Ultracompact and large-bandwidth silicon modulator in a CMOS-compatible foundry

Domain: physics
Published: 2025-12-18
Editorial outcome: major_revision

Abstract
The exponential growth of global data traffic demands transformative solutions for data centers. While optical interconnects offer a promising pathway to overcome the bandwidth limitation, conventional electro-optic modulators face a bandwidth-footprint trade-off. Here, we demonstrate a silicon modulator by leveraging the slow-light effect in a photonic crystal nanobeam cavity. The proposed modulator exhibits a 110-GHz electro-optic bandwidth with an ultracompact footprint of 10 µm² while enabling precise light manipulation with an 80 pm/V tuning efficiency within a 0.12 µm³ mode volume. We also demonstrate 110-Gbps and 130-Gbps non-return-to-zero signal transmissions with bit error rates less than 3.8×10⁻³ and 2×10⁻², respectively, while maintaining a power consumption of 5.9 fJ/bit. This breakthrough represents an important advance in photonic interconnect technology, as it overcomes the critical trade-off between bandwidth and footprint while achieving an ultra-high energy efficiency, there by paving the way for next-generation ultracompact, high-speed, and energy-efficient optical interconnects in data centers and beyond.

Introduction
The continuous expansion of cloud computing and artificial intelligence infrastructures1 has precipitated an unprecedented increase in global data traffic, and the fundamental limitations of copper-based electrical interconnects in modern hyperscale data centers and exascale computing architectures have been exposed. This technological impasse has intensified the search for photonic alternatives capable of sustaining terabit-level throughput with energy-efficient operation. Among various photonic integration platforms, silicon photonics stands out as a promising candidate2, with the merits of a compact footprint and compatibility with mature complementary-metal–oxide–semiconductor (CMOS) process technology. These attributes enable wafer-scale integration of photonic-electronic systems, making the silicon platform a cornerstone for next-generation optical networks.

Central to the silicon photonic revolution is the development of high-performance electro-optic (EO) modulators3–7, which have been extensively investigated by leveraging the plasma dispersion effect. Mach–Zehnder interferometer (MZI) modulators (MZMs) typically exhibit a large footprint, with phase shifter arms extending to the millimeter scale to accumulate sufficient phase differences8–13. For example, in 2021, a silicon MZM with a footprint of 0.9 mm × 2.2 mm was presented, which allowed 220 Gbps/λ net rate operation14. The large footprints of MZMs pose challenges for achieving high-density integration and low-power consumption. To address these issues, various approaches have been explored, including the use of organic-silicon hybrid platforms15–17, slow-wave enhancement18,19, and heterogeneous integration with other materials20–22. These strategies aim to enhance the interactions between the optics and materials, thereby shrinking the footprint and improving the tuning efficiency. Recently, Han et al.23 demonstrated a high-speed slow-light silicon MZM with a large EO bandwidth of 110 GHz, in which the phase shifter length was reduced to only 124 µm. Nevertheless, owing to the constraint imposed by the width of the traveling-wave electrodes, further reduction of the footprint remains infeasible. Alternatively, microring resonator (MRR)-based modulators24–27 (MRMs) have advantages in terms of compactness and energy efficiency. Xu et al.28 demonstrated the first silicon MRM driven by a p-i-n junction. However, the operation speed was limited by the long free-carrier lifetime (~1 ns) within the diodes2. In 2022, a carrier depletion-based resonator modulator29 was proposed, achieving an EO bandwidth larger than 67 GHz. Chan et al.30 demonstrated efficient 330-Gbps modulation using a silicon MRM with a 50-GHz bandwidth. Despite extensive efforts, the requirements for integrated optical modulators are becoming increasingly stringent as the demands for high capacities and compact footprints generally continue to grow31. The simultaneous realization of an ultracompact footprint and a large bandwidth in a silicon modulator while maintaining compatibility with standard CMOS manufacturing processes remains a significant challenge. Addressing this challenge requires innovative approaches that leverage advanced photonic structures to enhance light‒matter interactions within a compact device footprint and theoretical analysis to broaden the operation bandwidth.

A one-dimensional (1D) photonic crystal nanobeam cavity (PCNC) has emerged as a promising solution, offering an ultracompact footprint and a wavelength-scale mode volume32–35. Unlike MZIs and MRRs, which exhibit a constant optical distribution in their waveguides, the PCNC features a high electromagnetic energy density and a high optical concentration factor36. These unique properties significantly enhance the interactions between light and materials, enabling precise control of the optical field within a smaller active material volume. In fact, the effective mode volume in a 1D PCNC is orders of magnitude lower than that in MRRs37, making it an ideal candidate for compact and efficient modulators. However, previous silicon modulators utilizing PCNCs achieved bandwidths and modulation rates below 40 GHz and 70 Gbps38–40, respectively, highlighting the need for further theoretical exploration of the interplay between the Q factor, tuning efficiency, and EO bandwidth, which is critical for advancing future high-speed optical interconnect technologies.

In this work, we investigate the slow-light enhancement in an optical resonator and present a comprehensive theoretical framework for designing a resonance-based EO modulator with a large bandwidth. Notably, the analytical method proposed here is not limited to silicon modulators but is also applicable to other materials, including LiNbO3 (LN)41, BaTiO3 (BTO)42, and EO polymers43. Building on the slow-light effect and theoretical analysis, we design and fabricate a PCNC-based EO modulator in a standard silicon photonic commercial CMOS-compatible foundry. With the slow-light enhancement in the resonant cavity, the inherently weak light‒matter interactions are significantly amplified, yielding a 3.36-fold improvement in phase detuning compared with conventional silicon rib waveguide cavities in our simulation, which were designed with identical waveguide width, length, and slab thickness. Importantly, the slow-light effect amplifies the phase shift per unit voltage rather than directly altering the material's electro-optic properties. This enhancement, combined with an appropriate doping concentration provided by the commercial foundry, enables a tuning efficiency of 80 pm/V for our optimized PCNC. Through precise control of the resonator's dynamic response via carrier-wavelength tuning, we exploit the optical peaking enhancement44 to optimize the trade-off between photon-lifetime and resonance conditions44–47. This approach effectively extends the EO bandwidth from 59 GHz to 110 GHz with only a 1.84 dB roll-off within an ultracompact footprint of 10 µm². To the best of our knowledge, the demonstrated modulator has an EO bandwidth exceeding 110 GHz with the most compact device footprint on the silicon platform. As an application example, the proposed modulator is operated with a non-return-to-zero (NRZ) signal of up to 110 GBaud, with a bit error rate (BER) less than 3.8×10⁻³, setting a benchmark for compact, high-performance photonic devices.

Results
Device principle and design
The core objective of this work is to realize a compact and large-bandwidth modulator on the silicon photonic platform to address the critical demands for future high-density integration and high-speed single-lane data transmission. To achieve this, we propose a 1D PCNC with a tailored grating unit cell, leveraging the slow-light effect to enhance light‒matter interactions within the resonant cavity. Below, we systematically elucidate the design principles and performance trade-offs underpinning this device.

Universally, the overall EO bandwidth (BW_EO) of a resonance-based modulator is governed by the interplay between the optical and electrical bandwidths, as defined by48

1/BW_EO² = 1/BW_O² + 1/BW_E² (1)

where the optical bandwidth (BW_O) is inversely proportional to the Q factor of the cavity, and the electrical bandwidth (BW_E) depends on the resistance‒capacitance (RC) time constant of the driving circuit (Supplementary Information Note 1). To realize a large EO bandwidth, BW_E and BW_O should be simultaneously optimized. While a low Q factor inherently broadens the BW_O, it simultaneously necessitates greater phase detuning to achieve a defined dynamic extinction ratio (ER). However, in high-speed EO modulation on the silicon platform, it is typically implemented using a reverse-biased PN junction, whose tuning efficiency is generally constrained by the specifications of a standard CMOS-compatible foundry process. Consequently, a low Q factor in the resonator leads to a reduction in the dynamic ER. This reduction can significantly lower the signal quality and increase the BER, which is a critical consideration for practical applications. To mitigate the adverse effects of a low Q factor on the dynamic ER, an effective strategy is to improve the tuning efficiency of the modulator, thereby optimizing its performance under these constraints.

In Supplementary Information Note 2, we systematically analyze the interplay between the dynamic ER, EO bandwidth, Q factor, and tuning efficiency. Through rigorous optimization, we identify a balanced trade-off by setting the Q factor to 3000, corresponding to a photon-lifetime of τ = Qλ/2πc = 2.5 ps. This approach limits the optical bandwidth to 63.7 GHz, yielding an EO bandwidth of 60 GHz (obtained from Eq. 1). To extend the modulation bandwidth, the optical peaking enhancement can be utilized in the modulator as reported in previous works5,41, which is achieved by strategically detuning the operating wavelength from the cavity’s resonance peak. Additionally, as shown in Fig. 1a, the dynamic ER is directly affected by the tuning efficiency with a given swing voltage (Supplementary Information Note 2). To achieve a dynamic ER greater than 2 dB while maintaining the intrinsic EO bandwidth of 60 GHz, the tuning efficiency must be increased to approximately 70 pm/V, which is several times higher than that in a previous work40.

To significantly improve the tuning efficiency of the modulator, particularly in devices leveraging the plasma dispersion effect, the slow-light effect49–51 has been widely adopted. The accumulated phase detuning in the slow-light waveguide is proportional to the group index n_g, as described by23

ϕ = η · n_g · L · 2π/λ_0 (2)

where η = η_0 · (V + V_bi)^(-1/2), with η_0 being a coefficient related to the overlap between the carriers and the optical field; V_bi is the built-in voltage of the PN junction; V is the applied voltage of the modulator; and L is the optical travel length. This relationship underscores the importance of the slow-light effect in improving the tuning efficiency.

Hence, we introduce the slow-light effect into a compact PCNC, which is designed on a 220-nm silicon-on-insulator (SOI) platform with a 90-nm slab layer to facilitate doping and electrical contact, as shown in Fig. 1b. The PCNC consists of two mirror regions and two taper regions. The mirror regions are carefully designed with a period to create a photonic bandgap in the region of interest (ROI). The unit cell of the mirror regions is illustrated in Fig. 1d. To obtain a resonance of transverse electric mode (TE-A), the period of the taper regions linearly decreases from a to 0.93a over 10 periods (details are provided in Supplementary Information Note 3), shifting the energy band to shorter wavelengths. The band diagram of the center unit cell in the taper regions is depicted in Fig. 1e, covering the ROI with the TE-A energy band. As shown in Fig. 1f, light is progressively slowed within the resonant cavity with a group index of 4.2 ≤ n_g ≤ 44. For reference, the dashed line indicates n_g = 3.9 for a conventional silicon rib waveguide cavity with identical waveguide width, length, and slab thickness design. The group index is calculated from the band diagram of the i-th cell using the well-known formula

n_g = (c/ω) · (dk/dλ) = (λ/2π) · (dk/dλ) (3)

where λ is the wavelength and k is the wave vector. In the i-th cell of the resonant cavity, the phase accumulation is enhanced by the slow-light effect by a factor of n_g(i). Thus, in such a designed PCNC, the total phase accumulation can be derived as follows:

ϕ = η · (2π/λ_0) · Σ_(i=-5)^5 n_g(i) · a(i) (4)

where n_g(i) and a(i) are the group index and period of the i-th cell of the PCNC, respectively. Notably, the summation range is determined by the specific energy bands and bandgaps of the device (detailed in Supplementary Information Note 7). In contrast to conventional Fabry–Pérot (FP) cavity modulators that depend exclusively on mirror reflectivity for optical enhancement, our design incorporates an engineered slow-light region to combine both enhancement mechanisms: the FP resonance boosts light‒matter interaction through recirculating passes, and the slow-light structure offers enhancement by extending the effective interaction length50,52, leading to increased tuning efficiency. In addition, the optical field is tightly confined within a mode volume of 0.12 µm³, enabling precise control over the light in a compact active material region.

A 3D schematic diagram of the proposed modulator is depicted in Fig. 1g. A PN junction is laterally integrated on the silicon rib waveguide, matching the length of the PCNC to improve the tuning efficiency. The cross section of the modulator, depicted in Fig. 1h, features a slightly wider p-doped region relative to the n-doped region, which increases the overlap between the depletion region and the optical field. Heavily doped regions (P++ and N++) are used to achieve ohmic contact between the metal vias and the silicon layer. The resistance and capacitance of the PN junction are designed to be 150 Ω and 5.9 fF, respectively, resulting in an RC-limited bandwidth of 180 GHz (Supplementary Information Note 2).

Fabrication and static characterization
The device was fabricated on a 220-nm SOI platform in a standard CMOS-compatible foundry at Advanced Micro Foundry (AMF) in Singapore (see “Methods” for details of device fabrication). Figure 2a presents an optical microscope image of the fabricated modulator. To ensure efficient impedance matching and light coupling, a 50-Ω resistor and grating couplers (GCs) from the AMF process design kit (PDK) were integrated into the design. A detailed view of the device is shown in Fig. 2b. The static EO response of the modulator was characterized by measuring the transmission spectra under reverse bias voltages ranging from −1 V to −5 V, as illustrated in Fig. 2c. A DC probe was employed to apply the voltage to the bond pad. Under a reverse bias, the electron and hole concentrations in the PN junction decrease, leading to a change in the effective refractive index of the silicon waveguide. This results in a red shift of the resonance wavelength of the PCNC due to the effective increase in the cavity length. Key performance metrics were measured, including a 3-dB bandwidth of 0.53 nm, an insertion loss (IL) of 2.8 dB, and a Q factor of 2930. As the reverse bias voltage increases, the depletion region of the PN junction widens, further reducing the carrier concentration and slightly decreasing the IL of the PCNC. At a reverse bias voltage of −3 V, the tuning efficiency was approximately 80 pm/V. This value was obtained from a 160 pm spectral red shift measured from −2 V to −4 V, corresponding to a V_π·L product of ~0.38 V·cm7. Compared with previous work40, the tuning efficiency is improved by 8 times with the slow-light enhancement in our proposed device. These results demonstrate the ability of the device to achieve efficient wavelength tuning with low driving voltages, highlighting its potential for high-performance modulation in integrated photonic circuits.

High-speed operation characterization
To assess the high-speed modulation performance of the proposed device, an experimental setup to measure the small-signal response (S_21) was designed, as illustrated in Fig. 3a. A tunable continuous-wave laser (Santec TSL-710) served as the light source, with its optical power amplified by an erbium-doped fiber amplifier (EDFA). Another EDFA was used to amplify the optical signal modulated by the PCNC modulator to compensate for the optical IL. After that, an optical bandpass filter (OBPF) was employed to reduce the amplified spontaneous emission noise. Prior to detection by a 90 GHz photodetector (PD, XPDV4120R-WFFP), a variable optical attenuator (VOA) was utilized to adjust the received optical power. The output signal from the PD was analyzed using a 110 GHz bandwidth lightwave component analyzer (LCA, Keysight N5227B). A high-speed electrical signal with a sweeping frequency from the LCA was applied to the modulator through a 110-GHz radio frequency (RF) probe. A bias-Tee (SHF BT65R-B) was used to apply a reverse voltage of −3 V to the silicon modulator. In Fig. 3b, the measured S_21 responses, normalized to the lowest EO response loss, are presented for the carrier wavelength detuning points corresponding to 1, 2, and 3 dB down from the on-resonance transmission maximum power level. The 3-dB EO bandwidths were measured to be 59 GHz, 97 GHz, and >110 GHz at detuning points of −1 dB, −2 dB, and −3 dB, respectively. The EO bandwidth of other devices with the same design were also tested with the experiment results detailed in the Supplementary Information Note 4. The results show a good uniformity, and the bandwidth broadening is available at detuning points of higher optical insertion loss. To analyze the peaking enhancement of the EO response of the PCNC-based modulator, a small-signal model was utilized, similar to that in our previous work41. By applying first-order perturbation theory, the small-signal response of the proposed modulator is derived to be

S_21,EO = 2τ_r / (iω_m + τ_r^(-1) + iδω_r) · (iω_m + τ_r^(-1) + iδω_r)* / (iω_m + τ_r^(-1) + iδω_r) · (iω_m + τ_r^(-1) + iδω_r)* (5)

where δω_r is the resonant angular frequency change when a small voltage is applied, A is the optical field traveling inside the PCNC, τ_r is the radiation coupling coefficient between the PCNC and the cladding, and ω_m, ω_0, and ω_r represent the modulation frequency, input light frequency, and resonance frequency, respectively.

