1. INTRODUCTION

Angular anisoplanatic error describes the optical phase difference as a function of the angle relative to a reference direction (see Fig. 1). It arises from the volume effects of turbulence on an optical wave propagating through a random medium characterized by phase perturbations. In astronomical adaptive optics (AO) [1,2], it describes the difference in the wavefronts between a reference star and a nearby object of interest (science object). As the anisoplanatic error increases, the degree to which the measured reference wavefront allows correction decreases [2]. The limit of correction is generally taken as the angle when the anisoplanatic error reaches $1\;{\rm{ra}}{{\rm{d}}^2}$ and is referred to as the isoplanatic angle, ${\theta _0}$.

 

Fig. 1. Anisoplanatic error refers to the difference in the turbulence-induced phase perturbations between two separated sources. The error arises because the wavefronts propagate through different portions of the turbulence volume. Here sources are separated by an angle, $\theta$, or a spatial separation, $\rho$, and are related to $\theta = \rho /\bar h$, where $\bar h$ is the mean turbulence height. When the effects of a common aperture are considered, a portion of the turbulence volume will be common to both sources (shaded region). In this work, the anisoplanatic errors between two sources as observed by an aperture with diameter $D$ are considered with the aperture averaged piston and tilt removed. In this illustration the reference is assumed to align with the optical axis, and the anisoplanatic error is evaluated at an offset, $\theta$.

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The isoplanatic angle as described by Fried [1] is found by evaluating the wave structure function (WSF) as a function of angle rather than separation as in the definition of the Fried coherence length, ${r_0}$. Fried models the index of refraction and phase fluctuation using a simple structure function with a 5/3 power law. Using this model leads to a condition where the mean-squared phase difference with the angle reference grows without bound. This situation can be avoided by including an outer-scale ${L_0}$ in the index of refraction energy spectral density using the von Kármán or other models.

It was recognized early on that for large telescopes looking skyward the two approaches are effectively similar [3]. The reasoning is that the estimated outer scale lengths were thought to be much larger than the telescope diameter. In this regime the WSF expression and derived parameters are very well approximated by the simpler formulation in terms of the Fried parameter as ${D_w}(\rho) = 6.88(\rho /{r_0}{)^{5/3}}$, where $\rho$ is the separation. Similarly the anisoplanatic error can be defined in terms of the isoplanatic angle as $\sigma _\phi ^2(\theta) = (\theta /{\theta _0}{)^{5/3}}$, as shown in Fig. 1.

In practice, the isoplanatic patch size tends to be larger than prescribed using these formulas. One reason for this is that the definition of the isoplanatic angle relies on the WSF. The WSF describes the field statistics of a plane or spherical wave propagated through a turbulence volume and does not include aperture effects; the effect of a finite aperture is not considered. Of these effects, the aperture averaged phase has no effect on incoherent imaging systems. The aperture averaged tilt can be easily corrected by a fast steering mirror. To be clear this is not the same as the tilt as measured from a reference (see Fig. 1). Instead, the aperture averaged tilt applies to all points within the system’s field of regard.

As described by Stone et al. [4], if these effects are removed from the anisoplanatic error expression the effective isoplanatic patch size increases considerably. Stone’s work is presented in terms of aperture size for a fixed $C_n^2$ rather than a dimensionless parameter. Still, the work suggests that as ${r_0}$ increased relative to the aperture size there existed a point where the piston and tilt removed anisoplanatic error saturates.

Images of extended objects acquired over long horizontal paths through the atmosphere tend to be dominated by anisoplanatic tip and tilt distortions. These distortions are shift-variant and therefore violate the forward model used by most scene recovery techniques [5–8]. However, using both simulated [5] and field data [9] techniques like speckle imaging (SI) is shown to be effective. Following Stone [4], one possible explanation is that anisoplanatic error is not as severe under these conditions ignoring piston and aperture tilt.

Aperture sizes for horizontal imaging systems tend to be much smaller than those used in astronomy. So, while the seeing size may be only a few centimeters, the ratio of the Fried parameter to aperture size, $D/{r_0}$, is generally less than 10. At the same time the outer scale size is often estimated to be ${L_0} \sim 0.4h$ [10,11], where $h$ is the height above ground. However, the nature of the outer scale, its measurement, and impact on optical propagation is a subject of active research [12–14]. Similarly, it has become common to consider the effect of the change in the power law exponent often referred to as non-Kolmogorov turbulence [15–17]. In fact, it is possible that some observations of non-Kolmogorov turbulence can be attributed to scale sizes larger than ${L_0}$. Therefore, all of these effects are of interest in these scenarios.

In a previous work [18], we developed analytical expressions for the anisoplanatic error as a function of angle, for an arbitrary power law medium, with and without an outer scale. We also demonstrated that, for a plane wave source over a horizontal path with constant turbulence strength with the piston and aperture tilt removed, anisoplanatic error saturates to a value less than $1\;{\rm{ra}}{{\rm{d}}^2}$ for small values of $D/{r_0}$ in Kolmogorov turbulence. We also showed that when the outer scale is on the order of the aperture size, ${L_0}/D \sim 1$, the anisoplanatic error, also saturates. Anisoplanatic error was also found to be inversely correlated with the power law exponent in the range of $3 \lt \alpha \lt 4$.

