This letter proposes a novel direction-of-arrival (DoA) estimation approach which uses a lens-loaded, coded aperture antenna connected to a detector diode for millimeter-wave (mmWave) frequencies. The presented approach exploits three distinct, but closely related properties of mmWave communication front-end hardware: First, the quasi-random radiation modes created by a lens-loaded cavity antenna; Second, the frequency-diverse impulse response of over-sized metallic cavities; Third, the sensitivity of detector diodes at mmWave frequencies. Critically, we use a combination of the synthesis approach and experimental results to demonstrate that DoA estimation is possible with less complex hardware as compared to conventional methods, where an array of mmWave antennas is typically employed for DoA estimation. Bandwidth-to-accuracy trade-off is thoroughly studied, which is an important commercial matrix. It is demonstrated that the lens-loaded cavity antenna with a detector diode circuit can be an apt choice for DoA estimation. Finally, it is shown that the proposed approach provides fast and accurate DoA estimation, thus inherently suitable for DoA estimation in future mmWave systems.