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Optical turbulence, as determined by the widely accepted practice of profiling the temperature structure constant, CT2, via the measurement of ambient atmospheric temperature gradients, can be found to differ quite significantly when characterizing such gradients via thermal-couple differential temperature sensors as compared to doing so with acoustic probes such as those commonly used in sonic anemometry. Similar inconsistencies are observed when comparing optical turbulence strength derived via CT2 as compared to those through direct optical or imaging measurements of small fluctuations of the index of refraction of air (i.e., scintillation). These irregularities are especially apparent in stable atmospheric layers and during diurnal quiescent periods. Our research demonstrates that when care is taken to properly remove large-scale index of refraction gradients, the sonic anemometer-derived velocity structure constant, Cv2, coupled with the similarly derived turbulence-driven index of refraction and vertical wind shear gradients, provides a refractive index structure constant, Cn2, that can more closely match the optical turbulence strengths inferred by more direct means such as scintillometers or differential image motion techniques. The research also illustrates the utility and robustness of quantifying Cn2 from CT2 at a point using a single sonic anemometer and establishes a clear set of equations to calculate volumetric Cn2 data using instrumentation that measures wind velocities with more spatial/temporal fidelity than temperature.

Sonic anemometers are used to study the outer scale in near ground level turbulence. Turbulence is expected to obey a Kolmogorov power spectrum within some inertial range, where the temperature or index of refraction fluctuations decrease as the inverse 11/3rds power of the spatial wavenumber. Below this inertial range (that is for sufficiently small spatial wavenumbers, or equivalently sufficiently large scale sizes) the form of the power spectrum isn’t predicted by theory, but it is expected to roll off. A levelling off of the power spectrum at low spatial frequencies corresponds to a levelling off of the structure function at large spatial separations, and this is the signal sought in the data. Near the ground there is some evidence the outer scale size may be as small as the height above ground. Sonic anemometer data was collected in the summer of 2019 in conjunction with optical turbulence experiments. These experiments showed good agreement between different ways of monitoring turbulence. In these experiments, the sonic anemometers were mostly mounted 2.64 meters above the ground. In this work, the anemometer data is being revisited to study the outer scale. Outer scale effects are quite subtle with optical techniques, which are arranged to be most sensitive to variations in index of refraction within the inertial range precisely in order to avoid inner and outer scale effects. Sonic anemometry usually achieves this by including only nearest neighbor measurements in turbulence estimation, but here we examine the variance of temperature differences across a wide range of baselines in order to study the structure function itself.

Understanding how atmospheric turbulence is distributed along a path helps in effective turbulence compensation and mitigation. Phase-based techniques to measure turbulence have potential advantages when used over long ranges since they do not suffer from saturation issues as the irradiance-based techniques. In an earlier work, we had demonstrated a method to extract turbulence information along a path using the time-lapse imagery of a LED array from a pair of spatially separated cameras. By measuring the differential motion of pairs of LEDs of varying separations, sensed by a single camera or between cameras, turbulence profiles could be obtained. However, by using just a pair of cameras, the entire path could not be profiled. By using multiple spatially separated cameras, improvements can be made on the profiling resolution as well as the fraction of the path over which profiling is possible. This idea has been demonstrated in the present work by using a camera bank comprising of 5 identical cameras, looking at an arrangement of 10 nonuniformly spaced LEDs over a slant path. The differential tilt variances measured at a single camera and between all pairs of cameras have been used to obtain turbulence information. Profiling thus with elevated targets will ultimately help in a better understanding of how turbulence varies with altitude in the surface layer.

Knowledge of turbulence distribution along an experimental path can help in effective turbulence compensation and mitigation. Although scintillometers are traditionally used to measure the strength of turbulence, they provide a path-integrated measurement and have limited operational ranges. A technique to profile turbulence using time-lapse imagery of a distant target from spatially separated cameras is presented here. The method uses the turbulence induced differential motion between pairs of point features on a target, sensed at a single camera and between cameras to extract turbulence distribution along the path. The method is successfully demonstrated on a 511 m almost horizontal path going over half concrete and half grass. An array of Light-Emitting Diodes (LEDs) of non-uniform separation is imaged by a pair of cameras, and the extracted turbulence profiles are validated against measurements from 3D sonic anemometers placed along the path. A short-range experiment with a heat source to create local turbulence spike gives good results as well. Because the method is phase-based, it does not suffer from saturation issues and can potentially be applied over long ranges. Although in the present work, a cooperative target has been used, the technique can be used with non-cooperative targets. Application of the technique to images collected over slant paths with elevated targets can aid in understanding the altitude dependence of turbulence in the surface layer.

