physics Physics is the scientific study of matter, its Elementary particle, fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge whi ...

, X-waves are a type of wave

superposition In mathematics, a linear combination or superposition is an expression constructed from a set of terms by multiplying each term by a constant and adding the results (e.g. a linear combination of ''x'' and ''y'' would be any expression of the form ...

(or wavepacket) exhibiting a distinctive double-cone structure with its axis aligned along the propagation direction. The energy of the wave is concentrated along the vertex of this double cone, creating an "X" shape when intersected by a plane containing the propagation axis. These structures are also known as conical waves. X-waves possess several remarkable properties: # Weak Localization: They are weakly localized solutions of the wave equation, meaning they are extended both spatially and temporally. # Propagation Invariance: Under specific conditions, they maintain their shape during propagation, resisting the effects of

diffraction Diffraction is the deviation of waves from straight-line propagation without any change in their energy due to an obstacle or through an aperture. The diffracting object or aperture effectively becomes a secondary source of the Wave propagation ...

and

dispersion Dispersion may refer to: Economics and finance *Dispersion (finance), a measure for the statistical distribution of portfolio returns * Price dispersion, a variation in prices across sellers of the same item *Wage dispersion, the amount of variat ...

. This occurs when dispersion precisely balances diffraction, preventing the wavepacket from spreading. X-waves can manifest as

sound In physics, sound is a vibration that propagates as an acoustic wave through a transmission medium such as a gas, liquid or solid. In human physiology and psychology, sound is the ''reception'' of such waves and their ''perception'' by the br ...

electromagnetic In physics, electromagnetism is an interaction that occurs between particles with electric charge via electromagnetic fields. The electromagnetic force is one of the four fundamental forces of nature. It is the dominant force in the interacti ...

, or

gravitational wave Gravitational waves are oscillations of the gravitational field that Wave propagation, travel through space at the speed of light; they are generated by the relative motion of gravity, gravitating masses. They were proposed by Oliver Heaviside i ...

s. X-waves generalize the concept of diffraction-free solutions, exemplified by

monochromatic A monochrome or monochromatic image, object or palette is composed of one color (or values of one color). Images using only shades of grey are called grayscale (typically digital) or black-and-white (typically analog). In physics, mon ...

Bessel beams, to include wavepackets composed of multiple frequency components. To achieve propagation invariance, each frequency component must have the same velocity along the propagation axis. Therefore, an X-wave can be constructed as a non-monochromatic

Bessel beam Bessel may refer to: Mathematics and science * Bessel beam * Bessel ellipsoid * Bessel function in mathematics * Bessel's inequality in mathematics * Bessel's correction In statistics, Bessel's correction is the use of ''n'' − 1 in ...

s. Like ideal Bessel beams, ideal X-waves are mathematical constructs possessing

infinite Infinite may refer to: Mathematics *Infinite set, a set that is not a finite set *Infinity, an abstract concept describing something without any limit Music Performers *Infinite (group), a South Korean boy band *Infinite (rapper), Canadian ra ...

energy Energy () is the physical quantity, quantitative physical property, property that is transferred to a physical body, body or to a physical system, recognizable in the performance of Work (thermodynamics), work and in the form of heat and l ...

. However, finite-energy approximations have been experimentally observed in various physical contexts. X-wave pulses can exhibit superluminal

phase Phase or phases may refer to: Science *State of matter, or phase, one of the distinct forms in which matter can exist *Phase (matter), a region of space throughout which all physical properties are essentially uniform *Phase space, a mathematica ...

and

group velocity The group velocity of a wave is the velocity with which the overall envelope shape of the wave's amplitudes—known as the ''modulation'' or ''envelope (waves), envelope'' of the wave—propagates through space. For example, if a stone is thro ...

. Furthermore, X-wave solutions have been reported within a quantum mechanical framework in optics.A. Ciattoni and C. Conti
Quantum electromagnetic X-waves
''arxiv.org'' 0704.0442v1.

References

* J. Lu and J. F. Greenleaf, "Nondiffracting X waves: exact solutions to free-space scalar wave equation and their infinite realizations",

IEEE The Institute of Electrical and Electronics Engineers (IEEE) is an American 501(c)(3) organization, 501(c)(3) public charity professional organization for electrical engineering, electronics engineering, and other related disciplines. The IEEE ...

Trans. Ultrasonic Ferroelectric Frequency. ''

Control Control may refer to: Basic meanings Economics and business * Control (management), an element of management * Control, an element of management accounting * Comptroller (or controller), a senior financial officer in an organization * Controlling ...

'' 39, 19–31 (1992). * Erasmo Recami and Michel Zamboni-Rached and Hugo E. Hernandez-Figueroa,
Localized waves: A scientific and historical introduction
''arxiv.org'' 0708.1655v2. * Various authors in the boo
Localized Waves
edited by Erasmo Recami, Michel Zamboni-Rached and Hugo E. Hernandez-Figueroa

External links

The Virtual Institute for Nonlinear Optics (VINO)
, a research collaboration devoted to the investigation of X-waves and conical waves in general

page at th
nlo.phys.uniroma1.it
website. Wave mechanics {{classicalmechanics-stub

See also

References

External links