1. Introduction

The significant role of oxygen in geochemical and biological cycles necessitates its detection in various scientific and environmental settings [1–6]. Modern applications are increasingly stringent, requiring greater accuracy, stability and shorter acquisition times [1, 3]. Magnetodynamic detection of oxygen by exploiting its paramagnetic properties is one technique that has been explored to meet these requirements [7, 8], but is susceptible to mechanical vibrations and drifts. Alternate techniques include solid-state sensors for combustion diagnostics [9], and more recently miniature oxygen sensors, where alteration of the electrical or chemical properties of nanostructures is used [10, 11]. Despite substantial progress in oxygen detection technologies, in situ oxygen detection remains challenging due to issues involving environmental contamination, operating pressure constraints and interfering molecular species.

Optical spectroscopic systems have received strong interest due to their sensitivity and specificity [12–21] and are frequently built for low-power, field deployable operation. Sub-part-per-million (sub-ppmv) oxygen detection limits have been achieved with vacuum-ultraviolet absorption [22] and fiber-based techniques [23], although portability, dynamic range and in situ capabilities hinder their application. To date, we have not identified any sensor in literature that performs in situ atmospheric oxygen detection with the target sub-ppmv sensitivity required for biorespiratory diagnostics.

Recent work [24–26] has identified promise for sub-ppmv oxygen detection at atmospheric pressure using Faraday rotation spectroscopy (FRS) that targets the A-band of oxygen at 762 nm. In FRS, an applied magnetic field splits the ³Σ_g^‾ oxygen ground state, and quantum selection rules allow only ∆M_j = + 1 and −1 for rovibronic transitions in the A-band, which interact with right-handed and left-handed circularly polarized light respectively. This creates circular birefringence, which causes rotation of linearly polarized light as it travels through an oxygen sample, and is detected by projection onto a nearly-crossed polarizer. So et al. [25, 26] performed atmospheric pressure FRS on the oxygen ^PP₁(1) transition in the A-band [27] using a modulated (AC) magnetic field, yielding an AC-FRS detection limit of 10 ppmv·Hz^-1/2. Brumfield et al. [24] employed rare-earth magnets for static (DC) field generation and conventional balanced-detection FRS for fringe removal and intensity-noise suppression. This configuration was used to achieve a shot-noise limited sensitivity of 6 ppmv·Hz^-1/2.

In the present work we demonstrate further sensitivity enhancement using a distributed-feedback (DFB) laser diode and a hybrid-FRS system that combines balanced optical detection and optimization of the analyzer offset angle typical for conventional AC-FRS systems. The hybrid-FRS technique, which also uses a DC magnetic field, achieves a 10$\times$ enhancement beyond the sensitivity reported in [24].

2. Conventional FRS methods

There are two distinct FRS signal detection methods reported in literature: (1) 90°-method implemented with nearly-crossed polarizers [26, 28, 29] and (2) 45°-method employing balanced optical detection [29–31]. The 90°-method is more popular [28, 31] because it is simpler to implement, requiring only a single photodetector and two nearly-crossed polarizers (one before and after the gas sample) to measure the Faraday rotation of the light polarization [31]. In this nearly-crossed polarizer configuration, the laser noise is effectively suppressed, providing an improved signal-to-noise (SNR) ratio in the FRS measurement compared to direct absorption spectroscopy. The optimum analyzer uncrossing angle (α_opt.) is determined by equalizing the detector-noise equivalent power (NEP) with the laser intensity noise or quantum shot noise generated from the laser radiation incident on the photodetector. In the 45°-method, the laser noise suppression is performed electronically by splitting the laser beam emerging from the sample cell into two orthogonally polarized components (usually of equal power achieved with α = 45°) and performing balanced optical detection using two photodetector elements. In this configuration, the component FRS signals measured on both photodetector elements are out of phase, while the intensity-noise is in phase allowing for effective laser-noise suppression. Auto-balancing photodetectors (e.g. Nirvana auto-balancing photodetector by New Focus) that require other than a 50/50 split ratio have also been successfully used in balanced detection FRS systems (the split ratio can be conveniently adjusted by varying α) [24]. Prior experimental work has assumed the 45°-method and the 90°-method to be separate [28, 30, 31]; that is, the use of one method precludes the other. In what follows, we experimentally show that these two techniques can be used in a complementary fashion resulting in a hybrid-FRS method that can achieve a better SNR than either one alone. To identify the benefits of hybrid-FRS, we compare below the performance of hybrid-FRS with the conventional 45°-method and 90°-method.

