Precise measurement of ultra-narrow laser linewidths using the strong coherent envelope

Laser linewidth narrowing down to kHz or even Hz is an important topic in areas like clock synchronization technology, laser radars, quantum optics, and high-precision detection. Conventional decoherence measurement methods like delayed self-heterodyne/homodyne interferometry cannot measure such narrow linewidths accurately. This is because a broadening of the Gaussian spectrum, which hides the laser’s intrinsic Lorentzian linewidth, cannot be avoided. Here, we introduce a new method using the strong coherent envelope to characterize the laser’s intrinsic linewidth through self-coherent detection. This method can eliminate the effect of the broadened Gaussian spectrum induced by the 1/f frequency noise. We analyze, in detail, the relationship between intrinsic laser linewidth, contrast difference with the second peak and the second trough (CDSPST) of the strong coherent envelope, and the length of the delaying fiber. The correct length for the delaying fiber can be chosen by combining the estimated laser linewidth (Δfest) with a specific CDSPST (ΔS) to obtain the accurate laser linewidth (Δf). Our results indicate that this method can be used as an accurate detection tool for measurements of narrow or super-narrow linewidths.


Numerical simulation module details based on SDSHI
The detected power spectrum density can be summarized as the following equation from [1][2][3][4][5]. where P0 is the detected optical power by PD, f is the measurement frequency, f1 is the AOM frequency shift, τd (τd=L/c, L is the length of the delaying fiber, c is the speed of light) is the time delay of one path with respect to the other path, and Δf is the full-width-half-maximum (FWHM) of the power spectrum (Lorentzian linewidth). As for EqS1(c), when f ≠ f1, δ(f ±f1)=0, S3 = 0 and f = f1, S3=infinite, the power spectrum S can be simplified to be S(f, Δf)=S1S2 whereas the detected power spectrum is unstable at f = f1. All simulated normalized power spectra were computed using S(f, Δf)=S1S2.
From Eq. S1, we can see that the power spectrum S is the product of the Lorentzian spectrum S1 and the periodic modulation power spectrum S2. Figure S1 shows the simulated normalized power spectrum for S (brown line), S1 (red line), and S2 (blue line), respectively, with a 1 kHz laser linewidth (Δf=1 kHz) and 1500 m of delaying fiber (L=1500 m). If 500 km delaying fiber is used to detect the 1 kHz laser linewidth, the amplitude for S2 is so small that the power spectrum S is almost equal to Lorentzian spectrum S1, and this is the classical DSHI used to detect the laser linewidth. However, for the actual experiment, the use of such a long delaying fiber would induce a large Gaussian spectrum by 1/f noise for the center frequency [5][6][7][8][9][10][11][12][13][14] , which could mask the Lorentzian spectrum. This is why it is difficult to detect accurately the narrow linewidth using traditional DSHI. If τd is much smaller than the laser coherent time τc=1/ (2πΔf), the amplitude of S2 is too large to be neglected and it will be periodically superimposed on the Lorentzian line shape S1 to finally form S shown in figure S1.
Since the power spectrum S is the product of the Lorentzian spectrum S1 with the periodic modulation power spectrum S2, the Lorentzian linewidth Δf can also be reflected by its coherent envelope. This method can also eliminate the effect of the Gaussian spectrum for the center frequency.
From this point of view, our method provides a new way to accurately detect laser linewidths 5 .

Influence of the Gaussian linewidth
In reference [15], the Gaussian linewidth induced by the 1/f noise is about where k depends on the type of phase-locked loop and the acceptable phase-error variance [15][16][17][18][19][20][21][22] , and τd=nL/c. If we want to reduce the Gaussian linewidth induced by the 1/f noise to obtain the Lorentzian linewidth, the level k should be decreased and the delaying fiber L should be shortened. Since the level for k is difficult to detect and different lasers have different k values, the value for Glw is difficult to be determined. It has been shown that shortening the length of the delaying fiber is a valuable way to eliminate Gaussian linewidth. Since the power spectrum detected by DSHI is the convolution of the Lorentzian spectrum and the approximately Gaussian spectrum 15,23,24 , the linewidth of the detected power spectrum must be larger than the linewidth of the Gaussian spectrum. Therefore, in this letter, we use the power spectrum detected using DSHI to displace the Gaussian spectrum and determine the suitable length for the delaying fiber to eliminate the effect of the Gaussian spectrum induced by the 1/f noise.