Wideband multi-stage CROW filters with relaxed fabrication tolerances

: We present wideband and large free spectral range optical filters with steep passband edges for the selection of adjacent WDM communication channels that can be reliably fabricated with mainstream silicon photonics technology. The devices are based on three cascaded stages of coupled resonator optical waveguides loaded on a common bus waveguide. These stages differ in the number of resonators but are implemented with exactly identical unit cells, comprised of a matched racetrack resonator layout and a uniform spacing between cells. The different number of resonators in each stage allows a high rejection in the through port response enabled by the interleaved distribution of zeros. Furthermore, the exact replication of a unique cell avoids the passband ripple and high lobes in the stopband that typically arise in apodized coupled resonator optical waveguide based filters due to fabrication and coupling induced variations in the effective path length of each resonator. Silicon photonics filters designed for the selection of 9 adjacent optical carriers generated by a 100 GHz free spectral range comb laser have been successfully fabricated with 248 nm DUV lithography, achieving an out-of-band rejection above 11 dB and an insertion loss of less than 0.5 dB for the worst channels.


Introduction
Optical notch filters are a key component in optical communication systems. For example, they can serve for channel selection in a wavelength division multiplexed (WDM) communication system or to reduce amplified spontaneous emission (ASE) after optical amplification. Here, we aim at wideband notch filters for the selection of a predetermined number of optical carriers generated by a comb source in a WDM transmitter. In this system, several optical carriers are sourced by a single semiconductor mode-locked laser (MLL) on a 100 GHz grid and are further modulated by silicon photonics resonant ring modulators (RRMs) prior to being jointly amplified by a semiconductor optical amplifier (SOA) [1,2]. In this specific context, extinction of unused comb lines is of paramount importance to avoid excessive saturation of the SOA and to optimally allocate the available SOA output power to active channels. In order for the wideband optical notch filter to serve its purpose, it needs to have a wide enough passband to let through the selected optical carriers (900 GHz), while maintaining a transfer function with sufficiently steep edges and a wide enough stopband of at least 900 GHz to fully cut off adjacent unwanted MLL comb lines. Moreover, in order to avoid burdening the optical power link budget, maintaining low insertion losses is critical. Requirements on stopband extinction are discussed in more details towards the end of the paper, however it can already be said here that while a guaranteed extinction of at least −16 dB throughout the stopband would be ideal for the target application, the −11 dB worst case extinction shown in the experimental section proved fully adequate due to averaging of the transmitted power across the rejected comb lines. In order to monolithically integrate as much of the required optical functionality as possible on a single transceiver chip, the optical filters are developed in silicon photonics technology.
A number of approaches have been followed to implement integrated optical filters with a flat transfer function and/or steep passband edges. Athermal flattop filters have for example been obtained with multi-stage Mach-Zehnder interferometers with asymmetric waveguide cross-sections [3]. Grating based approaches have also yielded high performance [4][5][6], in particular in terms of very high rejection levels critical for quantum optics applications such as heralded photon generation [7], but tend to require lower critical dimensions and higher precision lithographic resolution and are thus most suited to fabrication with electron beam lithography. Moreover, they tend to be long devices, potentially increasing the required power levels for thermal tuning. Here, we base our approach on coupled resonator optical waveguides (CROW) consisting in several coupled racetrack resonators [8,9]. While this is a well-known approach for compact filter implementation, it is also associated with a number of difficulties in its reduction to practice: An important challenge associated with the implementation of optical filters in silicon photonics technology resides in the spectral shifting of the transfer function of individual filter elements, here the resonance frequency of individual rings, due to fabrication tolerances [10] or due to coupling-induced resonance frequency shifts [11]. While independent dynamic tuning of the ring's resonances can in principle address this problem [12], independent tuning of a large number of optical elements is a very complex problem. Just the acquisition of necessary feedback signals and interpreting them in terms of the required tuning adjustments can be very difficult. Independent thermal tuning of close by elements can furthermore be very challenging due to thermal cross-talk [13]. On the other hand, tuning of the entire filter transfer function with a single control signal, for example by globally heating up the filter, while being undesirable from the perspective of power consumption, remains a tractable problem for sufficiently compact devices.
