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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.

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.

Increasing the angular acceptance can collect higher spatial frequencies, but rays at large angles and away from the optical axis are harder to bring to the same focus. Spherical aberration, coma, astigmatism and field curvature can degrade contrast across a large exposure field. The depth of focus also decreases approximately as $\lambda /{\mathrm{NA}}^2$, making wafer height errors more important. A larger geometrical aperture is therefore not automatically better over the whole wafer field.

Immersion lithography raises $\mathrm{NA}=n\sin \theta$ by increasing the refractive index of the medium between the final lens and wafer. A suitable UV-transparent liquid permits a larger numerical aperture without increasing the ray angle by the same factor, easing the purely geometrical demand on the lens. Lens design and focus control remain necessary; immersion does not abolish aberrations. The liquid must also be compatible with the resist and free of bubbles and contamination, and absorption at $193\mathrm{nm}$ must be low.

Depth of focus is the axial tolerance over which the projected pattern remains acceptable. Wafer topography, resist thickness, wafer tilt and focus-setting errors can move part of the film outside that range, reducing contrast and changing printed linewidth. Increasing numerical aperture can improve lateral resolution while reducing this axial process window.

Using the supplied coefficient in the conventional expression,This is the range assigned to DOF by that convention. Whether an instrument reports a full allowable interval or a one-sided defocus tolerance must be stated separately; an additional factor of two cannot be inferred from the numerical coefficient alone.