1. Introduction

Phenomenon of slow light has long been a focus due to its many potential applications, such as ultrafast all-optical signal processing [1, 2], quantum computing [3, 4], and enhancement of light-matter interactions [5, 6]. Slow light propagation had been achieved by a variety of method, including electromagnetically induced transparency [2,7], coherent population oscillation [8], stimulated Brillouin scattering [9] and photonic crystals (PCs) [10]. As one kind of method to slow down the light speed, PC waveguides have attracted much attention recently because they have many merits, such as ability of being integrated on optical chip, operating at room temperature. Moreover, bandwidth and dispersion of slow light are crucial in telecommunications using short pulses and high-speed modulated slow-light signals, PC waveguides provide the ability of potential wide-bandwidth and dispersion-free propagation.

The extensive research about slowing down light in 2D triangular lattice PC slab line-defect waveguides have been done both in theory and experiment [10–19,21,22]. In the structure, there is an intrinsic interaction of even gap guided and index guided modes [11]. The interaction leads to an anticrossing between these two types of modes. The effect of such an interaction is similar to dispersion compensation [12]. It can be used to achieve slow light and control dispersion by designing the structure of waveguides. In previous literature, researchers have studied many waveguide structures. These researches show that some parameters can sensitively influence the dispersion to attain wide flat band and high group index of slow light. Such parameters include the waveguide width [13], the shape of holes in PC lattice [14], the hole size or period length of the first two rows of the W1 PC waveguides [15,16]and the positions of the first two rows of the W1 PC waveguides [17]. In addition, chirped PC waveguides structure can also be used to slow down light [18, 19]. However, to the best of our knowledge, the influence on dispersion by the angle between elementary lattice vectors has not been reported.

In this paper, we employ an oblique lattice structure to study slow light and its dispersion. Numerical results show that the available bandwidth of flat band slow light can be effectively increased as compared with a triangular lattice structure, when the angle between elementary lattice vectors of oblique lattice is designed suitably. The increasing of bandwidth is closely related to the shift of anticrossing point.

2. Design and numerical simulations

The studied structure is shown in Fig. 1 . There is a line defect in a Si air-bridge slab PC consisting of the oblique lattice. The lengths of two elementary lattice vectors of the oblique lattice are a1= a2= a, where a is the lattice constant. The angle θ between two elementary lattice vectors (ABELV) is 66°. The radius of air-hole is r = 0.32a. The thickness of the slab is h = 0.5328a, and n = 3.5 is the refractive index of Si. The entire structure is symmetrical along X and Z directions.

 

Fig. 1 Schematic diagram of the PC waveguide. The basic structure consists of a W0.85 waveguide in an oblique lattice. The waveguide width is $W0.85=1.7a\cdot \cos \left(\theta /2\right)$. Based on the basic structure, the first two rows of air-hole are shifted horizontally with distances of d1 and d2 respectively; and we define d1 and d2 to be positive if the air-holes are shifted along the directions denoted by the blue arrows.

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Due to the fact that flat band can be systematically designed by changing the positions of the first two rows of air-hole of PC waveguides [17], we modify the positions of the first two rows of air-hole adjacent to the line-defect based on the W0.85 waveguide whose width is $W0.85=1.7a\cdot \cos \left(\theta /2\right)$. d1 and d2 denote the shift distances of the first two rows of air-hole from their original positions of an unmodified structure, as shown in Fig. 1.

3D calculations with the plane wave expansion method are used here [20]. The group velocity v_g can be obtained by using the built-in function of the MIT Photonic Bands package. The group index is defined as n_g = c/v_g. Figure 2 shows the calculated dispersion relation and the group index of slow mode. Following previous literatures, the group index n_g is considered as constant within a ±10% range [13,15,17]. The simulation results show that the flat band have relatively large bandwidths in a wide range where the group index of slow mode changes from 30 to 150. For constant group indices of 30, 48.5, 80, 130, and 150, the bandwidths are 20nm, 11.8nm, 7.3nm, 3.8nm, and 2.95 nm, respectively. Also, we can see that the group index – bandwidth product n_g(Δω/ω) keeps almost constant around 0.37 for a range of n_g from 30 to 80, which is higher than previously reported values. Since the group index – bandwidth product n_g(Δω/ω) is proportional to the delay-bandwidth product per unit length [17], its higher value implies that the waveguides have better buffering capacity.

 

Fig. 2 (a) Dispersion curves with different parameter of d1 and d2. Λ= 2asin(θ/2), is the period length along the waveguide. (b) Group index correspond to curves in map (a). Δω present the bandwidth of flat band.

