Slice-Less Optical Arbitrary Waveform Measurement (OAWM) in a Bandwidth of More than 600 GHz Using Soliton Microcombs

We propose and demonstrate a novel scheme for optical arbitrary waveform measurement (OAWM) that exploits chip-scale Kerr soliton combs as highly scalable multiwavelength local oscillators (LO) for ultra-broadband full-field waveform acquisition. In contrast to earlier concepts, our approach does not require any optical slicing filters and thus lends itself to efficient implementation on state-of-the-art high-index-contrast integration platforms such as silicon photonics. The scheme allows to measure truly arbitrary waveforms with high accuracy, based on a dedicated system model which is calibrated by means of a femtosecond laser with known pulse shape. We demonstrated the viability of the approach in a proof-of-concept experiment by capturing an optical waveform that contains multiple 16 QAM and 64 QAM wavelength-division multiplexed (WDM) data signals with symbol rates of up to 80 GBd, reaching overall line rates of up to 1.92 Tbit/s within an optical acquisition bandwidth of 610 GHz. To the best of our knowledge, this is the highest bandwidth that has so far been demonstrated in an OAWM experiment.


Introduction
Optical arbitrary waveform measurement (OAWM) based on frequency combs [1][2][3][4][5][6][7] has the potential to unlock a wide variety of applications, ranging from reception of high-speed data signals [3][4][5][6][7][8][9][10][11] and elastic optical networking [6] to investigation of ultra-short events and photonic-electronic analog-to-digital conversion [12][13][14][15]. Previous demonstrations of OAWM relied on spectrally sliced coherent detection [1][2][3][4][5][6][7], where optical filters are used to decompose a broadband input signal into several spectral slices. These slices are then individually detected by an array of in-phase and quadrature (IQ) receivers using local-oscillator (LO) tones derived from a phase-locked frequency comb as a common reference, and the optical waveform is reconstructed by stitching of the received spectral slices through digital signal processing (DSP). Based on this concept, OAWM of a 228 GHz-wide signal was demonstrated using discrete components [2], and further work demonstrated a 320 GHz photonic-electronic analog-to-digital converter that combines spectrally sliced OAWM with high-speed electro-optic modulators [14]. However, all these schemes crucially rely on high-quality optical filters for spectral slicing of the optical signal and for separating the comb tones. While the IQ receivers can be efficiently integrated using readily available platforms such as silicon photonics (SiP) [16,17] or indium phosphide (InP) [18], high-quality slicing filters are much more challenging to implement on high index-contrast photonic integration platforms. Specifically, previous demonstrations of integrated OAWM receivers either relied on InP-based arrayed waveguide gratings (AWG), that required individual phase correction in the various arms [6], or on SiP coupled-resonator optical waveguide (CROW) structures [7,19], that need sophisticated control schemes for thermal tuning. In addition, most of the previously demonstrated OAWM schemes [1][2][3][4][5][6][7] were based on frequency combs generated by RF modulation of a narrowband laser tone. This approach requires complex sequences of broadband electro-optic modulators, each driven with a dedicated radio-frequency signal, and limits the number of achievable comb tones and thus the acquisition bandwidth of the OAWM scheme. In this paper, we propose and experimentally demonstrate an OAWM scheme that does not require optical slicing filtersneither for the signal nor for the LO [10,11] and that exploits chip-scale Kerr soliton combs as highly scalable multi-wavelength LO. For detection, our scheme relies on an array of IQ receivers, which are fed by the full optical waveform as well as by timedelayed copies of the full LO comb. The electrical output signals of the IQ receivers then contain superimposed mixing products of the various LO tones with the respective adjacent portions of the signal spectrum and allow to reconstruct the full-field information of the incoming waveform using advanced DSP. In a proof-of-concept experiment, we implement the scheme using a chipscale dissipative Kerr soliton comb as multi-wavelength LO. Our experiment relies on a precise frequency-domain model of the OAWM system that accounts for the complex-valued transfer functions of the various detection paths and that is instrumental for high-fidelity reconstruction of truly arbitrary waveforms. We calibrate our model by measuring these transfer functions using a femtosecond laser with a known pulse shape. The viability of the scheme is demonstrated by simultaneously acquiring multiple 16 QAM and 64 QAM wavelength-division multiplexed (WDM) data signals with symbols rates of up to 80 GBd, reaching overall line rates of up to 1.92 Tbit/s within an optical acquisition bandwidth of 610 GHz [10]. To the best of our knowledge, this is the highest bandwidth that has so far been demonstrated in an OAWM experiment.

Hardware and signal acquisition
The fundamental concept of a slice-less OAWM system is illustrated in Fig. 1. The scheme combines a chip-scale frequency comb generator such as a Kerr soliton comb [20][21][22][23][24][25][26][27], see Inset 2, with a receiver system, that does not contain any slicing filters and that is thus amenable to implementation on state-of-the-art high-index-contrast integration platforms such as silicon photonics [11]. The incoming optical arbitrary waveform .s S () at. with Fourier spectrum S () af is split into N copies, which are routed to an array of IQ receivers, labeled by subscripts (1) The 2N output signals of all IQ receivers are digitized with an array of analog-to-digital converters (ADC), and an estimate of the envelope of the arbitrary optical waveform . S () at. is reconstructed via DSP. Each IQ receiver (IQR) comprises a 90° optical hybrid (OH) and two pairs of balanced photodetectors (BPD) and is read out by two corresponding ADC, to obtain the inphase component () It  and the quadrature component () Qt  of the respective baseband signal, see Inset 1 in Fig. 1. All IQ receivers and ADC have a bandwidth of , B within which mixing products of the various LO tones with the respective adjacent portions of the signal spectrum can Fig. 1. Concept of slice-less optical arbitrary waveform measurement (OAWM) using a chip-scale frequency comb generator such as a Kerr soliton microcomb as a multi-wavelength LO. The scheme does not require any slicing filters and thus lends itself to integration on high-index-contrast photonic platforms such as silicon photonics. (a) The optical input signal with spectrum S () af and the LO comb with spectrum LO () af are split into N copies. The N copies of the LO comb, each comprising M narrowband tones spaced by a free spectral range FSR f , are then delayed by distinct delays   before being fed to N in-phase and quadrature receivers (IQR), where they act as LO for coherent detection of the respective signal portion. The in-phase ( I  ) and quadrature ( Q  ) outputs of the IQ receivers are read out by 2N corresponding analog-to-digital converters (ADC). Digital signal processing (DSP) is applied to recover an estimate (est) S () af of the optical input spectrum. Inset 1: Schematic of IQR ν with a 90° optical hybrid (OH), a pair of balanced photodetectors (BPD) and corresponding ADC (bandwidth B ), providing the in-phase and quadrature components () It  and () Qt  of the respective baseband signals. Inset 2: Illustration of a chip-scale Kerr soliton comb source, comprising a continuous-wave pump laser and a high-Q Kerr-nonlinear microresonator. Kerr soliton microcombs provide a multitude of optical tones with narrow linewidths that are typically spaced by for free-spectral range FSR f of tens of GHz, thus offering a highly compact attractive option for scaling the bandwidth of the scheme. (b) The composite baseband signals ( ) ( ) j ( ) contain a superposition of the individual mixing products of each LO tone with the spectrally adjacent portion of the signal within a bandwidth B to either side of the respective LO tone 12 . , , , M f ff  Exploiting the fact that the various portions of the LO are subject to different time delays   and that individual LO tones thus feature distinct phase differences, we can separate the superimposed mixing products by DSP. Note that the receiver bandwidth B is chosen slightly larger than half the FSR, FSR / 2, Bf  leading to 1 M − overlap regions 1 12 OR , OR ,...., OR M − (shaded stripes). Within these overlap regions, the same spectral portion of the signal is transferred to two distinct portions at the edges of the baseband signal spectrum close to 1 FSR . This leads to redundant information that can be used to estimate random phase drifts along the various detection paths. be detected. For simplicity, we assume for now that the overall transfer function for the in-phase () It  and quadrature () Qt  components are identical for each IQR ν, and that they have an ideal 90° phase relationship. In real systems, this is not necessarily the case, such that a more advanced system model is needed, see Section 1.1 of Supplement 1 for a more detailed description of a general system model.

