Abstract

We investigate the propagation of Airy beams in linear gradient index inhomogeneous media. We demonstrate that by controlling the gradient strength of the medium it is possible to reduce to zero their acceleration. We show that the resulting Airy wave beam propagates in straight line due to the balance between two opposite effects, one due to the inhomogeneous medium and the other to the diffraction of the beam, in a similar way as a solitary wave in a nonlinear inhomogeneous medium. Going even further we were able to invert the sign of the acceleration of the beam.

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  1. G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
    [CrossRef] [PubMed]
  2. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
    [CrossRef] [PubMed]
  3. J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
    [CrossRef] [PubMed]
  4. J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987).
    [CrossRef]
  5. D. DeBeer, S. R. Hartmann, and R. Friedberg, “Comment on “Diffraction-free beams”,” Phys. Rev. Lett. 59(22), 2611 (1987).J. Durnin, J. J. Miceli, and J. H. Eberly, “Durnin, Miceli, and Eberly reply,” Phys. Rev. Lett. 59(22), 2612 (1987).
    [CrossRef] [PubMed]
  6. S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).
  7. M. V. Berry and N. L. Balazs, “Nonspreading wave-packets,” Am. J. Phys. 47(3), 264–267 (1979).
    [CrossRef]
  8. M. A. Bandres, J. C. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29(1), 44–46 (2004).
    [CrossRef] [PubMed]
  9. C. López-Mariscal, M. Bandres, J. Gutiérrez-Vega, and S. Chávez-Cerda, “Observation of parabolic nondiffracting optical fields,” Opt. Express 13(7), 2364–2369 (2005).
    [CrossRef] [PubMed]
  10. D. Marcuse, “TE modes of graded index slab waveguides,” IEEE J. Quantum Electron. 9(10), 1000–1006 (1973).
    [CrossRef]
  11. C.-L. Chen, Foundations of guided-wave optics (Wiley, New Jersey 2006), Ch 3.
  12. D. N. Christodoulides and T. H. Coskun, “Diffraction-free planar beams in unbiased photorefractive media,” Opt. Lett. 21(18), 1460–1462 (1996).
    [CrossRef] [PubMed]
  13. S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104(25), 253904 (2010).
    [CrossRef] [PubMed]
  14. T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
    [CrossRef]
  15. I. Dolev, T. Ellenbogen, and A. Arie, “Switching the acceleration direction of Airy beams by a nonlinear optical process,” Opt. Lett. 35(10), 1581–1583 (2010). Ye, Zhuoyi; Liu, Sheng; Lou, Cibo; Zhang, Peng; Hu, Yi; Song, Daohong; Zhao, Jianlin; Chen, Zhigang, Quantum Electronics and Laser Science Conference (QELS) 2011 paper: JTuI32, OSA Technical Digest (CD).
  16. W. Liu, D. N. Neshev, I. V. Shadrivov, A. E. Miroshnichenko, and Y. S. Kivshar, “Plasmonic Airy beam manipulation in linear optical potentials,” Opt. Lett. 36(7), 1164–1166 (2011).
    [CrossRef] [PubMed]
  17. M. Born and E. Wolf, Principles of Optics, Seventh Ed., (Cambridge University Press, Cambridge 1999), Ch. 3.
  18. There exist several forms for producing a solution of this equation, for instance by using the Zassenhaus formula [19], or by simplifying the differential equation via a transformation (see for instance [20] for a quadratic term). The operators to disentangle the exponential of the sum of two operators can also be given in different orderings of the exponentials involved. Of course, they are all equivalent.
  19. R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8(4), 962–982 (1967).
    [CrossRef]
  20. H. Moya-Cessa and M. Fernández Guasti, “Coherent states for the time dependent harmonic oscillator: the step function,” Phys. Lett. A 311(1), 1–5 (2003).
    [CrossRef]
  21. W. H. Louissel, Quantum Statistical Properties of Radiation (Wiley-Interscience, New York 1990), Ch. 3.
  22. H. Moya-Cessa and F. Soto-Eguibar, Differential equations: an operational approach, (Rinton Press, New Jersey 2011), Ch. 2.
  23. G. N. Watson, A treatise on the theory of Bessel functions, Ch. VI, Cambridge University Press, Cambridge 1944).
  24. W. M. Strouse, “Bouncing light beams,” Am. J. Phys. 40(6), 913–914 (1972).
    [CrossRef]
  25. D. Ambrosini, A. Ponticiello, G. S. Spagnolo, R. Borghi, and F. Gori, “Bouncing light beams and the Hamiltonian analogy,” Eur. J. Phys. 18(4), 284–289 (1997).
    [CrossRef]
  26. SLM 512, Boulder nonlinear systems.

