A design for a one-dimensional photonic crystal (1D PhC) based on thin-film optics is proposed to improve the light extraction efficiency of phosphor-converted flip-chip white-light-emitting diodes (LEDs). The 1D PhC thin film consisting of alternating layers of $\mathrm{Ti}{\mathrm{O}}_2$ and $\mathrm{Si}{\mathrm{O}}_2$ is added between a phosphor layer and sapphire substrate of the LEDs. Monte Carlo simulation method is introduced to analyze performance of the proposed LEDs with a 1D PhC. Results show that the light extraction efficiency is higher than that of conventional LEDs by 16%, and the correlated color temperature becomes lower, below $4000\mathrm{K}$ . These improvements are attributed to the efficient extraction of backscattered light from the phosphor layer by using the 1D PhC thin film.