1. INTRODUCTION

Due to the exponential growth of computing needs for artificial intelligence and machine learning, the energy consumption of data centers has risen to levels that can significantly impact the environment [1]. It has been established that copackaged optical interconnects are required for data centers and high-performance computing systems to keep up with increasing bandwidth demands in an energy efficient manner [2]. Integrated Kerr frequency combs—when used in combination with wavelength-selective modulators and filters—present an attractive solution for driving massively parallel intensity-modulated direct-detection (IM-DD) dense wavelength-division multiplexing (DWDM) links compared with other platforms that rely on distributed feedback (DFB) laser arrays and supermode laser diodes, which have fundamental limitations on the number of wavelength channels they can produce [3–5]. However, achieving sufficient comb line power and conversion efficiency for line spacings ${\lt}100\;{\text{GHz}}$ remains challenging. Consequently, there is a practical lower limit on the achievable channel spacing in a Kerr comb laser [6,7]. As the maximum micro-resonator free spectral range (FSR) is limited by practical design constraints such as excess bend radiation losses [8,9], many proposed DWDM links are limited in their ability to scale in the wavelength dimension due to the required optical bandwidth and typically employ techniques such as mode-division multiplexing (MDM) to achieve  $\ge 1 \;{\text{Tb/s}}$ operation [10,11].

To address this challenge, the authors have recently demonstrated a scalable link architecture [12,13] that can significantly increase the achievable bandwidth density in Kerr comb-driven interconnects [Fig. 1(a)]. A key innovation of this architecture is the use of the multi-FSR regime to design the resonant modulators and filters. The standard approach for designing micro-resonators in DWDM systems, hereafter referred to as the single-FSR regime, requires that the micro-resonator FSR be larger than the total optical bandwidth of the link to avoid alias resonance-induced crosstalk on any of the target resonances [Fig. 1(b)]. The multi-FSR regime, by contrast, is a technique for avoiding aliasing using micro-resonators with FSRs smaller than the total optical bandwidth of the link [Fig. 1(c)]. Previous studies on the transmitter have shown that this regime allows designers to guarantee a minimum ${\lambda _{{\text{ag}}}}$ using resonator FSRs smaller than the total optical bandwidth, unlocking a path toward designing ultra-broadband links with Kerr comb sources [12,14].

 

Fig. 1. (a) Overview of the entire wavelength scalable link architecture. The link design is comprised of (i) group velocity dispersion (GVD) Kerr comb, (ii) and (v) de-interleaving stages, (iii) resonant modulators, (iv) interleavers, and (vi) resonant filter and photodiodes. (b) Single-FSR regime design versus (c) multi-FSR regime design. The channel and alias aggressor spacings are denoted by ${\lambda _{{\text{ch}}}}$ and ${\lambda _{{\text{ag}}}}$, respectively.

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Breaking the FSR limit, however, comes at the cost of reducing ${\lambda _{{\text{ag}}}}$, which increases signal degradation due to both loss and crosstalk [15]. The degradation in performance can be compensated by either increasing the power of the source or reducing the per-channel data rate. As Kerr comb sources rely on careful tuning to achieve and maintain the comb state, the output power per line cannot be simply increased as is possible in the case of a DFB array. However, without knowing the severity of this reduction in the maximum channel data rate, it is unclear how much the multi-FSR regime can increase aggregate bandwidth (${{\text{BW}}_{{\text{agg}}}}$) on a single DWDM channel without also requiring a larger link power budget. Further scaling of ${{\text{BW}}_{{\text{agg}}}}$ can be achieved with use of (de-)interleavers—devices that can reduce the number of micro-ring resonators (MRRs) that must be cascaded on a single bus by spectrally separating or combining groups of comb lines into different bus waveguides [16]. However, this can introduce additional loss, which must be compensated for in the link power budget, along with the accompanying power consumption and control complexity. It is of great interest, therefore, to define how much further the multi-FSR regime allows scaling in the wavelength dimension without necessitating an impractical number of interleaver stages.

