The authors of J. Opt. Soc. Am. B 42, 1804 (2025) generate a besinc pseudo-Schell-model source-a member of a class of partially coherent beams known as self-focusing nonuniformly correlated sources-via the incoherent sum of pseudo-modes. Here, we show how such a source can be synthesized more quickly using coherent modes. (c) 2026 Optica Publishing Group. All rights, including for text and data mining (TDM), Artificial Intelligence (AI) training, and similar technologies, are reserved.

Correlations between Z-tilt, G-tilt, and centroid motion as a function of thresholding are studied through simulations and analyses. The Z-tilt, or Zernike tilt, is the first moment of the phase distribution over the aperture, while the G-tilt, or gradient tilt, is the average phase gradient over the aperture. It is found that raw centroid motion best correlates with G-tilt, but that even a small amount of thresholding makes Z-tilt the stronger correlation. The Zernike decomposition of the G-tilt is also presented. Using this decomposition, a theoretical value for the correlation between Z-tilt and G-tilt is determined. It is further demonstrated that the quad-cell signal best correlates with Z-tilt even though no thresholding is involved.

We use a benchtop atmospheric turbulence simulator (ATS) and adaptive-optics (AO) system to experimentally evaluate AO performance using a Shack--Hartmann wavefront sensor (SHWFS) and digital-holographic wavefront sensor (DHWFS). Consistent with past results, we show that while the SHWFS yields performance improvement relative to no AO compensation, the SHWFS struggles in strong scintillation conditions. However, when we control our AO system using the least-squares component of our DHWFS measurements, we notice a substantial improvement in performance relative to the results we obtain with the SHWFS. We also leverage our DHWFS measurements to implement a post-processing congruence operation algorithm, referred to as the least-squares principal value (LSPV). We show that the LSPV-based results yield substantial AO performance improvements in strong scintillation conditions compared to only using the least-squares component of our DHWFS measurements. LSPV enables partial compensation of the hidden-phase component---a requirement to achieve better AO performance when scintillation is strong. These findings will be of interest to researchers exploring AO performance in deep turbulence and those who are interested in branch-point-tolerant AO approaches.

We present a method to numerically compute the coherent mode representations (CMRs) for partially coherent beams with separable phases. This special class of random light field has the ability to self-focus and is resistant to turbulence-induced degradation, making it potentially useful in applications such as optical communications. We validate our method by generating (in simulation) two such sources from the literature using their computed CMRs. Lastly, we conclude with a summary of our approach and a discussion of applications.

We developed a laboratory-scale testbed to simulate atmospheric turbulence effects and to investigate the role of turbulence on the quantum information content of a free-space qubit transport. We report the development, characterization, and calibration of the testbed for the statistical parameters of the turbulence employing two phase screens, which produced weak to strong turbulence replicating long distances in the atmosphere for single photon propagation. By taking a long-exposure image of the point-spread function of a pinhole and analyzing the associated modulation transfer function, we measured a D/r0 ranging from 1.04 to 23.27. Using irradiance fluctuations, we measured an on-axis scintillation index ranging from 0.25 to 2.02. The TASQ was designed to integrate a biphoton source, along with a single photon detection module to test the effect of turbulence on qubit transport. The calibrated setup will provide critical data for field experiments involving long-distance quantum networks.

In this paper, we derive single-integral solutions, applicable in the weak-to-moderate scintillation regime, for both the noise equivalent angle (NEA) due to scintillation and the scintillation-induced root mean squared error (RMSE) between gradient tilt (G-tilt) and centroid tilt (C-tilt). In practice, the NEA due to scintillation gives a measure of the scintillation-induced track error, whereas the scintillation-induced RMSE between C-tilt and G-tilt gives a measure of the C-tilt, G-tilt anisoplanatism due to scintillation. Assuming spherical-wave propagation, we fit closed-form expressions to the numerically integrated solutions. These closed-form expressions serve as "scaling laws," and we validate their use with wave-optics simulations. At large, we determine that the one-axis NEA due to scintillation scales as a function of aperture size, propagation distance, wavelength, and Rytov number, whereas the one-axis scintillation-induced RMSE between C-tilt and G-tilt scales proportionally to the Rytov number when normalized by the diffraction angle. These findings will aid in the design of active electro-optical systems, which inevitably experience the effects of scintillation when imaging through distributed-volume turbulence.

