Abstract

Reflection and refraction of a finite-power Airy beam at the interface between two dielectric media are investigated analytically and numerically. The formulation takes into account the paraxial nature of the optical beams to derive convenient field evolution equations in coordinate frames moving along Snell’s refraction and reflection axes. Through numerical simulations, the self-accelerating dynamics of the Airy-like refracted and reflected beams are observed. Of special interest are the cases of critical incidence at Brewster and total-internal-reflection (TIR) angles. In the former case, we find that the reflected beam achieves self-healing, despite the severe suppression of a part of its spectrum, while, in the latter case, the beam remains nearly unaffected except for the Goos–Hänchen shift. The self-accelerating quality persists even if the beam is trapped by multiple TIRs inside a dielectric film. The grazing incidence of an Airy beam at the interface between two media with close refractive indices is also investigated, revealing that the interface can act as a filter depending on the beam scale and tilt. We finally consider reverse refraction and perfect imaging of an Airy beam into a left-handed medium.

© 2012 Optical Society of America

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References

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  1. M. Berry and N. Balazs, “Non-spreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
    [CrossRef]
  2. G. Siviloglou and D. Christodoulides, “Accelerating finite energy Airy beams,” Opt. Lett. 32, 979–981 (2007).
    [CrossRef]
  3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Observation of accelerating Airy beams,” Phys. Rev. Lett. 99, 213901 (2007).
    [CrossRef]
  4. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, “Ballistic dynamics of Airy beams,” Opt. Lett. 33, 207–209 (2008).
    [CrossRef]
  5. Y. Kaganovsky and E. Heyman, “Wave analysis of Airy beams,” Opt. Express 18, 8440–8452 (2010).
    [CrossRef]
  6. J. Durnin, J. Miceli, and J. Eberly, “Diffraction-free beams,” Phys. Rev. Lett. 58, 1499–1501 (1987).
    [CrossRef]
  7. J. Gutiérrez-Vega, M. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000).
    [CrossRef]
  8. M. Bandres, J. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29, 44–46 (2004).
    [CrossRef]
  9. I. Besieris and A. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. 32, 2447–2449 (2007).
    [CrossRef]
  10. J. Broky, G. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16, 12880–12891 (2008).
    [CrossRef]
  11. X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. 36, 2701–2703 (2011).
    [CrossRef]
  12. Y. Hu, S. Huang, P. Zhang, C. Lou, J. Xu, and Z. Chen, “Persistence and breakdown of Airy beams driven by an initial nonlinearity,” Opt. Lett. 35, 3952–3954 (2010).
    [CrossRef]
  13. Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen, “Optimal control of the ballistic motion of Airy beams,” Opt. Lett. 35, 2260–2262 (2010).
    [CrossRef]
  14. J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photon. 2, 675–678 (2008).
    [CrossRef]
  15. D. Christodoulides, “Optical trapping: riding along an Airy beam,” Nat. Photon. 2, 652–653 (2008).
    [CrossRef]
  16. 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]
  17. J.-X. Li, W.-P. Zang, and J.-G. Tian, “Vacuum laser-driven acceleration by Airy beams,” Opt. Express 18, 7300–7306 (2010).
    [CrossRef]
  18. P. Polynkin, M. Kolesik, J. Moloney, G. Siviloglou, and D. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
    [CrossRef]
  19. A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35, 2082–2084 (2010).
    [CrossRef]
  20. 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]
  21. 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]
  22. N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35, 4045–4047 (2010).
    [CrossRef]
  23. I. M. Besieris and A. M. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E 78, 046605 (2008).
    [CrossRef]
  24. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, “Airy–Bessel wave packets as versatile linear light bullets,” Nat. Photon. 4, 103–106 (2010).
    [CrossRef]
  25. Y. Fattal, A. Rudnick, and D. M. Marom, “Soliton shedding from Airy pulses in Kerr media,” Opt. Express 19, 17298–17307 (2011).
    [CrossRef]
  26. C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum generation with self-healing Airy pulses,” in CLEO 2011—Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPC9.
  27. F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
    [CrossRef]
  28. M. Green, P. Kirkby, and R. S. Timsit, “Experimental results on the longitudinal displacement of light beams near total reflection,” Phys. Lett. A 45, 259–260 (1973).
    [CrossRef]
  29. G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic2001).
  30. J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
    [CrossRef]

2011 (5)

2010 (7)

2009 (1)

