Tracking bandwidth limitations for strong optical-turbulence conditions.

We derive a modified fundamental tracking frequency that is applicable for beam-control systems that do not employ adaptive-optics compensation. Specifically, we show that there are diminishing returns on tracking faster than the modified fundamental tracking frequency. Furthermore, when D/r0 > 4, where D is the aperture diameter and r0 is the Fried parameter, we show that increasing the track bandwidth alone will not improve system performance. These conclusions result from beam spreading being the dominant driver of decreased system performance, as opposed to beam jitter.

Speckle-free coherent imaging through deep turbulence.

We develop and validate a model-based iterative reconstruction framework for digitally correcting coherent images corrupted by deep turbulence. In general, this framework is applicable to coherent-imaging approaches that gain access to the complex-optical field; however, we demonstrate our approach with multi-shot digital holography data. To test our image correction framework, we generate calibrated deep-turbulence conditions from our laboratory testbed. Using the resulting data, we demonstrate groundbreaking performance in terms of speckle-free image correction in deep-turbulence conditions.

Scaled-laboratory demonstrations of deep-turbulence conditions.

This paper uses five spatially distributed reflective liquid-crystal phase modulators (LcPMs) to accurately simulate deep-turbulence conditions in a scaled-laboratory environment. In practice, we match the Fresnel numbers for long-range, horizontal-path scenarios using optical trombones and relays placed between the reflective LcPMs. Similar to computational wave-optic simulations, we also command repeatable high-resolution phase screens to the reflective LcPMs with the proper path-integrated spatial and temporal Kolmogorov statistics.

"Hidden phase" in two-wavelength adaptive optics.

Two-wavelength adaptive optics (AO), where sensing and correcting (from a beacon) are performed at one wavelength λ B and compensation and observation (after transmission through the atmosphere) are performed at another λ T , has historically been analyzed and practiced assuming negligible irradiance fluctuations (i.e., weak scintillation). Under these conditions, the phase corrections measured at λ 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 λ B relate to the compensation (or observation) wavelength λ T ? If the hidden phase is highly correlated across λ B and λ 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 λ B , λ 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 λ T . Over the range λ B ,λ T ∈[1,3]µm and in weak-to-moderate scintillation conditions (spherical-wave log-amplitude variance σ χ2∈[0.1,0.5]), we find the average hidden-phase correlation linewidth to be approximately 0.35 µm. Consequently, for |λ B -λ T | greater than this linewidth, including the hidden phase does not significantly improve AO performance over least-squares phase compensation.

Active-illumination extension to the Priest and Meier pBRDF.

This paper develops a 3D vector solution for the scattering of partially coherent laser-beam illumination from statistically rough surfaces. Such a solution enables a rigorous comparison to the well-known Priest and Meier polarimetric bidirectional reflectance distribution function (pBRDF) [Opt. Eng.41(5), 988 (2002)10.1117/1.1467360]. Overall, the comparison shows excellent agreement for the normalized spectral density and the degree of polarization. Based on this agreement, the 3D vector solution also enables an extension to the Priest and Meier pBRDF that accounts for the effects of active illumination. In particular, the 3D vector solution enables the development of a closed-form expression for the spectral degree of coherence. This expression provides a gauge for the average speckle size based on the spatial-coherence properties of the laser source. Such an extension is of broad interest to long-range applications that deal with speckle phenomena.

System-level noise performance of coherent imaging systems.

We provide an in-depth analysis of noise considerations in coherent imaging, accounting for speckle and scintillation in addition to "conventional" image noise. Specifically, we formulate closed-form expressions for total effective noise in the presence of speckle only, scintillation only, and speckle combined with scintillation. We find analytically that photon shot noise is uncorrelated with both speckle and weak-to-moderate scintillation, despite their shared dependence on the mean signal. Furthermore, unmitigated speckle and scintillation noise tends to dominate coherent-imaging performance due to a squared mean-signal dependence. Strong coupling occurs between speckle and scintillation when both are present, and we characterize this behavior by fitting a scale factor capable of generating variances in closed form. We verify each of these claims through a series of wave-optics simulations, and we see strong agreement in general between numerical results and theoretical predictions. Our findings allow us to confidently gauge signal-to-noise ratio (SNR) expectations when active illumination produces coherent noise.

Active-tracking scaling laws using the noise-equivalent angle due to speckle.

It is well known to system engineers that speckle imposes a limitation on active-tracking performance, but scaling laws that quantify this limitation do not currently exist in the peer-reviewed literature. Additionally, existing models lack validation through either simulation or experimentation. With these points in mind, this paper formulates closed-form expressions that accurately predict the noise-equivalent angle due to speckle. The analysis separately treats both well-resolved and unresolved cases for circular and square apertures. When compared with the numerical results from wave-optics simulations, the analytical results show excellent agreement to a track-error limitation of (1/3)λ/D, where λ/D is the aperture diffraction angle. As a result, this paper creates validated scaling laws for system engineers that need to account for active-tracking performance.

Measuring laser-speckle statistics in scaled-laboratory experiments.

