1. INTRODUCTION

Adaptive optics (AO) [1] has revolutionized numerous fields, including (but certainly not limited to) directed energy [2–4], vision science [5–7], and astronomy [8–10]. It is hard to overstate AO’s impact on the latter. Indeed, one could easily argue that if not for AO, the 30–50 m primary-mirror astronomical observatories [11,12] currently being designed and built would be impractical.

One of the early subjects of AO research concerned two-wavelength AO, where atmospheric sensing and correcting are performed at one wavelength ${\lambda _{B}}$, and compensation and observation (after transmission through the atmosphere) are performed at another ${\lambda _{T}}$. The impetus for this work was the development of artificial AO beacons, e.g., sodium laser guide stars and Rayleigh beacons [13–16]. In the case of the former, the artificial star or beacon is formed by the spontaneous emission of sodium atoms in the upper atmosphere and emits light at ${\lambda _{ B}} = 589.2\,{\rm nm}$. This light is used to sense and correct for the phase distortions caused by atmospheric turbulence. Astronomical compensation and observation are typically performed at longer wavelengths of ${\lambda _{ T}} \sim 1 - 2\,\unicode{x00B5}{\rm m}$. The primary purpose of the early two-wavelength AO work was to quantify the error (phase variance, Strehl ratio, etc.) inherent in this approach.

Via the works of Fante [17], Lukin [18], Hogge and Butts [19], and Holmes and Gudimetla [20], a consensus emerged: as long as ${\lambda _{ B}} \lt {\lambda _{ T}}$ and furthermore, ${\lambda _{ T}} - {\lambda _{ B}}$ was not too large (less than a few microns), the resulting error was generally negligible. In weak scintillation conditions, like those often encountered in astronomical viewing, this is indeed accurate, and researchers designing AO systems for astronomical telescopes shifted focus to correcting atmospheric aberrations over fields of view many times the isoplanatic angle (multiconjugate AO), as well as atmospheric refractive (dispersive) effects that are relevant under astronomical observing conditions [21–25].

In contrast to astronomy, directed energy systems operate in complementary conditions (i.e., with a small field of view and in the presence of significant irradiance fluctuations [26,27]). AO in strong scintillation is a classic problem in optics and has been studied by some of the brightest scientists in the field, including Tatarskii [28,29], Ishimaru [30], and Fried [16].

In the 1990s, Fried authored two seminal papers on phase compensation in the presence of scintillation, formally introducing the AO community to the phase singularities (caused by destructive interference) known as branch points and their corresponding branch cuts [31,32]. In the latter paper, by applying Helmholtz’s decomposition to the phase gradient, he showed that the least-squares estimate or reconstruction [33–35] of the pupil-phase function is blind to the $2\pi$ phase discontinuities caused by branch cuts [32]. These features of the pupil-plane phase, which Fried called the “hidden phase,” severely degrade AO performance even with perfect least-squares phase compensation. As a result, phase reconstruction or unwrapping in the presence of branch cuts continues to be a very active area of research to this day (see Refs. [36–46] for a small sampling).

Although both two-wavelength and strong-scintillation AO have an extensive history, important questions remain as it pertains to applying a two-wavelength AO system in weak-to-moderate scintillation conditions, i.e., the regime where AO system performance rapidly degrades using least-squares phase compensation [47,48]. For example, in directed energy applications and in contrast to astronomy, the beacon might have a longer wavelength than the primary transmit beam (i.e., ${\lambda _{B}} \gt {\lambda _{T}}$). As such, intuition dictates that the beacon will experience less scintillation than the transmit beam, which may provide a more robust least-squares phase estimate due to there being fewer branch cuts present in the pupil-phase function. Nonetheless, at what cost, for the quality of the transmit beam is likely to be poor because the hidden phases at ${\lambda _{B}}$ and ${\lambda _{T}}$ are almost certainly different.

In this paper, we attempt to answer under what conditions it makes sense (in other words, provides a benefit) to include the hidden phase ${\phi _{{\rm hid}}}$, obtained from the beacon, to precompensate for the primary transmit beam. To this end, we obtain an estimate for the wavelength range over which the hidden phases are correlated and consequently, worth correcting.

In the next section, we numerically propagate a point source—representing an AO beacon and spanning a range of wavelengths—through a single realization of atmospheric turbulence and examine the behavior of the hidden phase (versus ${\lambda _{ B}}$) to gain insight. We then perform a statistical analysis to obtain the average (over turbulence realizations) correlation coefficients of the received beacon field amplitude, least-squares phase ${\phi _{{\rm ls}}}$, and ${\phi _{{\rm hid}}}$ as functions of ${\lambda _{ B}}$ and ${\lambda _{T}}$. Focusing on ${\phi _{{\rm hid}}}$, we obtain a semi-empirical expression for the correlation coefficient as a function of ${\lambda _{ B}}$, ${\lambda _{ T}}$, and the log-amplitude variance $\sigma _\chi ^2$.

