Faraday rotation spectroscopy based on permanent magnets for sensitive detection of oxygen at atmospheric conditions

Brian Brumfield; Gerard Wysocki

doi:10.1364/OE.20.029727

1. Introduction

Molecular oxygen is a dominant component of the atmosphere that plays a vital role in life on Earth. Monitoring of molecular oxygen is useful for a variety of medical, environmental, and industrial applications. Given the diverse number of applications, a variety of oxygen sensors have been developed using electrochemical [1, 2], optical [3–6], mass spectrometric [7], and magnetodynamic detection methods [1, 8].

In the case of environmental monitoring, precise and accurate measurements of the oxidative ratio (O₂:CO₂) can provide useful information regarding terrestrial CO₂ exchange due to biotic and industrial processes [3, 9]. Knowledge of the oxidative ratios for different environments can provide useful data regarding the partitioning and uptake of terrestrial and anthropogenic CO₂ that are of importance for modeling the global carbon cycle. Monitoring of changes in the oxygen concentration associated with biotic processes requires a measurement precision on the same order as for CO₂, and ideally should target 1 ppmv (parts-per-million by volume) in a 1 s integration time [3, 9]. In contrast to measurement of CO₂, which is 600 times less abundant, environmental monitoring of O₂ is not a trace measurement and represents a significant challenge. In situ measurements of oxygen require a detection method with significant dynamic range capable of measuring a 1 ppmv change in a 208,000 ppmv concentration of atmospheric O₂. This level of sensitivity/dynamic range is extremely challenging and cannot be achieved using current commercially available electrochemical or magnetodynamic instrumentation. Moreover, the electrodes used in electrochemical-based sensors have a finite lifetime on the order of a year (or less), and magnetodynamic sensors are sensitive to mechanical vibration, illustrating their lack of robustness for field deployments.

In contrast, optical methods for O₂ sensing can provide robust systems that are useful for unattended field deployments. A variety of optical detection techniques have been applied for oxygen sensing, and these include interferometric [9], photoacoustic [4, 10], direct absorption [11, 12], wavelength modulation spectroscopy [5], and balanced detection techniques [6]. Laboratory and field studies have shown that interferometric [9] and vacuum ultraviolet absorption detection methods [12] are capable of detection limits equal to or less than 1 ppmv by averaging over minutes, however these systems are not small, portable, or inexpensive.

Laser-based absorption spectrometers using VCSELs or DFB diode lasers to target transitions in the A electronic band of oxygen can provide compact, portable, and cost-effective systems for in situ monitoring of changes in oxygen concentration. Laser-based sensing also opens up the possibility of monitoring the ¹⁶O¹⁸O isotopomer of molecular oxygen for isotope labeling techniques, providing another tool for exploring the evolution and consumption of oxygen and carbon dioxide during plant respiration and photosynthesis [13]. While many of the laser-based detection methods can reach a 1σ detection limit on the order of 1 ppmv for trace measurements, to the best of our knowledge, no study published to-date has been able to reach the desired dynamic range enabling in situ measurements of atmospheric oxygen levels performed at atmospheric pressure. Typically, laser-based direct absorption systems operate at a nominal absorption level of ≤10% (or ≤0.1 in absolute units, which assures operation within an approximately linear regime of Beer-Lambert’s Law) and the best systems can achieve minimum detectible absorption down to 3 × 10⁻⁶ at a 1 s integration time [14, 15]. This corresponds to a dynamic range of ~3 × 10⁴, which is an order of magnitude lower than the dynamic range required for oxygen monitoring in biotic processes. Therefore the best achievable precisions for laser absorption systems for atmospheric oxygen monitoring fall between 10 and 100 ppmv/Hz^1/2 range.

Faraday rotation spectroscopy (FRS) is another laser-based detection method that can be used for oxygen sensing [16–19] as well as for free radical detection [20]. In this method, the Faraday Effect is used to enhance the sensitivity for detecting molecular species whose ground or upper electronic states have a magnetic dipole moment. The ground electronic state of molecular oxygen (³Σ_g^-) has a permanent magnetic dipole moment because of unpaired electron spins. The application of a magnetic field breaks the degeneracy of the magnetic sublevels for the ground electronic state rotational energy levels (the Zeeman Effect). Laser light propagating co-linearly to the applied magnetic field can only promote transitions between the ground and excited electronic states if the change in the magnetic quantum number (m_J) is + 1 or −1. The Zeeman split transitions involving Δm_J = + 1 or Δm_J = −1 interact with right-handed circularly polarized (RHCP) or left-handed circularly polarized (LHCP) light respectively, which causes magnetically induced circular birefringence (MCB) in the vicinity of the target transition. If linearly polarized light is used, the MCB results in a rotation of the polarization plane of the transmitted light (the Faraday Effect). The polarization rotation can be converted into a measured intensity change by placing a linear polarizer after the sample. The sensitivity achievable with FRS for paramagnetic molecules is usually 2 to 3 orders of magnitude greater than what can be obtained with direct absorption spectroscopy techniques [21]. Most importantly, the dynamic range of concentration measurements in FRS is significantly improved because it is based on the measurement of optical dispersion, which is linear with concentration.

