Fast photodetectors, with high sensitivity and quantum efficiency are highly demanded elements for many applications, such as optical communication, optical interconnects, quantum key distribution, and infrared imaging. Therefore, it is crucial to optimize the quantum efficiency-bandwidth product, while maintaining a low noise. Integration of optical antennas with photodetectors leads to high quantum efficiency due to enhancing light - matter interaction as well as reducing the volume of the detector beyond the diffraction limit. Although, a direct consequence of shrinking the detection size is the noise reduction, the local density of state (LDOS) will be enhanced in the near-field of optical antenna. LDOS enhancement can lead to an increased noise current, by reducing the carrier lifetime due to the enhancement of the radiative recombination rate. This effect could become significant, especially for interband detectors, where the carrier lifetime is quite long compared with intersubband detectors. Here, we simulate and study this effect on the interband and intersubband infrared photodetectros using FDTD simulation.