1. Introduction

Traditional adaptive optics (AO) systems used for correction of the effects of atmospheric turbulence on imaging and laser-based systems have three main components: a wavefront sensor, a deformable mirror and a reconstruction computer. Wavefront sensorless AO disposes of the first element and relies on blind optimization algorithms to improve a suitable metric, such as image sharpness or the amount of delivered energy. This type of AO was first proposed by Muller and Buffington in their seminal paper on image sharpening [1]. Since then this approach has found several applications, especially in medical imaging and in microscopy [2–5], but also in free-space optical communications [6] and directed energy [7].

The most widely used algorithm in wavefront sensorless AO is the stochastic parallel gradient descent (SPGD). Traditionally, SPGD is used to perturb randomly the actuators on a deformable mirror and the image, or the laser beam quality metric is checked for improvement. This can become a highly-dimensional problem for modern deformable mirrors with many actuators. To reduce the number of degrees of freedom one can project the actuator space onto an orthogonal modal basis, e.g. Zernike polynomials and perturb these instead of actuators.

Gradient-descent-based sensorless wavefront correction aims to achieve the best possible compensation with the fewest possible correction steps. Here, convergence of the iterative algorithm is of paramount importance, because correction must keep up with the evolving optical aberrations. The correction window allowed by the atmosphere before conditions change is between 1 and 10 ms long, depending on turbulence strength, wind speed and wavelength of light being used [8]. There are two avenues which can be explored in order to make a wavefront sensorless AO system more effective: increase its speed, which is the hardware solution, or increase the convergence rate of the underlying algorithm, which is the software solution. The fact that deformable mirrors with frame rates above 40 kHz are currently not available on the market imposes an upper limit to the first approach.

In this work we focus on the second option by including a priori knowledge about the Kolmogorov spectrum of refractive index fluctuations and an optimal method to manage this knowledge over the course of a full correction cycle. It has already been shown that switching from zonal to modal correction significantly improves convergence of wavefront sensorless AO [9]. An alternative modal system is used here, which forms an optimal modal decomposition of atmospheric wavefronts, in the sense that the squared residual phase error over the aperture is minimum, when compared to any other orthogonal set with the same number of corrected modes [10,11]. Specifically, the so-called Karhunen-Loève (KL) modes overcome the inherent statistical dependence between Zernike modes and thus increase correction performance of the blind search algorithm, as will be shown later.

Starting in Section 2 we present a mathematical derivation of the KL modes and demonstrate their advantages compared to the Zernike modes. The procedure used to generate KL modes numerically with the help of Zernike modes is also described in this section. Section 3 describes modal gradient descent optimization and introduces the incorporation of a priori knowledge in order to maintain accurate spatial frequency representation of a wavefront with Kolmogorov statistics [9]. In Section 4 we present the simulation framework used throughout this paper and specify the turbulence conditions for our analysis. The analysis of Strehl ratio evolution of the tested methods in static turbulence conditions is given in this section as well as a detailed investigation of global correction optima. Applying the findings from the previous sections and emulating wind and turbulence boiling [12], we study the dynamic correction behavior of the tested modal bases in Section 5. A summary of the key findings and their consequences for the field of wavefront sensorless AO concludes this paper.

2. Mean-squared-error-minimized wavefront decomposition in Kolmogorov turbulence

Zernike polynomials are one of the most commonly used tools for modal decomposition of wavefront aberrations. They were initially introduced by Frits Zernike to describe diffraction effects when applying the knife-edge method on concave mirrors [13]. Noll [14] later used them to give a polynomial description of Kolmogorov’s turbulence statistics [15] by extending the work of Fried [16]. With Noll’s normalization they are defined on the unit circle according to:

Wavefront decomposition with Zernike polynomials enables the prediction of wavefront correction degree when removing the first $J$ modes from a distorted wavefront over a circular aperture of arbitrary radius $R$

