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Effects of square-wave magnetic fields on synchronization of nonlinear spin precession for sensitivity improvement of MX magnetometers
Accepted Manuscript Effects of square-wave magnetic fields on synchronization of nonlinear spin precession for sensitivity improvement of MX magnetometers M. Ranjbaran, M.M. Tehranchi, S.M. Hamidi, S.M.H. Khalkhali PII: DOI: Reference:
S0304-8853(17)31403-8 http://dx.doi.org/10.1016/j.jmmm.2017.06.084 MAGMA 62880
To appear in:
Journal of Magnetism and Magnetic Materials
Received Date: Revised Date: Accepted Date:
4 May 2017 15 June 2017 15 June 2017
Please cite this article as: M. Ranjbaran, M.M. Tehranchi, S.M. Hamidi, S.M.H. Khalkhali, Effects of square-wave magnetic fields on synchronization of nonlinear spin precession for sensitivity improvement of MX magnetometers, Journal of Magnetism and Magnetic Materials (2017), doi: http://dx.doi.org/10.1016/j.jmmm.2017.06.084
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Effects of square-wave magnetic fields on synchronization of nonlinear spin precession for sensitivity improvement of MX magnetometers M. Ranjbaran1, M. M. Tehranchi1,2, S. M. Hamidi1 , S. M. H.. Khalkhali3 1
: Laser and Plasma Research Institute, Shahid Beheshti University, Tehran, Iran. 2
: Physics Department, Shahid Beheshti University, Tehran, Iran. 3
: Physics Department, Kharazmi University, Tehran, Iran.
Abstract: Optically pumped atomic magnetometers have found widespread application in biomagnetic studies. Most of the studies utilize MX gradiometers as sensitive and simple arrangements. One the sensitivity improvement methods in the MX configurations is detection of magnetic resonance at higher harmonics due to nonlinear precession of spin polarization. To enhance the harmonic components, we have proposed square wave RF magnetic fields with various duty cycles as substitute for sinusoidal fields. Our results revealed that detection of the 5th harmonic of a 10% duty cycle square wave magnetic field, improved the magnetometer sensitivity by a factor of 4.5 respect to the first harmonic which could be a reliable option to generate high sensitivity MX magnetometers in the MCG applications. Keywords: MX optically pumped magnetometers, harmonic detection, square wave duty cycle, Bloch equations, biomagnetism
Introduction: Measurement of extremely weak biomagnetic fields has attracted great attention in basic and clinical applications [1]. The most important challenge in these precise measurements is the trade-off between high sensitivity and applicability of utilized magnetometers. To achieve the high sensitivity, superconducting quantum interference device (SQUID) magnetometers operate at cryogenic temperatures which restricts their applicability. Consequently, as applicable alternatives to SQUIDs, the uncooled, simple and highly sensitive atomic magnetometers have been introduced [2-4]. Presently, the most sensitive atomic magnetometers operate in the spin exchange relaxation free (SERF) regime which have reached to the highest level of sensitivity, 160 aT/Hz−1/2 [5], as compared to non-SERF mode magnetometers such as coherent population trapping (CPT) [6], electromagnetically induced transparency (EIT) [7], nonlinear magnetooptical rotation (NMOR) [8], Bell-Bloom [9] and MX [10] configurations.
Among the mentioned magnetometers, the MX gradiometers are the configuration of choice in most applications due to their simplicity and fast response [11-13]. Taking advantage of recent advances in design and performance of the MX configuration, could resulted in the sensitivity improvement of simple arrangements of room temperature magnetometers. Some of the advances include usage of special cell design [14], detuned high intensity pump laser [15], double light beams [16] phase-stabilization method [10] and harmonic detection of magnetic resonance technique [17]. In the last technique, the nonlinear evolution of transverse spin polarization (MX) which is caused due to broken rotational symmetry results in drastic increment in amplitude to line-width ratio of the magnetic resonance signal and so the sensitivity. To achieve the magnetic resonance signal, the spin precession of optically pumped atoms should be synchronized at the Larmor frequency. To synchronize the spin precession, various methods such as modulation of the amplitude [18], frequency [19] or polarization [20] of pumping light in the Bell-Bloom configuration and applying a driving sinusoidal radio frequency (RF) magnetic field in the MX configuration [21] have been developed. As a new insight into the Bell-Bloom configurations, square-wave amplitude modulated light has been used to optimize the sensitivity [22,23]. In this paper, with regards to the benefits of magnetic resonance measurement at the higher harmonics in the MX configurations, we have proposed a square wave RF magnetic field as a surrogate for the sinusoidal magnetic field. As the applied square wave RF field contains all harmonics of the resonance frequency, the sensitivity improvement by detection of the higher harmonics has been investigated. Duty cycle of the square wave used as an extra parameter to control the concentration of energy at the special harmonics. The validity of the results has been studied by simulation of the nonlinear dynamics of the transverse spin polarization with analysis of Bloch equations.
Experimental setup: The general principles of operation of MX optically pumped magnetometers have been presented in the earlier papers [11-17 & 24]. The core of set-up was an Octadecyltrichlorosilane (OTS) coated cylindrical (2.5×5cm) Pyrex glass cell containing 85Rb, filling with 10 Torr of N2 for quenching of excited atoms. The 85Rb atoms were spin polarized when a circularly polarized beam of a distributed feedback (DFB) laser passed through the cell. The laser locked to F=3→ F′=2 hyperfine components of the D1 transition (794.8 nm) by a Doppler-free method. Using an optical fiber, the transmitted light coupled to a photodetector. The spin polarization precessed around a static magnetic field (B0) at the Larmor frequency (ωL). The static magnetic field was produced by a 3D Helmholtz coil with a uniformity better than 0.4% across the cell. To synchronize the precession of spin polarization, an oscillating radio frequency (RF) magnetic field was applied transverse to the B0, while B0 was oriented at 45◦ with respect to the light beam (Fig. 1). Magnetic resonance would be detected when the RF frequency matched the Larmor frequency (ωL=ωRF). The RF magnetic field, was produced by a
pair of Helmholtz coils connected to a waveform generator (Function/Arbitrary waveform generator, Picotest, model G5100A). The photodetector output was fed into a lock-in amplifier (Stanford Research Systems, model SR830) for demodulation of the signal which was yielded to in-phase and quadrature components of the transverse spin polarization with reference to the frequency supplied by a scanning waveform generator. The experimental set-up has been depicted in Fig. 1. Note that, in order to suppress the environmental magnetic gradient field, the coils were installed inside three layers of high permeability µ-metal magnetic shielding. The residual field was compensated by the three axis (3D) Helmholtz coils.
