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Half Period Zones of Plane Wave

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.

 

Radii of Half Period Zone: It will be

 

OMl = √{(p + λ/2)2- P2} = √(pλ)

OM2 = √{(p + 2λ/2)2-p2} = √(2pλ)

Similarly,

OMn=√{(p+nλ/2)2 – p2}

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 – p2} – π { ((p+(n-1)λ/2)2 – p2}

= π { pλ+λ2(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

 

Rn = area of the zone/ distance of point P from zone * obliquity factor

 

= πλ (1 + cos θn)

 

If n increases, cos θn decreases and hence Rn 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 R1, R2, R3 and so on are the amplitudes at point P from various zones then the resultant amplitude at point  will be

R = R1-R2 + R3 –R4 … (-1)n-1 Rn

We can have

R2 = Rl +R3/2  R4 =  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                          Rn-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 = R2,

= R12/4

 

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