  Recently various laser-chirping schemes have been investigated with the goal
of reducing or eliminating ponderomotive line broadening in Compton or Thomson
scattering occurring at high laser intensities. As a next level of detail in
the spectrum calculations, we have calculated the line smoothing and broadening
expected due to incident beam energy spread within a one-dimensional plane wave
model for the incident laser pulse, both for compensated (chirped) and
unchirped cases. The scattered compensated distributions are treatable
analytically within three models for the envelope of the incident laser pulses:
Gaussian, Lorentzian, or hyperbolic secant. We use the new results to
demonstrate that the laser chirping in Compton sources at high laser
intensities: (i) enables the use of higher order harmonics, thereby reducing
the required electron beam energies; and (ii) increases the photon yield in a
small frequency band beyond that possible with the fundamental without
chirping. This combination of chirping and higher harmonics can lead to
substantial savings in the design, construction and operational costs of the
new Compton sources. This is of particular importance to the the widely popular
laser-plasma accelerator based Compton sources, as the improvement in their
beam quality enters the regime where chirping is most effective.
