= Critical modulation for a photoresist
{title2=$M_c=(D_c-D_0)/(D_c+D_0)$}
The <critical modulation for a photoresist> is the smallest <aerial-image modulation> that permits a common exposure time to clear bright regions while retaining full thickness in dark regions. Consider a <positive photoresist> with retention <exposure dose> $D_0>0$ and clearing <exposure dose> $D_c>D_0$. Write the extreme image <intensities> as
$$
I_{\max}=I_0(1+M),\qquad I_{\min}=I_0(1-M),\qquad 0\leq M\leq1.
$$
For exposure time $t$, the requirements are $tI_{\max}\geq D_c$ and $tI_{\min}\leq D_0$. A feasible $t$ exists exactly when
$$
\frac{1+M}{1-M}\geq\frac{D_c}{D_0},
$$
with $M=1$ handled directly since the dark-region <exposure dose> is zero. Rearrangement gives
$$
M\geq M_c=\frac{D_c-D_0}{D_c+D_0}.
$$
This derivation assumes <exposure dose> is <intensity> times exposure time and uses fixed retention and clearing thresholds. In the idealized linear contrast-curve model of <photoresist contrast>, $D_c/D_0=10^{1/\gamma}$, so
$$
M_c=\frac{10^{1/\gamma}-1}{10^{1/\gamma}+1}.
$$
For example, $D_c=4D_0$ gives $M_c=3/5$. Greater <photoresist contrast> lowers the required <aerial-image modulation> in this model.<Glossary of Lithography Terms by Chris A. Mack>[Entry Photoresist Contrast supplies the material-response definition; threshold derivation and example given here]{ref}
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