Early attempts at implementing CROWs with a large number of coupled resonators suffered from the mismatch in resonance frequencies arising as a consequence of fabrication tolerances. While the resulting frequency spread has been considerably improved by moving to more advanced technology nodes [10], the remaining resonance frequency variations still need to be addressed. A solution consists in increasing the coupling between the resonators to the extent that the resulting increase of the resonators' linewidths exceeds the spread in resonance frequencies [14]. In other words, the time required to couple light from one resonator to the next has to be shorter than the time required for the phase of light stored in two resonators to significantly dephase as a consequence of their resonance frequency mismatch. This requirement on resonator-to-resonator coupling strength constrains the minimal width of the CROW filter stop-or passband, depending on the configuration of the device (through or drop port configurations described below). This problem of resonance frequency repeatability is further exacerbated by systematic effects when the coupling strength between adjacent resonators is purposefully modified in order to obtain specific transfer functions [15][16][17][18][19][20]. Due to proximity effects associated to lithography or microloading effects during etching [21], modifications in the layout introduced to modify the coupling strength will also result in small systematic modifications in the waveguide width leading to a shift in the resonance frequency. Moreover, modified coupling strengths also result in the modification of the coupling induced resonance frequency shift [11]. While in principle these systematic biases can be determined and compensated for, this might not only require several experimental iterations to get the device right, but also an extremely stable process for this calibration procedure to converge. Filters relying on a single non-apodized CROW on the other hand, while featuring good extinction when operated in drop port configuration, as defined below, require a large number of resonators to achieve a steep rolloff. When operated in through port configuration, they feature the lowest passband insertion losses, but suffer from side lobes in the stopband limiting the maximum extinction. Several techniques can be applied to apodize the coupling strength between resonators, such as varying the spacing between resonators on a nanometric scale, or offsetting the resonators one relative to the other so as to vary the length of the coupling junction [22,23]. While the latter alleviates difficulties related to the necessity of shifting the resonator positions by very small amounts -placement accuracies being ultimately limited by the design grid on which the mask is defined -problems related to proximity effects and coupling induced resonance frequency shifts remain. An additional difficulty, a coupling strength dependent modification of the phase of the resonator-to-resonator coupling coefficient, is introduced, constraining design optimization [24]. While the high resonator-to-resonator coupling strengths required to achieve the wide pass-and stopbands targeted for our application help to overcome variability in resonance frequencies, in particular the residual non-systematic resonance shifts still present in our device, fairly large systematic resonance shifts need to be addressed by design.
The approach followed in this paper is to implement CROW filters without apodization, but to cascade several stages each composed of a different number of resonators. As will be described in the following, this results in a compact filter implementation small enough to be globally thermally tuned, while achieving the targeted transfer function without tuning of individual rings. Since all the resonators are exactly identical, down to the snapping on the design grid and including the layout of their nearest neighbors, proximity effects and coupling induced resonance frequency shifts become a non-issue. While local process variability remains, with the std. dev. of the resonance frequency shift for close by rings extracted to be on the order of 0.3 nm in the 248 nm deep ultra-violet (DUV) line used here, it is seen as being significantly smaller than the resonance frequency shifts observed here in test structures due to systematic effects resulting from layout changes introduced to vary the resonator-toresonator coupling strengths. While we are varying the resonator-to-resonator gap in order to apodize the coupling strength in these test structures and while the magnitude of systematic effects might be different when using the offsetting technique instead, such systematic effects are likely to remain a limiting factor in apodized CROWs if they are not addressed by design, particularly in view of progress made in intra-die (non-systematic) device repeatability with more advanced 193 nm lithography, with reported variability as low as 0.15 nm (std. dev.) for close by devices [10].
In section 2 we will review single stage CROW filters with and without resonator-toresonator spacing apodization, as already reported in the literature, and exemplify difficulties in their implementation associated to process biases and minimum feature size. In section 3, we will introduce our designs relying on exactly identical unit cells, exemplify how this facilitates design, and report experimental results. , so that the maximum achievable FSR is limited by bending losses and junction lengths required to achieve targeted coupling strengths. As depicted in Fig. 1(a), the racetrack resonator is comprised of two 180° waveguide bends with radius R connected by two straight sections with length S L that also constitute the directional couplers between the resonators. Thus, the total resonator length is given by

Single stage CROW filters
. In order to facilitate the filter implementation, all individual resonators in a CROW typically share a common length and therefore a similar FSR. Highest required coupling strength (corresponding to κ 1 = κ N+1 , the coupling between the bus waveguide and the first resonator as well as the last resonator and the drop waveguide), as a function of the ratio between the width of the passband (BW P ) and the Free Spectral Range (FSR) for the implementation of a 5 th order apodized CROW filter. BW P is shown both for a CROW filter in drop port configuration (lower x-axis) and a CROW filter in through port configuration (upper x-axis). The red, green and blue curves refer to Chebyshev responses with passband ripple levels of respectively 0.1 dB, 0.5 dB and 1 dB (-16.4 dB, -9.6 dB and -6.86 dB side lobe levels in through port configuration). The black curve corresponds to a Butterworth filter response with a maximally flat drop port passband response. (c) Spectral response over a whole FSR of the Chebyshev filters in drop (solid lines) or through (dashed lines) configurations with a ratio BW P /FSR of 0.3.
