Dynamics of the optical swallowtail catastrophe

Alessandro Zannotti; Falko Diebel; Cornelia Denz

doi:10.1364/OPTICA.4.001157

Dynamics of the optical swallowtail catastrophe

Alessandro Zannotti, Falko Diebel, and Cornelia Denz

Optica
Vol. 4,
Issue 10,
pp. 1157-1162
(2017)
•https://doi.org/10.1364/OPTICA.4.001157

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Alessandro Zannotti, Falko Diebel, and Cornelia Denz, "Dynamics of the optical swallowtail catastrophe," Optica 4, 1157-1162 (2017)

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Related Topics
Optics & Photonics Topics
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The topics in this list come from the OSA Optics and Photonics Topics applied to this article.
Previously assigned OCIS codes
- Paraxial wave optics (070.2580)
- Wave propagation (070.7345)
- Wave dressing of rays (080.7343)
- Singular optics (260.6042)

About this Article
History
- Original Manuscript: May 2, 2017
- Revised Manuscript: August 1, 2017
- Manuscript Accepted: August 3, 2017
- Published: September 25, 2017

Accessible

Open Access

Abstract

Perturbing the external control parameters of nonlinear systems leads to dramatic changes in their bifurcations. A branch of singular theory, the catastrophe theory, analyzes the generating function that depends on state and control parameters. It predicts the formation of bifurcations as geometrically stable structures and categorizes them hierarchically. We apply the approach to evaluate the catastrophe diffraction integral with respect to two-dimensional cross sections through the control parameter space and thus transfer these bifurcations to optics, where they manifest as caustics in transverse light fields. For all optical catastrophes that depend on a single state parameter (cuspoids), we analytically derive a universal expression for the propagation of all corresponding cuspoid beams. We show that the dynamics of the resulting cuspoids can be expressed by higher-order optical catastrophes with dynamically changing control parameters. We show analytically and experimentally that particular swallowtail beams develop caustics with geometrical structures corresponding to higher-order butterfly catastrophes during propagation, whereas differently tailored swallowtail beams decay to lower-order cusp catastrophes.

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References

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Year
|
Author
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Publication

R. Thom, Structural Stability and Morphogenesis (W. A. Benjamin, 1972).
V. I. Arnol’d and G. S. Wassermann, Catastrophe Theory, 3rd ed. (Springer, 2003).
M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[Crossref]
D. Gifford, C. Miller, and N. Kern, “A systematic analysis of caustic methods for galaxy cluster masses,” Astrophys. J. 773, 116 (2013).
[Crossref]
R. Höhmann, U. Kuhl, H.-J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: a microwave study,” Phys. Rev. Lett. 104, 093901 (2010).
[Crossref]
A. Mathis, L. Froehly, S. Toenger, F. Dias, G. Genty, and J. M. Dudley, “Caustics and rogue waves in an optical sea,” Sci. Rep. 5, 12822 (2015).
[Crossref]
H. Trinkhaus and F. Drepper, “On the analysis of diffraction catastrophes,” J. Phys. A 10, L11–L16 (1977).
[Crossref]
J. E. Sheridan and M. A. Abelson, “Cusp catastrophe model of employee turnover,” Acad. Manage. J. 26, 418–436 (1983).
[Crossref]
R. C. Méndez and F. F. Alfaro, “Fold, cusp, and swallowtail catastrophes in the evolution of planar closed orbits of hydrogen in crossed electric and magnetic fields,” Revista Colombiana de Fisica 45, 28–44 (2013).
M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Philos. Trans. R. Soc. London A 291, 453–484 (1979).
[Crossref]
M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 28, 257–346 (1980).
[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. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20, 18955–18966 (2012).
[Crossref]
A. Zannotti, M. Rüschenbaum, and C. Denz, “Pearcey solitons in curved nonlinear photonic caustic lattices,” J. Opt. 19, 094001 (2017).
[Crossref]
F. Diebel, B. M. Bokic, D. V. Timotijevic, D. M. Jovic Savic, and C. Denz, “Soliton formation by decelerating interacting Airy beams,” Opt. Express 23, 24351–24361 (2015).
[Crossref]
J. Baumgartl, M. Mazilu, and K. Dholakia, “Optically mediated particle clearing using Airy wavepackets,” Nat. Photonics 2, 675–678 (2008).
[Crossref]
M. Manousidaki, D. G. Papazoglou, M. Farsari, and S. Tzortzakis, “Abruptly autofocusing beams enable advanced multiscale photo-polymerization,” Optica 3, 525–530 (2016).
[Crossref]
A. Zannotti, F. Diebel, M. Boguslawski, and C. Denz, “Optical catastrophes of the swallowtail and butterfly beams,” New J. Phys. 19, 053004 (2017).
[Crossref]
M. V. Berry, “Stable and unstable Airy-related caustics and beams,” J. Opt. 19, 055601 (2017).
[Crossref]
J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (IOP Publishing, 1999).
A. E. Siegman, Lasers (University Science Books, 1990).
A. Szameit and S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B 43, 163001 (2010).
[Crossref]
M. V. Berry and N. L. Balazs, “Nonspreading wave packets,” Am. J. Phys. 47, 264–267 (1979).
[Crossref]
J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999).
[Crossref]
E. C. Zeeman, “Catastrophe theory,” in Structural Stability in Physics (Springer, 1979), pp. 12–22.
M. V. Berry, “Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces,” J. Phys. A 8, 566–584 (1975).
[Crossref]
Y. A. Kravtsov and Y. I. Orlov, “Caustics, catastrophes, and wave fields,” Usp. Fiz. Nauk 141, 591–627 (1983).
[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]

