Abstract

The Airy beams are analyzed in order to provide a cogent physical explanation to their intriguing features which include weak diffraction, curved propagation trajectories in free-space, and self healing. The asymptotically exact analysis utilizes the method of uniform geometrical optics (UGO), and it is also verified via a uniform asymptotic evaluation of the Kirchhoff-Huygens integral. Both formulations are shown to fully agree with the exact Airy beam solution in the paraxial zone where the latter is valid, but they are also valid outside this zone. Specifically it is shown that the beam along the curved propagation trajectory is not generated by contributions from the main lobe in the aperture, i.e., it is not described by a local wave-dynamics along this trajectory. Actually, this beam is identified as a caustic of rays that emerge sideways from points in the initial aperture that are located far away from the main lobe. The field of these focusing rays, described here by the UGO, fully agrees with the Airy beam solution. These observations explain that the “weak-diffraction” and the “self healing” properties are generated, in fact, by a continuum of sideways contributions to the field, and not by local self-curving dynamics. The uniform ray representation provides a systematic framework to synthesize aperture sources for other beam solutions with similar properties in uniform or in non-uniform media.

©2010 Optical Society of America

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References

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  1. G. A. Siviloglou, J. Brokly, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99 (2007).
    [Crossref]
  2. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
    [Crossref] [PubMed]
  3. M. A. Bandres and J. C. Gutierrez-Vega, “Airy-Gauss beams and their transformation by paraxial optical systems,” Opt. Express 15, 16719–16728 (2007).
    [Crossref] [PubMed]
  4. I. M. Besieris and A. M. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. 32, 2447–2449 (2007).
    [Crossref] [PubMed]
  5. M. A. Bandres, “Observation of accelerating parabolic beams,” Opt. Express 16, 12866–12871 (2008).
    [Crossref] [PubMed]
  6. M. A. Bandres, “Accelarating parabolic beams,” Opt. Lett. 33, 1678–1680 (2008).
    [Crossref] [PubMed]
  7. M. A. Bandres, “Accelerating beams,” Opt. Lett. 34, 3791–3793 (2009).
    [Crossref] [PubMed]
  8. M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
    [Crossref]
  9. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008).
    [Crossref] [PubMed]
  10. J. Broky, G. A. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16, 12880–12891 (2008).
    [Crossref] [PubMed]
  11. E. Heyman and L. B. Felsen, “Gaussian beam and pulsed beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Soc. Am. 181588–1611 (2001).
    [Crossref]
  12. D. Ludwig, “Wave propagation near a smooth caustic,” Bull. Amer. Math. Soc. 71, 776–779 (1965).
    [Crossref]
  13. Yu. A. Kravtsov, “A modification of the geometrical optics method,” Izv. VUZ Radiofiz. Engl. transl., Radiophys. Quantum Electron. 7, 664–673 (1964).
  14. R. M. Lewis and J. Boersma, “Uniform asymptotic theory of edge diffraction,” Math. Phys. 10, 2291–2305 (1969).
    [Crossref]
  15. S. W. Lee and G.A. Deschamps, “A uniform asymptotic theory of electromagnetic diffraction by a curved wedge,” IEEE Trans. Antennas Propag. AP-24, 25–34 (1976).
  16. L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, New Jersey, 1973).
  17. C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descenets,” Proc. of Cambridge Phi.Soc. 53, 599–611 (1957).
    [Crossref]
  18. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie “Nonlinear generation and manipulation of Airy beams,” Nature Photonics 3, 395–398 (2009).
    [Crossref]

2009 (2)

M. A. Bandres, “Accelerating beams,” Opt. Lett. 34, 3791–3793 (2009).
[Crossref] [PubMed]

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie “Nonlinear generation and manipulation of Airy beams,” Nature Photonics 3, 395–398 (2009).
[Crossref]

2008 (4)

2007 (4)

2001 (1)

E. Heyman and L. B. Felsen, “Gaussian beam and pulsed beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Soc. Am. 181588–1611 (2001).
[Crossref]

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

1976 (1)

S. W. Lee and G.A. Deschamps, “A uniform asymptotic theory of electromagnetic diffraction by a curved wedge,” IEEE Trans. Antennas Propag. AP-24, 25–34 (1976).

