A microscope is set up for Köhler illumination

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 microscope is set up for Köhler illumination. A lamp filament, treated as a uniform incoherent disc of diameter

4.0\ \mathrm{mm}

, sits in the front focal plane of a condenser lens of focal length

20\ \mathrm{mm}

, so that (as in the aperture-stop geometry of the paraxial system) the illumination half-angle at the specimen is set by

\mathrm{NA}_{\mathrm{cond}} \approx D_{\mathrm{source}}/(2f)

, with

D_{\mathrm{source}}

the source diameter. The objective has

\mathrm{NA}_{\mathrm{obj}} = 0.25

. The lamp is filtered to a

10\ \mathrm{nm}

band centred on

546\ \mathrm{nm}

. Small-angle (paraxial) approximation applies throughout, and the source may be treated as spatially incoherent. ((a)) Find

\mathrm{NA}_{\mathrm{cond}}

from the condenser geometry. ((b)) Using the van Cittert-Zernike theorem for a uniform circular incoherent source, find the transverse coherence radius (distance to the first zero of

|\mu_{12}|

) produced at the specimen plane. ((c)) Compute the coherence parameter

\sigma_{\mathrm{c}} = \mathrm{NA}_{\mathrm{cond}}/\mathrm{NA}_{\mathrm{obj}}

and the highest spatial frequency the microscope transmits. ((d)) Compute the coherence length of the filtered light, and check whether the largest path difference across the coherence width found in part (b) stays well inside it.

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