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In the field of biomagnetic measurement, optically-pumped atomic magnetometers (OPAMs) have attracted significant attention. With the improvement of signal response and the reduction of sensor noise, the sensitivity of OPAMs is limited mainly by environmental magnetic noise. To reduce this magnetic noise, we developed the optical gradiometer, in which the differential output of two distinct measurement areas inside a glass cell was obtained directly via the magneto-optical rotation of one probe beam. When operating in appropriate conditions, the sensitivity was improved by the differential measurement of the optical gradiometer. In addition, measurements of the pseudo-magnetic noise and signal showed the improvement of the signal-to-noise ratio. These results demonstrate the feasibility of our optical gradiometer as an efficient method for reducing the magnetic noise.
© 2015 Optical Society of America
1. Introduction
In the field of biomagnetic measurements, optically-pumped atomic magnetometers (OPAMs) [1–6] are expected to serve as alternative sensors to magnetometers based on superconducting quantum interference devices (SQUIDs). OPAMs are theoretically more sensitive than SQUID magnetometers [7–9]. In addition, cryogenic cooling is not required for OPAMs, unlike SQUIDs. Therefore, the running costs of the OPAMs could be smaller than those of SQUIDs and the miniaturization of OPAMs and multi-channel detections could be done easily.
To measure magnetoencephalograms [10] and magnetocardiograms [11], we have been developing OPAM systems [12–14]. To measure weak biomagnetic fields, high-sensitivity OPAMs are required. By suppressing the system noises and optimizing the operating conditions, we have improved the sensitivity of the OPAMs. The sensitivity of our OPAM system reached about 20 fTrms/Hz1/2 at around 10 Hz [15]. Although the sensitivity is already sufficient for magnetoencephalogram measurements, further improvements will enable the detection of weaker signals. For now, the sensitivity is limited by the environmental magnetic noise.
In general, to suppress magnetic noise, differential measurements with gradiometer configurations are commonly used [10]. In the first-order gradiometer, the differential output of two sensing positions is obtained. The magnetic noises at the two positions are assumed to be identical when the source is far enough from them and the base length is much smaller than the distance from the source. Therefore, differential output can remove the magnetic noises which are identical at the two positions. On the other hand, when the sensing positions are close to the signal source, such as a brain or heart, a part of the signal remains even after differential measurement because the strengths of the signals depend on the distance from the source. Therefore, with an appropriate configuration, the gradiometer measurements can improve the signal-to-noise ratio (SNR). For example, in an axial gradiometer, the base length between the two positions should be longer than the distance between the nearer position and the signal source [10, 16].
An improvement of the sensitivity by conducting differential measurements with two miniaturized OPAMs to cancel the environmental noise has been reported [17]. These OPAMs adopt a one laser beam arrangement, so that miniaturization is easy, but the magnetic modulation is required. Therefore, the flexible arrangement of multi-sensors would be limited and complex magnetic designs would be necessary to prevent the magnetic interference [18] that may arise in future multichannel detections. With the pump-probe arrangement, in which the pump and probe beams cross at a right angle, OPAMs can detect weak magnetic fields without any magnetic modulation, and differential measurements showed noise reductions using a multi-channel photodetector array and one glass cell [3, 8]. However, the base length was quite short and signals could be reduced greatly. To obtain a sufficient base length, expansion or separation of the probe beam is required. Therefore, the system requires a lot of optical components and is not suitable for miniaturization. In addition, the differential measurements in previous works [19–21] were carried out with independent OPAMs by electrical subtraction of individual outputs. However, multiple polarimeters are required for OPAMs. Furthermore, since their electrical noises are independent of each other, the electrical noises would increase with the differential measurements.
To address this issue, we developed an optical gradiometer for OPAMs with a base length of 30 mm. In the optical gradiometer, the difference between the magnetic fields at two sensing positions is obtained directly via the magneto-optical rotation of one probe beam. Therefore, the configuration of the optical gradiometer is simple and suitable for miniaturization. In addition, the optical gradiometer requires only one polarimeter and the electrical noise is not affected by the subtraction. On the other hand, the optimum operating conditions of the optical gradiometer are different from those of conventional OPAMs.
In this study, we theoretically and experimentally found the appropriate wavelength of the probe beam, at which the optical gradiometer operated efficiently. By operating in the appropriate conditions, we showed improved sensitivity. In addition, by using pseudo-magnetic noise and signal, we showed an improvement in the SNR, which was not mentioned explicitly in previous works.
