Optical pumping denotes the redistribution of population within atomic multiplets by state-selective optical excitation in an electronically excited state and the subsequent spontaneous emission. In most cases the considered multiplets consist of Zeeman substates. This allows to achieve huge nonlinear effects with very modest (down to micro Watts) power levels. The simplest case is the one of a *J*=1/2 to *J’*=1/2-transition with a two-fold degeneracy in both the ground and in the excited state.

Due to the selection rules for angular momenta circularly polarized light will couple only to one of the Zeeman substates of the ground state and thus the population of this level will be reduced. Since spontaneous emission occurs also into the unpumped sublevel, there will be a net accumulation of population in this sublevels. Optical pumping is particularly effective, if the population in the excited state is rapidly equalized between the Zeeman sublevels due to collisions with a buffer gas because then the spontaneous emission is isotropic. Note that the direction of the pumping will depend on the sign of the helicity of the pumping light. Linearly polarized light will not induce pumping because it contains sigma_{+} and sigma_ light of equal strength.

The induced population difference between the two Zeeman substates is called orientation and often denoted by *w* (normalized to [-1,1]). If the population of the excited state can be neglected, it obeys the following equation of motion:

The last term is the source term for the optical pumping. *P _{+/}*_ denotes a pump rate which is proportional to the intensity of the sigma

_{+}, respectively sigma_ component. As mentioned above, the two components pump in different directions. Gamma denotes relaxation due to collisions and is very small (of the order of s

^{-1}). The damping term proportional to the pump rate represents saturation. The diffusion term models the thermal motion of the sodium atoms in the buffer gas atmosphere.

The optical properties of the medium now depend on the orientation and – via *w*=*w*(*P*) – on the intensity of the light field. If the linear absorption coefficient is alpha_{0} and the linear refractive index 1+*n*_{0}, the nonlinear absorption coefficient is

and the nonlinear refractive index is

For circularly polarized light the vapor is bleached by the pump beam. For very high intensity, one sublevel will be completely empty, the orientation reaches one. Then the absorption drops to zero and the refractive index is one. The beam will essentially propagate as in vacuum. Note that an increase in optical density for the sigma_{+} component means a decrease for the sigma_ one and vice versa.

It turns out that the simple model of a homogeneously broadened *J*=1/2 to *J’ *=1/2-transition is not only of academic interest but a very appropriate description for the sodium D_{1}-line if a buffer gas of sufficient pressure (typically 200 to 300 hPa argon or nitrogen) is introduced so that the homogeneous broadening is larger than the hyperfine splitting and the Doppler broadening. It is simple enough to allow for analytical investigations as well as extensive numeral studies on state of the art workstations. Since these are prerequisites for a thorough understanding of spatially extended nonlinear system the *J*=1/2 model is used in most of our theoretical studies.