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SCI-D$^2$NN: An Optimization Framework for OAM-Multiplexed FSO Communications

Rui Deng, Renzhi Yuan, Xinyi Chu, Siming Wang, Chengzhi Liu, Zehao He, Haifeng Yao, Mugen Peng

arXiv:2608.30962v1eess.SP

TL;DR

OAM-multiplexed FSO detection is degraded by atmospheric turbulence, pointing errors, and photodetection noise, while existing D2NN schemes are not optimized for communication detection. The paper proposes SCI-D2NN with supervised decision-domain training, complex-amplitude crosstalk modeling, two detection schemes, and two training losses. SCI-D2NN improves BER over conventional D2NN, with the BD-based loss providing the strongest reported improvement.

  • Problem

    Existing D2NN compensation schemes are not specifically optimized for communication detection under turbulence, pointing errors, and photodetection noise.

  • Method

    SCI-D2NN combines projection and label branches with complex-amplitude crosstalk modeling, profile-likelihood and joint ML detection, and BD-based and ML-based training losses.

  • Results

    More than 3-dB BER improvement is achieved over the conventional D2NN baseline in most transmit-power regions, while the BD-based loss provides more than 10-dB improvement in the high-transmit-power region.

  • Takeaways & Limitations

    The BD-based loss achieves the lowest BER under different symbol-rate, turbulence-strength, diffractive-layer, and pointing-error settings.

Abstract

from arXiv · show

Orbital angular momentum (OAM) multiplexing can increase the capacity of free-space optical (FSO) communications, but its detection performance is strongly affected by impairments such as atmospheric turbulence, transmitter pointing errors, and photodetection noise. The diffractive deep neural network (D$^2$NN) can be used as an all-optical front end to mitigate turbulence-induced distortions before detection. However, existing D$^2$NN compensation schemes are not specifically optimized for communication detection. In this paper, we propose a supervised contrastive inspired D$^2$NN (SCI-D$^2$NN) framework for improving the detection performance of OAM-multiplexed FSO communications under these impairments. The proposed framework introduces two training branches: a projection branch that maps the optical field to low-dimensional decision domain samples, and a label branch that provides supervised labels to impose a separation constraint among decision domain samples. In addition, we characterize complex-amplitude crosstalk to obtain the receiver observation vector and formulate two detection schemes, namely single-port profile-likelihood detection and joint maximum-likelihood (ML) detection. We further design two SCI-D$^2$NN training losses called Bhattacharyya distance (BD) based loss and the ML based loss to improve decision domain separability and mitigate detection-performance degradation. Numerical results show that SCI-D$^2$NN achieves more than a 3-dB improvement in bit error rate (BER) over the conventional D$^2$NN baseline in most transmit-power regions. The BD based loss gives the lowest BER under different system parameters and provides more than a 10-dB BER improvement over the baseline in the high transmit power region.

I. INTRODUCTION

OAM multiplexing expands the spatial multiplexing dimension of FSO links, but turbulence, pointing errors, and photodetection noise degrade detection. SCI-D2NN addresses this gap with detection-oriented optical processing and achieves substantial BER improvements.

  • OAM modes provide an additional spatial degree of freedom for increasing FSO transmission capacity.
  • Atmospheric turbulence, pointing errors, and photodetection noise increase inter-mode crosstalk and impair receiver decision statistics.
  • Existing compensation methods either require electrical-domain processing or optimize optical restoration without explicitly targeting communication detection.
  • SCI-D2NN uses projection and label branches to map optical fields into low-dimensional, supervised decision-domain samples with improved separability.
  • The framework characterizes complex-amplitude crosstalk, formulates profile-likelihood and joint maximum-likelihood detection, and introduces BD-based and ML-based losses.
  • More than 3-dB BER improvement is achieved over the conventional D2NN baseline in most transmit-power regions.
  • More than 10-dB BER improvement is obtained with the BD-based loss over the conventional D2NN baseline in the high-transmit-power region.

II. OAM-MULTIPLEXED FSO COMMUNICATION SYSTEM MODEL IN TURBULENT CHANNELS

The modeled OAM-FSO system coherently multiplexes Laguerre-Gaussian branches, propagates them through turbulence and pointing errors, and uses SCI-D2NN before modal demultiplexing and photodetection.

