Achieving sub-pm wavelength regression via minimum-phase in a single-stream photonic IC

Domain: physics
Published: 2026-03-25
Editorial outcome: major_revision

Abstract
Photonic chips are powerful tools for measuring and analyzing light, but most compact spectrometers face a fundamental trade-off: improving resolution usually requires larger devices or sacrifices in signal quality. Here, we introduce a chip-scale architecture that overcomes this limitation by extracting phase information corresponding to the hidden timing of light waves using only simple intensity measurements. Our method generalizes earlier minimum phase designs to allow sparse and non-sequential optical delays, enabling accurate phase reconstruction on a single circuit. By engineering these delays, the device can determine the wavelength of an unknown laser with sub-picometer precision, all while using just one input and one output. This single-stream design reduces loss, improves robustness, and avoids the complexity of traditional spectrometers. The result is a compact, scalable platform that enables high-accuracy wavelength metrology and opens possibilities for on-chip sensing and computational spectroscopy.

Introduction
THE field of computational spectroscopy has recently received a great deal of attention in the photonic integrated circuit research community [1]–[3]. The main reason behind this is the ability to incorporate functionalities that, in the past, required large footprints [4]–[10]. Photonic integrated circuits (PICs) allow the miniaturization of all these bulky systems while maintaining their functionality and desired performance [11]. In addition to the size advantage, the ability to mass-produce these miniaturized spectrometers enables a more cost-effective solution for the consumer market.

Photonic integrated circuit (PIC) spectrometers have been implemented using three primary architectures: Mach–Zehnder interferometers (MZIs) with multiple optical path lengths [12]–[21], optical cavities that partition the spectrum into discrete channels [22]–[27], and dispersive elements that spatially spread the spectrum for reconstruction [28]–[33]. Each offers distinct strengths but faces fundamental trade-offs between resolution, bandwidth, footprint, and signal fidelity. Dispersive designs can access a broad bandwidth, but they require many spatial channels, which increases size and loss. Optical cavity spectrometers achieve high resolution with large quality factors, yet wide-band operation demands multiple cavities. MZI-based approaches are broadband, but resolution scales directly with path length, and intensity-only regression near interferometric nulls produces error bias. In all three cases, the multi-path nature of the architecture inherently reduces SNR, as optical power is divided among multiple channels or paths.

Using the light’s phase overcomes the drawbacks of interferometric fringes for spectroscopy. However, accessing the phase information in a photonic circuit is nontrivial. For example, it is common to utilize integrated 90-degree hybrids [13], but they require a large footprint and are highly susceptible to manufacturing variations. In contrast, Xu et al. showed that the phase in a photonic circuit can be extracted solely from intensity measurements using minimum-phase systems [34], [35]. Specifically, the phase is determined directly from the magnitude via their fundamental relationship in a minimum phase system, enforced by the Hilbert transform [36]–[38].

Here we generalize the work of Xu et al., who applied the minimum phase concept specifically to optical FIR (Finite-Impulse Response) filter implementations [34]. Our generalization extends the applicability of the minimum-phase approach to high-performance wavelength metrology and spectroscopy applications, as well as to general circuit parametric extraction (key for the development of PIC process design kits, an area of growing commercial importance). Specifically, Xu et al. demonstrated that the full complex response of a circuit can be reconstructed from intensity-only measurements by leveraging the minimum-phase Kramers-Kronig relationship [36]. However, their implementation was limited to sequential tap filter configurations (n=1,2,3,...). This conceptual limitation would prohibitively increase a spectrometer circuit’s footprint by n-fold (n delays paths needed to go from bandwidth-defining FSR to higher-resolution FSR). Here, we overcome this through the use of non-sequential tap increases, uncovering the necessary optical signal processing needed to achieve high-performance wavelength metrology. This work enables arbitrary and sparse tap filter configurations, where n can be any integer, removing the footprint requirements from n delay paths to any m < n. Enabling this arbitrary increase in delay length significantly enhances spectral resolution, proving particularly valuable for wavelength metrology applications. Specifically, in this work, we utilize large tap values to achieve sub-picometer (sub-pm) wavelength resolution in a single-stream circuit. As a result, our approach overcomes the fundamental trade-offs faced by traditional multi-path architectures, which must balance resolution, footprint, and signal/noise ratio.

In this work, we present a single-stream, minimum-phase photonic wavemeter that reconstructs optical phase from intensity measurements using a sparse multi-∆L architecture, enabling absolute wavelength inference with sub-picometer precision over user-defined windows. We derive the design rules that ensure minimum-phase operation in asymmetric Mach–Zehnder interferometers, and we show how a zero-phase FIR approach isolates the relevant delays while preserving phase fidelity. We develop wavelength-regression models that use the uniquely determined short-delay phase to order longer delays, and we incorporate a second-order dispersion term for high-accuracy inference at the largest ∆L. We validate the approach on fabricated photonic integrated circuits, and we quantify error and robustness within practical bandwidth limits. Section II-A details the theory and design; Section II-B presents the wavelength-regression methods for single and multi-∆L regression methods; Section II-C demonstrates absolute wavelength retrieval across user-defined windows with sub-pm error; Section III discusses limitations and future work; and Section IV describes fabrication and measurement.

Results
A. Minimum-Phase Mach-Zehnder Interferometer

We analyze the minimum phase condition for an integrated MZI by examining its transfer function from the schematic presented in Figure 1(a). The transfer matrix method is used to derive the MZI transfer function given in Equation (1).

H(ω) = 1/2 (√τ e−αL1e−jβL1 − √κ e−αL2e−jβL2) (1)

where κ and τ are the input coupler coefficients and are related by |τ|2+|κ|2 = 1, L1 and L2 are the MZI arm lengths, β is the wavevector and is given by β = 2πneff/λ with λ and neff being wavelength and the waveguide effective index, and α being the waveguide loss per unit length.

Following the convention defined in Supplement, Section 1, we let Href(ω) = √τ e−αL1e−jβL1 and Hcpu(ω) = √κ e−αL2e−jβL2, where L2 > L1, and we have assumed that the directional couplers are broadband in the wavelength window of interest. Using these substitutions, the MZI becomes a minimum phase system when |Hcpu(ω)/Href(ω)| < 1, which provides us with the minimum phase boundary for an MZI, as given in Equation (2)

|Hcpu(ω)/Href(ω)| < 1 =⇒ |κ|2 < 1 / (1+e−2α∆L) (2)

where ∆L = L2 − L1. This means that, to retrieve the phase of the MZI transfer function, the input coupling coefficient ought to be smaller than the sigmoid dependence on the optical path length difference of the two arms. As a result, if the input coupling coefficient meets this condition, we can retrieve the phase of the photo-detected intensity relationship (G(ω)) of the MZI shown by Equation 3.

H(ω) = e−jβL1e−αL1 (√τ/2 − √κ/2 e−α∆Le−jβ∆L) (3)
= e−jβL1G(ω)

where G(ω) has a constant DC factor (√τ/2) and a linear phase (e−jβ∆L) sans dispersion. We note that |G(ω)|2 is the photo-detected signal that is measured at the MZI output. Inspecting Equation (3) shows that the relationship between G(ω) and H(ω) is the global phase offset e−jβL1 that is lost due to photodetection being a complex magnitude operation.

