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Abstract
We propose and experimentally demonstrate a two-stage digital differential method for atomic spin precession detection. The first differential operation is carried out with a polarimeter module and subsequent digital differential. The second differential operation is achieved by orthogonally modulating the polarization direction of a linearly polarized probe light with a LiNbO3 electro-optic modulation module and by digitally demodulating the difference in the outputs corresponding to the positive and negative half periods of the modulation square-waves. This method is insensitive to the error of modulator and double sensitivity coefficient was obtained. The built detection system with the digital circuit was applied to a spin-exchange relaxation-free magnetometer, and the sensitivity coefficient, sensitivity, and bias instability were tested and compared with two other typical detection methods. The highest sensitivity and minimum bias instability and noise were achieved with the proposed method.
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1. Introduction
The spin-based atomic sensors have attracted considerable attention due to their ultrahigh sensitivity, high precision, and miniaturized structure [1,2]. In general, these sensors consist of two main parts: the preparation of highly sensitive atomic spin states and the detection of atomic spin precession. The spin-exchange relaxation-free (SERF) state of the atoms is prepared by applying an effective magnetic shield, in combination with precision temperature and magnetic field control. In this state, the atomic spin precession is very sensitive to external magnetic field and rotation, and can form highly sensitive magnetometers and gyroscopes [3,4]. To achieve the full capability of SERF state atomic sensors, the detection of atomic spin precession with high sensitivity and stability is one of the key issues. Most atomic spin precession detection schemes use linearly polarized light as the probe light, and the atomic spin precession is detected by measuring the polarization plane rotation (namely, optical rotation angle) after passing through the alkali metal vapor cell. A variety of spin precession detection techniques have been reported, including the polarization differential detection (PDD) method and polarization modulation detection (PMD) method. The PDD method has a simple optical structure, which can suppress the common path error effectively and has been widely used in SERF magnetometers [5,6] and gyroscopes [7–9]. However, there is a large low-frequency 1/f noise because the scheme generally works in the DC band. In order to suppress the low-frequency noise, modulation-based techniques were introduced. H. Yao et al. [10] inserted an acousto-optic modulator into the differential detection optical path, and the optical rotation angle of the SERF magnetometer is measured by extracting the first harmonic signal from the modulated signal and good signal-to-noise ratio and stability were obtained. L. Xing et al. [11] used a miniaturized liquid crystal phase retarder to control the light intensity and modulate the probe light polarization, which minimizes the low-frequency electronic noise and improves the long-term stability of SERF gyroscopes. The PMD method is usually implemented with a polarization modulation module, such as Faraday modulator [12], photo-elastic modulator [13,14], and electro-optic modulator [15], and the optical rotation angle was extracted by using a lock-in amplifier. This method is also a commonly used detection technique for atomic spin precession, which is easy to implement and can achieve high precision in laboratory. However, the modulation module itself has the problem of drift and it is susceptible to the fluctuation of ambient temperature, light source power and wavelength, etc.. Hence, complex parameter compensation and/or control techniques are needed to pursue better performance. S. A. Wan et al. proposed a dual-beam scheme with closed-loop Faraday modulation to reduce the effect of laser intensity noise and thermal noise generated by the Faraday modulator [16]. L. H. Duan et al. [17] proposed a feedback control method by extracting the second harmonic component of the detection signal, which effectively reduced the influence of the power fluctuation of the light source and the loss variation in the optical path. In 2017, the same research group used the optical differential detection method to suppress the common-mode noise of light source and light path [18]. However, these measures increase system complexity and alignment requirements.
In this paper, we propose and demonstrate a two-stage digital differential atomic precession detection method. The first stage differential operation is carried out with a polarimeter module and subsequent digital differential, and the second stage is achieved by orthogonally modulating the polarization direction of linearly polarized probe light with a LiNbO3 electro-optic modulation module and demodulating digitally the difference in the outputs corresponding to the positive and negative half period of the modulation square-waves. In section 2, the principle of the proposed method is described, the detection equation is formulated, and the errors due to modulation phase drift are analyzed. In section 3, the all-digital detection circuit is realized and applied in a SERF magnetometer. The performance of the system was tested and compared experimentally with the PDD and PMD methods. The method provides a practical solution for high sensitivity and low drift atomic spin precession detection and has great practical potential for use in both SERF magnetometers and SERF gyroscopes.
