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We study the method of Voigt profile fitting for ultra-narrow linewidth measurement. It filters out the effect of the spectrum broadening due to the 1/f frequency noise and extracts out the Lorentzian lineshape from the measured spectrum. The resolution is thus greatly promoted than the direct measurement from the self-heterodyne technique. We apply this method to an ultra-narrow-linewidth (~40 Hz by heterodyne beat technique) Brillouin/erbium fiber laser. The linewidth estimated from Voigt fitting method is indicated to be more accurate. In contrast, the linewidths estimated direct from the 3-dB and the 20-dB heterodyne-spectrum width are far over the true linewidth of the BEFL. The Voigt fitting method provides an efficient tool for ultra-narrow-linewidth measurement. And compared with heterodyne beat technique, it is applicable for all types of lasers.
© 2015 Optical Society of America
1. Introduction
Narrow-linewidth lasers have been drawing great research interest for their importance in many applications [1–3]. With the development of linewidth narrowing technique, lasers with several kilohertz linewidth is conveniently available [4]. Especially, a novel single-frequency Brillouin/erbium fiber laser (BEFL) has been proposed recently, whose linewidth is indicated to be sub-kilohertz [5–7]. The appearance and development of ultra-narrow-linewidth lasers calls for high-accuracy linewidth measurement techniques.
Self-heterodyne technique has been an important tool for linewidth measurement. Compared with the heterodyne technique, it does not need an extra narrow-linewidth local oscillator whose frequency should be close to the lasers under test. However, its requirement that the delay time should be over the coherent time of the lasers limits its usefulness to lasers with relatively broad linewidths. Theoretically, about 100 km delay fiber is required for 2 kHz instrument resolution. Obviously, such a long delay causes severe light absorption. Moreover, long delays result in considerable broadening of the self-heterodyne spectrum due to the 1/f frequency noise [8]. The convolution of 1/f noise leads to the overestimation of the natural linewidth of the laser. Fortunately, the 1/f noise broadening is mostly pronounced near the center of the spectrum. The estimation from the 20-dB width of the self-heterodyne spectrum is closer to the natural linewdith of the laser under test. This method performs well when the Lorentzian contribution is more than the 1/f contribution. For narrow-linewidth lasers, however, this method still much overestimates the laser linewidth.
The measured self-heterodyne spectrum is actually the Voigt profile, i.e., the Lorentzian spectrum convoluted with the Guassian spectrum. The Voigt function has been widely used in the study of stellar atmospheres, in atomic and plasma spectroscopy. A number of efficient and accurate algorithms have been presented for estimating the Voigt profile for any combinations of Lorentzian and Guassian contributions [9–12].
In this paper, we use Voigt fitting method for ultra-narrow linewidth measurement. We analyze the Voigt profile and the numerical Voigt fitting method for the self-heterodyne spectrum. We measure the spectrum of a single-frequency BEFL, in which 45-cm EDF acts as the hybrid gain media. We apply the Voigt fitting procedure to the actual spectrum, and then compare it with the methods of direct measurement from the 3-dB and 20-dB width of the spectrum. The cases with different fiber delays are studied. The effect of the fiber delay is discussed on the Guassian component, the Lorentzian component, and the accuracy of the linewidth estimate. The Voigt fitting method promises a powerful candidate for linewidth measurement for all types of lasers, especially for ultra-narrow-linewidth lasers.
