Can lenses and mirrors be described in terms of beamwidth and directivity?

Question

As a first background, I am an Electrical Engineer with experience in antennae design for microwave bands.

Lately, I have been interested by optical devices, and I notice one strange phenomenon: when reading about a lens or a parabolic reflector for light, nobody talks about their beamwidth or gain, as we would in case of describing an antenna or reflector for microwave bands. Hence, my doubts:

1. Does it even make sense to talk about, for example, the beamwidth of a Fresnel lens or a parabolic reflector for an specific wavelength in the visible light portion of the spectrum? For big enough apertures (far from the diffraction limit), is there any phenomenon (maybe diffraction) I am missing that would make this nonsense?

2. Do the main relationships for beamwidth and gain (e.g.:

$$D= \frac{4 \pi A}{\lambda^2}$$

where $D$ is the directivity and $A$ is the area of the aperture of the antenna; or:

$$D= \frac{4 \pi}{\theta_E \theta_H}$$

for the beamwidths $\theta$) still hold at visible light frequencies?

3. In case 1. and 2. hold true, wouldn't then the directivity of a lens vary a lot depending on whether we are working on the lower or the upper part of the visible light spectrum? Does this have any consequence in practice?

Thanks in advance!

Answer 1

Yes, they do, most of the time! There area a few caveats, though. Optical "antennas" are also made for thermal noise like sun light but RF engineers have not much use for such non-coherent signals. The higher the sidelobes the less practical use these formulas obviously would have, and note that the operation of optical systems are rarely limited by grating lobes the way a bad radar antenna might ruin the air-defense system, so the concept of beamwidth is less clear there.

In the directivity formula $D=k \frac {4 \pi}{\theta_E \theta _A}$ the factor $k < 1$ and in RF it is rarely above $0.7$, but optical radiators can be much more efficient. When the directivity is written as $D=\frac{4\pi A_{eff}}{\lambda^2}$ the formula defines $A_{eff}$ but the approximation that $A_{eff} \approx A_{geom}$ assumes that the phase distribution in the radiating aperture is smooth. This is always reasonable for optics but is not true for phased arrays and/or superdirective antennas.