A Plus or Minus Sign in the Fourier Transform?

Abstract

The sign of the exponential in the Fourier transform is something that we have been concerned with for many years. Of course, there are two conventions that have been used with almost equal frequency but we have been trying to get people to stick to one of them to avoid confusion. In our books and papers (Refs. 65,207, etc.) we have used the convention of the positive sign in the exponential for the forward transform which represents the Fraunhofer diffraction pattern for a real-space object. This is consistent with assuming that a plane wave, going in positive direction in real space is written exp $[i\,(\omega t - \overrightarrow k \overrightarrow r )]$ rather than a minus sign before the i, so that the phase advances with time. This is also consistent with making the Scherzer defocus negative, and also implies that the phase shift in a solid with a positive potential (accelerating the electrons) is negative so that the transmission function is exp [−iσÆ(x,y)]. If the other convention is to be used for the Fourier transform exponent sign, then all authors should be advised of all these other implications, which are not immediately obvious. Otherwise, we might find ourselves producing a treatment of positron holography! There is a summary of our conventions and definitions in the appendixes of Refs. 65 and 207.