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Abstract
In this paper, we present the development and testing of a balanced Zeeman spectroscopy method utilizing wavelength modulation for selective detection of paramagnetic molecules. We perform balanced detection via differential transmission measurement of right-handed circularly polarized and left-handed circularly polarized light and compare the performance of our system to the Faraday rotation spectroscopy technique. The method is tested using oxygen detection at 762 nm and can provide real-time oxygen or other paramagnetic species detection for a variety of applications.
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1 Introduction
Continuous monitoring of oxygen concentration is vital in many areas of research, including medical diagnostics, molecular biotechnology, industrial production monitoring, environmental analysis, and food packaging [1]. Optical methods for oxygen detection are attractive because they do not consume oxygen or suffer from electrical interference, and they are non-invasive, scalable, and relatively inexpensive [2]. In particular, spectroscopic methods leveraging magneto-optical properties of oxygen such as Faraday rotation spectroscopy (FRS) have been widely investigated in measuring paramagnetic gas concentrations due to their superior detection sensitivity, noise-suppression capabilities, and improved selectivity provided by their immunity to spectroscopic interference from non-paramagnetic molecular species [3].
Consider a simple case of a Zeeman split ro-vibrational transition. In this case, the two Zeeman components interact differently with left-handed circularly polarized (LHCP) and right-handed circularly polarized (RHCP) light respectively. The difference in absorption between the two components leads to magnetic circular dichroism (MCD), and the difference in dispersion leads to magnetic circular birefringence (MCB). FRS examines MCB induced by a paramagnetic species in an axial magnetic field via measuring the rotation of linear polarization caused by a difference in phase shift for RHCP and LHCP light. FRS is extremely effective in the suppression of laser noise often to the fundamental shot noise levels via either balanced detection (the so-called $45^\circ$-method [4]) or by optimizing the small analyzer uncrossing angle (the so-called $90^\circ$-method [5]) that maximizes signal-to-noise ratio. Zeeman spectroscopy (ZS), on the other hand, examines MCD, which is typically measured by probing the absorption of a single circular component (either RHCP or LHCP) [6–8]. In a conventional ZS system, circular polarization is produced using a quarter wave plate ($\lambda$/4) placed before a sample cell, and only one circular component interacts with the Zeeman split transitions of the paramagnetic species. This severely limits the method’s capabilities of laser noise suppression. A differential measurement of both circular components in a balanced detection scheme would give MCD laser noise suppression capabilities that are similar to those of FRS.
In this work we develop an oxygen detection system that implements a wavelength modulated Zeeman spectroscopy (WM-ZS) technique with a balanced detection of MCD via differential measurement of RHCP and LHCP absorption performed using a circular polarization analyzer setup. Similar balanced detection approaches employing the Zeeman Effect have been utilized for frequency stabilization of laser diodes [9–13]. However, to the best of our knowledge, a balanced MCD detection system has not been investigated for quantitative spectroscopic molecular sensing. We compare balanced WM-ZS signals obtained from our novel configuration to wavelength modulated FRS (WM-FRS) implemented using the same system components.
2 Theoretical background
An axial magnetic field, with respect to the beam propagation, applied to a sample containing paramagnetic molecules creates a Zeeman splitting of the molecular transition energy levels. In a simple case such as the oxygen A-band at low splitting energies, the transition is split into two transition components. One component interacts with the RHCP light, and the other interacts with the LHCP light.
The electric field of linearly polarized light interacting with a Zeeman-split transition can be generally described as an elliptical polarization defined as
As the laser wavelength is scanned across the Zeeman-split transition the MCB and MCD of the gas sample will cause the transmitted light to undergo polarization rotation (Faraday effect) as well as difference in attenuation for RHCP and LHCP light that will manifest itself as an elliptical polarization described by Eq. (1). Different polarization states of transmitted light are schematically shown in the spectrum plotted in Fig. 1. The conventional approach used for detection of polarization rotation due to MCB is typically implemented in Faraday rotation spectrometers based on linear polarization analyzers configured in a $90^\circ$-method [5,14,15], $45^\circ$-method [4,16], or a hybrid of the two [17]. A typical configuration of an FRS spectrometer using a $45^\circ$-method shown in Fig. 1 leverages a balanced detection to efficiently suppress the common-mode laser noise that often leads to shot-noise limited detection [17–19]. In this work we developed an alternative balanced detection method to measure MCD effects rather than MCB, which essentially performs a balanced Zeeman spectroscopy.
Fig. 1. Simplified balanced detection method schematic for FRS (top-left) and ZS (bottom-left). The polarization state at various locations in the setup is illustrated for two optical frequencies marked in green and purple. The rotation angle spectrum of the linearly polarized light just before the polarizing beam splitter (PBS) is modeled (right). The polarization state of the light just after the cell at various points in the laser frequency sweep are illustrated above the graph.
