1. INTRODUCTION

An idealized tracking scenario for directed-energy applications might place a point-source beacon at the desired aim point, operating at the same wavelength and measured in the same aperture as the scoring beam. If beacon tilt measurements were free of noise and error, if we could ignore the travel time of photons along the path, and if corrections could be applied to the scoring beam with infinite bandwidth and zero latency, then conventional wisdom would dictate that we could drive the scoring-beam jitter to zero. This conclusion is based on “tilt reciprocity”: the assumption that, in this situation, beacon tilts and scoring-beam centroid motion would track one another perfectly. Tilt reciprocity is a common unstated assumption in tracker analysis and design, though it is important to note that it is not among the phenomena that follow rigorously from Helmholtz reciprocity [1–4]. In this paper, we develop the concept of diffractive anisoplanatism and show that it causes a loss of tilt reciprocity between counterpropagating beacons and scoring beams, even in this idealized scenario.

It is well established that anisoplanatic effects can limit the performance of tracking and adaptive-optics corrections of atmospheric turbulence. The scenario described above, however, was specifically designed to eliminate all of the usual sources of anisoplanatism. Previous publications have described an observed upper limit to useful tracker bandwidth, even in geometries approaching this ideal [5–8]. That bandwidth limit has been seen to scale with the wind speed divided by the aperture diameter, and the idea of a “$v/D$ problem” has arisen in the tracking literature, most notably in the work of Dr. Paul Merritt. These previous studies, discussed in more detail in Section 4, have hypothesized that this phenomenon can be attributed to a combination of measurement noise and the decreasing tilt content of spatial wavelengths shorter than the aperture diameter. This noise-driven loss of tilt reciprocity presents a practical limitation that may dominate diffractive anisoplanatism but could in principle be avoided; this research was motivated by a desire to identify additional, fundamental-physics limits to tilt reciprocity. The term “diffractive anisoplanatism” was previously used in a qualitative discussion of adaptive optics, but the similarity between that work and this ends with the observation that diffraction can affect reciprocity [9].

Diffractive anisoplanatism arises from inescapably different sampling of the atmosphere by a point-source beacon and a scoring beam propagating to the target. We will demonstrate this analytically, first with a simple sinusoidal phase profile and then for full atmospheric turbulence, showing that it results in uncorrectable residual jitter and an upper limit $f_{\mathrm{S}}$ on useful tracker bandwidth, even for an idealized system. Though only one of a number of factors that may limit useful bandwidth, we believe that $f_{\mathrm{S}}$ is an important parameter to consider in tracker design.

This paper is self-contained but draws on the analytical methods developed in a previous article [10]. Relevant results will be cited and used here, but, because our focus here is the physical conclusions about diffractive anisoplanatism, we will not go through the full derivations. Please refer to the previous article for a detailed treatment on developing these kinds of analytic expressions for atmospheric propagation statistics.

2. DIFFRACTIVE ANISOPLANATISM

Anisoplanatism arises between two beams that sample the atmosphere differently. Three familiar forms of anisoplanatism arise when beams have lateral, angular, or temporal offsets from one another—examples which can all be understood with simple ray-optics pictures (e.g., [11], Figs. 3.7, 5.2, and 5.3). Diffractive anisoplanatism differs in that it requires us to look beyond the ray-optics limit of propagation. The two relevant effects, illustrated in Fig. 1, are the loss of phase information on the beacon light as diffraction converts phase to amplitude, and the diffractive spreading of the scoring beam outside of the cone of light sampled by the beacon. We will treat the beacon light as coming from an ideal point source at the aim point, which could be a cooperative beacon, a retroreflector, or a glint. In this section, we will develop the equations necessary to show the relevant differences between beacon tilt and scoring-beam centroid motion.

 

Fig. 1. Illustration of the propagation of point-source beacon light to an aperture (yellow cone) and a scoring beam to the target plane (red region), showing the two diffractive effects that lead to anisoplanatism. First is the conversion of beacon phase into amplitude, indicated by a change in color in the yellow cone, which results in a loss of phase information on the beacon light. Second is the spreading of the focused scoring beam outside of the cone sampled by the beacon, spoiling the geometric overlap of the two beams.