Waveform reconstruction
Based on the acquired in-phase () It  and quadrature () Qt  components obtained from the various IQ receivers, we construct the complex-valued composite baseband signals () Ut Importantly, as the optical arbitrary waveform S () at is simultaneously down-converted with all LO tones, each complex-valued composite baseband signal () Ut  contains a superposition of independent components, generated by mixing each LO tone with the spectrally adjacent portion of the signal, see Fig. 1   ( Note that the expressions () Gf  only account for electrical noise at the respective receiver, whereas the optical noise entering the receiver is regarded as part of the input signal S () af. The electrical receiver noise spectra () Gf  of the various receivers comprise thermal noise, shot noise, and quantization noise, and are assumed to have identical power spectral densities and to be statistically independent. These assumptions allow to use a simple least-square estimator as a special case of a minimum-variance unbiased estimator for reconstructing the incoming signal, see Section 1. 2   . In these relations, we disregard noise impairments of the incoming optical signalthey are considered part of the signaland we assume perfectly balanced photodiodes as well as an LO with a sufficiently large optical carrier-to-noise ratio (OCNR) such that optical noise contributions from the LO can be neglected. If this condition is not fulfilled then the reconstructed signal will additionally be impaired by multiplicative noise from the LO, see Section 7 of Supplement 1 for a detailed noise and distortion characterization. Note that the pseudo-inverse of the (M, N)-transfer matrix ( ) are impacted by amplitude and phase drifts of the comb lines and by random phase drifts in the setup. For Kerr combs pumped with highly stable lasers, the optical linewidth typically amounts to a few kHz, and mechanical vibrations in our setup are also limited to a similar frequency range [29]. The associated amplitude and phase changes hence occur on a time scale of hundreds of microseconds, and the transfer function ( ) f H may be considered constant during one recording with a typical length of a few microseconds. Therefore, we split each transfer-matrix element and into a time-variant, but frequency-independent part (t ) The time-variant, but frequency-independent matrix elements (t ) µ H  can be expressed by a product of a first ν-dependent complex-valued parameter (t) F, H  , which accounts for the time-dependent phase drift in the fiber leading to IQR ν, and of a second µ-dependent complex-valued parameter (t) LO, H  , which represents the amplitude and phase fluctuation of the -th  comb tone. These parameters are estimated by using the redundant information contained in the overlap regions of the various slices, see Section 1.4 of Supplement 1 for details. For measuring the time-invariant frequency-dependent parts ( ) of the overall transfer functions, we perform a one-time calibration by feeding the system with a known optical reference waveform (ORW). The ORW is derived from an ultra-stable femtosecond laser (MENHIR-1550) with well-defined pulse shape and an FSR R ORW FS f f , which is not an integer fraction of FSR f . A step-by-step description of the calibration procedure can be found in Section 4 of Supplement 1.

System conditioning and relation to time-domain optical sampling
To allow for a high-quality signal reconstruction according to Eq. (4), the transfer matrix ( ) f H must be well-conditioned, which can be achieved by choosing approximately equidistant time delays, see Section 1.5 of Supplement 1 for a more detailed discussion, LO LO FSR This relation can be intuitively understood when considering the special case of strictly equidistant time delays, , in combination with chirp-free LO signals, for which all comb lines have the same initial phases,  . In this case, the LO signals arriving at the various IQ receivers can be interpreted as short transform-limited pulses, and the mixing process with the signal can be understood as a time-interleaved linear optical sampling process [8,9,30], see Section 2 of Supplement 1 for details. In contrast to those techniques, however, the frequency-domain method presented in this paper inherently compensates for imperfectly chosen time delays   , for chirped LO pulses, for relative drifts of the optical phase between IQ receivers or LO comb lines, and for the exact frequency-dependent transfer characteristics of the various detection channels. This allows for measuring truly arbitrary waveforms without restriction to data signals with known structure, as was the case for previously demonstrated time-interleaved linear optical sampling techniques [8,9].

Experimental demonstration
To demonstrate the viability of the proposed OAWM scheme, we perform a proof-of-concept experiment using a setup based on discrete fiber-optic components, see Fig. 2 (a). The LO comb source (blue box) comprises a Kerr soliton comb with a pump laser L8, an erbium-doped fiber amplifier (EDFA) to boost the comb power, and a wavelength-selective switch (WSS) to select four adjacent comb lines. The free spectral range (FSR) of the comb is 110 GHz, see Fig. 2 (b) for the associated spectrum taken at point Ⓑ. The optical carrier-to-noise ratio (OCNR) of the individual LO comb lines was measured with respect to a reference bandwidth of 12.5 GHz, corresponding to a wavelength interval of 0.1 nm at a center wavelength of 1550 nm, and ranges from 23 dB to 24 dB. Variable optical delay lines are used to adjust the time delays   to the IQ receiver array. For the first IQ receiver (IQR 1) we use balanced photodiodes (BPD) with a nominal 3 dB bandwidth of 43 GHz and an actual 12 dB bandwidth of 80 GHz, while the nominal BPD bandwidth amounts to 100 GHz for the remaining IQ receivers (IQR 2 … IQR 4). The outputs of IQR 1 and IQR 2 are digitized with a 100 GHz oscilloscope (Keysight UXR1004A), while an 80 GHz oscilloscope (Keysight UXR0804A) was used for IQR 3 and IQR 4. Both oscilloscopes are synchronized. For the signal reconstruction, we digitally limit the RF frequency range to B = 80 GHz for all eight channels. The input signal is either given by a known optical reference waveform (ORW) Ⓒ, used for calibrating the system, or by a broadband test signal Ⓐ for demonstrating the viability of the scheme. The ORW is derived from a highly stable modelocked laser (Menhir 1550, Menhir Photonics AG) and features a smooth spectral amplitude and phase profile, see Fig. 2 (b) Ⓒ. The broadband test signal comprises seven wavelength-division multiplexing (WDM) data channels and is configured such that the full optical acquisition bandwidth opt B is occupied. The test signal is generated by modulating four optical carriers (L1 to L4) with a first and three optical carriers (L5 to L7) with a second IQ modulator. The driving (a) Experimental setup based on discrete fiber-optic components: A broadband optical signal is generated by modulating seven spectrally interleaved lasers (L1 to L7) with two different IQ modulators (IQM) that are driven with different signals, derived from a common arbitrarywaveform generator (AWG, Keysight M8194A). The output signals of the IQM are combined, and the resulting broadband optical test signal is then amplified and routed through a band-pass filter (BP) to the IQ receiver (IQR) array of the OAWM system. At the LO side, four coherent tones are isolated from a Kerr frequency comb (LO comb), split into four paths, delayed by distinct time intervals 14 ,   , and routed to the IQ receiver array. For calibrating the OAWM system, a known optical reference waveform (ORW) derived from a highly stable mode-locked laser is fed to the input of the OAWM system. The ORW features a smooth spectral amplitude and phase, see Subfigure (b) (b) Exemplary optical spectra of the broadband test signal at point Ⓐ, of the LO comb with FSR FSR 110GHz f = at point Ⓑ, and of the ORW at point Ⓒ. The ORW was generated by a solid-state mode-locked femtosecond laser (Menhir 1550), and the spectral amplitude and phase were measured by Menhir Photonics using frequency-resolved optical gating (FROG). All plots are shown with respect to the same center frequency cent 192.43 THz f = .
signals are generated by a 120 GSa/s arbitrary-waveform generator (AWG, Keysight M8194A) and amplified by electrical amplifiers.