2017 (3)

A. Zannotti, M. Rüschenbaum, and C. Denz, “Pearcey solitons in curved nonlinear photonic caustic lattices,” J. Opt. 19, 094001 (2017).
[Crossref]

A. Zannotti, F. Diebel, M. Boguslawski, and C. Denz, “Optical catastrophes of the swallowtail and butterfly beams,” New J. Phys. 19, 053004 (2017).
[Crossref]

M. V. Berry, “Stable and unstable Airy-related caustics and beams,” J. Opt. 19, 055601 (2017).
[Crossref]

2016 (1)

M. Manousidaki, D. G. Papazoglou, M. Farsari, and S. Tzortzakis, “Abruptly autofocusing beams enable advanced multiscale photo-polymerization,” Optica 3, 525–530 (2016).
[Crossref]

2015 (2)

F. Diebel, B. M. Bokic, D. V. Timotijevic, D. M. Jovic Savic, and C. Denz, “Soliton formation by decelerating interacting Airy beams,” Opt. Express 23, 24351–24361 (2015).
[Crossref]

A. Mathis, L. Froehly, S. Toenger, F. Dias, G. Genty, and J. M. Dudley, “Caustics and rogue waves in an optical sea,” Sci. Rep. 5, 12822 (2015).
[Crossref]

2013 (2)

D. Gifford, C. Miller, and N. Kern, “A systematic analysis of caustic methods for galaxy cluster masses,” Astrophys. J. 773, 116 (2013).
[Crossref]

R. C. Méndez and F. F. Alfaro, “Fold, cusp, and swallowtail catastrophes in the evolution of planar closed orbits of hydrogen in crossed electric and magnetic fields,” Revista Colombiana de Fisica 45, 28–44 (2013).

2012 (2)

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]

J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20, 18955–18966 (2012).
[Crossref]

2010 (2)

R. Höhmann, U. Kuhl, H.-J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: a microwave study,” Phys. Rev. Lett. 104, 093901 (2010).
[Crossref]

A. Szameit and S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B 43, 163001 (2010).
[Crossref]

2008 (1)

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

2007 (1)

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

1999 (1)

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999).
[Crossref]

1983 (2)

Y. A. Kravtsov and Y. I. Orlov, “Caustics, catastrophes, and wave fields,” Usp. Fiz. Nauk 141, 591–627 (1983).
[Crossref]

J. E. Sheridan and M. A. Abelson, “Cusp catastrophe model of employee turnover,” Acad. Manage. J. 26, 418–436 (1983).
[Crossref]

1980 (1)

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 28, 257–346 (1980).
[Crossref]

1979 (2)

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

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Philos. Trans. R. Soc. London A 291, 453–484 (1979).
[Crossref]

1977 (1)

H. Trinkhaus and F. Drepper, “On the analysis of diffraction catastrophes,” J. Phys. A 10, L11–L16 (1977).
[Crossref]

1976 (1)

M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[Crossref]

1975 (1)

M. V. Berry, “Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces,” J. Phys. A 8, 566–584 (1975).
[Crossref]

Abelson, M. A.

