Slow‐Light‐Based Dispersion Compensation of High‐Speed Data on a Silicon Nitride Chip

Transmission of optical data over fiber is subjected to signal degradation from dispersion impairments that worsen with increasing fiber lengths and higher data rates. Consequently, dispersion impairments significantly limit the data rates and reaches of transceivers used for data center communications. Herein, the design and experimental demonstration of a low loss, CMOS‐compatible on‐chip silicon nitride transmission grating device for dispersion compensation of high‐speed data are reported. The design, which involves inscribing a perturbation in the effective index profile of the grating, generates a considerable increase in the dispersion magnitude and enables up to 25 km of single‐mode fiber to be compensated for. Importantly, high‐speed measurements after dispersion compensation of a 20 km single‐mode fiber, using non‐return‐to‐zero data at rates up to 28.05 Gb s−1 show nine orders of magnitude reduction in BERs from 5  ×  10−1 to 1  ×  10−10, accompanied by significant improvements in the eye diagram. The underlying operating mechanism involves slow‐light effects that are designed to generate large dispersion while maintaining low losses. A pathway toward dispersion impairment‐free high‐speed data transmission over fiber using a compact device is demonstrated, which can be seamlessly integrated within transceiver chips, bringing the applicability of slow‐light devices firmly into the domain of commercial viability.


Introduction
Intensity-modulated direct detection (IMDD) data such as nonreturn-to-zero (NRZ) encode digital bits into a signal's amplitude level. These modulation formats that utilize direct detection have lower costs and require lower power compared to coherent modulation formats, but are susceptible to degradation from optical fiber dispersion. [1] Globally, transceiver companies serving the intradata center, interdata center, and core/metro markets need to continuously scale their bandwidths to match the burgeoning pace in cloud computing services, e-commerce, supercomputing, and future 5G networks. Multisource agreements (MSAs) among companies allow the alignment of technological strategies and hardware architectures. MSAs including the PSM4, [2] 100G Serial Lambda MSA/Ethernet Alliance, [3] and OpenEye, [4] are scaling up to achieve higher data rates. However, signal degradation from optical fiber dispersion limits the transmission of higher data rates. This problem worsens with longer fiber length. Without proper consideration or management of dispersion, the transmitted data suffer from a dispersion penalty. This is quantified as the power penalty (in dB) due to the closure of the NRZ eye. Left unmitigated, the dispersion impairment causes intersymbol interference leading to high bit error rates (BERs) at the receiver and eventual data loss.
In practice today, dispersion compensation in optical networks utilizes fiber-based or other nonintegrated solutions such as dispersion compensating fiber and dispersion compensating modules, [5][6][7][8] neither of which can be integrated on a chip. Though their ability to overcome dispersion impairments in the transmission of high-speed data has yet to be definitively proven, integrated dispersion-engineered devices have been proposed as a potential solution. Viable approaches for integrated dispersion management could allow significantly higher data rates to be adopted, especially in transceiver products to increase the reaches that they could serve. [9,10] Among the spectrum of integrated photonic devices, gratings are a promising option, having shown themselves to possess advantageous transmission [11][12][13][14][15][16][17][18] and phase properties. [11,12,17] The latter in particular is instrumental to the generation of dispersion. Since dispersion is essentially derived from the second derivative of phase with respect to frequency, minimal ripple is useful to achieve low noise. Minimizing transmission and group delay ripple in turn require proper implementation of apodization, without which strong spectral oscillations will be manifested. [17,18] The ability to tune the operating wavelength of on-chip grating devices may also enhance the number of wavelength channels that can be simultaneously compensated for. Methods such as thermo-optic tuning [19,20] and strain-induced methods [21,22] have been demonstrated. Furthermore, these approaches also allow for the compensation of dynamic factors that change the extent of dispersion experienced along the fiber. Nonoptimal compensation can also lead to residual dispersion in which the tolerable residual dispersion scales inversely with the data rate, and may be exacerbated by environmental factors such as temperature or stress-induced variations. [23] Despite the progress made in engineering dispersion on a chip, to date, none of the aforementioned approaches have demonstrated dispersion compensation of high-speed data. A significant, prohibitive additional drawback of the mechanisms in which dispersion is generated in prior work, is that grating-based devices generating dispersion operate in reflection, inevitably require the use of a circulator, which is in and off itself a tremendous disadvantage since it cannot be easily integrated with silicon-photonics-based transceiver systems. Hence, viable methods for dispersion compensation, which can overcome limitations of existing solutions that possess a fixed or limited magnitude of dispersion, determined by the waveguide geometries, index perturbation, and material are very much needed. With industry trends moving toward on-chip integration, [24] CMOS-compatible solutions which may be integrated within silicon-photonics-based transceivers are of great merit.
