Abstract

We study curved trajectory dynamics and design in discrete array settings. We find that beams with power law phases produce curved caustics associated with the fold and cusp type catastrophes. A parabolic phase produces a focus that suffers from spherical aberrations. More important, we find that by designing the initial phase or wavefront of the beam we can construct trajectories with pure power law caustics as well as aberration-free focusing of discrete waves.

© 2012 Optical Society of America

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  1. M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
    [Crossref]
  2. G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
    [Crossref]
  3. G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
    [Crossref]
  4. M. A. Bandres, Opt. Lett. 33, 1678 (2008).
    [Crossref]
  5. J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photon. 2, 675 (2008).
    [Crossref]
  6. P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
    [Crossref]
  7. A. Salandrino and D. N. Christodoulides, Opt. Lett. 35, 2082 (2010).
    [Crossref]
  8. P. Zhang, S. Wang, Y. Liu, X. Yin, C. Lu, Z. Chen, and X. Zhang, Opt. Lett. 36, 3191 (2011).
    [Crossref]
  9. A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
    [Crossref]
  10. A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photon. 4, 103 (2010).
    [Crossref]
  11. D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
    [Crossref]
  12. N. K. Efremidis and D. N. Christodoulides, Opt. Lett. 35, 4045 (2010).
    [Crossref]
  13. I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, Opt. Lett. 36, 1890 (2011).
    [Crossref]
  14. D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, Opt. Lett. 36, 1842 (2011).
    [Crossref]
  15. P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett. 36, 2883 (2011).
    [Crossref]
  16. E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
    [Crossref]
  17. M. V. Berry and C. Upstill, in Progress in Optics, E. Wolf, ed. (North-Holland, 1980), Vol. XVIII, pp. 259–323.
  18. Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer, 1999).
  19. F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
    [Crossref]
  20. I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, arXiv:1107.2992v1 (2011).
  21. U. Peschel, T. Persch, and F. Lederer, Opt. Lett. 23, 1701 (1998).
    [Crossref]
  22. R. El-Ganainy, K. G. Makris, M. A. Miri, D. N. Christodoulides, and Z. Chen, Phys. Rev. A 84, 023842 (2011).
    [Crossref]

2011 (7)

2010 (4)

A. Salandrino and D. N. Christodoulides, Opt. Lett. 35, 2082 (2010).
[Crossref]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photon. 4, 103 (2010).
[Crossref]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[Crossref]

N. K. Efremidis and D. N. Christodoulides, Opt. Lett. 35, 4045 (2010).
[Crossref]

2009 (1)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[Crossref]

2008 (3)

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[Crossref]

M. A. Bandres, Opt. Lett. 33, 1678 (2008).
[Crossref]

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photon. 2, 675 (2008).
[Crossref]

2007 (2)

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

1998 (1)

1979 (1)

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[Crossref]

Abdollahpour, D.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[Crossref]

Assanto, G.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[Crossref]

Balazs, N. L.

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[Crossref]

Bandres, M. A.

Baumgartl, J.

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photon. 2, 675 (2008).
[Crossref]

Berry, M. V.

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[Crossref]

M. V. Berry and C. Upstill, in Progress in Optics, E. Wolf, ed. (North-Holland, 1980), Vol. XVIII, pp. 259–323.

Broky, J.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Chen, Z.

Chong, A.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photon. 4, 103 (2010).
[Crossref]

Chremmos, I.

Christodoulides, D. N.

I. Chremmos, N. K. Efremidis, and D. N. Christodoulides, Opt. Lett. 36, 1890 (2011).
[Crossref]

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, Opt. Lett. 36, 1842 (2011).
[Crossref]

R. El-Ganainy, K. G. Makris, M. A. Miri, D. N. Christodoulides, and Z. Chen, Phys. Rev. A 84, 023842 (2011).
[Crossref]

P. Zhang, J. Prakash, Z. Zhang, M. S. Mills, N. K. Efremidis, D. N. Christodoulides, and Z. Chen, Opt. Lett. 36, 2883 (2011).
[Crossref]

N. K. Efremidis and D. N. Christodoulides, Opt. Lett. 35, 4045 (2010).
[Crossref]

A. Salandrino and D. N. Christodoulides, Opt. Lett. 35, 2082 (2010).
[Crossref]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photon. 4, 103 (2010).
[Crossref]

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[Crossref]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Dholakia, K.