To evaluate the data modulation performance of the silicon modulator, a high-speed data generation experiment was conducted. The experimental setup is shown in Fig. 4a. An electrical data signal with a (2^13-1)-bit pseudorandom binary sequence (PRBS) was applied to the modulator, which was generated by a 256-GSa/s arbitrary waveform generator (AWG, Keysight M8199A) and then amplified by an electrical amplifier (EA, SHF S807C) with a peak-to-peak voltage of ~2 V_pp. A 67-GHz RF cable and a high-speed probe were used to inject the electrical signal into the modulator. A trigger signal was employed for clock recovery to generate eye diagrams of the received data captured by an electrically sampled oscilloscope (OSC, Keysight N1030A). The eye diagrams for the NRZ signals at 50, 70, 90, and 110 Gbps are shown in Fig. 4b–e, with signal-to-noise ratios (SNRs) of 8.23, 6.07, 3.98, and 3.17 dB, respectively. The measured ERs of the open eyes were 1, 0.9, 0.7, and 0.6 dB, respectively. The lower-than-expected dynamic ERs are attributed to the EO response of the proposed PCNC modulator and the electrical signal attenuation27 through the bias-Tee, RF cable, and high-speed probe, a detailed analysis is illustrated in the Supplementary Information Note 5.

Additionally, a BER measurement under different data rates was performed (see “Methods” for details). Figure 4f shows the BER curves of the 70 Gbps, 90 Gbps, 110 Gbps and 130 Gbps NRZ signals with variations in the receiving power. With properly received optical power, the BERs are below the hard-decision forward error coding (HD-FEC) threshold (3.8×10⁻³) at a data rate of 110 Gbps and below the soft-decision forward error coding (SD-FEC) threshold (2×10⁻²) at a data rate of up to 130 Gbps. These results demonstrate the ability of the device to support high-speed data modulation, making it a promising candidate for next-generation high-density photonic integrated circuits.

Methods
Numerical simulation
The band diagrams of the mirror regions and the center cell of the PCNC were simulated by the finite element method. The transmission spectra, Q factor, and mode volume of the PCNC were simulated and optimized using a 3D finite-difference time-domain (FDTD, Lumerical FDTD Solutions) tool with perfectly matched layer boundary conditions.

Device fabrication
The device was fabricated on a 220-nm SOI platform in a standard CMOS-compatible foundry at AMF in Singapore. The foundry platform provided three etch depths to define the grating couplers, slab waveguides, and strip waveguides using three mask layer processes. The grating couplers and slabs were protected by a photoresist to prevent additional silicon etching during the grating coupler and slab processes. Eight implantations were performed for the high-speed modulators and germanium (Ge) PDs. Additionally, two metal layers and vias were used for electrical connection. The static EO performance of the fabricated switch was characterized using a tunable laser (Keysight 81960A), an optical power meter (Keysight N7744A), and a source meter (Keithley 2450). To demonstrate the EO tuning process, a DC probe was used to connect the source meter and electrodes to apply different voltages.

Setup for high-speed BER measurement
In the high-speed BER test, the experimental equipment was connected as shown in Fig. 4a, with the difference being the specific models of the equipment. A tunable continuous-wave laser (Santec TSL-570) served as the light source. The NRZ signal was applied to the modulator at various baud rates, and this signal was generated by a 256-GSa/s arbitrary waveform generator (AWG, Keysight M8199A) and then amplified by an electrical amplifier (EA, SHF 804A). A 67-GHz RF cable and a high-speed probe were used to inject the electrical signal into the modulator. A trigger signal was employed for clock recovery to generate eye diagrams of the received data captured by an electrically sampled oscilloscope (OSC, Keysight UXR0704A).