In this work, we demonstrate the design of a multi-Tb/s Kerr comb-driven photonic link using the multi-FSR regime. Assuming the link uses two (de)-interleaving stages, we show that operating in the multi-FSR regime can achieve a $1.94 \times$—$3.28 \times$ increase in DWDM channel capacity over equivalent single-FSR links, supporting a ${{\text{BW}}_{{\text{agg}}}}$ of $2.80 \;{\text{Tb/s}}$. Comprehensive wafer-scale analysis of 704 micro-resonators fabricated in a commercial complementary metal–oxide–semiconductor (CMOS) foundry shows that the FSR of our resonator design has an expected standard deviation of 14.63 GHz, which can support a $\ge 6\sigma$ yield. Higher aggregate bandwidths of 4.72 Tb/s and 3.76 Tb/s are shown to be possible if yield targets are relaxed ($1\sigma$ and $3\sigma$, respectively). If more FSR robust filter designs are implemented, a ${{\text{BW}}_{{\text{agg}}}}$ of 10.51 Tb/s is achievable. BER measurements validate that the expected drop in the maximum achievable per-channel data rate does not prevent the multi-FSR regime from providing significant gains in link ${{\text{BW}}_{{\text{agg}}}}$. Consequently, co-optimization of the resonator design, data rate, and ${\lambda _{{\text{ag}}}}$ limits the additional power penalties to achieve these bandwidths to just that of the off-resonance insertion (${{\text{IL}}_{{\text{off}}}}$) and routing losses. Finally, link-level implications on the transmitter requirements are discussed and analyzed. This work demonstrates the multi-FSR regime as a key strategy in achieving production-ready multi-Tb/s Kerr comb-driven optical interconnects that can support next-generation data centers and high-performance computing.

2. MICRO-RESONATOR DESIGN CONSIDERATIONS

In our scalable link architecture, well-designed resonant modulators and filters [17] are fundamental to successful execution and operation. The resonator extinction ratio, IL, full-width at half-maximum (FWHM), and FSR must all be balanced while ensuring compatibility with high-volume production at a commercial CMOS foundry. As the nominal and statistical variation of the resonator FSR determines the severity of the channel count versus ${\lambda _{{\text{ag}}}}$ trade-off in the multi-FSR regime, we now discuss the loss, reliability, and dispersion performance of FSRs for several micro-resonator designs.

 

Fig. 2. (a) Micrograph of a micro-disk test structure. (b) Micrograph of an ellipsoid micro-ring test structure. (c) Picture of 300 mm wafer fabricated with AIM Photonics. All the test structures measured were tested across all 64 reticles of the wafer. (d) Sample optical sweep of a micro-disk spectrum. (e) Statistical variation data taken from the wafer-scale measurements.

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A. Off-Resonance Insertion Loss

To achieve the highest possible bandwidth density in these links, it is necessary to maximize the number of channels on a single bus. As discussed in [18], this maximum channel count, represented by ${N_\lambda}$, is limited by two key factors of the MRR: the FSR and ${{\text{IL}}_{{\text{off}}}}$. The limit can be calculated as follows:

For most ultra-wide FSR resonators, achieving the required coupling strength to simultaneously meet the necessary FWHM and FSR for a target data rate is performed via phase-matched directional couplers. The optical mode needs to pass through a pair of S-bends in the through waveguide [17–19]. By matching the resonator curvature and engineering the width along the coupling path to optimally excite the resonator’s fundamental mode, the target coupling strength can be achieved. However, the engineered S-bends introduce significant IL, becoming even more significant with greater FSR due to the reduced bend radii [20]. This results in an ${{\text{IL}}_{{\text{off}}}}$ of approximately 0.1 dB for phase-matched coupling to a ${\sim}25\; {\text{nm}}$ FSR resonator [18]. In an effort to minimize ${{\text{IL}}_{{\text{off}}}}$ for wide-FSR resonators, recent research has explored constructing a MRR from a pair of 180° modified hybrid Euler bends instead of a standard radial bend [18,21,22]. In using this approach, one can achieve an ${{\text{IL}}_{{\text{off}}}} \le 0.02\;{\text{dB}}$ while maintaining a $\text{FSR} \gt 40\; \text{nm}$ [18]. This ultra-low ${{\text{IL}}_{{\text{off}}}}$ allows cascading of 50 to 100 resonators while only requiring an additional 1–2 dBm in the permitted link power budget, making quasi-ellipsoid rings particularly attractive as a link filter.

