Creating Airy Beams With

2D Airy beam, from physics.aps.org 
Bessel beam, from opticsinfobase.org 
Our results in this project so far very clearly illustrate the properties of a 1D Airy beam. In the future we hope to create Airy beams with mirrors rather than lenses, and to demonstrate the selfhealing property. We also hope to model how the the aberrations of lenses or mirrors are used to create Airy beams.
The existence of Airy beams was first predicted by Michael Berry and Nandor Balazs in 1979. Berry and Balazs described a wave packet (a localized "burst") that does not spread and curves as it propogates from its origin. This Airy packet is made up of particles and has a probability distribution described by an Airy function which evolves according to Schrodinger's equation. They gave a quantum mechanical description of this wave packet using semiclassical orbits. However Airy beams were only created experimentally for the first time almost 30 years later, when Siviloglou et al [2,3] recognized that an Airy beam is the Fourier transform of a normal Gaussian laser beam with a cubic phase modulation. They used a spatial light modulator to create the cubic phase variation, and a lens to create the Fourier transform. Their results led to considerable popular interest in Airy beams and related phenomena. The method of Papazoglou et al [4] doesn't require specialized equipment and can be used with very highpowered lasers that would damage an SLM. Also, unlike a custom phase plate specifically designed to create a cubic phase modulation, it is tunable (adjustable). Since Papazoglou et al published their method, others have successfully used lenses to create Airy beam, including Yalizay et al, who used parts of two cylindrical lenses bonded together to create a single cubic element and Pacebal et al, who used a single cylindrical lens to generate Airy beams.
I began learning about aberrations using Wikipedia, Hyperphysics, telescopeoptics.net, Principles of Optics by Born & Wolf, Basic Wavefront Aberration Theory for Optical Metrology, by James C. Wyant, and Useful Optics by Welford.
The image appears to have a "tail." This effect occurs when rays from an offaxis image enter the lens or a curved mirror at an angle. Path length, or phase, of the light is different depending on its positionon the optic. Thus the aberred wavefront is different from the original wavefront. The phase shift varies cubically with position in the optic (distance from its center).
Our setup is based on that of Papazoglou et al illustrated in the figure below[4]. A relatively wide, collimated beam of light is incident on one side of two adjacent lenses of equal and opposite focal length. The tilt in the first lens creates the desired cubic phase modulation (coma). The second lens corrects for other aberrations and roughly collimates the beam. We used a single pair of cylindrical lenses onedimensional in order to create w onedimensional Airy beam. Not show in the diagram, a subsequent Fourier transform lens brings the beam into the far field. The focal length of this lens sets the scale of the final Airy beam.
For all of our experiments our light source was a Cambridge Collimators LM635 fibercoupled diode laser, which produced a beam about 1.0 cm in diameter. The Airy beam images were observed and recorded with a Thorlabs DCC1545M CMOS camera, which was mounted on a rail. An ND 3.0 filter was mounted directly on the camera to attenuate the laser light to an appropriate level and suppress ambient light. The beam was about 1 cm in diameter. We aimed it on only one side of the lens in order to minimize certain aberrations.
At first, I used a 80mm cylindrical lens with a +100 mm cylindrical lens, since these were easy to find, and the difference in focal length made collimation painless. With this lens combination, however, the Airy beam was never quite right and spread out very quickly as it moved from the focal plane. Dr. Cohen suggested I either switch to two lenses of the same focal length or use a single lens as explained in this paper. I found a small, +80mm lense lying around on the optics table and used it. I have since generated much better Airy beams.
The difficulty with using two lenses of the same focal length is that one must closely space the lenses in order to collimate the light. In fact, if neither lens is tilted, then according to Gullstrand's equation there would have to be no space between the two. Tilting the negative cylindrical lens allows for spacing between the two lenses. Originally, with each lens in a clamp, I was unable to properly tilt and space the lenses properly. I had to remove the positive lens from the clamp and mount it on doublesided tape in order to allow for a greater range of motion. Camera Washing Out
Originally, I used a "naked" CMOS camera without a lens or filter. I set the pixel clock high
and the exposure time low, but the images I saw looked something like this:
At first I thought what I was seeing was the Airy beam. I was very confused when at the focal point the beam became a rectangular blob. Dr. Cohen pointed out that that the camera was actually getting overloaded with light. He suggested putting a neutral density filter over the camera. Currently, with an ND 3.0 the camera is no longer washing out and I am able to see the beam very clearly at the focal plane.
My final setup is shown in this picture. The lenses are 80 mm, +80 mm and +200 mm. The first two lenses are spaced approximately 2.5 cm apart with a tilt of 20 degrees for the negative lens. The separation varies based on the angle of the first lens. The camera records the beam at and near the focal length of the Fourier lens (20 cm).
This animation shows a sequence of 16 images recorded in 5 mm steps over an 8 cm distance centered on the image plane of the Fourier transform lens. The plots below show a slice through the image recorded at the focal plane and a plot of the square of the Airy function. The results do not exactly match the theoretical plot. This is partially due to ambient light, which prevents the intensity from nearing 0. 
On the right is a plot, created using ImageJ, of the intensity of the Airy beam at focus. The graph on the left is the square of an Airy function, the theoretical plot of an Airy function at focus.
The peak lobe of the generated Airy beam deflected parabolically in agreement with a theoretical trajectory. The slight discrepancy appears to be a linear component resulting from an angle between the path of the beam and the path of the camera.
The lobe width appears roughly constant within about 4 cm from the focal
plane. This shows the nonspreading property of the Airy beam. For
comparison, the divergence of a Gaussian beam with equal minimum width is
plotted as well.
The next steps we may take with this project include creating Airy beams using mirrors in place of the lenses, demonstrating the reconstructing property of the beam by blocking a portion of the peak lobe, and verifying that the curvature of the beam is not affected by objects opposite the side of the secondary lobes.
The authors show that waves which are described by the Airy function propagate along a parabolic path and do not spread.
The authors investigate the possibility of generating finite Airy beams.
Details the creation of optical Airy beams using an SLMgenerated phase mask and a Fourier transform lens.
Explains how Airy beams were generated with cylindrical lenses.
A useful reference describing current methods of generation and applications of Airy beams.
Authors used a single lens to generate an Airy beam.
Among other things, the authors derive the equation which can be used to calculate the acceleration of an Air beam based on the FWHM of the peak lobe at the focal plane.
Greenberger offers another explanation of the Airy packet using the equivalence of gravitational and inertial mass.
Parts of a positive and a negative cylindrical lens are cut and glued together to create a cubic element. The element is used to generate Airy beams.