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An Airy beam, is a propagation invariant wave whose main intensity lobe propagates along a curved parabolic trajectory while being resilient to perturbations (self-healing).

## Physical description

A cross section of an ideal Airy beam would reveal an area of principal intensity, with a series of adjacent, less luminous areas trailing off to infinity. In reality, the beam is truncated so as to have a finite composition.

As the beam propagates, it does not diffract, i.e., does not spread out. The Airy beam also has the characteristic of freely accelerating. As it propagates, it bends so as to form a parabolic arc.

## History

The term "Airy beam" derives from the Airy integral, developed in the 1830s by Sir George Biddell Airy to explain optical caustics such as those appearing in a rainbow.^{[1]}

The Airy waveform was first theorized in 1979 by M. V. Berry and Nándor L. Balázs. They demonstrated a nonspreading Airy wave packet solution to the Schrödinger equation.^{[2]}

In 2007 researchers from the University of Central Florida (United States) were able to create and observe an Airy beam for the first time in both one- and two-dimensional configurations. The members of the team were Georgios Siviloglou, John Broky, Aristide Dogariu, and Demetrios Christodoulides.^{[3]}

In one-dimension, the Airy beam is the only exactly shape-preserving accelerating solution to the free-particle Schrödinger equation (or 2D paraxial wave equation). However, in two dimensions (or 3D paraxial systems), two separable solutions are possible: two-dimensional Airy beams and accelerating parabolic beams.^{[4]} Furthermore, it has been shown^{[5]} that any function on the real line can be mapped to an accelerating beam with a different transverse shape.

In 2009 accelerating "Airy like" beams have been observed for the first time in material, notably a system with optical nonlinear behaviour, by a joint team of Pavia University and L'Aquila University (Italy); the members of the team were Jacopo Parravicini, Paolo Minzioni, (from Pavia), and Eugenio DelRe (from L'Aquila).^{[6]} Subsequently, this kind of beams has been investigated in 2011 and 2012 mainly by the teams of University of Central Florida.^{[7]}^{[8]}^{[9]} Later, Airy beams have been demonstrated for other types of equations such as Helmholtz equation, Maxwell's equations.^{[10]}^{[11]} Acceleration can also take place along a radial instead of a cartesian coordinate, which is the case of circular-Airy abruptly autofocusing waves^{[12]} and their extension to arbitrary (nonparabolic) caustics.^{[13]} Acceleration is possible even for non-homogeneous periodic systems.^{[14]}^{[15]} With careful engineering of the input waveform, light can be made to accelerate along arbitrary trajectories in media that possess discrete^{[16]} or continuous^{[17]} periodicity.
In 2018, researchers from Tel-Aviv university measured the cubic phase of Airy beams in an analogous system of surface gravity water-waves. They also managed to accelerate the Airy beam analog, using an external hydrodynamic linear potential and stop the self-accelerating front of the Airy beam. The members of the team associated with the experiment were Georgi Gary Rozenman, Ady Arie and Lev Shemer.
^{[18]}

## Mathematical description

The potential free Schrödinger equation:

Has the following Airy accelerating solution:^{[19]}

where

- is the Airy function.
- is the electric field envelope
- represents a dimensionless traverse coordinate
- is an arbitrary traverse scale
- is a normalized propagation distance

This solution is non-diffracting in a parabolic accelerating frame. Actually one can perform a coordinate transformation and get an Airy equation. In the new coordinates the equation is solved by the Airy function.

## Experimental observation

Georgios Sivilioglou, et al. successfully fabricated an Airy beam in 2007. A beam with a Gaussian distribution was modulated by a spatial light modulator to have an Airy distribution. The result was recorded by a CCD camera.^{[1]}^{[3]}

## Modified Airy beams

### Attenuation-compensation

Beams may encounter losses as they travel through materials which will cause attenuation of the beam intensity. A property common to non-diffracting (or propagation-invariant) beams, such as the Airy beam and Bessel beam, is the ability to control the longitudinal intensity envelope of the beam without significantly altering the other characteristics of the beam. This can be used to create Airy beams which grow in intensity at they travel and can be used to counteract losses, therefore maintaining a beam of constant intensity as it propagates.^{[20]}^{[21]}^{[22]} In temporal domain, an analogous modified dispersion-free attenuation-compensating Airy-based ("rocket") pulse was previously proposed and demonstrated in,^{[23]} designed to compensate media losses as it propagates through dispersive media.

