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Abstract
The single-beam comagnetometer working in the spin-exchange relaxation-free (SERF) state is being developed into a miniaturized atomic sensor with extremely high precision in rotation measurement. In this paper, we propose a reflective configuration for the single-beam SERF comagnetometer. The laser light simultaneously used for optical pumping and signal extraction is designed to pass through the atomic ensemble twice. In the optical system, we propose a structure composed of a polarizing beam splitter and a quarter-wave plate. With this, the reflected light beam can be separated entirely from the forward propagating one and realize a complete light collection with a photodiode, making the least light power loss. In our reflective scheme, the length of interaction between light and atoms is extended, and because the power of the DC light component is attenuated, the photodiode can work in a more sensitive range and has a better photoelectric conversion coefficient. Compared with the single-pass scheme, our reflective configuration has a stronger output signal and performs better signal-to-noise ratio and rotation sensitivity. Our work has an important impact on developing miniaturized atomic sensors for rotation measurement in the future.
© 2023 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The comagnetometer has a great potential for advancement as it offers an exceptional level of precision in measurement [1,2]. Comagnetometers utilise a thermal sample of alkali vapor and noble gas contained in a sealed glass cell to sense weak signals [3], and the measured result can be obtained from a probing laser that passes through the atomic ensemble [4]. Using optical pumping technique, the electron spins and nuclear spins can be polarized and brought to an operational state [5,6]. A study in 2002 found that under the condition of a high atomic number density of the alkali atoms and a near-zero magnetic field environment, the electron spins will operate at a state without the spin-exchange relaxation [7]. This condition is named as the spin-exchange relaxation-free (SERF) state and enables comagnetometers to achieve a higher measuring sensitivity. Later, a SERF comagnetometer scheme operating with a self-compensation capability is proposed [3]. By having the nuclear spins track and compensate for the low-frequency magnetic field fluctuations, the electron spins can remain unaffected by these disturbances, thereby preventing any signal drift in the output [8]. With the above techniques, the SERF comagnetometers have been widely concerned in various research areas, especially in exploring the frontiers of fundamental physics, such as Lorentz and CPT symmetry tests [9–11] and searching for exotic spin-dependent interactions [12]. Since the nuclear spins are also sensitive to rotation input, the comagnetometers can also be applied to rotation measurement and navigation, and they have proved their potential in this area [1,13–15].
While improving the measuring precision, researchers have also made efforts to miniaturize the SERF comagnetometer and optimize its performance. In 2022, a single-beam SERF comagnetometer using circularly polarized light was proposed and applied to rotation measurement, realizing a bias instability comparable to the traditional pump-and-probe scheme [16]. Later, a single-beam comagnetometer using elliptically polarized light was also proposed. Meanwhile, a phase-dependent signal decoupling method was proposed [17], which can effectively tackle the dual-axis output coupling problem and is more convenient to operate than the ones in previous work [18,19]. So far, the single-beam comagnetometer has realized dual-axis rotation measurement, which promotes miniaturization research. After the problem of signal coupling is solved, the practicability of the device is further improved, and the research in this area is also pushed to a new higher level.
Although the single-beam comagnetometer has been initially developed, there is still scope for improving its measuring performance, such as the signal-to-noise ratio and also the rotation sensitivity. The current configuration of the single-beam comagnetometer is still the single-pass scheme. The laser light only propagates one way through the glass cell, and the distance for the light interacting with atoms is restricted. In previous studies, in order to strengthen the optical signal, researchers proposed a multipass scheme for the traditional magnetometers with the pump-and-probe configuration [20,21]. It allows the probe beam for signal extraction to pass through the cell for multiple times, increasing the distance for the light to interact with the atoms. As of now, the reflective scheme has not yet been employed in single-beam comagnetometers for rotation measurement.