We perform a simulation to validate the minimum-phase boundary condition (Eq. (2)) by varying the values of ∆L and κ of the MZI. As described in Figure 1(b), the phase of H(ω) is recoverable from its intensity relationship in a given bandwidth when H(ω) is a minimum-phase system. As a result, we compare the retrieved phase given by

θ(ω) = H{log(|G(ω)|)} (4)

where θ(ω) is the G(ω) phase, and |G(ω)| is the intensity of G(ω), with the simulated phase arg(G(ω)) as a function of both ∆L and |κ|2. In order to validate the conditions under which minimum-phase occurs in Figure 1(c), we use the Pearson correlation coefficient to compare the extracted phase with the simulated phase. The correlation coefficient is maximized when the minimum phase boundary condition is met, as shown by Figure 1(c) (blue shaded region below the minimum phase boundary). Furthermore, we note that the correlation coefficient approaches −1 when ∆L is small and |κ|2 > 0.5. This happens because the partitioned paths are similar in length, causing κ to present a stronger minimum-phase condition than ∆L. As a result, the roles of reference and delay paths are reversed in this region, causing a phase reversal, i.e. a multiplication by −1, shown by the negative correlation coefficient. Consequently, we conclude that with MZI’s meeting the minimum-phase condition (Eq. 2) we can successfully recover the phase of G(ω) by using Equation (4).

B. Single ∆L Design

Here we demonstrate wavelength regression using the experimental process and the wavelength regression algorithm shown in Figure 2 and Algorithm 1, respectively. A PIC with four different MZI circuits with varying input coupling coefficients (90:10 and 50:50), and ∆L (0.5 cm and 1 cm) was designed to experimentally test the boundary condition of the minimum phase as shown in Figure 2(a) and indicated in Figure 1(c). The wavelength regression algorithm is based on calibrating the wavemeter and then testing it using an unknown laser to validate the prior calibration. The output of the MZI is directly connected to an on-chip photodetector, and we read the photo-generated current as a function of the MZI’s input wavelength, given by either a known value from the calibration laser (S1) or an unknown value given by the uncalibrated semiconductor tunable laser diode (S2). To assess the performance variance from chip-to-chip, two packaged PICs (P1 & P2 - one of the packages is shown in Figure 2(a)) are utilized to cross-correlate training and testing data.

The intermediate steps of the wavelength regression algorithm are shown in Figure 2. The algorithm starts by extracting the phase (ϕ) from the MZI’s intensity response by applying the Hilbert transform to the intensity (P). Subsequently, the discrete-time complex-valued response (h(n)) is constructed by taking the inverse Fourier transform of the constructed complex-valued frequency response. Finally, G(ω) is obtained by eliminating the n = 0 tap by using a window function (wmin(n)) that is zero at the n=0 point and one everywhere else. After obtaining G(ω), which corresponds to the ejβ∆L response, we use SciPy find peaks algorithm to find the peak present in the Fourier transform of ejβ∆L [39]. Using the peak information, we define a band-pass filter (wbpf(n)) centered around this peak. Subsequently, the filter is used to eliminate any spectral components that are not the linear phase found in G(ω) as shown by the apply filter stage. The phase of the filtered response is then unwrapped and used as the basis for linear regression using Numpy’s polyfit function [40] that produces the linear model coefficients C0 and C1 used for wavelength inference. To perform wavelength inference, we need to find the roots of our chosen polynomial function (linear in this case) and then convert the result into wavelength. We use frequency as our regression axis because the wavevector β expansion as a function of frequency is a more tractable function, as shown in Equation (5). While we expand β to the second term in the Taylor series (∝∆ω2), we only use the first term for the results presented in this section (the linear model). We will use the second term in the next section due to a need for modeling dispersion at large ∆L. Note that our results are all shown in wavelength by using the relation λ=c/f where c is the speed of light.

β ≈ β0 + 1/c (dn/dω) ω0 ∆ω + 1/2c (d2n/dω2) ω0 ∆ω2 (5)

where n is the material’s group index and β was Taylor expanded around ω0.

C. Sparse Filter Structure for Custom Wavelength Windows

The phase measurement of a single ∆L MZI design is not uniquely determined in the bandwidth of interest as β∆L = β∆L + n2π, where n is any integer. Therefore, this approach is constrained to cases where the testing laser wavelength window is known a priori (such as laser stabilization). To circumvent this limitation, we designed a circuit that enables the recovery of a uniquely determined phase in the wavelength window of interest. To do this, we take advantage of the fact that the phase is directly proportional to ∆L. As a result, we can define a value ∆L that will be uniquely determined in the operating bandwidth of the wavemeter. Furthermore, the addition of larger optical delay lines on top of the bandwidth-defining ∆L enables higher resolution measurements, achieving sub-pm resolution. The resulting optical structure is that of a sparse optical filter where the optical delays are non-subsequent integer increases of the bandwidth defining ∆L. The reason why we need non-subsequent integer increases of the bandwidth defining ∆L is to increase the resolution of the final wavelength regression at the largest ∆L stage. It has been consistently shown in the literature that larger ∆L values allow for finer resolution [13], [17], [42]. Thus, we wish to incorporate this functionality into our design while maintaining a single-input-single-output architecture. As a result, all optical paths that form the distinct ∆Li ∀i∈[1,2,...,N] paths need to be interferometrically coupled with each other at the output. This architecture resembles an FIR filter structure analyzed for filtering applications by Xu et al [34]. However, in their analysis, the optical delay lines are assumed to be subsequent integer increases of the bandwidth-defining (smallest) ∆L. Therefore, in the work presented here, this assumption has been eliminated in the analysis of this FIR filter structure, thereby generalizing that work. In this context, the resulting architecture is that of a sparse optical filter structure where the order of the filter is dictated by the largest ∆L and most of the filter taps are zero except for the present ∆L paths.

Here we present a high-resolution wavemeter using a single-stream asymmetric MZI circuit with four different ∆L paths of increasing value (Figure 3(a)). The minimum phase condition in this circuit is met by the use of a reference path that goes through a single-splitter operation instead of the two-splitter operations that the optical delay lines undergo. As a result, the power entering the reference path is guaranteed to exceed that entering the optical delay lines, ensuring the circuit meets the minimum phase condition.

As shown in Algorithm 2, the wavelength regression algorithm used for this design, follows the same procedure as the single ∆L design with a couple of adjustments needed to accommodate the extraction of the four different delay paths present in this structure. The extraction of G(ω) is carried out in the same way, while the calibration and wavelength regression models need to be adjusted. We note that we can perform the same mathematical treatment that was applied to the single ∆L design, where we can factor out the eiβ∆L1 term that provides a G(ω) comprised of four different eiβ∆Ln terms and a constant term. Similarly, we filter out the DC term by applying the same wmin(n) window function to extract G(ω). The calibration process begins with extracting G(ω) for the multi-path MZI structure. The extraction of each delay line occurs by zero-padding the resulting G(ω) to allow the sinc-interpolated Fourier space enough resolution between ∆L1 and ∆L2 since ∆L2/∆L1 ≈ 2. We note that this optical path length difference is not sufficient to allow us to properly discern between these two optical paths in the Fourier space without sufficient zero-padding. In addition, G copies is the result of duplicating the G(ω) vector to buffer the filtering process (hfilter) similar to the single ∆L scenario. Once we have the location of the index of the four peaks, we iterate over the found peaks, and for each peak, we follow a similar filtering process to that presented for the single ∆L design. We define a filter centered on the found i-th peak and apply the filter to the G copies. Finally, the calibrated phase is used to train the second-degree polynomial. Details regarding the need for a second-degree polynomial are shown in Supplement, Section 4.