In this article, we demonstrated dispersion compensation of high-speed NRZ data using a new type of dispersive grating device, not previously reported for potential use in dispersion compensation, importantly operating in transmission. The dispersive grating is implemented in CMOS-compatible silicon nitride and possesses low losses of À1.7 dB. Using a line rate of 28.05 Gb s À1 , high-speed measurements show that the devices can compensate for the dispersion in a 20 km long fiber, significantly reducing the BERs by nine orders of magnitude from 5 Â 10 À1 to 1 Â 10 À10 and restoration of the eye diagram. We further characterize the dispersion compensation of the grating devices as a function of the magnitude of dispersion provided, unequivocally corroborating that optimal dispersion compensation occurs when the magnitude of dispersion provided by the grating matches that in the 20 km fiber. The effectiveness of the dispersion compensation is further evidenced by high-speed characterization of the eye amplitude, eye height, and jitter. We further documented temperature-dependent measurements of the devices, which showcase the usefulness of thermo-optic tuning to provide a dynamic tuning mechanism for accessing various dispersion magnitudes for various fiber lengths. The low losses, small footprint, CMOS-compatibility, and favorable dispersive properties showcase the viability of this approach toward transceiver-integrated dispersion compensation which may be implemented either in the transmitter or receiver, thus augmenting the usable data rates and/or extending the fiber reaches served by the transceiver. Figure 1 shows the schematic diagram of the two devices reported in this manuscript for dispersion compensation. The effective index modulation of the gratings is created using sinusoidal sidewall modulation. Figure 1a shows a single grating device (SGD), possessing a single grating pitch, Λ 1 of 434 nm. Figure 1b shows an overlaid grating device (OGD). Its design is created by overlaying two gratings with dissimilar pitch (Λ 1 and Λ 2 are 434 and 440 nm, respectively). In both devices, the length (L), average width (W ), and height (H) of the gratings are 4 mm, 1.5, and 800 nm, respectively. A raised cosine apodization is implemented by gradually increasing the sinusoidal sidewall modulation amplitude, ΔW from zero at the ends of the grating to a maximum of ΔW ¼ 150 nm at the center of the grating. The coupling coefficient at the center of the grating where the sidewall modulation amplitude is at its maximum value is calculated to be %20 000 m À1 . Of note, the OGD design is intended to introduce some extent of perturbation to the apodization profile, so as to create regions of higher group delay variation.

Device Design and Optical Properties
We first characterize the optical properties of both devices using an optical vector analyzer. A tapered-lens fiber was coupled to the input and output of the devices under test. A polarization controller was used to optimize for transverse electric (TE) mode propagation in the devices. The grating devices are designed to operate in the transmission mode. We characterize the transmission, group delay ðpsÞ, and the dispersion profile ðps nm À1 Þ of the gratings. The dispersion, D ¼ À2πc where c is the speed of light, λ as the wavelength, and β 2 is the group-velocity dispersion parameter. Figure 1c,d shows the transmission spectra of the SGD and OGD, respectively. The SGD's transmission spectrum shows a single stopband at 1576 nm with minimal ripple in both group delay and transmission. Close to the stopband, the group delay is observed to increase rapidly. The origin of this rapid group delay variation is the interaction between the forward and backward propagating modes in the grating which arises from the artificial stopband and gives rise to a slow-light effect. The reduction in the group velocity of light varies rapidly with wavelength and generates regions of high dispersion. This phenomenon is distinctly different from chirped gratings operating in reflection, which rely on a differential propagation length into the grating which stems from the distributed Bragg wavelengths arising from the linear chirp. By virtue of the underlying mechanism by which dispersion is generated, as well as their operation in transmission, the extinction ratio of the SGD and OGD devices does not impact the dispersive properties; More importantly, a smaller extinction ratio does not lead to more loss and could in fact be advantageous if accessing dispersive regions within the stopband since less light will be reflected. Conversely in chirped gratings operating in reflection, the extinction ratio is critical to ensuring the light undergoing dispersion is reflected instead of being lost to the transmission port.
