CN117538021A - Narrow laser linewidth measurement method based on short fiber time delay self heterodyne interference - Google Patents
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Abstract
The invention belongs to the technical field of spectrum measurement, and discloses a narrow laser linewidth measurement method based on short fiber delay self heterodyne interference, which comprises the following steps: s1, acquiring a coherent envelope spectrum of laser to be tested based on a short fiber delay self-heterodyne interference device; s2, determining delay time according to the wave trough orders and the corresponding frequencies in the coherent envelope spectrum; s3, calculating an estimated laser linewidth; s4, calculating a coherent envelope demodulation spectrum according to the coherent envelope spectrum, the delay time and the estimated laser linewidth, performing Lorentz fitting, and calculating corresponding R 2 Coefficients; s5, taking a plurality of laser linewidths as new estimated laser linewidths at equal intervals on two sides of the estimated laser linewidth, repeating the step S4, and calculating corresponding R under different estimated laser linewidth values 2 Coefficients; s6, according to the estimated laser linewidth and R 2 And determining the corresponding relation of the coefficients and the actual line width value. The invention greatly reduces the measurement error and can be widely usedThe method is widely applied to line width measurement of narrow laser and ultra-narrow laser.
Description
Technical Field
The invention belongs to the technical field of spectrum measurement, and relates to a method for precisely measuring laser linewidth, in particular to a narrow laser linewidth measuring method based on short fiber delay self heterodyne interference.
Background
Narrow linewidth lasers have good coherence and are in need of many applications, such as optical fiber communication sensing and quantum precision measurement, so precision measurement of laser linewidth is the primary task for applications based on narrow linewidth lasers.
The traditional laser linewidth measurement method has limitation on spectral resolution, for example, the resolution of a commercial spectrometer with best performance working in visible light and near infrared bands is only about 0.01nm (the corresponding frequency resolution is of the order of GHz), and the resolution of a Fabry-Perot Luo Saomiao interferometer can only reach the MHz level. With the development of laser technology, a narrow linewidth laser of a kHz level is realized at present, and an ultra-narrow linewidth laser of a Hz level or even an mHz level can be obtained through a certain frequency locking technology. Therefore, the traditional measuring method is not suitable for the laser linewidth measurement requirement of the new generation of narrow linewidth lasers, and the new measuring method is required to provide more accurate and reliable technical means for performance evaluation and application of the narrow linewidth lasers.
The time delay self heterodyne interferometry based on optical fibers is a laser linewidth measurement method commonly used at present. The basic idea of the method is to divide the laser beam to be measured into two beams, so that the time delay of the two beams is far longer than the laser coherence time, and the Lorentz formula is adopted to fit the beat frequency signals of the two beams to obtain the laser linewidth. To ensure that the beat signal has a lorentzian line shape, this technique typically requires a fiber delay that is about 6 times longer than the coherence time of the laser. For example, for a laser with a linewidth of kHz, the delay fiber length needs to exceed 1000 km, which obviously makes the whole measurement scheme difficult to implement, so how to implement accurate measurement of the laser linewidth with a shorter fiber is of more practical value.
In 1986, L.E. Richter et al [ IEEE J.Quantum Electron 22,2070 (1986) ] have theoretically analyzed and deduced the principle of measuring laser linewidth by short fiber delay self heterodyning, indicating that a beat signal can form a coherent envelope spectrum under the condition that the fiber delay is less than the coherence time of the laser. How to determine the laser linewidth according to the coherent envelope spectrum of the beat signal becomes the key of the short fiber delay self heterodyne method laser linewidth measurement technology. The widely adopted method is to compare the spectral amplitude of a specific point of the coherent envelope spectrum with a theoretical value and calculate to obtain the laser linewidth [ patent CN105571830B (2017), IEEE Photonics technologies letters 28,759 (2016), sci.reports 7,41988 (2017) ]. This approach is more demanding on the fiber length because the fiber is too short with the effect of the noise floor and too long with 1/f noise. Moreover, this method only considers the amplitude difference of a few special points, so that there is a large measurement error.
Disclosure of Invention
The invention overcomes the defects existing in the prior art, and solves the technical problems that: the narrow laser linewidth measuring method based on the short fiber time delay self heterodyne interference is provided to realize accurate measurement of the narrow laser linewidth.
