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Fast surface integral equation (SIE) methods seem to be ideal for simulating 3-D nanophotonic devices, as such devices generate fields in both the interior device volume and in the infinite exterior domain. SIE methods were originally developed for computing scattering from structures with finite surfaces, and since SIE methods automatically represent the infinite extent of the exterior scattered field, there was no need to develop numerical absorbers. Numerical absorbers are needed when SIE methods are used to simulate nanophotonic devices that process or couple light, to provide nonreflecting termination at the optical ports of such devices. In this paper, we focus on the problem of developing an approach to absorbers that are suitable for port termination, yet preserve the surface-only discretization and the geometry-independent Green's function properties of the SIE methods. Preserving these properties allows the absorber approach to be easily incorporated in commonly used fast solvers. We describe our solution to the absorber problem, that of using a gradually increasing surface conductivity, and show how to include surface conductivity in SIE methods. We also analyze numerical results using our absorber approach to terminate a finite-length rectangular cross section dielectric waveguide. The numerical results demonstrate that our surface-conductivity absorber can easily achieve a reflected power of less than 10-7, and that the magnitude of the transition reflection is proportional to 1/L2d+2, where L is the absorber length and d is the order of the differentiability of the surface conductivity function.