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Abstract
Laser linewidths of the order of 100 Hz are challenging to measure with existing technology. We propose a simple, efficient method to measure ultra-narrow linewidths using dual-parameter acquisition based on partially coherent light interference. The linewidth is obtained using two parameters that are easily extracted from the power spectrum. This method reduces the influence of 1/f noise by utilizing a kilometer-order-length delay fiber and is independent of the fiber-length error for a general situation. Simulation results show that, for a length error less than 10%, the total linewidth measurement error is less than 0.3%. Experimental results confirm the feasibility and superior performance of this method.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Ultra-narrow-linewidth lasers exhibit low noise and large coherence lengths and are widely applied in many fields such as optical fiber communication [1,2], distributed optical fiber sensing [3,4], and high-resolution spectrum measurement [5,6]. In optical fiber communication systems, the laser linewidth directly affects the extraction of phase information. With the signal modulation format becoming increasingly advanced, the linewidth has progressively decreased from 1 MHz to 10 kHz or even 100 Hz [7,8]. To meet the demand for small linewidths, many lasers with linewidths less than 1 kHz have been reported [9–12]. Therefore, measurement technology for 100-Hz-order linewidths is very important.
Most laser-linewidth measurement methods can be classified into two types: methods based on the signal power spectrum and those based on the frequency noise power spectrum [13–15]. Methods based on the signal power spectrum have been widely applied owing to its simplicity and efficiency. These methods can be divided into methods using incoherent light interference and those using partially coherent light interference [16]. Conventional laser-linewidth measurement techniques are mostly based on incoherent light interference. The Lorentzian fitting method, which is the most frequently used among such methods [17], requires the length of the delay fiber in the system to be greater than the coherence length of the laser [18]. When the laser linewidth is very narrow, the delay fiber needs to be hundreds of kilometers in length, which not only causes great loss, but also expands the impact of 1/f noise, resulting in inaccurate measurement results [19]. Chen et al. [20] proposed a Voigt profile fitting method, which can reduce the influence of 1/f noise in linewidth measurement. Compared with methods based on incoherent light interference, the delay-fiber length required for methods based on partially coherent light interference is significantly lower [21]. However, the relationship between the laser linewidth and power spectrum in such methods is unclear. Some methods were proposed to overcome this problem. Huang et al. [22,23] proposed a method based on the amplitude difference comparison of coherent envelope. This method can measure ultra-narrow linewidths simply and efficiently. He et al. [24] proposed a coherent envelope demodulation method, which is based on iterative algorithm and shows high measurement accuracy.
In the present paper, we propose a simple and efficient method based on partially coherent light interference, which we call the dual-parameter acquisition (DPA) method. In this method, which applies the delayed self-heterodyne interferometer structure, the laser linewidth is calculated by extracting the power difference between the maximum and minimum points of the first-order sidelobe as well as the frequency difference between the zero-order minimum point and the center frequency from the power spectrum. We analyze the feasibility of this method through theory and simulation. The simulation results show that, when the fiber length is of kilometer order and the error in the fiber length is less than 10%, the total measurement error is less than 0.3%. In addition, an experimental system is set up and a laser with an ultra-narrow linewidth of the order of 100 Hz is measured by the Lorentzian fitting method, Voigt fitting method and DPA method. The simulation and experimental results demonstrate that this method can reduce the influence of 1/f noise by using a fiber of kilometer-order length, and it is independent of the length error of the delay fiber for a general situation. The superior performance of the proposed method makes it one of the best candidates for ultra-narrow-linewidth measurement.
2. Operating principle
A delayed self-heterodyne interferometer is used to measure laser linewidth. While the conventional Lorentzian fitting method is based on incoherent light interference, the DPA method is based on partially coherent light interference [16]. Figure 1 schematically shows laser linewidth measurement using both methods. The laser beam to be tested is split into two beams by a coupler (Coupler 1), wherein one laser beam is frequency-shifted by a frequency shifter and the other is delayed by a single-mode fiber (SMF). After combining these two beams using another coupler (Coupler 2), the photocurrent of the recombined beam is detected by a photodetector (PD) to obtain its power spectrum.
