## Abstract

Adaptive-optics (AO) systems correct the optical distortions of atmospheric turbulence to improve resolution over long paths. In applications such as remote sensing, object tracking, and directed energy, the AO system’s beacon is often an extended beacon reflecting off an optically rough surface. This situation produces speckle noise that can corrupt the wavefront measurements of the AO system, degrading its correction of the turbulence. This work studies the benefits of speckle mitigation via polychromatic illumination. To quantify the benefits over a wide range of conditions, this work uses a numerical wave-optics model with the split-step method for turbulence and the spectral-slicing method for polychromatic light. It assumes an AO system based on a Shack–Hartmann wavefront sensor. In addition, it includes realistic values for turbulence strength, turbulence distribution along the path, coherence length, extended-beacon size, and object motion. The results show that polychromatic speckle mitigation significantly improves AO system performance, increasing the Strehl ratio by 180% (from 0.10 to 0.28) in one case.

## 1. INTRODUCTION

Turbulence in the atmosphere causes optical-phase perturbations that degrade imaging and beam propagation over long paths. To improve performance, optical devices sometimes use adaptive-optics (AO) systems to measure and correct some of the optical-phase perturbations caused by turbulence. Such AO systems benefit the following applications: astronomy, remote sensing, LADAR, power beaming, object tracking, free-space optical (laser) communication, and directed-energy laser systems [1–8].

To correct for turbulence, the AO system must first measure the associated optical-phase perturbations. It does so by using a wavefront sensor to measure a reference wave that has passed through the turbulence. If the object is uncooperative and does not provide a reference wave, then the AO system must create a reference wave by projecting a focused laser beam [7,9]. Such a laser beam has significant width after propagating to the object. Thus, it is known as an extended beacon. The reflection of the extended beacon off the object provides the reference wave for wavefront sensing. Unfortunately, due to the width of the extended beacon, the object’s rough surface gives rise to significant speckle noise. This speckle noise can corrupt the wavefront measurements and degrade the AO system’s correction of the turbulence.

Speckle occurs when partially coherent light reflects off a rough surface. Because the roughness of most surfaces is greater than the wavelength of light, such surfaces add spatially random phase to the light upon reflection. After propagation, the reflected wave breaks up into regions of constructive and destructive interference, causing the bright and dark patches known as speckles. Notably, speckle occurs in both the aperture plane and the image plane of a wavefront sensor, and both can be problematic.

In this work, we focus on a single method for speckle mitigation. It is known as polychromatic speckle mitigation because it uses polychromatic illumination to reduce the speckle noise. Researchers sometimes call this approach wavelength diversity, linewidth broadening, temporal coherence reduction, or frequency compounding [10–15]. It is one of a number of methods for speckle mitigation, including angle diversity, polarization diversity, temporal integration, and image processing [10–12,16–18].

Polychromatic speckle mitigation offers several potential advantages for AO systems [15,19–22]. It does not reduce the spatial resolution or bandwidth of the wavefront measurements, which is beneficial for keeping up with the spatial and temporal changes in the turbulence effects [23]. In addition, it allows a near-diffraction-limited beacon spot size, which is important for the minimization of beacon anisoplanatism (see Section 2.C) [24–28]. For these reasons, polychromatic speckle mitigation might be the optimal type of speckle mitigation for AO systems that experience extended beacons.

This work quantifies the benefits of polychromatic speckle mitigation for AO systems with extended beacons. Such a quantification is not available in the literature. This work shows that polychromatic speckle mitigation significantly improves AO system performance. The specific conditions used here are aligned with directed-energy laser systems, but they are also directly applicable to object tracking and remote sensing. While we published some of the results shown here in a previous conference paper [19], this work considers a much wider range of turbulence conditions. These conditions include both weak and strong turbulence as well as different distributions of turbulence strength along the path. Further, we now analyze the results in terms of the number of coherence lengths per resolution cell (the smallest resolvable feature on the object under diffraction-limited conditions), which makes it easy to translate our results to other scenarios.

