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We have used high-resolution magnetic birefringence to study very dilute mixtures of the nonionic surfactant pentaethylene glycol mono-dodecyl-ether (C12E5) in water at fields to H=18 T. The induced refractive index anisotropy Δn shows distinct H dependencies in the Lα-lamellar and L3-‘‘sponge’’ phases, and demonstrates that these are separated by a two-phase coexistence region. In the L3 phase and for surfactant concentrations γ[bar_over_tilde:_approx._equal_to]0.02 (the vicinity of the Lα phase terminus), we observe a sharp change in the magnitude and H dependence of Δn. A quantitative analysis shows this to be consistent with a crossover from a highly-connected ‘‘symmetric sponge’’ to a droplet or ‘‘asymmetric sponge’’ structure as a function of γ, a result previously suggested by neutron scattering and electrical conductivity measurements [R. Strey, R. Schomacker, D. Roux, F. Nallet, and U. Olssen, J. Chem. Soc. Faraday Trans. II 86, 2253 (1990)]. Finally, for γ>=0.02 and at large H, we present evidence of a field-induced transformation of the L3 structure. © 1995 American Institute of Physics. [ABSTRACT FROM AUTHOR]