## Abstract

Recently a technique to optically eliminate the background residual amplitude modulation in 1f wavelength modulation spectroscopy was demonstrated, where perfect elimination throughout the scan range was not achieved due to the wavelength-dependence of couplers and that of the laser intensity modulation. This paper theoretically analyzes the technique and experimentally demonstrates that the elimination can be perfect for one of three possible experimental configurations, making this important for potential applications with some recently-developed laser sources. For the other configurations a non-zero background slope is predicted, experimentally verified, and the anomalous nature of signals is thereby explained. A common signal normalization method is devised that is independent of the signal slope, a fact that is important for industrial deployment of such systems.

©2010 Optical Society of America

## 1. Introduction

Near infrared (NIR) tunable diode laser spectroscopy (TDLS) [1–9], is a well-established technique for trace gas detection as well as quantitative analysis of various gas parameters that are of significant importance in industrial process control and environmental sensing applications. The technique essentially involves tuning the wavelength of a spectrally narrow probe laser across a strong, well-isolated absorption feature of a target gas and monitoring the transmitted intensity as a function of the wavelength. This basic approach is known as direct detection. The numerous advantages of this method along with the intrinsic adaptability has resulted in a continued interest in this field that in turn has led to great refinement of the theoretical understanding and practical implementation of this method. The availability of reliable, frequency-agile, and low-cost diode lasers, high-sensitivity photo-detectors, and various fiber-optic components has fuelled the rapid growth of this technology. The advent of the fiber-optic telecommunications age has made it possible to miniaturize previously bulky equipment, and realize remote and distributed sensing [10,11], using the optical fiber as a light guide and as a sensing element. To increase the detection sensitivity, various modulation formats have been used, of which a particular variant termed wavelength modulation spectroscopy (WMS) [2,4–8], has been very popular because it affords greater sensitivity than direct detection. In a generic WMS experiment, the centre wavelength of a diode laser is scanned across an absorption line at a slow rate (few Hz), while simultaneously the instantaneous laser wavelength is modulated by a high frequency (tens to hundreds of kHz) dither signal. For a given laser, the modulation of the injection current produces an associated intensity modulation (IM) of the laser along with the wavelength modulation (WM) with a frequency-dependent phase difference, ψ, between them. Since a DFB’s current-intensity relation is slightly nonlinear, a sinusoidal modulation of its injection current produces a corresponding and progressively smaller IM at increasingly higher harmonics of the modulation frequency. Fourier analysis [8,9], of the light intensity resulting from the interaction of the simultaneous IM and WM with a gas absorption feature shows that signals at various harmonics of the modulation frequency are produced, with the *n*
^{th} harmonic signal being dominated by the *n*
^{th} derivative of the absorption line shape with an associated asymmetry due to the IM. These signals are collectively known as the IM-WM components. Typically though, phase sensitive detection of only one harmonic component of the output signal is carried out for quantitative gas analysis. The interaction of the IM of the laser with the absorption line gives rise to signal components, widely known as the residual amplitude modulation (RAM) signals, which consist of concentration-independent background as well as concentration-dependent RAM components. The origin and distorting effects of the RAM on the various harmonics has been theoretically treated in great detail by several authors [5,7–9]. The concentration-independent high background RAM has long been a major limiting factor in WMS, particularly if signal recovery is implemented at the modulation frequency (i.e. if 1f detection is used). Signal amplification is limited by saturation of the detection electronics by the high background and there are significant sensitivity/resolution issues in the digital acquisition of signals. A concentration-dependent RAM signal that follows the absorption line shape and the conventional 1st derivative-like 1f WMS signal separated by ψ, are superimposed on this background RAM. The recent RAM technique [12], for 1f, calibration-free extraction of gas absorption line shapes, exploits the phase-sensitive detection capability of a lock-in amplifier (LIA) to recover a ψ–dependent projection of the two RAM signals on one detection axis in complete isolation from the conventional 1st derivative-like 1f WMS signal that is fully projected on the other LIA axis but is not recorded. In a very recent paper, a fiber-optic RAM nulling technique [13], was reported to optically eliminate the concentration-independent 1f RAM component due to the direct IM of the DFB laser. This is a significant step forward particularly for the RAM technique [12], as well as another 1f calibration-free technique namely, Phasor Decomposition method (PDM) [14], for the recovery of absolute gas absorption profiles. It is reiterated that the isolated conventional 1f WMS signal is irrelevant to the RAM technique and therefore does not figure in the rest of this paper.