The aim of this work is to identify the conditions where the anisoplanatic error saturates to less than $1\;{\rm{ra}}{{\rm{d}}^2}$. Limits are established and compared using two methods. To the best of my knowledge, these are the first attempts to find and evaluate asymptotic expressions for the anisoplanatic error function for finite apertures with an outer scale for both plane and spherical wave sources. This approach is repeated for the case where both the piston and tilt are removed. My second approach is to evaluate the anisoplanatic error function via numerical integration for a spherical source. Again, this is performed both with an outer scale and with and without compensation. The effect of non-Kolmogorov turbulence is also considered both in the asymptotic analysis and numerical integration. Comparison between varied power law exponents is enabled by pulling strength parameters outside the integral. This approach also aids in understanding the effect of varied power law and turbulence strength over the propagation path. All evaluations are done in terms of normalized parameters ($D/{r_0}$, $\theta /{\theta _0}$, ${L_0}/D$) so the results can be easily generalized. I find that the asymptotic approach broadly agrees with numerical integration except in cases where ${L_0}/D \gtrsim 6$.

The remainder of this paper is organized as follows: In the next section I provide some preliminaries and revisit the plane wave anisoplanatic error expression and evaluate it asymptotically. In Section 3 I use the same approach to evaluate the anisoplanatic error for a spherical wave with a finite outer scale. Results of the numerical integration are provided in the same section. Conclusions and directions for future work are included in Section 4. Appendices are included with details regarding the derivation of the asymptotic expressions in Section 3 and further numerical integration results.

2. BACKGROUND

In this work I am concerned with the propagation of optical fields through random media. In this case, the random media is the open atmosphere. Temperature differences in the atmosphere resolve themselves via the process of turbulent diffusion resulting in a spatial correlation in the distribution of variations in temperature and thereby index of refraction. For the purpose of this work I assume that these variations are described broadly by 3D power law using a power spectral density (PSD) model as

For a random medium to have both a structure function and PSD description the power law exponent is restricted to the range $3 \lt \alpha \lt 4$. When the value of $\alpha$ deviates from the Kolmogorov value of $\alpha = 11/3$ the turbulence is often referred to “non-Kolmogorov,” and that is the definition I use here.

Both the Fried parameter and the isoplanatic angle are defined by the wave structure function (WSF). For a plane wave in terms of the index of refraction PSD, the WSF is [11,19]

For plane wave propagation in Kolmogorov turbulence with constant turbulence Eq. (8) becomes the common handbook expression ${\theta _0} = 0.314{r_0}/L$. Following from Eq. (6), the anisoplanatic error can be written in terms of the isoplanatic angle and $\alpha$ as [21]

As I mentioned in the introduction, it is often useful to ignore both the average and gradient tilt of the phase. These contributions can be removed using the filter function approach popularized by Sasiela [22,23] and first used by Stone [4] in this context.

The filter function, ${F_0}(\kappa)$, for the piston across an aperture with diameter, $D$, is

The plane wave propagation case represents the worst case relative to phase error contributions in the aperture. Over slant and horizontal paths a spherical wave model is both more appropriate and less severe. This can be easily demonstrated by comparing the spherical ${r_{0,s}}$ and plane wave ${r_0}$ Fried parameters. These quantities are related by ${r_{0,s}}={(8/3)^{3/5}}{r_0} = 1.8{r_0}$ [2]. So, the spherical Fried length is almost twice that for a plane propagating in the same medium. In non-Kolmogorov turbulence this expression becomes ${r_{0,s}} = (\alpha - {1)^{\alpha - 2}}{r_0}$ [21].

The plane and spherical WSF expressions can be similarly related [19],

While it is likely strictly possible to find an analytic expression as a solution for this integral, they are likely just as cumbersome as those in Ref. [18], as in that work, evaluation via numerical integration is straightforward. However, it is aided in this instance by replacing the Bessel functions with their spherical counterparts using the relation ${j_n}(z) = \sqrt {\frac{\pi}{{2z}}} {J_{n + 1/2}}(z)$.

Equation (18) represents the main result of this section. In the work that follows I will expand the work in Ref. [18] providing an asymptotic result in the limit for large values of $\theta$ including for piston and tilt removed cases. That work is further extended to the spherical wave case, and the results are compared to numerical integration of Eq. (18).

