Peta-bit-per-second optical communications system using a standard cladding diameter 15-mode fiber

Data rates in optical fiber networks have increased exponentially over the past decades and core-networks are expected to operate in the peta-bit-per-second regime by 2030. As current single-mode fiber-based transmission systems are reaching their capacity limits, space-division multiplexing has been investigated as a means to increase the per-fiber capacity. Of all space-division multiplexing fibers proposed to date, multi-mode fibers have the highest spatial channel density, as signals traveling in orthogonal fiber modes share the same fiber-core. By combining a high mode-count multi-mode fiber with wideband wavelength-division multiplexing, we report a peta-bit-per-second class transmission demonstration in multi-mode fibers. This was enabled by combining three key technologies: a wideband optical comb-based transmitter to generate highly spectral efficient 64-quadrature-amplitude modulated signals between 1528 nm and 1610 nm wavelength, a broadband mode-multiplexer, based on multi-plane light conversion, and a 15-mode multi-mode fiber with optimized transmission characteristics for wideband operation.


9) Major: [In Page 5, line 109] …of a symbol stream, corresponding to the…
Could you explain why the random pickup from a symbol stream can be justified when calculate GMI? I guess that you assumed implicitly the bit-interleaved FEC coding over mode/polarization channels. If so, please explain it. 10) Major: [In Page 5, line 114] The total date rate, ….
In the ref. [21], the date rate of 402 Tb/s is defined as the capacity assuming a specific FEC scheme and its NGMI threshold. Therefore, the imperfection of the FEC is include. However, the date rate of 1.14 Pb/s in this paper is defined as GMI. As you know, GMI is the information rate when we assume the perfect binary FEC coding and bit-wise decoding.
Therefore, the GMI is always larger than actual system capacity. Roughly speaking, the typical difference is 10 %. It is note that, in your previous research [20], this point was clear and it was no doubts.
Please use "decoded data rate" [20] as the metric rather than GMI, for fair comparison.

11) Minor: [In Page 6, line 128] Odd-and even channels were…
Could you add the value of the decorrelation between odd and even channels? We would like to thank the reviewers for their generally positive assessment of our manuscript. Upon the recommendations, we performed further investigations of (1) the reason for the observation of the transmission system's change of total delay spread and (2) a further identification of (mode-dependent) loss sources, both by additional wavelength and mode resolved loss measurements but also by analyzing the channel matrix for backto-back measurements, with and without the mode-multiplexers.
Regarding the first point, we have found that the delay spread variations were introduced by the time-domain multiplexed receiver setup (dynamic skew of 3.4 and 6.8 km delay fiber) and are not an effect that occurred in the transmission fiber. Hence, we could re-calibrate the corresponding DSP parameters and now plot solely the impact of the transmission fiber, showing a minimum in delay spread at 1530 nm that increases steadily towards higher wavelength channels.
The additional measurements of mode-dependent losses allow a better understanding of the performance reduction for channels towards the high L-band as we could identify a strong wavelength dependence of the attenuation in the highest mode-group. We also performed additional loss measurements of one mode multiplexer and recorded images of mode profiles directly after the mode multiplexer and after transmission, giving further insighte into the stability of modes after propagation. An additional MDL analysis from channel matrix further allowed to show the impact of the TDM receiver setup (4dB MDL in back-to-back) as well as the MDL increase when only considering mode multiplexer and de-multiplexer, indicating that for low wavelength channels, the transmission fiber adds no MDL, while additional MDL is observed towards the high L-band channels. This is in agreement with the mode-and wavelength dependent fiber attenuation measurements.
In the following, we give detailed responses to all points raised by the reviewers. Changes in the manuscript are marked in blue.

Reviewer #1 (Remarks to the Author):
1) Major: [In Page 2, line 37] While promising... Although many transmission experiments utilizing standard-cladding and/or uncoupled MCF has been reported, why just one reference [18] is quoted here? For example, the ref.
[*] reported standard-cladding uncoupled 4CF transmission over 3,00 km. I'm also wondering why you don't quote other reports utilizing standard-cladding multi-mode fiber.

Response:
We added the suggested reference to the additional demonstration using a 4-core MCF. Further demonstrations with multi-mode fibers are referenced later in the manuscript.
2) Minor: [In Page 3, line 54] The theoretical loss... The method used to calculate the theoretical loss should be described. At least, some paper or textbook should be quoted.

