Classical Model of the Refractive Index

The forced oscillation of a damped oscillator with charge q, mass m, and damping coefficient b under the influence of the external force qE in the X direction [E = (E, 0, 0) = Eq exp(iwt)] is described in complex notation by the differential equation 

Assuming a solution of the form x = Xq exp(iwt), we find for the amplitude Xq the expression 

In a sample with N oscillators per unit volume the macroscopic polarization P which is the sum of all dipole moments per unit volume is therefore 

On the other hand, the polarization can be derived in classical electrodynamics from Maxwell s equations using the dielectric constant Cq or the susceptibility x  

The relative dielectric constant e is related to the refractive index n by 

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