How is antenna gain correlated to beam width? Let's say you have two dipole type antennas. Antenna A has a gain of 2.15 dBi, a horizontal beam width of 360 deg and a vertical beam width of 45 deg. Antenna B is similar to antenna A, but has a horizontal beam width of 360 deg and a vertical beam width of 42 deg. Can you use the ratio of the vertical beam widths to predict the gain of antenna B?
Note: In the application I'm asking about I'm not sure what method they used to calculate the beam widths. Maybe someone else knows which methods are most commonly used for dipole antennas.
 A: Antenna gain is often expressed in the following form,
$G = \frac{4\pi A_{e}}{\lambda^{2}}$,
where $A_{e}$ is the effective area of the antenna and $\lambda$ is the operating wavelength. However, using the antenna equation, the effective area can be expressed in terms of the main beam width (3dB width) $\Omega$,
$A_{e} = \frac{\lambda^{2}}{\Omega}$.
Assuming both antennas operate at the same wavelength, the following is true,
$G_{B} = G_{A}\frac{\Omega_{B}}{\Omega_{A}}$.
I hope this is helpful.
P.s. this essentially follows from the definition of gain.
A: If we apply few idealizations, then simple geometry will work.
The change in gain will be equal to ratio of the r' to r of 2 sectors (portions of full circle) maintaining equal area. One sector will have angle 45, another 42. When areas are equal, the sector with 42 will be of greater radius. Turn the ratio into dB and add to original gain of antenna with known gain. Ratio for r'/r=SQRT(45/42)= -> in logarithmic form it is +0.015 dB improvement.
The idealizations come from following assumprions:
- By design antennas have minimized losses, so assuming losses are zero for both antennas.
- As losses are zero, the volume of donuts for 2 dipoles will be same (idealization).
- Dissection of donuts are sectors, meaning every direction inside 45 or 42 degrees is at max gain, any other direction is sharply zero. So the cut donut looks like a goggles made of two sectors.