2. Operating principle
A schematic of the two-stage digital differential atomic precession detection scheme is shown in Fig. 1. Light from a probe laser (PL) passes through a polarizer (P) and becomes linearly polarized. A LiNbO3 electro-optic phase modulator (EOPM) is placed between P and a quarter-wave plate (QWP) with its principal polarization axes 45° from that of P. The polarimeter module consists of a half-wave plate (HWP), a polarization beam splitter (PBS), and two photo-detectors (PD1 and PD2), and is placed after the atomic vapor cell. As shown in Fig. 2, a modulation square-wave (Fig. 2(a)) generated by the digital differential circuit is applied on the EOPM through the driver to realize ±π/2 phase modulation and the incident linearly polarized light will be converted to the left and right circularly polarized light (Fig. 2(b)) respectively. They will become linearly polarized light (Fig. 2(c)) that is perpendicular to each other after passing through QWP. The modulated linearly polarized lights will rotate an angle, i.e. optical rotation angle θ (Fig. 2(d)), after passing through the atomic vapor cell, and detected by the polarimeter module. The output signals from PD1 and PD2 are collected in parallel by two AD converters and processed by two-stage digital differential operations in a FPGA. The first stage differential (FSD) and the second stage differential (SSD) undertakes, respectively, the operations of Eq. (8) and Eq. (10), which will be described in detail in the next paragraphs. The detection results are recorded by a computer.
Fig. 1. Schematic of the two-stage digital differential atomic precession detection scheme. The arrows above the optical path indicate the principal polarization axis of each device and (a) ∼ (e) are the marks of concern positions. PL is the probe laser, P is the linear polarizer, EOPM is the electro-optic phase modulator, QWP is the quarter-wave plate, HWP is the half-wave plate, PBS is the polarization beam splitter, PD1 and PD2 are two photo-detectors, FSD is the first stage differential, SSD is the second stage differential, SG is the signal generator, and I/O is the interface.
Fig. 2. The modulation and signal waves and polarization states at the concern positions shown in Fig. 1. (a) ± π/2 modulated square-wave, (b) Left and right circularly polarized lights, (c) Linearly polarized lights perpendicular to each other, (d) Linearly polarized lights after passing through the atomic vapor cell, (e) Output signal after the first stage digital differential operation.
The derivation process of polarization state of polarized light in the optical path can be described with Jones matrix. The probe beam is linearly polarized after the polarizer. Its Jones vector can be written as,
where E0 is the electric field amplitude of the incident beam, and the incident beam intensity can be noted as I0=|E0|2. ɛP2 is the extinction ratio of the polarizer.The azimuth of EOPM principal polarization axes was precisely adjusted to 45° from the incident polarized light. Assuming the misalignment angle of incident polarized light is α and the square-wave modulation phase is δ(t), the Jones matrix of the EOPM can be written as,
The Jones matrix of the QWP with its optic axis having 0° to the incident polarization direction can be written as,
The Jones matrix of the polarized vapor cell with optical rotation angle of θ can be written as,
The direction of linearly polarized light incident on the polarimeter module is precisely adjusted initially to 45° from the principal axis of the PBS and the transmitted and reflected lights were detected by PD1 and PD2. The Jones matrix of transmitted and reflected lights can be written as, respectively,
The Jones matrixes of the lights detected by PD1 or PD2 can be written as,
where i=1, 2. The error due to ɛP, α, β, γ, φ, ɛPBS can be ignored by selecting high extinction ratio device and adjusting finely, the light intensity detected by PD1 and PD2 can be expressed as,The output signal after the first stage digital differential operation can be described as follows,
when δ(t)= -π/2 and δ(t)= +π/2 during the negative and positive half period of the modulation square wave, and the corresponding output are noted as D1-, and D1+, as shown in Fig. 2(e), can be described as, respectively,Further, the second-stage differential operation will be carried out by demodulating the difference between D1-, and D1+,
Note that for small optical rotation angle θ, the approximation described by Eq. (10) is reasonable, and the four-times optical rotation angle is achieved after the two-stage differential operation.
It can be seen that the proposed detection technique eliminated effectively the low-frequency noise and the error caused by the fluctuation of the intensity of the light source. The last error factor to be considered may be only the phase modulation error. The EOPM made from LiNbO3 has high linearity and good retardation stability, but there is still a weak dependence on temperature [19]. Assuming the modulation phase error is δɛ, the modulation phase can be written as,
After a simple derivation, Eq. (10) can be rewritten as,
The relative detection error can be written as,
Referencing to [19], the temperature coefficient of phase δ of LiNbO3 EOPM can be η=-1.1×10−3 /°C. Assuming δɛ=0 at initial operating temperature of 25°C, the relative detection error of optical rotation angle can be calculated with Eq. (13) within a larger operating temperature range of -40 °C ∼ +60 °C. In this temperature range, the value of δɛ is in the range of +0.11 ∼ -0.06 rad when δ=±π/2, the relative detection error reD curve is shown in Fig. 3, the relative detection error is effectively suppressed due to the cosine response to the modulation phase error in Eq. (12), and the maximum of reD is about 6.3×10−3.