2. Optical gradiometer
2.1. Principle
In the optical gradiometer, a probe beam is reflected with mirrors and irradiates two positions in one glass cell which contains alkali metal atoms, as shown in Fig. 1(a). The electron spins at the two positions are polarized to the same direction by two pump beams directed along the z-axis, which consist of right-handed circularly polarized light. The magnetic field (along the y-axis) rotates the spins [22] and generates the spin component Sx, along the x-axis. In the presence of Sx, the magneto-optical rotation is generated on the linearly polarized probe beam which travels along the x-axis. When the angle of the polarization plane of the incident probe beam (point A in Fig. 1(a)) is 0 and the angle of the magneto-optical rotation at the lower position is θLower, the angle at the point B is θLower. If the polarization dependence of the reflectivity of the mirrors is small enough to be ignored, polarization state is preserved during the reflection and the angle at the point C is θLower as well as the point B since the probe beam is reflected to the right angle twice. When the magnetic fields at the two positions have the same directions, the directions of Sx at the two positions are also the same. However, the traveling directions of the probe beams at the two positions are reversed. In other words, the directions of Sx from the viewpoint of the probe beam are different at the two positions. Therefore, the polarization plane rotates oppositely. When the magnetic fields at the two positions are not only the same directions but also the same amplitudes, the amplitudes of Sx at the two positions are also the same and the rotation angles θLower and θUpper generated at the two positions are canceled each other out, as shown in Fig. 1(b). On the other hand, when the magnetic fields at the two positions are different, θLower and θUpper are different. Accordingly, the output angle, θDiff, reflects the difference between the magnetic fields at the two positions directly, as shown in Fig. 1(c).
Fig. 1 (a) Principle of an optical gradiometer. Two pump beams are made of right-handed circularly polarized light. The probe beam is linearly polarized. The polarimeter consists of a polarizing beam splitter and two photodetectors. (b) Transition of the polarization plane of the probe beam when the magnetic fields at the two positions are same. (c) Transition of the polarization plane of the probe beam when the magnetic fields at the two positions are different. The polarization planes at points A – D in (a) are drawn from the traveling direction of the probe beam. θLower and θUpper are the angles of the magneto-optical rotations generated at the lower and upper positions, respectively. θDiff is the difference between θLower and θUpper.
However, as a matter of course, there are some drawbacks of the optical gradiometer. The misalignment of beams and difference of the spin relaxations between the two positions can cause unbalanced characteristics at the two positions. With the reflection, the polarization states might be changed.
The optical gradiometer requires only one polarimeter and fewer optical components for the probe beam. Two pump beams are used to clearly separate the two areas. The two pump beams can be replaced by one expanded pump beam under the ideal conditions, in which the pump beam has a homogeneous profile. Therefore, the system of the optical gradiometer can be simple and made more suitable for miniaturization.
2.2. Signal response
In OPAMs, the signal response, Sout, is represented as follows [23]:
where η is the conversion efficiency from the beam power to the output voltage, Iprobe is the incident power of the probe beam, α is the absorption coefficient, l is the length of the glass cell along the probe beam, and θ is the rotation angle. nK is the atomic density of alkali-metal atoms (we used potassium), c is the speed of light, and re is the classical electron radius. In the optical gradiometer, the probe beam passes through the cell twice, that is, l → 2l. For the D1 line, f ≃ 1/3 is the oscillator strength, ν0 is the resonance frequency of the D1 line, νprobe is the frequency of the probe beam, lcross is the length along the x-axis of the measurement volume in which the pump beam and probe beam cross, and Γ is the linewidth of the pressure-broadened optical absorption caused by buffer and quenching gases. The spin component Sx is obtained by a solution of the Bloch equation as follows [24]: where D is the diffusion coefficient, S is the spin vector, γe is the gyromagnetic ratio of a bare electron, q is the slowing-down factor [25], B is the magnetic field applied to the alkali atoms, T2 is the transverse spin relaxation time, and ROP is the pumping rate [12]. In Eq. (5), RSD is the relaxation rate due to spin-destruction collisions [7] and is the transverse spin relaxation time due to spin exchange collisions [7,24]. Under the spin exchange relaxation free condition, we can assume that [7, 26]. RPR is the relaxation rate caused by the absorption of the probe beam. By using the cross-section of the probe beam Aprobe, RPR is represented as follows [12]:The rotation angles θLower and θUpper in Figs. 1(b) and 1(c) are obtained by using Eq. (3), and the angle of differential output, θDiff, is represented as follows:
3. Experimental methods
The arrangement of the optical gradiometer is shown in Fig. 2. In this study, we used potassium as the sensing atom. A cubic glass cell (side 50 mm), which contained potassium, helium, and nitrogen was surrounded by the heating unit and placed inside a three-layered magnetic shield (104 shielding factor at 10 Hz). Helium and nitrogen with a ratio of 10:1 were used as the buffer and quenching gases, and the total pressure was 108 kPa at room temperature. The glass cell was heated to 180°C. Three-axes coils were used to cancel the DC magnetic field which passed through the magnetic shield. To generate spatially-homogeneous magnetic field around the glass cell, the three-axes coils consisted of two saddle coil pairs for the x- and y-axes, and one Helmholtz coil pair for the z-axis. We also used two Golay coils (∂Bz/∂x, ∂Bz/∂y) to improve the homogeneity of the magnetic fields at the two positions. A reference coil was also equipped to apply a reference magnetic field along the y-axis. These coils were attached to an acrylic cylinder with a diameter of 800 mm. In addition, we placed a small coil (6 mm in diameter) above the cell as a signal source.
