J. Eur. Opt. Society-Rapid Publ. 21, 1( 2025) 5
Fig. 3. Phase profiles in the input plane for the input beam( top row) and estimated 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.
the redundant / unnecessary phase characteristics in the negative / central peaks. In the third step, we apply a twodimensional inverse fast Fourier transform, F fft
�1 { }, to the filtered output and multiplynthenresult by the phase o factor e j2pxo / K to give F �1 fft
F fft E~ 0 ðx o; y o Þ E~ 0 ðx o; y o Þ
U RS e j2pxo = K g. The phase factor here shifts the origin in reciprocal space to the centre of the positive peak and thus removes the fringe pattern that appeared in the imaged intensity distribution. In the fourth step, we compute the arctangent of the ratio of the last distribution’ s real component Refg and imaginary component Imfg, scale the horizontal dimension by f 1 / f 2, to undo any magnification incurred by the confocal primary and secondary lenses, and unwrap the phase. This gives an estimated beam phase profile of
See Equation( 4) at the bottom of the page
which will ideally depict the input beam phase profile / i( x i, y i). Branch point / phase discontinuities may arise from the unwrap { } function here, but strategies to remove them are shown elsewhere [ 22 – 24 ].
3 Results and discussion
We consider a beam entering the SRI wavefront sensor with a radius of x = 2.5 mm and an arbitrary input beam phase
profile, / i( x i, y i) in equation( 1). We then solve for the electric field of the output beam, E~ 0 ðx o; y o Þ in equation( 3), and apply image processing to its intensity distribution to extract the estimated beam phase profile / i( est)( x i, y i). The analyses of / i( est)( x i, y i) are had with the input beam phase profile / i( x i, y i) cast as a superposition of( orthogonal) Zernike polynomials enumerated by the( Noll) mode order J = 1, 2, 3,.... The characteristics underlying these mode orders are given in the Appendix, with details on their wavefront aberrations and symmetries.
3.1 Optical design
The performance of the SRI wavefront sensor’ s design is gauged by its ability to both pass the signal beam unperturbed through the system( aside from our negation and tilt on its phase) and image the reference beam in the output pupil plane with a flat phase. The diameter of the pinhole aperture is the key parameter in such efforts and is focused upon here. We consider four representative 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). Thefour phase profiles on the input beam( top row) and estimated beam( bottom row) are illustrated in Figures 3a and 3e, 3b and 3f, 3c and 3g, and3d and 3h, respectively. The resulting phase profiles on the signal beam( top row) and
0 0 n n n Im F �1 fft
F fft
E~ 0 ðx o; y o
/ iðestÞ ðx i; y i Þ ¼ unwrap@ arctan @ n n n Re F �1 fft
F ~ fft
E 0 ðx o; y o
Þ E~ 0 ðx o; y o Þ E~ 0 ðx o; y o
Þ
Þ o o
U RS
U RS oo oo e j2pxo = K
e j2pxo = K
11 AA; ð4Þ