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Abstract
We analyze and suppress the magnetic noise response in optical rotation detection system (ORDS) in atomic magnetometers in this study. Because of the imperfections of the optical elements, the probe light is actually elliptically polarized in ORDS, which can polarize the atom ensemble and cause the responses to the three-axis magnetic noise. We theoretically analyze the frequency responses to the magnetic noise, and prove that the responses are closely associated with the DC magnetic field. The values of the DC magnetic fields are calculated with special frequency points, called ‘break points’, in the transverse responses. We reveal the relationships between the DC magnetic field and the sensitivities of ORDS, and effectively suppress the magnetic noise responses with the residual magnetic field compensation. Finally, the sensitivity of ORDS is improved by approximately two times at 10-20 Hz.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
Optical rotation detection systems (ORDS) are widely used in various kinds of atomic sensors, such as atomic magnetometers, co-magnetometers, and nuclear magnetic resonance (NMR) gyroscopes [1–4]; and appear in many applications such as fundamental physics [5–7], biomagnetism [8–10], magnetism test [11,12], etc. In ORDS, linearly polarized light is utilized for probing the atomic spin polarization, whose rotation angle varies when interacting with the alkali-atom ensemble. The sensitivity of ORDS is affected by several noise sources [13–16]. The photon shot noise originates from the quantum properties of laser light, which determines the fundamental sensitivity limits. The technical noise sources include electronic noise, laser light intensity and frequency fluctuation noise, which can be suppressed using feedback control methods [4,17,18]. Moreover, the magnetic noise source could affect the sensitivity, especially at low frequencies and high probe light intensities. With higher light intensities, the pumping rate of the light gets larger, leading to larger responses to the magnetic noise. For instance, in photo-elastic modulator (PEM)-based ORDS of the spin-exchange relaxation-free (SERF) magnetometer, the magnetic noise source could account for more than one-third of the total noise at the optimal light intensities [16]. Hence, for sensitivity improvement, the influence of the magnetic noise should be suppressed.
Without the pumping beam, the ORDS in the atomic sensors has no magnetic field responses in ideal conditions because the linearly polarized light will not cause atom polarization. However, because of the imperfections of the polarizer and the vapor cell, and the interaction with the atom ensemble, the probe light in ORDS is elliptically polarized with a small ellipticity (usually less than 2°) [3,16,19,20]. This elliptically polarized light can polarize the atom ensemble, and thus the ORDS will respond to the magnetic field. The influences of the magnetic noise are determined by both the value of magnetic noise and the frequency responses of the ORDS [21]. The magnetic noise comes from the magnetic shield owing to the Johnson noise and magnetization noise, whose suppression requires ultra-low noise material and the ingenious design of the magnetic shield [21–23]. Besides, by reducing the frequency responses caused by the elliptically polarized probe light, the influences of the magnetic noise can also be suppressed and the sensitivities of ORDS are improved. With the decreasing of the ellipticity, the frequency responses are reduced, but the ellipticity cannot be decreased to zero because of the imperfection of the optical elements and the influences of the atoms [20]. Thus, other methods should be researched for frequency responses reduction in ORDS considering small ellipticity.
Frequency response analysis, including the amplitude-frequency response and the phase-frequency response, have been investigated in a number of studies of optically pumped atomic magnetometers. The ORDS considering elliptical polarization has similar structure with single-beam magnetometers using elliptically polarized light [24,25]. However, their frequency responses are distinct because of the absence of the magnetic modulation in ORDS. In some studies, frequency response along a certain axis is investigated [26–28]. This is not sufficient for magnetic noise response analysis because magnetic noise appears along the three axes [22,23]. Fan et al. considered three-axis frequency responses based on the state-space method [29]. They did not consider the triaxial DC residual magnetic field which have a significant impact on the frequency responses. Ichihara et al. solved the frequency responses to a transverse dynamic field with triaxial residual magnetic fields [30], but they did not obtain three-axis dynamic responses. In conclusion, to describe the magnetic noise responses, the frequency responses to three-axis dynamic magnetic fields should be obtained considering three-axis DC residual magnetic fields. The existing methods are not suitable for describing the magnetic noise responses caused by elliptically polarized light in ORDS.