In what follows, Section 2 provides the background needed to understand this work. It includes discussions of the object-subaperture Fresnel number, the impact of speckle on wavefront sensing, the effects of beacon anisoplanatism, and the performance metrics. Further, it introduces the concept of the number of coherence lengths per resolution cell. Section 3 then discusses the wave-optics model, including the parameters and assumptions, the spectral slicing method, and our treatment of object motion. Next, Section 4 shows the results. They indicate that polychromatic speckle mitigation can largely eliminate speckle noise under realistic conditions. The associated improvements to AO system performance are quite large.

## 2. BACKGROUND

This section reviews the following topics: object-subaperture Fresnel number, speckle in wavefront sensing, beacon anisoplanatism, performance metrics, and coherence lengths per resolution cell.

#### A. Object-Subaperture Fresnel Number

To understand the amount of speckle-induced noise present for a particular wavefront sensor geometry, it is useful to consider a metric known as the object Fresnel number ${N_F}$. It is defined as

where $D$ is the aperture or subaperture diameter (whichever is appropriate), $T$ is the width of the object or the width of the extended beacon at the object (whichever is smaller), $\lambda $ is the wavelength, and $Z$ is the distance between the object and the aperture [15]. In this work, we use the subaperture diameter for $D$ because we are dealing with a Shack–Hartmann wavefront sensor (SHWFS) [29,30]. Thus, we call ${N_F}$ the object-subaperture Fresnel number for clarity. The object-subaperture Fresnel number defines the number of resolution cells across the object under diffraction-limited imaging conditions. It also quantifies the number of speckles across each subaperture [15]. Therefore, it determines the amount of the speckle-induced noise, as we discuss further in the next section.#### B. Wavefront Sensing and Speckle

Perhaps the most common type of wavefront sensor is the SHWFS [29,30]. It measures local wavefront slopes by using an array of subapertures [31]. Unfortunately, the irradiance and phase variations of speckle cause noise in the measurements of the local wavefront slopes [20]. Much of this noise passes through the reconstructor and into the estimate of the total wavefront.

Higher ${N_F}$ results in more speckle noise as shown in Fig. 1 [20]. This figure shows the speckle-induced, root-mean-square (RMS) slope-measurement error versus ${N_F}$ for fully coherent beacon illumination. We created this plot using numerical wave-optics simulations as we discuss in Section 3. Further, the ${N_F}$ values shown use the subaperture diameter for $D$, as is appropriate when analyzing a SHWFS. The slope error increases with ${N_F}$. It is small for ${N_F} \ll 1$, and it approaches an asymptote for ${N_F} \gg {1}$. The knee in the curve occurs at about ${N_F} = {1}$, where the speckle width is equal to the subaperture diameter. Because of the strong relationship between the ${N_F}$ and the strength of the speckle-induced noise, we consider a range of values for the ${N_F}$ in the numerical experiments of this work.

#### C. Beacon Anisoplanatism

Speckle noise occurs when both the object and the beacon have significant width. This condition also produces beacon anisoplanatism. Thus, the results that we show later in Section 4 include both speckle and beacon anisoplanatism, and we now provide a brief explanation of the latter effect.

Beacon anisoplanatism degrades wavefront sensor measurements. Consider that the wavefront sensor’s subapertures accumulate energy from all illuminated points on the object. When the optical distortions caused by the turbulence change significantly over the width of the beacon, the light from each point of the extended beacon experiences a slightly different turbulent path. This condition is known as beacon anisoplanatism. When the light from all the points combines, it degrades the estimate of the phase relative to the estimate that one would obtain using an ideal point-source beacon. Such degradation of the measurements results in degraded AO compensation of the turbulence [24–28].