Next, we perform two-wavelength AO simulations to validate our expression. Like in the prior simulations, we propagate a point-source beacon (at multiple ${\lambda _{B}}$) through atmospheric turbulence; however, this time we use the pupil-phase function from the received beacon to determine phase corrections that enable us to precompensate for the transmit beam at ${\lambda _{T}}$. We then propagate the transmit beam to the beacon/observation plane and find Strehl ratios—using ${\phi _{{\rm ls}}}$ as well as ${\phi _{{\rm ls}}} + {\phi _{{\rm hid}}}$ [39–41,44]—to assess the quality of the AO corrections. As a final step, we find the average (over turbulence realizations) Strehl ratios and compare the results to Hogge and Butts’ phase variance expressions [19,49] using the extended Maréchal approximation [1–4,8,9].

We lastly conclude with summaries of our work, findings, and contributions. Before moving on to the next section, it is important to note that the results presented in this paper go well beyond those presented in a recent conference proceeding [50]. It was determined (after publication) that the preliminary wave optics simulations conducted in Ref. [50] suffered from aliasing. This is corrected here.

2. ${\phi _{{\rm hid}}}$ VERSUS ${\lambda _{B}}$

As discussed above, we performed a series of wave optics simulations to study the behavior of ${\phi _{{\rm hid}}}$ versus ${\lambda _{B}}$. In these simulations, we propagated a point source (modeling an AO beacon) at various ${\lambda _{ B}}$ through atmospheric turbulence and obtained the received field magnitude, ${\phi _{{\rm ls}}}$, and ${\phi _{{\rm hid}}}$. Before presenting and analyzing the results, we discuss the details of the simulations.

A. Simulation Details

The geometry consisted of a point source (located in the beacon/observation plane) emitting light that propagates through atmospheric turbulence and is received $Z = 10\,{\rm km}$ away by a transceiver of diameter ${D_{{\rm trans}}} = {D_{{\rm rec}}} = 75\,{\rm cm}$. The wavelength of the point-source beacon varied from ${\lambda _{B}}(\unicode{x00B5} {\rm m}) \in [{1,3}]$ in 101 equal steps.

With this $Z$ and ${\lambda _{B}}$, we chose the index of refraction structure constant $C_n^2 = 2 \times {10^{- 15}}\,{{\rm m}^{- 2/3}}$ so that the scintillation strength varied from weak at 3 µm to moderate at 1 µm [27]. In addition, we set the turbulence inner and outer scales equal to ${l_0} = 1\,{\rm mm}$ and ${L_0} = 100\,{\rm m}$, respectively. Table 1 gives the values for spherical-wave Fried’s parameter ${r_0}$ [1,8,9,51] $\sigma _\chi ^2$ [1,8,30], and the isoplanatic angle ${\theta _0}$ [1,8,9].

Table 1. Simulated Turbulence Parameter Values

View Table

We propagated the light emitted by the point source through the atmosphere using the split-step beam propagation method (BPM) [2,52,53]. The split-step BPM is a numerical solution to the parabolic wave equation, in which diffraction (due to wave propagation) and refraction (due to transmission through the atmosphere) are treated independently [2,52,54]. This solution is approximate (to second order—see Refs. [2,54]) due to the coupling of these phenomena in inhomogeneous media. Nevertheless, the split-step BPM has been utilized for decades to simulate wave propagation through atmospheric turbulence, remains the most computationally efficient method for doing so, and, most importantly, yields results that are consistent with experiment.

Applying the split-step BPM, we discretized the continuous turbulent path with 14 equally spaced phase screens. The phase screens were generated from the von Kármán phase power spectral density [2,28] using the spectral (Fourier) method and augmented with subharmonics [52,55,56]. The strengths of the phase screens were determined by finding the screen ${r_0}$ values that minimized the root-mean-square error between the discrete and continuous ${r_0}$ and $\sigma _\chi ^2$ [52].

A 14-screen instance or realization of atmospheric turbulence was synthesized using the ${r_0}$ and $\sigma _\chi ^2$ at $\min ({{\lambda _{B}}}) = 1\,\unicode{x00B5}{\rm m}$. Each phase screen was then divided by ${k_{{\max}}} = 2\pi /\min ({{\lambda _{B}}})$ to convert from radians to optical path length (OPL) in meters. This turbulence instance in OPL was used for all subsequent ${\lambda _{B}}$. Consequently, for each of the 101 beacon wavelengths, we multiplied the OPL screens by ${k_{ B}} = 2\pi /{\lambda _{B}}$ to effect the proper strength phase distortions on the propagating wave at that particular ${\lambda _{B}}$.