J. E. Sheridan and M. A. Abelson, “Cusp catastrophe model of employee turnover,” Acad. Manage. J. 26, 418–436 (1983).
[Crossref]

Alfaro, F. F.

Arnol’d, V. I.

V. I. Arnol’d and G. S. Wassermann, Catastrophe Theory, 3rd ed. (Springer, 2003).

Balazs, N. L.

M. V. Berry and N. L. 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, “Stable and unstable Airy-related caustics and beams,” J. Opt. 19, 055601 (2017).
[Crossref]

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 28, 257–346 (1980).
[Crossref]

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

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Philos. Trans. R. Soc. London A 291, 453–484 (1979).
[Crossref]

M. V. Berry, “Waves and Thom’s theorem,” Adv. Phys. 25, 1–26 (1976).
[Crossref]

M. V. Berry, “Cusped rainbows and incoherence effects in the rippling-mirror model for particle scattering from surfaces,” J. Phys. A 8, 566–584 (1975).
[Crossref]

Boguslawski, M.

A. Zannotti, F. Diebel, M. Boguslawski, and C. Denz, “Optical catastrophes of the swallowtail and butterfly beams,” New J. Phys. 19, 053004 (2017).
[Crossref]

Bokic, B. M.

F. Diebel, B. M. Bokic, D. V. Timotijevic, D. M. Jovic Savic, and C. Denz, “Soliton formation by decelerating interacting Airy beams,” Opt. Express 23, 24351–24361 (2015).
[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]

Campos, J.

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999).
[Crossref]

Christodoulides, D. N.

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

Cottrell, D. M.

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999).
[Crossref]

Courvoisier, F.

Davis, J. A.

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999).
[Crossref]

Dennis, M. R.

J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20, 18955–18966 (2012).
[Crossref]

Denz, C.

A. Zannotti, F. Diebel, M. Boguslawski, and C. Denz, “Optical catastrophes of the swallowtail and butterfly beams,” New J. Phys. 19, 053004 (2017).
[Crossref]

A. Zannotti, M. Rüschenbaum, and C. Denz, “Pearcey solitons in curved nonlinear photonic caustic lattices,” J. Opt. 19, 094001 (2017).
[Crossref]

F. Diebel, B. M. Bokic, D. V. Timotijevic, D. M. Jovic Savic, and C. Denz, “Soliton formation by decelerating interacting Airy beams,” Opt. Express 23, 24351–24361 (2015).
[Crossref]

Dholakia, K.

J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20, 18955–18966 (2012).
[Crossref]

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

Dias, F.

A. Mathis, L. Froehly, S. Toenger, F. Dias, G. Genty, and J. M. Dudley, “Caustics and rogue waves in an optical sea,” Sci. Rep. 5, 12822 (2015).
[Crossref]

Diebel, F.

A. Zannotti, F. Diebel, M. Boguslawski, and C. Denz, “Optical catastrophes of the swallowtail and butterfly beams,” New J. Phys. 19, 053004 (2017).
[Crossref]

F. Diebel, B. M. Bokic, D. V. Timotijevic, D. M. Jovic Savic, and C. Denz, “Soliton formation by decelerating interacting Airy beams,” Opt. Express 23, 24351–24361 (2015).
[Crossref]

Dogariu, A.

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

Drepper, F.

H. Trinkhaus and F. Drepper, “On the analysis of diffraction catastrophes,” J. Phys. A 10, L11–L16 (1977).
[Crossref]

Dudley, J. M.

A. Mathis, L. Froehly, S. Toenger, F. Dias, G. Genty, and J. M. Dudley, “Caustics and rogue waves in an optical sea,” Sci. Rep. 5, 12822 (2015).
[Crossref]

Farsari, M.

M. Manousidaki, D. G. Papazoglou, M. Farsari, and S. Tzortzakis, “Abruptly autofocusing beams enable advanced multiscale photo-polymerization,” Optica 3, 525–530 (2016).
[Crossref]

Froehly, L.

A. Mathis, L. Froehly, S. Toenger, F. Dias, G. Genty, and J. M. Dudley, “Caustics and rogue waves in an optical sea,” Sci. Rep. 5, 12822 (2015).
[Crossref]

Furfaro, L.

Genty, G.

A. Mathis, L. Froehly, S. Toenger, F. Dias, G. Genty, and J. M. Dudley, “Caustics and rogue waves in an optical sea,” Sci. Rep. 5, 12822 (2015).
[Crossref]

Gifford, D.