The wavelength model for the multi-∆L design requires the addition of the MZI order as we sweep from the smallest ∆L path (largest FSR) to the largest path (smallest FSR). To accomplish this, we follow a similar procedure as shown in [13], where the initial MZI (i == 0) has an unambiguous phase, as it is uniquely determined in the bandwidth of interest. As a result, the calculation of frequency/wavelength deviation (∆f) is also unambiguous. However, as we iterate over the subsequent delay paths (i > 0), we need to find the delay path order (∆m) referenced by the previous delay path frequency deviation approximation. To do this, we first calculate the FSR of the current MZI using the previously derived phase model and the previous frequency deviation approximation, noting that

FSR = 2π / (Ci∆f + Ci) ≈ 2π / (dβ/dω)−1 (6)

where Ci, and Ci are the model coefficients found by the unwrapped phase model regression, and the β expression is given by Equation (5). Next, we find the relative frequency calculation (δf) using the wrapped filtered phase found in Θ[i]. Finally, the order is found by comparing the current i-th filtered phase with the frequency deviation approximation found by the previous delay path normalized by the current i-th FSR. Subsequently, the absolute frequency deviation of the current i-th path is calculated by unwrapping the relative frequency using the calculated delay path order and FSR.

Methods
A. Single ∆L Experiment

We used AIM Photonics [43] to fabricate and package the designs shown in Figure 2(a). We designed 4 MZIs with different input coupling coefficients and ∆L to experimentally test the minimum-phase boundary, corresponding to the circuits marked by a red ”x” in Figure 1(c). For the integrated directional couplers, we used the AIM Photonics PDK components for both the 50:50 splitters and the 90:10 splitters. Achieving larger bandwidths can be accomplished by designing couplers with the intended operational bandwidth using alternatives such as MMI or adiabatic couplers [44]. A Keithley 2400 SMU is used to measure the photodetector current at the output of each K circuit at each wavelength using a Keysight 81606A tunable laser source (S1). The laser is swept from 1525 nm to 1530 nm in 1 pm increments, and the SMU is then used to measure the current at each wavelength point. We used a 128-point buffered measurement reporting the average. An uncalibrated semiconductor tunable laser diode (S2) and a Keysight DSOX1204g oscilloscope are used as the test laser source, as shown in Figure 2(a). The laser is set to continuously sweep the wavelength from 1525 nm to 1527.5 nm in 0.1 seconds. We put the scope capture window to 0.5 seconds, allowing us to capture multiple laser sweeps. We then note that the process used to retrieve the phase can also be used to discriminate the entire captured window, allowing us to identify the points where the data is valid. That is, the phase will be discontinuous where the wavelength jumps from 1527.5 nm to 1525 nm, so we use this discontinuity to limit the valid data window. On-chip loopbacks were used to measure coupling loss of around ∼ 7 dB per facet (which can be reduced by optimizing the fiber packaging process), and the propagation loss is estimated to be ∼1 dB/cm.

B. Sparse Filter Experiment

We use the Applied Nanotools SiN MPW process to design and fabricate the multi-∆L MZI circuit shown in Figure 3. In this case, we choose SiN as our material platform owing to its small thermo-optic coefficient when compared to Si. The latter allows us to be less cognizant of temperature variations while still enabling the study of the minimum-phase approach without confounding factors such as mixed group indices designs [13]. For the integrated directional couplers, we use the 50:50 splitter within the ANT PDK. Similar to the single ∆L case, we could design MMI or adiabatic couplers that maintain splitting ratios over a larger bandwidth than conventional directional couplers [44]. The chosen ∆L values are 121 µm, 242 µm, 1210 µm, and 12103 µm corresponding to FSR values of 10 nm, 5 nm, 1 nm, and 0.1 nm, respectively. Another MZI circuit was also designed with the following ∆L values: 12.1 µm, 15.12 µm, 60.51 µm, 121.03 µm, 302.58 µm, 605.16 µm, 1210.32 µm, 12103.27 µm with corresponding FSR values of 100 nm, 80 nm, 20 nm, 10 nm, 4 nm, 2 nm, 1 nm, 0.1 nm. We use an input and output coupler coefficient of 0.5 for both MZI structures. This is deemed sufficient for the minimum-phase condition, as we further split the delay lines 1 : 4, making the reference path have at least four times as much power as each delay line while also being the shortest. As a result, this is enough to meet the minimum-phase condition, as shown by both the simulation and the experimental results in the single ∆L case. We utilize edge coupling and a Maple Leaf automated test setup with a polarization-maintaining fiber array in combination with a Keysight 81606A tunable laser source and a Keysight N7744A photodetector. The coupling loss was measured at ∼9 dB per facet (which can be reduced with optimized edge coupler design), while the propagation loss was measured at ∼1 dB/cm. The laser source was swept from 1500 nm to 1600 nm in 1 pm steps. However, the data presented here are limited to the four delay lines circuit and the 10 nm window between 1545 nm and 1555 nm. Although the tunable laser supports sweeping from 1500–1600 nm, the demonstrated 10 nm operational bandwidth of the photonic IC is set by intentional delay-line design choices and by practical limitations of the directional couplers, rather than by the laser source itself. See Supplement, Section 3.A & 3.B for more discussion regarding this chosen circuit, wavelength window, and how to manage dispersion in the Fourier space.