3. RESULTS

A. Asymptotic Anisoplanatic Error

Following the development in Section 2 it is clear that total phase variance for a plane wave propagating in a turbulent medium described by a von Kármán PSD with outer scale ${L_0}$ is given by the first term in Eq. (12). Specifically, the expression

B. Asymptotic Piston Phase Variance: Plane Wave

In Appendix A I derive an expression for the asymptotic piston phase variance contribution to the anisoplanatic error. That expression is found to be

C. Asymptotic Tilt Phase Variance: Plane Wave

In Appendix B the asymptotic tilt phase variance contribution is found to be

D. Asymptotic Piston Phase Variance: Spherical Wave

In Appendix C the piston phase variance for a spherical wave is found to be

E. Asymptotic Tilt Phase Variance: Spherical Wave

The asymptotic tilt phase variance is identical to Eq. (22) except the argument for hypergeometric functions changes from ${q^2}$ to $q_s^2$.

 

Fig. 2. Isocurves for $\sigma _\phi ^2 = 1$ for turbulence with a finite von Kármán outer scale (solid), piston compensated (dashed), and piston and tilt removed (dotted). Shown here are the results for plane wave sources and three values of power law exponent, $\alpha$.

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F. Asymptotic Piston and Tilt Removed Phase Variance

Given the results above the piston and tilt removed phase variance can be found by subtracting Eqs. (21) and (B2) from Eq. (19) following Eq. (15) as

In Fig. 2 the conditions necessary for anisoplanatic error saturation are visualized for three values of $\alpha$. The solid lines indicate the boundary in terms of $D/{\hat r_0}$ and ${L_0}/D$ prescribed by Eq. (20), and the slope of each line is $2 - \alpha$. The region to the left of these lines represents the conditions where the isoplanatic angle is not defined; the anisoplanatic error saturates to a value less than $1\;{\rm{radia}}{{\rm{n}}^2}$. The plot also includes isolines for the case where either the piston or the piston and tilt are removed. As may be expected these aperture-related contributions have their largest effect, ${L_0}/D \gt 1$. When $\alpha = 11/3,{L_0}/D = 1$, the maximum $D/{r_0}$ where saturation occurs increases from $D/{r_0} = 2.8$ to $D/{r_0} = 3.2$ when removing only the piston and $D/{r_0} = 3.9$ with the piston and tilt. Subtracting these terms increases $D/{r_0}$ $12\%$ and $36\%$, respectively. However, as ${L_0}/D$ increases, total uncompensated anisoplanatic error continues to increase while the slope of the piston and piston slopes are much smaller. In fact, for ${L_0}/D \gg 1$ the piston and tilt corrected anisoplanatic error saturates when $D/{r_0} \lesssim 2.2$. For this case, the piston-only corrected phase variance saturates when $D/{r_0} \lesssim 0.7$. The piston and tilt corrected result suggests that when $D/{r_0} \lesssim 2.2$ increases, the outer scale contributes only to these corrected terms without increasing high order aberrations. However, I will note that this condition relies upon the outer scale following the von Kármán model. If there is additional energy in spatial scales larger than ${L_0}$ the results may be very different.

Figure 3 is the spherical wave counterpart to Fig. 2 and can be interpreted in the same way. Again, the solid lines represent the limit imposed by the outer scale and are the same as Fig. 2. Different in this case is the effect of compensating for the piston and tilt. At $\alpha = 11/3,{L_0}/D = 1$, anisoplanatic error saturates for $D/{r_0} \lt 4.3$ when the piston is removed and $5.8$ when both the piston and tilt are compensated. Compared to the plane wave a compensated spherical wave allows for a $34$ and $50\%$ higher values of $D/{r_0}$ for this condition. Both values also approach an asymptote for ${L_0}/D \gg 1$: $1.6$ for piston only correction and $4.0$ for the piston and tilt corrected case. Perhaps not surprisingly, these values are $1.8$ times their respective plane wave values.

 

Fig. 3. Isocurves for $\sigma _\phi ^2 = 1$ for turbulence with a finite von Kármán outer scale (solid), piston compensated (dashed) and piston and tilt removed (dotted). Shown here are the results for spherical wave sources and three values of power law exponent, $\alpha$. Note that the piston and piston and tilt removed values are 1.8 times larger than the plane wave results in Fig. 2 for $\alpha = 11/3$_.

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Fig. 4. Spherical wave anisoplanatic error ($\sigma _\phi ^2$) numerical integration of Eq. (18) (solid), with piston removed (dashed), and piston and tilt removed (dotted). Shown here are cases for $\alpha = 11/3$, ${L_0}/D = 1,2$, and $D/{r_0} = 3,5,9$. The horizontal axis is scaled to the isoplanatic angle (${\theta _0}$) for each case. Solid, light, gray lines indicate $\sigma _\phi ^2$ and $\theta /{\theta _0} = 1$, and a light blue solid line indicates the asymptotic value predicted by Eq. (19). Light blue dashed and dotted lines indicate the piston removed and piston and tilt removed asymptotes.

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G. Numerical Evaluation of Spherical Wave Anisoplanatic Error

The results in the previous section are arrived at by ignoring the position or angular dependence in Eq. (15). The filter functions in Eqs. (13) and (14) are only valid for separations of $\rho \le D$. Therefore, the anisoplanatic error function is limited to a range of $0 \le \theta \le (0.5D/\bar h = {\theta _{{\max}}})$. It is possible that in this range the anisoplanatic error function may exceed $1\;{\rm{ra}}{{\rm{d}}^2}$. It is also possible that $\sigma _\phi ^2({\theta _{{\max}}}) \le 1$.

