In order to know effect of bandwidth on the focused properties and focal shift of Hermite-Gaussian (H-G) beam, propagation formula of H-G beams passing through a dispersion lens system with polychromatic TEM11 mode and TEM22 mode was obtained by using diffraction integral. The focused intensity distribution was studied by numerical calculation. The effect of bandwidth on focal shift of both modes were analyzed. The results show that the focal shifts of H-G beams with TEM11 mode and TEM22 mode increase with the increase of the bandwidth and depend on the relative bandwidth. Focal shift of TEM22 mode is greater than that of TEM11 mode when relative bandwidth is smaller than 0.25 whereas the latter is somewhat greater than the former when relative bandwidth exceeds 0.25. The principle maximum intensity and the secondary maximum intensity of TEM22 mode on the axis compete each other with the vary of bandwidth and then the axial primary maximum intensity transits from one place to another. The results are helpful for further application of the polychromatic H-G beams.