Finite-aperture effects for Fourier transform systems with convergent illumination. Part I: 2-D system analysis

The Optical Fourier Transform (OFT) is one of the most fundamental operations in analogue Optical Signal Processing (OSP). There are many optical arrangements for implementing the OFT, however one which is particularly popular is the Scaled Optical Fourier Transform (SOFT) because it offers the user the ability to scale the output Fourier distribution. In this paper we study some of the practical limits introduced by using a converging spherical lens of finite aperture to produce the illuminating field in the implementation of the SOFT. By deriving simple rules of thumb, based on examining phase and intensity deviations from the ideal unapertured case, we define an area inside the geometric shadow, which we refer to as a sub-geometric shadow. Inside this sub-geometric shadow we show that the worst-case errors in the resulting SOFT, arising due to diffraction, can be quantified and avoided.