A modified method for determining the focal ratio degradation and length properties of optical fibres in astronomy

Focal ratio degradation (FRD) is a major contributor to throughput and light loss in a fibre spectroscopic telescope system. We combine the guided mode theory in geometric optics and a well-known model, power distribution model (PDM), to predict and explain the FRD dependence properties. We present a robust method by modifying the energy distribution method (EDM) with \emph{f-intercept} to control the input condition. This method provides a way to determine the proper position of the fibre end on the focal plane to improve energy utilization and FRD performance, which lifts the relative throughput up to 95\% with variation of output focal ratio less than 2\%. And this method can also help to optimize the arrangement of the position of focal-plane plate to enhance the coupling efficiency in a telescope. To investigate length properties, we modified PDM by introducing a new parameter, focal distance \emph{f}, into the original model to make it available for multi-position measurement system. The results show that the modified model is robust and feasible for measuring the key parameter \emph{d}$_0$ to simulate the transmission characteristics. The output focal ratio in the experiment does not follow the prediction trend but shows an interesting phenomenon that the output focal ratio increases at first to the peak, then decreases and remains stable finally with increasing fibre length longer than 15m, which provides a reference for choosing appropriate length of fibre to improve the FRD performance for the design of the fibre system in a telescope.