Abstract

Self-accelerating optical beams form as a direct outcome of interference, initiated by a predesigned initial condition. In a similar fashion, quantum mechanical particles exhibit force-free acceleration as a result of interference effects following proper preparation of the initial Schrödinger wave function. Indeed, interference is at the heart of such wave packets, and hence it is implied that self-accelerating wave packets must be coherent entities. Counter to that, we demonstrate theoretically and experimentally spatially incoherent self-accelerating beams, in both the paraxial and the nonparaxial domains. We show that in principle, the transverse correlation distance can be as short as a single wavelength, while a properly designed initial beam will give rise to shape-preserving acceleration for the same distance as a coherent accelerating beam propagating on the same trajectory. These findings expand the understanding of the relation between coherence and accelerating beams, and may have implications for the design of self-accelerating quantum wave packets with limited quantum coherence.

© 2015 Optical Society of America

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References

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2014 (3)

S. Jia, J. C. Vaughan, and X. Zhuang, “Isotropic three-dimensional super-resolution imaging with a self-bending point spread function,” Nat. Photonics 8, 302–306 (2014).
[Crossref]

T. Vettenburg, H. I. C. Dalgarno, J. Nylk, C. Coll-Lladó, D. E. K. Ferrier, T. Čižmár, F. J. Gunn-Moore, and K. Dholakia, “Light-sheet microscopy using an Airy beam,” Nat. Methods 11, 541–544 (2014).
[Crossref]

R. Schley, I. Kaminer, E. Greenfield, R. Bekenstein, Y. Lumer, and M. Segev, “Loss-proof self-accelerating beams and their use in non-paraxial manipulation of particles’ trajectories,” Nat. Commun. 5, 5189 (2014).
[Crossref]

2013 (6)

M. A. Bandres, M. A. Alonso, I. Kaminer, and M. Segev, “Three-dimensional accelerating electromagnetic waves,” Opt. Express 21, 13917–13929 (2013).
[Crossref]

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature 494, 331–335 (2013).
[Crossref]

I. D. Chremmos and N. K. Efremidis, “Nonparaxial accelerating Bessel-like beams,” Phys. Rev. A 88, 063816 (2013).
[Crossref]

Y. Hu, D. Bongiovanni, Z. Chen, and R. Morandotti, “Multipath multicomponent self-accelerating beams through spectrum-engineered position mapping,” Phys. Rev. A 88, 043809 (2013).
[Crossref]

I. Kaminer, J. Nemirovsky, K. G. Makris, and M. Segev, “Self-accelerating beams in photonic crystals,” Opt. Express 21, 8886–8896 (2013).
[Crossref]

Y. Dong, L. Zhang, J. Luo, W. Wen, and Y. Zhang, “Degree of paraxiality of coherent and partially coherent Airy beams,” Opt. Laser Technol. 49, 1–5 (2013).
[Crossref]

2012 (11)

H. T. Eyyuboğlu and E. Sermutlu, “Partially coherent Airy beam and its propagation in turbulent media,” Appl. Phys. B 110, 451–457 (2012).
[Crossref]

I. Kaminer, J. Nemirovsky, and M. Segev, “Self-accelerating self-trapped nonlinear beams of Maxwell’s equations,” Opt. Express 20, 18827–18835 (2012).
[Crossref]

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett. 37, 2820–2822 (2012).
[Crossref]

I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett. 108, 113903 (2012).
[Crossref]

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[Crossref]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
[Crossref]

I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, “Bessel-like optical beams with arbitrary trajectories,” Opt. Lett. 37, 5003–5005 (2012).
[Crossref]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. 108, 163901 (2012).
[Crossref]

F. Courvoisier, A. Mathis, L. Froehly, R. Giust, L. Furfaro, P. A. Lacourt, M. Jacquot, and J. M. Dudley, “Sending femtosecond pulses in circles: highly nonparaxial accelerating beams,” Opt. Lett. 37, 1736–1738 (2012).
[Crossref]