In this paper, we propose a reflective single-beam comagnetometer for rotation measurement. A circularly polarized light is simultaneously used for optical pumping and signal detection. The light beam is designed to propagate through the cell carrying the atomic ensemble twice with a mirror for reflection. In the optical system, we also design a structure composed of a polarizing beam splitter (PBS) and a quarter-wave plate (QWP). This structure can circularly polarize the forward-propagating light and separate all the backward light from its original route, realizing a total collection of all the reflected light and avoiding making light power loss. We theoretically explain the enhancement of the oscillating light signal while using our reflective configuration and analyze how the DC light component affects the photoelectric conversion rate of the photodiode and thereby affects the output. Our reflective scheme can improve the signal-to-noise ratio and the rotation sensitivity while maintaining the miniaturization advantage of the single-beam comagnetometer. Our research is of great significance for developing miniaturized atomic rotation sensors.
2. Principle of operation
In our comagnetometer, the glass cell contains both alkali metal droplets and noble gas. The alkali droplets are a mixture of K and Rb. A hybrid optical pumping technique is adopted to polarize the atomic ensemble [13,22,23], which can improve the homogeneity of the spin polarization. Under a heated condition, the saturated pressure of the alkali metal vapor will increase, forming an environment with high atomic number density in the closed cell. Combined with a zero magnetic environment provided by a magnetic shielding system, the electron spins can work at a SERF state, and the system can achieve a higher measuring sensitivity. First, we define the three-dimensional coordinates of the comagnetometer device. As shown in Fig. 1, the direction in which the laser beam travels through the cell is defined as the $z$ axis. The intersecting line of the plane perpendicular to the $z$ axis and the installation plane of the device is defined as the $y$ axis, and through the right-hand rule, the $x$ axis is obtained. Generally, the number density of Rb is much larger than that of K in the cell. Here, we define ${{\mathbf{P}}^e}$ as the electron spins and ${{\mathbf{P}}^n}$ as the nuclear spins. The time evolution process of ${{\mathbf{P}}^e}$ and ${{\mathbf{P}}^n}$ can be described with a set of Bloch equations [1]:
Fig. 1. The schematic diagram of our reflective single-beam SERF comagnetometer for rotation measurement. LG, a lens group for adjusting the beam size; HWP, half-wave plate; PBS, polarized beam splitter; LCVR, liquid crystal variable retarder; GT, Glan-Taylor polarizer; PD (PD1, PD2), photodiode; QWP, quarter-wave plate. The thick red line represents the incident light. The thick blue line represents the light reflected by the mirror, which travels in the opposite direction to the incident beam. The red dashed frame is the physical picture of the PBS and QWP optical structure.
Here, ${\gamma _e} = - 1.761 \times {10^{11}}\;{\textrm{rad}} \cdot {{\textrm{s}}^{ - 1}} \cdot {{\textrm{T}}^{ - 1}}$ and ${\gamma _n} = - 2.113 \times {10^7}\;{\textrm{rad}} \cdot {{\textrm{s}}^{ - 1}} \cdot {{\textrm{T}}^{ - 1}}$ are the gyromagnetic ratios of the electron spins ${{\mathbf{P}}^e}$ and nuclear spins ${{\mathbf{P}}^n}$, respectively. $Q\left ( {{P^e}} \right )$ is the slowing-down factor, which is a function of the spin quantum number and the electron spins polarization $P^e$ [24]. ${{{\mathbf{B}}}}$ is the external magnetic field sensed by the atoms. ${\mathbf{L}}$ is a fictitious magnetic field caused by the ac-Stark light shift from the pump beam [25]. ${{\mathbf{B}}^e}$ and ${{\mathbf{B}}^n}$ are the Fermi-contact fields generated by the electron spins and the nuclear spins when they are interacting with each other. The fields can be expressed as ${{\mathbf{B}}^e} = {\lambda {M^e}{{\mathbf{P}}^e}}$ and ${{\mathbf{B}}^n} = {\lambda {M^n}{{\mathbf{P}}^n}}$, where $\lambda = 8\pi {\kappa _0}/3$ is the geometrical factor containing the Fermi-contact enhancement factor ${\kappa _0}$ [6,26], ${M^e}$ and ${M^n}$ are the generalized magnetization density of electron spins and nuclear spins corresponding to full spin polarization. ${\mathbf{\Omega } }$ is the rotation input. ${{R_p}}$ is the pumping rate of the alkali atoms introduced by the pumping laser beam. ${{{\mathbf{s}}_p}}$ is the average photon spins with orientation to the laser beam propagating direction, and it also indicates the ellipticity of the laser beam.