As shown in Fig. 1, the Lorentzian fitting method adopts incoherent light interference, where the delay time of the SMF Δt is longer than the coherent time of the laser τ. The light after frequency-shifting and the light after delaying are incoherent, and the linewidth of the laser is half of the width of the power spectrum. On the other hand, the proposed DPA method adopts partially coherent light interference, where Δt is shorter than τ. The light after frequency-shifting and the light after delaying are partially coherent. We found that the linewidth of the laser Δν is related to the power difference between the maximum and minimum points of the first-order sidelobe ΔS and the frequency difference between the zero-order minimum point and the center frequency Δf0. In the measurement, ΔS and Δf0 are first extracted from the measured power spectrum and then brought into the formula to get Δν. Derivation process of formula can be found in the theoretical analysis part.
3. Theoretical analysis
When the delay time of the fiber is shorter than the coherent time of the laser, the white noise is dominant and the influence of 1/f noise is not obvious [21]. So only white noise is considered. At this time, the power spectrum S(f) can be expressed as follows [18,25]:
The delay time of the fiber is shorter than the coherent time of the laser, which could be expressed as [16],
The impulse function is ignored here because the power at the center of the power spectrum is not measured. Thus, S(f) consists of a Lorentzian function and a quasi-periodic function. The frequency difference Δfm between the minimum point of each order and the center frequency of the power spectrum is expressed as follows: where m is a natural number.The DPA method analyzes the power points of the first-order sidelobe. The frequency difference between these points and the center frequency are greater than the frequency difference between the zero-order minimum point and the center frequency Δf0. Then combined with Eq. (2), Eq. (1) can be simplified as follows:
The extreme points of the power spectrum are obtained by claculating the derivative of Eq. (4). The calculation yields a transcendental equation for which an analytical solution cannot be obtained. Therefore, we apply a simplified solution. Equation (4) consists of an exponential function and a quasi-periodic function. The approximate frequencies of the maximum and minimum points of the quasi-periodic function are considered to substitute for the maximum and minimum points of Eq. (4), respectively. The power difference between the first-order maximum point and the minimum point ΔS is expressed as follows [22]:
Because of the approximation in the above process, ΔS has the system error δ(L, Δν), which should be corrected. According to the simulation results, for different delay-fiber lengths and linewidths of the ultra-narrow-linewidth laser, the system error is distributed in a narrow range. The distribution is subjected to a normal fitting with a mean of 0.204 and variance of 4.465e – 6. The confidence interval of 99% is (0.199, 0.209). Thus, we have δ(L, Δν) = 0.204. Then by combining Eqs. (3), (4), and (5), we obtain:
In the Lorentzian fitting method, the delay-fiber length LD should be longer than the coherent length of the laser [14]. In the DPA method, if we assume that c/nLΔν in Eq. (2) is greater than 1 by two orders of magnitude, the delay-fiber length LS should be shorter than 100 times of the coherent length of the laser. Then we obtain:
Equation (7) indicates that for a laser with known linewidth, the delay-fiber length in the Lorentzian fitting method is 100 times that in the DPA method.Let ΔνR denote the true linewidth of the laser to be tested, LS + ΔLS be the true length of the delay fiber, where ΔLS is the length error of the delay fiber. In fact, because the DPA method uses a short fiber, ΔLS cannot be ignored. In the DPA method, the frequency difference between the frequency of the zero-order minimum point and the center frequency is expressed as:
Then, Eq. (5) should be corrected as:4. Simulation results
Figure 2(a) shows the system error δ(L, Δν), which was introduced in Eq. (5), for different delay-fiber lengths and laser linewidths. Figure 2(b) shows a frequency-distribution histogram of the system error. When the delay-fiber length does not satisfy Eq. (2), the DPA method is not applicable, leading to the blank part of Fig. 2(a). According to Fig. 2(b), the system error has an approximately normal distribution. A normal fitting of the distribution in Fig. 2(b) yields the mean $\bar\delta = 0.204 $ and variance s2 = 4.465e – 6; therefore, the 99% confidence interval is (0.199, 0.209). This range is relatively small compared to ΔS, which is generally greater than 20 dB. Therefore, we consider δ(L, Δν) ≈ $\bar\delta = 0.204 $ to correct the system error, as expressed in Eq. (6). It is concluded from Fig. 2(a) that this correction is applicable to lasers with linewidth of the order of 100 Hz.