Our previous work has illustrated the promise of a compact low-power VCSEL-based FRS sensor with a 1σ detection sensitivity of 30 ppmv-m/Hz^1/2 (normalized to the optical path and measurement bandwidth) when operating at a reduced pressure of 225 Torr [19]. This system featured compact low-power electronics (<0.3 W) for operation of the VCSEL, but carried a significant power burden because a vacuum pump was needed to maintain a reduced sample cell pressure (up to ~50 W). The solenoid coil used to generate the AC magnetic field added an additional ~30 W of power consumption. In this paper a new, low-power (~4 W total power consumption), and potentially field deployable FRS oxygen sensor is presented. The sensor operates at atmospheric pressure and provides a bandwidth and path-normalized sensitivity of 40.8 ppmv-m/Hz^1/2 with an ultimate minimum detection limit of 1.3 ppmv achievable after a 60 s averaging time. The low-power consumption was made possible by the application of permanent rare earth magnets that provide a strong static magnetic field (DC-field). Application of DC-fields in FRS (DC-FRS) has been demonstrated in previous work (e.g. by Brecha and Pedrotti [17], or McCarthy et al. [22]). However, sufficient suppression of both the laser noise and absorption effects required by high-dynamic range concentration measurements are difficult to realize with a conventional approach using a single detector and nearly-crossed polarizer configuration used in Ref [17]. Additional suppression of laser noise in DC-FRS based on a single detector configuration can be achieved by shifting the detection to higher frequencies where 1/f laser intensity noise is significantly reduced. This approach was demonstrated in Refs [22–25], which achieved near quantum shot noise limited performance (see Ref [24, 25].) at the cost of higher complexity for a system based on heterodyne detection. In this work we have focused on the development of a DC-FRS system based on a relatively simple optical arrangement comprised of a multi-pass cell (MPC) and a balanced photodetection unit that allowed for FRS signal retrieval and effective intensity noise suppression. The system achieves quantum shot noise limited performance with a simple optical setup while preserving low power consumption. The experimental details and theoretical modeling are discussed in the following sections.

2. Experimental layout

The layout for the DC-FRS experiment is schematically shown in Fig. 1 . Rovibronic transitions in the A electronic band of O₂ are targeted using a polarization-stable vertical cavity surface emitting laser (VCSEL) (Avalon Photonics). The integrated TEC and thermistor in the VCSEL package are used in conjunction with a benchtop temperature controller (ThorLabs TED200C) to stabilize the laser temperature. The laser wavelength can be scanned between 762.3 to 764.7 nm by varying the laser temperature between −15°C and 26°C. This wavelength coverage permits measurement of P and Q-branch O₂ molecular transitions in the A electronic band. The VCSEL laser current is controlled using a dedicated benchtop current controller (ThorLabs LC200CV). To enable rotation of the laser polarization plane without introduction of a half-wave plate in the optical layout, the VCSEL is housed in a rotatable cage mounting system (ThorLabs CRM1) that allows physical rotation of the laser package. A custom LabView program is used for data acquisition and synchronized control of the laser current and temperature via analog signals from a DAQ board (National Instruments USB-6529) provided to the modulation inputs on the temperature and current controller.

Fig. 1 (a) Experimental layout of a DC-FRS instrument. The components and connections that are displayed with dotted lines are only used when acquiring the O₂ FRS signal using the line-locked mode of operation. The experimental components in the diagram are labeled as followed: (PC) computer running LabView code for laser control and data acquisition, (LD) laser current driver, (TC) laser temperature controller, (VCSEL) vertical cavity surface emitting laser, (GT) Glan-Thompson polarizer, (MPC) cylindrical mirror MPC enclosed by octupole magnet arrangement, (WS) Wollaston polarizer, (ABD) auto-balancing photodetector with photodiodes 1 and 2 labeled (PD1) and (PD2) respectively, (LIA 1,2) lock-in amplifier 1, and 2, (PID) electronic proportional-integral-derivative controller, (FXG) function generator providing the sine wave for laser wavelength modulation and the reference frequency for phase sensitive detection. (b) The concept of polarization angle modulation (PAM) created through wavelength modulation (WM) over a spectrum of the static difference in indices of refraction for circularly polarized light components induced by the DC-magnetic field. The input laser light polarization before the MPC is shown with a plane of polarization at an arbitrary angle θ. The xy axes have been selected to coincide with the two orthogonal polarization components emerging from the Wollaston polarizer that are incident on PD1 and PD2.

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Divergent light emerging from the VCSEL is collimated using a glass aspheric lens and directed to a telescope composed of two AR-coated lenses with 100 mm and −100 mm focal lengths (ThorLabs LA-1509-B and LC1120-B). The focal properties of the VCSEL laser beam were optimized for coupling into the MPC by setting the separation between these lenses to 5 cm. The beam emerging from the telescope passes through an AR-coated calcite Glan-Thompson polarizer (FocTek GMP7210-AR760nm) used to clean-up the laser light polarization before coupling it into the MPC.

The MPC is composed of a pair of protected silver cylindrical mirrors with 100 mm focal lengths (Lambda Research Optics, Inc. CCPC-50.88-200) that are held in rotatable cage mounts (ThorLabs LCRM2) separated by ~110 mm. A cylindrical MPC was selected because the unique in-plane geometry of the input and output laser beams simplifies the system optical alignment. This cell design also provides a greater number of passes than a conventional Herriott cell in a relatively small active volume [26], which is ideal for optimizing the overlap between the laser beam and the relatively small region of high magnetic field produced by the permanent magnet arrangement. The beam is introduced to the MPC at a 7° angle to produce a spot pattern on the mirrors that is sufficiently spread out around the centered 4 mm diameter through-hole (schematically shown in Fig. 1) so that beam overlap or clipping is prevented. The number of passes in this MPC is controlled by the relative angle between the mirror’s cylinder axes and the ratio of the mirror separation distance to the mirror focal lengths [26]. In the current configuration of the cell, one of the mirrors was rotated until a 62-pass pattern was observed. This configuration provides 6.8 meters of active optical path length through the magnetic field.

The magnetic field within the MPC is produced by an octupole magnet arrangement composed of 20 cylindrical rare earth magnets (K&J Magnetics RX04X0) grouped into four magnetic dipoles with 5 elements each. The cylindrical magnets have ¼” diameter through holes that make it easy to install the segments onto the 6 mm dia. rods mounted in a 60 mm cage structure. The axial magnetic field profile along the center of the magnet arrangement has been measured using a Gaussmeter (Lakeshore Model 410) and is shown in Fig. 2 . The axial magnetic field approaches zero at the physical edges of the 127 mm long magnet arrangement. As shown in Fig. 2, the magnetic field produced by this magnet arrangement extends significantly beyond the region inside the magnets. A reversal in the direction of the axial magnetic field occurs at the physical edge of the permanent magnets and its intensity increases sharply outside the magnet arrangement, reaching a peak field of ~370 G at ~28 mm distance from either side. If atmospheric oxygen is present in the laser beam path outside the gas cell this opposite external magnetic field would also provide an additional Faraday rotation, but in the opposite direction with respect to the signal created inside the cell. This effect partially cancels the Faraday rotation produced inside the cell [18]. The MPC marginalizes this cancellation effect by significantly increasing the optical path inside the magnet arrangement where the average axial magnetic field strength over a 110 mm distance between the mirrors is 554 ± 64 G.

