If the pri sm is rotate d about the in comi ng beam axis, the be am is rotated, formi ng a conical scan pattern   of half angle δ. If the beam is then pa ssed thro ugh a sec ond id entical p rism, the sub sequent deviatio n  provide s a se cond degree of freedom. T his allo ws for a variety of scan patterns, includin g linear sca ns.  The net angular deviation  of a bea m passin g through a pair of identi cal prisms is the sum of the  two  individual p rism angul ar deviations.  We  define an gular be am devi ation abo ut two o rthogonal axes a s δx  and δy. Thus the d eviatio n of a  beam passing th rough a pair of identi cal prisms at arbitrary rotational   positio ns can be define d as: 

) cos( ) cos(2 1 A Ax δ δ δ + =   y ) sin( ) sin(2 1 A A δ δ δ + =  

Where  A1 = rotationa l position of first pri sm and A2 = rotationa l position of seco nd pri sm 

If A 1= - A 2, a linear scan al ong the X axis is realized, δy = 0  If A 1= 90° – A 2, a linear sca n of slope 1 i s reali zed, δx = δy  If A 1= 180° – A 2, a linear scan al ong the  Y axis is real ized, δx = 0  For continu ous maximum d eviation of 2 δ:  A1  =  A 2  For continu ous minimum o r nulled (ze ro) deviation:  A 1  =  A 2 + 180°  

CounterCounter --Rotation  (degr ees) Rotation  (degr ees) Bead(mrad) am  De lect ion  ( f00mr)Beam   Deflection ( -22+2+2 +180+180 --180180 0 0  - 

Figure 4.  Risle y Prism Principle and Theore tical Beam Dev iation   The maximu m angula r deviation of the prism pai r is 2 δ. Therefore, only the prism angl e, α, and index of  refraction, n, determi ne th e stroke of a risley prism pair scan ner. Erro rs in deviation o ccur if the p rism  angle s or ind ex of refracti on are n ot perfectly match ed. Differen ces in α or n of the prisms result in sm all  linear vari ations of δ. Angul ar resolutio n within the stroke  is limited  only by rotati onal po sitioni ng re solution  of the individual pri sms.  

Operation   Whenever th e wedge s are align ed such th at the wedge a ngles are 180° out of pha se, then in comi ng  radiatio n is not deviated. This i s called a nulled position of the prisms. Wh en the pri sms are counte r  rotated fro m this po sition the incomi ng laser radiation is then d eviated dep ending o n the amount of  counter rotation and the wedge a ngle p resent on the  prisms. The beam d eviation is alo ng a line that i s  parall el to the null positio n of the wedges. The refore, in order to  swee p out a cone it is re quire d to first  define the nul led po sition a ngle an d then  counte r rotat e the pri sms to move along  this angle to deviate the   incomi ng ra diation. The risley prism pri nciple of operation an d the theoreti cal optical be am deviations  given the parameters of the prisms u sed in the BAM are sho wn in Figure 4.   

The critical factor d efining pointing re solutio n is matchin g the geometri c and  optical prop erties of ea ch  prism. Risl ey pairs h ave several de sirable attribute s as applie d to pre cision pointing of lasers o r othe r  refrac tive s ystems :    (1) Pre cise angula r stee ring (2 arcse c or better) is possible without precision lo cation or align ment of the  supporting mechanism st ructure.    48