To evaluate these possibilities I evaluate Eq. (15) numerically as in [18] here for the spherical wave case for the conditions ${L_0}/D = 0.5,1,2,5{,}10{,}100$ and $D/{r_0} = 3,5,9{,}10$. The uncompensated, piston compensated, and piston and tilt compensated anisoplanatic errors were all evaluated, and the results compared. Representative plots of the anisoplanatic error as a function of angle normalized to the isoplanatic angle $\theta /{\theta _0}$, as defined in Eq. (8), are found in Fig. 4 for the case $\alpha = 11/3$. These plots also include the predicted asymptotic values for each case and over the range $0 \le \theta \le {\theta _{{\max}}}$. In this range I note that the anisoplanatic error does not exceed the asymptotic value. However, in some cases (see ${L_0}/D = 1$, $D/{r_0} = 5$ in Fig. 4), the compensated values do not reach $1\;{\rm{ra}}{{\rm{d}}^2}$.

In Table 1 I summarized these results for Kolmogorov turbulence with the piston and aperture tilt removed. Bold cells in the table indicates cases where the anisoplanatic error saturates. The values in these cells indicate the maximum anisoplanatic error in the range $0 \le \theta \le {\theta _{{\max}}}$. The other values in the table are the normalized isoplanatic angle, $\theta /{\theta _0}$, where $\sigma _\phi ^2(\theta) = 1\;{\rm{ra}}{{\rm{d}}^2}$. Similar tables for non-Kolmogorov turbulence power law exponents of $\alpha = 3.1$ and $\alpha = 3.9$ are found in Tables 2 and 3. Tables for the uncompensated case with a finite outer scale and piston removed are found in Appendix D.

Table 1. Normalized Isoplanatic Angle ($\theta /{\theta _0}$) for $\alpha = 11/3$ as Evaluated via Numerical Integration of the Anisoplanatic Error with Piston and Aperture Tilt Removed^a

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Table 2. Normalized Isoplanatic Angle ($\theta /{\theta _0}$) for $\alpha = 3.1$ as Evaluated via Numerical Integration of the Anisoplanatic Error with Piston and Aperture Tilt Removed^a

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Table 3. Normalized Isoplanatic Angle ($\theta /{\theta _0}$) for $\alpha = 3.9$ as Evaluated via Numerical Integration of the Anisoplanatic Error with Piston and Aperture Tilt Removed^a

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Strictly speaking, anisoplanatic error saturation is only observed in Table 1 for $D/{r_0} \le 5$ and ${L_0}/D \le 2$. However, in comparing these results to Fig. 3 it seems likely that the actual limit is slightly higher for both quantities. On the other hand, the numerical results also contradict the asymptotic results. For example, Fig. 3 suggests that $\sigma _\phi ^2(\theta) \lt 1\;{\rm{ra}}{{\rm{d}}^2}$ out to ${L_0}/D = 20$ for $D/{r_0} = 4$, but this is not supported by numerical evaluation. Also interesting is that the effective isoplanatic angle with the piston and tilt removed is $33\%$ larger than the handbook expression at large $D/{r_0}$ and outer scales.

 

Fig. 5. Spherical wave anisoplanatic error ($\sigma _\phi ^2$) numerical integration of Eq. (18) (solid), with piston removed (dashed), and piston and aperture tilt removed (dotted). Shown here are cases for $\alpha = 3.1{,}11/3,3.9$, ${L_0}/D = 1{,}10$, and $D/{r_0} = 3$ and 9. The horizontal axis is scaled to the isoplanatic angle (${\theta _0}$) for each case. Solid, light, gray lines indicate $\sigma _\phi ^2$ and $\theta /{\theta _0} = 1$.

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H. Non-Kolmogorov Turbulence

Figure 5 shows the spherical wave anisoplanatic error as a function of a normalized angle for two values of non-Kolmogorov turbulence $\alpha = 3.1$ and $3.9$. All of the cases here include a von Kármán model outer scale. The range of values included in Fig. 5 is near the extremes ${L_0}/D = 1$ and $10$ and $D/{r_0} = 1$ and $9$. The Kolmogorov case $\alpha = 11/3$ is also included for reference. For all values of $\alpha$ the piston compensated, as well as the piston and aperture tilt compensated, cases are represented by the dashed and dotted lines, respectively.

Examining Tables 2 and 3 together with the Fig. 5, a clear pattern emerges. Turbulence volumes characterized by power law exponents smaller than 11/3 have higher anisoplanatic error. Saturation behavior also onsets more quickly with the angle. As an example, in Fig. 5, for the case where ${L_0}/D = 10$ and $D/{r_0} = 9$ when the aperture tilt and piston are removed the anisoplanatic error has begun to saturate when $\theta /{\theta _0} = 3$. Also, observe in Fig. 5, where ${L_0}/D = 10$, ${L_0}/D = 1$, and $\alpha = 3.9$, the effect of changing ${L_0}$ in comparison to $\alpha = 3.1$. This is consistent with the plane wave case [24] and other works [25,26] that suggest that $\alpha \to 3$ fluctuations are uncorrelated and saturation occurs more quickly. Conversely, as $\alpha \to 4$, the larger spatial scales are dominant and governed by ${L_0}$.