In order to solve the technical problems, the invention adopts the following technical scheme: a narrow laser linewidth measurement method based on short fiber time delay self heterodyne interference comprises the following steps:
s1, acquiring a coherent envelope spectrum of laser to be tested based on a short fiber delay self-heterodyne interference device;
s2, determining delay time according to the wave trough orders and the corresponding frequencies in the coherent envelope spectrum;
s3, calculating an estimated laser linewidth according to the delay time and the contrast between the wave crest and the wave trough in the coherent envelope spectrum;
s4, calculating a coherent envelope demodulation spectrum according to the coherent envelope spectrum obtained in the step S1, the delay time and the estimated laser linewidth obtained in the step S2, performing Lorentz fitting, and calculating corresponding R 2 Coefficients;
s5, taking a plurality of laser linewidths as new estimated laser linewidths at equal intervals at two sides of the estimated laser linewidth obtained in the step S3, repeating the step S4, and calculating corresponding R under different estimated laser linewidth values 2 Coefficients;
s6, according to the estimated laser linewidth value and R 2 And determining the actual linewidth value of the laser to be measured according to the corresponding relation of the coefficients.
The short fiber time delay self heterodyne interference device comprises: the device comprises a light splitter, an acousto-optic modulator, a transmission optical fiber, a delay optical fiber, an optical fiber coupler, a detector and a spectrum analyzer;
the laser to be measured is divided into two beams by the beam splitter, one beam enters the transmission optical fiber after being shifted by the acousto-optic modulator, the other beam enters the delay optical fiber for time delay, the other ends of the transmission optical fiber and the delay optical fiber are connected with the optical fiber coupler, the two beams of light are subjected to reverse interference in the optical fiber coupler, and interference signals are detected by the detector and then sent to the spectrum analyzer to obtain the coherent envelope spectrum of the laser to be measured.
The length of the transmission optical fiber is 0-1 m, and the length of the delay optical fiber is 10-100 km.
In the step S2, the method for determining the delay time specifically includes:
acquiring frequencies corresponding to wave troughs of all orders;
performing linear fitting on the trough orders and the corresponding frequencies;
the inverse of the slope of the linear fit is taken as the delay time.
In the step S3, the calculation formula for estimating the line width of the laser is:
wherein DeltaS represents the contrast ratio of the first-order wave crest and the m-order wave trough in the coherent envelope spectrum, and tau d The delay time is represented, and Δf represents the estimated laser linewidth.
Where l=m=2.
In the step S4, a calculation formula of the coherent envelope demodulation spectrum is as follows:
wherein S is 1 Represents the coherent envelope demodulation spectrum, S 2 Representing the periodic modulation spectrum, S representing the coherent envelope spectrum.
In the step S4, the calculation formula of the periodic modulation spectrum is as follows:
wherein Δf represents the estimated laser linewidth τ d Indicating the delay time and f indicating the relative frequency of the beat signal.
In the step S4, the lorentz fitting formula is:
wherein S is 1 Represents the coherent envelope demodulation spectrum, P 0 Representing the power of the signal, Δf represents the predicted laser linewidth, and f represents the relative frequency of the beat signal.
In the step S6, the specific method for determining the actual line width value of the laser to be measured is as follows:
according to each estimated laser linewidth value and corresponding R 2 Coefficient values are subjected to polynomial function fitting to obtain a fitting curve;
r in the fitting curve 2 The estimated laser linewidth corresponding to the maximum coefficient is the actual linewidth of the laser to be measured.
Compared with the prior art, the invention has the following beneficial effects:
the invention provides an optical measurement method of narrow laser linewidth based on short optical fiber delay, which is characterized in that a Lorentz linear spectrum hidden in a coherent envelope spectrum is recovered by demodulating the short optical fiber delay self-heterodyne coherent envelope spectrum, and the linewidth corresponding to the optimal Lorentz linear spectrum fitting is selected as the laser linewidth. Compared with the method in the prior art that only the special point of the coherent envelope spectrum is compared with the theoretical value, the method has the advantages that the requirement on the length of the optical fiber is not high, the line width measurement is realized by integrally considering the coherent envelope demodulation spectrum, the measurement error is greatly reduced, and the method can be widely applied to the line width measurement of narrow laser and ultra-narrow laser.
