The Debye series decomposition of the partial-wave scattering amplitudes of a multilayer sphere is derived. The partial-wave transmission and reflection terms appearing in the Debye series are multiplescattering amplitudes written in terms of four basic quantities and combined together layer by layer in an identical way. The resulting expressions are then used to calculate the scattered intensity of a spherical Bragg grating covering a dielectric core particle and to analyze a number of new structures appearing in the scattered intensity.
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