Fig. 2. (a) System error with different delay-fiber lengths and laser linewidths. (b) Frequency-distribution histogram of model error.
For a certain linewidth, the DPA method and Lorentzian fitting method require different delay-fiber lengths. Figure 3 shows the model error of Eq. (6) for different delay-fiber lengths and laser linewidths. As per the figure, the delay-fiber length required for the DPA method is significantly less than that for the Lorentzian fitting method. For shorter delay-fiber lengths, the linewidth range covered by the DPA method is wider. When the linewidth is sufficiently large, the fiber required for the DPA method is extremely short and even less than 10 m when the laser linewidth is greater than 100 kHz. Thus, the DPA method is not appropriate for lasers with large linewidth. On the other hand, when the laser linewidth is very narrow, the model error of the Lorentzian fitting method is significantly greater than that of the DPA method. When the linewidth of the laser is less than 10 Hz, the model error of the Lorentzian fitting method starts to increase. However, the effect of 1/f noise is not considered here. Due to the long fiber, 1/f noise is already very large. Therefore, the actual error will be even larger for the Lorentzian fitting method. In contrast, the DPA method can maintain relatively high precision under this condition.
Fig. 3. Model error with different delay-fiber lengths and laser linewidths for different measurement methods.
In the measurement, owing to the measurement error in fiber length and the extra optical path introduced by other optical devices, an error must exist in the fiber length. This error is especially significant for short fibers. Figure 4 shows the total error of the laser with respect to linewidth for different lengths and length errors of the delay fiber. The total error includes the model error and fiber-length measurement error. According to the figure, for a delay fiber with a length of 10–5000 m, when the length error is less than 10%, the total measurement error is less than 0.3%. Additionally, the error of the measured linewidth tends to increase in both the narrow and wide areas of the linewidth for a certain fiber length. When the laser linewidth is very large, the condition of Eq. (2) is gradually approached, and the DPA method gradually fails. When the laser linewidth is very narrow, the measurement error tends to increase owing to the limitation of the power-spectrum resolution. However, for lasers with linewidth of the order of 100 Hz, an error of 0.3% is acceptable under most conditions.
Fig. 4. Total error of the DPA method with respect to the laser linewidth for different lengths and length errors of the delay fiber.
5. Experimental setup and results
Figure 5 shows the experimental system of ultra-narrow-linewidth measurement. The laser is split into two beams by a coupler, and one of the beams is frequency-shifted by an acoustic optical modulator (AOM, Gooch & Housego T-M200-0.1C2J-3-F2S) with an offset frequency of 200 MHz. The other beam is delayed by an SMF. The two beams are recombined by another coupler. The recombined optical signal is detected by a PD (Newport 1592), and the photocurrent is analyzed by a frequency spectrum analyzer (FSA, Agilent E4447A). The optical isolator (ISO) in the system is utilized to protect the ultra-narrow-linewidth laser, the variable optical attenuator (VOA) is used to match the optical power of the upper and lower beams, and the polarization controller (PC) is employed to match the polarization states of the upper and lower beams.
The ultra-narrow-linewidth laser applied here was manufactured by Accelink Technologies Corporation, China. Figure 6 shows the spectrum measured using a delayed self-heterodyne interferometer, Lorentzian fitting curve, and Voigt fitting curve. The length of the delay fiber is 80 km. The figure shows the laser linewidth is determined as 4.17 kHz with the Lorentzian fitting method. However, a long fiber introduces a large impact of 1/f noise. Thus, this fitted power spectrum is far from the experimentally obtained power spectrum. This result is consistent with a previous report [19]. In order to eliminate the influence of 1/f noise and achieve a more accurate result, we use the Voigt fitting method, which was proposed by Chen et al. [20], to correct the result. The measured laser linewidth is 458 Hz, and the bandwidth of 1/f noise is 10.11 kHz. Figure 6 explains that the power spectrum obtained with Voigt fitting is closer to the experimental measurement result. Therefore, we conclude that the linewidth of 458 Hz obtained with Voigt fitting is closer to the actual laser linewidth.
