1. INTRODUCTION

Astronomers have long used adaptive-optics systems with wavefront sensors (WFSs) to correct for atmospheric turbulence [1,2]. However, the use of adaptive optics for certain other applications is not as established, and some of the associated difficulties are not yet solved. In particular, deep turbulence and extended non-cooperative targets pose serious problems for tactical systems [3–6]. A number of applications can experience these conditions, including remote sensing, target tracking, free-space optical communication, power beaming, and laser weapons. [6–9]. Certain advanced concepts offer promise for these situations, but they are not yet ready for widespread use. One example is wavefront sensing using digital holography, which is currently in the early stages of development [6].

In the meantime, conventional wavefront sensing approaches, such as a Shack–Hartmann WFS, may be able to overcome the challenges of extended non-cooperative targets. (We do not address the additional challenge of deep turbulence in this work.) Most adaptive-optics systems rely on a source of light at the target known as a beacon [10]. This beacon passes through the turbulent atmosphere, thus providing a reference wave for wavefront measurement. However, a non-cooperative target does not provide a beacon. Instead, the adaptive-optics system must create its own beacon by focusing a beacon-illumination laser onto the target, such that the diffusely reflected or scattered light provides the reference wave [5]. Unfortunately, the spatial extent of such a beacon introduces the harmful effects of speckle and beacon anisoplanatism [11–15]. In this work, we focus on speckle. In practice, speckle is caused by the roughness of the target’s surface, which randomizes the phase of the diffusely reflected light, causing regions of constructive and destructive interference known as speckles. Speckle severely degrades WFS performance by causing phase and irradiance variations. However, polychromatic illumination mitigates speckle. Researchers sometimes call this technique wavelength diversity, linewidth broadening, or temporal-coherence reduction [16,17].

In this work, we quantify the benefits of polychromatic speckle mitigation for a Shack–Hartmann WFS. While the basic concept is not new, it remains unclear exactly how much benefit it provides. To investigate the benefits, we rely on the spectral-slicing approach to polychromatic wave-optics simulations, allowing us to obtain results over a wide range of conditions. We explore both vacuum and weak-turbulence regimes. In order to quantify the speckle noise, even when turbulence is present, we introduce a metric referred to here as racetrack-mode strength in slope-discrepancy space. This metric allows us to gauge speckle strength, and the results show that polychromatic illumination greatly reduces speckle noise under realistic conditions.

In Section 2, we briefly review the concept of speckle noise in Shack–Hartmann WFS measurements. We then introduce the concept of racetrack-mode strength in slope-discrepancy space. Next, Section 3 discusses the wave-optics model, including the sampling requirements as they apply to wavefront sensing from the illumination of a non-cooperative target. Then, Section 4 presents results, including both wavefront-measurement error and modal analysis.

2. BACKGROUND

This section provides the background material needed to understand the following work. It covers Shack–Hartmann WFSs, speckle noise, speckle mitigation, slope discrepancy, racetrack mode, and target Fresnel number. Further, it summarizes the spectral-slicing simulation method that we use here.

A. Shack–Hartmann Wavefront Sensors and Speckle Noise

A Shack–Hartmann WFS divides the reference wave into a number of small subapertures that span the total aperture. Each subaperture images the beacon spot on the target. Because the system tries to keep the beacon spot small to reduce beacon anisoplanatism, these images are usually unresolved [2,18]. Any wavefront slope across the subaperture causes the images to shift off axis. By measuring the image shifts, the Shack–Hartmann WFS measures the wavefront slopes, which it then reconstructs to form an estimate of the continuous wavefront [1].

Such slope measurements experience noise due to the irradiance and phase variations of speckle. That said, polychromatic illumination can reduce these variations. Speckle, in practice, is mitigated by the combination of the short coherence length of polychromatic light and the depth of the target [16]. To provide a conceptual understanding, consider the case of an illuminated target with enough depth relative to the line of sight to contain several coherence lengths. Each coherence length that fits within the depth of the target forms an independent coherence region, and each region contributes an independent speckle pattern. These patterns add incoherently, thus averaging out some of the speckle effects. Though not rigorous, this description concisely explains polychromatic speckle mitigation.

Numerous researchers have studied such polychromatic speckle mitigation, but until recently, they all assumed a well-resolved image of the target in their publications [16,19–24]. Because Shack–Hartmann WFSs typically produce unresolved images, almost none of the previous publications are directly applicable to such systems. Recently, one paper considered the case of unresolved imaging [17]. It used a lab experiment to show that two polychromatic wave-optics methods are valid even for such conditions. In this work, we use one of those methods, namely, the spectral-slicing approach.

