Abstract

A two-dimensional field that is a product of three Airy beams is proposed and investigated. It is shown that the Fourier image of this field has a cubic phase and a radially symmetric intensity with a super-Gaussian decrease. Propagation of the product of three Airy beams in a Fresnel zone is investigated numerically.

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References

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  1. M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).
  2. O. Vallée and M. Soares, Airy Functions and Applications to Physics (Imperial College Press, 2004).
  3. G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1945).
  4. E. C. Titchmarsh, Introduction to the Theory of Fourier’s Integrals (Oxford University, 1937).
  5. A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vols.  1–3 (McGraw-Hill, 1953).
  6. F. W. J. Olver, Asymptotics and Special Functions(Academic, 1974).
  7. M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Problem (SIAM, 1981).
    [CrossRef]
  8. M. V. Berry, J. F. Nye, and F. J. Wright, Phil. Trans. R. Soc. A 291, 453 (1979).
    [CrossRef]
  9. M. V. Berry, Proc. R. Soc. A 463, 3055 (2007).
    [CrossRef]
  10. G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
    [CrossRef] [PubMed]
  11. A. Torre, Appl. Phys. B 99, 775 (2010).
    [CrossRef]
  12. M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264(1979).
    [CrossRef]

2010 (1)

A. Torre, Appl. Phys. B 99, 775 (2010).
[CrossRef]

2007 (2)

1979 (2)

M. V. Berry, J. F. Nye, and F. J. Wright, Phil. Trans. R. Soc. A 291, 453 (1979).
[CrossRef]

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

Ablowitz, M. J.

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Problem (SIAM, 1981).
[CrossRef]

Abramowitz, M.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Balazs, N. L.

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

Berry, M. V.

M. V. Berry, Proc. R. Soc. A 463, 3055 (2007).
[CrossRef]

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

M. V. Berry, J. F. Nye, and F. J. Wright, Phil. Trans. R. Soc. A 291, 453 (1979).
[CrossRef]

Christodoulides, D. N.

Erdélyi, A.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vols.  1–3 (McGraw-Hill, 1953).

Magnus, W.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vols.  1–3 (McGraw-Hill, 1953).

Nye, J. F.

M. V. Berry, J. F. Nye, and F. J. Wright, Phil. Trans. R. Soc. A 291, 453 (1979).
[CrossRef]

Oberhettinger, F.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vols.  1–3 (McGraw-Hill, 1953).

Olver, F. W. J.

F. W. J. Olver, Asymptotics and Special Functions(Academic, 1974).

Segur, H.

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Problem (SIAM, 1981).
[CrossRef]

Siviloglou, G. A.

Soares, M.

O. Vallée and M. Soares, Airy Functions and Applications to Physics (Imperial College Press, 2004).

Stegun, I. A.

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

Titchmarsh, E. C.

E. C. Titchmarsh, Introduction to the Theory of Fourier’s Integrals (Oxford University, 1937).

Torre, A.

A. Torre, Appl. Phys. B 99, 775 (2010).
[CrossRef]

Tricomi, F. G.

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vols.  1–3 (McGraw-Hill, 1953).

Vallée, O.

O. Vallée and M. Soares, Airy Functions and Applications to Physics (Imperial College Press, 2004).

Watson, G. N.

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1945).

Wright, F. J.

M. V. Berry, J. F. Nye, and F. J. Wright, Phil. Trans. R. Soc. A 291, 453 (1979).
[CrossRef]

Am. J. Phys. (1)

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

Appl. Phys. B (1)

A. Torre, Appl. Phys. B 99, 775 (2010).
[CrossRef]

Opt. Lett. (1)

Phil. Trans. R. Soc. A (1)

M. V. Berry, J. F. Nye, and F. J. Wright, Phil. Trans. R. Soc. A 291, 453 (1979).
[CrossRef]

Proc. R. Soc. A (1)

M. V. Berry, Proc. R. Soc. A 463, 3055 (2007).
[CrossRef]

Other (7)

M. Abramowitz and I. A. Stegun, Handbook of Mathematical Functions (Dover, 1972).

O. Vallée and M. Soares, Airy Functions and Applications to Physics (Imperial College Press, 2004).

G. N. Watson, A Treatise on the Theory of Bessel Functions (Cambridge University, 1945).

E. C. Titchmarsh, Introduction to the Theory of Fourier’s Integrals (Oxford University, 1937).

A. Erdélyi, W. Magnus, F. Oberhettinger, and F. G. Tricomi, Higher Transcendental Functions, Vols.  1–3 (McGraw-Hill, 1953).