When transitioning to the OGD, it is observed that while the general profile of the transmission and group delay spectra is similar to the SGD, greater oscillations are introduced. The rapid change in the group delay observed close to the stopband gives rise to large magnitudes of dispersion. These oscillations are intentionally induced by conferring a less effective apodization through the designed overlay of two gratings with different pitch. The effective index mismatch from a perturbation in apodization results in further interactions between the incident and reflected optical fields, thus increasing the group delay. Serendipitously, these additional oscillations create regions of rapidly varying group delay, which equate to regions of high dispersion. Fundamentally, high group delay at the band edges is induced by the interaction between forward and backward propagating waves from the grating periodicity. Oscillations that arise from the perturbation in the apodization result from a less smooth change in the effective index.
We first analyze the dispersive properties of the SGD. As shown in Figure 1c, the group delay increases rapidly as the wavelength approaches both the blue and red sides of the stopband, generating strong dispersion in these regions. This profile has been leveraged for nonlinear optics in on-chip Bragg gratings, [25,26] as well as fiber Bragg gratings, [27] where typically the red (blue) side of the stopband just outside the grating stopband, exhibits normal (anomalous) dispersion. To access normal dispersion to compensate for the anomalous dispersion in single-mode fiber, we require regions where the group delay is decreasing with increasing wavelength. Figure 1c shows two such regions, highlighted in yellow and green. The green region highlights a high normal dispersion region which simultaneously possesses high transmissivity (low loss) since this region is outside of the grating stopband and does not incur insertion losses from the grating's extinction. The magnitude of normal dispersion available in this highlighted region is À175 ps nm À1 , insufficient for dispersion compensation of a 20 km single-mode fiber (SMF). A longer grating length could, however, provide sufficiently large normal dispersion to compensate for dispersion in 20 km of SMF in this wavelength region. Also, referring to Figure 1c, it is observed that there is a second region (highlighted in yellow), where the group delay decreases with increasing wavelength. The magnitude of normal dispersion here is larger, providing the required À320 ps nm À1 for dispersion compensation of 20 km of SMF. Consequently, we utilize the blue side of the grating stopband to access larger normal dispersion values. This region however lies within the stopband of the grating and comes at the expense of slightly higher losses (3.5 vs 1.7 dB) compared to the OGD, at the operating Figure 1. Schematic of the a) single grating device (SGD) and b) overlaid grating device (OGD). c) Measured transmission spectrum (dB) and group delay (ps) measurements for (c) the SGD and d) the OGD. The yellow highlighted regions denote where normal dispersion was accessed for dispersion compensation of 20 km of SMF. The green-highlighted region denotes the region where the grating dispersion and transmissivity are high and dispersion normal, but with a smaller magnitude than the yellow region.
www.advancedsciencenews.com www.adpr-journal.com wavelength. Since the BER is impacted by both insertion loss and dispersion impairments, the improvement from dispersion needs to exceed the marginal losses introduced by operating within the grating stopband. Figure 1d shows the transmission and group delay spectrum for the OGD. Oscillations in the yellow highlighted region are observed. As group delay dispersion is the derivative of the group delay, these oscillations give rise to multiple regions of normal dispersion. A more rapid decrease in the group delay with increasing wavelength will generate a larger dispersion. The normal dispersion magnitudes corresponding to local maxima are À103, À177, and À407 ps nm À1 to À297 ps nm À1 , occurring within a wavelength range of 1.4 nm. This range of dispersion allows for dispersion compensation for multiple fiber lengths in a single device and the required amount of dispersion may be accessed through thermo-optic tuning.