System calibration
For calibration our OAWM system, we independently measure the frequency-dependent part ( ) , see Eq. (6), using a known reference waveform. Note that the implementation of the OAWM scheme in our experiment deviates slightly from the simplified notation introduced in Sect. 2 above, in which the in-phase and quadrature components of the each IQ receiver were merged into a common complex-valued baseband signal, see Eq. (2). Specifically, this merge requires identical transfer functions and an ideal 90° phase relationship of the in-phase and the quadrature detection path, which, in practice, is usually not fulfilled since the underlying balanced detectors typically have slightly different characteristics. We therefore reformulate Eq. (4) in terms of the signals () It  obtained from the in-phase detector and the signals () Qt  obtained from the quadrature detector, see Section 1.1 of Supplement 1 for details. This reformulation leaves the overall concept unchanged but allows for compensating slight differences of the in-phase and quadrature detection paths during system calibration. In the following, the complex-valued frequency-dependent and time-invariant transfer functions of the in-phase and quadrature components are referred to as ( ) , respectively. These transfer functions are measured by feeding an ORW with a comb spectrum of known amplitude and phase to the OAWM system and by extracting the amplitudes and phases of the beat notes with the LO comb at the in-phase and the quadrature detectors of the various IQ receivers, see , that is associated with the in-phase detection path of the first IQ receiver (IQR 1) and the second LO comb line at frequency 2 f . To validate our calibration technique for the magnitude, we replace the ORW in Fig. 2 (a) by a tunable external cavity lacer (ECL), ECL1, and we substitute the LO comb by another fixed-frequency ECL2 emitting at frequency 2 f , see Section 4.1 of Supplement 1. We sweep ECL1 in discrete frequency steps over the full detection bandwidth of the IQ receivers and record the amplitudes of the generated sinusoidal signals. This allows to extract the magnitude of the frequency response, indicated by a green trace in Fig. 3 (a), which is only partially visible since it coincides very well with the results of the ORW calibration. Note that the quality the calibration measurements crucially relies on the linearity of the underlying photodetectors and that detector saturation should hence be avoided, see Sections 4.3 and 4.4 of Supplement 1 for a more detailed discussion. In Fig. 3 (b), we show zoomed-in sections of the amplitude and phase transfer functions between 70 GHz and 78 GHz, indicated by a red and blue box in Fig. 3 (a), and visualize the results from 58 individual calibration recordings that were taken over a period of 2.5 h. We observe stable frequency-dependent transfer characteristics, where the frequency-dependent fluctuations are indeed a time-invariant property of the OAWM detection system. This validates our assumptions associated with Eq. (6).

Demonstration of broadband waveform measurements
We finally demonstrate the capability of the slice-less OAWM system by measuring broadband test signals that comprise seven wavelength-division multiplexed (WDM) data channels. In a first experiment, we use an LO comb with a free spectral range of FSR 110 GHz f = and a test signal that consists of four 40 GBd 64 QAM signals modulated on carriers L1 to L4 and three 60 GBd 16 QAM signals modulated on the carriers L5 to L7, see Ⓐ in Fig. 2 (b). We successfully separate the superimposed mixing products associated with the various LO tones and reconstruct the optical spectrum (est) S () af , the normalized magnitude of which is plotted in Fig. 4   = , and the data channels are labeled according to the respective optical carrier L1 to L7. For reference, we separately record the acquisition noise acq () f G comprising the thermal noise and the quantization noise contributed by the various ADC by disconnecting all optical receiver inputs. The recordings of the acquisition noise are processed in the same way as the signal recordings. Note that this processing introduces a frequency dependence of the reconstructed noise power spectral density 2 G () af , which is caused by two effects: First, the photodetector responses are equalized by multiplying the spectrally white acquisition noise acq () are uncorrelated, the resulting noise power within the overlap regions is reduced by up to 3 dB [31]. This effect does not occur at the lower and upper edge of the overall spectral acquisition range, because no overlap with an adjacent slice exists, thus leading to a slightly increased noise floor, Fig. 4(a). Note that the shot noise, which is unavoidably present in the signal recordings, is significantly smaller than measured acquisition noise, such that we can assume with good approximation that

=
We find that the reconstructed data signals are slightly impaired by amplified spontaneous emission (ASE) noise in the LO signal, which leads to a pronounced blurring of the outer constellation points since the LO noise affects the signal in multiplicative form [34]. This ASE noise originates from the EDFA that is used to boost the power of the soliton comb, see Fig. 2 (a), which originally has a power of only -31 dBm per comb line, see Section 5 of Supplement 1 for an in-depth analysis of the LO comb. Because we use different filter bandwidths to suppress this ASE noise in between neighboring comb lines, the noise is more pronounced for the LO tones 1 f and 4 f and the corresponding data channels L1 and L4. This problem can be overcome by using dark or bright Kerr soliton combs with higher conversion efficiency and correspondingly higher per-line power, as demonstrated in recent experiments [35,36]. In a second experiment, we increase the optical acquisition bandwidth to opt 610 GHz B = by using an LO comb with an FSR of 150 GHz and again 4 M = comb tones for coherent detection. We also increase the bandwidth of the individual data channels such that the broadband test signal fills the full optical acquisition bandwidth. All data channels can again be recovered with good SNR, Fig. 4(b). Note that the periodic increase of the reconstructed and stitched acquisition noise G () af (gray) in the reconstructed spectrum is now more pronounced due to a stronger roll-off of approximately 10 dB of the bandwidth-limited IQR 1 (43 GHz photodetectors) at half the FSR, FSR 2 75 GHz, f = see Fig. 3 for the corresponding transfer function. Still, these results correspond to the highest bandwidth that has so far been demonstrated in an OAWM experiment [1-7] while offering a greatly simplified scheme that does not require any slicing filters [10]. In a last step, we perform an in-depth analysis of various impairments that are relevant for the proposed OAWM scheme. In this analysis, we benchmark the proposed OAWM system against channel-by-channel reception of WDM signals with a series of independent IQ receivers, investigate the impact of the peak-to-average power ratio (PAPR) of different signals, estimate the crosstalk associated with the digital separation of the superimposed mixing products in the detected composite baseband spectra () , and finally quantify the impact of multiplicative noise originating from the ASE noise in the LO spectrum. Further details and results of this study can be found in Sections 6-9 of Supplement 1.

Summary
We demonstrated a technique for optical arbitrary waveform measurement (OAWM) that exploits chip-scale Kerr soliton combs as highly scalable multi-wavelength local oscillator (LO) and that does not require any optical slicing filtersneither for the signal nor for the LO. This greatly simplifies the implementation and the operation of the scheme and paves a path towards efficient integration on readily available platforms such as silicon photonics (SiP). Our scheme allows for precise reconstruction of truly arbitrary waveforms and relies on an accurate frequency-domain model of the detection system that is measured by means of a highly stable femtosecond laser with known pulse shape. We demonstrated the viability of the approach in a proof-of-concept experiment by capturing an optical waveform that contains multiple 16 QAM and 64 QAM wavelength-division multiplexed (WDM) data signals with symbol rates of up to 80 GBd, reaching overall line rates of up to 1.92 Tbit/s within a detection bandwidth of 610 GHz. To the best of our knowledge, this is the highest bandwidth so far been demonstrated in an OAWM experiment.