D. Gifford, C. Miller, and N. Kern, “A systematic analysis of caustic methods for galaxy cluster masses,” Astrophys. J. 773, 116 (2013).
[Crossref]

Heller, E. J.

R. Höhmann, U. Kuhl, H.-J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: a microwave study,” Phys. Rev. Lett. 104, 093901 (2010).
[Crossref]

Höhmann, R.

R. Höhmann, U. Kuhl, H.-J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: a microwave study,” Phys. Rev. Lett. 104, 093901 (2010).
[Crossref]

Jacquot, M.

Jovic Savic, D. M.

F. Diebel, B. M. Bokic, D. V. Timotijevic, D. M. Jovic Savic, and C. Denz, “Soliton formation by decelerating interacting Airy beams,” Opt. Express 23, 24351–24361 (2015).
[Crossref]

Kaplan, L.

R. Höhmann, U. Kuhl, H.-J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: a microwave study,” Phys. Rev. Lett. 104, 093901 (2010).
[Crossref]

Kern, N.

D. Gifford, C. Miller, and N. Kern, “A systematic analysis of caustic methods for galaxy cluster masses,” Astrophys. J. 773, 116 (2013).
[Crossref]

Kravtsov, Y. A.

Y. A. Kravtsov and Y. I. Orlov, “Caustics, catastrophes, and wave fields,” Usp. Fiz. Nauk 141, 591–627 (1983).
[Crossref]

Kuhl, U.

R. Höhmann, U. Kuhl, H.-J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: a microwave study,” Phys. Rev. Lett. 104, 093901 (2010).
[Crossref]

Lacourt, P. A.

Lindberg, J.

J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20, 18955–18966 (2012).
[Crossref]

Manousidaki, M.

M. Manousidaki, D. G. Papazoglou, M. Farsari, and S. Tzortzakis, “Abruptly autofocusing beams enable advanced multiscale photo-polymerization,” Optica 3, 525–530 (2016).
[Crossref]

Mathis, A.

A. Mathis, L. Froehly, S. Toenger, F. Dias, G. Genty, and J. M. Dudley, “Caustics and rogue waves in an optical sea,” Sci. Rep. 5, 12822 (2015).
[Crossref]

Mazilu, M.

J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20, 18955–18966 (2012).
[Crossref]

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

Méndez, R. C.

Miller, C.

D. Gifford, C. Miller, and N. Kern, “A systematic analysis of caustic methods for galaxy cluster masses,” Astrophys. J. 773, 116 (2013).
[Crossref]

Moreno, I.

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999).
[Crossref]

Mourka, A.

J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20, 18955–18966 (2012).
[Crossref]

Nolte, S.

A. Szameit and S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B 43, 163001 (2010).
[Crossref]

Nye, J. F.

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Philos. Trans. R. Soc. London A 291, 453–484 (1979).
[Crossref]

J. F. Nye, Natural Focusing and Fine Structure of Light: Caustics and Wave Dislocations (IOP Publishing, 1999).

Orlov, Y. I.

Y. A. Kravtsov and Y. I. Orlov, “Caustics, catastrophes, and wave fields,” Usp. Fiz. Nauk 141, 591–627 (1983).
[Crossref]

Papazoglou, D. G.

M. Manousidaki, D. G. Papazoglou, M. Farsari, and S. Tzortzakis, “Abruptly autofocusing beams enable advanced multiscale photo-polymerization,” Optica 3, 525–530 (2016).
[Crossref]

Ring, J. D.

J. D. Ring, J. Lindberg, A. Mourka, M. Mazilu, K. Dholakia, and M. R. Dennis, “Auto-focusing and self-healing of Pearcey beams,” Opt. Express 20, 18955–18966 (2012).
[Crossref]

Rüschenbaum, M.

A. Zannotti, M. Rüschenbaum, and C. Denz, “Pearcey solitons in curved nonlinear photonic caustic lattices,” J. Opt. 19, 094001 (2017).
[Crossref]

Sheridan, J. E.

J. E. Sheridan and M. A. Abelson, “Cusp catastrophe model of employee turnover,” Acad. Manage. J. 26, 418–436 (1983).
[Crossref]

Siegman, A. E.

A. E. Siegman, Lasers (University Science Books, 1990).

Siviloglou, G. A.