Dispersion Compensation Measurements with High-Speed Data
Next, we perform high-speed experiments using NRZ data to study the effectiveness of dispersion compensation using both the SGD and OGD. Line rates between 8 and 28.05 Gb s À1 were used in the high-speed measurements. A standard SMF has anomalous dispersion of about 16 psðnm⋅kmÞ À1 at 1.55 μm. [28] With the given line rate (B) at wavelength, λ = 1.55 μm and the dispersion (D) of 16 ps nm À1 km À2 , the maximum propagation length ðL CD Þ before chromatic dispersion occurs is given by L CD ¼ πc=8λ 2 jDjB 2 . [23] Hence, a fiber length, which exceeds L CD (4.6 km), needs to be used to demonstrate the dispersion compensation performance of our devices.
In our experiments, the 20 km SMF fiber length is considerably longer than L CD . The compensation of a 20 km SMF requires normal dispersion with a similar magnitude but with an opposite sign ðÀ320 ps nmps nm À1 Þ, that we access at the appropriate regions in the grating devices. Hence, the result at the receiver should ideally have a zero sum by concatenating the accumulated dispersion from the 20 km SMF ðD F L F Þ and the dispersive device ðD D L D Þ, where D F,D and L F,D are the dispersion and length of the fiber and dispersive device respectively, such that The schematic diagram for the high-speed measurements is shown in Figure 2a. A tunable laser is used as the input source before propagation through a polarization controller to ensure that it was tuned for the TE mode. A Mach-Zehnder optical transmitter then modulated the continuous wave laser source with a pseudorandom binary sequence (PRBS) and 31 patterns (2 31 -1 bits) were generated by a pattern generator from the bit error rate tester (BERT). NRZ data signals with a line rate of 25.78125 Gb s À1 were used to generate the eye diagrams. The BERT was used to measure the BERs at several data rates that are also used in commercial small-form pluggable transceivers. Erbium-doped fiber amplifiers (EDFA) were used before and after propagation through the 20 km fiber to ensure that the optical power received at the photoreceiver was similar, to compare the result with and without dispersion compensation. The device under test (DUT) was placed after the 20 km fiber for dispersion compensation. At last, a PIN-TIA photodetector was used to demodulate the optical signals. The generation of the eye diagrams also included a fourth-order Bessel-Thompson filter response with a cutoff frequency of 75% of the data rate at the receiver. This is consistent with receiver measurement requirements as indicated by the Parallel Single Mode 4 (PSM4) agreement and 200G LR4 Open Eye Technical Specification to provide consistent measurement conditions rather than being used as a noise filter. [2,4] The BER was also measured as a function of detuning within the grating stopband by scanning the wavelength at 0.01 nm intervals. The BER values obtained were then averaged across acquisitions over a fixed time scale, after the real-time BER value had stabilized.
