Free-space optical interconnections are important in both massive digital optical computing and communication systems. The architectural features of three interconnection networks are analyzed and compared, and the optical butterfly interconnection is shown to have many advantages over other interconnections in implementing various basic logic functions such as addition, subtraction, multiplication, and fast Fourier transforms (FFTs). Starting with conventional Karnaugh maps and Boolean algebra, the characteristics of full addition and full subtraction are analyzed and compared. An n-bit parallel calculator that can implement both ripple carry full additions and ripple borrow full subtractions using multilayer butterfly interconnection networks is designed. Then the schematic and architecture of the full adder/subtractor, interconnection networks, and the patterns of key devices such as masks to implement AND and OR operations in this calculation are described in detail. The correct simulation results of several groups of multibit digits are provided. Finally, the development of the interconnections in implementing logic operations is discussed.