Fig. 1. Experimental setup of MX magnetometer: The frequency-locked laser passed through an optical fiber, a polarizer and a quarter wave plate to optically pump a heated Rb cell. The transmitted light coupled to a photodiode through another optical fiber. The photodiode output demodulated via a lock-in amplifier connected to a PC. The lock-in amplifier locked on the frequency of a waveform generator used to produce the square wave magnetic field via RF coils. A DC power supply used to compensate the residual magnetic field in a three-layer magnetic shielding.
To study the magnetometer response to very weak alternative magnetic fields, the in-phase component of magnetic resonance spectrum was of particular interest because the steep slope of this curve near zero crossing (at the Larmor frequency of the B0) could be used to detect small changes of the Larmor frequency (proportional to the changes of applied magnetic field along the static field direction). We verified that the slope of in-phase component (related to the magnetometer sensitivity) at the detection of higher harmonics of RF frequency was so greater than the first harmonic [17]. Consequently, when the static magnetic field (B0) was tuned on resonance, the modification of amplitude of in-phase signal was measured on the center of the slope for each harmonic. For the measurement of magnetic resonance at higher harmonics, the RF coils were driven by a square wave with adjustable duty cycle. To evaluate the frequency response of the
magnetometer, using another function/arbitrary waveform generator, multi-frequency sinusoidal magnetic fields were applied parallel to the B0. Then, to assess the feasibility of biomagnetic field detection with the magnetometer, it was exposed to a simulated cardiac magnetic field. Investigation of the MX magnetometer performance under different operating conditions, revealed that the optimum laser intensity was about 2 mW/m2, the optimum Rb cell temperature was about 44 0C and at 220nT bias field, the optimum RF amplitude was about 5-15 nT [17]. So all the measurements were carried out at these operating conditions.
Results and discussions: To study the improvement of magnetometer sensitivity by applying a square wave magnetic field, we should first analyze the harmonic content of applied square wave. Using fast Fourier transform (FFT), we have plotted the amplitude of harmonic terms contained in the square waveform as a function of its duty cycle (as shown in Fig. 2). It is shown that the amplitudes of harmonics vary with changing the duty cycle which is verified experimentally (Fig. 2). And also when the cross product of harmonic number (n) and duty cycle (D), n×D, has an integer value, the amplitude of nth harmonic is zero. For the special case of a 50% duty-cycle ideal square wave (D=0.5), the amplitudes of all even harmonics are zero. The nonzero harmonics of 50% duty-cycle square wave could be expressed as S =
4A
π
∑
∞ n =3,5,7...
1 sin(n ωRF t) . At this duty cycle, n
because of the 1 coefficient, the amplitudes get smaller as the harmonic number increases. n
Fig. 2. Harmonic content of square wave magnetic fields as a function of duty-cycle. When the cross product of harmonic number and duty cycle, has an integer value, the amplitude of nth harmonic is zero (1st harmonic: black squares, 2nd harmonic: red circles, 3rd harmonic: green up tringles, 4th harmonic: blue down tringles, 5th harmonic: pink diamonds). Solid lines are a guide for the eye.
From the perspective of harmonic contents there is no difference between the square wave with duty cycle D and its supplementary duty cycle (1-D) (for example 20% (D) and 80% (1-D), 30% (D) and 70% (1-D) duty cycles and so on). Consequently, the harmonic contents of the square wave magnetic field have been studied up to 50% duty cycle (D=0.5; d=50%) in Fig. 2.
π∆ν FWHM S According to the sensitivity expression, δ B = γ S / N where γ is the gyromagnetic ratio, the N signal-to-noise ratio and ∆νFWHM the spectral line width, improvement of the sensitivity is related to the slope of dispersive (in-phase) component of resonance signal. The slope is determined by the ratio of the magnetic resonance signal amplitude to its line-width ( ∆S ). ∆F
The dispersive component of the magnetic resonance signals as a function of detection harmonic, when the duty cycle of applied square wave RF magnetic field is 10%, has been depicted in Fig. 3. As can be seen, the resonant frequency of our vapor cell (the Larmor frequency (ωL)) is about 1380 Hz. So by recording the first harmonic response (1×ωL) we detect the resonance curve. But, When the frequency of RF field is swept around ωL/2, we detected the resonance by recording the second harmonic response (2×ωL/2= ωL) and when the frequency of RF field was swept around ωL/3, we detected the resonance by recording the third harmonic response (3×ωL/3=ωL) and so on. So the effective gyromagnetic ratio is not changed. The resonance signals which are centered about ωL/n, where n is the order of harmonic, get narrower line-shapes as the detection harmonic number increases. The narrower line-shapes at the detection of higher harmonics of RF frequency lead to sensitivity improvement which is our main concern. As discussed in our earlier paper [17], when the sinusoidal RF magnetic field is linearly polarized the cylindrical symmetry of the system gets broken and the tip of transverse spin polarization does not move along a circular trajectory. The nonlinear evolution of transvers spin polarization leads to emergence of higher harmonic components. However, the small amplitudes of higher harmonics signals make their detection problematic [25]. By applying the square wave magnetic field as a substitute for sinusoidal field, we could excite the higher harmonics and so intensify the amplitude of the detected resonance signals at higher harmonics.
Fig. 3. Measured dispersive components of magnetic resonance (in-phase lock-in signal) demodulated at 1st harmonic (black squares), 2nd harmonic (red circles), 3rd harmonic (green triangles), 4th harmonic (pink stars) and 5th harmonic (blue diamonds) of swept RF frequency. The 1st, 2nd, 3rd, 4th and 5th harmonic resonances are respectively centered about the 1380 Hz (ωL), 690 Hz (ωL/2), 460 Hz (ωL/3), 345 Hz (ωL/4) and 276 Hz (ωL/5). The amplitude (∆S) and the linewidth (∆F) of a resonance curve is specified. The duty cycle of applied square wave RF magnetic field is 10%.
To study how variations of the duty cycle of the square wave affect the magnetometer sensitivity ( ∆S ), the sensitivity dependence to the duty cycle percentage and the detection harmonic ∆F
number is presented in Fig. 4. This figure exhibits similar trend to the applied square wave magnetic field compared with Fig. 2 but dissimilar amplitude. It can be seen in Fig.2 that the first harmonic of the applied square wave RF field has the maximum amplitude compared with the higher harmonics but it has the lowest amplitude in the measured sensitivity as shown in Fig. 4. So we can conclude that because of the nonlinear response of the spin polarization, the magnetic resonance measurements at the higher harmonics are more beneficial than the first harmonic and the higher harmonics could be enhanced by applying the square wave RF magnetic field.