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A. Diffractive Anisoplanatism with a Sinusoidal Phase Profile

Our ultimate goal in analyzing diffractive anisoplanatism is to make a comparison between the tilts present on beacon light in the receive aperture and the centroid motion of the scoring beam in the target plane after both beams have propagated through the atmosphere. Following the techniques developed in [10], we will build up to this by first considering the effect of a single sinusoidal phase profile at position $z$ along the path, with transverse wave vector $\mathit{\kappa }$ and real-valued transverse phase offset $\alpha (\mathit{\kappa },z)$. Our normalized phase profile at transverse position $\mathit{r}$ and time $t$ is given by

For a scoring beam propagating from 0 to $L$, assuming a circularly symmetric irradiance profile $a^2(r,z)$ and total power $P$, the phase profile in Eq. (1) causes a centroid motion in the target plane of ([10], Section 5)

Tracking systems can be designed to respond to either the G tilt or the Z tilt on the incoming beacon light. Because the C tilt that determines scoring-beam centroid motion is more closely related to G tilt (the aperture-averaged phase gradient), we expect greater tilt reciprocity if we use G tilt for tracking. To demonstrate this difference, we consider expressions for both. The sinusoidal phase profile in Eq. (1) results in a G tilt measurement in the aperture of ([10], Eq. 5)

Because Eq. (2) depends on the details of the irradiance profile $a^2(r,z)$, direct comparison with Eqs. (3) and (4) is nontrivial. In order to make a simple comparison, in this section we will replace $a^2(r,z)$ with a flat irradiance profile of diameter $D_{\mathrm{S}}(z)$ at all $z$. We can mimic diffractive spreading of the scoring beam by allowing $D_{\mathrm{S}}(z)>D_{\mathrm{B}}(z)$, as indicated schematically with the red region in Fig. 1. With this simplification, we can evaluate the Fourier transform in Eq. (2), giving

Equations (3)–(5) have identical time dependence in a shared cosine term, and are therefore perfectly correlated and coherent with one another. To the extent that their amplitudes differ, however, tilt reciprocity is degraded. Plots of these three equations, with the shared cosine terms removed, are shown in Fig. 2. The $x$ axes are the dimensionless quantity $D_{\mathrm{B}}(z)\kappa$, which is equal to $2\pi$ for a sinusoidal phase profile with wavelength equal to $D_{\mathrm{B}}(z)$. The curves in Fig. 2(a) show that, for long spatial wavelengths ($D_{\mathrm{B}}(z)\kappa \ll 2\pi$), all three measurements are the same and we have tilt reciprocity. As the spatial wavelength gets shorter ($D_{\mathrm{B}}(z)\kappa \gtrsim 2\pi$), first Z tilt, then G tilt diverge from the scoring-beam jitter measurement, due to the conversion of beacon phase to amplitude. At a given wind speed, larger values of $\kappa$ correspond to higher temporal frequencies, so we would expect to see worse tilt reciprocity at high frequencies. Not only do the tilt measurements have different magnitudes than the centroid motion: for some $\kappa$ values they have different signs. As shown in Fig. 2(b), if we include spreading of the scoring beam, mimicking diffraction by allowing $D_{\mathrm{S}}(z)>D_{\mathrm{B}}(z)$ in Eq. (5), small differences in the sizes of the beams mean a loss of tilt reciprocity at even smaller values of $D_{\mathrm{B}}(z)\kappa$; the fact that the scoring beam does not maintain a flat irradiance profile as it propagates would further reduce tilt reciprocity.

 

Fig. 2. These plots illustrate the breakdown of tilt reciprocity between beacon G tilt and Z tilt and scoring-beam centroid motion, all due to single sinusoidal phase profiles of varying $\kappa$. The plots are of Eqs. (3)–(5), with the shared cosine terms removed because all three signals have identical time dependence. Our sinusoidal phase profiles have unit amplitude and are placed at the midpoint of a 3-km path, the fields have 1-μm wavelengths, and we have a 20-cm aperture. In panel (a), we show a comparison with $D_{\mathrm{S}}(z)=D_{\mathrm{B}}(z)=(1-z/L)·20\mathrm{cm}$, so the loss of tilt reciprocity is due entirely to diffractive conversion of beacon phase to amplitude. In panel (b), we take $D_{\mathrm{S}}(z)=D_{\mathrm{B}}(z)+(z/L)·1\mathrm{cm}$ to simulate the effect of diffractive spreading of the scoring beam. In panel (c), we take the differential tilts to show how the measurements increasingly differ in magnitude, sometimes in sign, as $\kappa$ increases.

