Re:Bessel Beam
by 0xfaded on Tuesday August 28, 2018 @11:10AM
(#17859367)
My first encounter with Bessel functions was from a professor who in his youth had built digital amplifiers -- before dsps could switch at 44k Hertz. (From memory he had a 1k pwm)
He did it by predicting the position of the magnet in the speaker coil by fitting a function over the last few known positions.
Bessel functions are othogonal, and their Fourier transforms are the base polynomials band limited between -pi and pi. His argument was that by fitting a sum of Bessel functions to a sequence captured periodic signals.
I've used this trick several times, and love the "what is this black magic" reaction I get when showing it to others. Guess the cat's out of the bag now...
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.
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.
He did it by predicting the position of the magnet in the speaker coil by fitting a function over the last few known positions.
Bessel functions are othogonal, and their Fourier transforms are the base polynomials band limited between -pi and pi. His argument was that by fitting a sum of Bessel functions to a sequence captured periodic signals.
I've used this trick several times, and love the "what is this black magic" reaction I get when showing it to others. Guess the cat's out of the bag now...
What does "unbounded" mean in this context?
What is an example of a "bounded" beam?
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...
So, unbounded width of the beam? As opposed to unbounded length, which is generally expected in light beams.
But light beams do get very long for very modest work, and properly focused they can propagate very long distances indeed.
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.