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The recent introduction by Belafhal et al. [Opt. and Photon. J. 5, 234-246 (2015)] of mth-order Olver beams as a novel class of self-accelerating nondiffracting solutions to the paraxial equation is a direct contradiction to the seminal work of Berry and Balazs who determined that the infinite-energy Airy wave packet is the only accelerating nondiffracting solution to the (1 + 1)D Schr ödinger equation. It is shown in this note that the work of Belafhal et al. is valid only for m=0, which coincides with the Airy solution.

Extensive studies of self-accelerating beams have been made recently. The basic such beam is the Airy solution

ψ ( x , z ) = A i [ x − ( z 2 ) 2 ] exp [ − i ( z 3 12 − x z 2 ) ] (1)

governed by the paraxial equation

i ∂ ∂ z ψ ( x , z ) + 1 2 ∂ 2 ∂ x 2 ψ ( x , z ) = 0 (2)

in free space. Here, x = X / X 0 and z = Z / ( k X 0 2 ) are, respectively, dimensionless transverse and longitudinal variables, defined in terms of the original variables X and Z, the wavenumber k, and a scaling factor X 0 with units of meters. The complexification z → z − i 2 a 1 , where a 1 is a positive parameter, ensures the square integrability (finite energy) of the input function ψ ( x , 0 ) and, hence, of ψ ( x , z ) for z > 0 . The finite-energy version of the solution given in Equation (1) was first formulated analytically by Siviloglou and Christodoulides [

Both bending Airy beams and accelerating Airy wavepackets are characterized by self-healing properties; they tend to reform in spite of the severity of imposed perturbations; this is due to the reinforcement of the main lobe by the side lobes. The robustness of such beams has been studied in the presence of material dispersion [

Belafhal et al. [

O m ( x ) = 1 2 π ∫ − ∞ ∞ d λ exp [ a ( i λ ) γ + i λ x ] ; γ = m + 3 , | a | = 1 m + 3 , (3)

which, in turn, are related to the solutions of the ordinary differential equation

d 2 d x 2 W ∓ ( x ) ∓ 1 4 n 2 x n − 2 W ∓ ( x ) = 0 ; n > 2. (4)

Specifically, the authors state that a novel class of nondiffracting solutions to Equation (2) is given as follows:

ψ m ( x , z ) = 1 2π ∫ − ∞ ∞ d λ exp [ a ( i λ ) m + 3 − i 1 2 λ 2 z + i λ x ] = exp [ i ( z x 2 − z 3 12 ) ] O m ( x − z 2 4 ) ; | a | = 1 m + 3 . (5)

The first equality above is correct. The integral expression does, indeed, satisfy the parabolic Equation (2) for all values of m, as the authors have shown in Appendix B of Ref. [

ψ ( x , z ) = exp [ i ( x z 2 − s 3 12 ) ] W ( x − z 2 4 ) (6)

into the parabolic Equation (2). As a result, one obtains the ordinary differential equation

d 2 d ξ 2 W ( ξ ) − ξ W ( ξ ) = 0 ; ξ = x − s 2 4 , (7)

a solution of which is the Airy function; specifically, W ( ξ ) = A i ( ξ ) = 2 π O 0 ( ξ ) . This is in contradiction to the second equality in Equation (5) in which the Olver functions are related to the solutions to Equation (4) for m > 0 .

The authors give in Equation (24) of Ref. [

ψ m ( x , 0 ) = O m ( x ) exp ( a x ) ↔ O ˜ m ( λ + i a ) , (8)

for m = 0, 1 and 2. Here, O ˜ m ( λ ) denotes the Fourier transform of the Olver function O m ( x ) . For z ≥ 0 , one obtains, then,

ψ 0 ( x , z ) = exp ( a x + i a 2 2 z ) ∫ − ∞ ∞ d λ exp [ i λ ( x + i a z ) ] exp ( − i λ 2 2 z ) exp ( i λ 3 3 ) , ψ 1 ( x , z ) = exp ( a x + i a 2 2 z ) ∫ − ∞ ∞ d λ exp [ i λ ( x + i a z ) ] exp ( − i λ 2 2 z ) exp ( − λ 4 4 ) , ψ 2 ( x , z ) = exp ( a x + i a 2 2 z ) ∫ − ∞ ∞ d λ exp [ i λ ( x + i a z ) ] exp ( − i λ 2 2 z ) exp ( i λ 5 5 ) . (9)

The wave function ψ 0 ( x , z ) is the well-known Airy beam and contains finite energy for a positive parameter a. The Fourier representations of the wavefunctions ψ m ( x , z ) given by the integral expressions above do not yield analytical solutions for m > 0 . The expressions for m = 1 and 2 could justifiably be called Olver beams (but not necessarily accelerating Olver beams) because they are associated with apodized fourth-order diffusion (super diffusion) and fifth-order Airy (hyper Airy) functions, respectively, at z = 0 . However, a careful examination is needed in order to establish whether the exponential apodization is sufficient for ensuring finite energy. It turns out that the Airy beam ψ 1 ( x , z ) contains finite energy only for a < 0 . The choice of appropriate apodization for ψ 2 ( x , z ) requires careful examination.

