Dynamics of an all-optical atomic spin gyroscope Jiancheng Fang, Shuangai Wan,* and Heng Yuan School of Instrument Science and Opto-Electronics Engineering, Beihang University, Beijing, China *Corresponding author: [email protected] Received 10 July 2013; revised 14 September 2013; accepted 18 September 2013; posted 23 September 2013 (Doc. ID 193611); published 11 October 2013
We present the transfer function of an all-optical atomic spin gyroscope through a series of differential equations and validate the transfer function by experimental test. A transfer function is the basis for further control system design. We build the differential equations based on a complete set of Bloch equations describing the all-optical atomic spin gyroscope, and obtain the transfer function through application of the Laplace transformation to these differential equations. Moreover, we experimentally validate the transfer function in an all-optical Cs–129 Xe atomic spin gyroscope through a series of step responses. This transfer function is convenient for analysis of the form of control system required. Furthermore, it is available for the design of the control system specifically to improve the performance of all-optical atomic spin gyroscopes. © 2013 Optical Society of America OCIS codes: (120.4640) Optical instruments; (120.3930) Metrological instrumentation; (020.5580) Quantum electrodynamics. http://dx.doi.org/10.1364/AO.52.007220
1. Introduction
Sensitive gyroscopes are used in a wide range of applications, particularly in inertial navigation applications [1]. In recent years, with the rapid development of the quantum manipulation technique, the advantages, such as high precision and compact size, offered by all-optical atomic spin gyroscopes have attracted more and more researchers to this field. An angular random walk (ARW) of 2 × 10−3 °∕h1∕2 and a bias stability of 4 × 10−2 °∕h were achieved by the alloptical 3 He–K atomic spin gyroscope when it was first demonstrated in 2005 [2]. Since then, an equivalent rotation resolution of 1.8 × 10−7 °∕s has been realized with an all-optical 21 Ne–Rb–K atomic spin gyroscope [3]. A sensitivity of 7 × 10−5 °∕s∕Hz1∕2 was achieved by an all-optical 129 Xe–Cs atomic spin gyroscope [4,5]. Theoretical analysis shows that an ARW of 7.2 × 10−8 °∕h1∕2 can be achieved in a 150 cm3 sense volume with an all-optical 21 Ne atomic spin gyroscope [6,7], which indicates that an all-optical atomic spin gyroscope is a highly sensitive and 1559-128X/13/307220-08$15.00/0 © 2013 Optical Society of America 7220
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compact gyroscope for future inertial navigation applications [8]. Currently, the performance of all-optical atomic spin gyroscopes is far below the theoretical performance. One of the ways to improve the performances of all-optical atomic spin gyroscopes is to analyze their dynamics and errors. The errors of all-optical atomic spin gyroscopes have been studied by Kornack [6] and Brown [9]. The effects of imperfect cancellation of external magnetic fields, spin-exchange collisions, and light shifts on steady-state errors were considered in their studies. However, less analytical dynamics have been developed in recent papers. In this paper, we investigated a set of differential equations and a transfer function describing the dynamics of all-optical atomic spin gyroscopes. The differential equation obtained is similar to the differential equation of a rotor angular rate gyroscope. The all-optical atomic spin gyroscope is therefore equivalent to a rotor angular rate gyroscope, and the analytical method used for a rotor angular rate gyroscope can be applied in an all-optical atomic spin gyroscope. Moreover, the transfer function obtained was analyzed and simplified. The simplified transfer function was validated by experimental test. This
transfer function is useful for the design of the control system specifically to reduce the effects of imperfections, and further to improve the performance of the all-optical atomic spin gyroscope. 2. Description of an All-Optical Atomic Spin Gyroscope
A general schematic of all-optical atomic spin gyroscope is shown in Fig. 1. The gyroscope consists of a vapor cell that contains alkali atoms (K, Rb, or Cs) and noble gas (3 He; 21 Ne, or 129 Xe). Alkali atoms are polarized by optical pumping along the z direction and transfer the polarization to the noble gas by spin-exchange collisions. Thus, alkali metal atoms obtain electron spin magnetic moment SA , and noblegas atoms obtain nuclear spin magnetic moment SB . An all-optical atomic spin gyroscope utilizes the electron spin magnetic moment of alkali metal atoms SA for angular velocity measurement. SA can keep pointing at the fixed direction in the inertial coordinate system. An external magnetic field Bc is set to the “compensation point” where Bc cancels the field created by the nuclear magnetization of noble gas SB . Under this situation, SA and SB can strongly interact with each other, and SB can automatically track and compensate for the changes of the external magnetic field, to isolate the external magnetic field sensed by SA . When the carrier rotates, SA will remain in its original pointing direction, while the angle between SA and a probe laser fixed in the carrier changes, which reflects the rotation only. Consequently, the angle can be measured by measuring the electron spin polarization along the direction of the probe laser propagation. Hence, the block diagram of the all-optical atomic spin gyroscope can be simplified to an input–output form, as shown in Fig. 1(c). Ω and S are the input and output of the all-optical atomic spin gyroscope,
