It is an interesting optical device, constructed on the basis of Fresnel half period hypothesis, which proves rectilinear propagation of light. To construct a zone plate number of concentric circles with the radii proportional to square root of natural numbers are drawn on a paper. The alternate zones are pointed black and a reduced photograph is taken on glass plate which is known as zone plate. If the plate odd zones are transparent then it is called positive zone plate and when even zones are transparent it is called negative zone plate as shown in Figure 5.2.

FIGURE 5.2 Zone plate

** **

**Theory:** Let S is the source, P is the point of observation and XY is the edge wise section zone plate as shown in Figure 5.3. Then

SM_{n} = (a^{2} + r_{n}^{2})^{1/2}

=a+ r_{n}^{2 }/2a

Similarly

PM_{n} =b+ r_{n}^{2 }/2b

Then path difference between two rays reaching at P

∆ = (SM_{n}P- SOP) = [(SM_{n }+ PM_{n})- (SO + OP)],

= [a+b+ r_{n}^{2} /(1/a+l/b)-(a+b)]

= r_{n}^{2} /2 (1/ a+ 1/b)

Here r _{n} is the radii of nth zone.

Since the difference between two nearby zones is λ/2 therefore

∆= r_{n}^{2} /2(1/ a+ 1/b) = nλ/2

r = √(abλ/(a+b) √n

= √n

The area of nth zone will be

=π γ ^{2}_{n} =πγ^{2}_{n-1} =πabλ/(a+b)

which is independent of n.

Then the resultant amplitude at P, will be

R =R _{1} – R_{2}+ R _{3} -R_{4} + … =R _{1} /2

R_{+} = R_{1} + R_{3} + R_{5} … positive plate

R_ = – (R_{2} + R_{4} + R_{6} + … )negative plate

We find R+ or R_ are very large in comparison with 1/2 R1 when all zones are exposed.

Further one can write

R^{2}_{N}/_{2} (1/A +1/B ) =Nλ/2

1/A +1/B +Nλ/R^{2}_{N} =0

Which can be compared with the lens formula

1/u + 1/v = 1/f

and hence one can have

1/ f = nλ/R^{2}_{N }or f= r^{2}_{n} /nλ

where f is known as principal focal length of zone plate

and thus it acts as a convergent lens with multiple foci

**FIGURE 5.3**