1. Introduction

Slow light technology shows an amazing future for optical buffering, optical logical gates and all-optical signal processing in the next-generation photonic networks and optical integrated circuits [1–5]. Up to now, there are mainly three methods to exploit slow light effects: electromagnetically induced transparency [6], coherent population oscillation [7], and optical coupled resonators or photonic crystal waveguides (PCW) [4,5]. Slow light in PCW has received more attention because it works at room temperature, provides larger bandwidths if compared with the other two solutions, and can be fabricated using standard semiconductor planar technologies.

Among all PCW devices, W1 waveguides formed by removing the central row of holes/rods in a perfect two-dimensional planar photonic crystal (PC) have been studied thoroughly [8,9]. These waveguides sustain guided mode inside photonic band gap and produce slow light at the band edge where the dispersion relation has a parabolic form. However, large group index in W1 waveguides is only obtained in extremely narrow bandwidth and large group velocity dispersion (GVD) effects accompany the slow light regime, with as a result causes large signal distortion during pulse propagation [10].

So far, two main solutions in PCW have been proposed to eliminate the drawbacks due to limited bandwidth and obtain dispersion free slow light. The first one is based on nearly-zero dispersion by adjusting the PC geometry, including changing the hole size [11] or hole position [12] of the first two rows, and introducing annular holes for the whole lattices [13] or only the first row [14,15]. The second one is based on dispersion compensation, achieved by adding two opposite dispersion regions in a chirped waveguide [16,17].

We explore in this paper a much simpler solution to provide slow waves in the largest possible bandwidth. The proposed solution is based on a single parameter change with respect to W1 waveguide but brings advantages to achievable bandwidth for large group indices and low GVD values.

This paper is organized as follows. Frequency band engineering of the proposed structure is studied firstly. The physical explanations of the related phenomena are then given by comparing the electric-field distributions of several modified PC waveguides. The ultra-low dispersion properties of the proposed structure are studied, showing that it is well suited to the versatile control of slow waves with chosen group index and bandwidth.

The new waveguides could be useful for future integrated photonic applications such as optical delay lines and dispersion compensating devices.

2. The new waveguide geometry

The proposed structure starts from a W1 waveguide obtained by removing one row of holes in a triangular PC lattice. Then, the two rows of air holes that border the waveguide axis are deliberately shifted by a δx distance. Standard silicon on insulator (SOI) is considered as the typical reference technology. The considered indices for air (top layer), silicon layer (middle layer), and silica (bottom layer) are 1.0, 3.45 and 1.45, respectively. The silicon layer of SOI wafer has a thickness of 283 nm, and the radius of the air hole is 0.286a, with a the PC lattice constant. The whole structure is depicted in Fig. 1 (a).

 

Fig. 1 a) Schematic picture of the proposed PC waveguide geometry, b) An example of “U” type group index- normalized frequency curve obtained for δx = 0.27a

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Obviously, PCW geometry can be treated as a one-dimensional waveguide. Yet, shifting the two bordering of holes introduces another one-dimensional corrugation, which drastically changes the slow light nature. We found the group index – normalized frequency curves turn to the form of letter “U” when δx is increased to appropriate values. Figure 1(b) shows the situation for δx = 0.27a as a typical example. In this very first situation, it is observed that a nearly flat group index with values above 100 is obtained in a fairly large bandwidth.

3. Physical analysis

To understand this surprising phenomenon, we studied in detail the influence of the δx shift quantity on the waveguide dispersion curves using the plane wave expansion method (PWE). The investigated guide modes are supposed to transverse-electric (TE)-like polarization, i.e. with magnetic field parallel to the vertical direction. Due to the computational cost of three-dimensional simulations, two-dimensional simulations with a slab effective index 2.98 were considered here.

Figure 2(a) shows the calculated dispersion diagrams of the proposed structure when δx is gradually varied from 0 to 0.5a. Results for δx in the range between 0.5a and a would be the same as those obtained for δx between 0 and 0.5a due to the waveguide symmetry in x direction, and are thus not depicted. When there is no shift (δx = 0, regular PCW) the waveguide modes show two distinct regions below the light line. These are the fast light region (index-guiding regime) and slow light region (photonic gap guiding regime) [8]. The two regions have a turning point, i.e. the slope of the curve changes from linear one to a quadratic form. When δx starts to increase, the waveguide mode shifts to upper frequencies. The band nature then changes drastically due to the different variations at the two areas around k = 0.36(2π/a) and k = 0.50(2π/a). For the area near the first Brillouin zone (BZ) edge (k = (0.5)2π/a), band moves upwards particularly fast, while for the area near the light line, the band moves less pronounced. As a result, the tailored band curve becomes obviously curl up. When δx is increased from 0a to 0.2a, bands shift little to the higher frequencies and became more and more flat, which means the group index ($n_g$) gradually increases. When δx is about 0.25a, the flattest band curve is obtained, corresponding to the largest$n_g$. After 0.25a, with continuous increase of δx, band curves are no longer flat. However, with continuous increase of δx, quasi-linear evolution curves are observed between the high moving speed part and the low moving speed part. This predicts a constant $n_g$over a large frequency range [11,12,14,15].

