An auto-encoder that optimizes a VCSEL-based fiber-optic system end-to-end and provides a 1.5dB sensitivity gain at higher temperatures is trained, utilizing a neural network that models the response of a VCSEL for a range of operating temperatures

H\ue4ger, Christian

Schr\uf6der, Jochen

Song, Jinxiang

Srinivasan, Muralikrishnan

Szczerba, Krzysztof

Wymeersch, Henk

English

Chalmers Research

Learning Optimal PAM Levels for VCSEL-based Optical InterconnectsDownloaded from: https://research.chalmers.se, 2023-09-08 04:34 UTCCitation for the original published paper (version of record):Srinivasan, M., Song, J., Häger, C. et al (2022). Learning Optimal PAM Levels for VCSEL-basedOptical Interconnects. 2022 European Conference on Optical Communication, ECOC 2022N.B. When citing this work, cite the original published paper.research.chalmers.se offers the possibility of retrieving research publications produced at Chalmers University of Technology.It covers all kind of research output: articles, dissertations, conference papers, reports etc. since 2004.research.chalmers.se is administrated and maintained by Chalmers Library(article starts on next page)Learning Optimal PAM Levels forVCSEL-based Optical InterconnectsMuralikrishnan Srinivasan(1), Jinxiang Song(1), Christian Häger(1),Krzysztof Szczerba(2), Henk Wymeersch(1), and Jochen Schröder(3) *(1) Dept. of Electrical Engineering, Chalmers Univ. of Technology, Gothenburg, Sweden(2) Juniper Networks, Sunnyvale, CA, United States(3) Dept. of Microtechnology and Nanoscience, Chalmers Univ. of Technology, Gothenburg, Swedenmursri@chalmers.seAbstract An auto-encoder that optimizes a VCSEL-based fiber-optic system end-to-end and providesa 1.5dB sensitivity gain at higher temperatures is trained, utilizing a neural network that models theresponse of a VCSEL for a range of operating temperatures.IntroductionThe uncooled GaAs-based Vertical-CavitySurface-Emitting Laser (VCSEL) is the lightsource of choice for short-reach optical intercon-nects (OIs)[1]. It can be directly modulated at highspeed and enables miniature footprint transceivermodules, and the most energy and cost-efficientinterconnect[2]. Intensity modulation formatssuch as 4-PAM can be used to provide highdata rates in OIs[3],[4]. However, the non-trivialnonlinear VCSEL response limits OIs modulatedwith equidistant PAM levels[5]. As optics movescloser to the board and OIs find uses in diverseapplications, and settings like consumer devices,cars, or planes, operation over broader temper-ature ranges, e.g., from −40 to 125°C becomesimperative. Therefore, it is essential to modelthe VCSEL and adapt the modulation format tomaximize the data throughput from the lowest tothe highest temperature for the OIs.Machine learning (ML) techniques have beenused to adapt the performance of OIs, espe-cially for equalization[6]–[8]. ML has also been ap-plied to optimize a specific function in the fiber-optic system, e.g., coding, modulation, and equal-ization[9]–[11]. End-to-end learning using autoen-coders (AE) for jointly optimizing the transmit-ter and receiver has the potential to achieve anoptimal performance[12]–[17]. However, there isscant literature on applying AE to VCSEL links,in particular, when including temperature depen-dence. The differential equations governing theVCSEL are not directly amenable to gradientback-propagation and depend on the differential*Presented at European Conference on Optical Communi-cation 2022, Early