Fig. 2 The axial magnetic field profile measured along the center of the octupole magnet arrangement. The area highlighted under the axial magnetic field profile emphasizes the 110 mm region enclosed by the MPC.

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Light exiting the MPC is split by a calcite Wollaston prism (ThorLabs WP10) into two orthogonal polarization components that are incident on a pair of silicon photodiodes in a balanced photodetection unit. A manually balanced photodetection unit (ThorLabs PDB210A) and an auto-balancing photodetection unit (Nirvana Model 2007) have been tested in the system presented in Fig. 1. For the manually balanced photodetection unit the Glan-Thompson must be carefully adjusted to 45° with respect to the optical axes of the Wollaston prism to achieve equal power in both beams incident on the photodetector elements. Precise balancing is necessary to achieve optimum cancellation of the laser technical noise through the common mode rejection ratio (CMRR) of the balanced photodetector unit. This process is simplified with an auto-balancing photodetector, which prevents any potential reduction in the CMRR due to imperfect balance of optical powers on both detector elements and relaxes the need for precise manual alignment of the system. The Nirvana unit allows for a CMRR of >30 dB over 125 kHz bandwidth. In the auto-balancing mode, the optimum power split ratio between the reference and signal photodiode of 1.6 to 2.0 is recommended by the manufacturer, which requires the Glan-Thompson polarizer to be set between 51.8° and 54.7° angle with respect to the incident polarization plane.

When the VCSEL frequency is not coincident with an oxygen transition the polarization state of the light is not altered by the gas sample and the resulting signal from the balanced photodetector is zero. As the laser frequency is scanned over an oxygen transition, the MCB of the sample results in rotation of the polarization plane of the linearly polarized laser light. This polarization rotation unbalances the optical powers incident on the balanced photodetector and generates a net signal that can be measured. To reduce the detection bandwidth and improve noise rejection, laser wavelength modulation is combined with phase sensitive signal detection in the DC-FRS system. The wavelength of the VCSEL is modulated using a 10 kHz sine wave from a function generator (Tektronix AFG2002) provided to the modulation input on the laser current driver. Since the MCB causing the Faraday Effect is wavelength dependent, the laser wavelength modulation causes a modulation of the polarization angle as schematically shown in Fig. 1(b). For a more detailed description of signal generation using phase-sensitive detection in DC-FRS please see Fig. 1 in Ref [19]. The FRS signal, which is proportional to the polarization angle, is then demodulated at the first, second, or third harmonic (1f, 2f, 3f) of the modulation frequency using a lock-in amplifier (Signal Recovery Model 7265). The in-phase (x) and quadrature (y) components measured by the lock-in are recorded by the computer. The system is capable of being operated in two modes: 1) spectral scanning mode performed by temperature scanning of the laser wavelength, and 2) a line-locked mode that allows for active stabilization of laser wavelength at the center of the target transition. Both modes have been used in this work as indicated later in the text.

3. Derivation of DC-FRS signals with balanced detection

To analyze the photodetected signals as a result of the sample MCB and MCD, a theoretical model is presented using Jones calculus [27]. In the first part of this section a general derivation for the optical powers incident on each photodiode for linearly polarized light with a plane of polarization at an arbitrary angle will be described.

The polarization of the light before the MPC and after the Wollaston prism is shown in Fig. 1. The plane of polarization of the linearly polarized VCSEL light after the Glan-Thompson polarizer can be described by the following Jones vector:

J_{0} = [\begin{matrix} E_{x, 0} \\ E_{y, 0} \end{matrix}] = E_{0} [\begin{matrix} \cos θ \\ \sin θ \end{matrix}]

The laser light enters the MPC where the oxygen sample exposed to an axial magnetic field induces wavelength dependent MCB and MCD. When the light frequency is resonant with a Zeeman split oxygen transition both the polarization state as well as the absorption of light interacting with the sample are altered. The combined effects of MCB and MCD can be expressed in a single Jones matrix by using the N-matrix formalism [28, 29]:

J_{s a m p l e} = \exp (- \frac{1}{4} (α_{L} + α_{R}) L) \times [\begin{matrix} \cos h (κ + i Θ) & - i \sin h (κ + i Θ) \\ i \sin h (κ + i Θ) & \cos h (κ + i Θ) \end{matrix}]

where the rotation of the polarization plane of the light due to sample MCB is

Θ = (π L / λ) \times (n_{L} - n_{R})

(Faraday rotation proportional to active optical path length, L, and the difference in the indices of refraction

(n_{L} - n_{R})

for LHCP and RHCP respectively). Though not explicitly shown, the Faraday rotation angle

Θ

depends on the strength of the applied magnetic field, the magnetic g-factor for the transition, and the absorption line intensity [30]. The MCD from the sample is

κ = (L / 4) \times (α_{L} - α_{R})

, where

α_{L}

and

α_{R}

describe the absorption coefficient for LHCP and RHCP light respectively.

α_{L}

,

α_{R}

,

n_{L}

, and

n_{R}

are all optical frequency dependent quantities that alter the propagation of light through the sample, and in the presence of wavelength modulation of the VCSEL all of these properties become functions of time. In the following derivation the wavelength modulation is not introduced for the sake of simplicity.

