  The ability to enhance resonant signals and eliminate the non-resonant
background is analyzed for Coherent Anti-Stokes Raman Scattering (CARS). The
analysis is done at a specific frequency as well as for broadband excitation
using femtosecond pulse-shaping techniques. An appropriate objective functional
is employed to balance resonant signal enhancement against non-resonant
background suppression. Optimal enhancement of the signal and minimization of
the background can be achieved by shaping the probe pulse alone while keeping
the pump and Stokes pulses in transform-limited-form (TLF). In some cases
analytical forms for the probe pulse can be found, and numerical simulations
are carried out for other circumstances. It is found that a good approximate
solution for the optimal pulse in the two-pulse CARS is a superposition of
linear and arctangent type phases for the pump. The well-known probe delay
method is shown to be a quasi-optimal scheme for background suppression. The
results should provide a basis to improve the performance of CARS spectroscopy
and microscopy.