 

Fig. 2 a) Band diagram along the waveguide propagation direction for various values of the δx tuning geometrical parameter, and Ey field distributions for different δx values at k-points: b) K = 0.36881(2π/a), c) K = 0.44059(2π/a), d) K = 0.495(2π/a).

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The evolution of the PCW dispersion diagram with δx can be interpreted from the analysis of field profiles within the structure for different wave vectors and different values of δx. Figsures 2(b) (c) (d) give the electric field distributions for seven δx values at three specific wave numbers: K = 0.36881(2π/a), K = 0.44059(2π/a), K = 0.495(2π/a), respectively.

As shown in Fig. 2(b), when the wave vector is near the light line, field leak through sideways is limited. This is due to the fact that the confinement of E-field is then dominated by the index-like effect [8]. However, when the wave vector is close to BZ edge as in Fig. 2(d), field is less confined within the central waveguide region because confinement of waves turns to be dominated by band gap effect [8]. As a result, the waveguide mode at the BZ edge is more sensitive to the shift of the air holes if compared with those near the light line. This explains why the dispersion diagram evolution when δx varies is small around the light line near K = 0.36881(2π/a). Moreover, the degree of the localization of the field within the air holes increases with increasing δx values. As waves at the BZ edge are more and more strongly confined inside air holes, the effective index seen by the field decreases, which means that bands are pushed towards higher values.

To better illustrate the waveguide properties, band diagram and group index curves are simultaneously plotted in Fig. 3 for three values of δx. The black line in Fig. 3 represents un-shifted structure (a normal PCW), the brown circled line and the blue triangle line represent the shifted structure with δx = 0.28a and 0.5a, respectively. Two inflection points “a” and “c” should be noticed because their group velocities both go down to zero but have different origins. Point “c” is the well-known slow light band edge formed by the interference of forward and backward waves [5]. However, point “a” is believed to be formed by shifting the boundary of the waveguide in our proposed structure. It is clearly shown in Fig. 3(b) that inflection point “a” corresponds to a new sharp peak in the middle of K-path (in the region K = 0.36(2π/a)~0.38(2π/a)). This means we now have a new slow light center in the middle of the K-path. Regarding to the slow light at the band edge, we have two slow light peaks in the K-path, which form this U type.

 

Fig. 3 Details about the δx = 0.5a waveguide: a) Dispersion curve, b) Group index variation.

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Somehow, the cost to pay to obtain these original dispersion curves lies in the fact that the new PhC waveguides are not strictly single-mode for all δx values. It is clear from Fig. 3 that two modes with the same even symmetry with respect to the waveguide axis are present at the same normalized wavelength: a/λ = 0.255 and δx = 0.5a can be typically considered to see this. It is yet emphasized here that one of these two modes is situated above the light line, making it intrinsically leaky. As shown in [18], such modes strongly decay when light propagation lengths are more than 100µm.

All PhC waveguide structures proposed in this work operate in single mode situation below the light line. This makes them suitable for the practical realization of slow wave devices.

Moreover, due to the same geometry as W1 waveguide, light coupling into the proposed slow light waveguides could be optimized using previous coupling approaches to W1 waveguides, e.g.: gradual lattice tapers [19,20] or efficient injectors smartly exploiting evanescent modes [21]. As one step in that direction, we will not focus hereafter on the issue of light coupling into the new proposed waveguides, but rather analyse their slow light performances in terms of group index, bandwidth, and GVD.

4. Slow light performance

4.1 Group index characteristics

The most important issue for slow light device is the group velocity $v_g$ = ∂ω/∂k or group index $n_g$ = c/$v_g$ that quantitatively describe how slow the light is [4,5].

The related $n_g$ curves with normalized frequency under different values of δx are plotted in Fig. 4 . To clearly illustrate all situations, larger $n_g$ curves are plotted in Fig. 4(a) and smaller $n_g$ curves are plotted in Fig. 4(b), respectively.

 

Fig. 4 Group index variation as a function of normalized frequency for various values of the δx tuning geometrical parameter.

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As shown in Fig. 4, all calculated $n_g$ curves turn to an unusual form of “$n_g$decrease—constant $n_g$— $n_g$ increase”, hereafter called the “U-type” $n_g$ curves. As a consequence, nearly constant $n_g$ over a wide bandwidth have been achieved due to the flatness of the U-type curve center.

Furthermore, a general trend can be drawn here: when δx increases, the average group index (${\tilde{n}}_g$) value in the constant region decreases, at the same time the corresponding bandwidth increases. This trend predicts there exists a trade-off between $n_g$ values and bandwidth values in our device, like in all other slow light devices.

