= Phase-contrast microscopy
{wiki}
<Phase-contrast microscopy> converts specimen-dependent <phase differences> into visible <light intensity> differences. It can reveal structures in living, unstained <biological cells>, allowing their changes to be followed without first making an absorbing preparation.<Phase contrast microscope by University of Groningen>[Opening description of the phase-contrast microscope and biological motivation]{ref} The key question is how an <optical microscope> can read a <phase> when an ordinary detector records <light intensity>.
Consider a transparent specimen with <complex amplitude> transmission $t(x,y)=e^{i\phi(x,y)}$. In ideal in-focus <coherent imaging> with all relevant <spatial frequencies> transmitted, its image has $I=|t|^2=1$: varying $\phi$ alone produces no brightness detail. This is a particular model, not a claim that all transparent specimens are invisible. <Diffraction>, defocus, <absorption (electromagnetic radiation)> and a finite <optical pupil> can already change a real image.
A <phase-contrast microscope> images an illuminating annulus onto a <phase plate> ring near the objective's rear <focal plane>. The aligned ring attenuates and shifts direct light relative to most diffracted light; their <interference> changes image brightness.<Introduction to Phase Contrast Microscopy by Nikon Instruments Inc.>[Figure 1; Phase Plates and Phase Annuli]{ref}
\Image[/phase-contrast-microscope-reference-ring.png]
{title=Separating the reference light before recombining the image}
{description=Original functional diagram of <phase-contrast microscopy>. The annulus and ring indicate their optical correspondence, not an exact ray prescription. The lower <complex amplitude> construction uses the original <weak phase object> model below, with chosen positive specimen <phase> and two opposite quarter-cycle reference shifts.}
The method is beautiful because the bright direct beam becomes its own reference: a small specimen-dependent <complex amplitude> can interfere with it and change the image at first order. Merely magnifying an invisible <phase> pattern would leave $|t|^2$ unchanged.
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