Supplementary Document
This document provides supplementary information to "Slice-Less Optical Arbitrary Waveform Measurement (OAWM) in a Bandwidth of More than 600 GHz Using Soliton Microcombs". In Section 1, we derive the detailed frequency-domain system model that forms the base for digital signal reconstruction and discuss the compensation of optical phase drifts at the receiver. Sections 2 and 3 compare the slices-less concept to previous demonstrations of time-interleaved optical sampling and spectrally sliced OAWM, respectively. The system calibration techniques are explained in Section 4, which also details the measured transfer functions of all IQ receiver channels used in the experiments. Details of our setup for generation of dissipative Kerr soliton combs and associated optical carrier-to-noise ratios (OCNR) are given in Section 5. In Section 6, we use our OAWM system to simultaneously acquire a multitude of wavelength-division multiplexed (WDM) data signals of different optical signal-to-noise ratios (OSNR), and we benchmark the resulting signal quality against channel-by-channel reception of individual WDM signals. Section 7 presents single-tone measurements along with an analysis of the associated distortions and noise in the setup, and Section 8 gives more details on the noise of the oscilloscopes used to digitize all waveforms. Section 9 describes how the simultaneous downconversion of a broadband signal with various LO tones affects the peak-to-average power ratio and how we can use amplitude clipping to maximize the overall constellation signal-to-noise ratio (CSNR) of acquired data signals.

System model
In this section we derive a mathematical description for the OAWM system illustrated in Fig.1 of the main manuscript, which is the basis for the signal reconstruction. Throughout our documents, we use lowercase letters () t  for signals oscillating at optical carrier frequencies and uppercase letters () t  for the associated complex-valued envelopes (baseband signals), which in many cases correspond to electrical signals obtained from in-phase/quadrature (IQ) receivers or fed to IQ modulators. Fourier transforms () f We assume that the recording length of our system is much shorter than the coherence time of the LO so that phase and amplitude noise of the LO can be neglected, i.e., LO, A  is assumed constant within a given recording.
The slice-less optical arbitrary waveform measurement (OAWM) scheme described in the main manuscript relies on simultaneous down-conversion of different spectral portions of the broadband signal using an LO comb with distinct tones at frequencies f  , 1,..., M  = , in combination with a multitude of optical receiver channels, characterized by their respective delay Fig. 1 of the main manuscript. The underlying system concept is illustrated in the block diagram depicted in Fig. S1, describing the reception of the broadband signal S () at by IQ receiver ν (IQR ν). The power splitter and the characteristic propagation delays   accumulated by the signal and LO on their ways to the 90° optical hybrid are modeled by the optical impulse responses S, () , respectively, where the factor 1 N accounts for the splitting of the signal and LO comb power into N receiver paths. The signals at the input of the ν-th optical hybrid are given by The optical hybrid splits the incoming signals S, () at  and LO, () at  into four identical copies S, ) 4 ( at  and LO, 4 () at  , which are then superimposed with four distinct phases. For simplicity, all these phases are assigned to the LO paths and described by the complex-valued coefficients (I+) , . For an ideal optical hybrid, these four coefficients are given by The wavelength-dependent transfer functions of the optical hybrids and of the optical fibers to the subsequent in-phase (I) and quadrature (Q) balanced photodetectors (BPD) are modeled by the optical impulse responses (I) () ht  and (Q) () ht  , respectively. The model for the two BPD in each IQ receiver includes a pair of optical impulse responses (I) BPD, () ht  and (Q) BPD, () ht  that describe the optical properties such as wavelength-dependent responsivities, as well as a pair of electrical impulse responses (I) BPD, () Ht  and (Q) BPD, () Ht  that model the electrical characteristics such as the RF transfer functions. As we only measure the difference current at the output of the BPD, we can neither digitally compensate for the imbalance between the paired outputs of the optical hybrid nor for the imbalance of the BPD. Therefore, we assume a common impulse for the paired balanced outputs of the optical hybrid, as well as common optical and electrical impulse responses, ( , respectively, for the two photodetectors inside each BPD. This simplification is well justified because the optical fibers connecting the optical hybrid to the subsequent BPD in our setup are well length matched and because the BPD are well balanced. According to Fig. S1 and the above simplifications, we calculate the signals (I+) at the various inputs of the BPD. To simplify the subsequent derivation, these signals are already convolved with the optical impulse responses (I) do not exist as physically accessible optical signals in the real system.
After balanced detection and analog-to-digital conversion, we obtain the digital in-phase signal () It  and the digital quadrature signal () Qt  , which contain additional noise contributions such as shot noise, thermal noise, or quantization noise. We summarize all such noise sources and model them by additive voltage noises (I) () In the next steps, we make use of Eqs. (S5), (S3), (S2) and of the Fourier transformation to relate the electrical signals () in Eq. (S6) to the optical signal S () at at the system input. As we already assumed perfectly balanced photodetectors with infinite common-mode rejection, we also neglect any self-beating terms of signal and LO. By substituting Eq. (S5) in Eq. (S6) and expanding the expression, we obtain for the in-phase component, By Fourier-transforming (S7), we eliminate most convolutions )) ) ( As a next step, we simplify Eq. (S9). To this end, we define a baseband transfer function ( The same derivation is used for the quadrature component, which leads to the baseband transfer Note that ( Similarly, the quadrature components () Qf  are found by using the corresponding transfer functions (Q,t) () Equations (S12) and (S13) can be reformulated in matrix-vector form. To this end, we define the in-phase vector ( , see Eq. (S11). This leads to Note that for the simplified explanation used in Eq. . This allows to simplify Eq. (S14) by constructing the composite baseband spectra ( ) that are the Fourier transforms of the complex-valued baseband signals ( ) (2) in the main manuscript. For the composite baseband spectra () f U we can reformulate Eq. (S14) to obtain a relation that does not contain the frequency-inverted complex-conjugate counterpart . ( )

Signal reconstruction
We use the system described in the previous section to calculate an estimate (est) S () af of the spectrum S () af of the original complex-valued input signal S () at. To this end, we first find the best estimate (est) is the autocovariance matrix of the noise vector where superscript "T" denotes the transpose of the respective vector. Note that Eq. (S17) is only valid for frequencies f within the receiver bandwidth B , where D is the identity matrix. Equation (S17) then turns into the relation for the least-square estimator The reconstructed vector comprises the signal vector (est) S () f A can hence either be determined by evaluating the upper half of Eq. (S18) for all frequencies or by evaluating the full equation for non-negative frequencies , i.e., the relation is overdetermined when evaluated in the full range of positive and negative frequencies, . However, because of the special structure of the pseudo-inverse of (t) both results are consistent. This can be better understood from the corresponding forward relation (S14), where the internal structure of the transfer matrix (t) Because of the structure of (t) both results are consistent, which becomes apparent if we replace f by f − in Eq. (S14) and take the complex conjugate. These operations leave the right-hand side of Eq. (S14) unchanged and thereby reproduce the symmetry relations, * ( ) ( ), ff −= II and * ( ) ( ). ff −= QQ Evaluating Eq. (S14) for all frequencies is hence equivalent to assuming real-valued time-domain signals for () It  and () Qt  and exploiting the symmetry relations for the associated spectra. As a consequence, it is also sufficient to either consider Eq. (S18) only for non-negative baseband frequencies , which yields both (est) , or, alternatively, to restrict the evaluation of Eq. (S18) to the upper half, i.e., the calculation of (est) . In the following, we pursue the second approach.
To simplify the subsequent notation, we define a reconstruction matrix ( in Eq. (S18). The reconstruction rule (S18) then simplifies to   f This reference frequency may, e.g., be chosen at the center of the optical signal spectrum or at the lower-frequency edge of the first spectral slice, as done in Fig. S2.  of the LO comb, because the signal slices need to be frequency-shifted back relative to their original position before being stitched. Since our experiment currently relies on a free-running dissipative Kerr soliton comb as an LO, the FSR is not locked to the ADC clock and must thus be estimated for each recording. To this end, we exploit the fact that the bandwidth of the IQ receivers exceeds half the FSR of the comb, FSR 2 Bf  , such that spectral components inside the overlap regions, see "OR" in Fig. S2 Figure S3 illustrates this concept for data recorded with an LO comb FSR of FSR 150.8 GHz. f = In this example, the optical input signal S () af consisted of seven data signals that were generated from only two independent modulators, see Fig. 2 in the main manuscript, leading to additional peaks in the autocorrelation ( ) Fig. S3 (b). Note that the signal spectrum is obtained from a discrete Fourier transform (DFT) with discrete frequency points spaced by obs 1 fT = , where obs T is the observation time. Consequently, the FSR FSR f of the free-running comb may fall between two such points. For a precise estimation of the exact FSR, we therefore use a sinc-type interpolation of the discrete measurement points in the spectrum and detect the peak of the interpolation, see Fig. S3 (c). The resulting frequency shifts that are applied to the various slices prior to stitching, see Fig. S2  150.82125 GHz f = . We interpolate the data to estimate the FSR f with high precision.

