By R. C. Hansen
Microwave Scanning Antennas. quantity 1: Apertures [Hardcover] [Jan 01, 1964] HANSEN, R; Equations; Charts and R. C. Hansen. the 1st in a 3 half sequence Coving All facets of studying, construction and Maintainance of Microwave Antennas and their Peripheral gear
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_R_ 20 2/ X Fig. 19. On-axis power density-uniform square aperture. 18 in the radiating near-field region, the main beam has started to bifurcate or become hollow in the same fashion as the response of an overcoupled circuit. 1 tJ. 4 07 R • 202/X Fig. 20. On-axis power density-uniform line source. 0 1. 07. As the observer moves in closer, the main beam reforms and splits repeatedly. The directivity in two dimensions which has produced the pencil beam is destroyed by the Fresnel phase smear in this region.
These two ways of looking at Fresnel fields have created confusion on how fields vary with distance. The close-in portion of the radiating near-field region requires higher terms of l/r n in the integrand, hence is both l/r n and 1/ Rn. 2. Range of Validity and Field Regions The question naturally arises concerning the region of validity of the Fresnel approximation as a function both of distance from the aperture and of angular displacement from the axis. If one considers angle, the small-angle Fresnel result is good to about 30° from the axis; the general Fresnel formulation should be used beyond that angle (Hansen and Bailin, 1959).
For large kR and small 0, the argument of C(x) and S(x) is near the origin. In this case, R» Land kR sin f} < v;kR. The far-field result can be obtained from Eq. (54), although it is obtained directly by omitting the term in Eq. (53) and integrating. If the above conditions are satisfied, the small argument approximations for C(x) and S(x) can be used. These are usually written C(x) ~ x and S(x) ~ x 3, but an alternative approximation is useful. For x« 1, we have e 2 C(x) ~ sin(1rx j2) (55) 1rxj2 This used with Eq.
Apertures by R. C. Hansen