A novel method to measure the scale factor for the alloptical atomic spin inertial measurement device (ASIMD) is demonstrated in this paper. The method can realize the calibration of the scale factor by a selfconsistent method with small errors in the quiescent state. At first, the matured IMU (inertial measurement unit) device was fixed on an optical platform together with the ASIMD, and it has been used to calibrate the scale factor for the ASIMD. The results show that there were some errors causing the inaccuracy of the experiment. By the comparative analysis of theory and experiment, the ASIMD was unable to keep pace with the IMU. Considering the characteristics of the ASIMD, the mismatch between the driven frequency of the optical platform and the bandwidth of the ASIMD was the major reason. An alloptical atomic spin magnetometer was set up at first. The sensitivity of the magnetometer is ultrahigh, and it can be used to detect the magnetization of spinpolarized noble gas. The gyromagnetic ratio of the noble gas is a physical constant, and it has already been measured accurately. So a novel calibration method for scale factor based on the gyromagnetic ratio has been presented. The relevant theoretical analysis and experiments have been implemented. The results showed that the scale factor of the device was 7.272 V/°/s by multigroup experiments with the maximum error value 0.49%.
I. INTRODUCTION
With the development of optical pumping technology
[1
,
2]
, spinpolarized atoms have been used in many practical applications, such as biomagnetism atomic sensors
[3]
, atomic magnetometers
[4]
, atomic gyroscopes
[5]
, and so on. The gyroscope is one of the necessary sensors for the Inertial Navigation System (INS), which plays an important role in many facilities which need accurate position, velocity, and orientation information. The inertial measurement device based on spinpolarized atoms has become a hot spot. Spinpolarized atoms will precess at an atomicspeciesdependent frequency when subjected to an external magnetic field
[6]
. The physical rotation of the device containing such atoms will modify the frequency in proportion to the rotation rate. The novel device makes use of detecting the precession frequency to measure the rotation rate
[7]
. According to the theoretical analysis and experimental results, the ASIMD has many advantages
[8

11]
. Moreover, the nature of the device’s structure is inherently immune to external vibration. Because the ASIMD replaces the mechanical rotor with spinpolarized atoms, it does not have any mechanical collisions and friction, and it offers promise for applications with high precision
[5]
. The anglerandom walk (ARW) of 0.002
Hz and the bias instability of 0.04 °/h based on the atomic spin gyroscope have been achieved and it has the potential advantage to achieve the best performance gyroscope in the future
[12]
.
The output signal of the IMU has a definite proportional relationship with the attitude angular velocity of the moving carrier. The degree of the IMU’s accuracy affects the precision of the INS directly. Due to the influence of many factors such as machining processes, the external environment and so on, the characteristics of the relationship between the gyroscope’s input and output signals is nonlinear. It leads to errors in the scale factor. At present, many scholars have made contributions in the calibration of the scale factor. Some researchers established the relevant mathematic model of scale factor on full range and put forward the corresponding compensated methods
[13
,
14]
. Others carried out their research aiming at the characteristics of scale factor error such as asymmetric positions, nonlinear problems and so on
[15
,
16]
. Some research works have been proved valid by the experimentally verified results, and the errors have been reduced in different degrees
[13

16]
. The method of using the high precision IMU as a standard has been generally adopted in these traditional methods to determine the scale factor, and most of the research aimed at the common gyroscopes and focused on the mathematical process without the considerations of the work principle of the device.
The ASIMD is a new kind of gyroscope, and it is still in the stage of development without official methods for scale factor calibration. In this paper, we have calibrated the scale factor for the device based on the mature product (IMU) at first. However, the experimental results have been influenced by many factors such as the precision control of the frequency for the optical platform, the precision adjustment of the amplitude for the platform, the decrease of system errors and so on. In the view of these problems, we have changed the test method. According to the characteristics of the device, we calibrated the scale factor based on the gyromagnetic ratio of the noble gas, and it has been tested accurately by the related theoretical analysis and experimental verification.
