A method is described to generate the exact twohyphen;dimensional quadrupole field by closed boundaries with continuously variable potential, especially with linearly variable potential, in a ideal case by simple square or rectangle boundaries with linearly variable potential. For the latter square or rectangle structure, the technique to realize linearly variable potential on the boundaries is described. Within the same square or rectangle boundaries, an exact twohyphen;dimensional homogeneous electric field can be superimposed without interaction with the quadrupole field. The electrode structure can be employed in quadrupole mass spectrometers and in electron optical systems to generate focusing and/or deflection within the same interior regions. Experiments with electron beam tubes show good agreement with the theory and the practicability of the proposed structures.
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