In the RAM nulling technique, the intensity-modulated laser output is symmetrically split in two parts and these are separately directed through a gas cell and a delay fiber, with the fiber length and the modulation frequency/delay suitably chosen to introduce a relative phase difference of π between the IM on the two components. The relative IM amplitudes of the signals are adjusted so that in the absence of gas the direct IM cancels precisely when combined through a fiber-optic 3dB coupler. The resulting dc output is rejected by an LIA used for harmonic detection and the output is essentially zero. In the presence of a gas, the absorption produces a concentration-dependent imbalance at the output, and it had been initially expected that for an optimized nulling arrangement, the RAM nulled absorption signal would appear on a perfectly zero baseline. However, repeated experiments have shown that there is a small sloping baseline signal for all concentrations as borne out by Fig. 1 that shows the concentration-dependent RAM component isolated from the conventional 1st derivative-like IM-WM component [12,13]. For relatively strong absorption signals the baseline is not immediately obvious because the low post-detection amplifier gains do not accentuate this feature adequately. It is only when the gain is increased for smaller absorptions that the baseline becomes prominent. The presence of a non-zero baseline means that contrary to the initial expectation, the RAM nulling condition does not hold precisely over the entire wavelength scan. This raises a fundamental question about the basic understanding of the technique. In the preliminary mathematical treatment [13], the interplay of the wavelength dependence of the coupling ratios of the 3-dB coupler and that of the IM of the DFB laser was not considered in detail for the sake of brevity and clarity, although it was recognized that they were expected to influence the nature of signals.

This paper considers a slightly modified experimental setup and presents a generalized treatment of the RAM nulling technique that explicitly takes into account the wavelength-dependence of the various terms. It is shown that of the four possible input-output configurations only three give rise to distinct expressions for the output, and of those there is one configuration that leads to a precisely zero baseline throughout the wavelength scan range of the laser. This is an important step forward with regard to the RAM nulling technique for the following three reasons. First, although it is straightforward, in principle, to remove the baseline through curve fitting and subtraction, this may not necessarily be practical for systems intended for automated real-time measurement. A second and more subtle but significant issue is that if the nature of the wavelength-intensity (or equivalently, current-intensity) curve for a given laser is significantly nonlinear over the scan range of the laser current, the baseline of the signals shown in Fig. 1 that is influenced by the variation of the IM with wavelength, denoted by Δ*I(λ)*, will have greater curvature and this will further complicate the baseline fitting. Recently some novel DFB laser structures for carbon monoxide sensing [15,16], and VCSELs [17,18], for oxygen and other trace gas sensing have been reported in which the lasers have been shown to have distinctly more nonlinear current-intensity characteristics than a telecom DFB over their typical scan range. The variation of Δ*I(λ)* for such lasers can be expected to be significantly more nonlinear than a typical telecom DFB. Finally, for noisy signals it is not easy to define suitable off-line regions of the curve to which a polynomial fit can be made to obtain the trend of the background. Ideally one would like to avoid the process of baseline fitting and subtraction totally to eliminate associated errors. A mathematical model of the RAM-nulling process is developed, and through simulations the generic nature of background signals due to the wavelength-dependence of the IM and that of the couplers is explained for the distinct configurations. Finally experimental results are presented that agree with the simulations to a good extent thereby validating the model.

