Past exam of the optics and electrodynamics course of the University of Cambridge / 2015 / Paper 1 / 1

Let ${\mathbf{u}}_{\theta }=(\cos \theta ,\sin \theta {)}^{\mathsf{T}}$ be the transmission direction. An ideal linear polarizer projects the incident electric field onto this direction, so its Jones matrix isSince ${\mathbf{u}}_{\beta }^{\mathsf{T}}{\mathbf{u}}_{\alpha }=\cos (\alpha -\beta )$, the amplitude transmitted by a sequence of polarizers is the product of the projections between successive axes. Here the four directions are $0,\omega t,-2\omega t,0$. After the first polarizer the unpolarized light input has intensity $I/2$ and is in linear polarization along $x$. The transmission through a sequence of linear polarizers therefore follows from successive applications of Malus's lawThe relative angle between the rotating polarizers is $3\omega t$, not $\omega t$. The first factor $1/2$ is an ensemble average over incident polarization: unpolarized light cannot be represented by a single deterministic Jones vector. Equivalently, its coherency matrix is $(I/2)I_2$; the total matrix $P(0)P(-2\omega t)P(\omega t)P(0)$ gives the same output intensity by the trace rule. The quasistatic optical-element approximation assumes the mechanical rotation is slow compared with the optical oscillation.

To first order in the applied fields, the carrier acceleration gives $m\dot{\mathbf{v}}=q\mathbf{E}$. The current density $\mathbf{j}=nq\mathbf{v}$ consequently obeys the first London equationTaking its curl and using Faraday's law yieldsFor an initially field-free, current-free preparation, the integration constant is zero. Equivalently, impose the second London equation appropriate to the Meissner state, $\nabla \times \mathbf{j}=-{ϵ}_0{\omega }_p^2\mathbf{B}$. In the final static state, Ampère's law and $\nabla \cdot \mathbf{B}=0$ then giveFor a planar half-space $z>0$ with a tangential surface field, the bounded solution isThe plasma-frequency form of London penetration depth therefore shows that an applied field is exponentially small several penetration depths inside the material.

The magnetic-history constant in an undamped plasma explains the required qualification. Undamped carrier motion alone fixes a time-independent value of $\nabla \times \mathbf{j}+{ϵ}_0{\omega }_p^2\mathbf{B}$; it does not force that value to vanish. A perfect conductor can retain a pre-existing field. The additional London equation describes London screening independently of that arbitrary ideal-conductor history. The original London paper explicitly distinguishes its description from acceleration theory.^[ref]

wave phase labels the stage of a physical oscillation at a spacetime event. Two observers comparing the same event must identify the same crest, trough or intermediate stage: a change of coordinates cannot change the number of completed cycles between physically specified wavefronts. The continuous wave phase can therefore be treated as a Lorentz scalar, up to a common choice of wave phase origin. This does not assert invariance of the measured angular frequency, which is the rate of change of that wave phase along the observer's trajectory.

For Lorentz invariance of plane-wave phase, writeusing the metric with signature $(+,-,-,-)$. The gradient $K_{\mu }={\partial }_{\mu }\Phi$ transforms as a covector because $\Phi$ is a scalar. Raising its index therefore gives the four-wavevector. Explicitly, substituting the inverse boost $t=\gamma (t^{\prime }+v{x}^{\prime }/c^2)$, $x=\gamma (x^{\prime }+v{t}^{\prime })$ givesThese are exactly the Lorentz transformation laws for $(\omega /c,\mathbf{k})$. In vacuum, $K_{\mu }{K}^{\mu }=0$; the Doppler effect and aberration of light are two consequences of transforming the same four-vector.