TE _n mode (whose electric field is zero in the normal direction of the boundaries between layers) and TM _n mode (whose magnetic field is zero in the normal direction of the boundaries between layers) are defined. Then conditions under which pure TE _n (TM _n ) modes may exist in multilayer waveguide structure with discontinuities are presented. E (H) step wavguides, ridged wavguides, microstrip lines and fin lines all satisfy the conditions, and hold for TE _n (TM _n ) mode. The conventional conclusion that ridged waveguides with inhomogeneous dielectric-slab loading, microstrip lines and fin lines only hold for hybrid modes is revised. Compared with hybrid modes, the number of unknown variations and matching equations is reduced by half for pure TE _n (TM _n ) modes, and the computation cost is decreased dramatically.