Laser linewidth measurement based on long and short delay fiber combination

Mingyuan Xue; Juanning Zhao

doi:10.1364/OE.428787

1. Introduction

Narrow linewidth lasers are widely used in high precision detection and nonlinear frequency conversion fields such as coherent detection, coherent optical communication, laser frequency doubling, etc. due to its good spectral purity and coherence [1–5]. The laser linewidth directly determines the detection range, sensitivity and nonlinear frequency conversion efficiency of the system [6–9], so it is of great significance to accurately measure the narrow linewidth laser linewidth.

Compared with the heterodyne method and the delay homodyne method, the delay self-heterodyne method does not need extra laser as the reference source, which avoids the strict requirements of the reference source. At the same time, the acousto-optic modulator (AOM) is used to reduce the influence of low-frequency noise on the measurement, so it is widely used in narrow linewidth laser linewidth measurement. However, the delay self-heterodyne method requires that the length of the delay fiber is longer than the coherent length of the laser. For a laser with a linewidth of 2 kHz, the coherent length is more than 100 km. Such a long fiber will cause the loss of optical power and the broadening of the measured power spectrum due to 1/f noise [10]. Therefore, the researchers put forward the methods of adding amplifiers to compensate for loss in the measurement system, and the use of cyclic delay fiber to shorten the fiber length [11–13]. However, these methods can not eliminate the Gaussian broadening caused by 1/f noise in long delay fiber. Reducing the length of the delay fiber can effectively suppress the broadening of the measured power spectrum. Therefore, the short delay fiber self-heterodyne method is proposed [14,15], which mainly compares the theoretical power spectrum with the measured power spectrum to obtain the laser linewidth, so the accurate value of the laser linewidth cannot be obtained. In 2015, Chen et al. proposed a laser linewidth measurement method based on Voigt fitting [16], which can obtain more accurate linewidth values than direct measurement from the 3-dB and 20-dB width of the spectrum. In 2017, Huang et al. introduced a short delay fiber self-heterodyne method, which uses the modulation envelope amplitude in the spectrum to obtain laser linewidth [17]. The power spectrum information is not fully utilized in the method.

In this paper, we propose a method combining the long and short delay fiber self-heterodyne to measure laser linewidth. Taking the Lorentz linewidth calculated by the long delay fiber measurement results as the initial value, the iterative algorithm is used to demodulate measurement results of the short delay fiber self-heterodyne to realize the accurate measurement of laser linewidth. The feasibility of this method is proved theoretically, which provides a new method for laser linewidth measurement.

2. Numerical computation combined with the iterative algorithms method

The measured spectrum obtained by delay self -heterodyne is no longer standard Lorentz profile, but superposition of Lorentz spectrum and Gaussian spectrum, which is Voigt profile. Since the Voigt function is the convolution of the Lorentz function and the Gaussian function, its expression can be expressed as the follow:

(1)$$V(\nu )= \int_{ - \infty }^\infty {G({\nu^{\prime}} )} L({\nu - \nu^{\prime}} )d\nu ^{\prime}.$$

in which $G(\nu )$ is the Gaussian lineshape, $L(\nu )$ is the Lorentz lineshape. The relationship of Half width at half-maximum (HWHM) between the Voigt spectrum and Lorentz spectrum and Gaussian spectrum is as follows [18]:

(2)$$\Delta {\nu _V} = \frac{1}{2} \times \left( {1.0692\Delta {\nu_L} + \sqrt {0.86639\Delta {\nu_V}^2 + 4\Delta {\nu_G}^2} } \right).$$

in which $\Delta {\nu _V}$ is the HWHM of the Voigt lineshape, $\Delta {\nu _L}$ is the FWHM of the Lorentz lineshape and $\Delta {\nu _G}$ is the HWHM of the Gaussian lineshape. The Voigt function is an integral form, and there is no easy-to-calculate analytical solution. The analytic formula of the maximum value of the Voigt function can be obtained by the Fourier transform method, and its expression is:

(3)$${V_{\max }} = \sqrt {\frac{{\ln 2}}{\pi }} \frac{1}{{\Delta {V_G}}}\exp {({\ln 2{{\Delta {V_L}} / {\Delta {V_G}}}} )^2}\left[ {1 - erf\left( {\sqrt {\ln 2} {{\Delta {V_L}} / {\Delta {V_G}}}} \right)} \right].$$

in which erf() is the error function. The half width $\Delta {\nu _V}$ and the maximum value ${V_{\max }}$ of Voigt spectrum can be obtained from the 3-dB spectrum width and amplitude of the power spectrum curve measured by delay self-heterodyne. At this time, the half width $\Delta {\nu _L}$ of Lorentz curve can be calculated by the Eqs. (2) and ((3)) to obtain the laser linewidth to be measured.