I. Non-Kolmogorov Turbulence and Slant Paths

One question that is recurring when considering non-Kolmogorov turbulence is how to account for the situation where the propagation path consists of sections where the power law exponent varies. It is generally believed that turbulence in most of the atmosphere follows Kolmogorov statistics. The idea is that non-Kolmogorov turbulence may exist in limited regions where conditions for well-developed turbulence are violated. How might we account for these situations, especially when path weighting is important in understanding anisoplanatic error? This question is addressed indirectly in this work. In Eq. (18), a generalized model for spherical wave propagation is provided in terms of a generalized Fried parameter, ${\hat r_0}$, and the mean turbulence height, $\bar h$. Recall that the path weighting for both anisoplanatic error and spherical wave propagation is $5/3$. In fact, these quantities are counterparts to each other [27,28]. When reframed in terms of turbulence moments as in [20,21] the contribution is both the average turbulence strength and the height at which it occurs. The weighting of each depends on the power law of the medium in each segment. In this way, when $(\alpha - 2) \lt 5/3$, the turbulence is weighted further from the aperture and closer to the aperture when $(\alpha - 2) \gt 5/3$. This relationship is observed in Fig. 5 where ${\hat r_0}$ is fixed and $\alpha$ varied. When $\alpha = 3.1$ the mean turbulence height increases, decreasing the isoplanatic angle via Eq. (8); the converse is true when $\alpha = 3.9$.

4. CONCLUSION

Incorporating a von Kármán outer scale model limits the anisoplanatic error to less than one radian squared when ${r_0}/{L_0} \lt 0.349$ in Kolmogorov turbulence and is reduced further when aperture effects are included. Using asymptotic methods, I estimated the conditions that result in the saturation of anisoplanatic error for both plane and spherical wave sources when piston and aperture tilt are removed. The resulting expressions predict that anisoplanatic error saturates to less than 1 radian squared when $D/{r_0} \lt 2.2$ for a plane wave source and $D/{r_0} \lt 4$ for spherical wave sources when ${L_0}/D \sim 20$. However, these findings are not supported by numerical integration of the compensated anisoplanatic error function. On the other hand, when ${L_0}/D = 1$, the asymptotic expression agrees with numerical method predicting $D/{r_0} \lt 3.9$ and $5.8$ for the plane and spherical wave cases as the turbulence strength at which the anisoplanatic error saturates. It is also reassuring that the plane and spherical wave values are related by a factor of 1.8. Comparing the tabulated results to Figs. 2 and 3 a reasonable limit of ${L_0}/D \sim 5$ can be estimated for the validity of the asymptotic approach. Identifying the exact ratio of $D$ and ${r_0}$ producing saturation will require either a more detailed asymptotic model or inverting the anisoplanatic error function via optimization methods.

In this work, I also compared the compensated anisoplanatic error between Kolmogorov and non-Kolmogorov turbulence. In the process, I developed a generalized spherical WSF that uses the generalized Fried parameter and mean turbulence height as strength parameters. This model was presented here in Section 2 and allows us to understand the effect of a change in power law for arbitrary path geometries. In general, relative to anisoplanatism, when $\alpha \gt 11/3$, the relative effect is to decrease the mean turbulence height increasing the isoplanatic angle for a fixed volume turbulence strength in terms of ${\hat r_0}$. Likewise, as $\alpha$ is decreased the mean turbulence height is increased, increasing anisoplanatic error and resulting in a smaller isoplanatic angle. These conclusions hold regardless of whether or not the anisoplanatic error is compensated. This work also supports other findings that anisoplanatic error is dominated by small scale fluctuations as $\alpha \to 3$ and large scale fluctuations governed by ${L_0}$ as $\alpha \to 4$.

It is important to emphasize that this work relies on a von Kármán outer scale model. The extent of the usefulness of the results depends heavily on the validity of this model, which has been recently called into question. At the same time, this work provides a basis for comparison and for understanding the effect of outer scale regions that may follow different power laws.