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FIG. 1 is a schematic diagram of a measurement device used in a narrow laser linewidth measurement method based on short fiber delay self heterodyne interference according to an embodiment of the present invention;
FIG. 2 is a coherent envelope spectrum of a laser to be measured obtained in an embodiment of the present invention;
FIG. 3 shows the number of waves Gu Jie and the corresponding frequency relationship in an embodiment of the present invention;
FIG. 4 is a graph showing the relationship between the contrast of the coherent envelope spectrum and the linewidth of the laser beam obtained in the embodiment of the present invention;
FIG. 5 is a schematic diagram of a coherent envelope demodulation spectrum and a Lorentz fitting curve corresponding to the coherent envelope demodulation spectrum according to an embodiment of the present invention;
FIG. 6 is a graph of R obtained in the examples of the present invention 2 The dependence of the coefficient on the estimated laser linewidth;
in the figure: 1 is laser to be measured, 2 is a beam splitter, 3 is an acousto-optic modulator, 4 is a transmission optical fiber, 5 is a delay optical fiber, 6 is an optical fiber coupler, 7 is a detector, and 8 is a spectrum analyzer.
Detailed Description
For the purpose of making the objects, technical solutions and advantages of the embodiments of the present invention more clear, the technical solutions in the embodiments of the present invention will be clearly and completely described below, and it is apparent that the described embodiments are some embodiments of the present invention, but not all embodiments; all other embodiments, which can be made by those skilled in the art based on the embodiments of the invention without making any inventive effort, are intended to be within the scope of the invention.
Example 1
The short fiber time delay self heterodyne coherent envelope spectrum is obtained based on beat frequency signals of two beams of partially coherent light interference, and the power spectrum function S can be expressed as follows:
S(f,Δf)＝S 1 ’S 2 ； (1)
wherein:
p in the formula 0 For the power of the beat signal, Δf0 is the laser linewidth, f=f 1 ―f 0 Is the relative frequency (f 1 And f 0 The frequency of the beat signal and the radio frequency driving the acousto-optic modulator, respectively). τ d For the time delay of one light propagation path of two light beams relative to the other light propagation path, it is determined by the length L of the delay fiber and the refractive index n of the fiber core, and the expression is tau d =nl/c (c is the speed of light in vacuum).
From the formulas (1) to (3), the total power spectrum S is Lorentz spectrum S 1 ' periodic modulation Spectrum S 2 Is a product of (a) and (b). Obviously Lorentz spectrum S 1 ' can be expressed as
Wherein the total power spectrum S can be obtained by measuring with a spectrometer, and modulating the signal S 2 Two parameters are included. One parameter is the delay time τ d Another parameter is the laser linewidth Δf0, which is also contained in S 1 In the expression, it is not yet determinable.
Based on the above, the embodiment of the invention provides a narrow laser linewidth measurement method based on short fiber delay self heterodyne interference, which focuses on the lorentzian line shape of coherent envelope demodulation spectrum: when the estimated laser linewidth is closest to the true value, the coherent envelope demodulation spectrum has the best fitting with the lorentz formula. The specific operation is as follows: by determining coefficients (i.e. R 2 Coefficient) to quantify the lorentz fitting degree of the obtained coherent envelope demodulation spectrum, and drawing the dependence relationship of the determined coefficient on a series of estimated laser linewidths, wherein the laser linewidth corresponding to the maximum determined coefficient is the laser linewidth to be measured. When it is needed, the series of estimated laser linewidth ranges should cover the laser linewidth corresponding to the maximum determination coefficient.
Specifically, the embodiment of the invention comprises the following steps:
s1, acquiring a coherent envelope spectrum of laser to be tested based on a short fiber delay self-heterodyne interference device.
As shown in fig. 1, the short fiber time delay self heterodyne interference device used in the present invention includes: a beam splitter 2, an acousto-optic modulator 3, a transmission optical fiber 4, a delay optical fiber 5, an optical fiber coupler 6, a detector 7 and a spectrum analyzer 8;
the laser 1 to be measured is divided into two beams by the beam splitter 2, one beam enters the transmission optical fiber 4 after being shifted in frequency by the acousto-optic modulator 3, the other beam enters the delay optical fiber 4 for time delay, the other ends of the transmission optical fiber 4 and the delay optical fiber 5 are connected with the optical fiber coupler 6, the two beams of light are subjected to reverse interference in the optical fiber coupler 6, and interference signals are detected by the detector 7 and then sent to the spectrum analyzer 8 to obtain a coherent envelope spectrum of the laser to be measured.
