Bessel Beam Calculator
Core radius, FWHM and non-diffracting range of a Bessel beam from its cone angle and input beam radius, with J₀ and J₁ profiles.
Core radius to the first dark ring
24.22 µm
r₀ = 2.405/k_r; core diameter 48.44 µm
Core FWHM
22.69 µm
Full width at half the peak intensity
Bessel range z_max = R/tan θ
500.0 mm
Where the cone of rays from the input edge crosses the axis
Gaussian with waist r₀
z_R = 2.912 mm
The Bessel core lasts 172 × longer (w₀ = r₀; a Gaussian matched to the core FWHM, w₀ = FWHM/√(2 ln 2), has a z_R about 1.6× shorter still)
Radial wavenumber k_r = k sin θ
99.29 rad/mm
θ = 0.5730°
Rings across the input ≈ k_r R/π
158
Each ring carries about the core's power, so the core holds a small share
E(r, z) = J₀(krr) eikzz, kr = (2π/λ) sin θ, kz = (2π/λ) cos θ: plane waves on a cone of half-angle θ.
An ideal Bessel beam carries infinite power; a real one, from an axicon or annular slit, keeps its core only up to zmax. For an axicon of index n and base angle γ, θ ≈ (n − 1)γ.
λ is the wavelength in the medium; above θ ≈ 0.3 rad the scalar (paraxial-amplitude) picture is only indicative.
Radial intensity
- J₀ beam
- J₁ beam (vortex, l = 1)
Model and references
Textbook approximation. A standard approximation from the literature (for example paraxial, plane-wave or Gaussian-beam), valid within the range the page states. About the tiers
Ideal scalar Bessel beam from plane waves on a cone. The range z_max = R/tan θ is the geometric estimate for an input of radius R; the core intensity along z of a real axicon beam is not modelled.
- Durnin J., Miceli J. J., Eberly J. H. (1987). Diffraction-free beams. Phys. Rev. Lett. 58, 1499–1501. doi:10.1103/physrevlett.58.1499
- McGloin D., Dholakia K. (2005). Bessel beams: diffraction in a new light. Contemp. Phys. 46, 15–28. doi:10.1080/0010751042000275259