We investigate the advantages of a device named the HARP/FEL (for Harmonic Amplifier/Free-Electron Laser), which may be described as a two-element, optical klystron FEL with the prebuncher stage oscillating at a frequency different from the output-stage frequency. In analysis based on the single particle treatment of Harvey and Palmer (where the one-dimensional, free-space theory is examined), if the prebuncher-wiggler period ((lambda) w1) differs from the output-coupler-wiggler period ((lambda) w2), then the gain and saturated power of the output coupler are at a strong maximum when ((lambda) w1/(lambda) w2) is an integer. Physically, this synchronism condition arises when the ratio of the bunching wavenumbers is also an integer, a conditions that ensures that both FEL modes are resonant and coherently coupled via the electron-beam bunching. The gain- enhancement mechanism is precipitated by injecting electron bunches into the output coupler with a period that is a subharmonic of the output coupler's ponderomotive potential. If the bunches are sufficiently localized, then each one will be confined to a single potential well and efficient energy coupling occurs between the electrons and the fields. Through integration of the FEL equations of motion, we have analyzed how the HARP's saturated power, saturation length, and susceptibility to e-beam energy spread compare to a free-electron laser and an optical klystron when operated at the same frequency with the same e-beam. Experimental evidence for the HARP mechanism will be published in a separate paper.