Typical specifications for CROW filter designs include the bandwidth of the passband BW P , the bandwidth of the stopband BW S (or equivalently the FSR, since FSR ≈BW P + BW S ), the maximum allowed insertion loss in the passband (including the ripple attenuation), the minimum rejection in the stopband (including the required side lobe suppression level) as well as the width of the transition band (related to the roll-off steepness and filter order) [9].
CROW filters can be used either in through port or in drop port configuration, respectively referring to the through port or to the drop port being connected to the downstream portion of the system. Both configurations have their strengths and weaknesses in terms of stopband rejection, passband ripples, insertion losses, and required maximum coupling strengths. In particular, in the through port configuration the width of the stopband scales with the coupling strength, while in the drop port configuration the width of the passband scales with the coupling strength. The through port configuration tends to have less insertion losses, as in the drop port configuration the passband suffers from ripples and the light has to pass through the CROW in order to be transmitted. In the through port configuration, however, the stopband suffers from high side lobes, characterized in the following by the side lobe suppression level (SSL).
At the drop port, CROWs with a uniform coupling strength between resonators (κ i 2 = κ 2 , with i = 1, 2, … N + 1, wherein the squared parameters represent power coupling coefficients [25]) present a large rejection in the stopband but significant ripples in the passband. On the other hand, the complementary spectral response at the through port exhibits a flat and low loss passband. However, the filter rejection level is then hindered by prominent side lobes in the stopband. The width of the filter passband (drop port) or the width of the filter stopband (through port), as well as the level of passband ripples (drop port) or side lobes (through port) are mainly determined by κ 2 and are almost independent of the number of resonators N forming the CROW (filter order). On the other hand, increasing the filter order reduces the spectral width of the passband ripples (drop port) or side lobes (through port) and leads to a steeper roll-off.

Tradeoffs in the design of single stage, apodized CROW filters and limitations in the largest achievable FSR and bandwidth
In this section, we will review the tradeoffs in the design of single stage apodized CROW filters. In particular, we will describe how increasing the FSR (requiring a reduction of the resonator circumference) and increasing the BW P /FSR or BW S /FSR ratios, respectively in drop or through port configurations (requiring a high coupling strength, a minimal junction length and thus constraining the minimum resonator length) result in opposite requirements. This is particularly constraining in the design targeted here, in which both BW P and BW S are required to be above 900 GHz (hence swapping between a drop port and through port configuration does not significantly modify the requirements on the maximum coupling strength, even though it does have important consequences in terms of insertion losses vs. worst case extinction). The CROW filter thus needs both a high FSR of at least ≈1800 GHz, while at the same time also requiring high coupling strengths so that BW P /FSR (respectively BW S /FSR) is on the order of ≈0.5. In the following modeling, waveguides are assumed to be dispersionless -a simplification justified here by the intent of exemplifying design tradeoffs rather than designing a concrete device -so that quantitative results can be simply described with frequencies / wavelengths given as a percent of FSR (since the FSR scales as 1/L and the splitting between the resonances scales as κ 2 /L, the ratio of bandwidths to FSR can be given as a function of κ 2 only, independently of the assumed FSR).
The established approach for simultaneously achieving a flat response in the passband and a high rejection in the stopband consists in setting different coupling strengths between resonators (coupling apodization). With this aim, several synthesis algorithms have been proposed to determine the coupling values that implement typical filter responses such as Butterworth or Chebyshev [9,15]. In general, for a filter order N (selected to meet a target roll-off at the transition band), the coupling coefficients between resonators follow a symmetric distribution with κ 1 = κ N+1 , κ 2 = κ N , etc., wherein κ 1 is the amplitude coupling coefficient between the (input) bus waveguide and the first resonator, κ 2 is the coupling between the first and the second resonator, κ N is the coupling between the before last and the last resonator and κ N+1 is the coupling between the last resonator and the drop waveguide. Furthermore, the strongest coupling is typically implemented between the external resonators and the bus and drop waveguides (κ 1 and κ N+1 ). Coupling coefficients are successively reduced as one moves deeper into the structure, until a minimum value is reached between the central resonators. Figure 1(b) shows the highest required coupling strengths (κ 1 2 = κ N+1 2 ) for 5th order filters as a function of the ratio between the passband width (BW P ) in either drop or through port configurations and the FSR. The different curves correspond to filters in drop port configuration with different values of ripple attenuation in the passband, respectively filters in through port configuration with different SSL. Given an initial CROW filter design in drop port configuration, the bandwidth of its passband can be widened by i) increasing the FSR and/or ii) increasing the coupling. Figure 1(c) shows that, alternatively, the passband ripples (drop port configurations) or the SSL (through port configuration) can be decreased by increasing the maximum coupling strength at a fixed BW P /FSR. Thus, in our target application requiring wide BW P and BW S , high coupling strengths are required both to achieve the targeted BW P /FSR ≈0.5 as well as to achieve low passband ripples (drop port configuration) or low SSL (through port configuration). In addition, a large FSR is required. This leads to a number of difficulties: First, the maximum FSR achievable by means of a reduction in R is limited by the bending loss level, which eventually penalizes the insertion loss at the drop port. The high confinement of fully etched single-mode waveguides in silicon-on-insulator technology (220 nm core thickness) allows very compact designs with radii down to 3 µm without a significant performance degradation due to bending losses at the scale relevant to the performance of the device specifications targeted here.
