An X-ray source, with: a- a cylindrical resonant cavity with a length, diameter, and a longitudinal axis extending from a first end of the resonant cavity to an opposite second end of the cylindrical resonant cavity; b- an electron gun located at the first end of the cylindrical resonant cavity; c- a metallic target coupled to the cylindrical resonant cavity, close to the second end of the cylindrical resonant cavity; d- a microwave field energizing system coupled to the cylindrical resonant cavity, the microwave field energizing system comprises two waveguides (2, 3) separated azimuthally by 90°, a coupling waveguide, two ferrite insulators (7, 8) and a microwave source, wherein each waveguide (2, 3) has an end coupled to the cylindrical resonant cavity and the other end coupled through the coupling waveguide via the ferrite insulators to the microwave source, and wherein the waveguides (2, 3) are coupled to the cylindrical resonant cavity in a plane at a distance of a quarter of the length of the cylindrical resonant cavity measured from its first end; e- at least one magnetic field source (13', 13" and 13′′′) that generates a magnetic field that increases along the longitudinal axis of the cylindrical resonant cavity, starting from the first end of the cylindrical resonant cavity to the second end of the cylindrical resonant cavity; and f- a window transparent to X-rays, the window being incorporated to the surface of the cylindrical resonant cavity; wherein the length and diameter of the cylindrical resonant cavity must meet the relationship described by the following expression: d = p 2 f / c 2 − 1.841 / πr 2 − 1 / 2 wherein: d is the length of the cylindrical resonant cavity; p is the subscript of the resonance mode of the cylindrical resonant cavity; f is the frequency of the microwave source; c is the speed of light in vacuum; and r is the diameter of the cylindrical resonant cavity/2; whereby, the cylindrical resonant cavity is configured to resonate in a TE 11p mode due to the waveguides (2, 3) which provide microwave energy in a TE 10 mode from the microwave source, wherein the waveguides (2, 3) and the coupling waveguide are configured to form two paths for microwave injection, the paths having lengths L and L+λ/4 respectively, where λ is the wavelength of the TE 10 mode, which produces a phase shift of π/2 to energize a right hand circular polarized TE 11p mode in the cylindrical resonant cavity. a- a cylindrical resonant cavity with a length, diameter, and a longitudinal axis extending from a first end of the resonant cavity to an opposite second end of the cylindrical resonant cavity; b- an electron gun located at the first end of the cylindrical resonant cavity; c- a metallic target coupled to the cylindrical resonant cavity, close to the second end of the cylindrical resonant cavity; d- a microwave field energizing system coupled to the cylindrical resonant cavity, the microwave field energizing system comprises two waveguides (2, 3) separated azimuthally by 90°, a coupling waveguide, two ferrite insulators (7, 8) and a microwave source, wherein each waveguide (2, 3) has an end coupled to the cylindrical resonant cavity and the other end coupled through the coupling waveguide via the ferrite insulators to the microwave source, and wherein the waveguides (2, 3) are coupled to the cylindrical resonant cavity in a plane at a distance of a quarter of the length of the cylindrical resonant cavity measured from its first end; e- at least one magnetic field source (13', 13" and 13′′′) that generates a magnetic field that increases along the longitudinal axis of the cylindrical resonant cavity, starting from the first end of the cylindrical resonant cavity to the second end of the cylindrical resonant cavity; and f- a window transparent to X-rays, the window being incorporated to the surface of the cylindrical resonant cavity; d is the length of the cylindrical resonant cavity; p is the subscript of the resonance mode of the cylindrical resonant cavity; f is the frequency of the microwave source; c is the speed of light in vacuum; and r is the diameter of the cylindrical resonant cavity/2; whereby, the cylindrical resonant cavity is configured to resonate in a TE 11p mode due to the waveguides (2, 3) which provide microwave energy in a TE 10 mode from the microwave source, wherein the waveguides (2, 3) and the coupling waveguide are configured to form two paths for microwave injection, the paths having lengths L and L+λ/4 respectively, where λ is the wavelength of the TE 10 mode, which produces a phase shift of π/2 to energize a right hand circular polarized TE 11p mode in the cylindrical resonant cavity.