1969 (1)

R. M. Lewis and J. Boersma, “Uniform asymptotic theory of edge diffraction,” Math. Phys. 10, 2291–2305 (1969).
[Crossref]

1965 (1)

D. Ludwig, “Wave propagation near a smooth caustic,” Bull. Amer. Math. Soc. 71, 776–779 (1965).
[Crossref]

1964 (1)

Yu. A. Kravtsov, “A modification of the geometrical optics method,” Izv. VUZ Radiofiz. Engl. transl., Radiophys. Quantum Electron. 7, 664–673 (1964).

1957 (1)

C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descenets,” Proc. of Cambridge Phi.Soc. 53, 599–611 (1957).
[Crossref]

Arie, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie “Nonlinear generation and manipulation of Airy beams,” Nature Photonics 3, 395–398 (2009).
[Crossref]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

Bandres, M. A.

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

Besieris, I. M.

Boersma, J.

R. M. Lewis and J. Boersma, “Uniform asymptotic theory of edge diffraction,” Math. Phys. 10, 2291–2305 (1969).
[Crossref]

Brokly, J.

G. A. Siviloglou, J. Brokly, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99 (2007).
[Crossref]

Broky, J.

Chester, C.

C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descenets,” Proc. of Cambridge Phi.Soc. 53, 599–611 (1957).
[Crossref]

Christodoulides, D. N.

Deschamps, G.A.

S. W. Lee and G.A. Deschamps, “A uniform asymptotic theory of electromagnetic diffraction by a curved wedge,” IEEE Trans. Antennas Propag. AP-24, 25–34 (1976).

Dogariu, A.

Ellenbogen, T.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie “Nonlinear generation and manipulation of Airy beams,” Nature Photonics 3, 395–398 (2009).
[Crossref]

Felsen, L. B.

E. Heyman and L. B. Felsen, “Gaussian beam and pulsed beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Soc. Am. 181588–1611 (2001).
[Crossref]

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, New Jersey, 1973).

Friedman, B.

C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descenets,” Proc. of Cambridge Phi.Soc. 53, 599–611 (1957).
[Crossref]

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie “Nonlinear generation and manipulation of Airy beams,” Nature Photonics 3, 395–398 (2009).
[Crossref]

Gutierrez-Vega, J. C.

Heyman, E.

E. Heyman and L. B. Felsen, “Gaussian beam and pulsed beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Soc. Am. 181588–1611 (2001).
[Crossref]

Kravtsov, Yu. A.

Yu. A. Kravtsov, “A modification of the geometrical optics method,” Izv. VUZ Radiofiz. Engl. transl., Radiophys. Quantum Electron. 7, 664–673 (1964).

Lee, S. W.

S. W. Lee and G.A. Deschamps, “A uniform asymptotic theory of electromagnetic diffraction by a curved wedge,” IEEE Trans. Antennas Propag. AP-24, 25–34 (1976).

Lewis, R. M.

R. M. Lewis and J. Boersma, “Uniform asymptotic theory of edge diffraction,” Math. Phys. 10, 2291–2305 (1969).
[Crossref]

Ludwig, D.

D. Ludwig, “Wave propagation near a smooth caustic,” Bull. Amer. Math. Soc. 71, 776–779 (1965).
[Crossref]

Marcuvitz, N.

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, New Jersey, 1973).

Shaarawi, A. M.

Siviloglou, G. A.

Ursell, F.

C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descenets,” Proc. of Cambridge Phi.Soc. 53, 599–611 (1957).
[Crossref]

Voloch-Bloch, N.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie “Nonlinear generation and manipulation of Airy beams,” Nature Photonics 3, 395–398 (2009).
[Crossref]

Am. J. Phys. (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]

Bull. Amer. Math. Soc. (1)

D. Ludwig, “Wave propagation near a smooth caustic,” Bull. Amer. Math. Soc. 71, 776–779 (1965).
[Crossref]

IEEE Trans. Antennas Propag. (1)

S. W. Lee and G.A. Deschamps, “A uniform asymptotic theory of electromagnetic diffraction by a curved wedge,” IEEE Trans. Antennas Propag. AP-24, 25–34 (1976).

Izv. VUZ Radiofiz. Engl. transl., Radiophys. Quantum Electron. (1)

Yu. A. Kravtsov, “A modification of the geometrical optics method,” Izv. VUZ Radiofiz. Engl. transl., Radiophys. Quantum Electron. 7, 664–673 (1964).