2. Voigt profile and the curve fitting method
The self-heterodyne spectrum of the laser under test is Voigt profile, which is the convolution of the Lorentzian spectrum associated with the white frequency noise and the approximately Guassian spectrum arising from the 1/f noise. The Voigt profile can be expressed as the following [13]:
where G is the normalized Guassian lineshape centered at ν = ν0, shown as the following:in which ΔνG is the full width at half-maximum (FWHM) of the Guassion lineshape. And L is the normalized Lorentzian lineshape centered at ν = ν0, as the following:in which ΔνL is the FWHM of the Lorentzian lineshape. The Lorentzian and Guassion components can be well estimated by fitting the Voigt profile to the measured self-heterodyne spectrum. Many efficient algorithms are presented to accurately estimate the Voigt profile for any combination of Lorentzian and Guassion contribution. These fitting procedures use the Voigt function and derivatives for each point in the spectrum and iterate until a minimum error is reached.We adopt the following method which requires fewer computations than the nonlinear least squares fitting procedures. The relationship between the Voigt spectrum and the Lorentzian and Guassian spectra is approximated by the following expression:
where ΔνV is the Voigt linewidths. Since the broadening effect of the 1/f noise is mostly pronounced near the center of the spectrum, the 3-dB width of the spectrum is strongly affected by the Guassian component, while the 20-dB width is dominated by the Lorentzian contribution. Hence, the Lorentzian linewidth can be more accurately estimated from the 20-dB width. It equals to the 20-dB spectrum width divided by 2. We use this estimation as the initial value of the Lorentzian linewidth. Then the Guassian component is estimated from the 3-dB linewidth and Eq. (4). Using the obtained Guassian and Lorentzian linewidth, a Voigt profile can be obtained from Eqs. (1)-(3). Comparing the Voigt profile to the measured spectrum, a new estimation of Lorentzian linewidth is achieved by iterating to found the value that produces the 20-dB width of the Voigt profile equal to the measured width. After this, a new Guassian linewdith is refined by the new Lorentzian linewidth. The steps are repeated until the estimates converge.3. Experimental setup
The experimental setup is shown in Fig. 1. The structure of the BEFL under test is shown in the dashed box. It is based on SBS in a length of commercialized EDF. Its principle and some characteristics have been presented in previous work [5–7]. Compared with the previous ones, the BEFL under test here is based on much shorter hybrid gain medium of only 45-cm. The cavity length is about 2 m. Its linewidth is demonstrated to be 40 Hz by heterodyne technique [7]. In the following experiments, the Brillouin pump power and the 980 nm pump power are 1 mW and 200 mW, respectively. The BEFL output power is about 3 mW. The BEFL emission is measured by the delayed self-heterodyne technique, as in the solid box. The BEFL output is split into two beams through a 3-dB coupler. One beam is 300-MHz frequency shifted by an acousto-optic modulator (AOM). The other beam is delayed by a long-haul fiber. An erbium-doped fiber amplifier (EDFA) is after the delay fiber to compensate for the severe light transmission attenuation. The two beams are recombined through another 3-dB optical coupler and injected into an optoelectronic detector. An electrical spectrum analyzer (ESA) is used to analyze the beat note of the two beams.
Fig. 1 Experimental setup: BP, Brillouin pump; EDF, erbium-doped fiber; BOF, bandpass optical filter; AOM, acousto-optic modulator; EDFA, erbium-doped fiber amplifier; ESA, electrical spectrum analyzer.
4. Results and discussions
The measured spectrum and the fitting curves are presented in Fig. 2, when the delay fiber is 25 km. The central frequency is normalized to the 300 MHz frequency shift. The FWHM of the measured spectrum is 5200 Hz, as the red curve shows. The lorentzian fitting curves with 5200 Hz and 300 Hz FWHM are shown as the purple and green curves. Apparently, neither of the two Lorentzian shapes can reproduce the heterodyne spectrum, as is expected for the frequency independent white noises. The Voigt fitting curve well reproduces the measured spectrum, as the blue curve. The FWHM of the Voigt fitting curve is 6277 Hz. The Lorentzian and Guassion linewidths of the spectrum are estimated to 50 Hz and 6250 Hz, respectively. The spectrum of this ultra-narrow-linewidth BEFL is dominantly contributed by the Guassian component. The obtained Lorentzian linewidth of 50 Hz agrees well with the previously measured value by heterodyne beat technique in [7].
The BEFL is also measured when the delay fiber is 50 km and 100 km. The measured spectra are shown in Fig. 3, together with that measured at 25 km delay fiber. The spectrum is broadened with the increase of the delay fiber. The 3-dB widths of the self-heterodyne spectra are 11050 Hz and 17000 Hz in the cases of 50 km and 100 km delay fibers, respectively. The spectrum broadening with the increase of the delay fiber is resulted from the dependence of the 1/f frequency noise on the delay time [8]. The measured linewidth of a certain laser is substantially upon the delay fiber used in the measurement and the level of the 1/f noise. The increase of the fiber delay broadens the Guassian lineshape of the BEFL. Some reports used short delay to reduce the adverse effect of the 1/f frequency noise so that the Lorentzian linewdith of lasers can be obtained [14]. This method only fits for the lasers with broad Lorenztian linewdith due to its limited resolution. The results in Fig. 3 also verify that due to the convolution of the 1/f frequency noise, the increase of the delay time does not improve the accuracy of the ultra-narrow-linewidth measurement in the heterodyne technique.