A dual-beacon Hartmann Turbulence Sensor (HTS) was used to estimate atmospheric turbulence profiles along a 1.8 km long slant path. This HTS uses a 16“ telescope and has 700 active subapertures. Its camera can take frames as fast as 8 kHz. The laser beacons were placed near the top of a 12 story high tower while the HTS telescope was located in a field 1848 meters away. While the field where the HTS was located was mostly flat for the first few hundred meters in the direction of the tower the ground then sloped up gradually gaining 60 meters by the time the path reached the tower. The terrain beneath the beam path was quite variegated with grassy fields, roads, forested areas and buildings giving the potential for changing turbulence along the path. The experiment was also conducted at different times of day and night and in different weather conditions to further vary the turbulence encountered. The techniques used to profile turbulence with this dual-beacon HTS will be reviewed and the results from this experimental campaign will be presented. Several other turbulence measurement systems were operating at locations near and along this path providing some other measurements for comparison.

Atmospheric turbulence profiles were estimated for a horizontal path based upon measurements made with a dual beacon Hartmann Turbulence Sensor (HTS) using simulation derived weighting functions. These results are compared to estimates made using a weighting functions computed from theory. These results are further compared to anemometer and scintillometer based turbulence estimates for the same path. The previously published theoretical weighting functions for this situation are based upon some presumptions of geometric optics and thus ignore both diffraction and scintillation effects. All of these weighting functions quantify how turbulence at different distances along the path contributes to the expected value of the differential tilt variances measured by the HTS. In the experiment, the HTS used a 16” Meade telescope with 700 subapertures along a 511 m path roughly 2 meters above the ground. Two HeNe lasers separated by 11 cm served as beacons, each was beam expanded to well overfill the telescope aperture. The same situation was simulated with wave optics. To create simulated weighting functions, a single (usually weak) random turbulence screen was inserted at a single plane perpendicularly to the propagation path. Light from one beacon was then numerically propagated to the telescope aperture where the tilts were computed over each subaperture and saved. This propagation was then carried out for the second beacon. This random phase screen was then inserted at a different propagation plane and this procedure was repeated. When all the desired positions along the beam path had been sampled a new random phase screen was generated and this whole procedure was repeated hundreds of times. The desired weighting functions were then generated by computing the differential tilt variance between the beacons and all pairs of horizontally separated subapertures for each path position. All equivalent subaperture separations within each range bin were then averaged together to produce weighting functions which depend on path position and subaperture separation distance. The weighting functions produced in this fashion showed some differences from the theoretical ones. They were a little weaker far from the telescope, and they showed a somewhat broadened notch where the beacons overlapped compared to the theoretical ones. The effect of these differences on the resulting turbulence profile estimates will be discussed.

Previous turbulence measurements along a near-ground, 500 m, horizontal path using two helium-neon laser beacons and a Hartmann Turbulence Sensor (HTS) yielded profiles of Cn^2 by measuring local aberrated wavefront tilts. The profiles were consistent with Cn^2 values collected along the same path by a BLS900 scintillometer. Further validation of the HTS profiling method is necessary to produce accurate optical turbulence profiles for wavefront correction. To add confidence to the HTS dual-beacon profiling method, four sonic anemometers were added along the path to indirectly measure values of Cn^2. Comparison of the independently measured data sets helps legitimize the HTS turbulence profiling method. Propagation over an equal parts grass and concrete path ensured the turbulence profile is more varied. Cn^2 profiles in this work derived from HTS data captured on 25 and 26 July 2019 agreed strongly with the collocated anemometer and BLS measurements.

Wave optics were used to simulate a dual beacon Hartmann Turbulence Sensor (HTS). The system simulated was used experimentally to measure turbulence profiles. These simulations were intended to help explain differences between the experimental results and theoretical predictions. The theoretically predicted results presume weak turbulence, a Kolmogorov power spectrum for the turbulence, and a geometric optics derived weighting of the turbulence along the path. The simulations carried out used a modified von Karman spectrum, with finite inner and outer scales, so the effects of these scales could be readily studied. A number of interesting results were obtained. The simulations resulted in lower tilt variances in the HTS subapertures than expected, but this had little end effect on the turbulence profiles produced. The effect of the inner and outer scales on this point will be discussed. The profiling technique proved to be powerful enough to sometimes resolve individual phase screens used in simulation. While this result is very interesting, it points to the challenges in simulating a system like this, rather than any difference between theory and experiment. Finally, while the geometric optics presumption is seen as ignoring diffraction, no conclusion on the differences between theory and experiment (or simulation) based upon this point was made. The simulations concentrated on simulating an actual HTS system with a 32 × 32 subaperture array on a 16″ telescope at a 1 km range.

