  Unidirectional reflectionless propagation (or transmission) is an interesting
wave phenomenon observed in many $\mathcal{PT}$-symmetric optical structures.
Theoretical studies on unidirectional reflectionless transmission often use
simple coupled-mode models. The coupled-mode theory can reveal the most
important physical mechanism for this wave phenomenon, but it is only an
approximate theory, and it does not provide accurate quantitative predictions
with respect to geometric and material parameters of the structure. In this
paper, we rigorously study unidirectional reflectionless transmission for
two-dimensional (2D) $\mathcal{PT}$-symmetric periodic structures sandwiched
between two homogeneous media. Using a scattering matrix formalism and a
perturbation method, we show that real zero-reflection frequencies are robust
under $\mathcal{PT}$-symmetric perturbations, and unidirectional reflectionless
transmission is guaranteed to occur if the perturbation (of the dielectric
function) satisfies a simple condition. Numerical examples are presented to
validate the analytical results, and to demonstrate unidirectional invisibility
by tuning the amplitude of balanced gain and loss.
