PaperlayerView exact revision on arXiv

Source-linked AI summary

Analytical Statistics of Vortex Beams in a Turbulent Channel for OAM-Multiplexed FSO Communications

Xinyi Chu, Bo Cao, Renzhi Yuan, Xiangtian Zhao, Haifeng Yao, Mugen Peng

arXiv:2609.04700v1eess.SP

TL;DR

Atmospheric turbulence causes OAM modal crosstalk and irradiance fluctuations, while receiver-plane irradiance statistics alone do not characterize demultiplexed communication performance. The paper derives analytical irradiance and port-power statistics, validates them against phase-screen simulations, and finds that SER performance is especially sensitive to mode spacing with small receiver apertures.

  • Problem

    Atmospheric turbulence degrades OAM orthogonality and communication performance, while existing approximations do not capture the fluctuations and correlations described by port-power variance and cross-port covariance.

  • Method

    The paper derives irradiance and demultiplexed port-power statistics using complex Gaussian expansions of LG vortex fields with extended Rytov and extended Huygens–Fresnel frameworks.

  • Results

    The derived irradiance and port-power statistics are consistent with phase-screen simulations across different turbulence strengths and beam parameters.

  • Takeaways & Limitations

    SER analysis links analytical channel statistics to OAM communication performance, including high-power SER floors from co-polarized coherent interference and stronger mode-spacing sensitivity with small apertures.

Abstract

from arXiv · show

Orbital angular momentum (OAM) multiplexing can increase the capacity of free-space optical (FSO) communications, whereas atmospheric turbulence causes modal crosstalk and irradiance fluctuations that degrade demultiplexing performance. Analytical modeling is therefore important for characterizing turbulence-induced propagation effects and the demultiplexed port-power statistics of OAM channels. In this paper, we first study the receiver-plane irradiance statistics of vortex beams after propagating through the turbulent channel. The average irradiance is derived using frequency-domain convolution, and a closed-form frequency-domain diffraction kernel is obtained based on extended Rytov theory to evaluate the scintillation index for moderate and strong turbulence. However, receiver-plane irradiance statistics alone are insufficient to describe the performance of OAM-multiplexed FSO communications. We therefore derive the demultiplexed port-power statistics. Specifically, we derive the average port power, modal crosstalk, port-power variance, and cross-port covariance based on the complex Gaussian expansion of general LG vortex fields and the extended Huygens-Fresnel framework. The demultiplexed port-power statistics are then used to evaluate the symbol-error rate (SER) of OAM-multiplexed FSO communications. Numerical results demonstrate that all derived statistics are consistent with those obtained by phase-screen simulations under different turbulence strengths and beam parameters. The resulting SER performance further shows that OAM-multiplexing performance is more sensitive to mode spacing for small receiver apertures than for large apertures.

I. INTRODUCTION

The paper addresses how atmospheric turbulence disrupts OAM-multiplexed FSO links and develops analytical statistics connecting vortex-beam propagation to demultiplexed communication performance. It derives irradiance and port-power statistics, validates them against phase-screen simulations, and studies SER sensitivity to polarization and mode spacing.

  • Motivation: Atmospheric turbulence distorts vortex beams, induces irradiance fluctuations, and degrades OAM orthogonality through modal crosstalk and signal fading.These effects motivate statistical characterization of turbulent OAM channels.
  • Research gap: Existing work characterized receiver-plane irradiance and demultiplexed OAM behavior, but port-power variance and cross-port covariance require fourth-order field moments.Independent additive Gaussian-noise approximations do not capture the resulting fluctuations and correlations.
  • Contributions: The paper derives average irradiance and scintillation, then average port power, modal crosstalk, port-power variance, and cross-port covariance for arbitrary beam parameters and turbulence strengths.The framework uses frequency-domain convolution, Hankel transforms, complex Gaussian LG-field expansions, extended Rytov theory, and extended Huygens–Fresnel formulations.
  • Communication performance: Coherent interference between co-polarized OAM channels acts as a multiplicative impairment and produces a high-power SER floor.Orthogonal polarization placement suppresses this impairment in the compared configurations.
  • Validation: The derived irradiance and port-power statistics agree with phase-screen simulations across different turbulence strengths and beam parameters.The results also associate VVB scintillation reduction with negative irradiance cross-covariance between orthogonal polarization components.
  • Communication performance: OAM-multiplexing performance is more sensitive to mode spacing for small receiver apertures than for large apertures.The paper therefore considers mode spacing jointly with receiver aperture in co-polarized transmission.

