We propose a slot-waveguide with high dispersion, in which a slot waveguide is coupled to a strip waveguide. A negative dispersion of up to -181520 ps/nm/km is obtained due to a strong interaction of the slot and strip modes. A flat and large dispersion is achievable by cascading the dispersive slot-waveguides with varied waveguide thickness or width for dispersion compensation and signal processing applications. We show -31300 ps/nm/km dispersion over 147-nm bandwidth with < 1 % variance.
© 2009 Optical Society of America
There is a potential advantage to achieve various on-chip optical functions. One of the basic issues in optical fiber communications is combating chromatic dispersion. Dispersion compensating fibers (DCFs) tend to be quite long and have a large bending radius. Although fiber Bragg gratings are much smaller, the total dispersion is somewhat traded off with bandwidth [1–3]. For high-speed signals, the deployed DCFs may not provide accurate compensation, which would raise the need for integrated receiver-based compensation for the residual dispersion that may be performed on a per-channel basis. A laudable goal would be to design on-chip dispersion compensators that are capable of producing a large and tailorable dispersion . Moreover, the dispersive elements could also be essential in various signal-processing functions, such as analog-to-digital conversion , tunable delays [6–7] and optical correlation . Reports of integrated dispersion elements include: (i) ring resonators that have bandwidths on the order of tens of GHz [9, 10], (ii) wideband silicon waveguides that have dispersions of ~4000 ps/nm/km [11, 12], and (iii) sidewall-modulated Bragg gratings that exhibit a large dispersion and good bandwidth in reflection port .
Recently, slot waveguides have been proposed  for on-chip applications, exhibiting enhanced birefringence , nonlinearity  as well as improved modulation efficiency . A slotted structure could provide some design freedom to tailor the waveguide dispersion . A potential application that would make use of the slot waveguides would be to form on-chip highly dispersive elements with a large dispersion-bandwidth product.
In this paper, we propose a dispersive slot waveguide based on a strongly wavelength-dependent coupling between a slot waveguide and a strip waveguide. On a SOI platform, the two waveguides are vertically coupled, producing a strong negative dispersion due to an anti-crossing of the two modes [19, 20]. Simulations show that a negative dispersion of up to -181520 ps/nm/km is obtained. Dispersion compensation for a 320 Gb/s RZ signal has an eye-opening penalty of ~4 dB after 10.87-km single-mode fiber transmission. The dispersion bandwidth is greatly increased to147 nm for a dispersion of -31300 ps/nm/km, with a <1% variance, which corresponds to a 6.3-ns tunable delay given by a 1-m-long slot waveguide.
2. Narrow-band highly dispersive slot waveguide
The proposed structure consists of a strip waveguide and a slot waveguide that are vertically coupled to each other. The effective index of quasi-TM mode (vertically polarized) in the strip waveguide decreases with wavelength faster than that in the slot waveguide, and a strong mode coupling occurs around a certain wavelength where the effective indices are close to each other, as conceptually shown in Fig. 1. At the crossing-point, formed composite modes, including symmetric and anti-symmetric modes, experience a sharp transition of mode shape from short to long wavelength in Fig. 1, which induces a high dispersion [19, 20]. The strip waveguide with a thickness of 255 nm and width of 500 nm is placed on the top of the slot waveguide, separated by a 500-nm-thick silica base layer. The low-index slot is a 40-nm silica layer, surrounded by two 160-nm silicon layers. Such a vertical placement allows for more accurate control of the structural parameters during fabrication and reduced propagation loss to several dB/cm . Since the symmetric mode is used, we have to design a mode converter to excite it . At crossing wavelength 1.489-μm, the index difference of the symmetric and anti-symmetric modes equals 0.019, which produces the coupling length from a slot mode to a strip mode for 100% coupling of 39.2 μm. To excite the symmetric mode, ~50% coupling is needed, and mode converter length is ~20 microns. We use a mode solver based on finite-element algorithm to obtain effective index with an element size equal to 3, 5, and 15 nm for slot, silicon and other parts. Material dispersion is considered using Sellmeier equations for both silicon and silica, and dispersion profiles are calculated from the effective index .
