Optical theorem for light scattering from a linear atomic chain

The radiation pattern of light scattered from a chain of atoms having $J=0$ ground states and $J=1$ excited states is calculated in the weak driving field approximation. It is shown that, in general, for arbitrary orientation of the driving field to the chain axis, it is not possible to treat the atoms in a two-level approximation owing to cooperative effects. In the limit of large detuning, however, all cooperative decay effects can be neglected in calculating the scattered spectrum. Nevertheless, even in this limit, in order to ensure conservation of energy, it is still necessary to include cooperative decay when calculating the energy lost by the driving field. For arbitrary detunings, conservation of energy can be expressed in the form of a generalized optical theorem.

Figure 1

Atom-field geometry. The propagation vector of the linearly polarized incident field lies in the $x\text{−}z$ plane and makes an angle ${\theta }_0$ with the $z$ axis. Two or more atoms are located on the $z$ axis and scattter the incident radiation into a direction $(\theta ,\varphi )$.

Figure 2

Graphs of the dimensionless differential scattered intensity as a function of $\theta$ for ${\theta }_0=0.5, \delta /\gamma =1, k_L{Z}_0=1$, and $\varphi =0$. The solid red curve is our result, the dashed blue curve is the two-level result, and the dotted black curve is (twice) the single-atom result.

Figure 3

Graphs of the dimensionless total scattered intensity as a function of $\delta /\gamma$ for ${\theta }_0=0.5$ and $k_L{Z}_0=1$. The solid red curve is our result and the dashed blue curve is the two-level result.

Figure 4

Dimensionless differential scattered intensity as a function of $\theta$ for $\delta /\gamma =5, \varphi =0, {\theta }_0=0.5$, and 10 atoms placed at random positions on a line having length $L=200/k_L$. The curve corresponds to an average over 400 trials. The dashed black curve is 10 times the single-atom scattered intensity.

Figure 5

Dimensionless total scattered intensity as a function of $\delta /\gamma$ for ${\theta }_0=0.5$ and 10 atoms placed at random positions on a line having length $L=5/k_L$. The curve corresponds to an average over 400 trials.