Abstract

We present a mathematical analysis of the finite-energy Airy beam with a sharply truncated spectrum, which can be generated by a uniformly illuminated, finite-sized spatial light modulator, or windowed cubic phase mask. The resulting “incomplete Airy beam” is tractable mathematically, and differs from an infinite-energy Airy beam by an additional oscillating modulation and the decay of its fringes. Its propagation can be described explicitly using an incomplete Airy function, from which we derive simple expressions for the beam’s total power and mean position. Asymptotic analysis reveals a simple connection between the cutoff and the region of the beam with Airy-like behavior.

© 2013 Optical Society of America

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References

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2012 (3)

2011 (1)

2010 (2)

2009 (2)

2007 (2)

1987 (2)

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987).
[CrossRef]

1980 (1)

M. V. Berry and C. Upstill, Prog. Opt. 18, 257 (1980).
[CrossRef]

1979 (1)

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

1969 (1)

L. Levey and L. B. Felsen, Radio Sci. 4, 959 (1969).
[CrossRef]

Acebal, P.

Balazs, N. L.

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

Bandres, M. A.

Barwick, S.

Berry, M. V.

M. V. Berry and C. Upstill, Prog. Opt. 18, 257 (1980).
[CrossRef]

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

Blaya, S.

Carretero, L.

Carvalho, M. I.

Chen, Z.

Christodoulides, D. N.

Cottrell, D. M.

Dartora, C. A.

Davis, J. A.

Dennis, M. R.

Dholakia, K.

Durnin, J.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

J. Durnin, J. Opt. Soc. Am. A 4, 651 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

Facão, M.

Felsen, L. B.

L. Levey and L. B. Felsen, Radio Sci. 4, 959 (1969).
[CrossRef]

Fimia, A.

Garcia, C.

Gutiérrez-Vega, J. C.

Hazard, T. M.

Howls, C. J.

J. D. Ring, J. Lindberg, C. J. Howls, and M. R. Dennis, J. Opt. 14, 075702 (2012).
[CrossRef]

Hu, Y.

Huang, S.

Levey, L.

L. Levey and L. B. Felsen, Radio Sci. 4, 959 (1969).
[CrossRef]

Lindberg, J.

Lou, C.

Madrigal, R.

Mazilu, M.

Miceli, J. J.

J. Durnin, J. J. Miceli, and J. H. Eberly, Phys. Rev. Lett. 58, 1499 (1987).
[CrossRef]

Mourka, A.

Murciano, A.

Nóbrega, K. Z.

Ring, J. D.

Siviloglou, G. A.

Upstill, C.

M. V. Berry and C. Upstill, Prog. Opt. 18, 257 (1980).
[CrossRef]

Xu, J.

Zamboni-Rached, M.

Zhang, P.

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Figures (3)

Fig. 1.
Fig. 1.

Comparison of infinite and incomplete Airy beams with M=0, W=4. (a) |Ai(X)|2, (b) |Ai0,4inc(X)|2 in blue, |Ai(X)|2 in dotted pink, (c) |Ai˜(ξ)|2, shading by hue indicates the phase, (d) |Ai˜0,4inc(ξ)|2, (e) |AiB(X,Z)|2, and (f) |AiB0,4inc(X,Z)|2, symmetric about Z=0; white dashed line shows Eq. (8), which meets the solid white parabola X=Z2 at Eq. (9) (white square); purple dots correspond to the complex planes of Figs. 2(a) and 2(b); dotted lines at Z1 and Z2 indicate slices plotted in Figs. 2(c) and 2(d); Eq. (5) is like Airy right of the white dashed line.

Fig. 2.
Fig. 2.

(a), (b) Complex planes with real contours of iξ3/3+iξ(Xa,bZ1,22), respectively, corresponding to lower and upper purple dots of Fig. 1(f). Gray squares are saddles of Eq. (7), white dots are endpoints, white dashed line shows the integration contour. (a) Airy-like configuration and (b) non-Airy-like configuration. (c), (d) Show intensity along dashed purple lines of Fig. 1(f) at Z1 and Z2, respectively, the vertical black, dashed lines show intercepts X1,2 of the line of Eq. (8) with Z1,2.

Fig. 3.
Fig. 3.

|AiBM,Winc(X,Z)|2 for M0; dashed lines are given by Eq. (8) and enclose the Airy-like region of the beam. (a) M=3 and W=4 with enhanced main lobe and (b) M=4 and W=3. Side plots show main lobe intensity against that of (dashed) AiB(XP,Z); XP is the X coordinate of the ideal beam’s peak intensity.

Equations (10)

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Ai(X)=12πdξexp[i(13ξ3+ξX)],
AiB(X,Z)=ei(XZ2Z3/3)Ai(XZ2),
AiM,Winc(X)12πMWM+Wdξexp[i(13ξ3+ξX)],
dX|AiM,Winc(X)|2=W/π.
AiBM,Winc(X,Z)=ei(XZ2Z3/3)AiMZ,Winc(XZ2).
X¯=W2/3M2+2MZ,
ξ±=±Z2X.
X±=2Zμ±μ±2.
(X,Z)=(μ+2,μ+),
E0,W(X,Z)=eiW2Zsin[WX+13W3]/πW2,

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