Fig. 4. Dependence of sensitivity ( ∆S ) to duty cycle percentage and harmonic number of the applied square wave magnetic ∆F field. 1st harmonic (black squares), 2nd harmonic (red circles), 3rd harmonic (blue up triangles), 4th harmonic (green down triangles) and 5th harmonic (pink left triangles) are represented. (The inset shows the variation of sensitivity as a function of harmonic number at 10% duty cycle)
The study revealed that when the nonlinear precession of the spin polarization was synchronized with the 5th harmonic of a 10% duty cycle square wave magnetic field, the magnetometer sensitivity improved by a factor of 4.5 respect to the first harmonic of sinusoidal magnetic field. The figure is plotted for the 1st to 5th harmonics because for larger harmonic number, the sensitivity does not improve anymore. In order to verify this assertion, the magnetometer sensitivity as a function of harmonic number for a square wave with 10% duty cycle has been plotted in the inset of this figure. As can be seen, the maximum value belongs to the 5th harmonic. To investigate the reason of the sensitivity improvement at detection of the 5th harmonic of a 10% duty cycle square wave magnetic field, the variation of linewidth (∆F) and amplitude (∆S) of the magnetic resonance line-shapes as a function of detection harmonic number and the square wave duty cycle have been considered individually. As the magnetic resonance linewidth (∆F) has a crucial role in improvement of the sensitivity [14], its dependency to the detection harmonic number at different duty cycles (d) has been plotted in Fig. 5. It can be seen that by increasing the harmonic number the linewidth of
magnetic resonance line-shapes decrease by a factor of 1/n. However, upper than the 5th harmonic the linewidth fails to decrease efficiently. It seems that in analogy to nonlinear optics, this effect can be denoted as multi-quantum absorption [25]. As the resonance linewidth is actually dependent on the strength of RF magnetic field and the number of atoms, in the multiRF quantum absorption the probability of contribution of atom-RF photons interactions in the resonance process decrease and so the line-width of multi-quantum transition is narrower than the line-width of the single-quantum transition. This behavior, which is independent of duty cycle percentage, is not noticeable for the harmonics larger than the 5th harmonic. So the 5th harmonic has been selected as an optimal harmonic.
Fig. 5. Dependence of measured magnetic resonance linewidth (∆F) to harmonic number at different duty cycles: The black squares at d=5%, red circles at d=10%, blue up triangles at d=20%, green down triangles at d=30%, pink left triangles at d=40% and yellow right triangles at d=50%, represent the linewidths obtained from the in-phase signal of the lock-in amplifier. The optimum harmonic (the 5th harmonic) is shown as a red arrow.
To prove the hypothesis, we have developed an approximate theoretical analysis based on the nonlinear time evolution of the spin polarization in a symmetry-broken system [17]. For simulation of transverse spin polarization (My) oscillations, we have employed the Bloch equations in the presence of a linear RF magnetic field, d dt
M x M y M z
= −γ
M x M y M z
B rf co s (ωrf t ) M x T 2 , − M y T2 ×0 M T B 0 z 1
where γ was the gyromagnetic ratio and T1 and T2 were longitudinal and transverse relaxation times respectively. To find the resonance behavior of the oscillations at higher harmonics, FFT of numerical solution of the Bloch equations was applied. The derivatives of phenomenological predicted resonance signals have been depicted in Fig. 6. To achieve the signals, we have utilized the measured T1 and T2 [26] when a 50% duty cycle square wave RF magnetic field was substituted for the sinusoidal RF magnetic field in the Bloch equations. It is worth to mention that as the magnetic resonance linewidth is independent of the duty cycle percentage, as a case study the 50% duty cycle square wave have been chosen.
Fig. 6. Derivative of predicted magnetic resonance line-shapes by the numerical solution of the Bloch equations as function of swept square wave RF frequency. 1st harmonic (black line), 2nd harmonic (red dash dots), 3rd harmonic (green dash dot dots) and 5th harmonic (blue short dashes) are simulated.
It can be seen in Fig.6 that although the excitation RF field does not contain even harmonics, due to the broken rotational symmetry, the resonance signals are detectable at even harmonics with low amplitude. However, the amplitude is too small at the 4th and upper harmonics. Also, the resonance signal at odd harmonics are reinforced considerably. The linewidth (∆F) of simulated resonance signals versus the harmonic number is represented in Fig. 7, which is in good correlation with the normalized experimental results of Fig. 5 at 50% duty cycle.
Fig. 7. Dependence of simulated magnetic resonance linewidth (∆F) to the number of detection harmonic at 50% duty cycle (blue line) predicted by numerical solution of the Bloch equations. Red squares are the normalized linewidth of in-phase lock-in signals as a function of harmonic number at the duty cycle.
The dependence of amplitude (∆S) of the experimental magnetic resonance line-shapes to the number of detection harmonic at various duty cycles has been measured and plotted in Fig. 8. It can be seen that the maximum value of ∆S at detection of the 5th harmonic belongs to 10% duty cycle square wave magnetic field. So, we could concentrate the RF energy on the desired harmonic components to acquire the best sensitivity.
Fig. 8. Dependence of magnetic resonance amplitude (∆S) to harmonic number at different duty cycles: the amplitude (peak to peak) of in-phase lock-in signals at 5% (black squares), 10% (red circles), 20% (blue up triangles), 30% (green down triangles), 40% (pink left triangles) and 50% (yellow right triangles) duty cycles are represented. The maximum amplitude of magnetic resonances which are demodulated at the 5th harmonic belongs to 10% duty cycle (red arrow).
To realize the potential of our magnetometer in the magnetocardiogram (MCG) recording in comparison to the classical MX magnetometers, the MCG data recorded at the 5th harmonic of resonance frequency without any noise cancellation or signal processing. It should be mentioned that the cardiomagnetic field is produced by one of the Helmholtz coil connected to an arbitrary waveform generator. It can be seen in Fig. 9. that the typical features of MCG trace, i.e. the QRS complex (ventricular depolarization) as well as the P (atrial depolarization) and T (ventricular repolarization) waves are clearly distinguishable in the measured data.
Fig. 9. Sample MCG data (down) and measured response of the magnetometer to the sample (up), recorded at 5th harmonic when a 10% duty cycle square-wave RF magnetic field is applied.
