# Bessel beam

Diagram of Axicon and resulting Bessel Beam
Cross-section of the Bessel beam and graph of intensity
Bessel beam re-forming central bright area after obstruction

A Bessel beam is a field of electromagnetic, acoustic or even gravitational radiation whose amplitude is described by a Bessel function of the first kind.[1][2][3] A true Bessel beam is non-diffractive. This means that as it propagates, it does not diffract and spread out; this is in contrast to the usual behavior of light (or sound), which spreads out after being focussed down to a small spot. Bessel beams are also self-healing, meaning that the beam can be partially obstructed at one point, but will re-form at a point further down the beam axis.

As with a plane wave a true Bessel beam cannot be created, as it is unbounded and would require an infinite amount of energy. Reasonably good approximations can be made, however, and these are important in many optical applications because they exhibit little or no diffraction over a limited distance. Approximations to Bessel beams are made in practice by focusing a Gaussian beam with an axicon lens to generate a Bessel-Gauss beam, by axisymmetric diffraction gratings,[4] or by placing a narrow annular aperture in the far field.[5]

## Properties

The properties of Bessel beams[6][7] make them extremely useful for optical tweezing, as a narrow Bessel beam will maintain its required property of tight focus over a relatively long section of beam and even when partially occluded by the dielectric particles being tweezed. Similarly, particle manipulation with acoustical tweezers may be feasible with a Bessel beam that scatters[8][9][10] and produces a radiation force resulting from the exchange of acoustic momentum between the wave-field and a particle placed along its path.[11][12][13][14][15][16][17][18][19]

The mathematical function which describes a Bessel beam is a solution of Bessel's differential equation, which itself arises from separable solutions to Laplace's equation and the Helmholtz equation in cylindrical coordinates. The fundamental zero-order Bessel beam has an amplitude maximum at the origin, while a high-order Bessel beam (HOBB) has an axial phase singularity along the beam axis; the amplitude is zero there. HOBBs can be of vortex (helicoidal) or non-vortex types.[20]

X-waves are special superpositions of Bessel beams which travel at constant velocity.

Mathieu beams and parabolic (Weber) beams [21] are other types of non-diffractive beams that have the same non-diffractive and self-healing properties of Bessel beams but different transverse structures.

### Acceleration

In 2012 it was theoretically proved [22] and experimentally demonstrated [23] that, with a special manipulation of their initial phase, Bessel beams can be made to accelerate along arbitrary trajectories in free space. These beams can be considered as hybrids that combine the symmetric profile of a standard Bessel beam with the self-acceleration property of the Airy beam and its counterparts. Previous efforts to produce accelerating Bessel beams included beams with helical [24] and sinusoidal [25] trajectories as well as the early effort for beams with piecewise straight trajectories.[26]