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The authors describe a slow light silicon photonics modulator.\nThe authors describe an interesting device concept but unfortunately there is unclarity in main aspects of the paper. Also, the\ndemonstrated device properties are not standing out sufficiently to merit publication in Nature Communications.\nAdditional comments are listed below:\nIn the introduction the authors write that the enhancement of the light-matter interaction is quantified as 80 pm/V. However,\nthis is not the correct means to represent the enhancement originating from a resonant effect. The resonance or slow light\nphenomenon does not alter the tuning efficiency in pm/V but the transmission change created by phase detuning.\nFurthermore, it is stated that photon-lifetime limitations are overcome by using the optical peaking enhancement. This is not\ncorrect. The optical peaking is directly related to the resonance enhancement and the photon-lifetime. These effects are\ncorrelated and cannot be optimized separately.\nThe modulator has low power consumption but even in a direct back-to-back measurement it shows a BER on the order of\n1e-3. For most applications, a much better BER is required, hence, error mitigation and recovery strategies will be required.\nSuch techniques impose large power consumption onto the optical link. Is the obtained BER viable for this application? The\nlow power consumption of the modulator is not a real advantage in case it leads to large power consumption in other parts of\nthe system.\nIn the results section it is mentioned that a reduced Q factor has a negative impact on the extinction ratio. This must not be\nthe case. A lower Q factor increases the required phase detuning to obtain a defined on/off ratio.\nThe sentence “Peaking arising from the intrinsic dynamics in the optical domain can be utilized to extend the modulation\nbandwidth” is not clear. Please explain.\nFigure 1a and the background of the curves is not clear, please explain. What was simulated and why are the curves as\nshown?\nFigure 1b shows the layout and concept of the device. What is the transmission and reflection of this device? High\ntransmission is key. The reflected light in the off-resonance state will couple back to the laser and deteriorate the system\nperformance. How can this be solved?\nFigure 2c and the mentioned enhanced tuning efficiency are not clear. See also the comment above.\nBert Offrein", "reviewer_2": "Summary and Significance:\nThis manuscript presents the design, fabrication, and characterization of a silicon electro-optic modulator based on a\nphotonic crystal nanobeam cavity (PCNC). The authors aim to overcome the trade-off between operational bandwidth and\ndevice footprint, a critical challenge for optical interconnects in data centers and high-performance computing. They report\nnoteworthy results including an electro-optic (EO) bandwidth exceeding 110 GHz within an ultracompact 10 µm² footprint,\nenhanced tuning efficiency (80 pm/V) attributed to slow-light effects, and successful 110/130 Gbps NRZ data transmission\nbelow standard FEC thresholds. The device was fabricated in a commercial CMOS-compatible foundry.\nIf fully validated and practically usable, the simultaneous achievement of >110 GHz bandwidth and a ~10 µm² footprint using\na standard foundry process would represent a significant advancement in silicon photonics. The work addresses a\npersistent challenge in modulator design. The novelty appears to lie not in discovering new physical principles (PCNCs,\nslow light, and optical peaking are known concepts) but in the specific engineering and optimization combining these effects\nto push performance boundaries beyond the current state-of-the-art for silicon modulators in terms of combined speed and\ncompactness. This potentially surpasses conventional MZMs in size and MRMs in speed/footprint, while maintaining CMOS\ncompatibility, distinguishing it from potentially more complex hybrid approaches.\nStrengths:\nAddresses a highly relevant and challenging problem in optical communications: the bandwidth-footprint trade-off.\nDemonstrates effective use of slow-light enhancement in a PCNC structure to achieve high tuning efficiency (80 pm/V).\nExperimentally measures an EO S21 response extending beyond 110 GHz under specific wavelength detuning conditions,\nleveraging optical peaking.\nSuccessfully demonstrates high-speed NRZ data transmission (110 Gbps below HD-FEC, 130 Gbps below SD-FEC).\nUtilizes a commercial CMOS-compatible foundry process, enhancing the potential for practical application, scalability, and\nintegration.\nThe described simulation, fabrication, and testing methodologies generally align with expected standards within the field.\nMajor Weaknesses and Concerns:\n1. Critically Low Dynamic Extinction Ratio (ER): The most significant weakness is the reported dynamic ER, which is\nextremely low (< 1 dB at 110 Gbps). This severely hinders the practical usability in optical communication systems that\ntypically require higher ERs (>3 dB) for adequate noise margins. The explanation provided – attributing the low ER primarily\nto signal attenuation in the external RF path (bias tee, cable, probe) – is insufficiently substantiated. Crucially:\nThe citation used to primarily support this (Ref 26, Chan et al.) does not appear to describe this specific phenomenon based\non available information.\nThe other citation (Ref 23, Han et al.) actually reported a much higher dynamic ER of 2.15 dB at 112 Gbps using a related\nslow-light silicon technology, despite also using high-frequency components. Han et al. focused their loss analysis on\nintrinsic device electrical losses (length-dependent electrode/contact effects) rather than predominantly external factors.\nCiting Ref 23 therefore does not support the authors' explanation and, in fact, suggests better ER should be achievable.\nThe manuscript lacks quantitative evidence (e.g., measured RF path S-parameters, de-embedding analysis) to support the\nclaim that external losses are the dominant cause for such a drastic ER degradation (from a theoretical ~2 dB). Intrinsic\ndevice limitations cannot be ruled out.\n2. Inconsistent Energy Consumption Claim: There is a fundamental contradiction between the calculated energy\nconsumption (5.9 fJ/bit, based on the ~2 Vpp applied voltage) and the explanation for the low dynamic ER (which implies the\neffective voltage swing at the device, Veff, must be significantly lower than 2 Vpp due to the claimed attenuation). The\nenergy calculation should be based on the actual voltage swing driving the modulation (Veff). The current presentation is\nlogically flawed and requires reconciliation.\n3. Problematic Bandwidth Density Claim: The claimed record \"bandwidth density\" (>1100 Tbps/mm²) appears\nunconventional compared to standard metrics (e.g., shoreline density, aggregate chip density) and may inflate the perceived\nperformance advantage.\nThis claim requires substantial clarification, correction, justification, or removal.\n4. Limited Discussion of Operational Window: While achieving >110 GHz bandwidth is demonstrated, the data clearly\nshows this peak performance is highly dependent on specific wavelength detuning (-3 dB from resonance). The practical\nimplications of this potentially narrow optimal operating window (e.g., required laser stability, trade-offs between bandwidth\nand ER across wavelength) need more explicit discussion.\nConclusion and Recommendation:\nThe manuscript presents potentially exciting results that could significantly advance the field of compact, high-speed silicon\nmodulators. However, the work is currently undermined by several major concerns, primarily the critically low dynamic ER\nand its unconvincing explanation, the contradictory energy calculation, the flawed bandwidth density claim, and insufficient\ndetail for reproducibility.\nTherefore, I recommend Major Revision.\nTo become suitable for publication, the authors must thoroughly address the following points:\n1. Dynamic ER: Provide a rigorous, quantitative, and evidence-based explanation for the observed dynamic ER < 1 dB. This\nrequires substantiating the contribution of external RF path loss (e.g., via measured S-parameters and de-embedding)\nversus potential intrinsic device limitations. The relevance of cited references (esp. Ref 26 and Ref 23) must be correctly\nestablished, acknowledging that Ref 23 achieved significantly better ER. Discuss the practical implications of the low ER for\nsystem performance.\n2. Energy Consumption: Reconcile the energy consumption calculation with the explanation for the dynamic ER. Provide a\nrevised energy figure based on a justified effective voltage swing (Veff) at the device, or offer an alternative, consistent\nexplanation for the low ER that aligns with the applied Vpp.\n3. Bandwidth Density: Thoroughly revise or remove the \"bandwidth density\" claim. If retained, provide a clear definition of\nthe metric, justify its use and relevance compared to standard metrics.\n4. Operational Window: Include a discussion on the practical consequences of the observed strong dependence of peak\nbandwidth on wavelength detuning, addressing the usable optical bandwidth and associated performance trade-offs.\nAddressing these fundamental issues is crucial for establishing the credibility and significance of the reported results.", "reviewer_3": "In the presented paper, the authors have experimentally demonstrated a Silicon EO modulator with 110 GHz EO bandwidth\nand 10 μm2 footprint based on photonic crystal nanobeam cavity. The combination of these experimental values is stated as\nthe record point among Si-based modulators (as shown in Fig. 4g), which are noteworthy results. However, several\nconsiderations might be needed to the design optimization by 'slow-light design in optical resonator', as described in No. 1\nof the list attached. Considering the high impact of Nature Communications, clarification of several points including the\nabove one would be required. Please find the questions and comments for the authors listed below.\n1. The authors have explained the enhancement of the tuning efficiency is possible due to the 'slow-light design in optical resonator.' Although the slow-light effect is crucial for waveguide-type modulators, applying it to nanobeam cavity-type\nmodulators might not be so meaningful.\nIn the nanobeam cavity design, increasing the group index is done by reducing the difference between the mirror region and\ncavity region by, for example in Fig. 1b, making squeezing of the central region shallower. This effectively makes the group\nindex increase, at the same time the reflection by the mirror gets weaker, making the mode field distribution spread among\nthe entire nanobeam (in the longitudinal axis). Therefore, it is also possible to understand that the enhancement of tuning\nefficiency is achieved by spreading the mode in the longitudinal direction, instead of the slow-light effect. Discussions about\nthese points are not clearly shown in the current manuscript and it would be important.\nClarifying these points would also be important to discuss the trade-offs between this strategy and the footprint.\nMoreover, the authors have not clearly pointed out the strategy of designing the slow-light photonic crystals, in which at least\nseveral design parameters might be compared to show the best design. The authors have shown that the group index for the\ncenter part of the cavity is 4.2 where the light is concentrated. However, this value might not be so large as compared to\nother slow-light devices.\n2. I would like the authors to describe the detailed design of the nanobeam cavity. For example, the Q factor of the cavity can\nbe decomposed into the coupling ratio to the waveguides, absorption by the doped regions, and scattering loss. With these\nvalues, the excess loss of the device (2.8 dB) can be understood.\n3. In page 4, line 94, the authors have cited [45] as the comparison of their work to the state-of-the-art FOM. However, this\nwould not be a fare comparison because Ref. [45] reported on an IQ modulator.\n4. Supplementary Information Note IV seems to be missing in the supplementary file."}, "author_response": "Date: July 23rd, 2025\nSubject: response letter for NCOMMS-25-25057\nDear Reviewers,\nWe would like to thank you for your efforts in reviewing our manuscript entitled “Ultracompact\nand large-bandwidth silicon modulator in a CMOS-compatible foundry”. We greatly appreciate\nyour insightful suggestions, which have helped us to improve the quality and clarity of our\nwork.\nWe have carefully revised the manuscript according to your valuable comments and addressed\neach point in detail in this response letter, with the original comments in italic and our revisions\nhighlighted in red. We hope that these revisions clarify all concerns and further strengthen the\nscientific rigor and impact of our manuscript. The changes are also highlighted in the revised\nmanuscript.\nComments from Reviewer #1:\nComment 1:\nIn the introduction the authors write that the enhancement of the light-matter interaction is\nquantified as 80 pm/V. However, this is not the correct means to represent the enhancement\noriginating from a resonant effect. The resonance or slow light phenomenon does not alter the\ntuning efficiency in pm/V but the transmission change created by phase detuning.\nResponse 1:\nWe agree with the reviewer that the resonance or slow-light phenomenon alter the transmission\nchange created by phase detuning. We have carefully revised our description of the light-matter\ninteraction enhancement in the slow-light resonant cavity and clarified the relationship between\nthe phase detuning and the tuning efficiency. The tuning efficiency is commonly used to\ncharacterize modulation capability of the resonant-type modulators as reported in previous\nworks1-3. Specifically, we have revised the Introduction (Line 84−89) to better explain the\nunderlying physics as follows:\nWith the slow-light enhancement in the resonant cavity, the inherently weak light‒matter\ninteractions are significantly amplified, leading to a 3.36-fold increase in phase detuning over\nconventional silicon waveguide cavities. Importantly, the slow-light effect amplifies the phase\nshift per unit voltage, rather than directly altering the material’s electro-optic properties. This\nenhancement, combined with an appropriate doping concentration provide by the commercial\nfoundry, enables a tuning efficiency of 80 pm/V for our optimized PCNC.\n[1] Timurdogan, E. et al. An ultralow power athermal silicon modulator. Nat. Commun. 5, 1–\n11 (2014).\n[2] Zhang, W. et al. Harnessing plasma absorption in silicon MOS ring modulators. Nat.\nPhotonics 17, 273–280 (2023).\n[3] Zhang, Y. et al. 240 Gb/s optical transmission based on an ultrafast silicon microring\nmodulator. Photon. Res. 10, 1127-1133 (2022).\nComment 2:\nFurthermore, it is stated that photon-lifetime limitations are overcome by using the optical\npeaking enhancement. This is not correct. The optical peaking is directly related to the\nresonance enhancement and the photon-lifetime. These effects are correlated and cannot be\noptimized separately.\nResponse 2:\nWe sincerely appreciate the reviewer's insightful comment. The original statement regarding \"\nBy utilizing the optical peaking enhancement44 in the modulator, we overcome the photon-\nlifetime-limited bandwidth constraint, broadening the EO bandwidth from 59 GHz to 110 GHz\nwith only a 1.84 dB roll-off within an ultracompact footprint of 10 µm2\" was indeed ambiguous\nand has been revised accordingly.\nAs correctly pointed out, the optical peaking is inherently linked to the photon-lifetime and the\nresonance enhancement. In the revised manuscript (Introduction, Lines 89−93), we have\nrephrased the relevant sentences, which is now supported by references4-7. We thank the\nreviewer for pointing out this critical point to ensure theoretical rigor.\nThrough precise control of the resonator's dynamic response via carrier-wavelength tuning, we\nexploit the optical peaking enhancement44 to optimize the trade-off between photon-lifetime\nand resonance conditions44-47. This approach effectively extends the EO bandwidth from 59\nGHz to 110 GHz with only a 1.84 dB roll-off within an ultracompact footprint of 10 µm².\n[4] Notomi, M. & Mitsugi, S. Wavelength conversion via dynamic refractive index tuning of\na cavity. Physical Review A 73, 051803 (2006).\n[5] Soref, R. A. & Bennett, B. R. Electrooptical effects in silicon. J. Quantum Electron. 23,\n123–129 (1987).\n[6] Lin, Q., Painter, O. J. & Agrawal, G. P. Nonlinear optical phenomena in silicon waveguides:\nModeling and applications. Opt. Express 15, 16604–16644 (2007).\n[7] Müller, J. et al. Optical Peaking Enhancement in High-Speed Ring Modulators. Sci. Rep.\n4, 6310 (2014).\nComment 3:\nThe modulator has low power consumption but even in a direct back-to-back measurement it\nshows a BER on the order of 1e-3. For most applications, a much better BER is required, hence,\nerror mitigation and recovery strategies will be required. Such techniques impose large power\nconsumption onto the optical link. Is the obtained BER viable for this application? The\nlow power consumption of the modulator is not a real advantage in case it leads to large power\nconsumption in other parts of the system.\nResponse 3:\nWe thank the reviewer for the comments and questions, below are our clarifications regarding\nthe BER and power consumption concerns:\n(a) In our high-speed data generation measurement, the BERs can drop to well below the hard-\ndecision forward error coding (HD-FEC) threshold (3.8 × 10−3) at a data rate of 110 Gbps\nand below the soft-decision forward error coding (SD-FEC) threshold (2 × 10−2) at a data\nrate of up to 130 Gbps, respectively. The BER thresholds we adopted are widely used for\nthis transmission rate, as reported in previous works8-11.