The constitutive relation for the refractive index of dilute solutions of flexible polymer molecules is obtained, and calculations of the intrinsic birefringence exhibited in uniaxial elongational flow are presented. The polymer molecules are modeled as chains of beads connected by finitely extensible nonlinear elastic springs under the Peterlin approximation. It is found that at very low elongation rates there is very little birefringence, but there is a critical elongation rate above which there is substantial birefringence. This critical elongation rate scales as the inverse square of the number of beads in the chain. At this elongation rate the individual springs in the chains are stretched and oriented with the flow, but the chains have not uncoiled. It is found that the stress-optic relation holds for elongation rates below the critical elongation rate. However, the stress-optic coefficient is not constant above the critical rate. In the instantaneous inception of the flow above the critical elongation rate, the stress optic coefficient initially takes on the value that it would have below the critical elongation rate, but it decays to a much lower value as the strain increases.