As mentioned in section 1, the PMD method can eliminate effectively the low-frequency noise too by using the modulation/demodulation technique. In general, the sinusoidal modulation wave is applied to the polarization modulation module, and the optical rotation angle θ is extracted by a lock-in amplifier [18]. The modulation retardation phase and the output can be written as, respectively,
where δm and ωm are the modulation amplitude and frequency respectively, I0 is the intensity of the probe laser. The typical value of δm is in the range of 0.026∼0.08 rad [15,17]. When θ<<1, the output single can be obtained approximately by extracted the amplitude of the first harmonic [18],The relative detection error can be expressed as,
where Δ(δm) is the error of δm. Obviously, the relative detection error is completely determined by the stability of the modulator and the error Δ(δm) can be calculated with its temperature coefficient when the temperature changes and Δ(δm)= ηΔTδm, where ΔT is temperature change. Δ(δm) is in the range of +0.0029∼-0.0015 rad when δm=0.04 rad and temperature ranges from -40 °C to +60 °C. The reM is calculated and shown in Fig. 3, the maximum of reM is about 0.072.The results shown in Fig. 3 indicate that the two-stage differential detection method is insensitive to the error of modulator and has a far better (more than an order of magnitude) stability than the PMD method under the same temperature environment. The low frequency noise and laser intensity noise are are also reduced by the high frequency periodic modulation in combination with the FSD and the SSD. Traditionally, an analog differential balanced photodetector is used to reduce effect of laser intensity noise. Here we performed the difference, sum, and division operations digitally with the FSD module in the FPGA, which makes the system output indpenedent of the laser intensity and hence improves the scale factor stability.
3. Experiment investigation and comparison
The experiment setup of a SERF magnetometer is shown in Fig. 4. The cube vapor cell with an 8 mm diameter is filled with a small amount of 87Rb alkali metal, 700 torr 4He, and 50 torr N2. It is heated by a non-magnetic heater to 150 °C and the 87Rb alkali metal density is about 1014 cm-3. Three pairs of Helmholtz coils, outside the heater, are used to compensate the residual magnetic fields in cell. The cell, heater, and coils are all housed inside in a three-layer permalloy magnetic shield. A pump laser of 795 nm is generated by a distributed Bragg reflector (DBR) laser diode (Photodigram Inc.) driven by a high-precision controller. The pump laser is circularly polarized along the z-axis with a QWP, which is used to realize the spin polarization of alkali metal vapor cell. The two-stage digital differential atomic spin precession detection system shown in Fig. 1 was applied to this SERF magnetometer to detect the optical rotation angle θ. The probe laser with power of ∼3 mW is detuned by 100 GHz from the D1 resonance frequency of Rb. The probe beam passes through the polarized vapor celleq along the x-axis, then injects into the polarimeter module. The EOPM with Vπ=260 V at 795 nm is used to produce phase modulation with frequency of ∼2 kHz. Two photodetectors are employed in this work to detect the two beams divided by the PBS. The optical rotation angle θ induced by the polarized vapor cell is extracted by the digital differential circuit shown in Fig. 1. Before the following experiment, all the detection systems are calibrated with the optical rotation input according to [20].
In the SERF regime, the electron spin evolution can be described by the classical Bloch equation [21],
The electron spin polarization along x-axis Pex is linearly related to the optical rotation angle θ and the atomic spin precession can be obtainedby detecting θ [22], and the relationship can be expressed,
The sensitivity coefficient of the proposed detection method was investigated by applying a DC bias magnetic field along the y-axis and detecting the spin precession component of the x-axis. The range of applied bias magnetic is ± 8 nT, the interval within the narrow linear region (± 1 nT) is 0.2 nT and 0.5 nT out of this range. The tested data and the linear fitting curve near zero are shown in Fig. 5 and the linear fitting sensitivity coefficient is listed in Table 1. The high sensitivity coefficient of 0.081 rad/nT is achieved.
Fig. 5. The response of magnetometer to DC magnetic field By with three detection methods. TDD: two-stage digital differential detection, PDD: polarization differential detection, PMD: polarization modulation detection.
Table 1. The experimental results with different detection methods
To get the sensitivity of the SERF with the proposed detection method, the frequency response of SERF magnetometer was tested first and fitted to theoretical equation y = A/(f2+B2)1/2 where f is the frequency, A is the coefficient, B is the bandwidth [23], and shown in Fig. 6. Then a sinusoidal magnetic field By with amplitude of 30 pT and frequency of 10 Hz was applied to the magnetometer as calibration fields by accurately setting the current in corresponding Helmholtz coils, and the output of the SERF magnetometer for 100 s was stored. The frequency spectrum was performed and averaged for each 0.5-Hz bin, as shown in Fig. 7. By comparing the signal amplitude with the noise level around 10 Hz, the detection sensitivity in terms of noise equivalent magnetic field is estimated to be 2.3 fT/Hz1/2.
