  We report the results of a recent search for the lowest value of thermal
noise that can be achieved in LIGO by changing the shape of mirrors, while
fixing the mirror radius and maintaining a low diffractional loss. The result
of this minimization is a beam with thermal noise a factor of 2.32 (in power)
lower than previously considered Mesa Beams and a factor of 5.45 (in power)
lower than the Gaussian beams employed in the current baseline design. Mirrors
that confine these beams have been found to be roughly conical in shape, with
an average slope approximately equal to the mirror radius divided by arm
length, and with mild corrections varying at the Fresnel scale. Such a mirror
system, if built, would impact the sensitivity of LIGO, increasing the event
rate of observing gravitational waves in the frequency range of maximum
sensitivity roughly by a factor of three compared to an Advanced LIGO using
Mesa beams (assuming all other noises remain unchanged). We discuss the
resulting beam and mirror properties and study requirements on mirror tilt,
displacement and figure error, in order for this beam to be used in LIGO
detectors.