## 2. Modeling of RAM-nulled output

Figure 2
shows a simplified version of the RAM nulling setup [13], in which only the relevant components have been retained. The laser can be connected to either of the input ports of the first coupler (therefore the grayed-out block at IP_{1}), and for each configuration, the output can be measured at either of the two output ports of the second coupler (therefore the grayed-out port OP_{2}). For the purpose of the model the two couplers are assumed to be identical. It is also assumed that for each coupler the relative distribution of power from any one input to the two outputs is the same for both input ports that is to say that each coupler is symmetrical with regard to its input and output ports. It is easy to source such couplers thanks to the highly mature manufacturing technology for various telecoms-grade fiber-optic components. The fractional self-coupling ratios *y _{1}(λ)* and

*y*for each coupler denote the fraction of launched power retained in the launch fiber, and that transferred to the coupled fiber respectively at the output of the coupler. In accordance with the RAM technique of signal recovery [12,13], the lock-in phase was set to isolate the IM component of the signal from the IM-WM, and therefore only the Δ

_{2}(λ)*I(λ)*term is considered in the analysis since the intensity term

*I(λ)*, contained entirely in the IM-WM component, is eliminated by phase-sensitive detection.

#### 2.1 Configuration 1: DFB connected to IP_{2} of coupler1; output taken from OP_{1} of coupler2

The RAM nulling condition is initially established with the gas cell evacuated, the current ramp switched off and only the 100 kHz modulation applied to the laser current. Throughout the analysis Δ*I(λ _{0}), y_{20}* and

*y*denote respectively the IM amplitude, the fractional cross-coupling and self-coupling values for the couplers at the starting wavelength λ

_{10}_{0}. The losses for the two signal paths along the gas cell arm and the delay fiber arm arise from insertion losses at the connectors, the propagation loss through the cell and the fiber, and the attenuation due to the variable optical attenuator (VOA). Therefore the signals

*y*Δ

_{20}*I(λ*and

_{0})*y*Δ

_{10}*I(λ*that emerge from OP

_{0})_{1}and OP

_{2}of coupler1 arrive at IP

_{1}and IP

_{2}of coupler2 having suffered usually unequal attenuation along their respective paths. For the analysis, the attenuation (loss) factor through the gas cell arm is denoted by α

_{L}. A variable

*x*is also defined as the ratio of the loss along the delay line arm to that along the gas cell arm. The factor

*x*can be varied by adjusting the attenuation factor of the VOA to achieve perfect balance (and hence cancellation) of the two IM signal components. It will be shown shortly that perfect RAM nulling for each input-output configuration requires a specific value of

*x*that can be expressed in terms of the fractional coupling ratios of the two couplers. It is also evident from the definition of

*x*that its value may be equal to, greater than or less than unity.

The IM signals at the inputs of coupler2 may therefore be written as,

*π*is due to the appropriate arrangement of the modulation frequency and fiber length given by $L=c/2nf$, where

*c*is the speed of light,

*n*is the effective index of the fiber and

*L*is the length of the fiber [13]. The expression for the output at

*OP*of coupler2 with no gas in the cell and before the ramp is applied can be written as,

_{1}*x*will not be unity by default and the output in the absence of gas will be non-zero. However by controlling the VOA’s attenuation the output can be adjusted to be zero by satisfying the RAM nulling condition given by,In other words, it is necessary and sufficient, in this case, to precisely balance the loss through the two signal paths to ensure a zero output in the absence of gas. The generalized expression for the output including the wavelength-dependence of the terms when the current ramp to sweep the laser wavelength is switched on can be written as,

*x*in conjunction with Δ

*I*(

*λ*), y

_{1}(

*λ*) and y

_{2}(

*λ*). The cases for which

*x ≠ 1*denotes partial nulling while that for which

*x = 0*denotes no nulling, with the output being taken with the delay fiber arm disconnected from coupler2. These different cases for each configuration are discussed in the simulated results presented later. Since the actual signal

*OP*is ideally zero in this case, no additional measurement or baseline fitting to an

_{1 no gas}*OP*is necessary. It will become evident shortly that for the other configurations the

_{1 gas}*OP*will not be zero, and therefore an additional measurement or baseline fitting will be necessary and associated inaccuracies will arise.

_{1 no gas}For signal normalization it is noted that with gas in the cell, the signal at the input *IP _{2}* of coupler2 gets multiplied by ${e}^{-\alpha (\lambda )Cl}$due to the gas absorption (α(λ) not to be confused with α

_{L}). The resulting generalized output

*OP*for any value of

_{1 gas}*x*is given by,

*x*is eliminated since it is not practical to determine its value experimentally. The relative transmission is given by,

*x*= 1, the

*OP*signal given by Eq. (5) becomes zero irrespective of the other terms in the expression, and the two equations above reduce to,

_{1 no gas}*OP*with the delay fiber arm disconnected from coupler2 (i.e.