For predicting and adjusting the cardio-magnetometer performance, it is necessary to study its frequency response. The atomic magnetometer responded differently to the magnetic fields of different frequencies. The input-output relation in the Fourier domain is denoted as the frequency response of the magnetometer. In this regard, we have varied the frequency of sinusoidal input signal over the low frequency range of interest for the cardiomagnetic studies (DC to 40 Hz) [27] and evaluated the resulting signal. The frequency response of our magnetometer at detection of
the 1st and 5th harmonics of scanning RF frequency, are depicted in Fig. 10. As it was predicted the improvement of magnetometer sensitivity has leaded to enhancement of magnetometer gain at detection of the 5th harmonic.
Figure 10. Experimental frequency response of the magnetometer (output to input ratio as a function of applied magnetic field frequency) at the detection of 1st (black squares) and 5th (red circles) harmonics when the square wave RF frequency is stabilized on 1380 Hz and 276 Hz, respectively.
In signal analysis, distortion is interpreted as the changing of signal shape. To have a nondistorted MCG signal, it is necessary to investigate the linearity of the magnetometer response. The linearity is a measure of the magnetometer ability to follow changes in the input parameters. Amplitude distortion characteristic and phase distortion characteristic of output signal are two important factors in determining the linearity.
Figure 11. The in-phase lock-in output as a function of applied magnetic field at the 5th harmonic when the square wave RF frequency is fixed on 276 Hz. The experimental data is shown as black squares and the fitted line as the red line. The inset shows the linear input-output relationship for the magnetic fields below 400pT, clearly.
The capability of the magnetometer to follow the changes in amplitude of applied magnetic field, in a fixed frequency, is shown in Fig. 11. It can be seen that there is a linear relationship between the amplitude of output and input signals. To clarify the capability of magnetometer at detection of extremely weak magnetic fields, below 400pT is magnified in the figure inset. Conclusion:
MX gradiometers are well-known magnetometers for biomagnetic measurements. One the respectable techniques for sensitivity improvement of these magnetometers is harmonic detection of magnetic resonance. In this technique, nonlinear spin dynamics due to broken symmetry results in appearance of higher harmonics of excitation frequency. Detection of magnetic resonance at the higher harmonics leads to enhancement of amplitude to linewidth ratio of the resonance spectrum. Benefiting from this technique, we have proposed substitution of square wave RF magnetic field for sinusoidal field. Usage of the square wave RF magnetic field with various duty cycle for synchronization of spin precession resulted in concentration of the RF energy on the desired harmonic components to improve the sensitivity. We found that the steepest slope of the dispersive signal is obtained for detection of the magnetic resonance at the 5th harmonic of a 10% duty cycle square wave field, which allowed the improvement of magnetic field sensitivity by a factor of 4.5 respect to the first harmonic. We verified the results, by numerical solution of Bloch equations to simulate the dependence of sensitivity to the harmonic number and the duty cycle separately. To assert the advantage of this technique in MCG recording, the frequency response of the magnetometer at detection of the 1st and 5th harmonics of scanning RF frequency have been studied. We showed that the enhancement of magnetometer gains at 5th harmonic measurement resulted in improvement of the magnetometer sensitivity in the detection of QRS complex. Acknowledgment This work was supported by the Cognitive Sciences and Technologies Council of Iran [grant number 124]. References: [1] G. Bison, N. Castagna, A. Hofer, P. Knowles, J.-L. Schenker, M. Kasprzak, H. Saudan, and A. Weis, A room temperature 19-channel magnetic field mapping device for cardiac signals, Appl. Phys. Lett. 95 (2009) 173701. [2] S. Knappe, O. Alem, D. Sheng, J. Kitching, Microfabricated optically-pumped magnetometers for biomagnetic applications, J. Phys. Conf. Ser. 723 (2016) 012055. [3] D. Robbes, Highly sensitive magnetometers—a review, Sens. Actuators A Phys. 129 (2006) 86–93. [4] D. Budker and M. Romalis, Optical magnetometry, Nat. Phys. 3 (2007), pp. 227–234 [5] H. B. Dang, A. C. Maloof, and M. V. Romalis, Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer. Appl. Phys. Lett. 97 (2010) 151110. [6] M. Huang and J. C. Camparo, Coherent population trapping under periodic polarization modulation: Appearance of the CPT doublet, Phys. Rev. A. 85 (2012) 012509.
[7] V. I. Yudin, A. V. Taichenachev, Y. O. Dudin, V. L. Velichansky, A. S. Zibrov and S. A. Zibrov, Vector magnetometry based on electromagnetically induced transparency in linearly polarized light, Phys. Rev. A. 82 (2010) 033807. [8] S. Pustelny, A. Wojciechowski, M. Gring, M. Kotyrba, J. Zachorowski, and W. Gawlik, Magnetometry based on nonlinear magneto-optical rotation with amplitude modulated light, J. Appl. Phys. 103 (2008) 063108. [9] H. Hai-Chao, D. Hai-Feng, H. Hui-Jie and H. Xu-Yang, Close-Loop Bell–Bloom Magnetometer with Amplitude Modulation, Chin. Phys. Lett, 32 (2015) 098503. [10] S. Groeger, G. Bison, J.-L. Schenker, R. Wynands, A. Weis, A high-sensitivity laser pumped MX magnetometer, Eur. Phys. J. D. 38 (2006) 239–247. [11] G. Bison, R. Wynands, A. Weis, A laser-pumped magnetometer for the mapping of human cardiomagnetic fields, Appl. Phys. B. 76 (2003) 325-328. [12] V. Tiporlini, K. Alameh, Optical magnetometer employing adaptive noise cancellation for unshielded magnetocardiography, Univ. J. Biomedical. Eng. 1 (2013) 16-21. [13] S. Groeger, G. Bison, P.E. Knowles, R. Wynands, A. Weis, Laser-pumped cesium magnetometers for high-resolution medical and fundamental research, Sens. Actuators A Phys. 129 (2006) 1-5. [14] D. Arnold, S. Siegel, E. Grisanti, J. Wrachtrup, and I. Gerhardt, A Rubidium MXmagnetometer for Measurements on Solid State Spins, Rev. Sci. Instrum. 88 (2017) 023103. [15] T. Scholtes, V. Schultze, R. IJsselsteijn, S. Woetzel and H. Meyer, Light-shift suppression in a miniaturized MX optically pumped Cs magnetometer array with enhanced resonance signal using off-resonant laser pumping, Opt. Express. 20 (2012) 29217. [16] A. K. Vershovskii, Project of Laser-Pumped Quantum MX Magnetometer, Tech. Phys. Lett. 37 (2011) 140–143. [17] M. Ranjbaran, M.M. Tehranchi, S.M. Hamidi, S.M.H. Khalkhali, Harmonic detection of magnetic resonance for sensitivity improvement of optical atomic magnetometers, J. Magn. Magn. Mater. 424 (2017) 284–290 [18] B. Patton, O. O. Versolato, D. C. Hovde, E. Corsini, J. M. Higbie and D. Budker, A remotely interrogated all-optical 87Rb magnetometer, Appl. Phys. Lett. 101 (2012) 083502. [19] R. Jiménez-Martínez, W. C. Griffith, Y. Wang, S. Knappe, Sensitivity Comparison of Mx and Frequency-Modulated Bell–Bloom Cs Magnetometers in a Microfabricated Cell, IEEE Trans. on Instrum. Meas. 59 (2010) 372-378. [20] I. Fescenko, P. Knowles, A. Weis and E. Breschi, A Bell-Bloom experiment with polarization-modulated light of arbitrary duty cycle, Opt. Express, 21 (2013) 15121-15130.