Phase-drift compensation
For a practical implementation of the OAWM system, all transfer functions (I,t) () Hf  and (Q,t) () Hf  as used in Eqs. (S12) and (S13) must be determined, which requires a calibration of the system. Importantly, the calibrated transfer functions might be impaired by amplitude and phase drifts of the comb lines, see Eqs. (S10) and (S11), and by random phase drifts in the setup. that is constant during one recording, but varies between recordings, Note that the complex-valued factor (t) which was found to lead to stable convergence. This cost function is minimized using the quasi-Newton method. For efficient implementation of the parameter estimation, it is imperative to reduce the number of free elements in the parameter vector p as much as possible and to minimize the computational effort of calculating the cost according to Eq. (S23) in each iteration. This can be accomplished by using additional assumptions, which allow for certain mathematical simplifications as explained in the following sections.
To reduce the number of free parameters, we first make use of the fact that the optical fibers contribute phase drifts, but do not cause any amplitude fluctuations. We may thus reduce the number of free parameters by assuming As an additional simplification, we may use the fact that the comb tones are strictly phase-locked and that the pulse shape of the LO comb is thus stable, rendering the delay  of the pulse with respect to the trigger point 0 t = of the oscilloscope the only free parameter of the LO comb, where We further want to reduce the computational effort for evaluating the cost function ( ) rec () f H based on the respective parameter vector p . To minimize the computational effort, we reformulate the associated reconstruction relation according to Eq. (S19) such that all processing steps that are independent of the parameter vector p can be moved outside the iteration loop and hence only need to be performed once. The various steps of this reformulation are described in the following sections.
In a first step, we need to overcome the problem that the general system model according to Eq. (S14) does not allow to separate the time-variant complex factors (t ) , see Eq. (S16). This is not fulfilled for our experimental setup, which can, e.g., be inferred from Fig. S13 and Fig. S14 in Sect. 4.3, which show distinct differences in the amplitudes and phases of the measured frequency-dependent transfer functions for in-phase and quadrature phase detection channels of the same IQR.
To overcome this problem, we re-formulate the system model to arrive at a representation that requires weaker assumptions than the simplified model in Eq. (S16), but still leads to a relation of the form of Eq. (S16) between the signal vector S () f A and a composite measurement vector () and which is assumed to represent the transfer function of the detector for all comb lines. We then add the results using the factor j to obtain the corresponding pre-corrected composite baseband spectrum () Uf  , Introducing Eqs. (S12) and (S13) into Eq. (S24) leads to a condition for which () Uf  becomes independent of the complex-conjugate frequency-inverted signal components * S () a f f  −+ , as in the representation of Eq. (S16), In this relation, we explicitly indicate the dependence of the diagonal matrices containing the measured pre-corrected baseband spectra () NM and may be understood as a quality metric of the measured signals. Note, however, that this ratio cannot be simply related to the frequency-dependent SNR of the various receiver channels, which is defined by the ratio of the individual signal and noise powers,  deviate from their ideal counterparts, the matrix condition number is increased. This can be seen from Fig. S4 (a), where the matrix condition number If we relax the condition formulated in Eq. (S35) allowing for transfer functions (f) () Hf  with non-identical frequency dependencies, then the condition number will become frequencydependent. We illustrate this effect in Fig. S4 (b) by plotting the condition number of the actually measured transfer matrix IQ () f H according to Eq. (S14) as a function of frequency f. Since the photodetectors used in our experiment have quite different amplitude responses, see Fig. S13 in Sect. 4.3, the relative magnitude of the elements within the same column of the transfer matrix IQ () f H varies with frequency, see Eqs. (S14) and (S15). This leads to a frequency-dependent condition number, Fig. S4 (b) Fig. S4 (b). This confirms that the delays   in path  , leading to equidistantly interleaved sampling pulses, see Fig. S5 for an illustration of an associated setup using 2 N = parallel IQ receivers [6]. Assuming that the optical sampling pulses are sufficiently narrow, the complex amplitude of the beat between the signal and the sampling pulses is proportional to the signal field at the center of the sampling pulse [4]. Consequently, the signal is optically sampled in each of the N parallel channels. After digitizing all signals, an estimate of the original signal is reconstructed by interleaving the complex-valued samples that have been acquired by the N parallel IQ receivers beforehand. However, as the optical phase in the different receiver paths is not stable, a digital phase-drift compensation is required before interleaving the acquired samples. In [6], this problem is addressed by running a multidimensional optimization procedure [7] during signal reconstruction, exploiting a-priori information about the measured QPSK signals. Time-interleaved optical sampling of arbitrary signals with unknown time dependence has not been demonstrated so far. This problem can be overcome by slice-less OAWM, which does not require a-priori knowledge about the signal. Instead, we exploit spectral overlap regions that allow for estimating all relevant phase parameters for arbitrary waveforms see Sect. 1.4. We further rely on a calibrated frequencydomain system model to compensate not only for imperfect delays   but also for the frequency dependent characteristics of the various receiver channels. Consequently, we can significantly suppress the crosstalk among different spectral slices that would otherwise impair the reconstructed waveform according to Eq. (S20).

Comparison to spectrally sliced OAWM
The description of the slice-less OAWM system can also be related to that of spectrally sliced OAWM systems [8][9][10]. By including slicing filters for the signal and LO, the frequencydependent time-invariant transfer matrix (f ) () f H in Eq. (S29) essentially simplifies to a diagonal matrix, and the time-variant diagonal matrices (t) F H and (t) LO H can be combined into a single diagonal matrix (t) H , ) (f 11

Calibration
The signal reconstruction in slice-less OAWM crucially relies on a precise calibration of the receiver system, which is obtained in several steps. In a first step, we measure and digitally compensate the IQ-skew of the IQ receivers, i.e., the time-delay differences of the in-phase and quadrature signals on their way from the optical hybrids to the respective BPD, see Sect. 4.1. In a second step, we tune the various delays in our system to obtain a well-conditioned transfer matrix according to Eq. (S37), see Sect. 4.2. Based on these pre-calibration steps, we finally measure the time-invariant frequency-dependent transfer functions (I) () Hf  , Eq. (S10), and (Q) () Hf  , Eq. (S11), by using a well-known optical reference waveform (ORW) generated by a highly stable femtosecond mode-locked laser (Menhir 1550), see Section 4.3 below.