I. Kaminer, E. Greenfield, R. Bekenstein, J. Nemirovsky, A. Mathis, L. Froehly, F. Courvoisier, and M. Segev, “Accelerating beyond the horizon,” Opt. Photon. News 23(12), 26 (2012).
[Crossref]

A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. 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]

2011 (11)

I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express 19, 23132–23139 (2011).
[Crossref]

R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express 19, 23706–23715 (2011).
[Crossref]

R. El-Ganainy, K. G. Makris, M. A. Miri, D. N. Christodoulides, and Z. Chen, “Discrete beam acceleration in uniform waveguide arrays,” Phys. Rev. A 84, 023842 (2011).
[Crossref]

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106, 213903 (2011).
[Crossref]

A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear Airy beams,” Phys. Rev. A 84, 021807 (2011).
[Crossref]

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106, 213902 (2011).
[Crossref]

L. Froehly, F. Courvoisier, A. Mathis, M. Jacquot, L. Furfaro, R. Giust, P. A. Lacourt, and J. M. Dudley, “Arbitrary accelerating micron-scale caustic beams in two and three dimensions,” Opt. Express 19, 16455–16465 (2011).
[Crossref]

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36, 2883–2885 (2011).
[Crossref]

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, “Generation and near-field imaging of Airy surface plasmons,” Phys. Rev. Lett. 107, 116802 (2011).
[Crossref]

P. Zhang, S. Wang, Y. Liu, X. Yin, C. Lu, Z. Chen, and X. Zhang, “Plasmonic Airy beams with dynamically controlled trajectories,” Opt. Lett. 36, 3191–3193 (2011).
[Crossref]

L. Li, T. Li, S. M. Wang, C. Zhang, and S. N. Zhu, “Plasmonic Airy beam generated by in-plane diffraction,” Phys. Rev. Lett. 107, 126804 (2011).
[Crossref]

2010 (2)

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010).
[Crossref]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010).
[Crossref]

2009 (3)

2008 (1)

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

2007 (2)

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

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

1997 (1)

M. Mitchell and M. Segev, “Self-trapping of incoherent white light,” Nature 387, 880–883 (1997).
[Crossref]

1996 (1)

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[Crossref]

1991 (1)

1989 (1)

J. A. Giannini and R. I. Joseph, “The role of the second Painlevé transcendent in nonlinear optics,” Phys. Lett. A 141, 417–419 (1989).
[Crossref]

1979 (1)

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

1971 (1)

B. Crosignani, “Light scattering by a rotating disk,” J. Appl. Phys. 42, 399 (1971).
[Crossref]

Abdollahpour, D.

A. Lotti, D. Faccio, A. Couairon, D. G. Papazoglou, P. Panagiotopoulos, D. Abdollahpour, and S. Tzortzakis, “Stationary nonlinear Airy beams,” Phys. Rev. A 84, 021807 (2011).
[Crossref]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010).
[Crossref]

Aleahmad, P.

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[Crossref]

Alonso, M. A.

Arie, A.

N. Voloch-Bloch, Y. Lereah, Y. Lilach, A. Gover, and A. Arie, “Generation of electron Airy beams,” Nature 494, 331–335 (2013).
[Crossref]

I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett. 108, 113903 (2012).
[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.

Baumgartl, J.

Bekenstein, R.

R. Schley, I. Kaminer, E. Greenfield, R. Bekenstein, Y. Lumer, and M. Segev, “Loss-proof self-accelerating beams and their use in non-paraxial manipulation of particles’ trajectories,” Nat. Commun. 5, 5189 (2014).
[Crossref]

I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. 108, 163901 (2012).
[Crossref]

I. Kaminer, E. Greenfield, R. Bekenstein, J. Nemirovsky, A. Mathis, L. Froehly, F. Courvoisier, and M. Segev, “Accelerating beyond the horizon,” Opt. Photon. News 23(12), 26 (2012).
[Crossref]

R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express 19, 23706–23715 (2011).
[Crossref]

Berry, M. V.

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

Bongiovanni, D.