APPENDIX A: ASYMPTOTIC EXPRESSION FOR THE PISTON PHASE VARIANCE: PLANE WAVE CASE

In a similar fashion, from Ref. [18], Appendix D, the plane wave piston phase variance can be expressed as

(A1)$$\sigma _{\phi ,P}^2(\theta ,\alpha ,{L_0}) = 4\pi B(\alpha){c_1}(\alpha)\hat r_0^{2 - \alpha}{I_P},$$

with

(A2)$${I_P} = \frac{{- 16{d^\alpha}}}{{{D^2}}}\int_0^\infty \frac{1}{\omega}{\omega ^{- \alpha}}\left({{{\rm{J}}_0}\left(\omega \right) - 1} \right){\rm{J}}_1^2\left({\omega /x} \right){\left[{1 + {{\left({\frac{y}{\omega}} \right)}^2}} \right]^{- \frac{\alpha}{2}}}{\rm d}\omega ,$$

where $d = \theta L$, $\omega = \kappa d$, $y = 2\pi d/{L_0}$, and $x = 2d/D$. As in Section 2, this integral can be expanded and terms with the zero-order Bessel function set aside leaving an expression for only the piston contribution to the overall phase variance,

(A3)$${I_{{P^\prime}}} = \left({\frac{{16{d^\alpha}}}{{{D^2}}}} \right)\int_0^\infty \frac{1}{\omega}{\omega ^{- \alpha}}{J_1}{\left({\frac{\omega}{x}} \right)^2}{\left({{{\left({\frac{y}{\omega}} \right)}^2} + 1} \right)^{- \frac{\alpha}{2}}}{\rm d}\omega .$$

This integral has a solution in terms of the generalized hypergeometric function, $_p{F_q}$, and its generalized counterpart, $_p{\tilde F_q}$, as

(A4)$$\begin{split}{{I_{{P^\prime}}}}& ={ - \frac{{8{d^\alpha}}}{{{D^2}}}{{\left({{y^2}} \right)}^{- \frac{\alpha}{2}}}}\\{}&\quad\times {\left({1{- _1}{F_2}\left[{\frac{1}{2};2,1 - \frac{\alpha}{2};\frac{{{y^2}}}{{{x^2}}}} \right]} \right.}\\{}&\quad\times{\left. { \Gamma \left[{\frac{{1 + \alpha}}{2}} \right]\Gamma {{\left[{\frac{{1 + \alpha}}{2}} \right]}_1}{{\tilde F}_2}\left[{\frac{{\alpha + 1}}{2};\frac{{\alpha + 2}}{2},\frac{{\alpha + 4}}{2};\frac{{{y^2}}}{{{x^2}}}} \right]} \right).}\end{split}$$

The piston phase variance contribution is shown in Eq. (21).

APPENDIX B: ASYMPTOTIC EXPRESSION FOR THE TILT PHASE VARIANCE: PLANE WAVE CASE

The contribution due to tilt can be found in much the same manner as above. Again, referencing [18], this time Appendix E, the integral to find the tilt only contribution, ${I^\prime _T}$, is given by

(B1)$${I^\prime _T} = \frac{{- 64{d^\alpha}}}{{{D^2}}}\int_0^\infty \frac{1}{\omega}{\omega ^{- \alpha}}{\rm{J}}_2^2\left({\omega /x} \right){\left[{1 + {{\left({\frac{y}{\omega}} \right)}^2}} \right]^{- \frac{\alpha}{2}}}.$$

The solution to this integral is similar to Eq. (A4),

(B2)$$\begin{split}{{I_{{T^\prime}}}}& ={ - \frac{{16{d^\alpha}}}{{{D^2}}}{{\left({{y^2}} \right)}^{- \frac{\alpha}{2}}}}{ \left({- 1 + {2_1}{F_2}\left[{- \frac{1}{2};3,1 - \frac{\alpha}{2};\frac{{{y^2}}}{{{x^2}}}} \right]}\right.}\\&\quad{\left.{{- _{\,2}}{F_3}\left[{- \frac{1}{2},2;1,3,1 - \frac{\alpha}{2};\frac{{{y^2}}}{{{x^2}}}} \right]} \right.}\\{}&\quad+{ \left({{{\left({\frac{{{x^2}}}{{{y^2}}}} \right)}^{- \alpha /2}}\Gamma \left[{2 - \frac{\alpha}{2}} \right]\Gamma \left[{\frac{{\alpha - 1}}{2}} \right]} \right.}\\{}&\quad{\left. {{\times _{\,2}}{{\tilde F}_3}\left[{- \frac{1}{2} + \frac{\alpha}{2},\frac{\alpha}{2}; - 1 + + \frac{\alpha}{2},1 + \frac{\alpha}{2},3 + \frac{\alpha}{2};\frac{{{y^2}}}{{{x^2}}}} \right]} \right)}\\{}&\quad\big/{\left. {\left({\sqrt \pi \Gamma \left[{1 + \frac{\alpha}{2}} \right]\Gamma \left[{3 + \frac{\alpha}{2}} \right]} \right)} \right).}\end{split}$$

The result of the asymptotic tilt contribution is shown in Eq. (21).

APPENDIX C: ASYMPTOTIC PISTON PHASE VARIANCE: SPHERICAL WAVE

The expression for the asymptotic piston phase variance for the spherical wave case proceeds from the same starting position as Appendix A. Via Eq. (17) to account for spherical wave propagation, $\kappa \rho$ terms must be include a propagation parameter $\gamma$. For spherical waves, $\gamma = z/L$ or $\gamma = 1 - z/L$. In Eq. (18), $\gamma$ is replaced by $u$ and integrated from 0 to 1 rather than 0 $L$.

