# 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] 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 either by focusing a Gaussian beam with an axicon lens to generate a Bessel–Gauss beam, by using axisymmetric diffraction gratings,[3] or by placing a narrow annular aperture in the far field.[4] High order Bessel beams can be generated by spiral diffraction gratings.[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 was achieved[8] with a Bessel beam that scatters[9][10][11] and produces a radiation force resulting from the exchange of acoustic momentum between the wave-field and a particle placed along its path.[12][13][14][15][16][17][18][19][20]

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.[21]

X-waves are special superpositions of Bessel beams which travel at constant velocity, and can exceed the speed of light.[22]

Mathieu beams and parabolic (Weber) beams[23] 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[24] and experimentally demonstrated[25] 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[26] and sinusoidal[27] trajectories as well as the early effort for beams with piecewise straight trajectories.[28]

## References

1. ^ Garcés-Chávez, V.; McGloin, D.; Melville, H.; Sibbett, W.; Dholakia, K. (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.
2. ^ McGloin, D.; Dholakia, K. (2005). "Bessel beams: diffraction in a new light". Contemporary Physics. 46: 15–28. Bibcode:2005ConPh..46...15M. doi:10.1080/0010751042000275259.
3. ^ Jiménez, N.; et al. (2014). "Acoustic Bessel-like beam formation by an axisymmetric grating". Europhysics Letters. 106 (2): 24005. arXiv:. Bibcode:2014EL....10624005J. doi:10.1209/0295-5075/106/24005.
4. ^ Durnin, J. (1987). "Diffraction-free beams". Physical Review Letters. 58 (15): 1499–1501. Bibcode:1987PhRvL..58.1499D. doi:10.1103/PhysRevLett.58.1499. PMID 10034453.
5. ^ Jiménez, N.; et al. (2016). "Formation of high-order acoustic Bessel beams by spiral diffraction gratings". Physical Review E. 94 (5): 053004. arXiv:. Bibcode:2016PhRvE..94e3004J. doi:10.1103/PhysRevE.94.053004.
6. ^ Fahrbach, F. O.; Simon, P.; Rohrbach, A. (2010). "Microscopy with self-reconstructing beams". Nature Photonics. 4 (11): 780–785. Bibcode:2010NaPho...4..780F. doi:10.1038/nphoton.2010.204.
7. ^ Mitri, F. G. (2011). "Arbitrary scattering of an electromagnetic zero-order Bessel beam by a dielectric sphere". Optics Letters. 36 (5): 766. Bibcode:2011OptL...36..766M. doi:10.1364/OL.36.000766.
8. ^ Hill, M. (2016). "Viewpoint: A One-Sided View of Acoustic Traps". Physics. 9 (3).
9. ^ Marston, P. L. (2007). "Scattering of a Bessel beam by a sphere". The Journal of the Acoustical Society of America. 121 (2): 753–758. Bibcode:2007ASAJ..121..753M. doi:10.1121/1.2404931.
10. ^ Silva, G. T. (2011). "Off-axis scattering of an ultrasound bessel beam by a sphere". IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control. 58 (2): 298–304. doi:10.1109/TUFFC.2011.1807.
11. ^ Mitri, F. G.; Silva, G. T. (2011). "Off-axial acoustic scattering of a high-order Bessel vortex beam by a rigid sphere". Wave Motion. 48 (5): 392–400. doi:10.1016/j.wavemoti.2011.02.001.
12. ^ Mitri, F. G. (2008). "Acoustic radiation force on a sphere in standing and quasi-standing zero-order Bessel beam tweezers". Annals of Physics. 323 (7): 1604–1620. Bibcode:2008AnPhy.323.1604M. doi:10.1016/j.aop.2008.01.011.
