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Abstract
This paper uses five spatially distributed reflective liquid-crystal phase modulators (LcPMs) to accurately simulate deep-turbulence conditions in a scaled-laboratory environment. In practice, we match the Fresnel numbers for long-range, horizontal-path scenarios using optical trombones and relays placed between the reflective LcPMs. Similar to computational wave-optic simulations, we also command repeatable high-resolution phase screens to the reflective LcPMs with the proper path-integrated spatial and temporal Kolmogorov statistics.
1. INTRODUCTION
The presence of distributed-volume atmospheric aberrations or “deep turbulence” presents unique challenges for beam-control applications which look to sense and correct for disturbances found along the laser-propagation path [1–4]. For example, the deep-turbulence conditions result in a phenomenon known as scintillation (i.e., the constructive and destructive interference that manifests from laser-beam propagation through distributed-volume atmospheric aberrations). The Rytov number, also referred to as the log-amplitude variance, gives us a gauge for the strength of scintillation. As the scintillation becomes appreciable (Rytov numbers greater than ${\sim}{0.1}$ [5]), total destructive interference gives rise to branch points in the pupil-phase function of an optical system [6–8].
Branch points, at large, add a rotational component [9–11] to the pupil-phase function that traditional least-squares phase reconstructors [12–14] cannot account for without modification. As such, the rotational component is often referred to as the hidden phase, thanks in large part to the foundational work of Fried [15]. The existence of branch points leads to unavoidable $2\pi$ phase discontinuities in the phase function known as branch cuts. Because of inter-actuator coupling, continuous-face sheet deformable mirrors (DMs) with high-power coatings are unable to fully compensate for the resultant branch cuts. Thus, in the presence of moderately strong scintillation (Rytov numbers greater than ${\sim}{0.5}$ [5]), the corresponding branch-point problem tends to be the “Achilles heel” to current beam-control solutions [16–18].
To accurately demonstrate deep-turbulence conditions in a scaled-laboratory environment, we developed an innovative atmospheric turbulence simulator (ATS). Of the different approaches to repeatable turbulence generation [19–26], liquid-crystal phase modulators (LcPMs) have been studied extensively [27–36]. Also known as liquid-crystal spatial light modulators, LcPMs can produce repeatable programmable sequences of phase screens.
As shown in Fig. 1, the developed ATS uses five spatially distributed reflective LcPMs. In practice, we can match the Fresnel numbers for long-range scenarios using optical relays and trombones placed between the LcPMs. Similar to computational wave-optic simulations ([37], Chapter 9), we can also command repeatable high-resolution phase screens to the reflective LcPMs with the proper path-integrated Kolmogorov statistics (both spatial and temporal).
Since deep-turbulence conditions can provide Greenwood frequencies on the order of 1 kHz for some beam-control applications [1–4], this scaled-laboratory setup does not allow for real-time demonstrations, since the max framerates for the commercial off-the-shelf LcPMs are on the order of 100 Hz for visible light; however, it does provide a flexible scaled-laboratory environment in which to test novel beam-control solutions, such as those which use branch-point-tolerant phase reconstructors [38–47].
In the remainder of this paper, we provide an overview of the five-layer ATS in Section 2. We then present our methodology for generating distributed-volume atmospheric aberrations in Section 3. The results follow in Section 4 and Appendix A, with experimental demonstrations that show the proper path-integrated spatial and temporal Kolmogorov statistics for three cases with horizontal-path assumptions. We briefly summarize these results in Section 5. Before moving on to Section 2, it is important to note that the results contained in this paper significantly build on the initial results presented in previous conference proceedings [48–50]—please refer to the Acknowledgment section for more details.
2. OVERVIEW OF THE FIVE-LAYER ATS
Guidestar Optical Systems in Longmont, CO [51], designed and built the five-layer ATS and delivered it to the Air Force Research Laboratory, Directed Energy Directorate in Albuquerque, NM. As shown in Fig. 1, the benchtop system consisted of multiple beam splitters (BSs); modular stages, each simulating a layer of the atmosphere with one reflective LcPM; a camera to measure the point spread function (PSF); and a digital holography (DH) sensor.
