  The free streaming length of dark matter particles determines the abundance
of structure on sub-galactic scales. We present a statistical technique,
amendable to any parameterization of subhalo density profile and mass function,
to probe dark matter on these scales with quadrupole image lenses. We consider
a warm dark matter particle with a mass function characterized by a
normalization and free streaming scale $m_{\rm{hm}}$. We forecast bounds on
dark matter warmth for 120-180 lenses, attainable with future surveys, at
typical lens (source) redshifts of 0.5 (1.5) in early-type galaxies with
velocity dispersions of 220-270 km/sec. We demonstrate that limits on
$m_{\rm{hm}}$ deteriorate rapidly with increasing uncertainty in image fluxes,
underscoring the importance of precise measurements and accurate lens models.
For our forecasts, we assume the deflectors in the lens sample do not exhibit
complex morphologies, so we neglect systematic errors in their modeling.
Omitting the additional signal from line of sight halos, our constraints
underestimate the true power of the method. Assuming cold dark matter, for a
low normalization, corresponding the destruction of all subhalos within the
host scale radius, we forecast $2\sigma$ bounds on $m_{\rm{hm}}$ (thermal relic
mass) of $10^{7.5} \ (5.0)$, $10^{8} \ (3.6)$, and $10^{8.5} \ (2.7) \
M_{\odot} \left(\rm{keV}\right)$ for flux errors of $2\%$, $4\%$, and $8\%$.
With a higher normalization, these constraints improve to $10^{7.2} \ (6.6)$,
$10^{7.5} \ (5.3) $, and $10^{7.8} \ (4.3) \ M_{\odot} \left(\rm{keV}\right)$
with 120 systems. We are also able to measure the normalization of the mass
function, which has implications for baryonic feedback models and tidal
stripping.
