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A novel Michelson interferometric hydrogen sulphide sensor coated with copper/reduced graphene oxide (Cu/rGO) composite membrane is proposed and fabricated. A section of endlessly photonic crystal fibre (EPCF) was sandwiched in two single-mode fibres (SMFs). One SMF was spliced and tapered with EPCF; the other SMF was connected with the Faraday rotator mirror to construct the Michelson structure. The cladding of the EPCF was coated by the Cu/rGO-sensing membrane, which was prepared by the dip-coating method. The obtained Cu/rGO-sensing film has a length of 25.0 mm. The fabricated sensing membrane is characterised by the scanning electron microscopy, X-ray photoelectron spectroscopy (XPS), Raman spectroscopy, and so on. Experimental results demonstrated that the Cu/rGO-sensing film has a 24.56-nm thickness with a compact and uniform appearance. The XPS and Raman spectra indicate that there are three elements (C, O, and Cu), which are consistent with the expected compositions of the Cu/rGO membrane. With the increase of concentration of hydrogen sulphide, the interference spectra appear red-shifted. The linearity of 0.97662 and the sensitivity of 13.23 pm/ppm are achieved. In addition, the dynamic response and recovery time of the sensor are approximately 70 and 88 s, respectively. The surface adsorption energies of the film are calculated by the density functional theory. The theoretical results are in good agreement with the experimental findings. This sensor has some key advantages of small size, simple structure, easy fabrication, and great applicability for detecting the trace hydrogen sulphide. [ABSTRACT FROM AUTHOR]