We consider the polarization of unstable type-IIB D0-branes in the presence of a background five-form field strength. This phenomenon is studied from the point of view of the leading terms in the non-abelian Born Infeld action of the unstable D0-branes. The equations have SO(4) invariant solutions describing a non-commutative 3-sphere, which becomes a classical 3-sphere in the large-N limit. We discuss the interpretation of these solutions as spherical D3-branes. The tachyon plays a tantalizingly geometrical role in relating the fuzzy S 3 geometry to that of a fuzzy S 4 .

Relationship between nonlinear refractivity and two-photon absorption has been studied for PbO -SiO 2 glasses using a nonlinear Kramers-Kronig relation. Nonlinear refractive indices, which are determined with z-scan measurements, are consistent with those which are calculated using the relation from two-photon absorption spectra. This consistency suggests that large intensity-dependent refractivity in this glass system arises from resonant two-photon electronic transitions from oxygen 2p to lead 6p states. © 2005 American Institute of Physics. ͓DOI: 10.1063/1.1891269͔With developments of optical fibers and devices, nonlinear properties in photonic glasses have attracted growing interests. This is because the optical nonlinearity in glasses, which may be smaller than that in crystals, appears in substantial magnitudes in fibers, 1,2 waveguides, [3][4][5] and microspheres 6 due to long propagation distances. In addition, the nonlinearity can be continuously changed by varying glass composition. [7][8][9] The nonlinearity can be induced at selected positions. 5 In some cases, the nonlinearity is responsible for photoinduced phenomena.10 Because of these features, several applications such as all-optical switches, optical limiters, and waveguide formation have been extensively explored. 4,11 However, the optical nonlinearity in glasses remains to be studied. Specifically, fundamental mechanisms of the intensity-dependent refractivity n 2 ͑n = n 0 + n 2 I + ...͒ in glasses have been less-well understood than those in simple systems such as two-level atoms 12 and crystalline semiconductors. 13 For glasses, only empirical or semiempirical relationships have been proposed, 14 most of which cannot provide spectral dependence. It seems to be very difficult to predict the magnitude of n 2 in glasses from an expectation value of polarization which is formulated quantum mechanically. 12 Here, another way to grasp the intensity-dependent refractivity is to relate it with nonlinear absorption. That is, we start with the absorption, which can be connected with photoelectronic transitions between some electronic energy levels. Then, using a nonlinear Kramers-Kronig relation, 15 we can derive the nonlinear refractivity explicitly. Such a procedure has provided satisfactory understanding for crystalline materials.13 It should also be noted that this procedure has an experimental advantage, since absorption spectra can be obtained more easily through transmittance measurements than refractivity spectra. To the authors' knowledge, however, such a procedure has not been applied to glasses.To obtain a unified insight into the optical nonlinearity in glasses, we explore the above method in the present work. As an example, a PbO system is taken, which is known to be a kind of heavy-metal oxide glasses. We measure the nonlinear refractivity n 2 using z-scan measurements, and compare the results with those derived by applying a nonlinear Kramers-Kronig relation to nonlinear absorption spectra. Since the absorption spectra can be directly ...