The parabolic functional differential equation $\frac{{\partial u}} {{\partial t}} = D\frac{{\partial ^2 u}} {{\partial x^2 }} - u + K(1 + \gamma \cos u(x + \theta ,t - T)) $ is considered on the circle [0, 2π]. Here, D > 0, T > 0, K > 0, and γ ∈ (0, 1). Such equations arise in the modeling of nonlinear optical systems with a time delay T > 0 and a spatial argument rotated by an angle θ ∈ [0, 2π) in the nonlocal feedback loop in the approximation of a thin circular layer. The goal of this study is to describe spatially inhomogeneous rotating-wave solutions bifurcating from a homogeneous stationary solution in the case of a Andronov-Hopf bifurcation. The existence of such waves is proved by passing to a moving coordinate system, which makes it possible to reduce the problem to the construction of a nontrivial solution to a periodic boundary value problem for a stationary delay differential equation. The existence of rotating waves in an annulus resulting from a Andronov-Hopf bifurcation is proved, and the leading coefficients in the expansion of the solution in powers of a small parameter are obtained. The conditions for the stability of waves are derived by constructing a normal form for the Andronov-Hopf bifurcation for the functional differential equation under study.