The influence of the structure imperfectness of a crystalline undulator on the emission spectrum

We study the influence of an imperfect structure of a crystalline undulator on the spectrum of the undulator radiation. The main attention is paid to the undulators in which the periodic bending in the bulk appears as a result of a regular (periodic) surface deformations. We demonstrate that this method of preparation of a crystalline undulator inevitably leads to a variation of the bending amplitude over the crystal thickness and to the presence of the subharmonics with smaller bending period. Both of these features noticeably influence the monochromatic pattern of the undulator radiation.