We investigate the effects of unparticles on Compton scattering, e gamma -> e gamma based on a future e^+e^- linear collider such as the CLIC. For different polarization configurations, we calculate the lower limits of the unparticle energy scale Lambda_U for a discovery reach at the center of mass energies sqrt(s)=0.5 TeV- 3 TeV. It is shown that, especially, for smaller values of the mass dimension d, (1 <d <1.3), and for high energies and luminosities of the collider these bounds are very significant. As a stringent limit, we find Lambda_U>80 TeV for d<1.3 at sqrt(s)=3 TeV, and 1 ab^(-1) integrated luminosity per year, which is comparable with the limits calculated from other low and high energy physics implications.
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