Abstract: In this paper, an exact single particle theory of free electron laser (FEL) with a circularly polarized wiggler field and an axial guide magnetic field in the tenuous beam limit has been developed. The unperturbed electron orbits, their stable conditions and stability boundary of guide field, and the single pass gain have been derived out. The gain is composed of three dominant terms. The first term, which is the gain of FEL instability, has been found to be in excellent agreement with the exponential growth rate obtained by the previous kinetic theories . And the second and third terms represent the gains of a couple of new instabilities. Without assumption on the axial electron velocity, this theory is valid for conventional FELs and,especially, for small period wiggler FELs in which the electron beam energy is relatively lower.Thus the assumption commonly used in previous single particle theories that the axial electron velocity is approximately equal to the speed of light no longer remains exact. Numerical analyses indicate that the new instabilities will probably affect the FEL operation when the guide field increases to its stability boundary for the highest gain. The results also reveal that it is possible to use relatively weaker wiggler field to generate high gain radiations at high frequency with the help of a relatively stronger guide field. At last, two groups of parameters of such FELs using bifilar helical wigglers are presented.