_{1}*x = 0*) and with no gas in the cell. Alternatively, a baseline fit to the signal with the delay arm disconnected can be taken if it is not feasible to take an actual “no-gas” signal. It is noted that this signal is heavily dominated by the Δ

*I(λ)*term which, for the DFB used, produces a monotonically varying, strong and clean signal of high average value and it is very easy to fit a baseline to it making it a convenient normalization signal. It is therefore clear that it is not necessary to separately and accurately determine the values of

*α*, Δ

_{L}*I(λ), y*and

_{1}(λ)*y*. It is reiterated that a highly useful feature of this configuration is that the RAM nulled output is zero irrespective of the wavelength-dependence of the IM and the couplers. This point will be developed further at a later stage once the other experimental configurations have been analyzed.

_{2}(λ)#### 2.2 Configuration 2: DFB connected to IP_{2} of coupler1; output taken from OP_{2} of coupler2

In the second distinct configuration the expressions for the signals at *OP _{2} and OP_{1}* of coupler1 and those for the signals at

*IP*and

_{2}*IP*of coupler2 remain unchanged. The output at

_{1}*OP*of coupler2 before the ramp is applied can be written as,

_{2}*OP*can be obtained by making a baseline fit to the gas signal given by the next equation if an actual no-gas signal is inconvenient to record. The signal in the presence of gas in the cell is given by,

_{2 no gas}*OP*with the delay fiber arm disconnected and the gas cell empty, and the same procedure for normalization can be followed. It is pointed out that even if the RAM nulling condition given by Eq. (11) is precisely satisfied, it cannot be assumed that the final output will be zero throughout the scan unless

_{2}*y*and

_{1}(λ)*y*are totally wavelength-independent. This is where the variation of Δ

_{2}(λ)*I(λ)*with

*λ*becomes important. If the

*I-λ*curve for a given laser is highly nonlinear [15–18], Δ

*I(λ)*, and therefore the baseline, will have greater curvature and this will further complicate the baseline fitting.

#### 2.3 Configuration 3: DFB connected to IP_{1} of coupler1; output taken from OP_{1} of coupler2

In the final distinct configuration the expressions for the signals at *OP _{2} and OP_{1}* of coupler1 and those for the signals at

*IP*and

_{2}*IP*of coupler2 are different. The signals at

_{1}*IP*and

_{1}*IP*of coupler2 before the ramp is applied may be written as,

_{2}*OP*of coupler2 before the ramp is applied is given by,

_{1}*OP*can be obtained by making a baseline fit to the gas signal given by the next equation if an actual no-gas signal is inconvenient to record. The signal in the presence of gas in the cell is given by,

_{1 no gas}*OP*with the delay fiber arm disconnected and the gas cell empty, and the same procedure for normalization can be followed.

_{1}It was initially assumed that once the RAM nulling condition (Eq. (11) and Eq. (18)) is established, it would hold over the entire wavelength scan since the wavelength-dependence of *y _{2}(λ)* and

*y*over the very small tuning range of the laser (typically 0.4nm) is weak, and it is well known in modulation spectroscopy that the wavelength-dependence of Δ

_{1}(λ)*I(λ)*for a given DFB is strong. Therefore it was assumed that Δ

*I(λ)*would be the dominant factor and that nulling it at the starting wavelength would be sufficient to ensure a zero background throughout. However, this assumption was proved to be inaccurate by the distinct slope on the RAM nulled signals. Simulations of the configurations are carried out to reveal the expected trend of signals for the DFB used in the experiments.