Due to their complexity, the wave functions ψ m ( x , z ) for m = 1 and 2 can be examined only numerically. The authors allude to such a program in Sec. 4, where they propose hologram masks. However, it is not clear whether the integrations given in Equation (9) above, which are required for z > 0 , have been carried out, and what is the basis for the plots in

Consider the second expression in Equation (5), viz.,

ψ m ( x , z ) = exp [ i ( z x 2 − z 3 12 ) ] O m ( x − z 2 4 ) . (10)

It has been shown in this note that this does not represent a solution of the parabolic Equation (2), unless m = 0 . Is it possible, however, that there exists another type of equation the solutions of which can be expressed as in Equation (10) for all values of m? Such an equation does exist and it is given as follows:

[ i ∂ ∂ z + 1 2 ∂ 2 ∂ x 2 + 1 2 ( x − z 2 4 ) − 1 8 m 2 ( x − z 2 4 ) m − 2 ] ψ m ( x , z ) = 0. (11)

Introducing the ansatz given in Equation (6) into Equation (11), one obtains the ordinary differential equation

d 2 d ξ 2 W m ( ξ ) − 1 4 m 2 ξ m − 2 W m ( ξ ) = 0 ; ξ = x − z 2 4 , (12)

which has been studied extensively by Olver [

W 3 ( ξ ) = O 0 [ ( 3 / 2 ) 2 / 3 ξ ] . (13)

Since Equation (12) above is identical to Equation (1) in Ref. [

Hennani et al. [

U 2 ( x 2 ) = ( i k 2 π B ) 1 / 2 ∫ − ∞ ∞ d x 1 U 1 ( x 1 ) exp [ − i k 2 B ( A x 1 2 − 2 x 1 x 1 + D x 2 2 ) ] , (14)

in order to determine the output due to the input Olver-Gauss function

U 1 ( x 1 ) = O n ( x 1 ω 0 ) exp ( a 0 x 1 ω 0 ) exp ( − b 0 x 1 2 ω 0 2 ) . (15)

Here, ω 0 is a normalization parameter with units of length, and a 0 , b 0 are positive dimensionless parameters used to ensure finite energy. The main result is given in Equation (7) of Ref. [

U 2 ( x 2 ) = 1 2 π ( i k 2 π B ) 1 / 2 ( π i k A 2 B + b 0 ω 0 2 ) 1 / 2 exp ( − i k D 2 B x 2 2 ) exp [ ( a 0 ω 0 + i k b x 2 ) 2 4 ( i k A 2 B + b 0 ω 0 2 ) ]

× exp [ 1 96 1 ω 0 6 ( i k A 2 B + b 0 ω 0 2 ) 3 ] exp [ 1 8 ( a 0 ω 0 + i k b x 2 ) ω 0 3 ( i k A 2 B + b 0 ω 0 2 ) 2 ] × O n [ a 0 ω 0 + i k b x 2 2 ω 0 ( i k A 2 B + b 0 ω 0 2 ) + 1 16 ω 0 4 ( i k A 2 B + b 0 ω 0 2 ) 2 ] . (16)

Unfortunately, this result is incorrect, in general, because it is based on a variation of the expression given in Equation (5); specifically,

1 2 π ∫ − ∞ ∞ d λ exp [ a ( i λ ) n + 3 − 1 2 λ 2 s + i λ x ] = exp ( s x 2 + s 3 12 ) O n ( x + s 2 4 ) . (17)

This formula and, as consequence, Equation (16), is valid only for n = 0 . In this case, the solution in Equation (16) corresponds to a finite-energy accelerating Airy-Gauss beam which has been studied previously (see, e.g., Ez-Zariy et al. [

The free-space version of Equation (16) corresponding to the optical ABCD matrix

( A B C D ) = ( 1 z 0 1 ) , (18)

with z the direction of propagation, has been used by Hennani et al. [

In the seminal work of Berry and Balazs [

The introduction by Belafhal et al. [

In closing, it should be pointed out that the Olver functions

O m ( x ) = 1 2π ∫ − ∞ ∞ d λ exp [ a ( i λ ) γ + i λ x ] ; γ = m + 3 , | a | = 1 m + 3 , (19)

can be considered as “incomplete” elementary catastrophe integrals; the latter are defined as

C n ( q → ) = ∫ R d λ exp [ i P n ( q → , λ ) ] ; P n ( q → , λ ) = λ n n + ∑ j = 1 n − 2 q j λ j j . (20)

C 3 ( q = x ) is precisely equal to O 0 ( x ) and corresponds to the fold catastrophe. C n ( q → ) for n = 4, 5 and 6 are the cusp, swallowtail and butterfly catastrophes, respectively. The dynamics of the cusp, swallowtail and the butterfly optical catastrophes have been studied in terms of solutions to the 3D paraxial equation [

The authors declare no conflicts of interest regarding the publication of this paper.

Besieris, I.M., Shaarawi, A.M. and Davis, B. (2019) A Note on Self-Accelerating Olver and Olver-Gauss Beams. Optics and Photonics Journal, 9, 1-7. https://doi.org/10.4236/opj.2019.91001