A bright-field (transmitted-light) microscope has an objective of numerical aperture…
PHYSICS · Multi-part · Fermi University Physics · Optics & Photonics · Fourier optics and image formation · Coherence and statistical optics
What an Instrument Can See, Ch Spatial coherence and the van Cittert-Zernike theorem
Problem
A bright-field (transmitted-light) microscope has an objective of numerical aperture NAobj=0.85 and is used at λ=520 nm. Its condenser diaphragm can set NAcond anywhere from 0 up to NAobj. The instrument is to be run at the usual compromise coherence parameter σc=NAcond/NAobj=0.70, which reduces edge ringing while still gaining resolution beyond the fully coherent limit. Separately, a stellar interferometer observes a red giant with a narrow filter centred on λ=500 nm, narrow enough that all optical path differences in the beam combiner are held well inside the coherence length. The two collecting apertures gather equal light, so the measured fringe visibility V is the real coherence factor μ12 of the symmetric disc, whose sign records whether the fringe pattern is in its normal phase or reversed. At baseline B=5.0 m the visibility is V=+0.074 (normal phase); at baseline B=5.6 m it is V=−0.021 (reversed phase, near the first null of the visibility curve). Model the star as a uniformly bright circular disc.
((a)) Find the condenser numerical aperture NAcond that gives σc=0.70, and the highest spatial frequency (in cycles per micrometre) the microscope transmits at this setting.
((b)) Estimate the baseline B0 of the star's first visibility null by linear interpolation between the two measurements.
((c)) Using the uniform-disc van Cittert-Zernike result θs=1.22λ/B0, find the star's angular diameter in milliarcseconds (1 mas =4.8481×10−9 rad). State in one sentence how a limb-darkened brightness profile, rather than a uniform disc, would change the fitted diameter.