J. Eur. Opt. Society-Rapid Publ. 21, 1( 2025) 9
the centre of the pinhole aperture, i. e., optical axis, which then reduces its transmission. Such deflection / reduction will be greatest for wavefront error manifesting in the loworder( tip-tilt) modes, with reducing effects from increasing orders. Thus, the correction imparted by the tip-tilt mirror in the overall AO system should be made as accurate as possible, to lessen the low-order( tip-tilt) wavefront error on the beam, and then the pinhole aperture diameter d should be selected for the net asymmetric wavefront error, including any residual low-order( tip-tilt) error and highorder( asymmetric) error. For example, given our primary lens with a focal length of f 1 = 100 mm and a representative net asymmetric wavefront error of dh = 10 lrad, we would expect the reference beam’ s focus to deflect off the optical axis by f 1 dh = 1lm. For the pinhole aperture diameters in our work, d = 15and75lm, this deflection would have little consequence, but the deflection could be a concern if a longer f 1 was used and / or a smaller diameter d was used. In such cases, it may be necessary to improve the correction had from the tip-tilt mirror, reduce the focal length f 1, and / or increase the pinhole aperture diameter d.
5 Conclusion
This work presented the design and development of an SRI wavefront sensor for implementation in an AO system that corrects for the effects of atmospheric turbulence in FSOC links. This was done with thought to the demands for wavefront sensing in such links under weak through strong turbulence conditions. For the sensor’ s optical design, we observed a trade-off for the pinhole aperture’ s diameter, whereby smaller diameters yield better uniformity / flattening across the reference beam’ s wavefronts and larger diameters better transmit the reference beam’ s power in the presence of asymmetric wavefront error. This is because such error deflects the focus off the centre of the pinhole aperture. In light of this trade-off, the tip-tilt mirror in the overall AO system should lessen the low-order( tip-tilt) wavefront error as much as possible, and then the pinhole aperture diameter d should be selected for the remaining net asymmetric wavefront error, which can include residual low-order( tip-tilt) error and high-order( asymmetric) error. For the sensor’ s image processing, we concluded that the fringe spacing K should be set at or above twice the pixel size on the image sensor and the reciprocal-space filter diameter should then be set at the separation between the central and positive peaks, 1 / K. Such conditions reduce the overall error and allow the system to function roughly independent of the fringe spacing. Overall, our analysed SRI wavefront sensor, with an aperture diameter of d = 15lm and a fringe spacing of K = 87 lm, gave an accurate representation of the input beam’ s phase profile. It is hoped that these analyses and insights can enable wavefront sensing with improved functionality in future FSOC links.
Funding
Portions of this work were supported by the Natural Sciences and Engineering Research Council of Canada, grant RGPIN-2017- 04073. The core project on which this report is based was funded by the German Federal Ministry of Education and Research under funding code 16KIS1265( QuNET). The authors are responsible forthecontentofthispublication.
Conflicts of interest The authors declare no conflicts of interest.
Data availability statement
The data presented in this paper may be obtained from the authors upon reasonable request.
Author contribution statement
Authors A. C. M., I. R. H., M. F. J., and J. F. H. contributed to the data analysis / processing and the interpretation of results. I. R. H., A. P. R., R. M. C., and J. F. H designed and implemented the experimental setup. A. C. M and J. F. H co-wrote the paper.
References
1 Nousiannen J, Rajani C, Kasper M, et al., Toward on-sky adaptive optics control using reinforcement learning, Astron. Astrophys. 664( A71), 1( 2022). https:// doi. org / 10.1051 / 0004-6361 / 202243311.
2 Davies R, Kasper M, Adaptive optics for astronomy, Annu. Rev. Astron. Astrophys. 50, 305. https:// doi. org / 10.1146 / annurev-astro-081811-125447.
3 Carrizo CE, Calvo RM, Belmonte A, Proof of concept for adaptive sequential optimization of free-space communication receivers, Appl. Opt. 58, 5397( 2019) https:// doi. org / 10.1364 / AO. 58.005397. 4 Carrizo CE, Calvo RM, Belmonte A, Intensity-based adaptive optics with sequential optimization for laser communications, Opt. Express 26, 16044( 2018). https:// doi. org / 10.1364 / OE. 26.016044.
5 Land JE, Aerosol absorption measurement by a Shack- Hartmann wavefront sensor, Appl. Opt. 62, 4836( 2023). https:// doi. org / 10.1364 / AO. 492066.
6 Kalensky M, Kemnetz MR, Spencer MF, Effects of shock waves on Shack-Hartmann wavefront sensor data, AIAA J 61, 2356( 2023). https:// doi. org / 10.2514 / 1. J062783. 7 Hutterer V, Neubauer A, Shatokhina J, A mathematical framework for nonlinear wavefront reconstruction in adaptive optics systems with Fourier-type wavefront sensing, Inverse Probl. 39( 035007), 1( 2023). https:// doi. org / 10.1088 / 1361-6420 / acb568.
8 Knapek M, Adaptive optics for the mitigation of atmospheric effects in laser satellite-to-ground communications, Technische Universität München( 2010).
9 Roddier F, Curvature sensing and compensation: a new concept in adaptive optics, Appl. Opt. 27, 1223( 1988). https:// doi. org / 10.1364 / AO. 27.001223.
10 Notaras J, Paterson C, Demonstration of closed-loop adaptive optics with a point-diffraction interferometer in strong scintillation with optical vortices, Opt. Express 15, 13745( 2007). https:// doi. org / 10.1364 / OE. 15.013745.
11 Crepp JR, Letchev SO, Potier SJ, et al., Measuring phase errors in the presence of scintillation, Opt. Express 28, 37721( 2020). https:// doi. org / 10.1364 / OE. 408825.
12 Thornton DE, Spencer MF, Perram GP, Deep-turbulence wavefront sensing using digital holography in the on-axis phase shifting recording geometry with comparisons to the self-referencing interferometer, Appl. Opt. 58, A179( 2019).