Derivation of optical conductivity σ:

Noticing that E=D-P= ε₀(εE-P), one gets

P=ε₀(ε-1)E (1)

The electric current density (considering E=E₀e^-iωt):

J=dP/dt=-iωε₀(ε-1)E (2)

Compare (2) with the Ohm's law:

J=σE (3),

one gets:

σ=-iωε₀(ε-1)=ωε₀ε₂-iωε₀(ε₁-1), therefore

σ₁=ωε₀ε₂ and
σ₂=-ωε₀(ε₁-1)