The wavelength response of the waveguide grating router plays an important role in optical networks, for it determines the maximum number of channels, the minimum channel separation, and the level of isolation between channels. One would like in general the router to approximate a rectangular response without exceeding a specified loss. Here, we show how the best approximation obtained for a given loss depends on the filter complexity, the total number of arms, and the specified tolerance to fabrication errors. We derive explicity the minimum loss required to produce a given approximation, and show that the filter design is primarily determined by two parameters, the ratio of channel spacing to passband width and the passband flatness. We determine explicitly the optimum design conditions. We show that the loss can be entirely eliminated by concatenating two routers, synchronized so as to produce a unitary transformation over a finite passband.
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