B. Statistical Characterization of FSR Robustness

To assess the scalability of the multi-FSR regime, it must be considered that the fabricated FSR of a micro-resonator will deviate from the target design. As any deviations from the target FSR will result in a reduced worst-case ${\lambda _{{\text{ag}}}}$, all practical sources of FSR errors must be included for rigorous design space characterization. For this work, two sources of FSR errors are considered: (i) micro-resonator dispersion, which will alter the device’s FSR across the link’s total optical bandwidth, and (ii) process variations, which stochastically change the FSR of each micro-resonator due to geometric variations from imperfect fabrication. To quantify each source of FSR error, optical sweeps were performed on micro-disk test structures fabricated through AIM Photonics, as shown in Fig. 2(a). The devices were tested using a ficonTEC TL1200 Wafer Level Tester (WLT), enabling wafer-scale measurement [Fig. 2(b)]. The optical response of 10 micro-resonators per reticle for all 64 reticles is measured, for a total of 704 measurements. The sweeps ranged from 1450 to 1650 nm at a 5 pm resolution using a tunable laser source (Keysight 8164B) and power meter (Keysight N7744A) [Fig. 2(c)]. Reference structures were measured on each reticle to remove the optical response of the grating couplers from the final sweep [Fig. 2(d)].

From these sweeps, the statistical variation of each resonator’s FSR and dispersion was measured and analyzed using the linear regression technique presented in [23]. Seven different micro-resonator designs were measured and analyzed: six micro-disks with radii ranging from 2 to 4.5 µm, as well as an ellipsoid MRR with an equivalent radius of $\approx 2.5\;{{\unicode{x00B5}\text{m}}}$ [Fig. 2(d)]. Outliers more than 1.5 inter-quartile ranges above and below 75% and 25% of the data, respectively, were removed, with all device sample sizes being $\ge 51$. The variation of the micro-disks’ FSR was found to be roughly proportional to the radius of the resonator being examined, with larger radii experiencing lower sensitivity to variations compared to smaller ones [Fig. 3(a)]. This can be explained by the increased confinement of the whispering gallery mode (WGM) as the radius is increased, resulting in less sensitivity to variations. The FSR standard deviation eventually reaches a floor of around 47 GHz. The aforementioned ellipsoid MRRs, on the other hand, showed much higher robustness to fabrication variations. The ellipsoid MRR FSR variance ${\sigma _{{\text{FSR}}}}$ from 100 samples was found to be 14.63 GHz. To construct a confidence interval of the FSR standard deviation of the measured micro-resonators, we use the commonly employed chi-squared (${\chi ^2}$) distribution. We found the ${\chi ^2}\, 99\%$ confidence interval of the true FSR standard deviation to range from 12.34 to 17.87 GHz.

 

Fig. 3. (a) Plot of the measured free spectral range (FSR) standard deviation (${\sigma _{{\text{FSR}}}}$) as a function of resonator radius. The performance of the nonheater disk and ellipsoid ring are nearly identical, seen by their data points nearly overlapping. (b) Optical spectra of the 2.5 µm disk with and without the doped heater. (c) Plot of measured dispersion (defined as change in FSR w.r.t. optical bandwidth) versus resonator radius.

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The authors attribute the discrepancy in performance between the disk and ring to the inclusion of doped heaters in the former. Micro-disks are highly multi-mode systems and support a plethora of higher-order WGMs, which can result in undesired spectral features leading to excess crosstalk and loss for nontarget channels [17]. Strategic doping was used to suppress higher-order modes in the micro-disk [Fig. 3(b)] but also decreased FSR robustness [Fig. 3(a)]. While successful, we see that this introduces another source of FSR variation as the micro-disk without a heater performed slightly better than the ellipsoid ring (${\sigma _{\text{FSR}}} = 12.38\;{\text{GHz}}$). This indicates that, with additional engineering, the FSR robustness of the disk resonators should match that of the ellipsoid rings. Additionally, the dispersion performance of the ellipsoid ring was found to be significantly improved, due to the tapered width of the ellipsoid ring geometry balancing the dispersion at slightly beyond the single-mode regime [24]. The ellipsoid rings showed a $6.3 \times$ reduction in dispersion compared to the most similar micro-disk [Fig. 3(c)]. The proposed link architecture design will therefore use ellipsoid rings, as their robustness and dispersion performance allow for more aggressive bandwidth scaling.