13. ^ Mitri, F. G.; Fellah, Z. E. A. (2008). "Theory of the acoustic radiation force exerted on a sphere by standing and quasistanding zero-order Bessel beam tweezers of variable half-cone angles". IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control. 55 (11): 2469–2478. doi:10.1109/TUFFC.954.
14. ^ Mitri, F. G. (2009). "Langevin acoustic radiation force of a high-order bessel beam on a rigid sphere". IEEE Transactions on Ultrasonics, Ferroelectrics and Frequency Control. 56 (5): 1059–1064. doi:10.1109/TUFFC.2009.1139.
15. ^ Mitri, F. G. (2009). "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". The European Physical Journal E. 28 (4): 469–478. Bibcode:2009EPJE...28..469M. doi:10.1140/epje/i2009-10449-y.
16. ^ Mitri, F. G. (2009). "Negative axial radiation force on a fluid and elastic spheres illuminated by a high-order Bessel beam of progressive waves". Journal of Physics A. 42 (24): 245202. Bibcode:2009JPhA...42x5202M. doi:10.1088/1751-8113/42/24/245202.
17. ^ Mitri, F. G. (2008). "Acoustic scattering of a high-order Bessel beam by an elastic sphere". Annals of Physics. 323 (11): 2840–2850. Bibcode:2008AnPhy.323.2840M. doi:10.1016/j.aop.2008.06.008.
18. ^ Mitri, F. G. (2009). "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. 56 (5): 1100–1103. doi:10.1109/TUFFC.2009.1143.
19. ^ Marston, P. L. (2006). "Axial radiation force of a Bessel beam on a sphere and direction reversal of the force". The Journal of the Acoustical Society of America. 120 (6): 3518–3524. Bibcode:2006ASAJ..120.3518M. doi:10.1121/1.2361185.
20. ^ Marston, P. L. (2009). "Radiation force of a helicoidal Bessel beam on a sphere". The Journal of the Acoustical Society of America. 125 (6): 3539–3547. Bibcode:2009ASAJ..125.3539M. doi:10.1121/1.3119625.
21. ^ Mitri, F. G. (2011). "Linear axial scattering of an acoustical high-order Bessel trigonometric beam by compressible soft fluid spheres". Journal of Applied Physics. 109 (1): 014916. Bibcode:2011JAP...109a4916M. doi:10.1063/1.3518496.
22. ^ Bowlan, P.; et al. (2009). "Measurement of the Spatiotemporal Electric Field of Ultrashort Superluminal Bessel-X Pulses". Optics and Photonics News. 20 (12): 42. doi:10.1364/OPN.20.12.000042.
23. ^ Bandres, M. A.; Gutiérrez-Vega, J. C.; Chávez-Cerda, S. (2004). "Parabolic nondiffracting optical wave fields". Optics Letters. 29 (1): 44. Bibcode:2004OptL...29...44B. doi:10.1364/OL.29.000044.
24. ^ Chremmos, I. D.; Chen, Z; Christodoulides, D. N.; Efremidis, N. K. (2012). "Bessel-like optical beams with arbitrary trajectories". Optics Letters. 37 (23): 5003–5. Bibcode:2012OptL...37.5003C. doi:10.1364/OL.37.005003. PMID 23202118.
25. ^ Juanying, Z.; et al. (2013). "Observation of self-accelerating Bessel-like optical beams along arbitrary trajectories". Optics Letters. 38 (4): 498–500. Bibcode:2013OptL...38..498Z. doi:10.1364/OL.38.000498. PMID 23455115.
26. ^ Jarutis, V.; Matijošius, A.; DiTrapani, P.; Piskarskas, A. (2009). "Spiraling zero-order Bessel beam". Optics Letters. 34 (14): 2129. Bibcode:2009OptL...34.2129J. doi:10.1364/OL.34.002129. PMID 19823524.
27. ^ Morris, J. E.; Čižmár, T.; Dalgarno, H. I. C.; Marchington, R. F.; Gunn-Moore, F. J.; 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.
28. ^ Rosen, J.; Yariv, A. (1995). "Snake beam: a paraxial arbitrary focal line". Optics Letters. 20 (20): 2042. Bibcode:1995OptL...20.2042R. doi:10.1364/OL.20.002042. PMID 19862244.