[21] P. D. D. Schwindt, B. Lindseth, S. Knappe, V. Shah and J. Kitching, Chip-scale atomic magnetometer with improved sensitivity by use of the Mx technique, Appl. Phys. Lett. 90 (2007) 081102. [22] Z. D. Gruji´ and A. Weis, Atomic magnetic resonance induced by amplitude-, frequency-, or polarization-modulated light, Phys. Rev. Lett. 88 (2013) 012508. [23] M. Wang, M. Wang, G. Zhang, and K. Zhao, Study of the optimal duty cycle and pumping rate for square-wave amplitude-modulated Bell–Bloom magnetometers, Chin. Phys. B. 25 (2016) 060701. [24] D. Budker and D. Kimball, Optical magnetometry, Cambridge University Press, 2013. [25] A. G. Gurevich, G. A. Melkov, Magnetization Oscillations and Waves, CRC, New York, 1996, pp. 231–242 [26] S.M.H. Khalkhali, M. Ranjbaran, D. Mofidi, S.M. Hamidi, M.M. Tehranchi, Improvement of the spin polarization lifetime in the 85Rb vapor cell by Octadecyltrichlorosilane coating, Chinese. J. Phys. 55 (2017) 301–309. [27] G. Bison, R. Wynands, A. Weis, Dynamical mapping of the human cardiomagnetic field with a room-temperature, laser-optical sensor, Opt. Express. 11 (2003) 904-909.
Highlights: • • • • •
Square wave RF magnetic field substituted for sinusoidal field in MX magnetometers. Amplitude of magnetic resonance signal reinforced at detection of higher harmonics. Narrower resonance signals achieved as the number of detection harmonic increased. Sensitivity of biomagnetic field detection improved respect to classical methods. The Bloch equations predicted the nonlinear dynamics of transverse spin polarization.
S0304-8853(17)31403-8 http://dx.doi.org/10.1016/j.jmmm.2017.06.084 MAGMA 62880
To appear in:
Journal of Magnetism and Magnetic Materials
Received Date: Revised Date: Accepted Date:
4 May 2017 15 June 2017 15 June 2017
Please cite this article as: M. Ranjbaran, M.M. Tehranchi, S.M. Hamidi, S.M.H. Khalkhali, Effects of square-wave magnetic fields on synchronization of nonlinear spin precession for sensitivity improvement of MX magnetometers, Journal of Magnetism and Magnetic Materials (2017), doi: http://dx.doi.org/10.1016/j.jmmm.2017.06.084
This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
Effects of square-wave magnetic fields on synchronization of nonlinear spin precession for sensitivity improvement of MX magnetometers M. Ranjbaran1, M. M. Tehranchi1,2, S. M. Hamidi1 , S. M. H.. Khalkhali3 1
: Laser and Plasma Research Institute, Shahid Beheshti University, Tehran, Iran. 2
: Physics Department, Shahid Beheshti University, Tehran, Iran. 3
: Physics Department, Kharazmi University, Tehran, Iran.
Abstract: Optically pumped atomic magnetometers have found widespread application in biomagnetic studies. Most of the studies utilize MX gradiometers as sensitive and simple arrangements. One the sensitivity improvement methods in the MX configurations is detection of magnetic resonance at higher harmonics due to nonlinear precession of spin polarization. To enhance the harmonic components, we have proposed square wave RF magnetic fields with various duty cycles as substitute for sinusoidal fields. Our results revealed that detection of the 5th harmonic of a 10% duty cycle square wave magnetic field, improved the magnetometer sensitivity by a factor of 4.5 respect to the first harmonic which could be a reliable option to generate high sensitivity MX magnetometers in the MCG applications. Keywords: MX optically pumped magnetometers, harmonic detection, square wave duty cycle, Bloch equations, biomagnetism
Introduction: Measurement of extremely weak biomagnetic fields has attracted great attention in basic and clinical applications [1]. The most important challenge in these precise measurements is the trade-off between high sensitivity and applicability of utilized magnetometers. To achieve the high sensitivity, superconducting quantum interference device (SQUID) magnetometers operate at cryogenic temperatures which restricts their applicability. Consequently, as applicable alternatives to SQUIDs, the uncooled, simple and highly sensitive atomic magnetometers have been introduced [2-4]. Presently, the most sensitive atomic magnetometers operate in the spin exchange relaxation free (SERF) regime which have reached to the highest level of sensitivity, 160 aT/Hz−1/2 [5], as compared to non-SERF mode magnetometers such as coherent population trapping (CPT) [6], electromagnetically induced transparency (EIT) [7], nonlinear magnetooptical rotation (NMOR) [8], Bell-Bloom [9] and MX [10] configurations.
Among the mentioned magnetometers, the MX gradiometers are the configuration of choice in most applications due to their simplicity and fast response [11-13]. Taking advantage of recent advances in design and performance of the MX configuration, could resulted in the sensitivity improvement of simple arrangements of room temperature magnetometers. Some of the advances include usage of special cell design [14], detuned high intensity pump laser [15], double light beams [16] phase-stabilization method [10] and harmonic detection of magnetic resonance technique [17]. In the last technique, the nonlinear evolution of transverse spin polarization (MX) which is caused due to broken rotational symmetry results in drastic increment in amplitude to line-width ratio of the magnetic resonance signal and so the sensitivity. To achieve the magnetic resonance signal, the spin precession of optically pumped atoms should be synchronized at the Larmor frequency. To synchronize the spin precession, various methods such as modulation of the amplitude [18], frequency [19] or polarization [20] of pumping light in the Bell-Bloom configuration and applying a driving sinusoidal radio frequency (RF) magnetic field in the MX configuration [21] have been developed. As a new insight into the Bell-Bloom configurations, square-wave amplitude modulated light has been used to optimize the sensitivity [22,23]. In this paper, with regards to the benefits of magnetic resonance measurement at the higher harmonics in the MX configurations, we have proposed a square wave RF magnetic field as a surrogate for the sinusoidal magnetic field. As the applied square wave RF field contains all harmonics of the resonance frequency, the sensitivity improvement by detection of the higher harmonics has been investigated. Duty cycle of the square wave used as an extra parameter to control the concentration of energy at the special harmonics. The validity of the results has been studied by simulation of the nonlinear dynamics of the transverse spin polarization with analysis of Bloch equations.