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

Stöckmann, H.-J.

R. Höhmann, U. Kuhl, H.-J. Stöckmann, L. Kaplan, and E. J. Heller, “Freak waves in the linear regime: a microwave study,” Phys. Rev. Lett. 104, 093901 (2010).
[Crossref]

Szameit, A.

A. Szameit and S. Nolte, “Discrete optics in femtosecond-laser-written photonic structures,” J. Phys. B 43, 163001 (2010).
[Crossref]

Thom, R.

R. Thom, Structural Stability and Morphogenesis (W. A. Benjamin, 1972).

Timotijevic, D. V.

F. Diebel, B. M. Bokic, D. V. Timotijevic, D. M. Jovic Savic, and C. Denz, “Soliton formation by decelerating interacting Airy beams,” Opt. Express 23, 24351–24361 (2015).
[Crossref]

Toenger, S.

A. Mathis, L. Froehly, S. Toenger, F. Dias, G. Genty, and J. M. Dudley, “Caustics and rogue waves in an optical sea,” Sci. Rep. 5, 12822 (2015).
[Crossref]

Trinkhaus, H.

H. Trinkhaus and F. Drepper, “On the analysis of diffraction catastrophes,” J. Phys. A 10, L11–L16 (1977).
[Crossref]

Tzortzakis, S.

M. Manousidaki, D. G. Papazoglou, M. Farsari, and S. Tzortzakis, “Abruptly autofocusing beams enable advanced multiscale photo-polymerization,” Optica 3, 525–530 (2016).
[Crossref]

Upstill, C.

M. V. Berry and C. Upstill, “Catastrophe optics: morphologies of caustics and their diffraction patterns,” Prog. Opt. 28, 257–346 (1980).
[Crossref]

Wassermann, G. S.

V. I. Arnol’d and G. S. Wassermann, Catastrophe Theory, 3rd ed. (Springer, 2003).

Wright, F. J.

M. V. Berry, J. F. Nye, and F. J. Wright, “The elliptic umbilic diffraction catastrophe,” Philos. Trans. R. Soc. London A 291, 453–484 (1979).
[Crossref]

Yzuel, M. J.

J. A. Davis, D. M. Cottrell, J. Campos, M. J. Yzuel, and I. Moreno, “Encoding amplitude information onto phase-only filters,” Appl. Opt. 38, 5004–5013 (1999).
[Crossref]

Zannotti, A.

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Fig. 1. Scheme of the experimental setup. BS, beam splitter; FF, Fourier filter; L, lens; MO, microscope objective; SLM, phase-only spatial light modulator.

Download Full Size | PPT Slide | PDF

Fig. 2. Propagation of the

Sw (X, Y, 0)

beam. Top: intensity volume evaluated according to Eq. (4). Bottom: experimentally obtained intensity volume with the same parameters.

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Fig. 3. Propagation of the

Sw (X, 0, Y)

beam. Top: intensity volume evaluated according to Eq. (8). Bottom: experimentally obtained intensity volume with the same parameters.

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Fig. 4. Dynamics of the swallowtail caustic. Two different perspectives to the 3D caustic surface during propagation. The initial swallowtail caustic (

z = 0

) decays to cusps

(z \neq 0)

. Representative

z

positions are highlighted with different colors and mapped as contours to the

x - y

plane.

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Equations (15)

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C_{n} (a) = \int_{R} \exp [i P_{n} (a, s)] d s, P_{n} (a, s) = s^{n} + \sum_{j = 1}^{n - 2} a_{j} s^{j} .

{\frac{\partial P_{n}}{\partial s} |}_{s = s_{i}} = 0 and {\frac{\partial^{2} P_{n}}{\partial s^{2}} |}_{s = s_{i}} = 0 .

C_{n} (a, z) = \frac{k}{2 π i z} \int_{R^{2}} C_{n} (\frac{x^{'}}{x_{0}}, \frac{y^{'}}{y_{0}}, a^{'}) \times \exp [\frac{i k}{2 z} ({(a_{α} - x^{'})}^{2} + {(a_{β} - y^{'})}^{2})] d x^{'} d y^{'} = \int_{R} \exp [i (P_{n} (a, s) - \frac{z}{z_{e x}} s^{2 α} - \frac{z}{z_{e y}} s^{2 β})] d s .