Additionally, we make use of one of the analytical solutions, even though it assumes a well-resolved target. Later, we show that this equation for well-resolved imaging is useful for normalization in the present work. We could also use the diffraction angle for such normalization, but we choose to use an equation from previous speckle mitigation research to align our work with what has been done in the past. In Ref. [24], the authors approximate the effectiveness of polychromatic speckle mitigation by computing the number of coherence lengths that fit within a diffraction-limited resolution cell. They determine the effective width of the resolution cell such that their approximate solution best agrees with more accurate theory. The target depth associated with this effective width is

 

Fig. 1. Wavefront sensing geometry. The Shack–Hartmann wavefront sensor consists of $N_s$ subapertures (a.k.a. lenslets) of width $d$ across the total aperture of diameter $D$. In this work, we set the target width $T$ to $N_b$ times the diffraction width of the total aperture. Further, the target possesses a slope angle $\theta$ relative to the imaging and illumination line of sight, thus introducing target depth and speckle mitigation when the illumination is polychromatic. Under these conditions, the target Fresnel number $N_F$ is simply $N_b/N_s$. Additionally, the figure shows $Z_{\mathrm{cell}}$, the effective depth of the target assuming diffraction-limited conditions and a well-resolved target. In this work, the target is not well resolved. However, we will use $Z_{\mathrm{cell}}$ as a normalization factor.

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B. Slope Discrepancy Space

When both speckle and turbulence are present, measuring speckle strength becomes difficult. Under such conditions, we use racetrack-mode strength in slope-discrepancy space. In 2000, Tyler introduced the elegance of slope discrepancy, which is the rotational component of the phase [25]. To find the slope discrepancy, we first reconstruct the wavefront from a set of slope measurements using a least-squares reconstructor. Then, we differentiate the reconstructed wavefront to find a new set of slopes, and compare the two. The difference is the slope discrepancy [25]. Mathematically, the slope discrepancy $\delta$ is given by

In general, the slope discrepancy is caused by factors such as read-out noise, fitting error, and branch points [25]. Branch points are particularly relevant to this work. A branch point forms where the irradiance equals zero in the propagating optical field, and it can be caused by either target-induced speckle or turbulence-induced scintillation [26]. Around the branch point, the phase in the optical field circulates from 0 to $2\pi$, and a Shack–Hartmann WFS often struggles to adequately resolve such a circulation [3].

Only deep turbulence causes enough scintillation to produce a significant number of branch points. In this work, we do not consider cases of deep turbulence. Thus, the branch points are predominately caused by speckle. Because branch points introduce slope discrepancy, the slope-discrepancy space will allow us to investigate the strength of the speckle noise separate from the turbulence effects.

C. Racetrack Mode

The strongest speckle mode in slope-discrepancy space takes a distinctive shape known as racetrack (a.k.a. vortex) mode [27]. In this work, we obtain the modes by first converting a set of slope measurements to slope-discrepancy space via Eq. (3). Next, we compute the covariance matrix. Finally, we apply a singular-value decomposition (SVD) to determine the modes (eigenvectors) and their strengths (singular values). The slope discrepancy is the rotational component of the phase, which is caused by factors such as the branch points introduced by speckle. Therefore, it is natural to expect that the portion of slope discrepancy caused by speckle will take the form of rotations. As illustrated in Fig. 2, the racetrack mode circulates around the WFS in a single track, making it the largest possible rotation. Racetrack mode tends to dominate the slope discrepancy caused by speckle. The figure shows a circular aperture with no obscuration, but the racetrack mode dominates even for a square aperture with a central obscuration. There is one caveat; specifically, if the target Fresnel number is large, the strongest speckle mode is not the racetrack mode. Rather, it involves a number of smaller rotations, as we will see later in Section 4.A.2.

 

Fig. 2. Racetrack mode in slope-discrepancy space. Each arrow represents the slope discrepancy in the measurement of a single subaperture of a Shack–Hartmann WFS. Arrow size and direction indicate the magnitude and direction of the discrepancy, respectively. The racetrack mode shown here has strength equal to the root-mean-square (RMS) value of racetrack mode for a target Fresnel number of 0.3 and a vacuum path. The WFS has 15 subapertures across the total aperture and no obscuration. Racetrack mode forms a continuous circulation around the aperture. The arrow to the right of the racetrack-mode plot shows the relative size of the RMS slope measurement for comparison.

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As we show later, the racetrack mode is one of the weakest modes induced by turbulence, at least for conditions of weak turbulence, which do not cause many branch points. Thus, it provides a way to differentiate between the effects of turbulence and speckle. In Section 4, we use the racetrack mode to investigate speckle strength as a function of the coherence of the beacon-illumination laser.

