Past exam of the optics and electrodynamics course of the University of Cambridge / 2019 / Section A

Take $u>0$ to mean that the mirror recedes along the incident plane electromagnetic wave. Put $\beta =u/c$, with $|\beta |<1$. The moving-mirror Doppler shift can be obtained directly by matching wave phase on the moving surface. Write the incident wave phase as $k_{i}z-{\omega }_{i}t$ and the reflected wave phase as $-k_{r}z-{\omega }_{r}t$, where $k_i={\omega }_i/c$ and $k_r={\omega }_r/c$ are positive wavenumbers. At the mirror, $z=z_0+ut$, the two phases must have the same time dependence; their constant offset incorporates the reflection phase shift. ThusConsequently the observed reflected angular frequency and ordinary frequency areThis is a redshift for a receding mirror and a blueshift for an approaching mirror. At small velocity, ${\omega }_r\simeq {\omega }_i(1-2u/c)$.

Equivalently, the relativistic Doppler effect gives ${\omega }_i^{\prime }=\gamma (1-\beta ){\omega }_i$ in the mirror rest inertial frame. Reflection there reverses propagation and preserves frequency. Transforming the backward-going reflected wave to the laboratory gives another factor $\gamma (1-\beta )$, so ${\omega }_r={\gamma }^2(1-\beta {)}^2{\omega }_i={\omega }_i(1-\beta )/(1+\beta )$. A single Doppler factor would miss the effect of the return path.

Use a transparent reciprocal wave plate at its design wavelength. For the return calculation, assume a mirror at normal incidence with the same reflection coefficient for both transverse components; this is the usual ideal double-pass quarter-wave plate arrangement.

Birefringence gives two orthogonal linear eigenpolarizations with different refractive indices, $n_f$ and $n_s$. They accumulate different wave phases through the same physical thickness $d$. Their relative phase retardation isHere ${\lambda }_0$ is the vacuum wavelength. For the thinnest positive-retardance plate, $d={\lambda }_0/[4(n_s-n_f)]$. The plate uses a difference in phase velocity, rather than differential wave absorption. Equal wave amplitudes along the two axes, obtained from input linear polarization at $45^{\circ }$ to them, acquire phase quadrature and become circular polarization. Reversing that conversion gives linear polarization from the corresponding circular input. A different input angle generally gives elliptical polarization on one pass.

Use the same fixed laboratory $x,y$ axes for the incident and returning Jones vectors, with $x$ along the fast axis. Remove a common phase factor so that the one-pass Jones matrix is $J=\mathrm{diag}(1,i)$. The reverse pass through this reciprocal plate accumulates the same relative phase retardation, not its inverse. The mirror has scalar reflection coefficient $r_m$ in these fixed axes. ThereforeThe effective retardance is $\pi$: the complete return acts as a half-wave plate. For initial linear polarization at angle $\alpha$ to the fast axis,In particular, the $45^{\circ }$ input used for linear-to-circular conversion returns in linear polarization perpendicular to its original line: $(1,1)/\sqrt{2}\mapsto (1,-1)/\sqrt{2}$. An input along either plate axis returns along that same axis.

The wording does not specify the initial polarization of waves. Thus “orthogonal linear return” requires the usual $45^{\circ }$ linearly polarized input; it is not a universal answer for arbitrary polarized input. The Jones matrix above supplies the general answer. A non-normal metallic reflection may introduce different complex reflection coefficients for the two components and would require additional geometry and optical data. Descriptions of the intermediate circular polarization must also account for the reversed propagation direction when assigning handedness.

The transverse and longitudinal quantities characterize different correlations: spatial optical coherence across a wavefront and temporal optical coherence between delayed fields. Their numerical values require a convention for how much correlation counts as coherent.

Convert the distances to SI units: $r=1.5\times 10^{11}\mathrm{m}$ and $D=1.4\times 10^9\mathrm{m}$. The angular solar diameter isTwo receivers separated by a transverse baseline $b$ see a spread of relative wave phases of order $2\pi b\Theta /\lambda$. Averaging over incoherent source directions therefore reduces interferometric visibility when $b$ reaches the transverse optical coherence length scale $\lambda /\Theta$. At $\lambda =500\mathrm{nm}$,If “coherence length” specifically means the first zero of uniform-disk stellar visibility, a uniformly bright circular solar disk gives $g^{(1)}(b)=2J_1(x)/x$, with $x=\pi b\Theta /\lambda$. The first zero then lies at $\boxed{b_0=1.21967\lambda /\Theta \simeq 65\mu \mathrm{m}}$. This more specific number assumes a uniform disk and a first-zero criterion; the characteristic estimate is approximately $50$–$60\mu \mathrm{m}$. Solar brightness variation across the disk changes the exact contrast curve.

The coherence length from a finite spectral band is most safely estimated by converting its endpoints to frequency, because the quoted band is broad:The reciprocal-bandwidth optical coherence time and corresponding longitudinal optical coherence length areThe given temperature is consistent with the spectral peak: Wien's displacement law gives ${\lambda }_{\mathrm{peak}}=2.898\times 10^{-3}\mathrm{m}\mathrm{K}/5778\mathrm{K}\simeq 502\mathrm{nm}$. Using the common narrow-band estimate ${\lambda }_{\mathrm{peak}}^2/\Delta \lambda$, with $\Delta \lambda =600\mathrm{nm}$, gives about $0.42\mu \mathrm{m}$. Its closeness here does not justify the narrow-band approximation for arbitrary broad spectra.

The $0.40\mu \mathrm{m}$ result is a bandwidth-scale estimate. A flat spectral density per unit frequency would make it the first-zero optical coherence length; a truncated Planck law spectrum is not flat and would have a different correlation function. Neither the stated endpoints nor the word “coherence” specifies an exact visibility threshold. Propagation to Earth sets the angular size and transverse scale, but does not by itself narrow the solar optical spectrum or lengthen this longitudinal scale.