Logic circuits based on a photonic integrated circuit (PIC) have attracted significant interest due to their ultra-high-speed operation. However, they have a fundamental disadvantage that a large amount of the optical signal power is discarded in the path from the optical source to the optical output, which results in significant power consumption. This optical signal power loss is called a garbage output. To address this issue, this paper considers a circuit design without garbage outputs. Although a method for synthesizing an optical logic circuit without garbage outputs is proposed, this synthesis method can not obtain the optimal solution, such as a circuit with the minimum number of gates. This paper proposes a cross-bar gate logic (CBGL) as a new logic structure for optical logic circuits without garbage outputs, moreover enumerates the CBGLs with the minimum number of gates for all three input logic functions by an exhaustive search. Since the search space is vast, our enumeration algorithm incorporates a technique to prune it efficiently. Experimental results for all three-input logic functions demonstrate that the maximum number of gates required to implement the target function is five. In the best case, the number of gates in enumerated CBGLs is one-half compared to the existing method for optical logic circuits without garbage outputs.