An experimental search for quasi-real photoproduction of possible {\Theta }⁺ and {\Theta }⁺⁺ resonances was performed using the decay modes pK_S and pK⁺ with the HERMES experiment at DESY. We report evidence for the existence of a {\Theta }⁺ peak with a statistical significance of 4 \sigma . The absence of a peak in the pK⁺ spectrum in the vicinity of the {\Theta }⁺ mass yields an estimate of the upper limit for the quasi-real photoproduction cross-section of the {\Theta }⁺⁺ resonance. In addition, a search for exotic baryon resonances with S=-2, Q=-2 or 0 has been performed in quasi-real photoproduction on a deuterium target by isolating the decay channel {\Xi }^- \pi ^- \to {\Lambda } \pi ^- \pi ^- \to p \pi ^- \pi ^- \pi ^- and {\Xi }^- \pi ⁺ \to {\Lambda } \pi ^- \pi ⁺ \to p \pi ^- \pi ^- \pi ⁺. No evidence for the previously reported {\Xi }^--(1860) and {\Xi }⁰(1860) resonances is found in the {\Xi }^- \pi ^- and {\Xi }^- \pi ⁺ invariant mass spectra, respectively. Upper limits for their photoproduction cross sections of 2.1 nb and 2.5 nb are found at the 90\% confidence level. The photoproduction cross section for the {\Xi }⁰(1530) resonance, which has the same decay mode as the hypothetical {\Xi }⁰(1860) peak, is found to be between 9 and 24 nb.

PACS numbers: 12.39.Mk, 13.60.--r, 13.60.Rj, 14.20.--c