Symbolic substitution is a method for manipulating binary data that depends on both the value of the data and its spatial location to realize logical operations [1,2]. A substitution system requires only a pattern recognizer, a nonlinear device, and a pattern scriber. The operation of both the recognition and scribing subsystems is based on the replication of an input object to produce several output images; the replicated images are then translated and overlayed. For optical implementation of symbolic substitution the recognizer and scriber systems can be constructed using classical and holographic optical elements; single-channel and dual-channel systems have been proposed . In a single-channel system holographic elements are used either to replicate an input object or combine several shifted images. The holgraphic elements in a dual-channel system perform both replication and translation. A zero-based representation of binary-phase gratings developed by Dammann  has been reported as one method for designing beam splitters and combiners [5,6]. Dammann's method assumes a symmetric one-dimensional display of the replicated images and determines the locations of phase changes based on the number of desired replicas. The application of this method to the design of systems for symbolic substitution requires an extension of Dammann's method to asymmetric displays  and an extension to two dimensions. These extensions are discussed herein and results are also presented. A brief review of the requirements for image replication and combination in an optical symbolic substitution system is presented in Section 2. Section 3 discusses Dammann's method and its extensions. Practical considerations with respect to design are discussed in Section 4 and results are presented. Some limitations of the implementation of Dammann's method as well as other concluding remarks are discussed in Section 5.