Chromatic dispersion is an essential property of nonlinear waveguides, which plays a critical role in many nonlinear processes, such as soliton formation,^[1–6] super-continuum generation,^[3,7–10] four-wave-mixing (FWM) based amplification,^[11] and wavelength conversion.^[12–16] Low and flat dispersion over a broad wavelength range is preferred for optimizing the phase matching condition.^[17] Recently, chalcogenide glass (ChG) waveguides have drawn much attention as promising candidates for integrated nonlinear photonic devices.^[18–29] Many ChG materials have high Kerr nonlinear indexes (n₂) and low two-photon absorption coefficients;^[30–32] hence, they have high figure of merit (FOM) of the third-order nonlinear optical materials.^[30,31] However, ChG materials usually have large negative dispersion at the near-infrared and mid-infrared bands, which requires a special waveguide design to compensate for the material dispersion by the waveguide dispersion.^{[17,22,23,33–37]}

Strip and rib waveguides are the most popular structures of ChG waveguides. The geometry parameters of these structures can be adjusted to tailor the waveguide dispersions. However, the bandwidth of low dispersion region is always limited. The introduction of a slot structure into the waveguide has been proven as an effective way to tailor the waveguide dispersion.^{[12,23,38–42]} Recently, a strip/slot hybrid As₂S₃ waveguide design has been proposed, which has a flat dispersion band of 249 nm and whose dispersion is limited in ±170 ps·nm⁻¹·km⁻¹.^[43] Although this design shows impressive dispersion characteristics as nonlinear waveguides, many difficulties can be expected in its fabrication. On the one hand, the commercial (NH₄OH)-based developer using in the UV-lithography or electron beam lithography (EBL) may lead to pinholes and film peeling in ChGs^[43–45] which impacts the quality of ChG waveguides. On the other hand, both the wet-etching and dry-etching processes on the ChG films are not easy due to the fragility of ChG materials.^[27,45–49] Recently, we proposed a reverse ridge/slot hybrid As₂S₃ waveguide structure with an ultra-flat dispersion profile.^[23] This structure can be fabricated by the micro-trench filling technique,^[50] which avoids the photo lithography process, wet or dry etching processes and lift-off processes on the As₂S₃ layer. However, to realize this waveguide structure, vertical slot structures of tens of nanometers should be fabricated by dry etching on the silica substrate, and the requirement of the accuracy of the structure parameters is very high to realize dispersion flattening, which is difficult in waveguide fabrication.

In this paper, we introduce a horizontal slot in a reversed-rib As₂S₃ waveguide. This waveguide can also be fabricated by the micro-trench filling technique; however, the slot in the horizon direction replaces the vertical slots in the previous work^[23] to tailor its dispersion property. Hence, the difficulties in the fabrication of the small slot structures in the silica substrate are avoided, leading to a simpler fabrication process. On the other hand, the dispersion of quasi-TM mode is adjusted by the horizontal slot in this structure, realizing an ultra-flat dispersion profile, which varies from −10.6 ps·nm⁻¹·km⁻¹ to +11.14 ps·nm⁻¹·km⁻¹ over a wavelength range of 1080 nm (from 1760 nm to 2840 nm).

Figure 1(a) shows the reversed-rib waveguide structure. Due to the low transformation temperature of As₂S₃,^[28,31] the fabrication process of this structure is relatively simple. Firstly, a groove was made on the silica substrate by wet etching or dry etching. Secondly, the As₂S₃ film was deposited by traditional vacuum deposition methods (thermal evaporation, chemical vapor deposition or sputtering). Then, the sample was annealed to melt the As₂S₃ material, which made the upper surface of the film flat to realize the reversed-rib structure. It can also be fabricated by the micro-trench filling method based on solution-processed ChG.^[27,29,51] Finally, a silica layer as the cladding of the waveguide could be fabricated by sputtering or the method based on the silicon based polymer. In this process, there is no photolithography or etch processes on the As₂S₃ film, which reduces the difficulty in the fabrication of ChG waveguides. It is worth noting that this structure can also be fabricated by the traditional top-down method, in which the ChG glass and silica materials are deposited layer by layer firstly, and then only the last ChG layer needs to be etched to form the ridge structure.

Fig. 1.

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	Fig. 1. Sketches of the proposed waveguide structures. (a) The reversed-rib waveguide without slot. (b) The reversed-rib waveguide with a silica slot.

To tailor the dispersion of the reversed-rib waveguide, we introduce a horizon slot above the rib, which is shown in Fig. 1(b). It can be realized by simply repeating the deposition of both As₂S₃ film and silica layer one more time.

