  Whispering gallery modes [WGM] are resonant modes displaying special
features: They concentrate along the boundary of the optical cavity at high
polar frequencies and they are associated with complex scattering resonances
very close to the real axis.
  As a classical simplification of the full Maxwell system, we consider
two-dimensional Helmholtz equations governing transverse electric [TE] or
magnetic [TM] modes. Even in this 2D framework, very few results provide
asymptotic expansion of WGM resonances at high polar frequency $m\to\infty$ for
cavities with radially varying optical index.
  In this work, using a direct Schr\"odinger analogy we highlight three typical
behaviors in such optical micro-disks, depending on the sign of an effective
curvature that takes into account the radius of the disk and the values of the
optical index and its derivative. Accordingly, this corresponds to abruptly
varying effective potentials (step linear or step harmonic) or more classical
harmonic potentials, leading to three distinct asymptotic expansions for ground
state energies.
  Using multiscale expansions, we design a unified procedure to construct
families of quasi-resonances and associate quasi-modes that have the WGM
structure and satisfy eigenequations modulo a super-algebraically small
residual $O(m^{-\infty})$.
  We show using the black box scattering approach that quasi-resonances are
$O(m^{-\infty})$ close to true resonances.