Fig. 2 Arrangement of the optical gradiometer. (a) Top view of the measurement system. CLs are cylindrical lenses. The K cell is surrounded by AlN and heaters for uniform heating, and covered by thermal insulators (TI). Four vacuum glass cells are used as windows for the beam paths. (b) Another view of the optical components for the pump beams. PBS is a polarizing beam splitter. One pump beam is expanded and separated. Two beams are circularly polarized by quarter wavelength plates and irradiate the K cell in (a) at different heights, respectively. (c) Another view of the optical components for the probe beam. The probe beam at the lower position is linearly polarized by a half wavelength plate and irradiates the cell. The probe beam is reflected in (d) and comes back from with different height. The polarization plane of the probe beam at the upper position is measured by the polarimeter (PM), which consists of a polarizing beam splitter and two photodetectors. The differential output of the photodetectors is amplified and recorded in the PC in (a). The half wavelength plate before the polarimeter rotates the polarization plane to 45° so that the output becomes zero when there is no magnetic field. (d) Another view of the mirrors for the probe beam. The probe beam is reflected to the right angle twice and raised 30 mm.
To avoid the misalignment of the beams, we configured the OPAM system strictly as follows. One pump beam was expanded with lenses to cross the probe beam easily at the glass cell. Subsequently, the pump beam was separated as shown in Fig. 2(b). Two pump beams were used to adjust their powers separately. When the upper position is not irradiated with the pump beam, only the lower position senses the magnetic field and vice versa. By not irradiating one of the positions, we can detect the magnetic fields at the two positions separately. The signal response is proportional to the length along the x-axis of the measurement volume in which the pump and probe beams cross. To obtain sufficient length, two pump beams were expanded and shaped as rectangles with cylindrical lenses and a slit. The length of the long and short axes of the pump beams were 30 and 10 mm, respectively. The cell was irradiated with two pump beams at 10 mm above the bottom and 10 mm below the top. The pump beams tuned to the D1 line of the potassium atoms (770.1 nm) were supplied by a cw Ti-Sapphire laser (Coherent, MBR-110).
The probe beam blue-detuned from the D1 line was supplied by a distributed feedback laser (TOPTICA Photonics, DL DFB). The cell was irradiated at 10 mm above the bottom with the probe beam (4 mm in diameter). The transmitted beam was reflected and raised 30 mm by mirrors as shown in Fig. 2(d), and the cell was irradiated again at 10 mm below the top. To minimize the change in the polarization state due to the reflection, we used dielectric multilayer mirrors (SIGMAKOKI, TFVM-20C05-780, AUTEX, BND-10) with small polarization dependence of the reflectivity and the angle of the polarization plane of the incident probe beam was set to perpendicular to the reflection plane. The polarization plane of the reflected beam was measured by the polarimeter, which consist of a polarizing beam splitter and two photodetectors. The differential output of the photodetectors was amplified and recorded in the PC in Fig. 2(a). The half wavelength plate was placed before the polarimeter, and the polarization plane was rotated to 45° so that the output becomes zero when there is no magnetic field.
In our arrangement, the signal coil was placed 33 mm above the center of the upper path. Considering the diameter of the probe beam, the minimum distance between the signal source and upper path was 31 mm, almost equal to the base length of 30 mm.
