A novel methodology to directly pre- determine the relative wavelength response of DFB laser in wavelength modulation spectroscopy

A novel methodology to directly pre-determine the relative wavelength response (RWR) of a DFB laser, in terms of a combined current linearly scanned wavelength response and current modulated wavelength response (CMWR), in wavelength modulation spectroscopy (WMS) is presented. It is shown that the assessed RWR can be used to mimic the measured response with standard deviation of discriminations that are below 3.4 × 10cm under a variety of conditions. It is also shown that its performance supersedes two commonly used assessment models of the CMWR but is slightly worse than that of the third model, however with the benefit of solely using a single fitting parameter (the concentration) instead of more. When the novel method is applied to the assessment of CO2 concentration in a Herriot-type multipass cell by using the technique of calibration-free WMS, the results show that there is virtually no difference compared to that by use of the best of the other methods. It is concluded that the novel method is more robust and simplifies the retrieval process of gas concentration. © 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Due to its simplicity and calibration-free properties, DAS is one of the most versatile detection techniques of TDLAS.However, in its normal mode of operation, its detection sensitivity is generally limited to 10 −3 region, primarily because of the excess noise (often called 1/f noise) of the measurement system [13].A means of overcome this is to scan the wavelength of the laser rapidly, normally in the kHz range [12,14].Alternatively, WMS can be used [15].In both cases, the absorption information is encoded and detected at an audio frequency where there is less noise, whereby a detection sensitivity of 10 −5 -10 −6 can be reached.Such techniques are more applicable than DAS [12,13].
When applying WMS in the field, the concentration of target gas can be retrieved by measuring the peak value of the absorption profile or by fitting a model function to the detected signal.The latter shows generally a better sensitivity and stability than the former since the full information of the absorption is used and the offset drifts are better accounted for.Fitting is especially needed when the calibration-free (CF) WMS in 2f/1f detection mode is used [16].In order to perform a lineshape fitting, the accurate model of WMS signal is indispensable, which requires an accurate model of the absorption profile and a good knowledge about laser characteristics, including laser intensity and relative wavelength response (RWR).The RWR consists of current scanned wavelength response (CSWR) and current modulated wavelength response (CMWR).Among all the requirements, spectral model and laser intensity response can be predetermined.As we known, for WMS, the scan is normally performed by a triangle (or saw) wave, however a sinusoidal scan can be adopted in some special situations such as for fast measurement.In this paper, we only analyze the former situation, i.e. with linear scan.Under this condition, the CSWR can be directly measured by performing a polynomial fit to the laser relative wavelength as a function of linearly scanned injection current (or time).The wavelength response of the laser can be calibrated by use of an external etalon since the adjacent transmission peaks of etalon are separated by its fixed free spectral range (FSR).The positions of these peaks, as a marker, therefore can be used to record the CSWR and CMWR, respectively, when the laser wavelength is only scanned or together with modulation.However, for the measurement of CMWR, a high data acquisition rate (in the order of MHz for a dozen of kHz modulation) is needed to get all the narrow etalon transmission peaks, meanwhile the spurious peaks should be carefully picked out.Since this is non-trivial to perform with the required accuracy, it is of importance to find alternative means to assess the CMWR accurately in efficient ways [17,18].
For a semiconductor laser, the simplest description of the wavelength response, ( ) t ν , when a scanned and modulated current is injected to the laser, is [19]: where 0 ( ) t ν represents the CSWR, the second term on the right side corresponds to the CMWR, where, in turn, a is the modulation amplitude and f is the modulation frequency.Here, the modulation amplitude is considered to be constant, which follows from an assumption that there is a linear response between wavelength and current modulation.This model is therefore here referred to "the constant method".[19,20].
In reality, however, the response is not linear; for example, it was found that the wavelength modulation amplitude changed from 0.26 to 0.815 GHz/mA during a single scan for a specific AlGaInAsSb DFB laser [21].J. Chen found that the wavelength modulation amplitude is proportional to the 1st derivative of the CSWR.This implies that the wavelength response can be expressed as [16,22,23]: This is referred to "the first derivative method".This model established a connection between the CSWR and the CMWR, and the a is introduced to describe the wavelength response to the different scan and modulation frequencies.Later Z. Qu has applied this idea successfully to CF-WMS [16,22,23].
Recently, in order to further consider the nonlinear response to the modulation, G. Zhao empirically suggested that the CMWR consists of two parts: a 1 st harmonic component with a linearly time-dependent amplitude and a nonlinear 2 nd harmonic component with a constant amplitude that can be expressed as [24]  θ is the phase difference between the first and second harmonic of frequency modulation.This is here referred to "the Zhao's method".
In [24], G. Zhao et al. presented a comparison between these three models when fitted to an experimental spectrum.The results showed that the Zhao's method, due to its more completed description, obtained a smaller residual than others.
In WMS, the expression of CMWR is used to the spectral fitting process for retrieval of the gas concentration.For the three aforementioned models, in addition to the concentration, additionally 1 to 4 free parameters, (in these cases, a, b, c and 2 θ ) need to be assessed, which can complicate the fitting process and introduce errors in the concentration, particularly if parts of the free parameters are partially dependent.Although, the free parameters in Eqs.
(1)-( 3) can be pre-determined, i.e. determined by some fitting-based characterization procedure before the actual concentration assessments, this is a process that, in general, requires a significant amount of time since the parameters of CMWR will vary under different experimental conditions.Therefore, it is necessary to find a simple and straightforward way to directly pre-determine the CMWR without fitting parameters included.In order to make the readers read smoothly, the aforementioned models are referred to as "the traditional methods".
In this paper, we propose a novel methodology to directly pre-determine the RWR, here defined as a joint description of the CMWR and the CSWR, based on the fitted CSWR of a DFB laser.The CSWR can be measured by using a low finesse etalon in real time.This implies that there is no need to separately assess any CMWR.The method is verified under a few experimental conditions and compared to the three traditional methods [given by the Eqs.( 1) -( 3)].Then the method is applied to an assessment of the CO 2 concentration by the use of Herriott cell assisted CF-WMS.This novel method is expected to simplify the process of gas concentration assessment by use of WMS.