Because the relationship between Voigt spectrum half width $\Delta {\nu _V}$, Lorentz spectrum half width $\Delta {\nu _L}$ and Gaussian spectrum half width $\Delta {\nu _G}$ is an empirical formula, there is still error between the calculated laser linewidth and the measured laser linewidth. In order to measure the laser linewidth more accurately, based on the calculated laser linewidth, the short delay self-heterodyne is used to measure the laser linewidth. The short delay self-heterodyne method can effectively suppress the influence of 1 / f noise on the measurement curve. When using short delay fiber to measure laser linewidth, the power spectrum of photocurrent is shown as the black curve in Fig. 1, which is composed of delta function curve ${S_1}$, red Lorentz curve ${S_2}$ and blue modulation curve ${S_3}$. Among them [19]:

(4)$${S_1} = \frac{1}{2}A_1^2A_2^2\pi \exp \left( { - \frac{{{\tau_d}}}{{{\tau_c}}}} \right)\delta ({\omega - \Omega } ).$$

Fig. 1. Photocurrent power spectra composition of delay self-heterodyne

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(5)$${S_2} = \frac{{A_1^2A_2^2{\tau _c}}}{{1 + {{({\omega - \Omega } )}^2}\tau _c^2}}.$$

(6)$${S_3} = 1 - \exp \left( { - \frac{{{\tau_d}}}{{{\tau_c}}}} \right)\left[ {\cos ({\omega - \Omega } ){\tau_d} + \frac{{\sin [{({\omega - \Omega } ){\tau_d}} ]}}{{({\omega - \Omega } ){\tau_c}}}} \right].$$

In order to obtain the laser linewidth, it is necessary to analyze the curve related to the laser linewidth, and then the laser linewidth can be obtained from this part of the curve. From the expressions of ${S_2}$ and ${S_3}$, we can see that both of them contain the amount related to the laser linewidth, that is, the coherence time ${\tau _c}$. Therefore, as long as ${S_2}$ or ${S_3}$ can be obtained by measuring the spectral line, the laser linewidth can be obtained, and for the same laser to be measured, the coherence time in ${S_2}$ and ${S_3}$ should be the same. The expression also shows that the modulation curve is periodic, and the period is ${1 / {{\tau _d}}}$, the period can be known by determining the length of delay fiber used in measurement. The amplitude of the modulation term ${S_3}$ is related to the laser linewidth. The amplitude of the modulation curve ${S_3}$ can be determined by using the laser linewidth calculated from the power spectrum curve measured by the long delay fiber, and then the modulation curve can be determined. At this time, Lorentz curve ${S_2}$ can be demodulated by short delay fiber measurement result and ${S_3}$. In order to get the laser linewidth more accurately, take the 20-dB spectral width of Lorentz curve ${S_2}$ to calculate the laser linewidth, and compare it with the linewidth of the initial modulation term curve. If they are inconsistent, reset the line width value according to the demodulated value until it is equal, then the set line width value is the line width of the laser to be measured. When the demodulated line width is within 2% of the set line width, it is considered to be consistent.

According to the analysis, the specific steps of measuring laser linewidth with long and short delay optical fibers can be summarized as shown in Fig. 2.

Fig. 2. Demodulated curve with different coherent times in modulation curve

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3. Simulation analysis and feasibility verification

Assuming the length of the delay fiber ${L_d} = 5$ km and the refractive index of the fiber $n = 1.5$, the corresponding delay time is:

(7)$${\tau _d} = \frac{{{L_d}}}{{{c / n}}} = 2.5 \times {10^{ - 5}}s.$$

Assuming that the linewidth of the laser to be measured is 1 kHz, the corresponding coherence time is:

(8)$${\tau _c} = \frac{1}{{\Delta \nu }} = 1 \times {10^{ - 3}}s.$$

For the convenience of analysis, suppose that the amplitudes of the two channels of light after delay and frequency shift are equal to 1, and the center frequency of the beat frequency and the working frequency of the acousto-optic modulator used in the experiment are the same as 100 MHz. In this case, the photocurrent power spectrum of the delay laser and the frequency shifted laser after beat frequency is as follows:

(9)$$\begin{aligned} S(\omega ) &= \frac{1}{2}\pi {\rm exp} \left( { - \frac{{{\tau_d}}}{{{\tau_c}}}} \right)\delta ({\omega - \Omega } )+ \\ &\frac{{{\tau _c}}}{{1 + {{({\omega - \Omega } )}^2}\tau _c^2}}\left\{ {1 - {\rm exp} \left( { - \frac{{{\tau_d}}}{{{\tau_c}}}} \right)\left[ {\cos ({\omega - \Omega } ){\tau_d} + \frac{{\sin [{({\omega - \Omega } ){\tau_d}} ]}}{{({\omega - \Omega } ){\tau_c}}}} \right]} \right\}. \end{aligned}$$

In the analysis, the infinitely narrow delta function is removed, in this case:

(10)$$\begin{aligned} S(\omega ) &= {S_2} \times {S_3} = \\ &\frac{{{\tau _c}}}{{1 + {{({\omega - \Omega } )}^2}\tau _c^2}}\left\{ {1 - \exp \left( { - \frac{{{\tau_d}}}{{{\tau_c}}}} \right)\left[ {\cos ({\omega - \Omega } ){\tau_d} + \frac{{\sin [{({\omega - \Omega } ){\tau_d}} ]}}{{({\omega - \Omega } ){\tau_c}}}} \right]} \right\}. \end{aligned}$$

where ${\tau _d} = 2.5 \times {10^{ - 5}}$s, ${\tau _c} = {10^{ - 3}}$s, $\varOmega = 100$MHz

The coherence time in the modulation curve is set to be less than that of the laser, which is set to be $0.28 \times {10^{ - 3}}$s, $0.50 \times {10^{ - 3}}$s and $0.83 \times {10^{ - 3}}$s respectively, and the corresponding linewidth is 3.6 kHz, 2 kHz and 1.2 kHz respectively. At this time, the demodulated results are shown in (a), (b) and (c) of Fig. 3, in which the red curve is the modulation power spectrum curve and the blue curve is the demodulated curve.

Fig. 3. Demodulated curve with different coherent times in modulation curve: (a) Demodulated curve when the coherence time in the modulation curve is set to 0.28×10⁻³s; (b) Demodulated curve when the coherence time in the modulation curve is set to 0.50×10⁻³s; (c) Demodulated curve when the coherence time in the modulation curve is set to 0.83×10⁻³s; (d) Demodulated curve when the coherence time in the modulation curve is set to 1×10⁻³s; (e) Demodulated curve when the coherence time in the modulation curve is set to 1.17×10⁻³s; (f) Demodulated curve when the coherence time in the modulation curve is set to 1.45×10⁻³s.

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When the coherence time in the modulation curve is less than the coherence time corresponding to the laser linewidth, the demodulated curve is not the standard Lorentz curve. The shorter the coherence time in the modulation curve is compared with the laser coherence time, the more serious the deviation of the demodulation curve from the standard Lorentz curve is, and the modulation envelope cannot be fully demodulated. The reason is that the coherence time set is less than the coherence time of the laser itself, so that the $\exp ({ - {{{\tau_d}} / {{\tau_c}}}} )$ value is less than the true value of the amplitude of the modulation curve. Therefore, the modulation term cannot be completely removed in the demodulation process. At this time, the laser linewidth value obtained by demodulation is greater than the real linewidth of the laser.

Set the coherence time in the modulation curve to $1 \times {10^{ - 3}}$s, $1.17 \times {10^{ - 3}}$s, and $1.45 \times {10^{ - 3}}$s respectively, and the corresponding line widths are respectively 1 kHz, 0.85 kHz, and 0.69 kHz. At this time, the demodulated results are shown in (d), (e), (f) of Fig. 3. It can be seen that when the coherence time of the modulation curve is less than that of the laser, the demodulated curve is no longer the standard Lorentz curve. The longer the coherence time of the modulation curve is compared with that of the laser, the more serious the deviation of the demodulation curve from the standard Lorentz curve is, and the linewidth value obtained is smaller than that of the laser.