Accessequations solver[18]. Therefore, incorporating therate equations in the end-to-end optimization ofAEs is challenging. Moreover, equivalent circuitmodels such as[19] are also not sufficient for in-corporating into the AE optimization. Therefore,to fully capture the nonlinear behavior of VC-SELs and optimize the PAM levels using learn-ing techniques, a neural network (NN) equivalentthat captures the dynamics of the rate equationsof VCSELs is highly advantageous. Initial workalong these lines was presented in[20] but failed toinclude the temperature dependence.Our contributions in this paper are two-fold:1. We first develop a VCSEL NN model (re-ferred to as VNet) that models the nonlin-ear dynamic response of the VCSEL to accu-rately reproduce the optical waveforms gen-erated by the VCSEL under different operat-ing temperatures.2. We incorporate the VNet in a low-complexityAE that models an end-to-end fiber-optic sys-tem, and we use it to determine optimized in-put PAM levels that provide improved bit errorrate performance.VCSEL Neural Network - VNetFor this first proof-of-concept demonstration, weutilize the single-mode laser rate equations of asimple VCSEL model[20]dNdt=IqV− cngeffgS − Nτn(1)dSdt= ΓcngeffgS + ΓβNτn− Sτp, (2)where N denotes carrier density, S the photondensity, I the injection current, V the active vol-ume, ngeff the effective refractive index, g theFC (21)ReLUSigmoidReLUReLUFC (14)FC (7)FC (3)Output power (1)Input current (20) Temperature (1)(a) VNet structurecomprising fully connected(FC) layers, rectified linearunits (ReLUs), and asigmoid. The numbers inthe parentheses representthe number of neurons in aspecific layer0 20 40 60 80 1000246SamplesOutputPower(mW)Rate equations -5◦CVNet -5◦CRate equations 95◦CVNet 95◦C(b) Snapshots of output waveform10 1510−310−1SNR (dB)BEREquations 5◦CVNet 5◦CEquations 95◦CVNet 95◦C(c) BER performanceFig. 1: VNet structure, and equivalence to the rate equationsin terms of response and BER.gain per unit length, Γ the internal quantum ef-ficiency, τn the carrier lifetime and τp the photonlifetime. Here we consider Γ, g, τn and τp to bethe temperature-dependent terms. Note that themodel does not account for parasitics, gain com-pression, and self-heating of the VCSEL.By generating input-current and output-powersamples obtained from (1)–(2) using a 28 GHzwhite Gaussian noise sequence with 8 mA meancurrent and 6 mA standard deviation for temper-atures −5°C to 95°C in steps of 5°C, the VNetNN (structure shown in Fig. 1a) is trained in a su-pervised fashion to capture the VCSEL dynam-ics in the relevant parameter ranges. The inputcurrent sequence is over-sampled by a factor 10and shaped into rectangular pulses. The cur-rent sequence is then partitioned into 20 consec-utive samples, which, along with the temperature,serve, after normalization, as the NN’s input, thusavoiding the need for retraining for different tem-peratures in[20].The trained network is tested with a 28 Gbaud4-PAM current sequence ranging between 2 mAand 12 mA, with equidistant modulated levels.A snapshot of the matching output power se-quences from the rate equations and the VNet fortwo different temperatures is shown in Fig. 1b. Tofurther validate the accuracy of VNet, the bit errorrate (BER) vs. the signal-to-noise ratio (SNR) ob-tained from VNet and rate equations for a 4-PAMinput sequence over an additive white Gaussiannoise (AWGN) channel are shown in Fig. 1c. Thes 2 M<latexit