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Analytical expressions for the power spectral densities of intensity and frequency noise of single-mode semiconductor lasers operating in a regime of injection locking are derived by appropriately taking into account the spontaneous emission processes into the lasing modes of both the master and slave lasers. They show how the noise spectra of the slave are influenced by the value of the injected power, by the difference between the emission frequencies of the master and slave optical cavities, and how they are correlated to the noise properties of both the master and the free-running slave. In particular, the very low frequency part of the frequency noise of the slave turns out to coincide with that of the master within a certain frequency region whose range increases as the values of the injected signal does, too. We also present measurements of the power spectral densities obtained by means of an experimental apparatus similar to that described in  and show how the experimental results are accounted for by the present theory.