= Coherent zero-mode model of phase contrast
The following original model isolates the conversion mechanism. Use a periodic image window, unit illumination, ideal unit <magnification>, and a <phase plate> that changes only the zero <spatial frequency>. Write $c_0=\langle e^{i\phi}\rangle$ for the spatial <mean>. Multiplying that single <Fourier coefficient> by $a e^{i\theta}$, with $0<a\leq1$, gives the exact image field in this model:
$$
E_{\rm out}=e^{i\phi}+(a e^{i\theta}-1)c_0,
\qquad I_{\rm out}=|E_{\rm out}|^2.
$$
This formula follows by splitting $e^{i\phi}=c_0+(e^{i\phi}-c_0)$: the second term has zero <mean> and is left unchanged. A real annular, partly coherent instrument alters a finite band of directions, so this calculation is a mechanism model rather than its full image-formation law.
For a <weak phase object> with $\langle\phi\rangle=0$, one has $c_0=1+O(\phi^2)$ and hence
$$
E_{\rm out}=a e^{i\theta}+i\phi+O(\phi^2),\qquad
\boxed{I_{\rm out}=a^2+2a\phi\sin\theta+O(\phi^2).}
$$
The cross term is $2\operatorname{Re}(a e^{-i\theta}i\phi)=2a\phi\sin\theta$. Thus $\theta=\pi/2$ makes positive $\phi$ brighter and $\theta=-\pi/2$ makes it darker under this stated <phase convention for complex amplitudes>. The sign comes from this convention and chosen plate; a manufacturer’s positive/negative contrast labels should not be inferred from the sign of $\theta$ alone.
For example, choose $a=0.5$ and a zero-mean periodic pattern whose peak <phase> is $0.084$. The first-order peak intensities are $0.25\pm0.084$, compared with the uniform ideal <bright-field microscopy> value $1$. The relative change about the reduced background is approximately $2\phi/a$ for a quarter-cycle shift. Attenuating the reference increases relative contrast while reducing the background photon count; it is not light amplification or a universal improvement of <signal-to-noise ratio>.
\Image[/phase-contrast-microscope-coherent-model.png]
{title=Making a transparent periodic phase pattern visible}
{description=Original calculation for $\phi(x,y)=0.06[\cos(2\pi x)+0.4\cos(2\pi y)]$ on a unit periodic window and $a=0.5$. The specimen pattern is synthetic, not a cell photograph. The top row compares ideal <bright-field microscopy> and the exact <coherent zero-mode model of phase contrast>; brightness is shown relative to the empty-field value for each mode. Lower plots compare exact and first-order results along $y=0$ and as the reference <phase> changes at the pattern maximum. These are calculations, not historical observations.}
Two checks expose what the model can and cannot measure. A spatially constant <phase> gives $E_{\rm out}=a e^{i\theta}e^{i\phi}$ and $I_{\rm out}=a^2$, so no absolute <phase> is recovered. Orthogonality of the constant and zero-mean parts also gives
$$
\langle I_{\rm out}\rangle=1-(1-a^2)|c_0|^2.
$$
For $a\leq1$ this cannot exceed the incident mean <light intensity>. The image redistributes brightness and loses some reference light rather than creating energy.
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