This paper presents a compact fully-differential distributed amplifier using a coupled inductor. Differential distributed amplifiers are widely required in optical communication systems. Most of the distributed amplifiers reported in the past are single-ended or pseudo-differential topologies. In addition, the differential distributed amplifiers require many inductors, which increases the silicon cost. In this study, we use differentially coupled inductors to reduce the chip area to less than half and eliminate the difficulties in layout design. The challenge in using coupled inductors is the capacitive parasitic coupling that degrades the flatness of frequency response. To address this challenge, the odd-mode image parameters of a differential artificial transmission line are derived using a simple loss-less model. Based on the analytical results, we optimize the dimensions of the inductor with the gradient descent algorithm to achieve accurate impedance matching and phase matching. The amplifier was fabricated in 0.18-µm CMOS technology. The core area of the amplifier is 0.27 mm², which is 57% smaller than the previous work. Besides, we demonstrated a small group delay variation of ±2.7 ps thanks to the optimization. the amplifier successfully performed 30-Gbps NRZ and PAM4 transmissions with superior jitter performance. The proposed technique will promote the high-density integration of differential traveling wave devices.