We use wave-optics simulations to investigate branch-point density (i.e., the number of branch points within the pupil-phase function) in terms of the grid sampling. The goal for these wave-optics simulations is to model plane-wave propagation through homogeneous turbulence, both with and without the effects of a finite inner scale modeled using a Hill spectrum. In practice, the grid sampling provides a gauge for the amount of branch-point resolution within the wave-optics simulations, whereas the Rytov number, Fried coherence diameter, and isoplanatic angle provide parameters to setup and explore the associated deep-turbulence conditions. Via Monte Carlo averaging, the results show that without the effects of a finite inner scale, the branch-point density grows without bound with adequate grid sampling. However, the results also show that as the inner-scale size increases, this unbounded growth (1) significantly decreases as the Rytov number, Fried coherence diameter, and isoplanatic angle increase in strength and (2) saturates with adequate grid sampling. These findings imply that future developments need to include the effects of a finite inner scale to accurately model the multifaceted nature of the branch-point problem in adaptive optics. |
1.IntroductionLaser-beam propagation through deep turbulence results in constructive and destructive interference known as scintillation. The Rytov number (aka log-amplitude variance) gives a gauge for the amount of scintillation. In turn, when the Rytov number increases above 0.1, the scintillation becomes severe and total-destructive interference gives rise to branch points in the pupil-phase function. These so-called branch points manifest where the real and imaginary parts of the complex-optical field equate to zero. Branch points, in practice, add a rotational component to the phase function. This rotational component gets mapped to the null space of traditional-least-squares phase reconstruction algorithms. Due to the foundational work of Fried,1 researchers appropriately refer to this rotational component as the “hidden phase.” With this last point in mind, the existence of branch points leads to unavoidable phase discontinuities known as branch cuts. These branch cuts become linked to positively and negatively charged branch points within the pupil-phase function. Because of interactuator coupling, continuous-face-sheet deformable mirrors are unable to fully compensate for the branch cuts.2–7 Thus, the branch-point problem in adaptive optics tends to be the “Achilles’ heel” to beam-control systems that perform deep-turbulence phase compensation. Fried and Vaughn were the first to study the existence of turbulence-induced branch points.8 Afterward, many works began investigating the effects of branch points (aka phase dislocations,9,10 screw dislocations,11,12 optical vortices,13,14 etc.), which typically require the use of high-fidelity and wave-optics simulations.15–22 For example, Voitsekhovich et al.23 were the first to perform a wave-optics investigation of branch-point density (i.e., the number of branch points within the pupil-phase function). They did so as a function of propagation distance with the effect of a finite inner scale but for a fixed grid sampling. As previously mentioned, Fried was the first to describe the branch-point problem in terms of the hidden phase.1 Since then, researchers have proposed several branch-point-tolerant phase reconstruction algorithms.24–29 These algorithms, at large, have had limited degrees of success due to the multifaceted nature of the branch-point problem in adaptive optics. Most recently, members of the Starfire Optical Range at the Air Force Research Laboratory investigated the aggregate behavior of branch points,30–37 and the goal being to relate the branch-point pairs received in a pupil-phase function to the upstream turbulence that created them. This work, in total, could inform the development of future branch-point-tolerant phase reconstruction algorithms. To increase the fidelity of the results, future developments need to include the effects of additive-sensor noise, low signal-to-noise ratios, and subaperture-sampling requirements.38–45 With this history in mind, reformulating the problem in terms of slope discrepancy,46,47 especially when accounting for the effects of speckle due to rough-surface scattering,48–51 could also inform these future developments. This paper builds on this aforementioned history. In particular, it uses wave-optics simulations to investigate the branch-point density as a function of the grid sampling. The goal for these wave-optics simulations is to model plane-wave propagation through homogeneous turbulence, both with and without the effects of a finite inner scale modeled using a Hill spectrum. In practice, the Rytov number, Fried coherence diameter, and isoplanatic angle help to setup and explore the associated deep-turbulence conditions. These parameters provide a gauge for the amount of scintillation, turbulence-limited resolution, and anisoplanatism, respectively, within the wave-optics simulations. On the other hand, the grid sampling provides a gauge for the amount of branch-point resolution within the wave-optics simulations. Via Monte Carlo averaging, the results show that without the effects of a finite inner scale, the branch-point density grows without bound with adequate grid sampling. Even so, as the inner-scale size increases, the results also show that this unbounded growth (1) significantly decreases as the Rytov number, Fried coherence diameter, and isoplanatic angle increase in strength and (2) saturates with adequate grid sampling. These findings are encouraging from the standpoint that they could readily improve the performance of existing branch-point-tolerant phase reconstruction algorithms, as well as inform the development of future algorithms. It is important