© 2026 Aqora Quantum S.A.S.

Terms Privacy Legal Notice

Acta Physica Sinica

Research on diffraction and self-acceleration of Airy beam

Yue Yang-Yang, Xiao Han, Wang Zi-Xiao, Wu Min

Published Jan 1, 2013• 7 citations

Abstract

In 1979, Berry and Balazs [M V Berry and N L Balazs 1979 Am. J. Phys. 47 264] obtained a strict solution of the Schrödinger equation with Airy function used as the initial condition, and described the wave function represented by such solution as the Airy wave-packets. They discovered that infinite Airy wave-packet has unique properties such as non-spreading and free acceleration, proving that it is the only nontrivial non-spreading solution of the time-dependent Schrödinger equation in one dimension. However, the observing of the finite Airy beam seems to be more meaningful since wave-packets in reality is inevitably band limited. A certain form of finite Airy beam was investigated by Siviloglou et al. in 2007 [Siviloglou G A, Broky J, Dogariu A, Christodoulides D N 2007 Phys. Rev. Lett. 99 213901; Siviloglou G A, Christodoulides D N 2007 Opt. Lett. 32 979]. They noted that the Airy wave packet still exhibits its most exotic feature, i.e., its trend toward free acceleration. While in the present paper we discuss the properties of Airy beam in a few steps further and propose several conclusions. On the one hand, a theoretic explanation is given to solve the matter of the centre of mass of infinite Airy beam. On the other hand, deeper research is conducted on the unique properties of finite Airy beam. Another form of finite Airy beam is discussed by reduction to absurdity, and its field distribution is put forward by numerical simulation. We find that the trajectory of the centroid holds its position, which means that the beam cannot accelerate freely as a whole. Ultimately, we have the conclusion that finite Airy beam can neither freely accelerate nor be non-diffractive.

DOI: 10.7498/aps.62.044205

Related papers

Dispersive Airy packets

Antonio Boulhosa Nassar, José Maria Filardo Bassalo, Paulo de Tarso Santos Alencar

We present the solution of Schrödinger’s equation for time-dependent quadratic potentials in terms of an Airy packet. We demonstrate new features of the Airy packet in light of its apparent unfamiliar properties discovered by Berry and Balazs (who found a nonspreading and accelerating Airy packet to be a unique solution to the free-particle Schrödinger equation). Our solution, found using space-time scale transformations, not only overcomes the packet’s tendency to accelerate, but also make it possible to spread or contract in time.

American Journal of Physics•Sep 1, 1995•9 citations

Airy wavepackets are coherent states

Vivek M. Vyas

Accelerating non-spreading wavepackets in nonrelativistic free particle system, with probability distribution having an Airy function profile, were discovered by Berry and Balazs (Am. J. Phys., 47, 264 (1979)), and have been subsequently realised in several optical experiments. It is shown that these wavepackets are actually coherent states. It is found that the Galilean invariance of the Schrodinger equation plays a key role in making these states unique and giving rise to their unusual propagation properties.

•Oct 27, 2017•0 citations

Nondispersive accelerating wave packets

Ioannis M. Besieris, Amr Mohamed Shaarawi, Richard W. Ziolkowski

Motivated by the work of Berry and Balazs and Greenberger on the 1-D Schrödinger equation, we have investigated a class of nonspreading solutions to the 3-D Schrödinger equation involving accelerating Airy envelopes. These solutions are characterized by an asymmetric structure, in contrast to recently derived spherically symmetric packets moving with constant velocities. The field of a characteristic Airy packet extends in an oscillatory fashion behind its peak amplitude, while it quickly disappears in front of the packet’s center. A particle modeled by such a packet seems to leave a wake of its field behind as it accelerates in a certain direction. On the other hand, a wave packet moving with a constant velocity has a field which is symmetrically distributed in all directions. Our work on Airy-type solutions to the 3-D Schrödinger equation has led us also to analogous solutions for the 3-D scalar wave equation.

American Journal of Physics•Jun 1, 1994•50 citations

Self-accelerating matter waves

C. Yüce

The free particle Schrödinger equation admits a nontrivial self-accelerating Airy wave packet solution. Recently, the Airy beams that freely accelerate in space was experimentally realized in photonics community. Here, we present self-accelerating waves for the Bose–Einstein condensate in a time-dependent harmonic oscillator potential. We show that parity and time reversal symmetries for self-accelerating waves are spontaneously broken.

arXiv:1505.01272•Modern Physics Letters B•Oct 1, 2015•11 citations

REFLECTION AND TRANSMISSION OF SELF-ACCELERATING WAVE PACKETS

Glenn Dusing

Airy beams have been studied in an optical context since their discovery in the late 1970s and have found numerous applications in technologies such as microscopy and optical trapping. Many of these applications are based on the wave packets’ unique features such as zero or minimal diffraction, self-acceleration, and self-healing. Recently, Airy beams have been produced using electrons and these matter waves exhibit many of the same unique characteristics of their optical counterparts. We use our Path Integral Quantum Trajectory (PIQTr) model to present a study of the reflection and transmission of matter Airy wave packets. Like the more familiar Gaussian wave packets, Airy beams have a Gaussian momentum distribution. However, despite this similarity, Airy beams show unique features for transmission and reflection.

