For the circular aperture case, the tilt-variance can be written as Display Formula
(41)where measures the turbulence strength and gauges the relative effects of diffraction through parameter involving the transition from 0.5 for to 1.0 for . It can be expressed approximately using the below equation: Display Formula
(42)where and is a spliced sigmoid function: Beginning with a standard sigmoidal function that transitions between and as transitions from large negative values to large positive values, a spliced form is constructed by joining two separate rescaled sigmoid representations at a transition point as Display Formula
(43)By requiring , both the value of the function and its slope will be contiguous at . This function approximates the curve plotted in Fig. 2 of Ref. 5. The current form has been tailored for the range .