D. Target Fresnel Number

A second metric, the target Fresnel number, provides insight into two types of speckle-induced noise: wavefront-measurement error and dropouts. It also indicates the resolvability of the target [16,17]. It is defined as

A large $N_F$ indicates that the target is well resolved by a diffraction-limited imaging system. If $N_F<1$, then the target cannot be resolved [16]. If we look at this metric from a different perspective, we can also say that when $N_F<1$, the speckle size is larger than the subapertures. In this case, only one speckle usually influences each subaperture, and if that speckle happens to be dark, then signal strength will be far below average, potentially causing a dropout.

Recall that a Shack–Hartmann WFS measures the wavefront slopes across each subaperture. The target Fresnel number influences the strength of the slope-measurement error due to speckle. For $N_F\ll 1$, the speckle size is much larger than a subaperture, and the speckle irradiance and phase are nearly constant across each subaperture. In this case, the slope-measurement error is very small. However, when $N_F$ is large, multiple speckles affect each subaperture, and the speckle noise can be severe. Figure 3 shows the relationship between $N_F$ and speckle-induced slope-measurement error. This plot displays numerical results from wave-optics simulations. Here, the illuminator is fully coherent. Note that the slope-measurement error increases rapidly when $N_F$ is small, but it increases more slowly when $N_F$ is large. The knee in the curve occurs around $N_F=1$. Additionally, error is small for $N_F\ll 1$.

 

Fig. 3. Speckle-induced slope-measurement error versus target Fresnel number. In this work, we compute the speckle-induced slope-measurement error by taking the root-mean-square (RMS) of the error in each subaperture’s measurement in both $x$ and $y$. In this figure, the illuminator is fully coherent. As $N_F$ approaches zero, so does the error. For large $N_F$, the error is quite large at greater than 0.2 waves RMS.

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Before leaving the subject of target Fresnel number, it is worth mentioning that most adaptive-optics systems operate at $N_F$ of $\sim 2$ or less [16]. Assuming a non-cooperative target, the system must project a beacon-illumination laser. The system will keep the resulting beacon spot as small as possible to minimize beacon anisoplanatism [11,13,14]. Generally, beacon size is a small multiple of the diffraction limit, thus restricting the target’s effective width. The WFS will also employ some finite number of subapertures across the total aperture, which limits the subaperture size and further restricts $N_F$.

To understand the consequences of these restrictions, we let the effective target width (the smaller of the beacon width and the target width) be $N_b$ times the diffraction limit, such that it is

E. Spectral Slicing Method

In this work, we use the spectral-slicing method for polychromatic wave-optics simulations [16,28]. The basic operation is quite simple. This method breaks the source’s spectrum into a series of discrete wavelengths, as shown in Fig. 4. Next, it separately propagates the light at each wavelength to the target, where the interaction between the light and the target’s depth depends on wavelength. The backscattered light then propagates to the aperture. From there, it proceeds to the image plane, where the speckled image is slightly different at each wavelength. These speckled images add incoherently, thus creating a partially speckled image of the target.

 

Fig. 4. Gaussian spectrum broken up into discrete wavelengths for the spectral-slicing method. Here, the center wavelength is 1.064 μm, while the bandwidth yields a coherence length of 1 cm.

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A recent paper validated this method for cases of unresolved imaging [17]. In fact, it is the most accurate of three numerically efficient approaches [17]. One could also perform a full time-evolving treatment of the situation [29], but doing so greatly increases run times, which are already several weeks when run in parallel on multiple high-end computers. Therefore, we use the more efficient spectral-slicing method. Notably, it does assume that the light is uncorrelated from wavelength to wavelength, but this assumption is usually justified [28,30]. Further, to produce accurate results, one must keep the spacing of the wavelengths small and meet several other sampling requirements, as discussed in [16].

3. MODEL SETUP AND EXPLORATION

This section defines the wave-optics model used to obtain the numerical results shown later in this work. It presents the model parameters, speckle and turbulence conditions, and Shack–Hartmann WFS details. Further, it defines the sampling requirements as they pertain to the specific conditions of wavefront sensing.

A. Model Parameters

The conditions that we use here are meant to represent a typical scenario involving a non-cooperative target and a Shack–Hartmann WFS. As such, the beacon wavelength is 1.064 μm, the aperture diameter is 65 cm, and the range to target is 5 km. There are 15 square subapertures across the total aperture, each with a width of 4.29 cm. Each subaperture sees an angle of $6\lambda /d$ and has 13 pixels in its image plane. Further, we use a threshold. The threshold is a cutoff type, which sets any value below 1/10th of the peak to 0 during post processing. Shack–Hartmann detectors usually employ such thresholds, as they reduce the impact of read-out noise and crosstalk. Further, the coherence length of the beacon illuminator is set to one of the following eight values: infinity, 6 cm, 4 cm, 2 cm, 1 cm, 5 mm, 2.7 mm, or 1.3 mm. For those coherence lengths, we use 1, 6, 8, 10, 20, 40, 76, and 150 spectral slices, respectively. The numerical grid consists of 512 points on a side with point-to-point spacing of 3.3 mm. Finally, to obtain a large number of independent speckle and turbulence realizations, we use 4,000 realizations of the target’s rough surface for the vacuum cases, while we use 1,000 realizations of both turbulence and surface roughness for the turbulence cases.

