Law of total covariance

An Airy beam is a non-diffracting waveform which gives the appearance of curving as it travels.

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.

The Airy waveform was first theorized in 1979 by M. V. Berry and N. L. Balasz. They demonstrated a nonspreading Airy wave packet solution to the Schrödinger equation.

In 2007 researchers from the University of Central Florida 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.

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. Furthermore, it has been shown that any function on the real line can be mapped to an accelerating beam with a different transverse shape.

Recently accelerating "Airy like" beams have been demonstrated for other types of equations such as Helmholtz equation, Maxwell's equations and even for non-linear systems. Acceleration can also take place along a radial instead of a cartesian coordinate, which is the case of circular-Airy abruptly autofocusing waves and their extention to arbitrary (nonparabolic) caustics. Acceleration is possible even for non-homogeneous periodic systems. With careful engineering of the input waveform, light can be made to accelerate along arbitrary trajectories in media that possess discrete or continuous periodicity.

Mathematical description

The potential free Schrödinger equation:

$i\frac{\partial \mathrm{\Phi }}{\partial \xi }+\frac{1}{2}\frac{{\partial }^2\mathrm{\Phi }}{\partial s^2}=0$

Has the following Airy nondispersive solution:

$\mathrm{\Phi }(\xi ,s)=\mathrm{A}\mathrm{i}(s-(\xi /2{)}^2)\exp (i(s\xi /2)-i({\xi }^3/12))$

where

• $\mathrm{A}\mathrm{i}$ is the Airy function.
• $\mathrm{\Phi }$ is the electric field envelope
• $s=x/x_0$ represents a dimensionless traverse coordinate
• $x_0$ is an arbitrary traverse scale
• $\xi =z/k{x}_0^2$ is a normalized propagation distance
• $k=2\pi n/{\lambda }_0$

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.

Applications

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. 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. 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 or medical laser treatments.

See also

• Bessel beam

Notes and references

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• Wave analysis of Airy beams