By definition the covariance matrix of $\mathbf {B}$ must be diagonal, since its components are uncorrelated. Calculating the covariance of Eq. (10) yields

Using the SVD of $\mathbf {C}$ and Eq. (10) the KL radial polynomials $R_p^q(r)$ can be obtained numerically. However, in doing so, the number of higher-order Zernike modes used in the computation must be limited to avoid numerical artifacts and diverging phase values close to the aperture boundary. Due to the absence of a closed-form description for radial KL polynomials the mode order $p$ is obtained simply from the ordinal number of the radial Zernike function. Since Zernike correlations only exist for modes exhibiting the same triangular functions (e.g. X-Tilt couples into X-Coma, because $m_{\textrm {Tilt}}=m_{\textrm {Coma}}=1$), all triangular Zernike contributions to one particular KL mode are identical ($q=m$). Therefore, the triangular functions $\varTheta _p^q(\vartheta )$ can be adopted from the incorporated $j$ Zernike modes and are factored out in Eq. (7). For the natural index $l$, the modes are ordered according to their eigenvalue magnitude taken from the diagonal of $\mathbf {S}$.

The residual KL wavefront errors can be calculated from the squared eigenvalues of the KL expansion on the diagonal of $\mathbf {S}$ (see Table 1 in Ref. [19]). Numerical values for the first 21 residuals were published by Wang and Markey [10], and higher-order values can be obtained by using a formula proposed by Dai [20].

3. Mean-squared-minimized wavefront sensorless correction

Modal wavefront sensorless AO has been proposed with the goal of reducing the optimization complexity of the zonal algorithms, which must, by definition, optimize as many control channels as there are actuators on the deformable mirror [9,11,21,22]. In the modal approach the number of degrees of freedom is equal to the number of modes $J$ (see Eq. (5)).

In this work we focus on the implementation of a KL-based modal SPGD method, where the control vectors, containing the individual mode coefficients $u_n^{m}=(a_n)$, are iteratively optimized in order to maximize the system performance metric $J(u_n^m)$. By randomly altering the sign $\gamma _n^m$ of small perturbations $\delta u$ to the current control vector of the $m$-th iteration,

The parameters $G$ and $\delta u$ are chosen empirically and affect convergence speed and accuracy of the whole process. Together with the number of incorporated modes $J$, a trade-off between these three parameters has to be found in order to meet desired application requirements.

In order to exploit the known power contributions of the incorporated modes to the wavefront variance, we weight the individual mode coefficients with their eigenvalues’ square roots $\sqrt {\langle a_j^2\rangle }$, replacing $\delta u$ with:

4. Numerical investigations in static turbulence conditions

4.1 Simulating modal sensorless wavefront correction

We set out to check the conjecture that the KL basis is more efficient than the Zernike basis by first using a purely spatial test. By efficiency we mean delivering better performance with the same number of modes, or alternatively, delivering the same performance with fewer modes (which for the type of AO considered in this paper translates to fewer optimization channels and faster convergence). We used the modal wavefront decomposition method described in Section 2, and the statistically independent KL basis from Eq. (8) to generate phase screens. Introduced by Roddier [17], the method is the most accurate way to simulate atmospheric phase screens, albeit the most time consuming. We restricted the KL decomposition to 1890 modes for the reasons stated in Section 2. The coefficients of each KL mode were created by generating Gaussian random variables and scaling them by the corresponding element of the matrix $\textbf {S}$ in Eq. (12). We simulated a scenario with a normalized turbulence strength of $D/r_0$ = 4, where the wavefront exhibits a spatial coherence length $r_0=5$ cm over an aperture $D$ of 20 cm. The phase screens were generated on a grid of 1080-by-1080 pixels to match the resolution of the spatial light modulator to be used in future experimental validation.