## 3. Simulation of RAM nulled output

In order to simulate the RAM nulling situation it is necessary to obtain the variation of Δ*I(λ)* over the typical range of a wavelength scan, and also the variation of the fractional coupling ratios of the 3-dB coupler. To estimate Δ*I(λ)* from the DFB output intensity *I(λ)*, the mean laser current, *i _{LD},* was scanned from 30 to 110mA corresponding to a typical scan range. The photodiode voltage was recorded for every current step of 5mA to give the variations of the mean laser intensity as a function of the mean laser current. A polynomial fit to the curve was obtained that represents the functional dependence of the intensity on the current. The variation of the IM is estimated programmatically by assuming a constant current dither Δ

*i*, and evaluating the polynomial at

*i*± Δ

_{LD}*i/2*to give the intensity excursion about each value of the mean current. The variation of intensity with current is shown in Fig. 3(a) where a slightly nonlinear variation is present although it is not immediately obvious. The laser wavelength at each mean current value was also measured using an Agilent optical spectrum analyzer to correlate the laser current with the wavelength. Using this correlation, Fig. 3(b) shows that the slope of the ∆

*I*(

*λ*) vs

*λ*trace is negative. This result has been experimentally verified by impressing a sinusoidal current modulation on the DFB and measuring the peak-to-peak variation of the IM as the mean laser current is stepped from 30 to 110mA. It was observed that the peak-to-peak value of the IM progressively reduced as the current (and equivalently the wavelength) was ramped up. This clearly proves that the slope of the ∆

*I*(

*λ*) vs

*λ*curve is indeed negative. This conclusion is also supported by the negative slope of the off-centre regions of the non-nulled, RAM signal (very similar in nature to signals shown later in Fig. 4(e) , Fig. 5(e) and Fig. 6(e) ) that is dictated by the ∆

*I*(

*λ*) term only. The approach of programmatically extracting the nature of variation of the IM has been preferred for the simulations so that a comparison can be made later between the DFB used and recently-reported laser structures with significantly more nonlinear laser characteristics. Although we do not yet have access to such lasers, a reasonable comparison can still be made by this method of simulating the variation of the IM from the nonlinear laser characteristics.

The wavelength-dependent fractional coupling ratios are estimated next. For a phase-matched case of a coupler the fractional coupling ratios *y _{1}*(

*λ*) and

*y*(

_{2}*λ*) are given by [19–25],

*κ*(

*λ*) is the wavelength-dependent coefficient of coupling between the two fibers, and

*L*is the coupling length that denotes the length for complete power transfer from one fiber to the other. It is reasonable to assume a phase-matched case since the wavelength range scanned by the current ramp is a fraction of a nanometer. The expression for the coupling coefficient for the case of coupling between two parallel identical single mode fibers [21,22] is given by,where, $\delta \text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}({n}_{1}^{2}-{n}_{2}^{2})\text{\hspace{0.17em}}/{n}_{1}^{2}$, and ${d}_{1}\text{\hspace{0.17em}}=\text{\hspace{0.17em}}\text{\hspace{0.17em}}d/a$, where

_{c}*d*,

*a*,

*n*

_{1}and

*n*

_{2}represent the fiber core separation, core diameter, core index and cladding index respectively. The parameters

*A*,

*B,*and

*C*are calculated from a set of coefficients and the V number of the fiber, and are given in Ref [21,22]. It is reiterated here that if calculated values of

*y*and

_{2}(λ)*y*are to be used for signal normalization to extract the relative transmission in an actual experiment, it would be necessary to accurately determine the value of

_{1}(λ)*κ(λ)*. However, since the signal normalization method does not explicitly need these values, the exact values of the 3-dB coupler parameters are not required, and calculated values may be used to simulate the nature of signals in order to illustrate the nulling mechanism. For the purpose of the model, typical values of a 3-dB coupler designed to operate at the centre wavelength of 1550nm have been assumed. For a typical telecom-grade 3-dB coupler with${n}_{1}=1.4532,\text{}$ ${n}_{2}=1.45$,$a=5\mu m,$and $d=12\mu m,$the coupling length,

*Lc*, of the coupler is estimated by assuming that the coupler is a perfect 3-dB coupler at the design wavelength of 1550nm. By substituting this wavelength value, $\kappa ({\lambda}_{design})$can be calculated and

*L*is calculated using the relation${L}_{c}=\pi /2{\kappa}_{design}$. The length of the coupler must be half this value for it to be a 3-dB coupler, and is constant for a given coupler. Assuming this coupler length it is now possible to estimate the coupler’s response by re-calculating values of

_{c}*y*and

_{1}(λ)*y*from Eq. (22) and Eq. (23) for the wavelength range scanned by the laser current. The wavelength-dependence of

_{2}(λ)*y*and

_{1}(λ)*y*for a typical scan range is shown in Fig. 3(c) and Fig. 3(d). The percentage changes in

_{2}(λ)*y*(

_{1}*λ*) and

*y*(

_{2}*λ*) over the wavelength range are only −0.046% and 0.0375% respectively, which shows that the wavelength-dependence of the coupling coefficients is extremely weak in comparison to that of Δ

*I*(

*λ*) that changes by −14% which justifies the initial assumption that the coupling ratios are constant.

