in the fast-light mode, and the energy of the center was much larger than the center of the oscillation period of the fast-light mode.

Figure 6. Modal field distribution of the APCW with R₁ = 0.38a and n_f = 1.95 for (a) fast-light mode and (b) slow-light mode.

Another important issue for slow-light devices is group-velocity dispersion (GVD), which causes pulse broadening and signal distortion [24]. The GVD, calculated by evaluating ∂²k/∂ω², is plotted in Fig. 7 for R₁ = 0.38a and n_f = 1.95. To show the low-GVD advantage, the GVD was on the order of 10⁵ (ps²/km) in the slow-light region seen in Fig. 5(b), which was extremely small compared to those in Refs. [1], [4], and [15].

Figure 7. Group-velocity dispersion (GVD) relationship calculated for an APCW with R₁ = 0.38a and n_f = 1.95.

To verify the PWE results, numerical simulations were performed using the two-dimensional FDTD method with a boundary treatment of perfectly matched layers. In the simulation, we studied the transmission characteristics of an optical pulse in the infiltrated waveguide of Fig. 5 with R₁ = 0.38a and n_f = 1.95. The transmission length was 100a. The Gaussian-pulse source was centered at 0.253 (2πc/a) with a full width at half maximum (FWHM) of Δω/ω₀ = 1/30. The average group index was calculated by the formula , where Δt and L are the delay time between input and output pulses and the propagation length respectively. We considered the case for . Figure 8 shows the normalized intensity in two monitors that were placed at the input and output ports of the waveguide. As illustrated, the peaks of the normalized field amplitude were observed at the input and output monitors at 1.3 ps and 3.01 ps respectively. Therefore, the time delay between the input pulse and the output pulse was equal to Δt = 2.71 ps. Consequently, the average group index could be calculated using the formula , which agreed well with the PWE simulation. Furthermore, the FWHM of the injected pulse was distorted from 0.567 ps to 0.68 ps over the 100a propagation length. In other words, the relative pulse distortion was obtained as 0.6% per micrometer, which confirmed the low dispersion of the proposed waveguide.

Figure 8. Optical-pulse profile in the time domain, when it was propagating through the APCW with .

In summary, we investigated the slow light in a line-defect APCW. By altering the outer radius of the second row of the annular air holes adjacent to the center of the waveguide, and selectively injecting optoliquid as a post-processing method, high-performance slow light with a wide bandwidth of 58.4 nm, large NDBP value of 0.502, and low GVD of 10⁵ (ps²/km) could be obtained in the optical-communication window for a silicon-on-insulator structure. Our numerical results demonstrate that this approach allows researchers to exploit the proposed structure for ultracompact low-dispersion devices in the slow-light regime. Admittedly, the current work does not provide an exhaustive parameter search. Other APCW geometries might provide better performance.