## Abstract

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

© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

## 1. Introduction

Traditional adaptive optics (AO) techniques use a wavefront sensor (WFS) to sense the dynamic and path-integrated phase aberrations that exist in the pupil of a directed-energy laser system. To correct for these phase aberrations, traditional AO techniques also use a deformable mirror (DM) to perform phase compensation on an outgoing high-energy laser (HEL) beam. In practice, however, there are many factors that degrade closed-loop performance using traditional AO techniques.

For up-looking scenarios, like those found in surface-to-space applications, the phase aberrations due to turbulence exist mostly near the pupil of the directed-energy laser system. We refer to such phase aberrations as being isoplanatic, since the different points on a distant object sample a single geometrical cone of turbulence in the atmosphere. For slant-path scenarios, like those found in surface-to-air applications, the phase aberrations due to turbulence become distributed throughout the entire propagation volume. In this case, we refer to the phase aberrations as being anisoplanatic, since the different points on a distant object sample multiple geometrical cones of turbulence in the atmosphere. As a result, researchers cannot use a single point-source-like beacon to sense and correct for the phase aberrations that exist across the entire field of view of a directed-energy laser system [1].

Given surface-to-space applications, researchers often use distant stars or artificial laser guidestars as their point-source-like beacons to correct for the effects of turbulence over an isoplanatic patch [2]. Here, the isoplanatic patch is simply the isoplanatic angle [3] multiplied by the propagation distance. For surface-to-air applications, however, a point-source-like beacon rarely exists within the field of view of the directed-energy laser system. To overcome this lack of a point-source-like beacon, one approach is to create an extended beacon by focusing the outgoing coherent light from a beacon illumination laser (BIL) onto a distant object and collecting the incoming scattered return in the pupil of the directed-energy laser system, again, for the purposes of closed-loop phase compensation on an outgoing HEL beam. Unfortunately, the outgoing light from the focused BIL is subject to the same distributed-volume phase aberrations that degrade the outgoing HEL beam, causing up-link scintillation [4]. Down-link speckle, resulting from rough surface scattering, also provides extraneous phase contributions that we do not want to sense and correct for using traditional AO techniques [5,6].

To be clear, in this paper the term scintillation refers to the constructive and destructive interference that results from the distributed-volume phase aberrations due to turbulence and should not be confused with the term speckle, which results from the constructive and destructive interference that arises from the extended beacon scattering off of an optically rough object. In effect, these phenomena motivate the desire for extended beacons that are as small as possible for two reasons: (1) smaller extended beacons will give rise to fewer speckles across the pupil of the directed-energy laser system, and (2) smaller extended beacons will more likely provide information from a single isoplanatic patch. In practice, extended beacons can span several isoplanatic patches, so that the incoming scattered return from the distant object samples multiple geometrical cones of turbulence in the atmosphere. This outcome causes anisoplanatic phase aberrations, as discussed above, and is less than ideal in terms of sensing and correcting for the distribted-volume phase aberrations due to turbulence using traditional AO techniques.

One potential approach to creating a smaller extended beacon is to use compensated-beacon adaptive optics (CBAO). Here, the outgoing light from the focused BIL used to create the extended beacon first reflects off of the DM before propagating to the distant object. Ideally, the resulting phase compensation provides a more point-source-like beacon, which ultimately results in (1) fewer speckles across the pupil of the directed-energy laser system and (2) more information from a single isoplanatic patch. In turn, CBAO will lead to better closed-loop performance when compared to uncompensated beacon adaptive optics (UBAO). Figure 1 displays several examples including (a) a diffraction-limited extended beacon, (b) an uncompensated extended beacon, and (c) a compensated extended beacon. Here, we can see that the compensated extended beacon has a smaller spatial extent than the uncompensated extended beacon in addition to a more localized central core. In terms of sensing and correcting for the distributed-volume phase aberrations due to turbulence, these attributes lead to better closed-loop performance using traditional AO techniques.

With the potential benefits of CBAO in mind, there are additional factors that degrade closed-loop performance. For example, given weak scintillation, the tried and true Shack-Hartmann wavefront sensor (SH-WFS) performs well; however, its ability to measure the phase aberrations from a compensated extended beacon rapidly degrades given strong scintillation [7]. Because strong scintillation results in total-destructive interference, two effects end up limiting the performance of the SH-WFS.

