A Bessel beam is a wave whose amplitude is described by a

Bessel function of the first kind Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...

Electromagnetic In physics, electromagnetism is an interaction that occurs between particles with electric charge. It is the second-strongest of the four fundamental interactions, after the strong force, and it is the dominant force in the interactions of a ...

, acoustic,

gravitational In physics, gravity () is a fundamental interaction which causes mutual attraction between all things with mass or energy. Gravity is, by far, the weakest of the four fundamental interactions, approximately 1038 times weaker than the strong ...

, and

matter In classical physics and general chemistry, matter is any substance that has mass and takes up space by having volume. All everyday objects that can be touched are ultimately composed of atoms, which are made up of interacting subatomic partic ...

waves can all be in the form of Bessel beams. A true Bessel beam is non-diffractive. This means that as it propagates, it does not

diffract Diffraction is defined as the interference or bending of waves around the corners of an obstacle or through an aperture into the region of geometrical shadow of the obstacle/aperture. The diffracting object or aperture effectively becomes a s ...

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 ''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 An optical axis is a line along which there is some degree of rotational symmetry in an optical system such as a camera lens, microscope or telescopic sight. The optical axis is an imaginary line that defines the path along which light propagat ...

. As with a

plane wave In physics, a plane wave is a special case of wave or field: a physical quantity whose value, at any moment, is constant through any plane that is perpendicular to a fixed direction in space. For any position \vec x in space and any time t, th ...

, a true Bessel beam cannot be created, as it is unbounded and would require an infinite amount of

energy In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...

. Reasonably good approximations can be made, however, and these are important in many

optical Optics is the branch of physics that studies the behaviour and properties of light, including its interactions with matter and the construction of instruments that use or detect it. Optics usually describes the behaviour of visible, ultraviole ...

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 In optics, a Gaussian beam is a beam of electromagnetic radiation with high monochromaticity whose amplitude envelope in the transverse plane is given by a Gaussian function; this also implies a Gaussian intensity (irradiance) profile. This ...

with an

axicon An axicon is a specialized type of lens which has a conical surface. An axicon transforms a laser beam into a ring shaped distribution. They can be convex or concave and be made of any optical material. The combination with other axicons or lenses ...

lens to generate a Bessel–Gauss beam, by using axisymmetric

diffraction grating In optics, a diffraction grating is an optical component with a periodic structure that diffracts light into several beams travelling in different directions (i.e., different diffraction angles). The emerging coloration is a form of structura ...

s, or by placing a narrow

annular Annulus (or anulus) or annular indicates a ring- or donut-shaped area or structure. It may refer to: Human anatomy * ''Anulus fibrosus disci intervertebralis'', spinal structure * Annulus of Zinn, a.k.a. annular tendon or ''anulus tendineus com ...

aperture In optics, an aperture is a hole or an opening through which light travels. More specifically, the aperture and focal length of an optical system determine the cone angle of a bundle of rays that come to a focus in the image plane. An opt ...

in the

far field The near field and far field are regions of the electromagnetic (EM) field around an object, such as a transmitting antenna, or the result of radiation scattering off an object. Non-radiative ''near-field'' behaviors dominate close to the ant ...

. High order Bessel beams can be generated by

spiral diffraction grating In mathematics, a spiral is a curve which emanates from a point, moving farther away as it revolves around the point. Helices Two major definitions of "spiral" in the American Heritage Dictionary are: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 with a Bessel beam that scatters and produces a radiation force resulting from the exchange of acoustic momentum between the wave-field and a particle placed along its path. The

mathematical Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

function which describes a Bessel beam is a solution of

Bessel's differential equation Bessel functions, first defined by the mathematician Daniel Bernoulli and then generalized by Friedrich Bessel, are canonical solutions of Bessel's differential equation x^2 \frac + x \frac + \left(x^2 - \alpha^2 \right)y = 0 for an arbitrary ...

, which itself arises from separable solutions to

Laplace's equation In mathematics and physics, Laplace's equation is a second-order partial differential equation named after Pierre-Simon Laplace, who first studied its properties. This is often written as \nabla^2\! f = 0 or \Delta f = 0, where \Delta = \nab ...

and the

Helmholtz equation In mathematics, the eigenvalue problem for the Laplace operator is known as the Helmholtz equation. It corresponds to the linear partial differential equation \nabla^2 f = -k^2 f, where is the Laplace operator (or "Laplacian"), is the eigenv ...

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. X-waves are special superpositions of Bessel beams which travel at constant

velocity Velocity is the directional speed of an object in motion as an indication of its rate of change in position as observed from a particular frame of reference and as measured by a particular standard of time (e.g. northbound). Velocity is a ...

, and can exceed the speed of light. Mathieu beams and parabolic (Weber) beams 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 proven and experimentally demonstrated 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 and sinusoidal trajectories as well as the early effort for beams with piecewise straight trajectories.

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 Bessel beams which grow in intensity as they travel and can be used to counteract losses, therefore maintaining a beam of constant intensity as it propagates.

Applications

Imaging and microscopy

In light-sheet fluorescence microscopy, non-diffracting (or propagation-invariant) beams have been utilised to produce very long and uniform light-sheets which do not change size significantly across their length. The self-healing property of Bessel beams has also shown to give improved image quality at depth as the beam shape is less distorted after travelling through scattering tissue than a Gaussian beam. Bessel beam based light-sheet microscopy was first demonstrated in 2010 but many variations have followed since. In 2018, it was shown that the use of attenuation-compensation could be applied to Bessel beam based light-sheet microscopy and could enable imaging at greater depths within biological specimens.

Acoustofluidics

Bessel beams are a good candidate for the selectively trapping because of the concentric circles of pressure maximum and minimum in the transverse planes.

References

External links

New microscope captures 3D movies of living cells
gizmag.com (switched Bessel beams used effectively in real-time microscopy)
'Tractor beam' is possible with lasers, say scientists

Ultrasound (zeroth-order) Bessel beam profile - Front cover image (April 2002 Issue of the IEEE Trans. Ultrason. Ferr. Freq. Ctrl.)
{{DEFAULTSORT:Bessel Beam Laser science