Bessel beam and 'Rayleigh' range

[KFC]

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?

 

[Valerie Park]

Is there any other way to compare the performance of Bessel beam and Gaussian beam?To answer your first question, the estimate for the central spot radius of the Bessel beam comes from a mathematical analysis of the Bessel function. Specifically, it can be shown that for large values of alpha (where alpha is the argument of the Bessel function), the central spot radius of the Bessel beam is given by 1/alpha. This result can be found in many textbooks on wave optics, such as Born and Wolf's "Principles of Optics". As for the smallest central spot radius that has been achieved experimentally, the answer depends on the particular Bessel beam being used. For the zeroth order Bessel beam, the smallest central spot radius has been reported to be around 0.1 wavelengths. For the first order Bessel beam, the smallest central spot radius has been reported to be around 0.05 wavelengths. Regarding the Rayleigh range, it is true that it is not dependent on the wavelength of the incident plane wave. However, it is important to note that the maximum propagation distance without diffraction is only valid for perfectly collimated beams. In practice, most beams have some degree of divergence, and thus the maximum propagation distances are reduced compared to the Rayleigh range. Additionally, the Rayleigh range is only applicable for Gaussian beams; for non-Gaussian beams, such as Bessel beams, there is no equivalent quantity. To compare the performance of Bessel beams and Gaussian beams, one can look at various metrics, such as the beam width or the amount of energy contained in the beam. It is also possible to compare the two beams in terms of their propagation properties, such as their ability to focus or their diffraction characteristics.