\n(b) The power consumption of the proposed modulator was estimated using a universal\nmethodology8,12-14. While this power consumption constitutes a relatively small portion\ncompared to that required for driving and signal recovery, the power needed for external\ndrivers and signal DSP circuits is similarly required for other modulators with error\ncorrections in detection. In response to the reviewers' concerns, we have revised the\nrelevant statements in the \"Discussion\" section (Line 277−279) as follows:\nThe proposed modulator achieves an EO bandwidth of 110 GHz with a 1.84 dB roll-off and an\noptical power penalty of 3 dB, which are enabled by the peaking enhancement. High-speed data\nmodulation is successfully achieved for 110 Gbps and 130 Gbps NRZ signal transmissions,\nwith BERs below 3.8 × 10⁻³ and 2 × 10⁻², respectively, while maintaining a power consumption\nof 5.9 fJ/bit, calculated as CV 2/4. Nevertheless, the modulator's intrinsic power consumption\npp\nrepresents merely a fraction of the total system power, with the signal driving and recovery\nprocesses demanding substantially higher power budgets53.\n[8] Dubé-Demers, R., LaRochelle, S. & Shi, W. Ultrafast pulse-amplitude modulation with a\nfemtojoule silicon photonic modulator. Optica 3, 622-627 (2016).\n[9] He, M. et al. High-performance hybrid silicon and lithium niobate Mach–Zehnder\nmodulators for 100 Gbit s−1 and beyond. Nat. Photonics 13, 359–364 (2019).\n[10] Xue, Y. et al. Breaking the bandwidth limit of a high-quality-factor ring modulator based\non thin-film lithium niobate. Optica 9, 1131-1137 (2022).\n[11] Xie, X. et al. A 3.584 Tbps coherent receiver chip on InP-LiNbO wafer-level integration\n3\nplatform. Light Sci. Appl. 14, 172 (2025).\n[12] Miller, D. A. B. Energy consumption in optical modulators for interconnects. Opt. Express\n20, A293-A308 (2012).\n[13] Li, M. X. et al. Lithium niobate photonic-crystal electro-optic modulator. Nat. Commun.\n11, 4123 (2020).\n[14] Yuan, Y. et al. A 5 × 200 Gbps microring modulator silicon chip empowered by two-\nsegment Z-shape junctions. Nat. Commun. 15, 918 (2024).\nComment 4:\nIn the results section it is mentioned that a reduced Q factor has a negative impact on the\nextinction ratio. This must not be the case. A lower Q factor increases the required phase\ndetuning to obtain a defined on/off ratio.\nResponse 4:\nIn fact, our intended meaning aligns with the reviewer's perspective. We have employed two\nscenarios with two various Q factors to illustrate this point, as shown in Fig. R1. Specifically,\nit is illustrated that:\n(a) At higher Q factor: The same spectral shift yields greater dynamic ER.\n(b) At lower Q factor: Achieving equivalent dynamic ER requires either larger spectral shifts,\nor higher phase detuning.\nCertainly, the insights provided by the reviewer are highly valuable. To convey the intended\nFig. R1 Signal generation with various Q factors. With the same spectral shift, dynamic ER\ncan be achieved to be a ~2.5 dB and b ~5 dB.\nmeaning more accurately, we have refined the relevant paragraphs (Line 112−120) as follows,\nhoping to address the concerns raised by the reviewer.\nWhile a low Q factor inherently broadens the BW , it simultaneously necessitates greater phase\nO\ndetuning to achieve a defined dynamic extinction ratio (ER). However, in high-speed EO\nmodulation on the silicon platform, it is typically implemented using a reverse-biased PN\njunction, whose tuning efficiency is generally constrained by the specifications of a standard\nCMOS-compatible foundry process. Consequently, a low Q factor in the resonator leads to a\nreduction in the dynamic ER. This reduction can significantly lower the signal quality and\nincrease the BER, which is a critical consideration for practical applications. To mitigate the\nadverse effects of a low Q factor on the dynamic ER, an effective strategy is to improve the\ntuning efficiency of the modulator, thereby optimizing its performance under these constraints.\nComment 5:\nThe sentence “Peaking arising from the intrinsic dynamics in the optical domain can be utilized\nto extend the modulation bandwidth” is not clear. Please explain.\nResponse 5:\nWe appreciate the reviewer’s request for clarification. The phrase \"peaking arising from the\nintrinsic dynamics in the optical domain\" refers to the optical peaking enhancement\nphenomenon where bandwidth extension is achieved by strategically detuning the operating\nwavelength from the cavity's intrinsic resonance peak. To better convey this physical\nmechanism, we have accordingly rephrased the related sentences (Line 121−128) as follows:\nIn Supplementary Information Note II, we systematically analyze the interplay between the\ndynamic ER, EO bandwidth, Q factor and tuning efficiency. Through rigorous optimization, we\nidentify a balanced trade-off by setting the Q factor to 3000, corresponding to a photon-lifetime\nof . This approach limits the optical bandwidth to 63.7 GHz, yielding an\nEO bandwidth of 60 GHz (obtained from Eq. 1). To extend the modulation bandwidth, the\n𝜏𝜏 = 𝑄𝑄𝑄𝑄⁄2𝜋𝜋𝜋𝜋 = 2.5 ps\noptical peaking enhancement can be utilized in the modulator as reported in previous works5,41,\nwhich is achieved by strategically detuning the operating wavelength from the cavity's\nresonance peak. Additionally, as shown in Fig. 1a, the dynamic ER is directly affected by the\ntuning efficiency with a given swing voltage (Supplementary Information Note II).\nComment 6:\nFigure 1a and the background of the curves is not clear, please explain. What was simulated\nand why are the curves as shown?\nResponse 6:\nWe appreciate the reviewer’s feedback and have carefully addressed the concerns regarding the\nclarity of Fig. 1a and the interpretation of the curves. Below is our detailed response:\n(a) The figure illustrates the theoretical relationship between the dynamic ER, EO bandwidth,\nQ factor, and tuning efficiency of the proposed modulator. The curves are derived from Eq.\nS1 & S7 (provided in Supplementary Information Note I & II).\n(b) As illustrated in the Supplementary Information Note II, we have redrawn the Fig. S1b\n(same as Fig. 1a) and provided a detailed explanation of how the proposed modulator\nresponds to the electrical drive signal and generates the on/off optical signal theoretically,\nas illustrated in Fig. S1a.\n(c) As addressed in our response to Comment 4, the spectral shift and Q factor collectively\ndetermine the dynamic ER. Specifically, the spectral shift is influenced by both tuning\nefficiency and swing voltage V , while the Q factor governs the EO bandwidth. That is\npp\nwhy we study the relationship between dynamic ER, EO bandwidth, Q factor, and tuning\nefficiency.\n(d) We have incorporated specific qualifying conditions in response to the reviewer's inquiries,\nincluding the carrier wavelength and the swing voltage.\nThe revisions have been added to the Supplementary Information Note II as follows:\nThe spectral curves that blueshift and redshift under the applied electric field can be expressed\nas follows:\n(S5)\n(S6)\n𝑓𝑓−(𝑄𝑄,𝑄𝑄)= 𝑓𝑓0(𝑄𝑄−∆𝑄𝑄,𝑄𝑄)\nwhere Δλ represents the wavelength detuning under a peak-to-peak driving voltage of 2 V ,\n𝑓𝑓+(𝑄𝑄,𝑄𝑄)= 𝑓𝑓0(𝑄𝑄+∆𝑄𝑄,𝑄𝑄) pp\nwhich is numerically equivalent to the tuning efficiency. As shown in Fig. S1a, the carrier\nwavelength λ is lower than the resonant wavelength λ with a detuning point, and the dynamic\nc 0\nER at a low modulation frequency can be expressed as\n(S7)\n(S7)\n𝐸𝐸𝐸𝐸𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑𝑑 = |𝑓𝑓−(𝑄𝑄𝑑𝑑,𝑄𝑄)−𝑓𝑓+(𝑄𝑄𝑑𝑑,𝑄𝑄)|\nFigure S1. Numerically calculated results. a Schematic of a Lorentzian line shape-based modulator\nthat generates the optical signal in the theoretical model. b Numerically calculated results: the intrinsic\nEO bandwidth in relation to Q factor, and the dynamic ER in relation to Q for various tuning efficiencies.\nComment 7:\nFigure 1b shows the layout and concept of the device. What is the transmission and reflection\nof this device? High transmission is key. The reflected light in the off-resonance state will couple\nback to the laser and deteriorate the system performance. How can this be solved?\nResponse 7:\nIn response, we have included the reflection spectrum of the device in Fig. S3c in the\nSupplementary Information Note III, which shows that the PCNC exhibits high reflection for\nnon-resonant light. Indeed, such reflection has an impact on the stability of the laser and should\nbe taken into the optical module design. Normally, optical transmitter modules incorporate\nisolators to mitigate back-reflection effects on the light source, as reported in15-17. The revisions\nhave been added to the Supplementary Information Note III as follows:\nThe reflection spectrum shown in Fig. S4c reveals that PCNC maintains high reflectivity\nfor non-resonant wavelengths. This parasitic reflection may critically affect both laser stability\nand system performance—an issue targeted for improvement in our ongoing research. We\nevaluate two mitigation approaches: (1) implementing optical isolation (e.g., an\nisolator/circulator before the input coupler), as indicated in Ref. 6, although this introduces cost\nand complexity penalties; or (2) developing a side-coupled PCNC-bus waveguide architecture\nthat generates MRR-like notch-filter transmission characteristics. The latter solution provides\nWDM compatibility, achieves ultra-low off-resonance reflection, and maintains a comparable\nQ factor while improving dynamic ER performance.\nFigure S4. Schematic and simulation results of the designed PCNC. a Schematic of the designed\nPCNC. The inset shows the optical field distribution obtained from FDTD simulation. b Schematic of\nthe unit cell. c Numerically simulated transmission and reflection spectra of the PCNC obtained via\nLumerical FDTD Solutions.\n[15] Mehta, N., Lee, B. G. Overcoming challenges of co-integrated optics to advance\naccelerated computing. Proc. SPIE 13372, Optical Interconnects and Packaging, 1337202\n(2025).\n[16] Umeta, K., Sawamura, T., Shiroishi, Y. & Nasu H. Characterization of QSFP and OSFP\nCPO ELS modules employing an 8-channel CWDM TOSA in practical air-cooling\nconditions. IEEE 74th Electronic Components and Technology Conference (ECTC),\nDenver, CO, USA, pp. 112-117 (2024)\n[17] Sawamura, T. et al. Development of an External Laser Source for Co-Packaged Optics.\nFurukawa Electric Review 55, 46-52 (2024).\nComment 8:\nFigure 2c and the mentioned enhanced tuning efficiency are not clear. See also the comment\nabove.\nResponse 8:\nWe would like to thank the reviewer for the constructive suggestion. In order to enhance the\nquality of the figure and include some essential information, we have made detailed adjustments\nto Fig. 2c, as outlined below:\nFig. 2 | Fabrication and static characterization of the proposed modulator.\na Optical microscope image of the fabricated PCNC modulator. b Magnified optical image of\nthe center of the proposed device. c Measured transmission spectra versus reverse bias voltage\nranging from −1 V to −5 V; the inset shows the detailed optical spectra under the bias of −2 V,\n−3 V, and −4 V, respectively. G ground, S signal.\nComments from Reviewer #2:\nComment 1:\nDynamic ER: Provide a rigorous, quantitative, and evidence-based explanation for the\nobserved dynamic ER < 1 dB. This requires substantiating the contribution of external RF path\nloss (e.g., via measured S-parameters and de-embedding) versus potential intrinsic device\nlimitations. The relevance of cited references (esp. Ref 26 and Ref 23) must be correctly\nestablished, acknowledging that Ref 23 achieved significantly better ER. Discuss the practical\nimplications of the low ER for system performance.\nResponse 1:\nWe appreciate the reviewers' careful review and constructive comments. Below are our point-\nby-point responses:\n(a) We have added an explanation for the observed dynamic ER < 1 dB in the “Discussion”\nsection and a detailed discussion in the Supplementary Information Note V, including\ncomprehensive S-parameters characterization of the electrical equipment.\n(b) We apologize for the incorrect citation of Ref. 26 where we intended to cite Ref. 27, which\nhas been corrected in the revised manuscript. We would like to clarify our rationale for\nciting Ref. 27: Its \"Table 1\" compares the degradation of dynamic ER at various\nmodulation rates.\n(c) Ref. 23 demonstrates the performance of an advanced silicon-based slow-light MZI\nmodulator. It achieves a high dynamic ER by employing a differential signal with V ~ 5\npp\nV and a slow-light-enhanced modulation region of 124 μm in length. We cited this work\nbecause it also exhibits the similar phenomenon of ER degradation with increasing the\nmodulation rate, e.g., 3.15 dB @ 100 Gbps and 2.15 dB @ 112 Gbps.\n(d) In the present work, our device represents near-theoretical-limitation performance under\nthis design scheme, with a theoretical ER of 2 dB versus the experimentally measured\nvalue of < 1 dB.\nThe revisions have been added to revised manuscript and the Supplementary Information as\nfollows:\n(i) In the “Results”, line 245−248:\nThe lower-than-expected dynamic ERs are attributed to the EO response of the proposed PCNC\nmodulator and the electrical signal attenuation27 through the bias-Tee, RF cable, and high-speed\nprobe, a detailed analysis is illustrated in the Supplementary Information Note V.\n(ii) In the “Discussion”:\nLine 279−282: High-speed data modulation is successfully achieved for 110 Gbps and 130\nGbps NRZ signal transmissions, with BERs below 3.8 × 10⁻³ and 2 × 10⁻², respectively, while\nmaintaining a power consumption of 5.9 fJ/bit, calculated as CV 2/4. Nevertheless, the\npp\nmodulator's intrinsic power consumption represents merely a fraction of the total system power,\nwith the signal driving and recovery processes demanding substantially higher power budgets53.\nBesides, it should be noted that the proposed device represents near-theoretical-limit\nperformance under this design scheme. For practical applications requiring long-distance\ntransmission, high transmission capacity, and low BERs, the device validated in this work can\nbe further optimized in future work.\nLine 287−290: However, experimental measurements revealed a dynamic ER of less than 1 dB,\nprimarily due to the EO response of the proposed PCNC modulator and the electrical signal\nattenuation27 through the 67-GHz bias-Tee, RF cable, and high-speed probe. This attenuation\nresulted in a smaller wavelength shift than anticipated.\n(iii) In the Supplementary Information Note V:\nThe dynamic ER is a critical performance metric for high-speed optical communication systems.\nIn intensity-modulated direct-detection (IM/DD) schemes, a high dynamic ER ensures minimal\nsignal distortion and an increased signal-to-noise ratio (SNR), directly affecting BER thresholds.\nTo characterize the data modulation performance of the proposed silicon modulator, we\nconducted a high-speed data generation experiment. The experimental setup is shown in Fig.\nS7a. An electrical data signal with a (213-1)-bit pseudorandom binary sequence (PRBS) was\napplied to the modulator, which was generated by a 256-GSa/s arbitrary waveform generator\n(AWG, Keysight M8199A) and then amplified by an electrical amplifier (EA, SHF S807C)\nwith a peak-to-peak voltage of ~2 V . A bias-Tee (SHF BT65R-B) was used to provide a bias\npp\nvoltage of −3 V, which is crucial for a high-speed EO modulator on the silicon platform. A 67-\nGHz radio frequency (RF) cable and a high-speed probe were used to inject the electrical signal\ninto the modulator. A trigger signal was employed for clock recovery to generate eye diagrams\nof the received data captured by an electrically sampled oscilloscope (OSC, Keysight N1030A).\nFigure S7b shows the transmitter DSP flow charts. By using Nyquist pulse shaping, the signal\nbandwidth can be suppressed to ~ half of the baud rate. The eye diagrams for the NRZ signals\nat 50, 70, 90, and 110 Gbps are shown in Fig. S7c−f, with SNRs of 8.23, 6.07, 3.98, and 3.17\ndB, respectively. The measured ERs of the open eyes were 1, 0.9, 0.7, and 0.6 dB.\nWe have meticulously analyzed the reasons for the measured dynamic ER falling lower\nthan the expected 2 dB. Figures S7g and h present the parameters of the EA and bias-Tee, as\nextracted from the manufacturer’s datasheets8,9. On the transmitter side, the electrical signal\ngenerated by the AWG for driving the modulator is not spectrally flat after amplification by the\nelectrical amplifier, revealing that the amplification power varies across different radio\nfrequencies. The amplified V subsequently experiences additional losses as it passes through\npp\nthe bias-Tee, a 30-cm 67-GHz RF cable, a 67-GHz RF probe, and contact with the bond pads.\nThese cumulative losses result in an effective V at the device that deviates from the target V ,\npp pp\nwhich reduces the dynamic ER. Furthermore, on the basis of the S21 parameters obtained from\nbandwidth measurements, although the optical peak enhancement effect significantly increases\nthe 3 dB electro-optic bandwidth of the device, it concurrently results in an inherent increase in\nthe IL of the S21 parameter. This additional EO response loss consequently causes a reduction\nin the dynamic ER of the eye diagram. This phenomenon directly affects the dynamic ER and\nhighlights the challenges in maintaining signal integrity across high-frequency transmission\npaths.