Optical ring cavities as tailored four-level systems: An application of the group U(2,2)

Publisher Website
Google Scholar
Abstract
We report on the experimental realization and theoretical analysis of four-level systems produced in passive optical ring cavities by introducing various intracavity elements. Birefringent elements couple waves with different polarizations and thus lift the polarization degeneracy of the cavity modes. Reflecting elements couple waves propagating in different directions and lift the propagation degeneracy. A combination of these elements leads, in general, to a splitting of every longitudinal cavity mode into a four-mode system. Under the restriction that the optical components are loss-free, the description of these ring cavities in a transmission-matrix formalism leads in a natural way to the study of the Lie group U(2,2) and its Lie algebra u(2,2). We associate each of the 16 generators of this algebra with a specific type of optical element, some of which are standard components, others not. From the commutation relations of the generators, we derive a recipe for the construction of the nonstandard components as a sequence of standard ones. It follows that the entire U(2,2) group can actually be realized. The number of independent parameters—16 for a general U(2,2) element—is shown to be reduced substantially if the optical components are selected out of a restricted number of types, provided that the corresponding generators define a subalgebra of u(2,2). In such cases, a subgroup of U(2,2) is realized and the number of independent parameters of the optical system is given by the number of generators of the subalgebra. A connection is established between the subalgebras and symmetry properties of the optical components in the cavity. We consider the influence of symmetries on the mode structure and consider our experimental results from this viewpoint. In particular, we discuss time-reversal invariance, leading to the symplectic group USp(2,2), and we identify antiunitary symmetries leading to Kramers’s degeneracy in the mode spectrum. We propose the application of the group U(2,2) in optics as a tool yielding direct practical results.