3. Experimental setup

The studies and experiments have been performed using the optical configuration shown in Fig. 1. The setup is similar to that reported by Brumfield et al. [24], but the VCSEL laser used in [24] has been replaced with a DFB diode laser (Sacher Lasertechnik, λ = 762 nm), with small modifications made to the polarization optics to enable hybrid-FRS measurement. The DFB laser targets the ^PP₁(1) O₂ transition in the A electronic band. The laser current is modulated at 6 kHz with a modulation depth optimized to maximize the second-harmonic (2f)

 

Fig. 1 Optical layout of the FRS setup used in this study. The DFB laser output power is attenuated using a NPP (NPP1), and polarized by the GTP prior to entering the MPC. The signal beam exiting the MPC passes through a WP and is detected by the BPD. The NPP2 attenuates the reference beam to prevent optical saturation of the reference-photodiode. Abbreviations: DFB: distributed-feedback; GTP: Glan-Thompson polarizer; MPC: multi-pass cell; WP: Wollaston prism; NPP: nano-particle polarizer; BPD: balanced-photodetector.

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FRS signal. The ambient laboratory air at room temperature and atmospheric pressure is used as the sample gas for our studies. The laser light is first transmitted through a Glan-Thompson polarizer (GTP) to establish a clean polarization state prior to entering a cylindrical-mirror multi-pass cell (MPC). The cell provides 6.8 m path length with 40% optical throughput. An array of rare-earth magnets generates a 554 Gauss axial magnetic field in the active region probed by the laser beam within the multi-pass cell. Light passing through the cell undergoes polarization rotation due to interaction with the sample and is split (after exiting the cell) into orthogonal components by a Wollaston prism (WP). Balanced detection and demodulation is performed using a Nirvana auto-balancing photodetector (New Focus, model 2007) and lock-in amplifier (Signal Recovery 7265), with automated data acquisition using a DAQ board (NI-USB-6529) and customized LabVIEW software. The auto-balancing function is employed with an optimal 2:1 reference- to signal-photodiode split-ratio (480 µW on the signal-photodiode), resulting in laser intensity-noise suppression of > 20 dB (the 21.6 dB common-mode rejection ratio (CMRR) of the balanced detector was determined experimentally). The system is used to perform detection using the 45°-method and hybrid-FRS methods. As the DFB laser is capable of providing up to 30 mW of output optical power, a nano-particle polarizer (NPP) (shown as NPP1 in Fig. 1) is placed just at the laser output to attenuate the laser radiation and avoid detector saturation in the 45°-method. In hybrid-FRS the NPP1 is set to maximum transmission and the detector saturation is avoided by combination of the appropriate α setting and attenuation of the reference branch using the NPP2 (see Fig. 1). Since the optical fringes introduced by the NPP2 are very stable, the auto-balancing circuit is capable of compensation of any slow optical power drift between the reference- and signal-photodiodes. It should also be noted that NPP2 is relatively thin resulting in a large fringe free spectral range (> 10$\times$ the ^PP₁(1) linewidth), so any parasitic intensity modulation induced by the laser wavelength modulation creates a negligible baseline offset in the measured FRS signal.

In order to assess the proposed hybrid FRS method and establish a performance baseline, conventional balanced detection FRS has been implemented using the same system components. In our prior work we used a VCSEL [24] capable of delivering 20 µW of optical power to the signal-photodiode. This resulted in a minimum detection limit (MDL) for O₂ of 6 ppmv·Hz^-1/2. Application of a more powerful DFB laser that can easily deliver up to 480 µW to the signal-photodetector (detector saturation occurs at 500 µW) is expected to result in a 4.9$\times$ improvement in sensitivity to oxygen. At P₀ ≈1.4 mW (which provides 480 µW to the signal detector at split ratio of 2:1), the MDL for O₂ measured with the conventional balanced FRS system using DFB laser is 1.8 ppmv·Hz^-1/2. This corresponds to an improvement of 3.3$\times$, or 70% of the expected enhancement. The 30% discrepancy has been attributed to the higher laser relative intensity-noise (RIN) of 5.3$\times$10⁻⁷ Hz^-1/2 (2$\times$ the VCSEL RIN), and an order of magnitude smaller CMRR. At these conditions the 45°-method operates roughly at 2$\times$ the quantum shot-noise limit. However, it should be noted that due to the detector saturation limit only a fraction of the available laser power (P₀ ≈1.4 mW vs. 8 mW maximum available after the MPC) could be used to perform this measurement.

