Analytical approach to X-phenomenon in alternating phase-shifting masks

An analytical approach to X-phenomenon in alternating phase-shifting masks is given in the framework of the thin-mask approximation. We present an analytical expression for the focus-dependent intensity imbalance between 0° and 180° phase regions when there exists relative phase error. It is shown that X-phenomenon results from the interference between 0th diffracted order, which originates from the phase error and has an in- or out-of-phase component with respect to the ±1st diffracted orders depending on the defocus directions, and the ±1st diffracted orders. Dependences of the intensity imbalance on the phase error and the duty ratio of the structure are given.