The constraints on the anomalous quartic WW gamma gamma and ZZ gamma gamma gauge boson couplings are investigated through the processes e gamma -> W-gamma nu(e) and e gamma -> Z gamma e. Considering the longitudinal and transverse polarization states of the final W or Z boson and incoming beam polarizations we find 95% confidence level limits on the anomalous coupling parameters a(0) and a(c) with an integrated luminosity of 500 fb(-1) and root s = 0.5, 1 TeV energies. Assuming the W+W-gamma gamma couplings are independent of the ZZ gamma gamma couplings we show that the longitudinal polarization state of the final gauge boson improves the sensitivity to anomalous couplings by a factor of 2-3 depending on energy and coupling. An extra enhancement in sensitivity by a factor of 1.3 comes from a set of initial beam polarizations.
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