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photon. 2, 675 (2008).
[Crossref]

Dogariu, A.

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

Efremidis, N. K.

El-Ganainy, R.

R. El-Ganainy, K. G. Makris, M. A. Miri, D. N. Christodoulides, and Z. Chen, Phys. Rev. A 84, 023842 (2011).
[Crossref]

Garanovich, I. L.

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, arXiv:1107.2992v1 (2011).

Greenfield, E.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[Crossref]

Janunts, N.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[Crossref]

Kivshar, Y. S.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[Crossref]

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, arXiv:1107.2992v1 (2011).

Klein, A. E.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[Crossref]

Kolesik, M.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[Crossref]

Kravtsov, Y. A.

Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer, 1999).

Lederer, F.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[Crossref]

U. Peschel, T. Persch, and F. Lederer, Opt. Lett. 23, 1701 (1998).
[Crossref]

Liu, Y.

Longhi, S.

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, arXiv:1107.2992v1 (2011).

Lu, C.

Makris, K. G.

R. El-Ganainy, K. G. Makris, M. A. Miri, D. N. Christodoulides, and Z. Chen, Phys. Rev. A 84, 023842 (2011).
[Crossref]

Mazilu, M.

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photon. 2, 675 (2008).
[Crossref]

Mills, M. S.

Minovich, A.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[Crossref]

Miri, M. A.

R. El-Ganainy, K. G. Makris, M. A. Miri, D. N. Christodoulides, and Z. Chen, Phys. Rev. A 84, 023842 (2011).
[Crossref]

Moloney, J. V.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[Crossref]

Neshev, D. N.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[Crossref]

Orlov, Y. I.

Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer, 1999).

Papazoglou, D. G.

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, Opt. Lett. 36, 1842 (2011).
[Crossref]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[Crossref]

Persch, T.

Pertsch, T.

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[Crossref]

Peschel, U.

Polynkin, P.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[Crossref]

Prakash, J.

Raz, O.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[Crossref]

Renninger, W. H.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photon. 4, 103 (2010).
[Crossref]

Salandrino, A.

Segev, M.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[Crossref]

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[Crossref]

Silberberg, Y.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[Crossref]

Siviloglou, G. A.

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

G. A. Siviloglou and D. N. Christodoulides, Opt. Lett. 32, 979 (2007).
[Crossref]

Stegeman, G. I.

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[Crossref]

Sukhorukov, A. A.

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, arXiv:1107.2992v1 (2011).

Suntsov, S.

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[Crossref]

Tzortzakis, S.

D. G. Papazoglou, N. K. Efremidis, D. N. Christodoulides, and S. Tzortzakis, Opt. Lett. 36, 1842 (2011).
[Crossref]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[Crossref]

Upstill, C.

M. V. Berry and C. Upstill, in Progress in Optics, E. Wolf, ed. (North-Holland, 1980), Vol. XVIII, pp. 259–323.

Walasik, W.

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[Crossref]

Wang, S.

Wise, F. W.

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photon. 4, 103 (2010).
[Crossref]

Yin, X.

Zhang, P.

Zhang, X.

Zhang, Z.

