Amplitude- and frequency-modulated pulses or the equivalent amplitude- and phase-modulated pulses based on adiabatic principles achieve their transformations of magnetization vectors over a large range of variation in B1 magnitude. The B1-insensitivity of the pulses depends on the particular modulation functions employed. In this paper, we examine the factors that govern the B1 tolerance of these pulses and illustrate that previously used amplitude/frequency modulation functions sin/cos, sech/tanh, or constant/tan are far less than optimum in achieving maximal B1 insensitivity on resonance. We describe a new method by which optimized modulation functions can be constructed to impart insensitivity to B1 inhomogeneities over a predetermined B1 range. This is accomplished by dictating that the pulses based on the new modulation functions fulfill the adiabatic condition over the predetermined B1 range and for the complete duration of the pulse. The new functions also provide the additional advantage of operating at lower pulse power. Experimental and theoretical data are presented to illustrate the superiority of the new modulation functions.