Single-frequency lasers' linewidth elegantly characterized with Sigmoid functions of observation time

Linewidth is the most important parameter for characterizing the coherence properties of a single-frequency laser, but unfortunately only the natural linewidth representing the contributions of the spontaneous emission or quantum noise can be described with an analytical expression known as the Schawlow-Townes-Henry formula. To the best of authors' knowledge, no analytical expression is formulized after 63 years since laser's invention for characterizing the effective linewidth of a single-frequency laser including the linewidth broadening caused by the flicker noises, which strongly depends on the measurement duration and is much larger than the natural linewidth. By carefully measuring the instantaneous frequency fluctuations of multiple commercial single-frequency lasers using a self-built optical frequency analyzer with ultra-high resolution and speed to obtain their linewidths with our time domain statistical analysis method, we discover and validate that the laser linewidths can be expressed as one or more Sigmoid functions of observation time. Not only the simple Sigmoid linewidth expression provides clear linewidth information of the laser, but also better understanding of the physical origins affecting the laser linewidths, which will benefit a large number of applications ranging from coherent distributed sensing to gravitational wave detection and therefore is worthy to be widely adopted to fully and elegantly characterize the linewidths of single-frequency lasers.