Dr Searle, Experiments on focal lines formed by a zone plate 333 



The emergent rays of order p, i.e. the normals to the emergent 

 wave front of order p, do not pass through a single point but 

 through two focal lines. The primary line passes through Q and 

 is perpendicular to the plane of Fig. 2, and the secondary line 

 through R is in that plane. 



If a slit illuminated with sodium light is placed at P, and if 

 the slit is perpendicular to the plane of Fig. 2, there will be a focal 

 line image of the slit at Q, and the image will be perpendicular 

 to that plane. If the slit is in the plane of Fig. 2 and perpendicular 

 to OP, there will be a focal line image at R, the image lying in the 

 plane of Fig. 2. 



§ 4. General case. Let OX (Fig. 3) be the axis of the zone plate 

 and OP^ the forward direction of the chief ray of the incident 



beam. Take OY perpendicular to OX 

 in the plane PfiX, and OZ perpen- 

 dicular to that plane. Then the zone 

 plate lies in the plane YOZ. Let 

 PfiX = dy. 



Let the refractive indices of th • 

 object and image spaces be /x^, ju., . 

 Let t'o be the velocity of light, and Aq 

 the wave length, in a vacuum, and 

 let T be the periodic time of the 

 vibration. Then rv^ = Xq . 



Take OP^ as the axis of r^ in a new set of axes Or^, Osy, Ot^, 

 such that Osi is in the plane XOY and Oti coincides with OZ. Let 

 the equation, referred to these axes, of the incident wave front, 

 when passing through 0, be 



r, = h_Srh'+W,s,t, + iT,t^ (4) 



Let Gn be a point on the zone plate on the circle of radius p„ , 

 and let G^OY = co. Then the x, y, z coordinates of (?„ are 0, 

 Pn cos CO, pn sin CO, and its r^, s^, t^ coordinates are 



^1 = Pn cos CO sin ^1 , §1 = pn cos CO cos ^1 , t^= Pn siu CO. 



If a straight line through (?„ parallel to OP^ cuts the incident wave 

 front ODj^D^ ... in Z)„, the second and third coordinates of D^, 

 referred to the axes of r^, s^, t^, are p„ cosco cos 0^ and ,o„ sin co. 

 Hence, by (4), the distance of D^ from the plane r^ = 0, which 

 touches the wave front at 0, is 



Pn^ (1^1 cos^ CO cos- 6^ + Wi cos CO sin co cos 6^ + hT^ sin- co), 



and the distance of G^ from the same plane is />,j cos co sin d^ . Hence 



D,fin ^ Pn COS CO sin 6 1 



— p^^ (^/Si cos- CO COS- dy + W^ cos CO sin co cos d^ + ^T^ sin- co). 



VOL. XX. PART III. 22 



