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Periodic optical lattices consisting of isolated-particle arrays in vacuum are treated with rigorous electromagnetics. These structures possess a wealth of interesting properties including perfect reflection across small or large spectral bandwidths depending on the choice of materials and design parameters. Pertinent spectral expressions have been observed theoretically and experimentally via one-dimensional (1D) and two-dimensional (2D) structures commonly known as resonant gratings, metamaterials, and metasurfaces. The physical cause of perfect reflection and related properties is guided-mode resonance mediated by lateral Bloch modes excited by evanescent diffraction orders in the subwavelength regime. Here, we review recent results on differentiation of local Mie resonance and guided-mode lattice resonance in causing resonant reflection by periodic particle assemblies. We treat a classic 2D periodic array consisting of dielectric spheres. To disable Mie resonance, we apply antireflection (AR) coatings to the spheres. Reflectance maps for coated and uncoated spheres demonstrate that perfect reflection persists in both cases. We find that the Mie scattering efficiency of an AR-coated sphere is greatly diminished. Additionally, in a 1D cylindrical rod-type lattice, we investigate and compare local field profiles in periodic assemblies and in the constituent isolated particles. In general, the lattice and particle resonance wavelengths differ. When the lateral leaky-mode field profiles approach the isolated-particle Mie field profiles, the resonance locus tends towards the Mie resonance wavelength. This correspondence is referred to as Mie modal memory. These fundamentals may help distinguish Mie effects and leaky-mode lattice effects in generating the observed spectra in this class of optical devices while elucidating the basic resonance properties across the entire spectral domain.