When $\left | {{{\mathbf{s}}_p}} \right |{\rm {\ =\ }}1$, the beam is circularly polarized, while when ${{\mathbf{s}}_p}{\rm { = 0}}$, the beam is linearly polarized. ${R_{se}^{en}}$ is the spin-exchange rate sensed by the electron spins from the nuclear spins and ${R_{se}^{ne}}$ is the one sensed by the nuclear spins from the electron spins. $R_{sd}^e$ and $R_{sd}^n$ are the spin-destruction rates of electron spins and nuclear spins. $R_{tot}^e$ and $R_{tot}^n$ are the total relaxation rates of electron spins and nuclear spins, which are defined as $R_{tot}^e = {R_p} + R_{se}^{en} + R_{sd}^e$ and $R_{tot}^n = R_{se}^{ne} + R_{sd}^n$, including all the rates mentioned above.
A function generator supplies power to the three-axis coils and generates a compensation magnetic field ${B_c} = - {B^e} - {B^n}$ along the $z$-axis direction. Under this compensation magnetic field, the comagnetometer can operate in the self-compensation state [3]. In this state, the nuclear spins will track and compensate for the low-frequency magnetic field fluctuations. These disturbances will not be felt by the electron spins, and a stable signal output can be obtained. It is also necessary to apply a modulating magnetic field ${{\mathbf{B}}_m} = {B_m}\sin (\omega t)\hat x + {B_m}\cos (\omega t)\hat y$ to the atomic ensemble for extracting the rotation information, and this modulating magnetic field is part of the external magnetic field ${{\mathbf{B}}}$ felt by the atomic ensemble in Eq. (1) and (2). The frequency of the modulating magnetic field is set in the order of kHz, high enough to avoid the response range of the nuclear spins. Therefore, it can be considered that the modulating magnetic field will only affect the electron spins. When rotation occurs, the nuclear spins will first sense the rotation input, deviate from its original orientation and start a process of precession. Eventually, the nuclear spins will point in a new direction and generate transverse polarization components $P_{x/y}^n$ in the $x$-$y$ plane. These components will introduce the transverse magnetic fields $B_{x/y}^n = \lambda {M^n}P_{x/y}^n$. At the same time, the electron spins feel these transverse magnetic fields, and combined with the modulating magnetic field, the amplitude of its oscillating $z$-axis component will carry the rotation information in the $x$-$y$ plane. The first harmonic of the $z$-axis electron spins can be expressed as [17,27],
The incident laser light simultaneously undertakes the duties of optical pumping and signal extraction. The incident light going into the glass cell is right-handed circularly polarized, and its polarization state can be analyzed with the Jones calculus. The incident light is adjusted from linearly polarized into circularly polarized with a combination of polarizing beam splitter (PBS) and a quarter-wave plate (QWP). Here, as shown in Fig. 2, the angle between the fast-axis of the QWP and the polarization axis of the PBS transmitted light is set to $\pi /4$. The polarization state of the incident light ${{\mathbf{E}}_i}$ can be expressed as ${{\mathbf{E}}_i} = \frac {{{E_0}}}{{\sqrt 2 }}{\left [ {1, - i} \right ]^{\textrm{T}}}$, where ${E_0}$ is the amplitude of the electric field component of the light.
