Past exam of the MPhil in micro- and nanotechnology enterprise course of the University of Cambridge / 2019 / Paper 1 / Section A(ii) / 4

Solve the stated depth of focus relation before substituting into the resolution expression:This uses the supplied empirical $k_1$ and $k_2$ unchanged and interprets $R$ as the minimum line or space width of an equal-line/space square grating. Under that convention its pitch is $2R=578.6\mathrm{nm}$. Other pattern shapes or a different definition of the quoted depth of focus would require different process factors.

An immersion liquid increases numerical aperture through $NA=n\sin \alpha$, if the optics can use the same ray angle $\alpha$ inside a transparent, chemically compatible liquid of refractive index $n$. With unchanged $k_1$ and vacuum wavelength, the width scales as $R_{\mathrm{imm}}=R_{\mathrm{air}}/n$ at fixed angle. If the two empirical formulae printed in the question are used literally with fixed $k_2$, they give $DO{F}_{\mathrm{imm}}=300/n^2\mathrm{nm}$.

For a physical axial displacement inside the medium, however, the familiar paraxial defocus range contains an additional index factor: $DOF\propto n{\lambda }_0/N{A}^2$. That convention gives $DO{F}_{\mathrm{imm}}/DO{F}_{\mathrm{air}}\simeq 1/n$ at fixed ray angle, or an approximately $n$-fold larger depth of focus at fixed $NA$ and resolution. Thus immersion can improve lateral resolution, while its focus benefit depends on what is held fixed and on the axial-focus convention. It cannot improve resolution merely by inserting $n$ into $NA$ while ignoring optical redesign, absorption or resist compatibility. No liquid index is supplied for a numerical prediction.

First, use a phase-shift mask: neighbouring openings transmit appropriately phased light, increasing the intensity slope and improving separation of neighbouring bright regions. The benefit is pattern dependent and phase conflicts or mask defects must be controlled. Second, use off-axis illumination matched to the grating orientation, so useful diffracted orders fall inside the projection pupil and contribute interference contrast at smaller pitch. Check orientation dependence and the reduced process window; it does not enlarge the geometrical pupil.

Third, use optical proximity correction and subresolution assist features to compensate the pattern-dependent aerial-image distortion at edges and corners. Verify the printed critical dimensions using a focus–exposure matrix and scanning electron microscopy. These are three adaptations of the existing exposure process that lower effective $k_1$; none promises arbitrary subwavelength structure. A thinner, higher-contrast photoresist and an antireflection underlayer can further improve pattern transfer, but do not change the wavelength.