High-speed measurements were done to characterize the eye diagrams after propagation of NRZ data through 20 km SMF, with and without dispersion compensation using the SGD and OGD. The comparisons without dispersion compensation and after dispersion compensation using the SGD and OGD are shown in Figure 2b,c, respectively. High-signal degradation from dispersion impairments can be observed in the eye diagrams after propagation through a 20 km SMF. However, there are significant improvements when dispersion compensation is applied using both grating devices, with an eye-opening observed for both grating devices. Note that the dispersion compensation is achieved by operating in the regions highlighted in yellow in Figure 1c,d. Figure 2d shows the BERs measured as a function of data rate. The attainable BERs after dispersion compensation with either grating device are significantly lower compared to the BERs after 20 km fiber without dispersion compensation. As expected, there is a general increase in the BER as data rate increases for the three cases, but there is a steeper increase in the BERs without dispersion compensation (blue triangles) as compared to the SGD (green circles) and OGD (red squares). The BER after dispersion compensation for both grating devices falls within the range of 10 À9 -10 À12 with some variations in the BERs measured. The BERs achieved after dispersion compensation with both devices is similar, which indicate that their effectiveness in dispersion compensation is comparable. In addition, we note that the BER improves significantly after dispersion compensation of the 20 km fiber for both grating devices, by seven and nine orders of magnitude at a data rate of 10 and 28.05 Gb s À1 , respectively. As the operating wavelength to compensate for the dispersion of a 20 km fiber are distinct for both devices, the optical power received at the photodetector is different. The insertion loss incurred from the respective operating wavelength is higher in the SGD as compared to the OGD. Additionally, the gain spectrum of the EDFA is also affected by the operating wavelength. Hence, for consistent qualitative comparisons between eye diagrams after 20 km of fiber and after the dispersion compensation device, both cases have the same received optical power (ROP), controlled using the fiber amplifier. However, we also perform additional characterization to confirm that the improvement in BER is attributed to the effectiveness of dispersion compensation rather than lower losses. Figure 2e shows the measured BER as a function of the optical power received at the photoreceiver at a data rate of NRZ 25.78125 Gb s À1 . The OGD and SGD achieve a significant improvement in the BER compared to when no dispersion www.advancedsciencenews.com www.adpr-journal.com compensation is used. At a BER of 5 Â 10 À5 , the forward error correction (FEC) limit, the SGD requires 4 dBm higher ROP than the OGD. Hence, lower optical power is needed to achieve equivalent BERs in data transmission with the OGD. It is further observed that the BER is lower for the OGD compared to the SGD across the ROP. In the absence of dispersion compensation, the BER improves marginally with increase in the ROP, indicating that better transmission alone does not significantly improve the BER but rather, is contingent on effective dispersion compensation.
The BER is also observed to be lower with increased ROP with dispersion compensation as compared to without dispersion compensation (Figure 2e). This further indicates that the improvement of BERs is influenced more by dispersion compensation rather than from higher transmission. We may also reach this conclusion by considering the additional insertion loss introduced by the gratings. For instance, the additional insertion loss introduced by the OGD is about 1 dB. At an ROP of À6 dBm, the BER without dispersion compensation is 10 À3 . The OGD introduces additional loss of 1 dB, which would equate to an ROP of À7 dBm. At this ROP, the BER is 10 À9 , a significantly lower value. This further confirms that even with the slightly higher insertion loss introduced by the gratings, there is a marked improvement in the BER from the amelioration of the fiber dispersion. Figure 3a,c shows the transmission spectrum at a stopband of the SGD and OGD, respectively. The wavelength range highlighted in yellow corresponds to the dispersion profile, as shown in Figure 3b,d. In Figure 3b,d, the measured BER as a function of wavelength is plotted together with the dispersion profile. It may be observed that for both devices, the BER varies as a function of wavelength. This is attributed to the dispersion provided by the gratings at those specific wavelengths. Approaching the exact compensation of dispersion in 20 km Figure 2. a) Setup used for the high-speed measurements. Measured eye diagrams for NRZ 25.78125 Gb s À1 data after transmission through 20 km of fiber without dispersion compensation (above), and with dispersion compensation (below) using b) the SGD and c) OGD. d) The measured bit error rates (BERs) as a function of the NRZ data rate. e) Bit error rate measured as a function of the received optical power at the photoreceiver.