Peer review rounds
[{"round": 1, "reviewer_comments": {"reviewer_1": "The paper introduces a single-stream, minimum-phase PIC architecture that uniquely reconstructs optical phase from intensity measurements via the Hilbert transform, eliminating the need for additional phase retrieval components. Additionally, the paper achieves sub-picometer root-mean-square wavelength regression error over arbitrary wavelength windows. The reviewer believes that several key issues remain that need to be clarified or justified, after which the manuscript can be considered for publication.\nQ1: The phrase \"Arbitrary Wavelength Windows\" is used in the title of Part IV. While the multi-ΔL design does address wavelength ambiguity and allows operation over a much wider window than the FSR of a single large ΔL, the term \"arbitrary\" might be too absolute. The actual operating window is still limited by factors such as the bandwidth of the directional coupler and waveguide losses. It is suggested to modify the wording in the title or at the beginning of the section to be more precise.\nQ2: The manuscript mentions \"computational spectroscopy\" in the abstract and introduction. However, all experiments in the paper are based on a single-wavelength, narrow-linewidth tunable laser. This demonstrates that the device can function as a high-precision wavemeter. Still, there is no evidence to support its capability in handling broadband sources, multi-line sources, or continuous spectrum analysis. It is recommended to clearly define the positioning as \"high-precision wavelength metrology\" and elaborate on the conditions and challenges of extending to complex spectra in the discussion.\nQ3: Quantitative comparisons with conventional systems and recently proposed nanophotonics-enabled wavemeters are necessary.\nQ4: The authors state in the introduction that conventional 90-degree hybrids \"require a large footprint and are highly susceptible to manufacturing variations.\" However, the manuscript provides no evidence or analysis to substantiate the claim that the proposed method offers any advantages in either footprint or fabrication robustness.", "reviewer_2": "This paper introduces a single-stream, minimum-phase photonic integrated circuit (PIC) architecture, with core innovations including the minimum phase conditions and the Hilbert transform for phase retrieval. These contributions demonstrate substantial theoretical value through a systematic experimental design and offer implications for advancing next-generation computational spectroscopy technologies.\n(1) Please clarify the novelty of the “generalized sparse tap configuration” compared to previous works. While the authors claim to generalize Xu et al.’s the minimum phase approach from sequential filter taps to arbitrary sparse taps, the distinction between the two could be made more explicit. Specifically, it is suggested to clearly explain how their design differs from existing works in terms of synthesis methods and functional advantages (including resolution, footprint, and signal/noise ratio). This will help to strengthen the manuscript’s contributions.\n(2) It is suggested to enhance the understandability of Hilbert transform-based phase retrieval. The key idea of the proposed research is the reliance on the minimum phase conditions and the Hilbert transform for phase retrieval, but the physical principles that enable unique optical phase reconstruction may be obscure.\n(3) As shown in Fig. 2, the experimental results for wavelength regression prove that the experimental data drives the results and the filter is not biasing the proposed regression algorithm. Please explain in detail how the dataset improves the performance of the regression algorithm and how the dataset was constructed. How to improve both the measurement accuracy and the generalization of the proposed regression algorithm?", "reviewer_3": "Rubio, et al. have reported single input/output minimum phase PICs to extract optical phase from intensity measurement. They have presented the theory as well as some experimental results for single and multiple delay MZIs. Overall, the paper is well-written, and the idea is interesting, although the basic concept has been reported before. Below are my comments:\n- Can the authors comment on using MMIs instead of directional couplers as they would make the system more wideband and less sensitive to fabrication variation (basically coupling ratio variations)? Especially for the sparse filter chip where 50/50 couplers are used.\n- I suppose the chip is temperature stabilized. Is that the case? How would temperature change affect the measurement accuracy?\n- There should be phase shifters within the MZIs to adjust/calibrate for phase mismatches and variations, but there is very little information on this. This should be explained in the paper, maybe in the methods.\n- The PICs shown in the paper are fabricated in a SiN platform. Can the authors please comment on what the pros and cons of implementing the system on a silicon platform will be (in terms of loss, noise, etc.)?\n- Some information about coupling and on-chip loss would be useful.\n- The paragraph on page 4 explaining Fig. 2d and the figure itself are hard to follow. I suggest revising the text. Also, I think the correspondence between Fig 2c and 2d (arrows) is not very clear. I suggest revising that, too.\n- I suggest changing the supplementary figure names to Fig. S1, S2, …, or something similar, to avoid confusion with the main text. Also, it would be clearer if the lasers were referred to differently than the delay lengths in Fig. 1 (L1, L2)."}, "author_response": "We thank you for the constructive feedback and the time you dedicated to reviewing our\nmanuscript. Inthepeerreviewresponsedocument,weprovideadetailedpoint-by-pointresponseto\nall of the comments. As you will see, we have addressed each concern thoroughly and implemented\nextensive revisions to the manuscript, which we summarize here:\n• Reviewer 1. Comment 1. Revised Section II.C title from “Sparse Filter Structure for\nArbitraryWavelengthWindows”to“SparseFilterStructureforCustomWavelengthWindows”\nto more accurately reflect realistic design constraints.\n• Reviewer 1. Comment 2. Added a new Supplementary Section IV titled “Potential Appli-\ncation: Complex Fourier Transform Spectroscopy”, which demonstrates how the experimental\ncalibration data from the main text can be used to reconstruct synthetic complex spectra. This\nillustrates the potential extension of the method to broadband and multi-line sources.\n• Reviewer1. Comment3. Expandedthequantitativecomparisontoincludeconventionaland\nrecently reported integrated wavemeters (adapted from Yao et al., Nat. Commun. 2023). The\nnew figure highlights the bandwidth–resolution trade-offs and shows that the proposed single-\nstream minimum-phase approach achieves sub-picometer resolution without requiring multiple\ndetection channels.\n• Reviewer 1. Comment 4. Addednewsimulationsofconventional90°hybridarchitecturesto\nassess sensitivity to phase imbalance. The results confirm that small fabrication deviations can\nseverelydegradehybrid-basedapproaches. Theproposedminimum-phasemethodeliminatesthe\nneedforphase-sensitivehybrids,offeringimprovedfabricationrobustnessandreducedfootprint.\n1\n• Reviewer 2. Comment 1. Clarified the novelty of the generalized sparse tap configura-\ntion compared to prior sequential designs (Xu et al, Nat. Phot. 2022). We added an explicit\nexplanation in the Introduction and Supplemental Section III, showing that sequential taps\nwould require ∼100 delay lines to achieve the same resolution, whereas our sparse configura-\ntion reduces footprint and complexity while maintaining sub-pm resolution. The revised text\nemphasizes functional advantages in resolution, and footprint.\n• Reviewer 2. Comment 2. Enhanced the description of Hilbert-transform-based phase re-\ntrieval by adding a clear explanation of the physical principles and citing Mecozzi’s work on\nnecessaryandsufficientconditionsforminimum-phasesystems. TheIntroductionnowexplicitly\nstates that phase recovery is guaranteed when zeros and poles lie inside the unit circle, enabling\nunique phase reconstruction from intensity measurements.\n• Reviewer 2. Comment 3. Revised Figure 2 for clarity and added a new Supplemental\nSection titled “II. Single ∆L: Calibration & Testing Algorithm”. This section details dataset\nconstruction, calibration workflow, and testing across 49 circuits from two packaged chips,\ndemonstrating robustness to manufacturing variability. Subsection II.B in the main text was\nrewritten to align with the updated figure.