On the other hand, stronger coupling coefficients have to be implemented by reducing the gap between the resonators if the FSR is not to be reduced. Here, the minimum value is limited by the resolution of the fabrication process. The CROW filters presented in this work have been designed to be compatible with fabrication with 248 nm DUV lithography and a minimum feature size of 180 nm. A thinner waveguide could also enhance the coupling strength, but at the price of higher radiative losses arising from the bending or complications resulting from adiabatic mode conversion inside the resonators [26].
Once both gap and waveguide width are fixed, the remaining alternative for a wider passband consists in increasing the coupling strength by elongating the coupler section length S L . As a drawback, this approach also limits the FSR which further increases the required κ 1 to reach a target passband width.
Since, in drop port configuration, the design performance in terms of passband ripple also depends on the coupling strength (see different colored curves in Figs. 1(b) and 1(c)), this leads to stringent limitations when trying to meet all specifications simultaneously. This is particularly constraining for applications requiring both large FSR values and a wide passband width. These limitations may seem to be less constraining in case of CROW filters in through port configuration since, additionally to the characteristic flat passband, their passband width scales inversely with the coupling strength. However, in this case the challenge typically consists in achieving the required stopband width, which again scales directly with the coupling strength. Furthermore, the reduction of the SSL for a higher stopband rejection by means of coupling apodization further increases the required coupling strength with the concomitant limitation in the maximum achievable FSR. Since in our case the requirement ended up being BW P /FSR ≈BW s /FSR ≈0.5, the choice of drop port vs. through port configuration would be driven by other considerations (compare both configurations in Fig. 1(b)), such as a minimization of passband insertion losses favoring a through port configuration.

Fabrication challenges related to coupling apodization and optical lithograhy
CROW filters with coupling apodization are typically implemented by changing the gap in the coupling sections between resonators while maintaining a common racetrack resonator layout [17][18][19][20] or by offsetting resonators relative to one another [22,23]. The common racetrack length is intended to ensure matching between the round-trip phases of each individual resonator, assuming deviations in the silicon device layer film thickness and process variations are sufficiently small across the device. This matching is essential in order to get the desired transfer function with the targeted distribution of filter poles and zeros. However, coupling strength apodization also results in systematic variations of the resonance wavelength due to either process proximity effects [10,21] or due to coupling-induced resonance frequency shifts [11], as well as, in case of the offsetting technique, variations in the resonator-to-resonator coupling phase [24], so that significant additional challenges arise that need to be addressed for a successful fabrication with optical lithography.
In both schemes, in case obtaining a large FSR is a high priority, the gaps for the first and last resonators may be set to the minimum value that can be safely resolved, allowing a minimization of the coupling length L S . In this section, we are taking a closer look at the random process variations and systematic resonance frequency shifts occurring when coupling apodization is obtained by varying the resonator-to-resonator spacing, in which case the resonator-to-resonator spacing is successively increased for the inner resonators.