to note that this paper also builds on the work contained in a recent conference proceeding.52 The main difference is that this paper includes the effects of a finite inner scale. Both papers show that the branch-point density grows without bound when neglecting the effects of a finite inner scale. This finding speaks to a preconceived notion within the atmospheric-propagation research community that the branch-point density grows without bound with increasing branch-point resolution. Nonetheless, the results of this paper ultimately show that if one includes the effects of a finite inner scale, then this unbounded growth (1) significantly decreases as the associated deep-turbulence conditions become more pronounced and (2) saturates with adequate branch-point resolution within the wave-optics simulations. These findings are novel and worth sharing with the atmospheric-propagation research community. In general, (1) and (2) imply that future developments need to include the effects of a finite inner scale to accurately model the multifaceted nature of the branch-point problem in adaptive optics. In what follows, Sec. 2 provides the background details needed to use the branch-point density as a metric of interest. Section 3 provides the wave-optics simulation details needed to setup and explore the associated deep-turbulence conditions, which assume plane-wave propagation through homogeneous turbulence. Results and discussion naturally follow in Sec. 4 with a conclusion in Sec. 5. 2.BackgroundIn the pupil plane of an optical system, the complex-optical field, , takes the following phasor form: where is the pupil-amplitude function and is the pupil-phase function, such thatHere, and are the real and imaginary components of . Substituting Eq. (1) into Eq. (2) results in a modulo- function that researchers often refer to as the “principle value” or “wrapped phase.” When , the argument of is indeterminate and therefore a multi-valued function.12 This outcome corresponds to the case where and is the reason branch points arise within due to total-destructive interference. Researchers can determine the location of a branch point using a contour integral around the the gradient of the pupil-phase function, .1 Specifically, when the following relationship holds true: is no longer homogeneous and is no longer a purely potential field. In Eq. (3), is the number of positive branch points and is the number of negative branch points within .The sign of the closed-loop contour integration in the clockwise direction determines the overall polarity in Eq. (3). Due to this convention, one can determine the number of branch points, , in an grid, viz, where and . Equation (4) says that if one first breaks the grid into a series of subgrids and sums up the phase difference around each set of four grid points (in the clockwise direction), then a positive value results in a positive branch point and a negative value results in a negative branch point. To determine , one then sums up the total number of positive and negative branch points.These aforementioned sums, as written in Eq. (4), include what Fried refers to as spurious branch points.1 In practice, spurious branch points are a pair of positive and negative branch points centered on immediately adjacent grid points. This outcome is undesirable because the associated branch cut remains unresolved and could be the result of an artifact within the wave-optics simulations. To address these concerns, one can use a simple search to remove the spurious branch points from the sums in Eq. (4). Empirical evidence says that spurious branch points make up a significant portion of the total number of positive and negative branch points— on average. Thus, in this paper, we present all results with spurious branch points removed. With results in mind, the branch-point density, is the metric of interest in this paper. As such where is the circular-pupil diameter. Simply put, is the number of branch points within the pupil-phase function.3.Setup and ExplorationThe wave-optic simulations performed in this paper made use of the WavePlex Toolbox for MATLAB.53 This toolbox uses the split-step beam propagation method (BPM) to simulate the propagation of monochromatic and polychromatic light through the atmosphere.15–22 In practice, the split-step BPM divides the atmosphere into independent volumes, such that a phase screen represents the atmospheric aberrations present in a volume. Angular-spectrum propagation to each phase screen (from the source plane to the pupil plane) then represents the propagation of light through the atmosphere. With the split-step BPM in mind, this section provides the setup and exploration needed to appreciate the results presented in the next section. 3.1.SetupTable 1 contains a summary of all the parameters of interest in the wave-optics simulations. The goal for these wave-optics simulations was to model plane-wave propagation through homogeneous turbulence. For this purpose, the wave-optics simulations made use of grids, where is the grid resolution. The physical side length, , was the same in both the source and pupil planes, allowing for unity scaling within the setup. With that said, the source plane was setup with a plane wave of unit irradiance and the pupil plane was setup with a circular-pupil diameter, . For the minimum grid resolution used within the wave-optics simulations, the setup also satisfied critical sampling,16 such that , where is the minimum grid resolution, is the wavelength, and is the propagation distance. The overall setup resulted in a substantial guard band ratio (GBR), where . Such a GBR helped in combating the effects of aliasing,15,16 which we determined to be visually negligible within the wave-optics simulations for all . The overall setup also resulted in a grid sampling, , where Table 1Parameters of interest in the wave-optics simulations.
