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Abstract
In the spin-exchange relaxation-free (SERF) magnetometer, the probe noise is a consequential factor affecting the gradiometric measurement sensitivities. In this paper, we proposed a new characteristics model of the probe noise based on noise separation. Different from noise analysis on single noise source, we considered most of the noise sources influencing the probe system and realized noise sources level measurement experimentally. The results demonstrate that the major noise type changes with the signal frequency. Below 10 Hz, the probe noise mainly comes from the sources independent of light intensity such as the vibration, which accounts for more than 50%; while at 30 Hz, the photon shot noise and the magnetic noise are the main origins, with proportion about 43% and 32%, respectively. Moreover, the results indicate that the optimal probe light intensity with highest sensitivity appears when the response of the magnetic noise is equal to the sum of the electronic noise and half of the shot noise. The optimal intensity gets larger with higher signal frequency. The noise characteristics model could be applied in modulating or differential optical systems and helps sensitivity improvement in SERF magnetometer.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
The optically pumped magnetometers (OPMs) operated with polarized alkali atoms achieved higher sensitivities compared with the classical magnetometers. Large range varieties of OPMs have been developed, among which the atomic magnetometer in spin-exchange relaxation-free (SERF) regime is the most sensitive magnetic field detecting equipment [1]. In recent years, the SERF magnetometer has been applicated in many fields such as the physics research [2,3], magnetoencephalography (MEG) [4], and magnetic material test [5,6]. With the elimination of the relaxation due to spin-exchange collisions, the magnetic noise and the noise from probe system have become the most influential factors to the sensitivity [7]. The magnetic noise comes from Johnson currents in the shield, which has definite sources and accurate calculation method [8,9]. In the OPMs, the magnetic noise could be suppressed with low-noise ferrite shield and difference method.
Different from the magnetic noise, the probe noise characteristic is not yet refined. The probe noise measured with the pumping light blocked is derived from several sources, including classical origins such as the environmental vibration or the electromagnetic noise, and quantum photon shot noise. The noise characteristics model should contain the influence of these origins, and demonstrate the level of each noise source. With the accurate value of noise level, the most influential noise can be emphasized, and thus the suppression method has more pertinence on the pivotal sources. For the classical origins, modulating or feedback methods are commonly used [10,11], and for the shot noise, the squeezed light is helpful [12]. In addition, the characteristics model should reveal the relationship between the probe noise and the probe system parameters such as the light intensity, which helps parameter optimization in the magnetometer. The probe noise characteristics are worthy researching for sensitivity improvement.
The commonly used noise analysis methods in atomic magnetometers are the direct measurement method and the theoretical analysis method. The direct measurement method is suitable for situations where single noise source stands out while other noises are suppressed. For example, when the probe laser is blocked, the electronic noise from the photodetector (PD) and the lock-in amplifier (LIA) can be measured [13]. Besides, the noise characteristic of some sources can be analyzed theoretically. Romalis et al. and Budker et al. [14,15] analyzed the photon shot noise in the SERF magnetometer and derived the quantum noise limit in the magnetometer. Fang et al. calculated the deviation azimuths of the optical elements and their effects to the probe noise [16,17]. However, they did not measure them experimentally. Li et al. showed the elliptically polarization effect of the vapor cell and its influence to the output signal [18], but ignored its influence to the noise. The methods above considered single noise source, but are not suitable for describing the noise characteristics with several noise sources. Krzyzewski investigated the characterization of noise sources and realized noise analysis in a microfabricated OPM based on direct measurement [19], but the method relies on the acousto-optic modulator (AOM) which is not suitable for SERF magnetometer.