The second ZS setup shown in Fig. 1 first converts the MCD into polarization rotation using a quarter-wave plate ($\lambda$/4). The rotation can then be measured using the same set of optical elements, i.e., a polarizing beam splitter (PBS) and two photodetectors. This combination of the $\lambda$/4-plate and the PBS forms a circular polarization analyzer that transforms the attenuation difference between the circular polarizations into a differential signal on the photodetectors.
In order to better understand the difference between FRS and ZS approaches, one can use an alternative representation of a polarized electric wave expressed as a combination of a single linear and single circular component as shown in Eq. (4). It can be shown that an arbitrary elliptical polarization can be decomposed into this alternative representation (Eq. (4)) from the conventional representation using two orthogonal circular polarizations (Eq. (1)), (see derivation in Supplement 1, A):
Now, if this elliptically polarized light is incident on the PBS used in the FRS setup in Fig. 1 at 45$^\circ$, the two resulting electrics fields for s and p are:
Therefore, the differential light intensity measured via the detectors is insensitive to the circular component that presents itself as a common mode signal on both detectors, while the Faraday rotation of the linear component is measured directly as:
Now, if the light before the PBS additionally passes through a $\lambda$/4 plate positioned such that its slow and fast axes are 45$^\circ$ shifted with respect to the s and p components of the PBS (see Fig. 2), then the electric field becomes:
Fig. 2. Elliptical polarization resulting from the difference in attenuation between RHCP and LHCP through the sample can be decomposed into a combination of a single circular component of the same handedness (top) and a linearly polarized wave (bottom). The $\lambda$/4 plate converts circular polarization into linear polarization, and an arbitrary linear polarization into an elliptical state aligned with the primary axes of the waveplate.
The linear component at an arbitrary angle $\Theta _{F}$ is transformed into an elliptical polarization with its minor and major axes aligned with the slow and fast axes of the $\lambda$/4 plate, which always results in equal power split between s and p channels, and thus the differential light intensity measured with the photodetectors becomes insensitive to this linear component. On the other hand, the circular component of an arbitrary handiness is transformed by the $\lambda$/4 plate into a linear polarization aligned either with s or p axes, resulting in a differential light intensity signal directly proportional to the $\Theta _{Z}$:
This transformation process used by the circular polarization analyzer is schematically depicted in Fig. 2 and enables differential ZS, which essentially allows us to measure the difference in attenuation between RHCP and LHCP ($\delta _{LH}-\delta _{RH}$) using a balanced detection that is highly efficient in suppressing the laser noise that occurs as the common mode on both detectors. By giving direct access to the absorption spectra and associated spectral line shapes (rather than dispersion line shapes measured by FRS), the ZS technique offers several advantages over FRS in terms of optimum signal strength and lower harmonic detection, as shown in the following sections.
3 Experimental methods and results
3.1. Set-up
The set-up for our balanced ZS detection is shown in Fig. 3. Using a two-mirror beam-steerer, the laser beam is directed through a multi-pass cell placed within a coil that produces a 270 Gauss magnetic field at 6.5A DC current. 21 passes across the 10 cm long multi-pass Herriott cell creates a path length of 2.1 m within the sample cell. The sample cell is filled with pure oxygen to a pressure of 220 Torr, which is the optimal pressure for FRS given the Zeeman split provided by the 270 Gauss magnetic field. This pressure is not optimal for ZS: improvements in signal amplitudes of ZS over FRS increase with increasing magnetic field, as shown in Fig. 8 and analyzed further in the Discussion. However, for the purpose of comparing our balanced WM-ZS measurements to the balanced WM-FRS measurements, we record both sets of measurements at the same target pressure. The diode laser (Cheetah, Sacher Lasertechnik) is swept over the P$_1$P$_1$ oxygen transition at 763 nm at a ramp rate of 2 Hz, and it is additionally modulated at a modulation frequency f. Each spectrum recorded is an average of 10 traces acquired at a sampling rate of 1.6 kHz to adequately sample the output of a lock-in amplifier set to an equivalent bandwidth of 500 Hz.