Dispersion tailoring is carried out by properly optimizing the structural parameters of the waveguide including the rib width (W₁), the rib height (H₁), and the thicknesses of the As₂S₃ layer on the subtract (H₂), the silica slot (H_s), and the As₂S₃ layer above the slot (H₃). In order to prevent the oxidation of the ChG glass, the waveguide is covered with a layer of SiO₂ with a thickness of H_cladding. In this paper, the effective indexes (n_eff) of the fundamental quasi-TE mode and quasi-TM mode with different wavelengths are calculated by the finite-element mode method. In the calculation, the material dispersions of As₂S₃^[52] and silica^[53] are taken into account using their Sellmeier equations. The dispersion can be calculated by

Fig. 2.

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Fig. 2. Dispersion of the fundamental quasi-TE and quasi-TM modes in the reversed-rib waveguide with/without the slot. The solid red and black lines are the dispersion of the quasi-TE and quasi-TM modes in the reversed-rib waveguide with a slot, respectively. The dashed red and black lines are the dispersion of the quasi-TE and quasi-TM modes in the reversed-rib waveguide without slot, respectively. The blue dashed-dotted line is the material dispersion of the As2S3.

The red and black solid lines in Fig. 2 are the dispersion of the fundamental quasi-TE and -TM modes in the reversed-rib waveguide with the slot (as shown in Fig. 1(b)). Here, the width and height of the rib (W₁, H₁), the thickness of the As₂S₃ layer above the substrate (H₂), and the silica cladding layer (H_cladding) are all the same as those of the waveguide without the slot. The thicknesses of the silica slot (H_s) and the As₂S₃ layer above the slot (H₃) are 93 nm and 230 nm, respectively. It can be seen that, for both modes, the dispersion shows flatter profiles with broader near-zero dispersion bands than that of the reversed-rib waveguide without the slot. The inset of Fig. 2 shows the detail of the dispersion of the quasi-TM mode in the reversed-rib waveguide with the slot. It can be seen that it has two ZDWs at 1849 nm and 2742 nm, respectively. The dispersion is limited between −10.6 ps·nm⁻¹·km⁻¹ and +11.14 ps·nm⁻¹·km⁻¹ over a band of 1080 nm (from 1760 nm to 2840 nm).

To show the mechanism of the dispersion tailoring effect of the silica slot, the electric field distributions of the quasi-TM modes with different wavelengths in the proposed waveguide with slots and without slots are calculated and shown in Figs. 3(a) and 3(b), respectively. It can be seen that the electric field concentrates in the rib region at the short wavelength. As the wavelength increases, the electric field gradually transfers into the SiO₂ slot. This mode field transfer provides a modification on the waveguide dispersion of the reversed-rib waveguide. Hence, the slot provides additional dispersion tailoring freedom.

Fig. 3.

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	Fig. 3. Quasi-TM mode evolution of the waveguide for various wavelengths (a) with slots and (b) without slots.

The slot structure provides enough degrees of freedom to tailor the waveguide dispersion. On the other hand, the variation of structure parameters, which is unavoidable in waveguide fabrication, would also impact the dispersion property of the waveguide. Figure 4 shows the influence of structural parameters on the dispersion of the fundamental quasi-TM mode by changing them around the optimized values.

Fig. 4.

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	Fig. 4. Dispersion profiles of the slot waveguides with different (a) slot thicknesses (Hs), (b) heights of groove (H1), (c) heights of the As2S3 layer (H2), (d) widths of groove (W1), and (e) heights of another As2S3 layer (H3).

Figure 4(a) shows the calculation results of dispersion with different thicknesses of the slot. The other structure parameters used in this calculation are the same as the optimized design used in Fig. 2 (This condition is also used in the calculations of Figs. 4(b)–4(e)). It can be seen that the dispersion curve with a larger slot thickness shows higher suppression at the long-wavelength band. By optimizing the slot parameters, a dispersion curve with a flat profile can be realized.

The dispersion profiles for different rib heights (H₁) are plotted in Fig. 4(b). It can be seen that the dispersion profile is almost unchanged. While it rises as a whole as H₁ increases from 704 nm to 724 nm, the maximum dispersion value rises from 0 ps·nm⁻¹·km⁻¹ to 21.43 ps·nm⁻¹·km⁻¹, with a rate about 1 ps·nm⁻¹·km⁻¹ per nanometer. A similar effect can be seen when the thickness of the As₂S₃ layer on the subtract (H₂) changes. Figure 4(d) shows the calculation results under different rib widths (W₁). When the width varies from 1390 nm to 1410 nm, the dispersion shape and slope at the short wavelength are almost unchanged. However, the dispersion in the longer wavelength region changes rapidly and the dispersion on the peak increases from 9.5 ps·nm⁻¹·km⁻¹ to 12.4 ps·nm⁻¹·km⁻¹ with a rate about 0.15 ps·nm⁻¹·km⁻¹ per nanometer.

The influence of the thickness of the As₂S₃ layer above the slot (H₃) is calculated and shown in Fig. 4(e). The dispersion profile drops with the increase of H₃. The maximum dispersion reduces with a rate of about 1 ps·nm⁻¹·km⁻¹ per nanometer and shifts to longer wavelength.