\n• Reviewer 2. Comment 4. Expanded discussion on how the dataset informs algorithm\nperformance. Clarified that the dataset does not bias regression but enables visualization of\ncalibration/testing combinations and RMSE trends, improving interpretability and validating\ngeneralization across chips.\n• Reviewer 3. Comment 1. Added discussion on using MMIs or adiabatic couplers to im-\nprove bandwidth and reduce sensitivity to fabrication variations. Methods section now includes\nsentences highlighting these alternatives for future designs.\n• Reviewer 3. Comment 2. Addressed temperature stabilization and its impact on accuracy.\nAdded explanation in Subsection II.B and Methods section noting that global phase shifts\nmanifest as small RMSE increases and can be corrected in post-processing. Discussed potential\nfor athermal designs using complementary materials (Si/SiN).\n• Reviewer 3. Comment 3. Clarified the role of on-chip phase shifters. While not required\nfor wavemeter operation, they enable advanced applications such as complex Fourier-transform\nspectroscopy(demonstratedinSupplementalSectionV).AddedexplanatorysentencesinMeth-\nods.\n• Reviewer 3. Comment 4. Compared Si vs. SiN platforms, outlining trade-offs in foot-\nprint, propagation loss, and thermal sensitivity. Added rationale for selecting SiN for this first\ndemonstration in Methods section.\n2\n• Reviewer 3. Comment 5. Included details on coupling and propagation losses, supported\nby experimental measurements using the minimum-phase framework (see new Figure R4 in\nSupplemental). Added numerical values for insertion and propagation losses in Methods.\n• Reviewer 3. Comment 6. Revised Figure 2 and corresponding text for clarity; added new\nsupplementalsectionforcalibration/testingworkflow. Updatedsupplementaryfigurenamingto\nFig. S1, S2, etc., and renamed laser labels in Fig. 1 to avoid confusion with delay-line notation.\nOverall, these changes have significantly enhanced the paper, and we hope you will agree that\nit is now suitable for publication in Nature Communications.\nRemarks to the Author:\nThe paper introduces a single-stream, minimum-phase PIC architecture that uniquely recon-\nstructs optical phase from intensity measurements via the Hilbert transform, eliminating the\nneedforadditionalphaseretrievalcomponents. Additionally,thepaperachievessub-picometer\nroot-mean-square wavelength regression error over arbitrary wavelength windows. The re-\nviewer believes that several key issues remain that need to be clarified or justified, after which\nthe manuscript can be considered for publication.\nResponse:\nWe thank the reviewer for their supportive feedback and insightful suggestions. We\nhave addressed each of their points below.\nReviewer Comment 1:\nThe phrase ”Arbitrary Wavelength Windows” is used in the title of Part IV. While the multi-\n∆L design does address wavelength ambiguity and allows operation over a much wider window\nthan the FSR of a single large ∆L, the term “arbitrary” might be too absolute. The actual\noperating window is still limited by factors such as the bandwidth of the directional coupler\nand waveguide losses. It is suggested to modify the wording in the title or at the beginning of\nthe section to be more precise.\nResponse:\nWe thank Reviewer 1 for their suggestion, and we agree that the term “arbitrary” may\nbe too absolute, as it suggests that the methodology applies to any wavelength window,\nregardless of component design or limitations. To clarify this, we have modified the\nwording in Section II.C. Specifically, the title has been changed from “Sparse Filter\nStructure for Arbitrary Wavelength Windows” to “Sparse Filter Structure for Custom\nWavelength Windows”, reflecting that the design methodology applies to wavelength\nranges achievable given the properties of the implemented circuit elements.\nReviewer Comment 2:\nThe manuscript mentions “computational spectroscopy” in the abstract and introduction.\nHowever, all experiments in the paper are based on a single-wavelength, narrow-linewidth\ntunable laser. This demonstrates that the device can function as a high-precision wavemeter.\nStill, there is no evidence to support its capability in handling broadband sources, multi-line\nsources, or continuous spectrum analysis. It is recommended to clearly define the positioning\n4\nas ”high-precision wavelength metrology” and elaborate on the conditions and challenges of\nextending to complex spectra in the discussion\nResponse:\nWe thank Reviewer 1 for their comment on the applicability of our work in the compu-\ntationalspectroscopyfield. Sincesubmittingthemanuscript,wehaveworkedtodemon-\nstrate the broader applicability of our approach to computational spectroscopy. Specif-\\ically, we have now applied the minimum-phase design framework to enable complex-\nvalued Fourier-transform spectroscopy (FTS) for reconstructing various types of spec-\ntra in the same multi-∆L single-stream circuit described in subsection II.C of the main\narticle. By utilizing minimum-phase, we overcome the manufacturing sensitivity of\nconventional FTS, which has hindered reconstruction robustness [1]. Specifically, like\nin the wavemeter algorithm, the minimum-phase design framework enables phase re-\ntrieval directly from the intensity measurement [2]. In doing so, we have directly fit\nthe spectral phase function, thereby absorbing manufacturing variations in our calibra-\ntion procedure. Furthermore, owing to the complex-valued nature of our approach, we\nachieved lossless interpolation of the interferogram alignment by leveraging phase slope\nmultiplication in the complex domain. Consequently, these features have enabled us to\nreconstruct both the amplitude and the phase of broadband sources.\nWehavehighlightedthisexampleofcomplex-valuedreconstructioncomputationalspec-\ntroscopy in a new section of the Supplemental Information (Section V. Application:\nComplex Fourier Transform Spectroscopy). It utilizes the experimental data from Sub-\nsection II.C of the main paper (Multi-Delta-L / Sparse Filter Structure for Custom\nWavelength Windows) to showcase the reconstruction of complex spectra produced by\na chirped Gaussian, Mach-Zehnder Interferometer, or Ring Resonator. As shown in\nFig. R1(a) (Fig. S6 in the supplemental), the experimental configuration closely mir-\nrors that of Figure 3 in the main article. In order to realize the complex-valued FTS\nfunctionality, thermo-optic tuning was performed using DC probes and a Keithley 2400\nSMU supplying 0–10 V in 0.5 V steps, with a wavelength sweep conducted at each\nvoltage level.\nUsing the circuit’s minimum-phase property, the phase response is derived from the\nmeasured intensity through a Hilbert transform (Fig. R1(b,c)). Digital filtering around\neachdelaylinethenisolatestheindividual∆Linterferometers(Fig.R1(d)). Thefiltered\noutputs reveal the spectral phase response as a function of the electrical power applied\nto each phase shifter (Fig. R1(e–h)). This experimental phase information allows direct\ncharacterizationoftheopticalpathdelaysweep, ∆L(P), whichissubsequentlymodeled\nusing a polynomial fit (Fig. R1). Based on this delay model, the interferograms are\nalignedbyapplyingthephaseresponseexpressedas∝ eβ∆Li(P) (Fig.R1(m–p)). Finally,\ntheFouriertransformoftheaveragedalignedinterferogramsproducesthereconstructed\nspectrometer calibration data using a Keysight tunable laser and PD combo with a Keithley 2400\nSource-Measure-Unit (SMU). (b) Photodetected power and (c) retrieved phase using the Hilbert\ntransform and leveraging the minimum-phase design of the resulting MZI system. (d) The inverse\nFourier Transform of the constructed complex-valued response allows us to filter each ∆L delay\nline for further post-processing. The application of each filter provides the phase of each ∆L (e-h).\ni\nSubsequently, each spectral phase is fitted using a linear model to obtain ∆L(P) (i-l). The data\nfrom this is then used to align each interferogram by using the phase factor ∝ ∆L(P)/c at each\nelectrical power value (m-p).\nspectrum. The reconstruction for the three cases (chirped Gaussian, Mach-Zehnder\nFigure S6 of the Supplemental Section V. Application: Complex Fourier Transform\nSpectroscopy.\nReviewer Comment 3:\nQuantitative comparisons with conventional systems and recently proposed nanophotonics-\nenabled wavemeters are necessary.