Theoretical analysis
In semiconductor lasers, following pumping by an injection current, the emission is obtained by recombination of the electrons and holes.However, a change of the injection current will not only affect the recombination rate, and thereby the output power, it will also affect the refractive index of the lasing medium.Such a change will therefore cause a combined variation of both the power and the wavelength of the laser output [17,18].In linearly scanned WMS, a triangle (or saw) wave with frequency in the Hz range and a sinusoidal wave with frequency in the tens kHz range are added to control the current of a DFB laser for scanning and modulation of the wavelength, respectively.
When no modulation is imposed, the injection current is given by ( ) 0 i t , which here is assumed to be linear with time.Under this situation, the RWR is a function of current (or time).The CSWR can be directly determined by a polynomial fit to the measured discrete frequency values calibrated by use of the transmission peaks of an external etalon and be expressed as where the ( 0,1, 2......, ) n A n = N are the polynomial coefficients.If also a modulation at a frequency of f is added, the injection current can be written as 0 ( ) where a i is the amplitude of the modulated current.
As is well known, there is not a constant relationship between the modulations of the wavelength and the current in a semiconductor laser.Instead, these can be related to each other by the use of a Fourier frequency dependent transfer function (which, in general, shows a pronounced low pass behavior).In order to consider this effect in the following analysis, it is suitable to define an effective current, ( ) e i t , that represents the part of the injection current that generates the wavelength variation as 0 ( ) ( ) cos( 2), e a a i t i t i ft where κ = κ a /κ 0 , where, in turn, κ a and κ 0 represent the amplitudes of the wavelength responses at the modulation and scan frequencies, respectively, while ψ a is the phase delay at frequency of f.In this case, the RWR can be described as a function of this effective current (and thereby time) as .
In order to compare the ability of this description to describe the CMWR with those of the three traditional ones, given by the Eqs.( 1) -(3), Eq. ( 7) can be rewritten [by the use of Eq. ( 6) and a Taylor series expansion] as where the expansion variable is the scan current.After some simple algebra, this expression can be rewritten as where 0 dt di is a constant that represents the reciprocal of the slope of the scan current.Since the CSWR, in general, can be described by a 2nd order polynomial in time, i.e. as and since the scan current is proportional to the time t, Eq. ( 9) can be expressed as Here, the second term is a constant and can solely produce a small offset of the optical frequency, which, for all practical purposes, can be neglected.If the third term is considered without its time dependence, Eq. ( 11) will revert to Eq. ( 1).If the entire third term is considered, it transposes to Eq. ( 2).If also the fourth term is included, it reverts to Eq. ( 3).This shows that all the transitional expressions have a solid base in processes in the laser, although included to various degrees.
More importantly, it shows that Eq. ( 11) constitutes a more general description of the RWR that provides a joint description of the CSWR and the CMWR.This implies that once the transfer function of the laser, i.e. the injection current and the CSWR, have been assessed, the CMWR is automatically given.This implies that no unknown fitting parameters need to be introduced when WMS is performed, which will reduce the risk for picking up of unknown effects.