## Applications

### Optical trapping and manipulation

Researchers at the University of St. Andrews have used Airy beams to manipulate small particles, moving them along curves and around corners. This may find use in fields such as microfluidic engineering and cell biology.^{[24]}
Significant theoretical works have been also undertaken by F.G. Mitri and his collaborators both in optics and acoustics, and related works can be found in these references: Airy acoustical–sheet spinner tweezers; Acoustics of finite asymmetric exotic beams: Examples of Airy and fractional Bessel beams; Pulling and spinning reversal of a sub-wavelength absorptive sphere in adjustable vector Airy light-sheets; Adjustable vector Airy light-sheet single optical tweezers: negative radiation forces on a subwavelength spheroid and spin torque reversal; Optical radiation force on a dielectric sphere of arbitrary size illuminated by a linearly polarized Airy light-sheet; Optical torque on an absorptive dielectric sphere of arbitrary size illuminated by a linearly-polarized Airy light-sheet; Circularly-polarized Airy light-sheet spinner tweezers and particle transport

(see also: Optical tweezers)

### Imaging and microscopy

Researchers at the University of St. Andrews have further utilised Airy beams to make a large field of view (FOV) while maintaining high axial contrast in a light-sheet microscope.^{[25]}^{[26]} This technique has been adapted to use multi-photon excitation^{[27]} and attenuation-compensated Airy beams^{[28]}^{[29]} to achieve imaging at greater depths within biological specimens.

### Micro-machining

The accelerating and diffraction-free features of the Airy wavepacket have also been utilized by researchers at the University of Crete to produce two-dimensional, circular-Airy waves, termed abruptly-autofocusing beams.^{[12]} These beams tend to focus in an abrupt fashion shortly before a target while maintaining a constant and low intensity profile along the propagated path and can be useful in laser microfabrication^{[30]} or medical laser treatments.