First, as a consequence of the chosen technique, the practical implementation of the targeted coupling strengths requires resolving the different gap sizes with an accuracy of a few nanometers. As an example, Fig. 2(a) shows the required coupling strengths for the implementation of a 7 th order Chebyshev filter with a maximum ripple of 0.1 dB, an FSR of 14 nm and a BW P /FSR ratio of 0.47. The corresponding required gaps (marked with green circles) have been determined with 3D FDTD simulations considering a resonator design that meets specifications with R = 3 µm, L S = 9.7 µm and a minimum gap of 200 nm (silicon-oninsulator waveguide with a 220 nm thickness and a 400 nm width, clad on all sides by SiO 2 ). Based on these simulation results, we determined that this CROW filter requires gaps of 200, 239, 275, 281, 281, 275, 239, and 200 nm. An accurate fabrication of these gaps is already challenging in terms of mask layout irrespectively of process biases, since structures have to be snapped onto the design grid if one wants to ensure minimal changes to the resonators. As already described above, the problem of required nanometric resolution can be alleviated by an alternative apodization scheme, the longitudinal offset technique, in which individual resonators are offset one relative to the other in order to vary the coupling strength [22,23]. However, this does not resolve other issues related to systematic resonance frequency shifts as proximity effects and coupling induced frequency shifts continue to play a role, while a new difficulty, phase offsets in the resonator-to-resonator coupling coefficients, is introduced [24].
The resolution of the waveguide width is affected by optical proximity effects, which introduce nanometric deviations that prevent the exact matching of the optical path lengths of the individual resonators. These process biases depend on whether the waveguide is isolated or in proximity to another one (line pair) [27]. Furthermore, in the case of line pairs, the waveguide cross-section is affected by spacing dependent process biases. Since the lengths of the line pairs vary from resonator to resonator in the offsetting technique, this problem remains relevant also in that case even though the resonator-to-resonator spacing remains constant.
In order to experimentally illustrate this problem, we have fabricated a set of test structures consisting in single racetrack resonators that have a common resonator length (R = 4 µm, L S = 6.7 µm) but different gaps between the bus/drop waveguide and the resonator. The test structures with gaps of 200, 250 and 300 nm were replicated and interspersed in order to differentiate the magnitude of the resonance shifts due to local process variations and local variations of the film thickness from the magnitude of systematic effects, so as to conclusively identify the dominant source of mismatch.
Test structures in 5 different chips from the same wafer were spectrally characterized with a tunable laser. They showed significant shifts in the resonance wavelength around 1550 nm as a function of the gap size. The results are plotted in Fig. 2(b). Resonance shifts are taken relative to the mean resonance wavelength for the structures with 200 nm gap averaged over the same chip, in order to normalize out the effect of longer range film thickness variations and process biases between different chips. Thus, the data serves to visualize the dominant sources of mismatch for nearby structures (same chip). The rationale behind this is that the global shift of the resonance across all the rings of a given device is not the issue here, as it is deemed to be fine-tuned by global thermal control across the entire device. Rather, the issue consists in mismatch between rings of a same filter stage, as a correction of the latter would require individual trimming of the rings.
The high magnitude of the measured systematic resonance shifts is attributed to variations in the effective path lengths resulting from optical proximity effects, as well as coupling induced resonance shifts. Assuming similar systematic biases, the implementation of the 7th order Chebyshev filter described above would result in a deviation of the individual resonances in a range of more than 4 nm, which would very significantly deteriorate the filter response ( Fig. 2(c)). The random variations recorded within a given device category on the same chip, in an area of 4 mm by 2 mm, is significantly smaller. Within an actual CROW stage, this random variation is expected to be even smaller as the rings are much closer to each other as compared to being spread across a chip. The resulting variation is estimated in section 3.2. Since mismatch between resonators featuring different gaps is deterministic across the different measured chips, it appears to be feasible to apply optical proximity correction in the mask for compensation of the process biases [27] combined with optical design for a correction of the coupling induced frequency shifts. However, it should also be noted that fine tuning such a compensation strategy might require several experimental iterations and an extremely stable process.
For all these reasons, it appeared highly desirable to find an alternative filter topology that allows keeping both the racetrack resonators and their spacing constant throughout the device, as this is undoubtedly the safest strategy when a small number of design iterations are allowable in the overall design of an already highly complex system such as an integrated transceiver chip [1,2].

Multi-stage CROW filters with constant spacing
In order to alleviate the stringent fabrication requirements associated to CROW filters with coupling apodization, we propose an optical filter that relies on the cascaded combination of three stages of CROWs loaded on a common bus waveguide. All three are implemented with exactly identical unit cells, comprised of a matched racetrack resonator layout with equal radius and racetrack length, and a uniform spacing (uniform coupling κ 2 ) between resonators, chosen for all the racetrack resonators to be equally snapped to the design grid (see Fig. 3(a)). Moreover, we find that for the specifications of our target application resulting in BW P /FSR ≈BW S /FSR ≈0.5, the required coupling strength κ 2 is reduced relative to the maximum required coupling strength 2 1 κ of an apodized single stage CROW filter of identical specifications, so that design constraints relative to maximum achievable FSR are reduced.