When the light passes through the glass cell for the first time, the light will sense a polarization component of the electron spins ${P^e_z}$ along its propagating direction. Similar to the birefringence effect when light passing through a crystal, the right-handed circular polarized light will experience the following refractive index $n_r$ when passing through the alkali vapors. The refractive index is a function of ${P^e_z}$ and can be expressed as ${n_r} = 1 + \frac {{n{r_e}{c^2}f}}{{4\pi \nu }}\left ( {1 + P_z^e} \right )\frac {1}{{\left ( {\nu - {\nu _0}} \right ) + i\Gamma /2}}$ [28]. Here, $n$ is the number density of the K atoms. $r_e$ is the classical electron radius. $c$ is the light speed. $f$ is the oscillator strength for the D1 transition of K atoms, and its value is about 1/3. $P_z^e$ is the electron spins polarization along the $z$ axis. $\nu$ is the laser frequency. A reference cell containing K atoms is used for frequency locking. In order to distinguish from this reference cell, we call the cell used in the comagnetometer for sensing and measuring the rotation the "core cell". $\nu _0$ is the central frequency of the K atoms in the core cell, and $\Gamma$ is the full width at half maximum of the broadening. As a stable system requires a stable laser frequency, frequency locking is indispensable. The locking method we adopt is the saturated absorption spectroscopy method, which is mature and has been widely used in the current comagnetometers [29]. Indeed, the laser frequency locked with the reference cell possesses a frequency shift of about 2 GHz from the central frequency of K in the core cell. However, since the K atoms in the core cell experience significant pressure broadening due to the high-pressure environment of about 8 GHz, the system can still be pumped and achieve a polarized working state, despite the presence of this frequency shift [30].
The light transmitted through the cell for the first time can be expressed as follows,
The backward outgoing light is left-handed circularly polarized, and the rotation information is carried within the oscillating light signal. The special feature of our device is that when the reflected left-handed circularly polarized light passes through the QWP, the light will become s-polarized from the perspective of the PBS. Therefore, the backward light will be completely reflected by the PBS to the bypass photodiode, realizing signal acquisition and avoiding the loss of light power.
The total collected light signal $I_{sum}$ is the sum of a DC component $I_{DC}$ and an oscillating component $I_{osc}$. The oscillating part contains rotation information in its amplitude, which can be used for signal extraction. The DC component is mainly responsible for polarizing the atomic ensemble, and it can also affect the light sensitivity of the photodiode, thereby affecting the output. Here, the light sensitivity $\mu$ is also known as the photoelectric conversion rate, defined as the derivative of the photodiode output voltage $U$ with respect to the light power $I_{PD}$ received on its surface. It can be expressed as $\mu \left ( {{I_{PD}}} \right ) = \Delta U/\Delta {I_{PD}}$ and is a function that will increase as ${I_{PD}}$ decreases. Since the oscillating light $I_{osc}$ is much smaller than the DC one ${I_{DC}}$, when $I_{sum}$ is received by the photodiode, the light sensitivity coefficient can be approximately expressed as $\mu \left ( {{I_{DC}}} \right )$. Then the output voltage of the oscillating light signal ${U_{osc}}$ can be expressed as,
Here, $I_0$ is the incident light power. $\kappa$ is the attenuation of the light power when passing through the glass cell. It is a product of light absorption and light transmittance coefficient. $N$ is the number of times that the beam travels through the atomic ensemble. For our reflective scheme, $N=2$, while for the traditional single-pass scheme, $N=1$. Combined with Eq. (3), by using a lock-in amplifier, the rotation information of ${\Omega _y}$ and ${\Omega _x}$ can be obtained. In general, in our reflective scheme, the increase of $N$ can enhance the oscillating light signal and raise the proportion of ${I_{osc}}$ in ${I_{sum}}$. The attenuation of the DC light ${I_{DC}}$ caused by the reflection can also increase the light sensitivity $\mu \left ( {{I_{DC}}} \right )$. Therefore, a larger oscillating light signal ${I_{osc}}$ and a bigger $\mu \left ( {{I_{DC}}} \right )$ coefficient together lead to the an improvement of the output signal and the signal-to-noise ratio.