respectively. Alkali atom spins align along the z direction by optical pumping. Thus, these spins aligned have an x projection under an inertial rotation rate input Ω along the y direction when the external magnetic field is set to the “compensation point.” This x projection of the spins is measured through optical rotation of a linearly polarized probe laser. A polarizer translates the optical rotation angle into the intensity of the probe laser, which is measured through a photodiode. This intensity is referred to as the output signal of all-optical atomic spin gyroscope S. 3. Modeling of All-Optical Atomic Spin Gyroscope Dynamics A. Differential Equations of All-Optical Atomic Spin Gyroscope Dynamics
1. Differential Equations A further technical description of an all-optical atomic spin gyroscope follows a complete set of Bloch equations [6,9]: ∂P⃗ γ n ⃗ × P⃗ e − Ω ⃗ × P⃗ e e B⃗ λM n P⃗ L ∂t Q ⃗ n ⃗ e Rp s⃗ p Ren se P − Rtot P ∕Q; e
∂P⃗ e n ⃗ × P⃗ n γ n B⃗ λM e P⃗ × P⃗ − Ω ∂t n ⃗ n ⃗ e Rne se P − R P ; n
tot
(1)
⃗ is the inertial rotation rate. P⃗ e ∕P⃗ n , γ e ∕γ n , where Ω λM e ∕λM n , and Rtot ∕Rntot are polarization, gyromagnetic ratio, polarization field factors, and the total spin relaxation rate of electron/nuclei, respectively. B⃗ and L⃗ are the net magnetic field and light shift experienced by electrons, respectively. Rp and Rm are
Fig. 1. Schematic of the all-optical atomic spin gyroscope. (a) Alkali atoms and noble-gas atoms are polarized along the bias field Bc. (b) SB is able to automatically track and compensate for changes in the external magnetic field, thus isolating the external magnetic field sensed by SA. (c) Simple block diagram of an atomic spin gyroscope in a control system. 20 October 2013 / Vol. 52, No. 30 / APPLIED OPTICS
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the pumping rates of the pump laser and the probe ne laser, respectively. Ren se and Rse are the pumping rates of the electrons and the nuclei by spin-exchange with each other. s⃗ p is the polarization of the pump photon and, for a circularly polarized pump laser, j⃗sp j 1. Q is the slowing-down factor. In this paper, the z axis is defined to be the direction of the circularly polarized pump laser and s⃗ p f 0 0 1 g. The x axis is defined as the direction of the linearly polarized probe laser, which measures the x component of the electron polarization Pex . The direction of the inertial rotation ⃗ f 0 Ω 0 g. We rate is along the y axis, that is, Ω define the z axis as the longitudinal direction and the x and y axes as the transverse axes. e n The angle α of the polarization vectors P⃗ and P⃗ with respect to the z axis is small enough so that cos α ≈ 1. Therefore, it is a good approximation to assume that the longitudinal components Pez and Pnz are not affected by the presence of the transverse components. Equations (1) can be described by the following 4 × 4 matrix form:
2
3
2
∂Pe 6 ∂tx 6 ∂Pey 6 6 ∂tn 6 ∂Px 6 ∂t 4 ∂Pny ∂t
7 7 7 7 7 7 5
6 − RQtot 6 6 γe Bz λM n Pnz Lz 6 Q 6 Rne 6 se 4 −γ λM e n Pz n
− γ n Pnz By − Pnz Ω − iγ n Pnz Bx : (5) Equation (5) is similar to the dynamic equation of a rotor angular rate gyroscope, which is given by [10] ̈ 2ζω0 U _ ω2 U ω2 k H ω; U 0 0 S
(6)
where U and ω are the output and input of a rotor angular rate gyroscope, respectively. ζ and ω0 are the damping factor and natural frequency, which indicate the dynamic performance of the gyroscope. The all-optical atomic spin gyroscope is therefore equivalent to a rotor angular rate gyroscope, and the analytical method used for a rotor angular rate
3
2
̈ _ P~ e − A DP~ e − BC − ADP~ e W; W γe e γe e e P B Ly − Pz Ω i P B Lx D Q z y Q z x
n n − γe Bz λMQ Pz Lz − RQtot γ n λM e Pnz Rne se
Ren se Q n e − γe λMQ Pz −Rntot γ n Bz λM e Pez
2 3 e 7 7 6 Pxe 7 7 6 Py 7 7×6 n7 6 7 e e −γ n Bz λM Pzz 7 7 4 Pxn 5 5 n Py −R γe n e Q λM Pz en Rse Q tot
3
γ e e 6 Qe Pz By Ly − Pz Ω 7 7 6 γe e 6 − Q Pz Bx Lx 7 7: 6 n n γ n Pz By − Pz Ω 7 6 5 4 n γ n P z Bx
(2)
It is convenient to write Eq. (2) as a system of equations for P~ e Pex iPey and P~ n Pnx iPny : " ~e # e γe e ∂P ~ A B Pz By Ly − Pez Ω Q1 Rp Rm sm i γQe Pez Bx Lx − Q1 Rp Rm sm P ∂t Q ; ∂P~ n C D P~ n γ n Pnz By − Pnz Ω − iγ n Pnz Bx
(3)
∂t
where A − RQtot i γQe Bz λM n Pnz Lz ; B RQse − i γQe λM n Pez : e n C Rne D −Rntot iγ n Bz λM e Pez se − iγ n λM Pz ; en
The output signal of the all-optical atomic spin gyroscope is proportional to the electron polarization Pex which is the real value of P~ e. The input signal of the atomic spin gyroscope is Ω. From Eq. (3), we can derive the simple differential equation of alloptical atomic spin gyroscope dynamics, which is given by 7222
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4
gyroscope can be applied in the all-optical atomic spin gyroscope. 2. Dynamic and Steady State Response The eigenfunction of Eq. (5) is r2 − A Dr − BC − AD 0:
(7)
The two eigenvalues, which determine the dynamical behavior of the atomic spin gyroscope, are
r1;2
1 γ γ λM n Pnz λM e Pez Γ1 − Rntot − e n ; Rtot 2 ω1 γ n Bz λM e Pez ;
p A D A D2 4BC − AD : 2
(8)
Γ2 − ω2
With reasonable approximations Rntot ≪ Rtot, λM e ≪ λM n , γ n ≪ γ e , γ e λM e Pez ≪ Rtot , and γ n λM n Pnz ≪ Rtot (a condition that is realized in atomic spin gyroscope operation, Rtot ∕Rntot ≈ 105 , Rtot ∕γ e λM e Pez ≈ 102 , Rtot ∕γ n λM n Pnz ≈102 , λM n ∕λM e ≈103 –104 , γ e ∕γ n ≈ 103 ), the square root in Eq. (8) can be written as A D2 4BC − AD 1 γ γ λM n Pnz λM e Pez Rtot − 2 e n Rtot Q 2 γe n n e e B λM Pz Lz − γ n Bz λM Pz : (9) −i Q z The two eigenvalues can be approximated to be 1 γ γ λM n Pnz λM e Pez r1 − Rntot − e n iγ n Bz λM e Pez ; Rtot 2 R γ γ λM n Pnz λM e Pez γ i e Bz λM n Pnz Lz : r2 − tot e n Q Rtot Q (10) Therefore, the general dynamical solution of Eq. (5) contains two separate oscillations with different frequencies (imaginary part of the eigenvalues) and decay rates (real part of the eigenvalues), and is given by P~ e t c1 er1 t c2 er2 t P~ e ;