## 4. Results of simulation and experiments

The simulated RAM nulled outputs and experimental data for the three configurations analyzed in Sec. 2.1 through Sec. 2.3 are shown in Fig. 4, Fig. 5 and Fig. 6. In each figure the three upper subplots show the simulated outputs for three values of *x* for that configuration. The lower subplots show the corresponding experimental data scaled by the LIA sensitivities. A representative absorption profile has been superimposed on the simulated baselines to illustrate the situation better, but it has no bearing on the nature of the baselines. A 1% mixture of methane in nitrogen was used for Fig. 4 and Fig. 6, while methane of 0.1% concentration was used for Fig. 5. The modulation index was chosen to be 0.75.

It is seen from Fig. 4(a) and Fig. 4(d) that if the RAM nulling condition is exactly satisfied for the first configuration given by Eq. (4), it is indeed possible to achieve a perfectly zero baseline. The absorption signal can then be selectively amplified to a greater desired extent without saturating the electronics. However, any deviation from this condition will produce a non-zero and sloping background as shown in Fig. 4(e) and Fig. 4(f) for two values of *x* nearly equal to 1. The extent of the deviation from 1 determines the level of the background. The signal Δ*I(λ)* was shown to have a downward slope in Fig. 3(c), and the product *y _{1}(λ)y_{2}(λ)* is easily verified to slope downward since

*y*and

_{1}(λ)*y*have slopes of opposite sign. Therefore for the non-nulled cases the value of (1-

_{2}(λ)*x*) determines the sign of the signal, while

**Δ**

*I*(

*λ*), being the dominant factor by far, determines the magnitude as well as the nonlinearity of the slope of the output signal. It is noted that the vertical axes of the simulations do not represent actual signal values since certain assumptions have been made regarding the ideal behaviour of the couplers, and the lock-in sensitivity cannot be incorporated in the model.

For the second and third configurations depicted in Fig. 5 and Fig. 6 respectively, the situation is different in an important respect. As explained earlier, it is not possible, in these cases, to obtain a perfectly zero baseline even with perfect RAM nulling. A residual slope will necessarily remain on the output signal. The variation of *y _{1}(λ)* and

*y*is seen to dominate the closer the system is tuned to the RAM nulling condition given by Eq. (11) and Eq. (18). As the system moves away from this condition the

_{2}(λ)**Δ**

*I*(

*λ*) term again begins to dominate thereby producing the large offset. The fourth configuration in which the DFB is connected to IP

_{1}of coupler1 and the output is measured at OP

_{2}of coupler2 is identical to configuration 1 and is therefore not treated separately. The simulations and experimental results pertaining to the three configurations clearly bring out the interplay between the wavelength-dependence of the couplers and the IM of the laser, and demonstrate that depending on the experimental configuration, a small residual slope may arise in the RAM nulled signals.

Finally the practical utility of this RAM nulling system for future work is demonstrated by the investigating the projected behaviour of the system for a laser with significantly more nonlinear characteristics than the DFB used here. Figure 7
shows a simulated representative current-intensity curve of such a laser and the corresponding Δ*I(λ)* (extracted in the same way as before) variation that is significantly more nonlinear than that of the DFB. It is pointed out that the nature of the current-intensity curve is a not unrealistic, but a good representation of actual laser characteristics as borne out by some recent work [15–18].

The output of the precisely nulled case for each of the three configurations is shown in Fig. 8
, in which the off-centre regions for only the first configuration are flat and zero while for the other two configurations there is a significantly nonlinear baseline due to the nature of the Δ*I(λ)* curve. The important implication of this simulation is that for 1f gas detection based on the new RAM technique (and PDM), the first configuration can effectively cancel the effect of significantly nonlinear variation of the Δ*I(λ)* Baseline fitting on the nulled signals will be obviated by the flat, zero-level nulled outputs.

