Bubble-Chamber Study of Photoproduction by 2.8- and 4.7-GeV Polarized Photons. I. Cross-Section Determinations and Production of ${\rho }^0$ and ${\Delta }^{++}$ in the Reaction $\gamma p\to p{\pi }^{+}{\pi }^{-}$

Photoproduction is studied at 2.8 and 4.7 GeV using a linearly polarized monoenergetic photon beam in a hydrogen bubble chamber. We discuss the experimental procedure, the determination of channel cross sections, and the analysis of the channel $\gamma p\to p{\pi }^{+}{\pi }^{-}$. A model-independent analysis of the ${\rho }^0$-decay angular distribution allows us to measure nine independent density-matrix elements. From these we find that the reaction $\gamma p\to p{\rho }^0$ proceeds almost completely through natural parity exchange for squared momentum transfers $\mathrm{}\left|t\right|<1\mathrm{}$ Ge${\mathrm{V}}^2$ and that the $\rho$ production mechanism is consistent with $s$-channel c.m. helicity conservation for $\mathrm{}\left|t\right|<0.4\mathrm{}$ Ge${\mathrm{V}}^2$. A cross section for the production of ${\pi }^{+}{\pi }^{-}$ pairs in the $s$-channel c.m. helicity-conserving $p$-wave state is determined. The $\rho$ mass shape is studied as a function of momentum transfer and is found to be inconsistent with a $t$-independent Ross-Stodolsky factor. Using a $t$-dependent parametrization of the ${\rho }^0$ mass shape we derive a phenomenological ${\rho }^0$ cross section. We compare our phenomenological ${\rho }^0$ cross section with other experiments and find good agreement for $0.05<\mathrm{}\left|t\right|<1\mathrm{}$ Ge${\mathrm{V}}^2$. We discuss the discrepancies in the various determinations of the forward differential cross section. We study models for ${\rho }^0$ photoproduction and find that the Söding model best describes the data. Using the Söding model we determine a ${\rho }^0$ cross section. We determine cross sections and nine density-matrix elements for $\gamma p\to {\Delta }^{++}{\pi }^{-}$. The parity asymmetry for ${\Delta }^{++}$ production is incompatible with simple one-pion exchange. We compare ${\Delta }^{++}$ production with models.