An examination of an iterative method for the solution of the phase problem in optics and electron optics: II. Sources of error

Abstract

In the evaluation of an iterative scheme that determines the amplitude-phase distribution of an image from two image intensity distributions recorded at two different lens defocus values, the effects of various sources of error on the solution for the phase distribution are examined, namely, a background signal superimposed on the image intensity, a mismatching of the two images and an error in determining the defocus difference between the two images. In electron optics, a background intensity, arising from (say) inelastic electron scattering, corresponding to 10% of the maximum image intensity has a significant effect on the calculated phase distribution. In high-resolution electron microscopy, achieving a potential image resolution of 01-03 nm, a mismatching of the two images by 01 nm is acceptable, corresponding to a misalignment of the two electron micrographs by 50 μm at an electron-optical magnification of 500 000; and a defocus error of 10 nm, in a total defocus difference of 100 nm between the two images, does not severely distort the solution for the phase distribution. The combined effect of photographic noise (10-20% of the maximum intensity), mismatching (01 nm), defocus error (10 nm) and a background (6% of the maximum intensity) on the calculated phase distribution gives an indication of the maximum error that can be tolerated in an experimental test of the method. The magnitude of the maximum error that can be tolerated may be scaled to the resolution required in the phase solution and the wavelength of the radiation used; in optics with a potential image resolution of 10 μm, a mismatching of images by 5-10 μm (at unit magnification or 05-10 mm at an optical magnification of 100) is acceptable and a corresponding defocus error of 100 μm for a defocus difference of 1 mm (depending on the numerical aperture of the objective lens).