In any case, the integration begins, as in Appendix A,

(C1)$$\begin{split}{{I_{{P_s}}} }&= {\frac{{- 16{d^\alpha}}}{{{D^2}{u^2}}}\int_0^\infty \frac{1}{\omega}{{\left({\omega u} \right)}^{- \alpha}}}\\{}&\quad\times {\left({{{\rm{J}}_0}\left({\omega u} \right) - 1} \right){\rm{J}}_1^2\left({\omega u/x} \right){{\left[{1 + {{\left({\frac{y}{\omega}} \right)}^2}} \right]}^{- \frac{\alpha}{2}}}{\rm d}\omega {\rm d}u.}\end{split}$$

This integral evaluates to

(C2)$$\begin{split}{{I_{{P_s}}}}& ={s \frac{{4{d^\alpha}}}{{{D^2}}}{{\left({{y^2}} \right)}^{- \alpha /2}}}\\{}&\quad\times {\left({- 2 + {2_1}{F_2}\left[{- \frac{1}{2};2,1 - \frac{\alpha}{2};\frac{{{y^2}}}{{{x^2}}}} \right]} \right.}\\{}&\quad+ {\frac{1}{{\sqrt \pi}}{{\left({\frac{{{y^2}}}{{{x^2}}}} \right)}^{\alpha /2}}\Gamma \left[{1 - \frac{\alpha}{2}} \right]\Gamma \left[{\frac{\alpha}{2} - \frac{1}{2}} \right]}\\{}&\quad{\left. {{\times _{\,1}}{{\tilde F}_2}\left[{\frac{\alpha}{2} - \frac{1}{2};\frac{{\alpha + 2}}{2},\frac{{\alpha + 3}}{2};\frac{{{y^2}}}{{{x^2}}}} \right]} \right)}\end{split}$$

and results in the expression in Section 3.F.

APPENDIX D: TABLE RESULTS

In this appendix, tabulated results are provided for the normalized isoplanatic angle as evaluated via numerical integration. Tables 4–6 indicate either the maximum anisoplanatic error (bold) for a given combination of ${L_0}/D$ and $D/{r_0}$ or the normalized angle $\theta / {\theta _{0}}$ where the ansisoplanatic error reaches $1\;{\rm{ra}}{{\rm{d}}^2}$. Tables 4–6 are for the vales $\alpha = 11/3$, $\alpha = 3.1$, and $\alpha = 3.9$, respectively. Tables 7–9 provide the same information as Tables 4–6 but for the case where piston has been removed.

Table 4. Normalized Isoplanatic Angle ($\theta /{\theta _0}$) for $\alpha = 11/3$ as Evaluated via Numerical Integration of the Spherical WSF with a Finite Outer Scale^a

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Table 5. Normalized Isoplanatic Angle ($\theta /{\theta _0}$) for $\alpha = 3.1$ as Evaluated via Numerical Integration of the Spherical WSF with a Finite Outer Scale^a

View Table | View all tables in this article

Table 6. Normalized Isoplanatic Angle ($\theta /{\theta _0}$) for $\alpha = 3.9$ as Evaluated via Numerical Integration of the Spherical WSF with a Finite Outer Scale^a

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Table 7. Normalized Isoplanatic Angle ($\theta /{\theta _0}$) for $\alpha = 11/3$ as Evaluated via Numerical Integration of the Spherical WSF with a Finite Outer Scale and Piston Removed^a

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Table 8. Normalized Isoplanatic Angle ($\theta /{\theta _0}$) for $\alpha = 3.1$ as Evaluated via Numerical Integration of the Spherical WSF with a Finite Outer Scale and Piston Removed^a

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Table 9. Normalized Isoplanatic Angle ($\theta /{\theta _0}$) for $\alpha = 3.9$ as Evaluated via Numerical Integration of the Spherical WSF with a Finite Outer Scale and Piston Removed^a

View Table | View all tables in this article

Funding

Air Force Office of Scientific Research (FA9550-17-1-0201, FA9550-23-1-062).

Acknowledgment

Any opinions, finding, conclusions, or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the United States Air Force.

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

REFERENCES

1. D. L. Fried, “Angular dependence of atmospheric turbulence effect in speckle interferometry,” in International Astronomical Union Colloquium (Cambridge University, 1979), Vol. 50, pp. 26-1–26-25.

