A simple quantitative analysis of phase contrast microscopy, not restricted to objects of very low retardation

A simple vector analysis of the diffraction of light by an ideal, extended phase grating suggests that irrespective of the amount of retardation imposed on the light passing through a transparent microscopic object the diffracted light leaving the object plane is precisely 90 degrees out of phase with the resultant direct (zero-order diffracted) light. In positive phase contrast microscopy, with a 90 degrees phase plate of transmittance T for the direct light, the image and halo respectively have intensities relative to the original illumination of (square root T.cos 1/2 phi - sin 1/2 phi)2 and (square root T.cos 1/2 phi + sin 1/2 phi)w. The empty background has an apparent intensity of T. Zero intensity of the image is seen if T = tan2(1/2 phi), and reversal of contrast if T less than tan2(1/4 phi). The same equations can be used to predict the intensities of the image and halo in negative phase contrast microscopy, if phi is replaced by (360 degrees - phi). Unlike the 'standard' description of phase contrast microscopy, exemplified by Barer's vector method, the present account is consistent with the conservation of energy and is not restricted to very small object retardations. Barer's method is, however, theoretically valid for objects of any shape, and the two approaches may perhaps be regarded as complementary.