Past exam of the physics 2 paper of the University of Oxford / 2013 / 9

Fermat's principle says that the travel time, or equivalently the optical path length, is stationary under small path variations at fixed endpoints. For a flat interface, let the ray cross at horizontal coordinate $x$, with source and detector at heights $h_1,h_2$ and horizontal separation $D$. Apart from $1/c$, its travel time isSetting $d\mathcal{L}/dx=0$ givesThe angles are measured to the normal. This is Snell's law; “least time” is more generally a stationarity condition.

Take a horizontal cross-section in units of $R$. The eye is at $E=(\sqrt{2},0)$, and the tadpole swims along the perpendicular diameter $T=(0,y)$ shown in the source. Since glass and water have the same red-light index, their internal interface causes no red-light refraction. Write an exit point on the circular wall as $P=(\cos \varphi ,\sin \varphi )$. An unobstructed outward ray to the eye has $|\varphi |\le \pi /4$.

For $\varphi \ge 0$, let $e$ be the magnitude of the emerging ray's angle to the outward normal and $i$ its internal angle. Geometry and Snell's law giveThe outgoing internal ray makes angle $\beta =\varphi -i$ to $+x$. Its backward continuation intersects the swim diameter atThe directly visible positive endpoint is the largest such intercept, including all single-pass refracted rays. Maximizing this one-variable expression gives ${\varphi }_{\ast }\simeq 38.07^{\circ }$ and $y_{\ast }\simeq 0.75744$. By symmetry the visible length is $2R{y}_{\ast }$, soChecking only the grazing exit at $\varphi =45^{\circ }$ gives $\sqrt{2}/[1+\sqrt{1.33^2-1}]\simeq 0.75350$, a close estimate but a slightly smaller fraction. The exact limiting ray occurs before that grazing ray, where two refracted paths merge at a caustic. This is a geometrical, single-pass visibility boundary; finite pupil size and detection thresholds would smooth the transition.

The result need not be the same. Optical dispersion changes the refractive index with wavelength. Under normal dispersion, the blue-light index is larger, changing the refracted rays and generally reducing the visible fraction in the one-interface model. The blue indices of both water and glass are not specified; they may also differ from each other, introducing refraction at their interface. Thus no unique numerical blue-light fraction follows from the red-light value alone.

For near-normal viewing through the flat top surface, the refracted ray appears to originate at an apparent depth $h_{\mathrm{app}}$. A ray at horizontal offset $x$ has $x=h\tan {\theta }_w=h_{\mathrm{app}}\tan {\theta }_a$, and Snell's law gives $n\sin {\theta }_w=\sin {\theta }_a$. In the paraxial limit,At an oblique viewing angle the exact ray-based apparent depth is $h\tan {\theta }_w/\tan {\theta }_a$; the simple ratio presumes the usual near-normal top view.