II. PRINCIPLE OF ALLOPTICAL ATOMIC SPIN INERTIAL MEASUREMENT DEVICE
The basic theory of the ASIMD is based on the effect of nuclear magnetic resonance
[10
,
12]
. The principle can be described as follows:
Here
is the polarization of nuclear spins belonging to the noble gas, and it contains three orthogonal components,
,
, and
is the spinpolarization of the outer most electrons belonging to alkaliatoms, and it contains three orthogonal components,
, and
is the spinpolarization of the outermost electrons belonging to alkaliatoms along the x direction, which can be used to describe the output signal of the equipment;
γ_{e}
is the gyromagnetic ratio of the electron;
γ_{n}
is the gyromagnetic ratio of the noble gas;
is the magnetic field detected by electrons belonging to alkaliatoms, and it contains three components,
,
, and
is the magnetic field detected by nuclei belonging to the noble gas;
B_{c}
is the magnetic field produced by the alkali atoms and the noble gas;
is the rotating angular velocity, which describes the relative motion between carrier coordinate system and inertial coordinate system, and it contains three orthogonal components, Ω
_{x}
, Ω
_{y}
, and
is the spinexchange rate between the electron spins of alkaliatoms and the nuclear spins of the noble gas;
is the total relaxation of the nuclear spins belonging to the noble gas;
is the total relaxation of the electron spins;
R_{P}
is the average rate at which unpolarized atoms absorb photons from the pump beam; the circular polarization degree of pump light is defined as
S_{P}
, and
S_{P}
= 1;
R_{m}
is the average rate at which unpolarized atoms absorb photons from the probe beam; the circular polarization degree of probe light is defined as
S_{m}
, and
is defined as light shift, which is equivalent to the virtual magnetic field, and it contains three orthogonal components,
L_{x}
,
L_{y}
, and
L_{z}
;
b
is defined as an anomalous field;
α
is the nonorthogonal angle between pump beam and probe beam, and when the probe beam is perpendicular to the pump beam, the value of
α
is zero. Because several parameter values are very small, the output signal can be simplified as follow.
Eq. (4) shows that the ASIMD is based on the pattern of the angular rate model. Because the device is a new type of inertial measurement device, there are no mature standards for the scale factor calibration. According to the features of the ASIMD, a test scheme is designed to calibrate the scale factor for the device. The scale factor is always used in the inertial technology, as shown in the following equation.
Here V is the output voltage signal of the ASIMD, and its unit is V; Ω is the angular rate signal of the carrier detected by the ASIMD, and it is the input signal with the unit °/s; K is the scale factor, and its unit is °/s/V.
The original output signal of the device is a voltage signal, which needs to be converted to an angular rate signal to match with the input signal. The inertial measurement unit (IMU) based on interference fiber optic gyros (IFOGs) can be used to assist the atomic spin inertial measurement device to implement this function. The atomic spin inertial measurement device and the IMU are fixed on the same optical platform. The IMU consists of three IFOGs based on angular rate model and three accelerometers, which can be used to measure the carrier motion information including translational motion and rotary motion in three perpendicular directions.
III. EXPERIMENTAL SETUP AND RESULTS
 3.1. Experimental Setup
The ASIMD (the size of the ASIMD is 1800 mm (Length) × 1500 mm (Width)) was installed on the optical platform. The optical platform can be supported through the six airbearing legs, which makes the device rotate around the support legs slightly. Because spaces available for motion of the gasbags of the legs are limited, the maximum rotation angle is about ±2°. In order to provide a measurement criterion of angular velocity for the atomic spin inertial measurement device, high precision fiber optic IMU was fixed on the optical platform together with the atomic spin inertial measurement unit. High precision fiber optic gyroscopes can measure the angular velocity of the device, playing an important part in calibrating the output signal of the atomic spin inertial measurement device. The test hardware structure diagram and the physical graph are shown as follows.
Schematic diagram of experimental device.
Photograph of experimental device.
 3.2. Results and Error Analysis
The optical platform was driven around the yaxis with small amplitude. The results of angular rate along the yaxis detected by the high precision optical fiber gyroscopes (the sampling frequency of the gyroscope is 200 Hz and the bias instability of the gyroscope is 0.03 °/h) and the ASIMD were shown as follows. The output results of the two kinds of devices were subjected to leastsquares model fitting, as shown in
Fig. 3
.
Scale factor calibration of ASIMD based on the IMU.