Regarding atmospheric turbulence, we use the common Hufnagel–Valley 5/7 profile. The imaging platform altitude is 3 m, and the target altitude is 1,500 m. Thus, the spherical-wave Fried coherence diameter is 7.4 cm, somewhat larger than a subaperture, while the spherical-wave and plane-wave Rytov numbers are 0.036 and 0.051, respectively, indicating that the turbulence is weak. Consequently, the simulated turbulence will cause little scintillation.

To approximate a compensated beacon while avoiding the extra assumptions and numerical expense of simulating closed-loop adaptive optics, we assume that the target is uniformly illuminated. Thus, we apply turbulence effects to the light only as it propagates back to the wavefront sensor. This approach allows us to approximate a compensated beacon (i.e., a beacon that is largely unaffected by turbulence while propagating to the target) while keeping the simulation run times reasonable. Further, it allows us to compute the target Fresnel number. With a precise and understandable definition of the target Fresnel number, we can quantify the dependence of speckle strength on that parameter. Therefore, we take a more theoretical but less practical approach in this work. Future work should expand upon this work by considering a true compensated beacon rather than the surrogate used here.

To control the target Fresnel number, we adjust the target’s size. Here, we use target Fresnel numbers of 0.35, 0.62, 1.05, and 2.0, all of which fall within the range of reasonable values found in Section 2.D. A realistic system could potentially operate at a lower $N_F$; however the sampling requirements do not permit lower $N_F$ values in these simulations without significantly increasing the run time (see Section 3.D). Additionally, the speckle noise is insignificant when $N_F$ is much below 0.35. Thus, we did not pursue lower $N_F$ values, but the range of $N_F$ used here is quite broad and covers much of the realistic parameter space.

The target is a square plate sloped at 5.27° relative to the illumination and imaging vectors. With such a small slope angle, the target’s depth is also very small, thus limiting the polychromatic speckle mitigation. Such a slope angle is quite pessimistic but reasonable. We chose this value to demonstrate that polychromatic speckle mitigation is effective, even under near worst-case conditions. Under more typical conditions of $\sim 45°$ slope, the amount of speckle mitigation increases by about a factor of 3. Thus, it is important to keep in mind that the conditions used here are something of a worst-case scenario for polychromatic speckle mitigation.

B. Speckle and Turbulence Models

We assume that the target’s surface is rough compared to the optical wavelength, as is usually the case. To be specific, we use a surface height standard deviation of 10 μm. Further, we assume that the target’s surface height is Gaussian distributed and uncorrelated from point to point in the numerical grid. Thus, we treat the roughness by simply generating a matrix of uncorrelated Gaussian random numbers with an appropriate standard deviation. To obtain results over many random realizations, we randomize the roughness from run to run, thus randomizing the speckle effects.

To treat the depth caused by the target’s slope angle, we compute the geometric depth at each grid point and add it to the depth due to roughness. Then, the optical path difference (OPD) experienced by the light is twice the physical depth due to the double-pass (a.k.a. out and back) geometry. To apply this OPD to the optical field, we divide the OPD by the wavelength and multiply by $2\pi$ to obtain the phase delay. Next, we apply the phase delay to the optical field. Because of the division by wavelength, the speckle-model results change with wavelength.

Because the target’s surface is rough compared to the optical wavelength, the reflection off the target is diffuse. Only a portion of the diffusely reflected light travels back to the aperture. The rest of the light travels in different directions. If such light reaches the edge of the numerical grid, the FFT-based propagator will wrap it around to the other side of the grid, where it will continue to travel at an angle. Such light is called stray light (a.k.a. aliased light). To avoid the numerical errors that can result from such stray light, we apply a filter to the reflected light. The filter transmits all light that falls within the inner 60% of the numerical grid (the inner 1.01 m). It removes all other light with a tapered filter that drops to zero transmission for the outermost 20% of the numerical grid. In this way, the numerical simulations avoid any errors that the stray light would otherwise cause.

