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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 asFor exposure time $t$, the requirements are $t{I}_{\max }\ge D_c$ and $t{I}_{\min }\le D_0$. A feasible $t$ exists exactly whenwith $M=1$ handled directly since the dark-region exposure dose is zero. Rearrangement givesThis 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, $D_c/D_0=10^{1/\gamma }$, soFor example, $D_c=4D_0$ gives $M_c=3/5$. Greater photoresist contrast lowers the required aerial-image modulation in this model.^[ref]

A fully clear opening requires the higher clearing threshold, not merely the onset of thinning. At $\tau =20\mathrm{s}$,The intensity curve crosses this level at about $x=0.25$ and $3.75\mu \mathrm{m}$. Therefore the clear opening is approximatelyThe longer exposure widens the opening because even the relatively dim image tails now accumulate enough exposure dose to clear the positive photoresist. The edge readings are approximate; interpolating a plotted curve does not support nanometre precision.

Use exposure dose $D(x)=I(x)\tau$, with $\mathrm{mW}\times \mathrm{s}=\mathrm{mJ}$. At approximately $x=1.0$ and $3.0\mu \mathrm{m}$, the positive resist just starts to thin. The intensity there is about $5\mathrm{mW}\mathrm{c}{\mathrm{m}}^{-2}$, soAs a consistency check, the fully cleared region begins at approximately $x=1.5$ and ends at $2.5\mu \mathrm{m}$, where the intensity is about $25\mathrm{mW}\mathrm{c}{\mathrm{m}}^{-2}$. Multiplying by $2\mathrm{s}$ gives the clearing dose $50\mathrm{mJ}\mathrm{c}{\mathrm{m}}^{-2}$ at both edges.

Photoresist contrast measures how sharply a photoresist changes its developed thickness as the exposure dose changes. For a positive photoresist, let $h(D)$ be the remaining thickness after a specified development process and $h_0$ the initial thickness. A contrast curve plots $h(D)/h_0$ against ${\log }_{10}D$. The measured photoresist contrast is the magnitude of the slope near clearing; it need not describe the entire curve.^[ref]

In an idealized model, the thickness decreases linearly with ${\log }_{10}D$ between a retention exposure dose $D_0$ and a clearing exposure dose $D_c$, where $0<D_0<D_c$. ThenOutside that interval the idealized thickness is clipped to $1$ or $0$. A large $\gamma$ means a smaller exposure dose ratio suffices to change from retention to clearing. This is why photoresist contrast influences the critical modulation for a photoresist; the inverse-logarithm formula assumes this particular contrast-curve model.