Open Access
Add to BibSonomyAbstract
In previous work, we presented theory of how atmospheric turbulence can impart orbital angular momentum to propagating optical waves. In this paper we provide the first experimental demonstration of the detection of orbital angular momentum from distributed volume turbulence through the identification of well-defined, turbulence-induced, optical vortex trails in Shack-Hartmann wave front sensor measurements.
© 2013 Optical Society of America
1. Introduction
Twenty years ago it was shown that optical beams carry orbital angular momentum (OAM) [1]. Since then, photonic orbital angular momentum (POAM) has been studied for applications from communication [2–5] to astronomy [6–9].
Our research with POAM began with characterizing atmospheric turbulence via pupil plane branch point measurements using the Atmospheric Simulation and Adaptive-optics Laboratory Testbed (ASALT) at the Air Force Research Laboratory, Directed Energy Directorate’s Starfire Optical Range (SOR). Our initial interest derived from the role of the “hidden phase” (i.e. branch points) in limiting the performance of adaptive optical systems [10]. However, we found that pupil plane branch point measurements provide a wealth of information on the three-dimensional turbulence of the atmosphere. These measurements led to four characteristics of branch point pairs in the pupil plane; density, separation, velocity and persistence [11]. The parameters of density, separation and velocity can be used to characterize the number, velocity, strength and distances of the layers of turbulence in the atmosphere.
The persistence of branch points [12] in wave front sensor (WFS) measurements was key to the determination of the other parameters. However, it was also the first indicator that this phenomena carried a significance greater than the scope of adaptive optics. That branch points are an enduring feature of the propagating wave revealed that they are more than aspects of the measured phase but also served as markers for photons with OAM in the propagating beam. Based on persistence, we showed that the appearance of branch points, in fact, indicates the transformation of a portion of the propagating wave from an m = 0 to an m = ±1 OAM state [13, 14].
At the heart of our work with branch points in atmospheric turbulence is the concept of trails [15]; “frozen” images of the motion of the optical vortices within the WFS data. These optical vortex trails define the presence of POAM within the propagating beam.
Our previous results [11] used either a two-layer, bench-top atmospheric turbulence simulator (ATS) [16] or wave optical simulations done in Waveprop [17]. In either case the turbulence was confined to one of more idealized layers. Here we present the first field results that confirm the formation of POAM in beams propagating through distributed atmospheric turbulence.
We begin with an overview of our branch point algorithms and the identification of vortex trails from WFS data (Section 2). We discuss the instrumentation (Section 3), the procedure for data handling (Section 4) and the measured atmospheric conditions under which our searches were conducted (Section 5). Then in Section 6, we present examples of turbulence-induced OAM through the presentation of well-defined, optical vortex trails from experimental results in distributed volume turbulence. We discuss these results and their implications in Sections 6 and 7 respectively.
2. Background
POAM beams (e.g. Laguerre-Gaussian beams [1]) are characterized by a Poynting vector that spirals about the optical axis as the beam propagates. This is known as an optical vortex, where the amplitude, A, is zero for r < R where R is a function of the order of the OAM state, m. The phase of the vortex beam, θ, has an azimuthal dependence; ∝ eimϕ.
Strong turbulence causes the formation of branch points in the phase function of the propagating wave. Branch points are circulations in the phase about points of zero amplitude. Therefore branch points are the turbulence-induced version of the laboratory created optical vortices like those found in Laguerre-Gaussian beams. Branch points, however, always form in pairs of opposite helicity [18] infinitesimally close together and drift apart as the wavefront propagates [11].
Optical vortices are identified in WFS measurements by their circulation in the phase. However, circulations in the phase can also be caused by noise. So while all branch points (i.e. turbulence-induced, optical vortices) are circulations in the phase, not all circulations in the phase are optical vortices (indications of POAM). To differentiate POAM from noise we rely on the temporal character of the circulations.
