▁for ▁a ▁b ire fr ing ent ▁material . ▁In ▁the ▁general ▁expression : ▁The ▁relative ▁phase ▁ret ard ation ▁induced ▁between ▁the ▁fast ▁axis ▁and ▁the ▁slow ▁axis ▁is ▁given ▁by ▁▁ ▁is ▁the ▁orientation ▁of ▁the ▁fast ▁axis ▁with ▁respect ▁to ▁the ▁x - axis . ▁ ▁is ▁the ▁circular ity . ▁ ▁Note ▁that ▁for ▁linear ▁ret ard ers , ▁ ▁= ▁ 0 ▁and ▁for ▁circular ▁ret ard ers , ▁ ▁= ▁ ± ▁/ 2 , ▁ ▁= ▁/ 4 . ▁In ▁general ▁for ▁elli pt ical ▁ret ard ers , ▁ ▁takes ▁on ▁values ▁between ▁- ▁/ 2 ▁and ▁/ 2 . ▁ ▁Ax ial ly ▁rot ated ▁elements ▁ ▁Assume ▁an ▁optical ▁element ▁has ▁its ▁opt ic ▁axis ▁ ▁per pend icular ▁to ▁the ▁surface ▁vector ▁for ▁the ▁plane ▁of ▁inc idence ▁and ▁is ▁rot ated ▁about ▁this ▁surface ▁vector ▁by ▁angle ▁ θ / 2 ▁( i . e ., ▁the ▁principal ▁plane , ▁through ▁which ▁the ▁opt ic ▁axis ▁passes , ▁makes ▁angle ▁ θ / 2 ▁with ▁respect ▁to ▁the ▁plane ▁of ▁polar ization ▁of ▁the ▁electric ▁field ▁of ▁the ▁incident ▁TE ▁wave ). ▁Rec all ▁that ▁a ▁half - wave ▁plate ▁rot ates ▁polar ization ▁as ▁twice ▁the ▁angle ▁between ▁incident ▁polar ization ▁and ▁opt ic ▁axis ▁( pr incipal ▁plane ). ▁Therefore , ▁the ▁Jones ▁matrix ▁for ▁the ▁rot ated ▁polar ization ▁state , ▁M ( θ ), ▁is ▁▁▁ ▁where ▁ ▁This ▁agre es ▁with ▁the ▁expression ▁for ▁a ▁half