1. Introduction

The basic principle of micro-resonator based optical frequency combs is cascade four-wave mixing (CFWM)^[1-3]. Nevertheless, because of the effect of two-photon absorption (TPA)^{[4, 5]} and free carrier absorption (FCA)^{[6, 7]} in a silicon waveguide, the CFWM efficiency will be depressed at the optical communication wavelengths of the C-band. However, silicon nitride does not suffer from TPA at these wavelengths^[8] and it possesses even higher thermal stability than that of silicon, which guarantees that the silicon nitride micro-resonator can work more stably at high pump power conditions for the long-term without suffering thermal deterioration or damage. Therefore, at the C-band a silicon nitride micro-resonator is an ideal platform for an optical frequency comb based integrated optical light source.

Currently, the mainstream methods for growing silicon nitride membrane are plasma enhanced chemical vapor deposition (PECVD)^[9] and low pressure chemical vapor deposition (LPCVD)^[10]. Because of the thermal expansion coefficient mismatch between the silicon nitride and the silica substrate, the tensile stress in the silicon nitride is very high and the thickness of the silicon nitride film deposited by LPCVD is relative thinner, usually less than 600 nm^[11-13]. Alternatively, a thicker silicon nitride film could be deposited by PECVD. However, compared with deposited by PECVD technology, the silicon nitride film grown by LPCVD is higher quality and the waveguide loss is much lower^[14]. In 2009, researchers at Cornell University were the first to introduce thermal cycling technology to successfully grow silicon nitride film with a thickness reaching 750 nm^{[15, 16]}. Although multiple depositions combined with annealing technology can produce thicker silicon nitride film with high optical quality, the process is much more complex and it is difficult to master the optimum technical parameters during the fabrication process.

Effective FWM must satisfy the phase-matching condition, which requires the waveguide to have anomalous dispersion. For a traditional rib or strip waveguide, the silicon nitride thickness must be larger than 700 nm to realize anomalous dispersion at the C-band. However, thick silicon nitride film is difficult to synthesize technically by LPCVD because of the thermal expansion coefficient mismatch induced strain in the film.

In this letter, we propose a new type of silicon nitride suspended membrane waveguide, also known as an air-bridge waveguide, which can realize anomalous dispersion with thin silicon nitride film. We investigate the relationship between the group velocity dispersion (GVD) properties and the key parameters of the waveguide by using the finite element method (FEM). With suitable waveguide parameter selection, a suspended silicon nitride membrane waveguide with a thickness of merely 400 nm can also realize anomalous dispersion at wavelengths of around 1550 nm.

2. Dispersion engineering and analysis

The total dispersion of a waveguide includes three parts: material dispersion, waveguide dispersion, and polarization mode dispersion. In a nanometer-scaled waveguide, the material dispersion is intrinsic and determinate. In general, the polarization state of the pump, signal, and idler light are the same in FWM, thus we take no account of polarization mode dispersion in this case. The waveguide dispersion can be tailored by changing the waveguide structure and dimensions. Therefore, we can obtain different dispersion profiles through tailoring the structural parameters and dimensions of the nanometer-scaled waveguides.

The phase-matching condition of the nonlinear process, such as FWM, is mainly influenced by the second order dispersion. So, engineering the second order dispersion of the waveguides is crucial. We usually use the group velocity dispersion (GVD) coefficient D to describe the second order dispersion, with the expression shown as follows:

$D=-\frac{\lambda }{c}\frac{{{\text{d}}^{2}}{{n}_{\text{eff}}}}{\text{d}{{\lambda }^{2}}},$

(1)

where λ is the wavelength of the light, c is the light speed in vacuum, and n_eff is the effective refractive index of the waveguide. When D>0, it is called anomalous dispersion, otherwise it is a normal dispersion.

Due to the fact that silicon nitride and silica are dielectric materials, and in order to obtain accurate results affected by the high order dispersion effects, it is necessary to take account of the material dispersion. We use the Sellmeier equations, shown as follows, to describe the material dispersion. For silicon nitride, the Sellmeier equation is^[17]:

$n_{\text{S}{{\text{i}}_{3}}{{\text{N}}_{4}}}^{2}=1+\frac{2.921{{\lambda }^{2}}}{{{\lambda }^{2}}-0.01648}.$

(2)

And for silica, the Sellmeier equation is^[18]:

$\begin{align} & \text{n}_{\text{Si}{{\text{O}}_{2}}}^{2}=1+\frac{0.6961663{{\lambda }^{2}}}{{{\lambda }^{2}}-0.0684043} \\ & +\frac{0.4079426{{\lambda }^{2}}}{{{\lambda }^{2}}-0.2262414}+\frac{0.8974794{{\lambda }^{2}}}{{{\lambda }^{2}}-9.8961663}, \\ \end{align}$

(3)

where the unit of the wavelength λ is μm.