C. Atmospheric Turbulence Model

The turbulence model represents the channel with a von Kármán refractive-index spectrum and long-exposure wave structure function. Average receiver-plane irradiance is computed by convolving unperturbed irradiance with an atmospheric point-spread function in the spatial-frequency domain.

  • Turbulence spectrum: The horizontal terrestrial FSO link uses a statistically homogeneous and isotropic refractive-index field with a von Kármán spatial spectrum.The model specifies an outer scale through κ0 = 2π/L0 and sets the inner scale to zero.
  • Turbulence spectrum: Under long-exposure conditions, time averaging sets the temporal filtering factor FK(κ) to 1.The resulting spectrum is denoted Φn(κ).
  • Wave structure function: Finite-waist beams are modeled under long-exposure conditions using the spherical-wave structure function.The structure function depends on receiver-plane separation, normalized propagation distance, a modified Bessel function, and the Gamma function.
  • Average irradiance: Under the Markov approximation, turbulence-averaged irradiance equals the convolution of unperturbed irradiance with the atmospheric point-spread function.This converts the propagation problem into a spatial-frequency-domain calculation.
  • Average irradiance: The spatial-frequency irradiance spectrum is F{I}(f) = F{I0}(f) · M(f), with M(f) = exp[−Dsp(λL|f|)/2].Here f is the transverse spatial frequency and Dsp is the spherical-wave structure-function term.
  • Average irradiance: The average irradiance is obtained by expanding Laguerre-polynomial terms, evaluating Hankel–Bessel integrals with Bessel–Gaussian identities, and applying an inverse Hankel transform.The total average irradiance is then formed as an incoherent sum over branches.

B. Scintillation Index and Irradiance Cross-Covariance

The section develops scintillation and irradiance cross-covariance statistics for vortex beams using extended Rytov theory and analytically evaluated spectral kernels.

  • Scintillation index: Extended Rytov theory is adopted because conventional Rytov theory diverges at vortex-beam dark rings under moderate and strong turbulence.The irradiance is modeled through independent large-scale and small-scale lognormal factors.
  • Scintillation index: The received field is represented as the unperturbed Fresnel-propagated field multiplied by an exponential Rytov perturbation.The first-order perturbation is used to obtain log-amplitude and correlation moments.
  • Scintillation index: A closed-form spectral transfer amplitude is obtained from Fresnel and Bessel-type integrations for the vortex-beam diffraction kernel.The kernel uses Fresnel scaling and an offset spectral vector to separate radial and angular dependencies.
  • Irradiance cross-covariance: Mixed correlation moments and a normalized correlation coefficient are used to form the saturated irradiance cross-covariance between modal branches.The cross-covariance is evaluated from correlation moments associated with the relevant mode pair.
  • Irradiance cross-covariance: The total vector-vortex-beam scintillation index combines single-mode indices with irradiance cross-covariance weighted by the left- and right-circular polarization branches.The same turbulence affects both polarization components, so their normalized irradiance cross-covariance is included.

IV. STATISTICAL MOMENTS OF THE DEMULTIPLEXED PORT POWERS

The paper derives statistical moments of demultiplexed OAM port powers to characterize turbulence-induced effects beyond receiver-plane irradiance statistics.

  • Statistical moments: The analysis derives average port power, modal crosstalk, port-power variance, and cross-port covariance for polarization-resolved OAM receiver ports.Port (σ, n) identifies a polarization branch and reference OAM mode.

A. Average Port Power and Modal Crosstalk

The average power coupled into an OAM-demultiplexer port is derived from turbulence-perturbed fields and used to distinguish desired-mode retention from modal crosstalk.

  • Port projection: Each demultiplexer port projects the received field onto its reference mode over a circular receiver aperture.The modal coupling power is the squared magnitude of the complex coupling coefficient.
  • Modal crosstalk: The zeroth azimuthal Fourier component of the coherence kernel determines signal retention, while nonzero harmonics describe crosstalk.This decomposition follows from analytically performing the angular integrations.
  • Modal crosstalk: For a mismatched transmitted mode, crosstalk ratio is defined as XT_n←q′ = ⟨I_n←q′⟩/⟨I_n←q_n⟩.The numerator is mismatched-mode power and the denominator is desired-mode power at port n.
  • Normalization: Active subchannels are normalized to unit launched power so different multiplexing configurations can be compared fairly.The instantaneous port power is defined over the set of OAM charges transmitted on each polarization branch.
  • Average port power: Azimuthal orthogonality removes cross-terms in the ensemble-averaged port power.Rotational invariance yields the Kronecker-delta selection rule for OAM charges.