In this work we chose to evaluate the Strehl ratio $SR$ of the residual wavefront as the performance metric. It denotes the ratio of the peak value of the short exposure image divided by the peak value of the diffraction-limited image and is defined here over a discrete pupil as:

Here $N_A$ gives the total number of pixels of the discrete wavefront phase $\varphi (k)$ inside the modeled circular aperture.

For KL mode generation we used the method described in Section 2. A total of 1325 Zernike modes were used for the SVD of Noll’s matrix, since we found that the incorporation of higher-order Zernike modes only yielded an increase in spatial frequency artifacts and phase divergence, while only marginally contributing to spatial frequency accuracy. In order to choose the most accurate representation of KL modes, we included the first $l$ KL modes from matrix $\mathbf {U}$ (see Eq. (11)), such that the smallest Zernike coefficient $(U_{jl})$ in the $l$-th row is less than 0.001. Modal SPGD (M-SPGD) correction was implemented using 105 of these pre-generated modes, and was used to correct a total of 100 Monte-Carlo turbulence screens with a fixed number of 1000 correction iterations per screen. Most high-order AO systems have a tilt pre-compensation subsystem and therefore here too we made the assumption of perfect tilt correction. We are of the opinion that SPGD is not a good approach for tilt correction and that a simple four-quadrant-detector is a reliable and practical solution to this problem. Additionally we assume a noise-free metric signal in Eq. (14). We tested four approaches: correcting with unweighted Zernike modes, correction with weighted Zernike modes according to Eq. (15), correction with unweighted KL modes and the correction with weighted KL modes. Furthermore, we extracted all modal coefficients from the final correction screens in order to perform further analysis.

4.2 Convergence analysis

M-SPGD correction performance in simulated static turbulence ($D/r_0=4$) is shown in Fig. 1. Here, the evolution of the Strehl ratio for all implemented configurations is compared with their respective theoretical (black lines) and numerical Maréchal approximations (colored dashed/dotted lines). The latter were calculated by evaluating the extended Maréchal approximation for the Strehl ratio

 

Fig. 1. Comparison of the Strehl ratio evolution for the four tested SPGD methods. Simulated SPGD corrections (solid lines) are compared to their respective theoretical (black lines) and numerical Maréchal approximations. The average Strehl ratio resulting from the tip-tilt corrected phase screens was 0.29.

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With the help of Eq. (6) we can approximate Eq. (17) as:

Although the benefit of using KL modes over Zernike modes only leads to a small increase in Strehl ratio, it comes at no additional cost in terms of system complexity and is essentially "free". After the initial pre-generation of modes, no additional measures must be taken. We see similarities between behaviors of weighted vs. non-weighted modal SPGD algorithms to the performance of modal vs. zonal SPGD [21]. In this previous work, we have demonstrated that modal SPGD converges significantly faster than zonal SPGD, which is due to the reduced number of optimization channels. The major difference now is that weighted SPGD does not incur a significant Strehl loss. The constraint on the accurate representation of the Kolmogorov model of refractive index fluctuations increases the convergence rate significantly, because the metric gradient assumes, on average, optimal values. Although unweighted SPGD still does converge and find the local optimum of the cost function even in the case of sub-optimal gradient estimations, this comes at the cost of convergence rate. Whereas for weighted SPGD, sub-optimal gradient estimates are of less significance to the instantaneous Strehl ratio than in the non-weighted configuration, since the correction capabilities are less affected by erroneous compensation. Here, the numerical Maréchal predictions should not be considered as a limit, but rather as an indicator for faster convergence capabilities. Although both weighting configurations achieve nearly the same final Strehl ratio, they differ in the required number of iterations to reach their final value. Furthermore, there is no unique solution to the problem of maximizing the Strehl ratio of a wavefront with an infinite sum of polynomials in a small aberration regime [23], thus small deviations from the ground truth modal coefficients still can lead to, on average, high Strehl ratios.