Y. Hu, D. Bongiovanni, Z. Chen, and R. Morandotti, “Multipath multicomponent self-accelerating beams through spectrum-engineered position mapping,” Phys. Rev. A 88, 043809 (2013).
[Crossref]

Broky, J.

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

Cannan, D.

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
[Crossref]

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett. 37, 2820–2822 (2012).
[Crossref]

Chen, Z.

Y. Hu, D. Bongiovanni, Z. Chen, and R. Morandotti, “Multipath multicomponent self-accelerating beams through spectrum-engineered position mapping,” Phys. Rev. A 88, 043809 (2013).
[Crossref]

P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
[Crossref]

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett. 37, 2820–2822 (2012).
[Crossref]

I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, “Bessel-like optical beams with arbitrary trajectories,” Opt. Lett. 37, 5003–5005 (2012).
[Crossref]

P. Zhang, S. Wang, Y. Liu, X. Yin, C. Lu, Z. Chen, and X. Zhang, “Plasmonic Airy beams with dynamically controlled trajectories,” Opt. Lett. 36, 3191–3193 (2011).
[Crossref]

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, “Trapping and guiding microparticles with morphing autofocusing Airy beams,” Opt. Lett. 36, 2883–2885 (2011).
[Crossref]

R. El-Ganainy, K. G. Makris, M. A. Miri, D. N. Christodoulides, and Z. Chen, “Discrete beam acceleration in uniform waveguide arrays,” Phys. Rev. A 84, 023842 (2011).
[Crossref]

M. Mitchell, Z. Chen, M. Shih, and M. Segev, “Self-trapping of partially spatially incoherent light,” Phys. Rev. Lett. 77, 490–493 (1996).
[Crossref]

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy-Bessel wave packets as versatile linear light bullets,” Nat. Photonics 4, 103–106 (2010).
[Crossref]

Chremmos, I. D.

Christodoulides, D. N.

I. D. Chremmos, Z. Chen, D. N. Christodoulides, and N. K. Efremidis, “Bessel-like optical beams with arbitrary trajectories,” Opt. Lett. 37, 5003–5005 (2012).
[Crossref]

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[Crossref]

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106, 213903 (2011).
[Crossref]

R. El-Ganainy, K. G. Makris, M. A. Miri, D. N. Christodoulides, and Z. Chen, “Discrete beam acceleration in uniform waveguide arrays,” Phys. Rev. A 84, 023842 (2011).
[Crossref]

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I. Kaminer, J. Nemirovsky, K. G. Makris, and M. Segev, “Self-accelerating beams in photonic crystals,” Opt. Express 21, 8886–8896 (2013).
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A. Mathis, F. Courvoisier, L. Froehly, L. Furfaro, M. Jacquot, P. A. Lacourt, and J. M. Dudley, “Micromachining along a curve: femtosecond laser micromachining of curved profiles in diamond and silicon using accelerating beams,” Appl. Phys. Lett. 101, 071110 (2012).
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P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
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R. El-Ganainy, K. G. Makris, M. A. Miri, D. N. Christodoulides, and Z. Chen, “Discrete beam acceleration in uniform waveguide arrays,” Phys. Rev. A 84, 023842 (2011).
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P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
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Y. Hu, D. Bongiovanni, Z. Chen, and R. Morandotti, “Multipath multicomponent self-accelerating beams through spectrum-engineered position mapping,” Phys. Rev. A 88, 043809 (2013).
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[Crossref]

P. Zhang, Y. Hu, D. Cannan, A. Salandrino, T. Li, R. Morandotti, X. Zhang, and Z. Chen, “Generation of linear and nonlinear nonparaxial accelerating beams,” Opt. Lett. 37, 2820–2822 (2012).
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I. Kaminer, J. Nemirovsky, K. G. Makris, and M. Segev, “Self-accelerating beams in photonic crystals,” Opt. Express 21, 8886–8896 (2013).
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I. Kaminer, E. Greenfield, R. Bekenstein, J. Nemirovsky, A. Mathis, L. Froehly, F. Courvoisier, and M. Segev, “Accelerating beyond the horizon,” Opt. Photon. News 23(12), 26 (2012).
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R. Schley, I. Kaminer, E. Greenfield, R. Bekenstein, Y. Lumer, and M. Segev, “Loss-proof self-accelerating beams and their use in non-paraxial manipulation of particles’ trajectories,” Nat. Commun. 5, 5189 (2014).
[Crossref]