For turbulence modeling, we use the common split-step beam propagation method (BPM) [31]. We lump the turbulence effects associated with each segment of the path into phase screens, while we generate the screens using a Kolmogorov spectrum [7]. We use five phase screens with equal-Rytov spacing. The split-step BPM propagates the light to the first screen and applies the phase effects of the first segment of the path. Next, it propagates the light to the second phase screen and applies the phase effects of the second segment. This process continues until the light reaches its destination. For the propagations themselves, we use the angular-spectrum method. The associated sampling requirements have been thoroughly investigated by authors such as Voelz and Schmidt [31,32].

C. Shack–Hartmann Model

The Shack–Hartmann model used here has 15 square subapertures in a subaperture-on-center configuration, as shown in Fig. 5. The outer aperture is circular, as is the central obscuration, which is quite large at 1/3rd the width of the outer aperture. This central obscuration slightly reduces noise in the measurements of the racetrack-mode strength.

 

Fig. 5. Wavefront-sensor geometry with square subapertures within a centrally obscured aperture.

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D. Sampling Requirements

Due to the specific conditions of Shack–Hartmann wavefront sensing, we needed to make small adjustments to the general sampling requirements for polychromatic wave-optics simulation. The adjusted requirements are listed in Table 1. They ensure that no single factor produces error greater than 1% in the image-plane speckle contrast. With wavefront sensing, the target width is usually smaller than the resolution cell. Thus, rather than controlling the number of samples per resolution cell, one needs to control the samples across the target. We found that 12 samples or more are sufficient to ensure accurate results, where we measured accuracy by comparison with simulations involving a very large number of samples. Also, there should be at least 12 samples across each subaperture. Further, accurate results require only two samples per speckle in the aperture plane but eight samples per speckle in the image plane. However, a typical Shack–Hartmann WFS uses only about 0.5 to two image-plane samples (pixels) per speckle to allow for the detection of weaker signals, thus extending the maximum range of operation. Consequently, we selected a value of 2.17 to provide more realism in the simulations.

Table 1. Sampling Requirements for Shack–Hartmann Wavefront Sensing of Non-Cooperative Targets

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4. RESULTS AND DISCUSSION

First, we present vacuum results that demonstrate the effectiveness of polychromatic speckle mitigation. Later, we include atmospheric turbulence to show that polychromatic speckle mitigation is unchanged by the turbulence, while racetrack mode provides an effective means to measure speckle strength.

A. Vacuum Results

In what follows, we present the vacuum results, which include speckle. In particular, we investigate slope-measurement error and perform a modal analysis with respect to the Shack–Hartmann WFS.

1. Slope-Measurement Error

To evaluate the impact of speckle on WFS measurements, we use root-mean-square (RMS) slope-measurement error. The slope-measurement error is the error in the estimate of the wavefront slope for each subaperture. In this work, we compute the slope-measurement error by taking the RMS of the error of each subaperture’s measurement in both $x$ and $y$. Because the read-out noise is zero here, the WFS should measure zero when no turbulence is present, and any non-zero values indicate that speckle noise is corrupting the measurements. Therefore, in this section, we define error as any non-zero slope measured by the Shack–Hartmann WFS under vacuum conditions.

Figure 6 shows the speckle-induced slope-measurement error versus the coherence for the conditions detailed in Section 3. We use Eq. (1) to normalize the coherence length. Therefore, the $x$ axis is $Z_{\mathrm{cell}}/l_c$, where $l_c$ is the coherence length. This parameter quantifies the number of coherence lengths within the effective depth of a diffraction-limited resolution cell. In (b), we normalize the results by the fully coherent values. The reduction in the slope-measurement error is quite significant by four coherence lengths per resolution cell (1 cm coherence length), and the error falls rapidly as the coherence continues to drop. With 15 coherence lengths per resolution cell (2.7 mm coherence length), the speckle-induced error drops by 56% for $N_F=0.35$. Notably, if the slope were 45° instead of only 5.27°, one could increase the coherence lengths by a factor of 10 and still obtain nearly the same benefits. Therefore, we conclude that reasonable numbers of coherence lengths per resolution cell greatly reduce the speckle-induced noise in wavefront measurements.

 

Fig. 6. Speckle-induced RMS slope-measurement error versus coherence. In (b), each curve is normalized by its maximum value. The target slope angle is only 5.27°, but error still falls rapidly as the number of coherence lengths per resolution cell ($Z_{\mathrm{cell}}/l_c$) increases above 4. Over the range of $N_F$ shown here, the impact of $N_F$ on the percentile reduction in error is rather weak. In fact, error reduction improves as $N_F$ drops from 1.05 to 0.35.