4.3 Static residual error analysis

The analysis of convergence rate is only one way of investigating the performance of M-SPGD. In this section, we investigate the accuracy of the spatial frequency representation of Kolmogorov turbulence. The extraction of the final mode coefficients for each corrected screen enables the calculation of the mean squared residual error according to Eq. (6). One exploits the orthonormality property of the modal basis and calculates the correction degree from Eq. (5) with the subsequent removal of each mode.By computing the ensemble averages of the squares of the modal coefficients, we obtain the residuals in Eq. (6). This was done for all implemented M-SPGD configurations.

The comparison between theoretical predictions for the residual phase variances according to Noll [14] and Wang and Markey [10] and the numerical optima are depicted in Fig. 2. The benefit of constraining the modal basis with a priori knowledge becomes apparent after the correction of just the first six modes (X/Y-Astigmatism and Defocus). We observe a final residual variance twice as large as the theoretical limit for the weighted modes and approximately 6 times larger for the unweighted bases. Unlike in a previous publication [19], where binned (12x12 pixel grid) modal bases were implemented, we do not observe large differences between KL and Zernike modes for high spatial resolution correction (1080x1080 pixel grid). Nevertheless, this is more in line with the predictions from theory as can be seen from the dashed and dotted black lines in Fig. 2. The region where the KL modes outperform Zernike modes to the highest extent is between 10 and 30 corrected modes. Some benefit of using the KL basis persists all the way to 105 corrected modes in the unweighted configuration, although the curves are almost flat after around 50 modes, indicating that adding new modes to the algorithm brings no performance improvement. For the weighted variant, the performance of the KL and Zernike bases is nearly identical after removal of more than 50 modes. Here, both curves start to flatten around 80 corrected modes.

 

Fig. 2. Static residual errors in comparison to their theoretical achievable limits. Please note that the first three modes are assumed to be corrected, hence the plot starts with mode 4.

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Obviously, weighting perturbed modes equally does limit the achievable minimum residual error, since sub-optimal mode guesses impact the correction by a larger margin, and thus force the algorithm to create a residual with a non-physical spatial frequency distribution. However this only affects the rate of convergence, and not the overall achievable Strehl ratio, as depicted in Fig. 1. Here, all configurations approach the theoretical predictions quite well, with unweighted KL modes slightly outperforming the rest. As already mentioned, Maréchal approximation of the Strehl ratio should not be used to predict a certain achievable limit, but rather to explain the gain in convergence rate of weighted modes. Still the question arises as to why the achieved ensemble Strehl ratios in Fig. 1 are very close to each other. First, the relation between ensemble phase variance and Strehl ratio follows a Gaussian function (Eq. (17)), leading to a slow change in Strehl ratio for small phase variances. Furthermore, the individual mode coefficients $a_j$ are optimized with discrete steps of perturbation size $\delta u$, and are combined in the gradient of each correction step. Since the solutions to maximizing the Strehl ratio for a particular turbulence realization are not unique (see previous section), the step gradient values are also not unique, meaning the metric enables the unweighted basis to compensate for the sub-optimal gradient values over the course of the whole correction cycle. This effect only becomes visible for a large number of iteration steps and sacrifices overall phase variance reduction for maximizing the Strehl ratio, posing a possible explanation for the discrepancy between the Maréchal approximation and the values obtained in simulations.

4.4 Relative KL vs Zernike compensation

We conclude the static performance analysis by evaluating the relative error between the square roots of the residual errors of KL ($\Delta _{\textrm {KL}}$) and Zernike ($\Delta _{\textrm {Z}}$) modes

This relative gain $G_{\textrm {rel}}$ enables the prediction of the individual benefit for each KL mode over its respective Zernike counterpart.In Fig. 3 the results for unweighted and weighted M-SPGD are compared to their theoretical limits. Three main observations from this comparison can be made:

 

Fig. 3. Correction gains for KL modes over Zernike modes. Please note that for the theoretical limit we used the fitted coefficients from Noll [14] and Wang and Markey [10] for mode indices larger than 21, resulting in the smooth slope after mode 21.