The first performance-limiting effect manifests because there is a reduction in phase reconstruction accuracy caused by subaperture-to-subaperture irradiance fades in the SH-WFS. Barchers *et al.* referred to this effect as centroid-tilt (C-tilt), gradient-tilt (G-tilt) anisoplanatism [7], which simply alludes to the fact that the SH-WFS measures C-tilt, as opposed to G-tilt, over each subaperture. As described by Holmes [8], G-tilt is the average phase gradient over a subaperture, assuming uniform illumination, whereas C-tilt is the average phase gradient weighted by the irradiance over a subaperture. Given subaperture-to-subaperture irradiance fades, C-tilt will not be equivalent to G-tilt. This detail leads to a reduction in accuracy using least-squares phase reconstruction, which assumes that the SH-WFS measures G-tilt. These irradiance fades are random in nature and highly dynamic because of the effects of strong scintillation. In turn, one can use clever thresholding schemes to combat this loss in accuracy, but factors such as unobservable modes (e.g., piston and waffle) tend to also limit performance [9].

The second performance-limiting effect is the result of branch points. In practice, branch points arise in the phase function where the real and imaginary parts of the complex-optical field equate to zero because of total-destructive interference caused by strong scintillation. Using least-squares phase reconstruction [10–12], the SH-WFS cannot sense the branch points and associated $2\pi$ phase discontinuities known as branch cuts. These branch points and cuts exist in what Fried referred to as the hidden-phase component of the overall phase function [13]. Mathematically, the hidden-phase component gets mapped to the null space of least-squares phase reconstructors. The inability to sense and correct for the hidden-phase component causes appreciable fitting error that degrades closed-loop performance. This outcome ultimately means that in the presence of strong scintillation, researchers cannot effectively sense and correct for the distributed-volume phase aberrations due to turbulence using traditional AO techniques.

The digital-holographic wavefront sensor (DH-WFS), on the other hand, potentially overcomes the two performance-limiting effects mentioned above with the use of a strong reference beam [14–16] and branch-point tolerant phase reconstruction [17]. In turn, the goal for this paper is to quantify the associated performance benefits of CBAO relative to UBAO using least-squares phase reconstruction. Such a quantification provides the baseline performance expectations needed to evaluate the potential benefits of branch-point tolerant phase reconstructors. In addition, these baseline performance expectations are instrumental to systems engineers who use Strehl-ratio-dependent scaling laws as part of link-budget analyses to perform system-level trade studies [9]. Depending on the directed-energy laser system of interest, future trade studies of this kind will allow systems engineers to determine whether the potential increase in hardware complexity associated with CBAO is worth implementing in comparison with UBAO. This paper, as a result, explores the use of a SH-WFS and a DH-WFS with similar subaperture sampling across the pupil [18–21]. It also explores the use of least-squares phase reconstruction without the effects of sensor noise. As such, this paper quantifies the benefits of CBAO relative to UBAO using wave-optics simulations that properly model the effects of weak to moderately strong scintillation.

In what follows, Section 2 reviews the background material needed to understand the subaperture sampling used in this paper, along with the least-squares phase reconstruction. Next, Section 3 covers the setup used for the wave-optics simulations. These wave-optics simulations make use of WavePlex, which is a MATLAB toolbox by Prime Plexus [22]. Section 3 also explores the trade space. Lastly, Section 4 presents results that quantify the benefits of CBAO relative to UBAO, and Section 5 concludes this paper.

## 2. Background material

This section describes the details behind the subaperture sampling and least-squares phase reconstruction used in this paper. In particular, we examine these details in terms of both the SH-WFS and the DH-WFS. Recall that our goal throughout is to quantify the benefits of CBAO relative to UBAO using wave-optics simulations with weak to moderately strong scintillation conditions.

#### 2.1 Subaperture sampling

For the SH-WFS, researchers often define the subaperture size as the width associated with a single lenslet. These lenslets, in practice, exist within a lenslet array located in a conjugate-pupil plane of the directed-energy laser system. It has been shown that one subaperture per Fried coherence diameter $r_0$ results in Strehl ratios of approximately 0.8 [18]. This analysis, however, did not include the effects of scintillation, which we do account for in this paper. Nonetheless, we will use this rule of thumb in the ensuing analysis to define subaperture sampling for the SH-WFS. One can reason that, for the SH-WFS, the wavefront will consist of mainly tip-tilt aberrations over each lenslet when the subaperture width is comparable to $r_0$. This outcome means that the centroid location of each spot will map directly to the slope of the wavefront over that subaperture.