4.2 Delay-bandwidth product

The concept of group index describes how slow the light is. However, for slow light device, only talking about $n_g$ is meaningless [4], because we must care about the bandwidth at the same time. The delay-bandwidth product (DBP) is a good indication of the highest slow light capacity that the device potentially provides [4]. We will focus on the normalized delay-bandwidth product (NDBP) value in this paper, because it is universal for comparison between devices having different lengths and different operating frequencies. The NDBP is defined by:

Up to now, there are many papers studying about flat band slow light [11–17,22–25]. However, it is difficult to make comparison between those papers, because different papers have given NDBP according to different flat judgments. For example, ref [12] has pointed out NDBP value of 0.24 within 10% $n_g$ variation with respect to their mean ${\tilde{n}}_g$ value, and ref [15] has pointed out value of 0.6 within $n_g$ variation much larger than 100% with respect to their mean ${\tilde{n}}_g$ value. It is not correct to say ref [15] has better NDBP performance than ref [12].

Before we start to investigate the NDBP, the concept of “flat ratio” is introduced for the first time to clearly define the flatness within the related bandwidth.

The flat ratio (FR) is defined as:

Table 1 summaries the values of average group index ${\tilde{n}}_g$ and NDBP values. In accordance with refs [11,12,23], the flat ratio considered here is 0.2 corresponding to a maximum of 10% $n_g$ variation with respect to the mean ${\tilde{n}}_g$ value.

Table 1. Overall of average group indices and NDBP values of our proposed structure obtained by tuning various the δx parameter under a 0.2 flat ratio.

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Results gathered in Table 1 show that NDBP values increase when constant ${\tilde{n}}_g$ decreases. This tendency is due to the fact that the corresponding bandwidth values increase more rapidly. In comparison with the previous works, it is observed that comparable NDPB values are obtained with a much simpler PC tuning geometry [11–14].

To the best of our knowledge, the largest constant $n_g$ with a bandwidth reported in PCs is 105 with a bandwidth 2.3 nm [23], but it requires a complicated structure. Our results provide even larger constant $n_g$with a similar delay-bandwidth product performance, for example: $n_g$ = 211 with ∆ω/ω = 0.000645 at δx = 0.25a, $n_g$ = 140 with ∆ω/ω = 0.001 at δx = 0.26a. It is underlined here that these results reveal the possibility to a versatile control of $n_g$ and bandwidth with only one simple structural parameter.

5. Dispersion performance

Another important issue for slow light devices is group velocity dispersion (GVD) effects [10,12,26] which cause drawbacks due to pulse broadening and signal distortion [12,26].

The GVD parameter D_λ is calculated as:

Figure 6 shows the calculated values of D_λ of the proposed structure. It is clearly shown from Fig. 6 that there exists a bandwidth of “zero” dispersion in each situation, and the bandwidth of “zero” dispersion increases rapidly when δx increases. These “zero” dispersion areas correspond to the constant $n_g$ areas as shown in Fig. 4.

 

Fig. 6 GVD dispersion obtained for different values of the δx waveguide geometrical parameter.

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For low $n_g$ situation, the corresponding GVD values are also low, and flat bandwidth with “zero” dispersion is thus observed. For high $n_g$ situations (0.25a ≤ δx ≤ 0.27a), GVD values are separately shown in the inset picture of Fig. 6, for clarity.

Interestingly, each GVD curve also presents a negative GVD frequency range and a positive GVD frequency range. This reveals the possibility of dispersion compensating applications [22] using our proposed structure. The slow light bandwidth could be effectively enlarged by cascading two sections of waveguides with opposite GVD values for the same wavelength. These aspects will be explored in a future work.

To show the advantage of our structure in dispersion, the bandwidth corresponding to absolute GVD value limited within 2 ps.nm⁻¹.mm⁻¹ are checked. We should note that this value is extremely small if compared with other papers [13–15,25,26]. However, even in this strict consideration, our proposed structure shows very good low dispersion properties. In the situation 0.35a ≤ δx ≤ 0.5a, it can be seen that most of the GVD values are smaller than 0.5 ps.nm⁻¹.mm⁻¹. The obtained bandwidth for low GVD is much larger than previous study [11,14,15,25], e.g.: in the situation δx = 0.5a, our structure provides as large as more than 40 nm bandwidth (centered at 1550nm).

6. Conclusion

In conclusion, a new photonic crystal waveguide geometry is proposed to obtain slow waves in the largest possible bandwidth and lowest group velocity dispersion. The proposed geometry is based only on the shift of the waveguide bordering rows of holes. Thus, it is characterized by its simplicity when compared to previously proposed solutions [11,12,14–16,24]. As fast waves dominated by the index guiding mechanism are almost unaffected by the introduced shift while slow waves mainly dominated by gap guided mechanisms are, the structure produces unusual “U” type $n_g$-frequency curves. Additionally, control of normalized delay-bandwidth product has been demonstrated (0.1363 for $n_g$ = 210 < NDPB < 0.3141 for $n_g$ = 10), and the obtained values are favorably compared with previous works. The GVD properties of the proposed structure have been also analyzed. It turns out that nearly-zero dispersion can be achieved within large bandwidth. Additionally, the new waveguide allows a possible control of positive or negative GVD values, opening opportunities for dispersion compensation devices.