1. Introduction

Various types of waveguide bends [1–5] form the basis for many integrated optical circuits and components, including optical delay lines, resonators, and interferometers. For circular waveguide bends [1,3,6–11], a bending radius as small as a few microns can be allowed by increasing index contrast [6–11], and a low bending loss has been demonstrated [8,10], with a bending radius reduced to even 1.5 μm [10]. Most of research attention has been paid to the properties of loss [8,10,11] and polarization [7]. However, chromatic dispersion in a curved waveguide can also be crucial to determine for nonlinear applications. When a ring resonator is made small for device compactness and nonlinearity enhancement, dispersion inside the cavity may vary greatly before the bending loss becomes too high. The sensitivity of dispersion to curvature is partially due to the second derivative of effective index of refraction with respect to wavelength. A report was recently published, examining the dispersion property for a toroid resonator [12], in which silica is surrounded by air and the bending radius is fairly large (i.e., hundreds of μms). However, that report did not consider curved waveguides with a large index contrast or a small radius of curvature.

Chromatic dispersion in optical cavities can be quite important for micro-resonator-based nonlinear applications [12–15], e.g., on-chip frequency comb generation enabled by cascaded four wave mixing with a low pump power. Flat and low in-cavity dispersion produces almost equidistant resonance peaks, which are desired not only for reducing pump power but also for improving measurement precision in comb-based frequency metrology. However, highly nonlinear waveguides or photonic wires typically have strong index contrast and waveguide dispersion, making it difficult to obtain equally spaced resonance peaks over a wide band. In-cavity dispersion of high-index-contrast micro-resonators needs to be engineered.

In this paper, we analyze chromatic dispersion in tightly curved silicon waveguides. It is shown that both strip and slot waveguides would have a dramatic dispersion change when the bending radius is reduced to a few microns. Both polarization states are examined, and they exhibit quite different sensitivity of dispersion to waveguide bending. The shift of zero-dispersion wavelength (ZDW) in these silicon waveguides could be up to 220 nm when the bending radius is 2 or 3 μm. Dispersion value at a given wavelength can have a change up to 1000 ps/(nm·km) for a different bending radius. This indicates that the dispersion change in bent waveguides is an important issue to be considered in device design. We also show that in-cavity dispersion in silicon microring resonators can be flattened, using a horizontal slot structure, over a 563-nm wavelength range from 1566 to 2129 nm. Free spectral range (FSR) of the resonator as a function of wavelength has a standard deviation reduced to 0.023 GHz, 460 times smaller than that produced by a strip waveguide resonator.

2. Dispersion in circular waveguide bends

When the waveguide is bent with a structural bending radius R₀ that is measured from the center of rotation to the waveguide center, the guided optical mode is extended outwards, as shown in Fig. 1 , and feels an “effective” bending radius R_eff that is greater than R₀. The increased path length of the guided mode is dependent on wavelength, which induces an additional dispersion. The effective bending radius R_eff also depends on waveguide geometry, especially waveguide width. To use the waveguide bends and ring resonators for nonlinear optics applications, one would have a ZDW close to a wavelength range of interest, near 1550 nm. From previously reported results [16,17], it can be seen that the waveguide size has to be relatively large. For silicon strip waveguides, we choose height H_l = 400 nm, and waveguide width W = 800 nm. For silicon horizontal slot waveguides, we set the heights H_u and H_l of two silicon layers to be 240 nm, and slot height H_s = 40 nm, width W = 500 nm. In both cases, 2-μm-thick SiO₂ is considered above the silicon substrate, and top cladding is air.

 

Fig. 1 A horizontal slot waveguide has a low-index slot between upper and lower high-index layers. When the upper layer and the slot are removed (H_u = 0, H_s = 0), the waveguide becomes a strip waveguide. In waveguide bends and micro-resonators, traveling optical mode feels an effective bending radius, R_eff, that is greater than structural bending radius, R₀.

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Simulations are carried out using finite-element-method software (COMSOL) with an element size of 5, 40 and 100 nm for the slot layer, silicon and other regions. Using a full-vector mode solver for circular waveguide bends [18], eigen-frequency f_eig of the guided mode is obtained iteratively at each wavelength, with corresponding R_eff measured from modal distribution [12], and then the effective index n_eff is determined by resonance condition 2πR_eff(f_eig)n_eff(f_eig) f_eig = mc, where m is the azimuthal mode order and c is the speed of light in vacuum. With material dispersion in silicon and silica taken into account using Sellmeier equations as in [19], overall dispersion in the resonator is calculated by taking the second order derivative of the effective index with respect to wavelength, i.e., D = -(c/λ)∙(d²n_eff/dλ²). The accuracy of this numerical model is confirmed by experimental data [12].

