1. Introduction

Chalcogenide glasses (ChGs) are infrared transmitting materials containing chalcogenide elements S, Se or Te, compounded with elements as As, Sb and Ge [1–3]. They are transparent at infrared wavelengths (0.8~20 μm), exhibiting high optical Kerr nonlinearity (the nonlinear index n₂ = 2~20 × 10⁻¹⁸ m²/W, 100~1000 times larger than that of silica), subpicosecond response time (t_r < 200 fs) as well as negligible two-photon absorption [4–6]. These characteristics make ChGs a promising platform for integrated all-optical nonlinear devices at infrared wavelengths.

Photonic crystal slab waveguides (PCSWs), which utilize the band gap of a periodic structure to confine light in the horizontal direction while guiding light vertically by total internal reflection, can enhance nonlinear effects by exploiting its remarkable slow light (large group index) effect. As the practical utilization of ultra-slow light in PCSWs is limited due to an inherent small bandwidth, the balance of group index and bandwidth should be considered. It has been demonstrated that the slow-light properties of SOI-based PCWs at λ = 1.55 μm can be altered via a structural tuning of the waveguide geometry, typically by changing the waveguide width or by introducing bi-periodicity [7–12]. However, the slow-light properties had rarely been investigated in chalcogenide PCSWs, though the enhancement of self-phase modulation and third harmonic generation were demonstrated at λ = 1.55 μm [13–15].

In this paper, we theoretically investigate the slow-light properties of Ge₂₀Sb₁₅Se₆₅ chalcogenide PCSWs at λ = 3.0 μm, and both of symmetric and asymmetric structure are studied via a simple method of tuning the hole radius in the first two rows next to the waveguide. Our results show perfect slow-light properties in asymmetric PCSWs and can be used as affordable reference for further research.

2. Theoretical method

The proposed chalcogenide PCSW is shown in Fig. 1(a), which consists of a two-dimensional triangular lattice of air holes and a line defect in a Ge₂₀Sb₁₅Se₆₅ layer, with air as the top and bottom cladding layers. The parameters of the chalcogenide PCSW are set as below: a - the lattice period, r – the radius of air holes, r_u – radius of one row air holes adjacent to the line defect waveguide, r_d - radius of another row air holes adjacent to the line defect waveguide, h – the thickness of slab, n - refractive index of Ge₂₀Sb₁₅Se₆₅ slab, and λ - operating wavelength. This kind waveguide structures can avoid large errors caused by shifting holes compared to PCSWs based on the lateral or longitudinal shift of the first two rows of holes adjacent to the waveguides, in which the distance of holes shift is very small, only dozens of nanometer, and therefore errors can be easily introduced just with not serious roughness [9,12,16]. In addition, with only the radius of holes changed, this kind slow-light waveguide structure can be very simple and easily calculated by plane wave expansion method. Table 1 shows the refractive indices of Ge₂₀Sb₁₅Se₆₅ glass fabricated in our laboratory, and n = 2.612 at λ = 3.0 μm is selected in this work. The three-dimensional plane wave expansion (PWE) method is used to calculate the dispersion relations ω(k) of quasi-TE mode(even mode) in the chalcogenide PCSW. As shown in Fig. 1(b), even modes exhibit a slab band gap where there is a guided mode under the light cone. Numerical result shows that PCSW with r = 0.3a and h = 0.5a can support a slab band gap large enough to contain guided modes.

 

Fig. 1 (a) Schematic diagram of the proposed photonic crystal slab waveguide with index of 2.612 fully embedded in air claddings and (b) calculated dispersion curve of the PCSW with r = 0.3a,h = 0.5a, r_u = 0.4a and r_d = 0.2a.

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Table 1. Refractive indices of Ge₂₀Sb₁₅Se₆₅ glass at different wavelength

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Slow light in the periodic waveguide is explained as an interaction between the forward and backward propagating modes [17]. The strongest interaction is at the edge of the Brillouin zone where the group velocity vanishes. When the guided mode is moved away from the zone boundary, the interaction becomes weaker, due to phase mismatch between the forward and backward propagating modes. The group velocity v_g of propagation light with frequency ω in an optical waveguide can easily be obtained from the dispersion relation:

3. Numerical simulation and discussion

The band diagram for even modes in PCSW with r = 0.3a and h = 0.5a is shown in Fig. 2(a). There are three guided modes in the slab band gap under the light cone, named band 1, band 2, and band 3 respectively. Clearly, the waveguide remains single mode because these guided modes do not overlap in the normalized frequency. From the figure, band 1 tends to leak into slab modes and enlarges the transmission loss eventually as it is closed to slab band gap edge. Figure 2(b) shows the corresponding group indices n_g calculated by relation (1) of these guided modes. Compared to band 1 and band 3, band 2 has a much larger group index with the same propagation constant, though its bandwidth is slightly narrower.

 

Fig. 2 (a) Dispersion relation and (b) group index of different modes in PCSW with r = 0.3a and h = 0.5a.

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It is known through calculation that by tuning the size of the air holes adjacent to the waveguide core, it is difficult to increase the group indices of band 1 or band 3(Our calculation shows that their n_g cannot be increased above ~12 within 20% with a certain bandwidth), whereas it is easy to change the bandwidth of band 2 to obtain relative large value both in bandwidth and group index. In order to obtain relative high value in both of bandwidth and group index, only the band 2 will be concerned in our following study. Next, desired PCSWs with large bandwidth and high group index will be demonstrated, which can be achieved by perturbing the holes adjacent to the waveguide core. To simplify matters, the work can be processed in two steps: first, investigating slow light properties of symmetric PCSWs by only changing r_s from 0 to 0.5a, where r_s = r_u = r_d; second, changing r_u and r_d respectively to study slow light properties of asymmetric PCSWs.