Lastly, applying the sampling analysis procedure described in Ref. [52], we discretized the transceiver and beacon/observation planes using $1900 \times 1900$ grids with 2.35 mm spacings in both planes. Note that the grid sizes and spacings in the intermediate or phase screen planes were also $1900 \times 1900$ and 2.35 mm, respectively. Propagations between the intermediate planes were performed by evaluating the convolution form of the Fresnel diffraction integral using fast Fourier transforms (also known as the angular spectrum method [52,53]).

Before proceeding, it is important to note that in the traditional split step, the phase screens that model the refractive effects of atmospheric turbulence are used at their “design” wavelength and therefore, the turbulence model’s accuracy at other wavelengths is immaterial. Here, however, we are using a “split-step atmosphere” designed to yield accurate ${r_0}$ and $\sigma _\chi ^2$ at 1 µm at other wavelengths. Indeed, we chose $\min ({{\lambda _{ B}}}) = 1\,\unicode{x00B5}{\rm m}$ as the design wavelength because the turbulence (and consequently, the screens) at that wavelength is the strongest. All other wavelengths being longer, we assumed that the 1 µm atmosphere (in particular, the 14 phase screens used to model it) remained valid. To our knowledge, using the split-step BPM in this manner is unverified. As we will show, the results qualitatively validate this approach; nevertheless, the accuracy of split-step atmospheres at non-design wavelengths is the main assumption of our work.

B. Results and Discussion

Figure 1 shows the results of a multiple-${\lambda _{ B}}$ point-source beacon propagated through a single realization of atmospheric turbulence. Figures 1(a) and 1(b) are comprised of two rows and five columns of images: (a) shows the magnitude $| U |$ and phase $\arg (U)$ of the received (collimated) beacon field, while (b) displays ${\phi _{{\rm ls}}}$ scaled by ${k_{ B}}$ and ${\phi _{{\rm hid}}}$ in the first and second rows, respectively. Row headings have been added to the figure to aid the reader.

 

Fig. 1. Point-source beacon at multiple ${\lambda _{B}}$ propagated through a single realization of atmospheric turbulence: (a) received beacon field magnitude $| U |$ (top) and phase $\arg (U)$ (bottom); (b) least-squares reconstructed/unwrapped ${\phi _{{\rm ls}}}$ (top) and hidden ${\phi _{{\rm hid}}}$ (bottom) phases; and (c) $| U |$, ${\phi _{{\rm ls}}}$, and ${\phi _{{\rm hid}}}$ correlation coefficients $r$ computed relative to those quantities at 2 µm. The white circles in (a) and (b) mark the edge of the transceiver aperture ${D_{{\rm trans}}} = {D_{{\rm rec}}} = 75\,{\rm cm}$. See Visualization 1 for a 20 s animation showing the results for all $101\,{\lambda _{\rm B}}$.

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In these results and those to follow, we assumed perfect knowledge of the transceiver plane field (i.e., we did not simulate a wavefront sensor), and we obtained ${\phi _{{\rm ls}}}$ and ${\phi _{{\rm hid}}}$ directly from $\arg (U)$. We found ${\phi _{{\rm hid}}}$ by [39–41,44]

The five images comprising each row in Figs. 1(a) and 1(b) are the results at different ${\lambda _{B}}$, whose values are included as a heading above the corresponding column in (a). In addition, the five images in each row are encoded using the same color scale defined by the bar at row’s end. The white circles in (a) and (b) mark the edge of the transceiver aperture (recall ${D_{{\rm trans}}} = {D_{{\rm rec}}} = 75\;{\rm cm} $).

Figure 1(c) plots the correlation coefficients $r$ versus ${\lambda _{ B}}$ for $| U |$, ${\phi _{{\rm ls}}}$, and ${\phi _{{\rm hid}}}$ over the transceiver aperture relative to those quantities at ${\lambda _{{\rm ref}}} = 2\,\unicode{x00B5}{\rm m}$. Note that $r$ for ${\phi _{{\rm ls}}}$ is

Although the results presented in Fig. 1 are only for a single turbulence realization, there are a couple aspects worth noting. One is that $| U |$ and ${\phi _{{\rm ls}}}$ are highly correlated over the entire range of ${\lambda _{ B}}$. This can be seen in the top rows of Figs. 1(a) and 1(b) as well as in 1(c). Since ${\phi _{{\rm ls}}}$ is blind to any branch points or cuts that might exist in the pupil-phase function [31,32] and furthermore, those phenomena are only (appreciably) present when $\sigma _\chi ^2 \gtrsim 0.25$ (an empirically determined rule [27]), ${\phi _{{\rm ls}}}$ is, in essence, the weak-scintillation phase of the received beacon field. Consequently, the two-wavelength AO analysis presented in Refs. [17–20] is germane and leads, quite naturally, to the finding that the OPL error [i.e., ${\phi _{{\rm ls}}}({{\lambda _{B}}})/{k_{B}} - {\phi _{{\rm ls}}}({{\lambda _{T}}})/{k_{T}}$]—and subsequent degradation in AO performance—is negligible.