A simple optical system is built to quickly realize partially coherent sources for directed energy applications. The apparatus implements a recently developed technique in which any uniformly correlated or Schell-model source can be generated by summing many statistically independent randomly tilted beams. This result means that only a tip-tilt or fast steering mirror (FSM) is needed to realize Schell-model sources, which in contrast to existing synthesis techniques, is both fast and flexible. A brief review of the theory necessary to understand and implement the tilt-only synthesis technique is presented. The FSM apparatus is then described. Lastly, experimental results of a Schell-model source are compared with theoretical predictions to validate both the synthesis technique and FSM apparatus.

Turbulence distribution along a path is estimated from differential motion of pairs of LEDs on a target board, sensed by a single camera and between two cameras. The method is compared to point measurements from sonic anemometers.

A time-lapse imaging method for turbulence profiling using targets of opportunity is demonstrated. The approach uses differential tilt variances between target features sensed by two cameras. These measurements can help understand altitude dependence of turbulence.

For effective turbulence compensation, especially in highly anisoplanatic scenarios, it is useful to know the turbulence distribution along a path. Irradiance-based techniques suffer from saturation when profiling turbulence over long ranges and hence alternate techniques are currently being explored. In an earlier work, a method to estimate turbulence parameters such as path weighted Cn2 and Fried’s coherence length r0 from turbulence induced random, differential motion of extended features in the time-lapse imagery of a distant target was demonstrated. A technique to measure the distribution of turbulence along an experimental path using the time-lapse imagery of a target from multiple cameras is presented in this work. The approach uses an LED array as target and two cameras separated by a few feet at the other end of the path imaging the LED board. By measuring the variances of the difference in wavefront tilts sensed by a single camera and between the two cameras due to a pair of LEDs with varying separations, turbulence information along the path can be extracted. The mathematical framework is discussed and the technique has been applied on experimental data collected over a 600 m approximately horizontal path over grass. A potentially significant advantage of the method is that it is phase based, and hence can be applied over longer paths. The ultimate goal of this work is to profile turbulence remotely from a single site using targets of opportunity. Imaging elevated targets over slant paths will help in better understanding how turbulence varies with altitude in the surface layer.

This study advances the benefits of previously reported 4D Weather Cubes towards creation of high fidelity cloud free line of sight (CFLOS) beam propagation for realistic assessment of auto-tracked/dynamically routed free space optical communication datalink concepts. 4D Weather Cubes are the product of efficient processing of large, computationally intensive, National Oceanic and Atmospheric Administration (NOAA) gridded numerical weather prediction (NWP) data coupled with embedded physical relationships governing cloud, fog, and precipitation formation to render highly realistic 4D cloud free line of sight analytical environments. The Weather Cubes accrue parameterization of optical effects and custom atmospheric resolution through implementation of the verified and validated Laser Environmental Effects Definition and Reference (LEEDR) atmospheric characterization and radiative transfer code. 4D Weather Cube analyses have recently been expanded to accurately assess Directed Energy weapons and sensor performance (probabilistic climatologies and performance forecasts) at any wavelength/frequency or spectral band in the absence of field test and employment data. The 4D Weather Cubes initialize the High Energy Laser End to End Operational Simulation (HELEEOS) propagation code, which provides a means to dynamically point the communication link. HELEEOS’ calculation of irradiance at the detector as a function of transmission, optical turbulence, and noise sources such as path radiance was the basis for comparative percentile performance binning of FSO communication bit error rates as a function of wide-ranging azimuth/elevation, earth-to-space uplinks. The aggregated, comparative bit error rate binning analyses for different regions, times of day, and seasons using a full year of data provided numerous occasions of clouds, fogs, and precipitation events, thus demonstrating the relevance of 4D Weather Cubes for adroit management of CFLOS opportunities to enhance performance analyses of point-to-point as well as evolving multilayer wireless network concepts.

