Past exam of the physics 2 paper of the University of Oxford / 2014 resit / 10

Fermat's principle states that the actual ray makes the travel time stationary under small variations of its path with endpoints fixed. Equivalently, for a stationary medium it makes the optical path length $\int nds$ stationary. “Stationary” allows minima, maxima, or other stationary paths; it does not require a strict minimum in every optical system.

In the thin-lens approximation, the geometric path length isThe glass replaces approximately length $s(d)$ of air, rather than adding a further geometric segment. Its excess optical path length is therefore $(n-1)s(d)$. With $s(d)=s_0-\alpha d^2$,The approximations retain terms quadratic in the small ray height and neglect products of lens thickness with the squared ray slope. The distinction between geometric and optical path length explains why the glass contribution contains $n-1$, not $n$.

For nonzero small $d$, stationarity of the travel time givesTo focus a bundle, this must hold for neighbouring off-axis rays, not merely for the automatically stationary axial ray $d=0$. Comparing with the thin lens equation, $1/a+1/b=1/f$, givesFor parallel incident rays $a\to \infty$, the image is at $b=f$. A centre-thick lens with $\alpha >0$ and $n>1$ is converging.

In wave optics, a finite lens aperture produces diffraction. Contributions interfere constructively near the geometric focus but form a finite point spread function, rather than a point of infinite intensity. For a uniformly illuminated circular aperture of diameter $D$, the paraxial focal pattern is an Airy disk, with first dark-ring radius approximately $1.22\lambda f/D$. Aberrations can broaden or distort it further; even an ideal lens retains the finite diffraction scale.