In this study, we analyzed and suppressed the magnetic noise responses caused by the elliptical polarization of the light in ORDS of atomic magnetometers in SERF regime. First, we linearized the Bloch equation based on the separation of the steady-state and dynamic responses, and then solved it using the Laplace transform. The three-axis frequency responses were obtained from the Laplace transform solutions. Then, the relationships between three-axis frequency responses were illustrated, from which the accurate values of the DC residual magnetic fields were calculated. The results showed that there were special frequency points appearing in the transverse responses, called ‘break points’. At these frequency points, the amplitude-ratios between the longitudinal response and the transverse responses decreased −3 dB, and the phase-differences between them were 45°. The break points were closely related to the DC magnetic fields, from which the residual magnetic fields were calculated and compensated accurately. Finally, we studied the angular sensitivity of ORDS with different DC magnetic fields. With only transverse DC fields present, the ORDS only responded to the transverse magnetic noise, whereas with only longitudinal DC fields, there were no magnetic noise responses. The sensitivity of ORDS was improved approximately two times at 10–20 Hz by reducing the frequency responses. The methods proposed in this paper are also suitable for sensitivity improvement in SERF co-magnetometers, NMR gyroscopes and other atomic magnetometers.
2. Principle
2.1 Frequency response analysis
In the ORDS of SERF magnetometers, elliptically polarized laser light polarizes the alkali-metal atom ensemble along the quantum spindle direction defined as the z-axis. With the magnetic field, the atomic energy levels split due to the Zeeman effect, leading to the Larmor precession of the atomic spin vector. This process is described with the Bloch equation,
The component along the z-axis ${S_z}$ is detected by the magnetometer, which varies with the magnetic field ${\textbf B}$. The magnetic field can be represented as ${\textbf B} = {{\textbf B}_0} + \tilde{{\textbf B}}$, where ${{\textbf B}_0}$ is the three-axis DC magnetic field, and $\tilde{{\textbf B}}$ is the dynamic magnetic field. Correspondingly, the spin vector is expressed as ${\textbf S} = {{\textbf S}_0} + \tilde{{\textbf S}}$. ${{\textbf S}_0}$ is the steady-state response, and $\tilde{{\textbf S}}$ is the dynamic response. When the dynamic response is much smaller than the steady-state response $|{\tilde{{\textbf S}}} |\ll |{{{\textbf S}_0}} |$, the ORDS with elliptically polarized light can be regarded as a linear system. The relationships between the input variables $\tilde{{\textbf B}} = {\left( {\begin{array}{*{20}{c}} {{{\tilde{B}}_x}(t )}&{{{\tilde{B}}_y}(t )}&{{{\tilde{B}}_z}(t )} \end{array}} \right)^T}$ and output variables ${\tilde{S}_z}(t )$ are described by the transfer functions ${G_i}(s)$, (see the Appendix)
Fig. 1. Schematic of the transfer functions of ORDS. The ORDS described by Eq. (2) is a linear system, whose characteristics are decided by the static magnetic field ${{\textbf B}_0}$ and steady-state response ${{\textbf S}_0}$. The input variables are the three dynamic magnetic fields ${\tilde{B}_x}(t )$, ${\tilde{B}_y}(t )$, and ${\tilde{B}_z}(t )$, and the output variable is the longitudinal spin vector component ${\tilde{S}_z}(t )$. The transfer functions are ${G_i}(s) = {{{\cal L}[{{{\tilde{S}}_z}(t )} ]} / {{\cal L}[{{{\tilde{B}}_i}(t )} ]}}$ ($i = x,y,z$).