2011 (1)

2010 (1)

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104(25), 253904 (2010).
[CrossRef] [PubMed]

2009 (1)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

2007 (2)

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[CrossRef] [PubMed]

2005 (1)

2004 (1)

2003 (1)

H. Moya-Cessa and M. Fernández Guasti, “Coherent states for the time dependent harmonic oscillator: the step function,” Phys. Lett. A 311(1), 1–5 (2003).
[CrossRef]

1999 (1)

S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).

1997 (1)

D. Ambrosini, A. Ponticiello, G. S. Spagnolo, R. Borghi, and F. Gori, “Bouncing light beams and the Hamiltonian analogy,” Eur. J. Phys. 18(4), 284–289 (1997).
[CrossRef]

1996 (1)

1987 (3)

D. DeBeer, S. R. Hartmann, and R. Friedberg, “Comment on “Diffraction-free beams”,” Phys. Rev. Lett. 59(22), 2611 (1987).J. Durnin, J. J. Miceli, and J. H. Eberly, “Durnin, Miceli, and Eberly reply,” Phys. Rev. Lett. 59(22), 2612 (1987).
[CrossRef] [PubMed]

D. DeBeer, S. R. Hartmann, and R. Friedberg, “Comment on “Diffraction-free beams”,” Phys. Rev. Lett. 59(22), 2611 (1987).J. Durnin, J. J. Miceli, and J. H. Eberly, “Durnin, Miceli, and Eberly reply,” Phys. Rev. Lett. 59(22), 2612 (1987).
[CrossRef] [PubMed]

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987).
[CrossRef]

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

1979 (1)

M. V. Berry and N. L. Balazs, “Nonspreading wave-packets,” Am. J. Phys. 47(3), 264–267 (1979).
[CrossRef]

1973 (1)

D. Marcuse, “TE modes of graded index slab waveguides,” IEEE J. Quantum Electron. 9(10), 1000–1006 (1973).
[CrossRef]

1972 (1)

W. M. Strouse, “Bouncing light beams,” Am. J. Phys. 40(6), 913–914 (1972).
[CrossRef]

1967 (1)

R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8(4), 962–982 (1967).
[CrossRef]

Ambrosini, D.

D. Ambrosini, A. Ponticiello, G. S. Spagnolo, R. Borghi, and F. Gori, “Bouncing light beams and the Hamiltonian analogy,” Eur. J. Phys. 18(4), 284–289 (1997).
[CrossRef]

Arie, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

Balazs, N. L.

M. V. Berry and N. L. Balazs, “Nonspreading wave-packets,” Am. J. Phys. 47(3), 264–267 (1979).
[CrossRef]

Bandres, M.

Bandres, M. A.

Berry, M. V.

M. V. Berry and N. L. Balazs, “Nonspreading wave-packets,” Am. J. Phys. 47(3), 264–267 (1979).
[CrossRef]

Borghi, R.

D. Ambrosini, A. Ponticiello, G. S. Spagnolo, R. Borghi, and F. Gori, “Bouncing light beams and the Hamiltonian analogy,” Eur. J. Phys. 18(4), 284–289 (1997).
[CrossRef]

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[CrossRef] [PubMed]

Chávez-Cerda, S.

Christodoulides, D. N.