3.1 Limitations of the 45°- and 90°-method

Based on the results obtained with the 45°-method in the previous section, one can realize that its main limitation is related to detector saturation, which limits the amount of total laser power that can be used in the FRS measurement. On the contrary, the 90°-method is free from this limitation, because the amount of light transmitted through a nearly crossed analyzer is significantly lower; however, laser-noise and electromagnetic interference limit its sensitivity. In order to discuss the main limitations in both techniques, an analysis of signal and noise in both methods is performed below.

In the 45°-method, the signal (V_45°) and noise (σ_45°) can be expressed as:

Therefore with an assumption of α < 10° in the 90°-method both techniques should provide comparable ultimate performance. However practical limitations exist, and result in SNRs significantly lower than SNR_SN in both techniques.

(A) Detector saturation limit in 45°-method

Given the total laser power on each detector element is limited by the specified saturation power P_sat, there is a limit to the total power P₀ that can be used in the FRS measurement. To understand this point, we may consider a high power (P_0,A) and low power (P_0,B) case of Malus’ law in Fig. 2(a), where conventional balanced detection may be used in the case of P_0,B (since both signal and reference branches are below saturation, Fig. 2(b)). Using small signal analysis and assuming shot-noise domination (Eq. (7)), it is clear that SNR enhancement will require increasing P_0,B, and is limited by detector saturation, i.e. P_0,B < P_sat. Thus despite excellent laser-noise suppression provided by the balanced detection that enables operation in the shot-noise regime, there is a hard limit in the maximum achievable SNR with this technique. Therefore application of more powerful lasers does not bring the P₀^1/2 improvement in SNR predicted by Eq. (7), but only allows for maximum power of P₀ = 2P_sat which limits the ultimate performance of the system.

 

Fig. 2 (a) Malus’ law for the signal- and reference-photodiodes, with a high-power (red curve, hybrid-FRS) and low-power (black curve, 45°-method) case. The low-power case (P_0,B) can be used for conventional balanced detection when both signal and reference branches are below detector saturation. (b) In conventional balanced detection using low-power, signal enhancement (black arrow) is achieved by increasing P_0,C to P_0,B (detector saturation, or intensity- and shot-noise crossover point). In hybrid-FRS, P_0,C increases to P_0,A (red arrow) with decreasing α at constant P_sig (thus constant noise). Signal ~1/α for small α, hence SNR ~1/α ~P₀^1/2. Hybrid-FRS avoids saturation and intensity-noise limitations by moving the operating point parallel rather than perpendicular to the α-axis.

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(B) Laser noise limit in 90°-method

As predicted by Eq. (8), the 90°-method should be capable of the same ultimate sensitivity as provided by the shot-noise limited 45°-method. However the lack of balanced detection that provides an additional 20 dB of CMRR makes it difficult to suppress the laser-noise in the FRS system using the 90°-method. As a result, the laser-noise suppression based solely on the optical suppression through a decrease in α may not be sufficient to achieve shot-noise limited operation (please note that the signal in Eq. (3) scales with sin(2α) while the laser-noise is proportional to sin²(α), which allows for improvement in SNR in the laser-noise dominated regime by decreasing α until the detector-noise floor is reached; in such a configuration the shot-noise limited operation can only be obtained with ultra-low noise photodetectors and high-quality polarizers). Therefore most of the FRS systems utilizing the 90°-method operate in the laser intensity-noise dominated regime. Using Eqs. (3) and (4) and the experimental parameters relevant to our setup with NEP = 5.4 × 10⁻¹² pW·Hz^-1/2, RIN = 5.3 × 10⁻⁷ Hz^-1/2, and P₀ = 8 mW (obtained from our 30 mW DFB diode after optical losses in the system primarilyfrom the MPC), the best SNR_90° is achieved at 2° as shown in Fig. 3(a).For ease of comparison, the SNR_90° values presented in Fig. 3(b) were normalized to the peak SNR of the hybrid-FRS method at P₀ = 8 mW.