Design
In this section, the main modeling tradeoffs are first exemplified. As in section 2.1, assuming a wavelength independent FSR (no waveguide dispersion) and coupling strength κ 2 , we obtain perfectly periodic transfer functions, so that these high-level modeling results are not reported at a specific wavelength, but rather stop-and passbands are reported as a function of the FSR. CROW filters are further modeled assuming waveguide losses of 20 dB/cm, consistent with excess losses extracted from single stage add-drop multiplexers based on the same racetrack geometry. These waveguide losses are ascribed to excess losses in the coupling junctions of ≈0.04 dB per junction, effectively split over the ring's circumference. The combined filter response can be easily determined by cascading the through port transfer functions of the three stages.
As shown in Fig. 3(b), the gradual increment in the CROW filter order of the cascaded stages leads to an interleaved distribution of the zeros inside the stopband (green, red and blue solid curves) which allows a significant reduction in both the side lobes' level and their spectral width in the combined filter response (black dashed curve). This results from the fact that the width of the stopband is a function of the coupling strength (κ 2 ) which is maintained constant throughout each of the stages, so that they have very similar stopbands. Since the transfer function zeroes are equally distributed throughout the stopband, and have a different number for each stage, this results in interleaving. While each stage has, individually, comparable SSL, once the interleaved transfer functions are combined a significantly improved SSL is obtained. In the proposed device, the exact replication of a unique cell avoids the deleterious effects (excess passband ripple or high lobes in the stopband) that typically arise in apodized CROWs due to unwanted systematic variations in the effective path length of each resonator. Non-systematic variations due to process and film thickness variations remain of course and are partially alleviated by the strong resonator-to-resonator coupling required to achieve the targeted pass-and stopbands.
Since the multi-stage CROW filter uses a cascaded through port configuration, there exists an inverse relation between the 3 dB passband width and the coupling strength (κ 2 ). Figure 4(a) shows the effect of increasing κ 2 on the combined filter response in a three stage CROW with N = 3, 4, and 5 resonators in each stage, respectively. On the one hand, a stronger coupling reduces both the SSL and the passband width. On the other hand, coupling strengths above 0.8 (BW P /FSR < 0.35) already deteriorate the flatness in the passband and introduce insertion losses above 0.5 dB at the center wavelength. Increasing the filter order of each of the stages could compensate for this last effect but at the price of a higher SSL (see Fig. 4(b)) as well as a larger, harder to globally tune structure. The introduction of an additional stage (for example filter with N = 3, 4, 5, and 6 resonators) could achieve both lower SSL and lower insertion loss at the center of the passband. These trade-offs are depicted in Figs. 4(c) and 4(d) where it can be seen that i) there is a minor dependence of the passband width on the filter order and the number of stages, ii) the SSL gets worse with higher filter orders but can be improved by cascading additional stages, iii) the insertion loss at the central wavelength depends on the coupling strength and is mainly given by the stage with the lowest order. All these characteristics make the proposed filter configuration particularly attractive for applications requiring ratios of BW P /FSR between 0.35 and 0.55, SSLs between −12 and −20 dB, as well as very low insertion loss in the passband.  We focused our design on a filter aimed at 950 GHz (7.6 nm) passband (BW P ), for the selection of 9 adjacent lines of a 100 GHz FSR comb laser (with the passband chosen somewhat above the minimum required to avoid incurring insertion losses due to the rollingoff of the transfer function at the outlying carriers). In order to introduce enough extinction in the rest of the comb lines on both sides of the passband, the FSR of the filter should verify 2⋅FSR -BW P > BW comb , with BW comb the width of the comb laser spectrum in which the lines exhibit significant power (> −10 dBm). Since the comb lasers of interest for our applications have typical values of BW comb between 15 and 20 nm, we designed the FSR of the CROW filter to be larger than 14 nm (≈1748 GHz).
We selected a CROW filter configuration based on three stages, consisting in 12 identical racetrack resonators with R = 4 µm and L S = 6.7 µm, distributed across three stages according to the orders N = 3, 4 and 5. As previously, 400 nm wide waveguides are etched into the 220 nm silicon device layer of Silicon-on-Insulator (SOI) material followed by the deposition of a PECVD oxide. The gaps between resonators are set to 200 nm in order to achieve the target coupling strength around a target wavelength of 1570 nm (around which the MLL spectrum is centered). These dimensions can be easily resolved by 248 nm DUV optical lithography. The filter achieves the required passband width (7.6 nm) by means of a coupling strength of κ 2 = 0.55 at 1565 nm wavelength, as well as an FSR of 14.5 nm / 1810 GHz (BW P /FSR = 0.53). Figures 4(c) and 4(d) show that the selected design presents negligible insertion loss at the center of the passband and an SSL better than −12.6 dB.