\nResponse:\nWeperformedaquantitativecomparisonwithpriorwork. Integratedwavemeterdemon-\nstrations are rare, and since our method introduces a new approach for high-precision\nwavelength metrology with broad relevance to computational spectroscopy, we com-\npared it against all conventional integrated spectrometer designs. Figure R2 adapts a\nrecent comparison by Yao et al. [3] (with additional publications as recent as of this\nwriting), highlighting different spectrometer types, including wavemeters, and showing\nthat our method achieves a new resolution regime.\nDirect comparisons to conventional computational spectroscopy can be misleading be-\ncause of fundamental differences in methodology. Traditional techniques generally fall\nintothreecategories: narrowbandfiltering, FourierTransformSpectroscopy(FTS),and\nreconstruction-based methods. These approaches segment the spectrum either spatially\n(narrowbandfiltering,reconstruction)ortemporally(FTS),imposingdesignconstraints\nsuch as multiple photodetectors or parallel measurement channels. For example, Yao et\nal. [3](Fig. 5(b))reportfootprintperchannel, whereasourmethodperformsregression\non retrieved spectral phase, eliminating per-channel footprint constraints. The spectral\nphase serves as a linear basis for regression, and the sweep rate and resolution define\nthe effective number of channels. Bandwidth and resolution remain useful metrics,\nand our approach notably surpasses the sub-pm resolution limit. Prior interferometric\nphase-based methods [4, 5] required multiple detection channels, while ours achieves\nsingle-stream operation with a higher bandwidth-to-resolution ratio.\nBecause these comparisons are somewhat arbitrary, we have excluded them from the\nrevised manuscript and supplementary material. Interested readers can access this\ninformation in this peer review file once the manuscript is published.\n[3] Fig. 5(a). Comparison of our methodology (red star) with current state-of-the-art works in\ncomputationalspectroscopywithapproachesencompassingFourierTransformSpectroscopy(FTS),\nspectrum reconstruction, and narrowband filtering (blue dots), along with phase-based wavelength\nmetrology (green dots).\nThe authors state in the introduction that conventional 90-degree hybrids ”require a large\nfootprint and are highly susceptible to manufacturing variations.” However, the manuscript\nprovides no evidence or analysis to substantiate the claim that the proposed method offers any\nadvantages in either footprint or fabrication robustness\nResponse:\nWe thank the reviewer for their thoughtful comment regarding the fundamental com-\nparison between 90-degree hybrids and our approach. Here, we will demonstrate that\n90-degree hybrids are sensitive to phase mismatches as small as 1.1◦. This limitation\nnarrows their applicability for use in wavelength metrology. As reported in [4], a phase\nimbalance exceeding 1.1◦ can significantly degrade the resolution of 90° hybrid–based\napproaches. To assess robustness, we simulated a conventional 90° hybrid architecture\nusing Lumerical Interconnect [6] under ideal conditions. We modeled three MZIs on a\nSiN platform — analogous to our multi-∆L design but employing 90° hybrids for phase\nextraction rather than minimum-phase reconstruction. We swept the hybrid phase im-\nbalance from 88◦ to 92◦. The results, shown in Fig. R3, highlight the sensitivity of this\napproachtosmallphasedeviations. Wealsonotethattheoscillatoryphase-retrievalbe-\nhaviorinducedbyphaseimbalancecanbeobservedinFig.8of[4]andwassubsequently\nidentified as a reproducible error source in their Fig. 9.\nExperimentally, prior work has demonstrated low-loss 90° hybrids [7]; however, their\nmeasured phase imbalances exceed the thresholds required for the high resolution re-\nported in [4]. In contrast, our minimum-phase methodology retrieves the spectral phase\ndirectlyfromintensitymeasurementsratherthanrelyingonphase-sensitiveI/Qoutputs,\nproviding intrinsic robustness to fabrication-induced phase errors. This robustness is\nfurther supported by Fig. 2 of the manuscript, which presents cross-chip calibration re-\nsults between two independently packaged chips (P1, P2). All calibration combinations\nmaintainaccuratewavelengthestimation, demonstratingthattheproposedmethodtol-\nerates manufacturing variation without the strict phase-matching requirements of 90°\nhybrids. As far as the footprint goes, 90° hybrids require, at a minimum, 4 couplers.\nIn our approach, we don’t need additional components to enable phase retrieval, as the\nminimum-phase approach allows us to obtain the phase from intensity measurements\ndirectly.\n9\n90° Hybrid With\n(a)\nIdeal Components\nL 2 +|•| 2\n}\nI\n-|•| 2\nONA\nIn +|•| 2\n}\nL Q\n1\nφ -|•| 2\n(b)\nFigure R3: 90-degree Hybrid Phase Noise Robustness Simulation.(a) Interconnect sim-\nulation setup using ideal components for the 90° hybrid component. The phase is calculated as\nθ = tan−1(Q/I). (b) simulation results showing three different MZI configurations corresponding\nto FSRs of 10 nm (121 µm), 5 nm (242 µm), and 1nm (1210 µm) with varying levels of phase\nimbalance (ϕ) to simulate manufacturing robustness under ideal conditions. The phase offset is\ncalculated ∆θ = θ −θ where θ is calculated at ϕ , and θ is at ϕ = 90°.\ni 90 i i 90\nRemarks to the Author:\nThis paper introduces a single-stream, minimum-phase photonic integrated circuit (PIC) ar-\nchitecture, with core innovations including the minimum phase conditions and the Hilbert\ntransform for phase retrieval. These contributions demonstrate substantial theoretical value\nthrough a systematic experimental design and offer implications for advancing next-generation\ncomputational spectroscopy technologies.\nResponse:\nWe thank the reviewer for their thoughtful evaluation of our work and for recogniz-\ning the theoretical significance of our design methodology. We appreciate the positive\nassessment of the core innovations presented in the manuscript. Below, we provide\ndetailed responses to each of the reviewers’ comments.\nReviewer Comment 1:\nPlease clarify the novelty of the “generalized sparse tap configuration” compared to previous\nworks. While the authors claim to generalize Xu et al.’s the minimum phase approach from\nsequential filter taps to arbitrary sparse taps, the distinction between the two could be made\nmore explicit. Specifically, it is suggested to clearly explain how their design differs from\nexisting works in terms of synthesis methods and functional advantages (including resolution,\nfootprint, and signal/noise ratio). This will help to strengthen the manuscript’s contributions.\nResponse:\nWe thank the reviewer for your comment regarding the novelty of our work. Here,\nwe will demonstrate the distinction by reviewing Xu’s work and showing that, if we were\nto use sequential taps, it would require 100 delay lines to realize the same functionality\nas in our manuscript. Specifically, Xu et al. focused solely on using the minimum-\nphase approach with the intention of synthesizing FIR filters leveraging programmable\nphotonics as their framework. This limits the scope of their demonstration to the case\nwhere we have an interference pattern that is composed of a sum given by\n(cid:88)\nH(ω) = einω,∀n ∈ [0,1,2,3,...,N] (1)\nn\nwhere ω is the radial frequency. We observe that this is a sum that converges to\nH(ω) = (1−eiNω)/(1−eiω). This fact makes the tap extraction process for their cir-\ncuit a simple Fourier transformation. No preprocessing is needed, as the mathematical\ndefinition automatically accounts for the sequential delay increases. In contrast, in our\nwork, the delay lines need to increase in a non-sequential manner. This is due to the\n11\nneed to increase our delay lines to ensure a higher resolution wavelength measurement.\nAs shown in the Supplemental in Section III, we had to condition our signal properly\nsuchthatwecouldextractourfiltertaps. Inaddition, wealsodemonstrate, inthesame\nSection, that the non-sequential filter taps need to follow a specific structure for proper\nsignal reconstruction and modeling of the resulting G(ω). Specifically, Fig. S4 shows\nthat the fabricated 8-tap filter does not fulfill the set of necessary conditions for proper\nsignal reconstruction as opposed to our 4-tap filter design. This work is a fundamen-\ntal generalization of Xu’s work, as this analysis was not provided in their work due to\nthe nature of their demonstration. For our case, following their approach would have\nresulted in a circuit that requires at least 100 delay paths to achieve the same reported\nsub-pm resolution. To address all these points, we have modified the Introduction of\nthe main paper. Specifically, we have added: “This conceptual limitation would pro-\nhibitively increase a spectrometer circuit’s footprint by n-fold (n delays paths needed\nto go from bandwidth-defining FSR to higher-resolution FSR). Here, we overcome this\nthrough the use of non-sequential tap increases, uncovering the necessary optical signal\nprocessing needed to achieve high-performance wavelength metrology. This work en-\nablesarbitraryandsparsetapfilterconfigurations,wherencanbeanyinteger,removing\nthe footprint requirements from n delay paths to any m < n. Enabling this arbitrary\nincrease in delay length significantly enhances spectral resolution, proving particularly\nvaluable for wavelength metrology applications.”