The light exiting the sample is then passed through a Wollaston polarizer that will split the linearly polarized light into its orthogonal components. The orientation of the Wollaston polarizer in the current coordinate system results in orthogonal polarization components only along the x or y-axis. Each photodiode in the balanced detection system will then only see light from one of these polarization components. From the perspective of each photodiode, the Wollaston polarizer is acting as a linear polarizer that selectively separates electric field vectors oscillating either along the x or y direction. Therefore the resulting Jones vectors for light incident on the photodiodes can be described by:

J_{P D} = [\begin{matrix} E_{x} \\ E_{y} \end{matrix}] = J_{W o l l a s t o n} \times J_{S a m p l e} \times J_{0}

J_{P D 1} = [\begin{matrix} E_{x} \\ 0 \end{matrix}] = E_{0} β \times [\begin{matrix} 1 & 0 \\ 0 & 0 \end{matrix}] \times [\begin{matrix} \cos h (κ + i Θ) & - i \sin h (κ + i Θ) \\ i \sin h (κ + i Θ) & \cos h (κ + i Θ) \end{matrix}] \times [\begin{matrix} \cos θ \\ \sin θ \end{matrix}]

J_{P D 1} = [\begin{matrix} 0 \\ E_{y} \end{matrix}] = E_{0} β \times [\begin{matrix} 0 & 0 \\ 0 & 1 \end{matrix}] \times [\begin{matrix} \cos h (κ + i Θ) & - i \sin h (κ + i Θ) \\ i \sin h (κ + i Θ) & \cos h (κ + i Θ) \end{matrix}] \times [\begin{matrix} \cos θ \\ \sin θ \end{matrix}]

where

β

is a substitution for the exponential term (

\exp (- (α_{L} + α_{R}) L / 4)

) in

J_{W o l l a s t o n}

that describes the attenuation of the electric field by the sample. The matrix multiplication provides the electric field amplitudes seen by each photodiode:

E_{x, P D 1} = E_{0} β [\cos θ \cos h (κ + i Θ) - i \sin θ \sin h (κ + i Θ)]

E_{y, P D 2} = E_{0} β [\sin θ \cos h (κ + i Θ) - i \cos θ \sin h (κ + i Θ)]

The light intensity is given by the product of the electric field amplitude with its complex conjugate I = E∙E*. This operation can be conveniently performed through the use of DeMoivre’s theorem combined with hyperbolic trigonometric identities to express the electric field amplitudes with separated real and imaginary parts. Because the intensity is proportional to the optical power

(P \propto I)

, the power incident on each photodiode is:

P_{P D 1} = P_{0} β^{2} \times (\frac{\cos h 2 κ}{2} + \frac{\cos 2 Θ \cos 2 θ}{2} + \frac{\sin 2 Θ \sin 2 θ}{2})

P_{P D 2} = P_{0} β^{2} \times (\frac{\cos h 2 κ}{2} - \frac{\cos 2 Θ \cos 2 θ}{2} - \frac{\sin 2 Θ \sin 2 θ}{2})

and the photocurrent generated by each photodiode can be calculated as:

A_{P D} = \frac{η e}{h ν} P_{P D}

where the photodiode responsivity (in A/W) given by (ηe/hν) is composed of the quantum efficiency of the detector (η), the charge of an electron (e in Coulombs), and the energy of the measured photon hν (where h is the Planck’s constant and ν is photon frequency).

At 𝜃 = 45° a differential measurement of the two photocurrents results in the following expression for the signal measured at the output of the balanced photodetector unit:

V_{s i g n a l - 45^{\circ}} = R_{V} (A_{P D 1} - A {}_{P D 2}) = R_{v} \frac{η e}{h ν} P_{0} β^{2} \times \sin 2 Θ

where R_v is the transimpedance gain (in V/A) applied to the differential photocurrent. From Eqs. (8) and (9) it can be seen that theoretically the MCD effects are cancelled regardless of the value of 𝜃. However, in practice all common mode signals are suppressed to the level defined by the CMRR of the detection unit and care must be taken in assuring that all unwanted contributions are sufficiently reduced. With this technical detail in mind and with an assumption of small

Θ

, the observed signal is linearly proportional to the rotation angle induced by the Faraday Effect associated with the oxygen transition:

V_{s i g n a l - 45^{\circ}} = R_{v} \frac{η e}{h ν} P_{0} \exp (- \frac{1}{2} (α_{L} + α_{R}) L) \times 2 Θ

where the exponential term describing sample absorption has been substituted back into the Eq. in place of

β^{2}

. Equation (12) is in agreement with a derivation of the differential FRS signal using the model presented by Adams et al. [31] if the sample absorption term is neglected. For the experiments presented in this paper, the total absorption for the ^pP₁(1) transition of O₂ is around 5% for 6.8 meters of optical path length in open air, which effectively reduces the signal. Therefore in high dynamic range FRS measurements in which sample absorption can be significant this term should not be completely omitted.

The expression for the signal measured with the auto-balanced photodetector unit requires a small modification compared to the specialized case of 𝜃 = 45° (valid only for the manually balanced configuration). The auto-balancing circuit compensates for imbalances in the currents generated by the photodiodes through an active feedback loop that reduces the current of one of the photodiodes by a factor of r allowing for measurements at 𝜃≠45°. The photodiode to which this fractional current gain r is applied is referred to as the reference photodiode, while the other photodiode is termed the signal photodiode. The active feedback loop is able to compensate for changes in photocurrents up to a cut-off frequency whose maximum value is dependent on the ratio of the splitting of the optical powers and the total optical power incident on the signal photodetector. The highest cut-off frequency determined by the available optical power from the VCSEL is only 1.5 KHz. Since the VCSEL wavelength is modulated at 10 KHz, which is above the cut-off, the measured signal is not affected and the feedback loop of the detector is primarily compensating for the difference in the DC photocurrents from both detectors. By defining PD1 and PD2 as the signal and reference photodiodes, the fractional current gain, r, applied to the reference photodiode is the ratio of the DC photocurrents for PD1 and PD2. The DC photocurrents estimated from Eq. (8) and Eq. (9) with an assumption that static MCD/MCB effects are negligible $(κ \approx 0 and Θ \approx 0)$ yield the following expression for r:

r = \frac{A_{P D 1 - D C}}{A_{P D 2 - D C}} = \frac{\cos^{2} θ}{\sin^{2} θ}

Since the PD2 photocurrent is multiplied by this gain, the balance is automatically restored and the common mode intensity noise within the entire detector bandwidth (DC-125 KHz) is effectively suppressed through the differential measurement. It should be noted that this factor was calculated with an assumption of the optical power incident on PD1 (used as a signal photodiode) to be smaller than optical power on PD2 (reference photodiode). Therefore this and further Eqns. are valid only for 45°<𝜃<90°.

