Coupled dark state magnetometer for the China Seismo-Electromagnetic Satellite

The coupled dark state magnetometer excites two magnetic field-dependent CPT resonances in parallel to determine the surrounding magnetic field. To fulfill the criteria of CPT resonance excitation [13] and for simplicity this is established by a double modulated light field [9, 17]. A third modulation is required to enable the detection with frequency modulation spectroscopy techniques [9, 17, 32].

The vertical-cavity surface-emitting laser (VCSEL) diode (vacuum wavelength $\lambda_{\rm Laser}=\lambda_{\rm FS} \approx 794.978$ nm) is frequency modulated (FM) by a microwave signal ($f_{\rm MW}=\frac{1}{2}\nu_{\rm HFS}\approx 3.417$ GHz). Both first-order sidebands of this FM spectrum fit the HFS energy levels F  =  1, m_F  =  0 and F  =  2, m_F  =  0 of the 5²S_1/2 ground state of the rubidium D₁ line in figure 2. According to the Zeeman effect [33], the energy levels with the magnetic quantum numbers m_F  =  −2, −1, 1, 2 are dependent on the magnetic field³. In figure 2 this is indicated by the shifted black lines while grey lines mean zero magnetic field. In order to establish permanent CPT resonances in presence of a magnetic field, the microwave frequency f_MW is used as carrier for a phase modulation (PM) whose sidebands fit and excite the magnetic field-dependent energy levels.

The modulation frequency f_B of this PM is limited to 2.1 MHz by the design which corresponds to a detectable magnetic field strength of approx. 150 μT for $n=\pm2$ and 100 μT for $n=\pm3$ via the corresponding Zeeman-shifting factors of approx. 14 and 21 Hz nT⁻¹. It is generated in the field programmable gate array (FPGA) using direct digital synthesis (DDS). In principle, a digital tuning word specifies the output frequency as a fraction of the clock frequency while the resolution is a function of the register width [34, 10]. The frequency component f_B is carrier for an additional third modulation of the light field with the modulation frequency ${f_{\rm ref}}_1$ which enables FM spectroscopy and will be discussed below.

The digital FM sinusoidal output of the DDS is transformed to the analogue domain with a parallel digital-to-analogue converter (DAC) and fed as modulation signal to the microwave generator in order to establish the PM.

In the bias-T the modulated microwave signal is superposed with the signal of the current source which for now is assumed to be constant. The output is fed to the VCSEL diode as injection current in which the superposition is transformed to the first FM. The vertical cavity design of VCSELs has the characteristic of low parasitic capacity and therefore can be modulated by microwave signals with good efficiency [35]. Furthermore, certain VCSELs have inherent single mode capability [35] and durability [36].

The result is an overall triple-modulated light field spectrum whose spectral components fit to the appropriate transitions as indicated by either the red solid or green dotted arrows in figure 2. In this way a permanent excitation of one of the magnetic field-dependent CPT resonance superpositions $n=\pm2$ or $\pm3$ is achieved. The light is guided through a 50 μm graded index multimode fibre to the sensor unit where a polarizer and a quarter-wave plate establish a circular polarized excitation field.

In the rubidium-filled glass cell the CPT resonances are acting as frequency discriminator and convert the DDS generated FM (${f_{\rm ref}}_1$) of the multi-chromatic light field spectrum into an amplitude modulation of the transmitted laser intensity. Further discussion on the interaction process was carried out in [32, 10].

Afterwards, the transmitted light field is guided to the photodiode via a 400 μm step index multimode fibre. The photodiode converts the amplitude modulated intensity, which carries the rubidium-interfered information of the excited CPT resonances, to an equivalent photo current. A pre-amplifier converts the ac component of the current into a voltage signal, which is fed to the digital lock-in amplifier via an analogue-to-digital converter (ADC).

