Fresnel diffraction pattern

Introduction :
Diffraction in the case of waves refers to their bending round the obstacles. The diffraction phenomena is more predominant when the size of he obstacle is small and is comparable with the wavelength of he incident light.

Fresnel diffraction:  In this approach source of light, the obstacle and the screen are relatively close and are at finite distances. The waves are spherical or cylindrical. The wave fronts that reach the obstacle and proceed on to illuminate the screen at any point on it are not plane ones; i.e., the rays involved are not parallel. Therefore Fresnel type of investigation of diffraction is a general one. No lenses are required to observe the diffraction pattern.
Spherical or cylindrical wave fronts are divided into large number of zones, the wavelets emanating from which superimpose to yield the intensity distribution on the screen. The amplitudes and relative phases of all the zones are taken into account to calculate the intensity distribution. So, mathematical treatment for  Fresnel diffraction is  quite complicated.

Fresnel Zones

In the above figure , S is a point source. It ends spherical wavefront in forward direction . Let the radius of the spherical wave front be 'a' after time 't'. The effect of this wavefront at P is determined by dividing the wavefront into annular or ring  shaped zones. The distances from the edges of two successive zones to point P differ by    `(lambda)/(2)`  . The annular zones having this property are known as Fresnel zones. The distance of the zeroth zone from point P is b₀ .
The first zone is at a distance       b₁    =    b₀    +    `(lambda)/(2)`.
The second zone is at a distance  b₂   =   b₀   +    `(2lambda)/(2)` 
The third zone is at a distance        b₃    =    b₀  +  `(3lambda)/(2)`
The m^th  zone is at a distance         b_m  =  b₀  +   `(mlambda)/(2)`

Conclusion to fresnel diffraction pattern:

These zones are also known as half period zones as the path difference of  `(lambda)/(2)`   corresponds to a phase difference of 180⁰  which in turn corresponds to half a period. The areas of Fresnel zones are approximately the same when m in not too great and hence an equal quantity of light energy will be transmitted through each of the zones.