It should also be noted that in order to reach BW P /FSR ≈0.5 as required in our application (constrained by the fact that the FSR cannot be increased indefinitely in order to compensate for a low BW P /FSR), a coupling strength κ 2 of ≈0.5 is needed, while in the apodized CROW filter designs described in section 2.1 a coupling strength of κ 2 ≈0.85 would have been required to obtain a similar (and slightly worse) SSL of −9.6 dB (see the green curve in Fig.  1(b) corresponding to 0.5 dB ripple in the drop port passband and −9.6 dB SSL in the through port configuration). Thus, while this does not allow to conclude on a general comparison of the maximum required coupling strength in the most general case, one can conclude that for our specific wideband filter application the cascaded CROW filter configuration relaxes the requirements on the maximum coupling strength, deconstraining either the minimum circumference of the rings (allowing a higher FSR) or the minimum spacing between the rings (facilitating fabrication).

Experimental results
The devices were fabricated in the standard silicon photonics fabrication process of IME A*STAR with 248 nm DUV optical lithography. Figure 5 shows micrographs of (a) a chip with the fabricated passive filter and (b) a chip in which the resonators have been overlaid with TiN heaters allowing independent tuning of the three CROW filter stages (but not independent tuning of individual rings). As can be seen in (a), not only were the racetrack unit cells and the distance between racetrack resonators kept constant throughout the device, a number of dummy structures were added in the layout of the waveguiding layer in order to minimize process micro-loading effects associated e.g. to etching [21]. Gratings couplers were connected to all ports of the device (including drop ports of individual stages) for complete optical characterization. All measurements reported in the following were made with the chip temperature stabilized to 25°C with a Peltier element.
First, we measured the spectral response of 5 passive filters corresponding to 5 different chips (D1-D5) randomly selected from the same wafer. The individual stages were measured with help of the monitor ports connected to the drop waveguides (labeled in Fig. 5), so that they could be independently characterized even though they are cascaded with each other on the main waveguide bus. Since the light is injected through one monitor port and collected through another monitor port, the transfer function of the stage is collected in through port configuration, with a transfer function that is nominally identical to the one obtained if it had been individually accessed through the main bus waveguide (assuming that non-uniform process variations throughout the CROW stage does not significantly impact the symmetricity of the transfer functions; even then, it would not change the overall statistics of the analysis).
For each structure, the optical characterization of the three individual stages showed a good alignment between the central wavelengths of their respective passbands, with the overall std. dev. of the wavelength misalignment evaluated as 0.15 nm. This number is below the std. dev. of the resonance frequency of individual rings, since the transfer function of entire stages already corresponds to an average. Factoring in the number of rings in each stage, this corresponds to a ring resonance std. dev. of ≈0.3 nm (close by rings within one device). This fact indicates a good reproducibility in the fabrication of the resonators with uniform gap in contrast to the fabrication of conventional CROW filters with gap apodization. Nevertheless, a residual mismatch between resonators still led to some penalization of the SSL (−8.2 dB in the worst measured chip) in comparison with the design value (−12.6 dB). The reduction in SSL seen in the experimental data can be modeled well assuming the resonances of the individual rings to be misaligned with the 0.3 nm std. dev. estimated above. On the other hand, the measured FSR and BW P were consistent with the expected design values. Variations in the central wavelength of the passband across chips are mainly attributed to the non-uniformity in the wafer core layer thicknesses. The measurement results are summarized in Table 1. Next, we measured the spectral response of five CROW filter structures with TiN heaters, corresponding to another five different chips (D6-D10). The initial characterization of the individual stages without thermal tuning showed a larger misalignment between the central wavelengths of their passbands (0.32 nm standard deviation) in comparison to the previous passive filter structures. We attribute these increased misalignments to larger nonuniformities resulting from the larger distance between the stages in the filter layout as made necessary to fit the electrical signal lines (see Fig. 5) as well as to the additional processing steps that may further increase variability. As depicted in Fig. 6(a), the wavelength misalignments between stages increase the SSL (e.g. −6 dB for the filter in D9). In these structures, the thermal phase shifters allowed for the correction of the fabrication induced deviations between stages. The heaters exhibit a resistance of around ≈80 Ω per resonator, with measured resistances of respectively 24.24 Ω, 20.88 Ω and 17.57 Ω for the 3, 4, and 5 resonator stages. The required tuning power is ≈2.6 mW/nm/resonator and 31 mW/nm for the whole structure. The measurement results of the filter structures with heaters are summarized in Table 2. It should be noted that, even after thermal tuning of the three CROW stages, the residual SSL is still slightly worse than the −12.6 dB design value. This points to the fact that, after tuning, the SSL remains limited by variability of the ring wavelengths inside a single CROW stage, that is not compensated for. As previously, the assumed 0.3 nm ring resonance std. dev. within a single stage allows mimicking the experimental results.