Experimental setup: The general principles of operation of MX optically pumped magnetometers have been presented in the earlier papers [11-17 & 24]. The core of set-up was an Octadecyltrichlorosilane (OTS) coated cylindrical (2.5×5cm) Pyrex glass cell containing 85Rb, filling with 10 Torr of N2 for quenching of excited atoms. The 85Rb atoms were spin polarized when a circularly polarized beam of a distributed feedback (DFB) laser passed through the cell. The laser locked to F=3→ F′=2 hyperfine components of the D1 transition (794.8 nm) by a Doppler-free method. Using an optical fiber, the transmitted light coupled to a photodetector. The spin polarization precessed around a static magnetic field (B0) at the Larmor frequency (ωL). The static magnetic field was produced by a 3D Helmholtz coil with a uniformity better than 0.4% across the cell. To synchronize the precession of spin polarization, an oscillating radio frequency (RF) magnetic field was applied transverse to the B0, while B0 was oriented at 45◦ with respect to the light beam (Fig. 1). Magnetic resonance would be detected when the RF frequency matched the Larmor frequency (ωL=ωRF). The RF magnetic field, was produced by a
pair of Helmholtz coils connected to a waveform generator (Function/Arbitrary waveform generator, Picotest, model G5100A). The photodetector output was fed into a lock-in amplifier (Stanford Research Systems, model SR830) for demodulation of the signal which was yielded to in-phase and quadrature components of the transverse spin polarization with reference to the frequency supplied by a scanning waveform generator. The experimental set-up has been depicted in Fig. 1. Note that, in order to suppress the environmental magnetic gradient field, the coils were installed inside three layers of high permeability µ-metal magnetic shielding. The residual field was compensated by the three axis (3D) Helmholtz coils.
Fig. 1. Experimental setup of MX magnetometer: The frequency-locked laser passed through an optical fiber, a polarizer and a quarter wave plate to optically pump a heated Rb cell. The transmitted light coupled to a photodiode through another optical fiber. The photodiode output demodulated via a lock-in amplifier connected to a PC. The lock-in amplifier locked on the frequency of a waveform generator used to produce the square wave magnetic field via RF coils. A DC power supply used to compensate the residual magnetic field in a three-layer magnetic shielding.
To study the magnetometer response to very weak alternative magnetic fields, the in-phase component of magnetic resonance spectrum was of particular interest because the steep slope of this curve near zero crossing (at the Larmor frequency of the B0) could be used to detect small changes of the Larmor frequency (proportional to the changes of applied magnetic field along the static field direction). We verified that the slope of in-phase component (related to the magnetometer sensitivity) at the detection of higher harmonics of RF frequency was so greater than the first harmonic [17]. Consequently, when the static magnetic field (B0) was tuned on resonance, the modification of amplitude of in-phase signal was measured on the center of the slope for each harmonic. For the measurement of magnetic resonance at higher harmonics, the RF coils were driven by a square wave with adjustable duty cycle. To evaluate the frequency response of the
magnetometer, using another function/arbitrary waveform generator, multi-frequency sinusoidal magnetic fields were applied parallel to the B0. Then, to assess the feasibility of biomagnetic field detection with the magnetometer, it was exposed to a simulated cardiac magnetic field. Investigation of the MX magnetometer performance under different operating conditions, revealed that the optimum laser intensity was about 2 mW/m2, the optimum Rb cell temperature was about 44 0C and at 220nT bias field, the optimum RF amplitude was about 5-15 nT [17]. So all the measurements were carried out at these operating conditions.
Results and discussions: To study the improvement of magnetometer sensitivity by applying a square wave magnetic field, we should first analyze the harmonic content of applied square wave. Using fast Fourier transform (FFT), we have plotted the amplitude of harmonic terms contained in the square waveform as a function of its duty cycle (as shown in Fig. 2). It is shown that the amplitudes of harmonics vary with changing the duty cycle which is verified experimentally (Fig. 2). And also when the cross product of harmonic number (n) and duty cycle (D), n×D, has an integer value, the amplitude of nth harmonic is zero. For the special case of a 50% duty-cycle ideal square wave (D=0.5), the amplitudes of all even harmonics are zero. The nonzero harmonics of 50% duty-cycle square wave could be expressed as S =
4A
π
∑
∞ n =3,5,7...
1 sin(n ωRF t) . At this duty cycle, n
because of the 1 coefficient, the amplitudes get smaller as the harmonic number increases. n
Fig. 2. Harmonic content of square wave magnetic fields as a function of duty-cycle. When the cross product of harmonic number and duty cycle, has an integer value, the amplitude of nth harmonic is zero (1st harmonic: black squares, 2nd harmonic: red circles, 3rd harmonic: green up tringles, 4th harmonic: blue down tringles, 5th harmonic: pink diamonds). Solid lines are a guide for the eye.
From the perspective of harmonic contents there is no difference between the square wave with duty cycle D and its supplementary duty cycle (1-D) (for example 20% (D) and 80% (1-D), 30% (D) and 70% (1-D) duty cycles and so on). Consequently, the harmonic contents of the square wave magnetic field have been studied up to 50% duty cycle (D=0.5; d=50%) in Fig. 2.
π∆ν FWHM S According to the sensitivity expression, δ B = γ S / N where γ is the gyromagnetic ratio, the N signal-to-noise ratio and ∆νFWHM the spectral line width, improvement of the sensitivity is related to the slope of dispersive (in-phase) component of resonance signal. The slope is determined by the ratio of the magnetic resonance signal amplitude to its line-width ( ∆S ). ∆F
The dispersive component of the magnetic resonance signals as a function of detection harmonic, when the duty cycle of applied square wave RF magnetic field is 10%, has been depicted in Fig. 3. As can be seen, the resonant frequency of our vapor cell (the Larmor frequency (ωL)) is about 1380 Hz. So by recording the first harmonic response (1×ωL) we detect the resonance curve. But, When the frequency of RF field is swept around ωL/2, we detected the resonance by recording the second harmonic response (2×ωL/2= ωL) and when the frequency of RF field was swept around ωL/3, we detected the resonance by recording the third harmonic response (3×ωL/3=ωL) and so on. So the effective gyromagnetic ratio is not changed. The resonance signals which are centered about ωL/n, where n is the order of harmonic, get narrower line-shapes as the detection harmonic number increases. The narrower line-shapes at the detection of higher harmonics of RF frequency lead to sensitivity improvement which is our main concern. As discussed in our earlier paper [17], when the sinusoidal RF magnetic field is linearly polarized the cylindrical symmetry of the system gets broken and the tip of transverse spin polarization does not move along a circular trajectory. The nonlinear evolution of transvers spin polarization leads to emergence of higher harmonic components. However, the small amplitudes of higher harmonics signals make their detection problematic [25]. By applying the square wave magnetic field as a substitute for sinusoidal field, we could excite the higher harmonics and so intensify the amplitude of the detected resonance signals at higher harmonics.
