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## Banach Center Publications

1999 | 50 | 1 | 169-177
Tytuł artykułu

### Topology and geometry of caustics in relation with experiments

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Treść / Zawartość
Warianty tytułu
Języki publikacji
EN
Abstrakty
EN
Caustics of geometrical optics are understood as special types of Lagrangian singularities. In the compact case, they have remarkable topological properties, expressed in particular by the Chekanov relation. We show how this relation may be experimentally checked on an example of biperiodic caustics produced by the deflection of the light by a nematic liquid crystal layer. Moreover the physical laws may impose a geometrical constraint, when the system is invariant by some group of symmetries. We show, on the example of polyhedral caustics, how the two constraints force degenerate umbilics of integer index to appear and determine their spatial organization.
Słowa kluczowe
Czasopismo
Rocznik
Tom
Numer
Strony
169-177
Opis fizyczny
Daty
wydano
1999
Twórcy
autor
• Laboratoire de Physique des Solides, Université de Paris-Sud, bât. 510, 91 405 Orsay cedex, France
Bibliografia
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• [9] M. Born and E. Wolf, Principles of Optics, Pergamon Press, Oxford, 1980.
• [10] A. Cayley, On the centro-surface of an ellipsoid, Trans. Cambridge Philos. Soc. 12 (1873), Part I, 319-365.
• [11] Yu. V. Chekanov, Caustics in geometrical optics (in Russian), Funktsional. Anal. i Prilozhen. 20 (1986), 66-69, 96; English transl.: Funct. Anal. Appl. 20 (1986), 223-226.
• [12] G. Darboux, Sur la forme des lignes de courbure dans le voisinage d'un ombilic, in: Leçons sur la théorie générale des surfaces, Vol. IV, Gauthier-Villars, Paris, 1896, 448-465.
• [13] P.-G. de Gennes and J. Prost, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1993.
• [14] J. Guckenheimer, Caustics and non-degenerate Hamiltonians, Topology 13 (1974), 127-133.
• [15] S. Janeczko and M. Roberts, Classification of symmetric caustic I: Symplectic equivalence, in: Singularity Theory and its Applications, Part II, M. Roberts and I. Stewart (eds.), Lecture Notes in Math. 1463, Springer, Berlin, 1991, 193-219.
• [16] S. Janeczko and M. Roberts, Classification of symmetric caustic II: Caustic equivalence, J. London Math. Soc. (2) 48 (1993), 178-192.
• [17] A. Joets, M. Monastyrsky and R. Ribotta, Ensembles of singularities generated by surfaces with polyhedral symmetry, Phys. Rev. Lett. 81 (1998), 1547-1550.
• [18] A. Joets and R. Ribotta, Hydrodynamic transitions to chaos in the convection of an anisotropic fluid, J. Physique 47 (1986), 595-606.
• [19] A. Joets and R. Ribotta, Structure of caustics studied using the global theory of singularities, Europhys. Lett. 29 (1995), 593-598.
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• [21] A. Joets and R. Ribotta, Experimental determination of a topological invariant in a pattern of optical singularities, Phys. Rev. Lett. 77 (1996), 1755-1758.
• [22] M. È. Kazarian, Umbilical characteristic number of Lagrangian mappings of 3-dimensional pseudooptical manifolds, in: Singularities and Differential Equations, S. Janeczko, W. Zajączkowski and B. Ziemian (eds.), Banach Center Publ. 33, Warsaw, 1996, 161-170.
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• [25] J. F. Nye and J. H. Hannay, The orientations and distortions of caustics in geometrical optics, Optica Acta 31 (1984), 115-130.
• [26] D. Panov, private communication.
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• [28] R. Ribotta and A. Joets, Pinching instability of convective rolls in an anisotropic fluid: first step to chaos, J. Physique 47 (1986), 739-743.
• [29] M. Roberts and V. M. Zakalyukin, Symmetric wavefronts, caustic and Coxeter groups, in: Singularity Theory, D. T. Lê, K. Saito and B. Teissier (eds.), World Sci. Publ., River Edge, 1995, 594-626.
• [30] J. Sotomayor and C. Gutiérrez, Structurally stable configurations of lines of principal curvature, Astérisque 98-99 (1982), 195-215.
• [31] A. Thiaville, Extensions of the geometric solution of the two-dimensional coherent magnetization rotation model, J. Magn. Magn. Mater. 182 (1997), 5-18.
• [32] A. Thiaville, Coherent rotation of magnetization in three dimensions: a geometrical approach, to be published.
• [33] R. Thom, Les singularités des applications différentiables, Ann. Inst. Fourier (Grenoble) 6 (1956), 43-87.
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