Broadband mode converters are essential devices for space-division and wavelength-division multiplexing systems. There are great challenges in the generation of higher-order modes above the third order with low loss and high mode purity employing all-fiber devices. In this paper, an all-fiber LP₄₁ mode converter is proposed and fabricated by tapering a nine-core single-mode fiber bundle. Experimental results indicate that this all-fiber LP₄₁ mode converter is low-loss, high-purity, and ultrabroadband. The insertion loss is less than 0.4 dB. The purity of odd LP₄₁ at 1310 nm is 95.09%, and the operating bandwidth exceeds 280 nm.