sha1_base64="frJqiZ+jGJm9sWn8QDZnCqRiNCw=">AAAB+nicbVDLSsNAFL3xWesr1aWbwSK4KkkVdFl040aoYB/QhDKZTtqhk0mYmSgl9lPcuFDErV/izr9x0mahrQcGDufcyz1zgoQzpR3n21pZXVvf2Cxtlbd3dvf27cpBW8WpJLRFYh7LboAV5UzQlmaa024iKY4CTjvB+Dr3Ow9UKhaLez1JqB/hoWAhI1gbqW9XFPKYQF6E9Yhgnt1O+3bVqTkzoGXiFqQKBZp9+8sbxCSNqNCEY6V6rpNoP8NSM8LptOyliiaYjPGQ9gwVOKLKz2bRp+jEKAMUxtI8odFM/b2R4UipSRSYyTyiWvRy8T+vl+rw0s+YSFJNBZkfClOOdIzyHtCASUo0nxiCiWQmKyIjLDHRpq2yKcFd/PIyaddr7lmtfndebVwVdZTgCI7hFFy4gAbcQBNaQOARnuEV3qwn68V6tz7moytWsXMIf2B9/gDUgpO6</latexit>mapFC (4)ReLUFC (1)SigmoidRescaleencoderOne-hot vectorMemory bufferVNetMeasure PAM levels hereTemperatureFiberNoiseFC (4)Softmaxarg maxŝ 2 M<latexit sha1_base64="flvZlNcbjI9qU3CsNuAsbRsJvAQ=">AAACAHicbVBNS8NAEN3Ur1q/oh48eFksgqeSVEGPRS9ehAr2A5pQNtttu3SzCbsToYRc/CtePCji1Z/hzX/jps1BWx8MPN6bYWZeEAuuwXG+rdLK6tr6RnmzsrW9s7tn7x+0dZQoylo0EpHqBkQzwSVrAQfBurFiJAwE6wSTm9zvPDKleSQfYBozPyQjyYecEjBS3z7yxgRSnWGPS+yFBMaUiPQu69tVp+bMgJeJW5AqKtDs21/eIKJJyCRQQbTuuU4MfkoUcCpYVvESzWJCJ2TEeoZKEjLtp7MHMnxqlAEeRsqUBDxTf0+kJNR6GgamMz9RL3q5+J/XS2B45adcxgkwSeeLhonAEOE8DTzgilEQU0MIVdzciumYKELBZFYxIbiLLy+Tdr3mntfq9xfVxnURRxkdoxN0hlx0iRroFjVRC1GUoWf0it6sJ+vFerc+5q0lq5g5RH9gff4AyQqWhw==</latexit>decoderFig. 2: AE structure containing two encoder layers, thetransmission blocks and a single decoder layer. The numbersin the parentheses represent the number of neurons in aspecific layer.performance again matches at different tempera-tures (shown only for 5°C and 95°C, for clarity).Autoencoder for Optimized PAM LevelsThis section implements the fiber-optic commu-nication system and transmission chain, includ-ing transmitter, receiver, and channel, as a com-plete end-to-end AE. All the components of thetransceiver chain, as well as the channel model,are depicted in detail in Fig. 2. A message s ∈[1, . . . ,M ] ≜ M is encoded into a one-hot vectorof size M (here, M = 4), denoted by x, wherethe s-th element equals 1 and the other elementsare 0. Then the encoder is applied (see Fig. 2, leftside). The output of the sigmoid layer, which isbetween [0, 1], is appropriately scaled and shiftedto the chosen dynamic range, here chosen to be[2, 12]mA, which ensures that the input to VNetis above the VCSEL threshold current for all am-bient temperatures and avoids that the AE arbi-trarily increases bias current, which would be pe-nalized by self-heating in a real system[20]. Theoutput of the encoder is over-sampled by a fac-tor of 10 (additional low-pass filtering can be ap-plied here) and stored in memory buffers to accu-mulate 20 samples, which corresponds to 2 sym-bol durations, concatenated with the temperature,and passed to the VNet (described in Fig. 1). Thefiber channel is modeled by AWGN, though theproposed architecture can be readily extendedto account for additional low-pass filtering andother forms of dispersion and complex circuitry,like output driver circuits. The photodiode out-put is processed using a single fully-connectedlayer with softmax activation. The output prob-ability vector y = [y1, . . . , yM ]⊤, and estimatedmessage is ŝ = argmaxi[y]i. The loss functionminimized is the categorical cross-entropy, givenTab. 1: AE determined PAM levels (in mA) vs temperature,where the first row corresponds to equidistant PAM.T ° C PAM-1 PAM-2 PAM-3 PAM-4– 2.0 5.33 8.67 12.05 2.0 5.06 8.56 12.025 2.0 5.05 8.50 12.045 2.0 5.05 8.41 12.065 2.0 4.86 8.21 12.085 2.0 4.76 8.15 12.095 2.0 4.73 7.87 12.0-35 -20 0 20 35Time (ps)024Power (mW)Eye Diagram (a) For Equidistant PAM-35 -20 0 20 35Time (ps)012345Power (mW)(b) For AutoEncoder PAMFig. 