The frequency responses can be obtained from the transfer functions with $s = j\omega$, where $j = \sqrt { - 1}$ [31]. The frequency responses include the amplitude-frequency responses, $|{G(\omega )} |$, which are the modulus of $G({s = j\omega } )$; and the phase-frequency responses, $\varphi (\omega )$ which are the arguments. As shown in Eq. (2), the frequency responses $G({s = j\omega } )$ are proportional to the photon polarization ${s_{\textrm{PR}}} = \sin 2\varepsilon$. With the increase of the ellipticity $\varepsilon$, the frequency responses become larger. In the ORDS, the ellipticity is usually less than 2°. At this range, the frequency responses are linear scaled with the ellipticity $G({s = j\omega } )\propto {s_{\textrm{PR}}} = 2\varepsilon$.
In the transverse frequency responses ${G_x}(\omega )$ and ${G_y}(\omega )$, special frequency points appear, which are ${\omega _{1x}},{\omega _{1y}}$ as shown in Table 1 and are called ‘break points’ [31]; while in the longitudinal response ${G_z}(\omega )$ there is no break point. The break points are important parameters which reveal the differences and relationships between the three-axis frequency responses. The ratios of the amplitude responses are,
It is shown in Eq. (3) that the break points ${\omega _{1x/1y}}$ are the −3 dB bandwidth of the amplitude-ratios between the longitudinal response and the transverse responses. Moreover, the differences between the phase responses are,
Table 1. Elements in Transfer Functions a
It is shown in Eq. (4) that the phase differences are 45° when the frequency satisfied $\omega = {\omega _{1x/1y}}$.
The break points ${\omega _{1x}},{\omega _{1y}}$ are closely associated with the three components of the magnetic field vector ${B_{0x}}$, ${B_{0y}}$, and ${B_{0z}}$, which can be used for the DC magnetic field vector calculation. With parameters ${\omega _0}$, ${\omega _{1x/1y}}$, $|{{A_z}/{A_{x/y}}} |$ which are all related to the components ${\omega _{0x}}$, ${\omega _{0y}}$, ${\omega _{0z}}$, the optimum solutions of the precession frequencies ${{\mathbf \omega }_0}$ are obtained, and the three-axis magnetic fields are given as follows,
2.2 Magnetic noise response
The magnetic noise $\delta {\textbf B}(\omega )$ is a type of dynamic magnetic field, whose response can be described by Eq. (2). The magnetic noise is induced by the magnetization noise and the eddy current noise from the magnetic shield. The longitudinal and transverse magnetic noises are uncorrelated variables and are numerically similar [21]. Because of the linear system, the measured noise spectrum of ${\tilde{S}_z}$ is the sum of the responses along the three axes ${\tilde{S}_z}(\omega )= \delta B(\omega )\cdot \sqrt {\sum\limits_{i = x,y,z} {{{|{{G_i}(\omega )} |}^2}} }$. The spin vector component ${\tilde{S}_z}(\omega )$ brings about the magneto-optical rotation angle of the elliptically polarized probe light because of the circular birefringence effect. Thus, magnetic noise $\delta {\textbf B}(\omega )$ is introduced to the angular noise spectrum $\delta \theta$. With larger ellipticity $\varepsilon$, the frequency responses become larger, which means that the magnetic noise responses ${\tilde{S}_z}(\omega )$ are linearly scaled with the photon polarization. With the increase of the ellipticity, the magnetic noise responses get larger, and the sensitivity of ORDS become worse.
To reduce the magnetic noise influence, the three-axis responses $|{{G_i}(\omega )} |$ ($i = x,y,z$) should be suppressed. According to Table 1, elements $|{{G_{1i}}(\omega )} |$ are associated with the Larmor frequency of the DC residual magnetic fields ${\omega _{0x}}$, ${\omega _{0y}}$, ${\omega _{0z}}$, which could be decreased by the magnetic field compensation. For example, if magnetic field along z-axis has been compensated ${\omega _{0z}} \to 0$, the parameter ${A_z}$ which is proportional to ${\omega _{0z}}$ approaches zero, leading to the longitudinal response $|{{G_z}(\omega )} |$ also approaches zero. Thus, the ORDS does not respond to the longitudinal magnetic noise. The DC residual magnetic fields can be accurately compensated according to Eq. (5). At this situation, all three-axis responses ${G_i}(\omega )\to 0$, the magnetic noise influence will be decreased to the greatest extent.
