Investigation of the bosonic spectral density in highly under-doped YBa$_{2}$Cu$_{3}$O$_{6.35 }$

We studied the doping dependence of the bosonic spectral function in nearly optimally-doped La$_{2-x}$Sr$_{x}$CuO$_{4}$, ortho-II YBa$_{2}$Cu$_{3}$O$_{6.35 }$and highly under-doped YBa$_{2}$Cu$_{3}$O$_{6.35}$ single crystals by optical spectroscopy. With fixed oxygen content, the hole doping of the YBCO system can be fine-tuned by varying the degree of oxygen ordering. After annealing and quenching, we were able to make oxygen less ordered and obtain a highly under-doped YBa$_{2}$Cu$_{3}$O$_{6.35 }$ sample with a very low transition temperature around 18K (about 20{\%} of the optimal T$_{c})$. The a-axis reflectance data of this sample at nine temperatures between 30K and 295K were measured with an infrared spectrometer between 60 and 40 000 cm$^{-1}$ with the aid of three different infrared and optical polarizers. The optical properties of the highly under-doped YBCO sample show dramatic changes compared to the ortho-II YBCO sample. The strong sharp mode in the bosonic spectral function \textit{$\alpha $}$^{2}$\textit{F($\Omega )$ }in the ortho-II YBCO is absent in the highly under-doped sample.