(11)
where c1 and c2 are constants determined from the boundary and continuity conditions. P~ e is a special solution of Eq. (5) and given by
Rtot γ e γ n λM n Pnz λM e Pez ; Q Rtot
γe B λM n Pnz Lz ; Q z
Pez γ e Rtot R2tot γ 2e Bz λM n Pnz Lz 2 Bz λM n Pnz Lz By × Bz λM e Pez γ Ly e Bz λM n Pnz Lz Lx Rtot γ e Bz λM n Pnz Lz 2 1 1 Bx − Q Ω ; Rtot Bz λM e Pez γn γe
ReP~ e
(15) where ω1 and ω2 are the eigenfrequencies, and Γ1 and Γ2 are the damping factors. For the best sensitivity of the gyroscope to the angular rate, it is reasonable to design its sensitive element in such a way that its values of the eigenfrequencies ω1 and ω2 are as close to each other as possible [11]. At the ideal point of all-optical atomic spin gyroscope operation, the longitudinal field is set to the compensation point and can be written as Bz −λM e Pez − λM n Pnz :
P
W : AD − BC
(12)
By combining Eqs. (10)–(12), the simplified solution for the measured signal Pex (real part of P~ e ), which is the output of the all-optical atomic spin gyroscope, can be written compactly: Pex t ReP~ e t c1 eΓ1 t cosω1 t c2 eΓ2 t cosω2 t ReP~ e ; (13) where
(16)
In general, γ n λM n Pnz and γ e λM e Pez ∕Q are of the order of several hertz and nearly equal in the atomic spin gyroscope, that is, γ n λM n Pnz ≈
γe λM e Pez : Q
(17)
With Lz properly zeroed and Eq. (17), ω1 and ω2 in Eq. (13) become ω1 ω2 −γ n λM n Pnz :
~ e
(14)
(18)
Thus, measured signal Pex contains two separate oscillations with the same frequency and different decay rates. In addition, it can be known that the all-optical atomic spin gyroscope has the best sensitivity to the angular rate in this condition according to gyroscope theory, which was proposed by Apostolyuk and Tay [11]. From Eq. (13), we can see that ReP~ e is the steadystate equation of an all-optical atomic spin gyroscope. The error effects in an all-optical atomic spin gyroscope are mainly caused by various imperfections, such as external magnetic fields, spinexchange collisions, and light shifts [12]. 20 October 2013 / Vol. 52, No. 30 / APPLIED OPTICS
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s1;2 Γ1 jω1 ;
B. System Transfer Function of All-Optical Atomic Spin Gyroscope
1 γ γ λM n Pnz λM e Pez − Rntot − e n ; Rtot 2
Through application of the Laplace transformation to Eq. (5) under zero initial conditions and the omission of imperfections of transverse magnetic fields and light shifts, one can obtain − A Ds − BC − ADP~ e s −DPez Pnz Ωs: (19)
Considering the angular rate as an input, the system transfer function is Gs
Pex s ~ ReGs; Ωs
(20)
ω2
γe B λM n Pnz Lz : Q z
(24)
Rtot γ e γ n λM n Pnz λM e Pez 1 n Rtot Q Rtot 2 n n e e Rtot γ n λM Pz γ e λM Pz 1−Q ≃− · ≪ 0; (25) Q Rtot Rtot
Γ2 − Γ1 −
−DPez Pnz P~ e s ~ 2 Gs Ωs s − A Ds − BC − AD −DPez Pnz : (21) s − r1 s − r2 Substituting Eqs. (4) and (10) into Eq. (21) results in the following equation:
Pex s ~ ReGs Ωs N s − Γ1 2 ω21 s − Γ2 2 ω22
Gs
Rntot Pez Pnz s − z1 s − z2 ; s − Γ1 2 ω21 s − Γ2 2 ω22
Rtot γ e γ n λM n Pnz λM e Pez ; Q Rtot
By analyzing Eq. (24), it is easy to see that the alloptical atomic spin gyroscope is inherently stable, and s1;2 are the predominant poles that are the most essential for the transient process, as a result of Γ2 ≪ Γ1 :
where
ω1
γ n Bz λM e Pez ; Γ2 −
s2
s3;4 Γ2 jω2 ; Γ1
where Q 4–6 for K and Q 8–22 for Cs, and Rtot Rtot 5 ≈ 10 ∼ 102 ; n ≈ 10 ; Rtot γ e λM e Pez
Rtot ≈ 10 ∼ 102 : γ n λM n Pnz (26)
Zeros of the transfer function given by Eq. (22) are as follows: (22) z1;2 −γ n Bz λM e Pez ω1 ω2 − Γ1 Γ2 Rntot Pez 2Rntot Pez Pnz
where N Rntot Pez Pnz s2 γ n Bz λM e Pez ω1 ω2
;
(27)
− Γ1 Γ2 Rntot Pez s
where
Γ1 Γ2 − ω1 ω2 Rntot Pez Pnz
Δ γ n Bz λM e Pez ω1 ω2 − Γ1 Γ2 Rntot Pez 2
− γ n Bz λM e Pez ω1 Γ2 ω2 Γ1 ; 1 γ γ λM n Pnz λM e Pez ; Γ1 − Rntot − e n Rtot 2
−4Rntot Pez Pnz Γ1 Γ2 − ω1 ω2 Rntot Pez Pnz − γ n Bz λM e Pez ω1 Γ2 ω2 Γ1
ω1 γ n Bz λM e Pez ; R γ γ λM n Pnz λM e Pez ; Γ2 − tot e n Q Rtot γ ω2 e Bz λM n Pnz Lz ; Q
p Δ
γ n Bz λM e Pez ω1 ω2 2 Γ1 − Γ2 Rntot Pez 2 4Rntot Pez 2 ω1 ω2 2γ n Rntot Pez Bz λM e Pez (23)
× ω2 − ω1 Γ1 − Γ2 ≻0:
(28)
Thus, z1 and z2 are real numbers, and where z1 and z2 are the two eigenvalues of equation N 0. Both stability and unit-step transient process quality depend on position of the system poles in the real–imaginary plane. Poles of the transfer function in Eq. (22) are as follows: 7224
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γ n Bz λM e Pez ω1 ω2 − Γ1 − Γ2 Rntot Pez 2 ≺Δ≺γ n Bz λM e Pez ω1 ω2 Γ1 − Γ2 Rntot Pez 2 ; (29)
jγ n Bz λM e Pez ω1 ω2 p − Γ1 − Γ2 Rntot Pez j≺ Δ≺γ n Bz λM e Pez ω1 ω2 Γ1 − Γ2 Rntot Pez ;
(30)
z 1 ≺ Γ1 ;
z 2 ≺ Γ2 :
(31)
It is therefore apparent from the above analysis that the dynamic performance of all-optical atomic spin gyroscopes is mainly determined by Γ1 and ω1 , corresponding to the transient process of the noble-gas nuclei. The transfer function can be simplified as follows: Gs
Rntot Pez Pnz Rntot Pez Pnz : s − Γ1 2 ω21 s2 − 2Γ1 s Γ21 ω21
(32)
Therefore, an all-optical atomic spin gyroscope can be simplified asqa second-order system. The damping
factor ζ −Γ1 ∕ Γ21 ω21 and natural frequency ωn q Γ21 ω21 can be deduced from Eq. (29) and the standard transfer function of a second-order system is given by the following form: Gs
kω2n : s2 2ζωn s ω2n
(33)
4. Experimental Test and Discussion A.
Experimental Setup
In order to validate the transfer function described above, the experiment was set up. The experimental apparatus of an all-optical atomic spin gyroscope is depicted in Fig. 2. A cubic glass cell with a 2 cm side was filled with a droplet of Cs metal, 20 Torr of 129 Xe,
Fig. 3. Test apparatus for the transfer function of the all-optical atomic spin gyroscope.
