Axicon

An axicon is an optical element with a conical surface, most often a plano-conical lens. Where a spherical lens bends rays by an amount that grows with their height, so that they meet at one point, an axicon bends every ray by the same angle toward the axis, like a prism rotated about the optical axis. Rays from each annulus of the input beam cross the axis at a different distance, and the light forms a line focus along the axis. Its transverse profile is close to a zeroth-order Bessel function, with a narrow central core that stays nearly constant in width over a length far exceeding the Rayleigh range of a Gaussian focus of the same size. Refractive axicons are commonly made of fused silica with base angles of about 0.5–20°; diffractive and reflective versions also exist.

Cone angle and Bessel core

For a plano-conical lens of index $n$ and base angle $\alpha$ (the angle between the cone surface and the flat face), a collimated beam entering the flat side leaves at an angle $\beta$ to the axis given by

$$\sin(\alpha + \beta) = n \sin\alpha ,$$

which for small angles is $\beta \approx (n-1)\alpha$. The rays converge on a cone of half-angle $\beta$, and their interference produces the transverse intensity

$$I(r) \propto J_0^2(k r \sin\beta), \qquad k = 2\pi/\lambda .$$

The first zero of $J_0$ is at 2.405, so the central core has a radius to its first dark ring of

$$r_0 = \frac{2.405}{k \sin\beta} ,$$

and a full width at half maximum of $2.253/(k\sin\beta)$. The line extends to a distance set by the input beam radius $w$:

$$z_\text{max} \approx \frac{w}{\tan\beta} .$$

Worked example. A fused-silica axicon ($n = 1.4570$ at 632.8 nm) with $\alpha = 1°$ deflects light by $\beta = 0.457°$. The core radius is $r_0 = 30.4$ µm and its FWHM 28.4 µm. With an input beam of radius $w = 2.5$ mm the focal line is about 313 mm long. A Gaussian beam focused to a waist of the same 30.4 µm radius stays within $\sqrt{2}$ of its waist size over only $2z_R = 9.2$ mm. Doubling the base angle to 2° halves both numbers: $r_0 = 15.2$ µm and $z_\text{max} = 157$ mm.

What the Bessel line does and does not provide

The long depth of focus is paid for in power distribution. Each ring of the Bessel profile carries roughly the same power as the core, so with many rings present the central lobe holds only a small fraction of the total, and the on-axis intensity at any plane is supplied by a thin annulus of the input beam. For a Gaussian input the on-axis intensity rises from the tip, peaks partway along the line, and falls toward $z_\text{max}$. The core is also "self-healing": an obstacle on the axis is filled in after a short distance because the light arrives from the side along the cone. The description as non-diffracting applies only within $z_\text{max}$; a true Bessel beam would need infinite power and aperture.

Rings and measurement

Placing a lens of focal length $f$ after the axicon transforms the cone into a thin ring of radius $f \tan\beta$ in the lens focal plane, 0.80 mm for the 1° example with $f = 100$ mm. This is used to generate annular illumination and to measure $\beta$. The focal line itself is characterized by scanning a camera along the axis and recording the core width and peak intensity versus distance; the core usually needs magnification, since it may span only a few pixels.

Applications

• Laser machining. Bessel-like foci of ultrashort pulses cut and drill transparent materials such as glass, because the long focal line modifies the full thickness in one pass.
• Optical coherence tomography and microscopy. Axicon illumination extends the lateral-resolution depth range in OCT and forms thin illumination lines in light-sheet microscopy.
• Ring beams. Axicon pairs convert a round beam into a collimated annulus, and axicon-lens combinations produce ring foci for trapping and material processing.

Pitfalls

The cone tip is never perfectly sharp. A rounded tip acts as a weak lens over the central part of the beam and produces on-axis intensity oscillations near the start of the line, so the first part of the focal line is often discarded or the tip region is blocked. A tilted or decentred axicon produces an asymmetric core, and because $\beta$ depends on the index, it changes with wavelength. Finally, diffraction at the edge of a hard-clipped input beam adds modulation along the line, so the input beam is best kept well inside the clear aperture.

Common questions

What is an axicon used for?

To create a long, narrow line focus or a ring: for laser processing of glass, extended-depth imaging, light-sheet microscopy, optical trapping and straightness references.

What is the difference between an axicon and a lens?

A lens deflects rays by an angle proportional to their height, bringing them to one focal point. An axicon deflects all rays by the same angle, so each radius focuses at a different distance and the light forms a line along the axis.

Is a Bessel beam really non-diffracting?

Only over the finite length $z_\text{max}$. Within it, the core width stays constant because the interfering plane waves all travel at the same angle to the axis; beyond it, the cone of light has passed the axis and the core disappears.

References: J. H. McLeod, J. Opt. Soc. Am. 44, 592 (1954); J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987); J. Durnin, J. J. Miceli, J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987); B. E. A. Saleh, M. C. Teich, Fundamentals of Photonics, 3rd ed. (Wiley, 2019).