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In Fig. 6(b), the curves seem to follow a very similar trend regardless of $N_F$, except for $N_F=1.05$. That curve shows a statistically significant deviation from the trend. The 95% confidence intervals are small here, too small to plot, with a maximum range of $\pm 0.34\%$ (full coherence) and a minimum range of $\pm 0.07\%$ (minimum coherence). Therefore, the deviation for $N_F=1.05$ is statistically significant.

Figure 6(b) shows only a weak dependence on $N_F$. As $N_F$ decreases from 2 to 1.05, we see less benefit from polychromatic speckle mitigation. However, the benefit returns to near the original value as $N_F$ decreases further, first to 0.62 and then to 0.35. This finding is somewhat surprising given the fact that target depth decreases with $N_F$, which often means that polychromatic illumination will not provide as much benefit. However, for wavefront sensing, it is important to keep in mind that speckle causes irradiance variations in addition to phase variations, both of which contribute to slope-measurement error. For $N_F<1$, the speckles are larger than the subapertures, and error mostly comes from irradiance variations and abrupt changes in phase known as branch cuts [26]. Because polychromatic illumination reduces both quite rapidly, the benefits remain large even for small $N_F$. Although we did not test the smallest values of $\sim 0.03$ in this work, speckle noise becomes insignificant for $N_F$ much below 0.35 (see Fig. 3). Therefore, the benefits of polychromatic illumination change little over reasonable values of $N_F$, at least as long as those values are large enough to indicate significant speckle noise.

The key to understanding Fig. 6 involves the fact that we change $N_F$ only over a certain range, and we do so by changing the target width. Apparently, under these conditions, adjusting the target depth via the target width barely affects the percentage reduction in slope-measurement error. Of course, there are other ways to change the target depth, such as by adjusting the target’s slope angle. Thus, the results indicate that while the target width is not a significant parameter, two things are significant: the coherence length and the portion of the target depth separate from the target width. It is possible to separate target depth from width by computing it via Eq. (1), which assumes that the target width is infinite but effectively limited by the diffraction-limited resolution cell. Therefore, regardless of the actual target width, Eq. (1) yields a constant result. Yet, that equation does depend on subaperture size, range, wavelength, and slope, meaning that it quantifies depth as separate from width. By normalizing the coherence length on the $x$ axis in Fig. 6 by the depth within a resolution cell from Eq. (1), we capture all pertinent information regarding speckle mitigation in a single metric that we call $Z_{\mathrm{cell}}/l_c$. This metric defines the number of coherence lengths that fit within a diffraction-limited resolution cell.

One might inquire as to how much of the speckle-induced slope measurement error will be filtered out by the reconstructor in an adaptive-optics system. Because there are many types of reconstructors, and because the focus of this work is the WFS, we will not attempt to fully investigate that question here. However, we can draw one useful conclusion. Adaptive-optics systems often use a least-squares reconstructor [25]. A least-squares reconstructor ignores all of the slope discrepancy. The ratio of the standard deviation of the slope discrepancy over the standard deviation of the total slope-measurement error is about 0.59, which means that about 35% of the energy is in the slope discrepancy. A least-squares reconstructor removes that portion of the error. Therefore, a least-squares reconstructor removes at least $\sim 35\%$ of the energy in the speckle-induced slope measurement error. Later, we show how this percentage varies slightly with the target Fresnel number and the polychromatic speckle mitigation (see Section 4.B.2 and Fig. 11).

2. Modal Analysis

Figure 7 shows the three strongest speckle modes in slope-discrepancy space. The conditions are fully coherent illumination and $N_F=0.35$. We obtained this result by converting a set of slope measurements to slope-discrepancy space via Eq. (3), computing the covariance matrix, and then applying SVD to determine the modes (eigenvectors) and their strengths (singular values). The strongest mode is racetrack mode. The next two both involve a few large rotations. Because all the errors here are caused by speckle, these results show that the strongest speckle modes involve large rotations.

 

Fig. 7. Three strongest speckle modes (left to right) in slope-discrepancy space. Here, the illumination is fully coherent, while $N_F=0.35$. The strongest mode is racetrack mode, and the next two also involve large rotations.

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Figure 8 shows the strongest speckle mode for each target Fresnel number. At $N_F=0.35$ in Fig. 8(a), the racetrack mode is well defined. It becomes less well defined as $N_F$ increases, and by $N_F=2$ in Fig. 8(d), the strongest mode is no longer racetrack, but rather it involves several large rotations, similar to the second- and third-strongest modes at $N_F=0.35$. Notably, the SVD did not even identify the racetrack mode as one of the modes for $N_F=2$. However, the racetrack mode still possesses significant strength. We provide evidence to support this claim in Section 4.B.2 and Fig. 12.

 

Fig. 8. Strongest speckle mode for each of the four target Fresnel numbers (increasing from left to right) with full coherence. By $N_F=2$ in (d), the strongest mode is no longer racetrack.