3. Experimental setup
The schematic diagram of our reflective single-beam comagnetometer for rotation measurement is shown in Fig. 1. The cell containing atomic ensemble is made of GE180 glass, which is chemically stable to alkali metal vapors in a heated environment. The diameter of this spherical glass cell is 10 mm. The alkali metal droplets composed of K and Rb are sealed into the cell. 18 torrs of quenching gas N$_2$ and 2 atm of noble gas $^{21}$Ne are also filled into the cell. This cell is installed in a boron nitride oven, and its outside is wrapped with a heating film made of twisted-pair resistance wires. A 100 kHz sinusoidal current is used to heat the resistance wires. The temperature is controlled at 453.15 K with a circuit system. The atomic number density ratio of K to Rb is measured approximately at 1:101 by using the spectral absorption method. A coil system consisting of two sets of saddle coils and a set of Lee-Whiting coil is used to generate magnetic fields in three directions. Here, the saddle coils provide magnetic fields in the $x$ and $y$ directions, and the Lee-Whiting coil generates magnetic fields along the $z$ direction. A magnetic shielding system composed of a layer of Mn-Zn ferrite and three layers of $\mu$-metal cylinders plays the role of blocking the magnetic field of the environment. The magnetic shielding system and the three-axis coils compensate for the remanence and create a near-zero magnetic-field environment for the atoms, which is one of the necessary conditions for creating the SERF state. In addition, thermal insulation materials are pasted around the ferrite for improving the temperature stability.
Fig. 2. The schematic diagram of polarization state of light when passing through PBS and QWP structure. The red line represents the incident light before entering the glass cell, and the blue line is the backward light that has passed the cell two times. For the red line, after passing through the PBS and the QWP, the light is switched from linearly polarized to right-handed circularly polarized. For the blue line, the reflected light is left-handed circularly polarized. After going through the QWP, it becomes an s-polarized light from the PBS perspective. Therefore, the PBS can reflect all the backward light, and light power loss can be avoided.
Our reflective single-beam comagnetometer uses circularly polarized light to pump the atomic ensemble. The incident laser light is generated with a distributed Bragg reflector (DBR) laser. The light will travel through an optical fiber emitted from a collimator and enters the optical system. Its frequency is locked at the D1 line of the K atom. The beam size is expanded with a group of lenses so that the light spot is large enough to cover the entire cross-section of the cell. The laser power is controlled with an intensity stabilization module. This module consists of two PBS, a liquid crystal variable retarder (LCVR) and a circuit system. Before going into the cell, the light will pass through a PBS and a QWP. With this structure, the light will switch from linearly polarized to fully right-handed circularly polarized. After it enters the cell, the atomic ensemble will experience the effect of optical pumping, and the transmitted light will carry the rotation information. Then, the light will continue propagating and be reflected by a mirror. It will propagate backward and re-enter the cell again. The backward light will be switched to left-handed circularly polarized due to the reflection $\pi$ phase change happening on the mirror surface. The backward light will travel through the cell again, and the outgoing light will be completely redirected to a bypass photodiode when it reaches the structure composed of the PBS and QWP, as shown in Fig. 2. The advantage of this structure is that it does not require additional components to achieve a complete reflection of the backward light beam, which is convenient for signal acquisition and avoids the loss of light power. When rotation occurs, the angular velocity information will be transferred from the nuclear spins to the electronic spins, and is finally carried within the light signal. With a lock-in amplifier, the dual-axis rotation information can be obtained from the signal collected by the photodiode. Our reflective scheme can effectively strengthen the output signal and realize a better rotation sensitivity.
4. Results and discussion
The experiments are implemented on a reflective single-beam SERF comagnetometer. The performance of the reflective scheme is compared with the single-pass scheme under different incident light powers. Before the experiment starts, we need to set the parameters for the modulating magnetic field. From the definition of the Bessel function ${{J_n}\left ( u \right )}$, an extreme value of the electron spins precession signal can be found by comprehensively adjusting the frequency and the amplitude of the magnetic field. In addition, we consider to avoid the frequency point of 1000 Hz, because the voltage with this frequency is also used in the modulation of LCVR in the laser intensity stabilization module. Eventually, a magnetic field with a frequency of 1233 Hz and an amplitude of 114 nT is determined for modulating the electron spins. The incident light is adjusted to right-handed circularly polarized through a PBS and a QWP. The laser light will simultaneously pump the atomic ensemble and be used to extract signals. When the system reaches a steady state, the rotation experiments can be started.
