Advanced crosstalk reduction in an N x N two-stage AWG-based switch with odd N

When multiple signals of the same wavelength cross an arrayed waveguide grating (AWG) at the same time, crosstalk of the same wavelength can severely degrade the quality of the signals. An x N two-stage AWG-based switch architecture was proposed in an earlier study to tackle the crosstalk problem, where each scheduling decision is represented by a permutation, and a permutation is called k-legal if signals with the same wavelength appear at most k times. The study showed that each permutation can be decomposed into two 4-legal permutations. In addition, for an N x N switch with 4 <= N <= 12, it showed that each permutation can be decomposed into two-legal permutations, pi 1 and pi(2), where pi(1) is chosen from a small set Pi(1). But no specific method was proposed to show how to generate Pi(1). In this paper, we further reduce the crosstalk of the 2-stage AWG-based switch for every odd N by lowering the number of signals using the same wavelength in each decomposed permutation. We derive various sufficient conditions for permutations to be decomposed into two permutations, tau(1) and tau(2), where tau(1) is 1-legal and tau(2) is 2-legal. In addition, to decompose each permutation into permutations tau(1) and tau(2), with the proviso that tau(1) and tau(2) exist, we propose an algorithm to generate a more compact set than Pi(1), even less than half, choose tau(1).