A. Overview of the Modular Stages
Figure 2 shows a detailed overview of a modular stage within the five-layer ATS.
As shown in Fig. 2, two 25 cm focal length lenses formed an image of the LcPM onto a raised cosine apodization window (RCAW) through an iris. In general, the RCAW and iris helped to mitigate any higher-order diffraction effects caused by the nonuniform fill factor of the LcPMs. A second pair of 25 cm lenses then relayed an image of the RCAW to a location past the fourth lens. In practice, we adjusted the optical path difference (OPD) by a pair of mirrors on an optical rail, forming a trombone. If the trombone was set such that the distance from the RCAW to the third lens was 25 cm, then we formed an image of the RCAW 25 cm past the fourth lens, which was at the location of the LcPM on the next stage of the five-layer ATS. However, if we moved the trombone such that the OPD was greater than 25 cm, then the image of the RCAW formed in front of the subsequent LcPM. This outcome produced an effective propagation distance between the two LcPMs.
The goal here was to simulate tactical scenarios with propagation distances of up to several kilometers with only several centimeters of available space between the various stages of the five-layer ATS. To accomplish this task, we wanted to maintain the same Fresnel number for the benchtop propagation as for the simulated, real-world propagation. In support, we defined the Fresnel number as $F = {{{D^2}} / {({\lambda Z})}}$, where $D$ is the diameter of the aperture stop, $\lambda$ is the wavelength, and $Z$ is the propagation distance. With this last point in mind, the five-layer ATS used a 3 mm aperture stop to simulate a 30 cm tactical aperture. The ratio of the apertures squared was then 10,000. This outcome indicated that the propagation distance on the benchtop system scaled by 1/10000 compared to the simulated tactical system. Thus, if the trombone on a given stage was set to produce an image of an LcPM 10 cm in front of the next LcPM, as shown in Fig. 2, then the corresponding simulated propagation distance was 1 km.
B. Overview of the DH Sensor
Recent studies have shown that digital-holographic detection is scintillation insensitive [52–54]. As shown in Fig. 3, we used off-axis pupil plane recording geometry (PPRG). In turn, we interfered a collimated signal (with the path-integrated effects of the distributed-volume atmospheric aberrations) with a tilted reference (split off from the collimated and spatially filtered 532 nm laser beam as input to the first LcPM). We digitized the resultant hologram with a camera [55].
As shown in Fig. 3, the digital hologram (that was read out from the camera and processed by a computer) contained diagonal fringes. These fringes were a carrier of the complex optical field, which contains both amplitude and wrapped phase information. We used a straightforward demodulation process consisting of a fast Fourier transform (FFT), high-pass filtering, and an inverse FFT to gain access to a ${48} \times {48}$ pixel estimate of the complex optical field with adequate subaperture sampling [55–59]. These estimates will be used in the following section.
3. METHODOLOGY FOR GENERATING DISTRIBUTED-VOLUME ATMOSPHERIC ABERRATIONS
Generating distributed-volume atmospheric aberrations with a finite number of discrete thin phase screens requires approximations. Our approach is that outlined by Schmidt ([37], Chapter 9) and starts with a continuous model based on the path-dependent refractive index structure constant $C_n^2(z)$. The relevant atmospheric parameters assume plane-wave propagation and Kolmogorov statistics [1–5], such that
is the Fried parameter,Our approach continues with a multilayer approximation to this continuous model with nearly the same values for the atmospheric parameters. Obviously, more layers provide a better approximation; however, recall that our benchtop system is limited to five layers. With this last point in mind, our approach assumes that the $i$th layer is $\Delta {z_i}$ thick. The ${r_0}$ value for the $i$th layer is then
As such, the atmospheric parameters can be rewritten in terms of ${r_{{0_i}}}$, such thatA. Methodology for Generating Time-Evolving Phase Screens
Given the approximations above, we generated sequences of time-evolving phase screens with realistic spatial and temporal Kolmogorov statistics using an approach described in Srinath et al. [60]. The first step was to generate a static phase screen using a standard Fourier transform power spectrum filter function approach, wherein we multiplied a Kolmogorov PSD filter function with a random draw of a circular Gaussian random process ([37], Chapter 9).