When $\sigma _\chi ^2 \gtrsim 0.25$, however, things are quite different. In the second row of Fig. 1(b) and particularly in (c), we observe that ${\phi _{{\rm hid}}}$, when present [i.e., not flat like in the first two columns of (b)], is strongly correlated over only a narrow band of ${\lambda _{B}}$ centered on the reference wavelength (2 µm in this case). This is shown quite clearly in the ${\phi _{{\rm hid}}}$ images in Visualization 1. Qualitatively speaking, these results strongly imply that sensing and correcting ${\phi _{{\rm hid}}}$ are not worth the effort unless, of course, $| {{\lambda _{ B}} - {\lambda _{ T}}} |$ is small. How small is addressed in the next section.

C. ${\phi _{{\rm hid}}}$ Correlation Linewidth $\Delta {\lambda _{{\rm hid}}}$

To obtain a physically relevant value for the correlation linewidth of ${\phi _{{\rm hid}}}$, we need to average the ${\phi _{{\rm hid}}}$ $r$ in Fig. 1(c) over independent turbulence realizations. Figure 2 includes this result: (a), (b), and (c) show the average correlation coefficients $\langle {r({{\lambda _{T}},{\lambda _{B}}})} \rangle$ for $| U |$, ${\phi _{{\rm ls}}}$, and ${\phi _{{\rm hid}}}$, respectively. These $\langle r \rangle$ were computed from 1000 statistically independent turbulence realizations. Figure 2(d) shows the horizontal slices through (a), (b), and (c) at ${\lambda _{B}} = 2\,\unicode{x00B5}{\rm m}$. The bars on the traces in (d) extend plus and minus one standard deviation ${\sigma _r}$ from the $\langle r \rangle$ values. Generally speaking, the single realization $r$ in Fig. 1(c) is representative of the average results in Fig. 2.

 

Fig. 2. Average correlation coefficient $\langle {r({{\lambda _{T}},{\lambda _{B}}})} \rangle$ computed over 1000 independent turbulence realizations: (a) $| U |$, (b) ${\phi _{{\rm ls}}}$, (c) ${\phi _{{\rm hid}}}$, and (d) horizontal slices, i.e., ${\lambda _{B}} = 2\,\,\unicode{x00B5}{\rm m}$, through (a), (b), and (c). The bars in (d) are ${\pm}{\sigma _r}$ (one standard deviation).

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Focusing on ${\phi _{{\rm hid}}}$, the shapes of the image in (c) and the curve in (d) are well approximated by

The exponential power $p$ is found by taking the natural log of Eq. (4), followed by the absolute value, and lastly, another natural log producing

The slope of the resulting line is $p$. Performing this computation for every diagonal slice (along the main diagonal and every sub and super diagonal) of (c) yields $p = 0.4891 \pm 0.1795$. We set $p = 0.4891$ in the analysis to follow.

With $p$, we can now find $\delta$ by fitting Eq. (4) to every diagonal slice through Fig. 2(c). As stated above, mathematically speaking, $\delta$ is a function of $\bar \lambda$. Nevertheless, physically, it is more appropriate to write it as a function of scintillation strength (i.e., the log-amplitude variance $\sigma _\chi ^2$), as will become evident shortly.

The correlation radius is shown in Fig. 3(a), which plots $\delta$ (solid blue trace) versus the average wavelength $\bar \lambda = ({{\lambda _{T}} + {\lambda _{B}}})/2$ and $\sigma _\chi ^2$ on a log-log scale. Note that the $\delta$ curve is comprised of three regions and marked by the dotted red-, green-, and purple-line segments. The slopes of these line segments (with respect to $\sigma _\chi ^2$) are approximately ${-}10$, ${-}1/2$, and ${-}2$, respectively, and are annotated on the plot.

 

Fig. 3. (a) Correlation radius $\delta$ versus the average wavelength $\bar \lambda = ({{\lambda _{T}} + {\lambda _{B}}})/2$ (top abscissa) and corresponding spherical-wave log-amplitude variance $\sigma _\chi ^2$ (bottom abscissa). (b) Simulated ${\phi _{{\rm hid}}}\,\langle r \rangle$ from Fig. 2(c). (c) ${\phi _{{\rm hid}}}\,\langle r \rangle$ computed using Eq. (4) and $\delta$ in (a).

