Past exam of the physics A course of the University of Cambridge / 2012 / Paper 2 / Section B / B7

The angle in the source ray sketch is inside the gap. Two extra reflections add a full round trip. Comparing adjacent emerging rays on a common outgoing phase plane subtracts the lateral-path contribution, so the optical path difference is $2\mu d\cos \theta$, rather than $2\mu d/\cos \theta$. Thusat transmission maxima. The two internal reflection phases add to an integer full cycle in the stated ideal symmetric-coating model. Snell's law, $\mu \sin \theta ={\mu }_0\sin {\theta }_{\mathrm{out}}$, converts to the external direction. A fixed internal angle defines a cone and hence an equal-inclination ring in a focal plane.

For identical lossless mirrors the first transmitted amplitude has magnitude $(1-R)$ times the incident amplitude. Each additional round trip multiplies it by $R{e}^{i\delta }$, because $R$ is an intensity reflection coefficient for one mirror but the product of the two amplitude reflection magnitudes. Summing the geometric series givesThe maximum is one in this symmetric lossless limit; absorption, unequal mirrors or additional incoherent reflections change it. At fixed angle and negligible material dispersion, $m\lambda =2\mu d\cos \theta$ and adjacent orders differ by one. For $m\gg 1$,

order-safe angular scanning of a Fabry–Pérot spectrum should cover the approximately $38\mathrm{nm}$ spectral span within one order while making the instrumental width smaller than $1\mathrm{nm}$. Near normal incidence, requiring the smallest approximate FSR to exceed the span gives $\mu d\lesssim 5.3\mu \mathrm{m}$, and $\mathcal{F}$ should exceed roughly $40$. These are design bounds, not a unique optimum. One suitable choice isThe physical spacing is $d=5.0\mu \mathrm{m}/\mu$; an air gap gives $d\simeq 5\mu \mathrm{m}$. Approximate FSRs near normal incidence are $40.1\mathrm{nm}$ at $633\mathrm{nm}$ and $44.9\mathrm{nm}$ at $670\mathrm{nm}$, with widths about $0.65$ and $0.73\mathrm{nm}$. For a concrete angular scan choose order $m=14$. The $670\mathrm{nm}$ pair lies near internal $\theta =20.3^{\circ }$ and the $633\mathrm{nm}$ line near $27.6^{\circ }$; their widths within this scan remain below $1\mathrm{nm}$. Restricting to this calibrated order window avoids the other-order $633\mathrm{nm}$ ring near $18.3^{\circ }$. Index dispersion and unknown extra source lines would need calibration or order-sorting filters.

The resolving power is approximately $m\mathcal{F}$: many coherent cavity round trips create a much longer effective optical delay than one traversal. A Michelson instrument with the same available physical delay has less resolving power unless its mirror travel provides a correspondingly long maximum optical path difference. Mirror separation errors spread the round-trip phase; keeping them below a linewidth requires roughlyFor this design the scale is about $6\mathrm{nm}$ in an air gap. Both surface flatness and parallelism over the illuminated area, as well as angular collimation and coating uniformity, must support the intended finesse.