  We report on a survey for H$\alpha$ bow shock emission around nearby
$\gamma$-detected energetic pulsars. This survey adds three Balmer-dominated
neutron star bow shocks to the six previously confirmed examples. In addition
to the shock around {\it Fermi} pulsar PSR J1741$-$2054, we now report
H$\alpha$ structures around two additional $\gamma$-ray pulsars, PSR J2030+4415
and PSR J1509$-$5850. These are the first known examples of H$\alpha$ nebulae
with pre-ionization halos. With new measurements, we show that a simple
analytic model can account for the angular size and flux of the bow shocks'
apices. The latter, in particular, provides a new pulsar probe and indicates
large moments of inertia and smaller distances than previously assumed, in
several cases. In particular we show that the re-measured PSR J0437$-$4715
shock flux implies $I = (1.7\pm 0.2) \times 10^{45}/(f_{HI} {\rm sin}i) {\rm
g\,cm^2}$. We also derive a distance $d\approx 0.72$kpc for the $\gamma$-ray
only pulsar PSR J2030+4415 and revised distances for PSRs J1959+2048 (1.4kpc)
and J2555+6535 ($\sim 1$kpc), smaller than the conventional DM-estimated
values. Finally we report upper limits for 94 additional LAT pulsars. An
estimate of the survey sensitivity indicates that for a warm neutral medium
filling factor $\phi_{WNM}\sim 0.3$ there should be a total of $\sim$ nine
H$\alpha$ bow shocks in our LAT-targeted survey; given that seven such objects
are now known, a much larger $\phi_{WNM}$ seems problematic.
