Dr Searle, Experiments on focal lines formed by a zone plate 335 

 Hence, since cos^ co + sin^ w — 1, 



i (jU'2'^2 COS^ 02 — fliSi COS^ 9i) COS^ CO 



and thus, by (7), /X2pf2 cos ^2 ~ /^i^i cos ^i = 0. (10) 



Since 6^ is known by (6), the last three equations determine S^, W2 

 and T2" 



In the ordinary use of the zone plate, the medium on either 

 side of the zone plate is air of refractive index /x, and thus 

 H-z — H'l ^ A*-) ^^^ ^2 = ^1 == ^- If tlic wave length of the light in 

 air is A, we have /xA = A^. If the "power" of the zone plate in air 

 is Fj, , 2pX/k'^ = i^j, , and then the equations giving S^i ^2 ? ^2 

 become 



^2=^i^^sec2^ + >Si, TF2=W^i> T2 = F^+T^. ...(11) 



If the incident beam is due to a luminous point at a distance 

 u from in the object space, 8^^= T^ = — 1/u, Wi = 0. Then 



,^2 = sec2 e/f^ - l/u, If 2 = 0, T2 = 1//^ - l/u. ...(12) 



Since IF2 = 0, the focal lines of the emergent beam are in and 

 perpendicular to the plane XOY. If their distances from are 

 c and b respectively, where b, c are positive when the focal lines 

 are in the image space, we have S2 = l/b, T2 == 1/c, and then 



l/u + l/b = sec2 d/f^, l/u + 1/c = 1/fp, (13) 



as was found in § 3. 



§ 5. The principal curvatures of the emergent wave front. Let the 

 principal planes of the incident front at (Fig. 4) intersect the 



tangent plane at in Orji, Oi^i. 

 Take these lines, with Oii along OP^ , 

 as axes for the front. Let the radii 

 of curvature of the sections of the 

 front by O^it]^, O^^^i be B^^-^ and 

 Cj~^, counted positive when the 

 sections are concave towards P^. 

 The equation to the incident front 

 referred to these axes is then 



Let Ot^i make an angle ifj^ with Os^, as in Fig. 4. Then 



^1 = ^1 ' Vi^ ^1 cos j/»i + f 1 sin i/»i , ^1 = — Si sin tpi + t^ cos if/i , 

 and hence (14) is equivalent to 



ri = |J5i (Sj cos iff I + «i sin ipi)^ + ^C^ (- Si sin ifj^ + ^ cos </fi)2. . . .(15) 



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