At the crossing wavelength 1.489-μm shown in Fig. 2(a), the symmetric mode has -181520 ps/nm/km dispersion. It is obtained over a relatively small wavelength range of 3.5 nm for a 1% dispersion variation. The anti-symmetric mode has almost the same amount of positive dispersion. Keeping all other parameters the same, we show in Fig. 2(b) that an increment in slot thickness (ST) shifts the dispersion profile towards a longer wavelength, and the symmetric mode becomes less dispersive. Fig. 3(a) shows that dispersion peak wavelength almost linearly shifts by 77 nm, as the slot increases from 40 to 60 nm. Accordingly, the dispersion value decreases by 38%. This can be attributed to the fact that the increased ST lowers the effective index of the slot mode and red-shifts the crossing point, where the indices of the two guided modes have closer slopes over wavelength. This causes a less dispersive symmetric mode. The silicon layers surrounding the slot also modify the dispersion properties of the proposed structure. In Fig. 3(b), an increment (from 150 to 170 nm, with a 56-nm slot) in the thickness of the silicon layers blue-shifts the dispersion by 103 nm and makes dispersion more negative, from -79683 to -182800 ps/nm/km. It is important to mention that the dispersion profile becomes wider, while the symmetric mode is less dispersive.
Although changing the ST or silicon-layer thickness can slightly modify the peak value of dispersion, it might be desirable to dramatically change dispersion for various applications. The thickness of the silica base between the strip and slot waveguides plays a critical role in tuning the dispersion while almost keeping the dispersion peak wavelength. In Fig. 4(a), the dispersion decreases greatly from -181520 to -28473 ps/nm/km, as the base thickness varies from 500 to 200 nm. To keep the same peak wavelength, one has to change the thickness of the strip waveguide to 255, 255.5, 257 and 259 nm for the silica base of 500, 400, 300 and 200 nm, respectively. As the base decreases, effective indices of the slot and strip waveguides increase, causing a small shift of the crossing wavelength. The strip thickness is thus changed to balance this effect. We note a trade-off  between peak dispersion and dispersion’s full width at half maximum (FWHM), from Fig. 4(b). The dispersion’s FWHM drops from 177 to 26 nm as the waveguide is more dispersive. A thicker base helps separate two modes well, and thus the strong interaction between them occurs at a smaller wavelength band where their effective indices are very close to each other. The symmetric mode thus experiences a sharper transition from the strip mode to slot mode and higher dispersion.
It may be desirable to design a dispersion-shifted profile without changing its peak value and bandwidth. One can achieve this by thickening both the slot and the silica base. A thickened slot makes the dispersion profile red-shifted but reduces its peak value as well, while a larger silica base makes the dispersion deeper and mostly keeps its peak wavelength. In Fig. 5(a), the dispersion profile is shifted relative to the original 40-nm slot by 37 and 77 nm, with ST changed to 48 and 60 nm, respectively. The base thicknesses are 500, 555 and 606 nm, and the maximum dispersion values are -181520, -182585, and -182028 ps/nm/km, respectively. Fig. 5(b) shows that shrinking the waveguide width from 500 to 200 nm can also vary the dispersion, but it is less sensitive to the width change as compared to changing the vertical dimensions, since horizontal light confinement for the y-polarization state is not as tight as in the vertical direction for the 500-nm width.