As such, provided a gauge for the amount of branch-point resolution within the wave-optics simulations. While Table 1 contains all the parameters of interest in the wave-optics simulations, Table 2 makes use of several path-integral expressions to define the deep-turbulence conditions. These path-integral expressions were the subject of a recent conference proceeding that discusses the limitations of deep turbulence.54 For compactness, Table 2 contains a representative subset of these deep-turbulence conditions, which consisted of Rytov numbers ranging from 0.1 to 10.0 in increments of 0.1. Table 2Representative subset of the deep-turbulence conditions.
Recall that the Rytov number provides a gauge for the amount of scintillation. Given plane-wave propagation,15 the path-integral expression takes the following form: where is the path-dependent refractive index structure coefficient and is the angular wavenumber. With homogeneous turbulence, the path-integral expression reduces to a closed-form expression, whereAlso recall that when the Rytov number increases above 0.1, the scintillation becomes severe and total-destructive interference gives rise to branch points in the pupil-phase function. Thus, when , , and , one is left with weak-to-moderate, moderate-to-strong, and strong scintillation conditions, respectively.54 For completeness in defining the deep-turbulence conditions, Table 2 also includes values for the Fried coherence diameter and the isoplanatic angle. Given plane-wave propagation through homogeneous turbulence,15 the path-integral expressions reduce to closed-form expressions, such that for the Fried coherence diameter, and for the isoplanatic angle. For all intents and purposes, the Fried coherence diameter helps in parameterizing resolution, whereas the isoplanatic angle helps in parameterizing anisoplanatism.55 Thus, when and , one is left with turbulence-limited resolution and anisoplanatic aberrations, respectively.54In what follows, we present an exploration of the deep-turbulence conditions provided in Table 2, both with and without the effects of a finite inner scale. In particular, we use the well-known Kolmogorov spectrum when the wave-optics simulations do not include the effects of a finite inner scale. This spectrum is an idealization with an inner-scale size, , of zero and an outer-scale size, , of infinity. Such a spectrum does not give an accurate representation of the actual energy distribution in a turbulent volume.15 This shortcoming can be remedied by using a modified spectrum such as the Hill spectrum.56 Specifically, this paper makes use of the Hill spectrum because it includes a small rise at high wavenumbers near , and it is readily available within the WavePlex Toolbox for MATLAB. 3.2.ExplorationFigures 1 and 2 show the numerical log-amplitude variance, , as a function of the analytical Rytov number, . In particular, Fig. 1 corresponds to the first half of the deep-turbulence conditions setup in Table 2, whereas Fig. 2 corresponds to all of the deep-turbulence conditions setup in Table 2. Both figures include this scintillation-strength exploration for the minimum grid sampling, , and the maximum grid sampling, , within the wave-optics simulations (cf. Table 1). Both figures also include this scintillation-strength exploration for values of 3.1, 6.2, 12.4, and 24.8 mm for the inner-scale size, . These values represent finite inner scales equal to one, two, four, and eight times , respectively. Due to the limitations of the Rytov approximation,57 the strength of the scintillation is often split into two regimes with respect to . Specifically, the weak-scintillation regime occurs when , and the strong-scintillation regime occurs when . One can clearly see both regimes in Figs. 1 and 2 with serving as an inflection point with respect to . Beyond this inflection point, heads into a saturated regime.58 This saturation process reaches another inflection point when . Beyond this second inflection point, heads into a supersaturated regime.58 It is important to note that Fig. 1 highlights