A Hartmann Turbulence Sensor (HTS) system has been used to study the outer scale of turbulence. The atmospheric turbulence power spectrum is usually presumed to obey the Kolmogorov power law within some inertial range, while at spatial frequencies outside this range, the power spectrum is expected to fall away from this curve. The outer scale is the spatial frequency where the low frequency side of this roll-off occurs. In length units the outer scale is just the inverse of this spatial frequency. In the free atmosphere, this outer scale is presumed to be on the order of a hundred meters, but near the ground, the outer scale is expected to be on the order of the height above ground. The HTS used for this study has an aperture of 16“ and employed a beam path which was around 5’ above the ground, thus the effects of the outer scale are expected to be minimal within the telescope aperture. However by relying on the cross wind to move turbulence across the telescope aperture much longer baselines can be achieved and outer scale effects can be sought. The presumption that the dominant temporal variation in turbulence is wind driven translation is called the Taylor Frozen Flow Hypothesis. When an outer scale is introduced into the Kolmogorov power spectrum the resulting power spectrum is called the von Karman power spectrum. The wave structure functions due to these two power spectra are very different, as the Kolmogorov spectrum leads to a structure function which increases without bound as the separation between points increases, whereas the structure function due to the von Karman spectrum rolls-over near the outer-scale and becomes constant. Unfortunately, the structure function itself isn't measured by the HTS. Instead, the HTS can observe the tilt differences between subapertures separated in space or time. The Taylor Frozen Flow Hypothesis can then be used to switch between time and space. It is clear in the experimental data that this presumption is largely correct for some of the cases studied. Some of the data sets were collected with the fortuitous condition that the wind was approximately perpendicular to the path, with product between the wind speed and frame rate nearly matching the subaperture spacing. The expected differential tilt variance between subaperture pairs rolls over as the subaperture spacing increases and approaches a constant value for both the Kolmogorov and von Karman power spectra, however in the case of the von Karman spectrum this roll-over happens more clearly and the differential tilt variance exhibits a broad weak peak near the outer scale. Also, in the case of the von Karman spectrum the constant value approached is smaller. Comparisons between measured differential tilt variances and those predicted by theory allows some estimates to be made about the size of outer scale.

This study investigates use of a novel data aggregation and interrogation tool, 4D Weather Cubes, and High Performance Computing (HPC)to further enlighten the ongoing debate regarding the potential for terrestrial free space optical (FSO)communications and benefits that might accrue on implementation of hybrid FSO architectures with a millimeter wave backup link. The 4D Weather Cubes were originally developed to accurately assess Directed Energy weapons and sensor performance (at any wavelength/frequency or spectral band)in the absence of field test and employment data. 4D Weather Cubes are the product of efficient processing of large, computationally intensive, National Oceanic and Atmospheric Administration (NOAA)gridded numerical weather prediction (NWP)data coupled to the verified and validated Laser Environmental Effects Definition and Reference (LEEDR)atmospheric characterization and radiative transfer code. The 4D Weather Cubes, inclusive of both conventional meteorological parameters, as well as optical features such as atmospheric transmission and turbulence, initialized the High Energy Laser End to End Operational Simulation (HELEEOS)propagation code. HELEEOS provided an additional tier of aggregation through development of comparative percentile performance binning of FSO communication bit error rates as a function of wide-ranging azimuth/elevation, earth-to-space uplinks. The aggregated, comparative bit error rate binning analyses for different regions, times of day and seasons are relevant to point-to-point as well as evolving multi-layer wireless network concepts.

At the Air Force Institute of Technology, we have developed a technique to estimate atmospheric turbulence parameters from the turbulence-induced random, differential motion of features in the time-lapse imagery of a distant target. The variance of differential motion is a path-weighted integral of the refractive index structure constant, ${\boldsymbol{C_{n}}}^{2}$ . The path weighting functions drop to zero at either ends of the path, their peak locations depending on feature sizes and separations. These weighting functions form a rich set and can be linearly combined to approximate a desired weighting function, such as that of a scintillometer or Fried's coherence diameter, $\boldsymbol{r}_{0}$ . The time-lapse measurements can thus mimic the measurements of any turbulence measuring instrument. Since this is a phase-based technique, it has the potential to estimate turbulence over long paths where irradiance based techniques suffer from saturation issues. The method has been validated earlier against scintillometer measurements over a 7 km path. In the present work, the method is used to obtain a direct estimate of $\boldsymbol{r}_{0}$ from the time-lapse imagery of a LED array. The $\boldsymbol{r}_{0}$ estimates are compared to those obtained from a co-located turbulence profiling instrument.