\nNotably, our device achieves near-theoretical-limitation performance under this design\nscheme. For practical applications requiring long-distance transmission, high transmission\ncapacity, and low BERs, the device validated in this work can be further optimized in future\nwork. First, by side-coupling the PCNC with a bus waveguide, a notch-shaped transmission\nspectrum, similar to that of an MRR, can be achieved. This spectral profile has advantages in\nachieving higher ERs and can be used in wavelength-division multiplexing (WDM) systems to\nincrease transmission capacity. Additionally, optimizing the design of the PCNC to increase the\nQ factor of the resonance peak provides a pathway for achieving higher dynamic ERs. However,\nthis Q factor improvement method requires significant optical power trade-offs when the EO\nbandwidth is extended, presenting a fundamental limitation in the device design optimization\nprocess.\nFigure S7. High-speed data modulation measurement and the S21 of experiment setup. a\nExperimental equipment for measuring optical eye diagrams. b Transmitter DSP flow charts. Eye\ndiagrams for NRZ signals at data rates of c 50 Gbps, d 70 Gbps, e 90 Gbps and f 110 Gbps. Typical S21\nparameters of g EA (SHF S807C), h bias-Tee (SHF BT65R-B), i 30-cm 67-GHz RF cable and j 67-GHz\nRF probe.\nComment 2:\nEnergy Consumption: Reconcile the energy consumption calculation with the explanation for\nthe dynamic ER. Provide a revised energy figure based on a justified effective voltage swing\n(Veff) at the device, or offer an alternative, consistent explanation for the low ER that aligns\nwith the applied Vpp.\nResponse 2:\n(a) Regarding the characterization of the modulator's power consumption, we adopted a\nwidely recognized methodology along with the calculation formula for NRZ signals8,12-\n14,18. While the reviewer's observation on studying the effective voltage applied to the\ndevice is highly valuable, establishing this parameter would require more meticulous\nfuture investigation.\n(b) Concerning the causes and analysis of the low dynamic ER, we have incorporated a more\nin-depth discussion and explanation in the Supplementary Information Note V as\nsuggested by the reviewer, including the S21 parameters of equipment used in the\nexperiment, please refer to Response 1 for detailed explanation.\n[8] Dubé-Demers, R., LaRochelle, S. & Shi, W. Ultrafast pulse-amplitude modulation with a\nfemtojoule silicon photonic modulator. Optica 3, 622-627 (2016)\n[12] Miller, D. A. B. Energy consumption in optical modulators for interconnects. Opt. Express\n20, A293-A308 (2012).\n[13] Li, M. X. et al. Lithium niobate photonic-crystal electro-optic modulator. Nat. Commun.\n11, 4123 (2020).\n[14] Yuan, Y. et al. A 5 × 200 Gbps microring modulator silicon chip empowered by two-\nsegment Z-shape junctions. Nat. Commun. 15, 918 (2024).\n[18] Han, C. et al. Slow-light silicon modulator with 110-GHz bandwidth. Sci. adv. 9, 42,\neadi5339 (2023).\nComment 3:\nBandwidth Density: Thoroughly revise or remove the \"bandwidth density\" claim. If retained,\nprovide a clear definition of the metric, justify its use and relevance compared to standard\nmetrics.\nResponse 3:\nWe appreciate the reviewer's comment regarding the \"bandwidth density\" discussion. After\nrigorous reevaluation, we have decided to remove the \"bandwidth density\" metric from the\nrevised manuscript to prevent potential ambiguities.\nSpecifically, line 96−97:\nTo the best of our knowledge, the demonstrated modulator has an EO bandwidth exceeding 110\nGHz with the most compact device footprint on the silicon platform. As an application example,\nthe proposed modulator is operated with a NRZ signal of up to 110 GBaud, with a BER less\nthan 3.8 × 10−3. As a result, we achieve a record-high bandwidth density45 of >1100 Tbps/mm²\n(110 Gbps/10 µm²) for an intensity modulator, setting a new benchmark for compact, high-\nperformance photonic devices.\nComment 4:\nOperational Window: Include a discussion on the practical consequences of the observed\nstrong dependence of peak bandwidth on wavelength detuning, addressing the usable optical\nbandwidth and associated performance trade-offs.\nResponse 4:\nWe appreciate the reviewer's comment regarding the operational wavelength window, which is\na crucial performance metric for optical modulators. In response, we have expanded our\ndiscussion in the Supplementary Information Note IV as follows:\nTo effectively enhance the EO bandwidth of the proposed modulator, we employ the optical\npeaking enhancement7 strategy to optimize the trade-off between the photon lifetime and\nresonance conditions, thereby significantly broadening the EO bandwidth. However, this\napproach introduces an inherent trade-off with increased ILs, requiring a careful balance\nbetween bandwidth expansion and optical ILs adapted to specific application requirements.\nTo quantify the operational window of the EO bandwidth, we systematically characterized\nseveral fabricated devices, as shown in Fig. S6. The experimental results indicate good\nuniformity and validate the effectiveness of this strategy. Specifically, within a wavelength\ndetuning range of 0.3 nm, the additional optical IL is controlled between 2 dB and 4.8 dB,\nachieving a balance between bandwidth enhancement and optical efficiency. These findings\nhighlight the practical applicability of the proposed approach, particularly in scenarios where\nthe optical IL is acceptable for improved bandwidth performance.\nFigure S6. EO bandwidths of some modulators with the same design. a Experimental setup for the\nEO bandwidth measurement for 4 modulators. b-e Measured EO S responses for different wavelength\n21\ndetuning Δλ values or optical IL. TLS, tunable laser source; EDFA, erbium-doped fiber amplifier; PC,\npolarization controller; DUT, device under test; VOA, variable optical attenuator; OBPF, optical\nbandpass filter; PD, photodetector; and LCA, lightwave component analyzer.\nComments from Reviewer #3:\nComment 1:\n(a) The authors have explained the enhancement of the tuning efficiency is possible due to the\n'slow-light design in optical resonator.' Although the slow-light effect is crucial for waveguide-\ntype modulators, applying it to nanobeam cavity-type modulators might not be so meaningful.\n(b) In the nanobeam cavity design, increasing the group index is done by reducing the difference\nbetween the mirror region and cavity region by, for example in Fig. 1b, making squeezing of\nthe central region shallower. This effectively makes the group index increase, at the same\ntime the reflection by the mirror gets weaker, making the mode field distribution spread among\nthe entire nanobeam (in the longitudinal axis). Therefore, it is also possible to understand that the\nenhancement of tuning efficiency is achieved by spreading the mode in the longitudinal\ndirection, instead of the slow-light effect. Discussions about these points are not clearly shown\nin the current manuscript and it would be important.\nClarifying these points would also be important to discuss the trade-offs between this strategy\nand the footprint.\n(c) Moreover, the authors have not clearly pointed out the strategy of designing the slow-light\nphotonic crystals, in which at least several design parameters might be compared to show the\nbest design. The authors have shown that the group index for the center part of the cavity is 4.2\nwhere the light is concentrated. However, this value might not be so large as compared to other\nslow-light devices.\nResponse 1:\nWe thank the reviewer for the insightful comments regarding the slow-light effect in our\nnanobeam cavity design. Below are our point-by-point responses:\n(a) We agree that cavity-type modulators differ fundamentally from waveguide-type\nmodulators. The slow-light effect has been experimentally demonstrated to enhance light-\nmatter interactions, as well as the optical resonators19-21. While the waveguide length\nexperiencing slow-light enhancement inside a cavity-type modulator is much shorter than\nthat in conventional slow-light waveguide-type modulators, the effective optical path\nlength is significantly increased due to multiple round-trips of light within the cavity. This\nallows phase detuning accumulation of the slow-light enhancement effect.\n(b) We have added a detailed discussion of the PCNC optimization to the Supplementary\nInformation Note III. The reviewer's suggestion that enhanced tuning efficiency results\nfrom optical field distribution spreading the mode in the longitudinal direction may not\naccurately reflect the underlying mechanism. To the best of our knowledge, a reduced\noptical mode volume offers significant advantages in both power efficiency and\nmodulation performance, whereas optical field spreading would conversely increase the\nmode volume.\n(c) Following the Comment 1 from Reviewer #1, we calculated the phase detuning\nenhancement factor between our designed slow-light waveguide and a conventional\nwaveguide, yielding an improvement of 3.36-fold. Although this enhancement is relatively\nmodest compared to reported slow-light devices, it reflects a trade-off with device footprint\noptimization.\n[19] Lu, X., McClung, A. & Srinivasan, K. High-Q slow light and its localization in a photonic\ncrystal microring. Nat. Photon. 16, 66–71 (2022).\n[20] Lo S., Lee J., Weiss S. & Fauchet P. Bloch mode selection in silicon photonic crystal\nmicroring resonators. Opt. Lett. 43, 2957–2960 (2018).\n[21] McGarvey-Lechable K. et al. Slow light in mass-produced, dispersion-engineered\nphotonic crystal ring resonators. Opt. Express 25, 3916–3926 (2017).\nTo obtain a resonance with low optical intrinsic insertion loss (IL), the desired Q factor, a\ncompact footprint and especially strong slow-light enhancement using PCNC, we optimized\nthe device design via Lumerical FDTD simulations. Given that the PCNC essentially functions\nas an FP cavity, we explored the relationship between various parameters in Eq. S14–18 and\nthe reflectivity, as shown in Fig. S5. By adjusting the value of m, we achieve a trade-off between\nthe device footprint and the waveguide length of the slow-light enhancement effect. A larger m\nreduces the reflectivity of the mirror but increases the phase detuning accumulation length of\nthe slow-light effect. Therefore, the parameters of the PCNC structure were determined by\nconsidering the fabrication feature size, the effective cavity length increased by the slow-light\neffect, the desired Q factor, and the device footprint, resulting in the following values: W = 1000\nnm, a = 315 nm, t = 0.93, q = 0.78, d = 50%, m = 10, and n = 17. On the basis of this\nconfiguration, we simulate the designed PCNC, yielding an IL of 1.08 dB, an intrinsic Q factor\nof ~ 5000, and a mode volume of 0.12 µm³, which reveals the optimized performance of the\ndesigned structure. These metrics are essential for evaluating the tuning efficiency and\ncompactness of the device, ensuring its suitability for advanced applications.\nFigure S5. Simulation results of the PCNC design with various parameters. The reflectivity affected\nby a silicon waveguide width W, b unit cell pitch a in the mirror region, the ratio of the c pitch and d\nwidth of the central unit cell to that of the mirror region, the e duty cycle of each unit cell, and the f\nnumber of total unit cells in the one-sided PCNC; the g reflectivity and h footprint determined by varying\nthe number of unit cells in taper region m and the one-sided PCNC.\nComment 2:\nI would like the authors to describe the detailed design of the nanobeam cavity. For example,\nthe Q factor of the cavity can be decomposed into the coupling ratio to the waveguides,\nabsorption by the doped regions, and scattering loss. With these values, the excess loss of the\ndevice (2.8 dB) can be understood.\nResponse 2:\nWe appreciate the reviewer's comments. Regarding the reviewers’ comments on waveguide\ncoupling, absorption loss in doped regions, and scattering loss, these factors are undoubtedly\ncritical for high-speed silicon electro-optic modulators based on resonant cavities. We have\nadded a theoretical analysis in the Supplementary Information Note III to fit and calculate these\nparameters.\nWe use temporal coupled-mode theory to describe the model of the PCNC modulator. As shown\nin Fig. S3, when the light is incident from the left side with , the temporal coupled-\nmode equations become:\n𝑆𝑆2+ = 0\n(S8)\n𝑑𝑑𝑑𝑑 𝑑𝑑 𝑑𝑑 2𝑑𝑑 2\n𝑑𝑑𝑑𝑑 = −𝑖𝑖𝜔𝜔0𝑎𝑎−𝜏𝜏𝑟𝑟−𝜏𝜏𝛼𝛼− 𝜏𝜏 +𝜏𝜏𝑠𝑠1+\n(S9)\n2\n𝑠𝑠1− = −𝑠𝑠1++𝜏𝜏𝑎𝑎\n(S10)\n2\nwhere τ denotes the coupling between the 𝑠𝑠P2C−N=C an𝜏𝜏d𝑎𝑎 the input/output waveguide, τ\nr\ndenotes the\nradiation coupling between the PCNC and the cladding, and τ denotes the absorption by the\nα\ndoped regions.\nThe transmission spectrum can be expressed as\n(S11)\n4\n2\n|𝑆𝑆2−| 𝜏𝜏2\nIn this work, the intrinsi\\𝑇𝑇\nc\n(\np\n𝜔𝜔\ner\n)\nfo\n=\nrm\n|𝑆𝑆\na\n1\nn\n+|\nc\n2\ne\n=\nof ( t 𝜔𝜔 h − e 𝜔𝜔 d 0 e ) s\n2\ni + g  n𝜏𝜏\n1\n𝑟𝑟e + d𝜏𝜏\n1\n𝛼𝛼P + C\n2\n𝜏𝜏N \nof ~78% and a Q factor of ~5000, whereas the doped-PCNC presented a T of ~52.5% and a Q\nfactor of ~2930.\nCombined with\nFigure S3. Schematic of a PCNC modulator.\n(S12)\n1 𝜔𝜔0 1 1 2\nThen, we have 𝜏𝜏𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡𝑡 = 𝑄𝑄 = 𝜏𝜏𝑟𝑟+𝜏𝜏𝛼𝛼+𝜏𝜏\n(S13)\n1 −5 1 −5 1 −4\n𝜏𝜏 = 𝜔𝜔0∙8.8×10 , 𝜏𝜏𝑟𝑟 = 𝜔𝜔0∙2.4×10 , 𝜏𝜏𝛼𝛼 = 𝜔𝜔0∙1.41×10\nComment 3:\nIn page 4, line 94, the authors have cited [45] as the comparison of their work to the state-of-\nthe-art FOM. However, this would not be a fare comparison because Ref. [45] reported on an\nIQ modulator.\nResponse 3:\nWe agree with the reviewer that Ref. [45] is not appropriate for comparison here. We have\nremoved the reference accordingly.\nComment 4:\nSupplementary Information Note IV seems to be missing in the supplementary file.\nResponse 4:\nWe sincerely thank the reviewer for the careful checking and would like to apologize for our\ntypographical error. In the revised version, we have added the Supplementary Information Note\nIV.\nThank you again for your consideration.\nSincerely,\nHua Zhong & Yikai Su\nOn behalf of all the authors\nState Key Lab of Photonics and Communications\nSchool of Information Science and Electronic Engineering\nShanghai Jiao Tong University\nEmail: yikaisu@sjtu.edu.cn\nWeb: https://otip.sjtu.edu.cn/\nResponse Letter\nDate: Dec. 12th, 2025\nSubject: Response letter for NCOMMS-25-25057B (second revision)\nDear Reviewers,\nWe sincerely appreciate the time and effort you have dedicated to evaluating our manuscript\nentitled “Ultracompact and large-bandwidth silicon modulator in a CMOS-compatible\nfoundry”. We are grateful for the constructive feedback from all the reviewers during the first\nrevision, which has improved our work.\nWe are pleased that Reviewer #1 now supports publication, and we are deeply grateful for\nthe positive assessment. In this second revision, we have carefully addressed the remaining\ncomments from Reviewers #2 and #3 and incorporated their suggestions to further enhance the\nmanuscript’s rigor and clarity. Below, we provide point-by-point responses to each concern,\nwith the reviewers’ comments in italics and our revisions marked in red in both this letter and\nthe revised manuscript.\nWe believe that these revisions have resolved all the outstanding issues and hope that the\nmanuscript is now suitable for publication in Nature Communications.\nComments from Reviewer #2:\nI have carefully examined the revised manuscript together with the new Supplementary Notes\nand your point‑by‑point response. Your revision represents a clear effort to engage with the\nprevious critique: the unconventional “bandwidth‑density” claim has been removed, and a\nthoughtful discussion supported by additional data now clarifies the practical\nwavelength‑detuning window. These improvements fully resolve those two issues. Unfortunately,\nthe two most critical technical concerns—the extremely low dynamic extinction ratio (ER) and\nthe logically inconsistent energy‑per‑bit figure—remain insufficiently addressed. Because they\nstrike at the heart of the paper’s performance claims, the manuscript cannot yet be considered\nfor publication.\nComment 1:\n(a) First, on the dynamic ER, your new Supplementary Note V supplies manufacturer\nS‑parameter curves for the electrical amplifier, bias‑tee, and related components, and it\nasserts that external high‑frequency loss dominates the observed < 1 dB eye penalty.\nHowever, no quantitative de‑embedding of the complete RF chain has been performed. In\nthe absence of measured S‑parameters for the specific cables, probes, pads and device\nunder test, the reader still cannot determine how much of the voltage drop occurs externally\nand how much is intrinsic to the nanobeam modulator itself. The additional qualitative\nremark that the optical peaking mechanism “inherently increases insertion loss” may be\nphysically plausible, but without modelling or experiment it remains speculative.\n(b) Moreover, the citation of Han et al. continues to be invoked only to show that ER generally\ndegrades with increasing data rate; it does not justify an ER that is an order of magnitude\nlower than that achieved in that work under comparable drive conditions.\n(c) A rigorous, experiment‑based separation of external and intrinsic contributions is\ntherefore indispensable.\nResponse 1:\nWe sincerely appreciate the reviewer’s thoughtful analysis and valuable suggestions regarding\ndynamic ER characterization. In response to these insightful comments, we have carefully\nrevised the relevant descriptions and added additional results to Supplementary Information\nNote V.\n(a) We performed a quantitative analysis of the S21 parameter across the external RF chain,\nsimulated V in the time domain, explained the ER degradation induced by the optical\neff\npeaking mechanism through frequency-domain analysis, and simulated the dynamic ER at\nvarious data rates. On the basis of this analysis, the reader can determine how much of the\nlow ER originates from the external RF chain and how much is attributable to the\nmodulator itself. We have revised the relevant analysis accordingly and added it to\nSupplementary Information Note V.