I. Kaminer, J. Nemirovsky, K. G. Makris, and M. Segev, “Self-accelerating beams in photonic crystals,” Opt. Express 21, 8886–8896 (2013).
[Crossref]

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[Crossref]

I. Kaminer, J. Nemirovsky, and M. Segev, “Self-accelerating self-trapped nonlinear beams of Maxwell’s equations,” Opt. Express 20, 18827–18835 (2012).
[Crossref]

P. Aleahmad, M.-A. Miri, M. S. Mills, I. Kaminer, M. Segev, and D. N. Christodoulides, “Fully vectorial accelerating diffraction-free Helmholtz beams,” Phys. Rev. Lett. 109, 203902 (2012).
[Crossref]

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[Crossref]

I. Dolev, I. Kaminer, A. Shapira, M. Segev, and A. Arie, “Experimental observation of self-accelerating beams in quadratic nonlinear media,” Phys. Rev. Lett. 108, 113903 (2012).
[Crossref]

I. Kaminer, E. Greenfield, R. Bekenstein, J. Nemirovsky, A. Mathis, L. Froehly, F. Courvoisier, and M. Segev, “Accelerating beyond the horizon,” Opt. Photon. News 23(12), 26 (2012).
[Crossref]

E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106, 213902 (2011).
[Crossref]

I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106, 213903 (2011).
[Crossref]

I. Kaminer, Y. Lumer, M. Segev, and D. N. Christodoulides, “Causality effects on accelerating light pulses,” Opt. Express 19, 23132–23139 (2011).
[Crossref]

R. Bekenstein and M. Segev, “Self-accelerating optical beams in highly nonlocal nonlinear media,” Opt. Express 19, 23706–23715 (2011).
[Crossref]

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E. Greenfield, M. Segev, W. Walasik, and O. Raz, “Accelerating light beams along arbitrary convex trajectories,” Phys. Rev. Lett. 106, 213902 (2011).
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D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, “Spatiotemporal Airy light bullets in the linear and nonlinear regimes,” Phys. Rev. Lett. 105, 253901 (2010).
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[Crossref]

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I. Kaminer, M. Segev, and D. N. Christodoulides, “Self-accelerating self-trapped optical beams,” Phys. Rev. Lett. 106, 213903 (2011).
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I. Kaminer, R. Bekenstein, J. Nemirovsky, and M. Segev, “Nondiffracting accelerating wave packets of Maxwell’s equations,” Phys. Rev. Lett. 108, 163901 (2012).
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P. Zhang, Y. Hu, T. Li, D. Cannan, X. Yin, R. Morandotti, Z. Chen, and X. Zhang, “Nonparaxial Mathieu and Weber accelerating beams,” Phys. Rev. Lett. 109, 193901 (2012).
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Science (1)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
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Other (1)

L. Mandel and E. Wolf, Optical Coherence and Quantum Optics (Cambridge University, 1995).

Supplementary Material (1)

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Figures (4)