Generating fully developed speckle in a repeatable way is of interest to ongoing scaled-laboratory experiments. Such experiments often look to validate theoretical and numerical predictions for numerous laser-based applications. Unfortunately, experimental constraints such as camera-pixel sampling, residual-sensor noise, and cover-glass etaloning limit one's ability to match the statistics of fully formed speckle in a straightforward way. In this paper, we develop expressions for the speckle probability density function (PDF) and speckle contrast, which account for the effects of camera-pixel sampling (relative to the size of the speckles), as well as Gaussian-distributed additive noise. We validate these expressions using wave-optics simulations, which also account for the separate effects of cover-glass etaloning. Next, we set up an experiment that limits the effects of the cover-glass etaloning (as much as possible). The results show excellent agreement with the expressions we develop for the speckle PDF and speckle contrast. This agreement will enable future scaled-laboratory experiments to match the statistics of fully developed speckle in a straightforward way.

Applications of Lasers for Sensing and Free Space Communications: introduction to the feature issue.

This feature issue highlights the latest developments in laser-based sensing and free space communications. In total, 15 papers were published in Applied Optics, including an invited review paper that celebrates the legacy of David L. Fried.

Deep-turbulence phase compensation using tiled arrays.

Tiled arrays use modulo-2π phase compensation and coherent beam combination to correct for the effects of deep turbulence. As such, this paper uses wave-optics simulations to compare the closed-loop performance of tiled arrays to a branch-point-tolerant phase reconstructor known as LSPV+7 [Appl. Opt.53, 3821 (2014)10.1364/AO.53.003821]. The wave-optics simulations make use of a point-source beacon and are setup with weak-to-strong scintillation conditions. This setup enables a trade-space exploration in support of a power-in-the-bucket comparison with LSPV+7. In turn, the results show that tiled arrays outperform LSPV+7 when transitioning from weak-to-strong scintillation conditions. These results are both encouraging and informative for those looking to tackle the branch-point problem in adaptive optics.

Pulse-quality metric for nonstationary partially coherent fields.

This paper generalizes a pulse-quality metric referred to as P2, i.e., the time analogue of Siegman's beam quality factor M2, to include pulsed (nonstationary) random fields of any state of coherence. The analysis begins with the derivation of a general P2 relation, which we then specialize to the important cases of coherent and Schell-model pulsed beams. As examples, we derive the P2 for two stochastic sources: (1) a cosine Gaussian-correlated Schell-model pulsed beam and (2) a nonuniformly correlated pulsed beam. For both of these sources, we generate (in simulation) random instances of each and compare the simulated (Monte Carlo) P2, i.e., computed directly from its definition, to the theoretical quantity. The agreement is excellent, thereby validating our P2 analysis.

Pulsed laser source digital holography efficiency measurements.

In this paper, a 1064 nm pulsed laser source and a short-wave IR (SWIR) camera are used to measure the total system efficiency associated with a digital holography system in the off-axis image plane recording geometry. At a zero path-length difference between the signal and reference pulses, the measured total system efficiency (15.9%) is consistent with that previously obtained with a 532 nm continuous-wave laser source and a visible camera [Appl. Opt.58, G19 (2019)APOPAI0003-693510.1364/AO.58.000G19]. In addition, as a function of the temporal delay between the signal and reference pulses, the total system efficiency is accurately characterized by a component efficiency, which is formulated from the ambiguity function. Even with multimode behavior from the pulsed laser source and substantial dark current noise from the SWIR camera, the system performance is accurately characterized by the resulting ambiguity efficiency.

Estimation of atmospheric optical turbulence strength in realistic airborne environments.

In this paper, atmospheric optical turbulence strength is estimated for realistic airborne environments using a modified phase-variance approach, as well as a modified slope-discrepancy approach. Realistic airborne environments are generated using wave-optics simulations of a plane wave propagating through increasing strengths of homogeneous atmospheric optical turbulence, both with and without aero-optical contamination (from in-flight wavefront sensor data) and additive-measurement noise. In comparison to the modified phase-variance approach, the results show that the modified slope-discrepancy approach more accurately estimates atmospheric optical turbulence strength over a wide range of conditions. Such results are encouraging for realistic airborne environments because they can be scaled to different freestream conditions as long as the boundary layer is considered canonical.

Achieving the shot-noise limit using experimental multi-shot digital holography data.

In this paper, we achieve the shot-noise limit using straightforward image-post-processing techniques with experimental multi-shot digital holography data (i.e., off-axis data composed of multiple noise and speckle realizations). First, we quantify the effects of frame subtraction (of the mean reference-only frame and the mean signal-only frame from the digital-hologram frames), which boosts the signal-to-noise ratio (SNR) of the baseline dataset with a gain of 2.4 dB. Next, we quantify the effects of frame averaging, both with and without the frame subtraction. We show that even though the frame averaging boosts the SNR by itself, the frame subtraction and the stability of the digital-hologram fringes are necessary to achieve the shot-noise limit. Overall, we boost the SNR of the baseline dataset with a gain of 8.1 dB, which is the gain needed to achieve the shot-noise limit.

