Airy Beams and Gaussian Beams
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Airy Beams and Gaussian Beams
Was thinking of Miles' recently released papers and remembered this "Airy Beams" read.... thought to post it here:

An Airy beam is a nondiffracting 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.[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 twodimensional configurations. The members of the team were Georgios Siviloglou, John Broky, Aristide Dogariu, and Demetrios Christodoulides.[3]
In onedimension, the Airy beam is the only exactly shapepreserving accelerating solution to the freeparticle Schrödinger equation (or 2D paraxial wave equation). However, in two dimensions (or 3D paraxial systems), two separable solutions are possible: twodimensional 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 nonlinear systems by a joint team of Pavia University and L'Aquila University (Italy)[6] and again they have 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 circularAiry abruptly autofocusing waves [12] and their extension to arbitrary (nonparabolic) caustics.[13] Acceleration is possible even for nonhomogeneous 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.
https://en.wikipedia.org/wiki/Airy_beam
Gaussian Beam
https://en.wikipedia.org/wiki/Gaussian_beam
Beam parameters
The geometric dependence of the fields of a Gaussian beam are governed by the light's wavelength λ (in the dielectric medium, if not free space) and the following beam parameters, all of which are connected as detailed in the following sections.
Transverse mode of electromagnetic radiation
https://en.wikipedia.org/wiki/Transverse_mode
Bessel Beam
https://www.phys.ksu.edu/reu2014/joshuanelson/BesselBeam1.pdf
https://en.wikipedia.org/wiki/Bessel_beam
A Bessel beam is a wave whose amplitude is described by a Bessel function of the first kind.[1][2][3]. Electromagnetic, acoustic, gravitational, and matter waves can all be in the form of Bessel beams. A true Bessel beam is nondiffractive. 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 focused down to a small spot. Bessel beams are also selfhealing, meaning that the beam can be partially obstructed at one point, but will reform at a point further down the beam axis.
Top Hat Beam
https://en.wikipedia.org/wiki/Tophat_beam
In optics, a tophat (or tophat) beam such as a laser beam or electron beam has a nearuniform fluence (energy density) within a circular disk. It is typically formed by diffractive optical elements from a Gaussian beam. Tophat beams are often used in industry, for example for laser drilling of holes in printed circuit boards.
Airy beam

An Airy beam is a nondiffracting 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.[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 twodimensional configurations. The members of the team were Georgios Siviloglou, John Broky, Aristide Dogariu, and Demetrios Christodoulides.[3]
In onedimension, the Airy beam is the only exactly shapepreserving accelerating solution to the freeparticle Schrödinger equation (or 2D paraxial wave equation). However, in two dimensions (or 3D paraxial systems), two separable solutions are possible: twodimensional 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 nonlinear systems by a joint team of Pavia University and L'Aquila University (Italy)[6] and again they have 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 circularAiry abruptly autofocusing waves [12] and their extension to arbitrary (nonparabolic) caustics.[13] Acceleration is possible even for nonhomogeneous 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.
https://en.wikipedia.org/wiki/Airy_beam
Gaussian Beam
https://en.wikipedia.org/wiki/Gaussian_beam
Beam parameters
The geometric dependence of the fields of a Gaussian beam are governed by the light's wavelength λ (in the dielectric medium, if not free space) and the following beam parameters, all of which are connected as detailed in the following sections.
Transverse mode of electromagnetic radiation
https://en.wikipedia.org/wiki/Transverse_mode
Bessel Beam
https://www.phys.ksu.edu/reu2014/joshuanelson/BesselBeam1.pdf
https://en.wikipedia.org/wiki/Bessel_beam
A Bessel beam is a wave whose amplitude is described by a Bessel function of the first kind.[1][2][3]. Electromagnetic, acoustic, gravitational, and matter waves can all be in the form of Bessel beams. A true Bessel beam is nondiffractive. 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 focused down to a small spot. Bessel beams are also selfhealing, meaning that the beam can be partially obstructed at one point, but will reform at a point further down the beam axis.
Top Hat Beam
https://en.wikipedia.org/wiki/Tophat_beam
In optics, a tophat (or tophat) beam such as a laser beam or electron beam has a nearuniform fluence (energy density) within a circular disk. It is typically formed by diffractive optical elements from a Gaussian beam. Tophat beams are often used in industry, for example for laser drilling of holes in printed circuit boards.
Airy beam
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