Experimental setup
The experimental setup is shown in Fig. 1.A triangle wave, with a frequency of 32 Hz and an amplitude of 0.5 V, and a sine wave, with a frequency of 16 kHz and an amplitude of 0.35 V, are produced by a function generator (Tektronix, AFG3102).These are summed and sent into a laser driver (ILX Lightwave, LDC-3724C) to scan and modulate the wavelength of a DFB laser (NTT, NLK1L5EAAA).The output wavelength of the laser is around 1578 nm which covers the target transition of CO 2 .The laser output is split into two parts in a fiber beam splitter with a 1:1 power ratio.One beam passes through a homemade Herriott cell whose output is detected by photodetector (PD1, Thorlabs, PDA10CS-EC).The other beam passes through an etalon, with a FSR of 0.0449 cm −1 and a finesse of 200, and is detected by photodetector (PD2).Both detected signals are finally sent to the PC and recorded by the use of a 10 MHz, 12 bits data acquisition (DAQ) card (NI Corporation, PCI-6115).Data processing, including demodulation and signal fitting, is performed by a homemade LabVIEW program.In order to accurately determine the effective optical path length of the Herriott cell, a phase based laser ranging method is adopted [25].For this, a fiber coupled acousto-optic modulator (AOM, AA Opto Electronic, MT110-IIR20-Fio-PM0.5-J1-A) and the photodetector (PD3) (both in gray dashed boxes) were temporarily inserted to the beam path.By this methodology, the length of the cell could be assessed to 11.73 m with an uncertainty of 0.01 m.For the determination of the injection current to the DFB laser, a 10 ohms high precision resistor was connected in series between the laser driver and current input pin of laser and the voltage across it was monitored.The gas analyzed was a standard gas containing 3.98(1)% of CO 2 in N 2 , obtained from Beijing AP BAIF gases industry Co., Ltd.  Figure 2 shows, in the upper panel, the measured injection current to the DFB laser (by black dots) together with the fit (by red curve), and, in the bottom panel, the residual.For clarity, every tenth measured data point is displayed in upper panel.The insert shows a zoom of the region around 0.006s.The theoretical model was based on Eq. ( 5) and a 1st order polynomial was used to simulate the scan current.The fitted result was given by ( ) 0.0799 2.3423 0.0091 cos(2 ),

Measurement of the DFB injection current
where the modulation frequency f was 16 kHz.The lower panel shows that the residual is limited to 2% with no evident drift.This demonstrates that the injection current, ( ) 0 i t , solely has a linear dependence on time, as was assumed above.In order to get the effective current given by Eq. ( 6) in terms of the a κ , 0 κ and a ψ parameters, the transfer function of the DFB laser (from current to optical frequency) needs to be assessed.This assessment followed the method suggested by J. L. Hall in which an etalon is used to calibrate the optical frequency [26].In short, the transfer function was measured by applying a sinusoidal modulation to the injection current and observing the etalon transmission, with the transmitted modes serving as frequency markers.The amplitude response was measured by adjusting the amplitude of the sinusoidal modulation so as to yield a known peak-to-peak laser frequency response.The phase shift was measured by comparing where the turning point of the laser frequency modulation was with respect to the reference sinusoidal signal.The measured results are shown in Fig. 3, which clearly shows a low-pass response.The upper panel shows that the amplitude response (for simplicity normalized to unity at 1 Hz) at 32 and 16 kHz, at which 0 κ and a κ are defined, is 0.962 and 0.418, respectively.The lower panel shows that the phase lag, a ψ , is 0.553 rad.This implied that the effective current, defined by Eq. ( 6), could be assessed to

Determination of the κ and the a ψ
The phase lag of 0.065 at a scan frequency of 32 Hz is not considered in this equation since it is automatically included by the direct measurement of CSWR.

