= Grazing-incidence X-ray optics
<Grazing-incidence X-ray optics> redirect <X-rays> by shallow <specular reflections> along mirror surfaces. <Riccardo Giacconi>'s lecture describes coaxial paraboloidal and hyperboloidal mirrors using two reflections, and the later Einstein observatory's imaging capability. Focusing separates collecting area from detector area: the collected source signal can remain large while the detector background is drawn from a smaller region.<The Dawn of X-Ray Astronomy by Riccardo Giacconi>[sections 3.2–4, pp. 123–124, Fig. 13]{ref}
An ideal lossless <geometrical optics> calculation explains the shallow angle. Take vacuum <refractive index> one and a material with real <refractive index> $n=1-\delta$, where $0<\delta\ll1$. Let $\theta$ be the grazing angle measured from the surface. <Snell's law> written for angles from the normal requires $\cos\theta=n\sin\theta_t$. A propagating transmitted ray exists only if $\cos\theta\leq n$. Thus the ideal threshold is
$$
\theta_c=\arccos(1-\delta)\simeq\sqrt{2\delta}.
$$
The last step uses the <Taylor series> $\cos\theta=1-\theta^2/2+O(\theta^4)$. For a chosen $\delta=2\times10^{-5}$, $\theta_c\simeq0.006325$ <radians>, or $0.3624^\circ$. Absorption makes the <refractive index> complex and softens this ideal threshold; surface roughness and alignment errors also reduce throughput or worsen the <point spread function>. These are reasons to calibrate <X-ray effective area> and <angular resolution>, rather than assume that reflection is perfect.
The benefit can be quantified without specifying a historical telescope. Suppose a source retains fraction $f$ of its counts inside an extraction region, while a uniform detector background contributes $q a t$ counts over detector area $a$. In the background-dominated limit of the same independent on-off model,
$$
Z\simeq\frac{f F A_{\mathrm{eff}}\sqrt t}{\sqrt{(1+\alpha)q a}}.
$$
Reducing $a$ by a factor of 100 at fixed $f$ and <X-ray effective area> improves this ideal ratio tenfold. A <collimator> restricts admitted directions; focusing optics bring selected rays together. An image also separates nearby sources and identifies clean background regions, connecting optical design directly to a better constrained <inverse problem>.
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