At the same time, a considerable number of frontier and interdisciplinary applications, such as optical microparticles manipulation [4], formation of Airy beam lasers [5], generation of optical bullets [6], routing of surface plasmon polaritons [7], generation of curved plasma channels [8], and creating curved electron beams without any external field [9], have been took advantage of the Airy function and Airy beam technology.

where Ai[??] denotes the Airy function, [x.sub.0] and [y.sub.0] are beams horizontal size and vertical size, respectively, [a.sub.0] and [b.sub.0] represent exponential modulation factors, and E[??] is electric field intensity on incident surface.

The

Airy function v([zeta] - [nu]), which represents the incident plane wave in (6), is replaced with the Whittaker function [M.sub.it,n](-i[chi][tau]), while the

Airy function [w.sub.1]([zeta] - [nu]) which represents the secondary wave outgoing from the surface is replaced with the Whittaker function [W.sub.it,n](-i[chi][tau]).

An argument of

Airy function equation (31) can be further expanded, taking into account large values of [mu] ~ [[mu].sub.0] [much greater than] 1:

where [x.sub.k] is the kth zero of the

Airy function and the quantities [gamma] and [beta] are two scaling and fitting parameters, respectively, that are used to take into account both the effects of anisotropy ([gamma] parameter) and the other displacement effects ([beta] parameter).

The specific form of the wave packet can be finally expressed by using some special functions, slay, in terms of oscillating exponentials,

Airy function, Fresnel integral, Pearcey- type integral, etc.

Note that the allowed values of energy E implicitly enter in (6) in the form of the

Airy function arguments in (5).

where Ai([rho](x)) is the

Airy function and its complement is Bi([rho](x)) [16].

Similar, but thinner shapes, also part of the

Airy function, extended parallel to the main parabola, the team reports in the Nov.

On the other hand, if I = 3 the integral is described asymptotically by the

Airy function. In this case the integral in (4) has typically an asymptotic series expansion which is a linear combination of terms of order ([[parallel]([n.sub.0], n)[parallel].sup.-l-1/3)] l[greater than or equal to]0 and also of order ([[parallel] ([n.sub.0],n)[parallel].sup.-l-2/3)]l[greater than equal to] 0.

where Ai(t) and Bi(t) are the

Airy functions defined by Miller [18].

As the 3j, 6j, 9j (angular momenta) and

Airy functions chapters were no more dense than the above two, I must have started with the worst!