In the all-optical logical AND gate based on cascaded two-stage semiconductor optical amplifiers (SOAs), the output signal from the first stage has notable impact on the final result. When the bulk material SOA with a slow carrier-recovery process is used in the first-stage conversion, the conversion speed beyond 10 Gibt/s can not be reached perfectly, so the speed of all-optical logical AND operation is limited. The first-stage output signal will be improved when a delayed interferometer (DI) is cascaded after the first-stage SOA, and the second-stage all-optical logical AND gate operation will also benefit from the DI cascade. The function of the DI in the revised scheme is illustrated and simulated. In experiments, inverted-signals of high-speed return-to-zero (RZ) signals and non return-to-zero (NRZ) signals are realized by the modified scheme, so a high-speed all-optical logical AND gate is achieved. All-optical logical AND gate operation at 20 Gbit/s for pseudo-random RZ signals and NRZ signals is demonstrated experimentally, and the results at 40 Gbit/s are also analyzed.