We have combined a survey at 24 μm to 0.3 mJy with the Multiband Imaging Photometer (MIPS) on the Spitzer Space Telescope, a 20 cm A-configuration Very Large Array (VLA) survey covering 0.5 deg2, and an existing 172 ks Chandra X-Ray Observatory exposure to investigate the nature of optically faint radio and X-ray sources in the NOAO Deep Wide-Field Survey (NDWFS) in the Bootes field. A catalog of 392 radio sources is defined with optical identifications or magnitude limits. We find little overlap between the radio- and infrared-selected populations: only 9% of the infrared sources are detected in the radio, and only 33% of the radio sources are detected in the infrared. Thirty-six (10%) of the 377 compact radio sources lack optical counterparts in the NDWFS BW, R, and I images. We refer to these objects as optically invisible radio sources (OIRSs). Only four (13%) of the 31 OIRSs observed with MIPS are detected at 24 μm. Comparisons of the radio and infrared properties of the OIRSs with various galaxy spectral energy distributions demonstrate that most of these sources are powered by active galactic nuclei (AGNs) rather than starbursts. Similarly, 11 X-ray sources observed by both MIPS and the VLA are classified as optically invisible X-ray sources (OIXSs). None is detected at 24 μm or 20 cm. All seven OIXSs detected in Chandra's 0.5-2 keV band have infrared to X-ray flux ratios consistent with their luminosity being dominated by an unobscured AGN. From these results we conclude that both the optically invisible radio and X-ray source populations are primarily AGNs, relatively unaffected by dust, and most likely at z 1. No OIRSs are detected in X-ray emission, and no OIXSs are detected at 20 cm. However, given the wide range in radio and X-ray properties of known AGNs and the size of our samples, this lack of overlap does not necessarily imply different AGN source populations.
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