the saturated regime when , whereas Fig. 2 highlights the supersaturated regime when . These regimes are nominally the same for both and and for all values of . Thus, Figs. 3 and 4 show the normalized irradiance (top row), wrapped phase (middle row), and branch-point density (bottom row) for a single realization of Kolmogorov turbulence with (left column) and (right column). Figure 3 corresponds to the first inflection point, where , and Fig. 4 corresponds to the second inflection point, where . Figures 3 and 4 show how the branch-point density, , increases as a function of increasing Rytov number, , and as a function of increasing grid sampling, . Comparing both figures together illustrates how increases from the first inflection point, , to the second inflection point, , whereas comparing the left and right columns in each figure separately illustrates how increases from the minimum grid sampling, , to the maximum grid sampling, . These outcomes confirm that provides a gauge for the amount of scintillation within the wave-optics simulations. On the other hand, provides a gauge for the amount of branch-point resolution within the wave-optics simulations. 4.Results and DiscussionThe results presented in this section show that without the effects of a finite inner scale, the branch-point density grows without bound with adequate grid sampling. In addition, the results show that as the inner-scale size increases this unbounded growth (1) significantly decreases as the Rytov number, Fried coherence diameter, and isoplanatic angle increase and (2) saturates with adequate grid sampling. To make these findings manifest, we discuss the relevant trends in the figures that follow. In support of (1), Figs. 5Fig. 6–7 show the branch-point density, , as a function of the Rytov number, , the circular-pupil diameter relative to the Fried coherence diameter, , and the isoplanatic angle relative to the diffraction-limited half angle, , respectively. The legends in each subplot denote the grid resolutions, , setup in Table 1. Each subplot also uses the same inner-scale size relative to the minimum grid sampling, , explored in Figs. 1 and 2. In Figs. 5Fig. 6–7, the plotted lines represent the Monte Carlo averages associated with 100 turbulence realizations, and the error bars represent the standard deviations. Similar to Figs. 1 and 2, the width of the error bars are small (at most 4% of the mean) and thus we believe that 100 Monte Carlo realizations are adequate in quantifying the behavior of in terms of , , and . In Fig. 5(a), where , we see that as increases, increases linearly without bound when . Figures 5(b)–5(e), on the other hand, show that when increases, this unbounded linear growth significantly decreases. In particular, when and , the results for tend to overlap, whereas when and , the results for tend to saturate. In Fig. 6(a), where , we see that as increases, increases exponentially without bound when . Figures 6(b)–6(e), on the other hand, show that when increases, this unbounded exponential growth significantly decreases. In particular, when and , the results for tend to overlap, whereas when and , the results for tend to saturate. In Fig. 7(a), where , we see that as decreases, increases exponentially without bound when . Figures 7(b)–7(e), on the other hand, show that when increases, this unbounded exponential growth significantly decreases. In particular, when and , the results for tend to overlap, whereas when and , the results for tend to saturate. The aforementioned saturation in Figs. 5–7 when is due to inadequate sampling of the turbulent volume and is not physical. One rule of thumb says that the Fried coherence diameter relative to the grid sampling needs to be greater than ten (i.e., ). Simply put, when , we do not always satisfy this rule of thumb within the wave-optics simulations, as shown in Figs. 5–7. In support of (2), Fig. 8 shows the branch-point density, , as a function of the grid resolution, . The legends in each subplot denote the Rytov numbers, , setup in Table 2. Each subplot also uses the same