2.1. The helicity array
The detection of POAM begins with the identification of circulations in the WFS data. The measurements returned by a Shack-Hartmann WFS are a collection of M frames of images of the system aperture as sampled by an NxN lenslet array. This provides estimates of the x and y gradients (Gx, Gy) by the displacement of the centroid in the image for each sub-aperture. For each 2×2 block of gradients the magnitude of the circulation [19] is calculated by
where the sub-aperture position in the array is indicated by the row and column indices, i ∈ [1,2,...N −1] and j ∈ [1,2,...N −1] for a given frame, f ∈ [1,2,...M].To focus on the optical vortices, we look for where the magnitude of the calculated circulation from Eq. 1 is 2π. With this information we create a new array, H, for each data set, which will represent the locations through time (x,y,t) and their helicity (positive or negative). This helicity array is given by
2.2. Optical Vortex Trails
The persistence of branch points [12] separates turbulence-induced optical vortices from noise. The helicity array contains information on the pupil plane branch point motion [15] that is displayed through what we refer to as the “projection”. The projection of the helicity array is given by
The projections of H capture the position vs. time behavior of the circulations in the WFS measurement with respect to one axis of the lenslet array.An example case depicting the result produced by Eq. 3 from a bench-top experiment is shown in Fig. 1[15]. The turbulence was generated using a two-layer atmospheric turbulence simulator (ATS) [16] and the 200 frames of WFS data were collected with a 256x256 temporal self-referencing interferometer (SRI) [20].
Fig. 1 Table-top laboratory demonstration of turbulence-induced, optical vortex trails using a two-layer atmospheric turbulence simulator
The turbulence layer velocities appear as two sets of lines with different slopes in the x-t projection of the helicity array calculated from the SRI WFS data. Repeated measurements of the same optical vortex in time creates the linear trails as the turbulence moves across the aperture. The presence of two sets of lines demonstrates that branch points persist both in time and through additional turbulence.
3. Instrumentation
The distributed volume turbulence experiments were conducted using the Starfire Optical Range Turbulence Sensor (SORTS) [21]. This instrument is a 16 inch, Meade telescope fitted with a Shack-Hartmann WFS. The WFS consists of a 32x32 lenslet array conjugate to the telescope pupil with a Phantom camera at the image plane of the array.
The SORTS experiments were conducted at two locations. A two-mile near horizontal path between the top of Mount Fugate and a mountainside near SOR and a low altitude (approximately 1.5 meters above the ground), 55 meter path at the Chestnut site roughly 6 miles from SOR. For the two-mile tests a HeNe laser located on the mountainside provided illumination, while at the Chestnut site the source was a red LED.
The work was done as part of a project to characterize the Earth’s boundary layer [22]. The goal was to measure the turbulence conditions for validation of their baseline atmospheric model, testing its predictions as well as some intermediate calculations, which was based on the Hutt model [23]. As a result the paths included additional instrumentation alongside the SORTS system. At both of these sites are weather stations for measuring the horizontal wind speed and direction. The two-mile path has a weather station located near either end of the path while the Chestnut site has a single weather station near the source. We refer to the horizontal axis of the SORTS camera array as the x-axis throughout this paper.
4. Methodology
SORTS was used to collect 2000 frames of data at an open loop frame rate of 8639 Hz over the two mile path or 8000 Hz over the 55 meter path, capturing approximately 0.25 seconds for each data set.
The SORTS phase gradients are processed to extract the circulations in the measurements, as was discussed in Section 2.1. In this way, a helicity array was obtained for each data set collected from the two sites,
For each helicity array, the density, ρ, was calculated. The density is defined to be the mean of the number of circulations per frame divided by the area of the telescope.
In those cases where there is a non-zero density, the helicity array is examined for optical vortex trails as discussed in Section 2.2. As the projections of the helicity array are images of position vs. time, the slopes of the trails are a measure of the velocity of the optical vortex crossing the telescope aperture.
Measurement noise
The helicity array contains the positions in time of all identified circulations within a given data set, some portion of these are due to noise effects. Noise circulations in Shack-Hartmann data can be grouped into two types. “Random noise” circulations are equally likely to appear anywhere in the data. If a circulation is detected at one sub-aperture often, maybe even continuously throughout the data set it is part of the “fixed pattern” noise circulations. Fixed pattern noise appears as one or more horizontal lines in the projection of the helicity array.
Neither of these effects mimic the well defined sloping lines created by an optical vortex traveling through the data set in the projection. Therefore the identification of optical vortex trails is the definitive method of isolating those circulations that are indicators of POAM.
5. Test and test conditions
This paper includes all data along the two mile path (collected in March, April and May of 2012) and data from the beginning of work at the Chestnut site up to October 31, 2012 (July, August, September and October of 2012). As the data collection is ongoing, this choice was driven by the time of the writing of this paper and is otherwise arbitrary. All mention of data therefore refers to these time frames. This includes more than 400 data sets, collected on 11 days, from the two mile path and more than 600 data sets in the first four months of operation at the Chestnut site, over 140 hours spanning 31 days.