  • Multiple Laguerre-Gaussian OAM branches are coherently combined into a coaxially multiplexed optical field.
  • The multiplexed field encounters atmospheric turbulence and pointing errors before entering the SCI-D2NN optical front end.
  • SCI-D2NN output is demultiplexed by modal projection, and the projected components are photodetected to form the receiver observation vector.
  • During training, projection and label branches provide receiver-side samples and supervised labels that construct the decision-domain geometry for the loss.
  • The system uses a finite set of OAM modes with radial index p = 0 and relies on orthogonality under ideal propagation for modal projection.
  • Turbulence and pointing errors cause optical power to leak from intended OAM branches into other branches, producing modal crosstalk.

B. Turbulent Effects and Pointing Errors for OAM-Multiplexed Signals

The channel model combines split-step turbulence propagation with random pointing-error displacement, producing the distorted complex field supplied to SCI-D2NN.

  • Free-space propagation between phase screens is implemented with the angular spectrum method.
  • The turbulence model uses von Karman refractive-index statistics to generate random phase screens for split-step propagation.
  • Pointing errors are modeled as random transverse angular misalignment between the transmitted-beam axis and receiver optical axis.
  • The numerical study uses a zero-mean isotropic pointing-error model unless otherwise specified.
  • Under the small-angle approximation, pointing-error angles produce receiver-plane displacement that combines with turbulence-distorted propagation.
  • The resulting distorted complex field is used as the SCI-D2NN input, whose trainable optical mapping produces the output field.

C. Mode Crosstalk of OAM Modes

The paper characterizes OAM crosstalk in the complex-amplitude domain and derives the receiver intensity vector used for subsequent detection analysis.

  • Complex-amplitude crosstalk is characterized because coherent superposition makes relative phases relevant to receiver intensities.
  • The crosstalk matrix records each transmitted branch’s complex-amplitude response across all receiver ports.
  • Diagonal matrix entries represent self-mode coupling, while off-diagonal entries represent leakage into other receiver ports.
  • For a transmitted modulation state, projected branch responses are coherently superposed to form the receiver-side complex-amplitude vector.
  • The receiver intensity vector depends on both modal-leakage magnitudes and relative phases through coherent cross terms.
  • The resulting intensity vector supplies the basis for the paper’s subsequent detection analysis.

III. SINGLE-PORT PL AND JOINT ML DETECTION

This section models receiver observations under OOK with state-dependent photodetection noise and formulates single-port profile-likelihood and joint ML detection.

  • Observation model: OOK assigns each OAM branch a binary symbol, forming a joint bit state b with 2^M candidate states.The mth branch state is b_m ∈ {0,1}, and the candidate set is B = {0,1}^M.
  • Observation model: The receiver observation vector converts photodetected intensities through responsivity and additive thermal-plus-shot noise under a Gaussian approximation.The approximation applies when the collected photoelectron count is sufficiently large.
  • Observation model: Modal crosstalk makes each receiver-port intensity depend on the complete transmitted state, producing state-dependent shot-noise variance.Each port receives its corresponding branch contribution plus leakage from other branches.
  • Joint ML detection: Joint ML detection instead uses the full multi-port observation vector to compare all candidate joint states under the state-dependent Gaussian model.This formulation evaluates likelihoods over the complete set B rather than profiling over hidden bits.
  • Single-port profile-likelihood detection: Single-port profile-likelihood detection estimates one branch bit from a scalar port observation while treating the other M−1 bits as nuisance states.Each binary hypothesis is scored by maximizing likelihood over its 2^(M−1) compatible joint states.
  • Single-port profile-likelihood detection: The single-port detector evaluates candidate-state likelihoods using the scalar Gaussian observation model and averages conditional BER over channel realizations.The reported expectations and probabilities are estimated by Monte Carlo simulation.

C. Joint ML Detection

Joint ML detection compares complete joint-state hypotheses using all receiver ports, addressing the overlap and separability limits of single-port detection.

  • Motivation: Single-port detection can suffer overlap between intensity sets for bit hypotheses because unobserved OAM branches create hidden states.This overlap limits single-port separability and motivates joint detection.
  • Joint ML formulation: Joint ML detection performs likelihood comparison over the complete 2^M-state set using the full observation vector.For an M-branch OAM-OOK system, the candidate joint-state set has |B| = 2^M states.
  • Joint ML formulation: For each candidate state, the joint detector evaluates its conditional density under the receiver-side intensity vector and state-dependent covariance.The ML decision selects the candidate with the greatest likelihood, equivalently the lowest negative log-likelihood metric.
  • Performance evaluation: The resulting heteroscedastic metric compares likelihoods of all candidate joint bit states for a fixed receiver-side intensity vector.Decision regions define conditional transition probabilities, which are averaged to obtain SER and BER.
  • Comparison: Joint ML detection exploits complete multi-port observations and provides better separability among joint bit states than single-port detection.The SCI-D2NN framework is consequently designed to improve joint-detection performance.

