Reflector backed submillimeter-wave slot antennas on substrate lenses are promising candidates for novel astronomical applications in integrated receivers for the far infrared region. Using an adjustable reflector, considerable resonance frequency tuning can be achieved, but with the risk of exciting parasitic parallel plate wave modes. Employing a surface/volume integral equation method and advanced spectral domain integration techniques, a complete radiation efficiency analysis of such kind of receivers is performed. Convenient integration path deformations in complex wavenumber planes allow a very fast evaluation of the spectral domain integrals in combination with efficient database and asymptotic extraction techniques. Saddle point and residue calculation techniques are employed to accurately determine the power of each parallel plate mode and the space wave contributions. The simulations reveal an appropriate set of parameters reducing the radiation loss to less than 10 percent for doubleslot and quadratic slot geometries preserving high tuning capability and optimized radiation patterns as well.