3. Experiment
The experimental setup shown in Fig. 2 is an ORDS of the SERF magnetometer. The alkali metal K in natural abundance is contained in the central spherical cell, with 50 Torr N2 as the quenching gas and 2 atm 4He as the buffer gas. The cell is approximately 25 mm in diameter and is placed in an oven heated to around 200 °C for obtaining high-density alkali atomic vapor. At the periphery, five-layer cylindrical µ-metal magnetic shields were utilized. The three-axis coils driven by function generators (Keysight, 33500B) can produce uniform magnetic fields.
Fig. 2. Schematic of the Experimental setup. The polarizer is used for linearly polarizing the light which becomes elliptically polarized after passing through the cell. The output signal is detected by the PD, and demodulated with the lock-in amplifier. The PEM is used for low frequency noise suppression. The azimuths between the transmission axis of the polarizer and the fast axis of PEM is 45°. QWP, quarter-wave plate; PEM, photo-elastic modulator; PD, photodetector.
The light source is a distributed Bragg reflection (DBR) laser. A polarizer (Glan-Taylor prism) is used to generate linearly polarized light. The ellipticity $\varepsilon $ is adjusted to its minimum using a quarter-wave plate (QWP) before the cell. The ellipticity is measured with a polarimeter (PAX1000, Thorlabs), which is less than 0.1° before the vapor cell and approximately 1.2° after the cell. After the cell, a QWP and a PEM (Model-100, Hinds Instruments) were used for laser signal modulation to suppress low-frequency electronic noise. Another polarizer transfers the polarization rotation signal to the laser power signal which is detected by a photodetector (PD). The signal is demodulated by the lock-in amplifier (LIA) (HF2TA, HF2LI, Zurich Instrument) with the reference frequency to the LIA from the PEM controller. In the demodulated signal, the first harmonic component is proportional to the atomic spin vector along the z-direction ${\tilde{S}_z}(t )$.
In the experiment, sinusoidal magnetic fields were applied along the x, y, and z directions, respectively, and frequency responses Gx, Gy and Gz near the peak points were collected. The three responses were fitted for the frequency ${\omega _0}$ and linewidth $\Delta \omega$. The amplitudes ratios $|{{A_z}/{A_{x/y}}} |$ and break points ${\omega _{1x/1y}}$ were obtained from the relationships of the frequency responses. The three-axis DC magnetic fields were calculated according to these parameters. Finally, with different DC magnetic fields, the ORDS sensitivities are measured, and then the optimal sensitivity is revealed with magnetic field compensation.
4. Results and discussion
The magneto-optical rotation angle $\theta $ is detected by the ORDS as shown in Fig. 2. The rotation angle is expressed as [26],
To verify the correctness of the model in Section 2, the frequency responses with a certain DC magnetic field ${{\textbf B}_0}$ were studied first. A sinusoidal magnetic field ${\tilde{B}_x}(t )$ with an effective value of approximately 100 pT and frequency f was applied along the x-axis. Because of the linearization, the output voltage signal ${U_1}({{{\tilde{S}}_z}} )$ is also a sinusoidal signal with frequency f, whose effective value is the amplitude-frequency response $|{{G_x}(f )} |$ and the phase is the phase-frequency response ${\varphi _x}(f )$. Similarly, the frequency responses along the y and z axes were collected, as shown in Fig. 3. The data were fitted using Eq. (2) $G(f) = {U_0}\prod\limits_{m = 1}^4 {{G_m}(f)} + e$, the elements ${G_m}(f)$ shown in Table 1 with $\omega = 2\pi f$. The fitting results are listed in Table 2. The average value of ${f_0}$ along the three axes is (56.7 ± 0.2) Hz. The magnetic linewidths $\Delta f$ influence the full width at half maximum, and the estimated value is (4.7 ± 0.3) Hz.
