Far-field retarded potentials

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Antenna models — reading order

Introduction

In the Fraunhofer region the amplitude goes as \(1/r\), and the phase is \(e^{-jkr}\) times a linearised path through the source. The angular part factors into a radiation vector \(\mathbf{F}(\mathbf{k})\)—the object that array theory multiplies.

Motive

We need something an element can contribute as its pattern, and that an array factor can multiply. That something is the radiation vector \(\mathbf{F}\).

Technical development

Under the far-field approximation \(R\approx r-\hat{r}\cdot\mathbf{r}’\) in the phase and \(R\approx r\) in the amplitude,

\[\mathbf{A}\propto\frac{e^{-jkr}}{r}\int \mathbf{J}(\mathbf{r}’)e^{j\mathbf{k}\cdot\mathbf{r}’}\,dV’ = \frac{e^{-jkr}}{r}\mathbf{F}(\mathbf{k}),\]

with \(\mathbf{k}=k\hat{r}\). The integral is a spatial Fourier transform of the current—the radiation vector.

How this achieves the motive

Shift a current distribution by \(\mathbf{l}_0\) and \(\mathbf{F}\) picks up a factor \(e^{j\mathbf{k}\cdot\mathbf{l}_0}\). That one property is what produces array factors in the next article.

Sample simulation

import numpy as np

I0_L = 1.0
F = np.array([0.0, 0.0, I0_L])  # z-directed Hertzian moment at the origin
print("F =", F, " |F| =", np.linalg.norm(F))

theta = np.deg2rad(60.0)
k = 2 * np.pi
z_prime = 0.1
phase = np.exp(1j * k * z_prime * np.cos(theta))
print("phase e^{j k z' cos theta} =", phase)

Sample results

The Hertzian radiation vector is angle-independent in this idealisation; the \(\sin\theta\) factor of the electric field appears after the far-field projection. The printed phase factor is the same term that appears when elements are stacked along \(z\).

Next topic

Array pattern multiplication: sum shifted copies of \(\mathbf{F}\) with feed weights.

References

[1] Orfanidis. [2] Balanis.

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