In this paper, we assumed that frozen flow gives rise to translation in the phase screens, which we introduced with a circular shift algorithm. We then used an auto-regressive algorithm to introduce temporal evolution. After translating the phase screen, we updated the shifted phase screen $\Phi {_{\rm{shifted}}}$ with a new phase screen ${\Phi _{\rm{new}}}$ generated using the same Kolmogorov PSD filter function, but with a new random draw, namely,
We controlled the amount of temporal evolution in the updated phase screen ${\Phi _{\rm{updated}}}$ by the auto-regressive parameter $\alpha$. In what follows, we generated sequences of 100 phase screens for each layer of the five-layer ATS.B. Three Cases with Horizontal-Path Assumptions
To set up a three-case parameter space, we assumed a long-range, horizontal-path scenario such that $Z = 4\; {\rm km}$ and $C_n^2(z) = C_n^2$. In turn, we used Eqs. (5)–(7) to calculate the path-integrated Fried parameter, Rytov number, and Greenwood frequency, respectively. By adjusting the trombones on the different LcPM stages, the simulated distance can be set between each phase screen and the aperture stop of the benchtop system to 0, 1, 2, 3, and 4 km. As shown in Table 1, we set up three cases using different phase-screen strengths at each of the layers. We also set each phase screen to translate in a different direction resulting in path-integrated effects that were boiling rather than slewing. The transverse wind speed at each layer was the same for all three cases, ${v_w} = [{2.16,1.54,1.54,1.54,2.16}]\; {\rm m/s}$.
Table 1. Atmospheric Parameters for Three Long-Range, Horizontal-Path Cases
Fig. 4. Wrapped (left) and unwrapped (right) phase estimates obtained from the DH sensor for the three cases (a)–(c) given in Table 1.
Figure 4 shows DH sensor measurements for the three cases given in Table 1. As shown in the left column of Fig. 4, the DH sensor demodulation process (see Fig. 3) produces wrapped phase estimates that are modulo ${2}\pi$. Both wrapping cuts and branch cuts are visible in these estimates. As shown in the right column of Fig. 4, these cuts go away with traditional least-squares phase unwrapping [41]. This outcome speaks to the fact that branch cuts get mapped to the null space of this mathematical operation. It also speaks to the reasoning Fried had in referring to this rotational component of the pupil-phase function as the “hidden phase.” [10]
With Fig. 4 in mind, Fig. 5 shows the corresponding hidden phase (left column) and the sum of the unwrapped phase with the hidden phase (right column) for the three cases given in Table 1. Note that we calculated the hidden phase by taking the principal value of the unwrapped phase and subtracting it from the wrapped phase [6,7,41,42,61].
Fig. 5. Hidden (left) and unwrapped-plus-hidden (right) phase estimates obtained from the DH sensor for the three cases (a)–(c) given in Table 1.
The hidden phase in the left column of Fig. 5 gives a visualization of the locations of the branch points and cuts in the pupil-phase function. As the Rytov number increases from Figs. 5(a)–5(c), so does the branch-point density [8]. The sum of the unwrapped phase with the hidden phase in the right column of Fig. 5 gives a better representation of the pupil-phase function than just the unwrapped phase alone. In turn, this representation will be used in the following section when computing the phase structure function and the phase power spectrum.
4. RESULTS
In this section, we demonstrate that the five-layer ATS simulated the proper path-integrated spatial and temporal Kolmogorov statistics for the three cases set up in the previous section (see Table 1).