Response:
We added a discussion on the expected and measured loss, including mode-dependent loss of the multiplexers: While the transformation can be loss-less when assuming ideal phase masks, fabrication limitations such as pixelated phase masks and a discretization of phase values lead to a theoretical loss of each multiplexer of approximately 0.3 dB. In addition, an excess loss of 0.25 dB can be assumed for each reflection on a phase mask due to scattering of light to higher order modes that are not supported by the transmission system, blurring of neighboring pixel values and limited reflection on the dielectric mirror, for a total loss of 3.5 dB [24].
3) Minor: [In Page 3, line 60] It was thus possible... This sentence seems too long to read at one time. The author should explain why the DMGD kept be smaller.

Response:
We changed the entire abstract describing the fiber and hope that the new version is more precise.

Response:
We added further references and comparisons to previously reported fibers guiding 15 modes: The lowest DMGD, measured for a fiber from this manufacturing batch was 78 ps/km at 1550 nm wavelength [28], to date the lowest reported DMGD in MMF supporting 15 modes [29,30].

5) Minor: [In Page 4, line 79]
The total temporal spread... Could you explain the relation between the temporal spread and DMGD described in the previous page?
Response: As this fiber operates in a weak coupling regime, the total delay spread should be approximately the product of the DMGD and the lengths of the fiber. Thus, it should be below 2.3 ns, for the wavelength with shortest DMGD. We added the following text to the manuscript for clarification: This is in agreement with previous measurements and confirms the design target of the fiber [28] reaching a DMGD of less than 100 ps/km at 1550 nm wavelength, yielding an accumulated delay spread of less than 2.3 ns.

Response:
We added a reference to a previously reported DMGD measurement of this fiber. Response: After adding more data points to Fig (now 4) (d), we added the ranges of C and L bands but found the resulting graph to be not as clear and thus prefer to leave the figure as is. 9) Major: [In Page 5, line 109] ...of a symbol stream, corresponding to the... Could you explain why the random pickup from a symbol stream can be justified when calculate GMI? I guess that you assumed implicitly the bit-interleaved FEC coding over mode/polarization channels. If so, please explain it.
Response: Indeed, the used method implies interleaving over modes and time. We extended the signal quality assessment part in the methods section to clarify this point: To assess the signal quality after transmission and sub-subsequently the data rates that can be achieved in each wavelength channel, we have chosen two metrics: generalized mutual information (GMI) [34] as well as the data rate after decoding as detailed in [35]. In both cases, code interleaving was assumed over time and modes within one wavelength channel, hence forming spatial-super channels. We assume that the interleaving is removing any residual memory seen in the channel. While GMI gives the highest data rate assuming ideal codes and bit-wise decoding, the implemented coding scheme gives a more realistic data rate. For the coding scheme, we apply a Monte Carlo approach, where random binary patterns are generated and encoded using low-density parity check (LDPC) codes with different rates from the DVB-S2 standard [36] in conjunction with code rate puncturing for a code rate granularity of 0.01. We then performed a bit-to-symbol mapping before randomly selecting matching symbols from a data-set constructed from the experimentally received symbol streams from all modes for the specific wavelength channel. The code rate is iteratively decreased until a post-FEC BER of less than 2.18·10−5 minus a 10% margin is reached. A minimum of 2 · 106 symbols were used in the coding scheme for a sufficient error statistic. We then assume an additional outer hard-decision FEC scheme with an overhead of 2.8% [39] to ensure error-free performance. The post-FEC data rate is calculated by deducting the overhead of both the inner and outer FEC from the raw data rate. [21], the date rate of 402 Tb/s is defined as the capacity assuming a specific FEC scheme and its NGMI threshold. Therefore, the imperfection of the FEC is include. However, the date rate of 1.14 Pb/s in this paper is defined as GMI. As you know, GMI is the information rate when we assume the perfect binary FEC coding and bit-wise decoding. Therefore, the GMI is always larger than actual system capacity. Roughly speaking, the typical difference is 10 %. It is note that, in your previous research [20], this point was clear and it was no doubts. Please use "decoded data rate" [20] as the metric rather than GMI, for fair comparison.