Besides the static MCD/MCB effects, the κ and $Θ$ also contain time-varying components as a result of wavelength modulation. These time-varying components are the target of the FRS measurement. A small signal approximation can be performed to extract those signals from Eqs. (8) and (9). With κ and Θ being small and close to zero, a first order approximation using a Maclaurin series (only the zero- and first-order terms) is used to derive the effective photocurrents from the reference and signal detector that are delivered to the differential stage. With the PD2 photocurrent value modified to reflect the autobalancing correction by r, both signal and reference photodetector photocurrents can be calculated as:

A_{s i g} = A_{P D 1} = \frac{η e}{h ν} P_{0} β^{2} \times (\cos^{2} θ + Θ \sin 2 θ)

A_{r e f} = r \times A_{P D 2} = r \times \frac{η e}{h ν} P_{0} β^{2} \times (\sin^{2} θ - Θ \sin 2 θ)

A differential measurement of the two photocurrents yields a signal at the output of the balanced photodetector unit that can be expressed as:

V_{s i g n a l} = R_{v} (A_{s i g} - A_{r e f}) = R_{v} \frac{η e}{h ν} P_{0} \exp (- \frac{1}{2} (α_{L} + α_{R}) L) (1 + r) \times Θ \sin 2 θ

As expected, the auto-balancing circuit leads to a complete subtraction of the DC photocurrents. Similarly to the 𝜃 = 45° case, the output signal is linearly proportional to the rotation angle induced by the Faraday Effect associated with the oxygen transition. However it should be noted that at 𝜃≠45° the factor of sin2𝜃 that relates the FRS signal to the analyzer offset angle is less than unity and thus must be considered in the auto-balancing configuration. This factor is consistent with derivations of FRS signals for 90° FRS systems that work with nearly crossed analyzers [32]. Also, there is a slight reduction of the signal due to the fractional gain r that affects the photocurrent from the reference element of the balanced photodetector. Therefore, instead of factor of 2 that is present in Eq. (12), the signal in the auto-balanced case is multiplied by a factor of (1 + r), which can be interpreted as a reduction of the effective optical power available for the measurement. By taking into account both pre-factors for the auto-balanced signal voltage, there is a factor of (1 + r)sin2𝜃 reduction in the FRS signal for 𝜃 between 45° and 90°. The experimental verification has been performed and confirmed that the measured FRS signal for the ^pP₁(1) of oxygen at 𝜃 = 52.5° is 26.8% smaller than at 𝜃 = 45°. This agrees reasonably well with the 23.3% signal loss predicted by Eq. (16). Despite the small loss in the signal amplitude, auto-balanced detection is the preferred mode of operation because it removes the need for precise manual alignment of the Glan Thompson polarizer and assures optimum intensity noise cancelation and immunity to slow opto-mechanical drifts that can unbalance the photodetectors.

3. Experimental results

A typical 1f DC-FRS spectrum of atmospheric oxygen acquired at atmospheric pressure using the cylindrical mirror MPC is shown in in Fig. 3 . The spectrum was obtained using the manually balanced Thorlabs PDB210 photodetection unit, which required careful balancing of the optical powers on the detectors. The VCSEL temperature was gradually decreased over the duration of the scan to acquire a spectrum over an 8 cm⁻¹ range. The VCSEL was wavelength-modulated using a 10 kHz sinusoidal waveform with a modulation depth that was optimized empirically to maximize the 1f and 2f DC-FRS signals for the ^pP₁(1) transition of ¹⁶O₂. This corresponded to an optimum modulation depth of ~0.1 and ~0.2 cm⁻¹ for the 1f and 2f DC-FRS signals respectively. The ^pQ₃(2) 1f signal has an opposite phase with respect to the two P-branch transitions because the g-factors associated with the energy levels in the Q-branch transitions in molecular oxygen have g-factors of opposite sign. There is also a noticeable asymmetry in the 1f DC-FRS signals for the ^pP₃(3) and ^pQ₃(2) lines in comparison to the ^pP₁(1) line. This asymmetry has been observed with manual and auto-balanced detection and cannot be justified by changes in the VCSEL power during the scan. The analysis of this asymmetry is currently under investigation and will be a focus of future work on accurate modeling of the DC-FRS signals.

Fig. 3 (a) Example 1f DC-FRS spectrum containing three lines of O₂ in the A electronic band acquired at atmospheric pressure (detection bandwidth 0.5 Hz, active optical path 6.8 m). (b) Trace A shows an expanded view of the spectrum shown in Fig. 3(a) plotted together with another spectrum recorded 75 minutes later under identical conditions (Trace B). The subtraction of the two spectra is shown in Trace C. DC-FRS signals from the minor ¹⁶O¹⁸O isotopologue are clearly visible and can be used for isotopic content quantification.

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The quality of the linear polarization of light before the cell was measured experimentally and showed an extinction ratio between 50,000:1 to and 80,000:1. After passing through the cell, the extinction ratio was reduced to ~10:1. Despite this deterioration of the polarization quality strong DC-FRS signals were observed, which illustrates the robustness of the balanced detection FRS method to polarization degradation due to multiple reflections off mirror surfaces [33]. Optimization of the optical arrangement by minimizing the number of elements used to couple the light into and out of the MPC improved the extinction ratio to ~100:1 (measured for the entire polarizer/MPC system). This optical arrangement was then used for further measurements.