In the optical gradiometer, the intensities of the probe beam passing through the two positions are different due to absorption by the potassium and loss caused by glass walls and mirrors. Furthermore, the absorption of the probe beam induces the relaxation of the spin polarization. Therefore, the relaxation rates of the probe beam in Eq. (6) at the two positions are different, and hence lead to the difference of signal responses and a low efficiency of differential measurements. In addition, since the optimum incident power densities of the pump beams, which maximize the signal responses, depend on the spin relaxation rates, the difference in the relaxation rates of the probe beam has to be minimized. The relaxation rates of the probe beam depend on the incident power and wavelength of the probe beam. The incident power of the probe beam was fixed at the minimum value, which was determined based on the noise of photodetectors of the polarimeter. Although the absorption can be minimized by detuning the probe wavelength, the signal response will be changed as well. Therefore, the appropriate wavelength, at which the optical gradiometer operates efficiently, has to be determined.
To determine the appropriate wavelength, we measured the signal response as a function of the wavelength of the probe beam. By not irradiating one of the positions, we measured the signal responses at the two positions separately. We applied a 10-Hz sinusoidal wave with an amplitude of 3 pT by using the reference coil. The incident power of the probe beam was fixed at 0.5 mW. The incident power densities of the pump beams were changed to maximize the signal response at each wavelength. In the OPAMs, the bandwidth and the center frequency of the signal response can be tuned to the desired frequency range by controlling the incident power density of the pump beam and the magnetic field along the pump beam. In addition, the maximum amplitude of the signal response can be obtained with the narrowest bandwidth at the center frequency. We tuned the center frequency around 10 Hz and the bandwidth was minimized to be in the same situation of our previous work [14, 15].
Next, we measured the sensitivity of the optical gradiometer with the wavelength we determined above. The incident power densities of the pump beams at the upper and lower positions were 0.43 and 0.44 mW/cm2, respectively, which were almost identical. We measured sensitivity with differential measurement and single measurements, in which one of the positions was not irradiated. In addition, we measured the pseudo-magnetic noise and signal to obtain the SNR. In the optical gradiometer, the upper, lower, and differential outputs were measured with different timings. To evaluate the effect of differential measurement accurately, we applied pseudo-magnetic noises with the same spectra in the upper, lower, and differential measurements. The pseudo-magnetic noise was white Gaussian noise, and it was applied by the reference coil. To distinguish the pseudo-magnetic noise from the environmental magnetic noise, the mean value of the pseudo-magnetic noise was set to about 140 fTrms/Hz1/2, which was about six times bigger than the typical magnetic noise at 10 Hz in our experimental environment. In addition, we applied a 10 Hz sinusoidal wave current with an amplitude of 0.02 mA to the coil above the cell as a pseudo-magnetic signal source.
4. Results and discussion
Figure 3 shows the measured signal responses at the upper and lower positions with calculated values obtained by Eqs. (1)–(6). In our experimental conditions, η was 0.44 ×106 V/W, l was 50 mm, lcross was 30 mm, nK was 1.20 ×1019 m−3, and Γ was 15.1 GHz. Because the first term of Eq. (4) could not be solved with a cubic glass cell, we obtained theoretical values with numerical calculations according to Ref. [27]. The measured and calculated values showed good agreement. With the change in the wavelength, the signal responses changed greatly due to the absorption of the probe beam as show in Eqs. (1) and (2) and the dispersion feature to the wavelength as shown in Eq. (3). As the wavelength approaches to the wavelength of the D1 line of the potassium more, signal responses decrease. Figure 4 shows the ratio of the signal responses at the lower and upper positions, obtained from Fig. 3. The ratio at 769.7 nm was 0.96 and it was highest in the measured range. By detuning the wavelength farther, the absorption would be smaller and the ratio would approach to 1. On the other hand, the signal responses would decrease gradually according to the dispersion feature in Eq. (3).
Fig. 3 Signal responses as functions of the wavelength of the probe beam. The blue circles and the red squares denote measured results at the upper and lower positions, respectively. The blue solid line and red broken line denote calculated results obtained by Eqs. (1)–(6).
Fig. 4 Ratio of signal responses of the upper and lower positions as functions of the wavelength of the probe beam.
