Explain that for a greater distance,the spreading due to diffraction dominates over that due to ray optics. $OR$ Explain the importance of Fresnel distance.

(N/A) An aperture of size $a$ illuminated by a parallel beam sends diffracted light into an angle $\theta$,where the angular width is $\theta \approx \frac{\lambda}{a}$.
In travelling a distance $z$,the diffracted beam acquires a width $w = z \theta = \frac{z \lambda}{a}$ due to diffraction.
The Fresnel distance $z_F$ is defined as the distance at which the spreading due to diffraction becomes equal to the size of the aperture $a$.
Setting $a = \frac{z_F \lambda}{a}$,we get $z_F = \frac{a^2}{\lambda}$.
For distances $z \ll z_F$,the spreading due to diffraction is negligible compared to the size of the aperture,and the beam travels in a straight line,which is consistent with ray optics.
For distances $z > z_F$,the spreading due to diffraction dominates over the spreading due to ray optics. Thus,ray optics is an approximation valid only when the wavelength $\lambda \to 0$ or for distances $z \ll z_F$.