www.advancedsciencenews.com www.adpr-journal.com of SMF would require a dispersion magnitude of À320 ps nm À1 . From Figure 3b, the lowest BER obtained by the SGD coincides with the amount needed to compensate for a 20 km SMF. As the magnitude of normal dispersion decreases beyond this value, it is observed that the BER increases. This is also seen in the OGD at 1578.5 nm (Figure 3d), where the BER is about 10 À3 in the region of anomalous dispersion. In the range of 1578.5À1578.55 nm, the BER reaches a local minimum before increasing again as the extent of normal dispersion exceeds À320 ps nm À1 . Consequently, deviations from this value would result in improvements in BERs but to a lesser extent. Even as insertion loss increases, the BER reduces as the amount of dispersion approaches À320 ps nm À1 . This indicates that for both devices, the improvement in the BER is not significantly influenced by better transmission but is dominated by the effectiveness of the dispersion compensation. This range of dispersion can be appropriately harnessed in the future when combined with thermo-optic tuning to provide dynamic dispersion compensation. The positive thermo-optic coefficient in silicon nitride would cause a red shift in both the transmission and group delay spectrum of the grating when subject to an increase in temperature, thus allowing access to specific dispersion magnitudes. As may be observed from the dispersion profiles plotted in Figure 3b,d above and corresponding to the respective transmission spectra, the insertion loss to access a normal dispersion of À320 ps nm À1 is different for both devices. For the SGD, the operating wavelength to compensate for 20 km SMF is at about 1576.48 nm, resulting in the lowest BER of about 3.4 Â 10 À9 which comes with an insertion loss of À3.5 dB. For the OGD, the lowest BER of 2 Â 10 À9 is achieved when the optical data is tuned to an operating wavelength of about 1578.55 nm, resulting in an insertion loss of À1.7 dB. Hence, dispersion compensation with low BER can be achieved with lower insertion loss in the OGD. Figure 4 shows eye diagram parameters, including the eye amplitude, eye height, and jitter, measured as a function of wavelength within the yellow region highlighted in Figure 3a,c (0.01 nm intervals). These are normalized such that values obtained are compared to the maximum value for the corresponding parameter. The dotted vertical line indicates the point where the dispersion compensation is optimized such that the eye height is at its maximum.
The eye amplitude is measured using the photoreceiver. For the SGD, the eye amplitude (cyan squares) in Figure 4a is shown to be descending along the wavelength range scanned. This is also consistent with the transmission spectrum shown in Figure 3a where insertion loss increases as wavelength increases in the blue side of the stopband. In contrast to the eye amplitude for the OGD, as shown in Figure 4b, the eye amplitude maintains an almost constant value across the wavelength range and this is www.advancedsciencenews.com www.adpr-journal.com also consistent with Figure 3c where low insertion loss is maintained across the two dips. The eye height (magenta circles) measures the extent of the eye opening and it reaches a maximum at the operating wavelength (shown as dotted black lines) with optimal dispersion compensation. The operating wavelengths with maximum eye height for both devices are similar to the operating wavelength with the lowest BERs measured in Figure 3b,d. For the SGD, the eye height decreases, resulting in a gradual eye closing away from the operating wavelength. For the OGD, the maximum value of the eye height coincides with the dispersion curve that has the lowest BER value at 1578.54 nm as shown in Figure 3d. The eye height decreases thereafter and increases again at about 1578.73 nm, showing the same result as the BER measurement. In contrast to the SGD, the eye opening is maintained across the wavelength range for the OGD.
The amount of jitter has an inverse relationship with the eye height such that it converges to a minimum value near the optimal dispersion compensation wavelength. This is observed in Figure 4a,b. However, similar to the eye height parameter, there is a greater variance between the maximum and minimum normalized value with the SGD as compared to the OGD. There is slightly higher jitter in the OGD, as may also be observed in the eye diagram shown in Figure 2b.
In Figure 4c, the dispersion profiles (green-SGD and red-OGD) are plotted with the transmission spectrum. The dispersion profiles (dotted lines) and transmission spectra (single lines) are plotted as a function of normalized wavelength such that the maximum normal dispersion value (λ D¼min Þ for the respective devices is aligned. With a normal dispersion of À320 ps nm À1 , the insertion loss for the OGD is lower than the SGD. Furthermore, the dispersion gradient for the OGD (red dotted lines) is steeper than the SGD (green dotted lines). Hence, the power required to thermo-optically tune the operating wavelength would be smaller with the OGD as compared to the SGD. At last, we plot the various dispersion regions as a function of wavelength detuning from their zero dispersion value (λ D¼0 Þ in Figure 4d. The colored (icon) labels of the plots correspond with their colored (icon) labels shown in the inset of Figure 4d, where it is observed that dispersion profiles with varying gradients exist. As expected, the bandwidth has a tradeoff with the magnitude of normal dispersion accessible in each region.