We examine the effective bending radius R_eff as a function of wavelength in Fig. 2(a) , for the horizontal polarization state (quasi-TE mode) in the strip waveguide with the geometry parameters given above, as the structural bending radius R₀ is increased from 3 to 16 μm. R_eff is always greater than R₀ and decreases with wavelength. For a smaller R₀, the radial mode shift (R_eff - R₀)/R₀ is larger, and R_eff decreases more quickly with wavelength. At R₀ = 3 μm, (R_eff - R₀) drops from 149 to 84 nm as wavelength is changed from 1300 to 1800 nm. Figure 2(b) shows similar trends for the vertical polarization state (quasi-TM mode). However, the mode shift is larger because the quasi-TM mode has continuity of electric field at the outer sidewall of the strip waveguide. For comparison, at R₀ = 3 μm, (R_eff - R₀) for the quasi-TM mode decreases from 189 to 118 nm as wavelength is changed from 1300 to 1800 nm.

 

Fig. 2 In curved strip waveguides, effective bending radius R_eff, relative to structural bending radius R₀, changes with wavelength, producing additional waveguide dispersion. (a) quasi-TE mode (b) quasi-TM mode. Dispersion curves with a different bending radius R₀ from 1300 to 1800 nm. (c) quasi-TE mode (d) quasi-TM mode. ZDW is shifted, relative to that in straight strip waveguides, by waveguide bending. (e) quasi-TE mode (f) quasi-TM mode.

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Dispersion curves in the waveguide bends are shown in Figs. 2(c) and 2(d). For the quasi-TM mode, the anomalous waveguide dispersion, induced by light confinement, is strong enough to over-balance the normal material dispersion in silicon, which causes a ZDW at a wavelength close to 1300 nm. Since the strip waveguide with a large width produces a relatively loose light confinement in horizontal direction, the quasi-TE mode has a smaller waveguide dispersion compared to the quasi-TM mode and has ZDWs at longer wavelengths. When R₀ is reduced from 4.5 to 2 μm, the dispersion curves are shifted to longer wavelengths for both polarization states. At a given wavelength, dispersion value changes rapidly. For example, at 1600 nm wavelength, the quasi-TE mode has a dispersion value increasing from −77 to 660 ps/(nm·km), while the dispersion changes from 1674 to 2614 ps/(nm·km) for the quasi-TM mode. Such a large dispersion change may greatly affect nonlinear interaction efficiency of optical waves traveling inside a ring resonator [14].

The position of ZDW in a dispersion curve is critical in nonlinear parametric processes. Figures 2(e) and 2(f) show that the ZDW of the curved waveguides shifts relative to that of a straight waveguide, as a function of R₀ for different waveguide widths. With a width of 800 nm, the ZDW of the quasi-TE mode is blue-shifted first as R₀ is reduced from 32 to 8 μm and then rapidly red-shifted by >220 nm, from −47 to 175 nm, as R₀ is reduced from 8 to 2 μm. All results are relative to the ZDW in the straight waveguide at 1448 nm.

This trend can be explained as follows. With a waveguide width of 800 nm, the quasi-TE mode in a straight waveguide has relatively loose confinement. Reducing the bending radius basically causes two effects. First, the modal field is pushed towards the outer sidewall of the waveguide and is slightly squeezed, which is essentially equivalent to a tighter mode confinement compared to that in a straight waveguide. This induces a stronger anomalous waveguide dispersion and makes the ZDW shift to a shorter wavelength. Second, since the field extends to the waveguide cladding, the effective index of the mode becomes smaller, and substrate leakage is thus stronger. Such leakage can produce significant normal waveguide dispersion [19] and reduce the overall dispersion value, which causes a red-shifted ZDW. As R₀ is reduced from 32 to 8 μm, the first effect is stronger, while the second becomes dominant for a further reduced R₀. With smaller waveguide widths of 700 and 600 nm, the ZDW of a straight waveguide is located at 1388 and 1334 nm, respectively, where material dispersion becomes more dominant. The ZDW shift induced by waveguide bending is thus more limited. For the quasi-TM mode, we note from Fig. 2(d) that the ZDW is very close to 1300 nm and it is tolerant to waveguide bending. As shown in Fig. 2(f), the ZDW shift is less than 60 nm.

Bending-induced dispersion change in a horizontal slot waveguide is examined, with the structural parameters given above. Figure 3(a) shows that the change in R_eff is relatively small as compared to the strip waveguides, because of a small waveguide width of 500 nm. At R₀ = 3 μm, (R_eff - R₀) for the slot mode decreases from 44 to 21 nm as wavelength is changed from 1300 to 1800 nm. As a result, dispersion profiles are quite tolerant to waveguide bending in Fig. 3(b). We vary the slot height from 30 to 50 nm, when keeping other parameters the same, and plot the ZDW shift versus R₀ in Fig. 3(c). The ZDW is blue-shifted as R₀ decreases.