IQ skew calibration and measurement of amplitude response
For the IQ skew calibration, we connect two external-cavity lasers (ECL) to the IQ receiver array and sweep their frequency difference f in discrete stepsan exemplary sketch for IQR ν is shown in Fig. S6 (a). This leads to sinusoidal output signals with frequency f , () It  and () Qt  at BPD-ν, Fig. S6 (b). From these output signals we extract the phase difference IQ  of the inphase and quadrature components as well as the respective amplitudes I A  and Q A  by fitting sinusoidal model functions to the measurement data. In Fig. S6 (c) we plot the extracted IQ phases as a function of frequency. The time delay differences between the I and Q signal, are obtained from the slope of the phase difference over frequency, accessible through a linear fit. The skews are then compensated by shifting the recorded time-domain waveforms accordingly. Note that the measurement technique used here only allows to extract the group delay differences between the I and the Q component of the same IQ receiver, but not the group delay differences between different IQ receivers or the associated phase transfer functionsthese measurements require a broadband optical reference waveform (ORW) with a precisely known frequency-dependent phase, see Sect.   S6. Measurement of the time-delay differences (IQ skew) between in-phase and quadrature signals. (a) Measurement setup: Two external-cavity lasers (ECL) are connected to an IQ receiver (IQR), consisting of a 90° optical hybrid (OH) with balanced photodetectors (BPD) and analog-to-digital converters (ADC). The frequency difference between ECL 1 and ECL 2 is swept in discrete steps, leading to sinusoidal output signals of the respective difference frequency at the two photodetectors. (b) Sketch of the time-domain trace for the in-phase I(t) and quadrature-phase Q(t) components. The frequency f , the amplitudes I A  and Q A  , and the IQ phases IQ  are extracted by fitting a sinusoidal model function to the measurement data. (c) IQ phase error IQ π2  − as a function of frequency f for all four IQ receivers. The time delay differences between the I and Q signal, are obtained from the slope of the phase difference over frequency, accessible through a linear fit.

Optical delay tuning and compensation
The goal of the second step is to measure and numerically equalize the time-delay difference , as required for a well-conditioned transfer matrix. To this end, we first generate a random broadband data signal using an IQ modulator and send it through the signal paths, while an external-cavity laser (ECL2) is connected to the LO input, switch position A in Fig. S7 (a). We extract the relative signal delays S, S,1   − by detecting the peak of the modulus of the temporal complex cross-correlation between the received baseband signals . For a sufficiently large unambiguity range, the signal must not repeat within twice the expected maximum path delay mismatch. We numerically compensate the time delay differences in the signal path so that S, S,1 Next, we send the broadband signal through the LO paths and connect ECL2 to the signal input, switch position B in Fig. S7 (a). The goal now is to adjust the relative delays  Figure S7 (b) shows the measured delay differences LO, as a function of the recording number having set the optimal equidistant delays for an LO comb with FSR 150 GHz f  . The delays associated with receiver three (yellow) and receiver four (purple) in Fig. S7 (b) show a common fluctuation relative to receiver one (blue) and two (orange). This fluctuation is presumably related to an imperfect synchronization of the two oscilloscopes (OSC 1 and OSC 2) and does not represent drifts of the optical delays. We do not expect these fluctuations to have any significant impact on the measurements presented in this work. Consecutive measurements of the LO-path differences after equalizing the signal delays and after tuning the delay lines (DL) according to Eq. (S37) for an LO with an FSR of 150 GHz. The delays associated with receiver three (yellow) and receiver four (purple) show a common fluctuation relative to receiver one (blue) and two (orange), which we attribute to an imperfect synchronization of the two oscilloscopes (OSC 1 and OSC 2). The software implementation of this calibration procedure requires several processing steps that are illustrated in Fig. S10. In the following, we give a more detailed explanation of the signal processing techniques that are used in each of these steps. Step (1): Estimation of FSR of ORW First, we estimate the FSR ORW f of the ORW using the clock of the recording oscilloscopes (sampling interval ss 1/ tf = ). To this end, we calculate the autocorrelation function of one of the baseband spectra measured at the in-phase or quadrature output of any the IQ receivers, see Fig. S11 for the autocorrelation e.g., 11   Step (2): Choice of processed samples In Step (2) Step ( will nevertheless notably suppress one set of RF sub-combs, dashed combs in Fig. S9 (b), relative to the respective other set, solid combs in Fig. S9 (b). This is sufficient for the identification of the respective set of lines, leading to one identified RF subcomb per LO tone, i.e., M identified RF sub-combs Only these identified sub-combs are considered further.

Saturation of photodetectors under illumination with a fs-laser
We investigate the saturation behavior of different photodetectors by sending the ORW to a single detector of the BPD, and by recording the pulse amplitudes A of the generated pulse train, Fig. S15 (a), (b). We observe a significantly lower saturation input power for the 100 GHz BPDs (Fraunhofer Heinrich-Hertz Institute, HHI; Berlin, Germany) as compared to their 43 GHz counterparts (Finisar 43 GHz Balanced Photodetector BPDV21x0R), see Fig. S15 (c). The BPD with the lowest saturation power (#W0103) also shows the highest discrepancy of the amplitude transfer function measured with a pair of external-cavity lasers (ECL) and with the known optical reference waveform (ORW), Fig. S13.

Local-oscillator comb
The four-tone local oscillator (LO) used for the experiment discussed in the main manuscript is derived from a dissipative Kerr soliton (DKS) comb that is generated using the setup depicted in Fig. S16 (a), see [12] for a more detailed description on how to tune into a low-phase-noise soliton state. A single-tone laser is amplified and injected into a high Q micro-resonator to generate the DKS. At the output, the remaining pump line is suppressed by a notch filter. The optical spectrum for the 110 GHz comb before (Ⓐ) and after (Ⓑ) a 5 nm optical bandpass (BP) is shown in Fig. S16 (b). The limited output power per line at point Ⓑ directly translates into an ASE-noiselimited optical carrier-to-noise ratio (OCNR) after amplification Ⓒ. For the comb with a 110 GHz FSR shown in Fig. S16 (b), the OCNR in a reference bandwidth of 12.5 GHz (0.1 nm at a center wavelength of 1550 nm) was measured to be 23. ff are isolated using a wavelengthselective switch (WSS). Since the WSS features a fixed grid of 12.5 GHz-wide switchable frequency "pixels" that does not perfectly coincide with the 110 GHz FSR of the comb, it was necessary to "open" two pixels for the outer comb lines at frequency 1 f and 4 f , leading to wider transfer functions for the underlying spectrally white noise. The rightmost plot in Fig. S16 (b) shows the four comb tones of interest along with the measured filter function of the WSS (green). Due to the low OCNR of the LO comb, the width of the individual filter functions for each of the four lines is important for the overall system performance. The transmitted LO noise imposes an upper limit on the SNR that can be achieved after coherent detection, see discussion of signal distortions in Sect. 7. When measuring data signals, the multiplicative noise originating from the LO increases the noise for the outer constellation points more than for the inner ones and is most noticeable for the part of the signal that was down-converted with lines at frequencies 1 f or 4 f , for which two pixels had to be "opened" in the WSS, see constellation diagrams recorded from channels L1 and L4 in Fig. 4(a) in the main manuscript. In Fig. S16 (c), we show a photograph of the micro-resonator and the associate optical package that was used for generation of the 110 GHz Kerr comb in our experiment. The LO comb with a 150 GHz FSR was derived from a native comb with a 50 GHz line spacing. In this case, the WSS was configured to select only every third tone. Since the 150 GHz FSR is well aligned with the 12.5 GHz grid of the WSS, it is now sufficient to "open" only a single WSS pixel per line. As a consequence, the outermost WDM channels received with the 150 GHz LO comb are less impacted by multiplicative noise than the ones received with the 110 GHz comb, see Fig. 4

