 Solution

An on-axis point illuminator increases spatial coherence. It is not a point object, and it does not remove diffraction. For an ideal aberration-free pupil, coherent imaging is linear in field amplitude and has cutoffFor a real amplitude grating $U(x)=a[1+m\cos (2\pi \nu x)]$, with $0\le m\le 1$ and all three field orders inside this pupil, the intensity contrast is $C=2m/(1+m^2)$. Flat coherent field transfer preserves this contrast; weak input modulation does not itself meet a high resist contrast threshold. Assume a sufficiently high-contrast amplitude mask: for example, $m=1$ gives $C=1$, exceeding both resist thresholds. Under this assumption, both resists have the same limiting equal-line/space widthThis is an approach to the cutoff from below; finite dose latitude, aberrations and a real binary mask can make usable resolution worse. A general coherent intensity image cannot be obtained by applying an incoherent intensity MTF unchanged: $I=|h\ast U{|}^2$, rather than $I=|h{|}^2\ast |U{|}^2$.^[ref]

With ${\nu }_0=2\mathrm{NA}/\lambda$, this coherent cutoff is $0.5{\nu }_0$: the low-contrast resist improves from roughly $302$ to $241\mathrm{nm}$, while the high-contrast resist worsens from roughly $201$ to $241\mathrm{nm}$. Coherence improves contrast within its passband but reduces the ultimate bandwidth relative to incoherent illumination.

With the literal Rayleigh-spacing normalization ${\nu }_0=1/R_R$, the same physical cutoff is instead $0.61{\nu }_0$, not $0.5{\nu }_0$. The corresponding widths from the preceding part would improve from roughly $368$ and $245\mathrm{nm}$ toward $241\mathrm{nm}$. The cutoff in physical units is unambiguous under the stated ideal on-axis model; the claimed direction and size of a change inferred from the plotted normalized axis depend on its definition. Off-axis illumination, partial coherence and arbitrary phase objects require a fuller imaging model.