## See also

## Notes and references

- ^
^{a}^{b}"Scientists make first observation of Airy optical beams" **^**Berry, M. V.; Balázs, Nándor L. (1979). "Nonspreading wave packets".*American Journal of Physics*.**47**(3): 264–267. Bibcode:1979AmJPh..47..264B. doi:10.1119/1.11855.- ^
^{a}^{b}Siviloglou, G. A.; Broky, J.; Dogariu, A.; Christodoulides, D. N. (2007). "Observation of Accelerating Airy Beams".*Phys. Rev. Lett*.**99**(21): 213901. Bibcode:2007PhRvL..99u3901S. doi:10.1103/PhysRevLett.99.213901. PMID 18233219. **^**Bandres, M.A. (2008). "Accelerating parabolic beams" (PDF).*Opt. Lett*.**33**(15): 1678–1680. Bibcode:2008OptL...33.1678B. doi:10.1364/OL.33.001678. PMID 18670501.**^**Bandres, M. A. (2009). "Accelerating beams".*Opt. Lett*.**34**(24): 3791–3793. Bibcode:2009OptL...34.3791B. doi:10.1364/OL.34.003791. PMID 20016615.**^**Parravicini, Jacopo; Minzioni, Paolo; Degiorgio, Vittorio; DelRe, Eugenio (15 December 2009). "Observation of nonlinear Airy-like beam evolution in lithium niobate".*Optics Letters*.**34**(24): 3908–10. Bibcode:2009OptL...34.3908P. doi:10.1364/OL.34.003908. PMID 20016654.**^**Kaminer, Ido; Segev, Mordechai; Christodoulides, Demetrios N. (30 April 2011). "Self-Accelerating Self-Trapped Optical Beams" (PDF).*Physical Review Letters*.**106**(21): 213903. Bibcode:2011PhRvL.106u3903K. doi:10.1103/PhysRevLett.106.213903. PMID 21699299.**^**Kaminer, Ido; Nemirovsky, Jonathan; Segev, Mordechai (1 August 2012). "Self-accelerating self-trapped nonlinear beams of Maxwell's equations" (PDF).*Optics Express*.**20**(17): 18827–35. Bibcode:2012OExpr..2018827K. doi:10.1364/OE.20.018827. PMID 23038522.**^**Bekenstein, Rivka; Segev, Mordechai (7 November 2011). "Self-accelerating optical beams in highly nonlocal nonlinear media" (PDF).*Optics Express*.**19**(24): 23706–15. Bibcode:2011OExpr..1923706B. doi:10.1364/OE.19.023706. PMID 22109397.**^**Kaminer, Ido; Bekenstein, Rivka; Nemirovsky, Jonathan; Segev, Mordechai (2012). "Nondiffracting accelerating wave packets of Maxwell's equations" (PDF).*Physical Review Letters*.**108**(16): 163901. arXiv:1201.0300. Bibcode:2012PhRvL.108p3901K. doi:10.1103/PhysRevLett.108.163901. PMID 22680719.**^**Courvoisier, F.; Mathis, A.; Froehly, L.; Giust, R.; Furfaro, L.; Lacourt, P. A.; Jacquot, M.; Dudley, J. M. (15 May 2012). "Sending femtosecond pulses in circles: highly nonparaxial accelerating beams".*Optics Letters*.**37**(10): 1736–8. arXiv:1202.3318. Bibcode:2012OptL...37.1736C. doi:10.1364/OL.37.001736. PMID 22627554.- ^
^{a}^{b}Efremidis, Nikolaos; Christodoulides, Demetrios (2010). "Abruptly autofocusing waves" (PDF).*Optics Letters*.**35**(23): 4045–7. Bibcode:2010OptL...35.4045E. doi:10.1364/OL.35.004045. PMID 21124607. **^**Chremmos, Ioannis; Efremidis, Nikolaos; Christodoulides, Demetrios (2011). "Pre-engineered abruptly autofocusing beams".*Optics Letters*.**36**(10): 1890–2. Bibcode:2011OptL...36.1890C. CiteSeerX 10.1.1.714.588. doi:10.1364/OL.36.001890. PMID 21593925.**^**El-Ganainy, Ramy; Makris, Konstantinos G.; Miri, Mohammad Ali; Christodoulides, Demetrios N.; Chen, Zhigang (31 July 2011). "Discrete beam acceleration in uniform waveguide arrays".*Physical Review A*.**84**(2): 023842. Bibcode:2011PhRvA..84b3842E. doi:10.1103/PhysRevA.84.023842.**^**Kaminer, Ido; Nemirovsky, Jonathan; Makris, Konstantinos G.; Segev, Mordechai (3 April 2013). "Self-accelerating beams in photonic crystals" (PDF).*Optics Express*.**21**(7): 8886–96. Bibcode:2013OExpr..21.8886K. doi:10.1364/OE.21.008886. PMID 23571979. Archived from the original (PDF) on 2013-10-16. Retrieved 2013-10-11.