Response:
As recommended by the reviewer, we added a second metric for the data rate assessment, as in our ECOC 2020 publication to allow a better comparison to previously published data rate records. We further added a section in the methods part where we explain the implementation of the coding scheme. 11) Minor: [In Page 6, line 128] Odd-and even channels were... Could you add the value of the decorrelation between odd and even channels?
Response: Odd-and even channels were decorrelated by 150 ns. We added a comment to the text: Odd-and even channels were optically de-correlated by 150ns. Response: indeed, the time window of the MIMO equalizer is approximately 5.7 ns. The MIMO equalizer window needs to be at least as long as the accumulated delay spread, which is in this system the product of the DMGD and the link distance, or around 2.3 ns at 1530 nm wavelength. We added the following comment in the text:

12) Minor
…with half symbol-duration-spaced taps, corresponding to a total time interval of 5.73 ns. and This is in agreement with previous measurements and confirms the design target of the fiber [28] reaching a DMGD of less than 100 ps/km at 1550 nm wavelength, yielding an accumulated delay spread of less than 2.3 ns.
13) Minor: [In Page 7, line 157] The equalizer was initialized... Could you quote the reference about the algorithm for updating MIMO taps? It might be least mean square (LMS) or radius directed equalizer (RDE).

Response:
The algorithm for updating the data aided and the decision-directed equalizers was LMS. We added a comment in the text: The equalizer was initialized in a supervised, data-aided mode before switching into a decision-directed mode for signal performance assessment, while both equalizers used the least-mean squares (LMS) algorithm to update the equalizer taps.

Response:
We used the channel matrix to calculate MDL. We do so, instead of using the equalization matrix, to stay in line with MDL definitions used by other authors. We added clarifying comments to the section and also added a reference to the extended paper by Ospina et al. (JLT, 2020). In Ref [**], it is found that MDL estimates from MIMO DSP equalizers can have an error if the SNR is low and the MDL is high. In the present experiment, we operate in a moderate SNR and moderate MDL regime where we can assume the error from MIMO-DSP based MDL estimation as acceptable. We changed the text to: The impulse response and MDL estimations can be estimated from the channel matrix as detailed in [40]. While the MIMO DSP used for signal quality calculation estimates the inverse of the channel matrix, we calculate the channel matrix by running the MIMO equalizer in the reverse direction. This is done by using the non-distorted, transmitted signal as the input to the MIMO equalizer and using the distorted, received signal as reference to calculate the LMS equalizer error. By using this fully supervised equalizer we assume to receive a high-quality estimate of the channel matrix.

Reviewer #2 (Remarks to the Author):
This paper reports state-of-the-art transmission experiment that exceeds 1 Pb/s.The Reviewer think the paper is published as it is.
The impulse response length in Figure 3(b) recovers evert morning. The Reviewer would like to know whether the setup was adjustment every morning or not.

Response:
No optical adjustments were made in between the measurements. However, we also found that the day-to-day variations of the impulse response durations where an artefact from the dynamic skew of the delay fibers in the TDM receiver setup that could be removed in the DSP, as can be seen in Figure 4(b) of the revised manuscript.

Reviewer #3 (Remarks to the Author):
1. (Line 54) Related to Fig. 3(d), theoretical value of MDL is helpful to guess the MDL source.

Response:
A theoretical estimation of the MDL is not straight forward and also goes beyond the scope of this paper. However, we added further measurements of the mode-resolved losses of one multiplexer and the fiber. We also added a discussion on the theoretically expected total loss from the multiplexer: While the transformation can be loss-less when assuming ideal phase masks, fabrication limitations such as pixelated phase masks and a discretization of phase values lead to a theoretical loss of each multiplexer of approximately 0.3 dB. In addition, an excess loss of 0.25 dB can be assumed for each reflection on a phase mask due to scattering of light to higher order modes that are not supported by the transmission system, blurring of neighboring pixel values and limited reflection on the dielectric mirror, for a total loss of 3.5 dB [24]. Figure 2(b) shows the measured wavelength-dependent average, minimum and maximum insertion loss for all 15 ports of one mode-multiplexers. The lowest average insertion loss was 9.2 dB at 1555 nm wavelength. We assume that the discrepancy between expected and measured loss stems from optical misalignment within the modemultiplexer. Figure 2(b) also shows the loss difference between the highest and lowest loss mode, being between 2.5 dB at 1610 nm wavelength and 3.2 dB at 1530 nm wavelength. While this metric can serve as an indication of the mode-dependent loss (MDL) behavior, it should not be confused with MDL calculated from the transfer matrix, as presented later in this manuscript. A photo of one of the two used multiplexers is shown in figure 2(b).

(Line 55)
The authors should show the loss of multiplexer before and after transporting the device. Without these values, the authors could not claim the transport caused the misalignment. At least, the readers may be curious about the best value of the loss for fabrication. If the authors are not confident in 100%, this sentence should be rephrased.

Response:
As we don't have multiplexer losses measured before shipment, we have to admit that our claim was speculative. We thus removed it and instead added a wavelength dependent loss measurement for one of the multiplexers.