We simulate the dispersion compensation of high-speed RZ on-off-keying signals after 10.87-km single-mode fiber transmission, using a 1-m-long waveguide. Data rate ranges from 160 to 400 Gb/s. The waveguide width is 500 nm, and the strip waveguide thickness is 255 nm. The slot is 48-nm thick, surrounded by two 160-nm silicon layers. The silica base layer is 555 nm. Signal carrier is aligned to the peak dispersion wavelength, where the single mode fiber has a dispersion of 14.4 ps/nm/km. In Fig. 6, the dispersion compensator works for the 160 Gb/s signal with almost no penalty. However, with the increased bit rate, the signals suffer from the third-order dispersion in the fibers, which would not be compensated by the waveguide in this case. An eye-opening penalty of 4 dB is induced to a 320 Gb/s RZ signal, relative to the case at 160 Gb/s.
3. Dispersion-flattening slot waveguide
Broadband and strong dispersion could be useful in both telecom systems and optical signal processing, e.g., for achieving multi-channel dispersion compensation or a tunable optical delay [6–7]. Dispersion can be flattened by a cascade of the waveguide sections with the modified structural parameter, each with a shifted dispersion profile. As an example, Fig. 7(a) shows that the strip waveguide has a slightly tailored waveguide thickness (WT) by depositing silicon, and the dispersion curve shifts over wavelength, as shown in Fig. 7(b). Similar trend can be seen in Fig. 2(b) and Fig. 5(b) as other structural parameters are changed. The length of each modified waveguide section is calculated by solving the following linear equations:
where Di(λ) (i=1, 2, …, n) is the dispersion profile of the i th waveguide section; D0 is the desired dispersion value after flattening. Length coefficients c1, c2, … , and cn are solved to determine the length ratio of each modified section to the total waveguide. This forms a dispersion profile with n dispersion values clamped to D0 at wavelengths λ1, λ2, … , and λn.
The waveguide width is 500 nm. The slot is 40-nm thick and surrounded by two 150-nm-thick silicon layers. The silica base is 500 nm. There are six waveguide sections. Figure 7(b) shows the shifted dispersion profiles when WT are 246, 249, 252, 255, 259 and 265 nm, respectively. It is important to choose the WT values according to the clamping wavelengths to make the overall dispersion as flat as possible. The length ratios are 15%, 11%, 12%, 12%, 17% and 33% accordingly. In Fig. 7(c), a dispersion of -31300 ps/nm/km is obtained over 147 nm bandwidth, with a variance of 305.5 ps/nm/km, <1% of the mean dispersion. Such dispersive media could be used to introduce a tunable optical delay of 6.3 ns/m for on-chip signal processing by converting wavelength from 1450 to 1680 nm, as shown in Fig. 7(c).
Flat dispersion can also be obtained by varying the waveguide width. With the same vertical dimensions of the slot-waveguide and silica base, we keep WT=255 nm and vary the width from 565 to 500, 445, 390 and 340 nm, and flat dispersion of -46100 ps/nm/km is from 1473 to 1564 nm. The length ratios are 26%, 17%, 17%, 7% and 33%, respectively, with a variance is 623 ps/nm/km, <1.4% of the mean dispersion. Varying the waveguide width is more fabrication friendly and can be easily realized by lithography .
The dispersion bandwidth is extendable if adding more sections. The overall dispersion can be tailored in other ways, e.g., changing with wavelength linearly when the structural parameters are chosen appropriately. Nevertheless, it is still challenging in current fabrication technology to make signal loss in a long waveguide (a few meters) acceptably low. It might be inevitable to adopt optical amplification  in the waveguides to mitigate this problem.
We have proposed the highly dispersive slot waveguides that can be used for telecom and on-chip signal processing applications. Negative dispersion of -181520 ps/nm/km is introduced by the strong interaction between the slot and strip modes in the waveguide coupler. We have shown that a dispersion-flattening slot waveguide can be designed by cascading a series of the waveguide sections with a slightly modified strip waveguide thickness or waveguide width. -31300 ps/nm/km is obtained over 147-nm wavelength range, with a <1% variance.
The authors would thank Prof. Michal Lipson, Jacob Levy and Ling Lu for helpful discussions. This work is sponsored by DARPA (under contract numbers FA8650-08-1-7820 and N66001-08-1-2058) and HP Laboratories.
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