We systematically investigated the effects of InAs coverage variation, two-step annealing and an asymmetric InGaAs quantum well (QW) on the structural and optical characteristics of InAs quantum dots (QDs) by using atomic force microscopy (AFM), transmission electron microscopy (TEM) and photoluminescence (PL) measurement. The transition of size distribution of InAs QDs from bimodal to multi-modal was noticeably observed with increasing InAs coverage. By means of two-step annealing, it is found that significant narrowing of the luminescence linewidth (from 132 to 31 meV) from the InAs QDs occurs together with about 150 meV blueshift, compared to as-grown InAs QDs. Finally, the InAs QDs emitting at longer wavelength of $1.3;{mu}m$ with narrow linewidth were grown by an asymmetric InGaAs QW. The excited-state transition for the InAs QDs with an asymmetric InGaAs QW was not noticeably observed due to the large energy-level spacing between the ground states and the first excited states. The InAs QDs with an asymmetric InGaAs QW will be promising for the device applications such as $1.3;{mu}m$ optical-fiber communication.