Q.The resolving power of a microscope is inversely proportional to the wavelength λ used to illuminate the object. Can we have a source with a very small wavelength compared to that of visible light, to resolve finer detail?

For a microscope viewing two point objects separated by distance a, with the objective subtending half-angle $\alpha$ and immersed in a medium of refractive index n, geometry gives the path difference between extreme rays as $2a\sin\alpha$. For NON-luminous (externally illuminated) objects, Rayleigh's criterion gives $a_{min}=\lambda/(2n\sin\alpha)=\lambda/(2\,\text{N.A.})$, where N.A. $=n\sin\alpha$ is the numerical aperture. For SELF-luminous point objects, applying Abbe's fuller Airy-disc theory gives instead $a_{min}=0.61\lambda/\text{N.A.}$, with resolving power $R=1/a_{min}=\text{N.A.}/(0.61\lambda)$. …