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Consider a second-order nonlinear medium in which perfect phase-matching can be achieved at one particular output wavelength for the process of the terahertz generation from a train of ultrafast laser pulses based on difference-frequency generation. We assume that the coherence lengths for the terahertz generation are sufficiently long within a wide bandwidth around this perfect phase-matching wavelength due to a slight dispersion in the terahertz region (i.e., broadband phase-matching). In this theoretical work, we show that quasi-single-cycle terahertz pulses can be efficiently generated. An efficient conversion for the terahertz generation is made possible not only by utilizing the broadband phase-matching but also by optimizing the pulse width for each peak terahertz frequency. We have investigated the regime of the strong pump depletion and found the limits to the conversion efficiencies. We have ruled out the significant contributions due to the effects of multiphoton absorption, free-carrier absorption, and nonlinear refractive indices.