This paper proposes a novel all-optical fiber mode converter for mode conversion from LP01 to LP02 and vice versa. The mode converter is formed by connecting a single-mode fiber, a taper-core multi-mode fiber, and a few-mode fiber (FMF) together. The taper fiber core is designed to convert LP01 mode to LP02. The few-mode fiber is used to cut off the modes that are higher than LP02. It is shown that the proposed mode converter provides 20 dB extinction ratio and low insertion loss in a very broad optical bandwidth of 200 nm, from 1465 to 1665 nm. By further optimizing the FMF radius only for each of the O-, E-, S-, C-, U-, etc. bands, the mode converter can apply to any band of interest with higher performance. It is found that the performance of the mode converter has a large tolerance to structural parameters. The mode converter has the same performance in reciprocal operation, i.e., LP02 to LP01.