Fig. 3. General schematic of balanced detection via differential measurement of RHCP and LHCP. The following elements are used in the figure: ($\lambda$/2) half wave plate, (P) polarizer, ($\lambda$/4) quarter wave plate, (PBS) polarizing beam splitter, (LD) laser current driver, (TA) variable gain high-speed transimpedance amplifier (Femto DHPCA-100), (LIA) lock-in amplifier (Zurich MFLI) with differential inputs, (PC) computer running LabOne for data acquisition, (FG) Tektronix function generator for ramping the laser
A polarizer is placed before the cell such that the light is linearly polarized when it enters the sample cell. A $\lambda$/4-plate followed by a PBS were inserted after the multipass cell to implement a circular polarization analyzer to directly measure MCD effects in a balanced configuration. Two parabolic mirrors direct the two resultant beams onto two photodetectors (Thorlabs SM05PD3A), whose outputs are fed into two transimpedance amplifiers (Femto DHPCA-100) with gain set to 10$^3$ V/A. The transimpedance amplifiers are connected to the differential inputs of a lock-in amplifier (Zurich MFLI). A half wave plate ($\lambda$/2) was placed into the system before the initial polarizer so that the input polarization angle could be set while maintaining maximum light intensity. The PBS was set to 45$^\circ$ so that the light was split equally between the two detectors. The system can be easily converted between the ZS and FRS set-ups by removing the $\lambda$/4 such that s- and p-polarizations are measured directly. To record measurements, we interfaced the lock-in amplifier via the LabOne software (Zurich Instruments) running on PC.
With this set-up, we acquired balanced WM-ZS and WM-FR spectra by taking a difference between measurements acquired with a forward DC magnetic field and zero DC magnetic field. Before each measurement, we optimized laser wavelength modulation depth, alignment, initial polarizer angle, and demodulator phases such that the 2f-wavelength modulation spectroscopy (WMS) signal taken with the magnetic field off (mf off) was minimized. Due to the difference in lineshapes, shown in Fig. 1, the demodulated signals of interest are measured at different harmonics with 1f demodulation used for WM-ZS and 2f for WM-FRS. Therefore, for the purpose of direct comparison and to assure similar noise conditions, we decided to set the signals of interest to be demodulated at the same frequency. With this in mind, we present WM-ZS measurements taken by modulating the laser at f = 16kHz with demodulation set to 1f and WM-FRS measurements taken by modulating the laser at f = 8kHz and demodulating at 2f. Note, the modulation bandwidth of the laser driver is 40MHz and should not have any negative effects on the modulating signals in the kHz range.
3.2. Results and analysis
We are interested in comparing the 1f-WM-ZS signal to the 2f-WM-FRS signal, since both lineshapes are symmetric with a centered peak, which supports frequency locked operation at the transition peak. Since noise is generally greater at lower frequencies due to 1/f noise, by modulating the laser at different frequencies, such that f = 16kHz for the 1f-WM-ZS signal and f = 8kHz for the 2f-WM-FRS signal, we can perform a comparison of demodulated signals and an evaluation of signal-to-noise ratio while assuring the same noise characteristics.
Figure 4 shows the WM-FRS and WM-ZS demodulated signals at the first and second harmonics. WM-ZS and WM-FRS signals have been acquired as the differences between the magnetic field on and magnetic field off signals demodulated by the lock-in amplifier at the given harmonic and modulation frequency f.
Fig. 4. Data for 1f- and 2f-differential WM-FRS and WM-ZS signals in our balanced detection system. Laser modulation f = 8 kHz for WM-FRS measurements. Laser modulation f = 16 kHz for WM-ZS measurements.
We compare the 1f-WM-ZS signal to the 2f-WM-FRS signal, shown in Fig. 5. In the static spectrum presented in Fig. 1 that was simulated at the experimental conditions (220 Torr and 270 Gauss) the peak-to-peak MCD signal peak is a factor of $\sim$1.2 larger as compared to FRS. This favorable difference increases further for the modulated case, as performed in this work, where the maximum amplitude of the 1f-WM-ZS signal is approximately 1.5 times greater than the 2f-WM-FRS signal amplitude. For the modulated case, the wavelength modulation across the odd symmetry ZS lineshape in Fig. 1 produces more WM-ZS signal power in the first harmonic after demodulation than an equivalent second harmonic WM-FRS signal derived from the even-symmetry FRS lineshape. This is indicated in the Fast Fourier Transform of the time domain signal acquired at the center of the transition shown in Fig. 6. Additionally, the higher harmonics of WM-ZS fall to lower values than WM-FRS, indicating slightly higher spectral purity of the 1f-WM-ZS signal. It should also be noted that the ideal wavelength modulation depth for WM-FRS is greater than for WM-ZS, which has been properly optimized before each measurement.
Fig. 5. Comparison of 1f-WM-ZS (f = 16kHz) and 2f-WM-FRS (f = 8kHz) signals. The models are a simulated using HITRAN data and the same wavelength modulation parameters as those used in measurements.
Fig. 6. Magnitude of Fast Fourier Transform calculated for a time-domain signal measured at the transition center for WM-FRS and WM-ZS respectively. The plot has been generated as a function of a relative frequency with respect to the modulation frequency f.