2. M. C. Roggemann, B. M. Welsh, and B. R. Hunt, Imaging through Turbulence (CRC Press, 1996).

3. A. Consortini, L. Ronchi, and E. Moroder, “Role of the outer scale of turbulence in atmospheric degradation of optical images,” J. Opt. Soc. Am. 63, 1246–1248 (1973). [CrossRef]  

4. J. Stone, P. Hu, S. Mills, et al., “Anisoplanatic effects in finite-aperture optical systems,” J. Opt. Soc. Am. A 11, 347–357 (1994). [CrossRef]  

5. J. P. Bos and M. C. Roggemann, “Robustness of speckle-imaging techniques applied to horizontal imaging scenarios,” Opt. Eng. 51, 083201 (2012). [CrossRef]  

6. G. E. Archer, J. P. Bos, and M. C. Roggemann, “Comparison of bispectrum, multiframe blind deconvolution and hybrid bispectrum-multiframe blind deconvolution image reconstruction techniques for anisoplanatic, long horizontal-path imaging,” Opt. Eng. 53, 043109 (2014). [CrossRef]  

7. G. Archer, J. P. Bos, and M. C. Roggemann, “Reconstruction of long horizontal-path images under anisoplanatic conditions using multiframe blind deconvolution,” Opt. Eng. 52, 083108 (2013). [CrossRef]  

8. B. J. Thelen, R. G. Paxman, D. A. Carrara, et al., “Overcoming turbulence-induced space-variant blur by using phase-diverse speckle,” J. Opt. Soc. Am. A 26, 206–218 (2009). [CrossRef]  

9. J. P. Bos and M. C. Roggemann, “Blind image quality metrics for optimal speckle image reconstruction in horizontal imaging scenarios,” Opt. Eng. 51, 107003 (2012). [CrossRef]  

10. R. K. Tyson, Introduction to Adaptive Optics (SPIE, 2000), Vol. 41.

11. V. I. Tatarski, Wave Propagation in a Turbulent Medium (Courier Dover, 2016).

12. S. Basu and A. A. Holtslag, “Revisiting and revising Tatarskii’s formulation for the temperature structure parameter (ct 2) in atmospheric flows,” Environ. Fluid Mech. 22, 1107–1119 (2022). [CrossRef]  

13. D. Sprung, E. Sucher, and A. M. van Eijk, “Investigation of vertical profiles of the outer scale L₀ in the atmospheric surface layer,” in Propagation Through and Characterization of Atmospheric and Oceanic Phenomena (Optica Publishing Group, 2023), paper PTh2F-2.

14. G. Kermarrec and S. Schön, “On the determination of the atmospheric outer scale length of turbulence using GPS phase difference observations: the Seewinkel network,” Earth Planets Space 72, 184 (2020). [CrossRef]  

15. I. Toselli, L. C. Andrews, R. L. Phillips, et al., “Free-space optical system performance for laser beam propagation through non-Kolmogorov turbulence,” Opt. Eng. 47, 026003 (2008). [CrossRef]  

16. I. Toselli and S. Gladysz, “On the general equivalence of the fried parameter and coherence radius for non-Kolmogorov and oceanic turbulence,” OSA Contin. 2, 43–48 (2019). [CrossRef]  

17. J. P. Bos, M. C. Roggemann, and V. R. Gudimetla, “Anisotropic non-Kolmogorov turbulence phase screens with variable orientation,” Appl. Opt. 54, 2039–2045 (2015). [CrossRef]  

18. J. R. Beck and J. P. Bos, “Angular anisoplanatism in non-Kolmogorov turbulence over horizontal paths,” J. Opt. Soc. Am. A 37, 1937–1949 (2020). [CrossRef]  

19. A. D. Wheelon, Electromagnetic Scintillation: Volume 1, Geometrical Optics (Cambridge University, 2001).

20. J. W. Hardy, Adaptive Optics for Astronomical Telescopes (Oxford University, 1998), Vol. 16.

21. J. P. Bos and V. R. Gudimetla, “Path dependency and angular anisoplanatism in non-Kolmogorov turbulence,” in IEEE Aerospace Conference (IEEE, 2015), pp. 1–6.

22. R. J. Sasiela, “Electromagnetic wave propagation in turbulence: evaluation and application of Mellin transforms,” in SPIE Press Monograph (SPIE, 2007), Vol. PM171.

23. R. J. Sasiela and J. D. Shelton, “Transverse spectral filtering and Mellin transform techniques applied to the effect of outer scale on tilt and tilt anisoplanatism,” J. Opt. Soc. Am. A 10, 646–660 (1993). [CrossRef]  

24. J. Beck and J. P. Bos, “Saturation of the anisoplanatic error in horizontal imaging scenarios,” Proc. SPIE 10408, 104080X (2017). [CrossRef]  

25. B. E. Stribling, “Laser beam propagation in non-Kolmogorov atmospheric turbulence,” Tech. Rep. (Air Force Inst of Tech Wright-Patterson AFB OH School of Engineering, 1994).

26. J. P. Bos, V. R. Gudimetla, and J. D. Schmidt, “Differential piston phase variance in non-Kolmogorov atmospheres,” J. Opt. Soc. Am. A 34, 1433–1440 (2017). [CrossRef]  

27. D. L. Walters and L. W. Bradford, “Measurements of r 0 and θ 0: two decades and 18 sites,” Appl. Opt. 36, 7876–7886 (1997). [CrossRef]  

28. L.-K. Yu, Z.-H. Hou, S.-C. Zhang, et al., “Isoplanatic angle in finite distance: experimental validation,” Opt. Eng. 54, 024105 (2015). [CrossRef]