# Linewidth Measurement of a Narrow-Linewidth Laser: Principles, Methods, and Systems
## 2 Basic Principles
### Linewidth Calculation Method Based on Frequency Noise  
The above method achieves accurate linewidth characterization through measuring the beat frequency power spectrum of the measured signal, but the lower limit and measurement accuracy of linewidth measurement through methods such as the delay  
Figure 3: Self\(-\)heterodyne spectrum of the Brillouin/erbium fiber laser (BEFL) measured with a 25 km delay fiber and the fitting curves [49]. self-heterodyne methods are limited by the length of the delay fiber and can only achieve a single linewidth expression [61; 62; 63]. Phase and intensity disturbances are introduced during the stimulated radiation process due to spontaneous emission, which cannot be eliminated in the active region, and fluctuations in quantum noise restrict the lower limit of the linewidth. The impact of frequency noise and phase noise on the linetype should be considered in some application scenarios, and the distribution of the linewidth at the complete Fourier frequency should be analyzed. The measurement of laser noise is also aimed at characterizing the laser linewidth, including the Lorentz linewidth and integral linewidth. The two linewidths are affected by white noise in the high-frequency region and \(1/f\) noise in the low-frequency region of the frequency noise power spectrum. The frequency noise spectrum can be obtained by the phase noise power spectrum, and the calculation formula between the two is as follows [64; 65]:  
\[S_{\rm{SV}}(f)\Big{[}Hz^{2}/Hz\Big{]}=2f^{2}L(f) \tag{18}\]  
where \(S_{\rm{SV}}(f)\) is the power spectral density of the frequency noise and \(L(f)\) is the power spectral density of the phase noise [66]. A partially intuitive power spectral density function curve is obtained to characterize the phase noise characteristics of the laser in detail by performing corresponding operations on the output light field of the laser [67; 68]. The phase noise is mainly related to the following parameters: the complex amplitude \(E(t)\), the power spectral density function \(S(f)\) of \(E(t)\), the phase difference \(\Delta\varphi_{\tau}(t)\), the FM noise spectrum \(S_{\rm{F}}(f)\), and the phase difference variance \(\Delta_{\varphi}(\tau)^{2}\). Their interrelationships are shown in Figure 4, and the derived formulas are as follows:  
\[\Delta\varphi_{\tau}(t)=\varphi(t)-\varphi(t-\tau) \tag{19}\]  
\[S_{\Delta\varphi_{\tau}}(f)=4\bigg{[}\frac{\sin(\pi t\tau)}{f}\bigg{]}^{2}S_{ \rm{F}}(f)E(t) \tag{20}\]  
\[\delta(\tau)^{2}=\Big{\langle}\Delta\varphi_{\tau}(t)^{2}\Big{\rangle} \tag{21}\]  
\[L(f)=\Big{\langle}\left|\Im[E(t)]\right|^{2}\Big{\rangle} \tag{22}\]  
\[\delta(\tau)^{2}=4\int_{0}^{\infty}\bigg{[}\frac{\sin(\pi t\tau)}{f}\bigg{]}^{ 2}S_{\rm{F}}(f)df \tag{23}\]  
\[L(f)=\Im\Bigg{\{}exp\bigg{[}-\frac{\delta(\tau)^{2}}{2}\bigg{]}\Bigg{\}} \tag{24}\]  
where \(\Im[\bullet]\) is the Fourier transform and \(<\bullet>\) is the statistical mean. The frequency linewidth of the laser can be directly determined by the frequency noise and phase noise. The power spectral density of the frequency noise can be integrated using the \(\beta\) isolation line algorithm to evaluate the laser frequency linewidth [69]. The laser linetype can be determined by the Fourier transform of the autocorrelation function of the light field according to the Wiener-Khinchin theorem, which are the Lorentz and Gaussian  
Figure 4: The relationship between commonly used representations of phase noise. linetypes. The boundary point between these two types in \(S_{\Delta v}(f)\) is determined by the \(\beta\) isolation line:  
\[S_{Sv}(f)=8 ln(2)f/\left(\pi^{2}\right) \tag{25}\]  
In the area where \(S_{\Delta v}(f)\) is higher than the \(\beta\) isolation line, the laser power spectrum is Gaussian shaped, and \(S_{\Delta v}(f)\) directly determines the frequency linewidth.