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It must be noted that neither the zero-crossing point (not visible in Fig. 3), nor the crossing point of the two curves mean that SPGD started introducing aberrations rather than correcting them. As can be seen in Fig. 2, the algorithm was still delivering improvements. Figure 3 in fact shows the limitation in achievable gains by using weighted and unweighted KL modes, when realistic AO is simulated.

The last observation actually suggests that the highest $J$ to be incorporated in the KL algorithm is slightly underestimated when applying mode weighting. This might be related to the drop of the mode coefficient below a certain sensitivity level in the metric calculations, when the weighting coefficients multiplied by the perturbation parameter give diminishing returns. It might be beneficial to drop the weighting of the modes which are above this $J$ in order to increase the relative gain obtained when using weighted KL modes even more.

5. Optimal mean-squared-minimized dynamic correction

The simulations in Section 4 were carried out in a static turbulent environment. When dealing with realistic atmospheric turbulence, one does not correct subsequent independent phase screens; instead phase distortions evolve continuously over the collecting aperture, and therefore the AO system has to adapt and follow these correlated changes to successfully apply correction.

In Ref. [22] the authors already exploited temporal correlations in the turbulent atmosphere and analyzed phase compensation regardless of whether or not full correction was already achieved. In this section we extend the previous work and combine the findings from Section 4 in order to further improve modal sensorless compensation. We first investigate M-SPGD correction in simulated single-layer turbulence before evaluating atmospheric conditions where turbulence boiling is present (see Section 5.3).

5.1 Simulating dynamic SPGD correction

Single-layer dynamic turbulence conditions are simulated by generating a long "phase-strip" including subharmonics [24] with the vertical dimension $Y$=1080 pixels and the horizontal dimension given by

Here $X$ was determined using the total number $N_i$ of desired turbulence realizations, the turbulence sampling frequency $f$, and the phase screen pixel spacing $dx$. Again we consider a turbulence scenario with $D/r_0=4$ with $D=20$ cm and $r_0=5$ cm, according to [16]:

In order to emulate turbulence boiling, the single-layer, pupil-plane simulations were extended by generating ten independent phase strips. Each of these ten phase layers was then translated from left to right according to a wind speed given by a random number drawn from a Gaussian distribution $\sim \mathcal{N}\left(\bar{v}_{\text {wind}}, \sigma_{b}\right)$, where $\bar{v}_{\text {wind}}$ denotes the chosen average wind speed and the standard deviation $\sigma _b$ determines the strength of the boiling effect. For higher values of $\sigma _b$ one would expect more boiling along the propagation path, and accordingly SPGD performance should suffer. The final pupil-plane screen was reassembled by the summation of all screens along the path of propagation. In all single-, and multi-layer simulations $D/r_0$ was always equal to 4.

The goal of the multi-layer model adopted here was to increase the realism of the simulations. Other works treating iterative turbulence correction have assumed frozen turbulence [25,26]. In frozen turbulence, the best performing SPGD variant will always be the zonal SPGD configuration [21], but in reality this approach might fail to converge within the atmospheric coherence time

Here we have chosen Gaussian statistical distribution of $v_{\textrm {wind}}$ along the path and the delta correlation between layers. This model has not been verified in field experiments. There exist many measurements of vertical profiles of $v_{\textrm {wind}}$ coming from astronomical site testing [e.g. 27,28] but none of them are applicable to terrestrial, horizontal-path laser communications, which is considered here. The multi-layer simulations we have performed span a range of 2.6 ms < $\tau _0$ < 12 ms, which make them challenging for wavefront sensorless AO. Table 1 lists the atmospheric coherence times corresponding to the dynamic simulations we have performed. Please note that in simulations the coherence time must be calculated according to:

Various frames rates of the deformable mirror, which we denote as $FR$, are simulated by replicating any given screen $N$ times in order to match a desired bandwidth. For example, in order to simulate a DM working at 40 kHz every pupil plane screen is replicated 10 times, for a 100 kHz DM the screens are replicated 25 times, etc. We note here that SPGD requires two iterations per single iteration loop, therefore $FR$ is twice the SPGD loop rate. Similar to the static simulations the empirical estimation of both SPGD parameters $G$ and $\delta u$ (see Eq. (14)) were taken out in advance, whereas the total number of modes remained unchanged with 105 modes.