The signal-to-noise ratio (SNR) for the 1f DC-FRS signals can be explored by examining the noise in the baseline of the spectrum in Fig. 3(a.) Fig. 3(b) shows an expanded view of Fig. 3(a) between the ^pQ₃(2) and ^pP₁(1) transitions (Trace A). A baseline structure which contains both spectral features as well as optical fringes is clearly noticeable.

In general, optical fringing is expected to be suppressed by using the balanced detection method. However, in the case of strong optical fringes and a finite CMRR of the balanced photodetection unit, the cancellation might be incomplete and some residual fringing could be visible in the DC-FRS spectrum. The examination of the free spectral range of the observed etalon fringes suggests that the Glan-Thompson polarizer is the likely source of fringes.

To determine the system SNR the stability of the baseline was evaluated. Another spectrum (Fig. 3(b) Trace B) was acquired under the same conditions 75 minutes after the first spectrum (Fig. 3(b) Trace A). The subtraction of the two spectra shown as Trace C in Fig. 3(b) provides evidence of residual optical fringes, which suggest a drift of the fringe pattern observed in Trace A. The standard deviation of Trace C provides an estimate of the baseline stability over 75 min interval. The ratio of the peak-to-peak amplitude of the ^pP₁(1) to the calculated standard deviation provides an estimated SNR of 18520:1 at a 0.5 Hz detection bandwidth. Assuming a stable 208,000 ppmv concentration of oxygen in the air, this SNR yields a bandwidth normalized 1σ detection limit of 1f DC-FRS of 16 ppmv/Hz^1/2.

4. Continuous O₂ monitoring

To gain a better understanding of the system performance in target applications, continuous monitoring of O₂ has been implemented by locking the laser frequency to the peak of the oxygen transition. The FRS signal is monitored using the 2f DC-FRS signal for the ^pP₁(1) transition of ¹⁶O₂. As shown in Fig. 4 Trace B, the maximum of the 2f DC-FRS spectrum occurs at the center of the O₂ line. The zero-crossing of the 3f DC-FRS spectrum, shown in Fig. 4 as Trace A, coincides with the center of the transition and can be used as an error signal for active stabilization of laser frequency. A pair of lock-in amplifiers (Signal Recovery Model 7265) are used to demodulate the DC-FRS signals at 2f and 3f. The laser modulation depth was optimized to maximize the 2f signal, and the 3f signal was fed into an electronic PID (proportional-integral-derivative) controller that adjusted the laser frequency by modifying the injection current to the VCSEL to maintain the 3f signal at zero. The injection current was controlled by adding an offset voltage to the laser wavelength modulation signal waveform provided to the laser current driver modulation input. These studies were carried out using the system configuration shown in Fig. 1.

Fig. 4 Example of 2f (Trace B) and 3f (Trace A) DC-FRS spectra of the ^pP₁(1) transition of ¹⁶O₂ acquired simultaneously as the VCSEL temperature is gradually tuned over the oxygen transition. The analog voltage outputs from the lock-in amplifiers (detection bandwidths of 2.5 Hz) are plotted as a function of the laser temperature acquired at the temperature controller output. The data was acquired using the Nirvana photodetector in auto-balance mode.

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While actively locking to the peak of the 2f DC-FRS signal, the measured FRS signal was recorded continuously over 10,000 seconds. The detection bandwidth was set to 2.5 Hz and data points were acquired with time intervals of 0.28 seconds. A typical time series of the 2f peak signal is shown in Fig. 5 Trace A. Simultaneously the optical power was recorded using an output signal from one of the photodiodes in the Nirvana balanced photodetection unit. Short-term drifts in the monitored optical power are well correlated with changes in the 2f signal as shown in Fig. 5. By normalizing the 2f signal to the monitored optical power the short-term drifts can be effectively reduced (Trace C in Fig. 5). It should be noted that this measurement is performed with an open MPC in the laboratory environment, which makes the discrimination between the actual oxygen concentration change and system drift difficult. A long-term drift in the measured 2f signal corresponds to a gradual (166 ppmv or relative 0.08%) increase in the oxygen concentration over a period of 2 hours, which could be caused by changes in the building ventilation rates. The Allan variance deviation plot of the power-normalized 2f signal depicted in Fig. 6 as Trace A shows an ultimate detection limit of 6 ppmv (1σ) obtained after 20 seconds of integration. The bandwidth normalized 1σ sensitivity estimated from the line-locked data corresponds to 11 ppmv/Hz^1/2 of O₂ in the air.

Fig. 5 Time series of the 2f signal peak for the ^pP₁(1) transition of ¹⁶O₂ (Trace A) while actively line-locking to the zero-crossing voltage of the 3f DC-FRS signal. Trace B shows optical power measured at the signal photodiode output in the Nirvana balanced photodetection unit. Trace C is the peak 2f signal from Trace A normalized to the optical power in Trace B.

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Fig. 6 Comparison of the Allan deviation obtained by monitoring the 2f signal at the center of the ^pP₁(1) transition of oxygen with active line-locking engaged (Trace A) and by measuring the value of the 2f signal in the baseline away from the target transition without active wavelength locking (Trace B). The right y-axis provides the voltage noise measured with the lock-in amplifier. The 2f DC-FRS signal was converted into the concentration units (ppmv) shown on the left y-axis. The initial detection limits at 0.28 s integration time are provided for both traces in the labels on the left, and the ultimate minimum detection limit values are provided in the labels above the minima observed in the Allan plots.

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To eliminate the influence of sample concentration drifts the ideal test for the ultimate system stability should be carried out with the entire MPC system enclosed in an environment with a known and stable oxygen concentration, or devoid of oxygen. Because this was difficult to realize with the current optical benchtop prototype, the ultimate system sensitivity was evaluated by monitoring the 2f baseline stability away from the oxygen transition. The baseline stability test is expected to provide a better estimate of the ultimate system minimum detection limit than the test obtained from the measurements of the ^pP₁(1) 2f peak signal using the laboratory air as the sample gas.