3: Eye Diagrams of output power for 95°C.by −∑Mi=1 xi log(yi). This loss can be related toan achievable information rate using argumentsfrom mismatched decoding[15].Performance AnalysisThe training is performed over a dataset of 2.5 ×104 randomly chosen message symbols with abatch size of 50 for an SNR of 21 dB for 1500epochs. At the end of the training procedure, theoptimized PAM levels are measured at the outputof the rescaling layer.The PAM levels obtained from the AE for vari-ous temperatures are shown in Tab. 1 along withequidistant PAM levels. At 5°C, the PAM levels areclose to that of the equidistant PAM. However, at95°C, the spacing between the first two levels is2.73 mA, and between the second and the third is3.14 mA. At higher temperatures, the AE obtainsan almost equal vertical eye-opening of the outputpower for all four levels by compressing the lowerlevels. The equal eye-opening for all the levelsfor the AE-optimized PAM can also be observedin the eye diagrams shown in fig. 3. The chosendriver current range is only for proof-of-concept,and the AE can be trained to obtain a differentextinction ratio.For a fair comparison of the cross-entropy per-formance of the AE-obtained PAM levels againstthat of the equidistant PAM levels, we havetrained a stand-alone decoder. The cross-entropyvs. SNR results of the AE and the equidistant-decoder (ED) are plotted in Fig. 4. The cross-entropy of AE is lower than that of ED for alltemperatures. For example, at 95°C, the cross-entropy of AE is 5 times smaller than that of ED.Even at lower temperatures, the cross-entropy ofAE is 2 times smaller than that of ED. The BERperformance of both AE and ED are shown in10 20 3010−410−2100SNR (dB)CrossEntropyEquidistant 5◦CAutoEncoder 5◦CEquidistant 95◦CAutoEncoder 95◦CFig. 4: Cross Entropy vs SNR for AE and ED.10 15 20 2510−610−410−2100SNR (dB)BEREquidistant 5◦CAutoEncoder 5◦CEquidistant 95◦CAutoEncoder 95◦CFig. 5: BER vs SNR of AE and ED.Fig. 5. The advantage of AE is that even thoughthe loss function minimized is the cross-entropy,the BER of AE is also lower than that of EDat higher temperatures. At 95°C, the sensitivityimprovement is about 1.5dB, validating the ideathat equidistant PAM levels do not attain an op-timal BER. Also, the training and computationalcomplexity of the proposed AE method are lowwhen compared to existing ML-based nonlinearequalization approaches[8]. Also, in a real sys-tem, training would likely be performed in a cali-bration step, and optimized PAM levels could beintegrated into the driver and receiver hardware.ConclusionsWe demonstrated AE-based end-to-end optimiza-tion of VCSEL-based fiber-optic system underdifferent operating temperatures utilizing a NN-based VCSEL model . The optimized 4-PAM in-put levels to the VCSEL are shown to have about1.5 dB sensitivity improvement when compared tothat of equidistant 4-PAM, at higher temperatures.Also, the optimized PAM levels are shown to havean improved cross-entropy across a range of op-erating temperatures.AcknowledgementsThis work was supported by the Swedish Foun-dation for Strategic Research (SSF, HOT-OPTICSProject). The work of C. Häger was also sup-ported by the Swedish Research Council (grantno. 2020-04718).References[1] D. Mahgerefteh et al., “Techno-economic Comparisonof Silicon Photonics and Multimode VCSELs,” Journalof Lightwave Technology, vol. 34, no. 2, pp. 233–242,2016.