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104(25), 253904 (2010).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[CrossRef] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[CrossRef] [PubMed]

D. N. Christodoulides and T. H. Coskun, “Diffraction-free planar beams in unbiased photorefractive media,” Opt. Lett. 21(18), 1460–1462 (1996).
[CrossRef] [PubMed]

Coskun, T. H.

DeBeer, D.

D. DeBeer, S. R. Hartmann, and R. Friedberg, “Comment on “Diffraction-free beams”,” Phys. Rev. Lett. 59(22), 2611 (1987).J. Durnin, J. J. Miceli, and J. H. Eberly, “Durnin, Miceli, and Eberly reply,” Phys. Rev. Lett. 59(22), 2612 (1987).
[CrossRef] [PubMed]

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[CrossRef] [PubMed]

Durnin, J.

J. Durnin, “Exact solutions for nondiffracting beams. I. The scalar theory,” J. Opt. Soc. Am. A 4(4), 651–654 (1987).
[CrossRef]

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

D. DeBeer, S. R. Hartmann, and R. Friedberg, “Comment on “Diffraction-free beams”,” Phys. Rev. Lett. 59(22), 2611 (1987).J. Durnin, J. J. Miceli, and J. H. Eberly, “Durnin, Miceli, and Eberly reply,” Phys. Rev. Lett. 59(22), 2612 (1987).
[CrossRef] [PubMed]

Eberly, J. H.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

D. DeBeer, S. R. Hartmann, and R. Friedberg, “Comment on “Diffraction-free beams”,” Phys. Rev. Lett. 59(22), 2611 (1987).J. Durnin, J. J. Miceli, and J. H. Eberly, “Durnin, Miceli, and Eberly reply,” Phys. Rev. Lett. 59(22), 2612 (1987).
[CrossRef] [PubMed]

Ellenbogen, T.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

Fernández Guasti, M.

H. Moya-Cessa and M. Fernández Guasti, “Coherent states for the time dependent harmonic oscillator: the step function,” Phys. Lett. A 311(1), 1–5 (2003).
[CrossRef]

Fleischer, J. W.

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104(25), 253904 (2010).
[CrossRef] [PubMed]

Friedberg, R.

D. DeBeer, S. R. Hartmann, and R. Friedberg, “Comment on “Diffraction-free beams”,” Phys. Rev. Lett. 59(22), 2611 (1987).J. Durnin, J. J. Miceli, and J. H. Eberly, “Durnin, Miceli, and Eberly reply,” Phys. Rev. Lett. 59(22), 2612 (1987).
[CrossRef] [PubMed]

Ganany-Padowicz, A.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

Gori, F.

D. Ambrosini, A. Ponticiello, G. S. Spagnolo, R. Borghi, and F. Gori, “Bouncing light beams and the Hamiltonian analogy,” Eur. J. Phys. 18(4), 284–289 (1997).
[CrossRef]

Gutiérrez-Vega, J.

Gutiérrez-Vega, J. C.

Hartmann, S. R.

D. DeBeer, S. R. Hartmann, and R. Friedberg, “Comment on “Diffraction-free beams”,” Phys. Rev. Lett. 59(22), 2611 (1987).J. Durnin, J. J. Miceli, and J. H. Eberly, “Durnin, Miceli, and Eberly reply,” Phys. Rev. Lett. 59(22), 2612 (1987).
[CrossRef] [PubMed]

Jia, S.

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104(25), 253904 (2010).
[CrossRef] [PubMed]

Kivshar, Y. S.

Lee, J.

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104(25), 253904 (2010).
[CrossRef] [PubMed]

Liu, W.

López-Mariscal, C.

Marcuse, D.

D. Marcuse, “TE modes of graded index slab waveguides,” IEEE J. Quantum Electron. 9(10), 1000–1006 (1973).
[CrossRef]

Miceli, J.

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

Miceli, J. J.

D. DeBeer, S. R. Hartmann, and R. Friedberg, “Comment on “Diffraction-free beams”,” Phys. Rev. Lett. 59(22), 2611 (1987).J. Durnin, J. J. Miceli, and J. H. Eberly, “Durnin, Miceli, and Eberly reply,” Phys. Rev. Lett. 59(22), 2612 (1987).
[CrossRef] [PubMed]

Miroshnichenko, A. E.