and 700 Torr of N2. Cs atoms were polarized by the pump beam along the z direction and transferred the polarization to the 129 Xe by spin-exchange collisions. The nitrogen serves as a quencher to enhance the efficiency of the optical pumping and to keep the atomic system in the ground state. The cell was placed in a nonmagnetic oven and heated to 110°C by flowing hot air. The oven was located inside a four-layer set of cylindrical magnetic shields with compensating magnetic coils driven by a precision current source. The compensating magnetic coils were also used to apply magnetic fields to the cell. After setting all three components of the external magnetic field to as near zero as possible, a bias magnetic field using the compensating magnetic coils was added in the z direction and its value was set to approximately cancel the field created by the nuclear magnetization. 129 Xe was spin-exchange optically pumped by Cs for 30 min. Then, the bias magnetic field was carefully adjusted to compensate the magnetic field of approximately 200 nT in our experiment. This allows high-sensitivity operation of the all-optical atomic spin gyroscope that is primarily sensitive to the inertial rotation rate. The experimental apparatus of the all-optical atomic spin gyroscope was then fixed on the optical table and the whole system was located at the rotation platform, as shown in Fig. 3. The sensing axis of the all-optical atomic spin gyroscope was in the y direction. The optical table was rotated clockwise and counterclockwise alternately using the rotation platform, which causes the all-optical atomic spin gyroscope to obtain alternate step inputs. The step responses of the all-optical atomic spin gyroscope were collected using a data acquisition system from the National Instruments Corp., USA. B. Results and Discussion
Fig. 2. Experimental apparatus of the all-optical atomic spin gyroscope. The pump laser transmits in the z direction with frequency detuned to the D1 line of Cs. The probe laser propagates in the x direction from the D2 line of Cs, and the Faraday modulation method is used for probing the direction of atomic spin to measure the output signal of the all-optical atomic spin gyroscope.
The step responses of the all-optical atomic spin gyroscope are shown in Fig 4(a). There are eight step responses and the elaborate first step response is shown in Fig. 4(b). The values of the relevant parameters are listed in Table 1. We define the amplitudes of the first oscillation and the second oscillation in each step response as A1 and A2 , respectively. d is defined as the slowdown factor, and T d is defined as the oscillation periods. According to the step response theory of a second-order system, the following equations can be obtained: 20 October 2013 / Vol. 52, No. 30 / APPLIED OPTICS
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Fig. 4. Step responses of the all-optical atomic spin gyroscope. (a) The eight total step responses according to alternate step input. (b) The elaborate first step response of the eight total step responses shown in (a).
Table 1.
Values of Relevant Parameters of the Eight Total Step Responses Shown in Fig. 4a
No.
A1 (V)
A2 (V)
d A1 ∕A2
D ln d
T d (s)
Damping Factor ζ
Natural Frequency ωn (Hz)
1 2 3 4 5 6 7 8
0.410 0.245 0.562 0.542 0.489 0.441 0.506 0.385
0.333 0.206 0.421 0.442 0.414 0.398 0.429 0.297
1.231 1.189 1.335 1.226 1.181 1.108 1.179 1.296
0.208 0.173 0.289 0.204 0.1665 0.1026 0.1651 0.2595
0.414 0.420 0.433 0.420 0.425 0.420 0.420 0.425
0.0331 0.0275 0.0459 0.0324 0.0265 0.0163 0.026 0.0412
2.4141 2.3801 2.3070 2.3797 2.3521 2.3806 2.3801 2.3509
The damping factors ζ and natural frequencies ωn are obtained according to Eqs. (34)–(36).
a
d
A1 ; A2
2πζ D ln d p ; 1 − ζ2 q ωd ωn 1 − ζ2 ;
(34)
(35)
(36)
where ωd is the oscillation frequency and equal to 1∕T d . ωn is the natural frequency of the system. By combining Eqs. (34)–(36), we can obtain system damping factor ζ 0.0311 0.0093 and natural frequency ωn 2.368 0.032 Hz. The results of each step response are listed in Table 1. In these operating conditions, the Cs electron spins experience a magnetic field generated by the 129 Xe magnetization of approximately λM n Pnz 200 nT. The 129 Xe spins experience a magnetic field generated by electron magnetization of about λM e Pez 4–8 nT. The total 129 Xe relaxation rate Rntot was approximately 0.024–0.067 1∕s, determined by free induction decay nuclear magnetic resonance. The total spin relaxation rate of electron Rtot was approximately 20; 000 1∕s. According to Eq. (24), we obtained Γ1 −0.02838–0.04988 1∕s and ω1 2.3868–2.4336 Hz. Furthermore, the damping factor ζ 0.0117–0.021 and nature frequency ωn 2.3868–2.4336 Hz, according to the theoretical 7226
transfer function [Eq. (32)] presented in this paper. These results were in agreement with the experimental results ζ 0.0311 0.0093 and ωn 2.368 0.032 Hz. Therefore, the simplified transfer function presented in this paper can be applied to describe the dynamics of all-optical atomic spin gyroscopes.
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5. Conclusion
We presented a transfer function for an all-optical atomic spin gyroscope that is the basis for further closed-loop control system design. We built the differential equations based on a complete set of Bloch equations describing an all-optical atomic spin gyroscope and then obtained the transfer function through application of the Laplace transformation to these differential equations. The transfer function has been analyzed, simplified, and validated in an all-optical Cs–129 Xe atomic spin gyroscope by a series of step responses. The transfer function by experiment was in agreement with the theoretical transfer function. This transfer function is convenient for analysis of the form of control system required, and for the design of the control system specifically to reduce the effects of imperfections and improve the performance of all-optical atomic spin gyroscopes. This work is supported by the Key Programs of the National Natural Science Foundation of China under
Grant Nos. 61227902, 61273067, 61074171, and 61004140, and by the SAST Foundation of China. References 1. N. Barbour, “Inertial sensor technology trends,” IEEE Sens. J. 1, 332–339 (2001). 2. T. W. Kornack, R. K. Ghosh, and M. V. Romalis, “Nuclear spin gyroscope based on an atomic comagnetometer,” Phys. Rev. Lett. 95, 230801 (2005). 3. M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, “New test of local Lorentz invariance using a 21NeRb-K comagnetometer,” Phys. Rev. Lett. 107, 171604 (2011). 4. J. C. Fang and J. Qin, “Advances in atomic gyroscope: a view from applicationfor inertialnavigation,” Sensors 12, 6331–6346(2012). 5. J. C. Fang, J. Qin, S. A. Wan, Y. Chen, and R. J. Li, “Atomic spin gyroscope based on 129Xe-Cs comagnetometer,” Chin. Sci. Bull. 58, 1512–1515 (2013).
6. T. W. Kornack, “A test of CPT and Lorentz symmetry using a K-3He comagnetometer,” Ph.D. dissertation (Princeton University, 2005). 7. S. J. Seltzer, “Developments in alkali-metal atomic magnetometry,” Ph.D. dissertation (Princeton University, 2008). 8. R. Stoner and R. Walsworth, “Collisions give sense of direction,” Nat. Phys. 2, 17–18 (2006). 9. J. M. Brown, “A new limit on Lorentz- and CPT-violating neutron spin interactions using a K-3He comagnetometer,” Ph.D. dissertation (Princeton University, 2011). 10. B. Yu, Navigation Technology (Aeronautic Industry, 1987). 11. V. Apostolyuk and F. E. H. Tay, “Dynamics of micromechanical coriolis vibratory gyroscopes,” Sensor Lett. 2, 252–259 (2005). 12. J. C. Fang, S. A. Wan, Y. Chen, and R. J. Li, “Light-shift measurement and suppression in atomic spin gyroscope,” Appl. Opt. 51, 7714–7717 (2012).