F. W. J. Olver, Asymptotics and Special Functions(Academic, 1974).

M. J. Ablowitz and H. Segur, Solitons and the Inverse Scattering Problem (SIAM, 1981).
[CrossRef]

Supplementary Material (1)

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Figures (4)

Fig. 1
Fig. 1

Intensity and phase of the beam funcFR l [ t Ai ( ρ ; b , c ) ] ( r ) for 3 2 / 3 c = a 1 and various values of l. The frame number N is related to the detection plane l by the relation 2 b 2 l / k = tan ( π N / 40 ) , i.e., N = 0 and 20 correspond to the initial and the Fourier planes, respectively. The frame N = 0 is shown in the square [ B , B ] × [ B , B ] , where B = 1.2 / b .

Fig. 2
Fig. 2

Same as Fig. 1, but for 3 2 / 3 c = a 3 and B = 4.0 / b (Media 1).

Fig. 3
Fig. 3

Same as Fig. 1, but for 3 2 / 3 c = a 3 and B = 4.5 / b .

Fig. 4
Fig. 4

Comparison of radial profiles of the Airy function Ai ( a 1 + r 2 ) / Ai ( a 1 ) and its Gaussian approximation.

Equations (21)

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Ai ( x ) = 1 2 π R exp ( i ξ 3 3 + i x ξ ) d ξ ,
( x 2 + y 2 + 2 i k l ) F = 0.
F 0 ( r , l ) | l = 0 = F 0 ( r ) ,
F ( r , l ) = FR l [ F 0 ( ρ ) ] ( r ) = k 2 π i l R 2 exp ( i k 2 l | r ρ | 2 ) F 0 ( ρ ) d ρ ,
F 0 ( r ) = 1 n 2 exp ( γ s n ) Ai ( s n + c n ) ,
F ( r , l ) = 1 n 2 exp ( γ s n + i R n l 2 k [ s n + c n + γ 2 ] γ R n 2 l 2 2 k 2 i R n 3 l 3 12 k 3 ) × Ai ( s n + c n + i γ R n l k R n 2 l 2 4 k 2 ) ,
    a 1 a 2 + b 1 b 2 = 0 ,
F 0 ( r ) = 1 n 3 exp ( γ s n ) Ai ( s n + c n )
F 0 ( r ) = 1 n 3 Ai ( s n + c n )
s 1 + s 2 + s 3 = 0 { a 1 + a 2 + a 3 = 0 , b 1 + b 2 + b 3 = 0.
t Ai ( r ; b , c ) = Ai ( b x 3 y 2 + c ) × Ai ( b x 3 y 2 + c ) Ai ( b y + c ) .
F [ t Ai ( ρ ; b , c ) ] ( r ) = 1 3 5 / 6 π b 2 exp ( 2 i r 3 sin 3 ϕ 27 b 3 ) Ai ( 3 2 / 3 c + 2 r 2 3 4 / 3 b 2 ) ,
F [ f ( ρ ) ] ( r ) = 1 2 π R 2 exp [ i ( r · ρ ) ] f ( ρ ) d ρ
Ai ( r 2 ) 1 2 π r exp ( 2 r 3 3 ) ( r ) .
f ( r ) = n , m = 0 f n , m ( w 0 ) H n , m ( r w 0 ) , f n , m ( w 0 ) = 2 1 n m π n ! m ! w 0 2 R 2 f ( r ) H n , m ( r w 0 ) d r ,
FR l [ f ( ρ ) ] ( r ) = 1 σ exp ( 2 i l r 2 k w 0 4 | σ | 2 ) × n , m = 0 f n , m ( w 0 ) e i ( n + m ) arg σ H n , m ( r w 0 | σ | ) ,
funcFR l [ f ( ρ ) ] ( r ) = n , m = 0 f n , m ( w 0 ) e i ( n + m ) arg σ H n , m ( r w 0 )
funcFR 0 [ f ( ρ ) ] ( r ) = f ( r ) , funcFR [ f ( ρ ) ] ( r ) = 2 w 0 2 F [ f ( ρ ) ] ( 2 r w 0 2 ) .
F [ exp ( ρ 2 w 0 2 ) ] ( r ) = w 0 2 2 exp ( w 0 2 r 2 4 ) .
Ai ( a 1 + 2 r 2 3 4 / 3 b 2 ) Ai ( a 1 ) exp ( γ 2 r 2 3 4 / 3 b 2 ) .
w 0 2 4 = 2 γ 3 4 / 3 b 2 w 0 1.027 b ,

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