In this paper, we proposed a novel probe noise characteristic model demonstrating the relationship with the intensity and the signal frequency. In the model, noise sources considered are classified to three kinds, the noises having negative correlation, positive correlation, and unrelated with the light intensity. Then, the total noise is expressed as their arithmetic square root. The model is verified experimentally in the photo-elastic modulator (PEM) modulated system in the SERF magnetometer. The results showed the major noise sources at different signal frequency. Moreover, the optimal probe light intensity with highest sensitivity varies with the frequency, which appears when the response of the magnetic noise is equal to the sum of the electronic noise and half of the shot noise. The method could be used in noise analysis with sources having different changing trend of light intensities, and helps the noise suppression and the sensitivity improvement in the OPMs.
2. Principle
The SERF magnetometer senses the magnetic field with the polarized alkali-mental atoms. In the experiment, the electron spin vector S of alkali atoms ensemble is pumped along z axis, and the atomic magnetic moment performs Larmor precession with the magnetic field B. The process can be described by the Bloch equation [20],
In SERF magnetometer, the magneto-optical rotation angle $\theta$ is less than $10^{-6}$ rad in sub-femtotesla magnetic field, thus an ultra-high sensitivity detection system is required for rotation angle detection. The commonly used detection systems are the modulating systems with Faraday modulator or PEM, etc., where the output voltage signal U is proportional to the rotation angle $\theta$ [20],
where ${I_0}$ is the incident light intensity, and $\textrm{OD} = {\pi }{ln}{r_\textrm{e}}cf \cdot \textrm{Re}[{\cal V}({\nu _\textrm{PR}} - {\nu _\textrm{0}})]$ represents the optical depth of the vapor cell.Several noise origins could influence the detection signal. In this paper, the magneto-optical rotation angular noises $\delta \theta$ [21] are calculated with the power spectral density (PSD), and classified into three kinds according to their characteristics with the incident light intensity, the angular noises having negative correlation, positive correlation, and unrelated with the light intensity as shown in Table 1.
Table 1. Noise sources in the probe system of SERF magnetometer
The angular noises having negative correlations with the light intensity include the photon shot noise and the electronic noise. The photon shot noise comes from the quantum characteristics of the coherent laser field. In coherent states, the photon number follows Poisson distribution whose standard deviation is proportional to the square root of the mean value. In the SERF magnetometer, the uncertainty of the photon number reflects in the fluctuation of the incident light intensity called photon shot noise, which depends linearly on the square root of the mean intensity, $\delta {U_\textrm{shot}} \propto \sqrt {h\nu {I_0}}$ [12,22]. Moreover, the shot noise of the PD from the uncertainty of the excitation electron has similar relations, $\delta {U_\textrm{shot}} \propto \sqrt {4e{I_0}}$ [13,14]. Considering the system response and the light absorption, according to Eq. (3), the photon shot noise can be written as $\delta {\theta _\textrm{shot}} = {k_\textrm{shot}}/\sqrt {{e^{ - \textrm{OD}}}{I_0}}$ [14,15], where ${k_\textrm{shot}}$ is the ratio of the shot noise unrelated with the light intensity.
The electronic noise can be measured directly when the probe light is blocked, which is mainly from the PD and LIA. According to Eq. (3), the equivalent rotation angular noise can be represented as $\delta {\theta _\textrm{e}} = {k_\textrm{e}}/({e^{ -\textrm{OD}}}{I_0})$.
The magnetic noise could enlarge the probe noise, which has linear dependence with the intensity. In the SERF magnetometer, the probe light contains circularly-polarized components because of the effect of the vapor cell or the imperfection of the optical elements [23,24]. With the circularly-polarized components, the probe laser can slightly pump the alkali-mental atoms, thus the atoms are able to sense the external magnetic field. The pumping rate of the probe laser beam is [14]
With the increases of the incident light intensity, the pumping rate enlarges, leading to larger response to the magnetic field. The magnetic field represents as $\textbf{B} = {\textbf{B}_0} + \delta \textbf{B}$ where ${\textbf{B}_0}$ is the residual magneto-static field in shield and $\delta \textbf{B}$ is the magnetic noise which satisfies $\left | {\delta \textbf{B}} \right | \ll \left | {{\textbf{B}_0}} \right |$. The magnetic noise has frequency characteristic $f^{-1/2}$ [8]. According to Eq. (1), the steady-state solution without pump beam satisfied [24,25]
For ${R_\textrm{rel}} \gg {R_\textrm{PR}}$, the angular noise $\delta {\theta _\textrm{B}} \propto {S_{\textrm{x}\_\textrm{PR}}} \propto {R_\textrm{PR}}$ can be regarded as proportional to the probe light intensity. With frequency response $G(f)$ to the magnetic field, the response of the magnetic noise can be written as $\delta {\theta _\textrm{B}} = {k_\textrm{B}}{I_0}\left ( {G(f) \cdot {f^{ - 1/2}}} \right )$.