\n(b) We would like to clarify the fundamental differences between the proposed modulator and\nthe implementation in Ref. 23. While Ref. 23 employs an MZI configuration that relies on\ninterference between two phase-modulated arms, our approach utilizes resonant spectral\nshifting of the cavity transmission spectrum under an applied voltage bias—an entirely\ndistinct operating principle. Our design framework inherently targets a dynamic ER of 2\ndB, which represents near-theoretical-limitation performance for the design principle, as\ndetailed in Supplementary Information Note II. With respect to the driving conditions, Han\net al. used a 5 V drive voltage, whereas we used a 2 V drive voltage.\npp pp\n(c) We systematically simulated the influence of the external RF chain on the input driving\nsignals and the dynamic ER at the transmitter, which enables clear discrimination between\nexternal and intrinsic contributions to ER degradation.\nLines 293–295: However, experimental measurements revealed a dynamic ER of less than 1\ndB, primarily because of the EO response of the proposed PCNC modulator and the electrical\nsignal attenuation27 through the 67-GHz bias-Tee, RF cable, and high-speed probe (discussed\nin Supplementary Information Note V).\nIn Supplementary Information Note V:\nThe dynamic ER is a critical performance metric for high-speed optical communication systems.\nIn intensity-modulated direct-detection (IM/DD) schemes, a high dynamic ER ensures minimal\nsignal distortion and an increased signal-to-noise ratio (SNR), directly affecting BER thresholds.\nTo characterize the data modulation performance of the proposed silicon modulator, we\nconducted a high-speed data generation experiment. The experimental setup is shown in Fig.\nS8a. An electrical data signal with a (213-1)-bit pseudorandom binary sequence (PRBS) was\napplied to the modulator, generated by a 256-GSa/s arbitrary waveform generator (AWG,\nKeysight M8199A) and then amplified to ~2 V by an electrical amplifier (EA, SHF S807C).\npp\nThe voltage amplitude of ~2 V was verified by an electrical back-to-back measurement. A\npp\nbias-Tee (SHF BT65R-B) was used to provide a bias voltage of −3 V, which is crucial for a\nhigh-speed EO modulator on the silicon platform. A 67-GHz radio frequency (RF) cable and a\nhigh-speed probe were used to inject the electrical signal into the modulator. A trigger signal\nwas employed for clock recovery to generate eye diagrams of the received data captured by an\nelectrically sampled oscilloscope (OSC, Keysight N1030A). The transmitter digital signal\nprocessing (DSP) flowcharts are shown in Figure S8b. By using Nyquist pulse shaping, the\nsignal bandwidth can be suppressed to ~ half of the baud rate while unclear transient patterns\n2\nare introduced in the eye diagrams, as shown in Fig. S9a−d. The measured eye diagrams for the\nNRZ signals at 50, 70, 90, and 110 Gbps are shown in Fig. S8c–f, with SNRs of 8.23, 6.07,\n3.98, and 3.17 dB, respectively. The measured ERs of the open eyes were 1, 0.9, 0.7, and 0.6\ndB.\nTo quantitatively investigate the discrepancy between the measured dynamic ER and\ntheoretical expectations, we performed frequency- and time-domain studies. Figures S8g–j\npresent the measured S21 parameters of the EA, bias-Tee, and RF cable and that of the probe,\nas extracted from the manufacturer’s datasheets8,9,10. The unnormalized and normalized S21\nparameters of the RF chain are presented in Figures S8k and S8m, respectively, and the\nquantitative values across specific frequencies are provided in Table S1. The measured\nelectrical ILs of the RF chain demonstrate frequency-dependent characteristics: 1.83 dB @ 25\nGHz, 2.42 dB @ 35 GHz, 4.66 dB @ 45 GHz, and 3.88 dB @ 55 GHz. These cumulative losses\nresult in a lower-than-expected electrical driving power being delivered to the pads,\nconsequently leading to a reduced effective voltage (V ) below 2 V. In accordance with the\neff\ntransmitter DSP and the measured S21 parameters of the RF chain, we simulated V at various\neff\ndata rates via inverse Fourier transform, yielding the values of 1.80 V @ 50 Gbps, 1.65 V @\nFurthermore, on the basis of the S21 parameters obtained from bandwidth measurements\nperformed with the PCNC modulator, although optical peaking enhancement significantly\nincreases the 3-dB EO bandwidth of the modulator, it concurrently increases the low-frequency\nIL of the S21 response. In the frequency domain, this additional low-frequency attenuation\nleads to power degradation, which directly reduces the dynamic ER in the time-domain eye\ndiagram. By combining the transmission spectra under various voltage biases and the S21\ncharacteristics of the PCNC modulator with the simulated V , we further simulated the\neff\ndynamic ER at various detuning points and data rates, as shown in Table S2. Specifically, the\nsimulated eye diagrams of the modulated optical signals at the −3 dB detuning point are shown\nin Fig. S9i–m. On the basis of the simulated eye diagrams, the dynamic ER at the transmitter\nwas observed to decrease to approximately 1 dB during the high-speed data generation\nexperiment. Further degradation occurred because of the receiver's frequency response,\nultimately yielding a measured ER below 1 dB at the receiver output. Similar dynamic ERs\nhave been reported in Ref.11,12 by using the MZM configuration.\nNotably, the proposed device achieves near-theoretical-limitation performance with the\ndesign scheme. For practical applications requiring long-distance transmission, high\ntransmission capacity, and low BERs, the device validated in this work can be further optimized\nin future work. First, by side-coupling the PCNC with a bus waveguide, a notch-shaped\ntransmission spectrum, similar to that of an MRR, can be achieved. This spectral profile has\nadvantages in achieving higher ERs and can be used in wavelength-division multiplexing\n(WDM) systems to increase transmission capacity. Additionally, optimizing the design of the\nPCNC to increase the Q factor of the resonance peak provides a pathway for achieving higher\ndynamic ERs. However, this method of improving the Q factor requires significant optical\npower trade-offs when the EO bandwidth is extended, presenting a fundamental limitation in\noptimizing the device design.\nExperimental equipment for measuring optical eye diagrams. b Transmitter DSP flow charts. Eye\ndiagrams for NRZ signals at data rates of c 50 Gbps, d 70 Gbps, e 90 Gbps, and f 110 Gbps. S21 parameters for the g EA\n(SHF S807C, Ref.8), h bias-Tee (SHF BT65R-B, Ref.9), i 30-cm RF cable (measured), and j high-speed\nRF probe (Ref.10), k unnormalized S21 of the RF chain, and m normalized S21 of the RF chain.\nEA, bias-Tee, RF cable, and high-speed probe.\nFreq (GHz)\n5 10 20 25 35 45 55\nS21 (dB)\nEA 23.1 24.1 23.4 23.1 22.8 22.1 22.5\nBias-Tee -0.6 -0.62 -0.71 -0.60 -0.60 -0.67 -0.87\nRF Cable -0.56 -1.18 -1.60 -1.66 -2.01 -3.30 -2.75\nProbe -0.16 -0.23 -0.32 -0.41 -0.43 -0.56 -0.55\nTotal 21.78 22.07 20.77 20.43 19.76 17.57 18.33\nTable S2. Simulation results for dynamic ER at various detuning points and data rates.\nData rates (Gbps)\n50 70 90 110\nDetuning points\n−1 dB 1.27 dB 1.17 dB 1.09 dB 1.00 dB\n−2 dB 1.20 dB 1.13 dB 1.08 dB 1.03 dB\n−3 dB 1.16 dB 1.09 dB 1.06 dB 1.02 dB\nsignals. The unclear transient patterns are introduced by Nyquist pulse shaping. e–h Electrical signals\nafter they pass through the external RF chain. i–m Modulated optical signals at the detuning point of\n−3 dB.\nSecond, the energy‑per‑bit calculation still employs the full 2 Vpp applied at the probe tips,\neven though your own explanation for the low ER presumes a much smaller effective swing at\nthe device terminals. By continuing to quote 5.9 fJ/bit without an experimentally validated V_eff,\nthe manuscript presents mutually inconsistent statements: a large, unquantified RF attenuation\nis invoked to explain the ER, yet the energy figure silently ignores that same attenuation. Simply\nnoting that a careful study of V_eff is “future work” does not resolve the contradiction; it leaves\nthe central efficiency metric misleading and unverifiable.\nResponse 2:\nWe sincerely appreciate the reviewer's comments regarding the power consumption\ncalculations. The power consumption formula we adopted is a benchmark calculation method\nused for optoelectronic devices1-5. When evaluated at a 2 V operating voltage, this\npp\nmethodology provides an upper-bound estimation of the modulator's intrinsic power\nconsumption without considering the RF chain loss. We calculated the power consumption\nusing V , which was simulated using the experimentally measured S21 responses of the\neff\nexternal RF chain and included a discussion in Supplementary Information Note VI.\n[1] Dubé-Demers, R., LaRochelle, S. & Shi, W. Ultrafast pulse-amplitude modulation with a\nfemtojoule silicon photonic modulator. Optica 3, 622-627 (2016).\n[2] Yuan, Y. et al. A 5 × 200 Gbps microring modulator silicon chip empowered by two-\nsegment Z-shape junctions. Nat. Commun. 15, 918 (2024).\n[3] Miller, D. A. B. Energy consumption in optical modulators for interconnects. Opt. Express\n20, A293-A308 (2012).\n[4] Hsu, WC., Nujhat, N., Kupp, B. et al. Sub-volt high-speed silicon MOSCAP microring\nmodulator driven by high-mobility conductive oxide. Nat. Commun. 15, 826 (2024).\n[5] Geravand, A., Zheng, Z., Shateri, F. et al. Ultrafast coherent dynamics of microring\nmodulators. Nat. Photon. 19, 740–750 (2025).\nIn Supplementary Information Note VI:\nBy using the simulated V , as detailed in Supplementary Information Note V, the calculated\neff\npower consumption tends to decrease with increasing data rates: 4.78 fJ/bit @ 50 Gbps, 4.02\nfJ/bit @ 70 Gbps, 3.50 fJ/bit @ 90 Gbps, and 2.97 fJ/bit @ 110 Gbps. Under ideal conditions\nwith RF losses excluded, the intrinsic power consumption of the proposed modulator reaches\nan upper bound of 5.9 fJ/bit. Notably, the RF amplifier accounts for a major portion of the\npower consumption in the transmitter11.\nComments from Reviewer #3:\nComment 1:\nAlthough the authors have answered the reviewer's questions, there still are several points that\nneed to be clarified. The main concern in the current manuscript is that the physical effect of\nthe slow light is not described in sufficient detail.\n(a) Firstly, could the authors describe the physical meaning of Eq. (4) especially in connection\nwith the tuning efficiency in unit [pm/V]?\n(b) Also, it is required to explain the reason why the sum was taken from -5 to 5, and whether\nthe mirror region is included or not. Since the edge of the taper region should have very\nlarge n_g, which may go infinite, Eq. (4) may diverge in that case. Moreover, n_g cannot\n7\nbe obtained in the mirror region. Because Eq. (2) from Ref. 23 is derived for the waveguide-\ntype modulator, mathematical derivation of the accumulated phase for Fabry-Perot (FP)\ncavity would be required.\n(c) In addition, it is required to describe the physical meaning of using the slow-light effect. In\nsingle-pass devices such as waveguide modulators, the slow-light effect may be the only\nway to enhance the light-matter interaction with fixed length. However, in multi-pass\ndevices such as FP cavities, the light reflects between mirrors, and the light-matter\ninteraction is enhanced even without the slow-light effect. Therefore, to enhance the light-\nmatter interaction in FP cavities, there would be at least two ways: One is to increase the\nreflectance of the mirrors, and another is to use the slow-light effect. These explanations\nwould be required in the manuscript to clearly state the physical meaning of the slow-light\neffect in the proposed device.\nResponse 1:\n(a) Eq. 4 quantifies the phase accumulation within the resonant cavity. In FP cavities, the\nresonance wavelength is fundamentally governed by the phase-matching condition, whose\nsensitivity to the material’s refractive index varies with cavity design parameters.\nFor modulators, the input is an electrical voltage signal, while the output is an optical signal.\nIn resonance-based silicon modulators, signal modulation arises from spectral shifting, as\nillustrated in Fig. R1. The applied voltage modulates the carrier distribution in the PN junction,\ninducing refractive index changes that perturb the cavity's phase-matching condition. This\nphase-matching perturbation directly controls the resonance wavelength shift, which is our\nprimary optimization target.\nThe [pm/V] tuning efficiency precisely characterizes how effectively the electrical signal\nperturb the phase-matching condition. We achieve increased sensitivity through the engineered\nslow-light enhancement effect, as shown by the red solid curve in Fig. R1. Moreover, the [pm/V]\ntuning efficiency has been widely adopted as a standard metric for characterizing device\nmodulation performance, as shown in Table 1 of the manuscript.\nWe contend that the existing discussion (Lines 84−90) adequately captures this physical\nmechanism.\nFigure R1. Schematic illustration of the physical principles of the resonant modulator. The red\nsolid curve indicates the slow-light enhancement effect in this study.\nmatter interactions are significantly amplified, yielding a 3.36-fold improvement in phase\ndetuning compared with conventional silicon rib waveguide cavities in our simulation, which\nwere designed with identical waveguide width, length, and slab thickness. Importantly, the\nslow-light effect amplifies the phase shift per unit voltage rather than directly altering the\nmaterial’s electro−optic properties. This enhancement, combined with an appropriate doping\nconcentration provided by the commercial foundry, enables a tuning efficiency of 80 pm/V for\nour optimized PCNC.\n(b) The range of −5 to 5 corresponds to the 10 linearly graded lattice periods on each side in\nour design, where some lattices exhibit a photonic bandgap at the target wavelength. Our\nanalysis specifically considers the phase accumulation from in-band lattices within the cavity\nregion, explicitly excluding contributions from the mirror regions. To increase clarity, we have\nincorporated a detailed discussion of this methodology in the manuscript and provided\ncomprehensive theoretical derivations in Supplementary Information Note VII.\nLines 155–160: Thus, in such a designed PCNC, the total phase accumulation can be derived\nas follows:\n(4)\n2𝜋𝜋 5\nwhere are the group i𝜙𝜙nd=ex𝜂𝜂 an∙d\n𝜆𝜆0\np∙e∑rio𝑖𝑖=d− o5f𝑛𝑛 th𝑔𝑔e( 𝑖𝑖i)-t∙h𝑝𝑝 c(e𝑖𝑖l)l of the PCNC, respectively.\nNotably, the summation range is determined by the specific energy bands and bandgaps of the\n𝑛𝑛𝑔𝑔(𝑖𝑖) and 𝑝𝑝(𝑖𝑖)\ndevice (detailed in Supplementary Information Note VII).\nThe following can be found in Supplementary Information Note VII:\nThe accumulated phase detuning in the slow-light waveguide is proportional to the group index\nn , as described by13:\ng\n(S19)\n2𝜋𝜋\nwhere , with 𝜙𝜙 b=ein𝜂𝜂g∙ a𝑛𝑛 c𝑔𝑔o∙e𝐿𝐿ff∙ic𝜆𝜆i0ent related to the overlap between\nthe carriers and the optical field; V is the built-in voltage of the PN junction; V is the applied\n𝜂𝜂 = 𝜂𝜂0𝑉𝑉+𝑉𝑉𝑏𝑏𝑖𝑖 −𝑉𝑉𝑏𝑏𝑖𝑖bi 𝜂𝜂0\nvoltage of the modulator; and L is the optical travel length.\nThe proposed PCNC structure incorporates a slow-light region with linearly graded lattice\nperiodicity. In this design, each unit cell exhibits a unique group refractive index n and lattice\ng\nlength L, thereby leading to the derivation of Eq. S19 as an integral over the slow-light region:\n(S20)\n2𝜋𝜋\nThe slow-light cavity behaviour is fund\\𝜙𝜙\n=\n𝜂𝜂\n∙\n𝜆𝜆\n0\n∙\n∫\n𝑐𝑐\n𝑐𝑐\n𝑖𝑖\n𝑛𝑛\n𝑔𝑔b(y\n𝑥𝑥\nt)\n𝑑𝑑\n𝑥𝑥\nam\nen\nta\nl\nl\ny\no\nv\ne\nr\nc\n𝑐𝑐\ned\n𝑛𝑛\ngb(y\nt)h\nd\ne\nx\nengineered energy bands and\nbandgaps, as evidenced in the band diagrams of both the mirror and slow-light regions, as\nshown in Fig. S10a–b, respectively. The progressive light slowing effect within the resonant\ncavity is shown in Figure S10c, with a group index range of 4.2 ≤ n ≤ 44. For reference, the\ng\ndashed line indicates the group index of n = 3.9 for a conventional silicon rib waveguide cavity\ng\n(with identical waveguide width, length, and slab thickness design). Notably, unit cells with i >\n5 and i < −5 fall within the photonic bandgap at the target wavelength and are consequently\nexcluded from the integration domain. Thus, Eq. S20 is derived as follows:\n9\n(S21)\n2𝜋𝜋 5\n𝜙𝜙 = 𝜂𝜂∙ 𝜆𝜆0 ∙∑𝑖𝑖=−5𝑛𝑛𝑔𝑔(𝑖𝑖)∙𝑝𝑝(𝑖𝑖)\nFigure S10. Band diagrams of a the mirror regions and b the unit cell in the taper regions. c\nGroup index n ≥4.2 within the cavity along the x-axis. The dashed line indicates the group\ng\nindex of a conventional silicon rib waveguide for comparison.\n(c) We sincerely appreciate the reviewer’s insightful comments. The reviewer is correct in\nnoting that in FP cavities, light‒matter interactions can be enhanced by increasing the\nreflectance of the mirrors and by introducing the slow-light effect. In response to this valuable\ncomment, we have expanded the manuscript to include a detailed discussion explaining how\nour design incorporates a carefully engineered slow-light region to synergistically combine\nboth enhancement mechanisms.\nLines 160–164: In contrast to conventional Fabry–Pérot (FP) cavity modulators that depend\nexclusively on mirror reflectivity for optical enhancement, our design incorporates an\nengineered slow-light region to combine both enhancement mechanisms: the FP resonance\nboosts light‒matter interaction through recirculating passes, and the slow-light structure offers\nenhancement by extending the effective interaction length50,52, leading to increased tuning\nefficiency.