Near-field – far-field : Where is the boundary ?

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Antenna models — reading order

Introduction

Antenna fields change character with distance: reactive storage near the element, Fresnel diffraction at intermediate range, and spherical-wave radiation farther out. Link-budget and pattern formulas almost always assume that last region. So where are the boundaries?

Motive

Suppose you measure a dish pattern at a few metres and someone asks: is that the far field? The usual rule of thumb is the Fraunhofer distance \(2l^2/\lambda\), together with Balanis’s reactive and Fresnel partitions. We will put numbers on a few apertures and see why millimetre-wave ranges have to be long.

Technical development

For a maximum path-length error of about \(\lambda/16\) (phase \(\pi/8\)) across an aperture of largest dimension \(l\), the Fraunhofer distance is

\[r_{\mathrm{ff}} = \frac{2 l^2}{\lambda}.\]

A common partition also names a reactive near-field zone roughly \(r \lt 0.62\sqrt{l^3/\lambda}\) and a radiating near-field (Fresnel) zone between that and \(r_{\mathrm{ff}}\). Beyond \(r_{\mathrm{ff}}\) the parallel-ray approximation is accepted for pattern work.

Field regions
Figure 2: Schematic of reactive, Fresnel, and Fraunhofer regions.

How this achieves the motive

Figure 2 is the map. The listing below puts numbers on the map for a few practical apertures.

Sample simulation

import numpy as np

def field_region_bounds(l_m, freq_hz, c=299_792_458.0):
    lam = c / freq_hz
    r_reactive = 0.62 * np.sqrt(l_m ** 3 / lam)
    r_ff = 2.0 * l_m ** 2 / lam
    return lam, r_reactive, r_ff

for l, f in ((0.5, 1e9), (1.0, 2.4e9), (2.0, 28e9)):
    lam, r_rx, r_ff = field_region_bounds(l, f)
    print("l=%.2f m  f=%.1f GHz  lambda=%.3f m  reactive~%.2f m  Fraunhofer=%.2f m" % (
        l, f/1e9, lam, r_rx, r_ff))

Sample results

At 28 GHz a 2 m aperture has a Fraunhofer distance of hundreds of metres—why mmWave antenna ranges are long. At 1 GHz a half-metre antenna is already in the far field within a few metres.

Next topic

In that far region the retarded integrals simplify: far-field retarded potentials and the radiation vector.

References

[1] Balanis. [2] Orfanidis.

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