 

Fig. 3 Hybrid-FRS and 90°-method SNR calculation. (a) SNR for 90°-method at P₀ = 8 mW. For small α, noise ~α² and signal ~α. (b) SNR for hybrid-FRS at the same P₀. CMRR suppression of intensity-noise gives significant SNR increase. V_90° is normalized to the V_hyb maximum, σ_hyb is normalized to the σ_90° maximum, and SNR_90° is normalized to maximum of SNR_hyb. (c) Multiple SNR plots for P₀ = 1 mW, 3 mW, 8 mW and 15 mW. From the curves it is clear that SNR_hybrid ≥ SNR_90°. The shaded red regions indicate the disallowed operating regime due to the limitation of detector saturation.

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4. Hybrid-FRS

Theoretically, the benefit of hybrid-FRS can be explained using SNR calculations. The hybrid-FRS signal equation is derived as balanced detection for an arbitrary α (0 < α < 45°), with the NPP2 polarizer (Fig. 1) and auto-balancing circuit suppressing P_ref by γ = tan²(α) and analyzing the difference P_sig - γP_ref, which yields [24]:

4.1 Comparison of hybrid-FRS to 45°- and 90°-methods

The impact of laser noise suppression through balanced detection in hybrid-FRS is clearly noticeable in Figs. 3(a) and 3(b) which compare the SNR_hybrid with SNR achievable with the conventional 90°-method using experimental parameters relevant to our optical setup. For easier comparison all values of the SNR presented in the Fig. 3 are normalized to the maximum SNR achieved with the hybrid method at P₀ = 8 mW. For the maximum P₀ of 8 mW, the 90°-method achieves optimum operating conditions at α_opt. = 2° with the SNR nearly ½ of that achievable with hybrid-FRS. The hybrid method achieves an optimal SNR at α_opt. = 7.3°, indicating a lower effective laser noise achieved with the balanced detection that permits opening the analyzer further to increase the laser power on the photodetector. Similar calculations were performed for other laser powers with an assumption of the same RIN and NEP. Figure 3(c) shows that in all cases the hybrid method demonstrates better performance than the 90°-method. It is also clear that for low-power lasers (e.g. VCSELs, see the P₀ = 1 mW plot) there may be no optimum point on the SNR plot and the α_opt. becomes 45°, making hybrid-FRS equivalent to the conventional 45°-method.

In the case of low laser power such that the incident power on the signal-photodiode remains always below detector saturation even at an analyzer uncrossing angle of α = 45° (P_sig = P₀/2) (Fig. 3(c)), the 45°-method is usually expected to provide better performance than the 90°-method. This is due to efficient laser noise suppression by balanced detection while operating the FRS system at the point of maximum signal generated at α = 45°. This is also indicated in Fig. 3(c) that shows SNR_90° and SNR_hybrid at P₀ = 1 mW (effectively SNR_45°) as a function of α for low power. If the conventional 45°-method is used the SNR_45° can only be improved through an increase of optical power, but once the detector saturation is reached, further increase of signal in the 45°-method is not possible (see black line in Fig. 2(b)). Therefore at higher laser power it is beneficial to use hybrid-FRS and decrease α to achieve an increase in FRS signal while keeping the P_sig constant and below detector saturation (see red line in Fig. 2(b)). If the laser noise is effectively suppressed by the balanced detector, the shot-noise limited operation can be maintained and SNR follows the P_o^1/2 increase predicted by Eq. (12).

In conclusion hybrid-FRS eliminates the detector-saturation limitation of 45°-method and reduces intensity-noise through balanced detection, thus enabling shot-noise limited operation that is rarely achieved in practice with the conventional 90°-method.

4.2 Power constraints

We now consider minimum and maximum power limitations in hybrid-FRS. It has been observed in Fig. 3(c) for the given experimental parameters, that when P₀ is low (~1 mW), the SNR curve increases monotonically with no local maximum. In this case we see that there exists a minimum power P_0,min under which no benefit is derived from decreasing α, and conventional balanced detection is superior (i.e. one should work as close as possible to 45° when operating below saturation of the balanced-photodetector). One can solve for this condition as the minimum laser power which satisfies d(SNR)/dα = 0 for 0° < α < 45°. The angle α where d(SNR)/dα = 0 is the optimal crossing angle α_opt., which is plotted for varyingP₀ in Fig. 4(a).We calculate that below P_0,min = 1.5 mW our system requires conventional balanced detection, while for P₀ > 1.5 mW hybrid-FRS becomes optimal.

