  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 mesoscopic superposition of the generalized cat state is
correlated with the corresponding state overlap function, controlled by the
sub-Planck structures arising from phase space interference. The asymptotic
expression of this overlap function is evaluated and the validity of large
phase space support and distinguishability of the constituent states, in which
context the asymptotic limit is achieved, are discussed in detail. For large
number of coherent states, uniformly located on a circle, the overlap function
significantly matches with the diffraction pattern for a circular ring source
with uniform angular strength. This is in accordance with the van
Cittert-Zernike theorem, where the overlap function, similar to the mutual
coherence function, matches with a diffraction pattern. The physical situation
under consideration is delineated in phase space by utilizing Husimi-Q
function.
