  An accurate method for designing transmissive metasurfaces is presented that
provides perfect transmission while transforming the amplitude and phase of the
wavefront. The designed metasurfaces consist of three spatially-varying,
electric impedance sheets separated by two dielectric substrates. The design
method uses integral equations to account for interactions within and between
the impedance sheets, allowing for accurate design. In this paper, a comparison
between the integral equation method and the local periodicity approximation is
presented. The comparison includes one design example for a transmitted field
of uniform phase and amplitude. The design using integral equations provides
better collimation. Two other examples involving an amplitude tapered
transmitted field are reported to show the versatility of the proposed design
technique. In all the examples, the metasurface is $7.35\lambda_0$ wide, the
focal length is $4\lambda_0$, and has an overall thickness of $0.1355 \lambda_0
$ at the operating frequency of 5GHz. The designs are verified using a
commercial finite element electromagnetic solver.