In adaptive optics (AO), the tracker and wavefront sensor commonly measure irradiance centroids (in their respective focal planes) to estimate the turbulence-degraded wavefront in the pupil plane. Several factors affect the accuracy of these centroid measurements, including noise, speckle, and scintillation. The centroid-tilt or “C-tilt” errors due to these factors have been studied by numerous researchers; however, to our knowledge, closed-form expressions for the C-tilt error due to scintillation have not been found.In this paper, we derive such expressions, assuming spherical-wave illumination of the pupil, a path-invariant index of refraction structure constant Cn2, and Kolmogorov turbulence. We compare the analytically predicted C-tilt values to those obtained by wave-optics simulations. The agreement is quite good over a wide range of conditions. Researchers and engineers will find this analysis useful when quantifying the performance of centroid-based trackers and wavefront sensors, both of which are critical AO components.

We design, build, and validate an optical system for generating light beams with complex spatial coherence properties in real time. Beams of this type self-focus and are resistant to turbulence degradation, making them potentially useful in applications such as optical communications. We begin with a general theoretical analysis of our proposed design. Our approach starts by generating a Schell-model (uniformly correlated or shift-invariant) source by spatially filtering incoherent light. We then pass this light through an optical coordinate transformer, which converts the Schell-model source into a nonuniformly correlated field. After the general analysis, we discuss system engineering, including trade-offs among system parameters and expected performance. Finally, we test and validate the system by comparing experimental results to theoretical predictions. We conclude with a brief summary and a discussion of future work.

We derive a simple, physical, closed-form expression for the optical-path difference (OPD) of a two-wavelength adaptive-optics (AO) system. Starting from Hogge and Butts’ classic OPD variance integral expression [J. Opt. Soc. Am. 72, 606 (1982)], we apply Mellin transform techniques to obtain series and asymptotic solutions to the integral. For realistic two-wavelength AO systems, the former converges slowly and has limited utility. The latter, on the other hand, is a simple formula in terms of the separation between the AO sensing (i.e., the beacon) and compensation (or observation) wavelengths. We validate this formula by comparing it to the OPD variances obtained from the aforementioned series and direct numerical evaluation of Hogge and Butts’ integral. Our simple asymptotic expression is shown to be in excellent agreement with these exact solutions. The work presented in this paper will be useful in the design and characterization of two-wavelength AO systems.

We derive the cross-spectral density (CSD) function for a twisted vortex partially coherent beam at the output of a general ABCD system in terms of multidimensional Hermite polynomials (MDHPs). MDHPs offer notational and computational advantages over prior CSD function representations that use common (one-dimensional) Hermite polynomials. We explain how to compute MDHPs using the recurrence relation given in the literature and include MATLAB code to generate MDHPs of any order. Lastly, we validate our work experimentally by comparing the measured spectral density of a twisted vortex beam at the output of an asymmetric optical system to predictions from our theoretical CSD function.

In this paper, we use wave-optics simulations to explore the limitations of beam-control compensation. We evaluate performance in terms of the normalized power in a diffraction-limited bucket for the cases of no beam-control compensation, perfect phase compensation, and perfect full-field compensation. From these results we are able to arrive at the following conclusions: (1) without any form of beam-control compensation, performance begins to degrade when D/r0>1; (2) with perfect phase compensation, performance begins to degrade when D/r0>1 and (λ/r0)/θ0>1; and (3) with perfect full-field compensation, performance begins to degrade when D/r0>1 and (λ/D)/θ0>1. Here, D is the aperture diameter, r0 is the Fried parameter, λ is the wavelength, and θ0 is the isoplanatic angle. We show (1)--(3) to be true for varying aperture sizes, uniformly distributed turbulence, and varying turbulence profiles. These findings will inform the development of future laser systems that need to sense and correct for the effects of atmospheric turbulence.