Making use of scale factor calibration coefficient, the output signal of ASIMD can be transformed into the angular velocity from the voltage signal, which is in contrast to IMU in
Fig. 4
. There are many factors causing the errors and they can be mainly divided into two categories.
Comparisons of output signal the between IMU and ASIMD after scale factor calibration.
Firstly, because the experimental platform is supported by the airbearing legs, the driven amplitude of the optical platform is very small. The maximum driven amplitude of the optical platform was about 0.2°, as seen in
Fig. 4
. Under this condition, the random errors have a great influence on the results of the experiment, and it led to the inaccuracy of the scale factor.
Secondly, the driven frequency of the platform cannot be controlled accurately. According to the amplitudefrequency curve, most vibration frequencies of the optical platform were less than 5 Hz, and the driven frequency was mainly focused on 1.133 Hz, as shown in
Fig. 5
. The bandwidth of the ASIMD needs to be confirmed, which may influence the experimental results. According to the
Fig. 6
, the adjustment time of the system is about 0.4s. The bandwidth (0 dB) of the ASIMD is less than 2.5 Hz. The signal amplitude whose frequency was higher than 2.5 Hz has been attenuated because of the bandwidth limit of the ASIMD. Under this condition, the scale factor calibration of the ASIMD may be inaccurate.
Vibration frequency of the optical platform.
Response to the step signal from the ASIMD.
 3.3. Experimental Improvement
According to the Eq. (1) to (3), the right hand side of the Eq. (3) can be simplified and merged. Thus, the output signal of the ASIMD can be simplified into the following expression.
According to the Eq. (5), the scale factor can be described with the letter
K
. The magnetic shield barrels shield the external magnetic field and the magnetic coils further offset the residual magnetic field inside the barrels. So the residual magnetic field in three different directions (
B_{x}
=
B_{y}
=
B_{z}
≈ 0
L_{x}
=
L_{y}
=
L_{z}
≈ 0) approaches zero under the actual condition. According to the Eq. (4) to (5), the scale factor
K
can be described with the Eq. (7).
According to the output signal of the ASIMD which is shown in Eq. (4), we took its partial respect to the magnetic field
B_{y}
, as shown in Eq. (8). The actual data of the left hand side in Eq. (8) can be obtained by the experiments.
Based on the above theories, we carried out related experimental research work. The core sensitive header of the device was a square glass cell with a side of length 2 cm. The cell, containing a droplet of cesium, 5 torr xenon 129 and 1 atm nitrogen, was heated to 100 ℃ with average relative error of 0.5 ℃. The power of the pumping light was 100 mW. The power of the probe light was 4 mW. After about one hour, the system had reached steady working condition. In order to get the signal
, we modulate the magnetic field
B_{z}
under the square wave magnetic field
B_{y}
with the amplitude of 3 nT and frequency of 90 mHz to get the output signal. The experimental results of the situation
are shown in
Fig. 7(a)
. Because the situation
is similar to the situation
we only displayed the situation
. The experimental results of the situation
are shown in
Fig. 7(b)
. The whole experimental results of the scale factor calibration for the device are shown in
Fig. 7(c)
. The scale factor can be got from the fit curve in
Fig. 7(c)
.
response to dB_{y} under the situation of modulating B_{z} step by step. (a) The situation , (b) The situation , (c) The experimental results of selfconsistent scale factor calibration for ASIMD.
According to the Eq. (7) to (8), the experimental test points can be used to fit a curved line. The scale factor can be determined by the fitting coefficients
a
b
, and
c
, as shown in Eq. (9) to (10).
The fitting results of the experiments are shown in
Table 1
. Choosing coefficient of determination (Rsquare) as the judging standard, the best results of the two important fitting coefficients were 5159, 94.07 for
a
and
b
, respectively. The gyromagnetic ratio belonging to xenon129 is 4.2397 °/ s /
nT
, and the calibration result for scale factor was 7.272
V
/ ° /
s
by multigroup experiments with the maximum error value 0.49%.
The experimental results for parameters fitting
The experimental results for parameters fitting
IV. CONCLUSION
In this paper, we calibrated the scale factor for the ASIMD device by using a new method, which utilized its selfsignal rather than other calibration equipment to determine the scale factor with small errors. At first, the IMU was used to calibrate the scale factor for the ASIMD. However, the output signal of the ASIMD always fell behind the output signal of the IMU because of the limits of the ASIMD’s bandwidth. Then, the method to determine the scale factor in a selfconsistent way by using the gyromagnetic ratio of the noble gas was presented, and it has been verified by multigroup experiments.