After applying the Voigt fitting procedure, the linewidths of the Voigt profile and the two separate components are obtained for the spectra measured with 50 km and 100 km, respectively. The results are shown in Table 1 together with those obtained when the delay fiber is 25 km. Interestingly, the measured spectrum is well fitted by the Guassian lineshape when the delay fiber is 100 km, as shown in Fig. 4. It can be inferred that the natural linewidth of the BEFL is actually much less than 50 Hz. Comparing the fitting results at different delay fibers, it can be found that the Lorentzian linewidth is decreased with the increase of the delay. The component of the spectrum due to the white frequency noise is exactly Lorentzian in the limit of large delay times. The increase of the delay fiber permits the white frequency noise to fully generate the Lorentzian lineshape. Insufficient delay time leads the Lorentzian part broadened and oscillatory [14]. It is true that longer fiber delay is more favorable for Lorentzian component measurement in Voigt fitting method.
Table 1. Estimated linewidths of the Voigt fitting and the direct measurement methods
Fig. 4 Self-heterodyne spectrum of the BEFL measured with 100 km delay fiber and its Guassion fitting curve.
The linewidth estimated from the direct 3-dB and 20-dB width measurements are also presented in Table 1, which are symbolized as Δν3-dB and Δν20-dB. Take the case of 25 km delay fiber for example, if we use the value of the 3-dB width of the spectrum for linewidth estimation, the obtained BEFL linewidth turns to 2600 Hz, which is far more than the value of 50 Hz obtained by the Voigt fitting. The 20-dB width of the spectrum is 16600 Hz. The linewidth of the BEFL is 830 Hz if estimated from the 20-dB width. The 20-dB measurement method indeed suppresses the broadening effect from the 1/f frequency noise to some extent. It is relatively more accurate than the 3-dB width measurement, since the effect of 1/f noise is not that pronounced away from the center. Nevertheless, the 20-dB width measurement still much overestimates the natural linewidth of the ultra-narrow-linewidth fiber laser, for the Lorentzian component is far smaller than Guassian component. The adverse effect from 1/f noise becomes even more severe with the increase of the delay fiber length. Shortening the delay fiber to several tens of meters is a possible way to reduce the effect from 1/f noise, but the instrument resolution is thus limited. Hence direct 3-dB measurement is applicable for broadband laser like semiconductor lasers with more than several megahertz Lorentzian linewidth. The Voigt fitting method realizes efficient filtering out of the 1/f noises. This method does not suffer adverse effect from long delays. It presents great superiority in ultra-narrow-linewidth measurements.
We have previously demonstrated that the linewidth of the BEFL is about 40 Hz through heterodyne beat technique. Considering the speciality of the BEFL, heterodyne beat technique can be used conveniently with two different Brillouin pump light injected into the cavity. But for other types of lasers, the main difficulty to use heterodyne technique is that two lasers must be used. And the linewidth of the reference laser must be narrower than the laser under test. Meanwhile, the frequency of the reference laser should be very close to the laser under test, which is limited by the finite bandwidth of the photoelectric detector. Contrarily, the Voigt fitting method is suitable for all types of lasers with a wide range of different frequencies.
5. Conclusion
In summary, the Voigt profile of self-heterodyne spectrum of lasers is studied. A simple but effective numerical Voigt fitting method is proposed. The Lorentzian lineshape of a laser can be accurately recovered from the measured self-heterodyne spectrum through this fitting procedure. Compared with the direct measurement method, it filters out the adverse effect of spectrum broadening from long delays due to the 1/f frequency noise. After applying Voigt fitting to the actual spectra of an ultra-narrow-linewidth BEFL, the obtained linewidth agrees well with that measured by heterodyne beat technique. It presents great superiority in ultra-narrow linewidth estimation. Meanwhile, this method is applicable for all types of lasers with a wide range of different frequencies and linewidths. It provides a powerful candidate for accurate ultra-narrow linewidth measurement.
Acknowledgment
This work is supported by the National Natural Science Foundation of China under Grant No. 61177073 and the Major Applied Basic Research Project in the Research Program of National University of Defense Technology under Grant No. ZDYYJCYJ20140701.
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