IV. SUPERVISED CONTRASTIVE INSPIRED D2NN TRAINING FRAMEWORK

SCI-D2NN combines diffractive optical processing with supervised decision-domain training, using label-guided losses to separate joint-state receiver statistics for joint ML detection.

  • A. Forward Propagation of SCI-D2NN: SCI-D2NN propagates a turbulence-distorted complex field through K trainable diffractive planes separated by free-space intervals.Each plane modulates the incident field through a complex transmittance before subsequent propagation.
  • A. Forward Propagation of SCI-D2NN: Each diffractive unit acts as a secondary wave source, and layer propagation is represented using complex fields and Fourier-domain Fresnel operations.The output field is formed after sequential transmittance modulation and inter-layer propagation.
  • B. SCI-D2NN Training Structure: The encoder processes the distorted field, while a projection branch maps the output field to low-dimensional receiver-side decision-domain samples.The mapping uses modal projection, coherent complex-field superposition, and photodetection.
  • B. SCI-D2NN Training Structure: The joint bit state serves as the label-branch supervision for receiver-side samples, imposing separation among different decision-domain classes.The constraint targets receiver-side intensity-vector geometry because it determines the observation statistics used by the ML detector.
  • C. Training Losses: The BD-based loss enlarges Bhattacharyya distance for Hamming-1 state pairs, whose one-bit differences are linked to BER events.Pairs below the target distance receive larger softplus margin penalties.
  • C. Training Losses: The ML-based loss separates all candidate joint states by penalizing incorrect states whose ML detection scores approach or exceed the true state's score.It uses the joint ML negative log-likelihood metric under a short quasi-static channel assumption.

C. Computational Complexity Analysis

Training complexity is dominated by FFT-based optical propagation for the evaluated three-branch system, while both decision-oriented losses scale exponentially with branch count.

  • Loss-evaluation cost: The BD-based loss scales as O(M 2^(2M−1)), whereas the ML-based loss scales as O(M4^M) per channel realization.The BD loss evaluates Hamming-1 pairs, while the ML loss evaluates every transmitted-state and candidate-state combination.
  • Propagation cost: For one complex-field input, each free-space propagation interval costs O(N_xN_y log(N_xN_y)) using a two-dimensional FFT and inverse FFT.The grid dimensions are N_x × N_y.
  • Propagation cost: Crosstalk-matrix construction across M branches and K+1 propagation intervals costs O(M(K+1)N_xN_y log(N_xN_y)) per channel realization.Backward propagation has comparable computational cost.
  • Overall training cost: For the three-branch configuration, FFT-based angular-spectrum propagation is the main training cost because of the large complex-field grid.This identifies numerical propagation as the dominant cost in the considered configuration.
  • Scaling limitation: As M increases, loss evaluation grows exponentially while propagation grows linearly with M, so loss evaluation can eventually dominate training cost.This comparison applies for fixed field-grid size and network depth.

D. Scalability Discussion

The framework’s scalability is constrained by exponentially increasing loss-computation costs as OAM multiplexing grows, motivating reduced-complexity training strategies. Extensions to higher-order modulation and practical channels further increase training and deployment demands.

  • Low-complexity training: Groupwise training reduces each step’s joint-state dimension by optimizing smaller OAM-branch groups.The complete multiplexing problem is partitioned into smaller optimization problems.
  • Low-complexity training: List-based training retains the full OAM dimension but evaluates losses only for representative or competing joint states.Both groupwise and list-based strategies reduce loss-evaluation cost.
  • Complexity scaling: Loss-function complexity grows exponentially with the number of OAM branches, while numerical-propagation complexity increases linearly with M.This creates a performance-complexity trade-off for larger OAM-multiplexed systems.
  • Practical deployment: Practical deployment requires stronger channel modeling, robust training against device imperfections, receiver-side processing, and accurate diffractive-layer calibration.The proposed front end can be physically implemented with spatial light modulators or diffractive optical elements, but calibration is needed to match training geometry.