In the future, we intend to implement an inertial measurement with different spinpolarized noble gases and to find the most suitable noble gas for our application.
Acknowledgements
The authors would like to thank Profs. Jiancheng Fang and Xiyuan Chen for valuable suggestions and helpful discussions. This work is supported by the scientific research foundation of graduate school of Southeast University YBJJ1536 and the special fund for basic research on scientific instruments of the national science foundation of China under Grant No. 61227902.
Allmendinger F.
,
Heil W.
,
Karpuk S.
,
Kilian W.
,
Scharth A.
,
Schmidt U.
,
Schnabel A.
,
Sobolev Y.
,
Tullney K.
2014
“New limit on lorentzinvariance and CPTviolating neutron spin interactions using a freespinprecession He3Xe129 comagnetometer,”
Phys. Rev. Lett.
112
110801 
DOI : 10.1103/PhysRevLett.112.110801
Kim S. J.
,
Noh J. J.
,
Kim M. S.
,
Lee J. S.
,
Yu H.
,
Kim J. B.
2014
“Dynamics of a BoseEinstein condensate on changing speeds of an atomchip trap potential,”
J. Opt. Soc. Korea
18
633 
638
DOI : 10.3807/JOSK.2014.18.6.633
Kim K.
,
Begus S.
,
Xia H.
,
Lee S.
,
Jazbinsek V.
,
Trontelj Z.
,
Romalis M. V.
2014
“Multichannel atomic magnetometer for magnetoencephalography: A configuration study,”
NeuroImage
89
143 
151
DOI : 10.1016/j.neuroimage.2013.10.040
Fang J.
,
Qin J.
2012
“Advances in atomic gyroscopes: A view from inertial navigation applications,”
Sensors
12
6331 
6346
DOI : 10.3390/s120506331
Tazartes D.
2014
“An historical perspective on inertial navigation systems,”
Proc. The Inertial Sensors and Systems (ISISS), 2014 International Symposium on
Laguna Beach, CA, USA
Groves P. D.
2013
“The PNT boom: Future trends in integrated navigation,”
Inside GNSS
8
44 
49
Allred J. C.
,
Lyman R. N.
,
Kornack T. W.
,
Romalis M. V.
2002
“Highsensitivity atomic magnetometer unaffected by spinexchange relaxation,”
Phys. Rev. Lett.
89
130801 
DOI : 10.1103/PhysRevLett.89.130801
Budker D.
,
Gawlik W.
,
Kimball D.
,
Rochester S.
,
Yashchuk V.
,
Weis A.
2002
“Resonant nonlinear magnetooptical effects in atoms,”
Reviews of Modern Physics
74
1153 
1201
DOI : 10.1103/RevModPhys.74.1153
Larsen M. S.
,
Abbink H. C.
,
Walker T. G.
,
Bulatowicz M. D.
2013
“Nuclear magnetic resonance probe system,”
Google Patents
Kornack T. W.
,
Ghosh R. K.
,
Romalis M. V.
2005
“Nuclear spin gyroscope based on an atomic comagnetometer,”
Phys. Rev. Lett.
95
23080123 
Moeller R. P.
,
Burns W. K.
,
Frigo N. J.
1989
“Openloop output and scale factor stability in a fiberoptic gyroscope,”
IEEE J. Lightwave Technol.
30
262 
269
Fang J. C.
,
Zhang X.
,
Li J. L.
2010
“A compensation method for MEMS gyro scale factor error,”
Acta Aeronautica et Astronautica Sinica
in Chinese
02
350 
355
Shen C.
,
Chen X. Y.
2012
“Analysis and modeling for fiberoptic gyroscope scale factor based on environment temperature,”
Appl. Opt.
51
2541 
DOI : 10.1364/AO.51.002541
Chung H.
,
Ojeda L.
,
Borenstein J.
2001
“Accurate mobile robot deadreckoning with a precisioncalibrated fiberoptic gyroscope,”
IEEE Transactions on Robotics and Automation
17
80 
84
DOI : 10.1109/70.917085