Atmospheric optical turbulence can be a significant source of image degradation, particularly in long range imaging applications. Many turbulence mitigation algorithms rely on an optical transfer function (OTF) model that includes the Fried parameter. We present anisoplanatic tilt statistics for spherical wave propagation. We transform these into 2D autocorrelation functions that can inform turbulence modeling and mitigation algorithms. Using these, we construct an OTF model that accounts for image registration. We also propose a spectral-ratio Fried parameter estimation algorithm that is robust to camera motion and requires no specialized scene content or sources. We employ the Fried parameter estimation and OTF model for turbulence mitigation. A numerical wave-propagation turbulence simulator is used to generate data to quantitatively validate the proposed methods. Results with real camera data are also presented., 32 pages, 23 figures, Copyright 2021 Optical Society of America. One print or electronic copy may be made for personal use only. Systematic reproduction and distribution, duplication of any material in this paper for a fee or for commercial purposes, or modifications of the content of this paper are prohibited. https://doi.org/10.1364/AO.418458

We advance the benefits of previously reported four-dimensional (4-D) weather cubes toward the creation of high-fidelity cloud-free line-of-sight (CFLOS) beam propagation for realistic assessment of autotracked/dynamically routed free-space optical (FSO) communication datalink concepts. The weather cubes accrue parameterization of optical effects and custom atmospheric resolution through implementation of numerical weather prediction data in the Laser Environmental Effects Definition and Reference atmospheric characterization and radiative transfer code. 4-D weather cube analyses have recently been expanded to accurately assess system performance (probabilistic climatologies and performance forecasts) at any wavelength/frequency or spectral band in the absence of field tests and employment data. The 4-D weather cubes initialize an engineering propagation code, which provides the basis for comparative percentile performance binning of FSO communication bit error rates (BERs) as a function of wide-ranging azimuth/elevation, earth-to-space uplinks. The aggregated, comparative BER binning analyzes for different regions, times of day, and seasons applying a full year of 4-D weather cubes data provided numerous occasions of clouds, fogs, and precipitation events. The analysis demonstrated the utility of 4-D weather cubes for adroit management of CFLOS opportunities to enhance performance analyses of point-to-point as well as evolving multilayer wireless network concepts.

An experiment was conducted to study turbulence along a 149-km path between the Mauna Loa and Haleakala mountain tops using digital cameras and light-emitting diode (LED) beacons. Much of the path is over the ocean, and a large portion of the path is 3 km above sea level. On the Mauna Loa side, six LED beacons were placed in a roughly linear array with pair spacings from 7 to 62 m. From the Haleakala side, a pair of cameras separated by 83.8 cm observed these beacons. Turbulence along the path induces tilts on the wavefronts, which results in displacements of the LED spots in the images. The image motion is caused by unwanted noise sources such as camera platform motion. Differential motion between spots cancels much of this noise, and this differential motion is weighted by the turbulence along the path in different ways depending on the geometry between the sources and the cameras. A camera motion insensitive weighting function is developed to deal with this observational issue. A linear combination of these weighting functions is then used to generate a composite weighting function, which better rejects turbulence near the sources and receivers and is most sensitive to turbulence in the portion of the path out over the ocean. This technique is used to estimate turbulence in this region. The long range involved caused very strong scintillation in the image, which added new challenges to the data processing. A resulting estimate for Cn2 of 4 × 10 − 17 m − 2 / 3 is in good agreement with the Hufnagel–Valley HV5/7 model and the results of numerical weather modeling.

The Hartmann Turbulence Sensor (HTS) is an optical system capable of estimating several atmospheric turbulence parameters, such as Greenwood frequency, Fried’s coherence diameter and inner scale of turbulence. It primarily comprises of a 40 cm Meade telescope, a 32 x 32 Shack- Hartmann lenslet array, and a high-speed camera. The HTS estimates the turbulence parameters by measuring the local tilts of the aberrated wavefront coming from a laser source and incident at the pupil plane of the telescope. At the Air Force Institute of Technology (AFIT), a technique has been developed to measure the distribution of turbulence along an experimental path using the HTS and two laser sources of the same wavelength. By measuring the variances of the difference in wavefront tilts due to the two sources sensed by a pair of Hartmann subapertures with varying separations, turbulence information along the path can be extracted. The method relies on deriving a set of weighting functions, each weighting function dipping to zero at a range where the two sensing paths from the beacons to the subapertures intersect, thus canceling out the effect of turbulence at this location on the differential tilt signal. The analytical expression for the path weighting functions has been derived here. The technique has been applied to experimental data collected over a 500 m grassy path and the profiling results have been compared to a co-located scintillometer. This work will eventually aid in obtaining a better understanding of turbulence in the lower atmosphere and how it varies with height.