To quantify the amount of beacon anisoplanatism, we use the isoplanatic angle. The isoplanatic angle defines the angular extent over which the turbulence effects are approximately constant. It is defined as

#### D. Performance Metrics

To assess the performance of the AO compensation of the turbulence, we use the following two metrics: Strehl ratio and normalized power in the bucket (PIB). Strehl ratio is a common metric in the imaging and beam projection communities. It compares the performance of a realistic optical system (degraded by effects such as turbulence, speckle-induced noise, and beacon anisoplanatism) to that of a diffraction-limited optical system [33]. In the context of beam projection, Strehl ratio $ S $ is defined as

where ${I_{\rm on \hbox{-} axis, real}}$ is the mean on-axis irradiance at the object for a beam projected from the realistic optical system and ${I_{\rm on \hbox{-} axis, diff \hbox{-} lim}}$ is the mean on-axis irradiance for a beam projected from the equivalent diffraction-limited optical system. A Strehl ratio of unity indicates ideal performance, while values below unity indicate degraded performance.The second metric is normalized PIB. It is another common metric in the area of beam projection. Non-normalized PIB measures the power entering a region of defined width (i.e., the bucket) centered on the optical axis at the object [29]. In this work, we normalize the PIB by dividing by the PIB of an equivalent diffraction-limited system. Thus, we define the normalized PIB as

where ${{\rm PIB}_{\rm real}}$ is the mean PIB of the realistic system and ${{\rm PIB}_{\rm diff \hbox{-} lim}}$ is the mean PIB of the equivalent diffraction-limited system. This quantity is sometimes called encircled energy, but that term is more often associated with imaging than with beam projection [34]. In this work, we use a circular bucket shape and define its diameter in terms of the diffraction angle as ${2.5}\lambda Z/D$.Notably, these performance metrics apply only to the scoring beam, not to the laser beam that creates the extended beacon. Unlike the scoring beam, we assume that the extended beacon is not compensated by the AO system. Thus, the metrics do not apply to it.

#### E. Coherence Lengths in a Resolution Cell

To make our results more universal, we plot them against the number of
coherence lengths within the effective depth of a diffraction-limited
resolution cell. This unitless metric provides an estimate of the
strength of the speckle when polychromatic speckle mitigation is used.
Through empirical analysis, Van Zandt *et al.* found the effective depth of a diffraction-limited
resolution cell that yields the smallest error in estimated speckle
strength over a wide range of conditions [13]. The number of coherence lengths $ N_c $ within that effective depth is

## 3. NUMERICAL MODEL

This section discusses the wave-optics model, the polychromatic model, and the parameters used in this work.

#### A. Wave-Optics Model

Analytical treatments for speckle noise are limited. No single theory includes the effects of polychromatic speckle mitigation, turbulence, and beacon anisoplanatism together [15]. To include all of those effects, we use numerical wave-optics simulations in this work. We employ the common split-step method with the angular-spectrum propagator [33]. Thus, we use a set of phase screens to add the effects of turbulence. Further, we use the spectral-slicing method to include polychromatic light (see Section 3.B or Ref. [15]).

The angular-spectrum propagator and the spectral-slicing method both impose a number of sampling requirements that a simulation must meet to produce accurate results. Authors such as Schmidt define the sampling requirements of the angular-spectrum propagator [33]. In this work, our parameter selections meet those requirements with one exception. In wave-optics simulations, the nature of the fast Fourier transform (FFT) algorithm can allow some light to “wrap around” the propagation grid and enter the region of interest from the opposite side. The community calls this effect aliasing. It is a nonphysical artifact of the simulations, and one of the sampling requirements ensures that it does not occur. However, instead of meeting that sampling requirement, we spatially filter the reflected light in the object plane to minimize the risk of aliasing in the simulations [33,35].

The sampling requirements for the spectral-slicing method for partially mitigated speckle are defined in Refs. [15] and [20]. We meet all of those requirements and thus ensure that the simulation error due to any single effect is less than 1% (in the image-plane speckle contrast). Again, there is one exception. We slightly relaxed the requirement on the number of spectral slices. Our selections are sufficient to keep the associated error less than 2% rather than less than 1%. Specifically, the numbers of spectral slices used are 151, 75, 39, 21, 11, 9, 7, and 3 for the coherence lengths defined in Section 3.C, in the same order. Further, the spectra are Gaussian. We place all of the spectral slices within a span that is 3 times the full width at half-maximum (FWHM) of the spectra. In this way, we ensure sufficient accuracy of the results.

