Fraunhofer diffraction and the beam quality β factor of a laser beam with sinusoidal wavefront

The effect of the amplitude and spatial frequency of a sinusoidal wavefront over a square aperture on the far-field diffraction pattern is considered for a laser beam. An analytical solution of the Fraunhofer diffraction problem for a 2D sinusoidal wavefront over a rectangular aperture with an arbitrary integer number of periods along each side is derived using an original technique. The result coincides with high accuracy with a numerical solution obtained using the Fast Fourier Transform. It has been shown that after correcting the second-order Zernike modes by a stacked array piezoelectric deformable mirror with different numbers and patterns of control actuators, the residual wavefront aberrations may be considered approximately sinusoidal. Therefore, the results for an ideal sinusoidal wavefront could be applied to a real residual wavefront in problems of adaptive optics. For sinusoidal aberrations, the so-called beam quality β factor has been shown to increase sharply as the root-mean-square wavefront error (RMS) passes certain values. A new criterion of wavefront quality for sinusoidal and residual aberrations over a square aperture has been proposed.