Based on the above analysis, we first calculate the dependence of the effective refractive index on the wavelength for the quasi-TE mode by finite-element method (FEM), and the software we used is COMSOL Multiphysics. Then, we use Equation (1) to calculate the GVD coefficient D.

The structure of the silicon nitride suspended membrane waveguide is shown in Figure 1(a). The height of the rib waveguide is H, the width of the waveguide is W, the slab height is h, and the ratio of the rib height to the slab height is defined as r=h/H. A higher value of r means that the etched depth is shallower. The suspended structure is designed and fabricated as follows. First, two lines of periodical holes far away from the rib waveguide region are patterned and fully etched at both besides of the rib waveguide, and the distance between the holes and the rib waveguide should be large enough to avoid the interaction of the waveguide modes with these holes. The BOX layer underneath the rib waveguide can then be removed by the chemical wet-etching method^{[19, 20]}. Compared to the dry-etching method, the wet-etching method does not require expensive equipment, and it can also provide a very smooth bottom surface and sidewalls^[21]. Therefore, we use the wet-etching method to remove the BOX layer with these holes acting as access for liquid etchant.

We first study the dependence of the dispersion properties of the waveguide on the structural parameters. In Figure 2 the GVD coefficient D for different waveguide dimensions is shown at wavelengths ranging from 1200 to 1800 nm. For comparison, the material dispersion of silicon nitride is also shown. We set the following structural parameters as our design baseline: H=400 nm, W=1000 nm, r=0.1. We investigate the influences of the etched depth ratio, the silicon nitride rib height, and the waveguide width on the GVD in turn. As shown in Figure 2, at the short wavelength end, the contribution of the waveguide dispersion shows the opposite dispersion property compared with that of the silicon nitride material dispersion, and the GVD curves are located above that of silicon nitride material dispersion, which illustrates that both the GVD and waveguide dispersion are anomalous. As the wavelength increases, the values of GVD coefficient gradually decrease, even changing to negative with some certain dimensions, which means that the waveguide dispersion is moving in the direction of normal dispersion. At the cross point between the GVD and the material dispersion curves, the contribution of the waveguide dispersion is zero. In Figure 2(a), we can see that as the values of etched depth ratio increase from 0.05 to 0.3, corresponding to the etched depth becoming shallower, the wavelength range of anomalous dispersion becomes narrower. Besides, the zero-dispersion wavelength (ZDW) moves to the shorter wavelength as r increases, and the corresponding ZDWs are 1554.9, 1502.7, 1390.7, and 1267.2 nm, respectively. So we can conclude that with deeper etched waveguide it is easier to realize anomalous dispersion in this wavelength range. Figure 3(b) shows the influence of H on the dispersion profile. Similarly, with the increase of the silicon nitride height, the scope of anomalous dispersion increases. This illustrates that a thicker silicon nitride thickness makes it easier to realize anomalous dispersion. Figure 2(c) depicts the influence of waveguide width on the dispersion profile. We can see that there are some cross points among the GVD curves with different waveguide widths, which means that the dispersion for different waveguide width would exhibit the same GVD coefficient. With the increase of the silicon nitride waveguide width, the wavelength range for anomalous dispersion becomes narrower and narrower, and even disappears.

We also study the dependence of the dispersion properties of the rib and strip waveguide on the structural parameters as a comparison, and the structures are shown in Figures 1(b) and 1(c). Like the suspended membrane waveguide, the key parameters of the rib waveguide are the rib height H, the waveguide width W, and the etched depth ratio r. When r=0, it corresponds to a strip waveguide; otherwise, it corresponds to a rib waveguide. The design baseline is: H=750 nm, W=2000 nm, r=0.2. The dispersion properties of the rib waveguide are similar with that of the suspended waveguide, as shown in Figure 3. We can see that the rib waveguide with deeper etch depth or larger silicon nitride height is easier to realize anomalous dispersion. Besides, the dispersion profiles of the waveguide with wider width are flatter but move to the direction of normal dispersion. When $r=0$, namely strip waveguide, the anomalous dispersion range is largest, as shown in Figure 3(a). The strip waveguide dispersion properties are shown in Figure 4. The waveguide height is H and the width is W. Similarly, a waveguide with larger height is easier to realize anomalous dispersion, and a waveguide with wider width makes the dispersion profile flatter. According to the results, only when the height of rib or strip waveguide is larger than 700 nm can the anomalous dispersion appear around the communication wavelength.