#### B. Polychromatic Model

A number of methods can add the effects of polychromatic light to wave-optics simulations, including depth slicing, spectral slicing, Monte Carlo averaging, and full time-evolving treatment [15,36]. As stated previously, we use the spectral-slicing method in this work. The spectral slicing method breaks the light’s spectrum into a series of discrete wavelengths. It propagates the light at each wavelength to the object, where the object’s macroscopic depth (due to its macroscopic shape) introduces deterministic phase shifts that depend on the specific wavelength [15]. The object’s surface is also microscopically rough (relative to the wavelength), which introduces additional phase shifts onto the reflected light. When the reflected light propagates back to the optical system, it forms a speckle pattern. Because the phase shifts imparted by the object depend on the wavelength, the speckle pattern also varies with the wavelength [15].

The spectral slicing method assumes that the light at each wavelength is incoherent with the light at all other wavelengths [12]. Thus, it forms the speckle pattern in the measurement plane as the weighted sum of the speckle irradiances at the various wavelengths, where the weighting comes from the spectrum. In this way, it mitigates the speckle strength appropriately for the given amounts of spectral width, object macroscopic depth, and object microscopic roughness.

#### C. Parameters

We selected the simulation parameters to represent a reasonable scenario for directed-energy, object tracking, or remote sensing applications. Table 1 summarizes the parameters. The wavelength is 1 µm, and the distance between the object and the aperture is 3 km. The total aperture diameter is 30 cm, while there are 10 subapertures (of 3 cm diameter each) across the total aperture. We apply a cutoff-type threshold to the SHWFS measurements. The threshold zeros out any portion of the measured irradiance that is below 10% of the peak value for each subaperture. Such thresholds are common with SHWFSs because they reduce the impacts of sensor noise and crosstalk.

To reconstruct the total wavefront from the local slope measurements, we use a least-squares reconstructor [4]. Then we use a closed-loop controller with a leaky integrator to apply the reconstructed wavefront to the deformable mirror (DM) [37]. The leak and gain parameters are 0.99 and 0.4, respectively. Because the wavefront sampling rate is 2 kHz, the closed-loop 3 dB bandwidth is 111 Hz.

We set the object slope angle to 5.22°. We choose that value because it provides round numbers of coherence lengths per resolution cell (see Table 1). It is also close to a worst case. Because greater slope produces more object depth, the slope angle has a direct impact on the effectiveness of polychromatic speckle mitigation [21]. More depth means more mitigation. While it is reasonable to presume that a typical object will exhibit a slope angle of about 45° on average, we use an angle of only 5.22° to provide something of a worst-case scenario for polychromatic speckle mitigation. As we show in Section 4, the results are still quite positive.

To provide results over a realistic range of conditions, we vary three parameters. They are the coherence length, the object-subaperture Fresnel number, and the object motion. We vary the coherence length over the following values: 1, 2, 4, 8, 16, 32, 64, and 1000 mm. Here we use the definition of coherence length employed by Mandel and Goodman (see Eq. 5.1-28 in [17]). Notably, the shortest coherence length of 1 mm requires a Gaussian spectrum of only 0.66 nm FWHM. Modern-day laser illuminators are capable of achieving that spectral width. Also note that the longest coherence length of 1000 mm is effectively full coherence under the conditions of this work.

We assume that the object is flood illuminated by an extended beacon that is not compensated by the AO system. Under this assumption, we can precisely compute the object-subaperture Fresnel number, which we adjust by changing the object’s width. We consider $ N_F $ values of 0.275, 0.525, 1.025, and 2.025. To understand the upper limit on this range of values, first consider that the SHWFS works well if $D \lt {r_0}$ [4], where $D$ is the subaperture diameter and ${r_0}$ is the Fried coherence diameter [32]. The SHWFS still functions for smaller ${r_0}$, but it suffers degraded performance. Next, consider that the width of an uncompensated beacon $T$ is approximately [29]