 

Fig. 3 Dispersion change induced by waveguide bending in horizontal slot waveguides. (a) effective banding radius, (b) dispersion profile and (c) ZDW shift.

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3. Flattening of in-cavity dispersion

It was shown above that curving waveguides may cause a large change in dispersion inside a cavity, which provides a potentially effective method to shift ZDWs, enabling more freedom in device design, but dispersion flatness does not change significantly. For ultra-wideband nonlinear applications, flat and low in-cavity dispersion over a wide wavelength range would be beneficial to produce equidistant resonance frequencies and increase efficiency of cascaded four wave mixing and frequency comb generation in micro-cavities. We propose a horizontal slot waveguide that forms a cavity with 460 times smaller standard deviation of FSRs than a silicon strip waveguide with the same size and without a slot layer. Structural parameters are: upper silicon height H_u = 286 nm, silica slot height H_s = 32 nm, lower silicon height H_l = 510 nm, and waveguide width W = 500 nm. Quasi-TM mode in the straight slot waveguide has a flat dispersion of ± 15 ps/(nm·km) over a 563-nm wavelength range, from 1566 to 2129 nm. The flattened dispersion results from an anti-crossing effect when the guided mode transits from a strip mode at short wavelength to a slot-like mode at long wavelength, which causes a slightly negative waveguide dispersion [20] to balance the convex dispersion in strip waveguides without a slot structure. Figure 4(a) shows in-cavity dispersion profiles as R₀ is reduced from ∞ to 3 μm. Little change in the dispersion is seen for R₀ >11 μm, but the average dispersion shifts by >100 ps/(nm·km) as R₀ is from 6 to 3 μm, since in this range more optical field extends outwards, which causes strong wavelength dependence of the effective index.

 

Fig. 4 (a) Dispersion inside the cavity is changed as R₀ is reduced from 16 to 3 μm. Both averaged value and flatness of the dispersion profiles are varied. (b) FSR's standard deviation changes rapidly with R₀.

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One can obtain the resonance frequencies that satisfy 2πR_eff(f_m)n_eff(f_m)f_m = mc and also the FSR as a function of frequency. Figure 4(b) shows the standard deviation of the FSRs that changes with R₀ for a single-ring resonator with the proposed slot structure. Due to the flat and low in-cavity dispersion, the FSR is nearly constant with a standard deviation of 0.023 GHz for R₀ = 32 μm from 1566 to 2129 nm wavelength. The averaged FSR is 376.3 GHz, and the averaged resonance linewidth is 2.97 GHz. In contrast, if the slot structure is taken out, a microring resonator (its bending radius is 32 μm) formed by a silicon strip waveguide with a height of 510 nm and a width of 500 nm produces a fast-changing FSR over the same wavelength band. The FSR standard deviation is 10.53 GHz, which is 460 times larger, and the linewidth is 2.79 GHz. In both cases, we assume 2 dB/cm waveguide loss. To illustrate the uniformity of the resonance frequencies, we plot a frequency-domain “eye-diagram”. As shown in Fig. 5 , the frequency axis is discretized with an interval equal to the averaged FSR, and these spectrum pieces are plotted on top of each other. If the FSR changes with frequency, the resonance peaks will be shifted relative to each other in the “eye-diagram”. As shown in Fig. 5, the resonance peaks produced by the strip-waveguide resonator drift by multiple resonance linewidths over a 100-nm bandwidth. The slot-waveguide resonator generates the well-aligned peaks in the “eye-diagrams” over a 563-nm bandwidth. This great improvement of FSR uniformity is enabled by engineering in-cavity dispersion with the proposed slot structure, allowing a large number of equidistant spectral lines determined by cascaded four wave mixing to be on resonance, which would help increase nonlinear efficiency and obtain octave-spanning spectral coverage of a frequency comb generated on a chip.

 

Fig. 5 Spectral response of a single-ring resonator is sliced with a frequency spacing equal to the averaged FSR. All the spectrum pieces are plotted together to show the uniformity of the resonance peaks. Proposed slot-waveguide resonator has well-aligned resonance peaks over a 563-nm wavelength band, in contrast with a strip-waveguide resonator.

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4. Conclusion

We have analyzed the dispersion change induced by tight waveguide bends in high-index-contrast silicon waveguides. Both strip and slot waveguides are simulated with a bending radius as small as a few microns. The dispersion is changed dramatically, which could raise an important issue in the design of micro-resonator devices for cavity-enhanced nonlinear optics applications. By engineering the in-cavity dispersion using a slot structure, we have improved uniformity of resonance peaks in frequency response of a silicon micro-cavity, which would be beneficial to generate a mid-infrared frequency comb on a chip.