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When optical pulses pass through the PC waveguide, they will be broadened and distorted as the result of nonzero dispersion [21,22]. Therefore, for the applications of small group velocities, the group velocity dispersion of the PC waveguides should be as small as possible. So we consider a stricter limit for group index that the fluctuation range of the group index is only ±1% in later discussions.

Figure 3 shows three sets group index curves. The left part and the right part correspond to θ = 66° and θ = 60°, respectively. There exists a constant group index for each curve within certain frequency range. And the constant group index is almost equal for the left part and the right part of each set curve. It is obvious that the bandwidth of flat band is increased for different group indices when θ varies from 60° to 66°. Δω₁ and Δω₂ are the bandwidths of flat band for θ = 60° and θ = 66°, respectively. For n_g are about 32, 70, and 204, the ratios of Δω₂/Δω₁ are 1.37, 1.2, and 1.68, respectively.

 

Fig. 3 Group index for θ = 66° and θ = 60°. The curves on the left of figure correspond to θ = 66°, while the curves on the right correspond to θ = 60°.

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3. Analysis

The reason of bandwidth broadening can be illustrated as following. The flat band slow light is related to the anticrossing effect [11,12], as result of this effect, the group velocity rapidly changes near the anticrossing point. As shown in Fig. 4(a) , the crossing red dash lines denote the wave number of anticrossing point A. The group velocity is very small in the range from the anticrossing point to the edge of the first Brillouin zone. In the following, we refer to this range as slow-light-zone. If the structure parameters are properly chosen, a segment of the dispersion curve will became flat within the slow-light-zone. When the slow-light-zone extends, the flat band segment do a similar extension accordingly.

 

Fig. 4 (a) The dispersion curves of PC waveguides with different ABELVs. For θ = 56°, The slow-light-zone is marked by the blue line and arrows. The crossing red dash lines denote the wave number of anticrossing point A. For all calculated θ, d1 = 0, d2 = 0. At the first Brillouin zone edge of each dispersion curve, the right endpoints of dispersion curves with different θ are denoted by dots with different color respectively. (b) The group velocity curves corresponding to (a). In Fig. 4, other parameters used are as follows: r = 0.32a, h = 0.5328a, n = 3.5, effective refractive indices are 2.7, 2.746, 2.775 and 2.8 for θ = 70°, 66°, 60°, 56°, respectively.

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Figure 4 show the dispersion and the group velocity curves calculated by using the two-dimensional model with the effective index approximation. It should be noted that the width of the first Brillouin zone along the waveguide direction will become smaller with increasing θ, as shown in Fig. 4(a). In order to compare slow-light-zone of waveguides with different θ, we have translated the group velocity curves along the horizontal axis to make the right endpoints of them coinciding at wavevector $k=0.5\cdot 2\pi /a$ in Fig. 4(b). In Fig. 4(b), the group velocity curves show that the anticrossing point shifts to smaller wave numbers with increasing θ, which result in an extended slow-light-zone. In this paper, we do not consider larger θ, because flat band likely come into the region where energies in waveguide are lost by coupling with air, with the anticrossing point shifts to smaller wave numbers. In fact, we can get wider flat band from our result calculated for θ = 70°.

To show the influence of slow-light-zone on bandwidth of flat band, group velocity curves of two waveguide structures with θ = 60° and θ = 66° are shown in Fig. 5 . During the flat band range, they have the same group velocity (v_g = c/27). It can be seen that the extension of slow-light-zone makes the flat band become wider.

 

Fig. 5 The group velocity curves of two waveguides with different θ. For θ = 60°, d1 = 0.152, d2 = 0.06; while for θ = 66°, d1 = 0.128a, d2 = −0.002a. Other parameters used are as follows: r = 0.32a, h = 0.5328a, n = 3.5, effective refractive indices are 2.746, 2.775 for θ = 66° and 60° respectively.

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4. Conclusion

In conclusion, we have demonstrated that near constant group index bandwidth can be increased by adopting oblique lattice structure with a proper ABELV for a PC waveguide. For n_g = 30, n_g = 48.5 and n_g = 80, the bandwidth is 20 nm, 11.8 nm and 7.3 nm, respectively. When the fluctuation of the group index within flat band is in a ± 1% range, the calculated results suggest that the ratio of available bandwidth of flat band can reach 1.68 between an oblique lattice structure (θ = 66°) and a triangular one (θ = 60°). The method of adjusting the ABELV can be combined with other design approaches to tailor group velocity and dispersion, which enhance the possibilities of practical applications for PC waveguide.