E. Traditional D2NN for Comparison

The comparison uses a diaphragm-equipped field-restoration D2NN as the conventional baseline against SCI-D2NN. Unlike SCI-D2NN, this baseline supervises the output field against a turbulence-free reference rather than directly optimizing decision-domain detection metrics.

  • Baseline configuration: The conventional baseline is a field-restoration D2NN equipped with a diaphragm to suppress stray light.The diaphragm follows the field-restoration configuration used for comparison.
  • Training objective: The baseline supervises the output complex field against an ideal turbulence-free reference field.A scalar phase factor removes sample-dependent global phase ambiguity before penalizing residual field mismatch.
  • Training objective: The field-restoration loss does not directly act on projected intensity vectors or the ML decision metric.This distinguishes the baseline objective from SCI-D2NN’s detection-oriented training.
  • Baseline selection: The diaphragm-equipped traditional D2NN is used as the comparison baseline because the version without a diaphragm yields higher BER under both joint ML and single-port PL detection.The reported comparison evaluates detection performance under the stated OAM-FSO simulation conditions.
  • Simulation setup: The five-layer SCI-D2NN uses 800,000 trainable phase parameters, with training epochs taking approximately 28.0 s for LBD and 30 s for LML.These implementation values describe the numerical comparison setup.
  • Detection protocol: The three-port joint ML detector consistently outperforms the corresponding single-port detector.The simulations therefore use the three-port joint ML detector as the main detection protocol.

SIMULATION PARAMETERS

The simulations compare detection-oriented SCI-D^2NN training with conventional D^2NN across detection schemes, symbol rates, layer depths, diffraction loss, turbulence, and pointing errors. SCI-D^2NN generally improves BER, with the BD-based loss delivering the strongest and most robust gains.

  • Diffractive layers: More than a 3-dB transmit-power gain over the baseline is achieved in most transmit-power regions with SCI-D^2NN under the reported layer comparisons.With three layers, the BD-based design already achieves good detection performance and outperforms the ten-layer baseline in the high-power region.
  • Error composition: dH = 2 transitions account for the largest BER fraction, while dH = 3 transitions contribute relatively little because they require three simultaneous bit errors.The BER composition is evaluated for Hamming distances dH = 1, 2, 3 at Pavg = 10 dBm.
  • Training losses: BD-based training corrects weakly separated state pairs most effectively and achieves the lowest BER over most transmit-power regions; ML-based training is more balanced at low power.The two losses target different geometries: BD focuses on the weak-separation tail, whereas ML improves the overall multi-class likelihood geometry.
  • Detection protocol: SCI-D^2NN improves BER over the raw distorted field and conventional D^2NN, while joint ML detection exploits multidimensional receiver observations.The simulations use the three-port joint ML detector as the main detection protocol.
  • Loss-aware depth: Five-layer SCI-D^2NN achieves the lowest BER when each layer has intensity diffraction efficiency η = 0.85, balancing optical processing capability against accumulated loss.Three layers preserve more optical power but process less effectively, while ten layers incur accumulated diffraction loss.

VI. CONCLUSION

The SCI-D2NN framework targets communication detection in OAM-multiplexed FSO links by combining low-dimensional projection, supervised labels, receiver crosstalk modeling, and detection-oriented losses. It improves BER over conventional D2NN compensation, with the BD based loss performing best across tested conditions.

  • SCI-D2NN addresses the limitation that existing D2NN compensation objectives are not designed for communication detection.
  • The framework uses projection and label branches to map optical fields into low-dimensional receiver-side samples and improve decision-domain separability.The label branch uses transmitted joint symbol states as supervised labels, while the projection branch simplifies the training task.
  • SCI-D2NN characterizes receiver-side complex-amplitude crosstalk and formulates single-port profile-likelihood and joint ML detection schemes.The framework also derives corresponding BER and SER criteria.
  • More than a 3-dB BER improvement over the conventional D2NN baseline is achieved in most transmit-power regions.The improvement is attributed to reducing training-sample dimensionality and aligning the optimization objective with final detection performance.
  • The BD based loss achieves the lowest BER across different system parameters and provides more than a 10-dB BER improvement in the high-transmit-power region.Its advantage is linked to improving the separability of weakly separated samples.
  • The results support using D2NNs to improve communication-system performance.
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