The baseline stability test was performed by setting the VCSEL wavelength in the spectral region between the ^pP₁(1) and ^pQ₃(2) transitions of ¹⁶O₂, such that it does not overlap with any of the weaker lines from the other oxygen isotopologues. The VCSEL wavelength was not actively stabilized. Given the system performance is primarily limited by the optical fringes in the background, this test represents a worst case scenario because the drift of the 2f signal will reflect a combination of the VCSEL wavelength drift and the variations in the parasitic optical fringes. Similar to a conventional wavelength modulation spectroscopy (WMS) measurement, the stability of the optical fringes should set the ultimate long-term minimum detection limit of the system. In this configuration the baseline 2f signal was recorded over a 3000 s. The resulting Allan variance deviation plot from the 2f baseline data is shown in Fig. 6 as Trace B.

Allan plots resulting from the line-locked mode and from the baseline stability studies show clear differences. First, the detection limit at short integration times for the line-locked mode is a factor of two higher than the one from the baseline stability measurements. This is a result of a residual laser frequency jitter observed in the active line-locked mode that is translated into noise in the 2f DC-FRS signal. This has been verified by comparing the measured noise in the 2f signal with a 2f noise modeled based on noise observed in the 3f signal (used for line-locking). Good agreement was found between the two 2f noise levels. Therefore the laser frequency jitter acts as a source of proportional noise that scales with the oxygen concentration and becomes dominant for high atmospheric levels of oxygen. Although the deterioration is not severe, to suppress this noise an improvement in the precision of the line-locking feedback loop must be implemented. Second, the minimum detection limit achieved for the baseline stability measurement after 60 seconds of integration is 4.6 times smaller than what was observed in the line-locked mode. As mentioned previously, the drift of the signal in the line-locked mode is likely a result of real changes in the oxygen concentration, while the 1.3 ppmv detection limit found through baseline stability analysis represents the ultimate minimum detection limit of the system. This yields the ultimate bandwidth-normalized 1σ minimum detection limit of 6 ppmv/Hz^1/2.

The system sensitivity can also be expressed in terms of a noise equivalent polarization rotation angle $(Θ_{N E A})$ , which is a measure of a minimum detectible polarization rotation introduced by the sample. This figure of merit allows for easy comparison with other FRS systems in the literature. The $Θ_{N E A}$ can be estimated by setting the signal in Eq. (16) to be equal to the voltage noise measured by the lock-in amplifier and solving for $Θ_{N E A}$ :

Θ_{N E A} = \frac{σ_{v}}{(1 + r) \sin 2 θ P_{0} R_{v} \frac{η e}{h ν} G_{L I A}}

where G_LIA is the lock-in-amplifier gain (in V/V). The small loss in signal due to absorption that is present in Eq. (16) is negligible and has not been included in this estimate. Using Eq. (17)

Θ_{N E A}

can now be calculated using the observed system noise and detection system parameters. From the Allan deviation measurements of the baseline stability the noise at 1s integration time is σ_v = 1.8 × 10⁻⁴ V. The P₀ for the measurement was 59.4 µW (and is calculated from the measured optical power on the signal detector of 22 µW and θ = 52.52°). Based on these quantities the estimated

Θ_{N E A}

is 8 x 10⁻⁸ rad/Hz^1/2. This optical measurement of

Θ_{N E A}

can be compared with another estimate provided by modeling of the polarization rotation corresponding to the 1σ concentration error, which is calculated using spectroscopic parameters of the target transition. To perform this comparison the FRS spectrum was numerically modeled accounting for 62 passes through the inhomogeneous axial magnetic field profile shown in Fig. 2. A 1σ concentration uncertainty of 6 ppmv at atmospheric pressure and room temperature corresponds to a

Θ_{N E A}

of 9 x 10⁻⁸ rad/Hz^1/2, which is in excellent agreement with the

Θ_{N E A}

estimate based on the system noise.

5. System noise analysis

Understanding the origin of system noise can provide guidance on what modifications could be made to the system to further improve the detection limit. Within the random noise dominated regime (short integration times) the total noise of the system (σ_total) can be represented by:

σ_{t o t a l} = {[σ_{\det}^{2} + σ_{l a s e r}^{2} + σ_{s h o t}^{2}]}^{1 / 2}

where the detector, laser intensity, and quantum shot noise are given by

σ_{\det}

,

σ_{l a s e r}

, and

σ_{s h o t}

respectively. The detector noise measured by at the output of the lock-in can be expressed as:

σ_{\det} = N E P \times R_{v} \frac{η e}{h v} G_{L I A} \sqrt{Δ f}

and is a function of the noise equivalent power (NEP in W

{Hz}^{- 1 / 2}

), measurement bandwidth

(\sqrt{Δ f})

,detector transimpedance gain

R_{v}

, detector responsivity

(η e / h ν)

, and the lock-in amplifier gain (G_LIA in V/V). An experimentally measured NEP of 2.8 x 10⁻¹² W

{Hz}^{- 1 / 2}

was used for the Nirvana balanced photodetector, which is in agreement with the manufacturer specification of <3 x 10⁻¹² W

{Hz}^{- 1 / 2}

.

The laser intensity noise measured by the balanced photodetector can be expressed using the relative intensity noise $(σ_{R I N})$ of the laser light incident on the signal photodiode of the balanced detector suppressed by the CMRR as:

σ_{l a s e r} = \frac{\cos^{2} θ P_{0} R_{v} \frac{η e}{h ν} G_{L I A} σ_{R I N} \sqrt{Δ f}}{10^{C M R R / 10}}

The

σ_{R I N}

for the VCSEL has been experimentally measured to be 3 x 10⁻⁷

{Hz}^{- 1 / 2}

.