Table 1. Atmospheric coherence times $\tau _0$ resulting from the multi-layer, dynamic simulations.

View Table

5.2 Single-layer correction

In Fig. 4 the mean Strehl ratio over a full correction cycle of 500 ms is given for all M-SPGD configurations, when phase screens from single-layer turbulence simulations are corrected. Surprisingly, unweighted Zernike SPGD appears to be the best performing configuration for a wind speed of 1 m/s when high SPGD loop rates (12-20 kHz) are available. In general, both unweighted configurations perform better than the weighted ones, when wind speeds are slow and the correction speed is maximal, with Zernike SPGD outperforming its KL counterpart. If, however, wind speed increases, or the provided correction bandwidth decreases, the weighted configurations clearly excel and weighted KL SPGD yields the best overall correction performance. Furthermore, we can observe that the efficiency of the weighted configurations at maximum loop rate decreases slower with increasing wind speed, while the performance of the equally weighted basis drops faster for wind speeds of 3 m/s and above. Even with a single correction step per turbulence frame (4 kHz SPGD loop rate) weighted KL modes are able to provide a mean Strehl ratio of 0.6 in a common scenario for near-ground horizontal laser beam propagation at wind speeds of 3 m/s. This loop rate is achievable with a variety of commercially DMs with frame rates of 8 kHz. For wind speeds of 6 and 10 m/s, SPGD loop rates of 8 and 12 kHz respectively are required to keep the Strehl ratio above 0.5. The best results are obviously achieved with a SPGD loop rate of 20 kHz, however this is the maximum bandwidth which can be provided by off-the-shelf DMs. It seems that weighted Zernike modes pose the best option to correct turbulence in strong wind scenarios (6 and 10 m/s) with rather slow DMs (4 kHz), but since the weighted KL configuration is only slightly worse at 1 kHz bandwidth for 6 and 10 m/s wind, this could be related to a minor sub-optimal parameter choice for weighted KL and is negligible when taking the high variance in the instantaneous Strehl ratio evolution into account.

The mean residual errors for weighted KL SPGD in simulated single-layer turbulence are shown in Fig. 5. Here the temporal evolution of each individual mode coefficient from the respective polynomial expansion was recorded and used to calculate a correction degree vector for every phase screen realization according to Eq. (6). Temporally averaging the squared vector elements yields a mean residual error for each mode for a full correction cycle of 500 ms. Although not identical to Noll’s residuals in Eq. (7), we still can compare these two quantities using the latter as a lower boundary for optimal correction.

 

Fig. 4. Comparison of the single-layer correction performance for the four tested SPGD configurations in different wind speed scenarios. SPGD loop rate ranges from 20 kHz (light colored) to 4 kHz (dark colored).

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Fig. 5. Mean residual errors for weighted KL SPGD in single-layer turbulence. The residual errors for different wind speeds are depicted vs. consecutive mode indices (following Noll’s convention) when a certain SPGD loop rate is available. Please note the reversed color scale with respect to earlier figures in this paper (20 kHz case is dark colored, 4 kHz case is light colored. Horizontal dashed bars give the theoretical residuals according to theory [18]. Here modes $j={10,28,55,78,105}$ represent the final modes for radial orders $n ={3,6,9,11,13}$.