3. MEASURING MAXIMUM DATA RATE VERSUS AGGRESSOR SPACING

We now move to discussing signal degradation due to the reduced ${\lambda _{{\text{ag}}}}$ in the link. As mentioned in Section 1, there will be a power penalty incurred by the reduction in ${\lambda _{{\text{ag}}}}$. If the overall link power budget is to remain fixed, the maximum data rate in each channel must be reduced to compensate for the additional power penalty. To investigate this, we quantify this trade-off at the receiver side of the link assuming these results will inform the limits of the transmitter design as well (see Section 5.B). For a DWDM receiver the expected loss-dependent power penalty associated with reducing the channel spacing will be at its worst for the last resonator on the bus. There will also be a power penalty associated with introducing crosstalk due to overlapping optical spectra, whose magnitude will be dependent on both transmitter and receiver characteristics [15,25]. Due to the inability to speak generally on the magnitude of this crosstalk power penalty, we limit our experimental study strictly to assessing the former of the two penalties. The loss-dependent power penalty will nonetheless have strong correlation with the crosstalk power penalty, implying that the relationship between the former and maximum per-channel data rates for the overall link will be proportional to the latter (though the exact coefficient of proportionality will depend on the specific transmitter/receiver implementations).

A. Receiver Bus Test Structure

To characterize the maximum data rate allowed for a given channel spacing, data transmission experiments were performed by modulating a continuous-wave (CW) laser with high-speed data and investigating the power penalty for different receiver configurations. To perform this experiment, we fabricated several waveguide buses with three cascaded MRRs with a radius of 5 µm [Fig. 4(a)]. Each bus of MRRs was fabricated with different coupling gaps from 130 to 200 nm [Fig. 4(b)]. The chip was fabricated in the same 300 mm AIM Photonics wafer as in Fig. 2(c). Micro-ring resonance positions were thermally tuned via a doped radial silicon heater placed in the center of the ring, shown in the inset of Fig. 4(b). To investigate the worst-case excess loss due to optically adjacent resonators capturing a target signal’s power, the filters were tuned such that the last filter resonance (${\lambda _2}$) on the bus was centered in between the first two [Fig. 4(c)]. The resonance location of ${\lambda _2}$ was set to be 1542.14 nm. The heaters of the other two filters were tuned such that their resonances can be placed arbitrarily close to one another by the applied voltage [Fig. 4(d)]. The power penalty was assessed at different ${\lambda _{{\text{ag}}}}$ from 100 to 5 GHz.

 

Fig. 4. (a) Diagram of the test structure: (i) input port of the de-multiplex (MUX) bus, (ii) drop port of the final resonant filter, and (iii) output port at the end of the de-multiplex bus. (b) Micrograph of a sample test structure used in the experiments. Inset: micrograph of a micro-ring resonator with a doped heater for resonance tuning. (c) Optical spectrum taken at each drop port of the resonant filters on the receiver bus. The rings were tuned such that the last ring on the bus was centered between the other two, to simulate a worst-case power penalty. (d) Optical spectrum at the output port of the de-multiplex bus with the filters set to a 200 GHz spacing.

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B. Description of Experiment

The experimental setup is illustrated in Fig. 5. To ensure that the fundamental transverse electric mode was launched into the chip edge couplers, the polarization of the CW laser was adjusted with a polarization controller. A 32 Gb/s Anritsu pulse pattern generator produced a pseudo-random bit sequence (PRBS) signal, which was then sent to a linear transmitter (Thorlabs MX35E) for amplification and modulation. The resulting optical signal was amplified using an erbium-doped fiber amplifier (AEDFA-23-B-FA) and transmitted through the bus of MRRs for filtering. The filtered light was coupled off the chip into single-mode fiber and sent to a variable optical attenuator (VOA) (Thorlabs EVOA1550A) to regulate the optical power sent into the receiver. The received light was detected by a photodetector (PD) and transimpedance amplifier unit (Thorlabs RXM40AF) to convert it back to an electrical signal. The electrical signal was then sent either to a real-time oscilloscope (Keysight Infiniium Z-Series) for eye characterization or back to the bit-error rate (BER) tester for BER evaluation. A set of DC power supplies (Agilent E3633A) provided all DC biases for controlling the receiver filters and VOA. The optical power received at the PD was monitored using a DC multimeter (Agilent 34401A). To eliminate the possibility of pattern-dependent behavior and provide the most rigorous test pattern, all data transmission experiments used a PRBS of length ${2^{31}} - 1$. The monitor and control signals were coordinated using a Python script on a laptop via General Purpose Interface Bus (GPIB) protocols.

