The theory of the formation of an image by a plane band grating used in the soft X-ray region

This paper describes an investigation of the reflection of light from a plane grating illuminated by a cylindrical wave diverging from a Huyghens line source. It is shown that for pencils sufficiently narrow the isophasics (properly parabolas) become circles with a virtual focus as centre. The position of this virtual focus is found. A formula is given for the intensity of the light along the reflected isophasic. It has a Fresnel integral as a factor. It is shown that if the dimensions of the grating are chosen so that the modulus of the Fresnel integral has its maximum value, the intensity along the isophasic varies after the fashion of an Airy image and the maximum possible intensity occurs on the axis of the reflected pencil - in other words, the reflected light is automatically focused. The critical (optimum) length of the grating for automatic focusing is determined by the condition that the quadratic term in the expansion for the optical path in powers of distance measured along the grating face from its centre must be three-eighths of a wave-length. Formulae for the dispersion and resolution of an optimum grating are given and the resolution turns out to be exactly the same as for an ordinary grating, namely the total number of lines on the grating multiplied by the spectral order. Some numerical examples are given. Third-order effects have been considered and it is shown that in the conditions contemplated in the use of these optimum gratings, the third-order term affects the length of the optical path by something like one 650th part of a wave-length, and is consequently negligible.