J. Opt. Soc. Am. (1)

E. Heyman and L. B. Felsen, “Gaussian beam and pulsed beam dynamics: complex-source and complex-spectrum formulations within and beyond paraxial asymptotics,” J. Opt. Soc. Am. 181588–1611 (2001).
[Crossref]

Math. Phys. (1)

R. M. Lewis and J. Boersma, “Uniform asymptotic theory of edge diffraction,” Math. Phys. 10, 2291–2305 (1969).
[Crossref]

Nature Photonics (1)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie “Nonlinear generation and manipulation of Airy beams,” Nature Photonics 3, 395–398 (2009).
[Crossref]

Opt. Express (3)

Opt. Lett. (5)

Phys. Rev. Lett. (1)

G. A. Siviloglou, J. Brokly, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99 (2007).
[Crossref]

Proc. of Cambridge Phi.Soc. (1)

C. Chester, B. Friedman, and F. Ursell, “An extension of the method of steepest descenets,” Proc. of Cambridge Phi.Soc. 53, 599–611 (1957).
[Crossref]

Other (1)

L. B. Felsen and N. Marcuvitz, Radiation and Scattering of Waves (Prentice-Hall, New Jersey, 1973).

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Figures (6)

Fig. 1.
Fig. 1. Ray description of the Airy beam plotted on a background of the intensity of the Airy beam in Eq. (3) (in Colors). As discussed after Eq. (7), the z and x axes are normalized with respect to zF and W, respectively. Here and in all other figures, λ = 0.5μm, x0 = 200λ, giving W = 139.4λ and zF = 2kW2 = 24.4104λ.

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Fig. 2.
Fig. 2. The intensity ∣U1∣2 of the field radiated by the main lobe of the Airy beam initial distribution (solid blue line), compared with the intensity ∣U∣2 of the total field calculated from the paraxial closed-form solution in Eq. (3) (solid magenta line), and the intensity ∣UGB∣2 of the Gaussian beam that has the same effective width as the main lobe of the Airy beam initial distribution (see discussion after Eq. (7); dashed green line). All intensities are normalized with respect to the maximal intensity of the close-form solution. The fields are calculated at several ranges from the aperture: (a) z = 0; (b) z = √3zF; and z = √8zF. At z = √3zF the effective width of the GB doubles whereas the effective width of the U1 field increases by a factor of 2.3. At z = √8zF the effective width of the GB and the U1 field increase by a factor of 3 and 3.6, respectively.

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Fig. 3.
Fig. 3. Definition of the ray coordinates (ρ1,2,x′1,2,θ1,2) for rays 1 and 2 (green lines) that are shown in Fig. 1. Also shown (in magenta) are the coordinates of the diffracting ray (ρd,xep,θd) and (in black) the shadow boundaries angles θSB± defined in Sec. 4.2. ρ are the distances traversed by the rays, θ are the ray exit angles and x′ are the ray exit points.

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Fig. 4.
Fig. 4. Exit points of the rays corresponding to certain observation points at the observation plane z = 10zF. The observation points are tagged by their location in the 1st, 2nd, or 3rd lobes of the field at this observation plane. As a reference, we also show the intensity of the initial field distribution on the z = 0 plane.

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Fig. 5.
Fig. 5. The intensities of the fields radiated by the main lobe of the initial distribution (∣U1∣2 – blue line) and by the rest of the initial distribution (∣U2∣2 – red line), calculated at the observation plane z = 10zF. The intensity of their sum ∣U1+U2∣2 (black line) agrees extremely well with the intensity of the total field ∣U∣2 calculated from the paraxial closed-form solution in Eq. (3) (magenta line); a difference is seen only near the peak of the main lobe. The locations of the caustic and of the artificial shadow boundary are indicated by tags on the x-axis.

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Fig. 6.
Fig. 6. The intensity ∣UUGO∣2 of the UGO field (green line) compared to the intensity ∣U∣2 of the total field calculated from the paraxial closed-form solution in Eq. (3) (magenta line), at the observation plane z = 10zF. The two results are indistinguishable within the scale of the figure. The location of the caustic is indicated by a tag on the x-axis.

