Haloscopes are resonant cavities that serve as detectors of dark matter axions when they are immersed in a strong static magnetic field. In order to increase the volume and improve space compatibility with dipole or solenoid magnets for axion searches, various haloscope design techniques for rectangular geometries are discussed in this study. The volume limits of two types of haloscopes are explored: those based on single cavities and those based on multicavities. In both cases, possibilities for increasing the volume of long and/or tall structures are presented. For multicavities, 1D geometries are explored to optimise the space in the magnets. Also, 2D and 3D geometries are introduced as a first step in laying the foundations for the development of these kinds of topologies. The results prove the usefulness of the developed methods, evidencing the ample room for improvement in rectangular haloscope designs nowadays. A factor of three orders of magnitude improvement in volume compared with a single cavity based on the WR-90 standard waveguide is obtained with the design of a long and tall single cavity. Similar procedures have been applied for long and tall multicavities. Experimental measurements are shown for prototypes based on tall multicavities and 2D structures, demonstrating the feasibility of using these types of geometries to increase the volume of real haloscopes.This work was performed within the RADES group. We thank our colleagues for their support. In addition, this work has been funded by the grant PID2019-108122GB-C33, funded by MCIN/AEI/10.13039/501100011033/ and by “ERDF A way of making Europe”. JMGB thanks the grant FPI BES-2017-079787, funded by MCIN/AEI/10.13039/501100011033 and by “ESF Investing in your future”. Also, this project has received partial funding through the European Research Council under grant ERC-2018-StG-802836 (AxScale)
