Optical normalization of Fresnel integrals ($C(w)+iS(w)={\int }_0^w{e}^{i\pi u^2/2}du$)

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 $\sqrt{2}$ in diffraction calculations.