Analytic properties of the contrast transfer function in high-resolution electron microscopy

To calculate high-resolution images it is necessary to convolute the wavefunction generated by scattering from the specimen with the microscope objective-lens wavefront aberration function. This is usually done by a multiplication of the transfer function and the specimen exit-surface wavefunction in reciprocal space followed by a numerical integration over all scattering wave vectors. Examination of the analytic behaviour of the wave-front aberration function in the complex plane shows that, for simple scattering functions, it is possible to perform the integral analytically using the method of stationary phase. Analytic results for the imaging of disordered planes of atoms are compared with fast Fourier transform calculations as a function of defocus. The limitations of stationary-phase integration are also discussed.