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Starting in the region with the dotted red-line segment, we would expect ${\phi _{{\rm hid}}}$ to be relatively flat [see the ${\lambda _{ B}} = 2.5\,\unicode{x00B5}{\rm m}$ and 3 µm ${\phi _{{\rm hid}}}$ images in Fig. 1(b)] and therefore, $\delta$ to be large. Indeed, in the absence of phase discontinuities (or irradiance fluctuations), $\sigma _\chi ^2 \to 0$ and $\delta \to \infty$. Consequently, the appearance of any branch-point pairs (and their associated branch cuts) will result in a drastic decrease in $\langle r\rangle$ and $\delta$. Note that this is precisely what we see in (a), where $\delta$ falls very rapidly with increasing $\sigma _\chi ^2$.

In the dotted green-line segment/region, the hidden phase contains only a few branch-point pairs like the ${\lambda _{B}} = 2\,\unicode{x00B5}{\rm m}$ ${\phi _{{\rm hid}}}$ image in Fig. 1(b). Subsequently, the ${\phi _{{\rm hid}}}$—over this relatively small range of $\sigma _\chi ^2$—qualitatively appear similar and $\delta$’s decline significantly slows. Nevertheless, it is important to note that although $\delta$’s slope is much greater in the prior region of (a) (dotted red-line segment/region), what truly matters are the values of $\delta$, which are very small in this dotted green-line segment/region, namely, $\delta \lt 0.1\,\unicode{x00B5}{\rm m}$ for the conditions simulated here.

As we progress into the dotted purple-line segment/region, the hidden phase contains more and more branch-point pairs, connected by branch cuts, forming patterns of increasing complexity. The ${\phi _{{\rm hid}}}$ images in the fourth and fifth columns of Fig. 1(b) show this quite nicely. As a consequence, the slope of $\delta$ increases and its value asymptotically approaches zero as $\sigma _\chi ^2$ increases. Physically, $\delta$ should converge to a nonzero value when the scintillation saturates. This $\delta$ is likely to be very small, such that Eq. (4) can be approximated by a Dirac delta function.

 

Fig. 4. AO through a single turbulence realization with ${\lambda _{ T}} = 1.5\,\unicode{x00B5}{\rm m}$ using corrections obtained from a point-source beacon at multiple ${\lambda _{B}}$: (a) no-correction normalized irradiance ${\hat I_{{\rm nc}}}$, (b) ${\phi _{{\rm ls}}}$-corrected normalized irradiance ${\hat I_{{\rm ls}}}$, (c) ${\phi _{{\rm ls}}} + {\phi _{{\rm hid}}}$-corrected normalized irradiance ${\hat I_{{\rm ls} + {\rm hid}}}$, and (d) $\epsilon = {\hat I_{{\rm ls} + {\rm hid}}} - {\hat I_{{\rm ls}}}$. The white circles in (a)–(d) mark the edge of the Airy disk (i.e., ${D_{{\rm dl}}} = 2.44{\lambda _{T}}Z/{D_{{\rm trans}}}$). See Visualization 2 for a 20 s animation showing the results for all 101 ${\lambda _{B}}$.

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Although $\delta$ is heavily dependent on $\sigma _\chi ^2$, the ${\phi _{{\rm hid}}}$ correlation linewidth $\Delta {\lambda _{{\rm hid}}}$ can be estimated over a range of ${\lambda _{ B}}$ and ${\lambda _{ T}}$ by $\Delta {\lambda _{{\rm hid}}} \approx 2\bar \delta$, where $\bar \delta$ is the average correlation radius. Here, $\Delta {\lambda _{{\rm hid}}} \approx 0.35\,\unicode{x00B5}{\rm m}$. To obtain this value, we computed the average correlation radius directly from Fig. 3(a) simply by averaging the $\delta$ values in the plot. Clearly, this procedure returns a $\bar \delta$ that is driven by the weak-turbulence $\delta$ and therefore, provides (in most cases) an optimistic $\Delta {\lambda _{{\rm hid}}}$. Nevertheless, it is a simple computation and returns a rough estimate of the maximum ${\lambda _{ B}}$-${\lambda _{ T}}$ separation where sensing and correcting ${\phi _{{\rm hid}}}$ in a two-wavelength AO system are advantageous.

Lastly, Fig. 3(c) shows the ${\phi _{{\rm hid}}}\,\langle r \rangle$ computed using Eq. (4) and the $\delta$ line segments in (a). For the reader’s convenience, we have included the simulated ${\phi _{{\rm hid}}}\,\langle r \rangle$ from Fig. 2(c) as Fig. 3(b). As expected, the agreement between (b) and (c) is excellent.

Before proceeding, it is important to note that although our ${\phi _{{\rm hid}}}\,\langle r \rangle$ and $\delta$ semi-empirical analysis is rigorously valid only over the conditions we simulated here, we believe our findings will hold more generally because 1) the simulations covered a wide range of scintillation strengths, from weak-to-moderate, and 2) because of the physical nature of the results—in particular, the relationship between $\sigma _\chi ^2$ and $\delta$.

