  We present a formalism to compute Brownian thermal noise in functional
optical surfaces such as grating reflectors, photonic crystal slabs or complex
metamaterials. Such computations are based on a specific readout variable,
typically a surface integral of a dielectric interface displacement weighed by
a form factor. This paper shows how to relate this form factor to Maxwell's
stress tensor computed on all interfaces of the moving surface. As an example,
we examine Brownian thermal noise in monolithic T-shape grating reflectors. The
previous computations by Heinert et al. [Heinert et al., PRD 88 (2013)]
utilizing a simplified readout form factor produced estimates of thermal noise
that are tens of percent higher than those of the exact analysis in the present
paper. The relation between the form factor and Maxwell's stress tensor implies
a close correlation between the optical properties of functional optical
surfaces and thermal noise.
