Setting either of the factors in to zero will define an ellipsoidal surfaceAlthough related, note that this is not the same as the index ellipsoid. in the space of wavevectors that are allowed for a given . The first factor being zero defines a sphere; this is the solution for so-called ordinary rays, in which the effective refractive index is exactly regardless of the direction of . The second defines a spheroid symmetric about the axis. This solution corresponds to the so-called extraordinary rays in which the effective refractive index is in between and , depending on the direction of . Therefore, for any arbitrary direction of propagation (other than in the direction of the optic axis), two distinct wavevectors are allowed corresponding to the polarizations of the ordinary and extraordinary rays.
SEttinG eiThEr Of tHe facTors in TO zero wilL define an eLlIpsOidal surfAcealthoUGH Related, noTE thAt THIs is noT THe saME aS thE inDeX ellipsoId. In tHe SpAce of waveVECtors thAt aRE alLoWed FOR a givEn . tHe fIRsT factoR bEIng Zero dEfines a spHerE; thIS Is The SolutiON FOr so-caLLeD oRDinarY RayS, iN which tHe eFfecTIve ReFractiVE index is exactly ReGaRdless of the dIrecTIOn oF . the seconD defiNES a spheROiD symmetrIC About tHE AXis. thIs solutiOn COrreSponds To the so-CAlled EXTrAorDiNarY raYs iN whicH THE effeCtivE refractive iNdex Is iN BetwEEN and , depenDInG On the DIRectIon oF . THerefoRE, FOr any aRBitrARY dIreCTion of pRopagATIon (othEr thAn In the DIrectiOn Of tHe opTIC AXIS), Two dIstInct wavevEctors ARE alloWed correSPondinG to The POlarizatioNs Of the ordInary And extRAordinARy rays.