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This finding is reasonable, because large $N_F$ means that speckle size is small. The size of the speckles is strongly correlated to the separations between the branch points, and it is the branch points that cause all the slope discrepancy here (in the absence of any noise, fitting error, or turbulence). Thus, we expect some correlation between the size of the speckles and size of the circulations in the strongest speckle mode. Therefore, while racetrack mode is not always the strongest speckle mode, it is for many wavefront sensing cases (when $N_F<1$). Further, even when the strongest mode is not racetrack, it is still similar in form.

It is also useful to determine whether polychromatic illumination produces similar modes. Figure 9 shows the strongest modes for 30 coherence lengths per resolution cell (1.3 mm coherence length). Note that the modes are still racetrack or similar, except that they are squished vertically. This vertical compression is caused by the fact that the target is a square plate tilted along the $x$ axis. Thus, the speckles shift along the $x$ axis as the wavelength changes across the source’s spectrum. Because speckles produce branch points, and branch points produce circulations in slope-discrepancy space, this shift in $x$ accentuates the horizontal features, while the vertical features tend to average out.

 

Fig. 9. This figure matches Fig. 8, except that the coherence length is now only 1.3 mm (30 coherence lengths per resolution cell). The strongest modes are still racetrack mode or similar, but they are squished vertically due to the shifting of the speckles in $x$ caused by the combination of target slope and polychromatic illumination.

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B. Turbulence Results

Having presented the vacuum results, we now add turbulence. The level of turbulence used here is quite significant, but it is best classified as weak turbulence, as it does not produce much scintillation.

1. Slope-Measurement Error

Figure 10 shows the slope-measurement error from both vacuum and turbulence simulations. To allow direct comparisons between the two, we remove the effects of turbulence by assuming that turbulence and speckle are independent. Under this assumption, we estimate the RMS error due to turbulence alone, $E_T$, by removing the speckle-only, fully coherent result, $E_{S,\mathrm{coh}}$, from the result with both turbulence and speckle included, $E_{T+S,\mathrm{coh}}$, in a root-sum-square (RSS) fashion as

 

Fig. 10. Slope-measurement error due to speckle versus the number of coherence lengths within each resolution cell’s depth. Both vacuum and turbulence-removed cases are shown. The two agree very well, indicating that turbulence does not significantly change the benefits of polychromatic illumination. However, for the two smallest target Fresnel numbers, some differences are visible at short coherence. In fact, the three smallest data points all exhibit differences that exceed the 95% confidence intervals (omitted for clarity). Thus, these results indicate a bit of interaction between speckle and turbulence.

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Even so, there are differences. Specifically, for short coherence with the two smallest $N_F$ values, the turbulence-removed results are noticeably lower than the vacuum results. These specific cases also involve the smallest errors shown with RMS values less than 0.1 waves. For this figure, the confidence intervals (omitted for clarity) are larger than they were for Fig. 6 due to the two RSS operations used to remove the turbulence effects. The differences between the vacuum and turbulence results usually fall within the 95% confidence intervals, which rise sharply to a maximum of $\pm 0.021$ waves when the error is at its smallest, which occurs at $N_F=0.35$ and $Z_{\mathrm{cell}}/l_c=30$. Still, three data points exhibit discrepancy greater than the confidence intervals. Interestingly, those three points are the three smallest values shown, which indicates that there is some small interaction between speckle and turbulence. Further, the differences show the same smooth trend at both $N_F=0.35$ and $N_F=0.62$, but this outcome could be caused by the fact that we use the same set of turbulence phase screens for all cases. Thus, within the confidence intervals of these simulations, turbulence seems to influence the effectiveness of polychromatic speckle mitigation, but only by a very small amount. The remaining uncertainty highlights the usefulness of a metric that inherently separates turbulence and speckle, such as racetrack-mode strength.

2. Modal Analysis

To directly separate speckle from turbulence, we turn to racetrack mode. First, it is important to note that racetrack mode exists in slope-discrepancy space, while most systems utilize either slope consistency (the irrotational component of phase) or both consistency and discrepancy [25,26]. If polychromatic illumination reduces one more than the other, then racetrack mode will not tell the whole story, and it will not be very useful as a metric for speckle strength. However, the results shown in Fig. 11 indicate that polychromatic illumination reduces both nearly equally. This figure shows the RMS slope-measurement error in discrepancy space divided by the total RMS error. Because this fraction is nearly constant with coherence, polychromatic illumination reduces both slope consistency and slope discrepancy almost evenly. Therefore, racetrack mode should provide a good indication of total speckle mitigation.