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There are a few additional insights that we can take away from the plots in Fig. 2. As we move from the center of these plots to the edges, either towards larger or smaller $D_{\mathrm{B}}(z)\kappa$, we see the amplitude of tilts decreasing. The amplitude of the sinusoidal phase profile being applied is constant, so this observation reflects a $\kappa$-dependent decrease in the tilt content of the sinusoidal phase: when $D_{\mathrm{B}}(z)\kappa \ll 2\pi$, the wavelength of the phase is much larger than $D$ and the phase looks flat over the aperture; when $D_{\mathrm{B}}(z)\kappa \gg 2\pi$, the phase has many periods over the aperture and contributes mostly to higher-order modes rather than to tilt. The peak in tilt content is near $D_{\mathrm{B}}(z)\kappa \sim 2\pi$, where the turbulence wavelength is close to the aperture diameter. Further, if there is a cross wind of velocity $v(z)$, then the frequency with which our tilt will change is given by $v(z)\kappa /2\pi$, so this peak in tilt comes at a frequency around $v(z)/D_{\mathrm{B}}(z)$. This is related to diminishing returns to tilt correction above $v/D$, which we will discuss in Section 4.

B. Diffractive Anisoplanatism with Atmospheric Turbulence

In the previous section, we demonstrated with simple sinusoidal phase profiles that tilt reciprocity worsens at high frequencies. For a more relevant illustration of this, we can look at the power spectral densities (PSDs) of the beacon G tilt and scoring-beam centroid jitter for propagation through full atmospheric turbulence; because of the clearly worse tilt reciprocity seen for Z tilt, we will focus on tracking with G tilt. Given the expressions for the centroid motion and beacon tilt due to a single sinusoidal phase profile, it is straightforward to calculate variances and PSDs for propagation through the atmosphere. First, we need the variances found by ensemble averaging our tilts over $\alpha (\mathit{\kappa },z)$, assuming uniform distribution of this random phase, with Eq. (2) giving us

From the path-integrated variance in Eq. (8) we can get a PSD in temporal frequency $f$ via a change in variables ([10], Section 2.E),

 

Fig. 3. These plots shows a comparison between the PSDs calculated with our analytic expressions for scoring-beam centroid jitter in Eq. (11) and beacon G tilt in Eq. (12), as well as results of wave-optics modeling, shown for $\widehat{\mathit{m}}\parallel \mathit{v}$ in panel (a) and for $\widehat{\mathit{m}}\perp \mathit{v}$ in panel (b). The theoretical curves agree well with wave optics, and both show the predicted disagreement between beacon and scoring beam at high frequencies. The scenario chosen was propagation over a 3-km path with a constant 0.2 m/s cross wind, Rytov variance of 0.01, constant $C_n^2\approx 4\times {10}^{-16}{\mathrm{m}}^{-2/3}$, and Kolmogorov turbulence. The scoring beam is a 20-cm-diameter top-hat beam at 1-μm wavelength, and the beacon is a point source at the same wavelength and measured in the same aperture. With these parameters, the characteristic frequency $v/D$ is 1 Hz. We have modeled weak turbulence here to show that diffractive anisoplanatism is not a strong-turbulence phenomenon; in the weak-turbulence regime, each PSD is linear in $C_n^2$.

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3. IMPLICATIONS FOR TRACKER PERFORMANCE

As we discussed in the previous section, the result of diffractive anisoplanatism will be a reduction in our ability to correct the scoring beam using tilt measurements from a point-source beacon. To find the residual jitter for a tracking system with no measurement errors or latency, we apply the beacon tilt measurements as corrections to the scoring beam by directly subtracting the measured tilt from the scoring-beam motion. The tracker described here does not have any direct knowledge of the scoring-beam motion, but instead relies on the assumption of tilt reciprocity between the two quantities; without a priori knowledge of any differences between the two quantities, this subtraction as a correction is the best that one could do. The difference between Eqs. (2) and (3),

A. Residual Jitter Variance

One measure of the severity of an anisoplanatic effect is the jitter that remains after we make our correction. Substituting Eq. (14) into Eq. (8) gives us the variance of the differential jitter,

 

Fig. 4. In this plot, we compare the variance of differential jitter in Eq. (15) to the scoring-beam centroid jitter and beacon G tilt in Eqs. (16) and (17). The calculations were performed for the same scenario used in Fig. 3, except that the aperture diameter was varied to change the Fresnel number of the propagation. The residual jitter variance is linear in the turbulence strength, and thus easily scaled from these values.

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For anisoplanatic effects, it is typically obvious what parameter we can adjust to reduce the severity of the effect in a given scenario; for example, we would reduce the effects of angular anisoplanatism by reducing the apparent angle between the beacon and the aim point. For diffractive anisoplanatism, the analog is that we can reduce the effect by reducing the importance of diffraction in our scenario, which is why we explored the dependence on the Fresnel number in Fig. 4. The Fresnel number, though, is not a parameter that is easy to modify because the range to target, wavelength of light, and aperture size are typically fixed by other considerations. This means that the residual jitter imposed by diffractive anisoplanatism is largely unavoidable.