The accuracy of our rotating platform is 0.001 $^{\circ }$/s. The $y$ axis of our reflective single-beam comagnetometer is adjusted to coincide with the rotation axis of the platform, vertically pointing upward. We set the incident light power at 30 mW and apply different rotation inputs from $\pm 0.1$ to $\pm 0.5$ $^{\circ }$/s along the $y$ direction. By demodulating the light power signal collected by the photodiode with a lock-in amplifier, the signal response can be obtained. The real-time curves of the signal (red) and the corresponding angular velocity (blue) are shown in Fig. 3. From the moment the system senses the rotation input, it takes approximately 20 s to reach a new steady state, and the settling time depends mainly on the atomic relaxation and its working condition [13]. This result verifies the feasibility of our reflective single-beam comagnetometer for rotation measurement, and its superiority compared with the single-pass scheme will be discussed in the following part.
Fig. 3. Graphs showing the relationship between the signal response and the rotation input. We input angular velocities from $\pm 0.1$ to $\pm 0.5$ $^{\circ }$/s under the condition of 30 mW incident light intensity. The red line is the signal response, and the blue line is the corresponding input angular rate. This validates the feasibility of our reflective single-beam comagnetometer in rotation measurement.
Under different incident light power conditions (20/30/40/50 mW), we examine the rotation measuring performance of the comagnetometer with either the reflective scheme or the single-pass scheme. The system usually takes about 3-4 hours to reach a stable state, and each round of testing begins when the device system has stabilized. In the experiments, except for the changes of the above two schemes and light powers, the rest of the test conditions remain the same, including the settings of the lock-in amplifier and the data acquisition module, ensuring the comparability of the experiment data. Now, we first need to calibrate the phase shift of the reference signal used in the demodulation process so that the strongest output signal can be obtained and the output channel can be determined [17]. This step can be achieved by applying a fixed rotation input of ${\Omega _y}{\rm { = }}0.1 ^{\circ }$/s to the system and recording the ${{\Omega _y}}$ scale factor under different demodulation phases. The scale factor is a parameter of signal response voltage divided by the input angular velocity, representing the signal strength. The results are shown in Fig. 4. The solid lines of different colors represent the calibrations tested on the reflective single-beam comagnetometer, and the dashed lines represent the ones from the single-pass scheme. Figure 4 shows the relationship between the demodulation phase and the ${{\Omega _y}}$ scale factor of each test condition. The output is determined by adopting the phase at its maximum point of each curve, and the corresponding signal of each light condition will be the strongest.
Fig. 4. The process of calibrating the ${{\Omega _y}}$ output channels. The diagram shows the relationship between the ${{\Omega _y}}$ scale factors under different demodulation phases. In this process, a rotation input of ${{\Omega _y}}$=0.1 $^{\circ }$/s is applied to the comagnetometer, and the scale factors under different phases are recorded. The phase of the maximum point of each curve is then used for demodulation. Furthermore, the output for each light condition is determined.
We test the relation between the signal responses and the rotation inputs of the reflective scheme and the single-pass scheme, respectively, and the results are shown in Fig. 5. We apply rotation inputs along the $y$ direction from -1.5 $^{\circ }$/s to +1.5 $^{\circ }$/s, and the step length is set at $0.1\;^{\circ }$/s. Since the linearity of the ${{\Omega _y}}$ scale factor is better in the range from - 0.5 $^{\circ }$/s to + 0.5 $^{\circ }$/s, we select data of this section for fitting, and the results of the ${{\Omega _y}}$ scale factor under each light power condition are presented in Table 1. The least square method is adopted for this fitting, and the standard deviations of the scale factor estimators are listed in Table 1, which have the same unit as the scale factors and can intuitively quantify the fitting uncertainty. The R-squares of the fit are also calculated, and the values are all greater than 0.9990, indicating that the linear model can be nicely used to describe the signal-rotation relationship, and the linearity within this range can be regarded as satisfactory. Regardless of the different light powers, the ${{\Omega _y}}$ scale factors of the reflective scheme (red circles in Fig. 5) are all larger than those of the single-pass scheme (blue triangles in Fig. 5), which proves that the signal strength of the reflective scheme is stronger compared to the single-pass scheme. The output signal of the reflective scheme should have a better signal-to-noise ratio.