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In contrast to the results in static turbulence, here the residual errors do not decrease monotonically. Even for the lowest wind speed we observe a point after which the residual errors increase for higher order modes. Moreover, high-order mode correction in wind speeds of 3 m/s and above does require SPGD loop rates above 8 kHz in order to capitalize on the incorporation of modes beyond index 10, since the residual error when correcting with 4 kHz cannot be brought below the tip/tilt correction. This effect becomes more severe as wind speed increases, demanding higher and higher correction bandwidths in order to not worsen the correction below the ideal tip/tilt compensation limit. For both strong wind scenarios of 6 and 10 m/s any incorporation of modes above index 10 requires at least 16 if not 20 kHz to have any impact on the correction degree. However, it seems that there is a sweet spot for the optimal number of incorporated modes around index 28, also indicated by the peak in relative gain from static weighted KL over weighted Zernike correction in Fig. 3. Only for a wind speed of 1 m/s does the correction of up to 55 modes becomes useful, if SPGD loop rates of 20 kHz are available. It might be worthwhile to reduce the number of corrected modes for all tested scenarios in order to use the provided SPGD loop rates to their full potential. Although this would reduce the theoretical achievable maximum Strehl ratio, the convergence rate due to the smaller number of optimization channels can be boosted. The increased convergence speed could then be used to correct the modes with higher impact to a larger degree when facing medium and strong wind scenarios. This is not in contradiction to the results from the static residual analysis, since larger high-order mode residuals do not improve the Strehl ratio rather they make it worse. The correction of up to 105 modes and the related gain in Strehl ratio is only hypothetical, since the provided loop rates are not high enough to match the required individual temporal response of each mode to actually realize the potential improvements in Strehl ratio. The trade-off between mode number and correction speed will be examined in future work.

5.3 Multi-layer correction

In Fig. 6 correction performance in terms of the Strehl ratio for all M-SPGD configurations is depicted, when phase screens from multi-layer turbulence simulations are corrected. As already mentioned in the first part of Section 5, turbulence boiling is taken into account and thus a more realistic scenario is evaluated.

In contrast to the single-layer results, now weighted KL modes provide the best correction performance in terms of Strehl ratio, regardless of wind speed, boiling strength or provided SPGD loop rate. When correcting at the maximum SPGD loop rate of 20 kHz, we observe a Strehl ratio that is 87$\%$ of the theoretical maximum in the least challenging turbulence conditions, which decreases to 69$\%$ in the most severe turbulence scenario. Again the weighted KL configuration utilizes the provided bandwidth most efficiently, maintaining a higher mean Strehl ratio with 4 kHz loop rates for $\bar{v}_{\text {wind}}$ = 6 m/s, $\sigma _{\textrm {b}}$ = 1 m/s than both unweighted bases with 20 kHz loop rate. When coupled with DMs capable of providing mid-range SPGD loop rates, weighted KL modes are still able to keep the average Strehl ratio between 60 and 70$\%$ for typical turbulence conditions in horizontal laser beam propagation ($\bar{v}_{\text {wind}}$ = 3 m/s, $\sigma _{\textrm {b}}$ = 1 or 2 m/s).

 

Fig. 6. Comparison of the multi-layer correction performance for the four tested SPGD configurations in different wind speed and turbulence boiling scenarios. SPGD loop rates range from 20 kHz (light colored) to 4 kHz (dark colored).