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We present the transfer function of an all-optical atomic spin gyroscope through a series of differential equations and validate the transfer function by experimental test. A transfer function is the basis for further control system design. We build the differential equations based on a complete set of Bloch equations describing the all-optical atomic spin gyroscope, and obtain the transfer function through application of the Laplace transformation to these differential equations. Moreover, we experimentally validate the transfer function in an all-optical Cs–129 Xe atomic spin gyroscope through a series of step responses. This transfer function is convenient for analysis of the form of control system required. Furthermore, it is available for the design of the control system specifically to improve the performance of all-optical atomic spin gyroscopes. © 2013 Optical Society of America OCIS codes: (120.4640) Optical instruments; (120.3930) Metrological instrumentation; (020.5580) Quantum electrodynamics. http://dx.doi.org/10.1364/AO.52.007220
1. Introduction
Sensitive gyroscopes are used in a wide range of applications, particularly in inertial navigation applications [1]. In recent years, with the rapid development of the quantum manipulation technique, the advantages, such as high precision and compact size, offered by all-optical atomic spin gyroscopes have attracted more and more researchers to this field. An angular random walk (ARW) of 2 × 10−3 °∕h1∕2 and a bias stability of 4 × 10−2 °∕h were achieved by the alloptical 3 He–K atomic spin gyroscope when it was first demonstrated in 2005 [2]. Since then, an equivalent rotation resolution of 1.8 × 10−7 °∕s has been realized with an all-optical 21 Ne–Rb–K atomic spin gyroscope [3]. A sensitivity of 7 × 10−5 °∕s∕Hz1∕2 was achieved by an all-optical 129 Xe–Cs atomic spin gyroscope [4,5]. Theoretical analysis shows that an ARW of 7.2 × 10−8 °∕h1∕2 can be achieved in a 150 cm3 sense volume with an all-optical 21 Ne atomic spin gyroscope [6,7], which indicates that an all-optical atomic spin gyroscope is a highly sensitive and 1559-128X/13/307220-08$15.00/0 © 2013 Optical Society of America 7220
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compact gyroscope for future inertial navigation applications [8]. Currently, the performance of all-optical atomic spin gyroscopes is far below the theoretical performance. One of the ways to improve the performances of all-optical atomic spin gyroscopes is to analyze their dynamics and errors. The errors of all-optical atomic spin gyroscopes have been studied by Kornack [6] and Brown [9]. The effects of imperfect cancellation of external magnetic fields, spin-exchange collisions, and light shifts on steady-state errors were considered in their studies. However, less analytical dynamics have been developed in recent papers. In this paper, we investigated a set of differential equations and a transfer function describing the dynamics of all-optical atomic spin gyroscopes. The differential equation obtained is similar to the differential equation of a rotor angular rate gyroscope. The all-optical atomic spin gyroscope is therefore equivalent to a rotor angular rate gyroscope, and the analytical method used for a rotor angular rate gyroscope can be applied in an all-optical atomic spin gyroscope. Moreover, the transfer function obtained was analyzed and simplified. The simplified transfer function was validated by experimental test. This
transfer function is useful for the design of the control system specifically to reduce the effects of imperfections, and further to improve the performance of the all-optical atomic spin gyroscope. 2. Description of an All-Optical Atomic Spin Gyroscope
A general schematic of all-optical atomic spin gyroscope is shown in Fig. 1. The gyroscope consists of a vapor cell that contains alkali atoms (K, Rb, or Cs) and noble gas (3 He; 21 Ne, or 129 Xe). Alkali atoms are polarized by optical pumping along the z direction and transfer the polarization to the noble gas by spin-exchange collisions. Thus, alkali metal atoms obtain electron spin magnetic moment SA , and noblegas atoms obtain nuclear spin magnetic moment SB . An all-optical atomic spin gyroscope utilizes the electron spin magnetic moment of alkali metal atoms SA for angular velocity measurement. SA can keep pointing at the fixed direction in the inertial coordinate system. An external magnetic field Bc is set to the “compensation point” where Bc cancels the field created by the nuclear magnetization of noble gas SB . Under this situation, SA and SB can strongly interact with each other, and SB can automatically track and compensate for the changes of the external magnetic field, to isolate the external magnetic field sensed by SA . When the carrier rotates, SA will remain in its original pointing direction, while the angle between SA and a probe laser fixed in the carrier changes, which reflects the rotation only. Consequently, the angle can be measured by measuring the electron spin polarization along the direction of the probe laser propagation. Hence, the block diagram of the all-optical atomic spin gyroscope can be simplified to an input–output form, as shown in Fig. 1(c). Ω and S are the input and output of the all-optical atomic spin gyroscope,
respectively. Alkali atom spins align along the z direction by optical pumping. Thus, these spins aligned have an x projection under an inertial rotation rate input Ω along the y direction when the external magnetic field is set to the “compensation point.” This x projection of the spins is measured through optical rotation of a linearly polarized probe laser. A polarizer translates the optical rotation angle into the intensity of the probe laser, which is measured through a photodiode. This intensity is referred to as the output signal of all-optical atomic spin gyroscope S. 3. Modeling of All-Optical Atomic Spin Gyroscope Dynamics A. Differential Equations of All-Optical Atomic Spin Gyroscope Dynamics
1. Differential Equations A further technical description of an all-optical atomic spin gyroscope follows a complete set of Bloch equations [6,9]: ∂P⃗ γ n ⃗ × P⃗ e − Ω ⃗ × P⃗ e e B⃗ λM n P⃗ L ∂t Q ⃗ n ⃗ e Rp s⃗ p Ren se P − Rtot P ∕Q; e
∂P⃗ e n ⃗ × P⃗ n γ n B⃗ λM e P⃗ × P⃗ − Ω ∂t n ⃗ n ⃗ e Rne se P − R P ; n
tot
(1)