The SORTS system measures phase gradients. These gradients are used to estimate the turbulence conditions along the propagation path via post processing routines. Specifically, the system returns estimates of the Fried parameter, r0, the Greenwood frequency, fG, and the inner scale, ℓ0, using assumptions of Kolmogorov turbulence. Here, we present an overview of the data collected and the measured turbulence parameters.
Figure 2 shows the measured test conditions; the coherence length, r0, inner scale, ℓ0 and the vortex density, ρ, versus the time of day. In Figs. 2(a)–2(c), data from the 2-mile path is shown in red while data from the Chestnut site is given in blue. In Figs. 2(d)–2(f) all data is shown again as gray data points with datasets containing optical vortices highlighted green.
Fig. 2 Measured test conditions of all of the SORTS data. In the top row (a) r0, the coherence length (Fried’s parameter), (b) ℓ0, the inner scale, and (c) ρ, the branch point density along the propagation path are shown against the time of day. Red indicates data taken over the 2-mile propagation path at SOR while blue indicates data taken over the 55 m path at the Chestnut site. In the bottom row, the data is shown again for (d) r0, (e) ℓ0 and (f) ρ vs time of day. Those data sets that were identified to have optical vortex trails are shown in green against the rest of the data shown in grey.
Figures 2(a) and 2(b) show r0 and ℓ0 are smallest in the middle of the day when the turbulence is strongest, as is to be expected. These are the conditions where branch point formation is strongest and thus the highest measured densities are found, as shown in Fig. 2(c). However, Fig. 2(c) also shows cases where the branch point density is low during the middle of the day. The formation of branch points depends on both the strength of the turbulence and the propagation distance. Strong turbulence close to the detector will results in a low density measurement. The distances along the path might be calculated from the the branch point density and separation [11] but that is left for a future paper.
6. Analysis
We found that 40% of the data collected by the SORTS instrument shows circulations in the phase.
Separating optical vortices from the noise is done by locating their trails in the helicity array, as was shown in Fig. 1. Approximately 24% of those sets that contain phase circulations demonstrate optical vortex trails, or about 14% of all of the SORTS data collected thus far. In the next set of graphics we show a sampling of data sets from our two SORTS experiments showing optical vortex trails.
6.1. Two mile
Table 1 lists the measured parameters for selected data sets over the 2 mile path with examples of optical vortex trails. The measured parameters are listed according to date and time and the examples were chosen to cover the range of data collected as completely as possible. The x-t projections of the helicity array for these example data sets are shown in Figs. 3 and 4.
Table 1. Initial 2-mile SORTS optical vortex trail examples’ measured turbulence parameters according to the date and time the data was collected.
Fig. 3 Optical vortex trail examples, 1–12, from the 2-mile path at Starfire Optical Range. The measured turbulence parameters for each example are listed on Table 1 along with the time and date. Horizontal lines, as in 10 and 11, are “fixed-pattern” noise.
Fig. 4 Optical vortex trail examples, 13–24, from the 2 mile path at Starfire Optical Range. The measured turbulence parameters for each example are listed on Table 1 along with the time and date the data was collected.
All projections show a 231.5 msec snapshot of the circulations in the phase of the beam following propagation through turbulence over the two mile path (8639 Hz). These represent a wide range of turbulence strengths and branch point densities. All cases show well-defined, optical vortex trails.
Noise is evident in a number of examples. Some examples, Figs. 3–5 and 3–6, demonstrate the noise is strongly localized. In these cases probably along a series of sub-apertures at the outer edge of the aperture causing the apparent band of circulations across the top half of the projection. In others, noisy sub-apertures are revealed by long horizontal lines in the projections like Fig. 3–11, these are part of the ”fixed pattern” and are usually attributed to camera noise.
In some examples, like Figs. 4–15 and 4–22, the branch point density is such that discerning the individual optical vortex trails is becoming difficult. Sufficiently high densities of optical vortices can lead to cases where their trails cannot be seen in the projections of the helicity array. Roughly 21% of the data has densities greater than 390 pts/m2, which, judging by Fig. 3–22, is approaching the limiting density that the projection of the helicity arrays can be used to discern optical vortex trails within SORTS data.
6.2. Chestnut
Next we present examples of optical vortex trails from the 55 meter propagation path at the Chestnut site. Table 2 lists the measured parameters arranged by date and time for the example data sets whose x-t projections are displayed in Figs. 5 and 6.