The dispersion relations of the guided modes in symmetric PCSWs with r_s ranging from 0 to 0.5a can be seen from Fig. 3(a). In the figure, the guided modes with different r_s represented by the colorized curves are almost in the slab band gap except the mode with r_s = 0.5a, which demonstrates that the chosen value of r_s is acceptable to achieve a large bandwidth and a high group index for symmetric PCSWs, with r = 0.3a and h = 0.5a. Obviously, the frequency of the guided mode decreases when r_s decreases, as the effective index increase.

 

Fig. 3 (a) Curves of the guided modes for different r_s and (b) curves of the group indices for different r_s

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As can be seen from Fig. 3(b), PCSWs have large bandwidth Δω/ω₀ with small r_s (0.1a ~0.25a) and high refractive index with large r_s (0.3a ~0.4a). As a result, we are only concerned about modes with 0.2a ≤ r_s ≤ 0.4a because they have a relative large group index above 15, and some of them have a wide band. There is a phenomenon in Fig. 3(b) that guided mode will has a larger bandwidth when r_s is away from the radius of photonic crystal slab air holes(0.3a), as the disorder degree increase. When r_s < 0.3a, the guided mode will have a relatively smaller group index, and vice versa. From the analysis above, we can tune dispersion properties properly to achieve the desired group velocity with a relative large bandwidth, by increasing r_u and decreasing r_d simultaneously, which brings out an asymmetric structure.

The results of asymmetric PCSWs are depicted in Fig. 4, which shows the guided mode group index as a function of the normalized frequency ω with r_u and r_d varied from 0.2a to 0.4a. For comparison, the group indices of symmetric PCSWs with r_u = r_d = 0.2a, 0.3a and 0.4a are also included in this figure.

 

Fig. 4 the curves of the group index for different r_u and r_d.

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As can be seen in Fig. 4, most curves have inflection points where a relative large bandwidth will be achieved, and thus the perturbed PCSW can support the desired group index with a certain bandwidth. It is very clear in Fig. 4 that the group index and bandwidth can hardly be increased simultaneously, for increasing group index will result in the decrease of bandwidth, and vice versa. Keeping r_u = 0.2a and increasing r_d from 0.2a to 0.4a, the group index at inflection point is increased from ~15 to 35 within 20% over bandwidth of ~30 nm to 10 nm at λ = 3.0 μm, while keeping r_u = 0.4a and decreasing r_d from 0.4a to 0.2a, the group index at inflection point is increased rapidly from ~40 to 120 within 20% with a relatively narrow bandwidth. The detailed data are shown in Table 2. Comparing to symmetric ones, the asymmetric PCSWs can increase guided mode group index in a larger range, especially for the structure with r_u = 0.4a. We also notice that the NDBPs are not changed remarkably by breaking the symmetry of PCSWs in this situation, and so we can selectively tailor the slow-light properties of a PCSW with specific NDBP ranging from 0.1 to 0.2.

Table 2. Slow light properties of symmetric and asymmetric PCSWs at λ = 3 μm

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In the structure designation, disorders of period (a = λ*ω) and radius (r_u and r_s) are very critical and have great impacts on middle normalized frequency or the position of guided mode, resulting in the change of the bandwidth and group index of slow light, according to Fig. 4. With laser of a certain linewidth (Δλ), the tolerance to disorder of the period (Δa) can be simply achieved by:

In practical applications, the slow light loss should be considered. Compared to the common 1550nm band, mid-infrared wavelengths (e.g. λ = 3.0 μm) can generally reduce slow light loss according to equations below [19, 20]:

On the other hand, for given operating wavelength, slow-light loss is proportional to n_g when n_g is small, and is proportional to n_g² when n_g is large. This can guide us to select suitable geometry parameters of PCSWs in practical applications. If relative small group index can introduce nonlinearity large enough, then it is not necessary to select lager group index, for slow-light loss would increases rapidly when group index become larger. In this situation, symmetric PCSW with r_u = r_d = 0.2a or asymmetric PCSW with r_u = 0.2a and r_d = 0.25a ~0.30a can be suitable choices. If we want to achieve super large nonlinearity enhanced by slow-light, then asymmetric PCSW with r_u = 0.4a and r_d = 0.2a ~0.3a would be a good choice.

4. Conclusion

In this work, we present a photonic crystal slab slow-light waveguide based on Ge₂₀Sb₁₅Se₆₅ chalcogenide glass at a mid-infrared wavelength of λ = 3.0 μm. By changing the size of the first two rows of air holes adjacent to the line defect waveguide, two kinds of structures, symmetric and asymmetric waveguide, can be achieved. These waveguides have very similar NDBP between 0.1 and 0.2. The symmetric waveguides has a range of group indices, typically between 16 and 43 within 20% over bandwidth of 30 nm to 10 nm, and the group index can be increased up to 130 in asymmetric waveguides, which is more attractive in structure design. Our work shows the possibilities of using chalcogenide PCSWs in the slow-light regime for practical applications, especially in the enhancement of nonlinear effects.