## Bessel beam and 'Rayleigh' range

In some papers and text, it is said that the central spot radius of the Bessel beam (take zeroth order $$J_0(\alpha \rho), \rho = \sqrt{x^ + y^2}$$ as example) can be estimated to be $$1/\alpha$$. In wonder how to obtain this relation? And does anyone know what's the smallest central spot radius of J0 and J1 can be obtained experimental so far?

In addition, it is well-known that perfect Bessel beam is non-diffracting but in practical case we can only have the approximated Bessel beam propagate for finite distance without diffraction, the maximum distance can be estimated by the so called 'Rayleigh range' (it from Gaussian beams, don't know if I should call it with same term), which given by J. Durnin. He have a plane wave shining on a ring slit and the outgoing wave through a len to form Bessel beam on the Fourier plane (focal plane of the len). Hence, we can get the 'Rayleigh range' as

$$z_{max} = \frac{fR}{r}$$

where f is the focal length of the len, R is the apecture (raidus) of the length and r is the radius of the ring slit. It seems that the maximum distance without diffraction is not related the incident wavelength? So all plane wave with any wavelength will propagate the same maximum distance without diffraction? But for Gaussian beam, the Rayleigh range is related to wavelength and spot size, so how do we compare how good Bessel beam with Gaussian beam if Rayleigh range for one are wavelength and spot size dependent but other not?

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