 

Fig. 11. RMS slope-measurement error in discrepancy space divided by the total RMS error versus coherence. These results show little change in this fraction. Thus, any metric that measures only slope consistency or discrepancy (such as racetrack mode) will still provide a good estimate of the overall trends.

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Figure 12 shows racetrack-mode strength versus coherence. To estimate strength, we compute the dot product of racetrack mode with the slope discrepancy measurement from each realization of speckle and turbulence. Then, we take the RMS over all realizations. Like speckle-induced slope-measurement error, racetrack-mode strength increases with $N_F$ and drops with coherence. As before, the knee in the curve occurs around four coherence lengths per resolution cell. Therefore, racetrack-mode strength correlates well with speckle noise, just as we expect.

 

Fig. 12. Racetrack mode strength versus coherence. The plot shows results both with and without turbulence for all $N_F$'s. The vertical bars about the turbulence data are 95% confidence intervals. Mode strength decreases quite significantly as we increase the number of coherence lengths per resolution cell. The trends appear identical whether or not turbulence is included, providing more evidence that turbulence does not change the benefits of polychromatic speckle mitigation.

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Further, the reduction in racetrack-mode strength is statistically equivalent for both vacuum and turbulence cases. Unfortunately, the turbulence results involve a large amount of uncertainty, as indicated by the size of the 95% confidence intervals in the figure, but the general trends appear identical to those of the vacuum results. Recall that for the vacuum runs, we used 4,000 realizations of the target’s rough surface, while for the turbulence runs, we used 1,000 realizations of both turbulence and surface roughness. To avoid clutter, we show only the confidence intervals about the turbulence data. Those about the vacuum data are about half that size. Because the confidence intervals are still quite broad after 1,000 realizations, we would need a very large number of realizations to keep them small. Even so, these results provide more evidence that turbulence does not alter the benefits of polychromatic speckle mitigation, at least not significantly.

Finally, we look at the strengths of all the modes in slope-discrepancy space, not just racetrack mode. Fig. 13 shows such results for three states of coherence. The vacuum results include only speckle effects, while the turbulence results include both speckle and turbulence. Here, we see that the several strongest modes are considerably stronger than all other modes both with and without turbulence. In fact, the strengths of the strongest modes change very little when turbulence is added. On the other hand, the weakest modes increase considerably due to turbulence, especially when coherence is short and speckle is weak. These findings support our previous assertion that turbulence primarily impacts the weakest modes while leaving the strongest modes nearly unchanged. Thus, the strongest modes, including racetrack mode, do indeed provide a way to assess speckle strength while largely ignoring the effects of turbulence.

 

Fig. 13. Mode strengths in slope-discrepancy space for three coherence states. Here, $N_F=0.35$. In vacuum, most of the energy is contained in the strongest modes. These strongest modes change little when turbulence is added, because most of the turbulence energy goes into the weaker modes.

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5. CONCLUSION

In this work, we quantified the benefits of polychromatic speckle mitigation for Shack–Hartmann wavefront sensing of extended non-cooperative targets. For fully coherent illumination, the speckle noise varies only with the target Fresnel number, while the target depth becomes a factor when the coherence length is finite. We considered target Fresnel numbers of 0.35–2.0, which span much of the range encountered by Shack–Hartmann devices in practice. Over this range, target width was not a significant factor. Therefore, we normalized the coherence length by the depth within a diffraction-limited resolution cell. The slope-measurement error showed very significant improvements as coherence dropped. At $N_F=0.35$, speckle-induced error fell by 56% with 15 coherence lengths per resolution cell. Under the conditions used here, 15 coherence lengths per resolution cell required a coherence length of 2.7 mm, which is attainable. Since we assumed near worst-case target depth (slope of 5.27°), the benefits of polychromatic illumination are usually even greater.

Further, we showed that racetrack mode is the strongest speckle-induced mode in slope-discrepancy space when $N_F$ is small, and it can be used to assess speckle reduction, even when weak turbulence is present. Plots of mode strength showed that weak turbulence affects the weakest modes, while leaving the strongest modes nearly unchanged. Thus, racetrack mode separates the effects of speckle from those of weak turbulence. Because the reduction in racetrack mode strength with coherence is nearly the same whether or not weak turbulence is present, we conclude that weak turbulence does not impact the effectiveness of polychromatic illumination, at least not significantly.

This work should find use in a variety of wavefront sensing applications, including remote sensing, target tracking, and laser weapons. Future work could expand upon these results by considering a true compensated beacon rather than the surrogate used here. Additionally, future work could consider cases of deep turbulence. Two interesting cases might involve Rytov number above 0.25 and isoplanatic angle smaller than the diffraction angle. For the former case, Shack–Hartmann performance usually degrades rapidly, but polychromatic illumination might help. The latter case is even more stressing.