## 3 Experimental Design and Analysis of Results
### Comparison of Line Width and Noise Characteristics of FP-ECDL with Two F-P Etalon Precision Levels
Assuming a 50:50 splitting ratio for the fiber coupler that spectrally separates the test laser in the system, the optical fields of the signal light after the AOM and the reference light through the delayed fiber can be represented as follows:  
\[E_{S}(t)=\frac{1}{\sqrt{2}}\sqrt{P_{S}}cos((\omega_{S}-\omega_{A})t+\varphi(t)) \tag{4}\]  
\[E_{D}(t)=\frac{1}{\sqrt{2}}\sqrt{P_{S}}cos((\omega_{S}(t-\tau_{d})+\varphi(t- \tau_{d})) \tag{5}\]  
where \(\omega_{A}\) is the frequency shift of the AOM, \(\tau_{d}\) is the delay introduced by the delay fiber, and the optical field after the second fiber coupler for the two optical paths is given by  
\[E_{B}(t)=E_{S}(t)+E_{D}(t) \tag{6}\]  
The photodetector (PD) converts the optical signal into a photocurrent \(I(t)\):  
\[I(t)=E_{B}(t)E_{B}^{*}(t) \tag{7}\]  
The autocorrelation function of the photocurrent is  
\[R_{I}(t)=<E_{B}(t)E_{B}^{*}(t)E_{B}(t+\tau_{d})E_{B}^{*}(t+\tau_{d})> \tag{8}\]  
According to the Wiener-Khinchin theorem, the Fourier transform of the autocorrelation function of the photocurrent is the power spectral density function \(S_{i}(\omega)\)[24]:  
\[S_{i}(\omega)=\frac{\frac{1}{2}P_{S}^{2}\tau_{c}}{1+[\tau_{c}(\omega_{S}\pm \omega_{A})]^{2}}\{1-[\cos(\omega_{S}\pm\omega_{A})\tau_{d}+\frac{\sin(\omega_ {S}\pm\omega_{A})\tau_{d}}{(\omega_{S}\pm\omega_{A})\tau_{c}}]\mathrm{e}^{- \tau_{d}/\tau_{c}}\}+\frac{1}{2}P_{S}^{2}\pi\mathrm{e}^{-\tau_{d}/\tau_{c}} \delta(\omega_{S}\pm\omega_{A}) \tag{9}\]  
When the delay of the self-heterodyne \(\tau_{d}\) is sufficiently large (the delay fiber is long enough), \(\tau_{d}\gg\tau_{c}\), the coherence time of the measured laser and the delay of the self-heterodyne can satisfy \(\exp(-\tau_{d}/\tau_{c})\ll 1\). Equation (9) can be simplified to  
\[S_{i}(\omega)=\frac{\frac{1}{2}P_{S}^{2}}{1+\tau_{c}(\omega_{S}\pm\omega_{A}) ^{2}} \tag{10}\]  
Figure 8:0 Delay self-heterodyne linewidth measurement system. In this case, the beat frequency line shape presents a Lorentzian shape, and its 3 dB bandwidth is twice the linewidth of the measured laser [25]. The measurement results of the two FP-ECDLs with different F-P etalon precisions under the same operating conditions are presented in Figure 8:1 It is evident from the broadening of the beat frequency spectrum lines that the high-F-P-etalon-precision FP-ECDL exhibits a narrower linewidth for a single longitudinal mode laser output. Next, the lasers with two different F-P etalon precisions were subjected to phase noise and frequency noise characteristic tests under the same operating conditions using a noise test instrument. The basic structure of the noise test instrument is shown in Figure 8:2  
Figure 8:3 Basic structure of the phase noise measurement system. Figure 8:4 Delay self-heterodyne (30 km OPD) beat frequency testing curves of two F-P etalon precision FP-ECDLs. In Figure 11, FRM stands for the Faraday rotating mirror, PZT is the piezoelectric ceramic used to finely tune the interference arm length, and A/D represents the analog-to-digital converter, while DAQ refers to the data acquisition module. The laser under test, when introduced into the fiber Michelson interferometer with a delay of \(\tau_{\mathrm{d}}\) through the circulator, generates an interference signal. This signal is converted into an analog electrical signal through photoelectric conversion then transformed into a digital signal by the analog-to-digital converter and inputted into the DAQ. Specialized algorithms are applied to process the digital signal, extracting the phase difference information of the laser under test from the raw data. Subsequently, Fourier transformation is employed to obtain the power spectral density of the phase noise. Since frequency is the derivative of phase, the spectral density function \(S_{\varphi}(f)\) of phase noise and the spectral density function \(S_{\varphi}(f)\) of frequency noise are related as follows:  
Figure 8:5  
The spectral density function of frequency noise can be obtained by differentiating the phase noise.