\nReviewer Comment 2:\nIt is suggested to enhance the understandability of Hilbert transform-based phase retrieval.\nThe key idea of the proposed research is the reliance on the minimum phase conditions and\ntheHilberttransformforphaseretrieval, butthephysicalprinciplesthatenableuniqueoptical\nphase reconstruction may be obscure.\nResponse:\nWe thank the reviewer for highlighting the need to clarify the physical principles un-\nderlying our Hilbert-transform-based phase retrieval approach. We acknowledge that\nthe existing literature on this topic is relatively limited, which may contribute to the\nperceived obscurity of the concept. To address this, we have incorporated a reference to\nthe work by Mecozzi[8], which rigorously derives the necessary and sufficient conditions\nfor accurate phase reconstruction from intensity measurements. As discussed in our\nmanuscript, these conditions ensure that when both the zeros and poles of the system\nlie inside the unit circle, the system is minimum phase, enabling phase recovery via the\nHilbert transform. We have added this reference to the Introduction and revised the\ntext to explicitly state: “Specifically, the phase is determined directly from the magni-\ntude via their fundamental relationship in a minimum phase system, enforced by the\nHilbert transform.[8, 9]”.\nAs shown in Fig. 2, the experimental results for wavelength regression prove that the experi-\nmental data drives the results and the filter is not biasing the proposed regression algorithm.\nPlease explain in detail how the dataset improves the performance of the regression algorithm\nand how the dataset was constructed. How to improve both the measurement accuracy and\nthe generalization of the proposed regression algorithm?\nResponse:\nWe thank the reviewer for their insightful comment regarding the clarification of our\nwavelength regression algorithm, as presented in Figure 2 of the main manuscript. To\nimprove clarity, we have extensively revised Figure 2 to streamline the workflow and\nadded a detailed description of the calibration and testing procedures in a new section\nof the Supplemental Material titled: “II. Single ∆L: Calibration & Testing Algorithm”.\nAdditionally, Subsection B of Section II (Results) has been rewritten to align with the\nupdatedfigure. Regardingthereviewer’sspecificquestion,thedatasetisnotintendedto\nenhance measurement accuracy or generalization of the proposed algorithm. Instead,\nit serves as a structured representation of data collected from 49 circuits across two\npackaged chips (each containing four chips, with one circuit (K in P ) excluded due\n4 2\nto a short circuit). The matrix format enables us to visualize calibration and testing\nacross different circuit combinations (rows correspond to calibration circuits P −K ,\ni j\ncolumns to testing circuits), allowing an assessment of chip-to-chip variation and, more\nimportantly, the robustness of the algorithm to manufacturing variability.\nRemarks to the Author:\nRubio, et al. have reported single input/output minimum phase PICs to extract optical phase\nfrom intensity measurement. They have presented the theory as well as some experimental\nresults for single and multiple delay MZIs. Overall, the paper is well-written, and the idea is\ninteresting, although the basic concept has been reported before. Below are my comments:\nResponse:\nWe appreciate the reviewer’s time and effort in evaluating our work. We are grateful\nfor the positive feedback regarding the clarity of our manuscript and the interest in\nour research. Below, we provide detailed responses to each comment in the following\nsections.\nReviewer Comment 1:\nCan the authors comment on using MMIs instead of directional couplers as they would make\nthe system more wideband and less sensitive to fabrication variation (basically coupling ratio\nvariations)? Especially for the sparse filter chip where 50/50 couplers are used\nResponse:\nWe thank the reviewer for their comment regarding the design complexity that arises\nfrom bandwidth-limited devices such as directional couplers. We used PDK (Process\nDesignKit)devicesforthecouplersinbothcircuitsinthemanuscript,thesingle∆Land\nthe multi-∆L architectures. Overall, broadband devices such as MMIs and adiabatic\ncouplers[10] would allow us to maintain a larger bandwidth where minimum-phase is\nmaintained,therebyenablinglargeroperatingbandwidthsforthewavemeter. Weadded\nthe following sentence to Subsection A of the Methods section of the main paper: “For\nthe integrated directional couplers, we used the AIM Photonics PDK components for\nboth the 50:50 splitters and the 90:10 splitters. Achieving larger bandwidths can be\naccomplished by designing couplers with the intended operational bandwidth using\nalternatives such as MMI or adiabatic couplers”. For Subsection B of the Methods\nsection, we added the following sentence: “For the integrated directional couplers, we\nuse the 50:50 splitter within the ANT PDK. Similar to the single ∆L case, we could\ndesignMMIoradiabaticcouplersthatmaintainsplittingratiosoveralargerbandwidth\nthan conventional directional couplers”.\nI suppose the chip is temperature stabilized. Is that the case? How would temperature change\naffect the measurement accuracy?\nResponse:\nWe thank the reviewer for their comments regarding temperature stabilization and the\nimplications on measurement accuracy. We did use temperature stabilization for both\nchips. However, global phase shifts due to temperature variations become apparent in\nthe results shown in Figure 2.(d) from the main paper. The slight RMSE increases\nfound in the matrices when both the calibration and testing data match in ∆L, i.e.\nK = K and K = K are due to this temperature instability. We added the the\n1 3 2 4\nfollowing sentence in Subection II.B to point this out: “Further evidence of this effect is\nshown in Figure 2(c), where the matched ∆L testing data (red arrow) presents a small\nglobal phase offset when compared to the calibrated data (blue arrow), producing an\nRMSE increase”.\nInthemulti-∆Ldesign,thephaseisuniquelydeterminedoverthebandwidthofinterest,\nso temperature changes manifest only as a global phase offset that can be corrected in\npost-processing. Amorerobustapproachwouldbetousematerialswithcomplementary\nthermo-optic coefficients, such as Si and SiN, to achieve true athermal operation [4].\nHowever, implementing such a scheme could introduce additional complexity due to\ndiffering group indices, potentially confounding the demonstration of the minimum-\nphase approach. Since Stern et al. already demonstrated athermal operation [4], our\nfocus here is on showcasing the capabilities of the minimum-phase methodology for\nhigh-precision wavelength metrology. We have addressed this point in conjunction with\nComment 4 (below) by adding the following sentences to Subsection B in the Methods\nsection of the main paper: “In this case, we choose SiN as our material platform owing\nto its low loss and small thermo-optic coefficient when compared to Si. The latter\nallows us to be less cognizant of temperature variations while still enabling the study of\nthe minimum-phase approach without confounding factors such as mixed group indices\ndesigns.”\nReviewer Comment 3:\nThere should be phase shifters within the MZIs to adjust/calibrate for phase mismatches and\nvariations, but there is very little information on this. This should be explained in the paper,\nmaybe in the methods.