We assume an extreme case of $D = {2}{r_0}$, solve for ${r_0}$, and plug the result into Eq. (6) to get an expression for $T$. Next, we use $T$ to solve for ${N_F}$ via Eq. (1). The result is ${N_F}={2}$. Therefore, the largest value for ${N_F}$ that a SHWFS will encounter during operation is about 2 (cf. Fig. 1). This outcome defines the upper limit for the range of values that we consider in this work.Consider now the low end of the range of possible values for ${N_F}$. In Fig. 1, we showed that ${N_{F}} \ll {1}$ produces very low speckle noise. Such a case should not need speckle mitigation. Thus, we do not consider ${N_F} \lt {0.275}$, and the overall range for ${N_F}$ that we do consider is limited to 0.275–2.025.

Table 2 summarizes the four turbulence cases used in this work. All assume Kolmogorov turbulence. They include a weak-turbulence case, a strong-turbulence case, a down-looking case, and an up-looking case. Here, we use the definition that strong turbulence is Rytov ${\rm number} \gt {0.25}$. Note that the Rytov number is also known as the log-amplitude variance and the Rytov parameter [32,38]. In all cases, the Fried coherence diameter is 3 cm or larger. Therefore, the 3 cm subapertures resolve the turbulence-induced phase perturbations reasonably well. In addition to the Rytov number and the Fried coherence diameter, Table 2 also lists the isoplanatic angle and the Greenwood frequency for each case. We will refer back to these parameters to explain the results in Section 4.

We assume a constant crossing wind of 5 m/s. This assumption produces a maximum Greenwood frequency [23,29] of 126 Hz (for the strong-turbulence case). Because the bandwidth of the AO system is 111 Hz, which is about equal to the maximum Greenwood frequency, the AO system offers acceptable correction for the temporal variations in the turbulence distortions [23,29].

#### D. Object Motion

In addition to polychromatic speckle mitigation, this work also allows for speckle mitigation from temporal integration. This section discusses our model for the object’s motion and the associated speckle mitigation from temporal integration. When the object is moving relative to the optical system, the speckle will change over time. If the effective integration time of the optical system is long enough, then the system will integrate multiple speckle patterns together, mitigating the speckle noise.

Consider an AO system observing a moving object. The angle of the object’s surface relative to the line of sight changes over time. This change in the surface angle effectively changes the wavefront tilt of the light that reflects directly back towards the AO system [15]. Tilt in the object plane produces a translation of the speckle field in the aperture plane, which causes the image-plane speckle to boil. This boiling of the speckle eventually results in a speckle pattern that is independent of the original pattern. Thus, the speckle pattern decorrelates over time when the object is moving. Because an AO system is a closed-loop control system with finite bandwidth, it has an effective integration time that is much longer than the integration time of the sensor. If the speckle changes over the effective integration time of the AO system, then there will be significant speckle mitigation via temporal integration.

To investigate such speckle mitigation in this work, we must make some assumptions about the object’s motion. We assume that the object is an unmanned aerial system (UAS) flying at 30 m altitude with a speed of 40 m/s. If the optical system is at 2 m altitude and the UAS flies directly towards the optical system (at 3 km range), then the angular rate of the UAS is 0.125 milliradians/second relative to the line of sight. Alternately, if the UAS starts from the same position but flies perpendicular to the line of sight rather than towards the optical system, then its angular rate is much higher at 13.3 milliradians/second. Note that the object’s motion during the 30 ms simulations is only 1.2 m, which is negligible except for the decorrelation of the speckle over time due to the angular rate. The two angular rates defined above form the lower and upper bounds, respectively, for this scenario. While this scenario does not exactly match any of the scenarios defined in Table 2, it allows us to place plausible bounds on the object’s angular rate. We then model the object’s motion by adding the appropriate time-dependent tilt in the object plane, allowing us to assess the impact of this type of speckle mitigation (see Section 4.A).