In OPMs, several noise sources are unrelated to the laser intensity, including the environmental vibration noise, the polarization error, the laser frequency noise, the heating temperature noise, etc. The vibration noise refers to the azimuth deviation of the optical elements caused by the environment vibration and the air perturbation. In the probe system, the azimuth deviation has the same response to the magnetometer with the magneto-optical rotation angle, thus the equivalent rotation angular noise of the vibration noise is independent with the intensity [16,17]. In the modulated probe system, the polarization error mainly induced by the imperfection of the polarizer is not sensitive to the laser intensity [26]. The fluctuation of the laser frequency is unaffected by the intensity when the changing of the intensity is implemented with the combination of the half-wave plate and polarization beam splitter (PBS) [27]. The heating temperature noise has impact on the atomic density and the rotation signal, which is controlled by the heating procedure but not the light intensity. In conclude, these intensity-unrelated noise source could be expressed as $\delta {\theta _{\textrm{un}}} = {k_{\textrm{un}}}$. At low-frequency, the frequency characteristic of intensity-unrelated noise is $f^{-1}$ approximately.
For all the noises are uncorrelated variables, the total noise can be represented as
The total noise has a minimum value with the light intensity when ${\left ( {\delta \theta _{\textrm{shot}}^{\textrm{ 2}} + \delta \theta _\textrm{e}^{\textrm{ 2}}} \right )^\prime } + {\left ( {\delta \theta _\textrm{B}^{\textrm{ 2}}} \right )^\prime } = 0$. The derivatives are $(\delta \theta _{\textrm{shot}}^{\textrm{ 2}})' = 2\delta \theta _{\textrm{shot}}\delta {\theta '_{\textrm{shot}}} = 2\delta \theta _{\textrm{shot}} \cdot \left ( { - \frac {{{k_{\textrm{shot}}}}}{{2\sqrt {{e^{ - \textrm{OD}}}{I_0}} {I_0}}}} \right ) = - \frac {1}{{{I_0}}}\delta \theta _{\textrm{shot}}^{\textrm{ 2}}$ for the shot noise, $(\delta \theta _\textrm{e}^{\textrm{ 2}})' = 2\delta \theta _\textrm{e}^\textrm{ }\delta {\theta '_\textrm{e}} = 2\delta \theta _\textrm{e}^\textrm{ } \cdot \left ( { - \frac {{{k_\textrm{e}}}}{{{e^{ - \textrm{OD}}}{I_0}^2}}} \right ) = - \frac {2}{{{I_0}}}\delta \theta _\textrm{e}^{\textrm{ 2}}$ for the electronic noise, and ${\left ( {\delta \theta _\textrm{B}^{\textrm{ 2}}} \right )^{\prime } } = 2\delta \theta _\textrm{B}\delta \theta '_\textrm{B} = 2\delta \theta _\textrm{B} \cdot {k_\textrm{B}} = \frac {2}{{{I_0}}}\delta \theta _\textrm{B}^{\textrm{2}}$ for the magnetic noise. Thus, the optimal probe light intensity with highest sensitivity appears when the response of the magnetic noise is equal to the sum of the electronic noise and half of the shot noise $\delta {\theta _\textrm{B}}\left ( {G(f) \cdot {f^{ - 1/2}}} \right ) = \sqrt {\frac {1}{2}\delta \theta _{\textrm{shot}}^{\textrm{ 2}} + \delta \theta _\textrm{e}^{\textrm{ 2}}}$. The shot noise and the electronic noise can be regarded as white noise, while the response of the magnetic noise decreases with the frequency increases. Considering their characteristic with the light intensity, the optimal intensity gets larger with higher frequency.