Finally, Fig. 9 shows the theoretical (based on HITRAN) and experimental relative transmission profiles for a sample of methane in nitrogen balance at a concentration of 0.1017%. The main source of noise in this experiment was the etalon fringes that tend to corrupt mainly the off-centre regions. The experimental data was filtered using a finite impulse response (FIR) digital filter with a very sharp roll-off to greatly suppress the etalon fringes whose power spectral density (PSD) nearly overlaps with that of the actual absorption signal. It is expected that the power spectra can be better separated and the filtering made more effective by designing a longer gas cell that would give rise to etalon fringes with a PSD that does not overlap with that of the gas absorption signal. However even with the present system it is clearly possible to extract concentration and pressure information with fairly high accuracy although the data towards the right is corrupted to a greater extent than the data towards the left. This demonstrates the potential of the RAM nulling method for quantitative gas measurements for industrial applications.

## 5. Conclusion

A mathematical analysis of the RAM nulling technique has been presented that shows, through simulations and experiments, that the apparently anomalous slope on the output signals is in fact an inherent attribute of the system and that it depends on the actual experimental configuration. However it turns out that for one configuration the absorption signal appears on an essentially zero baseline. This has a three-fold significance for practical industrial applications. First, it is not necessary to fit baselines to the nulled signals to obtain the *OP _{2 no gas}* and

*OP*as in Eq. (12) and Eq. (19). For relatively strong absorption signals, etalon fringes from the gas cell degrade the offline regions more than they do the actual absorption signal. This makes identifying suitable off-line leading and trailing regions to obtain a proper fit prone to error. The fact that the gas signal appears on a zero background obviates the need for baseline fitting and associated inaccuracies may be avoided. Second, 1f RAM-nulled applications using other near IR laser sources such as VCSELs [17], that have highly nonlinear characteristics should benefit from the same configuration. There is also reason to be cautiously optimistic about the extension of this technique to longer wavelength regions (around 2-3μm) as the fledgling mid-IR optical fiber technology (currently limited by the availability of suitable materials) matures. Finally, this is also important for automatic closed-loop feedback control of the nulling process (as opposed to manual tuning of the VOA) to dynamically compensate for departures from an initially established nulled condition due to slow system drifts. Feedback control can be used to re-establish the nulled condition by periodically adjusting the relative signal strengths at the input of the second coupler to bring the offline regions close to zero. To do this accurately, however, the value for the offline region of the nulled output for perfect RAM nulling should be known a priori. For configuration 1 this offline level is flat and zero. Therefore it is sufficient for the feedback algorithm to try and bring the average value of the offline region of the RAM-nulled gas absorption signal as close to zero as possible without having to take into consideration any signal slope. For a sloping nulled signal it is not possible to define a suitable offline region because such a signal will not have a flat region. It is therefore clear that from the point of view of practical signal processing it is very useful to know for which configuration the off-line background will be flat and zero. Finally gas measurement 0.1017% methane at approximately atmospheric pressure shows that the performance of the system is mainly affected by the etalon fringes due to the gas cell. For such expected pressures for typical applications, the nearly overlapping PSDs of the absorption and the fringes were resolved by a significantly optimized digital filter. A longer and optimized cell under test is expected to lead to smaller etalon fringes with a PSD that is spectrally well-separated from that of the absorption signal, thereby leading to higher noise rejection with a digital filter. The long-term stability of the system is excellent with the system capable of operation over several hours without fine tuning of the polarization controllers. The delay line is selected to be much longer than the coherence length of the laser that is further reduced from its un-modulated value by the modulation on the DFB current. This arrangement along with the use of polarization controllers ensures that optical interference noise remains stably below the receiver noise for long periods of time. In conclusion it may be said that the model presented in this paper not only enhances the theoretical understanding of the nulling technique but also provides practical information that enables the selection of a particular configuration of the nulling process that simplifies signal recovery, post-processing and extraction of information from experimental data.

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