A. Spatial Statistics
One can use the wave structure function $D({\Delta \rho})$ to characterize the spatial statistics. Assuming Kolmogorov statistics ([37], Chapter 9),
which follows a 5/3 power law with respect to the radial separation normalized by the Fried parameter ${{\Delta \rho} / {{r_0}}}$. With that said, we can calculate $D({\Delta \rho})$ from the data obtained from the PSF and DH cameras (see Fig. 1) in three different ways.The first approach uses the log-amplitude and phase structure functions. Recall that the wave structure function is the summation of the log-amplitude and phase structure functions, such that
One can numerically calculate ${D_\chi}({\Delta \rho})$ and ${D_\phi}({\Delta \rho})\,$ from the DH data using the Fourier methods taught by Schmidt ([37], Chapter 3).The second approach uses the mutual coherence function and the modulus of the complex degree of coherence (MDOC). Recall that the MCF is the 2D correlation of the complex optical field, and the MDOC is the amplitude associated with the normalized MCF. One can calculate the wave structure function from the MDOC as
where $\mu ({\Delta \rho})$ is the MDOC. One can also numerically calculate $\mu ({\Delta \rho})$ from the DH data using the Fourier methods taught by Schmidt ([37], Chapter 3).The third approach uses the modulation transfer function (MTF). As shown in Fig. 1, the benchtop system includes a PSF camera, which measures the far-field irradiance pattern of the system’s aperture stop. Averaging multiple frames together produces a long-exposure PSF. Recall that the MTF is the amplitude associated with the 2D Fourier transform of the PSF. One can calculate the wave-structure function from the MTF as
Figure 6 shows the results for the three approaches (numerical, MDOC, and MTF) outlined in Eqs. (10)–(12) for the three cases given in Table 1. We display these results relative to the analytical predictions from Eq. (9). The wave structure function calculations approximately follow the theoretical 5/3 power law of the analytical wave structure function up to a certain point, and then they start to deviate. We believe this behavior is due to the limited spatial extent of the system’s aperture stop, as well as the filter function used to generate the phase screens. Nonetheless, these results are consistent with those obtained from computational wave-optics simulations, as shown in Appendix A.
Fig. 6. Wave structure function calculations for the three cases (a)–(c) given in Table 1. These results are from the experimental demonstrations and complement Fig. 8. The black lines represent the analytical wave structure function, whereas the blue lines represent the wave structure function calculated numerically from the log-amplitude and phase structure functions, the red lines represent the wave structure function calculated from the MDOC, and the green lines represent the wave structure function calculated from the MTF. These structure functions are plotted against the radial separation normalized by the Fried parameter.
B. Temporal Statistics
One can use an asymptotic phase power spectrum to characterize the temporal statistics. Assuming Kolmogorov statistics [62,63],
which follows a minus 8/3 power law with respect to temporal frequency $\nu$. With that said, we can calculate the phase power spectrum from the unwrapped-plus-hidden phase estimates obtained from the DH sensor using the Fourier methods taught by Merritt and Spencer ([4], Chapter 7). We can then fit the mid-frequency asymptote to obtain an estimate of the Greenwood frequency ${f_G}$.Figure 7 shows the results for the phase power spectrum for the three cases given in Table 1. We display these results relative to the analytical predictions from Eq. (13). The phase power spectrum calculations approximately follow the theoretical minus 8/3 power law with respect to the mid-frequency asymptote up to a certain point and then they start to deviate. We believe this behavior is due to the limited number of temporal samples collected by the DH sensor. Nonetheless, these results are consistent with the Greenwood frequencies calculated in Table 1 using Eq. (7).
Fig. 7. Phase power spectrum calculations for the three cases (a)–(c) given in Table 1. These results are from the experimental demonstrations. The red lines represent an asymptotic phase power spectrum for the Greenwood frequencies calculated for each simulation case, whereas the blue lines represent the phase power spectrum calculated from the unwrapped-plus-hidden phase data obtained from the DH sensor.