 

Fig. 5. Experimental setup for characterizing the measurement of the BER of through the MRR for different full-width at half-maximum (FWHM) and data rates. TX, transmitter; EDFA, erbium-doped fiber amplifier; PC, polarization controller; LF, lensed fiber; DUT, device under test; VOA, variable optical attenuator; PD + TIA, photo-detector and trans-impedance amplifier; DC, direct current; BERT, bit-error rate tester; OSC, oscilloscope.

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C. Results and Discussion

An example set of the extracted power penalty results is shown in Fig. 6(a) for ${\lambda _{{\text{ag}}}}$. Aggressor spacing-dependent loss was included by recording the relative power at different ${\lambda _{{\text{ag}}}}$ relative to when ${\lambda _{{\text{ag}}}} = 200 \;{\text{GHz}}$. By then using linear interpolation and adding this loss back to the measurement data, the required power at the receiver ${P_{{\text{rec}}}}$ as a function of channel spacing and data rate can be plotted [Fig. 6(b)]. Typically it is necessary for the FWHM of the employed resonant filter to be twice the data rate of the non-return-to-zero on-off-keying (NRZ-OOK) signal [26]. Thus, for the micro-ring bus with a FWHM of 52 GHz, a ${\text{BER}} \le {10^{- 10}}$ was not possible for data rates faster than 22.5 Gb/s. As mentioned in Section 1, the power per line of any laser source cannot be increased arbitrarily. Therefore, it is of interest to investigate the evolution of the maximum per-channel data rate versus ${\lambda _{{\text{ag}}}}$ under the constraint that the required receiver power is minimized.

 

Fig. 6. (a) Minimum required ${P_{{\text{rec}}}}$ for bit-error rate (BER) $\le {10^{- 10}}$ operation versus aggressor spacing $({\lambda _{{\text{ag}}}})$. All data points at a given ${\lambda _{{\text{ag}}}}$ represent a family of BER measurements repeated for different data rates (DRs) from 15 to 30 Gb/s. (b) Sample BER versus received power (${P_{{\text{rec}}}}$) curve for micro-resonators with a FWHM of 66 GHz.

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Figure 6(a) shows that as ${\lambda _{{\text{ag}}}} \to \infty$, there is a difference between the power required to transmit data for ${\text{BER}} \le {10^{- 10}}$ between 10 and 30 Gb/s. At the same time, additional losses due to the overlapping spectra of the resonant filters on the bus are introduced as ${\lambda _{{\text{ag}}}}$ decreases. If the operating data rate is allowed to vary, the marginal power penalty (${{\text{PP}}_{{\text{margin}}}}$) due to data rate can be balanced with the power penalty due to loss to achieve the same BER for smaller ${\lambda _{{\text{ag}}}}$. To illustrate this, ${P_{{\text{rec}}}}$ is chosen to be ${-}11\; \text{dBm}$ as this is the lowest power required to receive 30 Gb/s with a ${\text{BER}} \le {10^{- 10}}$. From this we plot the maximum achievable data rate versus ${\lambda _{{\text{ag}}}}$, giving us a curve that specifies pairs of data rates and spacings that share a similar Q-factor and, therefore, power penalty [Figs. 7(a)–7(c)]. This constant power penalty curve enables us to explore highest theoretical throughput a multi-FSR receiver can achieve under the constraint that the overall power penalty is to remain constant. The maximum number of usable channels on the bus as a function of minimum ${\lambda _{{\text{ag}}}}$ can be calculated for the FSR of the ellipsoid ring (i.e., 4.83 THz) measured in Section 2.B assuming a channel spacing of 100 GHz and two de-interleaving stages, i.e., ${\lambda _{{\text{ch}}}} = 400 \; {\text{GHz}}$.

 

Fig. 7. (a) Plot of maximum achievable data rate (${{\text{DR}}_{{\max}}}$) versus aggressor spacing (${\lambda _{{\text{ag}}}}$). For a full-width at half-maximum (FWHM) of 52 GHz, ${\text{BER}} \le {10^{- 10}}$ was not achievable beyond 17.5 Gb/s at the specified power penalty (PP). Eye diagrams of (b) 30 Gb/s and (c) 20 Gb/s signals with equivalent bit-error rate (BER) performance and same loss-based PP. (d) Theoretical maximum bandwidth of the receiver (assuming no dispersion and process variations).