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Equations (31)

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∂ x 2 ϕ + 2 i k ∂ z ϕ = 0 ,
ϕ | z = 0 ≡ ϕ 0 ( x ) = Ai ( x / x 0 ) ,
U ( r ) = ϕ ( r ) e ikz = Ai ( x ̃ − z ̃ 2 / 4 ) exp ( ikz + i x ̃ z ̃ / 2 − i z ̃ 3 / 12 ) ,
U ( r ) = 2 ∫ − ∞ ∞ d x ′ ϕ 0 ( x ′ ) ∂ n ′ G ( r , r ′ ) ,
U 1 ( r ) = 2 ∫ x ep ∞ d x ′ ϕ 0 ( x ′ ) ∂ n ′ G ( r , r ′ ) , U 2 ( r ) = 2 ∫ − ∞ x ep d x ′ ϕ 0 ( x ′ ) ∂ n ′ G ( r , r ′ ) ,
W = [ 1 I ∫ x ep ∞ d x ∣ ϕ 0 ( x ) ∣ 2 ( x − 〈 x 〉 ) 2 ] 1 / 2 .
〈 x 〉 = 1 I ∫ x ep ∞ d x x ∣ ϕ 0 ( x ) ∣ 2 , I = ∫ x ep ∞ d x ∣ ϕ 0 ( x ) ∣ 2 .
ϕ 0 ( x ) ≈ ( − π 2 x ̃ ) − 1 / 4 sin [ 2 / 3 ( − x ̃ ) 3 / 2 + π / 4 ]
= A 0 + exp ( i k ψ 0 + ) + A 0 − exp ( i k ψ 0 − )
≡ ϕ 0 + ( x ) + ϕ 0 − ( x ) ,
k ψ 0 ± ( x ) = ∓ [ 2 / 3 ( − x ̃ ) 3 / 2 + π / 4 ] , A 0 ± ( x ) = ± ( i / 2 ) ( − π 2 x ̃ ) − 1 / 4 .
θ ± ( x ′ ) = sin − 1 [ ∂ x ′ ψ 0 ± ( x ′ ) ] = ± sin − 1 [ ( − x ′ / x 0 ) 1 / 2 / ( k x 0 ) ] .
( x , z ) = ( x ′ + ρ sin θ ( x ′ ) , ρ cos θ ( x ′ ) ) ,
U GO + ( r ) = A + ( r ) exp [ i k ψ + ( r ) ] .
ψ + ( r ) = ψ 0 + ( x ′ ) + ρ ,
A + ( r ) = A 0 + ( J 0 / J ) 1 / 2 = A 0 + [ 1 + ρ ( ∂ x ′ 2 ψ 0 + / cos 2 θ ) ] − 1 / 2 = A 0 + [ 1 − ρ / ρ c ] − 1 / 2 ,
∂ x ′ ψ 0 + ( x ′ ) = sin θ ( x ′ ) = ( x − x ′ ) / ρ .
U UGO + = exp ( ik φ 0 ) [ g Ai ( − σ k 2 / 3 ) + i h k − 1 / 3 Ai ′ ( − σ k 2 / 3 ) ] ,
φ 0 = 1 2 ( ψ 1 + + ψ 2 + ) , σ = [ 3 4 ( ψ 1 + + ψ 2 + ) ] 2 / 3
g = 1 2 σ 1 / 4 ( A 1 + + A 2 + ) , h = 1 2 σ − 1 / 4 ( A 2 + − A 1 + ) .
U d ± = A 0 d ± exp ( ik ψ 0 d ± ) D ± G ( ρ d ) ,
D ± = − 2 i cos θ d / ( sin θ SB ± − sin θ d ) ,
U UAT + = U d + + [ F ( η ) − F ̂ ( η ) ] U GO 2 + ,
F ( η ) = exp ( i π / 4 ) π − 1 / 2 ∫ η ∞ d t exp ( − it ) 2 , F ̂ ( η ) = ( 2 η π ) − 1 exp ( − i η 2 − iπ / 4 ) ,
η = sgn ( θ SB + − θ 2 ) [ k ( ψ 2 + − ψ d + ) ] 1 / 2 .
U 2 ± = ∫ − ∞ x ep d x ' A ± ( x ' ) exp ( ik ψ ± ( x ' ) )
A ± = A 0 ± cos θ exp ( − i π / 4 ) ( k / 2 π ρ ) 1 / 2 , ψ ± = ρ + ψ 0 ± ,
U s 1,2 + = A ± ( x ′ ) [ − 2 π / i k ∂ x ′ 2 ψ + ( x ′ ) ] 1 / 2 exp ( ik ψ + ( x ′ ) ) | x ′ = x ′ 1,2 .
U ep ± = A ± ( x ′ ) [ ik ∂ x ′ ψ ± ( x ′ ) ] − 1 exp ( ik ψ ± ( x ′ ) ) | x ′ = x ep .
U ~ U 1 + U 2 = U 1 + U GO 1 + + U GO 2 + + U d + + U d − ,
U ~ U UGO + .

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