 

Fig. 4 (a) Minimum power constraint. Optimum crossing angle α_opt. vs. P₀ on balanced-photodetector (solid black line). For P₀ < P_0,min no SNR local maximum exists, and conventional balanced detection is superior. The red circle is our experimental operating point. (b) Maximum power constraint. Finite polarization extinction ratio will cause light to leak through the nearly-crossed GTP and WP for large P₀ and small α. The leakage optical power introduces more noise than would otherwise be present. Each curve corresponds to traveling along a line of P_sig = 160 μW in Fig. 5(c), and SNR values are normalized to the maximum SNR at P₀ of 10⁵ mW calculated for the ideal polarizers.

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To optimize system performance, the operating point should lie upon the solid black line in Fig. 4(a), while maximizing available laser power P₀. Our experimental operating point is indicated by the red circle, with α = 8.3° at P₀ = 7.6 mW, which is close to the theoretical α_opt. = 7.5°. Slight deviations of α do not affect the SNR as each point on Fig. 4(a) represents a zero-derivative SNR condition.

To estimate the maximum power limit one has to consider polarization extinction ratio (R_ext.), which has been neglected so far but will play a significant role for large P₀ and small α, as light leakage will be comparable to the total power transmitted through an ideal polarizer. This effect of finite R_ext. manifests itself as a transformation of the reference-photodiode suppression factor γ → γ ∙ ζ in Eq. (9), and signal power P₀ ∙ sin²(α) → P₀ ∙ sin²(α) ∙ ζ in Eq. (10) where ζ = 1 + (γR_ext) ⁻¹. The effect of the non-ideal polarizers is shown in Fig. 4(b) which shows SNR as a function of P₀ for different R_ext. The SNR indeed increases as P₀^1/2 but only until the laser-noise leaking through the polarizers begins to dominate and outweighs any benefit derived from increasing P₀. In our experiment, although the GTP and WP have R_ext. ~10⁴, due to depolarization in the MPC an effective extinction ratio of R_ext. ~10³ has been determined (corresponding to the red curve in Fig. 4(b)). Therefore in our current system the maximum power limit is ~100 mW, beyond which little or no SNR improvement is expected.

5. Hybrid-FRS: measurement results

Hybrid-FRS measurements were performed using the system in Fig. 1. The hybrid system is optimized subject to limitations of detector-noise (dominating at P_sig < 20 µW), intensity-noise (dominating at P_sig > 660 µW), detector saturation (occurring at P_sig = 500 µW) and laser power available for measurement of P₀ = 8 mW. Figure 5(a) shows a plot of total noise as a function of optical power P₀ and analyzer offset angle α derived from experimental measurements of laser- and detector-noise. Since each constraint is delimited by a given P_sig, the curves determining the operation area of the system follow the form of P₀ = P_sig/sin²(α). The laser intensity-noise is calculated assuming a CMRR of 21.6 dB ± 1.8 dB, determined as an average from numerous measurements over different optical powers and split ratios.

 

Fig. 5 (a) Operating parameter space for the hybrid-FRS method. α is the crossing angle between the WP and GTP. P₀ is the total power incident upon the balanced detector. The desired shot-noise region of operation is indicated by the dashed lines. Each of the curves corresponds to a line of constant P_sig; P_sig < 20 µW in detector-noise regime, P_sig > 500 µW for detector saturation and P_sig > 660 µW in intensity-noise regime. The red horizontal line defines the power output limit of the DFB laser after accounting for optical losses of the system. (b) Calculated signals with measurements superimposed (black points), which lie upon a line of constant P_sig = 160 µW (dashed curve). The inset shows four measured 2f DC-FRS signals at varying α. (c) Calculated SNR for hybrid-FRS parameter space. Black points indicate measurements along P_sig = 160 µW. An MDL of 0.6 ppmv·Hz^-1/2 is achieved at α = 8.3° (α_opt. = 7.5°), which is 1.4$\times$ the shot-noise limit. The inset shows a comparison of measured detection limits and those calculated from the SNR at P_sig = 160 µW (the size of the red circles indicate the measurement error).

