Effective dispersion in an inhomogeneously broadened single mode laser

Recently, we have been investigating the development of a superluminal ring laser, where finely tuned anomalous dispersion leads to an enhancement in the sensitivity of the laser frequency to a change in the cavity length, by as much as six orders of magnitude, for applications such as hypersensitive rotation sensing and accelerometry, as well as for gravitational wave detection. For such a laser, as well as other lasers that are used for precision metrology, the effective dispersion – manifested in the manner in which the lasing frequency varies as a function of a change in the cavity length due to the index induced by the medium under saturated gain corresponding to steady-state lasing – is of utmost significance. In determining the effective dispersion, the role of inhomogeneous broadening (IB) must be taken into consideration carefully. In this work, we consider an inhomogeneously broadened gain medium in a single mode optical cavity, and study the effective dispersion experienced by the lasing field. It is well known that the steady state index for such a laser cannot be expressed analytically. Previous studies have employed approximate models to interpret the effective dispersion, in two limits: IB is much larger than homogeneous broadening (HB), and IB is insignificant compared to HB. Here, we use an iterative but quickly converging numerical code to determine the exact behavior of the effective dispersion under all conditions, and show that the results agree with the expected behavior in these two limits. This technique paves the way for taking into account the effective dispersion in any inhomogeneously broadened laser, including the superluminal laser, in determining accurately its sensitivity to change in cavity length, as well as it quantum noise limited linewidth.