Benchmarking of the OAWM scheme
We evaluate the performance of the OAWM system by measuring broadband test signals with bandwidths of approximately 490 GHz and 610 GHz, both consisting of several WDM channels with different modulation formats and symbol rates, which are simultaneously received. While this offers the possibility to receive signals with ultra-high symbol rates along with utmost flexibility and agility in terms of software-defined wavelength assignment, it is also important to understand which impairments are introduced by the OAWM scheme and how OAWM-based detection compares to channel-by-channel reception of WDM signals using a series of independent IQ receivers with dedicated LO laser tones. To investigate these effects, we use the 490 GHz-wide test signal, which comprises seven wavelength-division multiplexing (WDM) data channels, generated by modulating four optical carriers (L1 to L4) using a first IQ modulator and three additional optical carriers (L5 to L7) using a second IQ modulator, see Fig. S17 (a) below as well as Sect. 3 in the main manuscript for more details on the underlying experimental setup. The optical signal-to-noise power ratio (OSNR) of these signals can be artificially reduced by adding spectrally white noise, generated by an amplified spontaneous emission (ASE) noise source. The power level of the added ASE noise is controlled using a variable optical attenuator (VOA). The optical spectrum of the test signal at point Ⓐ is shown in Fig. S17 (c). Fig. S17. Benchmarking the performance of the broadband multi-channel OAWM receiver system against a channel-by-channel reception using a conventional single-channel IQ receiver (IQR). (a) OAWM setup: A broadband multi-channel optical WDM signal is generated by modulating seven spectrally interleaved lasers (L1 to L7) with two different IQ modulators (IQM) that are driven with different signals, derived from a common arbitrary-waveform generator (AWG, Keysight M8194A). The output signals of the IQM are combined, and the resulting broadband optical test signal is then amplified. Spectrally white noise is added, and the signal is routed through a band-pass filter (BP) to the IQR array of the OAWM system. The LO comb with FSR 110GHz f = is split into four paths, delayed by different time intervals 14 ,   , and routed to the IQR array. (b) Reference measurement with a conventional single-channel IQR: The various WDM channels contained in the broadband optical test signal are individually received with IQR 2, using the external-cavity laser L8, that previously served as a pump for the Kerr comb. The WDM channel of interest is isolated by a WSS, see spectrum Ⓓ in Subfigure (c). (c) Exemplary optical spectra of the broadband multi-channel WDM test signal at point Ⓐ and of an exemplary isolated WDM channel modulated onto the optical carrier generated by Laser L6 at point Ⓓ.
In our experiments, we benchmark our OAWM system against a single-channel IQ receiver. As the single IQ receiver is not sufficiently broadband to capture all seven data signals at once, we measure them sequentially by using a programmable WSS to isolate a single WDM channel along with a tunable external-cavity laser (L8) as LO for coherent detection, Fig. S17 (b). As an example, we additionally show in Fig. S17 (c) the optical spectrum recorded at point Ⓓ after isolating WDM channel L6, containing 60 GBd 16 QAM data modulated on a carrier generated by laser L6. After receiving the signal using the single-channel IQ receiver IQR 2, we obtain the power spectrum 2 (SCh) 2 () Uf of the composite baseband signal (SCh) Fig. S18 (a). The lines apparent in the spectrum originate from ADC clock tones or higher harmonics at integer multiples of 16 GHz. Additional clock tones from the arbitrary waveform generator (AWG) at the transmitter appear at around 30 GHz and are hidden within the signal in Fig. S18 (a). In addition, we plot the corresponding acquisition noise (SCh) 2 () Gf , gray curve in Fig. S18 (a), as measured by disconnecting all optical receiver inputs, see Sect. 8 below for details. In Fig. S18 (b) () It fills the full range without significant clipping. For the signal (SCh) 2 () It , we measure a peak-to-average power ratio (PAPR) of 10.3 dB. Figure S18 (c) shows the constellation diagram of the 60 GBd 16 QAM data modulated on the laser tone of L6, corresponding to the spectrum 2 (SCh 2 Fig. S18 (a). In a next step, we receive the full broadband signal consisting of all seven data channels using our OAWM system. In this case, each baseband signal ()  where the overbar denotes a temporal average. This choice was found to lead to a good trade-off between signal clipping on the one hand and excessive impact of acquisition noise on the other hand, and limits the PAPR of signal 2 () It to 12.6 dB, see Fig. S26 and the related discussion in Sect. 9 below. In the following, we briefly discuss why the superposition of multiple spectral components, as it is the case for the slice-less OAWM system, can lead to an increased PAPR. Assuming that the down-converted spectral portions originating from different WDM channels are statistically independent, we would expect the resulting superposition in Fig. S18 (e) to assume a more Gaussian-like histogram than an individual WDM channel, Fig. S18 (b), which is confirmed by our measurements. Disregarding noise and making use of the fact that the timedomain data signals in the various WDM channels have finite maximum amplitudes dictated by the respective transmitters, we would further expect that the PAPR of the superposition signal 2 () It is larger than that of the individual data signals observed in the output signal (SCh)   Fig. S17 (a) and channel-by-channel reception using a single IQ receiver (IQR 2) according to the setup in Fig. S17 (b). All spectra are plotted with a resolution bandwidth of 100 MHz. (a) Exemplary power spectrum above, which increases the noise for high frequencies, i.e., further away from the center frequency of each slice. Second, the uncorrelated complex-valued noise amplitudes of adjacent slices are weighted and averaged in the overlap regions of neighboring slices, see Eq. (S21) in Sect. 1.2, leading to a local reduction of the noise floor up to 3 dB, gray curve in Fig. S18 (g). Consequently, the resulting stitched receiver noise 2 G () af is highest at the lower and upper limit of the overall covered spectral range, where noise averaging from an adjacent slice does not occur. An exemplary constellation diagram for the 60 GBd signal L6 is given in Fig. S18 (f).
Finally, we compare the constellation SNR (CSNR) of the signals received with the OAWM system with those obtained from individual channel-by-channel IQ reception for various optical signal-to-noise ratios (OSNR), see red and blue curves in Fig. S18 (h) . The OSNR is adjusted by artificially adding increasing levels of ASE noise, see Fig. S17 (a) and (b), which eventually dominates over the native noise of the receiver system. For low OSNR levels, the CSNR hence increases in proportion to the OSNR, while, for larger OSNR, the CSNR reaches a plateau because the combined effect of electrical transmitter noise (DAC and RF amplifier), of LO noise, and of electrical receiver noise (RF amplifier and ADC) is still present, even of no additional noise is introduced. For high OSNR, we observe a slightly lower CSNR performance for the broadband OAWM system compared to the single-channel IQ receiver, which can be explained by the following effects: 1. The 3dB bandwidth of 43 GHz of IQR 1 is insufficient for reception of a 55 GHz-wide spectral slice, requiring strong "digital" amplification of high-frequency components in the reconstruction process. This unavoidably leads to digital amplification of acquisition noise for high RF frequencies and thus reduces the resulting CSNR especially for the signals L1 and L4 as these are located in the roll-off region. This effect is not present for the single-channel IQ receiver (IQR 2), for which we used photodetectors with a bandwidth of 100 GHz in combination with an LO tone tuned to the center of the respective WDM channel such that the down-converted composite baseband spectrum appears approximately symmetrically around DC as shown in Fig. S18 (a).
2. Due to the larger PAPR, Fig. S18 (b) and (e), and the limited dynamic range of the ADC, the SNR of the digitized signals is reduced for the OAWM system as compared to the single-channel IQ receiver [16]. This effect is analyzed in more detail in Sect. 9 below.
3. The LO comb lines have a lower OCNR than the tone emitted by L8, which is used as an LO for channel-by-channel reception, see Sect. 5. The OAWM system uses only 4 N = IQ receivers to measure the ch 7 N = WDM data channels comprised in the broadband test signal, whereas the single-channel IQ receiver is dedicated to one specific WDM channel only. This leads to an additional penalty of approximately 10 ch 10 10log ( / ) 10log (4 / 7) 2.4dB NN = = − for the OAWM system, which can be best understood by considering the simplified scenario depicted in Fig. S19. One receiver either detects a single WDM channel Fig. S19 (a), or simultaneously detects two WDM channels Fig. S19 (b). For simplicity, we assume that the overall waveforms acquired by the two receivers have the same PAPRdisregarding for the moment the fact that the superposition of two WDM signals usually leads to an increased PAPR in comparison to a single WDM channel, see Section 9 below. Consequently, the overall SNR of both waveforms is identical when integrating the signal and noise power spectral density over the full receiver bandwidth, see Eq. (S45) below. This result holds independent of the overall signal power P , because any increase of the overall signal power will lead to an increase of the acquisition noise by approximately the same amount, see Section 9. We thus may assume without loss of generality that that the two overall waveforms shown in Fig. S19 (a) and Fig. S19 (b) have also the same overall signal power P and are impaired by acquisition noise with the same power spectral density. Consequently, we obtain a lower per-channel power, a lower per-channel SNR, and thus a lower CSNR for the two-channel reception, Fig. S19 (b), as compared to its single-channel counterpart, Fig. S19 (a). In general, the more WDM channels ch N are simultaneously received by a single IQ receiver, the lower the power associated with a single WDM channel will be, and the same applies to the resulting CSNR. Conversely, if more than one receiver is used to measure the same WDM channel, we can reduce the acquisition noise by averaging, and improve the resulting CSNR. In case of the OAWM system, both effects occur simultaneously as four receivers are used to measure seven WDM channels. Consequently, if the acquisition noise dominates over other noise sources, the CSNR is reduced by 10 10log (4 / 7) 2.4dB =− compared to a single-channel reception. Note that in case the number of received channels exceeds that of the receivers, ch NN  , a performance improvement for the OAWM system compared a single-channel IQ receiver is expected. This was e.g., observed in [10,17], where OAWM systems receiving a single broadband data channel performed better than a single IQ receiver with the same total bandwidth. , where noise P is obtained by integrating the noise power spectral density over the signal bandwidth. (b) Reception of two WDM channels with total power P and individual power 2 P . Assuming that the power spectral density of the noise is the same as in (a), the SNR of an individual WDM channel is 3 dB lower compared to the single-channel reception in (a).

