# Linewidth Measurement of a Narrow-Linewidth Laser: Principles, Methods, and Systems
## 3 Research Progress
### Delayed Self-Heterodyne Method  
#### 3.3.1 MZI-Type Delayed Self-Heterodyne Method  
In 2013, [PERSON] reported a delayed self-heterodyne method for 100 Hz linewidth measurements [90]. The device is shown in Figure 6a, with a delay fiber length of 10 km to ensure that the delay time is much shorter than the laser coherence time. Narrow-linewidth measurements below 100 Hz can be achieved by fitting the Lorentz curve with bandwidths of \(-20\) dB or \(-40\) dB. In 2015, [PERSON] et al. proposed a method of separating the intrinsic linewidth and \(1/f\) noise of lasers using multiple fiber delayed self-heterodyne methods [91]. The spectral bandwidths corresponding to different intrinsic linewidth and \(1/f\) noise are extracted, and the mean square error (MSE) is calculated for each combination. The best fit of the intrinsic linewidth and \(1/f\) noise can be obtained from the global minimum value of the surface by least squares fitting. The method can eliminate the \(1/f\) noise effect and obtain a much smaller linewidth than traditional measurement methods. In 2023, [PERSON] et al. improved the linewidth measurement system based on the traditional MZI delayed self-heterodyne method [92], as shown in Figure 6b. The half of the delay fiber required for the same optical path is reduced by introducing a ringer and FRM. The polarization states of the two combined beams of light are perpendicular to each other, reducing the influence of thermal and mechanical disturbances on the polarization state in the fiber. The system achieves a linewidth of 2.58 kHz for an 852 nm narrow-linewidth external cavity diode laser (ECDL). The delayed self-heterodyne method can avoid the zero-frequency noise of the self-homodyne method and improve testing stability. However, the narrower the linewidth  
Figure 6: (**a**) Diagram of the self-heterodyne measurement for a narrow-linewidth laser [90]. BS: beam splitter, AOM: acousto-optic modulator, APD: Si avalanche photodiode, FP: Fabry-Perot interferometer. (**b**) Schematic of the laser linewidth measurement [92]. is, the longer the delay required for fiber decoherence. Excessive fiber length will result in additional losses, and \(1/f\) noise will cause spectrum broadening and phase noise jitter, which limits the detection accuracy and lower limit of linewidth characterization. Therefore, researchers have explored the use of short-delay optical fibers to construct a delayed self-heterodyne linewidth measurement. In 2016, [PERSON] et al. proposed a coherent envelope amplitude difference comparison method based on short-delay self-heterodyne interferometry for laser linewidth measurements at 100 Hz-1 kHz [93]. The contrast difference between the second peak and the second valley in the strongly coherent envelope (CDSPST) (\(\Delta\)S) is used to characterize the linewidth, and the system needs to select an appropriate length of optical fiber to maintain \(\Delta\)S between 10 dB and 30 dB. In 2017, the team further systematically studied a method for accurately measuring ultra-narrow-linewidth based on a high-coherence envelope [94]. The laser linewidth was calculated by analyzing the relationship between the intrinsic linewidth of the laser and the second peak/valley difference in the coherent envelope spectrum, as well as the length of the delay fiber. The correspondence between the laser linewidth and specific CDSPST combinations is calculated, the power spectrum is demodulated, the actual CDSPST values are modified, and the laser linewidth is obtained. The measured linewidth is 150 Hz, with a resolution of 10 Hz. In 2022, [PERSON] et al. reported a linewidth measurement scheme based on delayed self-homodyne difference technology [95]. The second peak-valley difference (SPVD) of the power spectrum is used to fit the laser linewidth, avoiding the multiple calculation processes of iterative algorithms. In 2020, [PERSON] et al. reported an ultra-narrow-linewidth measurement scheme based on partially coherent light interference and dual-parameter acquisition [96].