Moya-Cessa, H.

H. Moya-Cessa and M. Fernández Guasti, “Coherent states for the time dependent harmonic oscillator: the step function,” Phys. Lett. A 311(1), 1–5 (2003).
[CrossRef]

Neshev, D. N.

Ponticiello, A.

D. Ambrosini, A. Ponticiello, G. S. Spagnolo, R. Borghi, and F. Gori, “Bouncing light beams and the Hamiltonian analogy,” Eur. J. Phys. 18(4), 284–289 (1997).
[CrossRef]

Shadrivov, I. V.

Siviloglou, G. A.

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104(25), 253904 (2010).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[CrossRef] [PubMed]

G. A. Siviloglou and D. N. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32(8), 979–981 (2007).
[CrossRef] [PubMed]

Spagnolo, G. S.

D. Ambrosini, A. Ponticiello, G. S. Spagnolo, R. Borghi, and F. Gori, “Bouncing light beams and the Hamiltonian analogy,” Eur. J. Phys. 18(4), 284–289 (1997).
[CrossRef]

Strouse, W. M.

W. M. Strouse, “Bouncing light beams,” Am. J. Phys. 40(6), 913–914 (1972).
[CrossRef]

Voloch-Bloch, N.

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

Wilcox, R. M.

R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8(4), 962–982 (1967).
[CrossRef]

Am. J. Phys. (2)

M. V. Berry and N. L. Balazs, “Nonspreading wave-packets,” Am. J. Phys. 47(3), 264–267 (1979).
[CrossRef]

W. M. Strouse, “Bouncing light beams,” Am. J. Phys. 40(6), 913–914 (1972).
[CrossRef]

Eur. J. Phys. (1)

D. Ambrosini, A. Ponticiello, G. S. Spagnolo, R. Borghi, and F. Gori, “Bouncing light beams and the Hamiltonian analogy,” Eur. J. Phys. 18(4), 284–289 (1997).
[CrossRef]

IEEE J. Quantum Electron. (1)

D. Marcuse, “TE modes of graded index slab waveguides,” IEEE J. Quantum Electron. 9(10), 1000–1006 (1973).
[CrossRef]

J. Math. Phys. (1)

R. M. Wilcox, “Exponential operators and parameter differentiation in quantum physics,” J. Math. Phys. 8(4), 962–982 (1967).
[CrossRef]

J. Mod. Opt. (1)

S. Chávez-Cerda, “A new approach to Bessel beams,” J. Mod. Opt. 46, 923–930 (1999).

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

Nat. Photonics (1)

T. Ellenbogen, N. Voloch-Bloch, A. Ganany-Padowicz, and A. Arie, “Nonlinear generation and manipulation of Airy beams,” Nat. Photonics 3(7), 395–398 (2009).
[CrossRef]

Opt. Express (1)

Opt. Lett. (4)

Phys. Lett. A (1)

H. Moya-Cessa and M. Fernández Guasti, “Coherent states for the time dependent harmonic oscillator: the step function,” Phys. Lett. A 311(1), 1–5 (2003).
[CrossRef]

Phys. Rev. Lett. (4)

D. DeBeer, S. R. Hartmann, and R. Friedberg, “Comment on “Diffraction-free beams”,” Phys. Rev. Lett. 59(22), 2611 (1987).J. Durnin, J. J. Miceli, and J. H. Eberly, “Durnin, Miceli, and Eberly reply,” Phys. Rev. Lett. 59(22), 2612 (1987).
[CrossRef] [PubMed]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99(21), 213901 (2007).
[CrossRef] [PubMed]

J. Durnin, J. Miceli, and J. H. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58(15), 1499–1501 (1987).
[CrossRef] [PubMed]

S. Jia, J. Lee, J. W. Fleischer, G. A. Siviloglou, and D. N. Christodoulides, “Diffusion-trapped Airy beams in photorefractive media,” Phys. Rev. Lett. 104(25), 253904 (2010).
[CrossRef] [PubMed]

Other (8)

C.-L. Chen, Foundations of guided-wave optics (Wiley, New Jersey 2006), Ch 3.