Noise and distortion characterization
To further understand the impairments associated with the OAWM system, we perform measurements of well-defined single-frequency laser tones and identify the associated noise contributions and distortions. The setup for this experiment is depicted in Fig. S20(a). Note that a single-tone signal does not have spectral components in all overlap regions between neighboring spectral slices, see Fig. 1 (b) in the main manuscript for a visualization of the overlap regions, which is required to estimate the time dependent parameters (t) LO, H  and (t) F, H  , see Sect. 1.4 above. To circumvent this issue, we add pilot tones, P1 to P4, to the signal prior to detection. An example for the optical spectrum after adding the pilot tones is shown in the inset of Fig. S20. The power of the pilot tones is low compared to the signal power.

Oscilloscope noise
In this section, we quantify the noise contributions (I) acq, () Gt  and (Q) acq, () Gt  associated with the respective input channels of the Keysight UXR oscilloscopes used in our experiments [20]. To this end, we assume a sinusoidal input signal in () St and use a simplified noise model, Fig. S22, to describe the dependence of the noise level on the full-scale input voltage FS U of the oscilloscope. The full-scale voltage (ADC)

U
of the internal ADC is fixed. The variable power gain  between the oscilloscope input and the internal ADC relates both quantities by S (ADC) F F S UU =  , so that the full-scale input voltage scales inversely with the amplitude gain  . Note that the variable power gain 12  =   is provided by an adjustable electromechanical attenuator of power gain 1 1,   followed by an electrical amplifier of power gain 2 1   . We assume a constant internally added noise power 2 ADC  , which is effective at the input of the ADC. The variable input power gain is compensated digitally at the output for an overall gain of one. Therefore, the acquisition noise power 2 acq  after digital gain compensation scales inversely with the net gain  , and thus the acquisition noise amplitude acq  scales linearly with the full-scale voltage FS U , Simplified noise model for an analog-to-digital conversion unit ("oscilloscope", dashed box). We assume that the added noise voltage ADC () Gt is independent of the input power gain 12  =    , which is provided by an adjustable electro-mechanical attenuator of power gain 1 1   , followed by an electrical amplifier of power gain 2 1   . The acquisition noise of the ADC is modeled by a constant internally added noise power 2 ADC  , which is effective at the input of the ADC. As the net power gain  is digitally compensated at the output for an overall equivalent gain of one, the noise power 2 acq  of the (digital) noise "voltage" acq () Gt in the output signal out () St scales inversely with the net gain  of the input amplifier stage. (ADC) FS U is the internal fullscale voltage of the analog-to-digital converter, and FS U is the corresponding full-scale voltage at the oscilloscope's input.
We now compare this simplified model to actual measurements of the digitally represented acquisition noise acq () Gt output of the oscilloscope. To this end, we disconnect all inputs from the oscilloscope such that in ( )  (S46) Fig. S23 (a) shows the power spectra 2 (I) acq,1 () Gf of the noise recorded with the first oscilloscope channel for full-scale voltages FS 1 U  between 40 mV and 480 mV. The sharp decrease of the noise power spectral density above 100 GHz is caused by a digital filter that is built into the oscilloscope. Figure S23 (b) shows the relationship between acq  and FS U for all eight oscilloscope channels, the first four channels having a nominal bandwidth of 100 GHz and the last four a nominal bandwidth of 80 GHz. The discrete steps result from discrete settings of the variable attenuator and lead to a stepwise approximation of , where the overbar indicates an expectation value. Comparing the red trace to the green or the green trace to the blue, we find that the superposition of several data signals or of parts thereof leads to more Gaussian-like histograms with longer tails and tentatively higher PAPR. (b) Peak-to-average power ratio (PAPR) of clipped signals as a function of the empirical clipping probability. The star indicates the optimum empirical clipping probability of 0.06% for the superposition signals, providing an ideal trade-off between acquisition noise on the one hand and clipping-induced distortions on the other hand, see Fig. S26.
We further measure the PAPR of the various signals, which, in presence of noise and other distortions in the experiment, is difficult, as the extracted PAPR strongly depends on rarely occurring outliers. To overcome this problem, we consider clipped signals, where the highest occurring amplitudes are simply cut off. Specifically, we assume a certain full-scale voltage FS U of the ADC and we clip the signal to the associated acquisition range  

22
UU − . We then measure the PAPR of the clipped signals as a function of the empirical clipping probability, which is the ratio clipped sampels NN of the number of clipped samples clipped N and total number of samples samples N . Figure S25 (b) shows the corresponding traces for the seven WDM data channels that were individually recorded with a single-channel IQ receiver (blue), the time-domain waveforms ( , associated with the individual spectral slices of the OAWM experiment (green), as well as the eight associated superposition signals () It  and ( ), Qt  1,...,4  = (red). We can observe that for low clipping probability, the PAPR associated with the superposition signals (red) is 5.5 dB higher than the PAPR of a single WDM channel (blue), which is in accordance with the fact that the normalized histograms of the superposition signals in Fig. S25 (a) (red) appears broader the than the ones for the individual WDM channels (blue). Similarly, for a give clipping ratio, the PAPR of the superposition signal is bigger than that of the individual slices, in accordance with the shape of associated the histograms in Fig. S25 (a). Due to the longer tail of the red histogram, clipping is expected to be more effective for the superposition signals, as a significant PAPR reduction can be obtained when clipping a only small number of samples.
Note that clipping does not only lead to a reduced PAPR and an associated higher SNR, see Eq. (S45) [21], but also introduces signal distortions. One should hence expect an optimum clipping ratio, leading to an ideal trade-off between clipping-induced signal distortions, which dominate at high clipping ratios, and excessive acquisition noise at low clipping ratios. We investigate this aspect for the superposition signals obtained in our OAWM experiment. and we evaluate the constellation signal-to-noise-ratio (CSNR) of the seven received WDM signals. Figure S26 shows the CSNR of the four 40 GBd 16QAM and the three 60 GBd 16QAM WDM channels as a function of the normalized full-scale voltage, . Using this full-scale voltage, less than 0.06% of all recorded samples are clipped and the PAPR is limited to ~12 dB, indicated by a star in Fig. S25 (b).