3. TWO-WAVELENGTH AO SIMULATIONS

In this section, we perform two-wavelength AO simulations to assess how ${\phi _{{\rm hid}}}$’s poor wavelength correlation affects AO performance versus the least-squares-only correction.

A. Setup

The general simulation details—atmospheric parameters, number of phase screens, grid sizes, pixel pitches, procedure, etc.—are the same as those discussed in Section 2.A. The key difference here is that we applied ${\phi _{{\rm ls}}}$ or ${\phi _{{\rm hid}}}$ (obtained from the beacon field) to the transmit beam, reversed the order of the 14 screens modeling the atmosphere, and then propagated the transmit beam through those 14 screens to the beacon/observation plane.

Note that for the reasons discussed in Ref. [20], ${\phi _{{\rm ls}}}$ was OPL scaled, such that ${\phi _{{\rm ls}}}({{\lambda _{T}}}) = {\phi _{{\rm ls}}}({{\lambda _{B}}}){k_T}/{k_B}$. OPL scaling must be applied to reconstructed or unwrapped phase; otherwise, the cuts in the scaled wrapped phase will not be at the proper $2\pi$ height. Adjusting the scaled wrapped phase to have proper $2\pi$ cuts is equivalent to reconstruction. Because it is multivalued, ${\phi _{{\rm hid}}}$ cannot be reconstructed (or unwrapped); consequently, we did not OPL scale ${\phi _{{\rm hid}}}$.

Like in the simulations presented and discussed above, we assumed perfect knowledge of the received (transceiver-plane) beacon field. Furthermore, we assumed the ability to perfectly apply phase corrections to the transmit beam. Including the effects of wavefront sensors and correctors (deformable mirrors) is left to future work.

B. Results and Discussion

Figure 4 shows results of a single turbulence realization, two-wavelength AO simulation with ${\lambda _{T}} = 1.5\,\unicode{x00B5}{\rm m}$ using phase corrections obtained from a point-source beacon at multiple ${\lambda _{ B}}$. Figures 4(a)–4(c) display the normalized, beacon/observation-plane no-correction ${\hat I_{{nc}}}$, ${\phi _{{\rm ls}}}$-corrected ${\hat I_{{\rm ls}}}$, and ${\phi _{{\rm ls}}} + {\phi _{{\rm hid}}}$-corrected ${\hat I_{{\rm ls} + {\rm hid}}}$ irradiances for five ${\lambda _{B}}$, respectively. The normalized irradiance is defined as

The AO results in Fig. 4 and Visualization 2 bolster the findings revealed in Figs. 1 and 2. We observe little difference between the ${\phi _{{\rm ls}}}$- and ${\phi _{{\rm ls}}} + {\phi _{{\rm hid}}}$-corrected $\hat I$ until ${\lambda _{B}} \approx {\lambda _{T}}$. Using the analysis from Section 2.C, the ${\phi _{{\rm hid}}}$ correlation linewidth at ${\lambda _{T}} = 1.5\,\unicode{x00B5}{\rm m}$ is approximately 0.1 µm. While the 0.5 µm ${\lambda _{B}}$ steps in Fig. 4 are far too coarse to observe changes in ${\hat I_{{\rm ls}}}$ and ${\hat I_{{\rm ls} + {\rm hid}}}$ over this linewidth, we can see this in Visualization 2. From the video, we see ${\hat I_{{\rm ls} + {\rm hid}}}$ begin to significantly deviate from (provide a significant correction benefit over) ${\hat I_{{\rm ls}}}$ at about ${\lambda _{\rm B}} \approx 1.7\,\unicode{x00B5}{\rm m}$. Once ${\lambda _{B}} \lt {\lambda _{T}}$, the ${\hat I_{{\rm ls} + {\rm hid}}}$ spot begins to drift off center and its peak value falls rapidly. Nevertheless, it still provides a correction benefit over ${\phi _{{\rm ls}}}$ until ${\lambda _{B}} \approx 1.3\,\unicode{x00B5}{\rm m}$.