With the sampling criterion for the SH-WFS in mind, we specifically used a $20\times 20$ lenslet array in the ensuing wave-optics simulations. This choice led to a SH-WFS with $20\times 20$ subapertures. When using a SH-WFS in the presence of strong scintillation, Barchers *et al.* showed that one desires 4 subapertures across the Fried coherence diameter $r_0$ in order to better sample the resulting phase aberrations [7]; however, radiometrically speaking, smaller subapertures have difficulty collecting enough photons, especially given irradiance fades. This outcome creates a balancing act that is especially difficult in the presence of strong scintillation, which causes irradiance fades in the simulated pupil plane. Since we did not include the effects of sensor noise in the ensuing wave-optics simulations, we were driven more so by the goal of adequately sampling $r_0$ than collecting enough photons. To that end, we always kept the subaperture width smaller than $r_0$ in all turbulence scenarios simulated, as that led to better phase reconstruction accuracy in the wave-optics simulations.

For the DH-WFS, it has been shown that three pixels across the Fried coherence diameter $r_0$ results in Strehl ratios of 0.8 [21]. These pixels, in practice, exist within a conjugate-pupil plane of the directed-energy laser system. Based on this result, one can define a DH-WFS subaperture as a $3\times 3$ pixel region. Here, one desires a $3\times 3$ pixel region over each coherence area in order to sense the tip-tilt aberrations over that region.

With the sampling criterion for the DH-WFS in mind, we specifically obtained wrapped-phase estimates in the pupil plane of $60\times 60$ pixels in the ensuing wave-optics simulations. This choice led to a DH-WFS with $20\times 20$ subapertures. For digital holography in the off-axis image plane recording geometry (IPRG) [14], one estimates the wrapped phase in a conjugate-pupil plane of the directed-energy laser system by Fourier transforming the hologram that is formed in the image plane. There are many DH recording geometries [14–16], but we chose the off-axis IPRG in the ensuing wave-optics simulations for its potential ease of implementation and straightforwardness when windowing wrapped-phase estimates in the Fourier plane.

We expected the simulated performance associated with both noiseless sensors to be fairly similar. With that said, as described by Barchers *et al.* [7], the performance of the SH-WFS degrades in the presence of strong scintillation even without the effects of sensor noise because of the fact that the centroid calculation associated with each lenslet is weighted by the irradiance across the subaperture. This irradiance weighting results in C-tilt, G-tilt anisoplanatism, which degrades the accuracy associated with the reconstructed phase. As such, in the presence of strong scintillation, we expect to see a slight decrease in performance for the SH-WFS when compared to the DH-WFS, even without the effects of sensor noise.

#### 2.2 Least-squares phase reconstruction

Provided phase gradients, one can obtain an estimate of the unwrapped phase through least-squares phase reconstruction. For the SH-WFS, a lenslet array focuses light in the pupil plane to an image plane. In so doing, the local tip-tilt terms of the wavefront over each lenslet give rise to spatial shifts in the corresponding focused spots on the focal-plane array of the SH-WFS. Digital processing then calculates the centroid of each spot and, hence, a local tip-tilt measurement over each corresponding lenslet, which is akin to a phase gradient. In contrast, the DH-WFS produces estimates of the wrapped phase in the pupil plane, and one can digitally shear these wrapped-phase estimates in both the $x$ and $y$ directions to obtain gradient-like information in the form of finite-phase differences.

