Abstract

We generate a broadband “white light” Airy beam and characterize the dependence of the beam properties on wavelength. Experimental results are presented showing that the beam’s deflection coefficient and its characteristic length are wavelength dependent. In contrast the aperture coefficient is not wavelength dependent. However, this coefficient depends on the spatial coherence of the beam. We model this behaviour theoretically by extending the Gaussian-Schell model to describe the effect of spatial coherence on the propagation of Airy beams. The experimental results are compared to the model and good agreement is observed.

© 2009 Optical Society of America

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References

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  1. J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-Free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
    [CrossRef] [PubMed]
  2. J. Durnin, "Exact Solutions for Nondiffracting Beams.1. The Scalar Theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
    [CrossRef]
  3. Q1. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy beams," Phys. Rev. Lett. 99 (2007).
    [CrossRef]
  4. M. V. Berry and N. L. Balazs, "Non-Spreading Wave Packets," Am. J. Phys. 47, 264-267 (1979).
    [CrossRef]
  5. Q2. J. Baumgartl, M. Mazilu, and K. Dholakia, "Optically mediated particle clearing using Airy wavepackets," Nature Photonics 2, 675-678 (2008).
    [CrossRef]
  6. H. I. Sztul and R. R. Alfano, "The Poynting vector and angular momentum of Airy beams," Opt. Express 16, 9411-9416 (2008).
    [CrossRef] [PubMed]
  7. Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207 - 211 (1998).
    [CrossRef]
  8. 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]
  9. G. A. Siviloglou and D. N. Christodoulides, "Accelerating finite energy Airy beams," Opt. Lett. 32, 979-981 (2007).
    [CrossRef] [PubMed]
  10. A. E. Siegman, Lasers (University Science Books, 1986).
  11. M. A. Bandres and J. C. Guti’errez-Vega, "Airy-Gauss beams and their transformation by paraxial optical systems," Opt. Express 15, 16719-16728 (2007).
    [CrossRef] [PubMed]
  12. Q. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
    [CrossRef]
  13. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Ballistic dynamics of Airy beams," Opt. Lett. 33, 207-209 (2008).
    [CrossRef] [PubMed]
  14. P. Fischer, C. T. A. Brown, J. E. Morris, C. L’opez-Mariscal, E. M. Wright, W. Sibbett, and K. Dholakia, "White light propagation invariant beams," Opt. Express 13, 6657-6666 (2005).
    [CrossRef] [PubMed]

2008 (4)

2007 (3)

2005 (1)

1998 (1)

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207 - 211 (1998).
[CrossRef]

1988 (1)

Q. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

1987 (2)

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-Free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

J. Durnin, "Exact Solutions for Nondiffracting Beams.1. The Scalar Theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

1979 (1)

M. V. Berry and N. L. Balazs, "Non-Spreading Wave Packets," Am. J. Phys. 47, 264-267 (1979).
[CrossRef]

Alfano, R. R.

Balazs, N. L.

M. V. Berry and N. L. Balazs, "Non-Spreading Wave Packets," Am. J. Phys. 47, 264-267 (1979).
[CrossRef]

Bandres, M. A.

Baumgartl, J.

Q2. J. Baumgartl, M. Mazilu, and K. Dholakia, "Optically mediated particle clearing using Airy wavepackets," Nature Photonics 2, 675-678 (2008).
[CrossRef]

Berry, M. V.

M. V. Berry and N. L. Balazs, "Non-Spreading Wave Packets," Am. J. Phys. 47, 264-267 (1979).
[CrossRef]

Bouchal, Z.

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207 - 211 (1998).
[CrossRef]

Broky, J.

Brown, C. T. A.

Chlup, M.

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207 - 211 (1998).
[CrossRef]

Christodoulides, D. N.

Dholakia, K.

Q2. J. Baumgartl, M. Mazilu, and K. Dholakia, "Optically mediated particle clearing using Airy wavepackets," Nature Photonics 2, 675-678 (2008).
[CrossRef]

Dogariu, A.

Durnin, J.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-Free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

J. Durnin, "Exact Solutions for Nondiffracting Beams.1. The Scalar Theory," J. Opt. Soc. Am. A 4, 651-654 (1987).
[CrossRef]

Eberly, J. H.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-Free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Fischer, P.

Friberg, A. T.

Q. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

Guti’errez-Vega, J. C.

He, Q.

Q. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

Mazilu, M.

Q2. J. Baumgartl, M. Mazilu, and K. Dholakia, "Optically mediated particle clearing using Airy wavepackets," Nature Photonics 2, 675-678 (2008).
[CrossRef]

Miceli, J. J.

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-Free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Morris, J. E.

Siviloglou, G. A.

Sztul, H. I.

Turunen, J.

Q. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

Wagner, J.

