Let a source 5 emits a plane wave front –ABCD travelling from left to right and has wavelength λ. Now, we wish to see the effect of wavefront-point – P at a distance p from the wavefront. Let us now divide the wavefront into Fresnel’s zones with P as a centre and radii equal top+ n λ/2 (n = 1, 2, 3 … ).

Draw concentric spheres on the wavefront as shown in Figure 5.1. The area between two spheres is called zone. The secondary waves from any two consecutive zones reach the point P with a path difference of A./2 ( = T /2) that is why the name half O period zones. Here T stands for period. The point O is called the pole of the wavefront with respect to point P.

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**Radii of Half Period Zone:** It will be

OM_{l} = √{(p + λ/2)^{2}- P^{2}} = √(pλ)

OM_{2} = √{(p + 2λ/2)^{2}-p^{2}} = √(2pλ)

Similarly,

OM_{n}=√{(p+nλ/2)^{2} – p^{2}}

Thus radii are proportional to the square roots of natural numbers.

**Area of Half Period Zone**: The area of nth zone will be

= π { (p+nλ/2)^{2} – p^{2}} – π { ((p+(n-1)λ/2)^{2} – p^{2}}

= π { pλ+λ^{2}(2n-1)/4} = πpλ

which says that area of each half period zone is nearly the same.

**The distance of point P from half period zone:** It is

= (p + nλ/2) + (p + (n – 1) λ/2) / 2 = p + (2n- 1) λ/ 4)

**Amplitude at point P due to one zone:** It is given as

R_{n }= area of the zone/ distance of point P from zone * obliquity factor

= πλ (1 + cos θ_{n})

If n increases, cos θ_{n} decreases and hence R_{n} also decreases.

**Resultant amplitude of point P due to whole wavefront:** As the path difference between the two consecutive zones is A/2 so they are reaching in opposite phase. If R_{1}, R_{2}, R_{3} and so on are the amplitudes at point P from various zones then the resultant amplitude at point will be

R = R_{1}-R_{2} + R_{3} –R_{4} … (-1)^{n-1} R_{n}

We can have

R_{2} = R_{l} +R_{3}/2 R_{4} = R3 + R5/2 and so on

R = R 1/2 +R n/2 , for n to be odd

= R1/2 +Rn/2 -Rn for n to be even

taking R_{n-1 }= Rn as n is very large, then

R= R1/2 + Rn/2 = R1/2

Thus the amplitude due to a large wavefront at a point is just half that due to first half period Fresnel zone. The intensity will be

I = R^{2},

= R_{1}^{2}/4

i.e. one fourth that due to the first half period zone.