Fig. 1.
Fig. 1. Propagation dynamics of coherent and incoherent accelerating beams at wavelength λ=532nm. (a) Propagation of a coherent nonparaxial accelerating beam with radius parameter α=800. The width of the main lobe is approximately 2 μm. (b) Propagation of an incoherent nonparaxial accelerating beam with the radius parameter uniformly distributed in the range α[784,816]. The calculated bending angle in both cases is 160°, and the width of the main lobe is approximately 9 μm. (c) Spatial correlation function |μ(x1,x2)|2 at z=0 for the incoherent beam presented in (b). The spatial correlation distance at z=0 is calculated to be lc2μm.
Fig. 2.
Fig. 2. Finite-size effects on coherent and incoherent accelerating beams. (a) Propagation of a coherent paraxial accelerating beam with an exponential truncation factor of a=0.03, (b) propagation of an incoherent paraxial accelerating beam with exponential truncation factor of a=0.03 and transverse correlation distance of lc1.5x0 at z=0. The incoherent beam is described by the Gaussian Schell-model of Eq. (9). (c) Spatial correlation function |μ(x1,x2)|2 at z=0 for the incoherent beam presented in (b); (d) acceleration distance zzcc versus coherence length lc for paraxial incoherent beams, with a=0.03, for beams obeying the Schell model (blue) and for beams constructed by the modal decomposition of Eq. (8) (green). In the coherent case, zacc=15z0. In the Schell beams the acceleration dynamics deteriorates only when lc is on the order of 1.5x0, which corresponds to one half of the main lobe. The incoherent beam generated using modes that accelerate along the same trajectory continues to accelerate even though the coherence length is smaller than x0.
Fig. 3.
Fig. 3. Experiments with partially spatially incoherent accelerating beams. (a) Experimental setup. A laser beam of wavelength 532 nm is reflected from an SLM that imposes the appropriate phase profile. When the SLM alternates among these phase profiles faster than the integration time of our camera, the camera measures only the time-average intensity distribution. A cylindrical lens Fourier transforms the beam, which is tightly focused and imaged into a CCD camera using two objective lenses (X60, NA=0.85). (b) Measured propagation dynamics of a coherent nonparaxial accelerating beam, with radius parameter α=800. The bending angle is 65°, and width of the main lobe is approximately 2 μm. (c),(d) Measured propagation dynamics of two examples of incoherent nonparaxial accelerating beams, generated by temporally modulating the phase profiles on the SLM corresponding to the radius parameters in the ranges α[786,816] and α[730,880], respectively. The beams in both examples consist of 16 modes. The beam in (c) has a bending angle of 70° and width of the main lobe of approximately 10 μm, with a transverse correlation distance at z=0 of lc2μm, while the beam in (d) has a bending angle of 65° and width of the main lobe of approximately 20 μm, with a transverse correlation distance at z=0 of lc0.7μm.
Fig. 4.
Fig. 4. Experiments with two-dimensional incoherent paraxial accelerating beams. (a) Experimental setup, where L stands for lens and BS for beamsplitter. The incoherent beam is generated by a rotating diffuser followed by an SLM, with x0=75μm and a=0.075. (b)–(g) Transverse intensity patterns taken at propagation distance z=0 [(b), (d), (f)] and at z=20cm [(c), (e), (g)] for a coherent Airy beam [(b), (c)], an incoherent Airy beam with lc215μm[(d), (e)], and an incoherent accelerating beam with lc115μm [(f), (g)]; (h)–(j) side view of beam propagation for the three cases (coherent, lc215μm, lc115μm), respectively; (k) deflection of the main lobe of the accelerating beams as a function of propagation distance for the three cases. It is evident that the incoherence does not affect the acceleration for the distances presented.

Equations (9)

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U(r⃗,t)=ncn(t)ψn(r⃗),
W(r⃗1,r⃗2)=nanψn*(r⃗1)ψn(r⃗2).
2E⃗+k2E⃗=0,
Ey(x,z)=kkeiαsin1(kx/k)1(kx/k)2eikxxeizk2kx2dkx,
iψz=12k2ψ,
ψ˜(x˜,z˜)=Ai(x˜(12z˜)2iaz˜)×exp(ax˜12az˜2+112iz˜312ia2z˜12iz˜x˜),
E(x,z=0,t)=ncn(t)En(x,z=0)=ncn(t)kkeiαnsin1(kx/k)1(kx/k)2eikxxdkx.
ψ(x,z=0,t)=ncn(t)ψn(x,z=0)=ncn(t)Ai(xΔnx0)ea(xΔn)/x0,
W(k1,k2)=I0exp((k1k2)22σ2)×exp((k12+k22)w2i3c03(k13k23))

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