Figure 4 is the corresponding modulation curve when the coherence time in the modulation curve is $2.5 \times {10^{ - 3}}$s, $1 \times {10^{ - 3}}$s, and $0.30 \times {10^{ - 3}}$s respectively. The blue curve, the red curve and the black curve are the modulation curves when the coherent time is $2.5 \times {10^{ - 3}}$s, $1 \times {10^{ - 3}}$s and $0.30 \times {10^{ - 3}}$s respectively. As can be seen in Fig. 4, when the coherence time in the modulation curve is less than the laser coherence time, the amplitude of the modulation curve is less than its true value, so the influence of the coherence curve cannot be completely removed in the demodulation process, and the corresponding laser Lorentz curve cannot be obtained. When the coherence time of the modulation curve is greater than the laser coherence time, the amplitude of the modulation curve is greater than the real value, especially at the minimum amplitude of the modulation curve, the greater the deviation from the true value, so the demodulated curve at this time is not the standard Lorentz curve, and two peaks appear at the center frequency. When the coherence time in the modulation curve is set to $1 \times {10^{ - 3}}$, that is, the linewidth is 1 kHz, the 20-dB width of the demodulated curve is 19.6 kHz, and the corresponding laser linewidth is 0.98 kHz, which is basically the same as the laser linewidth value. At this time, the demodulated curve is shown in Fig. 3(b), which is the standard Lorentz curve. It can be seen from the analysis that when the linewidth value is set accurately, the linewidth value corresponding to the demodulated curve is consistent with the real linewidth of the laser, and the laser Lorentz curve can be restored.

Fig. 4. Periodic modulation curve with different linewidths

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Figure 5 shows the laser linewidth values demodulated with different modulation curve linewidth. It can be seen that only when the set coherence time is equal to the corresponding coherence time of laser linewidth, the demodulated linewidth value can be consistent with the real laser linewidth value.

Fig. 5. Demodulated linewidth with different linewidths in modulation curve

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The linewidth of the NKT narrow linewidth laser was measured and calculated. The result using the 50 km delay fiber is shown in Fig. 6. Its 20dB width is 15.4 kHz. Through simultaneous Eqs. (2), (3), the linewidth is calculated to be 620.3Hz, and the calculated linewidth is used as the initial value of the iterative calculation of the short-delay fiber measurement result. The laser linewidth is demodulated to be 151 Hz, and the calculated result is consistent with the nominal value. At this time, the demodulated curve is shown in Fig. 7. The blue curve is the demodulated curve, and the black dotted line is the standard Lorentz curve. It can be seen that the demodulated curve is basically a standard Lorentz curve, and the curve away from the central frequency has deviation, which is due to the deviation of the modulation curve far from the central frequency compared with the theoretical curve in the measured power spectrum. As shown in Fig. 8, the measured power spectrum by the self-heterodyne method with a delay of 5 km and the ideal power spectrum curve at 151 Hz are shown. The black curve is the measured curve, and the red curve is the ideal power spectrum. It can be seen that the amplitude of the modulation curve will decrease far away from the center frequency, and it is no longer a constant amplitude difference. Even at the center frequency, compared with the ideal power spectrum, the measured curve has a certain deviation. Therefore, the amplitude of the demodulated curve becomes smaller far away from the center frequency, especially at the frequency corresponding to the maximum amplitude of the modulation curve, so that the demodulated curve deviates from the standard Lorentz curve far away from the center frequency, but it basically coincides with the Lorentz curve near the center frequency.

Fig. 6. 50 km delay self-heterodyne measurement result.

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Fig. 7. 5km delay fiber measurement power spectrum de-modulation result

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Fig. 8. The measured power spectrum and the ideal power spectrum of 5 km delay fiber

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4. Conclusion

In summary, a new method to measure the linewidth of narrow linewidth laser is studied, and numerical calculation combined with iterative algorithm is proposed. By this method, the Lorentz line shape and the corresponding linewidth can be obtained from the power spectrum of delay self-heterodyne measurement. Compared with the direct measurement method, it can suppress the broadening of power spectrum induced by 1/f noise in long time delay measurement, and the accuracy and feasibility of this method are theoretically analyzed and proved. Therefore, this method can provide accurate linewidth measurement for narrow or ultra-narrow linewidth laser.

Funding

Natural Science Basic Research Program of Shaanxi Province (2021JC-47); Key Research and Development Program of Shaanxi (2021ZDLGY08-07); National Natural Science Foundation of China (61861024, 61871259, 62031021).

Disclosures

The authors declare no conflicts of interest.

Data availability

Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.