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207 - 211 (1998).
[CrossRef]

Am. J. Phys. (1)

M. V. Berry and N. L. Balazs, "Non-Spreading Wave Packets," Am. J. Phys. 47, 264-267 (1979).
[CrossRef]

J. Opt. Soc. Am. A (1)

Nature Photonics (1)

Q2. J. Baumgartl, M. Mazilu, and K. Dholakia, "Optically mediated particle clearing using Airy wavepackets," Nature Photonics 2, 675-678 (2008).
[CrossRef]

Opt. Commun. (2)

Z. Bouchal, J. Wagner, and M. Chlup, "Self-reconstruction of a distorted nondiffracting beam," Opt. Commun. 151, 207 - 211 (1998).
[CrossRef]

Q. He, J. Turunen, and A. T. Friberg, "Propagation and imaging experiments with Gaussian Schell-model beams," Opt. Commun. 67, 245-250 (1988).
[CrossRef]

Opt. Express (4)

Opt. Lett. (2)

Phys. Rev. Lett. (2)

J. Durnin, J. J. Miceli, Jr., and J. H. Eberly, "Diffraction-Free Beams," Phys. Rev. Lett. 58, 1499-1501 (1987).
[CrossRef] [PubMed]

Q1. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, "Observation of accelerating Airy beams," Phys. Rev. Lett. 99 (2007).
[CrossRef]

Other (1)

A. E. Siegman, Lasers (University Science Books, 1986).

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Figures (4)

Fig. 1.
Fig. 1.

On the left is the experimental set up. A visible mirror M (Comar 25MF02) was used to select the visible wavelength range from the source. On the right is the spectrum taken after L5. Wavelength components were selected using interference filters F (Comar, 3nm bandwidth). L1 and L2 are achromatic lenses used to expand the beam with focal lengths of 50mm and 500mm respectively. The grating was imaged onto a 10° prism using achromatic lenses L3 (f=500mm) and L4 (f=200mm). An aperture was used to select the first order diffracted beam. The dispersion compensated Airy beam was viewed on a CCD camera via achromatic lens L5 (f=250mm).

Fig. 2.
Fig. 2.

Shown is the profile of a dispersion compensated “white light” Airy beam, the main lobe is shown by the dashed line and is taken from the cross section of an unsaturated picture. The insert shows an image of the beam at z=0 with the red line indicating where the cross-section was taken.

Fig. 3.
Fig. 3.

Plotted are the data points showing how the position of the main beam lobe varied along the propagation direction, the lines indicate the parabolic fit to these points. The inserted table displays the values of the deflection coefficients b 0. On the right are pictures of the beams taken at z=0, (a) 515nm, (b) 546nm, (c) 578nm and (d) 633nm.

Fig. 4.
Fig. 4.

A graph of how the aperture coefficient a 0 for the horizontal lobes varies with the different wavelength beams, a dashed blue line is shown to guide the eye. Error bars of 10% are shown. The red (square) points show the values calculated using Eq. 20.

Tables (2)

Tables Icon

Table 1. Beam parameter x 0 determined using different methods.

Tables Icon

Table 2. Beam parameter a 0 determined using different methods.

Equations (25)

Equations on this page are rendered with MathJax. Learn more.

x2u0(x,y,z)+y2u0(x,y,z)2ikzu0(x,y,z)=0
A=u0 (x,y,z) exp (ikz) x̂
E=ikAik(·A)
H=ε0μ0×A,
2ikzux+x2ux=0
2ikzuy+y2uy=0.
ux (x,z=0)=Ai (xx0)exp(a0xx0)
ûx (kx,z=zslm)=x0 exp (a0x02kx2) exp (i3(x03kx33a02x0kxia03))
ux (x,z)= ik2πz exp (ik2z(x122x1x+x2))ux(x1,z=0)dx1
= ik2πz exp (ik2z(x122x1x+x2)) Ai (x1x0)ea0x1x0dx1
=Ai (xxm(z)x0+ia0zkx02) exp (a0(xxm(z))x0iz312k3x06+ia02z2kx02+izx2kx03)
u2 (x2)= ik2πB exp (ik2B(Ax122x1x2+Dx22)) u1 (x1) dx1
u2 (x2)= ik2πB exp (ik2B(Ax122x1x2+Dx22)) Ai (x1x0) ea0x1x0 dx1
=1AAi(x2Ax0B24A2k2x04+ia0BAkx02)exp(ikC2Ax22)
exp (a0x2Ax0a0B22A2k2x04iB312A3k3x06+ia02B2Akx02+iBx22A2kx03)
xm(z)=(4k2x03)1 (B+Dz)2 (A+Cz)1
W (r1,r2,z0)=u (x1,y1,z0)u(x2,y2,z0)μ(r1r2)
μ (r1r2)=exp (r1r22(2σμ2))
u0 (x,y,0)=exp ((x2+y2)ω02) exp (ic03(x3+y3)3)
x0=c0fk,
a0=ω02+2σμ22c02ω02σμ2,
b0=14k2x03.
M2=π ω02 (λzr)
a0=1+(M2)22c02ωslm
b0z2=2λ216π2x03z2

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