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By linearly interpolating the data plotted in Fig. 7(a), we can estimate ${{\text{BW}}_{{\text{agg}}}}$ as a function of ${\lambda _{{\text{ag}}}}$ by multiplying the maximum data rate by the associated ${N_\lambda}$. The results of this operation are shown in Fig. 7(d). We see that, while the per-channel data rate decreases, the associated increase in the number of channels on the bus still leads to a net increase in ${{\text{BW}}_{{\text{agg}}}}$. This co-optimization allows an aggressive increase in the number of channels per bus before having to resort to adding another (de)-interleaving stage, thereby minimizing excess losses and power consumption. There is a limit, however, to this co-optimization, resulting in a maximum achievable receiver throughput when ${\lambda _{{\text{ag}}}}$ equals FWHM/2. For ${\lambda _{{\text{ag}}}}$ smaller than this limit, the rate at which channels are added per bus cannot compensate for the drop in the maximum achievable data rate corresponding to the fact that alias-driven loss becomes significant below this spacing and can no longer be appropriately compensated by the aforementioned power penalty margin in the data-dependent power penalty. This results in a theoretical maximum ${{\text{BW}}_{{\text{agg}}}}$ of 10.51 Tb/s and 9.76 Tb/s for FWHM of 66 GHz and 51 GHz, respectively [Fig. 7(d)]. While reducing the FWHM allows a higher number of channels to be accommodated, there is a reduction in the per-channel data rate. We therefore reason that the FWHM could be further optimized to further maximize the highest achievable ${{\text{BW}}_{{\text{agg}}}}$. The maximum achievable data rate at the inflection point for the $\text{FWHM} = 66\; \text{GHz}$ resonator is $\approx 20\; {\text{Gb}}/{\text{s}}$. We therefore set both our minimum ${\lambda _{{\text{ag}}}}$ and maximum data rates according to this limit.

4. MULTI-FSR ARCHITECTURE DESIGN SPACE

A. Review of Multi-FSR Design Formalism

As previously mentioned, there are hard physical lower bounds on the achievable radius of MRRs due to bend radiation loss, which therefore imposes limits on designing MRRs with FSRs greater than the comb bandwidth. While any FSR larger than the optical bandwidth will work in the single-FSR regime, only particular combinations of FSR and channel aggressor values can result in a valid multi-FSR design. A multi-FSR channel arrangement scheme thus aims to put resonance aliases in between the modulated channels while maintaining an adequate spacing between each channel and its nearest aggressor for crosstalk minimization. In [17], two auxiliary variables were defined for choosing a particular FSR design:

B. Aggressor Spacing Reduction Due to FSR Error

Ideally, the fabricated micro-resonator structures would match the designed FSR exactly over the entire operating spectrum. However, in reality, as discussed in Section 2, several factors can cause the FSR to deviate from its designed value. This only reduces the ${\lambda _{{\text{ag}}}}$ of the first and last channels in the single-FSR regime, and can easily be mitigated by designing the resonator FSR with a sufficient guard band. But as the multi-FSR regime only guarantees a certain ${\lambda _{{\text{ag}}}}$ for a particular resonator FSR, the sensitivity of ${\lambda _{{\text{ag}}}}$ to FSR errors must be assessed. Thus, a relationship between the two quantities is needed. A small change in FSR can be considered as equivalent to a different link design that targets a different ${\lambda _{{\text{ag}}}}$. We will refer to this corresponding design as the “actual” and the original design as the “target.” While the actual and target designs will accommodate the same number of channels, the actual design is associated with a much larger ${\cal S}$. This is because the FSR error can be thought of as the alias resonances occupying arbitrarily small frequency slots, and small frequency spacings are associated with larger ${\cal S}$. The correspondingly larger ${\cal S}$ for the actual design will result in a ${\lambda _{{\text{ag}}}}$ that is much smaller than the target. But since most of the frequency slots in the actual design are unoccupied, the change in worst-case ${\lambda _{{\text{ag}}}}$ will still be small. By calculating how ${\lambda _{{\text{ag}}}}$ changes for an arbitrarily large value of ${\cal S}$ and accounting for the number of unoccupied frequency slots, we can track how the worst-case ${\lambda _{{\text{ag}}}}$ is reduced for smaller FSR variations.