One can map discrete 2D phase values to finite-phase differences (an approximation to the phase gradient which we will hold as an equivalence hereafter) via

where $\mathbf {g}$ (in bold because it represents a vector field) is the $x$-$y$ gradient of the phase, $\phi$. With both the SH-WFS and the DH-WFS, we measure $\mathbf {g}$ and want to obtain $\phi$. The first thought would be to invert the matrix $\Gamma$ to obtain $\phi$; however, there are generally more elements in the vector $\mathbf {g}$ than $\phi$ (i.e., the system is overdetermined) meaning that $\Gamma$ is a non-square matrix. Instead, one can take the Moore-Penrose inverse (or least-squares inverse) of $\Gamma$ to obtain where $\phi _{LS}$ is the phase obtained by applying the least-squares reconstructor $\mathcal {R}$ to $\mathbf {g}$ and the superscript $T$ denotes a transpose.As laid out by Fried [13], the overall phase function of a complex-optical field can be expressed as the sum of two components known as the least-squares phase, $\phi _{LS}$, and the hidden phase, $\phi _{hid}$, whose namesake originates from the fact that the hidden phase gets mapped to the null space of least-squares phase reconstructors. One can reason the existence of these components, as explained in Ref. [13], by decomposing $\mathbf {g}$ into

In the ensuing wave-optics simulations, we used least-squares phase reconstruction, as defined in Eqs. (1)–(5), with the phase gradients obtained from both the simulated SH-WFS and DH-WFS.

## 3. Setup and exploration

This section describes the setup used for the wave-optics simulations performed in this paper. It also explores the associated trade space through the use of a performance metric often referred to as the peak Strehl ratio. Within the wave-optics simulations, we performed closed-loop phase compensation on a focused HEL scoring beam using: (1) the phase gradients obtained from either a SH-WFS (in the Fried geometry) or a DH-WFS (in the Hudgin geometry), (2) a least-squares phase reconstructor [cf. Eq. (2)], (3) a continuous-face-sheet DM, and (4) the feedback from a leaky integrator control law. We sensed and corrected for the dynamic and path-integrated phase aberrations induced by the simulated turbulence using the null-seeking servo offered by the leaky integrator control law. Also note that with CBAO, the focused HEL scoring beam was identical to the outgoing focused BIL beam, but this simplification was not the case for UBAO, where we only performed closed-loop phase compensation on the focused HEL scoring beam.

#### 3.1 Setup of the wave-optics simulations

In the wave-optics simulations, the distant object illuminated by the outgoing focused BIL beam was an optically rough ${0.256}\,\textrm{m}$-wide, flat-square plate. As such, the simulated complex-reflectance function, grid point to grid point, was a unit-amplitude phasor with uniformly distributed phase from -$\pi$ to $\pi$ within the dimensions of the plate and zero elsewhere. We assumed that the resultant down-link speckle was uncorrelated from frame to frame; thus, we changed the seed on the random-number generator used to establish the aforementioned complex-reflectance function with each simulated time step. We used the split-step beam propagation method [23] to simulate distributed-volume phase aberrations due to turbulence via five equally-spaced Kolmogorov phase screens over a distance of $z= {4}\,\textrm{km}$. WavePlex generated these phase screens analogous to the approach presented by Schmidt with no added subharmonics or low-order tilt [23]. The propagation grids were all $N\times N$ arrays with $N=1000$, and all had unity scaling (meaning that every array had the same physical width of $S={2}\,\textrm{m}$). We also satisfied Fresnel scaling (or critical sampling of the angular spectrum transfer function [24]), such that $z=S^{2}/(\lambda N)$, where $\lambda$, the wavelength, was ${1}\;\mathrm{\mu}\textrm{m}$.

Starting with an outgoing focused BIL beam, we initialized the DM to a flat configuration, which resulted in an extended beacon on the distant object (cf. Fig. 1). A 0.5 m diameter circular pupil collimated the scattered return from the extended beacon and sent the light to either a SH-WFS or a DH-WFS. This circular pupil also served as the simulated beam director for the outgoing focused BIL beam, meaning that the diffraction-limited beacon was an airy pattern [cf. Fig. 1(a)]. The simulated beam director also projected an analogous focused flat-top, top-hat coherent beam for scoring performance, which we describe in the next section.

Recall that the SH-WFS consisted of $20\times 20$ subapertures in the Fried geometry [9,10]. We placed these subapertures in the simulated pupil plane, which focused light to a $160\times 160$ pixel focal-plane array in an image plane. In accordance with the off-axis IPRG [14], the DH-WFS consisted of a $160\times 160$ pixel focal-plane array in an image plane which provided $60\times 60$ pixels across the wrapped-phase estimate in the subsequent Fourier plane. Also recall that we considered each subaperture for the DH-WFS to be a $3\times 3$ pixel region, so for the DH-WFS there were $20\times 20$ subapertures in the Hudgin geometry [11,17].