Then, the long-term bias stability of the SERF magnetometer with the TDD method was investigated without input magnetic field and the outputs for one hour are shown in Fig. 8(a). Its Allan variance curve was calculated, as shown in Fig. 9, the bias instability of optical rotation angle θ over long times is about 1.9×10−7 rad. Low noise and drift are achieved due to the two-stage differential and modulation technique and the noise of the output of the SERF magnetometer is caused mainly by the polarization noise of atomic gas cell.
Fig. 8. Output of the SERF magnetometer for 1 hour with three detection methods. Minimal noise and drift are achieved with proposed TDD method. The large noise with PDD method is mainly caused by low frequency 1/f noise. With PMD method, due to the modulation drift and laser intensity noise, the output noise is medium and presents a slight drift.
In order to compare the performance of the TDD method, the PDD and PMD methods were respectively realized by changing the detection optical path and signal processing method and applied in the same SERF magnetometer platform as shown in Fig. 4. Their sensitivity coefficient, sensitivity, and long-term bias stability were all measured and investigated with the same experimental conditions as above.
The PDD method was implemented by removing the EOPM and QWP in Fig. 4 and setting initially the direction of linearly polarized light incident on the polarimeter module to 45° from the principal axis of the PBS by adjusting the HWP precisely. The output signal from PD1 and PD2 are collected in parallel by two AD converters and the differential operation was carried out and output directly to the computer. The experimental results of sensitivity coefficient, sensitivity, and long-term bias stability were plotted in Fig. 5, Fig. 7, Fig. 8(b), and Fig. 9 and listed in Table 1 respectively. The large noise with PDD method is mainly caused by low frequency 1/f noise.
The schematic diagram of PMD method is similar to that of the literature [18] and shown in Fig. 10. The two-stage digital differential circuit in Fig. 4 is replaced by the lock-in amplifier (Stanford Research SR850) and the output signal was acquired by the data acquisition card (NI USB-6289). The frequency of the sinusoidal wave added to the modulator is 5 kHz and the amplitude is 3.31 V. The sensitivity coefficient, sensitivity, and long-term bias stability were tested experimentally and shown in Fig. 5, Fig. 7, Fig. 8(c), and Fig. 9 and listed in Table 1 respectively. The output noise is medium and presents a slight drift due to the modulation drift and laser intensity noise.
The experimental investigation and comparison results are all listed in Table 1. It can be seen that the proposed two-stage digital differential atomic spin precession detection method achieves the maximum sensitivity, the best stability, and the lowest noise. The experimental results are consistent with the theory and model well.
In order to verify the modulation error suppression capacity of proposed TDD method, the voltage applied to the modulator is manually adjusted to simulate δɛ change due to temperature in the range of -40 °C to +60 °C as discussed in section 2. The corresponding adjustment range of δɛ is +0.11 ∼ -0.06 rad and the interval of about 0.016 rad. The output of SERF magnetometer was tested without input magnetic field and the 100-second average is taken at each testing point as the output. The tested results are shown in Fig. 11, the drift is about 6.3×10−7 rad. As a comparison, the same test with PMD method was also performed. The corresponding adjustment range of Δ(δm) is in the range of +0.0029 ∼ -0.0015 rad and the interval of about 0.0004 rad. And the tested results are shown in Fig. 11 too, and there is an obvious trend of drift, and the drift is 1.1×10−5 rad. The comparison experimental results show that the proposed two-stage digital differential atomic spin precession detection method has excellent temperature stability and strong anti-interference ability.
4. Conclusion
In order to improve the sensitivity and suppress the bias drift and noise, a two-stage digital differential atomic spin precession detection method is proposed and demonstrated experimentally and theoretically. And unlike other modulation detection schemes, it is insensitive to the modulation phase error of modulator and reduced the effects of the laser intensity noise and low-frequency 1/f noise. The experimental comparison with the other two traditional detection methods is carried out in the same SERF magnetometer platform. The proposed method has double sensitivity coefficient, lowest noise, and lowest bias drift by taking digital modulation /demodulation technology. The sensitivity of SERF magnetometer with the proposed method reached 2.3 fT/Hz1/2, the bias drift was 1.9×10−7 rad, and the drift was about 6.3×10−7 rad under simulation temperature of -40 °C to +60 °C. This work makes full use of differential and digital detection techniques and provides a practical solution for high sensitivity and low drift atomic spin precession detection, which has great practical potential in both SERF magnetometers and SERF gyroscopes.
Funding
National Natural Science Foundation of China (61227902, U1637106); National Key Research and Development Program of China (2018YFC1503703); Program for Innovative Research Team in University (IRT 1203).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
Data availability
Data underlying the results presented in this study are not publicly available at this time but may be obtained from the authors upon a reasonable request.
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