Demonstration of a general scaling law for far-field propagation.

This paper conducts experiments that demonstrate the utility of a general scaling law (GSL) for far-field propagation. In practice, the GSL accurately predicts the diffraction-limited peak irradiance in a far-field plane, regardless of the beam shape in a near-field plane. Within the experimental setup, we use a reflective, phase-only spatial light modulator to generate various beam shapes from expanded and collimated laser-source illumination, including both flattop and Gaussian beams with obscurations, in addition to phased arrays with these beam shapes. We then focus the resulting near-field source plane to a far-field target plane and measure the peak target irradiance to compare to the associated GSL prediction. Overall, the results show excellent agreement with less than 1% error for all test cases. Such experiments present a convenient and relatively inexpensive approach to demonstrating laser-system architectures (of varying complexity) that involve far-field propagation.

Subaperture sampling for digital-holography applications involving atmospheric turbulence.

Using wave-optics simulations, this paper defines what subaperture sampling effectively means for digital-holography applications involving atmospheric turbulence. Throughout, we consider the on-axis phase shifting recording geometry (PSRG) and off-axis PSRG, both with the effects of sensor noise. The results ultimately show that (1) insufficient subaperture sampling manifests as an efficiency loss that limits the achievable signal-to-noise ratio and field-estimated Strehl ratio; (2) digital-holography applications involving atmospheric turbulence require at least three focal-plane array (FPA) pixels per Fried coherence length to meet the Maréchal criterion; and (3) off-axis PSRG is a valid and efficient implementation with minor losses, as compared to on-axis PSRG. Such results will inform future research efforts on how to efficiently use the available FPA pixels.

3D multi-plane sharpness metric maximization with variable corrective phase screens.

Sharpness metric maximization is a method for reconstructing coherent images that have been aberrated due to distributed-volume turbulence. This method places one or more corrective phase screens in the digital-propagation path that serve to increase overall sharpness of the image. As such, this study uses sharpness metric maximization on 3D irradiances obtained via frequency-diverse digital holography. We vary the number of corrective phase screens in the propagation path and sharpen images of a realistic, extended object via multi-plane sharpness metric maximization. The results indicate that image reconstruction is possible when using fewer corrective screens than aberrating screens, but that image quality increases with a greater number of corrective screens.

Wave-optics simulation of dynamic speckle: I. In a pupil plane.

This two-part paper demonstrates the use of wave-optics simulations to model the effects of dynamic speckle. In Part I, we formulate closed-form expressions for the analytical irradiance correlation coefficient, specifically in the pupil plane of an optical system. These expressions are for square, circular, and Gaussian scattering spots and four different modes of extended-object motion, including in-plane and out-of-plane translation and rotation. Using a phase-screen approach, we then simulate the equivalent scattering from an optically rough extended object, where we assume that the surface heights are uniformly distributed and delta correlated from grid point to grid point. For comparison to the analytical irradiance correlation coefficient, we also calculate the numerical irradiance correlation coefficient from the dynamic speckle after propagation from the simulated object plane to the simulated pupil plane. Overall, the analytical and numerical results definitely demonstrate that, relative to theory, the dynamic speckle in the simulated pupil plane is properly correlated from one frame to the next. Such validated wave-optics simulations provide the framework needed to model more sophisticated setups and obtain accurate results for system-level studies.

Wave-optics simulation of dynamic speckle: II. In an image plane.

This two-part paper demonstrates the use of wave-optics simulations to model the effects of dynamic speckle. In Part II, we formulate closed-form expressions for the analytical irradiance correlation coefficient, specifically in the image plane of an optical system. These expressions are for square, circular, and Gaussian limiting apertures and four different modes of extended-object motion, including in-plane and out-of-plane translation and rotation. Using a phase-screen approach, we then simulate the equivalent scattering from an optically rough extended object, where we assume that the surface heights are uniformly distributed and delta correlated from grid point to grid point. For comparison to the analytical irradiance correlation coefficient, we also calculate the numerical irradiance correlation coefficient from the dynamic speckle after propagation from the simulated object plane to the simulated image plane. Overall, the analytical and numerical results definitely demonstrate that, relative to theory, the dynamic speckle in the simulated image plane is properly correlated from one frame to the next. Such validated wave-optics simulations provide the framework needed to model more sophisticated setups and obtain accurate results for system-level studies.

Compensated-beacon adaptive optics using least-squares phase reconstruction.

In this paper, we quantify the benefits of compensated-beacon adaptive optics (CBAO) relative to uncompensated-beacon adaptive optics (UBAO) using wave-optics simulations. Throughout, we present results for both the Shack-Hartmann wavefront sensor (SH-WFS) and the digital-holographic wavefront sensor (DH-WFS). Given weak to moderately strong scintillation conditions, the results show that the two noiseless sensors offer similar performance in terms of the peak Strehl ratio when using similar subaperture sampling and least-squares phase reconstruction. Specifically, CBAO leads to an average performance boost of 17% for the SH-WFS and 26% for the DH-WFS relative to UBAO for the turbulence scenarios studied here.