[2] A. Larsson et al., “VCSEL Modulation Speed: Sta-tus and Prospects,” in Vertical-Cavity Surface-EmittingLasers XXIII, SPIE, vol. 10938, 2019, p. 1 093 802.[3] K. Szczerba et al., “30 Gbps 4-PAM Transmission over200m of MMF using an 850-nm VCSEL,” Optics ex-press, vol. 19, no. 26, B203–B208, 2011.[4] F. Karinou et al., “IM/DD vs. 4-PAM using a 1550-nmVCSEL over Short-range SMF/MMF links for Optical In-terconnects,” in OFC, 2013.[5] C.-Y. Chuang et al., “Employing Deep Neural Net-work for High Speed 4-PAM Optical Interconnect,” in2017 European Conference on Optical Communication(ECOC), IEEE, 2017, pp. 1–3.[6] L. Ge et al., “Compressed Neural Network Equal-ization based on Iterative Pruning Algorithm for 112-Gbps VCSEL-enabled Optical Interconnects,” Journalof Lightwave Technology, vol. 38, no. 6, pp. 1323–1329,2020.[7] F. Tian and C. Yang, “Deep Belief Network-hiddenMarkov Model based Nonlinear Equalizer for VCSELbased Optical Interconnect,” Science China InformationSciences, vol. 63, no. 6, pp. 1–9, 2020.[8] Q. Zhou et al., “Low Computationally Complex Re-current Neural Network for High Speed OpticalFiber Transmission,” Optics Communications, vol. 441,pp. 121–126, 2019.[9] J. Thrane et al., “Machine Learning Techniques for Op-tical Performance Monitoring from Directly DetectedPDM-QAM Signals,” Journal of Lightwave Technology,vol. 35, no. 4, pp. 868–875, 2016.[10] D. Zibar et al., “Machine Learning Techniques in Opti-cal Communication,” Journal of Lightwave Technology,vol. 34, no. 6, pp. 1442–1452, 2015.[11] F. N. Khan et al., “Modulation Format Identificationin Heterogeneous Fiber-optic Networks using Artifi-cial Neural Networks,” Optics express, vol. 20, no. 11,pp. 12 422–12 431, 2012.[12] B. Karanov et al., “End-to-end Deep Learning of OpticalFiber Communications,” Journal of Lightwave Technol-ogy, vol. 36, no. 20, pp. 4843–4855, 2018.[13] W. Jiang et al., “End-to-end Learning based Bit-wiseAutoencoder for Optical OFDM Communication Sys-tem,” in Asia Communications and Photonics Confer-ence, Optical Society of America, 2021, T4A–64.[14] T. O’Shea and J. Hoydis, “An Introduction to DeepLearning for the Physical Layer,” IEEE Transactionson Cognitive Communications and Networking, vol. 3,no. 4, pp. 563–575, 2017.[15] S. Li et al., “Achievable Information Rates for Nonlin-ear Fiber Communication via End-to-end AutoencoderLearning,” in 2018 European Conference on OpticalCommunication (ECOC), IEEE, 2018, pp. 1–3.[16] O. Jovanovic et al., “End-to-end Learning of a Constel-lation Shape Robust to Channel Condition Uncertain-ties,” Journal of Lightwave Technology, 2022.[17] V. Aref and M. Chagnon, “End-to-end Learning of JointGeometric and Probabilistic Constellation Shaping,” in2022 Optical Fiber Communications Conference andExhibition (OFC), IEEE, 2022, pp. 1–3.[18] R. T. Chen et al., “Neural Ordinary Differential Equa-tions,” Advances in neural information processing sys-tems, vol. 31, 2018.[19] A. Grabowski et al., “Large-Signal Equivalent Circuit forDatacom VCSELs,” Journal of Lightwave Technology,vol. 39, no. 10, pp. 3225–3236, 2021.[20] K. Szczerba and C. Kocot, “Behavioral Modeling of VC-SELs for High-speed Optical Interconnects,” in Vertical-Cavity Surface-Emitting Lasers XXII, International So-ciety for Optics and Photonics, vol. 10552, 2018,p. 1 055 204.

Learning Optimal PAM Levels for VCSEL-based Optical Interconnects

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Learning Optimal PAM Levels for VCSEL-based Optical Interconnects

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