We designed a normal-conducting (NC) 1.5 GHz harmonic cavity that can be used in a future synchrotron light source at High Energy Accelerator Research Organization (KEK). By utilizing a higher TM020 mode for beam acceleration, this cavity has lower R/Q and higher unloaded Q as compared to those of conventional NC cavities. These features are advantageous in reducing the fluctuations in harmonic RF voltages when the bunch gaps are introduced. Harmful parasitic modes other than the TM020 mode can be heavily damped using coaxial slots located at the node of the TM020 mode. We made a concerted effort to lower the coupling impedances of parasitic modes while maintaining high Q value of the accelerating mode. We found that maintaining the axial symmetry of the cavity is essential for minimizing the leakage power of the accelerating mode into the coaxial slots. In this article, we present the optimization of the cavity shape, low-power measurement, and the basic design of a high-power model.