Two-wavelength adaptive optics (AO), where sensing and correcting (from a beacon) is performed at one wavelength $\lambda_\text{B}$ and compensation and observation (after transmission through the atmosphere) is performed at another $\lambda_\text{T}$, has historically been analyzed and practiced assuming negligible irradiance fluctuations (i.e., weak scintillation). Under these conditions, the phase corrections measured at $\lambda_\text{B}$ are robust over a relatively large range of wavelengths, resulting in a negligible decrease in AO performance. In weak-to-moderate scintillation conditions, which result from distributed-volume atmospheric aberrations, the pupil-phase function becomes discontinuous, producing what Fried called the ``hidden phase'' because it is not sensed by traditional least-squares phase reconstructors or unwrappers. Neglecting the hidden phase has a significant negative impact on AO performance even with perfect least-squares phase compensation. To the authors' knowledge, the hidden phase has not been studied in the context of two-wavelength AO. In particular, how does the hidden phase sensed at $\lambda_\text{B}$ relate to the compensation (or observation) wavelength $\lambda_\text{T}$? If the hidden phase is highly correlated across $\lambda_\text{B}$ and $\lambda_\text{T}$, like the least-squares phase, it is worth sensing and correcting; otherwise, it is not. Through a series of wave optics simulations, we find an approximate expression for the hidden-phase correlation coefficient as a function of $\lambda_\text{B}$, $\lambda_\text{T}$, and the scintillation strength. In contrast to the least-squares phase, we determine that the hidden phase (when present) is correlated over a small band of wavelengths centered on $\lambda_{\text{T}}$. Over the range $\lambda_\text{B},\lambda_\text{T} \in \left[1,3\right] \text{ } \mu\text{m}$ and in weak-to-moderate scintillation conditions (spherical-wave log-amplitude variance $\sigma_\chi^2 \in \left[0.1,0.5\right]$), we find the average hidden-phase correlation linewidth to be approximately $\text{0.35} \text{ } \mu\text{m}$. Consequently, for $\left|\lambda_\text{B}-\lambda_\text{T}\right|$ greater than this linewidth, including the hidden phase does not significantly improve AO performance over least-squares phase compensation.

We derive a closed-form expression for the mutual coherence function (MCF) of a twisted spatiotemporal optical vortex (STOV) beam after propagating a distance z in a linear dispersive medium. A twisted STOV beam is a partially coherent optical field that possesses a coherent STOV and a stochastic twist coupling its space and time dimensions. These beams belong to a special class of space–time-coupled light fields that carry transverse (to the direction of propagation) orbital angular momentum, making them potentially useful in numerous applications including quantum optics, optical manipulation, and optical communications. After presenting the derivation, we validate our new general MCF by showing that it simplifies to the MCFs for two example beams from the literature. Lastly, we present and analyze space–time beam profiles and complex arguments (phases) of the MCF at multiple z and differing amounts of material dispersion, field correlation or coherence, and beam twist to gain insight into how twisted STOV beams evolve in dispersive media. We conclude with a brief summary.

To study the effects of atmospheric turbulence on the quality of entanglement of photons propagating from a ground station to a satellite and to verify if coincidence measurements of entangled photons can be used to back out atmospheric turbulence parameters for long-distance propagation or moderate-to-strong turbulence , we built an atmospheric turbulence simulator (ATS) in a laboratory setting. The ATS comprised of two afocal systems with a Lexitek phase wheel and a Meadowlark spatial light modulator representing discrete layers of atmospheric turbulence. The ATS could represent propagation distances on the order of 1km and could theoretically simulate Rytov variances as high as 5.16 and Fried parameters as low as 0.5cm for a 1m telescope. The design parameters, numerical simulations, and experimental setup is detailed in this proceeding.

In this appendix, we present a method to simulate spatially incoherent or “near” incoherent sources. Much like simulating point sources in coherent wave-optics simulations, the method presented here reduces the source’s spatial bandwidth via filtering, such that the field emitted by the filtered (or spatially bandlimited) source, after propagating a distance z, has the same cross-spectral density (CSD) function over a specified region of interest as the field emitted by the incoherent source. In the analysis below, we assume frequency ω domain fields and omit the ω dependence of quantities, such as the CSD function, for brevity. Furthermore, we stipulate that all random fields are wide-sense stationary (WSS). We begin by assuming that our source is scalar and quasi-homogeneous; we specialize the result to a spatially incoherent source in the end.

We analyze three non-stationary partially coherent sources whose coherent modes are spatiotemporal optical vortex (STOV) beams. Using spatiotemporal (ST) Bessel–Gauss and Laguerre–Gauss beams (STOV-carrying solutions to the space-time paraxial wave equation) as eigenfunctions in the coherent-modes representation of the mutual coherence function, we derive the ST versions of J 0 -Bessel-correlated, I n -Bessel-correlated, and twisted Gaussian Schell-model beams. We model, in simulation, these ST random beams via their coherent-modes expansions, compare and contrast the simulated results to theory, and analyze/discuss their free-space propagation characteristics. The work presented in this paper will be useful for simulating or physically generating these ST beams for use in applications or future studies.