Fig. 3. Experimental results with DC magnetic field ${{\textbf B}_0}$. (a) The amplitude-frequency responses. The data are fitted with $G(f) = {U_0}\prod\limits_{m = 1}^4 {{G_m}(f)} + e$, fitting results shown in Table 2. The curves show the resonance frequency ${\omega _0} = 2\pi {f_0}$ and the magnetic linewidth $\Delta \omega = 2\pi \Delta f$. (b) The phase-frequency responses.
Table 2. Fitting results of the frequency responses
For the break points, the ratios of the amplitude-frequency responses $|{{G_z}(f)/{G_x}(f)} |$ and $|{{G_z}(f)/{G_y}(f)} |$, and the differences in the phase-frequency responses ${\varphi _z}(f) - {\varphi _x}(f)$ and ${\varphi _z}(f) - {\varphi _y}(f)$, are utilized. The results are shown in Fig. 4. The ratios were fitted using Eq. (3), and the fitting results are presented in Table 3. According to Eq. (4), the differences ${\varphi _z}(f) - {\varphi _x}(f)$ are an increasing function of f when ${f_{1x}} < 0$. At the break point, the difference is ±45°±180°, from which the estimated value is ${f_{1x}} \approx{-} ({22 \pm 1} )$ Hz. The additional phase 180° comes from the initial phase difference of applied magnetic field, or the opposite sign of ${A_x}$ and ${A_z}$. Similarly, the differences ${\varphi _z}(f) - {\varphi _y}(f)$ are a decreasing function, and the break point is ${f_{1y}} \approx{+} ({26 \pm 1} )$ Hz.
Fig. 4. Relationships between the three responses. (a) Ratios of the amplitude-frequency responses described by Eq. (3). The data are fitted, results showing in Table. 3. The break points ${f_{1x/1y}}$ appear at −3 dB bandwidth. (b) Differences of the phase-frequency responses described by Eq. (4). The break points ${f_{1x/1y}}$ appear when the differences are ±45°±180°. The additional phase 180° comes from the initial phase difference of applied magnetic field, or the opposite sign of ${A_x}$ and ${A_z}$. The break points are positive when the differences are decreasing function or the sign of 45° is minus.
Table 3. Fitting results of the frequency responses ratios
The resonance frequency ${f_0}$, break points ${f_{1x/1y}}$ and ratios $|{{A_z}/{A_{x/y}}} |$ are closely related to the three components of DC magnetic field. With the values in Tables 2 and 3, the optimum solutions of the magnetic field vectors are estimated based on the least squares method. The results were ${B_x} = ({6.65 \pm 0.09} )$ nT, ${B_y} = ({8.77 \pm 0.075} )$ nT, and ${B_z} = ({5.13 \pm 0.12} )$ nT.
As a confirmatory study, the coil constants were calibrated, as shown in Fig. 5. In Fig. 5(a), in addition to the sinusoidal magnetic field, we applied an additional DC magnetic field with a DC current using the x-axis coils ${I_x}$. Thus, the total DC magnetic field sensed by the atom ensemble is ${{\textbf B}_0} + {C_{\textrm{coil}}}{I_x}\vec{x}$. The variation in the DC magnetic field changes the resonance frequency ${f_0}$ and the break points ${f_{1x/1y}}$. These parameters were fitted again, and the new DC field was calculated. For different values of ${I_x}$, the DC magnetic field ${B_{0x}}$ obtained was different, as shown in Fig. 5(a). The coil constant is ${C_x} = ({\textrm{0}\textrm{.0695} \pm 0.0011} )$ nT/µA. The other coil constants were ${C_y} = ({\textrm{0}\textrm{.1845} \pm 0.0047} )$ nT/µA along y-axis and, ${C_z} = ({\textrm{0}\textrm{.0771} \pm 0.0010} )$ nT/µA along z-axis, respectively.