Specifically, in this embodiment, the beam splitter 2 is composed of a half-wave plate (λ/2) and a polarization beam splitter Prism (PBS). The length of the transmission optical fiber 4 is 0-1 m, and the length of the delay optical fiber 5 is 10-100 km.
Specifically, in this embodiment, two beams of light after passing through the beam splitter 2 are diffracted by the acousto-optic modulator 3, the zero-order light generated by diffraction is not blocked, and the positive first-order light (or the negative first-order light may be used) is transmitted to the transmission optical fiber 4 through two mirrors and a coupling lens. The frequency offset of the positive primary light with respect to the incident light is determined by the driving frequency of the acousto-optic modulator 3, which is equal to 80MHz in the embodiment. The other beam of light is directly transmitted to the delay optical fiber 5 through two reflectors and a coupling lens, and the length of the delay optical fiber 5 is longer. If the laser linewidth is measured by using the traditional delay self-heterodyne interferometry, the delay time generated by the delay fiber is required to be far longer than the laser coherence time. For example, assuming a laser linewidth of 100Hz and a fiber optic medium refractive index of 1.5, the required delay fiber length would be up to 2000km. The length of the delay fiber 5 in the device can be only 1/100 of that of the traditional fiber delay method, and the length of the fiber used in the embodiment is 20.33km. Two beams of light are coupled into the same detector 7 by using a1×2 optical fiber coupler 5 to detect photocurrent of beat frequency signals, and then connected to a spectrum analyzer 8 to obtain coherent envelope spectrum of laser light to be detected.
Specifically, the detector 7 used in this example was model 1554-B from New Focus, and the spectrum analyzer 8 was model FSVA13 from Rode and Schwarz.
As shown in fig. 2, the coherent envelope spectrum of the laser to be measured obtained by the spectrum analyzer 8 in the embodiment of the present invention is a typical short fiber delay self-heterodyne coherent envelope spectrum. For ease of display over the spectral scale, fig. 2 shifts the values on the abscissa by the drive frequency of the AOM (80 MHz). The spectral intensities were normalized with respect to a maximum near the center, with the ordinate units being logarithmic units. In the figure, there are a plurality of peaks and valleys, the peaks and valleys increase in order from the center to the two sides, and the contrast between the second peak and the second valley is marked, in this embodiment, the contrast between the second peak and the second valley is 18.6dB.
S2, determining delay time according to the wave trough orders and the corresponding frequencies in the coherent envelope spectrum.
Specifically, in the embodiment of the present invention, in the step S2, the method for determining the delay time specifically includes:
acquiring frequencies corresponding to wave troughs of all orders;
performing linear fitting on the trough orders and the corresponding frequencies;
and obtaining the slope of the linear fitting, and calculating delay time, wherein the delay time is the reciprocal of the slope.
From the correlation envelope spectrum in fig. 2, the order of each trough and peak, and its corresponding frequency, can be determined. A linear relationship of the trough order to the corresponding frequency can be obtained from fig. 2, as shown in fig. 3. The linear dependence of the valley order on the corresponding frequency reflects the periodic nature of the modulation spectrum. The fiber delay time can be deduced from the slope fitted by the linear equation. In this embodiment, the slope k=1/τ obtained by fitting d Corresponding delay time τ = 10049.8 ±6.4Hz d 99.504 + -0.063 μs.
In this embodiment, the delay time is directly obtained through the coherent envelope spectrum, so that compared with the scheme of calculating the delay time through the specific length of the optical fiber and the refractive index of the optical fiber medium in the prior art, the operation is simpler, and the result is more accurate.
S3, calculating the estimated laser linewidth according to the delay time and the contrast of the wave crest and the wave trough in the coherent envelope spectrum.
In the step S3, the calculation formula for estimating the line width of the laser is:
wherein DeltaS represents the contrast ratio of the first-order wave crest and the m-order wave trough in the coherent envelope spectrum, and tau d The delay time is represented, and Δf represents the estimated laser linewidth. The parameter i=2, 3, 4..m=1, 2, 3..the orders of peak and trough positions, respectively. The smaller the order, the closer to the center of the spectrum, and the larger the order, the further from the center of the spectrum. Because there is a large disturbance near the center of the spectrum, and the high order may be affected by the noise floor, the present embodiment uses the contrast Δs between the second peak and the second trough (i.e. l=m=2) to obtain the estimated laser linewidth.