inner-scale size relative to the minimum grid sampling, , explored in Figs. 5–7. In Fig. 8, the plotted lines represent the Monte Carlo averages associated with 100 turbulence realizations, and the error bars represent the standard deviations. Similar to Figs. 5Fig. 6–7, the width of the error bars are small (at most 4% of the mean) and thus we believe that 100 Monte Carlo realizations are adequate in quantifying the behavior of in terms of . In Fig. 8(a), where , we see that as increases, increases without bound for all . Figures 8(b)–8(e), on the other hand, show that when increases, this unbounded growth saturates. In particular, when and , the results for tend to roll over, whereas when and , the results for tend to grow monotonically. This monotonic growth increases as increases, which again is most likely caused by insignificant sampling of the turbulent volume. Figure 8 ultimately shows that as increases, increasingly saturates when . This result disagrees with a preconceived notion within the atmospheric-propagation community that the branch-point density grows without bound with increasing branch-point resolution. As shown in Fig. 8(a), this preconceived notion is the result of using the well-known Kolmogorov spectrum within the wave-optics simulations. Recall that this spectrum is an idealization with an inner-scale size, , of zero and an outer-scale size, , of infinity. Such a spectrum leads to energy distribution beyond the inertial sub range, which is not physical. Also recall that this shortcoming can be remedied using a modified spectrum such as the Hill spectrum.56 In so doing, the wave-optics simulations more accurately model the energy distribution in a turbulent volume and lead to the results presented in Figs. 8(b)–8(e). With this last point in mind, Ref. 55 recently showed that wave-optics simulations need to include the effects of a finite outer scale to accurately model the effects of anisoplanatism. 5.ConclusionsThis paper used wave-optics simulations to investigate the branch-point density in terms the grid sampling. The goal for these wave-optics simulations was to model plane-wave propagation through homogeneous turbulence, both with and without the effects of a finite inner scale modeled using a Hill spectrum. In practice, the Rytov number, Fried coherence diameter, and isoplanatic angle provided parameters to setup and explore the associated deep-turbulence conditions within the wave-optics simulations. The grid sampling, on the other hand, provided a gauge for the amount of branch-point resolution within the wave-optics simulations. Via Monte Carlo averaging, the results showed that without the effects of a finite inner scale, the branch-point density grew without bound. Nevertheless, the results also showed that as the inner-scale size increased, this unbounded growth (1) significantly decreased as the associated deep-turbulence conditions became more pronounced and (2) saturated with adequate branch-point resolution within the wave-optics simulations. The results of this paper imply that future developments need to include the effects of a finite inner scale to accurately model the multifaceted nature of the branch-point problem in adaptive optics. Recall that this problem tends to be the “Achilles’ heel” to beam-control systems that perform deep-turbulence phase compensation. Thus, the results of this paper are encouraging from the standpoint that they could readily improve the performance of existing branch-point-tolerant phase reconstruction algorithms. The results of this paper could also inform the development of future branch-point-tolerant phase reconstruction algorithms, which take into account the effects of additive-sensor noise, low signal-to-noise ratios, and subaperture-sampling requirements. These future developments are a critical next step to improving the performance of beam-control systems that perform deep-turbulence phase compensation. AcknowledgmentsThe authors would like to thank the Joint Directed Energy Transition Office for sponsoring this research. Approved for public release; distribution is unlimited. Public Affairs release approval #AFRL-2022-0153. ReferencesD. L. Fried,
