Abstract
Antenna gain (here the directivity, without loss) states how many times the power sent in one direction exceeds what it would be if spread evenly over all directions. This paper confirms what gain averages to over all directions, and counts how much the factor that estimates gain from the half-power beamwidth changes with the shape of the pattern. No new theorem or law is claimed.
First, gain averages to $1$ over all directions: for a short dipole, a half-wave dipole, four cosine-power patterns and two circular apertures alike, averaging gain over the $4\pi$ of solid angle gave $1$, within $4.6\times10^{-15}$. The short dipole has $1.500000$ and the half-wave dipole $\mathbf{1.640922}$ (Section 2).
Second, the factor that turns the half-power beamwidth into gain changes with the pattern: gain times the square of the half-power beamwidth (in square degrees) fell from $48600.0$ for cosine to the power $2$ to $39131.5$ for the power $10$, $36686.7$ for $100$ and $36435.5$ for $1000$ (Section 2).
Third, shapes without skirts come down to the Gaussian limit: the Gaussian limit is $\mathbf{36407.5}$ square degrees, $0.882542$ of the $41252.961$ square degrees in $4\pi$ steradians (Section 2).
Fourth, a circular aperture, which has skirts, has a factor below the Gaussian limit: a uniform circular aperture gives $34841.9$ at $ka=10$ and $\mathbf{34392.7}$ at $ka=30$. A circular aperture sends $\mathbf{0.162215}$ of its power outside the first dark ring (Section 3).
Fifth, the separator is the fraction of power outside the main beam: gain is the share of $4\pi$ given to the main beam, so at the same half-power beamwidth the gain is smaller the more power leaves the main beam. The estimate of $41253$ divided by the square of the beamwidth was off by factors from $1.178097$ to $0.833703$ across the patterns (Section 4).
Keywords: antenna gain, directivity, half-power beamwidth, solid angle, sidelobes, circular aperture.
Scope (what is not claimed)
No new theorem or law is claimed: the definition of directivity, the patterns and directivities of dipoles, the diffraction pattern of a uniform circular aperture, and the rules of thumb estimating directivity from half-power beamwidth are all standard. No antenna is designed: feeding, matching and values for real products are not treated. No loss is included: gain is taken equal to directivity, without reduction by efficiency. Which rule of thumb is right is not judged: whether $41253$ or another factor should be used is not stated; the values for each pattern are only set side by side. Only rotationally symmetric patterns are used: beams of different widths in two planes are not treated. The circular aperture and the cosine-power patterns are taken to radiate into the forward hemisphere only. Relation to earlier papers: Paper 384, in its scope note, did not enter the theory of antenna radiation resistance or directivity; this paper sits in that seat of directivity (no earlier occupant). Paper 335 showed that for radar the round trip changes the power of distance. Here, the gain that enters that equation is counted as the ratio that shares out the $4\pi$ of solid angle so that its average over all directions is $1$. What is added: confirming for eight patterns that gain averages to $1$ over all directions, setting down the factor that turns half-power beamwidth into gain, showing that the factor for the cosine-power patterns comes down to the Gaussian limit of $36407.5$ square degrees, finding that the factor for a circular aperture, which has skirts, is below that limit, and placing the separator on “the fraction of power outside the main beam”.
1. Notation and the Question
Setting: let $U$ be the radiation intensity in direction $(\theta,\phi)$. The directivity $D$ (gain without loss) is as follows, and by definition its average over all directions is $1$:
$$D(\theta,\phi)=\frac{4\pi\,U(\theta,\phi)}{\oint U\,d\Omega},\qquad \frac{1}{4\pi}\oint D\,d\Omega=1$$With half-power beamwidth $\theta_{\mathrm{HP}}$ in degrees, the factor $K$ of the estimate $D\approx K/\theta_{\mathrm{HP}}^2$ is found from the actual $D$ as $K=D \theta_{\mathrm{HP}}^2$. The check integrates rotationally symmetric patterns over the sphere, and for the circular aperture pattern $(2J_1(u)/u)^2$ with $u=ka\sin\theta$, finds $J_1$ from its integral representation.
Question: what does gain increase? What decides the factor that turns half-power beamwidth into gain?
2. The Average over All Directions, and Shapes Without Skirts
Computation 2.1: eight patterns.
| pattern | directivity $D$ | average over all directions | half-power beamwidth (deg) | $K$ (sq deg) |
|---|---|---|---|---|
| short dipole | $1.500000$ | $1.000000000$ | (broad in all directions) | - |
| half-wave dipole | $1.640922$ | $1.000000000$ | (broad in all directions) | - |
| $\cos^2\theta$ | $6.000000$ | $1.000000000$ | $90.0000$ | $48600.0$ |
| $\cos^{10}\theta$ | $22.000000$ | $1.000000000$ | $42.1747$ | $39131.5$ |
| $\cos^{100}\theta$ | $202.000000$ | $1.000000000$ | $13.4765$ | $36686.7$ |
| $\cos^{1000}\theta$ | $2001.999999$ | $1.000000000$ | $4.2661$ | $36435.5$ |
| circular aperture $ka=10$ | $100.672828$ | $1.000000000$ | $18.6035$ | $34841.9$ |
| circular aperture $ka=30$ | $901.400125$ | $1.000000000$ | $6.1770$ | $34392.7$ |
The average over all directions differed from $1$ by at most $4.6\times10^{-15}$, and the dipoles differed from $1.5$ and $1.6409$ by $2.8\times10^{-14}$ and $3.8\times10^{-7}$.
