1 Introduction

The optical frequency combs based on microresonators [1–6] which have attractive features such as wide spectral range, high repetition rate and integration have gained considerable attention in astronomy, spectroscopy, arbitrary waveform generator, optical clocks and telecommunication [6, 7]. However, for a high-quality optical frequency comb, the broader coverage in wavelength and less variation in amplitude are of great importance. Microresonator-based frequency combs are generated owning to cascaded four-wave mixing effect, which is seriously influenced by the dispersion and nonlinearity [8]. Thus, the dispersion tailoring and high nonlinearity of microresonator are vital for the optical frequency comb generation. In the dispersion aspect, the appropriate dispersion profile could produce desirable phase matching of the four-wave mixing responsible for the comb generation and efficiently regulate the ultimate spectral coverage and flatness of the combs [9, 10].

For a broader and flatter frequency combs generation, the key factor is to acquire a lower and flatter dispersion curve over a wide wavelength range of the microresonator. Recently, the waveguide structures with a strip [11], single-slot [12–15] and double-slot waveguide [16, 17] have been proposed to tailor the dispersion profile. Due to the unique capability of dispersion engineering, the single-slot waveguide was explored in a microresonator for a flattened dispersion [18, 19]. Zhang et al. reported that dispersion variation of the single-slot microring resonator was fluctuated from 0 to 17 ps/(nm km) [18]. More recently, Bao and Zhang found the dispersion varied between 0 and 7.2 ps/(nm km) over 718-nm bandwidth in the same waveguide structure, corresponding to the optical frequency combs spectrum with 10-dB bandwidth of 630 nm [19].

In this paper, the ultra-flat and broad optical frequency combs with a 7-dB power variation of comb lines from 1855 to 3010 nm wavelength are successfully achieved by simulation on the basis of our original microring resonator. The proposed resonator is formed of arsenic tri-sulfide (As2S3) double-slot waveguides and shows an ultra-flat and low dispersion of 0–3.8 ps/(nm km) over a 1150 nm wavelength range. Furthermore, the impact of different structural parameters of microring waveguide on the dispersion profile is analyzed.

2 Waveguide structure and dispersion property

Compared with the type of single-slot structure, double-slot waveguide has more dimensions to design; meanwhile, ultra-broadband, flat and low-dispersion profiles can be achieved and shaped by tailoring structural parameters [16, 17]. Here, the double-slot waveguide is combined with microring resonator in the first time for the flatter and broader optical frequency comb generation, to the best of our knowledge.

Figure 1a shows the basic structure of the designed double-slot waveguide-based microring resonator. The bending radius (R) of the microring resonator is 117 μm, corresponding to the free spectral range (FSR) of 200 GHz. The pump light is launched from a straight waveguide and coupled into the microring resonator. As shown in Fig. 1b, the waveguide has two horizontal SiO2 slots which are surrounded by three As2S3 layers. Additionally, the substrate of the waveguide is 4-μm-thick SiO2, and the upper cladding is air. Moreover, W, Hl1, Hs1, Hl2, Hs2 and Hl3 represent the waveguide width, bottom As2S3 height, lower SiO2 slot thickness, central As2S3 height, upper SiO2 slot thickness and upper As2S3 height, respectively.

Fig. 1
figure 1

a Double-slot waveguide-based microring resonator. b Cross section and material of double-slot microring waveguide. c The mode evolution of quasi-TM mode (vertical direction) at different wavelengths, respectively

Full size image

Figure 1c shows the mode evolution of quasi-TM mode (vertical direction) at different wavelengths. The principle of dispersion tailoring of the double-slot waveguide can be explained as follows: The anticrossing effect that the guided mode transits from a strip mode at short wavelength to a slot-like mode at long wavelength could lead to the flattened dispersion and finally produce a slightly negative waveguide dispersion for balancing the convex dispersion in strip waveguides without a slot structure [17].

The dispersion profiles D(λ) are performed by using the finite element method. The dispersion of quasi-TM mode (vertically polarized) is calculated by the second derivative of the effective index (n eff ) with respect to the wavelength (λ). It is defined as follows:

$$D = - \left( {\lambda /c} \right) \cdot \left( {\partial^{2} n_{eff} /\partial \lambda^{2} } \right) ,$$
(1)

where c is the speed of light in vacuum. In the calculation, the wavelength step and effective index precision are 10 nm and 16 float digits, respectively. In this work, material dispersion of As2S3 and SiO2 is also taken into account [20, 21].

