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Thick films (3 $\sim$ 4 $\mu$m) of nanoclustered SiN$_x$:H on p-type Si (100) substrates were fabricated by using an rf plasma-enhanced chemical vapor deposition (PECVD) method under flows of hydrogen-diluted silane and nitrogen gas. The photoluminescence (PL) of the SiN$_x$:H films at room temperature showed broad emission bands modulated by the Fabry-Perot interference patterns at a wavelength of 400 $\sim$ 600 nm. The Fabry-Perot interference characteristics in the visible ranges was discussed in view of the refractive index variations of the interface layers between films and the Si substrates and of the near-air surface layers of the thick films. A consideration of the thick films as Fabry-Perot microcavity structures is quite successful in explaining the modulated PL emission spectra, where the refractive indices of the thick films are observed to be in the range of 1.8 $\sim$ 2.5 for SiN$_x$:H films with $x$ = 1.0 $\sim$ 1.2. These results suggest that nanoclustered SiN$_x$:H films would be a probable material for further applications such as microcavities, resonators, filters, and gain media over wide ranges of visible wavelengths.