Dual-frequency solid-state lasers are attractive for a large number of applications such as microwave photonics, spectroscopy, and metrology. As far as dual-frequency lasers are considered, the intensity noise spectrum of each eigenmode exhibits two peaks lying at the well-known in-phase and anti-phase eigen-frequencies of two coupled oscillators. The in-phase noise, which corresponds to the standard relaxation oscillations of the laser, can be reduced either electronically or optically using feedback loops. However, the anti-phase noise, which is related to a resonant exchange of energy between the two laser modes, is very difficult to circumvent because the reduction of this noise would require additional servo-loop acting on the difference of the intensities of the two modes or two servo-loops acting independently on the intensity of each mode. An optimal dual-frequency laser in terms of intensity noise and beat frequency stability would be a single axis laser in which the population inversions related to each mode are independent. We experimentally demonstrate how the proper design of a two polarization dual-frequency solid-state laser allows to get rid of the anti-phase noise in the simplest possible architecture and without using any electronic or optical feedback loop. This design is based on an appropriate choice of the active medium cut and orientation in order to assign two almost independent families of active atoms to the two laser modes