3. Experiment
The experiment is performed in the probe system of the K atomic magnetometer working in the SERF regime with linewidth about 10 Hz, as shown in Fig. 1. The central sensing unit is a spherical vapor cell, 25 mm in diameter, containing a droplet of K alkali metals in nature abundance, 50 Torr N${_2}$ as quenching gas and 3 atm ${^4}$He as buffer gas. The cell is placed in an oven heated with alternating current at 100 kHz which can suppress the low-frequency electronic noise. The oven with cell is suspended in the center of a vacuum chamber which is wrapped with water cooling. The whole sensing system is enclosed with five layers cylindrical $\mu$-metal magnetic shields with diameter 280 mm and height 420 mm of the innermost layer to attenuate the earth magnetic field, and cylindrical ferrite with diameter 140 mm and height 245 mm for lower magnetic noise. Finally, the residual magnetic field is around 1 nT without compensation which can be further compensated to around 10 pT with the three-axis coil. The magnetometer is pumped by circularly polarized light at K D1 resonance line. The pump beam diameter is 25 mm to cover the whole cell and the power is optimized. The probe laser source is a distributed Bragg reflection (DBR) laser with wavelength (769.8500 $\pm$ 0.0001) nm for maximum signal strength, detuning 128.5 GHz from the center which changes less than 0.1% during the experiment. The half-wave plate (HWP) and the polarization beam splitter (PBS) combination is used to control the probe laser power. After that, the beam is expended to 5 mm in diameter. As the probe system shown in Fig. 1, the probe light is polarized by the polarizer (Glan-Taylor prisms) and modulated with PEM (Model-100, Hinds Instruments) whose modulation frequency is about 50 kHz. After sensing the magnetic field, the laser is detected by the two-channel PD (S5980, Hamamatsu) whose photosensitivity is 0.55 A/W at 770 nm and noise equivalent power is 1.4$\times$10${^{-14}}$ W$\cdot$Hz$^{-1/2}$. The current signals are demodulated with two-channel LIA (HF2TA, HF2LI, Zurich Instrument) respectively with reference frequency from the PEM controller. The maximum input of LIA is 10 $\mu$A at transimpedance gain 100 kV/A. The single channel signal from the output of LIA realizes the magnetometric measurements, while the gradiometric measurements are realized with two-channel signal difference. The probe noise is from the single channel signal when the pumping beam is blocked.The whole equipment is installed on the vibration isolation platform (BM-1, minus-K) with transmissibility -50 dB at 30 Hz in vertical and horizontal, and 10 dB at 0.3 Hz in vertical and horizontal.
Fig. 1. Schematic of the K SERF magnetometer. The probe system is shown in the middle. The combination of HWP and the PBS controls the probe light power. The azimuths between transmission axis of the polarizer and the fast axis of PEM, the fast axis of QWP and the transmission axis of the analyzer are 45$^{\circ }$, 0$^{\circ }$, 90$^{\circ }$, respectively. The lock-in amplifier demodulates 1${^\textrm{st}}$ and 2${^\textrm{nd}}$ harmonic components from the output signal of PD with reference frequency from PEM controller. With multi $\mu$-metal and ferrite shields, the magnetic noise is about 1 fT$\cdot$Hz$^{-1/2}$. The whole equipment is installed on vibration isolation platform to reduce the vibration. The x-axis is along probe laser while the z-axis is along pump laser. HWP, half-wave plate; PBS, polarization beam splitter; QWP, quarter-wave plate; PEM, photo-elastic modulator; PD, photodetector; BE, beam expender.