We note that the extent of the BER and eye opening is dependent on the amount of dispersion compensation and the insertion loss. Although a small deviation from the operating wavelength in the OGD results in a greater change in dispersion compared to insertion loss, we are operating at a low-loss regime, www.advancedsciencenews.com www.adpr-journal.com and hence achieve slightly lower BER compared to the SGD. This is also highlighted in Figure 4a where the SGD is observed to exhibit a greater variance in the maximum and minimum values compared to the OGD; In the SGD, the eye closes as the wavelength deviates away from the optimal operating wavelength, whereas the eye opening is maintained for the OGD. This difference originates from the operating regimes: the SGD requires operation within the grating stopband to achieve the required magnitude of normal dispersion, whereas the OGD operates at the band edge where the transmission is high. The functionality of the devices is enhanced with thermo-optic tuning, where the positive thermo-optic coefficient of silicon nitride results in a red shift in the grating stopband with an increase in temperature. The devices were thermally tuned with a thermoelectric controller from 20 to 70°C and the devices were placed on a Peltier module. As shown in Figure 5a,b for the SGD and OGD devices, respectively, the group delay spectrum has a red shift as temperature increases. The wavelength shift measured was 34 pm°C À1 for both devices and the total wavelength shift from 20 to 70°C was 1.7 nm. With thermal tuning, it allows for tuning of the dispersion magnitude within the dynamic range provided by the devices as well as accessing the required normal dispersion for compensation at a wavelength channel. As shown in Figure 5c, marginal thermo-optic tuning of the device (AE0.5°C) enables the access of the full range of dispersion (anomalous to normal dispersion), with the OGD requiring a lesser amount of temperature tuning as compared to the SGD to access the normal dispersion required for dispersion compensation.
At last, the effectiveness of dispersion compensation with a different fiber length is demonstrated to showcase the ability of the device to compensate for the dispersion in various fiber lengths within the device's maximum dispersion magnitude. In Figure 5d, high-speed measurements were done with a similar setup as shown in Figure 2a. The BER was measured as a function of data rates from 8.5 to 28.05 Gb s À1 after the data propagated through a 6 km SMF. It may be observed that there was a pronounced improvement in the BER when there was dispersion compensation from the SGD and OGD, compared to without dispersion compensation. The BER measured at a data rate of 28.05 Gb s À1 with and without dispersion compensation were 10 À11 and 10 À5 , respectively. This indicates that dispersion compensation improved the BER by six orders of magnitude. Also, as shown in Figure 5e, the BER measured after dispersion compensation with the OGD at various operating wavelengths indicates near error-free data transmission. With data transmission of 25.78125 Gb s À1 through a 6 km SMF, the normal dispersion required for dispersion compensation is about À96 ps nm À1 and the BERs measured after dispersion compensation fall within 3 to 4 Â 10 À12 . These results demonstrate that effective dispersion compensation is provided by the SGD and OGD for fiber lengths within their dynamic range of normal dispersion. Figure 5. Differential group delay (ps) with thermo-optic tuning from 20 to 70°C for the a) SGD and b) OGD. c) Dispersion as a function of temperature tuning for both devices. d) Measured bit error rate as a function of data rate (Gb s À1 ) after a 6 km SMF without dispersion compensation (blue triangles) and with the SGD (green circles) and OGD (red squares). e) Bit error rate measured at various operating wavelengths after dispersion compensation with the OGD with 25.78125 Gb s À1 through a 6 km SMF.