The third modulation in combination with the lock-in amplifier enables FM spectroscopy capable of sensitive and rapid measurement of the narrow and weak spectral features [32]. Since the modulation frequency ${f_{\rm ref}}_1$ is large compared to the linewidth of CPT resonances (typically  <2 kHz), the spectral feature is probed by a single sideband [32]. With a modulation index around one, the in-phase output of the lock-in amplier can be interpreted as absorption signal and the quadrature-phase output as dispersion signal of the CPT interaction with the rubidium atoms in the glass cell [32].

Due to the narrow spectral linewidth, which is a consequence of the strict constraints of the CPT mechanism, the slope at zero crossing of the quadrature-phase signal is steep (see figure 3). The point of zero crossing corresponds to the exact excitation of the magnetic field-dependent distinct hyperfine energy levels by the double modulated light field. The only variable parameter of the modulated light field is the frequency f_B of the DDS signal while the laser carrier wavelength $\lambda_{\rm Laser}$ and the microwave frequency f_MW are for now assumed to be constant. The zero crossing in combination with the Zeeman controller in figure 4 is used to track changes of the magnetic field induced Zeeman energy levels by adapting the DDS frequency f_B. The actuating variable of this controller corresponds to f_B of the DDS signal and, therefore, it is a direct measurement for the magnetic field by applying an appropriate form of the Breit–Rabi formula (see equation (1)).

The controller operates permanently at frequency values which are related to the resonance superposition $n=\pm2$ via the Zeeman-shifting factor of approx. 14 Hz nT⁻¹. In case of switching the magnetic field detection to the resonance superposition $n=\pm3$, the control loop does not need to be unlocked in order to find and follow the related zero crossing. The controller just scales the actuating variable f_B by the factor 1.5 for $n=\pm3$ which corresponds with sufficient accuracy to the Zeeman-shifting factor of approx. 21 Hz nT⁻¹. A detailed discussion can be found in [30].

In scientific space missions a scalar reference magnetometer is always operated in combination with vector magnetometers. The measurements of the latter can be used for a rough calculation of the angle between the magnetic field and the optical axis of the sensor as well as the magnitude of the magnetic field by applying, e.g. coordinate rotation digital computer (CORDIC) algorithms [37]. The instrument controller can use this information to select the suitable resonance superposition $n=\pm2$ or $\pm3$ in order to ensure a CDSM operation in all directions of the sensor with respect to the magnetic field and to rapidly find the CPT resonances at start-up. In principle, both functions can also be achieved without an additional vector magnetometer but have not been implemented yet. The procedure for independent resonance switching is discussed in [30]. The resonances can be found in a scanning mode but this approach takes more time.

Without control of the VCSEL temperature and current, the carrier frequency and, consequently, the entire triple-modulated light field spectrum would drift. This would cause a so-called two-photon detuning [38] of the CPT resonance excitation process and would lead to a malfunction of the magnetometer. For a reliable operation of the CDSM the carrier frequency must fit to the optical transition 5²S_1/2 $\rightarrow5^2$P_1/2 of the ⁸⁷Rb D₁ line (full width at half maximum (FWHM) typically 600 MHz) within a tolerance of  <100 MHz. Therefore, additional control loops [10] are required to stabilize the VCSEL carrier frequency which is strongly related to the dc injection current and the laser diode temperature via the frequency shifting coefficients, e.g.  −121 GHz mA⁻¹ and  −28 GHz K⁻¹ (@ $\lambda_{\rm VCSEL}=794.978$ nm), respectively.

The temperature of the VCSEL is set by a bi-directional pulse width modulated (PWM) current through a thermo-electric cooling (TEC) element and is monitored by a thermistor. The temperature signal is quantized by an ADC and fed to the laser temperature controller which calculates the duty cycle required to keep the actual temperature at the set temperature (see figure 4).

The laser current control loop sets the injection current of the VCSEL in order to fit the output frequency to the maximum of the spectral feature of the optical transition. The source supplies the VCSEL diode and, additionally, two adjustable voltage controlled current sinks in parallel. One of them sets the dc current through the laser diode. The control voltage of this sink corresponds to the actuating variable of the laser current controller in the FPGA. The other current sink is driven by a sinusoidal signal with the frequency ${f_{\rm ref}}_3$. It enables an FM spectroscopy technique which is used to stabilize the VCSEL carrier frequency onto the spectral feature. Consequently, the current at the output of the adjustable current source is a superposition of a dc and an ac component. The subsequent signal path is similar to that of the magnetic field control loop in section 3.1.

