  A novel diagnostic of cosmic-ray modified shocks by polarimetry of H $\alpha$
emissions is suggested. In a cosmic-ray modified shock, the pressure of cosmic
rays is sufficiently high compared to the upstream ram pressure to force the
background plasma to decelerate (measured in the shock rest frame).
Simultaneously, a fraction of the hydrogen atoms co-existing in the upstream
plasma collide with the decelerated protons and undergo charge-exchange
reactions. As a result, hydrogen atoms with the same bulk velocity of the
decelerated protons are generated. We show that when the shock is observed from
edge-on, the H $\alpha$ radiated by these upstream hydrogen atoms is linearly
polarized with a sizable degree of a few per cent as a result of resonant
scattering of Ly $\beta$. The polarization direction depends strongly on the
velocity modification; the direction is parallel to the shock surface for the
case of no modification, while the direction is parallel to the shock velocity
for the case of a modified shock.
