  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.