Fig. 3. Measured dispersive components of magnetic resonance (in-phase lock-in signal) demodulated at 1st harmonic (black squares), 2nd harmonic (red circles), 3rd harmonic (green triangles), 4th harmonic (pink stars) and 5th harmonic (blue diamonds) of swept RF frequency. The 1st, 2nd, 3rd, 4th and 5th harmonic resonances are respectively centered about the 1380 Hz (ωL), 690 Hz (ωL/2), 460 Hz (ωL/3), 345 Hz (ωL/4) and 276 Hz (ωL/5). The amplitude (∆S) and the linewidth (∆F) of a resonance curve is specified. The duty cycle of applied square wave RF magnetic field is 10%.
To study how variations of the duty cycle of the square wave affect the magnetometer sensitivity ( ∆S ), the sensitivity dependence to the duty cycle percentage and the detection harmonic ∆F
number is presented in Fig. 4. This figure exhibits similar trend to the applied square wave magnetic field compared with Fig. 2 but dissimilar amplitude. It can be seen in Fig.2 that the first harmonic of the applied square wave RF field has the maximum amplitude compared with the higher harmonics but it has the lowest amplitude in the measured sensitivity as shown in Fig. 4. So we can conclude that because of the nonlinear response of the spin polarization, the magnetic resonance measurements at the higher harmonics are more beneficial than the first harmonic and the higher harmonics could be enhanced by applying the square wave RF magnetic field.
Fig. 4. Dependence of sensitivity ( ∆S ) to duty cycle percentage and harmonic number of the applied square wave magnetic ∆F field. 1st harmonic (black squares), 2nd harmonic (red circles), 3rd harmonic (blue up triangles), 4th harmonic (green down triangles) and 5th harmonic (pink left triangles) are represented. (The inset shows the variation of sensitivity as a function of harmonic number at 10% duty cycle)
The study revealed that when the nonlinear precession of the spin polarization was synchronized with the 5th harmonic of a 10% duty cycle square wave magnetic field, the magnetometer sensitivity improved by a factor of 4.5 respect to the first harmonic of sinusoidal magnetic field. The figure is plotted for the 1st to 5th harmonics because for larger harmonic number, the sensitivity does not improve anymore. In order to verify this assertion, the magnetometer sensitivity as a function of harmonic number for a square wave with 10% duty cycle has been plotted in the inset of this figure. As can be seen, the maximum value belongs to the 5th harmonic. To investigate the reason of the sensitivity improvement at detection of the 5th harmonic of a 10% duty cycle square wave magnetic field, the variation of linewidth (∆F) and amplitude (∆S) of the magnetic resonance line-shapes as a function of detection harmonic number and the square wave duty cycle have been considered individually. As the magnetic resonance linewidth (∆F) has a crucial role in improvement of the sensitivity [14], its dependency to the detection harmonic number at different duty cycles (d) has been plotted in Fig. 5. It can be seen that by increasing the harmonic number the linewidth of
magnetic resonance line-shapes decrease by a factor of 1/n. However, upper than the 5th harmonic the linewidth fails to decrease efficiently. It seems that in analogy to nonlinear optics, this effect can be denoted as multi-quantum absorption [25]. As the resonance linewidth is actually dependent on the strength of RF magnetic field and the number of atoms, in the multiRF quantum absorption the probability of contribution of atom-RF photons interactions in the resonance process decrease and so the line-width of multi-quantum transition is narrower than the line-width of the single-quantum transition. This behavior, which is independent of duty cycle percentage, is not noticeable for the harmonics larger than the 5th harmonic. So the 5th harmonic has been selected as an optimal harmonic.
Fig. 5. Dependence of measured magnetic resonance linewidth (∆F) to harmonic number at different duty cycles: The black squares at d=5%, red circles at d=10%, blue up triangles at d=20%, green down triangles at d=30%, pink left triangles at d=40% and yellow right triangles at d=50%, represent the linewidths obtained from the in-phase signal of the lock-in amplifier. The optimum harmonic (the 5th harmonic) is shown as a red arrow.
To prove the hypothesis, we have developed an approximate theoretical analysis based on the nonlinear time evolution of the spin polarization in a symmetry-broken system [17]. For simulation of transverse spin polarization (My) oscillations, we have employed the Bloch equations in the presence of a linear RF magnetic field, d dt
M x M y M z
= −γ
M x M y M z
B rf co s (ωrf t ) M x T 2 , − M y T2 ×0 M T B 0 z 1
where γ was the gyromagnetic ratio and T1 and T2 were longitudinal and transverse relaxation times respectively. To find the resonance behavior of the oscillations at higher harmonics, FFT of numerical solution of the Bloch equations was applied. The derivatives of phenomenological predicted resonance signals have been depicted in Fig. 6. To achieve the signals, we have utilized the measured T1 and T2 [26] when a 50% duty cycle square wave RF magnetic field was substituted for the sinusoidal RF magnetic field in the Bloch equations. It is worth to mention that as the magnetic resonance linewidth is independent of the duty cycle percentage, as a case study the 50% duty cycle square wave have been chosen.
Fig. 6. Derivative of predicted magnetic resonance line-shapes by the numerical solution of the Bloch equations as function of swept square wave RF frequency. 1st harmonic (black line), 2nd harmonic (red dash dots), 3rd harmonic (green dash dot dots) and 5th harmonic (blue short dashes) are simulated.
It can be seen in Fig.6 that although the excitation RF field does not contain even harmonics, due to the broken rotational symmetry, the resonance signals are detectable at even harmonics with low amplitude. However, the amplitude is too small at the 4th and upper harmonics. Also, the resonance signal at odd harmonics are reinforced considerably. The linewidth (∆F) of simulated resonance signals versus the harmonic number is represented in Fig. 7, which is in good correlation with the normalized experimental results of Fig. 5 at 50% duty cycle.
