We compare the Lorentz violation terms of the pure photon sector between two field theory models, namely, the minimal standard model extension (SME) and the standard model supplement (SMS). From the requirement of the identity of the intersection for the two models, we find that the free photon sector of the SMS can be a subset of the photon sector of the minimal SME. We not only obtain some relations between the SME parameters but also get some constraints on the SMS parameters from the SME parameters. The CPT-odd coefficients (kAF)α of the SME are predicted to be zero. There are 15 degrees of freedom in the Lorentz violation matrix Δαβ of free photons of the SMS related with the same number of degrees of freedom in the tensor coefficients (kF)αβμν, which are independent from each other in the minimal SME but are interrelated in the intersection of the SMS and the minimal SME. With the related degrees of freedom, we obtain the conservative constraints (2σ) on the elements of the photon Lorentz violation matrix. The detailed structure of the photon Lorentz violation matrix suggests some applications to the Lorentz violation experiments for photons.