Sw (X, Y, a_{3}, Z) = \exp [i ψ] \cdot Sw (C_{1}, C_{2}, C_{3}),

C_{1} = X + 2 p Y Z + p (3 a_{3} p - 2) Z^{2} - 15 p^{4} Z^{4}, C_{2} = Y + (3 a_{3} p - 1) Z - 20 p^{3} Z^{3}, C_{3} = a_{3} - 10 p^{2} Z^{2}, ψ = p X Z + p^{2} Y Z^{2} + p^{2} (a_{3} p - 1) Z^{3} - 4 p^{5} Z^{5},

Sw (X, Y, a_{3}, Z) = {Sw}^{*} (X, - Y, a_{3}, - Z),

\tilde{S} w (k_{x}, k_{y}, 0) = \tilde{P} e (k_{x}, k_{y}) \cdot \exp [i ({(x_{0} k_{x})}^{5} - {(x_{0} k_{x})}^{4})] .

Sw (\frac{x}{x_{0}}, \frac{y}{y_{0}}, 0) = Pe (\frac{x}{x_{0}}, \frac{y}{y_{0}}) * \frac{δ (y)}{x_{0}} \exp [i (\frac{x}{5 x_{0}} - \frac{4}{3125})] Sw (\frac{x}{x_{0}} - \frac{3}{125}, - \frac{4}{25}, - \frac{2}{5}) .

Sw (X, a_{2}, Y, Z) = \frac{1}{γ} \exp [i ψ] \cdot Bu (B_{1}, B_{2}, B_{3}, B_{4}),

B_{1} = \frac{1}{γ} (X + 2 (Z - a_{2}) \frac{q}{Z} + 3 Y {(\frac{q}{Z})}^{2} + 4 {(\frac{q}{Z})}^{4}), B_{2} = \frac{1}{γ^{2}} (a_{2} - Z + 3 Y \frac{q}{Z} + \frac{15}{2} {(\frac{q}{Z})}^{3}), B_{3} = \frac{1}{γ^{3}} (Y + \frac{20}{3} {(\frac{q}{Z})}^{2}), B_{4} = - \frac{1}{γ^{4}} \frac{5}{2} \frac{q}{Z}, and ψ = - X \frac{q}{Z} + (Z - a_{2}) {(\frac{q}{Z})}^{2} + Y {(\frac{q}{Z})}^{3} + \frac{5}{6} {(\frac{q}{Z})}^{5}

Sw (X, a_{2}, Y, Z) = \frac{1}{γ} \exp [i ψ] \cdot {Bu}^{*} (B_{1}, B_{2}, B_{3}, B_{4}),

B_{1} = - \frac{1}{γ} (X - 2 (Z - a_{2}) \frac{q}{Z} + 3 Y {(\frac{q}{Z})}^{2} + 4 {(\frac{q}{Z})}^{4}), B_{2} = - \frac{1}{γ^{2}} (a_{2} - Z + 3 Y \frac{q}{Z} + \frac{15}{2} {(\frac{q}{Z})}^{3}), B_{3} = - \frac{1}{γ^{3}} (Y + \frac{20}{3} {(\frac{q}{Z})}^{2}), B_{4} = - \frac{1}{γ^{4}} \frac{5}{2} \frac{q}{Z}, and ψ = X \frac{q}{Z} - (Z - a_{2}) {(\frac{q}{Z})}^{2} + Y {(\frac{q}{Z})}^{3} + \frac{5}{6} {(\frac{q}{Z})}^{5} .

Sw (X, a_{2}, Y, Z) = {Sw}^{*} (X, - a_{2}, Y, - Z),

\tilde{S} w (k_{x}, a_{2}, k_{y}) = \tilde{B} u (k_{x}, a_{2}, k_{y}, a_{4}) \cdot \exp [- i ({(x_{0} k_{x})}^{6} - {(x_{0} k_{x})}^{5} + a_{4} {(x_{0} k_{x})}^{4})] .

Bu (\frac{x}{x_{0}}, 0, \frac{y}{y_{0}}, 0) = Sw (\frac{x}{x_{0}}, 0, \frac{y}{y_{0}}) * \frac{δ (y)}{x_{0}} \exp [i (\frac{x}{6 x_{0}} - \frac{5}{6^{6}})] Bu (\frac{x}{x_{0}} - \frac{1}{324}, - \frac{5}{144}, - \frac{5}{27}, - \frac{5}{12}),