The structure parameters of the waveguide including W, Hl1, Hs1, Hl2, Hs2 and Hl3 can be properly tuned for tailoring the dispersion when needed. As seen from Fig. 2, by optimizing structure parameters (structure I) as W = 933 nm, Hl1 = 330 nm, Hs1 = 114 nm, Hl2 = 960 nm, Hs2 = 112 nm, Hl3 = 328 nm and R = 117 μm, a saddle-shaped dispersion profile with two zero-dispersion wavelengths (ZDWs) is achieved. The ultra-flat and low-dispersion value D varies in 0–3.8 ps/(nm km) from 1855 to 3005 nm over a 1150-nm bandwidth. Correspondingly, the second-order dispersion coefficient β 2 (β 2 = −λ 2D/2πc) changes within 0 ~ −12.6 ps2/km. Compared with the single-slot waveguide-based microring resonator in Reference [18] which showed D varies in 0–17 ps/(nm km), accordingly β 2 varies from 0 to 16 ps2/km, the dispersion profile of the proposed method is lower and flatter (see above).

Fig. 2
figure 2

Dispersion and second-order dispersion coefficient profile of the optimum waveguide structure I

Full size image

3 Principle of dispersion tailoring

The dispersion can be tailored by tuning the structural parameters of double-slot microring resonator waveguide. So the effect of each structural parameter on the dispersion properties is investigated by changing it around the optimum value while the other parameters are kept the same. The dispersion profiles of the waveguide as shown in Figs. 3 and 4 are effectively tailored by properly tuning the parameters W, Hl2, Hl1 and Hs1.

Fig. 3
figure 3

Dispersion profiles with the variation of a waveguide width W, b central As2S3 height Hl2, respectively, in the structure I

Full size image
Fig. 4
figure 4

Dispersion profiles with the variation of a bottom As2S3 height Hl1, b upper As2S3 height Hl3, c lower SiO2 slot thickness Hs1, d upper SiO2 slot thickness Hs2, respectively, in the structure I

Full size image

As shown in Fig. 3a, changing the waveguide width W could adjust the flatness of dispersion profile while other structural parameters are constant. Dispersion values in long wavelength range are more sensitive to fluctuations of W, and both the shape and slope of dispersion profile are efficiently changed. Meanwhile, the sign of third-order dispersion (TOD) could be effectively regulated with changing the W. During optimizing the dispersion profile, the change of the central As2S3 height Hl2 could flexibly adjust the fluctuation of the entire dispersion curves. As shown in Fig. 3b, when the Hl2 decreases from 968 to 952 nm with other parameters unchanged (structure I), the dispersion moves up from anomalous dispersion to normal dispersion regime with almost the same shape and slope, and the dispersion value change is 0.7 ps/(nm km) per nm. The number of ZDWs firstly changes from 2 to 4 and then decreases to 0 as the Hl2 decreases from 968 to 952 nm. The decrease in Hl2 promotes mode transition from strip mode to slot mode. Obviously, negative waveguide dispersion by this transition moves down the dispersion curves.

There is a similar trend in both situations when the bottom As2S3 height Hl1 or upper As2S3 height Hl3 is decreased and the central As2S3 height Hl2 is increased. As shown in Fig. 4a, b, the reduction in Hl1 and Hl3 causes the moving of the dispersion profile from normal to anomalous dispersion regime with a dispersion value change of 0.5 ps/(nm km) per nm. Moreover, the number of ZDWs firstly changes from 2 to 4 and then decreases to 0 with the increase in Hl1 and Hl3. Furthermore, the flatness of total dispersion profiles can be improved. The sign of third-order dispersion (TOD) dispersion slope can be finely controlled by properly tuning the lower SiO2 slot thickness Hs1 and upper SiO2 slot thickness Hs2 as shown in Fig. 4c, d, respectively. The dispersion curves imply that the long wavelength part goes down while the short wavelength part almost remains consistent with Hs1 increasing from 104 to 124 nm, and the dispersion curves nearly rotate at 2230 nm. Meanwhile, the dispersion curves have the same trend when Hs2 increases from 102 to 122 nm, and the dispersion curves nearly rotate at 2020 nm in Fig. 4d. Therefore, the slope can be flexibly modified by tailoring the lower and upper slot thickness of waveguides.