By comparing the results in Fig. 4, it can be seen that the slot thickness is the most sensitive structure parameter for the dispersion profile. Hence, its fine control in the fabrication process is the key to obtain a flat and near-zero dispersion profile.

The As₂S₃ glass has the third-order nonlinear index (n₂) of 2.92 × 10⁻¹⁸ m²/W and a low TPA coefficient of 6.2 × 10⁻¹⁵ m/W at the telecom band. Its nonlinear FOM is as high as 312, which is much higher than that of the silicon waveguide (0.77).^[31] On the other hand, the proposed waveguide structure provides an effective way to realize a flat dispersion profile, which is preferred in many third-order parametric nonlinear processes. Hence, the proposed waveguide has great potential as a nonlinear medium for developing integrated nonlinear devices. To show this more clearly, we calculated the third-order nonlinear coefficient of the fundamental quasi-TM modes in the proposed waveguide with different wavelengths by

Fig. 5.

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	Fig. 5. The effective area Aeff (blue circles) and the nonlinear coefficient γe (black squares) of the fundamental quasi-TM mode in the proposed reversed-rib waveguide with a slot. The structure parameters used in the calculation are the same as those used in Fig. 2.

To demonstrate the potential of this waveguide in the third-order parametric processes, we calculated the linear phase mismatching of the degenerate four-wave mixing by Δβ = β_s + β_i − 2β_p, where β_s, β_i, and β_p are propagation constants of the signal, idler, and pump lights, respectively. The quasi-TM mode with the ultra-flat profile is considered in this calculation. The calculation results are shown in Fig. 6. Figure 6(a) is the linear phase mismatching under different pump wavelengths λ_pump and signal-pump wavelength differences Δ_λ. Different colors in the figure represent the values of linear phase mismatching under different λ_pump and differences Δ_λ. It can be seen that if the pump light is set close to the two ZDWs, small linear phase mismatching could be realized over large regions of signal-pump wavelength difference Δ_λ.

Fig. 6.

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Fig. 6. Calculation results of linear phase mismatching of the degenerate FWM process in this waveguide. (a) Linear phase mismatching Δβ under different pump wavelengths and signal-pump wavelength differences Δλ, (b) linear phase mismatching under different signal wavelengths when the pump wavelength is set near the shorter ZDW, (c) linear phase mismatching under different signal wavelengths when the pump wavelength is set near the longer ZDW.

To show this more clearly, the linear phase mismatch profiles when the pump wavelength is set near the shorter ZDW (1849 nm) and longer ZDW (2742 nm) are plotted in Figs. 6(b) and 6(c), respectively. It can be seen that when pump wavelength is set to 1853 nm, which is close to the shorter ZDW (1849 nm), the linear phase mismatching profile is like an upward parabola with a zero point at Δ_λ = 0 and a broad near-zero band. When the pump wavelength decreases, e.g., to 1833 nm, the parabola-like profile shrinks, leading to a smaller near-zero band. On the other hand, when the pump wavelength increases, e.g., to 1863 nm, two regions with negative linear phase mismatching appear at two sides, which would be deeper and wider if the pump wavelength increases further. The profile of the linear phase mismatching when the pump light is near the longer ZDW (2742 nm) is similar with the case that it is near the shorter ZDW. However, its variation with the pump wavelength is different. It can be seen that the two regions with negative linear phase mismatching appear when the pump wavelength is shorter than the longer ZDW, with a broader near-zero band.

It is well known that a small negative linear mismatching is preferred to the degenerate fourwave mixing because it can be compensated by the nonlinear mismatching term under a proper pump level. Hence, to realize the broad band degenerate fourwave mixing in the quasi-TM mode of the proposed waveguide, the pump light should have a wavelength a little longer than the shorter ZDW or a little shorter than the longer ZDW.

In this paper, we proposed a reversed-rib As₂S₃ waveguide with a slot to tailor its dispersion. By properly designing its structure parameters, an ultra-flat and near zero dispersion profile can be realized, in which the dispersion is limited between −10.6 ps·nm⁻¹·km⁻¹ and +11.14 ps·nm⁻¹·km⁻¹ over a band of 1080 nm. The slot structure provides a dispersion tailing effect to extend the low dispersion band, which is due to the electric field transfer process from the reversed-rib waveguide to the slot as the wavelength increases. To demonstrate its potential as a nonlinear waveguide for integrated nonlinear optical devices, its nonlinear coefficient and the linear phase mismatching profile for the degenerate FWM are also calculated, showing that the waveguide can support broadband degenerate FWM in near infrared and middle infrared band if the pump wavelength is set a little longer than the shorter ZDW or a little shorter than the longer ZDW. The proposed As₂S₃ waveguide can be fabricated without photolithography and etching processes on the As₂S₃ film, which can reduce the difficulty in the fabrication of ChG waveguides, showing great potential in applications of nonlinear photonic devices.