B. Normalized Variance of the Modal Coupling Power

The normalized variance of modal coupling power is derived from fourth-order field moments within the extended Huygens–Fresnel framework, with Monte Carlo integration used when no closed form is available.

  • Variance definition: The normalized variance measures desired-mode power fluctuations when q = q_n and turbulence-induced crosstalk fluctuations when q ≠ q_n.Both cases use ensemble averages over turbulence realizations.
  • Huygens–Fresnel formulation: The extended Huygens–Fresnel framework expresses the fourth-order moment using free-space fourth-order fields, spherical-wave structure functions, and complex log-amplitude correlations.The structure function depends on transverse point separations, while the correlation term captures log-amplitude coupling.
  • Correlation kernel: The complex log-amplitude correlation includes Bessel-function dependence, beam-size evolution, and Fresnel weighting for phase-to-amplitude conversion.These terms depend on transverse separation and the normalized beam-size factor along propagation.
  • Fourth-order moment: Port-power variance requires a fourth-order modal moment formed by projecting four propagation fields onto the reference mode.The conjugation pattern matches the fourth-order power expression.
  • Numerical evaluation: Because the fourth-order expression couples four two-dimensional receiver-plane coordinates, its evaluation generally uses importance-sampled Monte Carlo integration.Independent truncated Gaussian proposals sample points within the circular receiver aperture.

C. Port-Power Covariance Matrix in OAM-Multiplexed Transmission

The covariance analysis decomposes port-power fluctuations into turbulence-induced modal correlations and coherent interference, then determines them through fourth-order modal moments. The resulting covariance matrix covers variances and cross-port covariances for arbitrary physical receiver-port pairs.

  • Covariance decomposition: Port-power covariance combines individual modal power products with cross-products arising from coherent interference before square-law detection.The decomposition separates turbulence-induced modal power correlation from coherent interference contributions.
  • Covariance decomposition: Coherent interference remains zero between different polarization branches because of polarization orthogonality.
  • Fourth-order formulation: The complete port-power covariance is determined by fourth-order modal moments within the extended Huygens–Fresnel framework.The fourth-order mutual coherence function is projected onto the receiver reference modes to obtain the modal moments.
  • Normalized covariance matrix: For M-channel transmission, the normalized covariance matrix includes diagonal port-power variances and off-diagonal covariances for same-branch and cross-branch port pairs.

V. COMMUNICATION PERFORMANCE METRICS

The communication-performance analysis models conditional port-power statistics for composite OOK symbols and uses them to construct joint detection densities and SER. It accounts for modal coupling, coherent interference, detector noise, and separate maximum-likelihood decisions.

  • Conditional port-power statistics: Two co-polarized OAM channels are represented by four composite OOK symbols because modal crosstalk and coherent interference couple their port-power statistics.The active-mode sets are empty, single-mode, or two-mode depending on the transmitted symbol.
  • Conditional port-power statistics: The conditional average port power and variance are obtained from modal coupling coefficients for each composite OOK symbol.The coupling coefficient g_n←q describes transmission from mode q to port n.
  • Conditional PDF model: Conditional port-power PDFs are modeled with bivariate lognormal or Gaussian distributions for the normalized port-power vector.The model uses the conditional means and covariance matrix of the two-port power vector.
  • Conditional PDF model: For ℓ1 = 1 and ℓ4 = 4 under moderate turbulence with C2_n = 1.70 × 10^-15 m^-2/3, the (0, 1) PDF is lognormal, while the (1, 0) and (1, 1) PDFs are Gaussian.
  • Detection and SER: The received-current model combines scaled optical port power with electrical noise, approximating Poisson shot noise as Gaussian at sufficiently high received optical power.The joint conditional PDF assumes conditionally independent noise samples at the two receivers.
  • Detection and SER: Separate maximum-likelihood decisions use bit-conditioned PDFs, scalar decision regions, and the four-symbol correct-detection probabilities to evaluate SER.The SER is evaluated numerically by Monte Carlo sampling with the separate ML decisions.

VI. NUMERICAL RESULTS

Numerical results validate the analytical model against phase-screen simulations across turbulence strengths and examine irradiance and scintillation behavior. Increasing turbulence progressively destroys vortex-ring structure, while scintillation depends on mode charge magnitude, mode overlap, and power allocation.

