The possibility of utilizing plasma undulators and plasma accelerators to produce compact ultraviolet and X-ray sources, has attracted considerable interest for a few decades. This interest has been driven by the great potential to decrease the threshold for accessing such sources, which are mainly provided by a few dedicated large-scale synchrotron or free-electron laser (FEL) facilities. However, the broad radiation bandwidth of such plasma devices limits the source brightness and makes it difficult for the FEL instability to develop. Here, using multi-dimensional particle-in-cell (PIC) simulations, we demonstrate that a plasma undulator generated by the beating of a mixture of high-order laser modes propagating inside a plasma channel, leads to a few percent radiation bandwidth. The strength of the undulator can reach unity, the period can be less than a millimeter, and the number of undulator periods can be significantly increased by a phase locking technique based on the longitudinal tapering. Polarization control of such an undulator can be achieved by appropriately choosing the phase of the modes. According to our results, in the fully beam loaded regime, the electron current in the plasma undulator can reach 0.3 kA level, making such an undulator a potential candidate towards a table-top FEL.