B. Upper Bound for Useful Track Bandwidth

Analysis of anisoplanatic effects often ends with the calculation of a Strehl ratio or residual variance, as we found for diffractive anisoplanatism in the previous section. However, as highlighted in Fig. 3, the breakdown in tilt reciprocity that leads to this variance does not occur equally at all frequencies. The consequences of this frequency-dependent behavior manifest most clearly in a comparison between the PSDs for differential jitter and uncorrected scoring-beam motion.

We find the PSD of the differential jitter by substituting Eq. (14) into Eq. (10), giving us

 

Fig. 5. This plot shows the comparison between the PSDs of scoring-beam centroid jitter, beacon G tilt, and the differential jitter between them, calculated for the same scenario used in Fig. 3. We show the two-axis jitter, found by summing the PSDs parallel and perpendicular to the wind. We see that there is a frequency $f_{\mathrm{S}}\sim 12\mathrm{Hz}$ above which the differential jitter [yellow curve, from Eq. (18)] has more power than the scoring-beam motion [blue curve, from Eq. (11)], meaning corrections above that frequency will increase jitter on target. At frequencies above $f_{\mathrm{S}}$, the differential-jitter curve lies on top of the beacon G tilt [red curve, from Eq. (12)]. The characteristic frequency $v/D$ is again 1 Hz.

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We have added the $x$ and $y$ PSDs to get the two-axis results, and have labeled the point where those two lines intersect as the scoring-beam crossover frequency $f_{\mathrm{S}}$. This is the maximum frequency at which a tracking system should attempt to correct atmospheric turbulence in this scenario; diffractive anisoplanatism causes corrections above $f_{\mathrm{S}}$ to degrade performance. Though one could in principle define separate, directionally dependent values for $f_{\mathrm{S}}$ from the parallel and perpendicular PSDs, we have combined them to provide a single value, in keeping with other bandwidth considerations such as the Tyler and Greenwood frequencies [12,13].

By defining $f_{\mathrm{S}}$ as the crossing point between two PSDs, we have an implicit definition that, in practice, requires a numeric search. For preliminary assessment, one might prefer an approximate scaling-law expression for $f_{\mathrm{S}}$. For a path with constant $v$ and $C_n^2$, we find that one can estimate $f_{\mathrm{S}}$ with the empirical relationship

 

Fig. 6. The plot in panel (a) shows the calculated $f_{\mathrm{S}}$ values for a wide range of scenarios with varying aperture size, wind speed, and Fresnel number, with $v/D$ ranging from 0.01 to 100 Hz and Fresnel number ranging from 0.3 to 100, but all with constant $C_n^2$ and wind speed along the path. We see good agreement with our scaling law over more than 6 orders of magnitude of $f_{\mathrm{S}}$, with an RMS error of less than 7%. When we consider cases with variable wind speed (up to $10\times$ along the path) and $C_n^2$ (up to $100\times$ along the path), or Fresnel numbers less than 0.3, we see deviation from the scaling law; 95% of the cases run lie within the bounds indicated in panel (b) and still clearly follow the trend of the scaling law when we use the maximum wind speed along the path.

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Equation (19) has greater error in estimating $f_{\mathrm{S}}$ when $C_n^2$ and $v$ are allowed to vary along the path, as shown in Fig. 6(b). The results still follow the trends of our simple scaling law if we replace $v$ with $\max [v(z)]$, and the scaling law allows for a reasonable initial assessment. There are a number of additional effects beyond the scope of this article that could alter the behavior of $f_{\mathrm{S}}$, including modified turbulence statistics (e.g., adding inner and outer scale), extended beacons, conjugating the tilt measurement to a different location along the path, or interactions of the PSDs with realistic tracker error-rejection curves.

4. COMPARISON TO OTHER BANDWIDTH LIMITATIONS

This work on diffractive anisoplanatism was initially motivated by a desire to understand the $v/D$ problem, an observation of an upper limit to useful track bandwidth for reducing atmospheric-turbulence-induced jitter in some directed-energy applications. For reasons outlined above, diffractive anisoplanatism appears to be an independent, fundamental-physics limitation on track bandwidth, which highlights an important point: a number of different effects place upper limits on the useful tracker bandwidth, and the existing literature does not always clearly delineate them. We have found that a useful way to discuss these considerations is to classify them as follows:

In previous studies, especially experimental observations, it is likely that all four of these effects influenced the results. The dominant effect will be application- and scenario-dependent, and all of them should be considered in the balance of cost and complexity versus performance when designing a tracking system.