The simulated SH-WFS and DH-WFS ultimately provided the phase gradients needed to perform least-squares phase reconstruction. For the SH-WFS, the least-squares phase reconstructor resulting from Eq. (2) provided the smoothly-varying, least-squares phase estimate for direct application to the simulated $21\times 21$ actuator continuous-face-sheet DM via a leaky integrator control law. On the other hand, for the DH-WFS, the least-squares phase reconstructor resulting from Eq. (2) provided a higher-resolution estimate. After 2D linear interpolation, the resultant smoothly-varying, least-squares phase estimate was applied to the simulated $21\times 21$ actuator continuous-face-sheet DM via a leaky integrator control law.

As previously stated, we used the null-seeking servo offered by a leaky integrator control law to command the DM [9]. Here, the incoming light from the extended beacon first reflected off of the DM before being sensed by the SH-WFS or the DH-WFS; therefore, the received wavefront was close to being nulled out as the leaky integrator control law reached steady-state performance. The difference equation used to command the DM took the following form:

where $n$ is the current timestep (unitless), $y[n]$ is the output DM command, $x[n]$ is the actuator input command based on phase compensation, $a$ is a servo leakage coefficient, and $b$ is a servo gain coefficient. Upon examining Eq. (6), we see that the difference equation retains the previous DM commands with the leakage coefficient $a$ and adds the new commands with gain $b$. Ideally, with a null-seeking servo, the new inputs $x[n]$ should be close to zero as Eq. (6) reaches steady-state performance.Once the leaky integrator control law determined the DM commands, the focused HEL scoring beam reflected off of the DM and focused down to the object through the same simulated turbulence, where we scored performance. With CBAO, each subsequent timestep used the focused HEL scoring beam itself to create the compensated extended beacon [cf. Fig. 1(c)], but with UBAO, each subsequent timestep used an uncompensated extended beacon [cf. Fig. 1(b)]. In both cases, the scattered return from the extended beacon propagated back through the simulated turbulence, and WavePlex evolved the simulation by one timestep afterwards in the Fourier domain by fractions of a grid spacing. Assuming Taylor’s frozen flow, we performed these updates by shifting the Kolmogorov phase screens across the propagation path in the $x$ direction with transverse wind speed $v_w = {5}$ m/s. This outcome meant that at the next timestep the phase of the scattered return differed slightly when compared with the previous value. We captured these differences using the leaky integrator control law given in Eq. (6). For all of the wave-optics simulations, we set $a=0.998$ and $b=0.5$ and used a sampling frequency, $f_S$, of ${1500}\;\textrm{Hz}$.

Provided the setup outlined above, we studied seven unique turbulence scenarios in this paper. The scenarios progressively increased in scintillation strength, as shown in Table 1. Recall that the primary goal here was to set up a trade space with weak to moderately strong scintillation conditions.

From left to right in Table 1, we signify the strength of the constant turbulence with the index of refraction structure parameter $C_n^{2}$, and we denote the log-amplitude variance (or Rytov number) for a spherical wave with $\sigma _\chi ^{2}$. The log-amplitude variance gives a gauge for the amount of scintillation caused by the path-integrated aberrations due to the simulated turbulence [23]. Consequently, we can relate this parameter to the number of branch points and the structure of the hidden-phase component of the overall phase function. Since we make no effort to sense and correct for the hidden-phase component in the wave-optics simulations, we expect performance to decrease as a function of $\sigma _\chi ^{2}$. Here, we define the weak-scintillation regime as $0\leq \sigma _\chi ^{2}<0.25$ and the moderately strong scintillation regime as $0.25\leq \sigma _\chi ^{2}<0.5$ [25,26]. The next parameter, $D/r_0$, determines how many times $r_0$ will span the diameter of the circular pupil. Similarly, $d/r_0$ denotes how many times $r_0$ will span a subaperture width $d$, which is always 0.025 m in these simulations. In effect, these parameters give us a gauge for the severity of the path-integrated phase aberrations due to the simulated turbulence [23] and for how well the sensors can sample the aberrations [18,21]. We tabulate the isoplanatic angle, $\theta _0$, relative to the diffraction-limited angular resolution, $\lambda /D$, in the next column. This parameter determines how isoplanatic the path-integrated phase aberrations are due to the simulated turbulence [23], with $\theta _0/(\lambda /D)=1$ meaning that the point-spread function is different for every diffraction-limited resolution element in the object plane. Finally, $f_G$ is the Greenwood frequency, which determines the rate at which the path-integrated phase aberrations change due to the simulated turbulence [3]. Note that we tabulate $f_G$ in terms of the sampling frequency, $f_S$, and the 3 dB bandwidth of the null-seeking servo, $f_{3dB}$, which is a gauge for the temporal frequency limit of the DM response. Empirically determined rules of thumb typically state that $f_S>10f_G$ and $f_{3dB}>1.5f_G$ yield good closed-loop performance.

