Employing equation (7) to calculate the laser linewidth is based on the fact that the S₁ item of the laser line shape is above the standard Lorentz line shape. However, according to the semiconductor laser theory, the laser line shape is unstable due to the influence of various noises and is not the standard Lorentz line shape. Therefore, adopting equation (7) to calculate the laser linewidth will cause some errors, which also explains the reason why utilizing equation (1) as a fitting function in the experiment cannot completely match the experimental data. According to the line-shape broadening theory of semiconductor lasers, there will be a certain Gaussian line-shape component in the line-shape after the laser broadening, and the laser line-shape can be abstracted into a Voigt function under the joint action of the two. Additionally, this line shape can be employed as S₁ to the fit beat frequency signal spectrum for increasing the accuracy of laser linewidth measurements.

Visible single-frequency lasers have important applications in optical precision measurement and frequency standards. As an important parameter to determine laser coherence, linewidth guarantees the contrast of spatial interference fringes and directly determines the accuracy of the measurement system. In the visible light band, common measurement methods for laser linewidth are employing spectrometers and F-P cavities, but the measurement accuracy of these methods can only reach GHz and MHz levels, which cannot meet the current requirements of kHz or even Hz levels for linewidth measurement accuracy. As a new linewidth measurement method, the short-fiber delay self-heterodyne method can realize kHz linewidth measurement in communication bands, but the applications in visible light bands are rarely studied. Since the short-fiber delay self-heterodyne method can obtain high-precision linewidth without adopting too long optical fiber, it is a potential means to measure the laser linewidth in visible light bands.

We propose a measurement method of visible single-frequency lasers based on the short-fiber self-heterodyne method, which introduces the short-fiber delay self-heterodyne method in the communication bands into the visible light bands and realizes linewidth measurement of high-precision lasers in the visible light bands. The principle of the proposed method is that the laser beams are split by the beam splitter (BS), one path is time-delayed by the delay fiber, and the other path's frequency is shifted by the acousto-optic modulator (AOM). The two laser beams are combined by a beam combiner (BC) to obtain a beat signal. Since the optical path difference introduced by the delay fiber is much smaller than the laser coherence length, an interference envelope will appear around the center frequency in the spectrum of the beat frequency signal, and the sidelobe of the envelope contains the laser linewidth information. We design a short-fiber delay self-heterodyne optical path as shown in Fig. 1 to interpret the sidelobes of the interference envelope and obtain the second peak-valley value ΔS₁₀ of the sidelobes. According to the spectrum expression of the beat frequency signal, we obtain the equation [equation (7)] about the laser linewidth, and the laser linewidth can be obtained by solving this equation. Due to the low signal-to-noise ratio (SNR) of the beat signal, we design a data smoothing method based on wavelet transform and outlier elimination. Meanwhile, we adopt the solution of equation (7) as the initial value, and utilize the nonlinear least squares method to fit the smoothed curve to obtain the accurate linewidth. Additionally, we set up a visible single-frequency laser linewidth test system as shown in Fig. 5, employ different lengths of delay fibers to test the same laser, and compare the test results with the traditional double-beam heterodyne method. Finally, the linewidth of an external cavity semiconductor laser with a center wavelength of 635 nm is measured.

We put forward a linewidth measurement method of visible single-frequency lasers based on the short-fiber delay self-heterodyne method, and build a short-fiber delay self-heterodyne system that can be adopted for the laser linewidth measurement in the visible light bands. An external cavity diode laser with a center wavelength of 635 nm under the 127 m long delay fiber is measured, and the measured beat signal spectrum is shown in Fig. 7, where the blue curve is the original data, the green curve is the smoothed curve, and the red curve is the curve after fitting the smooth data. Several laser measurements show that the average laser linewidth is about 29.42 kHz with a standard deviation of 1.36 kHz. We employ the 500 m system delay fiber and the measured spectrum data are shown in Fig. 8. After measuring the laser linewidth several times by the 500 m fiber, the measured average laser linewidth is about 31.46 kHz, with a standard deviation of 2.24 kHz. Additionally, we leverage a laser of the same type as the laser under test to beat each other. The experimental device and the measured beat signal spectrum are shown in Fig. 6 and Fig. 9 respectively, and the average laser linewidth calculated by multiple measurements is 53.87 kHz, with the standard deviation is 4.51 kHz. Considering that the laser frequency instability will affect the beat frequency signal during the test, the measurement results of this measurement method are close to those of the short-fiber delay self-heterodyne method.

To sum up, we build a set of short-fiber delay self-heterodyne systems that can be adopted for laser linewidth measurement in the visible light bands. The short-delay fiber avoids high loss in the visible light bands and also reduces the low-frequency noise caused by fiber delay. Meanwhile, we design the corresponding smoothing and fitting methods of the beat-frequency signal spectrum to increase the low signal-to-noise ratio of the delay self-heterodyne spectrum in the visible light bands. Finally, the linewidth of a 635 nm single-frequency external cavity semiconductor laser is measured. This scheme has consistency under different lengths of delay fibers and is close to the traditional double-beam heterodyne measurement results. We prove that the short-fiber delay self-heterodyne method in linewidth parameter measurement of narrow linewidth lasers in the visible light bands is feasible.