References

1. B. Abbott, R. Abbott, T. Abbott, and M. Abernathy, “Observation of gravitational waves from a binary black hole merger,” Phys. Rev. Lett. 116(6), 061102 (2016). [CrossRef]

2. S. Cui, L. Zhang, H. Jiang, W. Pan, X. Yang, and Y. Feng, “High efficiency frequency doubling with a passive enhancement cavity,” Laser Phys. Lett. 16(3), 035105 (2019). [CrossRef]

3. Z. Zhang, J. Sun, B. Lu, J. Li, R. Zhu, and X. Hou, “Costas optical phase lock loop system design in inter- orbit coherent laser communication,” Chinese Lasers 42(8), 0805006 (2015). [CrossRef]

4. J. Dong, X. Zeng, S. Cui, J. Zhou, and Y. Feng, “More than 20 W fiber-based continuous-wave single frequency laser at 780 nm,” Opt. Express 27(24), 35362 (2019). [CrossRef]

5. M. Kwon, P. Huft, C. Young, M. Ebert, and P. Yang, “Generation of 14.0 W of single-frequency light at 770 nm by intracavity frequency doubling,” Opt. Lett. 45(2), 339–342 (2020). [CrossRef]

6. B. Argence, B. Chanteau, O. Lopez, D. Nicolodi, and A. Amy-Klein, “Quantum cascade laser frequency stabilization at the sub-Hz level,” Nat. Photonics 9(3), 1079–1083 (2015). [CrossRef]

7. T. Zhu, Q. He, X. Xiao, and X. Bao, “Modulated pulses based distributed vibration sensing with high frequency response and spatial resolution,” Opt. Express 21(3), 2953–2963 (2013). [CrossRef]

8. N. C. Pixley, T. L. Correll, D. Pappas, O. I. Matveev, and J. D. Winefordner, “Tunable resonance fluorescence mono-chromator with sub-Doppler spectral resolution,” Opt. Lett. 26(24), 1946–1948 (2001). [CrossRef]

9. L. Zhang, H. Jiang, S. Cui, J. Hu, and F. Yan, “Versatile Raman fiber for sodium laser guide star,” Laser Photonics Rev. 8(6), 889–895 (2014). [CrossRef]

10. L. B. Mercer, “1/f frequency noise effect on self-heterodyne linewidth measurements,” J. Lightwave Technol. 9(4), 485–493 (1991). [CrossRef]

11. X. Chen, H. Ming, Y. Zhu, D. Bo, and A. Wang, “Implementation of a loss-compensated recirculating delayed self-heterodyne interferometer for ultra-narrow laser linewidth measurement,” Appl. Opt. 45(29), 7712–7717 (2006). [CrossRef]

12. C. Albert, M. Andrew, C. John, H. John, F. Simon, H. Wang, and M. L. Aslund, “A Comparison of Delayed Self-Heterodyne Interference Measurement of Laser Linewidth Using Mach-Zehnder and Michelson Interferometers,” Sensors 11(10), 9233–9241 (2011). [CrossRef]

13. H. Tsuchida, “Laser frequency modulation noise measurement by recirculating delayed self-heterodyne method,” Opt. Lett. 36(5), 681–683 (2011). [CrossRef]

14. H. Ludvigsen, M. Tossavainen, and M. Kaivola, “Laser linewidth measurements using self-homodyne detection with short delay,” Opt. Commun. 155(1-3), 180–186 (1998). [CrossRef]

15. P. Yu, “A novel scheme for hundred-hertz linewidth measurements with self-heterodyne method,” Chinese Phys. Lett. 30(8), 1–12 (2013).

16. M. Chen, M. Zhou, J. Wang, and W. Chen, “Ultra-narrow linewidth measurement based on Voigt profile fitting,” Opt. Express 23(5), 6803–6808 (2015). [CrossRef]

17. S. Huang, T. Zhu, M. Liu, and W. Huang, “Precise measurement of ultra-narrow laser linewidths using the strong coherent envelope,” Sci. Rep. 7(1), 41988 (2017). [CrossRef]

18. J. J. Olivero and R. L. Longbothum, “Empirical Fits to the Voigt Line Width: A Brief Review,” J. Quant. Spectrosc. Radiat. Transfer 17(2), 233–236 (1977). [CrossRef]

19. , “Laser Linewidth Measurement Based on Amplitude Difference Comparison of Coherent Envelope,” IEEE Photonics Technol. Lett. 28(7), 759–762 (2016). [CrossRef]

Laser linewidth measurement based on long and short delay fiber combination

Abstract

1. Introduction

2. Numerical computation combined with the iterative algorithms method

3. Simulation analysis and feasibility verification

4. Conclusion

Funding

Disclosures

Data availability

References

Data availability

Cited By

Figures (8)

Equations (10)

Optics Express