Diffusive radiation generated when a charged particle crosses a disordered stack of plates in the infrared region is considered. The main mechanism causing radiation is multiple scattering of electromagnetic field that is more effective in a medium with near to zero average dielectric permittivity. To obtain such a system we suggest using a stack of plates with a negative dielectric constant that, with a positive vacuum value, makes the average dielectric constant near to zero. Numerical estimates for realistic systems are presented.