⃗ is the inertial rotation rate. P⃗ e ∕P⃗ n , γ e ∕γ n , where Ω λM e ∕λM n , and Rtot ∕Rntot are polarization, gyromagnetic ratio, polarization field factors, and the total spin relaxation rate of electron/nuclei, respectively. B⃗ and L⃗ are the net magnetic field and light shift experienced by electrons, respectively. Rp and Rm are
Fig. 1. Schematic of the all-optical atomic spin gyroscope. (a) Alkali atoms and noble-gas atoms are polarized along the bias field Bc. (b) SB is able to automatically track and compensate for changes in the external magnetic field, thus isolating the external magnetic field sensed by SA. (c) Simple block diagram of an atomic spin gyroscope in a control system. 20 October 2013 / Vol. 52, No. 30 / APPLIED OPTICS
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the pumping rates of the pump laser and the probe ne laser, respectively. Ren se and Rse are the pumping rates of the electrons and the nuclei by spin-exchange with each other. s⃗ p is the polarization of the pump photon and, for a circularly polarized pump laser, j⃗sp j 1. Q is the slowing-down factor. In this paper, the z axis is defined to be the direction of the circularly polarized pump laser and s⃗ p f 0 0 1 g. The x axis is defined as the direction of the linearly polarized probe laser, which measures the x component of the electron polarization Pex . The direction of the inertial rotation ⃗ f 0 Ω 0 g. We rate is along the y axis, that is, Ω define the z axis as the longitudinal direction and the x and y axes as the transverse axes. e n The angle α of the polarization vectors P⃗ and P⃗ with respect to the z axis is small enough so that cos α ≈ 1. Therefore, it is a good approximation to assume that the longitudinal components Pez and Pnz are not affected by the presence of the transverse components. Equations (1) can be described by the following 4 × 4 matrix form:
2
3
2
∂Pe 6 ∂tx 6 ∂Pey 6 6 ∂tn 6 ∂Px 6 ∂t 4 ∂Pny ∂t
7 7 7 7 7 7 5
6 − RQtot 6 6 γe Bz λM n Pnz Lz 6 Q 6 Rne 6 se 4 −γ λM e n Pz n
− γ n Pnz By − Pnz Ω − iγ n Pnz Bx : (5) Equation (5) is similar to the dynamic equation of a rotor angular rate gyroscope, which is given by [10] ̈ 2ζω0 U _ ω2 U ω2 k H ω; U 0 0 S
(6)
where U and ω are the output and input of a rotor angular rate gyroscope, respectively. ζ and ω0 are the damping factor and natural frequency, which indicate the dynamic performance of the gyroscope. The all-optical atomic spin gyroscope is therefore equivalent to a rotor angular rate gyroscope, and the analytical method used for a rotor angular rate
3
2
̈ _ P~ e − A DP~ e − BC − ADP~ e W; W γe e γe e e P B Ly − Pz Ω i P B Lx D Q z y Q z x
n n − γe Bz λMQ Pz Lz − RQtot γ n λM e Pnz Rne se
Ren se Q n e − γe λMQ Pz −Rntot γ n Bz λM e Pez
2 3 e 7 7 6 Pxe 7 7 6 Py 7 7×6 n7 6 7 e e −γ n Bz λM Pzz 7 7 4 Pxn 5 5 n Py −R γe n e Q λM Pz en Rse Q tot
3
γ e e 6 Qe Pz By Ly − Pz Ω 7 7 6 γe e 6 − Q Pz Bx Lx 7 7: 6 n n γ n Pz By − Pz Ω 7 6 5 4 n γ n P z Bx
(2)
It is convenient to write Eq. (2) as a system of equations for P~ e Pex iPey and P~ n Pnx iPny : " ~e # e γe e ∂P ~ A B Pz By Ly − Pez Ω Q1 Rp Rm sm i γQe Pez Bx Lx − Q1 Rp Rm sm P ∂t Q ; ∂P~ n C D P~ n γ n Pnz By − Pnz Ω − iγ n Pnz Bx
(3)
∂t
where A − RQtot i γQe Bz λM n Pnz Lz ; B RQse − i γQe λM n Pez : e n C Rne D −Rntot iγ n Bz λM e Pez se − iγ n λM Pz ; en
The output signal of the all-optical atomic spin gyroscope is proportional to the electron polarization Pex which is the real value of P~ e. The input signal of the atomic spin gyroscope is Ω. From Eq. (3), we can derive the simple differential equation of alloptical atomic spin gyroscope dynamics, which is given by 7222
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4
gyroscope can be applied in the all-optical atomic spin gyroscope. 2. Dynamic and Steady State Response The eigenfunction of Eq. (5) is r2 − A Dr − BC − AD 0:
(7)
The two eigenvalues, which determine the dynamical behavior of the atomic spin gyroscope, are
r1;2
1 γ γ λM n Pnz λM e Pez Γ1 − Rntot − e n ; Rtot 2 ω1 γ n Bz λM e Pez ;
p A D A D2 4BC − AD : 2
(8)
Γ2 − ω2
With reasonable approximations Rntot ≪ Rtot, λM e ≪ λM n , γ n ≪ γ e , γ e λM e Pez ≪ Rtot , and γ n λM n Pnz ≪ Rtot (a condition that is realized in atomic spin gyroscope operation, Rtot ∕Rntot ≈ 105 , Rtot ∕γ e λM e Pez ≈ 102 , Rtot ∕γ n λM n Pnz ≈102 , λM n ∕λM e ≈103 –104 , γ e ∕γ n ≈ 103 ), the square root in Eq. (8) can be written as A D2 4BC − AD 1 γ γ λM n Pnz λM e Pez Rtot − 2 e n Rtot Q 2 γe n n e e B λM Pz Lz − γ n Bz λM Pz : (9) −i Q z The two eigenvalues can be approximated to be 1 γ γ λM n Pnz λM e Pez r1 − Rntot − e n iγ n Bz λM e Pez ; Rtot 2 R γ γ λM n Pnz λM e Pez γ i e Bz λM n Pnz Lz : r2 − tot e n Q Rtot Q (10) Therefore, the general dynamical solution of Eq. (5) contains two separate oscillations with different frequencies (imaginary part of the eigenvalues) and decay rates (real part of the eigenvalues), and is given by P~ e t c1 er1 t c2 er2 t P~ e ;
(11)
where c1 and c2 are constants determined from the boundary and continuity conditions. P~ e is a special solution of Eq. (5) and given by
Rtot γ e γ n λM n Pnz λM e Pez ; Q Rtot
γe B λM n Pnz Lz ; Q z
Pez γ e Rtot R2tot γ 2e Bz λM n Pnz Lz 2 Bz λM n Pnz Lz By × Bz λM e Pez γ Ly e Bz λM n Pnz Lz Lx Rtot γ e Bz λM n Pnz Lz 2 1 1 Bx − Q Ω ; Rtot Bz λM e Pez γn γe
ReP~ e
(15) where ω1 and ω2 are the eigenfrequencies, and Γ1 and Γ2 are the damping factors. For the best sensitivity of the gyroscope to the angular rate, it is reasonable to design its sensitive element in such a way that its values of the eigenfrequencies ω1 and ω2 are as close to each other as possible [11]. At the ideal point of all-optical atomic spin gyroscope operation, the longitudinal field is set to the compensation point and can be written as Bz −λM e Pez − λM n Pnz :
P
W : AD − BC
(12)
By combining Eqs. (10)–(12), the simplified solution for the measured signal Pex (real part of P~ e ), which is the output of the all-optical atomic spin gyroscope, can be written compactly: Pex t ReP~ e t c1 eΓ1 t cosω1 t c2 eΓ2 t cosω2 t ReP~ e ; (13) where
(16)
In general, γ n λM n Pnz and γ e λM e Pez ∕Q are of the order of several hertz and nearly equal in the atomic spin gyroscope, that is, γ n λM n Pnz ≈
γe λM e Pez : Q
(17)
With Lz properly zeroed and Eq. (17), ω1 and ω2 in Eq. (13) become ω1 ω2 −γ n λM n Pnz :
~ e
(14)
(18)