Fig. 5. The relationship between the rotation inputs and the signal responses. The following data sets correspond to four test conditions with different incident light powers. At each condition, angular rates from - 1.5 $^{\circ }$/s to 1.5 $^{\circ }$/s are input along the $y$ direction, and the voltage values of the output signals are recorded respectively. When the incident light power is set at 20 mW, the signal response of the reflective scheme is the largest.
Table 1. Fitting result of the ${{\Omega _y}}$ scale factors
We analyze the reason why the scale factors of the reflective scheme are larger than those of the single-pass scheme and think that there are two main causes. One is that the reflected light has passed through the atomic ensemble twice, making $N$ change from 1 to 2 in Eq. (6). From the expression form of ${U_{{{osc}}}}$, we know that this change directly leads to an enhancement in the oscillating component of the light signal, and it is exactly this part that carries the rotation information. Another cause is related to the light sensitivity of the photodiode. For the photodiode, the output voltage $U$ and the light power hitting on its surface ${{I_{PD}}}$ are positively correlated, but not linearly, shown as the blue line in Fig. 6. The square data points of the blue line are directly measured from the photodiode that we use in our experiments. The light sensitivity of the photodiode is the difference quotient of the output voltage with respect to the light power. The circle data points represent the light sensitivity calculated from the square measured data, and the red line is the curve fitted with the polynomial method. From the red line, we know that when the DC light power is below 3 mW, the photodiode has a notable light sensitivity and can make a strong response to the oscillating light signal. However, under such condition, the power of the corresponding incident light is too small to polarize the nuclear spins, and the comagnetometer can not function properly to make measurements. With the light power increasing, the light sensitivity tends to decrease. During the test of the reflective scheme and the single-pass scheme, we mark down the DC light power values collected by the photodiode and calculate their corresponding light sensitivities, and the results are shown in Table 2. At the same incident light condition, the light sensitivities $\mu \left ( {{I_{DC}}} \right )$ of the photodiode in the reflective scheme are all greater than those in the single-pass scheme. In all, the proportion of the oscillating light component carrying the rotation information in the total light signal is increased, combined with the enlargement of $N$ and $\mu \left ( {{I_{DC}}} \right )$ in Eq. (6), which finally leads to a stronger output and a larger scale factor of the reflective scheme compared to the single-pass one.
Fig. 6. The relation graph of the output voltage $U$ and the light sensitivity of the photodiode $\Delta U/\Delta {I_{PD}}$ with respect to different light power ${I_{PD}}$. The blue line indicates that the output voltage $U$ increases with ${I_{PD}}$, but the slope is slowing down. The light sensitivity, defined as $\Delta U/\Delta {I_{PD}}$, characterizes the sensitivity of the photodiode reacting to small changes in light power. The red line shows the decrease in light sensitivity as ${I_{PD}}$ increases. It proves that the sensitivity of photodiodes to the oscillating light signal decreases when the DC component of the light signal is strong.
