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.
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