= Optical normalization of Fresnel integrals
{title2=$C(w)+iS(w)=\int_0^w e^{i\pi u^2/2}\,du$}
For the optical convention of <Fresnel integrals>, $C(w)=\int_0^w\cos(\pi u^2/2)du$ and $S(w)=\int_0^w\sin(\pi u^2/2)du$. Both functions are odd and tend to $1/2$ as $w\to+\infty$. If the unscaled convention uses phase $t^2$, multiply its argument by $\sqrt{\pi/2}$ and its value by $\sqrt{2/\pi}$ to obtain this convention. An aperture factor with phase $\pi x^2/(\lambda R)$ uses $u=x\sqrt{2/(\lambda R)}$. Stating the convention prevents missing factors of $\pi$ or $\sqrt2$ in <diffraction> calculations.
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