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.
@article{bwmeta1.element.bwnjournal-article-bcpv50z1p169bwm,
author = {Joets, Alain},
title = {Topology and geometry of caustics in relation with experiments},
journal = {Banach Center Publications},
volume = {50},
year = {1999},
pages = {169-177},
zbl = {0942.78001},
language = {en},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-bcpv50z1p169bwm}
}
Joets, Alain. Topology and geometry of caustics in relation with experiments. Banach Center Publications, Tome 50 (1999) pp. 169-177. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-bcpv50z1p169bwm/
[000] [1] V. I. Arnol'd, Singularities of Caustics and Wave Fronts, Math. Appl. (Soviet Ser.) 62, Kluwer Academic Publ., Dordrecht, 1990.
[001] [2] V. I. Arnol’d, Normal forms of functions near degenerate critical points, the Weyl groups of , , , and Lagrangian singularities (in Russian), Funktsional. Anal. i Prilozhen. 6 (1972), no. 4, 3-25; English transl.: Funct. Anal. Appl. 6 (1972), 254-272.
[002] [3] V. I. Arnol'd, Perestroikas of singularities of potential flows in collisionless media and metamorphoses of caustics in 3-space, J. Soviet Math. 32 (1986), 229-257.
[003] [4] V. I. Arnol'd, V. V. Goryunov, O. V. Lyashko and V. A. Vasil'ev, Singularity theory. II. Classification and applications, in: Dynamical Systems VIII, Encyclopaedia Math. Sci. 39, Springer, Berlin, 1993, 1-235.
[004] [5] M. V. Berry, Waves and Thom's theorem, Adv. in Phys. 25 (1976), 1-26.
[005] [6] M. V. Berry and J. H. Hannay, Umbilic points on Gaussian random surfaces, J. Phys. A 10 (1977), 1809-1821. | Zbl 0445.53002
[006] [7] M. V. Berry, J. F. Nye and F. J. Wright, The elliptic umbilic diffraction catastrophe, Philos. Trans. Roy. Soc. London Ser. A 291 (1979), 453-484.
[007] [8] M. V. Berry and C. Upstill, Catastrophe optics: Morphologies of caustics and their diffraction patterns, in: Progress in Optics, Vol. XVIII, E. Wolf (ed.), North-Holland, Amsterdam, 1980, 257-346.
[008] [9] M. Born and E. Wolf, Principles of Optics, Pergamon Press, Oxford, 1980.
[009] [10] A. Cayley, On the centro-surface of an ellipsoid, Trans. Cambridge Philos. Soc. 12 (1873), Part I, 319-365.
[010] [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.
[011] [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.
[012] [13] P.-G. de Gennes and J. Prost, The Physics of Liquid Crystals, Clarendon Press, Oxford, 1993.
[013] [14] J. Guckenheimer, Caustics and non-degenerate Hamiltonians, Topology 13 (1974), 127-133. | Zbl 0291.58010
[014] [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. | Zbl 0747.58015
[015] [16] S. Janeczko and M. Roberts, Classification of symmetric caustic II: Caustic equivalence, J. London Math. Soc. (2) 48 (1993), 178-192. | Zbl 0788.58005
[016] [17] A. Joets, M. Monastyrsky and R. Ribotta, Ensembles of singularities generated by surfaces with polyhedral symmetry, Phys. Rev. Lett. 81 (1998), 1547-1550. | Zbl 0949.58038
[017] [18] A. Joets and R. Ribotta, Hydrodynamic transitions to chaos in the convection of an anisotropic fluid, J. Physique 47 (1986), 595-606.
[018] [19] A. Joets and R. Ribotta, Structure of caustics studied using the global theory of singularities, Europhys. Lett. 29 (1995), 593-598.
[019] [20] A. Joets and R. Ribotta, A geometrical model for the propagation of rays in an anisotropic inhomogeneous medium, Optics Communications 107 (1994), 200-204.
[020] [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.
[021] [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. | Zbl 0881.57030
[022] [23] M. Kline and I. Kay, Electromagnetic Theory and Geometrical Optics, Interscience Publ., New York, 1965. | Zbl 0123.23602
[023] [24] L. Landau et E. Lifchitz, Électrodynamique des milieux continus, Éditions Mir, Moscou, 1969.
[024] [25] J. F. Nye and J. H. Hannay, The orientations and distortions of caustics in geometrical optics, Optica Acta 31 (1984), 115-130.
[025] [26] D. Panov, private communication.
[026] [27] I. R. Porteous, Geometric Differentiation for the Intelligence of Curves and Surfaces, Cambridge Univ. Press, Cambridge, 1994. | Zbl 0806.53001
[027] [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.
[028] [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. | Zbl 0944.58034
[029] [30] J. Sotomayor and C. Gutiérrez, Structurally stable configurations of lines of principal curvature, Astérisque 98-99 (1982), 195-215. | Zbl 0521.53003
[030] [31] A. Thiaville, Extensions of the geometric solution of the two-dimensional coherent magnetization rotation model, J. Magn. Magn. Mater. 182 (1997), 5-18.
[031] [32] A. Thiaville, Coherent rotation of magnetization in three dimensions: a geometrical approach, to be published.
[032] [33] R. Thom, Les singularités des applications différentiables, Ann. Inst. Fourier (Grenoble) 6 (1956), 43-87. | Zbl 0075.32104
[033] [34] R. Thom, Sur la théorie des enveloppes, J. Math. Pures Appl. (9) 41 (1962), 177-192. | Zbl 0105.16102
[034] [35] R. Thom, Topological models in biology, Topology 8 (1969), 313-335. | Zbl 0165.23301
[035] [36] V. M. Zakalyukin, Reconstructions of fronts and caustics depending on a parameter and versality of mappings, J. Soviet Math. 27 (1984), 2713-2735. | Zbl 0554.58011
[036] [37] Ya. B. Zeldovich, The desintegration of homogeneous matter into parts under the action of gravity, Astrofizika 6 (1970), 319-335.