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It is always desirable to operate in the shot-noise regime (shaded) which provides the ultimate sensitivity for a given power and allows for the ~P₀^1/2 increase in SNR. In other noise regimes the SNR either remains constant with increasing P₀ (intensity-noise regime) or is sub-optimal (detector-noise regime). It is important to note that the shaded area is the desirable parameter space accessible by the hybrid-FRS method, and measurement optimization occurs at the point with the highest SNR.

Within the same parameter space, the FRS signal from ^PP₁(1) O₂ transition is calculated using Eq. (9) and plotted together with the values measured experimentally (Fig. 5(b)). 2f DC-FRS peak signal levels were measured at P_sig = 160 µW, and the Fig. 5(b) inset shows examples of hybrid-FRS spectra of the ^PP₁(1) line acquired for varying α. Asymmetries in the line-shape are due to residual intensity modulation of the DFB laser, and peak signal measurements are in excellent agreement with calculations. In Fig. 5(c) an SNR map has been generated using the signal and noise from Fig. 5(b) and 5(a) respectively. It is clear from the figure that conventional balanced-detection that operates at a constant α (e.g. 45°) will not approach the highest SNR area through a simple increase of the optical power (the detector saturation level will be reached first). However, for hybrid-FRS which operates at constant P_sig = 160 µW, the highest SNR that coincides with the line of the available optical power can be conveniently approached. The minimum detection limit (MDL) of O₂ for each experimental measurement is labeled on the SNR plot and the measured MDL values are in reasonable agreement with the calculated MDL shown in the inset of Fig. 5(c). Taking horizontal cross-sections (fixed P₀) of the SNR map, we re-obtain SNR curves shown in Fig. 3(c).

The 0.6 ppmv·Hz^-1/2 MDL is 1.4$\times$ the shot-noise limit and corresponds to a noise-equivalent polarization rotation Θ_NEA = σ_υ/[(1 + γ)∙sin(2α)∙P₀∙G∙R_i∙H_2f] = 1.6$\times$10⁻⁸ rad·Hz^-1/2 (H_2f = 0.55 is signal reduction due to second harmonic detection). We may also calculate the noise-equivalent fractional absorption sensitivity (NEA) for comparison with other methods. For a 6.8 m path length we observe ~5% optical absorption, which translates to NEA = 1.5$\times$10⁻⁷ Hz^-1/2 and compares very favorably to sensitive absorption spectroscopy techniques [32].

6. Summary

We have demonstrated and characterized a hybrid-FRS system that combines optimization of the analyzer angle α with balanced detection. The system detects atmospheric oxygen with an MDL of 0.6 ppmv·Hz^-1/2 and consumes < 5 W due to application of permanent magnets for magnetic field generation.

We have established hybrid-FRS as a superset of the 90° and 45° FRS detection methods, and the parameter space of operation for our system has been defined. SNR improvements with varying α and P₀ are in good agreement with predictions.

For a given set of experimental parameters, hybrid-FRS will always outperform the 90°-method due to CMRR-suppressed intensity-noise, and eliminates the detector saturation limitation in FRS systems based on 45°-method.

For comparison, Table 1 summarizes alternative atmospheric oxygen detection techniques published in literature. Our result is a 10$\times$ enhancement beyond DC-FRS with a VCSEL emitting only 200 µW [24], and our MDL is also a significant improvement over alternate optical techniques (WMS, QEPAS). Furthermore, FRS is a dispersion-based measurement, thus providing inherently linear response to analyte concentration (in contrast to conventional absorption-based systems, in which the upper bound of dynamic range is limited to ≤ 10% absorption according to the Beer-Lambert Law) [28, 33]. For higher O₂ concentrations with absorption of > 10% (in our case this corresponds to > 40% O₂ levels for 6.8 m path length), power normalization of the FRS signal is required, which can be conveniently implemented using the DC optical signal measured with the photodetectors. Finally, the current hybrid-FRS system requires low power for operation (< 5 W), making it desirable for field deployable applications.

Table 1. Summary of alternate optical methods to detect atmospheric oxygen.

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