Figure 8 shows the ${\lambda _{{\text{ag}}}}$ versus FSR plot for a bus with 35 channels at 400 GHz effective channel spacing. The curve consists exclusively of regions with a constant slope. We therefore propose linearly interpolating between the calculated results to evaluate how an arbitrary change in FSR impacts ${\lambda _{{\text{ag}}}}$ around a particular solution of interest. The underlying relationship between ${\lambda _{{\text{ag}}}}$ and FSR is unique to the number of channels on the bus, and cannot easily be generalized in a closed-form manner. To begin the design space exploration, we can assume that initially there is a 1:1 correspondence between ${\lambda _{{\text{ag}}}}$ reduction and FSR error (i.e., ${\sigma _{{\text{ag}}}} = {\sigma _{{\text{FSR}}}}$), as typically a change in alias position will translate as a 1:1 reduction in spacing on some channel. Designs that include a larger number of alias resonances, however, can potentially have asymmetry in ${\lambda _{{\text{ag}}}}$ sensitivity to FSR errors depending on whether the FSR is larger or smaller than the design target (Fig. 8). In general, the expected yield should be verified by inputting expected statistical FSR data into the linear interpolation model, which we perform in Section 5. Including the resonator dispersion extracted in Fig. 3 and pessimistically adding it to the expected FSR variation, we expect a nominal ${\lambda _{{\text{ag}}}}$ of 133.33 GHz to accommodate a yield of $6\sigma$. This yield estimate is calculated assuming that resonators within the same chip will be highly correlated in their FSR performance [23,27].

 

Fig. 8. FSR versus ${\lambda _{{\text{ag}}}}$ for ${N_{{\text{ch}}}} = 35$ targeting a $6\sigma$ yield (${\lambda _{{\text{ag}}}} = 133.33 \;{\text{GHz}}$). ${\lambda _{{\text{ag}}}}$ displays a larger sensitivity to the resonator FSR being smaller than the target design (2:1) compared to being larger (1:1).

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5. AGGREGATE LINK BANDWIDTH IMPLICATIONS

Now that the design space has been comprehensively defined, we can investigate its implications on overall link bandwidth. As prior work has discussed link scalability from an energy efficiency and loss standpoint [12,13,17], this study restricts itself to investigating the additional bandwidth scalability of the multi-FSR regime in comparison to a single-FSR link with an equivalent number of (de-)interleavers.

A. Multi-Tb/s Link Design

To constrain our design exploration, we assume a target link design incorporating a 100 GHz comb laser source. The source is fed into two stages of even–odd interleavers resulting in an effective channel spacing, ${\lambda _{{\text{ch}}}}$, of 400 GHz in each bus. We consider a minimum spacing of half the FWHM of the micro-resonator since power requirements increase exponentially below this threshold (see Section 3). For the following analysis, we base our assessment on the performance of the resonator using a 66 GHz FWHM, as it allows for greater theoretical throughput. We simplify our approach by assuming that the ${{\text{BW}}_{{\text{agg}}}}$ is limited by the maximum achievable data rate of the poorest performing channel. After extracting both the dispersion and statistical variation (detailed in Section 2.B), we choose a value of 133 GHz for ${\lambda _{{\text{ag}}}}$ to target a $6\sigma$ yield. We then calculate the actual expected ${\sigma _{{\text{ag}}}}$ by entering the measured FSR variance into the interpolated ${\lambda _{{\text{ag}}}}$ model. In this design the ${\lambda _{{\text{ag}}}}$ can change by ${\sim}100 \;{\text{GHz}}$ before the crosstalk power penalty becomes unacceptable (as explained in Section 4), successfully allowing $\ge 6\sigma$ yield. The estimated upper bound for ${\sigma _{{\text{FSR}}}} = 17.87\;{\text{GHz}}$ implies ${\sigma _{\text{ag}}} = 18.47\;{\text{GHz}}$, which can still successfully enable  $\ge 5\sigma$ yield (Fig. 9).

 

Fig. 9. Plot of expected aggressor spacing (${\lambda _{{\text{ag}}}}$) statistical distribution based on extracted statistical free spectral range (FSR) ellipsoid micro-ring resonator (MRR) performance. A link yield of $6\sigma$ for a nominal ${\sigma _{{\text{FSR}}}} = 14.63\;{\text{GHz}}$ is expected, accounting for both process variations and dispersion. The FSR data in this plot was generated via 10,000 simulated samples using a normal distribution with the mean and variance extracted from Fig. 2(d).

