Computation of optimal beams in weak turbulence

TL;DR

This work addresses the problem of designing optimal beams that maximize received intensity while propagating through weak turbulent media. It develops a Fourier-space, perturbative framework for the paraxial wave equation, transforming the high-dimensional computation of the transfer kernel \mathcal{H} into a tractable eigenproblem for its Fourier counterpart \mathcal{H}_{\mathrm{f}}, and shows that, under standard turbulence assumptions (homogeneous statistics, small-length-scale cutoff, and Markov), the required integrals collapse from 8-fold to as few as 1-fold. The authors derive explicit perturbative expressions for the field and propagator up to second order (A_0,A_1,A_2 and g_0,g_1,g_2), formulate the associated kernels \mathcal{H}_{\mathrm{f},ij}, and demonstrate energy conservation within the truncation. Numerical examples validate the equivalence of physical- and Fourier-space formulations, show that optimal beams maintain near-full intensity with small divergence under turbulence, and highlight substantial computational savings from the proposed simplifications. Overall, the paper provides a practical route to compute general, coherent-optimal beams in weak turbulence with potential impact on optical communications and sensing through turbulent media.

Abstract

When an optical beam propagates through a turbulent medium such as the atmosphere or ocean, the beam will become distorted. It is then natural to seek the best or optimal beam that is distorted least, under some metric such as intensity or scintillation. We seek to maximize the light intensity at the receiver using the paraxial wave equation with weak-fluctuation as the model. In contrast to classical results that typically confine original laser beams to be from a special class, we allow the beam to be general, which leads to an eigenvalue problem of a large-sized matrix with each entry being a multi-dimensional integral. This is an expensive and sometimes infeasible computational task in many practically reasonable settings. To overcome this, we utilize an asymptotic expansion and transform the derivation to Fourier space, which allows us to incorporate some optional turbulence assumptions, such as homogeneous-statistics assumption, small-length-scale cutoff assumption, and Markov assumption, to reduce the dimension of the numerical integral. The proposed methods provide a computational strategy that is numerically feasible, and results are demonstrated in several numerical examples.

Figures (7)

• Figure 1: $\mathcal{H}$ computed using \ref{['eqn:H00_phys_space_simplification']} when the receiver size $R$ is $0.01$, $0.03$, $0.06$, and $0.09$ m. The transmitter size is fixed at $r=0.05$ m.
• Figure 2: In the panel on the left, we show the comparison of total intensity over various receiver sizes, and the focused beam solution. Receiver size is in meters. In the panel on the right, we show the agreement of the first nine eigenvalues given by $\mathcal{H}$ and $\mathcal{H}_\mathrm{f}$, taking $R=0.09$m.
• Figure 3: Eigenfunctions provided by computing $\mathcal{H}$ (blue line) and $\mathcal{H}_{\mathrm{f}}$ (red line) are on top of each other. The left and the middle panel show the real and imaginary parts of the first six engenvectors, and the panel on the right shows the light intensity received at the receiver with the $R = 0.09$m for each of the six eigenfunctions.
• Figure 4: The plot in blue is the relative error in Fourier space vs physical space calculations for $\mathcal{H}$ in a uniform medium (with respect to the Frobenious norm). The red plot shows the relative difference in the presence and absence of turbulence from Fourier space calculations. $\epsilon$ is fixed at $5\times 10^{-8}$ ($R$ is in meters)
• Figure 5: In the panel on the left, we show the initial intensity profile of optimal beam using $R=0.05$m. In the panel on the right, we show the comparison of total intensity at the receiver using optimal beams and focused beams under the homogeneous random medium assumption and small-length-scale cutoff assumption ($R$ is in meters).