For the circular aperture case, the tilt-variance can be written as Display Formula
$\u27e8a\xb7a\u27e9=2.0864J(S)G(Q)/D22,$(41)
where $S$ measures the turbulence strength and $G(Q)$ gauges the relative effects of diffraction through parameter $Q$ involving the transition from 0.5 for $Q<1$ to 1.0 for $Q\u22652$. It can be expressed approximately using the below equation: Display Formula$G(Q)=\Sigma P(q^,+0.908,+0.361,+0.092,1.311,+5.125,\u22120.201),$(42)
where $q^=log10(Q)$ and $\Sigma P$ is a spliced sigmoid function: Beginning with a standard sigmoidal function $\Sigma (x)=[exp(x)\u22121.0]/[exp(x)+1.0]$ that transitions between $\u22121$ and $+1$ as $x$ 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 $x=D$ as Display Formula$\Sigma P(x,A,B1,B2,C1,C2,D)={A+B1\Sigma [C1(x\u2212D)],x\u2264D,A+B2\Sigma [C2(x\u2212D)],x\u2265D.$(43)
By requiring $B1C1=B2C2$, both the value of the function and its slope will be contiguous at $x=D$. This function approximates the curve plotted in Fig. 2 of Ref. ^{5}. The current form has been tailored for the range $\u22122<q^<+2$.