In contrast to the FM spectroscopy of the CPT resonances the modulation parameters (modulation index, frequency and phase) are chosen differently for the laser carrier frequency control loop. Here, the frequency ${f_{\rm ref}}_3$ is lower than the linewidth of the spectral feature. In that case the in-phase signal is related to the first derivative of the absorption profile [39] and is preferably used as frequency discriminator in the control loop.

The HFS transition frequency $\nu_{\rm HFS}$ of the unperturbed ⁸⁷Rb atom is influenced by the buffer gas pressure (2.94 Hz Pa⁻¹ [28]), the buffer gas temperature (13 Hz K⁻¹; see figure 8)⁴ and in second order perturbation by the magnetic field (57.515 kHz mT⁻² [29]).

Under the assumption of negligible influence of the magnetic field, a mismatch $\Delta\nu_{\rm HFS_1}$ of the microwave generator induced sidebands and the HFS transition frequency causes a pair of CPT resonances (n  =  −2 and  +2 or n  =  −3 and  +3) which are split by the magnitude $\Delta\nu_{\rm HFS_1}$ instead of being perfectly superimposed. In case of $\Delta\nu_{\rm HFS_1}\leqslant \frac{\sqrt{3}}{6} \, \delta\nu_{\rm CPT}\, \approx 0.289\, \delta\nu_{\rm CPT}$ [9], where $\delta\nu_{\rm CPT}$ is the linewidth of the CPT resonance, this pair is not entirely resolved and thus observable as distorted, dispersive shaped signal. In case of equal CPT resonance line strengths (n  =  −2 and  +2 or n  =  −3 and  +3) the point of zero crossing is identical to the case $\Delta\nu_{\rm HFS_1}=0$. Therefore, a mismatch would not affect the magnetic field measurement of the CDSM [9]. In reality in most of the cases, the strengths of single positive and negative numbered CPT resonances are not symmetric. Hence, the systematic error induced by the mismatch is not entirely cancelled.

To avoid this error, the CPT resonance n  =  0 related to the HFS splitting transition $\nu_{\rm HFS}$ can be used as reference to tune the microwave generator frequency in order to reach $\Delta\nu_{\rm HFS_1}=0$ and thus achieve cancellation of the adverse systematic frequency shifts induced by the buffer gas. Additionally, the HFS transition control loop is capable of compensating drifts of the microwave generator.

The microwave generator is realized by a phase-locked loop which consists of a voltage-controlled microwave oscillator and a fractional n-counter frequency divider. The generator provides a constant radio frequency power level and a modulation bandwidth up to 1.5 MHz. The time base for the microwave generator is an adjustable reference oscillator which is tuned via a voltage input by the actuating variable of the microwave controller in figure 4. A sinusoidal signal with the frequency ${f_{\rm ref}}_2$ is superposed to the actuating variable and enables an FM spectroscopy technique for stabilizing the microwave oscillator. The subsequent signal path and the interpretation of the demodulation outputs of the lock-in amplifier are similar to the magnetic field detection control loop. Consequently, the quadrature-phase of the ${f_{\rm ref}}_2$ signal component is fed to the control unit as feedback signal.

Ideally, the HFS transition is permanently tracked in order to compensate the adverse systematic shifts mentioned above. However, the transition corresponds to n  =  0 in equation (3) and the amplitude of its spectral feature experiences the same dependence on the angle as the CPT resonances with $n=\pm2$. As consequence, the control loop cannot work reliable at angles $60^\circ<\beta<120^\circ$ and $240^\circ <\beta< 300^\circ$. For these alignment ranges, the controller state and the last actuating variable are preserved. Nevertheless, linear correction terms can be applied to adjust the reference oscillator as a function of sensor temperature and electronics board temperature.