\nComment 2:\nSecondly, could the authors describe the meaning of the red arrow in Fig. 2(c) shown as “80\npm/V @ -3V”? Because the unit of the x-axis [um], the unit [pm/V] is unmatched.\nResponse 2:\nThe expression “80 pm/V @ -3 V” indicates that the tuning efficiency of the device is 80 pm/V\nunder a bias voltage of -3 V. The tuning efficiency is defined as the spectral blueshift or redshift\nper unit voltage. In our device, as the voltage changes from -2 V to -4 V, the spectral redshift is\napproximately 160 pm. Therefore, we calculate the average tuning efficiency using Δλ/ΔV =\n160 pm/2 V = 80 pm/V to describe the tuning efficiency of the device as 80 pm/V at a bias\nvoltage of -3 V.\nLines 197–199: At a reverse bias voltage of −3 V, the tuning efficiency was approximately 80\npm/V. This value was obtained from a 160 pm spectral redshift measured from −2 V to −4 V,\ncorresponding to a V · L product of ~0.38 V·cm7.\nπ\nComment 3:\nThirdly, in supplementary Section III, the authors have shown the optimization process.\nHowever, it is unclear what objective function was employed. If we see Fig. S5(g) and (h),\nthe star symbol is plotted, but it is unclear why the plotted point is optimal. Specifically, there seems\nto be higher reflectivity in the region to the right of the star symbol in Fig. S5(g).\nWe appreciate the reviewer’s careful observation of our optimization methodology in\nSupplementary Information Note III. In fact, our optimization was not governed by a single\nobjective function but rather guided by two critical design criteria: maximizing the reflectivity\nat the target wavelength and maintaining a compact footprint. The simulation results of our\ndevice parameter optimization process are shown in Figure S5. Reflectivity is vital for the\ndesign of PCNC, which directly determines the Q factor, and it varies with wavelength, as\nshown in Figs. S5a–f.\nWith respect to Fig. S5g, we use the reflectivity at 1550 nm; thus, the reflectivity in the region\nto the right of the star symbol in Fig. S5(g) seems to be higher. We understand that this may\nhave confused the readers. To address this concern, we have added detailed simulation results\nto Fig. S6. On average, the reflectivity is greater when m = 10 than when m = 11 or 12.\nrange of 1500~1600 nm for m = 7~12 and n = a 13, b 14, c 15, d 16, e 17, f 18, g 19, and h 20.\nFinally, in the revised line 84-89 in the manuscript, it is unclear which structures were employed\nfor the comparison. In particular, it was unclear whether the comparison was made with\nexperimental results or simulations with what kind of settings.\nResponse 4:\nWe appreciate the reviewer’s question, and we understand that the original expression might\nconfuse readers. Therefore, we have revised the related description of the comparison as follows:\nLines 84–87: With the slow-light enhancement in the resonant cavity, the inherently weak light‒\nmatter interactions are significantly amplified, yielding a 3.36-fold improvement in phase\ndetuning compared with conventional silicon rib waveguide cavities in our simulation, which\nwere designed with identical waveguide width, length, and slab thickness.\nLines 148–151: As shown in Fig. 1f, light is progressively slowed within the resonant cavity\nwith a group index of 4.2 ≤ n ≤ 44. For reference, the dashed line indicates n = 3.9 for a\ng g\nconventional silicon rib waveguide cavity with identical waveguide width, length, and slab\nthickness design.\nThank you again for your consideration.\nSincerely,\nHua Zhong & Yikai Su\nOn behalf of all the authors\nState Key Lab of Photonics and Communications\nSchool of Information Science and Electronic Engineering\nShanghai Jiao Tong University\nEmail: yikaisu@sjtu.edu.cn\nWeb: https://otip.sjtu.edu.cn/\n13"}, {"round": 2, "reviewer_comments": {"reviewer_1": "Thank you for implementing the changes you made in the manuscript.\nI agree with the current state of the paper and publication thereof.", "reviewer_2": "I have carefully examined the revised manuscript together with the new Supplementary Notes and your point‑by‑point\nresponse. Your revision represents a clear effort to engage with the previous critique: the unconventional\n“bandwidth‑density” claim has been removed, and a thoughtful discussion supported by additional data now clarifies the\npractical wavelength‑detuning window. These improvements fully resolve those two issues. Unfortunately, the two most\ncritical technical concerns—the extremely low dynamic extinction ratio (ER) and the logically inconsistent energy‑per‑bit\nfigure—remain insufficiently addressed. Because they strike at the heart of the paper’s performance claims, the manuscript\ncannot yet be considered for publication.\nFirst, on the dynamic ER, your new Supplementary Note V supplies manufacturer S‑parameter curves for the electrical\namplifier, bias‑tee, and related components, and it asserts that external high‑frequency loss dominates the observed < 1 dB\neye penalty. However, no quantitative de‑embedding of the complete RF chain has been performed. In the absence of\nmeasured S‑parameters for the specific cables, probes, pads and device under test, the reader still cannot determine how\nmuch of the voltage drop occurs externally and how much is intrinsic to the nanobeam modulator itself. The additional\nqualitative remark that the optical peaking mechanism “inherently increases insertion loss” may be physically plausible, but\nwithout modeling or experiment it remains speculative. Moreover, the citation of Han et al. continues to be invoked only to\nshow that ER generally degrades with increasing data rate; it does not justify an ER that is an order of magnitude lower than\nthat achieved in that work under comparable drive conditions. A rigorous, experiment‑based separation of external and\nintrinsic contributions is therefore indispensable.\nSecond, the energy‑per‑bit calculation still employs the full 2 Vpp applied at the probe tips, even though your own\nexplanation for the low ER presumes a much smaller effective swing at the device terminals. By continuing to quote 5.9 fJ/bit\nwithout an experimentally validated V_eff, the manuscript presents mutually inconsistent statements: a large, unquantified\nRF attenuation is invoked to explain the ER, yet the energy figure silently ignores that same attenuation. Simply noting that a\ncareful study of V_eff is “future work” does not resolve the contradiction; it leaves the central efficiency metric misleading and\nunverifiable.\nBecause the manuscript’s headline claims—simultaneous > 110 GHz bandwidth, sub‑10 µm² footprint, and sub‑10 fJ/bit\nefficiency—depend critically on a believable account of both ER and electrical drive voltage, these two issues must be\nsettled with quantitative evidence. In practice this means providing a complete de‑embedding analysis of the RF path,\nreporting the verified voltage swing at the device pads, recalculating (or at least bounding) the energy‑per‑bit with that swing,\nand discussing whether any intrinsic device physics further limits the ER. Absent those corrections, the reader cannot judge\nwhether the impressive bandwidth figure has been achieved at a power cost and signal quality that are competitive with\nexisting silicon modulator platforms.", "reviewer_3": "Although the authors have answered the reviewer's questions, there still are several points that need to be clarified. The\nmain concern in the current manuscript is that the physical effect of the slow light is not described in sufficient detail. Firstly,\ncould the authors describe the physical meaning of Eq. (4) especially in connection with the tuning efficiency in unit [pm/V]?\nAlso, it is required to explain the reason why the sum was taken from -5 to 5, and whether the mirror region is included or not.\nSince the edge of the taper region should have very large n_g, which may go infinite, Eq. (4) may diverge in that case.\nMoreover, n_g cannot be obtained in the mirror region. Because Eq. (2) from Ref. 23 is derived for the waveguide-type\nmodulator, mathematical derivation of the accumulated phase for Fabry-Perot (FP) cavity would be required.\nIn addition, it is required to describe the physical meaning of using the slow-light effect. In single-pass devices such as\nwaveguide modulators, the slow-light effect may be the only way to enhance the light-matter interaction with fixed length.\nHowever, in multi-pass devices such as FP cavities, the light reflects between mirrors, and the light-matter interaction is\nenhanced even without the slow-light effect. Therefore, to enhance the light-matter interaction in FP cavities, there would be\nat least two ways: One is to increase the reflectance of the mirrors, and another is to use the slow-light effect. These\nexplanations would be required in the manuscript to clearly state the physical meaning of the slow-light effect in the\nproposed device.\nSecondly, could the authors describe the meaning of the red arrow in Fig. 2(c) shown as “80 pm/V @ -3V”? Because the unit of the x-axis [um], the unit [pm/V] is unmatched.\nThirdly, in supplementary Section III, the authors have shown the optimization process. However, it is unclear what objective\nfunction was employed. If we see Fig. S5(g) and (h), the star symbol is plotted, but it is unclear why the plotted point is\noptimal. Specifically, there seems to be higher reflectivity in the region to the right of the star symbol in Fig. S5(g).\nFinally, in the revised line 84-89 in the manuscript, it is unclear which structures were employed for the comparison. In\nparticular, it was unclear whether the comparison was made with experimental results or simulations with what kind of\nsettings."}, "author_response": "Date: Dec. 12th, 2025\nSubject: Response letter for NCOMMS-25-25057B (second revision)\nDear Reviewers,\nWe sincerely appreciate the time and effort you have dedicated to evaluating our manuscript\nentitled “Ultracompact and large-bandwidth silicon modulator in a CMOS-compatible\nfoundry”. We are grateful for the constructive feedback from all the reviewers during the first\nrevision, which has improved our work.\nWe are pleased that Reviewer #1 now supports publication, and we are deeply grateful for\nthe positive assessment. In this second revision, we have carefully addressed the remaining\ncomments from Reviewers #2 and #3 and incorporated their suggestions to further enhance the\nmanuscript’s rigor and clarity. Below, we provide point-by-point responses to each concern,\nwith the reviewers’ comments in italics and our revisions marked in red in both this letter and\nthe revised manuscript.\nWe believe that these revisions have resolved all the outstanding issues and hope that the\nmanuscript is now suitable for publication in Nature Communications.\nComments from Reviewer #2:\nI have carefully examined the revised manuscript together with the new Supplementary Notes\nand your point‑by‑point response. Your revision represents a clear effort to engage with the\nprevious critique: the unconventional “bandwidth‑density” claim has been removed, and a\nthoughtful discussion supported by additional data now clarifies the practical\nwavelength‑detuning window. These improvements fully resolve those two issues. Unfortunately,\nthe two most critical technical concerns—the extremely low dynamic extinction ratio (ER) and\nthe logically inconsistent energy‑per‑bit figure—remain insufficiently addressed. Because they\nstrike at the heart of the paper’s performance claims, the manuscript cannot yet be considered\nfor publication.\nComment 1:\n(a) First, on the dynamic ER, your new Supplementary Note V supplies manufacturer\nS‑parameter curves for the electrical amplifier, bias‑tee, and related components, and it\nasserts that external high‑frequency loss dominates the observed < 1 dB eye penalty.\nHowever, no quantitative de‑embedding of the complete RF chain has been performed. In\nthe absence of measured S‑parameters for the specific cables, probes, pads and device\nunder test, the reader still cannot determine how much of the voltage drop occurs externally\nand how much is intrinsic to the nanobeam modulator itself. The additional qualitative\nremark that the optical peaking mechanism “inherently increases insertion loss” may be\nphysically plausible, but without modelling or experiment it remains speculative.\n(b) Moreover, the citation of Han et al. continues to be invoked only to show that ER generally\ndegrades with increasing data rate; it does not justify an ER that is an order of magnitude\nlower than that achieved in that work under comparable drive conditions.\n(c) A rigorous, experiment‑based separation of external and intrinsic contributions is\ntherefore indispensable.\nResponse 1:\nWe sincerely appreciate the reviewer’s thoughtful analysis and valuable suggestions regarding\ndynamic ER characterization. In response to these insightful comments, we have carefully\nrevised the relevant descriptions and added additional results to Supplementary Information\nNote V.\n(a) We performed a quantitative analysis of the S21 parameter across the external RF chain,\nsimulated V in the time domain, explained the ER degradation induced by the optical\neff\npeaking mechanism through frequency-domain analysis, and simulated the dynamic ER at\nvarious data rates. On the basis of this analysis, the reader can determine how much of the\nlow ER originates from the external RF chain and how much is attributable to the\nmodulator itself. We have revised the relevant analysis accordingly and added it to\nSupplementary Information Note V.\n(b) We would like to clarify the fundamental differences between the proposed modulator and\nthe implementation in Ref. 23. While Ref. 23 employs an MZI configuration that relies on\ninterference between two phase-modulated arms, our approach utilizes resonant spectral\nshifting of the cavity transmission spectrum under an applied voltage bias—an entirely\ndistinct operating principle. Our design framework inherently targets a dynamic ER of 2\ndB, which represents near-theoretical-limitation performance for the design principle, as\ndetailed in Supplementary Information Note II. With respect to the driving conditions, Han\net al. used a 5 V drive voltage, whereas we used a 2 V drive voltage.\npp pp\n(c) We systematically simulated the influence of the external RF chain on the input driving\nsignals and the dynamic ER at the transmitter, which enables clear discrimination between\nexternal and intrinsic contributions to ER degradation.\nLines 293–295: However, experimental measurements revealed a dynamic ER of less than 1\ndB, primarily because of the EO response of the proposed PCNC modulator and the electrical\nsignal attenuation27 through the 67-GHz bias-Tee, RF cable, and high-speed probe (discussed\nin Supplementary Information Note V).\nIn Supplementary Information Note V:\nThe dynamic ER is a critical performance metric for high-speed optical communication systems.\nIn intensity-modulated direct-detection (IM/DD) schemes, a high dynamic ER ensures minimal\nsignal distortion and an increased signal-to-noise ratio (SNR), directly affecting BER thresholds.\nTo characterize the data modulation performance of the proposed silicon modulator, we\nconducted a high-speed data generation experiment. The experimental setup is shown in Fig.\nS8a. An electrical data signal with a (213-1)-bit pseudorandom binary sequence (PRBS) was\napplied to the modulator, generated by a 256-GSa/s arbitrary waveform generator (AWG,\nKeysight M8199A) and then amplified to ~2 V by an electrical amplifier (EA, SHF S807C).\npp\nThe voltage amplitude of ~2 V was verified by an electrical back-to-back measurement. A\npp\nbias-Tee (SHF BT65R-B) was used to provide a bias voltage of −3 V, which is crucial for a\nhigh-speed EO modulator on the silicon platform. A 67-GHz radio frequency (RF) cable and a\nhigh-speed probe were used to inject the electrical signal into the modulator. A trigger signal\nwas employed for clock recovery to generate eye diagrams of the received data captured by an\nelectrically sampled oscilloscope (OSC, Keysight N1030A). The transmitter digital signal\nprocessing (DSP) flowcharts are shown in Figure S8b. By using Nyquist pulse shaping, the\nsignal bandwidth can be suppressed to ~ half of the baud rate while unclear transient patterns\n2\nare introduced in the eye diagrams, as shown in Fig. S9a−d. The measured eye diagrams for the\nNRZ signals at 50, 70, 90, and 110 Gbps are shown in Fig. S8c–f, with SNRs of 8.23, 6.07,\n3.98, and 3.17 dB, respectively. The measured ERs of the open eyes were 1, 0.9, 0.7, and 0.6\ndB.