This result seems to imply that our correlation linewidth is too pessimistic. To test this further, we repeated the simulation in Fig. 4 500 times, each time with a statistically independent turbulence realization, and calculated the average Strehl ratios $\langle {\cal S} \rangle$. Figure 5 plots the results. The bars on the no-correction, ${\phi _{{\rm ls}}}$-corrected, and ${\phi _{{\rm ls}}} + {\phi _{{\rm hid}}}$-corrected $\langle {\cal S} \rangle$ extend plus and minus one standard deviation ${\sigma _{\cal S}}$ from $\langle {\cal S} \rangle$. In addition, the solid and dashed black traces (labeled pw and sw) are theoretical Strehl ratios obtained by using the extended Maréchal approximation with Hogge and Butts’ phase error or variance expressions for plane and spherical waves, respectively. For completeness, these are [19,49]

 

Fig. 5. Average Strehl ratios $\langle {\cal S} \rangle$ versus ${\lambda _{B}}$ computed over 500 independent turbulence realizations with ${\lambda _{T}} = 1.5\,\unicode{x00B5}{\rm m}$. The bars on the no-correction, ${\phi _{{\rm ls}}}$-corrected, and ${\phi _{{\rm ls}}} + {\phi _{{\rm hid}}}$-corrected results are ${\pm}{\sigma _{\cal S}}$ (one standard deviation). The pw and sw traces are theoretical Strehl ratios obtained by using the extended Maréchal approximation and Hogge and Butts’ phase error or variance expressions for plane and spherical waves, respectively. The dotted purple trace is the ${\phi _{{\rm hid}}}$ correlation coefficient $\langle r\rangle$ at ${\lambda _{T}} = 1.5\,\unicode{x00B5}{\rm m}$ computed using Eq. (4).

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Lastly, to show how our correlation coefficient analysis in Section 2.C applies to AO correction, we have also plotted the ${\phi _{{\rm hid}}}\,\langle r\rangle$ [Eq. (4) evaluated at ${\lambda _{ T}} = 1.5\,\unicode{x00B5}{\rm m}$] as the purple dotted trace.

One of the most noticeable aspects of the results in Fig. 5 is the significant overprediction of the Strehl ratio based on the Hogge and Butts’ equations. This outcome is somewhat expected as Hogge and Butts’ theoretical treatment is valid in weak scintillation. Nevertheless, this significant difference in predicted versus simulated Strehl ratios does provide justification for our work.

Note that Fig. 5 shows the average Strehl ratio for a ${\lambda _{T}} = 1.5\,\unicode{x00B5}{\rm m}$ transmit beam using corrections obtained at ${\lambda _{B}}$. At ${\lambda _{B}} = 3\,\unicode{x00B5}{\rm m}$, ${\phi _{{\rm hid}}}$ is likely flat; however, the transmit beam at ${\lambda _{T}} = 1.5\,\unicode{x00B5}{\rm m}$ does experience scintillation propagating back to the beacon/observation plane. This phase is not corrected (nor the field’s amplitude, i.e., we are considering a phase-only or phase-conjugate AO system), thus explaining the significant drop in $\langle {\cal S} \rangle$ for both the ${\phi _{{\rm ls}}}$- and ${\phi _{{\rm ls}}} + {\phi _{{\rm hid}}}$-corrected results. Indeed, if the transmit beam did not scintillate, both $\langle {\cal S} \rangle$ would be closer to the spherical-wave Hogge and Butts’ result. All of this is consistent with the $\langle r \rangle$ in Fig. 2.

Focusing on the ${\phi _{{\rm ls}}}$- and ${\phi _{{\rm ls}}} + {\phi _{{\rm hid}}}$-corrected results, for ${\lambda _{B}} \gt {\lambda _{T}}$, we observe a slow increase in the ${\phi _{{\rm ls}}} + {\phi _{{\rm hid}}}\,\langle {\cal S} \rangle$ relative to the least-squares $\langle {\cal S} \rangle$. This corresponds nicely with ${\phi _{{\rm hid}}}\,\langle r\rangle$’s long “leftward tail” in the dotted purple curve. Although the ${\phi _{{\rm ls}}} + {\phi _{{\rm hid}}}\,\langle {\cal S} \rangle$ is greater than the ${\phi _{{\rm ls}}}\,\langle {\cal S} \rangle$, both means are within one standard deviation ${\sigma _{\cal S}}$ of the other and consequently, the minor improvement obtained by including ${\phi _{{\rm hid}}}$ is inconsistent.

This changes when ${\lambda _{B}} = {\lambda _{T}}$, and as expected, including the hidden phase provides a statistically significant improvement over ${\phi _{{\rm ls}}}$. When ${\lambda _{ B}} \lt {\lambda _{T}}$, both the hidden phase and least-squares $\langle {\cal S} \rangle$ decline rapidly, with the former decreasing faster than the latter. Again, this corresponds nicely with the rapid rightward fall of $\langle r\rangle$. As a consequence of the significant improvement in Strehl ratio provided by ${\phi _{{\rm hid}}}$ at ${\lambda _{ B}} = {\lambda _{T}}$ and the subsequent decline of the least-squares $\langle {\cal S} \rangle$, the ${\phi _{{\rm ls}}} + {\phi _{{\rm hid}}}\,\langle {\cal S} \rangle$ remains above the ${\phi _{{\rm ls}}}$ bars until approximately ${\lambda _{B}} \approx 1.2\,\unicode{x00B5}{\rm m}$.