Discussion and Conclusion
Our high-speed experiments show that the SGD and OGD significantly reduce dispersion impairments from a 20 km SMF by nine orders of magnitude. It is further shown that the OGD provides a range of accessible dispersion values that could be useful for dynamic dispersion compensation. In the OGD, oscillations in the group delay spectrum arise from the designed perturbation in apodization, leading to a doubling in the magnitude of normal dispersion that may be accessed in the ultra-low-loss regime. Fortuitously, these rapid oscillations also generate additional regions of high normal dispersion of varying magnitudes, thus enhancing the ability of the OGD to compensate for dispersion in different fiber lengths, potentially simplifying the deployment of dispersion compensation systems in optical communications systems. Today's transceivers serve varying fiber reaches for applications spanning from intradata center to long-haul communications systems. [9] For example, the Open Eye Multisource Agreement (MSA)'s 200G-LR4, [4] PSM4, [2] and CWDM4 100G, [29] transceivers serve fiber reaches of up to 2 km whereas the Open Eye MSA's 200G LR4 specifies a 10 km reach. The appropriate magnitude of dispersion to compensate for dispersion in these different fiber reaches may be accessed by operating at the appropriate Bragg-detuning or designing the Bragg wavelength to coincide with the transmitter wavelength(s). Tunability in the magnitude of dispersion may also help mitigate effects temperature changes or fiber-stress factors. As data centers are sensitive to cost and power consumption, the amount of power needed for tunability is an important factor. The dispersion profile of the OGD may be varied with small changes in wavelength. This leads to a smaller degree of thermo-optic tuning or power required to tune the device to access the required dispersion.
At last, we note that silicon nitride Bragg gratings have been proposed for generating dispersion, but their use in compensation of high-speed data has not been successfully demonstrated. In addition, to compensate for a relatively long fiber of 20 km, the device must have a mechanism that enables large accessible normal dispersion while retaining low losses. The study by Choi et al., [30] Sahin et al., [31] Callahan et al., [32] and Xiang and Fu [33] reported chirped gratings, which provide dispersion that scales with the grating length. The underlying mechanism on which these gratings generate dispersion lies in the Bragg wavelengths being linearly distributed over the grating. This leads to different propagation distances into the grating for different wavelengths of light prior to them being reflected. Such gratings have two main drawbacks: 1) The gratings reported in Refs. [31][32][33] operate in reflection mode, and a circulator is needed. Circulators are generally not possible to integrate into silicon-photonics-based transceiver hardware. Aside from a circulator, the alternative is to use directional couplers, which would incur a 6 dB loss penalty since the light has to enter and exit upon reflection. 2) The magnitude of dispersion generated is considerably smaller than the transmission gratings we introduce here. Xiang and Fu, [33] for example, demonstrated a grating 13.8 cm in length with a dispersion of À156.5 ps nm À1 . Conversely, our gratings operate on slow light, which is a fundamentally different mechanism and allows larger magnitudes of normal dispersion with lengths two orders of magnitude smaller. Considering the length normalized dispersion provided by our devices compared to the state of the art, the OGD provides a dispersion of 8 Â 10 7 ps nm À1 km À2 , whereas the study by Callahan et al. [32] provides 1.1 Â 10 6 ps nm À1 km À2 , 70Â smaller. This difference lies in the distinct mechanism on which our gratings generate dispersion, that is, the interaction of forward and backward propagating modes arising from the artificial stopband, generating slow light with the extent of deceleration varying rapidly with wavelength.
We have demonstrated the use of two silicon nitride Bragg grating designs operating in transmission to successfully compensate for dispersion in 6 km and 20 km of single-mode fiber. A significant improvement in the eye diagram and the BERs at various data rates up to 28.05 Gb s À1 is demonstrated from the high-speed measurements when the gratings are used to compensate for the fiber dispersion. Additionally, we designed an OGD that has additional advantageous dispersion compensation features as compared to a SGD. The OGD utilizes an inscribed perturbation in the apodization to generate a twofold increase in the dispersion magnitude in the ultra-low-loss regime, and simultaneously creates a range of dispersion values, which may facilitate compensation of varying fiber lengths. We note further that the devices are ultra-low-loss, with a ninefold reduction in BER achieved using the OGD with an insertion loss of only À1.7 dB. Our work represents the first demonstration of a CMOS-compatible, slow-light-based transmission grating for dispersion compensation of high-speed data, at rates used in commercial products. The significance of this demonstration could pave the way for integrated on-chip dispersion compensation at either the transceiver's transmitter or receiver, potentially enabling far greater baud rates to be used and extend the reaches of existing transceivers.