Abstract

A recently derived Airy beam solution to the (1+1)D paraxial equation is shown to obey two salient properties characterizing arbitrary finite energy solutions associated with second-order diffraction; the centroid of the beam is a linear function of the range and its variance varies quadratically in range. Some insight is provided regarding the local acceleration dynamics of the beam. It is shown, specifically, that the interpretation of this beam as accelerating, i.e., one characterized by a nonlinear lateral shift, depends significantly on the parameter a entering into the solution.

© 2007 Optical Society of America

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References

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  1. G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
    [CrossRef] [PubMed]
  2. M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
    [CrossRef]
  3. D. Bohm, Phys. Rev. 85, 166 (1952).
    [CrossRef]

2007 (1)

1979 (1)

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

1952 (1)

D. Bohm, Phys. Rev. 85, 166 (1952).
[CrossRef]

Balazs, N. L.

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Berry, M. V.

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Bohm, D.

D. Bohm, Phys. Rev. 85, 166 (1952).
[CrossRef]

Christodoulides, D. N.

Siviloglou, G. A.

Am. J. Phys. (1)

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[CrossRef]

Opt. Lett. (1)

Phys. Rev. (1)

D. Bohm, Phys. Rev. 85, 166 (1952).
[CrossRef]

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Figures (2)

Fig. 1
Fig. 1

Composite of the graph ϕ S C ( s , ξ ) 2 versus s (black), a narrow square pulse centered at s = ξ 2 4 (red), and a narrow square pulse centered at s = s (blue) for different values of the normalized range ξ. (a) a = 5 × 10 2 ; (b) a = 2 .

Fig. 2
Fig. 2

Density plot of ϕ S C ( s , ξ ) 2 for four values of the parameter a.

Equations (21)

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i ξ ϕ ( s , ξ ) + 1 2 2 s 2 ϕ ( s , ξ ) = 0 ,
ϕ S C ( s , ξ ) = Ai [ s ( ξ 2 ) 2 + i a ξ ] exp [ a 3 3 + a s ( a ξ 2 2 ) ] × exp [ i ( ξ 3 12 ) + i ( a 2 ξ 2 ) + i ( s ξ 2 ) ] ; a > 0 .
ϕ B B ( s , ξ ) = exp [ i ( s ξ 2 ξ 3 12 ) ] Ai [ s ( ξ 2 ) 2 ] ,
ϕ ( s , ξ ) = ( 1 2 π ) d k exp [ i ( k s k 2 ξ 2 ) ] ϕ ̂ 0 ( k ) ,
ϕ ̂ 0 ( k ) = exp ( a k 2 ) exp [ i ( k 3 3 a 2 k i a 3 ) 3 ] .
s ( ξ ) ( 1 I ) d s s ϕ ( s , ξ ) 2 = S 0 + S 1 ξ ;
I = d s ϕ ( s , ξ ) 2 ,
S 0 = 1 2 π I d k Im { d d k ϕ ̂ 0 * ( k ) ϕ ̂ 0 ( k ) } ,
S 1 = 1 2 π I d k k ϕ ̂ 0 ( k ) 2 ,
σ 2 ( ξ ) ( 1 I ) d s [ s s ( ξ ) ] 2 ϕ ( s , ξ ) 2 = Σ 0 + Σ 1 ξ + Σ 2 ξ 2 s ( ξ ) 2
Σ 0 = 1 2 π I d k d 2 d κ 2 [ ϕ ̂ 0 * ( k + 1 2 κ ) ϕ ̂ 0 ( k 1 2 κ ) ] κ = 0 ,
Σ 1 = 2 π I d k k Im { d d k ϕ ̂ 0 * ( k ) ϕ ̂ 0 ( k ) } ,
Σ 2 = 1 2 π I d k k 2 ϕ ̂ 0 ( k ) 2 .
s ( ξ ) = 4 a 3 1 4 a ,
σ 2 ( ξ ) = ( a + 1 8 a 2 ) + 1 4 a ξ 2 .
ϕ ( s , ξ ) = R ( s , ξ ) exp [ i S ( s , ξ ) ] ,
d d ξ s ( ξ ) = v ( ξ ) , d d ξ v ( ξ ) = s Q ( s , ξ ) s s ( ξ ) ,
Q ( s , ξ ) 1 2 R ( s , ξ ) 2 s 2 R ( s , ξ ) .
d d ξ s ( ξ ) = v ( ξ ) , v ( ξ ) = [ s S ( s , ξ ) = Im { ϕ * ( s , ξ ) ( s ) ϕ ( s , ξ ) ϕ ( s , ξ ) 2 } ] s s ( ξ ) .
d d ξ s ( ξ ) = v ( ξ ) = ξ 2 .
d d ξ s ( ξ ) = v ( ξ ) = ξ 2 + Im { Ai [ i a ξ + s ( ξ ) ξ 2 4 ] Ai [ i a ξ + s ( ξ ) ξ 2 4 ] } ,

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