To simplify the analysis, we use a scaled value for the angular motion of the object when it is flying perpendicular to the line of sight. For this case, the phase perturbations caused by the turbulence change more rapidly (i.e., the Greenwood frequency is higher) than they do when the object flies directly towards the optical system. Thus, a real system would need to increase the AO bandwidth to keep up with the increased rate of turbulence changes. However, we are only concerned with speckle mitigation in this work. The speckle mitigation due to temporal integration only depends on the ratio between the AO bandwidth and the angular rate. Therefore, we avoid the complexity of changing the AO bandwidth for each case by instead scaling the object’s angular rate such that the ratio between the AO bandwidth and the object’s angular rate is correct. Thus, the speckle mitigation is correct. Given our assumption of a 5 m/s crossing wind for all cases, the scaled value for the angular rate for flight perpendicular to the line of sight is 4.17 milliradians/second. Therefore, the two angular rates that we use in this work are 0.125 milliradians/second and 4.17 milliradians/second.

## 4. RESULTS

This section provides the results of this work for the various conditions described previously in Tables 1 and 2.

#### A. Vacuum, Monochromatic

First, we present some results for vacuum conditions to serve as a baseline before we add turbulence. Figure 2 shows the average nPIB and Strehl ratio versus the object-subaperture Fresnel number for both of the object angular rates. The conditions are monochromatic (coherent) illumination and no turbulence. Note that in simulation, it is possible to achieve completely monochromatic (coherent) illumination. The bars about the data points indicate the 95% confidence intervals. The confidence intervals are very small here due to the large number of random realizations (1600) of the object’s roughness, which produce random speckle realizations. Additionally, we apply both ensemble and time averaging to reduce uncertainty [19]. Note that the nPIB in Fig. 2(a) falls off with increasing ${N_F}$ due to the increasing speckle noise (cf. Fig. 1). Further, the higher angular rate yields better performance than the lower angular rate due to the increased speckle mitigation from temporal integration. While the higher angular rate certainly improves performance, the nPIB is still well below the ideal value of unity. Therefore, the remaining speckle noise is significant.

Figure 2(b) is very similar to Fig. 2(a), but it shows the Strehl ratio instead of the nPIB. The trends are much the same as before, with performance dropping as ${N_F}$ increases but rising for the higher angular rate. Note that the Strehl ratio results are lower than the nPIB results. In general, the Strehl ratio is a more sensitive metric than nPIB. However, the relevance of each metric depends on the particular application, so we will continue to show both.

#### B. Vacuum, Polychromatic

Now we switch from monochromatic illumination to polychromatic illumination. Figure 3 shows the average nPIB and Strehl ratio versus ${N_F}$ for each of the eight coherence lengths. These results are for the slower object angular rate. To make the results more universal, we normalize the coherence length by dividing the effective depth within the diffraction-limited resolution cell by the coherence length as described in Section 2.E. This calculation yields the number of coherence lengths within a resolution cell, which allows one to interpret the results independently of the particular scenario used here [20]. For Fig. 3, we use only 30 speckle realizations due to the much longer run times associated with polychromatic illumination. Thus, the 95% conference intervals are much more visible than they were in the previous plots, but they are still small to moderate in size.

In Fig. 3(a), note that 2 or fewer coherence lengths per resolution cell (i.e., centimeter-scale coherence lengths) do little to improve nPIB, but 8 or more coherence lengths per resolution cell (i.e., millimeter-scale coherence lengths) yield considerable improvements. By 32 coherence lengths per resolution cell (1 mm coherence length), the nPIB is approaching the ideal value of unity, even for the largest ${N_F}$ of 2.025. Recall that the object surface slope angle is only 5.22°, which is essentially a worst-case scenario for polychromatic speckle mitigation. Even so, the polychromatic light has largely eliminated the speckle effects by an achievable value of 32 coherence lengths per resolution cell (1 mm coherence length).

Figure 3(b) presents the results in terms of the Strehl ratio. The 95% confidence intervals are now larger due to the increased sensitivity of this metric, which leads to increased variability from realization to realization. Still, the same general trend is apparent. Once again, using 8 or more coherence lengths per resolution cell produces considerable improvements in performance.