Measurement of the CSWR
In order to determine the CSWR of the DFB laser under a certain set of working conditions, the injection current is scanned by a triangle wave, which can be measured in real time by the voltage over the inserted 10 ohms resistor.The CSWR, obtained by a 2nd order polynomial fit to the transmitted peak positions of an external etalon, is shown in Fig. 4 (expressed in wave numbers).The black squares show the measured data at a laser temperature of 25.2 °C and a current offset of 99.23 mA.The zero position of the relative wavelength scale lies at the current of 117 mA.The result, shown by the red curve, was found to be with the adjusted R square close to unity (0.9998).By inserting the effective current of Eq. ( 13) to Eq. ( 14), the RWR of the DFB laser can be directly obtained as given in Eq. ( 8).Such simulations are shown by the red curves in Fig. 5.

Simulation of the RWR
The measured RWRs are shown by the black dots.The four panels represent four different conditions, as given in Table 1.The pertinent process parameters are listed in Table 2. From this, it can be seen that the standard deviation of the discriminations, σ, are, in all cases, smaller than 3.4 × 10 −3 cm −1 , which manifests the efficiency of this novel methodology.According to the analyses in section 2, the theoretical model has, ideally, nothing related to the laser working temperature and the current offset.However different current offsets or scanning ranges will introduce different errors to the fitting measurement of CSWR, thereby could cause different simulation error for RWR, which accounts for the little difference between the standard deviations of conditions 1 and 2. On the other hand, a larger modulation amplitude will also pick up more errors from the determination process of CSWR, which verified by the results of condition 3 and 4. In order to compare the performance of this novel method with that of the traditional three methods, i.e. the constant method, the first derivative method and the Zhao's method, an experiment based on the technique of CF-WMS is performed.In this process, the wavelength modulated direct absorption (WMDA) signal is measured and sent to the computer and then demodulated by a digital lock-in amplifier to get the measured 2f/1f signal according to the procedure given in Ref [16].This signal was then fitted by a theoretical lineshape with three CMWR models given by Eqs. ( 1)-( 3) with a, b, c θ 2 and gas concentration being free parameters.When analyzed by the method presented in this work, only the gas concentration was considered as a free parameter in the fit.In this process, the laser was working under the condition 1 in Table 1.

RWR determined by the traditional methods
In order to compare the performance of the three traditional methods, Fig. 6 shows the comparisons between the measured data (solid squares) with the obtained theoretical results (solid curves) for (a) the constant method, (b) the first derivative method, and (c) the Zhao's method.The figure clearly shows that the standard deviation of the discrimination becomes better as more terms for CMWR are considered, i.e. from (a), via (b), to (c).[27].The Voigt function is adopted to describe the absorption lineshape at the pressure of close 1 atm.The modulation current for this measurement is 9.1 mA, corresponding to the modulation index of 1.6.The parameter C displayed in each figure shows the retrieved concentration of CO 2 for that particular method.All the fitting parameters in Fig. 7(b)-7(d) are shown in Table 3.In this experiment, the Herriott cell was filled with CO 2 gas with a pressure of 0.92 atm, which corresponds to the local atmospheric pressure in Taiyuan.The concentration of the standard gas is 0.0398(1) which is verified by the supplier.

