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The permittivity ε(ω) becomes frequency-dependent due to the inertia of charges and damping mechanisms. This causes phase velocity to differ from group velocity and enables phenomena like dispersion, anomalous refraction, and material absorption.
From your study of dielectric susceptibility, you know that a static electric field polarizes a dielectric: the bound charges shift slightly, creating dipole moments that reduce the internal field. In the static case, the polarization P follows the field instantaneously because there is no dynamics to consider. But when the applied field oscillates at angular frequency ω, the charges need time to respond — and that inertia changes everything.
The simplest model is the Lorentz oscillator: treat each bound electron as a mass on a spring (the restoring force from the nucleus) subject to a driving force (the oscillating electric field) and a damping force (radiation and collisions). The equation of motion is exactly the driven damped harmonic oscillator from classical mechanics: mẍ + mγẋ + mω₀²x = eE(t). Solving in the frequency domain gives x(ω) ∝ E(ω) / (ω₀² − ω² − iγω). The polarization P = nex is proportional to x, so the susceptibility χ(ω) and therefore the permittivity ε(ω) = ε₀[1 + χ(ω)] inherit this complex frequency dependence. The real part of ε governs dispersion (how refractive index varies with frequency); the imaginary part governs absorption (how quickly a wave's amplitude decays as it propagates).
Three frequency regimes emerge. Far below resonance (ω ≪ ω₀), the electrons follow the field quasi-statically and ε is real and greater than ε₀ — normal transparent behavior. Near resonance (ω ≈ ω₀), the imaginary part peaks, meaning the material strongly absorbs that frequency. Far above resonance (ω ≫ ω₀), the electrons cannot keep up at all; their contribution to polarization vanishes, and in the extreme limit (as in X-rays through glass) ε approaches ε₀, effectively free space. This is why glass is opaque to UV despite being transparent to visible light: UV frequencies hit electronic resonances that X-rays pass right through.
Dispersion — the variation of refractive index n(ω) = √(ε(ω)/ε₀) with frequency — has two important consequences for wave propagation that you will need. The phase velocity v_p = c/n(ω) is the speed at which a pure monochromatic wave's phase fronts travel, and it varies with ω. The group velocity v_g = dω/dk is the speed at which a wavepacket (a superposition of nearby frequencies) travels, and it is this velocity that carries information and energy. In a dispersive medium v_g ≠ v_p, and a short pulse launched into a dispersive medium spreads out as its frequency components travel at different speeds — the phenomenon of pulse dispersion that limits bandwidth in optical fibers and is exploited in prisms to separate colors.