Optical homodyne detection in view of joint probability distribution

Optical homodyne detection is examined in view of joint probability distribution. It is usually discussed that the relative phase between independent laser fields are localized by photon-number measurements in interference experiments such as homodyne detection. This provides reasoning to use operationally coherent states for laser fields in the description of homodyne detection and optical quantum-state tomography. Here, we elucidate these situations by considering the joint probability distribution and the invariance of homodyne detection under the phase transformation of optical fields.