The reduction of the magnetic noise is effective when nonmagnetic noise, such as probe-beam noise, is smaller than the magnetic noise. To suppress the nonmagnetic noise, a higher signal response is required. On the other hand, to demonstrate the differential measurement of the optical gradiometer effectively, the ratio in Fig. 4 should be 1. Although the signal response of the upper position at 769.70 nm was about 40% smaller than that at 769.95 nm, at which the signal response was the strongest, the magnetic noise exceeded the probe-beam noise (as shown in Fig. 5). Therefore, in our experimental condition, we determined that the operating wavelength for the optical gradiometer was 769.70 nm, in order to minimize the difference in the signal responses between the two positions.
Fig. 5 Calibrated noise spectrum density of the optical gradiometer. The blue solid line denotes the magnetic noise floor of the single measurement at the upper position. The black solid line denotes the magnetic noise floor of the differential measurement. The gray broken line denotes the probe-beam noise floor.
Figure 5 shows the noise spectrum density of the optical gradiometer at a probe-beam wavelength of 769.70 nm. It was calibrated according to pre-measured signal responses with the same calibration method used in our previous study [14]. At low frequencies, the noise spectrum densities of upper position and differential measurement increased due to the low shielding factor. The probe-beam noise also increased at low frequencies due to the intrinsic laser noise and the beam fluctuation caused by the air along the path. In addition, since we tuned the center frequency around 10 Hz, the noise spectrum densities increased below 10 Hz. At around 10 Hz, the spectrum densities of the magnetic noise at the upper position and that of the probe-beam noise were 26.9 and 3.3 fTrms/Hz1/2, respectively. Therefore, the magnetic noise limited the sensitivity of the OPAM. On the other hand, the spectrum density of the differential measurement was 9.3 fTrms/Hz1/2. Although the inhomogeneous magnetic noise, which could not be removed with the first order gradiometer, still remained, the results demonstrated that the differential measurement of the optical gradiometer enabled the reduction of magnetic noise.
Next, we compared the SNR of the upper, lower, and differential outputs of the optical gradiometer. Figure 6 shows the noise spectrum density of the pseudo-magnetic noise and signal. The signal in Table 1 is the amplitude of the 10 Hz wave, and the noise is the mean value of the noise spectrum density around 10 Hz (8 to 12 Hz, except at 10 Hz) obtained from Fig. 6. The signal of the differential output was about 6100 fTrms/Hz1/2, which was slightly smaller than that of upper output (about 6600 fTrms/Hz1/2) due to subtraction. On the other hand, the noise at the upper and lower positions were around 140 fTrms/Hz1/2 and that of the differential output was 24.7 fTrms/Hz1/2. The pseudo-magnetic noise was reduced greatly as well as the environmental magnetic noise in Fig. 5. Therefore, the SNR greatly improved from 47.3 (upper) to 246.5 (differential).
Fig. 6 Calibrated noise spectrum density of the pseudo-magnetic noise and signal. The blue solid line and the red broken line denote the upper and lower output, respectively, and these lines are overlapped over the entire measured frequency range. The black solid line denotes the differential output.
Table 1. SNR of the optical gradiometer. The signal is the amplitude of 10 Hz wave. The noise is the mean value of the noise spectrum density from 8 to 12 Hz, except at 10 Hz. The unit of signal and noise is fTrms/Hz1/2.
These results demonstrate that the differential measurement of the optical gradiometer is an effective method for reducing the magnetic noise and improving the SNR of OPAMs.
5. Conclusion
We proposed and examined an optical gradiometer, in which the difference of the magnetic fields at two positions was obtained directly via the magneto-optical rotation of one probe beam. By operating at an appropriate probe-beam wavelength, the improvements of the sensitivity and SNR were confirmed. In this study, we demonstrated the feasibility of the first-order axial gradiometer which had two sensing positions along the y-axis. With repeating the reflections and a larger glass cell, the second- and third-order gradiometers could be realized. In addition, a planar gradiometer [10] could also be realized by reflecting the probe beam in the zx-plane and making two sensing positions along the z-axis. In the planar gradiometer, inhomogeneity of the spin polarization caused by the attenuation of the pump beam [12] could be a problem. By using hybrid OPAMs, in which we can polarize spins homogeneously over the cell [28], a planar gradiometer [10] can also be realized. The applicability of optical gradiometers might contribute to practical and reasonable OPAM systems for biomagnetic measurements.
Acknowledgments
This work was partially supported by the Innovative Techno-Hub for Integrated Medical Bioimaging of the Project for Developing Innovation Systems, by a Grant-in-Aid for Research ( 24240081 & 26560466), and by a Grant-in-Aid for JSPS Fellows ( 24-5350), all from the Ministry of Education, Culture, Sports, Science and Technology (MEXT), Japan.
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