The reflectivity $R (omega)$ of $ab$-oriented Mg$_{1-x}$Al$_x$(B$_{1-y }$C$_y$)$_2$ single crystals has been measured by means of infrared microspectroscopy for $1300<omega<17000$ cm$^{-1}$. An increase with doping of the scattering rates in the $pi$ and $sigma$ bands is observed, being more pronounced in the C doped crystals. The $sigma$-band plasma frequency also changes with doping due to the electron doping, while the $pi$-band one is almost unchanged. Moreover, a $sigmatosigma$ interband excitation, predicted by theory, is observed at $omega_{IB} simeq 0.47$ eV in the undoped sample, and shifts to lower energies with doping. By performing theoretical calculation of the doping dependence $omega_{IB}$, the experimental observations can be explained with the increase with electron doping of the Fermi energy of the holes in the $sigma$-band. On the other hand, the $sigma$ band density of states seems not to change substantially. This points towards a $T_c$ reduction driven mainly by disorder, at least for the doping level studied here. The superconducting state has been also probed by infrared synchrotron radiation for $30<omega<150$ cm$^{-1}$ in one pure and one C-doped sample. In the undoped sample ($T_c$ = 38.5 K) a signature of the $pi$-gap only is observed. At $y$ = 0.08 ($T_c$ = 31.9 K), the presence of the contribution of the $sigma$-gap indicates dirty-limit superconductivity in both bands.
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