In the experiment, with pumping light blocked, the output voltage signal of the LIA is collected and analyzed with PSD to show the probe system noise. The noise around 0.3 Hz, 1 Hz, 3 Hz, 10 Hz, 30 Hz, 90 Hz are researched for frequency characteristic demonstration. Firstly, the experiment is performed in room temperature condition where the alkali-mental atomic number is low and optical depth approaches 0. In this condition, the noise sources are shot noise, electronic noise, and intensity-unrelated noise. Secondly, the cell is heat to temperature 200$^{\circ }$C where the interaction between alkali-mental atoms and the light cannot be ignored. The influence of the laser frequency noise, the temperature noise and the magnetic noise appear. Because of the influence of these noise sources, the probe noise characteristics with the light intensity and the frequency change. At each condition, voltage signals in 100 seconds are collected in certain light intensity and analyzed with PSD to obtain the mean values and standard deviations at different frequency. Changing the incident light intensities, the noises are gathered and fitted with Eq. (6) to get the parameters k. From the fitting results, noise characteristics of the probe system are revealed including the noise levels of each sources and their frequency characteristics.
4. Results and discussion
As shown in Fig. 1, the probe system of SERF magnetometer is constituted by polarizers, PEM and quarter-wave plate. The linearly-polarized probe light is modulated to suppress low frequency electrical noise. When passing the vapor cell, the plane of polarization rotated the magneto-optical rotation angle $\theta$, which is detected with the PD and demodulated with the LIA. The output signal of the LIA contains the 1${^\textrm{st}}$ harmonic component U${_\textrm{1}}$ and 2${^\textrm{nd}}$ harmonic component U${_\textrm{2}}$, [11]
According to Eq. (7), the the 1${^\textrm{st}}$ harmonic component U${_\textrm{1}}$ is proportional to the rotation signal $\theta$. When blocking the pumping light, the output component $\delta$U${_\textrm{1}}$ reveals the probe system noise. Comparing Eq. (7) and Eq. (8), the corresponding angular noise $\delta \theta$ can be calibrated with voltage noise $\delta$U${_\textrm{1}}$ and the 2${^\textrm{nd}}$ harmonic component U${_\textrm{2}}$
The calibration process requires parameters U${_\textrm{2}}$ and peak retardation of PEM ${\alpha _0}$. The 2${^\textrm{nd}}$ harmonic component U${_\textrm{2}}$ are obtained from the LIA in the experiment and the results are shown in Fig. 2. The peak retardation can be in-situ calibrated using the compound Bessel function method [16], and the result is 0.23 rad in the experiment.
Fig. 2. Second harmonic components U${_\textrm{2}}$ with different probe light intensities. The black line shows results at room temperature 23$^{\circ }$C, and the red line is at heating temperature 200$^{\circ }$C. Because of the absorption of the light, the components U${_\textrm{2}}$ are lower at 200$^{\circ }$C with the same intensity. The result indicates that the absorption rate is about 0.7.
The noise level $\delta$U${_\textrm{1}}$ of different frequencies and intensities are obtained from PSD. At room temperature 23$^{\circ }$C, the OD is less than 10${^\textrm{-6}}$, meaning that the interaction between atoms and the probe light is weak. At this occasion, the total noises contain photon shot noise and electronic noise which have negative correlations with intensity, and intensity-unrelated noise including vibration noise, etc. When heating the cell to 200$^{\circ }$C, the alkali-mental atomic density in the vapor cell is about 10${^{14}}$/cm${^{3}}$, and the optical depth is OD$\approx$0.3. At this occasion, the interaction between alkali-mental atoms and probe light cannot be ignored.The experimental results of the probe noises at different frequency are shown in Fig. 3. With Eq. (6) and Eq. (9), the experimental data points are fitted and the results are shown in Table 2.
