Bessel Beam Generation
Introduction
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Bessel beams are a special type of light propagation that does not diffract. The light distribution of a Bessel beam maintains a tight focus with high irradiance over great distances. Bessel beams are also selfhealing, which means that the light pattern will regenerate after being partially obstructed. Such properties make this phenomenon useful for optical trapping and tweezing, highprecision drilling, and communication applications.
Description of a Bessel beam
Bessel beams are propagating light fields with a distribution described by a Bessel function of the first kind. The cross section of a Bessel beam consists of concentric rings. Each ring (including the central lobe) contains the exact same infinitesimal amount of energy.
Figure 1. Irradiance pattern along the crosssection of a Bessel beam. The Bessel function is shown in the upper right.
Higherorder Bessel beams are modified along the azimuthal direction according to the equation J_{l}(k_{r}r)exp(ilφ), where k_{r} determines ring spacing and l determines azimuthal phase variation. In the farfield, the Bessel beam takes the form of an annular ring.
Axicon generation of a Bessel beam
Practically speaking, the mathematically ideal Bessel beam is impossible to create: it contains an infinite amount of rings over an infinite extent. PseudoBessel beams, on the other hand, are confined to an aperture. The most straightforward way to create a Bessel beam is with an axicon (a coneshaped refractive material or reflective surface that transforms an incident plane wave into a selfinterfering cone of light). Selfinterference forms concentric fringes.
It is simple to create an axicon in FRED. Create a “circular cone” element primitive with Simple Glass material, Transmit coating, and Allow All raytrace control. Choose a base semiaperture of 1 mm and height of 0.1 mm. Next, create a Simplified Optical Source of the type Collimated Source (plane wave) and check the Coherent box to ensure selfinterference. Assign a wavelength of 500 nm to the source. Finally, place an absorbing surface with attached analysis surface 12 mm from the axicon.
Figure 2. Left: a collimated plane wave illumination passes through a glass axicon. Right: further along the axis, a detector is placed in a region of selfinterference where a pseudoBessel beam is generated.
To observe the Bessel beam, click Analysis > Coherent Scalar Wave Field. A graph of irradiance is shown. To see the light field, simply rightclick the graph and select Show Field Amplitude. Over a 0.2 mm diameter observation region, distinct Bessel rings are visible (Figure 3).


Figure 3. Distribution of irradiance (left) and light field (right) in selfinterfering region beyond an axicon illuminated by a plane wave with wavelength of 500 nm.
Conclusions
FRED is capable of modeling Bessel beam behavior in a practical way: by simulating the creation of pseudoBessel beams using coherent illumination and standard optical elements such as axicons, lenses, and annular slits.
References
[1] Dudly, A., Laverly, M., Padgett, M., and Forbes, A. “ Unraveling Bessel Beams.” Published 6/1/13. Accessed 11/17/15.
[2] " Bessel Beam" Wikipedia. January 18, 2016. Accessed January 21, 2016.
[3] Durnin, J. “Exact Solutions for Nondiffracting Beams. I. The Scalar Theory”, JOSA A 4, 651654 (1987).
[4] “' Tractor beam' is possible with lasers, say scientists”. Published 3/3/11. Accessed 1/21/16.
[5] Ridden, P. “ New microscope captures 3D movies of living cells.” Published 3/15/11. Accessed 1/21/16.