Computation 2.2: $4\pi$ steradians is $41252.961$ square degrees, and the Gaussian limit $16\ln2$ square radians is $\mathbf{36407.5}$ square degrees (ratio $0.882542$).
Observation 2.3 gain adds no power; it measures how the $4\pi$ of solid angle is shared out. For every pattern the average over all directions is $1$, and whatever exceeds $1$ in some directions falls below $1$ in others: for $\cos^n\theta$, which has no skirts, the factor $K$ came down to the Gaussian limit as the beam narrowed.
3. Shapes With Skirts
Computation 3.1: the factor for a uniform circular aperture is $34841.9$ at $ka=10$ and $\mathbf{34392.7}$ square degrees at $ka=30$, both below the Gaussian limit of $36407.5$ square degrees. The fraction of power outside the first dark ring ($u=3.8317$) is $J_0^2+J_1^2=\mathbf{0.162215}$ ($J_1$ there was $3.0\times10^{-12}$).
Observation 3.2 a shape with power in its skirts has less gain at the same half-power beamwidth. The half-power beamwidth sees only how narrow the main beam is: power that leaves the main beam reduces the main beam’s share of the average of $1$ over all directions.
4. The Separator
Observation 4.1 the separator is the fraction of power outside the main beam. Where that fraction is $0$, as for $\cos^n\theta$, the factor came down to the Gaussian limit; for the circular aperture it went below: the ratio of actual gain to the estimate of $41253$ over the squared beamwidth moved from $1.178097$ for $\cos^2\theta$ through $0.889310$ for $\cos^{100}\theta$ to $0.833703$ for the circular aperture at $ka=30$.
Observation 4.2 what is not said. Antenna design, loss, beams of different widths in two planes, and which rule of thumb should be used are not treated. What is shown is that gain is a ratio of how $4\pi$ is shared out, and that the factor turning beamwidth into gain changes with the power in the skirts.
5. Roll Call and Closing
- $D=4\pi U/\oint U d\Omega$ (Section 1): average $1$ over all directions.
- average differs by $4.6\times10^{-15}$ (Computation 2.1): eight patterns.
- half-wave dipole $\mathbf{1.640922}$ (Computation 2.1): the known value.
- $K$ from $48600.0$ to $36435.5$ (Computation 2.1): changes with the pattern.
- Gaussian limit $\mathbf{36407.5}$ sq deg (Computation 2.2): shapes without skirts.
- circular aperture $\mathbf{34392.7}$ sq deg, $\mathbf{0.162215}$ outside (Computation 3.1): a shape with skirts.
- fraction of power outside the main beam (Observation 4.1): the separator; the scope note of Paper 384; Paper 335.
Closing: Antenna gain averages to $1$ over all directions, and the factor that turns beamwidth into gain depends on the power in the skirts. For all eight patterns the average of gain over all directions was $1$, and gain measured how the $4\pi$ of solid angle is shared out without adding power: the factor that turns half-power beamwidth into gain came down to the Gaussian limit of $36407.5$ square degrees for shapes without skirts, and was smaller still, $34392.7$ square degrees, for a circular aperture that sends $0.162215$ outside its first dark ring. At the same half-power beamwidth, power that leaves the main beam reduces the main beam’s share: the separator is the fraction of power outside the main beam. Placed among the earlier papers: the directivity that Paper 384 did not enter was counted as the content of the gain in the round-trip equation of Paper 335. To be honest, which factor should be used is not said. That gain is a ratio of sharing out, and that the factor moves with the skirts, and no further.
References
- Balanis, C. A. (2016). Antenna Theory: Analysis and Design. Wiley.
- Kraus, J. D. (1988). Antennas. McGraw-Hill.
- Yamagishi, Y. (2026). The Round Trip Changes the Exponent [Paper 335]. YOI: https://www.yamagishiyuuki.com/yoi/335; DOI: 10.5281/zenodo.22681851
- Yamagishi, Y. (2026). “Free-Space Loss” Is Not a Loss: The Vacuum Absorbs Nothing [Paper 384]. YOI: https://www.yamagishiyuuki.com/yoi/384; DOI: 10.5281/zenodo.22699073
Reference
Yamagishi, Y. (2026). Antenna Gain Averages to $1$ over All Directions, and the Factor That Turns Beamwidth into Gain Depends on the Power in the Skirts [Paper 918]. YOI: https://www.yamagishiyuuki.com/yoi/918
ORCID: 0009-0006-8587-3450
PDF (English) · PDF (Japanese) · Date proof of the English PDF (OpenTimestamps .ots) · SHA256SUMS
The record page for this paper (in Japanese, with the PDFs in each language and the date proof): YOI 918. This paper is a preprint and has not been peer reviewed.