J. Eur. Opt. Society-Rapid Publ. 21, 1( 2025) 7
Fig
. 5. Phase profiles in the output 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 a pinhole aperture with a diameter of d = 15lm and a fringe spacing of K = 87lm.
Fig. 6. Residual wavefront error versus mode order J for weak( 1 rad of wavefront error, blue) and strong( 2 rad of wavefront error, red) turbulence conditions with tilt along x i( J = 2), defocus( J = 4), primary coma along x i( J = 8), and secondary coma along x i( J = 16). The pinhole apertures have diameters of d = 15lm( circles) and d = 75lm( squares).
is equal to this separation of 1 / K. Such scaling of the filter width and peak separation minimizes the encroachment of error from the central peak into the positive peak’ s passband. This error can also be reduced by making the fringe spacing as small as possible, and thus the separation as large as possible, but this must be done while considering the pixel size on the camera’ s image sensor. According to the fundamental Nyquist sampling theorem [ 25 ], the minimum fringe spacing resolved by the sensor will be two pixels wide, corresponding to a halved resolution, but larger fringe spacings are ideally used to fully resolve the fringes. Thus, we have used a fringe spacing of K = 87lminthisanalysis. This corresponds to the experimental fringe pattern displayed in Figure 2d and is roughly four pixels wide. Given these two parameters with an input beam subject 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 see strong agreement between the