Table 2. Chestnut SORTS optical vortex trail examples’ measured turbulence parameters.
Fig. 5 Optical vortex trail examples, 25–36, from the experiments conducted at the Chestnut site. The measured turbulence parameters for each example are listed on Table 2 along with the time and date the data was collected.
Fig. 6 Optical vortex trail examples, 37–48, from the experiments conducted at the Chestnut site. The measured turbulence parameters for each example are listed on Table 2 along with the time and date the data was collected.
As in the two mile data, all of the projections from the Chestnut data represent short duration images (0.25 sec) of the circulations in the atmosphere, but this time along a very short, low altitude path. These cases show many of the same traits as the two mile data except high density.
Less than 20% of all of the SORTS data showing optical vortex trails came from the Chestnut site. Some of those cases, Fig. 5–29 and 6–40, have densities so low that the vortex trails barely register. The gaps between detections in these cases and the fact that there is only a single circulation is likely caused by the pair separation being on the order of the sub-aperture size or smaller. The location of a single circulation within a 2×2 block of gradients influences the magnitude of the measured circulation [24]. As the pair moves together through a single sub-aperture they vanish. When they straddle two adjacent sub-apertures they may be detected but depending on their relative locations in the measurement they may provide different magnitudes to the circulation measurement, Eq. 1. Recall that our identification approach, Eq. 2, sets the identification of a circulation at |C| = 2π. In practice we use |C| ≥ 2π, but if the positions of the circulations were such that one of the pair was reduced below 2π in magnitude this approach would fail to identify that half of the vortex pair.
In the Chestnut data, random noise is more pronounced, though it is likely due to the lower density of optical vortices that makes the noise seem more prominent as opposed to any significant difference in the random noise.
The disappearance of optical vortex trails in Fig. 5–27, where the formation of optical vortices ceased around 700 frames into the data only to resume again after 1200 frames while maintaining the slopes of the trails, likely indicates that a high r0 region of air was carried into the path by the same wind that was driving the motion of the optical vortices. The size of the region might be estimated by the length of the time the optical vortices ceased appearing and the velocity of the slopes on either side of that period. It will be interesting to look for similar events in future work with the SORTS instrument.
6.3. High density
In some cases the branch point density becomes high enough that distinguishing the crowded optical vortex trails in the x-t projection becomes difficult. For these cases, we can look at a single x-t plane of the helicity array rather than the cumulative projection to see the vortex trails within the data. For example, on Apr 13 2012 at 13:46, a set of data was collected where the measured density (ρ = 510.85 pts/m2) was such that you can only just make out some of the vortex trails in the x-t projection, see Fig 7–49 (top). In this example the measured turbulence strength parameters were r0 = 1.33 cm and ℓ0 = 0.24 cm. If we compare this projection to a single x-t plane within the helicity array, Fig. 7–49 (bottom), we see that for higher density cases, like those found in some of the 2-mile data, we can still find vortex trails.
Fig. 7 Comparison of (top) the standard helicity x-t projection to (bottom) a single x-t frame of data for higher density data sets.
Now we re-examine some of the high density cases from the two-mile SORTS experiments using only a single x-t plane of the helicity array. The turbulence parameters of the selected high density examples are shown on Table 3 with the single x-t plane of the helicity arrays shown in Figs. 8 and 9.
Table 3. Optical vortex trail examples’ measured turbulence parameters for sets where a single x-t plane is used due to high branch point density.
Fig. 8 Optical vortex trail examples, 50–61, using a single x-t plane of the helicity array due to high branch point densities. The measured turbulence parameters for each example are listed on Table 3 along with the time and date the data was collected.
Fig. 9 Optical vortex trail examples, 62–73, using a single x-t plane of the helicity array due to high branch point densities. The measured turbulence parameters for each example are listed on Table 3 along with the time and date the data was collected.
7. Discussion
Interestingly, because of the short propagation path we did not expect to find optical vortices at the Chestnut site. As we have shown previously [25], the formation of optical vortices through interaction with the atmosphere is dependent on the strength of the turbulence and the propagation distance. Following a turbulence layer there is a minimum propagation distance, z0, that the wave must travel before optical vortices form. However, there were a number of cases at the Chestnut site where the turbulence was sufficiently strong to reduce the formation threshold distance to as little as 10 m.
