  The orthogonality of cat and displaced cat states, underlying Heisenberg
limited measurement in quantum metrology, is studied in the limit of large
number of states. The asymptotic expression for the corresponding state overlap
function, controlled by the sub-Planck structures arising from phase space
interference, is obtained exactly. For large number of coherent states,
uniformly located on a circle, it identically matches with the diffraction
pattern for a circular ring with uniform angular source strength. This is in
accordence with the van Cittert-Zernike theorem, where the overlap function,
similar to the mutual coherence function matches with a diffraction pattern.
Interestingly, the convergence to asymptotic limit is quite rapid, reminiscent
of the convergence of superposed incoherent sources.