As shown in Figs. 3 and 4, the optimal dispersion profile could be achieved when the structural parameters changed in double-slot microring resonator waveguides; meanwhile, the dispersion deviation is only ±20 ps/(nm km). However, compared with the single- or double-slot silicon waveguide [12, 16, 17] with ±200 ps/(nm km) dispersion deviation in the same variations of structural parameters, this double-slot arsenic tri-sulfide waveguide has less sensitivity to variations of the structural parameters and fabrication errors.

4 Flat optical frequency combs generation in double-slot microring resonator

From Reference [22], it can be predicted that the optical frequency comb is more easily generated when the dispersion is in anomalous area. Furthermore, the comb performance can be considerably influenced by the value of β 2 and the bandwidth (BW) between two zero-dispersion points (ZDPs) [19, 23]. Generally speaking, the lower the β 2, the flatter the comb generation, and the broader the BW between two ZDPs, the broader the comb bandwidth generation. Consequently, the dispersion profile of structure I in the simulation is adopted for flatter and broader frequency combs generation.

In order to simulate flatter and broader comb generation based on the optimized dispersion profile of the structure I, the Lugiato–Lefever equation [24–26] is utilized as follows:

$$t_{R} \cdot \frac{\partial E(t,\tau )}{\partial t} = \left[ { - \alpha - i\delta_{0} + iL\sum\limits_{k \ge 2} {\frac{{\beta_{k} }}{k!}\left( {i\frac{\partial }{\partial \tau }} \right)^{k} + i\gamma L\left| E \right|^{2} } } \right]E + \sqrt \theta E_{in} ,$$
(2)

where t R is the round-trip time and α describes the total cavity losses per round trip including the linear absorption inside the resonator and the power coupling coefficient θ. δ 0 means the cavity detuning, and L stands for cavity length. Other resonator parameters include the kth dispersion coefficient β k (β k  = d k β/dω k ) and the nonlinear coefficient γ (γ = n 2·ω0/(c · A eff ), n 2 the nonlinear refractive index, A eff the effective modal area of the resonator mode). E(t, τ) and E in are the intra-cavity field and input field (pump power P in  = |E in |2), respectively, where t and τ are the slow and fast times.

In our simulation, parameters are taken from the double-slot microring resonator of structure I, with the FSR = 200 GHz, under R = 117 μm, L = 735 μm and P in  = 100 mW. Here, α and θ are equal to 0.009 [27], the calculated cavity Q factor is approximate to 3 × 105, and calculated γ is about 8000 (1/W/km). The pump wavelength is close to 2.33 µm. Through turning the detuning as 0.086, the ultra-flat optical frequency combs with a 7-dB power variation of comb lines from 1855 to 3010 nm wavelength over 1155-nm bandwidth are gained and shown in Fig. 5. In general, the flatness or bandwidth of comb spectrum is measured in a 20-dB power variation with 1110-nm bandwidth [18] or 10-dB power variation with 630-nm bandwidth [19]. Meanwhile, compared with References [18, 19], the broader bandwidth of the output comb spectrum is obtained with the smaller power variation in our study. Specially, we note that there are two strong narrowband peaks in the output, which are the dispersive waves and around the second-order dispersion coefficient to zero, as mentioned in Reference [26].

Fig. 5
figure 5

Output frequency comb spectra of structure I with the given parameters

Full size image

As shown in Reference [28], the dispersion-induced change of the FSR could be minimized due to the low and flat cavity dispersion profile, which helps to generate a nearly equidistant frequency comb. Specially, in our design, the FSR standard deviation is as low as 0.052 GHz, providing potential for optical carriers of large capacity communication transmission. In addition, the optical frequency combs are located in mid-infrared range, which not only add tremendous value to existing applications including the optical communication in 2-μm waveband and precision spectroscopy of molecules, but also have a strong impact on emerging or unknown fields throughout many areas of research [29–31].

5 Conclusion

In conclusion, by tailoring the structural parameters of the proposed double-slot microring resonator based on an As2S3 waveguide, the ultra-flat and low dispersion over a wide bandwidth range is provided. The influence of the structure parameters on the dispersion curves is studied. Moreover, the optimized dispersion profile is of great importance for improving the performance of cascaded four-wave mixing and frequency comb generation in microresonator. Finally, an ultra-flat, broad and nearly equidistant frequency comb based on the optimized dispersion profile is generated with the Lugiato–Lefever equation in simulation, which is also potentially useful for both telecommunication and mid-infrared applications.