Am. J. Phys. (1)

M. V. Berry and N. L. Balazs, Am. J. Phys. 47, 264 (1979).
[Crossref]

Nat. Photon. (2)

J. Baumgartl, M. Mazilu, and K. Dholakia, Nat. Photon. 2, 675 (2008).
[Crossref]

A. Chong, W. H. Renninger, D. N. Christodoulides, and F. W. Wise, Nat. Photon. 4, 103 (2010).
[Crossref]

Opt. Lett. (9)

Phys. Rep. (1)

F. Lederer, G. I. Stegeman, D. N. Christodoulides, G. Assanto, M. Segev, and Y. Silberberg, Phys. Rep. 463, 1 (2008).
[Crossref]

Phys. Rev. A (1)

R. El-Ganainy, K. G. Makris, M. A. Miri, D. N. Christodoulides, and Z. Chen, Phys. Rev. A 84, 023842 (2011).
[Crossref]

Phys. Rev. Lett. (4)

E. Greenfield, M. Segev, W. Walasik, and O. Raz, Phys. Rev. Lett. 106, 213902 (2011).
[Crossref]

G. A. Siviloglou, J. Broky, A. Dogariu, and D. N. Christodoulides, Phys. Rev. Lett. 99, 213901 (2007).
[Crossref]

A. Minovich, A. E. Klein, N. Janunts, T. Pertsch, D. N. Neshev, and Y. S. Kivshar, Phys. Rev. Lett. 107, 116802 (2011).
[Crossref]

D. Abdollahpour, S. Suntsov, D. G. Papazoglou, and S. Tzortzakis, Phys. Rev. Lett. 105, 253901 (2010).
[Crossref]

Science (1)

P. Polynkin, M. Kolesik, J. V. Moloney, G. A. Siviloglou, and D. N. Christodoulides, Science 324, 229 (2009).
[Crossref]

Other (3)

M. V. Berry and C. Upstill, in Progress in Optics, E. Wolf, ed. (North-Holland, 1980), Vol. XVIII, pp. 259–323.

Y. A. Kravtsov and Y. I. Orlov, Caustics, Catastrophes and Wave Fields (Springer, 1999).

I. L. Garanovich, S. Longhi, A. A. Sukhorukov, and Y. S. Kivshar, arXiv:1107.2992v1 (2011).

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Figures (3)

Fig. 1.
Fig. 1.

Wave dynamics in the case of a wavefront given by Eq. (6). In (a)–(c) the amplitude profile is shown along with the caustic trajectory, while in (d)–(f) the corresponding ray picture is depicted. In particular, in (a) and (d) β=3/2, ξ0=40, A(ξ)=e0.02ξH(ξ); in (b) and (e) β=3, ξ0=40, A(ξ)=H(ξ+ξ0)H(ξ); in (c) and (f) β=2, ξ0=20, A(ξ)=e(ξ/ξ0)2.

Fig. 2.
Fig. 2.

Dynamics of waves that follow power law caustics [Eq. (11)] with A(ξ)=e0.02ξH(ξ). (a) Amplitude profile and the quadratic caustic trajectory, and (b) the corresponding ray picture for δ=2, ξ0=40; (c) amplitude profile and cubic caustic trajectory for δ=3, ξ0=40.

Fig. 3.
Fig. 3.

Discrete focusing of an optical wavefront given by Eq. (13) with zf=10 (a) amplitude profile and (b) the corresponding ray picture.

Equations (13)

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

iu˙n+κ(un1+un+1)=0
un(z)=12πm=um(0)02πeiq(nm)+2iκcos(q)zdq.
u(x,z)=12π02πA(ξ)eiΨdqdξ,
x=ξ+2κsin(q(ξ))z,
2zκcos(q(ξ))q(ξ)+1=0.
ϕ(ξ)=α(ξ)βH(ξ)/β,
z=(ξ)2β2α(β1)κcos(α(ξ)β1),
x=ξ+tan(α(ξ)β1)α(β1)(ξ)β2.
x=[(2β)/(β1)][2α(β1)κz]1/(2β).
x=ξ0+2κz.
x=f(z)=γzδ.
q(ξ)=R{arcsin[γ1/δδ2κ(ξ1δ)(δ1)/δ]}
ϕ=R{(2κzf)2ξ22κzf+ξarcsin(ξ2κzf)}.

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