#### C. Weak Turbulence

Up to this point, the results did not include the effects of turbulence. Now we add them. For the first set of turbulence conditions, the turbulence strength is constant along the path. It produces a Fried coherence diameter of 7.0 cm and a spherical-wave Rytov number of 0.11. Thus, it falls into the weak-turbulence category. Figure 4 shows the results for both the nPIB and the Strehl ratio. These plots include two new curves that did not appear in the previous plots, namely, “Point Source” and “Open Loop.” The open-loop results do not include any form of AO compensation, so they provide something of a lower bound on performance by indicating how bad it will be with no correction of the turbulence effects. On the other hand, the point-source results provide an upper bound. They assume that the beacon is a point source at the object, which is the ideal case. This case avoids the degradations caused by an extended beacon, namely, speckle noise and beacon anisoplanatism. Thus, the new curves effectively provide upper and lower bounds on performance.

Looking at the results, we see that the performance is noticeably worse than in previous plots due to the inability of the AO system to completely correct for the turbulence effects. There are many causes of the imperfect correction, including finite subaperture width (i.e., the spatial sampling of the turbulence effects), finite bandwidth, scintillation of the reflected light from the extended beacon, beacon anisoplanatism, and speckle noise. Note that the results with the point-source beacon are well below unity. Because the point-source beacon eliminates both the speckle and the beacon anisoplanatism, its results show the impact of the finite subaperture width, the finite bandwidth, and the beacon scintillation. The impact of those three effects is minor but significant.

Previously, the results of Fig. 3 showed that the effects of speckle are largely eliminated by using many coherence lengths per resolution cell. Yet, the differences between the point-source-beacon results and the extended-beacon results are significant in Fig. 4, even for 32 coherence lengths per resolution cell. The differences are mostly caused by beacon anisoplanatism. Therefore, they are particularly apparent when ${N_F}$ is large. Here, the isoplanatic angle is 7.3 µrad, so there are about 9 isoplanatic angles across the object when ${N_F}={2.025}$. For this case, using 32 coherence lengths per resolution cell increases the nPIB by about 69% (from 0.265 to 0.447) by largely eliminating the speckle. However, we would need to increase the nPIB by an additional 81% to reach the ideal performance of the point-source beacon. Consequently, we conclude that beacon anisoplanatism and speckle noise are about equally significant when ${N_F}={2.025}$.

Polychromatic speckle mitigation is particularly useful when ${N_F}$ is large. With ${N_F}={2.025}$, 32 coherence lengths per resolution cell yield a 69% increase in the nPIB and a 120% increase (from 0.0881 to 0.194) in the Strehl ratio relative to the fully coherent case. Without polychromatic illumination, the performance is comparable to the open-loop case, which means that the AO system has totally failed. If we were to increase $ N_F $ even further, we expect that the performance with AO would drop well below the open-loop performance due to stronger speckle noise and beacon anisoplanatism. Therefore, we conclude that the AO system might actually harm performance in some cases if polychromatic speckle mitigation is not used.

#### D. Strong Turbulence

For the strong-turbulence case, polychromatic speckle mitigation does not increase the performance as much as it did for the weak-turbulence case. Now the Fried coherence diameter is 3.0 cm, and the Rytov number is 0.45. The results are shown in Fig. 5. Even with an ideal point-source beacon, the nPIB and Strehl ratio are well below the ideal value of unity. The degradation is mostly due to scintillation. The presence of significant scintillation marks the crossover from weak turbulence to strong turbulence, and it harms the performance of a SHWFS [30]. Additionally, the Greenwood frequency is now higher, which also reduces the performance somewhat relative to the weak-turbulence case. Nevertheless, polychromatic speckle mitigation helps, especially when the object is small. For ${N_F}={0.275}$, reducing the coherence to 32 coherence lengths per resolution cell increases the nPIB by about 27% (0.220 to 0.280) and the Strehl ratio by about 82% (0.0736 to 0.134). Those increases are roughly half of what we saw for the weak-turbulence case.