**^**Efremidis, Nikolaos; Chremmos, Ioannis (2012). "Caustic design in periodic lattices".*Optics Letters*.**37**(7): 1277–9. Bibcode:2012OptL...37.1277E. CiteSeerX 10.1.1.713.7055. doi:10.1364/OL.37.001277. PMID 22466220.**^**Chremmos, Ioannis; Efremidis, Nikolaos (2012). "Band-specific phase engineering for curving and focusing light in waveguide arrays" (PDF).*Physical Review A*.**85**(63830): 063830. Bibcode:2012PhRvA..85f3830C. doi:10.1103/PhysRevA.85.063830.**^**G. G. Rozenman, A. Arie; W. P. Schleich, L. Shemer, M. Zimmerman, M. A. Efremov (2019). "Amplitude and Phase of Wave Packets in a Linear Potential".*Physical Review Letters*.**122**(12): 124302. doi:10.1103/PhysRevLett.122.124302. PMID 30978087. S2CID 111389900.CS1 maint: multiple names: authors list (link)**^**"Observation of Accelerating Airy Beams"**^**Schley, Ran; Kaminer, Ido; Greenfield, Elad; Bekenstein, Rivka; Lumer, Yaakov; Segev, Mordechai (2014). "Loss-proof self-accelerating beams and their use in non-paraxial manipulation of particles' trajectories".*Nature Communications*.**5**: 5189. Bibcode:2014NatCo...5.5189S. doi:10.1038/ncomms6189. PMID 25355605.**^**Preciado, Miguel A.; Dholakia, Kishan; Mazilu, Michael (2014-08-15). "Generation of attenuation-compensating Airy beams".*Optics Letters*.**39**(16): 4950–4953. Bibcode:2014OptL...39.4950P. doi:10.1364/ol.39.004950. hdl:10023/7244. PMID 25121916.**^**"GitHub Matlab/Octave code: Compensating Airy beam for diffractive light delivery control".**^**Preciado, Miguel A.; Sugden, Miguel (2012-12-01). "Proposal and design of Airy-based rocket pulses for invariant propagation in lossy dispersive media" (PDF).*Optics Letters*.**37**(23): 4970–4972. Bibcode:2012OptL...37.4970P. doi:10.1364/OL.37.004970. PMID 23202107.**^**"Light throws a curve ball"**^**Vettenburg, Tom; Dalgarno, Heather I C; Nylk, Jonathan; Coll-Lladó, Clara; Ferrier, David E K; Čižmár, Tomáš; Gunn-Moore, Frank J; Dholakia, Kishan (2014). "Light-sheet microscopy using an Airy beam" (PDF).*Nature Methods*.**11**(5): 541–544. doi:10.1038/nmeth.2922. hdl:10023/5521. PMID 24705473.**^**"Imaging turns a corner". Archived from the original on 2014-04-26. Retrieved 2014-04-26.**^**Piksarv, Peeter; Marti, Dominik; Le, Tuan; Unterhuber, Angelika; Forbes, Lindsay H.; Andrews, Melissa R. Andrews; Stingl, Andreas; Drexler, Wolfgang; Andersen, Peter E. (2017). "Integrated single- and two-photon light sheet microscopy using accelerating beams".*Scientific Reports*.**7**(1): 1435. Bibcode:2017NatSR...7.1435P. doi:10.1038/s41598-017-01543-4. PMC 5431168. PMID 28469191.**^**Nylk, Jonathan; McCluskey, Kaley; Preciado, Miguel A.; Mazilu, Michael; Yang, Zhengyi; Gunn-Moore, Frank J.; Aggarwal, Sanya; Tello, Javier A.; Ferrier, David E. K. (2018-04-01). "Light-sheet microscopy with attenuation-compensated propagation-invariant beams".*Science Advances*.**4**(4): eaar4817. arXiv:1708.02612. Bibcode:2018SciA....4R4817N. doi:10.1126/sciadv.aar4817. PMC 5938225. PMID 29740614.**^**Veettikazhy, Madhu; Nylk, Jonathan; Gasparoli, Federico; Escobet-Montalbán, Adrià; Hansen, Anders Kragh; Marti, Dominik; Andersen, Peter Eskil; Dholakia, Kishan (2020-05-15). "Multi-photon attenuation-compensated light-sheet fluorescence microscopy".*Scientific Reports*.**10**(1): 1–10. doi:10.1038/s41598-020-64891-8. ISSN 2045-2322.**^**Papazoglou, Dimitrios; Efremidis, Nikolaos; Christodoulides, Demetrios; Tzortzakis, Stelios (2011). "Observation of abruptly autofocusing waves".*Optics Letters*.**36**(10): 1842–4. Bibcode:2011OptL...36.1842P. doi:10.1364/OL.36.001842. PMID 21593909. S2CID 9384164.