Assuming that the noise in each system is comparable given that our measurements are taken at the same demodulation frequency (f = 16 kHz), this indicates that our balanced WM-ZS detection system offers a factor of 1.5 improvement in detection sensitivity compared to balanced WM-FRS detection.
3.3. Noise analysis
We then evaluated the noise in our balanced detection system by collecting detector noise measurements and off-resonance noise measurements.
For both sets of measurements, we enabled AC coupling and set the gain of the two transimpedance amplifiers to 10$^4$ V/A, and we set the acquisition for 200s and disabled averaging so that we could analyze the fluctuations of the difference in photocurrents. For the off-resonance noise measurements, the 2Hz sweep was disabled, and the laser was parked between two transitions by adjusting the laser temperature setting. The laser was still modulated at f. For detector noise measurements, both detectors were blocked.
Figure 7 shows the calculated shot noise, as well as the Allan deviation of the off-resonance noise and detector noise of the second harmonic differential signal for our system. As initially expected, we observed similar noise for the balanced WM-FRS and balanced WM-ZS configurations since both are acquired at the same demodulation frequency of f = 16 kHz. From this figure, we observe that the system is shot noise limited. We observe a slight drift in our noise measurement, which could be mitigated by future optimization for long term accuracy and drift in our experiment. Since for these proof-of-concept studies the magnetic coil in the system was not equipped with a proper heat removal hardware, we focus on comparison of short-term noise in the simplest configuration for balanced WM-ZS and balanced WM-FRS with a DC field. In the future, we will optimize for long term accuracy by introducing additional modulation of the magnetic field to reduce influence of the drift as well as a proper cooling system to prevent coil over-heatings with longer acquisition times. A minimum detection limit (MDL) of 1.5 ppmv$\cdot$Hz$^{-1/2}$ for O$_2$ is achieved in our WM-ZS system, which is a 1.5x improvement on the MDL of the WM-FRS system implemented with the same hardware.
4 Discussion
In order to identify optimum magnetic field for both ZS and FRS measurements we simulated polarization rotation angles for the systems presented in Fig. 1. Figure 8 models the peak-to-trough rotation angle as a function of applied magnetic field, which is representative of signal amplitudes achievable in both systems. As clearly visible in the figure, ZS is consistently providing higher peak-to-trough $\Theta _{Z}$ signal than $\Theta _{F}$, and this improvement increases further as the magnetic field is increased. Note that ZS also demonstrates more tolerance to a varying magnetic field. Figure 9 demonstrates how the lineshapes change at the higher magnetic fields. Since ZS is essentially a difference in transmission lineshapes for RHCP and LHCP, the peak-to-trough signal settles at a constant difference at high Zeeman splitting, while FRS that relies on difference in two dispersive profiles exhibits an optimum splitting condition.
Fig. 8. Model of peak-to-trough linear polarization rotation due to FRS and ZS techniques as a function of axially applied magnetic field strength. 210 cm pathlength, 220 Torr, $300^\circ$ K, and 100% O$_2$.
Fig. 9. Model of linear polarization rotation as a function of laser wavelength at 1500 Gauss. At this higher magnetic field strength, the splitting is so large that the maxima of dispersion signals of FRS no longer overlap, resulting in a peak FRS value that is less than the ZS peak value.
5 Conclusion and future directions
In this work, we present the development and testing of a novel method for oxygen detection based on Zeeman modulation and balanced detection that can be generalized to the detection of other paramagnetic gas species. We implement balanced wavelength modulated Zeeman spectroscopy via differential measurement of RHCP and LHCP absorption and compare the effectiveness of our system to balanced wavelength modulated Faraday rotation spectroscopy. We find that our balanced WM-ZS detection system results in $\sim$1.5x improvement in signal strength and minimum detection limit compared to a balanced WM-FRS implementation, with our WM-ZS system achieving an oxygen concentration-measurement precision of 1.5 ppmv$\cdot$Hz$^{-1/2}$. The noise analysis of the balanced WM-ZS system confirmed shot noise limited operation, an attribute that is much harder to achieve in conventional ZS systems probing only one circular polarization component. In the next stage, we aim to optimize our measurements for long term operation by improving thermal management of the coil while assuring the system accuracy and minimizing drift with the implementation of dual-modulation balanced WM-ZS detection.
Funding
2020 Innovation Fund for Industrial Collaborations, Princeton University; National Science Foundation (Graduate Fellowship); Princeton University Office of Undergraduate Research (2021 ReMatch+ Summer Program); Princeton University Department of Electrical and Computer Engineering. (Undergraduate Research Fund).
Disclosures
The authors declare that there are no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
Supplemental document
See Supplement 1 for supporting content.
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