As shown in FIG. 4, the delay time τ in the present embodiment is shown d In the known case, according to the contrast Δs between the second peak and the second trough in the coherent envelope spectrum and the relation curve between the laser linewidth, according to the contrast Δs=18.6 dB observed in fig. 3, the corresponding estimated laser linewidth is 77.5Hz. In this embodiment, the estimated value is smaller than the actual value due to the combination of the lorentz spectrum and the periodic modulation spectrum.
S4, calculating a coherent envelope demodulation spectrum according to the coherent envelope spectrum obtained in the step S1, the delay time and the estimated laser linewidth obtained in the step S2, performing Lorentz fitting on the coherent envelope demodulation spectrum, and calculating corresponding R 2 Coefficient (R) 2 The coefficients are determined coefficients).
Specifically, in the step S4, the calculation formula of the coherent envelope demodulation spectrum is:
wherein S is 1 Represents the coherent envelope demodulation spectrum, S 2 Representing the periodic modulation spectrum, S representing the coherent envelope spectrum. Periodic modulation spectrum S 2 The calculation formula of (2) is as follows:
wherein Δf represents the estimated laser linewidth τ d Represents the delay time, f represents the relative frequency of the beat signal, f=f 1 -f 0 。f 1 And f 0 The frequency of the beat signal and the frequency shift frequency of the acoustic optical modulator 3 are represented, respectively.
Specifically, in the step S4, the lorentz fitting formula is:
wherein S is 1 Represents the coherent envelope demodulation spectrum, P 0 Representing the power of the signal, Δf represents the predicted laser linewidth, and f represents the relative frequency of the beat signal.
By Lorentz fitting the coherent envelope demodulation spectrum, the corresponding R can be obtained simultaneously 2 Coefficients. R is R 2 The coefficients are between 0 and 1, reflecting the deviation of the data from the fitting formula, the larger the value, the closer the data to the fitting formula.
As shown in fig. 5, the coherent envelope demodulation spectrum obtained in the embodiment of the present invention and the lorentz fitting curve corresponding to the coherent envelope demodulation spectrum are shown. The fitting result obtained in this example is shown in the black curve in the figure, corresponding R 2 The coefficient is equal to 93.935%.
S5, taking a plurality of laser linewidths as new estimated laser linewidths at equal intervals at two sides of the estimated laser linewidth obtained in the step S3, repeating the step S4, and calculating corresponding R under different estimated laser linewidth values 2 Coefficients.
S6, according to the estimated laser linewidth value and R 2 And determining the actual linewidth value of the laser to be measured according to the corresponding relation of the coefficients.
In the step S6, the specific method for determining the actual line width value of the laser to be measured is as follows:
according to each estimated laser linewidth value and corresponding R 2 Coefficient values are subjected to polynomial function fitting to obtain a fitting curve;
r in the fitting curve 2 The estimated laser linewidth corresponding to the maximum coefficient is the actual linewidth of the laser to be measured.
In this example, R is plotted in the (30-200) Hz range at intervals of 10Hz 2 The dependence of the coefficients on the estimated laser linewidth is shown in fig. 6. The dependency shows R in this range 2 The coefficients have a maximum value, and the dependence is fitted by using a polynomial function, so that the optimal lorentz spectrum is obtained at the position of Δf=104.6Hz, and the value corresponds to the actual linewidth of the laser to be measured.
Finally, it should be noted that: the above embodiments are only for illustrating the technical solution of the present invention, and not for limiting the same; although the invention has been described in detail with reference to the foregoing embodiments, it will be understood by those of ordinary skill in the art that: the technical scheme described in the foregoing embodiments can be modified or some or all of the technical features thereof can be replaced by equivalents; such modifications and substitutions do not depart from the spirit of the invention.