## References

1. ^ Kishan Dholakia; David McGloin; Vene Garcés-Chávez (2002). "Optical micromanipulating using a self-reconstructing light beam". Retrieved 2007-02-06.
2. ^ V. Garcés-Chávez; D. McGloin, H. Melville, W. Sibbett and K. Dholakia (2002). "Simultaneous micromanipulation in multiple planes using a self-reconstructing light beam". Nature 419 (6903): 145–7. Bibcode:2002Natur.419..145G. doi:10.1038/nature01007. PMID 12226659.
3. ^ D. McGloin, K. Dholakia, Bessel beams: diffraction in a new light, Contemporary Physics 46 (2005) 15-28
4. ^ Jiménez, N.; V. Romero-García, R. Picó, A. Cebrecos, V. J. Sánchez-Morcillo, L. M. Garcia-Raffi, J. V. Sánchez-Pérez and K. Staliunas. (2 April 2014). "Acoustic Bessel-like beam formation by an axisymmetric grating". Europhysics Letters 106 (2): 24005. doi:10.1209/0295-5075/106/24005.
5. ^ Durnin, J.; J. J. Miceli, Jr. and J. H. Eberly (13 April 1987). "Diffraction-free beams". Physical Review Letters 58 (15): 1499–1501. Bibcode:1987PhRvL..58.1499D. doi:10.1103/PhysRevLett.58.1499.
6. ^ F. O. Fahrbach, P. Simon, A. Rohrbach, Microscopy with self-reconstructing beams, Nature Photonics 4 (2010) 780–785
7. ^ F. G. Mitri, Arbitrary scattering of an electromagnetic zero-order Bessel beam by a dielectric sphere. Optics Letters 36 (2011) 766-768
8. ^ P. L. Marston, Scattering of a Bessel beam by a sphere, J. Acoust. Soc. Am. 121 (2007) 753-758
9. ^ G. T. Silva, Off-axis scattering of an ultrasound bessel beam by a sphere. IEEE Trans. Ultrason. Ferroelectr. Freq. Control. 58 (2011) 298-304
10. ^ F. G. Mitri, G. T. Silva, Off-axial acoustic scattering of a high-order Bessel vortex beam by a rigid sphere, Wave Motion 48 (2011) 392-400
11. ^ F. G. Mitri, Acoustic radiation force on a sphere in standing and quasi-standing zero-order Bessel beam tweezers, Annals of Physics 323 (2008) 1604-1620
12. ^ F. G. Mitri, Z. E. A. Fellah, Theory of the acoustic radiation force exerted on a sphere by a standing and quasi-standing zero-order Bessel beam tweezers of variable half-cone angles, IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control 55 (2008) 2469-2478
13. ^ F. G. Mitri, Langevin acoustic radiation force of a high-order Bessel beam on a rigid sphere, IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control 56 (2009) 1059-1064
14. ^ F. G. Mitri, Acoustic radiation force on an air bubble and soft fluid spheres in ideal liquids: Example of a high-order Bessel beam of quasi-standing waves, European Physical Journal E 28 (2009) 469-478
15. ^ F. G. Mitri, Negative Axial Radiation Force on a Fluid and Elastic Spheres Illuminated by a High-Order Bessel Beam of Progressive Waves, Journal of Physics A - Mathematical and Theoretical 42 (2009) 245202
16. ^ F. G. Mitri, Acoustic scattering of a high-order Bessel beam by an elastic sphere, Annals of Physics 323 (2008) 2840-2850
17. ^ F. G. Mitri, Equivalence of expressions for the acoustic scattering of a progressive high-order Bessel beam by an elastic sphere, IEEE Transactions on Ultrasonics Ferroelectrics and Frequency Control 23 56 (2009) 1100-1103
18. ^ P. L. Marston, Axial radiation force of a bessel beam on a sphere and direction reversal of the force, J. Acoust. Soc. Am. 120 (2006) 3518-3524
19. ^ P. L. Marston, Radiation force of a helicoidal Bessel beam on a sphere, J. Acoust. Soc. Am. 125 (2009) 3539-3547
20. ^ F. G. Mitri, Linear axial scattering of an acoustical high-order Bessel trigonometric beam by compressible soft fluid spheres, J. Appl. Phys. 109 (2011) 014916
21. ^ M.A. Bandres, J.C. Gutiérrez-Vega, and S. Chávez-Cerda, "Parabolic nondiffracting optical wave fields," Opt. Lett. 29, 44-46 (2004).
22. ^ Chremmos, Ioannis; Zhigang Chen; Christodoulides; Demetrios, Efremidis, Nikolaos (2012). "Bessel-like optical beams with arbitrary trajectories". Optics Letters 37 (23): 5003. Bibcode:2012OptL...37.5003C. doi:10.1364/OL.37.005003.
23. ^ Juanying, Zhao; Zhang, Peng; Deng; Dongmei, Liu, Jingjiao; Gao, Yuanmei; Chremmos, Ioannis; Efremidis, Nikolaos; Christodoulides, Demetrios; Chen, Zhigang (2013). "Observation of self-accelerating Bessel-like optical beams along arbitrary trajectories". Optics Letters 38 (4): 498. Bibcode:2013OptL...38..498Z. doi:10.1364/OL.38.000498.
24. ^ Jarutis, Vygandas; Matijošius, Aidas; DiTrapani, Paolo; Piskarskas, Algis (2009). "Spiraling zero-order Bessel beam". Optics Letters 34 (14): 2129. Bibcode:2009OptL...34.2129J. doi:10.1364/OL.34.002129.
25. ^ Morris, JE; Čižmár, T; Dalgarno, HIC; Marchington, RF; Gunn-Moore, FJ; Dholakia, K (2010). "Realization of curved Bessel beams: propagation around obstructions". Journal of Optics 12 (12): 124002. Bibcode:2010JOpt...12l4002M. doi:10.1088/2040-8978/12/12/124002.
26. ^ Rosen, Joseph; Yariv, Amnons (1995). "Snake beam: a paraxial arbitrary focal line". Optics Letters 20 (20): 2042. Bibcode:1995OptL...20.2042R. doi:10.1364/OL.20.002042.