Summary: We consider a multi-frequency inverse scattering problem arising in the design of anti-reflection coatings. These thin films are deposited onto photovoltaic solar cells to enhance their performance. The objective is to determine the space-dependent refractive index in an inhomogeneous optical layer from the reflection coefficients at the surface. The relevant model yields a boundary value problem for the one-dimensional (1D) Helmholtz equation, which we formulate as an equivalent integral equation. The resulting inverse problem is nonlinear and ill-posed. We consider a series expansion of the field depending on the order of nonlinearity of the model. The first-order solution is obtained by using the Born approximation which is valid for weak scattering. Stronger scatterers are sought by considering a nonlinearity of higher order. The mathematical and numerical framework is provided by the (noniterative) method of the approximate inverse (AI) for nonlinear inverse problems. Numerical results are presented to attest the efficiency and stability of the method.