In addition to higher scintillation and Greenwood frequency, the beacon anisoplanatism is also greater in strong turbulence, and it tends to dominate over speckle noise. For the weak-turbulence case, the isoplanatic angle was 7.3 µrad. Now it is only 3.2 µrad. For ${N_F}={2.025}$, there are about 21 isoplanatic angles across the object. As such, there is a larger amount of beacon anisoplanatism, which tends to dominate over speckle noise for this case. Thus, the benefits of polychromatic speckle mitigation are now greatest for the smallest ${N_F}$, where the beacon anisoplanatism is much less severe.

#### E. Slant Paths

We also consider two slant paths, one up-looking path and one down-looking path. In both cases, the strength of the turbulence follows the Hufnagel–Valley 5/7 profile. The specific conditions are given in Table 2.

The first slant path is an up-looking path. The Fried coherence diameter is 3.0 cm, while the Rytov number is 0.10. For this case, the benefits of polychromatic speckle mitigation are large, as shown in Fig. 6. Using 32 coherence lengths per resolution cell increases the nPIB by about 83% (0.236 to 0.432) and the Strehl ratio by about 179% (0.0984 to 0.275) for ${N_F}={2.025}$. These benefits are even larger than what we saw for the previous weak-turbulence case. To understand this result, consider that the turbulence is now weaker near the object than it is near the optical system. Therefore, the isoplanatic angle is larger for this case than it was for the previous weak-turbulence case, and the effects of beacon anisoplanatism are less significant than the effects of speckle. Under such conditions, the benefits of polychromatic speckle mitigation are large.

The final case is down looking. Now the Fried coherence diameter is 7.1 cm, while the Rytov number is 0.21. Because the Rytov number is near the crossover between weak and strong turbulence, it can be considered moderate turbulence. However, the isoplanatic angle is only 3.5 µrad, which is close to the value for the strong-turbulence case. So, while this case does not fall into the strong-turbulence category, the beacon anisoplanatism tends to dominate over the speckle noise, just as it did for the strong-turbulence case.

Figure 7 shows that the benefits of polychromatic speckle mitigation for the down-looking case are roughly half of what they were for the weak-turbulence and up-looking cases. For ${N_F}={2.025}$, reducing the coherence to 32 coherence lengths per resolution cell improves the nPIB by about 41% (0.246 to 0.348) and the Strehl ratio by about 73% (0.0682 to 0.118). These findings are similar to what we observed for the strong-turbulence case. However, because the present case does not involve the large amount of scintillation that is characteristic of strong turbulence, the performance with an ideal point-source beacon is much higher than it was for the strong-turbulence case. Therefore, although the dominance of the beacon anisoplanatism limits the benefits of speckle mitigation, the peak performance is still quite good.

## 5. CONCLUSION

This work quantified the benefits of polychromatic speckle mitigation for closed-loop AO systems with extended beacons. It assumed a SHWFS and used wave-optics simulations to investigate the impacts of speckle noise and beacon anisoplanatism over a wide range of reasonable conditions. Unmitigated speckle noise was a moderate to severe degradation on the performance of the AO system. However, reducing the coherence to eight coherence lengths per resolution cell greatly mitigated the speckle noise, and 32 coherence lengths per resolution cell largely eliminated the speckle-noise effects. The latter case improved the Strehl ratio by up to 180% (from 0.1 to 0.28). For the reasonable conditions used here, 32 coherence lengths per resolution cell required a 1 mm coherence length (a spectral width of 0.66 nm FWHM at a central wavelength of 1 µm), which is achievable. This finding was particularly significant given that the object slope angle was only 5.22°, which was effectively a worst-case scenario for polychromatic speckle mitigation. For the strong-turbulence and down-looking cases, the benefits of speckle mitigation were roughly half of what they were for the weak-turbulence and up-looking cases due to the increased beacon anisoplanatism. However, even for the harsher cases, the benefits were significant.

## Acknowledgment

The authors thank Terry J. Brennan of PrimePlexus for many insightful discussions on adaptive-optics systems and wave-optics modeling.

The views expressed in this paper are those of the authors and do not reflect the official policy or position of the U.S. Air Force, the Department of Defense, or the U.S. Government.

## Disclosures

The authors declare that there are no conflicts of interest related to this paper.

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