All em radiation propagates spherically and at a far off place a small enough distance, when observed appears like a plane like our experience on the earth. So I tend to think that whether a wave on an aperture can be considered plane or not depends on how far the aperture is from the origin and how much is the aperture. So it should be only dependent on the angle made at the centre by the aperture i.e. apertures making more than a certain angle at the centre can be considered spherical, and otherwise can be considered plane. This is my wrong understanding. But in radar theory, this is not the criterion for considering plane and far field. A 2D^2/lambda formula is used. How is this formula relevant? How lambda is entering into the formula?
Rajeev Bansal in the appendix to this article explains the $2D/\lambda$ rule:
How does one arrive at the various criteria for the far-field zone? Basically, the criteria are guidelines to the boundaries where the fields start to approximate the “ideal” assumed characteristics. First of all, only the radiation ($1/r$) terms remain significant; higher order terms fade away. Second, in the far-field zone, the angular field distribution becomes independent of the distance . Third, only transverse field components remain, and the ratio of the electric and magnetic field components approaches the free-space impedance, 377 ohms [6 ,7]. Finally, for a receiving antenna, the incoming wave-front is nearly planar across the aperture. In fact, the $r = 2 D^2/\lambda$ formula corresponds to a phase error (due to the curvature of the actual spherical wave-front) of no more than $\pm 22.5$ degrees across the aperture, as compared with the ideal plane wave-front [6, 8].
IEEE Standard Dictionary of Electrical and Electronics Terms, IEEE standard 100-1984, 1984.
C. Paul, K. Whites, and S. Nasar, Introduction to Electromagnetic Fields, 3rd ed., McGraw-Hill, 1998.
Reference Data for Radio Engineers, 6th ed., SAMS, 1981.
G. Smith, An Introduction to Classical Electromagnetic Radiation, Cambridge, 1997.