  • Model validation: The analytical model agrees with phase-screen simulations using 16 screens, 4 subharmonic levels, and 4000 realizations per screen.
  • Average irradiance: At L = 3 km, increasing turbulence fills the central irradiance null and fades ring contrast until strong turbulence produces a monotonically decaying Gaussian-like profile.The compared turbulence strengths are C2_n = 1.7 × 10^-17, 1.7 × 10^-15, and 1.7 × 10^-13 m^-2/3.
  • Average irradiance: For C2_n ≤ 1.7 × 10^-15 m^-2/3, the modal structure is preserved and OAM subchannels remain demultiplexable.
  • Scintillation: Scintillation is mainly determined by |q|, while unequal |q| produces lower scintillation in the mode-overlap region.Configurations with equal |q| show similar scintillation behavior; unequal |q| does not.
  • Scintillation: Scintillation versus power-allocation angle has a U-shaped dependence, with minima near balanced allocation θ = π/2.The analysis adopts θ = π/2 for subsequent evaluation.

B. Demultiplexed Port-Power Statistics of OAM-Multiplexed Transmission

The paper derives and validates demultiplexed port-power statistics for turbulent OAM transmission, then connects them to SER and aperture-dependent mode-spacing trade-offs.

  • Port-power statistics: Analytical predictions agree closely with phase-screen simulations for crosstalk ratio, average port power, normalized port-power variance, and normalized cross-port covariance.The comparison covers two-channel scalar OAM transmission across mode spacings and turbulence strengths.
  • SER performance: Coherent interference between co-polarized OAM channels produces a high-power SER floor, while orthogonal polarization placement lowers SER.The difference between vector-vortex-beam and co-polarized scalar configurations becomes more pronounced as transmitted power increases.
  • Mode spacing and aperture: For smaller receiver apertures, SER is most favorable near mode spacing Δq = 3 but rises rapidly at larger spacings; larger apertures remain relatively stable.Increasing spacing reduces modal crosstalk but makes higher-order modes more spatially extended, increasing signal loss for small apertures.
  • Mode spacing and aperture: Mode spacing should be selected jointly with receiver aperture to balance reduced modal crosstalk against increased signal loss.This trade-off is especially important when polarization reuse requires co-polarized OAM channels.
  • Analytical framework: The framework connects receiver-plane irradiance statistics to demultiplexed port-power statistics and validates both against phase-screen simulations.The analytical statistics are used to evaluate communication-level SER under different turbulence strengths and beam parameters.

APPENDIX B CLOSED-FORM AVERAGE-IRRADIANCE SPECTRUM

Appendix B develops closed-form spectral expressions for average irradiance and related radial kernels using finite LG expansions, Fourier-Bessel identities, and complex Gaussian representations.

  • APPENDIX B CLOSED-FORM AVERAGE-IRRADIANCE SPECTRUM: The appendix introduces the azimuthally symmetric irradiance of one polarization branch and its zeroth-order Hankel transform.The branch irradiance is written from the propagated vortex field and its polarization weight.
  • APPENDIX B CLOSED-FORM AVERAGE-IRRADIANCE SPECTRUM: A finite expansion in powers of the radial variable is used to represent the Laguerre-Gaussian field coefficients.The expansion uses coefficients indexed by radial order and absolute topological charge.
  • APPENDIX B CLOSED-FORM AVERAGE-IRRADIANCE SPECTRUM: Standard Bessel-Gaussian integrals and Kummer’s transformation reduce each radial term to a Laguerre polynomial, enabling closed-form inverse-Hankel evaluation.The resulting analytic spectrum is constructed component by component.
  • A. Derivation of the Spectral Transfer Amplitude: The appendix then derives a spectral transfer amplitude by inserting the finite field expansion into the Rytov integral and introducing an offset spectral vector.Polar-coordinate conversion separates the source-plane integration into radial and azimuthal parts.
  • A. Derivation of the Spectral Transfer Amplitude: The offset-vector magnitude and azimuth parameterize the transfer kernel, whose radial integral is evaluated with Weber’s second exponential formula.Direct Fresnel propagation provides the corresponding free-space receiver-plane field.
  • B. Closed Form of the Single-Mode Radial Kernel: For the single-mode radial kernel, shifted spectral vectors produce cosine-dependent quadratic forms that support binomial expansion and termwise integration.The construction distinguishes the two quadratic forms associated with the shifted vectors.
  • B. Closed Form of the Single-Mode Radial Kernel: Complex Gaussian expansion terms combine with modified Bessel functions to produce the closed-form single-mode radial kernel.The parameters η+ and η− collect the Gaussian-width combinations used in the expression.
  • C. Mixed-Mode Kernels for the Irradiance Cross-Covariance: Mixed-mode kernels for irradiance cross-covariance are formed from Laurent-polynomial topological factors, whose harmonics integrate through a Bessel identity.The resulting mixed radial kernel is expressed in closed form with modified Bessel functions.
Loading 2609.04700v1…