“Branch point problem in adaptive optics,”
J. Opt. Soc. Am. A, 15
(10), 2759
–2768
(1998). https://doi.org/10.1364/JOSAA.15.002759 JOAOD6 0740-3232 Google Scholar
J. D. Barchers, D. L. Fried and D. J. Link,
“Evaluation of the performance of Hartmann sensors in strong scintillation,”
Appl. Opt., 41
(6), 1012
–1021
(2002). https://doi.org/10.1364/AO.41.001012 APOPAI 0003-6935 Google Scholar
J. D. Barchers, D. L. Fried and D. J. Link,
“Evaluation of the performance of a shearing interferometer in strong scintillation in the absence of additive measurement noise,”
Appl. Opt., 41
(18), 3674
–3684
(2002). https://doi.org/10.1364/AO.41.003674 APOPAI 0003-6935 Google Scholar
J. D. Barchers et al.,
“Performance of wavefront sensors in strong scintillation,”
Proc. SPIE, 4839 217
–227
(2003). https://doi.org/10.1117/12.457126 PSISDG 0277-786X Google Scholar
M. Vorontsov et al.,
“Comparative efficiency analysis of fiber-array and conventional beam director systems in volume turbulence,”
Appl. Opt., 55 4170
–4185
(2016). https://doi.org/10.1364/AO.55.004170 APOPAI 0003-6935 Google Scholar
M. F. Spencer and T. J. Brennan,
“Compensation in the presence of deep turbulence using tiled-aperture architectures [invited],”
Proc. SPIE, 10194 1019403
(2017). https://doi.org/10.1117/12.2258681 PSISDG 0277-786X Google Scholar
M. T. Banet and M. F. Spencer,
“Compensated-beacon adaptive optics using least-squares phase reconstruction,”
Appl. Opt., 28 36902
–36914
(2020). https://doi.org/10.1364/OE.409134 APOPAI 0003-6935 Google Scholar
D. L. Fried and J. L. Vaughn,
“Branch cuts in the phase function,”
Appl. Opt., 31
(15), 2865
–2882
(1992). https://doi.org/10.1364/AO.31.002865 APOPAI 0003-6935 Google Scholar
V. Tartakovsky,
“Phase dislocations and minimal–phase representation of the wave function,”
Atmos. Ocean. Opt., 8 231
–234
(1995). AOCOEK Google Scholar
V. Tartakovsky,
“Phase dislocations and focal spot,”
Atmos. Ocean. Opt., 9 926
–929
(1996). AOCOEK Google Scholar
J. F. Nye and M. V. Berry,
“Dislocations in wave trains,”
Proc. R. Soc. Lond. A. Math. Phys. Sci., 336
(1605), 165
–190
(1974). https://doi.org/10.1098/rspa.1974.0012 Google Scholar
V. P. Lukin and B. V. Fortes, Adaptive Beaming and Imaging in the Turbulent Atmosphere, SPIE Press, Bellingham, Washington
(2002). Google Scholar
G. J. Gbur, Mathematical Methods for Optical Physics and Engineering, Cambridge University Press, New York
(2011). Google Scholar
J. W. Goodman, Speckle Phenomena in Optics: Theory and Applications, 2nd ed.SPIE Press, Bellingham, Washington
(2020). Google Scholar
J. D. Schmidt, Numerical Simulation of Optical Wave Propagation with Examples in MATLAB, SPIE Press, Bellingham, Washington
(2010). Google Scholar
D. G. Voelz, Computational Fourier Optics: A MATLAB Tutorial, SPIE Press, Bellingham, Washington
(2011). Google Scholar
J. P. Bos and M. C. Roggemann,
“Technique for simulating anisoplanatic image formation over long horizontal paths,”
Opt. Eng., 51
(10), 101704
(2012). https://doi.org/10.1117/1.OE.51.10.101704 Google Scholar
R. C. Hardie et al.,
“Simulation of anisoplanatic imaging through optical turbulence using numerical wave propagation with new validation analysis,”
Opt. Eng., 56
(7), 071502
(2017). https://doi.org/10.1117/1.OE.56.7.071502 Google Scholar
N. R. Van Zandt et al.,
“Polychromatic wave-optics models for image-plane speckle. 1. Well-resolved objects,”
Appl. Opt., 57
(15), 4090
–4102
(2018). https://doi.org/10.1364/AO.57.004090 APOPAI 0003-6935 Google Scholar
N. R. Van Zandt et al.,
“Polychromatic wave-optics models for image-plane speckle. 2. Unresolved objects,”
Appl. Opt., 57
(15), 4103
–4110
(2018). https://doi.org/10.1364/AO.57.004103 APOPAI 0003-6935 Google Scholar
M. F. Spencer,
“Wave-optics investigation of turbulence thermal blooming interaction: I. Using steady-state simulations,”
Opt. Eng., 59
(8), 081804
(2020). https://doi.org/10.1117/1.OE.59.8.081804 Google Scholar
M. F. Spencer,