The quantum shot noise observed on both detectors is uncorrelated, so the CMRR of the balanced photodetection unit is not effective in suppressing this contribution. Because of the auto-balancing circuit, the photocurrent shot noise generated by the reference photodetector will be reduced to the equivalent current shot noise from the signal photodetector. As a result, the following expression represents the cumulative quantum shot noise and is the quadrature sum of noises from both detectors:

σ_{s h o t} = G_{L I A} R_{v} {(4 e \frac{η e}{h ν} \cos^{2} θ P_{0} Δ f)}^{1 / 2}

By inserting Eqs. (19)-(21) into Eq. (18) the total system noise can be calculated at a specified detection bandwidth, effective CMRR, and measured optical power incident on the signal photodetector. Figure 7 shows a modeled total system noise as a function of optical power incident on the signal photodiode at a detection bandwidth of 1.25 Hz. The system noise has been calculated assuming an effective CMRR of 30 dB. For comparison, a calculation assuming no suppression of the laser intensity noise (CMRR = 0 dB) is also provided. At a CMRR of 30 dB quantum shot noise clearly dominates the modeled total noise of the system.

Fig. 7 Output noise (1σ) as a function of optical power on the signal detector modeled for DC-FRS system with a measurement bandwidth of 1.25 Hz and a 𝜃 = 52.52°, and compared to experimentally measured noise for the same conditions. A vertical line indicates the optical power on the signal detector observed for the measurement of the ^pP₁(1) transition.

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To determine if indeed the DC-FRS system operated in a quantum shot noise dominated regime, the system noise was measured as a function of optical power. The optical power of the VCSEL was adjusted by changing the injection current of the laser. The system noise measurements in Fig. 7 are in excellent agreement with the modeled system noise that assumed a CMRR of 30 dB. For measurements of the ^pP₁(1) line of oxygen, the optical power incident on the signal photodiode is 17 μW. Under these conditions (marked with a vertical line on Fig. 7) the system is quantum shot noise dominated, with the quantum shot noise 2 times higher than the detector noise. The measured overall system noise at this power level is only 13% higher than the theoretical quantum shot noise limit.

6. Conclusions

A balanced-detection DC-FRS system has been designed for the measurement of atmospheric oxygen. The 1σ detection limits from line-locked and baseline stability measurements compare favorably with preliminary previous studies carried out in our laboratory using AC magnetic fields with balanced detection and a 3.5 meter Herriott cell [34], where a 10 ppmv ${Hz}^{- 1 / 2}$ sensitivity was achieved. While this sensitivity is comparable to what is achieved in the current DC-field study, the 150 Gauss RMS AC-field used in Ref [34]. requires driving a solenoid coil with a ~30 W power burden, which is significant in contrast to the zero-power consumption of the permanent magnets used in this work. Initial tests performed with compact low-power electronics developed in our laboratory for operation of the VCSEL by So et al. [19] show that the current permanent magnet DC-FRS system can achieve a total power consumption of ~4 W. This is determined primarily by the power requirement of the Nirvana photodetection unit, while less than 1 W is needed for the laser temperature and current control. Optimization of the photodetector circuitry should result in additional power savings.

The low-power consumption of the DC-FRS system is ideal for field deployments. The stability of the residual optical fringing that is present in DC-FRS systems will set an upper limit on the long-term system stability. For the current configuration, the Allan analysis in Fig. 6 demonstrates that drift becomes dominant at integration times that are comparable to those observed for systems based on WMS. An AC-field instrument is largely immune to the long-term drift in optical fringes because the FRS signal is generated via sample modulation and not frequency modulation of the laser source. As a result, AC-field FRS systems show system stabilities that permit effective integration times beyond 1000 seconds [32]. The long-term system stability is a key factor in dictating the time period between system calibrations, and the DC-field FRS system will need to be calibrated more frequently to achieve a specific accuracy than an AC-field system. Therefore further development of this technology will focus primarily on improving system stability and accuracy.

The ultimate detection limit of the system sensitivity is 6 times larger than the target 1 ppmv/Hz^1/2 desired for biotic respiration measurements. Because the current system is quantum shot noise dominated, little can be done to reduce the current system noise. The detection limit can only be improved by modifications that increase the signal. Increasing the optical path length, the magnetic field, and the available optical power are three viable ways to improve the DC-FRS SNR. Unfortunately the current 62-pass configuration of the MPC is optimized to balance the optical transmission loss with the FRS signal gain for the number of passes through the sample. An additional increase in the number of passes will reduce the sensitivity of the system. Similarly, the FRS signal enhancement through an increase in magnetic field (e.g. by using other potential permanent magnet arrangements [35, 36]) would provide only marginal improvement (the optimum magnetic field for the ^pP₁(1) transition is ~1400 G and would only yield a 1.5 times increase in FRS signal in comparison to the 554 G used in this work). Thus for this system configuration the most viable option to meet the target system sensitivity is to increase the laser optical power. DFB diode lasers operating around 760 nm are commercially available and provide optical powers on the order of 10 mW. This is more than adequate to reach the 500 μW saturation limit of the Nirvana photodetector. Assuming quantum shot noise limited performance can also be achieved with a DFB diode laser, then a 5.4 times $(\sqrt{500 μ W / 17 μ W})$ improvement in the system sensitivity is expected, which should result in a 1 s detection limit of 1.2 ppmv when targeting the ^pP₁(1) transition. This would then allow the system to reach the necessary sensitivity for real-time in situ biotic respiration measurements, but at the cost of a slight increase in system power consumption due to the power required for operation of the DFB diode laser.

Acknowledgments

This work was supported by the NSF Engineering Research Center MIRTHE and from a generous contribution by Lynn and Thomas Ou. The authors are grateful to Prof. Mark Zondlo and Kang Sun from Princeton University for their advice on the multi-pass cell.

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Faraday rotation spectroscopy based on permanent magnets for sensitive detection of oxygen at atmospheric conditions

Abstract

1. Introduction

2. Experimental layout

3. Derivation of DC-FRS signals with balanced detection

3. Experimental results

4. Continuous O₂ monitoring

5. System noise analysis

6. Conclusions

Acknowledgments

References and links

Cited By

Figures (7)

Equations (21)

Optics Express

Abstract

1. Introduction

2. Experimental layout

3. Derivation of DC-FRS signals with balanced detection

3. Experimental results

4. Continuous O2 monitoring

5. System noise analysis

6. Conclusions

Acknowledgments

References and links

Cited By

Figures (7)

Equations (21)

Optics Express

4. Continuous O₂ monitoring