Synthesis of Hadamard transformers by use of multimode interference optical waveguides

We propose a synthesis method of optical Hadamard transformer using multimode interference (MMI) couplers. By using the signal transfer matrix of 2 x 2, 4 x 4, and 8 x 8 MMI couplers, we show that sum and difference units of input signals can be synthesized. An interchange unit of two signals can also be synthesized. One synthesis method of Hadamard transformers is a combination of only 2 x 2 units, and the other is a combination of N x N(N > or = 4) units as well as 2 x 2 units. The design examples of operation units are shown, and the size and the output power of Hadamard transformers are estimated.