  The properties of a dual-frequency vertical external cavity surface emitting
laser (VECSEL), in which two linear orthogonal polarization modes are
oscillating simultaneously, are theoretically investigated. We derive a model
based on the ideas introduced by San Miguel et al. [San Miguel, Feng, and
Moloney, Phys. Rev. A \textbf{52}, 1728 (1995)], taking into account the spin
dynamics of the carriers inside the quantum well (QW) based gain medium of the
dual-frequency VECSEL. This model is shown to succeed in describing quite a few
properties of the dual-frequency VECSEL, such as the behavior of nonlinear
coupling strength between the modes, the spectral behavior of intensity noises
of the modes, and also the correlation between these intensity noises. A good
agreement is found with experimental data. The variables associated with the
spin dependent carrier dynamics can be adiabatically eliminated due to the
class-A dynamical behavior of the considered laser which is based on a cm-long
external cavity. This leads to a simple analytical description of the dynamics
of the dual-frequency VECSEL providing a better understanding of the physics
involved
