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J. Eur. Opt. Society-Rapid Publ. 21, 1( 2025)
input beam phase profiles, in Figures 3a – 3d, respectively, and our estimated beam phase profiles, in Figures 3e – 3h, respectively.
The overall functionality of the SRI wavefront sensor is encapsulated by Figure 6. The figure shows the residual wavefront error [ 18 ], as the root-mean-squared difference between the input beam phase profile and our estimated beam phase profile, versus the mode order J for weak( blue) and strong( red) turbulence conditions. Here, the conditions are defined by the wavefront error [ 18 ], as the root-meansquared difference between the input beam phase profile and its averaged phase across the profile, while the pinhole apertures have diameters of d = 15lm( circles) and 75 lm( squares). In following the foundational work of Noll [ 26 ], we define weak, moderate, and strong turbulence conditions as those with wavefront errors less than or equal to 1 rad, between 1 and 2 rad, and greater than or equal to 2 rad. The results in Figure 6 are shown for weak and strong turbulence conditions with a wavefront error of 1 and 2 rad, respectively. We can conclude from these results that the least residual wavefront error is had by the pinhole aperture with a diameter of d = 15lm, as its errors are less than 0.11 rad for all mode orders in weak and strong turbulence conditions. Nonetheless, it may still be possible to use the pinhole aperture with a diameter of d = 75lm, but the residual wavefront error here can only be kept below 0.95 rad in the weak turbulence conditions.
4 Limitations and recommendations
Our results from the prior section showed the SRI wavefront sensor’ s effectiveness, but its use is subject to limitations. The foremost six limitations and our corresponding recommendations are discussed here.
The first potential limitation of the SRI wavefront sensor relates to scalability. Our prior work [ 27 ] has shown that there is a fundamental relationship between the effects of atmospheric turbulence and the diameter of the telescope aperture under equivalent atmospheric turbulence conditions. Specifically, only simple low-order( tip-tilt) correction is typically required for diameters up to 5 cm, but when the system is scaled up and the diameter increases, the effects of atmospheric turbulence grow. The wavefront sensor must then be designed to characterize higher-order modes within its images.
The second potential limitation of the SRI wavefront sensor relates to the detection limits of its hardware. The camera is the greatest concern here, as its pixel sensitivity sets the minimum requirements for the beam powers( and signal-to-noise ratios) while its pixel size dictates the minimum resolvable spatial features( and thus the maximum measurable mode order). Ideally, the SRI wavefront sensor would be implemented with combined thought to its beam powers, which may demand optical amplification, and its upper limit for mode orders, which may necessitate the use of a high-resolution camera [ 28 ].
The third potential limitation of the SRI wavefront sensor relates to noise in its image processing. Such noise can manifest from sensor, manufacturing, and assembly errors [ 29, 30 ]. Fortunately, these errors can be mitigated through careful calibration [ 29 ]. It is also possible for quantization noise to arise from the fast Fourier transform in our image processing, due to rounding, floating-point representation, and truncation errors [ 31 ]. Such errors can also be mitigated [ 32, 33 ], but doing so comes at the cost of speed. Thus, the overall speed of the AO system, and specifically its control loop, should be considered while planning noise mitigation.
The fourth potential limitation of the SRI wavefront sensor relates to inefficiencies in its image processing. In particular, its phase unwrapping can become computationally intensive due to the emergence of branch points / cuts. Fortunately, challenges such as these are being met by recent advancements in machine and deep learning. Machine learning has led to improvements for wavefront sensing and turbulence characterizations via reward functions [ 1 ], wavefront estimations [ 34 ], and wavefront control [ 35 ]. Likewise, deep learning has advanced wavefront sensing via residual wavefront error rejection [ 20 ], convolutional neural networks [ 36 ], and sophisticated control models [ 37 ]. The image processing in our work could benefit from any number of these emerging technologies.
The fifth potential limitation of the SRI wavefront sensor relates to its speed. Here, we must recognize that wavefront errors exhibit both spatial variations, as defined by the mode orders, and temporal variations, as defined by the Greenwood frequency [ 38 ]. The speed of the SRI wavefront sensor, and the overall AO system’ s control loop, should then be made greater than the Greenwood frequency to mitigate any concern on temporal variations. Our SRI wavefront sensor was designed with spatial variations as the sole concern, as our overall AO system’ s control loop can function at speeds above the highest( real-world / realistic) Greenwood frequency. Specifically, given a wavelength of k 0 = 1550 nm, propagation length through the atmosphere of L = 10 km, and highest( real-world / realistic) wind velocity of v w = 30 m / s, the Greenwood frequency is
only 0.4v w /( k 0 L) 1 / 2 = 100Hz [ 38 ] while our system operates at a factor of 20 above this frequency, i. e., 2 kHz. This realtime speed is achieved by first training the system, whereby the tip-tilt / deformable mirrors are perturbed and wavefront errors are measured. This builds the loop’ s interaction matrix. We then apply the inverse of this interaction matrix between the inputs( from the wavefront sensor) and outputs( to the tip-tilt / deformable mirrors). Ultimately, the speed of any AO system’ s control loop should be designed with the Greenwood frequency in mind, to ensure that its wavefront errors can be sensed and mitigated solely in terms of their spatial variations, as done in this work.
The sixth potential limitation of the SRI wavefront sensor relates to trade-offs from its aperture diameter. Here, we recognize that smaller pinhole aperture diameters yield better uniformity / flattening across the reference beam’ s wavefronts, and thus improved estimates for the beam phase profiles, but they also give reduced power transmission when( azimuthally) asymmetric wavefront error exists across the beam. The reduction occurs because such asymmetric wavefront error deflects the beam’ s focus off