## 3 Experimental Design and Analysis of Results
### Comparison of Line Width and Noise Characteristics of FP-ECDL with Two F-P Etalon Precision Levels
The frequency noise spectrum of a semiconductor laser can be expressed as follows [25; 26]:  
Figure 8:6  
where \(f_{1}=K/S_{0}\), and \(f_{2}=\sqrt{K^{\prime}/S_{0}}\). Here, \(S_{0}\) represents white noise, and \(f_{1}\) and \(f_{2}\) are the center frequencies or cutoff frequencies for \(1/f\) noise (flicker noise) and \(1/f^{2}\) noise (random-walk noise), respectively. In the high-frequency range where white noise predominantly contributes, the laser linewidth exhibits a Lorentzian distribution, and the linewidth at this point is [26]:  
Figure 8:7  
The phase noise test results for FP-ECDLs with two different F-P etalon precisions are shown in Figure 8:8 The FP-ECDL with higher F-P etalon precision exhibits superior noise characteristics across the entire frequency range. The overall phase noise is more effectively suppressed, resulting in a lower noise spectrum. Whether in terms of \(1/f\) noise or white noise spectra, the FP-ECDL with higher F-P etalon precision demonstrates lower noise indicators. From the experimental results, it is evident that external cavity feedback significantly influences the noise characteristics of FP-ECDL. By increasing the precision of the etalon, external cavity feedback can be appropriately enhanced, further reducing the phase noise of the output laser. Next, a comparative analysis of frequency noise and Lorentzian linewidth for lasers with two different F-P etalon precisions is presented. The frequency noise curves in the 10 kHz to 1 MHz frequency range are shown in Figure 8:9  
Consistent with the analysis of phase noise characteristics, the high-F-P-etalon-precision FP-ECDL also exhibits a lower frequency noise power spectral density. Its white noise spectral density is less than 0.5 kHz, around 360 Hz\({}^{2}\)/Hz. In contrast, the low-precision standard-cavity FP-ECDL corresponds to a white noise spectral density of around 2 kHz. According to Equation (13), the Lorentzian linewidth can be calculated by frequency noise as follows: \(\delta f_{\mathrm{L}}(\mathrm{R}=90\%)=1.130\) kHz for the high-precision case, which is smaller than \(\delta f_{\mathrm{L}}(\mathrm{R}=70\%)=6.283\) kHz for the low-precision case. Therefore, the high F-P etalon precision FP-ECDL produces a laser output with a smaller Lorentzian linewidth, indicating a reduction of 5 kHz compared to the low-precision FP-ECDL. This is attributed to the enhanced external cavity feedback achieved by using a higher-coating-reflectivity etalon, effectively increasing the external cavity reflectivity and extending the photon lifetime in the cavity, further narrowing the laser's longitudinal mode linewidth. \[E_{i}(t)=\sqrt{P_{S}}e^{i(\omega t+\varphi(t))}=E(t)e^{i\omega t} \tag{3}\]0 Phase noise test curves for two FP-ECDLs with different F-P etalon precisions. \[E_{i}(t)=\sqrt{P_{S}}e^{i(\omega t+\varphi(t))}=E(t)e^{i\omega t} \tag{3}\]1 Frequency noise curves of two FP-ECDLs with different F-P etalon precisions. The single longitudinal mode laser power output of two FP-ECDLs at different operating currents is depicted in \[E_{i}(t)=\sqrt{P_{S}}e^{i(\omega t+\varphi(t))}=E(t)e^{i\omega t} \tag{3}\]2 The threshold current for the FP-ECDL is approximately in the range of 40-50 mA. Moreover, with an increase in the precision of the F-P etalon, there is a prolonged effect on the photon lifetime, leading to an enhancement in output power. Additionally, as the power within a certain range is increased, it can narrow the linewidth. Therefore, the results in Figure 15 also partly reflect the variations in the Lorentzian linewidth of FP-ECDLs with different etalon precisions, as shown in \[E_{i}(t)=\sqrt{P_{S}}e^{i(\omega t+\varphi(t))}=E(t)e^{i\omega t} \tag{3}\]3