Fig. 5. Coil constants calibration results of (a) x-axis, (b) y-axis, (c) z-axis. Currents were applied to the coils respectively and the three-axis DC magnetic fields were calculated with the method proposed in this paper.
With the value of the DC magnetic field, we studied the relationship between the DC field and the angular sensitivities, as shown in Fig. 6. The green line represented the sensitivity with the residual field compensated, meaning that ${{\mathbf \omega }_0} \to {\textbf 0}$. In Fig. 6(a), the longitudinal magnetic field was zero, ${B_{0z}} \to 0$, and the transverse magnetic field was applied. In this situation, only transverse response remained, and the sensitivities of ORDS were influenced by transverse magnetic noise. In contrast, in Fig. 6(b), a longitudinal magnetic field ${B_{0z}}$ was applied, while ${B_{0x}} \to 0$ and ${B_{0y}} \to 0$. Not only the transverse responses ${G_{x/y}}(f)$ but the longitudinal response ${G_z}(f)$ approached zero according to Table 1, and the sensitivities were not influenced by ${B_{0z}}$. As shown in Fig. 6(b), the angular sensitivities remained approximately unchanged with a distinct DC longitudinal magnetic field ${B_{0z}}$.
Fig. 6. Relationships between DC magnetic fields and the angular sensitivities of ORDS. (a) Longitudinal magnetic field ${B_{0z}}$ is compensated while transverse magnetic field ${B_\textrm{t}}$ is applied. The sensitivities become worse with the increase of .. because of the influence of transverse magnetic noise. (b) Longitudinal magnetic field ${B_{0z}}$ is applied while transverse magnetic fields are compensated. In this situation, the ORDS will not respond to dynamic magnetic fields, thus the sensitivities are unchanged.
Then, we studied the relationship between the ellipticity and the angular sensitivity, as shown in Fig. 7. In Fig. 7(a), the magnetic fields are compensated with different ellipticity. With DC magnetic fields approaches zero, the frequency responses approaches zero according to Eq. (2). Thus, the ORDS hardly responds to the magnetic noise, and the angular sensitivities have little relationship with the ellipticity. In Fig. 7(b), the transverse magnetic field is 1 nT. In this situation, the frequency responses are proportional to the photon polarization, and the angular sensitivities get worse with the ellipticities are larger.
Fig. 7. Relationships between the ellipticity $\varepsilon$ and the angular sensitivity of ORDS. The ellipticity is measured before the vapor cell. (a) The magnetic fields are compensated, and thus the frequency responses approaches zero. The sensitivities have little relationship with the ellipticity. (b) The longitudinal magnetic field is compensated while transverse magnetic field is 1 nT. The sensitivities become worse with the increase of the ellipticity.
The angular sensitivities of ORDS with the compensated and uncompensated residual magnetic fields are compared in Fig. 8. Below 40 Hz, where the magnetic noise influence is noteworthy, the sensitivities with magnetic field compensation show significant improvement. Above 40 Hz, the sensitivity is mainly affected by the photon shot noise source and the electronic noise source [16]. The magnetic field compensation method has little effect on the sensitivity improvement. At 10–20 Hz, the sensitivities improved by approximately two times.
