Theoretical multiplex gain in a UV/VIS Hadamard transform spectrometer utilizing a uniformly imperfect encoding mask

The effect of a uniformly imperfect (non-ideal) encoding mask on the signal-to-noise ratio improvement in a Hadamard transform spectrometer utilizing photon detection is theoretically studied. General equations are developed for the calculation of the multiplex gain (the Fellgett advantage) under conditions of limitation by shot noise and source-fluctuation noise. It is shown that for both cases, the multiplex gain depends on the transmission properties of the encoding mask, the UV/VIS spectrum impinging upon the mask, and the multiplex size, N. It is demonstrated that a uniformly imperfect encoding mask allows sufficient multiplex gain, and that photon-detection in Hadamard transform spectrometry can have advantages in some spectroscopic applications. In addition, comparisons are made between the multiplex gain advantages present in UV/VIS FTS and those present in UV/VIS HTS.