Optical waveguides carry bound modes which consist of a core E and H field, winch is oscillatory across the waveguide and evanescent in the waveguide cladding. Both the core and cladding component of each mode has the same frequency and propagation constant. When the frequency of the light earned by the waveguide matches an absorption transition of the material of the cladding, the mode loses optical power as it propagates due to the attenuation of the evanescent cladding portion of the mode. This process is called attenuated total reflection spectroscopy (ATR) or evanescent wave spectrophotometry. As in simple transmission spectrophotometry the absorbance of the mode is related to the interaction length of the waveguide with the absorbing cladding, the concentration of the absorbing species of the cladding and the fraction of the optical power in the evanescent waves of the various modes. This work firstly represents a theoretical analysis of the bound modes that can exist in a step index hollow cylindrical waveguide, their evanescent power fraction and the effective length of such a waveguide when located in an absorbing cladding material. The waveguide is found to have a normalized frequency or effective V number whose magnitude determines the total number of bound modes and influences the mean evanescent power fraction between modes. This effective V number reduces to that of the solid step index fiber waveguide in the limit of a zero radius inner cavity Likewise the expression for the mean evanescent mode power fraction becomes - in the limit of zero inner radius - identical to that of the fiber waveguide. The evanescent absorbance of such a hollow waveguide located in an absorbmg fluid is modeled in terms of the bulk absorption coefficient of the fluid and the waveguide dimensions. In the second part of the thesis a set of experimental absorbance values for ATR spectrophotometry using a hollow silica waveguide probe are reported. Good correspondence is found between the theoretical model and the experimental data.