For a monochromatic plane wave in a material v = 1/(με)^½ = c/n.  The index of refraction is n = c(με)^½.  For most dielectric materials μ ~ μ₀, and n = (ε/ε₀)^½. 
For many materials ε depends on ω.  Can we understand how ε(ω) depends on ω?

Let us look at a simple model of a lih material.
P = ε₀χ_eE = Nα_eE_eff = Np. (SI units)

Here  χ_e is the electric susceptibility, α_e is the atomic or molecular polarizability, N is the number of atoms or molecules per unit volume, and p is the dipole moment per atom or molecule.  E is the average macroscopic field in the medium.  E_eff is the microscopic field due to everything else except that particular atom or molecule at the location of that atom or molecule.
D = ε₀E + P = ε₀ (1 + χ_e)E = εE.

If in a material we model the electrons bound by a harmonic force to the atomic cores, and we include a damping force in the atomic oscillators, then the equation of motion for an electron in the presence of an electric field is
F_e = -q_eE_eff = m(d²r/dt² + γdr/dt + ω₀²r).

If the electric field varies sinusoidally in time and the wavelength is long enough so that spatial variations over the dimensions of an atom can be neglected, we write
E_eff = E₀exp(-iωt).

We find r = r₀exp(-iωt), with r₀ = -(q_e/m)E₀/(ω₀² - ω² - iγω).
The induced dipole moment of the atom is 
p = -q_er = (q_e²/m)E_eff/(ω₀² - ω² - iγω),
if the atom contains a single electron.

If there are f_j electrons per atom with binding frequencies ω_j and damping constants γ_j, then
p = (q_e²/m)E_eff∑_j (f_j/(ω_j² - ω² - iγ_jω)),  ∑_j f_j = Z.

Here Z is the total number of electrons per atom and the f_j are called the oscillator strengths.
The polarization P is now given by
P = Np =  (Nq_e²E_eff/m) ∑_j (f_j/(ω_j² - ω² - iγ_jω)) = ε₀χ_eE.

In substance with low density E_eff ~ E.  Then
χ_e = (Nq_e²/(ε₀m)) ∑_j (f_j/(ω_j² - ω² - iγ_jω))
and
ε = ε₀ + (Nq_e²/m) ∑_j (f_j/(ω_j² - ω² - iγ_jω)).

In general in dielectric materials ω_j >> γ_j and ε(ω) ~ real for most frequencies.
For metals some ω_j << γ_j and ε(ω) is a complex number.
If the angular frequency of the effective electric field ω is less than all the ω_j, then ε > ε₀, and if ω is greater than all the ω_j, then ε < ε₀.

• Normal dispersion: Re(ε(ω)) rises with ω.
• Anomalous dispersion: Re(ε(ω)) decreases with increasing ω.

Anomalous dispersion occurs when ω ~ ω_j, in the neighborhood of resonant frequencies.  Here Im(ε(ω)) becomes large.
A complex ε(ω) in the wave equation implies a complex wave number k for plane wave solutions.
k² = (ω²/v²) = μεω².
If we write k = β + iβ/2, then α is called the attenuation constant.
For a plane wave propagating along the z-direction with E = E_xi we have
E_x = E₀exp(i(kz - ωt)) = E₀exp(-(α/2)z) exp(i(βz - ωt)).
The intensity is proportional to E² and falls off as exp(-αz).

The index of refraction is n = (ε/ε₀)^½.
n = [1 + (Nq_e²/(ε₀m)) ∑_j (f_j/(ω_j² - ω² - iγ_jω))]^½.
If ε ~ ε₀ and n ~ 1, then we can write
n ≈ 1 + (Nq_e²/(2ε₀m)) ∑_j (f_j/(ω_j² - ω² - iγ_jω)).

The index of refreaction, in general, is a complex number.