\nResponse:\nWe thank the reviewer for their thoughtful consideration of the on-chip phase shifters\nforcalibratingforphasemismatches. Whilethiscouldbedone, itactuallyisn’trequired\n15\nforwavemeteroperation. Specifically, theminimum-phasereconstructionalgorithmcan\nidentify any phase imbalances within the circuit. As an example of this, in Section IV\nof the supplemental, we demonstrated that the bandwidth of the PDK coupler could be\nextractedbymonitoringthephaseimbalanceasafunctionofwavelength. Consequently,\nsince the reconstruction algorithm can identify these phases, an explicit thermo-optic\ncontrol isn’t required. With that said, in our response to Reviewer 1 (Comment 2),\nwe have now demonstrated that the on-chip phase shifters enable a new paradigm\nwithin Fourier Transform computational spectroscopy applications. Specifically, in the\nSupplemental, we have demonstrated how these phase shifters enable reconstruction of\narbitrary complex spectra, such as a Chirped Gaussian, Ring Resonator, and Mach-\nZehnder spectra. We plan to submit a detailed follow-up manuscript that will utilize\nthis concept.\nReviewer Comment 4:\nThePICsshowninthepaperarefabricatedinaSiNplatform. Cantheauthorspleasecomment\non what the pros and cons of implementing the system on a silicon platform will be (in terms\nof loss, noise, etc.)?\nResponse:\nWeappreciatethereviewer’sinsightfulquestionregardingtheadvantagesanddisadvan-\ntages of using the SiN/Si platforms. The experiments presented in the paper show both\nSi (single ∆L) and SiN (multi-∆L) material platforms. Implementing the multi-∆L on\nSi would generally allow for tighter integration and potentially smaller footprints due to\nthehigherindexcontrast, butwouldcomewithtrade-offs. Inparticular, Sitypicallyex-\nhibits higher propagation loss in the telecom band than SiN, and stronger thermo-optic\neffects, which could increase sensitivity to temperature variations. SiN, on the other\nhand, offers low propagation loss, wide transparency, and reduced thermal sensitivity,\nwhich are advantageous for high-precision wavelength metrology. Overall, the choice of\nplatform involves balancing footprint, loss, thermal stability, and fabrication maturity.\nHere, SiN was selected to optimize for low loss and stability because this is the first\ndemonstration of this methodology. We added the following sentence in Subsection B\nwithin the Methods section of the main paper: “In this case, we choose SiN as our\nmaterial platform owing to its small thermo-optic coefficient when compared to Si. The\nlatter allows us to be less cognizant of temperature variations while still enabling the\nstudyoftheminimum-phaseapproachwithoutconfoundingfactorssuchasmixedgroup\nindices designs.”\nSome information about coupling and on-chip loss would be useful.\nResponse:\nWe thank the reviewer for this suggestion. The on-chip devices exhibit typical propa-\ngation losses for the SiN platform, with waveguide losses of ∼1 dB/cm. Fiber-to-chip\ncoupling was achieved using edge couplers with insertion losses of 9dB per facet (which\ncanbeimproveduponwithoptimizationtothefiberpackagingprocess). Wehaveadded\nthis information to the Methods section to provide a clearer picture of the system per-\nformance. To confirm the waveguide loss, we actually investigated and found that our\nminimum-phase approach allowed us to obtain the SiN platform loss by measuring the\nintensity of the four taps in the discrete-time domain and fitting the loss as a function\noflength, asshowninFigureR4. Asaresultofthereviewer’sthoughtfulsuggestion, we\naddedthefollowingsentencesregardinglossinSubsectionAwithintheMethodssection\nof the main paper: On-chip loopbacks were used to measure coupling loss of around\n∼ 7[dB] per facet (which can be reduced by optimizing the fiber packaging process),\nand the propagation loss is estimated to be ∼ 1[dB/cm], and in Subsection B within\nthe Methods section of themain paper: The coupling loss was measured at∼ 9 [dB]per\nfacet (which can be reduced with optimized edge coupler design), while the propagation\nloss was measured at ∼ 1 [dB/cm]\n(a) (b)\nFigure R4: SiN Loss Measurement Using Minimum-Phase Framework.(a) Temporal re-\nsponse of 4-tap SiN chip after phase retrieval. (b) linear fit of the normalized tap energy vs tap\nlength, providing the chip loss per unit length.\nThe paragraph on page 4 explaining Fig. 2d and the figure itself are hard to follow. I suggest\nrevising the text. Also, I think the correspondence between Fig. 2c and 2d (arrows) is not\nvery clear. I suggest revising that, too.\nResponse:\nWe thank the reviewer for pointing out that the explanation of Fig. 2d and its cor-\nrespondence with Fig. 2c could be clearer. We have modified Figure 2 entirely to\nstreamline and simplify the presentation of our algorithm. In addition, we added a new\nsection in the supplemental titled “II. Single ∆L: Calibration & Testing Algorithm”\n(See Reviewer 2, Comment 3). In addition we have revised Subsection B in Section II\n(Results) of the main paper to reflect the streamlined Figure 2. These changes aim to\nmake the workflow and correspondence between the two panels more intuitive.\nReviewer Comment 7:\nI suggest changing the supplementary figure names to Fig. S1, S2, ..., or something similar,\nto avoid confusion with the main text. Also, it would be clearer if the lasers were referred to\ndifferently than the delay lengths in Fig. 1 (L1, L2).\nResponse:\nWe thank the reviewer for the helpful suggestion. We have updated the supplementary\nfigure labels to Fig. S1, Fig. S2, etc., to distinguish them from figures in the main text.\nIn addition, we renamed the laser labels in Fig. 1 from (L1, L2) to (S1, S2) to avoid\nconfusion with the delay-line notation (L1, L2), and we revised the caption accordingly\nfor clarity.\nReferences\n[1] Li,A.,Davis,J.,Grieco,A.,Alshamrani,N.&Fainman,Y. Fabrication-tolerantFouriertrans-\nform spectrometer on silicon with broad bandwidth and high resolution. Photonics Research\n8, 219–224 (2020). URL https://opg.optica.org/prj/abstract.cfm?uri=prj-8-2-219.\nPublisher: Optica Publishing Group.\n[2] Xu, X. et al. Self-calibrating programmable photonic integrated circuits. Nature Photonics\n16, 595–602 (2022). URL https://www.nature.com/articles/s41566-022-01020-z.\n[3] Yao, C. et al. 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Most of my previous concerns regarding the methodology and the positioning of the work have been addressed. However, there are two remaining points regarding the scalability and the operational constraints of the proposed architecture that require further clarification before the manuscript is suitable for publication.\n1. The authors demonstrate that resolution scales with the length and number of delay taps. However, the power of the reference arm must remain dominant to satisfy the stability of the Hilbert transform phase retrieval. As the number of delay taps increases, or as path lengths extend (leading to higher propagation losses), the power budget for each individual interferometric component is naturally squeezed. This inevitably constrains the system's dynamic range. It is recommended to discuss how the input power and the system’s Signal-to-Noise Ratio impact the accuracy of the wavelength regression.\n2. The manuscript notes that the tunable laser used in the experimental setup is capable of a wide sweep from 1500 nm to 1600 nm. However, the characterized operational range of the Photonic IC is limited to a 10 nm window. It is currently unclear whether this limitation is a fundamental design choice or a practical constraint.", "reviewer_2": "The authors have provided a comprehensive response to all comments and addressed all technical concerns. Their revisions enhance the manuscript's quality, resolving key issues and strengthening their contribution to the field of photonic integrated circuits (PICs).\n1. The authors clearly distinguished their approach from the prior work proposed by Xu et al., demonstrating that a sequential tap design would require 100 delay lines for the same functionality as in the manuscript.\n2. The manuscript introduces the work of Mecozzi et al., which provides the basic theory to link the minimum phase with the phase recovery from intensity measurements.\n3. The manuscript has been revised to optimize the proposed wavelength regression algorithm in Figure 2 and enhance readability.", "reviewer_3": "I would like to thank the authors for addressing my comments. I believe the manuscript is stronger now and I do not have any further comments except for the following:\nOn page 8 (Methods) there is inconsistency in the loss per facet (7 dB vs 9 dB). Please fix that."}, "author_response": ""}, {"round": 3, "reviewer_comments": {"reviewer_1": "I thank the authors for their comprehensive response to all my comments and addressed all the technical concerns. I think the paper can be accepted for publication now."}, "author_response": ""}]

Decision letter