P. Polynkin, M. Kolesik, J. Moloney, G. Siviloglou, and D. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef]

2008 (5)

I. M. Besieris and A. M. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E 78, 046605 (2008).
[CrossRef]

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

D. Christodoulides, “Optical trapping: riding along an Airy beam,” Nat. Photon. 2, 652–653 (2008).
[CrossRef]

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

J. Broky, G. Siviloglou, A. Dogariu, and D. N. Christodoulides, “Self-healing properties of optical Airy beams,” Opt. Express 16, 12880–12891 (2008).
[CrossRef]

2007 (3)

2004 (1)

2000 (2)

1987 (1)

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

1979 (1)

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

1973 (1)

M. Green, P. Kirkby, and R. S. Timsit, “Experimental results on the longitudinal displacement of light beams near total reflection,” Phys. Lett. A 45, 259–260 (1973).
[CrossRef]

1947 (1)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Agrawal, G.

G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic2001).

Ament, C.

C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum generation with self-healing Airy pulses,” in CLEO 2011—Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPC9.

Balazs, N.

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

Bandres, M.

Baumgartl, J.

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

Berry, M.

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

Besieris, I.

Besieris, I. M.

I. M. Besieris and A. M. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E 78, 046605 (2008).
[CrossRef]

Broky, J.

Chávez-Cerda, S.

Chen, Z.

Chong, A.

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

Christodoulides, D.

P. Polynkin, M. Kolesik, J. Moloney, G. Siviloglou, and D. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef]

D. Christodoulides, “Optical trapping: riding along an Airy beam,” Nat. Photon. 2, 652–653 (2008).
[CrossRef]

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

Christodoulides, D. N.

Chu, X.

Dholakia, K.

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

Dogariu, A.

Durnin, J.

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

Eberly, J.

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

Efremidis, N. K.

Fattal, Y.

Goos, F.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Green, M.

M. Green, P. Kirkby, and R. S. Timsit, “Experimental results on the longitudinal displacement of light beams near total reflection,” Phys. Lett. A 45, 259–260 (1973).
[CrossRef]

Gutiérrez-Vega, J.

Hänchen, H.

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Heyman, E.

Hu, Y.

Huang, S.

Iturbe-Castillo, M.

Janunts, N.

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]

Kaganovsky, Y.

Kirkby, P.

M. Green, P. Kirkby, and R. S. Timsit, “Experimental results on the longitudinal displacement of light beams near total reflection,” Phys. Lett. A 45, 259–260 (1973).
[CrossRef]

Kivshar, Y. S.

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]

Klein, A. E.

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]

Kolesik, M.

P. Polynkin, M. Kolesik, J. Moloney, G. Siviloglou, and D. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef]

Li, J.-X.

Liu, Y.

Lou, C.

Lu, C.

Marom, D. M.

Mazilu, M.

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

Miceli, J.

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

Mills, M. S.

Minovich, A.

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]

Moloney, J.

P. Polynkin, M. Kolesik, J. Moloney, G. Siviloglou, and D. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef]

Moloney, J. V.

C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum generation with self-healing Airy pulses,” in CLEO 2011—Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPC9.

Neshev, D. N.

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]

Pendry, J. B.

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[CrossRef]

Pertsch, T.

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]

Polynkin, P.

P. Polynkin, M. Kolesik, J. Moloney, G. Siviloglou, and D. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef]

C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum generation with self-healing Airy pulses,” in CLEO 2011—Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPC9.

Prakash, J.

Renninger, W. H.

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

Rudnick, A.

Salandrino, A.

Shaarawi, A.

Shaarawi, A. M.

I. M. Besieris and A. M. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E 78, 046605 (2008).
[CrossRef]

Siviloglou, G.

Siviloglou, G. A.

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

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

Tian, J.-G.

Timsit, R. S.

M. Green, P. Kirkby, and R. S. Timsit, “Experimental results on the longitudinal displacement of light beams near total reflection,” Phys. Lett. A 45, 259–260 (1973).
[CrossRef]

Wang, S.

Wise, F. W.

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

Xu, J.

Yin, X.

Zang, W.-P.

Zhang, P.

Zhang, X.

Zhang, Z.