\nTo quantitatively investigate the discrepancy between the measured dynamic ER and\ntheoretical expectations, we performed frequency- and time-domain studies. Figures S8g–j\npresent the measured S21 parameters of the EA, bias-Tee, and RF cable and that of the probe,\nas extracted from the manufacturer’s datasheets8,9,10. The unnormalized and normalized S21\nparameters of the RF chain are presented in Figures S8k and S8m, respectively, and the\nquantitative values across specific frequencies are provided in Table S1. The measured\nelectrical ILs of the RF chain demonstrate frequency-dependent characteristics: 1.83 dB @ 25\nGHz, 2.42 dB @ 35 GHz, 4.66 dB @ 45 GHz, and 3.88 dB @ 55 GHz. These cumulative losses\nresult in a lower-than-expected electrical driving power being delivered to the pads,\nconsequently leading to a reduced effective voltage (V ) below 2 V. In accordance with the\neff\ntransmitter DSP and the measured S21 parameters of the RF chain, we simulated V at various\neff\ndata rates via inverse Fourier transform, yielding the values of 1.80 V @ 50 Gbps, 1.65 V @\nFurthermore, on the basis of the S21 parameters obtained from bandwidth measurements\nperformed with the PCNC modulator, although optical peaking enhancement significantly\nincreases the 3-dB EO bandwidth of the modulator, it concurrently increases the low-frequency\nIL of the S21 response. In the frequency domain, this additional low-frequency attenuation\nleads to power degradation, which directly reduces the dynamic ER in the time-domain eye\ndiagram. By combining the transmission spectra under various voltage biases and the S21\ncharacteristics of the PCNC modulator with the simulated V , we further simulated the\neff\ndynamic ER at various detuning points and data rates, as shown in Table S2. Specifically, the\nsimulated eye diagrams of the modulated optical signals at the −3 dB detuning point are shown\nin Fig. S9i–m. On the basis of the simulated eye diagrams, the dynamic ER at the transmitter\nwas observed to decrease to approximately 1 dB during the high-speed data generation\nexperiment. Further degradation occurred because of the receiver's frequency response,\nultimately yielding a measured ER below 1 dB at the receiver output. Similar dynamic ERs\nhave been reported in Ref.11,12 by using the MZM configuration.\nNotably, the proposed device achieves near-theoretical-limitation performance with the\ndesign scheme. For practical applications requiring long-distance transmission, high\ntransmission capacity, and low BERs, the device validated in this work can be further optimized\nin future work. First, by side-coupling the PCNC with a bus waveguide, a notch-shaped\ntransmission spectrum, similar to that of an MRR, can be achieved. This spectral profile has\nadvantages in achieving higher ERs and can be used in wavelength-division multiplexing\n(WDM) systems to increase transmission capacity. Additionally, optimizing the design of the\nPCNC to increase the Q factor of the resonance peak provides a pathway for achieving higher\ndynamic ERs. However, this method of improving the Q factor requires significant optical\npower trade-offs when the EO bandwidth is extended, presenting a fundamental limitation in\noptimizing the device design.\nExperimental equipment for measuring optical eye diagrams. b Transmitter DSP flow charts. Eye\ndiagrams for NRZ signals at data rates of c 50 Gbps, d 70 Gbps, e 90 Gbps, and f 110 Gbps. S21 parameters for the g EA\n(SHF S807C, Ref.8), h bias-Tee (SHF BT65R-B, Ref.9), i 30-cm RF cable (measured), and j high-speed\nRF probe (Ref.10), k unnormalized S21 of the RF chain, and m normalized S21 of the RF chain.\nEA, bias-Tee, RF cable, and high-speed probe.\nFreq (GHz)\n5 10 20 25 35 45 55\nS21 (dB)\nEA 23.1 24.1 23.4 23.1 22.8 22.1 22.5\nBias-Tee -0.6 -0.62 -0.71 -0.60 -0.60 -0.67 -0.87\nRF Cable -0.56 -1.18 -1.60 -1.66 -2.01 -3.30 -2.75\nProbe -0.16 -0.23 -0.32 -0.41 -0.43 -0.56 -0.55\nTotal 21.78 22.07 20.77 20.43 19.76 17.57 18.33\nTable S2. Simulation results for dynamic ER at various detuning points and data rates.\nData rates (Gbps)\n50 70 90 110\nDetuning points\n−1 dB 1.27 dB 1.17 dB 1.09 dB 1.00 dB\n−2 dB 1.20 dB 1.13 dB 1.08 dB 1.03 dB\n−3 dB 1.16 dB 1.09 dB 1.06 dB 1.02 dB\nsignals. The unclear transient patterns are introduced by Nyquist pulse shaping. e–h Electrical signals\nafter they pass through the external RF chain. i–m Modulated optical signals at the detuning point of\n−3 dB.\nSecond, the energy‑per‑bit calculation still employs the full 2 Vpp applied at the probe tips,\neven though your own explanation for the low ER presumes a much smaller effective swing at\nthe device terminals. By continuing to quote 5.9 fJ/bit without an experimentally validated V_eff,\nthe manuscript presents mutually inconsistent statements: a large, unquantified RF attenuation\nis invoked to explain the ER, yet the energy figure silently ignores that same attenuation. Simply\nnoting that a careful study of V_eff is “future work” does not resolve the contradiction; it leaves\nthe central efficiency metric misleading and unverifiable.\nResponse 2:\nWe sincerely appreciate the reviewer's comments regarding the power consumption\ncalculations. The power consumption formula we adopted is a benchmark calculation method\nused for optoelectronic devices1-5. When evaluated at a 2 V operating voltage, this\npp\nmethodology provides an upper-bound estimation of the modulator's intrinsic power\nconsumption without considering the RF chain loss. We calculated the power consumption\nusing V , which was simulated using the experimentally measured S21 responses of the\neff\nexternal RF chain and included a discussion in Supplementary Information Note VI.\n[1] Dubé-Demers, R., LaRochelle, S. & Shi, W. Ultrafast pulse-amplitude modulation with a\nfemtojoule silicon photonic modulator. Optica 3, 622-627 (2016).\n[2] Yuan, Y. et al. A 5 × 200 Gbps microring modulator silicon chip empowered by two-\nsegment Z-shape junctions. Nat. Commun. 15, 918 (2024).\n[3] Miller, D. A. B. Energy consumption in optical modulators for interconnects. Opt. Express\n20, A293-A308 (2012).\n[4] Hsu, WC., Nujhat, N., Kupp, B. et al. Sub-volt high-speed silicon MOSCAP microring\nmodulator driven by high-mobility conductive oxide. Nat. Commun. 15, 826 (2024).\n[5] Geravand, A., Zheng, Z., Shateri, F. et al. Ultrafast coherent dynamics of microring\nmodulators. Nat. Photon. 19, 740–750 (2025).\nIn Supplementary Information Note VI:\nBy using the simulated V , as detailed in Supplementary Information Note V, the calculated\neff\npower consumption tends to decrease with increasing data rates: 4.78 fJ/bit @ 50 Gbps, 4.02\nfJ/bit @ 70 Gbps, 3.50 fJ/bit @ 90 Gbps, and 2.97 fJ/bit @ 110 Gbps. Under ideal conditions\nwith RF losses excluded, the intrinsic power consumption of the proposed modulator reaches\nan upper bound of 5.9 fJ/bit. Notably, the RF amplifier accounts for a major portion of the\npower consumption in the transmitter11.\nComments from Reviewer #3:\nComment 1:\nAlthough the authors have answered the reviewer's questions, there still are several points that\nneed to be clarified. The main concern in the current manuscript is that the physical effect of\nthe slow light is not described in sufficient detail.\n(a) Firstly, could the authors describe the physical meaning of Eq. (4) especially in connection\nwith the tuning efficiency in unit [pm/V]?\n(b) Also, it is required to explain the reason why the sum was taken from -5 to 5, and whether\nthe mirror region is included or not. Since the edge of the taper region should have very\nlarge n_g, which may go infinite, Eq. (4) may diverge in that case. Moreover, n_g cannot\n7\nbe obtained in the mirror region. Because Eq. (2) from Ref. 23 is derived for the waveguide-\ntype modulator, mathematical derivation of the accumulated phase for Fabry-Perot (FP)\ncavity would be required.\n(c) In addition, it is required to describe the physical meaning of using the slow-light effect. In\nsingle-pass devices such as waveguide modulators, the slow-light effect may be the only\nway to enhance the light-matter interaction with fixed length. However, in multi-pass\ndevices such as FP cavities, the light reflects between mirrors, and the light-matter\ninteraction is enhanced even without the slow-light effect. Therefore, to enhance the light-\nmatter interaction in FP cavities, there would be at least two ways: One is to increase the\nreflectance of the mirrors, and another is to use the slow-light effect. These\nexplanations\nwould be required in the manuscript to clearly state the physical meaning of the slow-light\neffect in the proposed device.\nResponse 1:\n(a) Eq. 4 quantifies the phase accumulation within the resonant cavity. In FP cavities, the\nresonance wavelength is fundamentally governed by the phase-matching condition, whose\nsensitivity to the material’s refractive index varies with cavity design parameters.\nFor modulators, the input is an electrical voltage signal, while the output is an optical signal.\nIn resonance-based silicon modulators, signal modulation arises from spectral shifting, as\nillustrated in Fig. R1. The applied voltage modulates the carrier distribution in the PN junction,\ninducing refractive index changes that perturb the cavity's phase-matching condition. This\nphase-matching perturbation directly controls the resonance wavelength shift, which is our\nprimary optimization target.\nThe [pm/V] tuning efficiency precisely characterizes how effectively the electrical signal\nperturb the phase-matching condition. We achieve increased sensitivity through the engineered\nslow-light enhancement effect, as shown by the red solid curve in Fig. R1. Moreover, the [pm/V]\ntuning efficiency has been widely adopted as a standard metric for characterizing device\nmodulation performance, as shown in Table 1 of the manuscript.\nWe contend that the existing discussion (Lines 84−90) adequately captures this physical\nmechanism.\nFigure R1. Schematic illustration of the physical principles of the resonant modulator. The red\nsolid curve indicates the slow-light enhancement effect in this study.\nmatter interactions are significantly amplified, yielding a 3.36-fold improvement in phase\ndetuning compared with conventional silicon rib waveguide cavities in our simulation, which\nwere designed with identical waveguide width, length, and slab thickness. Importantly, the\nslow-light effect amplifies the phase shift per unit voltage rather than directly altering the\nmaterial’s electro−optic properties. This enhancement, combined with an appropriate doping\nconcentration provided by the commercial foundry, enables a tuning efficiency of 80 pm/V for\nour optimized PCNC.\n(b) The range of −5 to 5 corresponds to the 10 linearly graded lattice periods on each side in\nour design, where some lattices exhibit a photonic bandgap at the target wavelength. Our\nanalysis specifically considers the phase accumulation from in-band lattices within the cavity\nregion, explicitly excluding contributions from the mirror regions. To increase clarity, we have\nincorporated a detailed discussion of this methodology in the manuscript and provided\ncomprehensive theoretical derivations in Supplementary Information Note VII.\nLines 155–160: Thus, in such a designed PCNC, the total phase accumulation can be derived\nas follows:\n(4)\n2𝜋𝜋 5\nwhere are the group i𝜙𝜙nd=ex𝜂𝜂 an∙d\n𝜆𝜆0\np∙e∑rio𝑖𝑖=d− o5f𝑛𝑛 th𝑔𝑔e( 𝑖𝑖i)-t∙h𝑝𝑝 c(e𝑖𝑖l)l of the PCNC, respectively.\nNotably, the summation range is determined by the specific energy bands and bandgaps of the\n𝑛𝑛𝑔𝑔(𝑖𝑖) and 𝑝𝑝(𝑖𝑖)\ndevice (detailed in Supplementary Information Note VII).\nThe following can be found in Supplementary Information Note VII:\nThe accumulated phase detuning in the slow-light waveguide is proportional to the group index\nn , as described by13:\ng\n(S19)\n2𝜋𝜋\nwhere , with 𝜙𝜙 b=ein𝜂𝜂g∙ a𝑛𝑛 c𝑔𝑔o∙e𝐿𝐿ff∙ic𝜆𝜆i0ent related to the overlap between\nthe carriers and the optical field; V is the built-in voltage of the PN junction; V is the applied\n𝜂𝜂 = 𝜂𝜂0𝑉𝑉+𝑉𝑉𝑏𝑏𝑖𝑖 −𝑉𝑉𝑏𝑏𝑖𝑖bi 𝜂𝜂0\nvoltage of the modulator; and L is the optical travel length.\nThe proposed PCNC structure incorporates a slow-light region with linearly graded lattice\nperiodicity. In this design, each unit cell exhibits a unique group refractive index n and lattice\ng\nlength L, thereby leading to the derivation of Eq. S19 as an integral over the slow-light region:\n(S20)\n2𝜋𝜋\nThe slow-light cavity behaviour is fund\\𝜙𝜙\n=\n𝜂𝜂\n∙\n𝜆𝜆\n0\n∙\n∫\n𝑐𝑐\n𝑐𝑐\n𝑖𝑖\n𝑛𝑛\n𝑔𝑔b(y\n𝑥𝑥\nt)\n𝑑𝑑\n𝑥𝑥\nam\nen\nta\nl\nl\ny\no\nv\ne\nr\nc\n𝑐𝑐\ned\n𝑛𝑛\ngb(y\nt)h\nd\ne\nx\nengineered energy bands and\nbandgaps, as evidenced in the band diagrams of both the mirror and slow-light regions, as\nshown in Fig. S10a–b, respectively. The progressive light slowing effect within the resonant\ncavity is shown in Figure S10c, with a group index range of 4.2 ≤ n ≤ 44. For reference, the\ng\ndashed line indicates the group index of n = 3.9 for a conventional silicon rib waveguide cavity\ng\n(with identical waveguide width, length, and slab thickness design). Notably, unit cells with i >\n5 and i < −5 fall within the photonic bandgap at the target wavelength and are consequently\nexcluded from the integration domain. Thus, Eq. S20 is derived as follows:\n9\n(S21)\n2𝜋𝜋 5\n𝜙𝜙 = 𝜂𝜂∙ 𝜆𝜆0 ∙∑𝑖𝑖=−5𝑛𝑛𝑔𝑔(𝑖𝑖)∙𝑝𝑝(𝑖𝑖)\nFigure S10. Band diagrams of a the mirror regions and b the unit cell in the taper regions. c\nGroup index n ≥4.2 within the cavity along the x-axis. The dashed line indicates the group\ng\nindex of a conventional silicon rib waveguide for comparison.\n(c) We sincerely appreciate the reviewer’s insightful comments. The reviewer is correct in\nnoting that in FP cavities, light‒matter interactions can be enhanced by increasing the\nreflectance of the mirrors and by introducing the slow-light effect. In response to this valuable\ncomment, we have expanded the manuscript to include a detailed discussion explaining how\nour design incorporates a carefully engineered slow-light region to synergistically combine\nboth enhancement mechanisms.\nLines 160–164: In contrast to conventional Fabry–Pérot (FP) cavity modulators that depend\nexclusively on mirror reflectivity for optical enhancement, our design incorporates an\nengineered slow-light region to combine both enhancement mechanisms: the FP resonance\nboosts light‒matter interaction through recirculating passes, and the slow-light structure offers\nenhancement by extending the effective interaction length50,52, leading to increased tuning\nefficiency.\nComment 2:\nSecondly, could the authors describe the meaning of the red arrow in Fig. 2(c) shown as “80\npm/V @ -3V”? Because the unit of the x-axis [um], the unit [pm/V] is unmatched.\nResponse 2:\nThe expression “80 pm/V @ -3 V” indicates that the tuning efficiency of the device is 80 pm/V\nunder a bias voltage of -3 V. The tuning efficiency is defined as the spectral blueshift or redshift\nper unit voltage. In our device, as the voltage changes from -2 V to -4 V, the spectral redshift is\napproximately 160 pm. Therefore, we calculate the average tuning efficiency using Δλ/ΔV =\n160 pm/2 V = 80 pm/V to describe the tuning efficiency of the device as 80 pm/V at a bias\nvoltage of -3 V.\nLines 197–199: At a reverse bias voltage of −3 V, the tuning efficiency was approximately 80\npm/V. This value was obtained from a 160 pm spectral redshift measured from −2 V to −4 V,\ncorresponding to a V · L product of ~0.38 V·cm7.\nπ\nComment 3:\nThirdly, in supplementary Section III, the authors have shown the optimization process.\nHowever, it is unclear what objective function was employed. If we see Fig. S5(g) and (h),\nthe star symbol is plotted, but it is unclear why the plotted point is optimal. Specifically, there seems\nto be higher reflectivity in the region to the right of the star symbol in Fig. S5(g).\nWe appreciate the reviewer’s careful observation of our optimization methodology in\nSupplementary Information Note III. In fact, our optimization was not governed by a single\nobjective function but rather guided by two critical design criteria: maximizing the reflectivity\nat the target wavelength and maintaining a compact footprint. The simulation results of our\ndevice parameter optimization process are shown in Figure S5. Reflectivity is vital for the\ndesign of PCNC, which directly determines the Q factor, and it varies with wavelength, as\nshown in Figs. S5a–f.\nWith respect to Fig. S5g, we use the reflectivity at 1550 nm; thus, the reflectivity in the region\nto the right of the star symbol in Fig. S5(g) seems to be higher. We understand that this may\nhave confused the readers. To address this concern, we have added detailed simulation results\nto Fig. S6. On average, the reflectivity is greater when m = 10 than when m = 11 or 12.\nrange of 1500~1600 nm for m = 7~12 and n = a 13, b 14, c 15, d 16, e 17, f 18, g 19, and h 20.\nFinally, in the revised line 84-89 in the manuscript, it is unclear which structures were employed\nfor the comparison. In particular, it was unclear whether the comparison was made with\nexperimental results or simulations with what kind of settings.\nResponse 4:\nWe appreciate the reviewer’s question, and we understand that the original expression might\nconfuse readers. Therefore, we have revised the related description of the comparison as follows:\nLines 84–87: With the slow-light enhancement in the resonant cavity, the inherently weak light‒\nmatter interactions are significantly amplified, yielding a 3.36-fold improvement in phase\ndetuning compared with conventional silicon rib waveguide cavities in our simulation, which\nwere designed with identical waveguide width, length, and slab thickness.\nLines 148–151: As shown in Fig. 1f, light is progressively slowed within the resonant cavity\nwith a group index of 4.2 ≤ n ≤ 44. For reference, the dashed line indicates n = 3.9 for a\ng g\nconventional silicon rib waveguide cavity with identical waveguide width, length, and slab\nthickness design.\nThank you again for your consideration.\nSincerely,\nHua Zhong & Yikai Su\nOn behalf of all the authors\nState Key Lab of Photonics and Communications\nSchool of Information Science and Electronic Engineering\nShanghai Jiao Tong University\nEmail: yikaisu@sjtu.edu.cn\nWeb: https://otip.sjtu.edu.cn/\n13"}, {"round": 3, "reviewer_comments": {"reviewer_2": "The authors have demonstrated a commendable commitment to addressing the reviewers' feedback. The two most critical technical issues have been resolved satisfactorily through detailed analysis and simulation. The manuscript presents a significant advance in silicon photonics, achieving an impressive combination of bandwidth (>110 GHz) and footprint (10 µm²).\nI am satisfied with the revisions and am pleased to recommend this manuscript for publication in Nature Communications."}, "author_response": ""}]

Decision letter