  Recent theoretical and experimental advances have shed light on the existence
of so-called `perfectly transmitting' wavefronts with transmission coefficients
close to 1 in strongly backscattering random media. These perfectly
transmitting eigen-wavefronts can be synthesized by spatial amplitude and phase
modulation. Here, we consider the problem of transmission enhancement using
phase-only modulated wavefronts. We develop physically realizable iterative and
non-iterative algorithms for increasing the transmission through such random
media using backscatter analysis. We theoretically show that, despite the
phase-only modulation constraint, the non-iterative algorithms will achieve at
least about 25$\pi$% or about 78.5% transmission assuming there is at least one
perfectly transmitting eigen-wavefront and that the singular vectors of the
transmission matrix obey a maximum entropy principle so that they are
isotropically random.
  We numerically analyze the limits of phase-only modulated transmission in 2-D
with fully spectrally accurate simulators and provide rigorous numerical
evidence confirming our theoretical prediction in random media with periodic
boundary conditions that is composed of hundreds of thousands of non-absorbing
scatterers. We show via numerical simulations that the iterative algorithms we
have developed converge rapidly, yielding highly transmitting wavefronts using
relatively few measurements of the backscatter field. Specifically, the best
performing iterative algorithm yields approx 70% transmission using just 15-20
measurements in the regime where the non-iterative algorithms yield
approximately 78.5% transmission but require measuring the entire modal
reflection matrix.