Because the tilt content of the atmosphere decreases as we increase frequency, at some point it is not worth the cost or effort to further increase track bandwidth. The diminishing returns for bandwidths above $v/D$ are a combined result of two effects. First is the turbulence spectrum of the atmosphere, which has decreasing power at higher frequencies. Second, as noted in our discussion of Fig. 2, is that temporal frequencies above $v/D$ come from short spatial wavelengths of turbulence that contribute primarily to higher-order Zernike modes. This second effect causes the change in slope around $v/D$ for PSDs shown in this paper and has been known in the literature for many years [15]. Diminishing returns are effectively the aspect of tracking dealt with by Siegenthaler’s Z-tilt gain function [8].

As tilt content decreases, there will be some limit beyond which any realistic tracking system is just measuring noise. The noise floor in tracker measurements is related to the decreasing tilt content as frequencies increase, and it makes sense that noise would come to dominate at some frequency close to $v/D$, given the change in slope and rapid falloff of the analytic PSDs above that frequency. Noise, however, is a practical hardware limitation and not a feature of the atmosphere. Merritt touches on this topic [7], in addition to the arguments from Siegenthaler, when he argues that it is hard to make accurate or repeatable measurements of the tilt due to higher spatial frequencies and postulates that the $v/D$ problem is the result of these measurement difficulties. It is also worth noting that Merritt includes results from wave-optics calculations ([7], Fig. 13.24) showing perfect coherence between jitter from a top-hat scoring beam and track-source tilt measurements when there is no noise. This observation is in complete accord with the results here; the coherence is expected, but, as mentioned in Section 2.A, does not in and of itself imply that subtracting one signal from the other will drive the residual to zero.

Centroid anisoplanatism calls into question whether the goal of tracker design should be driving scoring-beam centroid jitter to zero, because doing so can actually decrease the long-exposure Strehl ratio of the beam. This is the result of the fact that, in addition to the Zernike tilt term, all orders of Zernike coma contribute to the motion of the beam centroid [14]. Because a tracker uses tilt alone to reduce centroid jitter, we are trying to correct higher-order modes with tilt and as a result will increase the RMS phase error on the beam.

As for fundamental limitations on jitter correction, previous theoretical treatments of tracking have often begun with assumptions that preclude seeing the diffractive anisoplanatism discussed in this paper. Sasiela, for example, assumes that centroid jitter on target arises from Z tilt rather than C tilt and chooses to neglect diffraction in his analysis, thus guaranteeing perfect tilt reciprocity with a Z-tilt-driven tracker ([16], Section 4.7). Diffractive anisoplanatism results from fundamental properties of propagation that reduce tilt reciprocity between point-source-beacon tilt and scoring-beam jitter. This limits the possibility of reducing jitter by increasing bandwidth, even in an idealized system free of noise or other errors. We find uncorrectable jitter at all frequencies, and in particular there is a limiting frequency $f_{\mathrm{S}}$ above which attempted compensation of atmospheric tilt will actually increase jitter on target.

5. CONCLUSIONS

In this paper, we have described diffractive anisoplanatism, a phenomenon that has previously been omitted from discussions and analytic treatments of tracking. Diffractive anisoplanatism is the result of two effects: the conversion of phase to amplitude on the beacon light used to drive a tracker, and the spreading of the scoring beam outside of the geometric cone of atmosphere sampled by the beacon. These effects of diffraction lead to a loss of tilt reciprocity between beacon tilt measurements and scoring-beam centroid motion, even in an idealized tracking scenario specifically constructed to avoid previously treated anisoplanatic effects.

There are two consequences of diffractive anisoplanatism that need to be considered in the design and analysis of trackers for atmospheric compensation. The first is a residual jitter that we incur, even assuming an ideal geometry and a perfect tracker with no noise or latency. The second is the frequency $f_{\mathrm{S}}$ at which the PSDs for differential jitter and uncorrected scoring-beam jitter cross. Just as the Tyler frequency $f_{\mathrm{T}}$ provides a practical lower limit for bandwidth in tracker design, $f_{\mathrm{S}}$ provides an upper limit, as attempts to correct scoring-beam jitter with measured beacon tilt above $f_{\mathrm{S}}$ will increase jitter on target. In addition to these specific conclusions about diffractive anisoplanatism, we have described how tracker design constraints fall into practical or fundamental limitations, and can address either the desirability or possibility of jitter reduction. This framework should help in considering the place of diffractive anisoplanatism, a fundamental limitation on the possibility of jitter reduction, among all the other considerations when designing and analyzing tracking systems.