A Hartmann Turbulence Sensor (HTS) was used to quantify the atmospheric turbulence along a 1 km near ground-level path. This study examines the effect of removing the average tilt over all subapertures from each subaperture in the data analysis. The HTS captures a laser beam projected along a path of interest with a telescope; a lenslet array in the detector system breaks the beam up into 700 subapertures spread across the telescope pupil, and then forms images of the laser source from each of these subapertures onto a fast camera. Turbulence along the path induces tilts in the laser wavefront which are captured as centroid motion of the many laser spots in the camera image. This motion is used to quantify the turbulence. The raw spot positions contain undesired image motion due to telescope motion and vibration. This motion can be removed from the data by subtracting the average centroid motion of all subapertures from each subaperture. This subtraction changes the data in other ways, and this detail must be included in the analysis. The result can be exactly represented as weighted sum over the differential tilt variances between subapertures pairs. The tilt-removed variance averaged over all the subapertures is shown to be one-half of the average variance over all subaperture pairs. This work also resolves some discrepancies in previous results involving the expected variances of these differential tilts.

In recent times, there has been a growing interest in measuring atmospheric turbulence over long paths. Irradiance based techniques such as scintillometry, suffer from saturation and hence commercial scintillometers have limited operational ranges. In the present work, a method to estimate path weighted C n 2 from turbulence induced random, differential motion of extended features in the time-lapse imagery of a distant target is presented. Since the method is phase based, it can be applied to longer paths. The method has an added advantage of remotely sensing turbulence without the need for deployment of sensors at the target location. The imaging approach uses a derived set of path weighting functions that drop to zero at both ends of the imaging path, the peak location depending on the size of the imaging aperture and the relative sizes and separations of the features whose motions are being tracked. For sub-aperture sized features and separations, the peaks of the weighting functions are closer to the target end of the path. For bigger features and separations, the peaks are closer to the camera end. Using different sized features separated by different amounts, a rich set of weighting functions can be obtained. These weighting functions can be linearly combined to produce a desired weighting function such as that of a scintillometer or that of r 0 . The time-lapse measurements can thus mimic the measurements of a scintillometer or any other instrument. The method is applied to both simulated and experimentally obtained imagery and some validation results with a scintillometer is shown as well.

Measurements with Hartmann Turbulence Sensor revealed anisotropy in optical turbulence. The orientation of this anisotropy was found to be frequently correlated with the direction of the wind. A Hartmann sensor was employed in two data collection campaigns in the summers of 2016 and 2017. In 2016 data was collected on a 200 meter path over a grassy field; in 2017 a 1 km path over an asphalt runway was employed. Similar results were obtained in both cases. The sensor is based on a 16” telescope with a 32 × 32 subaperture array featuring 700 active subapertures and a camera operating as fast as 8000 frames/sec. A 2 mW HeNe operating at 632.8 nm was placed at the other end of the experimental path and was beam expanded to overfill the telescope aperture. The sensor recovered tilts across each subaperture from centroid shifts of the imaged HeNe spots. After global tilts were removed, anisotropy was gauged from the eccentricity and orientation of a bivariate Gaussian fit to this two-dimensional data. The anisotropy observed was modest in strength (

A method is demonstrated for accurately computing optical fields in cases where the Nyquist sampling criterion is violated. The Fast Fourier Transform (FFT) is routinely used to compute propagated optical fields; ordinarily this requires sampling the field at least twice per cycle in the directions perpendicular to the beam. Many physically interesting cases involve local surface tilts and curvatures which need exceedingly fine sampling to satisfy this requirement. This constraint can be relaxed by modifying the propagator to remove light lost from the beam where wavefront tilts are steep. When the range of angles of interest at the observation plane is narrow, this technique coupled with FFT based propagators gives good results. When accurate results are desired across a larger range of angles, the FFT can no longer be used, but the observation plane field can still be computed directly. This latter approach can sometimes still be faster than the FFT based approach with the Nyquist criterion satisfied by adding many more sampling points. This procedure works naturally with the split-step propagator used for atmospheric turbulence simulations and allows more varied surfaces and more extreme turbulence conditions to be handled.

In [Opt. Lett.42, 1512 (2017)OPLEDP0146-959210.1364/OL.42.001512], the authors present a new class of non-uniformly correlated sources with circular coherence. They also describe a basic experimental setup for synthesizing this class of sources, which uses the Van Cittert-Zernike theorem. Here, we present an alternative way to analyze these sources and a different way to generate them.

