Past exam of the optics and electrodynamics course of the University of Cambridge / 2021 / Section B / B5

Take $f(t)=a(t)e^{-i\omega t}$ at a fixed reference point, and define the full-field temporal optical coherence function byThe angle brackets denote an ensemble expected value, or a sufficiently long time average when justified. The factor $C$ puts $\Gamma$ in light intensity units for the chosen peak complex-field convention. The normalized first-order optical coherence is $g^{(1)}(\tau )=\Gamma (\tau )/\Gamma (0)$ for a stationary source.

“Independent of absolute time” means the second moment depends on the time difference $\tau$ but not on the starting time $t$: shifting both observation times by the same amount leaves it unchanged. Together with a constant mean this is second-order or weak stationarity; it does not mean that the random electric field is constant in time. Put $I_0=\Gamma (0)=C⟨|a{|}^2⟩$. The full-field convention is important: an envelope-only correlation omits $e^{-i\omega \tau }$ and must have that carrier factor restored before deriving optical interference fringes.

The given spatial and carrier factors combine into retarded copies of the same random field:On the $y$ axis, $t_1=ky/(\sqrt{2}\omega )$ and $t_2=-ky/(\sqrt{2}\omega )$. Expand the squared electric-field wave amplitude. Each self term contributes $I_0/2$, and the two cross terms together contribute the real part of the full-field correlation. Stationarity and $\Gamma (-\tau )=\Gamma (\tau {)}^{\ast }$ giveIn vacuum, $k=\omega /c$ and $\tau (y)=\sqrt{2}y/c$. Reversing the delay convention replaces $\tau$ by $-\tau$ without changing this real-part result. This $I$ is the electric-field light intensity convention of the question. For two nonparallel waves it should not automatically be identified with the magnitude of the net Poynting vector; part (d) gives that geometric conversion explicitly.

A line with lifetime broadening has a Lorentzian line shape. Write its half-width at half-maximum in angular-frequency units as $\kappa >0$. The Wiener-Khinchin theorem, or the Fourier transform of the Lorentzian, givesTherefore in vacuumThe sketch is a cosine fringe pattern about baseline $I_0$, bounded by $I_0[1\pm e^{-\sqrt{2}\kappa |y|/c}]$. Its central value is $2I_0$, its spatial interference fringe spacing is $\lambda /\sqrt{2}$, and its visibility decays on length $c/(\sqrt{2}\kappa )$. For an isolated transition whose population lifetime is $T_1$ and which has no extra dephasing, $\kappa =1/(2T_1)$, so the field coherence time is $2T_1$, not $T_1$. Introducing $\kappa$ avoids confusing population, wave amplitude and spectral-width conventions when no numerical linewidth is given.

For each beam in vacuum, its magnetic field is ${\mathbf{B}}_j={\hat{\mathbf{k}}}_j\times {\mathbf{E}}_j/c$. Write the two complex scalar contributions, including the beam splitter factor, as $u_1=f(t-t_1)/\sqrt{2}$ and $u_2=f(t-t_2)/\sqrt{2}$. Since their electric fields are both along $z$,The optical-cycle and ensemble-averaged Poynting vector isThe mean individual light intensities are equal, soThus energy flows along the bisector of the two beam directions, crossing a detector plane normal to $x$. Such a sensor can in principle record the variation across its $y$ coordinate. Its actual normal energy flux is $I(y)/\sqrt{2}$ in the normalization used above; this fixed factor does not change interferometric visibility.

There is a practical resolution condition in addition to the direction of energy flow. The interference fringe spacing $\lambda /\sqrt{2}$ is very small, so direct pixels would need spacing no larger than $\lambda /(2\sqrt{2})$ as required by the Nyquist–Shannon sampling theorem, or optical magnification and sufficient angular aperture to preserve both incident directions. An ordinary optical camera with much larger effective pixels averages the interference fringes. The Poynting-vector argument establishes the allowed measurement geometry, not guaranteed resolution with any optical camera.

On the $x$ axis the two geometric delays are identical:Their field wave amplitudes therefore add as equal copies at zero relative delay. Stationarity removes dependence on the common delay, givingindependent of $x$ in the ideal overlapping plane-wave region. The corresponding mean normal energy flux through a transverse plane is $\sqrt{2}{I}_0$. There is no increasing path difference along the bisector, so finite temporal optical coherence does not produce a longitudinal visibility envelope on this axis.

The $x$ axis lies along the mean Poynting vector. A two-dimensional optical camera sensor intended to sample many $x$ positions at once would have to lie along the energy flow, where it receives no normal flux in the ideal geometry; a side-view optical camera also sees no light from an empty nonscattering propagation region. Putting a detector normal to $x$ measures one transverse section instead, and absorbs the light before a downstream detector can sample the same rays.

One could translate a detector to successive $x$ positions or introduce a scattering screen, but that is a different measurement. Moreover the ideal result here is already the constant $I(x)=2I_0$, so it contains no longitudinal optical coherence structure for the optical camera to resolve. The difficulty is the longitudinal sampling geometry, rather than a hidden $x$-fringe period.

A physical delay is a time, whereas the stated $\lambda /(4\sqrt{2})$ is a length. First interpret it literally as an additional optical path length $\ell =\lambda /(4\sqrt{2})$ in arm 1, with the other arm unchanged. In vacuum this delays its full field by ${\tau }_d=\ell /c$. The exact stationarity argument from part (b) then givesDelaying arm 2 changes the plus sign before ${\tau }_d$ to a minus sign. As a function of $y$, the arm-1 result is the old pattern evaluated at $y+\ell /\sqrt{2}=y+\lambda /8$.

For a quasi-monochromatic source, ${\tau }_d$ is much shorter than the coherence time. The envelope correlation can then be treated as unchanged, while the carrier wave phase cannot:HenceThis literal optical-path delay is not a quarter-cycle wave phase shift. If the intended length instead specifies a displacement of the entire recorded fringe pattern by one quarter of its period, $\Delta y=\lambda /(4\sqrt{2})$, the equivalent delay is $\sqrt{2}\Delta y/c=\lambda /(4c)$ and the wave phase shift is $\pi /2$. Under that different interpretation the narrow-band result is $I_0+\mathrm{Im}\Gamma (\tau (y))$, with its sign reversed for the opposite displacement. The exact expression is still a shifted argument of $\Gamma$, not merely a wave phase multiplier when the envelope varies. The original wording does not specify such a fringe-pattern displacement, so these interpretations must be distinguished.