•Jan 1, 2019•0 citations

Airy wavepackets are Perelomov coherent states

Vivek M. Vyas

Accelerating non-spreading wavepackets in a nonrelativistic free-particle system, with probability distribution having an Airy function profile, were discovered by Berry and Balazs [Am. J. Phys. 47(3), 264–267 (1979)], and have been subsequently realised in several optics experiments. It is shown that these wavepackets are actually Perelomov coherent states. It is found that the Galilean invariance of the Schrödinger equation plays a key role in making these states unique and gives rise to their unusual propagation properties.

arXiv:1710.10302•American Journal of Physics•Sep 19, 2018•6 citations

In 1979, Berry and Balazs [M V Berry and N L Balazs 1979 Am. J. Phys. 47 264] obtained a strict solution of the Schrödinger equation with Airy function used as the initial condition, and described the wave function represented by such solution as the Airy wave-packets. They discovered that infinite Airy wave-packet has unique properties such as non-spreading and free acceleration, proving that it is the only nontrivial non-spreading solution of the time-dependent Schrödinger equation in one dimension. However, the observing of the finite Airy beam seems to be more meaningful since wave-packets in reality is inevitably band limited. A certain form of finite Airy beam was investigated by Siviloglou et al. in 2007 [Siviloglou G A, Broky J, Dogariu A, Christodoulides D N 2007 Phys. Rev. Lett. 99 213901; Siviloglou G A, Christodoulides D N 2007 Opt. Lett. 32 979]. They noted that the Airy wave packet still exhibits its most exotic feature, i.e., its trend toward free acceleration. While in the present paper we discuss the properties of Airy beam in a few steps further and propose several conclusions. On the one hand, a theoretic explanation is given to solve the matter of the centre of mass of infinite Airy beam. On the other hand, deeper research is conducted on the unique properties of finite Airy beam. Another form of finite Airy beam is discussed by reduction to absurdity, and its field distribution is put forward by numerical simulation. We find that the trajectory of the centroid holds its position, which means that the beam cannot accelerate freely as a whole. Ultimately, we have the conclusion that finite Airy beam can neither freely accelerate nor be non-diffractive.

We present the solution of Schrödinger’s equation for time-dependent quadratic potentials in terms of an Airy packet. We demonstrate new features of the Airy packet in light of its apparent unfamiliar properties discovered by Berry and Balazs (who found a nonspreading and accelerating Airy packet to be a unique solution to the free-particle Schrödinger equation). Our solution, found using space-time scale transformations, not only overcomes the packet’s tendency to accelerate, but also make it possible to spread or contract in time.

Accelerating non-spreading wavepackets in nonrelativistic free particle system, with probability distribution having an Airy function profile, were discovered by Berry and Balazs (Am. J. Phys., 47, 264 (1979)), and have been subsequently realised in several optical experiments. It is shown that these wavepackets are actually coherent states. It is found that the Galilean invariance of the Schrodinger equation plays a key role in making these states unique and giving rise to their unusual propagation properties.

Motivated by the work of Berry and Balazs and Greenberger on the 1-D Schrödinger equation, we have investigated a class of nonspreading solutions to the 3-D Schrödinger equation involving accelerating Airy envelopes. These solutions are characterized by an asymmetric structure, in contrast to recently derived spherically symmetric packets moving with constant velocities. The field of a characteristic Airy packet extends in an oscillatory fashion behind its peak amplitude, while it quickly disappears in front of the packet’s center. A particle modeled by such a packet seems to leave a wake of its field behind as it accelerates in a certain direction. On the other hand, a wave packet moving with a constant velocity has a field which is symmetrically distributed in all directions. Our work on Airy-type solutions to the 3-D Schrödinger equation has led us also to analogous solutions for the 3-D scalar wave equation.

Airy beams have been studied in an optical context since their discovery in the late 1970s and have found numerous applications in technologies such as microscopy and optical trapping. Many of these applications are based on the wave packets’ unique features such as zero or minimal diffraction, self-acceleration, and self-healing. Recently, Airy beams have been produced using electrons and these matter waves exhibit many of the same unique characteristics of their optical counterparts. We use our Path Integral Quantum Trajectory (PIQTr) model to present a study of the reflection and transmission of matter Airy wave packets. Like the more familiar Gaussian wave packets, Airy beams have a Gaussian momentum distribution. However, despite this similarity, Airy beams show unique features for transmission and reflection.

Accelerating non-spreading wavepackets in a nonrelativistic free-particle system, with probability distribution having an Airy function profile, were discovered by Berry and Balazs [Am. J. Phys. 47(3), 264–267 (1979)], and have been subsequently realised in several optics experiments. It is shown that these wavepackets are actually Perelomov coherent states. It is found that the Galilean invariance of the Schrödinger equation plays a key role in making these states unique and gives rise to their unusual propagation properties.