Multi-beam interference (MBI) represents a method of producing one-, two-, and three-dimensional submicron periodic optical-intensity distributions for applications including micro- and nano-electronics, photonic crystals, metamaterial, biomedical structures, optical trapping, and numerous other subwavelength structures. Accordingly, numerous optical configurations have been developed to implement MBI. However, these configurations typically provide limited ability to condition the key parameters of each interfering beam. Constraints on individual beam amplitudes and polarizations are systematically considered to understand their effects on lithographically useful MBI periodic patterning possibilities. A method for analyzing parametric constraints is presented and used to compare the optimized optical-intensity distributions for representative constrained systems. Case studies are presented for both square and hexagonal-lattices produced via three-beam interference. Results demonstrate that constraints on individual-beam polarizations significantly impact patterning possibilities and must be included in the systematic design of an MBI system.