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Planar lightwave circuits allow complex optical devices to be made in just a few lithography steps. One of the best ways to make gratings and lenses in planar circuits is to use waveguide arrays. Waveguide arrays typically terminate in the free-space regions of star couplers. The most famous example is the waveguide grating router (WGR), also called the PHASAR or arrayed waveguide grating. We will define waveguide gratings as those that have mainly asymmetric path length distributions, resulting in mainly chromatic angular dispersion, and lenses as those that have mainly symmetric path length distributions, resulting in mainly chromatic radial dispersion.