Fig. 3. Results of voltage noises $\delta$U${_\textrm{1}}$ at frequency 0.3 Hz (a), 1 Hz (b), 3 Hz (c), 10 Hz (d), 30 Hz (e), 90 Hz (f). The black lines and the red lines represent results at room temperature and heating temperature 200$^{\circ }$C respectively. The data points are fitted with Eq. (6) and Eq. (9), and the fitting results are shown in Table 2. The abscissas are adjusted with the second harmonics for equal transmitted light intensities ${I_0}{e^{ - \textrm{OD}}}$.
Table 2. Fitting results.a
Table 2 shows the fitting parameters ${k_{\textrm{shot}}}$, ${k_{\textrm{e}}}$, ${k_{\textrm{un}}}$, ${k_{\textrm{B}}}$in Eq. (6), representing the noise level of the photon shot noise, the electronic noise, the intensity-unrelated noise, and the magnetic noise. The electronic noise is measured directly with the probe light blocked, and other parameters are from data fitting with Eq. (6). The noise sources with larger ratio in the total noise have smaller uncertainties in the consequence. For instance, at 0.3 Hz, the photon shot noise accounts for less than 0.1, and it is described with the upper limit rather than accurate values. The intensity-unrelated noise are major noise sources, with uncertainty about 3%. By contrast, at 90 Hz, the uncertainty of the shot noise is about 3% while the intensity-unrelated noise or the magnetic noise is inaccurate.
As shown in Table 2, the parameters vary with OD and the frequency. Firstly, at higher temperature conditions, the strength of interaction increases because of the larger atomic density number. Thus, the magnetic noise source is more pronounced, and the intensity-unrelated noise is significantly changed because of the new technical noise sources induced in the probe system. Secondly, the intensity-unrelated noise and the magnetic noise are colored noise sources, thus the total noise are frequency related. Compared with the total probe noise shown in Fig. 3, the frequency and intensity characteristics of separated sources are shown in Fig. 4.
Fig. 4. Frequency and intensity characteristics of four different noise sources. The figure is drawn according to Eq. (6) and the fitting parameters in Table 2, with shadow representing the uncertainties. The intersection points of the sum of the electronic noise and half of the photon shot noise $\sqrt {\frac {1}{2}\delta \theta _{\textrm{shot}}^{\textrm{ 2}} + \delta \theta _\textrm{e}^{\textrm{ 2}}}$ (black) and the magnetic noise $\delta {\theta _\textrm{B}}$ (green) indicate the optimal probe light intensities with highest sensitivity.
According to Fig. 4, the photon shot noise remains unchanged with the increase of the frequency, which shows the white noise characteristic of the shot noise source. Budker et al. calculated that in the case of "ideal polarimeter", the photon shot noise is about $\textrm{2} \times {10^{ - 8}}\textrm{ rad} \cdot \textrm{H}{\textrm{z}^{\textrm{ - 1/2}}}$ with 0.5 mW transmitted light intensity[15], similar with our noise separation results. Above 30 Hz, the photon shot noise is one of the main sources. In the future, it could be a critical factor for sensitivity improvement that the photon shot noise is suppressed. Similarly, the electronic noise is also white noise source, which decrease rapidly with the increase of the light intensity.
The intensity-unrelated noise decreases with the rise of the signal frequency. In room temperature condition where OD approaches 0, the environmental vibration has appreciable impact on the probe system. The vibration noise source is $f^{-1}$ noise at low frequency, causing the intensity-unrelated shows $f^{-1}$ characteristic when OD$\approx$0. After heating the cell, the intensity-unrelated increases significantly because of the influence of the laser frequency noise and the temperature fluctuation. The intensity-unrelated noise sources are dominant in frequency lower than 10 Hz, but decreases significantly in the high frequency. Therefore, the influence of the intensity-unrelated noise sources should be considered particularly in the low frequency measurement.