Am. J. Phys. (1)

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

Ann. Phys. (1)

F. Goos and H. Hänchen, “Ein neuer und fundamentaler Versuch zur Totalreflexion,” Ann. Phys. 436, 333–346 (1947).
[CrossRef]

Nat. Photon. (3)

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

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

D. Christodoulides, “Optical trapping: riding along an Airy beam,” Nat. Photon. 2, 652–653 (2008).
[CrossRef]

Opt. Express (4)

Opt. Lett. (12)

N. K. Efremidis and D. N. Christodoulides, “Abruptly autofocusing waves,” Opt. Lett. 35, 4045–4047 (2010).
[CrossRef]

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

X. Chu, “Evolution of an Airy beam in turbulence,” Opt. Lett. 36, 2701–2703 (2011).
[CrossRef]

Y. Hu, S. Huang, P. Zhang, C. Lou, J. Xu, and Z. Chen, “Persistence and breakdown of Airy beams driven by an initial nonlinearity,” Opt. Lett. 35, 3952–3954 (2010).
[CrossRef]

Y. Hu, P. Zhang, C. Lou, S. Huang, J. Xu, and Z. Chen, “Optimal control of the ballistic motion of Airy beams,” Opt. Lett. 35, 2260–2262 (2010).
[CrossRef]

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

A. Salandrino and D. N. Christodoulides, “Airy plasmon: a nondiffracting surface wave,” Opt. Lett. 35, 2082–2084 (2010).
[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]

J. Gutiérrez-Vega, M. Iturbe-Castillo, and S. Chávez-Cerda, “Alternative formulation for invariant optical fields: Mathieu beams,” Opt. Lett. 25, 1493–1495 (2000).
[CrossRef]

M. Bandres, J. Gutiérrez-Vega, and S. Chávez-Cerda, “Parabolic nondiffracting optical wave fields,” Opt. Lett. 29, 44–46 (2004).
[CrossRef]

I. Besieris and A. Shaarawi, “A note on an accelerating finite energy Airy beam,” Opt. Lett. 32, 2447–2449 (2007).
[CrossRef]

Phys. Lett. A (1)

M. Green, P. Kirkby, and R. S. Timsit, “Experimental results on the longitudinal displacement of light beams near total reflection,” Phys. Lett. A 45, 259–260 (1973).
[CrossRef]

Phys. Rev. E (1)

I. M. Besieris and A. M. Shaarawi, “Accelerating Airy wave packets in the presence of quadratic and cubic dispersion,” Phys. Rev. E 78, 046605 (2008).
[CrossRef]

Phys. Rev. Lett. (4)

J. B. Pendry, “Negative refraction makes a perfect lens,” Phys. Rev. Lett. 85, 3966–3969 (2000).
[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]

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

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

Science (1)

P. Polynkin, M. Kolesik, J. Moloney, G. Siviloglou, and D. Christodoulides, “Curved plasma channel generation using ultraintense Airy beams,” Science 324, 229–232 (2009).
[CrossRef]

Other (2)

G. Agrawal, Nonlinear Fiber Optics, 3rd ed. (Academic2001).

C. Ament, P. Polynkin, and J. V. Moloney, “Supercontinuum generation with self-healing Airy pulses,” in CLEO 2011—Laser Applications to Photonic Applications, OSA Technical Digest (CD) (Optical Society of America, 2011), paper PDPC9.

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Figures (9)

Fig. 1.
Fig. 1.

Problem geometry and the different coordinate systems used. Axis xrx is not shown.

Fig. 2.
Fig. 2.

(a) Reflected and (b) refracted beam amplitude for a p-polarized finite-power Airy beam (x0=20μm, a=0.08) impinging at an angle θ1=30° on a glass–air interface (n1=1.5, n2=1.0). The incident beam has propagated for one diffraction length before it reaches the interface. According to the defined coordinate systems, in (a) zr is the depth inside the dielectric and χr measures the horizontal position with respect to Snell’s reflection axis; in (b) z is the height in the air and χt measures horizontal position with respect to Snell’s refraction axis (θ2=48.6°). Dotted curves are the caustics of Eq. (16).

Fig. 3.
Fig. 3.

(a) Reflected beam amplitude for the Airy beam of Fig. 2, impinging at Brewster’s angle θ1=33.69° on the same glass–air interface. The dotted curve is the caustic. (b) Beam amplitude (a.u.) versus position χr from Snell’s reflection axis at different indicated depths inside the glass.

Fig. 4.
Fig. 4.

Reflected beam amplitudes for the Airy beam of Fig. 2, impinging at angle (a) θ1=θB+0.25°=33.94° and (b) θ1=θB0.25°=33.44° on the same dielectric–air interface. The dotted curves are the caustics.