In a previous paper [Opt. Express22, 31691 (2014)] two different wave optics methodologies (phase screen and complex screen) were introduced to generate electromagnetic Gaussian Schell-model sources. A numerical optimization approach based on theoretical realizability conditions was used to determine the screen parameters. In this work we describe a practical modeling approach for the two methodologies that employs a common numerical recipe for generating correlated Gaussian random sequences and establish exact relationships between the screen simulation parameters and the source parameters. Both methodologies are demonstrated in a wave-optics simulation framework for an example source. The two methodologies are found to have some differing features, for example, the phase screen method is more flexible than the complex screen in terms of the range of combinations of beam parameter values that can be modeled. This work supports numerical wave optics simulations or laboratory experiments involving electromagnetic Gaussian Schell-model sources.

A Hartmann Turbulence Sensor was employed along a 200 meter path over a grassy field to investigate turbulence anisotropy. This sensor was based on a 16″ telescope with a 32 × 32 subaperture array featuring 700 active subapertures and a camera operating as fast as 8000 frames/sec. Data was recorded at different times of the day over the course of a 5-week period in 3–12 hour capture periods. Anisotropy was gauged by both differences between horizontal and vertical tilt variances observed on subapertures and also on variances of tilt differences between subaperture pairs. This latter differential tilt approach is expected to be largely immune to base disturbances. Unsurprisingly, during the day the turbulence appeared to be largely isotropic. During the evening neutral events, (at times near sunset) the turbulence showed a statistically significant anisotropy with tilt variances and differential tilt variances in the vertical direction exceeding those on the horizontal by roughly 5%. The anisotropy observed is consistent with the atmosphere being arranged into a vertically layered structure. At times near sunrise, where limited data was collected, this effect appeared to be weaker. No clear conclusions are drawn for the nighttime case, where while the turbulence was generally lower than that seen during the day, the measures of anisotropy appeared to be quite noisy. The results of this experiment are in general agreement with those of Sanchez et al. (referenced herein), who used a largely identical instrument at Kirtland AFB, NM, but with an environment and collection geometry which were quite different. Most models of surface layer turbulence presume isotropic conditions, this research shows that this presumption isn't always correct and a small correction is all that would be needed to create more realistic models.

Investigating the time-evolution of partially-coherent sources is necessary for certain optical coherence effects. Several simulation approaches have been developed, many of which can only treat cross-spectrally pure sources. However, some significant source types are not cross-spectrally pure. This paper reviews two methods for the synthesis of time-evolving sources which need not be pure. Both involve filtering matrices of uncorrelated Gaussian random numbers. One method requires control of both amplitude and phase, while the other only requires phase control. The utility of the methods for non-cross-spectrally pure sources is demonstrated for the first time. The source is generated by passing coherent light through two different diffusers which move at the same speed but in opposite directions. Simulation results for the time-evolving field are shown. Further, the coherence functions of the synthesized field are compared to theory for validation.

A time-lapse imaging method to estimate turbulence parameters such as path-weighted Cn2 and Fried’s coherence length r0 from the differential motion of features on a distant target is demonstrated here. The time-lapse measurements can even mimic the results of a scintillometer.

We present a numerical wave propagation method for simulating imaging of an extended scene under anisoplanatic conditions. While isoplanatic simulation is relatively common, few tools are specifically designed for simulating the imaging of extended scenes under anisoplanatic conditions. We provide a complete description of the proposed simulation tool, including the wave propagation method used. Our approach computes an array of point spread functions (PSFs) for a two-dimensional grid on the object plane. The PSFs are then used in a spatially varying weighted sum operation, with an ideal image, to produce a simulated image with realistic optical turbulence degradation. The degradation includes spatially varying warping and blurring. To produce the PSF array, we generate a series of extended phase screens. Simulated point sources are numerically propagated from an array of positions on the object plane, through the phase screens, and ultimately to the focal plane of the simulated camera. Note that the optical path for each PSF will be different, and thus, pass through a different portion of the extended phase screens. These different paths give rise to a spatially varying PSF to produce anisoplanatic effects. We use a method for defining the individual phase screen statistics that we have not seen used in previous anisoplanatic simulations. We also present a validation analysis. In particular, we compare simulated outputs with the theoretical anisoplanatic tilt correlation and a derived differential tilt variance statistic. This is in addition to comparing the long- and short-exposure PSFs and isoplanatic angle. We believe this analysis represents the most thorough validation of an anisoplanatic simulation to date. The current work is also unique that we simulate and validate both constant and varying C n 2 ( z ) profiles. Furthermore, we simulate sequences with both temporally independent and temporally correlated turbulence effects. Temporal correlation is introduced by generating even larger extended phase screens and translating this block of screens in front of the propagation area. Our validation analysis shows an excellent match between the simulation statistics and the theoretical predictions. Thus, we think this tool can be used effectively to study optical anisoplanatic turbulence and to aid in the development of image restoration methods.