Fig. 6. Spectrum of the ultra-narrow-linewidth laser measured using the delayed self-heterodyne interferometer with a delay fiber of 80 km length and the fitting curves.
Figure 7 shows the measurement results obtained using the DPA method for different delay-fiber lengths. The inset of Fig. 7 shows the power spectrum obtained when the delay-fiber length is 2.055 km. The delay-fiber lengths considered in the measurement are 511 m, 775 m, 1.047 km, 1.558 km, 2.055 km, 2.566 km, 3.102 km, and 4.11 km. For delay fibers with lengths of 1.047, 1.558, 2.055, and 2.566 km, the measurement results are close to those obtained with the Voigt fitting method. When the delay-fiber length is less than 1 km, the measurement result is larger. The reason for this increase is that when the delay-fiber length is short, the first-order minimum point is so low that the noise floor of the system cannot be ignored. When the fiber length is less than 500 m, the first-order minimum point is completely submerged by the system noise floor, which leads to the underestimation of ΔS and thereby larger measured linewidths. When the length of the delay fiber is greater than 2.5 km, the measurement results gradually increase. The reason for this increase is that when the fiber length is long, the period of the quasi-periodic function in Eq. (1) is so small that the two power points and two frequency points to be acquired are close to the center of the power spectrum. Consequently, ΔS and Δf0 may be misestimated owing to the 1/f noise, increasing the measured linewidths. In summary, when measuring ultra-narrow linewidths using the DPA method, the delay-fiber length should be of the kilometer order.
Fig. 7. Linewidth measured using the DPA method for different delay-fiber lengths. Inset: spectrum measured when the delay-fiber length is 2.055 km.
In order to verify that the measurement result of the DPA method is independent of the delay-fiber length error for a general situation, an error fiber is introduced, and the laser linewidth is measured using the system shown in Fig. 5. The lengths of the delay fibers used in this measurement are 1.558 and 2.055 km, respectively, as shown in Fig. 8. The figure shows that the measurement results are unchanged for different errors in the fiber length. Combined with the simulation results, these results prove that the measurement results of the DPA method are unrelated to the delay-fiber length error for a general situation. Therefore, the ultra-narrow linewidth of the laser can be accurately measured without accurate information on the delay-fiber length. The DPA method is highly robust, which is of great significance for engineering applications.
Fig. 8. Measured laser linewidth with respect to the delay-fiber length error for different delay-fiber lengths.
6. Conclusions
In summary, we propose a simple and efficient method for ultra-narrow-linewidth measurement based on partially coherent optical interference. We analyze the characteristics of this method through theory and simulation. The simulation results show that when the fiber length is of kilometer order and the fiber-length error is less than 10%, the measurement error is less than 0.3%. An ultra-narrow-linewidth laser of 100-Hz-order linewidth is experimentally measured using the Lorentzian fitting method, Voigt fitting method and DPA method. The experimental results confirm the feasibility of the proposed method. The method can reduce the influence of 1/f noise by using a fiber of kilometer-order length, and it is independent of the delay-fiber length error for a general situation. Therefore, the proposed method makes it one of the best candidates for ultra-narrow-linewidth measurement.
Funding
National Natural Science Foundation of China (61975063).
Acknowledgments
The authors are grateful to those who are fighting on the frontlines during the epidemic, especially in Wuhan. It is because of their selflessness, bravery and perseverance that this paper could be finally finished. We firmly believe that the epidemic will come to an end soon. Come on, Wuhan! Come on, China!
Disclosures
The authors declare no conflicts of interest.
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