As mentioned, the HFS splitting transition and, consequently, the related CPT resonance n  =  0 depend on the magnetic field strength in second-order perturbation theory. In contrast, the lifted HFS sublevels involved in the CPT resonance superpositions $n=\pm2$ and $\pm3$ change with the magnetic field in first⁵ and second-order perturbation theory. All shifting coefficients are quantum number dependent and thus different for each CPT resonance component. The second-order frequency shifts of the CPT resonances are 57.515 kHz mT⁻² for n  =  0, 43.136 kHz mT⁻² for $n=\pm2$ and 21.568 kHz mT⁻² for $n=\pm3$. If the microwave generator perfectly matches the HFS splitting transition, the microwave signal is not in the center of the single CPT resonances n  =  −2 and  +2 or n  =  −3 and  +3. Therefore, the second-order shift contribution of the CPT resonances causes a mismatch $\Delta\nu_{\rm HFS_2}$ which has a similar impact as $\Delta\nu_{\rm HFS_1}$ discussed above.

For an accurate magnetic field measurement the tracking of the HFS transition must be interrupted, the controller state preserved and the latest adjustment voltage corrected. With the correction the microwave signal is shifted into the center of the single CPT resonances n  =  −2 and  +2 or n  =  −3 and  +3 and the mismatch $\Delta\nu_{\rm HFS_2}=0$. The correction term depends on the magnetic field strength and the selected CPT resonance superposition $n= \pm2$ or $n = \pm3$.

For sensor alignment ranges $60^\circ <\beta<120^\circ$ and $240^\circ<\beta<300^\circ$ the controller state and actuating variable are already preserved and the magnetic field-dependent correction term can be applied permanently. For angles outside these ranges, the controller operates in an alternating mode with an overall period of 1 s. First it tracks the HFS transition. Afterwards the state is preserved and the microwave signal is corrected for the rest of the second. This strategy allows one to cancel adverse systematic frequency shifts induced by the buffer gas or the ac Stark effect as well as drifts of the microwave generator and enables an accurate magnetic field measurement with an update rate of 1 Hz.

The strength of a CPT resonance depends, amongst other parameters, on the vapor pressure of the rubidium atoms in the glass cell and, consequently, on the temperature [29]. By increasing the temperature, more rubidium atoms are in the gaseous phase and available for interaction with the light field. The signal-to-noise ratio of a CPT resonance increases with the sensor temperature [17, 19, 23, 28] but starts to decrease again at a certain temperature when the optical thickness effect gets dominant.

With a neon vapor pressure of 5 kPa the sensor cell temperature must be between 18 °C and 55 °C for a reliable operation of the CDSM. Hence, no temperature control is required for the operation at room temperature (25 °C). For lower temperatures, which for instance occur during the CSES mission, a sensor temperature control loop is needed.

Similar to the laser temperature control loop, the temperature of the rubidium atoms is set by a bi-directional PWM ac current through a bifilar wound heating coil around the glass cell. The glass cell temperature is monitored by a non-magnetic thermistor (both not shown in figure 1). The thermistor signal is quantized by an ADC and fed to the controller which calculates the required PWM duty cycle.

Table 1 shows the mass and power consumption required for the CDSM flight model. The magnetometer is part of the HPM instrument package and the design was driven by the available power interface of $\pm12$ V and 5V. The power consumption consists of a steady part of 2836 mW and a dynamic part, which depends on the laser and sensor units' thermal environment. The significant driver is the power required for laser heating or cooling with the power-to-temperature coefficients of 39.8 mW K⁻¹ and 120.6 mW K⁻¹, respectively. A careful selection of the laser diode regarding its operating temperature is crucial.

For CSES in a temperature stable low Earth orbit, a nominal power consumption of 3394 mW was measured which includes 486 mW to set the laser temperature and 72 mW to heat the sensor unit to 25 °C.

Significant losses occur in the line regulators and operational amplifiers due to the limited available supply levels. For example, with a more efficient but still not optimal power interface of $\pm8$ V and 3.3 V, the steady power consumption would be reduced to 66%.

The instrument mass is 1672 g. This includes the mounting frame for the electronics with 596 g.