Quantum dot micropillar laser is an excellent research platform for photonic integration and secure communication, due to its μm-scale mode volume and unique dynamics. In this study, a quantum dot micropillar laser with external cavity optical feedback is used to generate a two-mode chaotic light field. The competition between strong mode and weak mode, time delay signature, and effective bandwidth of the quantum dot micropillar laser with optical feedback are examined under various pump currents and feedback intensities. The results demonstrate that when the mode competition between strong mode and weak mode is high, i.e., the output power bifurcation between strong mode and weak mode is obvious, the output power difference between the two modes decreases at first, then decreases, as the feedback intensity increases. The chaotic light field's effective bandwidth increases at first, then decreases, and increases again as the feedback intensity increases. When the pump current is 13 μA, and the feedback intensities of the strong mode and weak mode are 4 ns^-1 and 3 ns^-1, the effective bandwidth of the chaotic laser of the strong mode and weak mode increases to the first peak values of 8.91 GHz and 8.49 GHz, respectively. The time delay signature of the two-mode chaotic laser first increases, then decreases, and increases again as the feedback intensity increases. When the pump current is 13 μA, the feedback intensities of the strong mode and weak mode are 16 ns^-1 and 14 ns^-1, respectively, the time delay signatures of the coherence-collapse chaotic laser for the strong mode and weak mode are 0.083 and 0.092, and the corresponding effective bandwidths are 11.31 GHz and 11.02 GHz.