#### 3.2 Exploration of the wave-optics simulations

In the trade-space exploration that follows, we use the peak Strehl ratio, $S_{P}$, as a performance metric for the focused HEL scoring beam, and we calculate this metric via

With Eq. (7) in mind, we now explore the trade space setup in Table 1. For example, in Fig. 2 we plot peak Strehl ratio as a function of time for both the SH-WFS and the DH-WFS. These curves are for a variety of compensation techniques and a single realization of simulated turbulence with two different scintillation strengths. In particular, we simulated 150 timesteps of closed-loop phase compensation on the focused HEL beam with an ideal-point-source beacon, CBAO, and UBAO. We also present curves for an uncompensated focused HEL scoring beam. Here, 150 timesteps is equivalent to the time it takes for the transverse wind speed to sweep over the distance of one circular pupil diameter, which we refer to as the wind clearing time.

When looking at the results from Fig. 2, we see that both the CBAO and UBAO cases are bounded by the ideal-point-source beacon performance on the top and by the uncompensated focused HEL scoring beam on the bottom. The CBAO curves are, on average, higher than the UBAO curves, but there are timesteps where performance using UBAO is better. Both the CBAO and UBAO curves are more erratic over time when compared to the smoothly varying ideal-point-source and uncompensated curves. This outcome ultimately results from speckle, which corrupts the least-squares phase estimate in the CBAO and UBAO cases for both the SH-WFS and the DH-WFS. Recall that we simulated independent speckle realizations from frame to frame, which causes rapid changes in the least-squares phase estimate when compared with the more slowly varying path-integrated phase aberrations due to the simulated turbulence. Furthermore, the uncompensated and compensated extended beacons illuminate different areas of the object, in general; thus, they give rise to different speckle phase in the simulated pupil. This could explain why the performance for CBAO occasionally dips below that of UBAO.

A single realization of simulated turbulence can give us some trade-space information, as seen in Fig. 2, but ultimately one desires Monte Carlo averaging over multiple realizations. In Fig. 3, we see that the peak Strehl ratio reaches a steady-state value for all compensation techniques and for both the SH-WFS and the DH-WFS. The overall results are much clearer: the ideal-point-source beacon always yields the best performance and it is followed by CBAO and UBAO, respectively. The uncompensated focused HEL scoring beam, of course, has the worst performance. Comparing the two noiseless sensors, it appears that the ideal-point-source beacon yields better performance for the SH-WFS than the DH-WFS in weak scintillation conditions, but that the opposite is true in moderately strong scintillation conditions. The CBAO and UBAO results are better on average for the DH-WFS. We quantify these statements in more detail in Section 4.

With Fig. 3 in mind, recall that increasing the Rytov number $\sigma _\chi ^{2}$ ultimately results in an increase in the number of branch points and the structure of the hidden-phase component to the overall phase function. Since we make no effort to sense and correct for the hidden-phase component in the wave-optics simulations, the DM never compensates for a critical component of the overall phase function. In addition, comparing Fig. 3(a) to Fig. 3(b), we notice that in moderately strong scintillation conditions the ideal-point-source beacon offers modest performance with both the SH-WFS and the DH-WFS, but both CBAO and UBAO offer performance that is only marginally better than using an uncompensated focused HEL scoring beam. This is due to the fact that the outgoing focused BIL beams in CBAO and UBAO spread more with stronger up-link scintillation, which in turn leads to more speckles across the simulated pupil plane and less isoplanatic least-squares phase estimates. That, coupled with the uncompensated hidden-phase component, leads to poor performance.

