> What does "unbounded" mean in this context?

The radial intensity distribution of a Bessel beam is described by Bessel functions, which are not square integrable (integral is not finite), see https://upload.wikimedia.org/wikipedia/commons/thumb/5/5d/Be....

Realizing a true Bessel beam would thus require infinite power.

A true Bessel beam is created by interfering (in 2D) two plane waves, see https://upload.wikimedia.org/wikipedia/commons/thumb/b/b8/Be.... You can see in the image that the "Bessel beam" area is bounded, it is the conical shape where the beams overlap.

> What is an example of a "bounded" beam?

A "standard" Gaussian laser beam, see https://upload.wikimedia.org/wikipedia/commons/thumb/a/a9/La...

> The radial intensity distribution

So, unbounded width of the beam? As opposed to unbounded length, which is generally expected in light beams.

Well, for truly unbounded length you need to keep the source on for infinite time of course.

But light beams do get very long for very modest work, and properly focused they can propagate very long distances indeed.

So, to be clear here - is the answer yes or no? Are we talking about beam width - i.e. unbounded in a direction at right angles to the direction of travel?

Not an expert, but we would need to have infinite thickness beam and an infinitely big axicon, for the Bessel beam to be infinite (as in, reaching infinity "like" regular light). Otherwise the Bessel beam is constrained to the space covered by the overlap of the radiations of both of the original waves.

It makes sense since bessel beam-like radiations can be formed inside waveguides (or waveguide-like structures like optical fiber), which effectively make the wave unbound by means of bouncing it off.