Cryogenic cooling system

A cryogenic cooling method, comprising:

• one or more steps of compression of a cryogenic gas in a compressor and a step of cooling said compressed gas by a radiator after each compression step,
• passing of said gas through a first recuperator exchanger such as to cool the compressed gas to a first temperature,
• passing of the gas downstream of the first recuperator exchanger through an intermediate thermal exchanger, cooled by a cooling device, such as to cool the gas coming from the first recuperator exchanger to an intermediate temperature less than the first temperature,
• passing of said gas downstream of the intermediate exchanger through a second recuperator exchanger, such as to cool the gas coming from the intermediate thermal exchanger to a second temperature less than the intermediate temperature,
• expansion of said gas in an expansion member such as to cool said gas to a cryogenic temperature less than the second temperature,
• use of said gas for cooling an apparatus to be cooled,
• passing of said gas downstream of the apparatus to be cooled in the second recuperator exchanger and the first recuperator exchanger by recovering a thermal energy of the gas compressed in the circuit.