Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2021 / Paper 1 / 4

Reading the straight-line extrapolation in the printed photoresist contrast curve gives approximately $D_0=20\mathrm{mJ}\mathrm{c}{\mathrm{m}}^{-2}$ at unit remaining thickness and $D_c=100\mathrm{mJ}\mathrm{c}{\mathrm{m}}^{-2}$ at clearing. On the logarithmic dose axis,The critical image modulation follows by requiring one exposure time to give $D_{\max }\ge D_c$ in a bright region and $D_{\min }\le D_0$ in a dark region:For unit object modulation, this meansThe doses are graphical estimates, so extra significant figures would not represent greater measurement accuracy.

The printed aerial image has approximately $I_{\max }=2.0$ and $I_{\min }=0.5\mathrm{mW}\mathrm{c}{\mathrm{m}}^{-2}$. Thus its modulation, and its modulation transfer for a fully modulated object, isThe image does not meet the full-thickness printing criterion, because $0.60<0.667$. Equivalently, clearing the bright regions requires $t\ge 100/2=50$ s, while preserving the full thickness in the dark regions requires $t\le 20/0.5=40$ s. There is no common exposure time.

Partial thickness modulation may remain visible, but that is different from a clean grating with fully cleared spaces and retained full-thickness lines. Small ripple peaks in the source drawing do not change this conclusion.

For a spatially coherent, on-axis source, the coherent spatial-frequency cutoff isA line-space grating must place at least its first diffracted orders within the pupil. Its limiting pitch is therefore $\lambda /\mathrm{NA}$, and its equal line/space half-pitch isThe corresponding full pitch is 143 nm. Explicitly distinguishing pitch from half-pitch avoids a factor-of-two ambiguity in the phrase “feature size.” This is an ideal optical cutoff estimate; finite resist contrast and process requirements can make the usable feature size larger.

Decreasing the source diameter increases spatial optical coherence and moves the illumination towards the coherent on-axis limit. Here the grating frequency is onlyThe zero and first orders are comfortably within the pupil for on-axis illumination. Reducing an extended source can prevent some off-axis source points from losing diffraction orders at the pupil edge, and can therefore increase the grating modulation, or leave it approximately unchanged if the relevant orders were already fully passed. It need not give a universal numerical increase: higher orders, source shape, defocus and optical aberrations matter for a binary grating.

The resist's $\gamma$ and critical modulation $M_c$ in part (i) are material/process properties and do not change merely because the source becomes smaller. What may change is part (ii)'s comparison between the actual image modulation and $M_c$. If the improved image exceeds about 0.667, an exposure window opens; the source information alone does not establish a new numerical value.