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The resulting designs are shown in Table 1. We see that, at minimum, the ${{\text{BW}}_{{\text{agg}}}}$ can be increased by $1.94 \times$ targeting a per resonator $5\sigma$ yield. The primary barrier to achieving even higher data rates for this particular FWHM design is the minimum acceptable yield. If the yield requirements are relaxed, we can see that bandwidth multipliers of $2.61 \times$ and $3.28 \times$ can be achieved targeting $3\sigma$ and $1\sigma$ yields, respectively. These results emphasize the need to explore additional ways to reduce resonator FSR variability, as they can unlock additional gains in link throughput. It is likely that ${{\text{BW}}_{{\text{agg}}}}$ could be improved with further optimization of the resonator Q-factor, as the channel count and maximum data rate are positively and negatively correlated, respectively, with resonator FWHM (as discussed in Section 3.C). Link throughput can also be increased if the restriction to keep the overall loss-based power penalty constant is relaxed, though the associated trade-offs will be highly implementation and design specific.

Table 1. Comparison between Aggregate Link Bandwidth (${{\text{BW}}_{{\text{agg}}}}$) for Single and Several Multi-FSR Designs

View Table

B. Transmitter Design Implications

We now will discuss the relative scalability of the transmitter from our measurements on the disk resonators. While a discussion of the high-speed performance of disk modulators in the transmitter is beyond the scope of this work (see [12,17] for further discussion), we can nonetheless analyze the measured FSR robustness of the micro-disks and the expected effect on link scalability. Future investigations should prioritize improving the degree of micro-disk robustness while suppressing higher-order modes, given their potential as a modulator candidate with high achievable modulation efficiency [17]. To achieve $6\sigma$ performance with the assumed source and interleaver designs requires a guard band of roughly 282 GHz ($6 \times$ doping variance floor of 47 GHz), meaning at least one more interleaver stage must be added to achieve a similar yield to the receiver. While results from Section 2.B would indicate the robustness of the undoped micro-disk WGM [Figs. 3(a) and 3(b)] is sufficiently robust to match the ellipsoid rings, it is still important to suppress the higher-order WGM to avoid deleterious loss and crosstalk effects. In the short term, these results indicate that an asymmetric interleaver design could be employed, with the transmitter using more interleaver stages than the receiver [13]. In this way, the additional robustness possible at the receiver side can be leveraged to decrease overall power consumption, as every additional interleaver stage must be accounted for in terms of additional optical loss and energy costs of tuning. Interestingly, the ellipsoid rings seem to inherently be able to achieve similar robustness to the intrinsic WGM of the micro-disks while also achieving far better dispersion performance. Given these results, future work should also investigate the feasibility of designing a modulator using hybrid Euler bends. Though it is unlikely they could achieve the same modulation efficiency, this downside could be compensated for in the multi-FSR regime since they would allow for a larger number of usable channels through the improved dispersion and off-resonance IL.

6. CONCLUSION

In summary, we have demonstrated the design of several multi-Tb/s comb-driven silicon photonic links. The results anticipate $\ge 2.8 \;{\text{Tb/s}}$ with $\ge 6\sigma$ yield, enabled by the high robustness and low dispersion of the measured ellipsoid MRRs. This extremely high yield, supported by comprehensive measurements of 704 micro-resonator devices fabricated on a 300 mm wafer in a commercial foundry, indicate that this scalable link architecture is rapidly approaching technological maturity. Detailed spacing- and data rate-dependent BER measurements show co-optimization of the per-channel data rate with ${\lambda _{{\text{ag}}}}$ to be a promising strategy to scale link bandwidth, enabling the multi-FSR designs to limit the requisite additional power penalties to off-resonance ($\le\! 0.02\;{\text{dB}}$ per-channel) and routing losses. Additionally, many of the various WDM-compatible MDM architectures that have been demonstrated in the silicon photonics platform [10,11] can be trivially extended to accommodate the IM-DD DWDM architecture, leveraging spatial modes as an orthogonal axis for multiplicative DWDM and MDM scaling. The advances presented in this work illustrate a promising trajectory for the practical use of Kerr-driven silicon photonic interconnects in data centers.