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For the sake of completeness, we also show the results of the multi-layer residual error study of weighted KL SPGD in Fig. 7. Here we expect similar results to 5, but with higher applicability to future atmospheric experiments, since those simulations resemble the reality to a greater extent than the single-layer simulations. Again we extracted the mode coefficient evolution during the SPGD correction and calculated the temporal mean of the individual residuals of each phase screen. First of all, turbulence boiling affects the residual errors similar to how wind speed affects these errors, but with a smoother gradation. Here the benefit of incorporating higher-order modes is greater than for the single-layer simulation. Even the lowest loop rate of 4 kHz in the strongest wind scenario of 6 m/s does not worsen the correction degree below the theoretical tip/tilt limit. Furthermore, the provided bandwidth can be better utilized in almost every wind speed and boiling scenario, depicted by the smaller distance for the majority of residuals to the theoretical limit at a particular mode. For an improved correction degree in the most common scenario of 3 m/s it might be beneficial to limit the number of incorporated modes to 55 if not 28, especially when SPGD loop rates up to 8 kHz are available. In stronger wind conditions capping the mode count at 28 could enable a better utilization of provided loop rates up to 16 kHz. Again, the study of a potential trade-off between corrected mode numbers and correction bandwidth will be studied in the future.

 

Fig. 7. Mean residual errors for weighted KL multi-layer turbulence. The residual errors for different wind speeds and boiling strengths are depicted vs. the mode indices (following Noll’s convention) when a certain SPGD loop rate is available. Please note the reversed color scale with respect to earlier figures in this paper (20 kHz case is dark colored, 4 kHz case is light colored. Horizontal dashed bars give the theoretical residuals according to theory [18]. Here modes $j={10,28,55,78,105}$ represent the final modes for radial orders $n ={3,6,9,11,13}$.

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6. Conclusion

In this work a new modal configuration of the traditional gradient-descent-based wavefront sensorless algorithm SPGD was demonstrated. Instead of Zernike polynomials we choose to use numerically generated Karhunen-Loéve functions for modal wavefront decomposition in order to overcome correction limitations due to statistical dependence of Zernike modes. After diagonalizing Noll’s matrix to obtain both the eigenvalues of the KL expansion and the conversion matrix used to map from Zernike to KL modes, we used simulated static turbulence in order to compare the correction capabilities of both modal bases. Furthermore we adopted the optimized individual weighting of KL amplitudes as proposed in Ref. [9], ensuring a maximization of the step gradient estimation in Kolmogorov turbulence. This new version of modal sensorless wavefront correction was tested in both static and dynamic turbulence with a normalized turbulence strength of $D/r_0=4$. In both scenarios the performance was evaluated with respect to the Strehl ratio and the correction degree.

The inclusion of a priori knowledge took the spatial frequency distribution of the Kolmogorov spectrum of refractive index fluctuations into account and led to the reduction of sub-optimal guesses of mode amplitudes during the correction cycles. This measure reduced the convergence rate significantly for both modal SPGD configurations, with KL SPGD outperforming the Zernike version by about 3%. The residual error analysis in static turbulence highlighted the benefit of KL SPGD correction and enabled a preliminary estimate of the number of modes to be included for the unweighted and weighted basis configurations.

Single and multi-layered sub-harmonic phase screen simulations of dynamically evolving Kolmogorov turbulence enabled first predictions of KL SPGD performance in varying wind speed regimes. We found achievable Strehl ratios between 60 and 70% when correcting in wind speed scenarios of 3 and 6 m/s using DMs with 8 to 12 kHz frame rate. With the exception of correcting strong wind (6 and 10 m/s) single-layer turbulence with a loop rate of 4 kHz, weighted KL SPGD delivered consistently the highest Strehl ratio improvements for all wind speeds and SPGD loop rates. This version of modal SPGD was able to achieve the best results in terms of residual error minimization. Furthermore, we observed better utilization of the provided DM frame rate when incorporating modes of higher orders even for varying boiling conditions. However, we also found that there exists an optimum number of modes to be included in M-SPGD in order to match temporal wavefront statistics to the SPGD loop rates available for correction. It seems reasonable to concentrate on more power-efficient low-order modes and achieving a good degree of correction rather than incorporate high-order modes and fail to keep up with the temporal evolution of those modes.

These findings from simulated performance will be combined and tested in future work. We will transfer the acquired knowledge from simulations to laboratory experiments and analyze the performance of M-SPGD in the context of free-space laser communication.