Fig. 5.
Fig. 5.

(a) Reflected and (b) transmitted wave amplitudes for the Airy beam of Fig. 2, impinging at angle θ1=θc=41.81°. (c) The Gaussian amplitude of the incident beam’s spectrum in comparison with the amplitude (upper dashed curve) and phase (lower dashed curve) of the Fresnel reflection coefficient. For the phase curve, the ordinate is in radians, while for the amplitude curves, it is in arbitrary units (a.u.). (d) Intensity of the incident (solid) and reflected (dashed) beams at the interface, giving evidence to the Goos–Hänchen shift.

Fig. 6.
Fig. 6.

Transmitted wave amplitudes for the (a) Airy beam of Fig. 2, impinging at an angle θ1=θc0.5°=41.31° and (b) Airy beam of Fig. 2 with reversed direction of lobes [negative sign in Eq. (14)] impinging at an angle θ1=θc=41.81°.

Fig. 7.
Fig. 7.

(a) Setting in which an optical beam undergoes multiple TIRs inside a dielectric layer of width h. (b)–(d) Amplitude of the exiting beam after (b) N=5, (c) N=15, and (d) N=35 TIRs. The index and width of the dielectric slab are taken as 1.5 and 0.5mm, respectively.

Fig. 8.
Fig. 8.

Grazing incidence of an Airy beam at the interface between two media with indices n1=1.001, n2=1.0. The beam parameters are a=0.05, =30μm and (a) x0=6μm, v=0, (b) x0=10μm, v=0, (c) x0=10μm, v=5. The vacuum wavelength is λ=500nm.

Fig. 9.
Fig. 9.

Normal incidence of the Airy beam of Fig. 2 propagating in air (n1=1) on the interface (white dashed line) to a LHM with refractive index n2=1.5. The beam is nearly perfectly imaged at the height of 7.5 cm inside the LHM. In medium 1, the total field (incident + reflected) is depicted.

Equations (22)

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

ψi(xi,zi=0)=u0(xi)exp(iβ0xi),
u0(xi)f0(xicosθ1,0),
ψi(x,z)=12π+Ψ(k)exp(ikx+iq1(z+h))dk,
ψr(x,z)=12π+Ψ(k)R(k)exp(ikxiq1(zh))dk,
ψt(x,z)=12π+Ψ(k)T(k)exp(ikx+iq2z+iq1h)dk,
R(k)=ε2q1ε1q2ε2q1+ε1q2,T(k)=2ε2q1ε2q1+ε1q2,
Ψ(k)=+ψi(x,h)exp(ikx)dk=exp(iks)cosθ1U0(kβ0cosθ1),
q1(ξ)=k12(β0+ξ)2=q10β0q10ξk122q103ξ2β0k122q105ξ3,
ψr(xr,zr)ur(xrzrtanθ1,zr)exp(iβ0(xr+s)+iq10(h+zr)),
ur(χr,zr)=12πcosθ1+U0(ξcosθ1)R(β0+ξ)exp[iξχriξ2k12(zr+h)2q103]dξ.
ψt(x,z)ut(xztanθ2,z)exp[iβ0(x+s)+iq20z+iq10h],
ut(χt,z)=12πcosθ1+U0(ξcosθ1)T(β0+ξ)exp[iξχtiξ22(k22zq203+k12hq103)]dξ,
ψt(x,z)ut(x,z)exp(iβ0(x+s)+iq10h),
ut(x,z)=12πcosθ1+U0(ξcosθ1)T(β0+ξ)exp(iξx+iq2zik12h2q103ξ2)dξ.
utz=ik222q2032utχt2
i2k1ux+V(z)u+2uz2=0,x>0,
f0(xi,zi=0)=Ai(±xix0)exp(±axix0),
U0(k)=+f0(xi,zi=0)exp(ikxi)dxi=x0exp[i3(±kx0+ia)3].
χr=±(h+zr)24k12x03cos3θ1,χt=±14k12x03cos3θ1(h+k1cos3θ1k2cos3θ2z)2
Pr=+|U0(ξ/cosθ1)R(β0+ξ)|2dξ+|U0(ξ/cosθ1)|2dξ,
ur(xb,zb)=12π+U0(ξ)RN(β0+ξcosθ1)T˜(ξ)exp(iξxbiξ2(N+1)h2q10iξ2zb2k2)dξ,
u(x=0,z)=Ai(z+x0)exp(az+x0+ivzx0),

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