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J. Eur. Opt. Society-Rapid Publ. 21, 1( 2025)
Fig
. 4. Phase profiles in the focal plane for the signal beam( top row) and reference beam( bottom row). The profiles are shown for an input beam experiencing turbulence-induced distortion as tilt along x i( J = 2) in( a) and( e), defocus( J = 4) in( b) and( f), primary coma along x i( J = 8) in( c) and( g), and secondary coma along x i( J = 16) in( d) and( h). The phase is displayed as colours mapped from low( blue) to red( high), given pinhole apertures with diameters of d = 15 and 75 lm( seen in the bottom row as small and large black circles, respectively), and a fringe spacing of K = 87 lm.
reference beam( bottom row) are shown for the focal plane in Figures 4a and 4e, 4b and 4f, 4c and 4g, and4d and 4h, respectively, and the output pupil plane in Figures 5a and 5e, 5b and 5f, 5c and 5g, and5d and 5h, respectively. All of the results are illustrated as two-dimensional colourmaps of phase spanning from low( blue) to high( red). The pinhole aperture is shown on the reference beam in Figure 4 for a narrow aperture diameter, d = 15 lm( black circle), and a wide aperture diameter, d = 75 lm( black circle).
There are two key characteristics to note in the optical design. First, the presence of azimuthal asymmetry on the input beam phase profiles in Figure 3 deflects the signal and reference beams off their optical axes within their respective focal planes. Such deflections are of little consequence to the signal beam, which has fixed tilt already applied to it( from the beamsplitter) and unobstructed transmission through its focal plane( given its lack of an aperture). However, the deflections are of great concern for the reference beam, which deflects along the + x f direction with extents that are large in Figure 4e( J = 2), negligible in Figure 4f( J = 4), moderate in Figure 4g( J = 8), and small in Figure 4h( J = 16). Thesedeflections reduce the transmitted power of the reference beam through the pinhole aperture to a great degree for the narrow aperture diameter, d = 15lm, and a lesser degree for the wide aperture diameter, d = 75 lm. Only the input beam phase profile of Figure 4f( J = 4) escapes this deflection-induced reduction in power, as a result of its pure azimuthal symmetry. Second, we note that the reference beam phase profile in the output pupil plane should be sufficiently flat / uniform, as this will allow the signal beam phase profile to be accurately mapped onto the( superimposed) output beam phase profile. The results displayed in Figures 5e, 5f, 5g, and5h show that the reference beam can exhibit this flat / uniform phase profile – but only for an aperture diameter of d = 15 lm. The corresponding profile for the aperture diameter of d = 75lm( not shown) is far from flat / uniform. Such trends can be understood by the inverse Fourier transform relationship between the focal and output pupil planes, whereby a point aperture at the focus outputs a flat phase profile on the reference beam and a wide aperture at the focus outputs similar phase profiles on the reference and signal beams.
3.2 Image processing
The performance of the SRI wavefront sensor’ s image processing can be assessed by its ability to estimate the input beam phase profile from the intensity distribution on the image sensor. As such, we consider the aforementioned phase profiles on the input beam, corresponding to turbulence-induced tilt along x i( J = 2), defocus( J = 4), primary coma along x i( J = 8), and secondary coma along x i( J = 16). We then analyse the resulting phase profiles on the estimated beam, which are shown in Figures 3a and 3e, 3b and 3f, 3c and 3g, and3d and 3h, respectively. Here, we have used Fourier fringe analysis with the pinhole aperture having a diameter of d = 15lm and the fringe spacing of K = 87lm. This fringe spacing separates the positive and negative peaks off the central peak in the reciprocal space by 1 / K = 11.5mm �1. We then apply a bandpass filter around the positive peak with a diameter that