Abstract

We propose and demonstrate a two-dimensional (2D) Airy-like beam generation technique based on arrayed waveguides. We show that the output beams with quasi-Airy amplitude and cubic-like phase from an arrayed waveguide end have 2D Airy-like patterns. These beams have the ability to remain quasi-nondiffracting, transverse accelerating, and self-healing during propagation. We also analyze wave propagation along this arrayed waveguide using coupled-mode theory and the beam propagation method.

© 2013 Optical Society of America

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    [CrossRef]
  3. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
    [CrossRef]
  4. G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
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  6. G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008).
    [CrossRef]
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    [CrossRef]
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  13. Y. Jiang, K. Huang, and X. Lu, “Airy-related beam generated from flat-topped Gaussian beams,” J. Opt. Soc. Am. A 29, 1412–1416 (2012).
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  14. A. W. Snyder, “Coupled-mode theory for optical fibers,” J. Opt. Soc. Am. 62, 1267–1277 (1972).
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  15. W. Huang and C. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
    [CrossRef]

2012

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. Lacourt, and J. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012).
[CrossRef]

Y. Jiang, K. Huang, and X. Lu, “Airy-related beam generated from flat-topped Gaussian beams,” J. Opt. Soc. Am. A 29, 1412–1416 (2012).
[CrossRef]

2011

2010

2008

2007

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
[CrossRef]

G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

1996

K. Unnikrishnan and A. Rau, “Uniqueness of the Airy packet in quantum mechanics,” Am. J. Phys. 64, 1034–1035 (1996).
[CrossRef]

1993

W. Huang and C. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[CrossRef]

1979

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

1972

Akturk, S.

Balazs, N.

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

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[CrossRef]

Berry, M. V.

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

Broky, J.

Chen, H.

Chen, Z.

Christodoulides, D.

G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008).
[CrossRef]

G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Christodoulides, D. N.

Courvoisier, F.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. Lacourt, and J. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012).
[CrossRef]

Dholakia, K.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[CrossRef]

Ding, J.

Dogariu, A.

Dudley, J.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. Lacourt, and J. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012).
[CrossRef]

Froehly, L.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. Lacourt, and J. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012).
[CrossRef]

Furfaro, L.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. Lacourt, and J. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012).
[CrossRef]

Huang, K.

Huang, W.

W. Huang and C. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[CrossRef]

Jacquot, M.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. Lacourt, and J. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012).
[CrossRef]

Jiang, Y.

Lacourt, P.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. Lacourt, and J. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012).
[CrossRef]

Liu, Y.

Lu, C.

Lu, X.

Mathis, A.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. Lacourt, and J. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012).
[CrossRef]

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[CrossRef]

Rau, A.

K. Unnikrishnan and A. Rau, “Uniqueness of the Airy packet in quantum mechanics,” Am. J. Phys. 64, 1034–1035 (1996).
[CrossRef]

Salandrino, A.

Siviloglou, G.

G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008).
[CrossRef]

G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

Siviloglou, G. A.

Snyder, A. W.

Soylu, B.

Unnikrishnan, K.

K. Unnikrishnan and A. Rau, “Uniqueness of the Airy packet in quantum mechanics,” Am. J. Phys. 64, 1034–1035 (1996).
[CrossRef]

Wang, H. T.

Wang, S.

Xu, C.

W. Huang and C. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[CrossRef]

Yalizay, B.

Yin, X.

Zhang, B. F.

Zhang, P.

Zhang, X.

Zheng, Z.

Am. J. Phys.

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

K. Unnikrishnan and A. Rau, “Uniqueness of the Airy packet in quantum mechanics,” Am. J. Phys. 64, 1034–1035 (1996).
[CrossRef]

Appl. Opt.

Appl. Phys. Lett.

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. Lacourt, and J. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012).
[CrossRef]

IEEE J. Quantum Electron.

W. Huang and C. Xu, “Simulation of three-dimensional optical waveguides by a full-vector beam propagation method,” IEEE J. Quantum Electron. 29, 2639–2649 (1993).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Nat. Photonics

J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[CrossRef]

Opt. Express

Opt. Lett.

Phys. Rev. Lett.

G. Siviloglou, J. Broky, A. Dogariu, and D. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
[CrossRef]

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Figures (8)

Fig. 1.
Fig. 1.

Intensity profiles of (a) a 2D finite-energy Airy beam and (b) its main lobe and outside lobes at the input z=0μm. (c) and (d) Amplitude (dotted line) and phase distribution (solid line) of a 2D finite-energy Airy beam at input z=0μm, y=4.3μm and after propagating z=70μm, y=4.3μm, respectively.

Fig. 2.
Fig. 2.

Schematic diagram of (a) model A and (b) its waveguide distribution in cross section. (c) Refractive index profile of the horizontal (vertical) arrayed waveguide of model A.

Fig. 3.
Fig. 3.

Numerical results of wave propagation along model A using CMT. (a) Coupling electric field distribution in XZ cross section of model A, (b) electric field profile in z=9.7mm, (c) the x-axial amplitude (dotted line) and phase curve (solid line) of the electric field at z=9.7mm, and (d) the corresponding optical simulation results based on the BPM.

Fig. 4.
Fig. 4.

Cross-sectional profile of model B.

Fig. 5.
Fig. 5.

Optical field profile of output beam from model B at propagation distance (a) z=0μm, (b) z=200μm, (c) z=400μm, (d) z=600μm in free space, and (e)–(h) the corresponding ideal finite-energy Airy beams.

Fig. 6.
Fig. 6.

Amplitude of the ideal finite-energy 2D Airy beam (solid line) and the output beams from model A (circle ○) and model B (triangle ▵) as a function of propagation distance in free space.

Fig. 7.
Fig. 7.

Deflection of the ideal finite-energy 2D Airy beam (solid line) and the output beams from model A (circles) and model B (triangles) as a function of propagation distance in free space.

Fig. 8.
Fig. 8.

Self-healing of the output beam from model B when its main lobe is blocked. (a) A 2D Airy-like beam whose main lobe is blocked at the input z=0μm; calculated transverse power flow at (b) z=50μm, (c) z=150μm, and (d) z=250μm (the arrow denotes the direction of the transverse Poynting vector).

Equations (7)

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

iϕz+12k2ϕx2+12k2ϕy2=0,
φ(x,y,z)=m=x,yum(sm,ξm),
u(sm,ξm)=Ai[smξm24vmξm+iamξm]exp[amsmamξm22amvmξm+i(ξm312+(am2vm2+sm)ξm2+vmsmvmξm22)].
ddz[a1(z)a2(z)a9(z)]=j·[β1κ12κ19κ21β2κ29κ91κ92β9][a1(z)a2(z)a9(z)].
κpq=ω2Aq(εqε)ep·eqdA,
Ep(x,y,z)=ep(x,y)exp(iβpz).
Et(x,y,z)=pap(z)ep(x,y),

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