Scaling laws for the noise-equivalent angle and C-tilt, G-tilt anisoplanatism due to scintillation: errata (Applied Optics (2025) 64 (E11–E19) DOI: 10.1364/AO.553861)

We correct two errors in [Appl. Opt. 64, E11 (2025)] that do not alter the overall conclusions of the paper. The first error involves an erroneous factor of 1/2 in the log-amplitude covariance used to compute the noise-equivalent angle (NEA) due to scintillation and the root-mean-squared error (RMSE) between the centroid tilt (C-tilt) and gradient tilt (G-tilt). The second error is a missing normalization factor in the log-amplitude covariance for a spherical wave [see Eq. (20)]. In addition, we clarify an underlying assumption in our C-tilt definition. This last point is not an error, but is important when comparing the analytic solutions with the wave-optics simulation results and the formulated scaling laws. Despite these corrections, the scaling laws presented in the original work remain valid and continue to provide a good estimate of the NEA due to scintillation and the C-tilt, G-tilt RMSE. In Section 2C of Ref. [1], the final sentence should be omitted, as a factor of 1/2 should not have been included in the log-amplitude covariance when converting the two-axis expression to a one-axis expression. We obtain the one-axis expressions for the centroid tilt (C-tilt) variance [see Eq. (14)] and the mean-squared error (MSE) between the C-tilt and gradient tilt (G-tilt) [see Eq. (18)] by simply dividing the overall result by a factor of two. To reflect this correction, the second sentence of Section 2D should read as follows: “We start by converting the two-axis expressions for the phase-gradient correlation from Section 2C to one-axis formulas by dividing by two.” In Eq. (20) of Ref. [1], we omitted a scaling factor of cos(5π/12) from the expression for the spherical-wave log-amplitudecovariance.The correct expression is (Formula presented) With these corrections in mind, we numerically integrated the single-integral expressions for both the C-tilt variance and the C-tilt, G-tilt MSE. The resulting root-mean-squared error (RMSE) reveals a discrepancy between the analytic solution and the wave-optics simulation results. In what follows, we reconcile this discrepancy by identifying an underlying assumption in our expression for the angular location of the centroid in the focal plane (see Eq. (5) in Ref. [1]). We start with the expression for C-tilt given by Eq. (4) in Ref.[1] (Formula presented) where 8 denotes the total power in the pupil plane. To account for the nonuniform irradiance caused by scintillation, we express 8 as where U₀ is the amplitude of the point source. Substituting the above expression into Eq. (2), we find Ref. [1] and the previous literature assume that the denominator is approximately equal to the aperture area. However, this assumption only holds under sufficiently weak scintillation, where exp[2χ(r)] ≈ 1. As a result, the C-tilt expression given in Eq. (5) of Ref. [1] serves as a simplified C-tilt expression. This simplification provides a good approximation for the NEA due to scintillation, but not for the C-tilt, G-tilt RMSE. To derive a more general expression for the C-tilt variance, we substitute Eq. (4) into the definition of the C-tilt variance (see Eq. (6) in Ref. [1]), which gives (Formula presented) Assuming the mean of the quotient is equal to the quotient of the means, we can compute the expected value of the numerator and the denominator separately. We simplify both using the same methods outlined in Ref. [1]. The result is Figure 1 shows the updated results of the NEA due to scintillation, incorporating the modifications outlined in this errata. The numerically integrated NEA (solid curves) and the wave-optics simulation results (circles) continue to agree well with the formulated scaling law from Ref. [1] (dashed curves) as a function of both Rytov number R and Fresnel number N_f . Even with the simplified C-tilt expression from Ref. [1], the analytic solution does not significantly deviate from the results of the wave-optics simulations in the weak-to-moderate scintillation regime. (Figure presented) Using this modified expression for the C-tilt variance, we can substitute the expressions for the normalized aperture autocorrelation 9(ρ, D), the phase-gradient correlation C_∇φ, and the log-amplitude covariance B_χ(ρ) and numerically integrate in MATLAB. Figure 2(a) shows the updated results for the C-tilt, G-tilt RMSE, incorporating the modifications outlined in this errata. The numerically integrated RMSE (solid curves) clearly deviate from both the wave-optics simulation results (circles) and the formulated scaling law from Ref. [1] (dashed curve) in the weak-to-moderate scintillation regime. Figure 2(b) shows the updated results for the jitter Strehl ratio, incorporating the modifications outlined in this errata. Again, the numerically integrated jitter Strehl ratio (solid curves) deviates from both the wave-optics simulation results (circles) and the formulated scaling law from Ref. [1] (dashed curve). This deviation stems from the well-documented limitations of first-order Rytov theory and potential errors in the MATLAB numerical integration routines. Given the modifications outlined above, we find that the scaling laws from the original work still provide a good engineering estimate for both the NEA due to scintillation and the C-tilt, G-tilt RMSE. Systems engineers can use these estimates to appropriately design active electro-optical systems, which inevitably experience the effects of scintillation when imaging through distributed-volume turbulence. Closed-form solutions remain a topic for future work.