Thus, measured signal Pex contains two separate oscillations with the same frequency and different decay rates. In addition, it can be known that the all-optical atomic spin gyroscope has the best sensitivity to the angular rate in this condition according to gyroscope theory, which was proposed by Apostolyuk and Tay [11]. From Eq. (13), we can see that ReP~ e is the steadystate equation of an all-optical atomic spin gyroscope. The error effects in an all-optical atomic spin gyroscope are mainly caused by various imperfections, such as external magnetic fields, spinexchange collisions, and light shifts [12]. 20 October 2013 / Vol. 52, No. 30 / APPLIED OPTICS
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s1;2 Γ1 jω1 ;
B. System Transfer Function of All-Optical Atomic Spin Gyroscope
1 γ γ λM n Pnz λM e Pez − Rntot − e n ; Rtot 2
Through application of the Laplace transformation to Eq. (5) under zero initial conditions and the omission of imperfections of transverse magnetic fields and light shifts, one can obtain − A Ds − BC − ADP~ e s −DPez Pnz Ωs: (19)
Considering the angular rate as an input, the system transfer function is Gs
Pex s ~ ReGs; Ωs
(20)
ω2
γe B λM n Pnz Lz : Q z
(24)
Rtot γ e γ n λM n Pnz λM e Pez 1 n Rtot Q Rtot 2 n n e e Rtot γ n λM Pz γ e λM Pz 1−Q ≃− · ≪ 0; (25) Q Rtot Rtot
Γ2 − Γ1 −
−DPez Pnz P~ e s ~ 2 Gs Ωs s − A Ds − BC − AD −DPez Pnz : (21) s − r1 s − r2 Substituting Eqs. (4) and (10) into Eq. (21) results in the following equation:
Pex s ~ ReGs Ωs N s − Γ1 2 ω21 s − Γ2 2 ω22
Gs
Rntot Pez Pnz s − z1 s − z2 ; s − Γ1 2 ω21 s − Γ2 2 ω22
Rtot γ e γ n λM n Pnz λM e Pez ; Q Rtot
By analyzing Eq. (24), it is easy to see that the alloptical atomic spin gyroscope is inherently stable, and s1;2 are the predominant poles that are the most essential for the transient process, as a result of Γ2 ≪ Γ1 :
where
ω1
γ n Bz λM e Pez ; Γ2 −
s2
s3;4 Γ2 jω2 ; Γ1
where Q 4–6 for K and Q 8–22 for Cs, and Rtot Rtot 5 ≈ 10 ∼ 102 ; n ≈ 10 ; Rtot γ e λM e Pez
Rtot ≈ 10 ∼ 102 : γ n λM n Pnz (26)
Zeros of the transfer function given by Eq. (22) are as follows: (22) z1;2 −γ n Bz λM e Pez ω1 ω2 − Γ1 Γ2 Rntot Pez 2Rntot Pez Pnz
where N Rntot Pez Pnz s2 γ n Bz λM e Pez ω1 ω2
;
(27)
− Γ1 Γ2 Rntot Pez s
where
Γ1 Γ2 − ω1 ω2 Rntot Pez Pnz
Δ γ n Bz λM e Pez ω1 ω2 − Γ1 Γ2 Rntot Pez 2
− γ n Bz λM e Pez ω1 Γ2 ω2 Γ1 ; 1 γ γ λM n Pnz λM e Pez ; Γ1 − Rntot − e n Rtot 2
−4Rntot Pez Pnz Γ1 Γ2 − ω1 ω2 Rntot Pez Pnz − γ n Bz λM e Pez ω1 Γ2 ω2 Γ1
ω1 γ n Bz λM e Pez ; R γ γ λM n Pnz λM e Pez ; Γ2 − tot e n Q Rtot γ ω2 e Bz λM n Pnz Lz ; Q
p Δ
γ n Bz λM e Pez ω1 ω2 2 Γ1 − Γ2 Rntot Pez 2 4Rntot Pez 2 ω1 ω2 2γ n Rntot Pez Bz λM e Pez (23)
× ω2 − ω1 Γ1 − Γ2 ≻0:
(28)
Thus, z1 and z2 are real numbers, and where z1 and z2 are the two eigenvalues of equation N 0. Both stability and unit-step transient process quality depend on position of the system poles in the real–imaginary plane. Poles of the transfer function in Eq. (22) are as follows: 7224
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γ n Bz λM e Pez ω1 ω2 − Γ1 − Γ2 Rntot Pez 2 ≺Δ≺γ n Bz λM e Pez ω1 ω2 Γ1 − Γ2 Rntot Pez 2 ; (29)
jγ n Bz λM e Pez ω1 ω2 p − Γ1 − Γ2 Rntot Pez j≺ Δ≺γ n Bz λM e Pez ω1 ω2 Γ1 − Γ2 Rntot Pez ;
(30)
z 1 ≺ Γ1 ;
z 2 ≺ Γ2 :
(31)
It is therefore apparent from the above analysis that the dynamic performance of all-optical atomic spin gyroscopes is mainly determined by Γ1 and ω1 , corresponding to the transient process of the noble-gas nuclei. The transfer function can be simplified as follows: Gs
Rntot Pez Pnz Rntot Pez Pnz : s − Γ1 2 ω21 s2 − 2Γ1 s Γ21 ω21
(32)
Therefore, an all-optical atomic spin gyroscope can be simplified asqa second-order system. The damping
factor ζ −Γ1 ∕ Γ21 ω21 and natural frequency ωn q Γ21 ω21 can be deduced from Eq. (29) and the standard transfer function of a second-order system is given by the following form: Gs
kω2n : s2 2ζωn s ω2n
(33)
4. Experimental Test and Discussion A.
Experimental Setup
In order to validate the transfer function described above, the experiment was set up. The experimental apparatus of an all-optical atomic spin gyroscope is depicted in Fig. 2. A cubic glass cell with a 2 cm side was filled with a droplet of Cs metal, 20 Torr of 129 Xe,
Fig. 3. Test apparatus for the transfer function of the all-optical atomic spin gyroscope.
and 700 Torr of N2. Cs atoms were polarized by the pump beam along the z direction and transferred the polarization to the 129 Xe by spin-exchange collisions. The nitrogen serves as a quencher to enhance the efficiency of the optical pumping and to keep the atomic system in the ground state. The cell was placed in a nonmagnetic oven and heated to 110°C by flowing hot air. The oven was located inside a four-layer set of cylindrical magnetic shields with compensating magnetic coils driven by a precision current source. The compensating magnetic coils were also used to apply magnetic fields to the cell. After setting all three components of the external magnetic field to as near zero as possible, a bias magnetic field using the compensating magnetic coils was added in the z direction and its value was set to approximately cancel the field created by the nuclear magnetization. 129 Xe was spin-exchange optically pumped by Cs for 30 min. Then, the bias magnetic field was carefully adjusted to compensate the magnetic field of approximately 200 nT in our experiment. This allows high-sensitivity operation of the all-optical atomic spin gyroscope that is primarily sensitive to the inertial rotation rate. The experimental apparatus of the all-optical atomic spin gyroscope was then fixed on the optical table and the whole system was located at the rotation platform, as shown in Fig. 3. The sensing axis of the all-optical atomic spin gyroscope was in the y direction. The optical table was rotated clockwise and counterclockwise alternately using the rotation platform, which causes the all-optical atomic spin gyroscope to obtain alternate step inputs. The step responses of the all-optical atomic spin gyroscope were collected using a data acquisition system from the National Instruments Corp., USA. B. Results and Discussion
Fig. 2. Experimental apparatus of the all-optical atomic spin gyroscope. The pump laser transmits in the z direction with frequency detuned to the D1 line of Cs. The probe laser propagates in the x direction from the D2 line of Cs, and the Faraday modulation method is used for probing the direction of atomic spin to measure the output signal of the all-optical atomic spin gyroscope.