Based on the above analysis, we can see that compared with a traditional strip or rib waveguide, the silicon nitride suspended membrane waveguide can realize anomalous dispersion with thin silicon nitride film. Besides, the dispersion properties are affected crucially by the waveguide structural parameters. We further study the dispersion properties of the silicon nitride suspended waveguide at 1550 nm, the central wavelength of C-band, which is important in optical fiber communication.

We first study the effect of the waveguide width on GVD with the rib height of the silicon nitride suspended waveguide kept to 400, 500, and 600 nm. The impact of the etched depth on GVD is also investigated for each rib height. As shown in Figure 5(a), we can see that with the increase of W, the value of GVD coefficient D first rapidly increases and then gradually decreases. On the other hand, as the etched depth ratio increases, the dispersion curves move down in the direction of normal dispersion. In particular, for the waveguide with an etched depth ratio of 0.05, the waveguide exhibits anomalous dispersion for the waveguide width ranging from 990 to 1255 nm, and the maximum dispersion is 22.56 ps/(nm$\cdot$km). As r further increases, the dispersion property turns to a normal dispersion. So the silicon nitride suspended waveguide must be etched deep enough to realize the anomalous dispersion required for the phase-matching condition of the FWM. The dispersion properties of the silicon nitride suspended waveguide with rib height of 500 and 600 nm are shown in Figures 5(b) and 5(c), respectively, which display similar characteristics to those in Figure 5(a). However, by making a comparison we can conclude that the waveguide with thicker silicon nitride rib makes it is easier to realize anomalous dispersion for the same etched depth ratio.

Next, the effect of the silicon nitride rib height on GVD is also studied by a similar method, with the waveguide width set to 1000 and 1200 nm. The results are shown in Figure 6. The GVD coefficient increases as the silicon nitride rib height increases, moving from a normal dispersion to an anomalous dispersion. As r increases, the zero-dispersion point moves in the direction of a larger H, which also illustrates that a deeper etched depth (smaller $r)$ silicon nitride suspended waveguide makes it is easier to realize anomalous dispersion. The dispersion curves in Figures 6(a) and 6(b) are similar. This indicates that the parameter H has a more obvious impact on GVD than W. So adjusting the silicon nitride rib height is more effective for the control of the dispersion properties of the silicon nitride suspended waveguide.

The dispersion properties analyzed above use quasi-TE mode because the GVD of the quasi-TM mode in the silicon nitride suspended waveguide cannot realize anomalous dispersion around the C-band, the dispersion profile is shown in Figure 7. We can see that the GVD is normal in the wavelength ranging from 1200 to 1800 nm. The leakage loss of the quasi-TM mode is very large compared to that of the quasi-TE mode. The optical field distributions of the quasi-TE and quasi-TM mode at 1550 nm are shown in Figure 8, and the structure parameters set to $W=1000$ nm, $H=400$ nm, and $r=0.05$. Therefore, the quasi-TE mode in the waveguide is superior to the quasi-TM mode in the aspects of dispersion and leakage loss, and we only need to analyze the GVD of quasi-TE mode.

Based on the above analysis, we can conclude that the dispersion properties of the silicon nitride suspended membrane waveguide are strongly influenced by some of the key structural parameters, and we can make the dispersion engineering by changing the waveguide structural parameters. The simulation results show that a waveguide with thicker silicon nitride rib and deeper etched depth makes it is easier to realize anomalous dispersion. However, both the mechanical strength and heat dissipation of a waveguide whose etched depth is too deep, such as for the case H=400 nm, r=0.05, and h=20 nm, will become poor. Therefore, in practice, we can choose the structure parameters H=500 nm, r=0.3, h=150 nm as our device dimensions, which not only guarantee anomalous dispersion but also ensure higher mechanical strength and better heat dissipation.

3. Conclusion

The simulation results indicate that a specially designed suspended membrane waveguide with silicon nitride rib thickness ranging from 400 to 600 nm can realize anomalous dispersion by altering the waveguide width and etched depth. Especially, for a silicon nitride membrane thickness of only 400 nm and the waveguide width ranging from 900 to 1255 nm, when the etched depth ratio is 0.05, the suspended waveguide can exhibit anomalous dispersion at 1550 nm. These results mean that, although it is hard to grow the silicon nitride film with large thickness by LPCVD, we can still realize anomalous dispersion at the central wavelength of C-band with silicon nitride thin membrane by adopting a specially designed suspended waveguide structure, which has practical significance in the fabrication of high quality silicon nitride micro-resonator for generation of optical frequency combs.