  A hybrid computational method of plane-wave and cylindrical-wave expansions
for distributed Bragg-reflector (DBR) pillars is proposed. The plane-wave
expansion is employed to represent the one-dimensional periodic structure of
the DBR. The cylindrical-wave expansion is employed to describe the scattering
by circular pillars with the DBR structure inside. This formalism enables us to
calculate the radiation fields, $t$-matrices, scattering cross sections,
photonic band structures, and quality factors of the DBR pillars. Furthermore,
optical properties of arrayed DBR pillars are also investigated with the aid of
the multiple-scattering method. Using this formalism, we demonstrate explicitly
that high $Q$ photonic band modes including the so-called bound states in
continuum are obtained both in isolated and arrayed DBR pillars. We also
present a novel formation of gapless Dirac-cone surface states in a
three-dimensional photonic crystal composed of a two-dimensional periodic
arrangement of core-shell DBR pillars.