Table 2. The DC component values and the corresponding light sensitivities under the two schemes
Then we analyze the long-term stability of our system. The acquisition module for data collection is the National Instruments USB-4431, and the sample rate is selected at 200 Hz. We record the output signal and the signal background of both the reflective scheme and the single-pass scheme under different light power conditions. The Allan deviations (ADEV) of both schemes are calculated and compared. The data of the output signal is recorded at the normal measuring condition, while the signal background is acquired when a 2119 nT magnetic field is applied along the $z$ direction. In this case, the signal is no longer sensitive to the rotation input or magnetic field, representing the drift mainly caused by the background noise of the system. The ADEV comparison is shown in Fig. 7. Since the long-term output is our focus, the value of each curve at $\tau =100$ s is our major concern. The ADEV of the output signal (blue dashed line) starts to climb up from 7 s, and gradually deviates from the signal background (red solid line). This is because the output signal is sensitive to factors such as low-frequency magnetic noise and has a greater signal drift [32,33]. However, we focus more on the contrast between the red solid lines and the orange dash-dotted lines, and the ADEV of the signal background in the reflective scheme is significantly lower than that of the single-pass one. This is because the rotation signal is enhanced in the reflective scheme, and the signal-to-noise ratio of the output is improved, so that the proportion of background noise in the output can be effectively suppressed. The results are consistent at different light powers, indicating that the reflective single-beam comagnetometer has a smaller signal background drift, and the lower background drift provides more space for the device to improve its long-term stability.
Fig. 7. Graphs of the Allan deviations (ADEV) of the reflective scheme and the single-pass scheme under different light powers. The blue dashed lines are the ADEV curves of the output signals measured with the reflective scheme, and the red solid lines and orange dash-dotted lines represent the ADEV of the signal backgrounds of the reflective scheme and the single-pass scheme, respectively. The signal background of the reflective scheme is significantly better than that of the single-pass one.
We calculate the rotation sensitivities of both the reflective scheme and the single-pass scheme at each light power condition. Comagnetometer system commonly uses the power spectral density (PSD) to describe the rotation sensitivity. The output voltage signal is first converted into angular velocities with the $\Omega _y$ scale factor, and then the fast Fourier transform algorithm (FFT) is adopted to calculate the PSD. Each signal data set for calculation is 3600 s long, and the results are presented in Fig. 8. Since the demodulation system utilizes a low-pass filter with a time constant of 30 ms, each curve has a sudden drop at around the 3 Hz position. Overall, since the scale factors of the reflective scheme are larger than those of the single-pass one, the reflective scheme has a lower noise floor around the frequency from 0.1 to 1 Hz and presents a better rotation sensitivity. Among the results of the reflective scheme, the best rotation sensitivities are obtained under the incident light conditions of 20 mW and 30 mW because the system has the largest scale factors. Therefore, our proposed reflective scheme can improve the rotation sensitivity of the single-beam comagnetometer, which can lay a foundation for developing miniaturized atomic inertial sensors.
Fig. 8. Rotation sensitivities of the reflective scheme and the single-pass scheme under different light powers. The red solid line is the sensitivity measured with the reflective scheme, and the blue dash-dotted line is one of the single-pass scheme. The rotation sensitivity of the reflective scheme is better due to a stronger $\Omega _y$ output signal and its larger scale factor.
5. Conclusion
We propose a reflective (double-pass) configuration for the single-beam comagnetometer used in rotation measurement. The process of optical pumping and signal detection can all be realized with only one laser beam. The light beam in our system is designed to pass through the cell containing the atomic ensemble twice with a mirror for reflection. In the optical path, we propose a structure composed of a polarizing beam splitter and a quarter-wave plate. This structure can effectively separate the reflected light from its original route which overlaps with the forward light beam, and realize a full collection of all the reflected light power. The rotation information can then be read out by demodulating the collected oscillating light signal. We analyze the reasons that lead to the improvement of the output signal. Our reflective configuration extends the length of the light interacting with the atoms, and the light sensitivity of the photodiode is also improved due to the attenuation of the DC light component. Therefore, since the proportion of the oscillating light signal which carries the rotational information is increased in the total light signal, our reflective scheme can achieve a better rotation sensitivity and signal-to-noise ratio compared with the single-pass scheme. Our work advances the development of miniaturized atomic sensors for rotation measurement in the future.
Funding
National Natural Science Foundation of China (62103024, 61925301); Innovation Program For Quantum Science and Technology (2021ZD0300402).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
References
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