5. Conclusion
In this paper, we demonstrated the influence of elliptically polarized light on the sensitivity of ORDS in SERF magnetometers. The elliptically polarized probe light polarizes the alkali atom ensemble, which leads to the responses of the magnetic noise in the shield. Theoretically, we linearized the Bloch equation by separating of the static and dynamic responses ${{\textbf S}_0}$ and $\tilde{{\textbf S}}(t )$, and obtained the output variable ${\tilde{S}_z}(t )$ with three input variables $\tilde{{\textbf B}}(t )= {\left( {\begin{array}{*{20}{c}} {{{\tilde{B}}_x}}&{{{\tilde{B}}_y}}&{{{\tilde{B}}_z}} \end{array}} \right)^T}$. We revealed the dynamic responses along different axes, ${G_x}(s)$, ${G_y}(s)$, ${G_z}(s)$, and their amplitude-frequency responses and phase-frequency responses. It was found that the break points in the transverse responses ${G_x}(s)$, ${G_y}(s)$ can be used to precisely calculate the triaxial DC magnetic field. Experimentally, the DC magnetic field was measured by fitting the three-axis frequency response, and the three-axis coil constants were calibrated. The angular sensitivities of the ORDS with different DC fields were measured. With only a transverse magnetic field, the sensitivities improved as the DC field became smaller, while having only the longitudinal magnetic field, the sensitivities were unrelated to the DC field. Finally, we effectively reduced the magnetic noise responses by precise residual magnetic field compensation, and the sensitivity improved by a factor of two at 10–20 Hz. This method is beneficial for improving probe sensitivity in SERF magnetometers, co-magnetometers, and NMR gyroscopes.
Appendix
Here, the detailed solutions to Eq. (1) are provided. In ORDS, the three-axis DC magnetic field ${{\textbf B}_0}\textrm{ = }{\left( {\begin{array}{*{20}{c}} {{B_{0x}}}&{{B_{0y}}}&{{B_{0z}}} \end{array}} \right)^\textrm{T}}$ and three-axis dynamic magnetic field $\tilde{{\textbf B}}$ should be considered. At this situation, the system is nonlinear because of the coupling of the three axes. To find the relationship between ${\textbf B}$ and the spin vector ${\textbf S}$, the system should be linearized. It is linearized with the separation of the steady-state response and the dynamic response. The spin vector is expressed as ${\textbf S} = {{\textbf S}_0} + \tilde{{\textbf S}}$. ${{\textbf S}_0}$ is the steady-state response affected by the pumping rate and DC magnetic field, and $\tilde{{\textbf S}}$ is the dynamic response. Putting ${\textbf B} = {{\textbf B}_0} + \tilde{{\textbf B}}$ and ${\textbf S} = {{\textbf S}_0} + \tilde{{\textbf S}}$ into Eq. (1), we have the steady-state equation,
The solution to the steady-state equation is
The solution to the dynamic equation is obtained using Laplace transform. Take the response to the x-axis dynamic magnetic field ${\tilde{B}_x}(t )$ as an example. In the dynamic equation, there are $\tilde{{\textbf B}}(t )= {\left( {\begin{array}{*{20}{c}} {{{\tilde{B}}_x}(t )}&0&0 \end{array}} \right)^T}$ and $\tilde{{\textbf S}}(t )= {\left( {\begin{array}{*{20}{c}} {{{\tilde{S}}_x}(t )}&{{{\tilde{S}}_y}(t )}&{{{\tilde{S}}_z}(t )} \end{array}} \right)^T}$. The dynamic equation can be expressed as,
Then the Laplace transform is used. The Laplace transform of the response ${\tilde{S}_x}(t )$ is ${\cal L}[{{{\tilde{S}}_x}(t )} ]= {S_x}(s )$, and the other variables are similar. The dynamic response vector is zero when t = 0, $\tilde{{\textbf S}}(0 )= {\textbf 0}$. Thus, the derived function of ${\tilde{S}_x}(t )$ has a Laplace transform ${\cal L}[{{{\dot{\tilde{S}}}_x}(t )} ]= s{S_x}(s )$. The Laplace transform of the equations is,
The differential equations translate into algebraic equations which are easier to solve. The solution is given by Eq. (2).
Funding
National Natural Science Foundation of China (61903013); Beijing Municipal Natural Science Foundation (4191002); Major Scientific Research Project of Zhejiang Lab (2019MB0AE01).
Disclosures
The authors declare no conflict of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but maybe obtained from the authors upon reasonable request.
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