We present a surface integral algorithm, utilizing Fourier integrals to solve optical fields within a volume bounded by a complicated polygonal surface. The method enables the full electric field to be solved from electric field values on the bounding surface at any point within the volume. As opposed to FDTD and FEM methods, volume discretization and the need to iteratively solve the E-field at every discrete volume element is not needed with this method. Our new surface integral algorithm circumvents the limitations that exist in current surface methods. Namely, in present methods, the need to determine a Green's function only allows for simple bounding surfaces, and these methods generally use integrals that cannot utilize computationally fast Fourier integrals. Here, we prove the algorithm mathematically, show it with a numerical example, and outline important cases where the algorithm can be used. These cases include the design of free-form reflectors and near field optical scanning microscopy (SNOM). We then briefly analyze the algorithm's computational scaling.