Notably, the device on D8 exceeds SSL performance expectations. Since BW P is also an outlier for this device (7.3 m vs. 7.7 nm for the other devices), it appears very likely that a higher coupling strength is the root cause for both deviations in this specific instance.
Finally, Fig. 7 shows the normalized spectrum of a typical comb laser used as a multicarrier light source for WDM communications in [1,2] before (red dashed line) and after being filtered by the proposed three stage CROW filter (solid blue line). In this example, it can be seen that the filter allows for the selection of 9 adjacent lines and introduces a small insertion loss to each one (less than 0.5 dB in the worst channels closest to the edges of the passband). On the other hand, the comb lines outside of the passband undergo a rejection of higher than 12 dB (worst case extinction in the stopband). Since outer comb lines already carry less power by nature of being at the periphery of the MLL's gain spectrum (with a cumulative power −4.2 dB below the power carried by the 9 central lines), and since some lines also overlay with the zeros of the transfer function rather than with the side lobe maxima, after filtering the total power carried by the peripheral comb lines is actually 20 dB below the power carried by the 9 central lines intended to be used as carriers. Thus, the power overhead entering the SOA is significantly reduced.
While a 20 dB rejection might seem large, one has to take into account that the average power of the 9 central carriers undergoes an additional extinction of −13.8 dB after modulation by RRMs in the intended WDM system [2], as a consequence of low drive voltages of 2 V pp sourced by chip scale driver electronics combined with a requirement for high extinction. This results in the peripheral lines carrying −6 dB of the average power of the 9 optical carriers at the entrance of the SOA. A guaranteed worst case extinction of 16 dB applied uniformly across the entire stopband would also have resulted in a guaranteed overall power ratio slightly better than −6 dB at the entrance of the SOA and would thus be adequate even in the worst-case alignment. While the worst case −11.2 dB extinction of the present filter would result in an overall power ratio of -1.6 dB at the entrance of the SOA if applied across the entire stopband and would thus be marginal in a worst-case yield analysis, the actual extinction is much higher as described above. Moreover, this scenario, in which all or most of the comb lines coincide with the side lobe maxima of the filter, is actually impossible due to the comb spectrum FSR differing from the spacing between the zeroes. Moreover, it should be noted that some amount of unmodulated power entering the partially saturating SOA can be beneficial in order to stabilize its gain [2] and reduce cross-gain modulation, so that an overall power ratio of −6 dB is considered fully adequate here. Fig. 7. Normalized spectrum of a typical comb laser used as a multi-carrier light source for WDM communications in [1,2] before (red dashed line) and after transmission through the proposed three stage CROW filter (blue solid line). The filter response in also plotted with a black solid line.

Conclusions
We have shown novel wideband optical filters based on three cascaded CROW stages that combine low insertion losses, low passband ripples, steep passband edges and reasonable extinction without requiring independent thermal tuning of individual resonators. Moreover, the proposed solution avoids issues associated to proximity effects during fabrication and to coupling induced resonance shifts during the design phase that typically play an important role in apodized CROW filters. Fabrication of the demonstrated three stage CROW filter with 248 nm DUV optical lithography resulted in adequate performance for the targeted WDM architecture: The fabricated filters exhibit a wide passband of 7.7 nm and an FSR of 14.7 nm. Measured passive filters feature a side lobe suppression level better than −8.2 dB. Additionally, the incorporation of thermal tuners for the adjustment of the individual stages allows an improvement of the side lobe suppression level to better than −10 dB across all measured devices. These filters were developed specifically to be part of integrated transceivers using comb light sources for the selection of 9 adjacent carriers located on a 100 GHz grid. Very low loss is applied to the selected comb laser lines (<0.5 dB). Out-of-band comb lines are attenuated by an average −16 dB, since some of the rejected comb lines fall in between side lobe maxima (the FSR of the MLL differs from the spacing between zeros in the filter transfer function).
By varying the common coupling strength between resonators throughout the device and by varying the number of stages and/or the number of resonators per stage, different requirements in terms of passband width, edge steepness, and side lobe suppression levels can be reached without having to recharacterize process biases at every design cycle, reducing risk and development time.