## 4. Results

Provided the trade-space exploration performed in the previous section, here we extract the time-averaged behavior for both the SH-WFS and the DH-WFS, in addition to each compensation technique. In particular, we calculate a mean of the last half of each peak Strehl ratio sequence to obtain the time-averaged peak Strehl ratio, which assumes that the leaky integrator control law has reached steady-state performance. Examining Fig. 4, we clearly see that the time-averaged peak Strehl ratio decreases as Rytov number increases for the CBAO and UBAO cases for both the SH-WFS and the DH-WFS, which is to be expected. Recall that an increase in the Rytov number corresponds to an increase in the strength of scintillation. Additionally, the CBAO and UBAO cases are strongly affected by increasing Rytov numbers because of the fact that both compensation techniques use an extended beacon which itself has been subject to increasing strengths of up-link scintillation (cf. Fig. 1). This outcome results in more spread in the extended beacon and thus more speckles in the circular pupil, in addition to a less isoplanatic least-squares phase estimate.

Although the effect is small, it appears that the DH-WFS offers higher time-averaged peak Strehl ratios when compared to the SH-WFS, except in the ideal-point-source case with weak scintillation conditions. We attribute this outcome to the aforementioned C-tilt, G-tilt anisoplanatism discrepancy that exists for the SH-WFS [7]. For the ideal-point-source beacon, the irradiance fluctuations are due to down-link scintillation alone, and performance for the SH-WFS decreases more rapidly than for the DH-WFS. In particular, the SH-WFS begins to offer worse results than the DH-WFS at a Rytov number of around 0.25, which is when irradiance nulls start to appreciably appear in the ideal-spherical-wave return [7]. For the CBAO and UBAO cases, the scattered return always contains speckles and thus will always contain irradiance fluctuations resulting in C-tilt, G-tilt anisoplanatism. This last point is a possible explanation as to why the DH-WFS outperforms the SH-WFS on average for CBAO/UBAO.

In order to quantify the benefits of CBAO relative to UBAO, we calculated the percent gain for each turbulence scenario simulated (cf. Table 1). Here,

## 5. Conclusion

The results of this paper quantified the benefits of CBAO relative to UBAO in the presence of weak to moderately strong scintillation conditions using wave-optics simulations. Throughout, we presented results for both the SH-WFS and the DH-WFS. The performance associated with each noiseless sensor was largely comparable. We expected this outcome, since the sensors had similar subaperture sampling and both used least-squares phase reconstruction. In general, the DH-WFS performed marginally better than the SH-WFS except for the case of an ideal-point-source beacon with weaker scintillation conditions.

When comparing CBAO to UBAO across all the turbulence scenarios studied here, we found an average performance boost of 26% for the DH-WFS and 17% for the SH-WFS when using CBAO as opposed to UBAO. Specifically, CBAO offered the largest relative performance gains when $0.167\leq \sigma _\chi ^{2}\leq 0.300$ for the SH-WFS and $0.233\leq \sigma _\chi ^{2}\leq 0.367$ for the DH-WFS, where $\sigma _\chi ^{2}$ denotes the Rytov number. These results indicate that CBAO with traditional AO techniques is best suited for weak to moderately strong scintillation strengths. Such baseline performance expectations are promising and pave the way forward for future research efforts. As stated in the introduction to this paper, these research efforts will, for example, help to evaluate the potential benefits of branch-point tolerant phase reconstructors. These baseline performance expectations will also help systems engineers to perform future system-level trade studies via link-budget analyses, which again make use of Strehl-ratio-dependent scaling laws. Future trade studies of this kind will allow systems engineers to determine whether the potential increase in hardware complexity associated with CBAO, in comparison with UBAO, is worth implementing into a directed-energy laser system.

## Acknowledgments

The authors of this paper would like to thank the Joint Directed Energy Transition Office for sponsoring this research, as well as J. R. Crepp and D. E. Thornton for many insightful discussions regarding the results presented within.

## Disclosures

The authors declare no conflicts of interest.

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**22. **T. J. Brennan is the sole author of WavePlex with correspondence to the following address, Prime Plexus, 650 N Rose Drive, #439, Placentia, CA, USA 92870.

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**26. **M. F. Spencer, “Wave-optics investigation of turbulence thermal blooming interaction: II. Using time-dependent simulations,” Opt. Eng. **59**(8), 1–18 (2020). [CrossRef]