“Wave-optics investigation of turbulence thermal blooming interaction: II. Using time-dependent simulations,”
Opt. Eng., 59
(8), 081805
(2020). https://doi.org/10.1117/1.OE.59.8.081805 Google Scholar
V. V. Voitsekhovich, D. Kouznetsov and D. K. Morozov,
“Density of turbulence-induced phase dislocations,”
Appl. Opt., 37
(21), 4525
–4535
(1998). https://doi.org/10.1364/AO.37.004525 APOPAI 0003-6935 Google Scholar
W. W. Arrasmith,
“Branch-point-tolerant least-squares phase reconstructor,”
J. Opt. Soc. Am. A, 16
(7), 1864
–1872
(1999). https://doi.org/10.1364/JOSAA.16.001864 JOAOD6 0740-3232 Google Scholar
M. C. Roggemann and A. C. Koivunen,
“Wave-front sensing and deformable-mirror control in strong scintillation,”
J. Opt. Soc. Am. A, 17
(5), 911
–919
(2000). https://doi.org/10.1364/JOSAA.17.000911 JOAOD6 0740-3232 Google Scholar
D. L. Fried,
“Adaptive optics wave function reconstruction and phase unwrapping when branch points are present,”
Opt. Commun., 200
(1–6), 43
–72
(2001). https://doi.org/10.1016/S0030-4018(01)01546-2 OPCOB8 0030-4018 Google Scholar
T. M. Venema and J. D. Schmidt,
“Optical phase unwrapping in the presence of branch points,”
Opt. Exp., 16
(10), 6985
–6998
(2008). https://doi.org/10.1364/OE.16.006985 OPEXFF 1094-4087 Google Scholar
M. J. Steinbock, M. W. Hyde and J. D. Schmidt,
“LSPV+7, a branch-point-tolerant reconstructor for strong turbulence adaptive optics,”
Appl. Opt., 53
(18), 3821
–3831
(2014). https://doi.org/10.1364/AO.53.003821 APOPAI 0003-6935 Google Scholar
J. J. Kim, B. Fernandez and B. Agrawal,
“Iterative wavefront reconstruction for strong turbulence using Shack–Hartmann wavefront sensor measurements,”
J. Opt. Soc. Am. A, 38
(3), 456
–464
(2021). https://doi.org/10.1364/JOSAA.413934 JOAOD6 0740-3232 Google Scholar
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. Exp., 18
(21), 22377
–22392
(2010). https://doi.org/10.1364/OE.18.022377 OPEXFF 1094-4087 Google Scholar
D. J. Sanchez and D. W. Oesch,
“Orbital angular momentum in optical waves propagating through distributed turbulence,”
Opt. Exp., 19
(24), 24596
–24608
(2011). https://doi.org/10.1364/OE.19.024596 OPEXFF 1094-4087 Google Scholar
D. J. Sanchez and D. W. Oesch,
“Localization of angular momentum in optical waves propagating through turbulence,”
Opt. Exp., 19
(25), 25388
–25396
(2011). https://doi.org/10.1364/OE.19.025388 OPEXFF 1094-4087 Google Scholar
D. W. Oesch et al.,
“Aggregate behavior of branch points: characterization in wave optical simulation,”
Opt. Eng., 51
(10), 106001
(2012). https://doi.org/10.1117/1.OE.51.10.106001 Google Scholar
D. W. Oesch, D. J. Sanchez and C. M. Tewksbury-Christle,
“Aggregate behavior of branch points-persistent pairs,”
Opt. Exp., 20
(2), 1046
–1059
(2012). https://doi.org/10.1364/OE.20.001046 OPEXFF 1094-4087 Google Scholar
D. W. Oesch and D. J. Sanchez,
“Creating well-defined orbital angular momentum states with a random turbulent medium,”
Opt. Exp., 20
(11), 12292
–12302
(2012). https://doi.org/10.1364/OE.20.012292 OPEXFF 1094-4087 Google Scholar
D. W. Oesch et al.,
“Creation of photonic orbital angular momentum by distributed volume turbulence,”
Opt. Exp., 21
(5), 5440
–5455
(2013). https://doi.org/10.1364/OE.21.005440 OPEXFF 1094-4087 Google Scholar
D. Oesch and D. Sanchez,
“Photonic orbital angular momentum in starlight,”
Astron. Astrophys., 567 A114
(2014). https://doi.org/10.1051/0004-6361/201323140 AAEJAF 0004-6361 Google Scholar
B. M. Welsh and C. S. Gardner,
“Performance analysis of adaptive-optics systems using laser guide stars and sensors,”
J. Opt. Soc. Am A., 6
(12), 1913
–1923
(1989). https://doi.org/10.1364/JOSAA.6.001913 Google Scholar
B. M. Welsh et al.,
“Fundamental performance comparison of a Hartmann and a shearing interferometer wave-front sensor,”
J. Opt. Soc. of Am. A, 34
(21), 4186
–4195
(1995). https://doi.org/10.1364/AO.34.004186 JOAOD6 0740-3232 Google Scholar
T. A. Rhoadarmer and J. D. Barchers,
“Noise analysis for complex field estimation using a self-referencing interferometer wave front sensor,”
Proc. SPIE, 4825 215
–227
(2002). https://doi.org/10.1117/12.450472 PSISDG 0277-786X Google Scholar
