Rectilinear Propagation of Light
ABCD is a plane wavefront perpendicular to the plane of the paper and P is an external point at a distance b perpendicular to ABCD. To find the resultant intensity at P due to the wavefront ABCD, Fresnel’s method consists in dividing the wavefront into a number of half period elements or zones called Fresnel’s zones and to find the effect of all the zones at the point P.
With P as centre and radii equal to b + λ/2, b + 2λ/2, b + 3λ/2 etc. construct spheres which will cut out circular areas of radii OM1, OM2, OM3 etc. on the wavefront. These circular zones are called half period zones or half period elements. Each zone differs from its neighbour by a phase difference of π or a path difference of λ/2. Thus the secondary waves starting from the points O and M1 and reaching O will have a phase difference of π or a path difference of λ/2. A Fresnel half period zone with respect to an actual point P is a thin annular zone (or a thin rectangular strip) of the primary wavefront in which the secondary waves from any two corresponding points of neighbouring zones differ in path by λ/2.
O is the pole of the wavefront XY with reference to the external point P. OP is perpendicular to XY. In fig. 1, 2, 3 etc. are the half period zones constructed on the primary wavefront XY. OM1 is the radius of the first zone. OM2 is the radius of the second zone and so on. P is the point at which the resultant intensity has to be calculated.
OP = b, OM1 = r1, OM2 = r2, OM3 = r3 etc.
And M1P = b + λ/2, M2P = b + 2λ/2, M3P = b + 3λ/2 etc.
The area of the first half period zone is
πOM12 = π[M1P2 – OP2]
= πbλ approximately (i)
(As λ is small, λ2 term is negligible).
The radius of the first half period zone is
The radius of the second half period zone is
OM2 = [M2P2 – OP2]1/2
= [(b + λ)2 – b2]1/2
The area of the second half period zone
= π[OM22 – OM12]
= π[2bλ – bλ] = πbλ
Thus, the area of each half period zone is equal to πbλ. Also the radii of the 1st, 2nd, 3rd etc. half period zones are 
etc. Therefore, the radii are proportional to the square roots of the natural numbers. However, it should be remembered that the area of the zones are not constant but are dependent on (i) λ, the wavelength of light and (ii) b, the distance of the point from the wavefront. The area of the zone increases with increase in the wavelength of light and with increase in the distances of the point P from the wavefront.
As discussed, the effect at a point P will depend on (i) the distance of P from the wavefront, (ii) the area of the zone, and (iii) the obliquity factor.
Here, the area of each zone is the same. The secondary waves reaching the point P are continuously out of phase and in phase with reference to the central or the first half period zone. Let m1, m2, m3 etc represent the amplitudes of vibration of the ether particles at P due to secondary waves from the 1st, 2nd, 3rd etc. half period zones. As we consider the zones outwards from O, the obliquity increases and hence the quantities m1, m2, m3 etc. are of continuously decreasing order. Thus, m1 is slightly greater than m2 ; m2 is slightly greater than m3 and so on. Due to the phase difference of π between any two consecutive zones, if the displacements of the ether particles due to odd numbered zones is in the positive direction, then due to the even numbered zones the displacement will be in the negative direction at the same instant. As the amplitudes are of gradually decreasing magnitude, the amplitude of vibration at P due to any zone can be approximately taken as the mean of the amplitudes due to the zones preceding and succeeding it.
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