= Solution
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 $R_{\rm imm}=R_{\rm air}/n$ at fixed angle. If the two empirical formulae printed in the question are used literally with fixed $k_2$, they give $DOF_{\rm imm}=300/n^2\ \mathrm{nm}$.
For a physical axial displacement inside the medium, however, the familiar paraxial defocus range contains an additional index factor: $DOF\propto n\lambda_0/NA^2$. That convention gives $DOF_{\rm imm}/DOF_{\rm air}\simeq1/n$ at fixed ray angle, or an approximately $n$-fold larger <depth of focus> at fixed $NA$ and resolution. Thus \b[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.
Back to article page