A Hartmann Turbulence Sensor measured statistically significant, but weak, anisotropy near sunset times between the horizontal and vertical directions. Further analysis shows this anisotropy is not well oriented with respect to these axes.

Partially coherent beams (PCBs) of light can be highly directional, as from lasers, yet resistant to speckle and scintillation, as from LEDs. Generating PCBs with separate optical components for beam shaping and spatial coherence is complicated, and fundamentally limited. The authors control both shape and coherence using liquid-crystal spatial light modulators, and can produce two different classes of PCBs using the same simple optical setup. This work is immediately applicable in such diverse fields as medicine, directed energy, free-space optical communication, and manufacturing.

Accurate characterization of atmospheric turbulence is useful for performance assessment of optical systems operating in real environments and for designing systems to mitigate turbulence effects. Irradiance-based techniques such as scintillometry, suffer from saturation, and hence commercial scintillometers have limited operational ranges. A method to estimate turbulence parameters, such as path weighted Cn2 and Fried’s coherence diameter r0 from turbulence-induced random, differential motion of extended features in the time-lapse imagery of a distant target is presented. Since the method is phase-based, it can be applied to longer paths. It has an added advantage of remotely sensing turbulence without the need for deployment of sensors at the target location. The approach uses a derived set of path weighting functions that drop to zero at both ends of the imaging path, the peak location depending on the size of the imaging aperture, and the relative sizes and separations of the features whose motions are being tracked. Using different sized features separated by different amounts, a rich set of weighting functions can be obtained. These weighting functions can be linearly combined to approximate a desired weighting function such as that of a scintillometer or that of r0 in inhomogeneous turbulence. The time-lapse measurements can thus mimic the measurements of a scintillometer or any other instrument. The method is applied to images captured along two different paths, and the estimates are compared to co-located scintillometer measurements.

The near real-time synthesis of a non-uniformly correlated partially coherent source using a low-actuator-count deformable mirror is demonstrated. The statistical optics theory underpinning the synthesis method is reviewed. The experimental results of a non-uniformly correlated source are presented and compared to theoretical predictions. A discussion on how deformable mirror characteristics such as actuator count and pitch affect source generation is also included.

High-spatial-frequency optical fields or sources are often encountered when simulating directed energy, active imaging, or remote sensing systems and scenarios. These spatially broadband fields are a challenge in wave optics simulations because the sampling required to represent and then propagate these fields without aliasing is often impractical. To address this, two spatial filtering techniques are presented. The first, called Fresnel spatial filtering, finds a spatially band-limited source that, after propagation, produces the exact observation plane field as the broadband source over a user-specified region of interest. The second, called statistical or quasihomogeneous spatial filtering, finds a spatially band-limited source that, after propagation and over a specified region of interest, yields an observation plane field that is statistically representative of that produced by the original broadband source. The pros and cons of both approaches are discussed in detail. A wave optics simulation of light transiting a ground glass diffuser and then propagating to an observation plane in the near-zone is performed to validate the two filtering approaches.

A fast, flexible and efficient method for generating partially coherent sources is presented. It is shown that the Schell-model (uniformly correlated) and non-uniformly correlated sources can be produced quickly using a fast steering mirror and low-actuator-count deformable mirror, respectively. The statistical optics theory underpinning the proposed technique is presented and discussed. Simulation results of two Schell-models and one non-uniformly correlated source are presented and compared to the theory to test the new approach.

Time-evolving simulation of sources with partial spatial and temporal coherence is sometimes instructive or necessary to explain optical coherence effects. Yet, existing time-evolving synthesis techniques often require prohibitive amounts of computer memory. This paper discusses three methods for the synthesis of continuous or pulsed time-evolving sources with nearly arbitrary spatial and temporal coherence. One method greatly reduces computer memory requirements, making this type of synthesis more practical. The utility of all three methods is demonstrated via a modified form of Young's experiment. Numerical simulation and laboratory results for time-averaged irradiance are presented and compared with theory to validate the synthesis techniques.