The accelerating bucket of a laser-plasma accelerator (a cavity of electron density maintained by the laser pulse radiation pressure) evolves slowly, in lock-step with the optical driver, and readily traps background electrons. The trapping process can thus be controlled by purely optical means. Sharp gradients in the nonlinear refractive index produce a large frequency red-shift, localized at the leading edge of the pulse. Negative group velocity dispersion associated with the plasma response compresses the laser pulse into a relativistic optical shock (ROS), slowing the pulse (and the bucket), reducing the electron dephasing length, and limiting energy gain. Even more importantly, the ponderomotive force of the ROS causes the bucket to constantly expand, trapping copious unwanted electrons, polluting the electron spectrum with a high-charge, low-energy tail. We show that using a drive pulse with a bandwidth close to a one-half of the carrier wavelength provides effective dispersion compensation. The negatively chirped, ultrahigh bandwidth (up to 400 nm) drive pulse: prevents ROS formation through dephasing; extends the dephasing length; and almost completely suppresses continuous injection. High quality, GeV-scale electron beams can be thus produced with sub-100 TW lasers (rather than PW-class) in mm-scale (rather than cm-scale), high-density plasmas. ROS formation can be further delayed by using a plasma channel to suppress diffraction of the pulse leading edge, minimizing longitudinal variations in the pulse. At the same time, the combination of a bubble (a selfconsistently maintained, “soft” hollow channel) and a preformed wide channel forces transverse flapping of the laser pulse tail, causing oscillations of the bubble size. The resulting periodic injection produces a polychromatic beam that consists of a number of background-free quasi- onoenergetic components. The number of these components, their charge, energy, and energy difference can be controlled by changing the channel radius and acceleration distance, whereas negative chirp of the driver suppresses the background and boosts their energy. Such clean polychromatic beams can be an asset for tunable X-ray table-top sources.