M. Born and E. Wolf, Principles of Optics, Seventh Ed., (Cambridge University Press, Cambridge 1999), Ch. 3.

There exist several forms for producing a solution of this equation, for instance by using the Zassenhaus formula [19], or by simplifying the differential equation via a transformation (see for instance [20] for a quadratic term). The operators to disentangle the exponential of the sum of two operators can also be given in different orderings of the exponentials involved. Of course, they are all equivalent.

W. H. Louissel, Quantum Statistical Properties of Radiation (Wiley-Interscience, New York 1990), Ch. 3.

H. Moya-Cessa and F. Soto-Eguibar, Differential equations: an operational approach, (Rinton Press, New Jersey 2011), Ch. 2.

G. N. Watson, A treatise on the theory of Bessel functions, Ch. VI, Cambridge University Press, Cambridge 1944).

I. Dolev, T. Ellenbogen, and A. Arie, “Switching the acceleration direction of Airy beams by a nonlinear optical process,” Opt. Lett. 35(10), 1581–1583 (2010). Ye, Zhuoyi; Liu, Sheng; Lou, Cibo; Zhang, Peng; Hu, Yi; Song, Daohong; Zhao, Jianlin; Chen, Zhigang, Quantum Electronics and Laser Science Conference (QELS) 2011 paper: JTuI32, OSA Technical Digest (CD).

SLM 512, Boulder nonlinear systems.

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Figures (4)

Fig. 1
Fig. 1

Top row: Typical behavior of a rays propagating in linear GRIN media. The red dotted line shows qualitatively the gradient of the refractive index and the blue line the corresponding ray trajectory for: a) negative gradient and b) positive gradient. Bottom row: Free space propagation of the Airy mode at the exit of the corresponding inhomogeneous media (see Airy equation at the bottom of the figure) with negative gradient, c), and positive gradient d). The horizontal coordinate corresponds to the transverse coordinate and the vertical one to the evolution coordinate.

Fig. 2
Fig. 2

Propagation of an Airy beam in a medium with positive gradient as function of the value and sign of the acceleration factor, a) negative, b) zero or c) positive. For zero acceleration the beam propagates as an Airy non-diffracting wave field.

Fig. 3
Fig. 3

Components of the experimental setup: Laser, beam expander (BE), SLM, Fourier transforming lens (L), and solution container (SC).

Fig. 4
Fig. 4

Left side images: trajectories of the Gaussian beam for solution concentrations (a) C1, (c) C2, and (e) C3. Right side images: trajectories of the Airy beam for solution concentrations (b) C1, (d) C2, and (f) C3.

Equations (12)

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

d d s ( n d r d s ) = n ,
d 2 x d z 2 n 1 2 n 0 2 = 0 ,
i u z = 1 2 2 u x 2 + k 1 x u .
u ( x , z ) = exp ( i 3 k 1 p 3 ) v ( x , z ) ,
i v z = k 1 x 2 v
v ( x , z ) = exp ( i k 1 x z 2 ) v ( x , 0 ) .
v ( x , 0 ) = exp ( i 3 k 1 p 3 ) A i ( k x x ) .
A i ( s ) = 1 ( 2 π ) 2 exp ( i t 3 / 3 + s t ) d t
v ( x , z ) = 1 ( 2 π ) 2 exp ( i k 1 x z 2 ) exp [ i ( t 3 / 3 + k x 3 t 3 / 3 k 1 + k x x t ) ] d t
u ( x , z ) = e i Φ ( x , z ) A i [ k x x + k x z 2 2 ( k 1 k x 3 2 ) ] ,
Ψ ( x ) = exp [ i ( 2 π / α ) x 3 ]
{ Ψ ( x ) } = A i ( q k ) ,

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