The response of the magnetic noise decreases with the increasing of the frequency, which is from the frequency response of the probe system and the magnetic noise $f^{-1/2}$ characteristic. The system is calibrated with 100 pT magnetic field. With the incident light 4 mW/cm$^{2}$, the frequency responses of the probe system at x, y, z direction are shown in Table 3. Assuming that the magnetic noises in three directions are equal and uncorrelated [6,9], the magnetic noises are calculated with the division of the voltage magnetic noise in Table 2 and averaged responses in Table 3, results satisfying the $f^{-1/2}$ characteristic in the ferrite magnetic shield. The response of the magnetic noise increases with the increasing of the intensity, making it major noise source in high light intensity detection system.
Table 3. Frequency responses and the magnetic noise
The four noise sources discussed above shows different characteristics with light intensity. The photon shot noise and the electronic noise are negatively correlated with the intensity, while the response of the magnetic noise is positively correlated. The optimal probe light intensity with highest sensitivity appears when $\delta {\theta _\textrm{B}} = \sqrt {\frac {1}{2}\delta \theta _{\textrm{shot}}^{\textrm{ 2}} + \delta \theta _\textrm{e}^{\textrm{ 2}}}$, which increases with the signal frequency increases. As shown in Fig. 5, below 1 Hz the optimal intensity is around 2 mW/cm$^{2}$, where the major influence is the intensity-unrelated noise. Above 30 Hz, the optimal intensity is 10 mW/cm$^{2}$, where the photon shot noise and the magnetic noise are the main sources.
Fig. 5. (a) Frequency characteristics of the total probe noise. (b) Intensity optimization at different frequencies. At 30 Hz, the optimal probe light intensity appears at 10 mW/cm$^{2}$, with sensitivity 1.42$\times$10${^{-8}}$ rad$\cdot$Hz$^{-1/2}$. (c) Intensity characteristics of the total probe noise.
The frequency and intensity characteristics of the probe noise helps intensity optimization of the SERF magnetometer. With the pumping light, the frequency responses of the magnetometer are collected in different intensities, and the magnetic field probe and gradiometer sensitivities are calculated as shown in Fig. 6. For detection near direct current signal in magnetometer or co-magnetometer, lower intensities are required compared with the detection at higher frequency. According to the characteristics, the sensitivity of the SERF magnetometers can be improved with the decrease of the response of the magnetic noise and larger probe light intensity.
Fig. 6. Probe and gradiometer sensitivities in different probe light intensities. The strong spectral peaks at 30 Hz comes from the magnetic signal we applied in the SERF magnetometer, while the signal at 50 Hz is the power-line interference.
5. Conclusion
In this paper, we proposed a novel probe noise characteristic model with the light intensity and frequency, and verified experimentally in the SERF magnetometer. The model described several noise sources whose relationships with the light intensity are different. The noise sources considered are classified into three kinds, the noise having negative correlations with the light intensity including the photon shot noise and the electronic noise, the noise having positive correlations from the response to the magnetic noise, and the noises unrelated to the intensity which contains various sources. According to the model, we obtained noise characteristics at 0.3 Hz, 1 Hz, 3 Hz, 10 Hz, 30 Hz, 90 Hz, respectively. The results indicate that below 1 Hz the intensity-unrelated noises are major components which mainly comes from the environmental vibration or the laser frequency fluctuations, etc. Above 30 Hz, the photon shot noise and the magnetic noise are significant. It helps sensitivity improvement that the noise characteristic results elucidate the noise source components, making the suppression process more pointedly. Besides, the results show that the probe light intensities are optimized when $\delta {\theta _\textrm{B}} = \sqrt {\frac {1}{2}\delta \theta _{\textrm{shot}}^{\textrm{ 2}} + \delta \theta _\textrm{e}^{\textrm{ 2}}}$, which increases with the frequency increases. For higher signal frequency detection, larger intensity is required. This method suits for analyzing noises whose sources have different relationship with the intensity, and may be applied in modulating, differential, or absorption detection system in atomic magnetometers or co-magnetometers.
Funding
National Key Research and Development Program of China (2017YFB0503100); BAQIS Research Program (No.Y18G32); Natural Science Foundation of Beijing Municipality (4191002); Major Scientific Research Project of Zhejiang Lab (2019MB0AE01).
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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