C. Rytov Number
We also calculated the log-amplitude variance from the amplitude estimates obtained from the DH sensor. For comparison, Table 2 shows an expected Rytov number for each case calculated in Table 1 using Eq. (6). The results show very close comparisons, except for Case 1. We believe this discrepancy is due to measurement noise.
Table 2. Comparison of Measured and Theoretical Rytov Numbers
5. SUMMARY
The experimental demonstrations summarized in this paper used five reflective LcPMs on modular stages to accurately simulate deep-turbulence conditions in a scaled-laboratory environment. In practice, we matched the Fresnel numbers for three cases of interest using optical trombones and relays placed between the reflective LcPMs. Similar to computational wave-optic simulations, we commanded repeatable high-resolution phase screens to the reflective LcPMs with the proper path-integrated Kolmogorov statistics (both spatial and temporal). These results will enable future studies with different wavefront sensors and adaptive-optics configurations and control algorithms to better understand and mitigate the deleterious effects of laser-beam propagation through distributed-volume atmospheric aberrations.
APPENDIX A
We set up computational wave-optics simulations to model the five-layer ATS, DH sensor in the off-axis PPRG, and far-field PSF measurements. These simulations used angular spectrum propagation and the split-step beam propagation method ([37], Chapter 9). They also included a realistic noise model for the DH camera and the PSF camera [55]. We applied the same sequence of phase screens used in the hardware to these simulations. Please see Ref. [64] for more information on the simulations.
Figure 8 shows the results for the three approaches (numerical, MDOC, and MTF) outlined in Eqs. (10)–(12) for the three cases given in Table 1. We display these results relative to the analytical predictions from Eq. (9). Overall, these results are consistent with those obtained from the experimental demonstrations, as shown in Fig. 6.
Fig. 8. Wave structure function calculations for the three cases (a)–(c) given in Table 1. These results are from the computational wave-optics simulations and complement Fig. 6. The black lines represent the analytical wave structure function, whereas the blue lines represent the wave structure function calculated numerically from the log-amplitude and phase structure functions, the red lines represent the wave structure function calculated from the MDOC, and the green lines represent the wave structure function calculated from the MTF. These structure functions are plotted against the radial separation normalized by the Fried parameter.
Acknowledgment
The authors would like to thank the Joint Directed Energy Transition Office for sponsoring this research. As discussed in the introduction, the results of this paper go well beyond the initial results contained in three conference proceedings by the authors and several other contributors [48–50]. With that said, some of the verbiage used in this paper is appropriately similar. The views expressed in this article are those of the authors and do not necessarily reflect the official policy or position of the U.S. Air Force, the Department of Defense, or the U.S. Government. This paper serves as the last first-author contribution from D. Dayton, “Dave,” who submitted a draft version of this article to M. Spencer, “Mark,” just days before his passing in the fall of 2018. At the time, Dave was an in-house contractor at the Air Force Research Laboratory, Directed Energy Directorate (AFRL/RD) in Albuquerque, NM, and Mark was a civil servant at AFRL/RD. Mark has since moved on from this role and needed a few extra years to process this heart-breaking loss and fulfill his co-author duties. Dave was a senior scientist and technical fellow at Applied Technology Associates in Albuquerque, NM [65]. For over 25 years, he conducted important research for the U.S. Air Force and other organizations. Dave was a world-class expert in laser beam control, image processing, laser propagation, and other optical and laser applications. In total, he authored over 150 technical papers and presented his work at many conferences. Dave was a highly respected scientist and was always eager to share his extensive knowledge and experience with coworkers. He trained several younger scientists and engineers in his work. Dave was a great co-worker and mentor to those who had the opportunity to work with him but, more importantly, he was a good friend to all those who had the opportunity to work with him. Dave will be missed.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are unfortunately not available at this time due to the reasons stated in the Acknowledgement section.
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