Thus, our definition of the correlation linewidth is somewhat pessimistic. Nevertheless, the concurrence of our ${\phi _{{\rm hid}}}\,\langle r\rangle$ with the AO Strehl ratio results does support our main hypothesis: for two-wavelength AO systems operating in weak-to-moderate scintillation conditions and in contrast to the least-squares phase, Fried’s hidden phase is correlated over a narrow band of wavelengths centered on ${\lambda _{T}}$. Including hidden phase estimates/corrections obtained from AO beacons with ${\lambda _{B}}$ that fall outside ${\lambda _{T}} \pm \Delta {\lambda _{{\rm hid}}}/2$ provides little improvement over the least-squares-only correction.

4. CONCLUSION

Here, we studied Fried’s hidden phase and its utility in two-wavelength AO systems operating in weak-to-moderate scintillation conditions—the regime where AO system performance rapidly degrades using traditional least-squares phase compensation. Two-wavelength AO, where sensing is performed at one wavelength and observation or transmission is performed at another (e.g., sodium laser guide stars and astronomical observation), has traditionally been studied assuming negligible irradiance fluctuations. In such conditions, the pupil-phase function is highly correlated over a relatively large range of wavelengths (i.e., has a broad correlation linewidth), resulting, generally, in a negligible decrease in AO system performance. The primary purpose of this paper was to answer whether the same holds true in weak-to-moderate scintillation conditions, where, due to destructive interference, the pupil-phase function becomes discontinuous producing Fried’s hidden phase.

To answer this question, we performed a series of wave optics simulations where we propagated a point source (modeling an AO beacon) at multiple wavelengths through atmospheric turbulence. After finding the hidden phase from the simulated point-source/beacon field, we computed the correlation coefficient of the hidden phase as a function of the beacon wavelengths. Averaging the correlation coefficients over 1000 statistically independent turbulence realizations, we estimated the hidden-phase correlation linewidth and, in stark contrast to the phase in weak scintillation conditions, found it to be very narrow.

Applying our definition of the hidden-phase correlation linewidth, we obtained a $\Delta {\lambda _{{\rm hid}}} \approx 0.35\,\unicode{x00B5}{\rm m}$ over the 1–3 µm two-wavelength AO system considered here. Consequently, we concluded that unless the difference between the beacon and transmit beam wavelengths is less than $\Delta {\lambda _{{\rm hid}}}$, sensing and correcting Fried’s hidden phase in two-wavelength AO systems provide little benefit over traditional least-squares phase compensation.

We tested this finding by performing two-wavelength AO simulations in which we applied phase corrections obtained from the point-source AO beacon to precompensate for the transmit beam. We then propagated the transmit beam to the beacon/observation plane and found the Strehl ratio. We repeated this procedure for each beacon wavelength and averaged the resulting Strehl ratios over 500 independent turbulence realizations.

Overall, the results (presented in Fig. 5) supported our conclusion. For ${\lambda _{B}} \gt {\lambda _{T}}$ and $| {{\lambda _{B}} - {\lambda _{T}}} | \gt \Delta {\lambda _{{\rm hid}}}$, there was little difference in the ${\phi _{{\rm ls}}}$- and ${\phi _{{\rm ls}}} + {\phi _{{\rm hid}}}$-corrected Strehl ratios, with both $\langle {\cal S} \rangle$ being within one standard deviation of the other. This changed when $| {{\lambda _{B}} - {\lambda _{T}}} | \lt \Delta {\lambda _{{\rm hid}}}$, where there was a clear $\langle {\cal S} \rangle$ improvement when correcting ${\phi _{{\rm hid}}}$. For ${\lambda _{ B}} \lt {\lambda _{T}}$ and $| {{\lambda _{B}} - {\lambda _{ T}}} | \gt \Delta {\lambda _{{\rm hid}}}$, the ${\phi _{{\rm ls}}} + {\phi _{{\rm hid}}}$-corrected $\langle {\cal S} \rangle$ did outperform the ${\phi _{{\rm ls}}}\,\langle {\cal S} \rangle$; yet, it fell rapidly from its peak at ${\lambda _{B}} = {\lambda _{T}}$ toward the ${\phi _{{\rm ls}}}\,\langle {\cal S} \rangle$.

The work presented herein will be useful in the design of two-wavelength AO systems that will operate in weak-to-moderate scintillation conditions. Although we made every effort to keep the analysis presented in this paper general, there are several aspects that require further investigation. For example, it may be possible to provide a theoretical basis for $\delta$’s behavior in Fig. 3(a) using the strong turbulence phase statistics in Ref. [58]. Other aspects that require further study are the effects of transceiver (aperture) size, turbulence inner scale, wavefront sensors and correctors, anisoplanatism, and extended or speckled beacons.