We found optical vortex trails under turbulence conditions where 0.37 cm ≤ r0 ≤ 4.07 cm and 0.24 cm ≤ ℓ0 ≤ 0.6 cm; with densities ranging from 0.43 pts/m2 ≤ ρ ≤ 2212.19 pts/m2. In most cases optical vortices were located in data sets collected during the afternoon. This is also the time of day when the highest densities were found. However, there is a collection of low density cases that were found in the early part of the day, between dawn and noon, but none in the evening see Fig. 2(c). This is more likely a result of the fact that the collecting of data was more heavily done in the morning and afternoon times and given the low frequency of conditions that form POAM those events were simply missed. We expect to find examples of optical vortices in the later part of the day similar to those found in the morning as data collection continues at the Chestnut site.
Going forward
A new optical system, the Turbulence Layer Sensor (TuLS), has been developed to test optical vortices in profiling distributed volume turbulence over the two mile path at SOR. This instrument will address some of the issues regarding the limited spatial sampling of the SORTS system for identifying optical vortex trails while also including the expanded capability to investigate the role of propagation distance on the formation of optical vortices in distributed turbulence. TuLS is equipped with two high density (48×64) Shack-Hartmann WFSs. The effective aperture of the TuLS system is comparable to that of the SORTS instrument so with its higher density lenslet array it will provide better sampling of the wavefront for investigations of density and separation. Additionally, one WFS follows an optical trombone in the optical path that can be used to add propagation distance. This additional propagation allows the two sensors to measure the created optical vortices at two distances simultaneously, which provides a probe of the functional dependence for density and separation on distance as was done in our previous research [11].
8. Summary
Optical vortex trails forming as a part of wave propagation through distributed volume turbulence, as shown in Section 6, verifies that the random index fluctuations of the atmosphere can indeed lead to the formation of well-defined orbital angular momentum states [13, 14, 26]. The examples shown are from 25 days of testing spread out over eight months along two different paths. Therefore, we can conclude that, the transformation of a portion of propagating beams from an m = 0 to an m = ±1 state is a normal result of the interactions of the electromagnetic field with the distributed turbulent atmosphere.
The correlations between the formation of optical vortices in distributed volume turbulence and the relative strength of the turbulence, as determined through the measured parameters of r0 and ℓ0, see Figs. 2(a)–2(c), corroborates the earlier laboratory and simulation work by the ASALT research group [11]. This provides justification for the development of the turbulence-induced, optical vortex parameters of density, separation and velocity in profiling distributed volume turbulence.
Acknowledgments
We express our gratitude to the Air Force Office of Scientific Research for their support in funding this research.
References and links
1. L. Allen, M. W. Beijersbergen, R. J. C. Spreeuw, and J. P. Woerdman, “Orbital angular momentum of light and the transformation of laguerre-gaussian laser modes,” Phys. Rev. A 45, 8185–8189 (1992) [CrossRef] [PubMed] .
2. G. Gibson, J. Courtial, M. J. Padgett, M. Vasnetso, V. Pas’ko, S. M. Barnett, and S. Franke-Arnold, “Free-space information transfer using light beams carrying orbital angular momentum,” Opt. Express 12, 5448–5456 (2004) [CrossRef] [PubMed] .
3. C. Paterson, “Atmospheric turbulence and orbital angular momentum of single photons for optical communication,” Phys. Rev. Lett. 94, 153901 (2005) [CrossRef] [PubMed] .
4. G. A. Tyler and R. W. Boyd, “Influenece of atmospheric turbulence on the propagation of quantum states of light carrying orbital angular momentum,” Opt. Lett. 34, 142–144 (2009) [CrossRef] [PubMed] .
5. F. Tamburini, E. Mari, A. Sponselli, B. Thidé, A. Bianchini, and F. Romanato, “Encoding many channels on the same frequency through radio vorticity: first experimental test,” New J. Phys. 14, 033001 (2012) [CrossRef] .
6. W. F. Thi, E. F. van Dishoeck, G. A. Blake, G. J. van Zadelhoff, and M. R. Hogerheijde, “Detection of H2 pure rotational line emission from the gg tauri binary system,” Astrophys. J. 521, L63 (1999) [CrossRef] .
7. J. S. Bary, D. A. Weintraub, and J. H. Kastner, “Detection of molecular hydrogen orbiting a ‘naked‘ T Tauri star,” Astrophys. J. 576, L73–L76 (2002) [CrossRef] .
8. N. M. Elias II, “Photon orbital angular momentum in astronomy.” A&A 492, 883–922 (2008) [CrossRef] .
9. N. M. Elias II, “Photon orbital angular momentum and torque metrics for single telescopes and interferometers,” A&A 541, 1–15 (2012) [CrossRef] .
10. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998) [CrossRef] .
11. D. W. Oesch, C. M. Tewksbury-Christle, D. J. Sanchez, and P. R. Kelly, “The aggregate behavior of branch points - characterization in wave optical simulation,” Opt. Eng. 51, 106001 (2012) [CrossRef] .
12. D. W. Oesch, D. J. Sanchez, and C. M. Tewksbury-Christle, “The aggregate behavior of branch points - persistent pairs,” Opt. Express 2, 1046–1059 (2012) [CrossRef] .
13. D. J. Sanchez and D. W. Oesch, “The localization of angular momentum in optical waves propagating through atmospheric turbulence,” Opt. Express 19, 25388–25396 (2011) [CrossRef] .
14. D. J. Sanchez and D. W. Oesch, “Orbital angular momentum in optical waves propagating through distributed atmospheric turbulence,” Opt. Express 19, 24596–24068 (2011) [CrossRef] [PubMed] .
15. D. W. Oesch, D. J. Sanchez, and C. L. Matson, “The aggregate behavior of branch points - measuring the number and velocity of atmospheric turbulence layers,” Opt. Express 18, 22377–22392 (2010) [CrossRef] [PubMed] .
16. S. V. Mantravadi, T. A. Rhoadarmer, and R. S. Glas, “Simple laboratory system for generating well-controlled atmospheric-like turbulence,” in “Advanced Wavefront Control: Methods, Devices, and Applications II,”, M. K. Giles, J. D. Gonglewshi, and R. A. Carerras, eds. (SPIE Press, 2004), Vol. 5553, pp. 290–300.
17. T. Brennan, (2001). WaveProp® is a wave optics propagation code written in the MATLAB® scripting language.
18. D. J. Sanchez, D. W. Oesch, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - the creation and evolution of branch points,” in “Advanced Wavefront Control: Methods, Devices, and Applications VII,”, R. A. Carerras, T. A. Rhoadarmer, and D. C. Dayton, eds. (SPIE Press, 2009), Vol. 7466, pp. 0501–0512 [CrossRef] .
19. M. Chen, F. S. Roux, and J. C. Olivier, “Detection of phase singularities with a shack-hartmann wavefront sensor,” J. Opt. Soc. Am. A 24, 1994–2002 (2007) [CrossRef] .
20. T. A. Rhoadarmer, “Development of a self-referencing interferometer wavefront sensor,” in “Advanced Wavefront Control: Methods, Devices, and Applications II,”, M. K. Giles, J. D. Gonglewshi, and R. A. Carerras, eds. (SPIE Press, 2004), Vol. 5553, pp. 112–126 [CrossRef] .
21. T. J. Brennan and D. C. Mann, “Estimation of optical turbulence characteristics from shack-hartmann wavefront measurements,” Proc. SPIE 7816, 781602 (2010) [CrossRef] .
22. T. C. Farrell, D. J. Sanchez, J. C. Smith, J. Holzman, P. R. Kelly, T. Brennan, A. Gallegos, D. W. Oesch, and D. Kyrazis, “Understading the physics of optical deep turbulence at the earth’s boundary layer,” in “Unconventional Imaging and Wavefront Sensing VIII,”, J. J. Dolne, T. J. Karr, V. L. Gamiz, and D. C. Dayton, eds. (SPIE Press, 2012), Vol. 8520, pp. 85200H.
23. D. L. Hutt, “Modeling and measurements of atmospheric optical turbulence over land,” Opt. Eng. 38, 1288–1295 (1999) [CrossRef] .
24. M. Chen and F. S. Roux, “Dipole influence on shack-hartmann vortex detection in scintillated beams,” J. Opt. Soc. Am. A 25, 1084–1090 (2008) [CrossRef] .
25. D. W. Oesch, D. J. Sanchez, C. M. Tewksbury-Christle, and P. R. Kelly, “The aggregate behavior of branch points - branch point density as a characteristic of an atmospheric turbulence simulator,” in “Advanced Wavefront Control: Methods, Devices, and Applications VII,”, R. A. Carerras, T. A. Rhoadarmer, and D. C. Dayton, eds. (SPIE Press, 2009), Vol. 7466, pp. 0601–0610.
26. D. W. Oesch and D. J. Sanchez, “Creating well-defined orbital angular momentum states with a random turbulent medium,” Opt. Express 20, 12292–12302 (2012) [CrossRef] [PubMed] .