Claims (10)
1. A narrow laser linewidth measurement method based on short fiber time delay self heterodyne interference is characterized by comprising the following steps:
s1, acquiring a coherent envelope spectrum of laser to be tested based on a short fiber delay self-heterodyne interference device;
s2, determining delay time according to the wave trough orders and the corresponding frequencies in the coherent envelope spectrum;
s3, calculating an estimated laser linewidth according to the delay time and the contrast between the wave crest and the wave trough in the coherent envelope spectrum;
s4, calculating a coherent envelope demodulation spectrum according to the coherent envelope spectrum obtained in the step S1, the delay time and the estimated laser linewidth obtained in the step S2, performing Lorentz fitting, and calculating corresponding R 2 Coefficients;
s5, taking a plurality of laser linewidths as new estimated laser linewidths at equal intervals at two sides of the estimated laser linewidth obtained in the step S3, repeating the step S4, and calculating corresponding R under different estimated laser linewidth values 2 Coefficients;
s6, according to the estimated laser linewidth value and R 2 And determining the actual linewidth value of the laser to be measured according to the corresponding relation of the coefficients.
2. The method for measuring the line width of the narrow laser based on the short fiber time delay self heterodyne interference according to claim 1, wherein the short fiber time delay self heterodyne interference device comprises: the device comprises a beam splitter (2), an acousto-optic modulator (3), a transmission optical fiber (4), a delay optical fiber (5), an optical fiber coupler (6), a detector (7) and a spectrum analyzer (8);
the laser to be measured (1) is divided into two beams by the beam splitter (2), one beam enters the transmission optical fiber (4) after being subjected to frequency shift by the acousto-optic modulator (3), the other beam enters the delay optical fiber (4) for time delay, the other ends of the transmission optical fiber (4) and the delay optical fiber (5) are connected with the optical fiber coupler (6), the two beams of light are subjected to anti-interference in the optical fiber coupler (6), and interference signals are detected by the detector (7) and then sent to the spectrum analyzer (8) to obtain coherent envelope spectrum of the laser to be measured.
3. The method for measuring the line width of the narrow laser based on the short fiber time delay self heterodyne interference according to claim 2, wherein the length of the transmission fiber (4) is 0-1 m, and the length of the delay fiber (5) is 10-100 km.
4. The method for measuring the line width of the narrow laser beam based on the short fiber time delay self heterodyne interference according to claim 1, wherein in the step S2, the method for determining the delay time is specifically as follows:
acquiring frequencies corresponding to wave troughs of all orders;
performing linear fitting on the trough orders and the corresponding frequencies;
the inverse of the slope of the linear fit is taken as the delay time.
5. The method for measuring the line width of the narrow laser beam based on the short fiber time delay self heterodyne interference according to claim 1, wherein in the step S3, the calculation formula for estimating the line width of the laser beam is as follows:
wherein DeltaS represents the contrast ratio of the first-order wave crest and the m-order wave trough in the coherent envelope spectrum, and tau d The delay time is represented, and Δf represents the estimated laser linewidth.
6. The method for measuring the line width of the narrow laser based on the short fiber time delay self heterodyne interference as claimed in claim 5, wherein l=m=2.
7. The method for measuring the line width of the narrow laser beam based on the short fiber time delay self heterodyne interference according to claim 1, wherein in the step S4, a calculation formula of a coherent envelope demodulation spectrum is as follows:
wherein S is 1 Represents the coherent envelope demodulation spectrum, S 2 Representing the periodic modulation spectrum, S representing the coherent envelope spectrum.
8. The method for measuring the line width of the narrow laser beam based on the short fiber time delay self heterodyne interference according to claim 7, wherein in the step S4, a calculation formula of the periodic modulation spectrum is as follows:
wherein Δf represents the estimated laser linewidth τ d Indicating the delay time and f indicating the relative frequency of the beat signal.
9. The method for measuring the line width of the narrow laser beam based on the short fiber time delay self heterodyne interference according to claim 1, wherein in the step S4, the lorentz fitting formula is as follows:
wherein S is 1 Represents the coherent envelope demodulation spectrum, P 0 Representing the power of the signal, Δf represents the predicted laser linewidth, and f represents the relative frequency of the beat signal.
10. The method for measuring the line width of the narrow laser beam based on the short fiber time delay self heterodyne interference according to claim 1, wherein in the step S6, the specific method for determining the actual line width value of the laser beam to be measured is as follows:
according to each estimated laser linewidth value and corresponding R 2 Coefficient values are subjected to polynomial function fitting to obtain a fitting curve;
r in the fitting curve 2 The estimated laser linewidth corresponding to the maximum coefficient is the actual linewidth of the laser to be measured.
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2023
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