  Together with one of its variants, the recently proposed phase-merging
enhanced harmonic generation (PEHG) free-electron lasers (FELs) have been
systematically studied in this paper. Different form the standard-HGHG, a
transverse gradient undulator is employed for introducing a phase-merging
effect into the transversely dispersed electron beam in PEHG. The analytical
theory of the phase-merging effect and the physical mechanism behind the
phenomenon were presented. Using a representative and realistic set of beam
parameters, intensive start-to-end simulations for soft x-ray FEL generation
were given to illustrate the performance of PEHG. Moreover, some practical
issues that may affect the performance of PEHG were also discussed.
