The crossings of Saturn's ring plane by Earth were observed in the near infrared on May 22 and August 10, 1995, from the 2.2-m telescope of the University of Hawaii, the 2-m telescope at Pic du Midi, France, and with the Adonis adaptive optics camera at the 3.6-m telescope of the European Southern Observatory in Chile. Images from the Hubble Space Telescope, obtained in August 1995, are also reanalyzed. The radial brightness profiles of the rings indicate that the outer and usually faint F ring dominates the edgeon brightness of the system, thus hiding the vertical structure of the main rings within a few hours around the ring plane crossing. The photometric behaviors of the A, B, and C rings and of the Cassini Division are analyzed, using a radiative transfer code which includes the illuminations by the Sun and by the planet. The F ring is modeled as a physically thick ribbon of height H, composed of large particles embedded in dust of fractional optical depth f. The observed profiles, combined with previous results, can be explained if the F ring is both optically thick (radial optical depth ∼0.20) and physically thick (H = 21 ± 4 km). We suggest that this vertical distribution results from the interactions between ring particles and shepherding satellites and/or from gravitational stirring by large bodies. The dust particles dominate the F ring's photometric behavior even in backscattered light ( f > 0.80). Constraints on the particle properties of the other rings are also derived.