The step responses of the all-optical atomic spin gyroscope are shown in Fig 4(a). There are eight step responses and the elaborate first step response is shown in Fig. 4(b). The values of the relevant parameters are listed in Table 1. We define the amplitudes of the first oscillation and the second oscillation in each step response as A1 and A2 , respectively. d is defined as the slowdown factor, and T d is defined as the oscillation periods. According to the step response theory of a second-order system, the following equations can be obtained: 20 October 2013 / Vol. 52, No. 30 / APPLIED OPTICS
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Fig. 4. Step responses of the all-optical atomic spin gyroscope. (a) The eight total step responses according to alternate step input. (b) The elaborate first step response of the eight total step responses shown in (a).
Table 1.
Values of Relevant Parameters of the Eight Total Step Responses Shown in Fig. 4a
No.
A1 (V)
A2 (V)
d A1 ∕A2
D ln d
T d (s)
Damping Factor ζ
Natural Frequency ωn (Hz)
1 2 3 4 5 6 7 8
0.410 0.245 0.562 0.542 0.489 0.441 0.506 0.385
0.333 0.206 0.421 0.442 0.414 0.398 0.429 0.297
1.231 1.189 1.335 1.226 1.181 1.108 1.179 1.296
0.208 0.173 0.289 0.204 0.1665 0.1026 0.1651 0.2595
0.414 0.420 0.433 0.420 0.425 0.420 0.420 0.425
0.0331 0.0275 0.0459 0.0324 0.0265 0.0163 0.026 0.0412
2.4141 2.3801 2.3070 2.3797 2.3521 2.3806 2.3801 2.3509
The damping factors ζ and natural frequencies ωn are obtained according to Eqs. (34)–(36).
a
d
A1 ; A2
2πζ D ln d p ; 1 − ζ2 q ωd ωn 1 − ζ2 ;
(34)
(35)
(36)
where ωd is the oscillation frequency and equal to 1∕T d . ωn is the natural frequency of the system. By combining Eqs. (34)–(36), we can obtain system damping factor ζ 0.0311 0.0093 and natural frequency ωn 2.368 0.032 Hz. The results of each step response are listed in Table 1. In these operating conditions, the Cs electron spins experience a magnetic field generated by the 129 Xe magnetization of approximately λM n Pnz 200 nT. The 129 Xe spins experience a magnetic field generated by electron magnetization of about λM e Pez 4–8 nT. The total 129 Xe relaxation rate Rntot was approximately 0.024–0.067 1∕s, determined by free induction decay nuclear magnetic resonance. The total spin relaxation rate of electron Rtot was approximately 20; 000 1∕s. According to Eq. (24), we obtained Γ1 −0.02838–0.04988 1∕s and ω1 2.3868–2.4336 Hz. Furthermore, the damping factor ζ 0.0117–0.021 and nature frequency ωn 2.3868–2.4336 Hz, according to the theoretical 7226
transfer function [Eq. (32)] presented in this paper. These results were in agreement with the experimental results ζ 0.0311 0.0093 and ωn 2.368 0.032 Hz. Therefore, the simplified transfer function presented in this paper can be applied to describe the dynamics of all-optical atomic spin gyroscopes.
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5. Conclusion
We presented a transfer function for an all-optical atomic spin gyroscope that is the basis for further closed-loop control system design. We built the differential equations based on a complete set of Bloch equations describing an all-optical atomic spin gyroscope and then obtained the transfer function through application of the Laplace transformation to these differential equations. The transfer function has been analyzed, simplified, and validated in an all-optical Cs–129 Xe atomic spin gyroscope by a series of step responses. The transfer function by experiment was in agreement with the theoretical transfer function. This transfer function is convenient for analysis of the form of control system required, and for the design of the control system specifically to reduce the effects of imperfections and improve the performance of all-optical atomic spin gyroscopes. This work is supported by the Key Programs of the National Natural Science Foundation of China under
Grant Nos. 61227902, 61273067, 61074171, and 61004140, and by the SAST Foundation of China. References 1. N. Barbour, “Inertial sensor technology trends,” IEEE Sens. J. 1, 332–339 (2001). 2. T. W. Kornack, R. K. Ghosh, and M. V. Romalis, “Nuclear spin gyroscope based on an atomic comagnetometer,” Phys. Rev. Lett. 95, 230801 (2005). 3. M. Smiciklas, J. M. Brown, L. W. Cheuk, S. J. Smullin, and M. V. Romalis, “New test of local Lorentz invariance using a 21NeRb-K comagnetometer,” Phys. Rev. Lett. 107, 171604 (2011). 4. J. C. Fang and J. Qin, “Advances in atomic gyroscope: a view from applicationfor inertialnavigation,” Sensors 12, 6331–6346(2012). 5. J. C. Fang, J. Qin, S. A. Wan, Y. Chen, and R. J. Li, “Atomic spin gyroscope based on 129Xe-Cs comagnetometer,” Chin. Sci. Bull. 58, 1512–1515 (2013).
6. T. W. Kornack, “A test of CPT and Lorentz symmetry using a K-3He comagnetometer,” Ph.D. dissertation (Princeton University, 2005). 7. S. J. Seltzer, “Developments in alkali-metal atomic magnetometry,” Ph.D. dissertation (Princeton University, 2008). 8. R. Stoner and R. Walsworth, “Collisions give sense of direction,” Nat. Phys. 2, 17–18 (2006). 9. J. M. Brown, “A new limit on Lorentz- and CPT-violating neutron spin interactions using a K-3He comagnetometer,” Ph.D. dissertation (Princeton University, 2011). 10. B. Yu, Navigation Technology (Aeronautic Industry, 1987). 11. V. Apostolyuk and F. E. H. Tay, “Dynamics of micromechanical coriolis vibratory gyroscopes,” Sensor Lett. 2, 252–259 (2005). 12. J. C. Fang, S. A. Wan, Y. Chen, and R. J. Li, “Light-shift measurement and suppression in atomic spin gyroscope,” Appl. Opt. 51, 7714–7717 (2012).
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