J. D. Barchers and T. A. Rhoadarmer,
“Evaluation of phase-shifting approaches for a point-diffraction interferometer with the mutual coherence function,”
Appl. Opt., 41 7499
–7509
(2002). https://doi.org/10.1364/AO.41.007499 APOPAI 0003-6935 Google Scholar
M. F. Spencer et al.,
“Deep-turbulence wavefront sensing using digital-holographic detection in the off-axis image plane recording geometry,”
Opt. Eng., 56
(3), 031213
(2016). https://doi.org/10.1117/1.OE.56.3.031213 Google Scholar
M. T. Banet, M. F. Spencer and R. A. Raynor,
“Digital-holographic detection in the off-axis pupil plane recording geometry for deep-turbulence wavefront sensing,”
Appl. Opt., 57
(3), 465
–475
(2018). https://doi.org/10.1364/AO.57.000465 APOPAI 0003-6935 Google Scholar
D. E. Thornton, M. F. Spencer and G. P. Perram,
“Deep-turbulence wavefront sensing using digital holography in the on-axis phase shifting recording geometry with comparisons to the self-referencing interferometer,”
Appl. Opt., 58
(5), A179
–A189
(2019). https://doi.org/10.1364/AO.58.00A179 APOPAI 0003-6935 Google Scholar
D. E. Thornton, M. T. Banet and M. F. Spencer,
“Subaperture sampling for digital-holography applications involving atmospheric turbulence,”
Appl. Opt., 60
(25), G30
–G39
(2021). https://doi.org/10.1364/AO.427038 APOPAI 0003-6935 Google Scholar
G. A. Tyler,
“Reconstruction and assessment of the least-squares and slope discrepancy components of the phase,”
J. Opt. Soc. of Am. A, 17
(10), 1828
–1839
(2000). https://doi.org/10.1364/JOSAA.17.001828 JOAOD6 0740-3232 Google Scholar
T. J. Brennan,
“Anatomy of the slope discrepancy structure function: characterization of turbulence,”
Proc. SPIE, 5078 103
–114
(2003). https://doi.org/10.1117/12.487749 PSISDG 0277-786X Google Scholar
N. R. Van Zandt, M. F. Spencer and S. T. Fiorino,
“Speckle mitigation for wavefront sensing in the presence of weak turbulence,”
Appl. Opt., 58
(9), 2300
–2310
(2019). https://doi.org/10.1364/AO.58.002300 APOPAI 0003-6935 Google Scholar
N. R. Van Zandt and M. F. Spencer,
“Improved adaptive-optics performance using polychromatic speckle mitigation,”
Appl. Opt., 59
(4), 1071
–1081
(2020). https://doi.org/10.1364/AO.379972 APOPAI 0003-6935 Google Scholar
D. J. Burrell et al.,
“Wave-optics simulation of dynamic speckle: I. In a pupil plane,”
Appl. Opt., 60
(25), G64
–G76
(2018). https://doi.org/10.1364/AO.427963 APOPAI 0003-6935 Google Scholar
D. J. Burrell et al.,
“Wave-optics simulation of dynamic speckle: II. In an image plane,”
Appl. Opt., 60
(25), G77
–G90
(2018). https://doi.org/10.1364/AO.427964 APOPAI 0003-6935 Google Scholar
J. R. Beck et al.,
“Investigation of branch-point density using traditional wave-optics techniques,”
Proc. SPIE, 10772 1077206
(2018). https://doi.org/10.1117/12.2319871 PSISDG 0277-786X Google Scholar
M. F. Spencer,
“Limitations of the deep-turbulence problem [invited],”
in Proc. Opt.,
PW3F.1
(2021). https://doi.org/10.1364/PCAOP.2021.PW3F.1 Google Scholar
J. R. Beck and J. P. Bos,
“Angular anisoplanatism in non-Kolmogorov turbulence over horizontal paths,”
J. Opt. Soc. Am. A, 37
(12), 1937
–1949
(2020). https://doi.org/10.1364/JOSAA.401613 JOAOD6 0740-3232 Google Scholar
R. Hill and S. Clifford,
“Modified spectrum of atmospheric temperature fluctuations and its application to optical propagation,”
J. Opt. Soc. Am., 68
(7), 892
–899
(1978). https://doi.org/10.1364/JOSA.68.000892 JOSAAH 0030-3941 Google Scholar
R. J. Sasiela, Electromagnetic Wave Propagation in Turbulence: Evaluation and Application of Mellin Transforms, 2nd ed.SPIE Press, Bellingham, Washington
(2007). Google Scholar
G. R. Osche, Optical Detection Theory for Laser Applications, Wiley, Hoboken, New Jersey
(2002). Google Scholar
BiographyJeffrey R. Beck is a recent graduate from the Department of Electrical and Computer Engineering at Michigan Technological University and is currently seeking a postdoctoral position. He is a member of SPIE. Jeremy P. Bos is an assistant professor of Electrical and Computer Engineering at Michigan Technological University. He is a senior member of SPIE. |