(a) Write down the relationship between the magnetic field $\boldsymbol{B}$ and the vector potential $\boldsymbol{A}$.
(b) Give the units of the vector potential in terms of the fundamental SI units for mass, length, time and electric charge.
(c) A laser beam, which can be considered spatially uniform in this question, is described by a vanishing scalar potential and a time-varying vector potential $\boldsymbol{A}=$ $\boldsymbol{A}{\text {laser }}(t)$. What is the electric field $\boldsymbol{E}{\text {laser }}$ associated with this beam?
(d) An electron with charge $e$ at position $r$ is illuminated by the laser. Show that a valid description of this situation is obtained if we simultaneously set $\boldsymbol{A}=0$ and modify the scalar potential energy of the electron by an amount $\boldsymbol{d}$. $\boldsymbol{E}_{\text {laser }}$, where $\boldsymbol{d}=e \boldsymbol{r}$ is the dipole moment produced by the electron.

Propagation of a light beam through a thin transparent optical element can be described by modifying the transverse spatial phase of the electric field of the beam.
(a) For the case of a thin plano-convex spherical lens with radius of curvature $R$ and refractive index $n$, derive – with the aid of a diagram – an expression for the transverse phase $\varphi_{\text {lens }}(x)$ imparted to a field with angular frequency $\omega$, as a function of transverse displacement $x$, in the paraxial approximation (small divergence angles and small displacements from the optical axis).
(b) Give, and justify, an expression for the focal length $f$ of the lens, in terms of $R$ and $n$.

A planar waveguide is formed from two large parallel perfectly conducting metal plates in vacuum, separated by a distance $d$ along the $y$-axis.
(a) Find an expression describing the transverse profile $E(y)$ of the electric field amplitude as a function of $y$ for the $m^{\text {th }}$ TE mode (polarised along the $x$ direction). Sketch the transverse intensity distribution of the fundamental TE mode with $m=1$.
(b) Show that the product of the phase and group velocities of this mode is a constant.
(c) Show that the maximum bit rate for uncorrupted optical data transmission with carrier frequency $\omega=\sqrt{2} \pi c / d$ over a distance $L$ through this waveguide is of order $c \sqrt{\pi / L d}$.