Far-field retarded potentials — formula, phase delay t−R/c, antenna radiation

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Antenna models — reading order

Introduction

In one sentence: Retarded potentials evaluate sources at the delayed time $latex t-R/c$ so far-field radiation carries the correct phase from each current element.

Engineering use: Use these far-field / retardation formulas when you move from near-zone EM to radiated fields and array factors — continue the antenna path on this site.

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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