Fig. 7. Dependence of simulated magnetic resonance linewidth (∆F) to the number of detection harmonic at 50% duty cycle (blue line) predicted by numerical solution of the Bloch equations. Red squares are the normalized linewidth of in-phase lock-in signals as a function of harmonic number at the duty cycle.
The dependence of amplitude (∆S) of the experimental magnetic resonance line-shapes to the number of detection harmonic at various duty cycles has been measured and plotted in Fig. 8. It can be seen that the maximum value of ∆S at detection of the 5th harmonic belongs to 10% duty cycle square wave magnetic field. So, we could concentrate the RF energy on the desired harmonic components to acquire the best sensitivity.
Fig. 8. Dependence of magnetic resonance amplitude (∆S) to harmonic number at different duty cycles: the amplitude (peak to peak) of in-phase lock-in signals at 5% (black squares), 10% (red circles), 20% (blue up triangles), 30% (green down triangles), 40% (pink left triangles) and 50% (yellow right triangles) duty cycles are represented. The maximum amplitude of magnetic resonances which are demodulated at the 5th harmonic belongs to 10% duty cycle (red arrow).
To realize the potential of our magnetometer in the magnetocardiogram (MCG) recording in comparison to the classical MX magnetometers, the MCG data recorded at the 5th harmonic of resonance frequency without any noise cancellation or signal processing. It should be mentioned that the cardiomagnetic field is produced by one of the Helmholtz coil connected to an arbitrary waveform generator. It can be seen in Fig. 9. that the typical features of MCG trace, i.e. the QRS complex (ventricular depolarization) as well as the P (atrial depolarization) and T (ventricular repolarization) waves are clearly distinguishable in the measured data.
Fig. 9. Sample MCG data (down) and measured response of the magnetometer to the sample (up), recorded at 5th harmonic when a 10% duty cycle square-wave RF magnetic field is applied.
For predicting and adjusting the cardio-magnetometer performance, it is necessary to study its frequency response. The atomic magnetometer responded differently to the magnetic fields of different frequencies. The input-output relation in the Fourier domain is denoted as the frequency response of the magnetometer. In this regard, we have varied the frequency of sinusoidal input signal over the low frequency range of interest for the cardiomagnetic studies (DC to 40 Hz) [27] and evaluated the resulting signal. The frequency response of our magnetometer at detection of
the 1st and 5th harmonics of scanning RF frequency, are depicted in Fig. 10. As it was predicted the improvement of magnetometer sensitivity has leaded to enhancement of magnetometer gain at detection of the 5th harmonic.
Figure 10. Experimental frequency response of the magnetometer (output to input ratio as a function of applied magnetic field frequency) at the detection of 1st (black squares) and 5th (red circles) harmonics when the square wave RF frequency is stabilized on 1380 Hz and 276 Hz, respectively.
In signal analysis, distortion is interpreted as the changing of signal shape. To have a nondistorted MCG signal, it is necessary to investigate the linearity of the magnetometer response. The linearity is a measure of the magnetometer ability to follow changes in the input parameters. Amplitude distortion characteristic and phase distortion characteristic of output signal are two important factors in determining the linearity.
Figure 11. The in-phase lock-in output as a function of applied magnetic field at the 5th harmonic when the square wave RF frequency is fixed on 276 Hz. The experimental data is shown as black squares and the fitted line as the red line. The inset shows the linear input-output relationship for the magnetic fields below 400pT, clearly.
The capability of the magnetometer to follow the changes in amplitude of applied magnetic field, in a fixed frequency, is shown in Fig. 11. It can be seen that there is a linear relationship between the amplitude of output and input signals. To clarify the capability of magnetometer at detection of extremely weak magnetic fields, below 400pT is magnified in the figure inset. Conclusion:
MX gradiometers are well-known magnetometers for biomagnetic measurements. One the respectable techniques for sensitivity improvement of these magnetometers is harmonic detection of magnetic resonance. In this technique, nonlinear spin dynamics due to broken symmetry results in appearance of higher harmonics of excitation frequency. Detection of magnetic resonance at the higher harmonics leads to enhancement of amplitude to linewidth ratio of the resonance spectrum. Benefiting from this technique, we have proposed substitution of square wave RF magnetic field for sinusoidal field. Usage of the square wave RF magnetic field with various duty cycle for synchronization of spin precession resulted in concentration of the RF energy on the desired harmonic components to improve the sensitivity. We found that the steepest slope of the dispersive signal is obtained for detection of the magnetic resonance at the 5th harmonic of a 10% duty cycle square wave field, which allowed the improvement of magnetic field sensitivity by a factor of 4.5 respect to the first harmonic. We verified the results, by numerical solution of Bloch equations to simulate the dependence of sensitivity to the harmonic number and the duty cycle separately. To assert the advantage of this technique in MCG recording, the frequency response of the magnetometer at detection of the 1st and 5th harmonics of scanning RF frequency have been studied. We showed that the enhancement of magnetometer gains at 5th harmonic measurement resulted in improvement of the magnetometer sensitivity in the detection of QRS complex. Acknowledgment This work was supported by the Cognitive Sciences and Technologies Council of Iran [grant number 124]. References: [1] G. Bison, N. Castagna, A. Hofer, P. Knowles, J.-L. Schenker, M. Kasprzak, H. Saudan, and A. Weis, A room temperature 19-channel magnetic field mapping device for cardiac signals, Appl. Phys. Lett. 95 (2009) 173701. [2] S. Knappe, O. Alem, D. Sheng, J. Kitching, Microfabricated optically-pumped magnetometers for biomagnetic applications, J. Phys. Conf. Ser. 723 (2016) 012055. [3] D. Robbes, Highly sensitive magnetometers—a review, Sens. Actuators A Phys. 129 (2006) 86–93. [4] D. Budker and M. Romalis, Optical magnetometry, Nat. Phys. 3 (2007), pp. 227–234 [5] H. B. Dang, A. C. Maloof, and M. V. Romalis, Ultrahigh sensitivity magnetic field and magnetization measurements with an atomic magnetometer. Appl. Phys. Lett. 97 (2010) 151110. [6] M. Huang and J. C. Camparo, Coherent population trapping under periodic polarization modulation: Appearance of the CPT doublet, Phys. Rev. A. 85 (2012) 012509.
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Highlights: • • • • •
Square wave RF magnetic field substituted for sinusoidal field in MX magnetometers. Amplitude of magnetic resonance signal reinforced at detection of higher harmonics. Narrower resonance signals achieved as the number of detection harmonic increased. Sensitivity of biomagnetic field detection improved respect to classical methods. The Bloch equations predicted the nonlinear dynamics of transverse spin polarization.