Comparison between the performance of the novel and the traditional methodologies to retrieve the CO 2 concentration
As can be seen in Fig. 7, the method presented in this work displays a residual that is smaller than both the constant method and the first derivative method.The cause of this is that none of these methods includes a second harmonic component of the modulation frequency in the CMWR.However, the residual of the Zhao's method is still smaller than that of the novel method presented in this work.This is expected since the two methods are based on the same fundamental model, given by the Eqs.( 3) and (11), and the method presented here does not make use of any free parameter for the line shape fitting, it only has the concentration as the free parameter, while the Zhao's method uses a fitting routing that has four free parameters.However, when it comes to assessment of gas concentrations, it is not clear that the Zhao's method would provide a more accurate assessment than the method presented in this work.The reason is that a fit based on a multitude of free parameters has a large ability to compensate for any temporarily appearing noise or disturbance; any such noise or disturbance (e.g.appearing in the form of background signals) might, by such a method, incorrectly and unnoticed be interpreted as gas.This risk is minimized for a technique that has a minimum of free fitting parameters.Yet other advantages of the method presented here are that it simplifies the retrieval process of gas concentration and that it is more robust.
Over all, the descriptions of CSWR for four methods are identical since they are from the same group of data.However the CSWR for this new method is as a function of injection current, but those of others are as a function of time since a linear current scan is performed.The fitting values of the free parameters for each traditional method are shown in Table 3 and the comparisons with the measured data for RWRs are shown in Fig. 6.The comparison for this new method is shown in Fig. 5(a).As can be concluded that the determined RWR descriptions of the constant method and the first derivative method are worse than the new method, but that of the Zhao's method has the same standard deviation with the new method, which means that the first two methods introduce more errors to the concentration retrieval.On the other hand, the Zhao's method will also possibly introduce more errors to the concentration retrieval than this new provided method since four more free parameters are included.

Conclusion
We propose a novel methodology to accurately and directly pre-determine the relative wavelength response (RWR) of a DFB laser in linearly scanned wavelength modulation spectroscopy (WMS) in general, and CF-2f/1f WMS in particular, based on its current scan wavelength response and the laser transfer function.The methodology is suitable for any laser working condition, including an arbitrary laser temperature, current offset, modulation frequency, etc.The performance to assess the RWR of this new method is compared with three other methods, above referred to as the constant method, the first derivative method, and the Zhao's method.The accuracy of the novel method for retrieval of the RWR is better than that of the two first methods but slightly worse than that of the Zhao's method.However, a fit with a smaller residual might not necessarily provide a more accurate assessment of gas concentrations; this is supported by the fact that there is virtually no difference between the ability of the novel method and Zhao's method regarding assessment of gas concentration.Most importantly, however, this novel method does not make use of any free parameter for line shape fitting in WMS, which simplifies the retrieval process of gas concentration.It is therefore expected to be more resistant to disturbances and thereby sturdier than the other.By this, it is assumed it can improve the accuracy and save the time for the trace gas detection based on the technique of linearly scanned WMS.For the sinusoidal scanned WMS, according to the theoretical analyses, the new provided method should show the same accuracy compared to that in the linearly scanned WMS.We will focus on this situation and field applications in our next work.Meanwhile, based on the analyses from Eq. ( 9) to (11), the first derivative method cannot be applied to the sinusoidal scanned WMS since the high order derivatives of 0 ( ) t ν is neglected in this situation.

Fig. 2 .
Fig. 2. Measured and fitted injection current to the DFB laser as functions of time.

Fig. 3 .
Fig. 3.The transfer function from injection current to wavelength of the DFB laser; (a) the amplitude response; (b) the phase delay.

Fig. 4 .
Fig. 4. The measured and fitted CSWR as functions of injection current.

Fig. 5 .
Fig. 5.The measured and simulated RWRs at four laser working conditions; the lower panels show the corresponding discriminations.
The conditions for the data in panel (b) differs from those in panel (a) by the laser temperature and current offset, while those of panel (c) and panel (d) differs also with regards to the scan frequency, modulation frequency and amplitude.The inset in each upper panel shows the fractional zoom, highlighted by the red box.The lower panel shows the discriminations between the measured data and the simulated curves.

Fig. 6 .
Fig. 6.The measured (black dots) and retrieved theoretical (red line) RWR by the three traditional methods; the lower panel shows the corresponding deviation.

Fig. 7 .Figure 7
Fig. 7.The measured and fitted 2f/1f signals of the CO 2 absorption line based on four different determination methods of RWR, i.e.(a) the new method, (b) the Constant method, (c) The First derivative method, (d) The Zhao's method; the lower panel shows the corresponding fitting residual.Table 3. Fitting results of free parameters of three traditional methods

National
Key R&D Program of China (Grant No. 2017YFA0304203), Changjiang Scholars and Innovative Research Team in University of Ministry of Education of China (Grant No. IRT_17R70), the Fund for Shanxi "1331 Project" Key Subjects Construction, 111 project (Grant No. D18001), the National Natural Science Foundation of China (Grant Nos.61675122, 61875107, 61875108, 11704236, 61475093 and 61775125), the Research Project